CE579 Half Course Summary

7/26/2019 CE579 Half Course Summary

1/192

CE 579: STRUCTRAL STABILITY AND DESIGN

Amit H. Varma

Assistant Pro!ssor

S"#oo$ o Ci%i$ En&in!!rin&P'r('! Uni%!rsit)

P#. No. *7+5, -9+ -/9

Emai$: a#%arma01'r('!.!('

2i"! #o'rs: 3446 9:4//: a.m.
mailto:[email protected]:[email protected]


2/192

C#a1t!r /. Intro('"tion to Str'"t'ra$ Sta8i$it)

2UTLINE D!inition o sta8i$it)

T)1!s o insta8i$it)

3!t#o(s o sta8i$it) ana$)s!s Eam1$!s sma$$ (!$!"tion ana$)s!s

Eam1$!s $ar&! (!$!"tion ana$)s!s

Eam1$!s im1!r!"t s)st!ms

D!si&n o st!!$ str'"t'r!s


3/192

STABILITY DE6INITI2N

C#an&! in &!om!tr) o a str'"t'r! or str'"t'ra$ "om1on!nt'n(!r "om1r!ssion r!s'$tin& in $oss o a8i$it) to r!sist $oa(in&

is (!in!( as instability in t#! 8oo;.

Insta8i$it) "an $!a( to "atastro1#i" ai$'r!m'st 8! a""o'nt!(

in (!si&n. Insta8i$it) is a str!n&t#4r!$at!( $imit stat!.

#) (i(

Sta8i$it) is not !as) to (!in!. E%!r) str'"t'r! is in !?'i$i8ri'm stati" or ()nami". I it is not in

!?'i$i8ri'm@ t#! 8o()


4/192


Str'"t'r! is in sta8$! !?'i$i8ri'm


5/192



6/192

BUCLING Vs. STABILITY

C#an&! in &!om!tr) o str'"t'r! 'n(!r "om1r!ssion t#atr!s'$ts in its a8i$it) to r!sist $oa(s "a$$!( instability.

Not tr'! t#is is "a$$!( buckling.

Buckling is a 1#!nom!non t#at "an o""'r or str'"t'r!s 'n(!r

"om1r!ssi%! $oa(s. T#! str'"t'r! (!orms an( is in sta8$! !?'i$i8ri'm in stat!4/.

As t#! $oa( in"r!as!s@ t#! str'"t'r! s'((!n$) "#an&!s to

(!ormation stat!4 at som! "riti"a$ $oa( P"r.

T#! str'"t'r! 8'";$!s rom stat!4/ to stat!4@


7/192


8/192

BUCLING Vs. STABILITY

T#'s@ t#!r! ar! t

Sta8i$it) o !?'i$i8ri'm As t#! $oa(s a"tin& on t#! str'"t'r! ar!

in"r!as!(@


9/192

TYPES 26 INSTABILITY

Str'"t'r! s'8F!"t!( to "om1r!ssi%! or"!s "an 'n(!r&o:/. B'";$in& 8i'r"ation o !?'i$i8ri'm rom (!ormation stat!4/ to

stat!4. Bi'r"ation 8'";$in& o""'rs or "o$'mns@ 8!ams@ an( s)mm!tri"

ram!s 'n(!r &ra%it) $oa(s on$)

. 6ai$'r! ('! to insta8i$it) o !?'i$i8ri'm stat!4/ ('! to $ar&!

(!ormations or mat!ria$ in!$asti"it) E$asti" insta8i$it) o""'rs or 8!am4"o$'mns@ an( ram!s s'8F!"t!(

to &ra%it) an( $at!ra$ $oa(s.

In!$asti" insta8i$it) "an o""'r or a$$ m!m8!rs an( t#! ram!.

!


10/192

TYPES 26 INSTABILITY

BI6URCATI2N BUCLING 3!m8!r or str'"t'r! s'8F!"t!( to $oa(s. As t#! $oa( is

in"r!as!(@ it r!a"#!s a critical%a$'!


11/192

SY33ETRIC BI6URCATI2N

Post48'";$in& $oa(4(!orm. 1at#s ar! symmetrica8o't $oa( ais. I t#! $oa( "a1a"it) in"r!as!s at!r 8'";$in& t#!n sta8$! s)mm!tri"

8i'r"ation.

I t#! $oa( "a1a"it) (!"r!as!s at!r 8'";$in& t#!n 'nsta8$!

s)mm!tri" 8i'r"ation.


12/192

ASY33ETRIC BI6URCATI2N

Post48'";$in& 8!#a%ior t#at is as)mm!tri" a8o't $oa( ais.


13/192

INSTABILITY 6AILURE

T#!r! is no 8i'r"ation o t#! $oa(4(!ormation 1at#. T#!(!ormation sta)s in stat!4/ t#ro'o't

T#! str'"t'r! stin!ss (!"r!as!s as t#! $oa(s ar! in"r!as!(.

T#! "#an&! is stin!ss is ('! to $ar&! (!ormations an( or

mat!ria$ in!$asti"it). T#! str'"t'r! stin!ss (!"r!as!s to !ro an( 8!"om!s n!&ati%!.

T#! $oa( "a1a"it) is r!a"#!(


14/192

INSTABILITY 6AILURE

6AILURE 26 BEA34C2LU3NSP

P

3

3

K

K=0

K


15/192

INSTABILITY 6AILURE

Sna14t#ro' 8'";$in&P

Snap-through


16/192

INSTABILITY 6AILURE

S#!$$ B'";$in& ai$'r! %!r) s!nsiti%! to im1!r!"tions


17/192



T)1!s o insta8i$it)

3!t#o(s o sta8i$it) ana$)s!s Eam1$!s sma$$ (!$!"tion ana$)s!s

Eam1$!s $ar&! (!$!"tion ana$)s!s

Eam1$!s im1!r!"t s)st!ms



18/192

3ETH2DS 26 STABILITY ANALYSES

Bi'r"ation a11roa"# "onsists o


19/192

STABILITY ANALYSES

Ea"# m!t#o( #as its a(%anta&!s an( (isa(%anta&!s. In a"t@

)o' "an 's! (i!r!nt m!t#o(s to ans

"riti"a$ 8'";$in& $oa( or a *1!r!"t, s)st!m s'8F!"t!( to $oa(s. T#! (!ormations ar! 's'a$$) ass'm!( to 8! sma$$. T#! s)st!m m'st not #a%! an) im1!r!"tions. It "annot 1ro%i(! an) inormation r!&ar(in& t#! 1ost48'";$in& $oa(4

(!ormation 1at#.

T#! !n!r&) a11roa"# is t#! 8!st


20/192

STABILITY ANALYSIS

T#! ()nami" m!t#o( is %!r) 1o

stin!ss. T#is is si&nii"ant )o' #a%! not s!!n it 8!or!.

#at #a11!ns


21/192

STABILITY ANALYSIS

FOR ANY KIND OF BUCKLING OR STABILITY ANALYSIS

NEED TO DRAW THE FREE BODY DIAGRAM OF THE DEFORMED

STRUCTURE.

WRITE THE EQUATION OF STATIC EQUILIBRIUM IN THE

DEFORMED STATE

WRITE THE ENERGY EQUATION IN THE DEFORMED STATE TOO.

THIS IS CENTRAL TO THE TOPIC OF STABILITY ANALYSIS

NO STABILITY ANALYSIS CAN BE PERFORMED IF THE FREE

BODY DIAGRAM IS IN THE UNDEFORMED STATE


22/192

BI6URCATI2N ANALYSIS

A$


23/192


rit! t#! !?'ation o stati" !?'i$i8ri'm in t#! (!orm!( stat!

T#'s@ t#! str'"t'r!


24/192


P

k

L

PL

L (1-cos)

L cos

Lsin

k L sinO

Eam1$! 4 Ri&i( 8ar s'11ort!( 8) trans$ationa$ s1rin& at !n(

Ass'm! (!orm!( stat! t#at a"ti%at!s a$$ 1ossi8$! (.o..

Dra< 6BD in t#! (!orm!( stat!


25/192


rit! !?'ations o stati" !?'i$i8ri'm in (!orm!( stat!

PL

L (1-cos)

L cos

Lsin

k L sinO

LkL

LkP

nsdeformatiosmallFor

L

LkP

LPLLkM

cr

o

===

=

=+=

2

2

sin

sin

sin

0sin)sin(0

JT#'s@ t#! str'"t'r!


26/192


Eam1$! T#r!! ri&i( 8ar s)st!m


27/192


rit! !?'ations o stati" !?'i$i8ri'm in (!orm!( stat!PP

A D

kk

LL

1

2

L sin 1L sin 2

BC(1 2)

(1 2)

P

A

L

1

B

L sin 1

1+(12!

0)2(0sin)2(0 121121 === LPkLPkMB

k(212!

P

D

k

L2L sin

2

C

2(1 2)

k(221!

0)2(0sin)2(0 212212 =+=+= LPkLPkMC


28/192


E?'ations o Stati" E?'i$i8ri'm

T#!r!or! !it#!r /an( ar! !?'a$ to !ro or t#! (!t!rminant o t#!

"o!i"i!nt matri is !?'a$ to !ro. #!n /an( ar! not !?'a$ to !ro t#at is


29/192


T#is is t#! "$assi"a$ !i&!n%a$'! 1ro8$!m. ([K]-[I]){x}={0}.

! ar! s!ar"#in& or t#! !i&!n%a$'!s *, o t#! stin!ss matri K.T#!s! !i&!n%a$'!s "a's! t#! stin!ss matri to 8!"om! sin&'$ar Sin&'$ar stin!ss matri m!ans t#at it #as a !ro %a$'!@


30/192


Ea"# !i&!n%a$'! or "riti"a$ 8'";$in& $oa( *P"r, "orr!s1on(s to a 8'";$in& s#a1!

t#at "an 8! (!t!rmin!( as o$$o

21

21

121

121

0

0)2(

0)2(

==

=

==

=

kk

kk

LkPPLet

LPk

cr

PP

A D

kk

LL

1

2"1

B C

21

21

212

212

0

0)2(

0)2(

==

=+

==

=+

kk

kk

LkPPLet

LPk

cr


31/192


S!"on( !i&!n%a$'!

R!m!m8!r t#at t#is is a sma$$ (!$!"tion ana$)sis. So@ t#! %a$'!s ar!n!&$i&i8$!. #at

B

C

21

21

212

212

0

03)2(

3

0)2(

==+

=+

==

=+

kk

kk

LkPPLet

LPk

cr


32/192


Hom!


33/192



T)1!s o insta8i$it)

3!t#o(s o sta8i$it) ana$)s!s

Bi'r"ation ana$)sis !am1$!s sma$$ (!$!"tion ana$)s!s

En!r&) m!t#o( Eam1$!s sma$$ (!$!"tion ana$)s!s

Eam1$!s $ar&! (!$!"tion ana$)s!s Eam1$!s im1!r!"t s)st!ms



34/192

ENERGY 3ETH2D

!


35/192

ENERGY 3ETHD

T#! !n!r&) m!t#o( is t#! 8!st or !sta8$is#in& t#! !?'i$i8ri'm

!?'ation an( !aminin& its sta8i$it) T#! (!ormations "an 8! sma$$ or $ar&!.

T#! s)st!m "an #a%! im1!r!"tions.

It 1ro%i(!s inormation r!&ar(in& t#! 1ost48'";$in& 1at# i $ar&!

(!ormations ar! ass'm!(

T#! maFor $imitation is t#at it r!?'ir!s t#! ass'm1tion o t#!

(!ormation stat!@ an( it s#o'$( in"$'(! a$$ 1ossi8$! (!&r!!s o

r!!(om.


36/192

ENERGY 3ETH2D

Eam1$! / Ri&i( 8ar s'11ort!( 8) rotationa$ s1rin&

Ass'm! sma$$ (!$!"tion t#!or)

St!1 / 4 Ass'm! a (!orm!( s#a1! t#at a"ti%at!s a$$ 1ossi8$! (.o..

Ri&i( 8ar s'8F!"t!( to aia$ or"! P

Rotationa$$) r!strain!( at !n(

Pk

L

L P

L cosL (1-cos)

k


37/192

ENERGY 3ETH2D S3ALL DE6LECTI2NS

rit! t#! !?'ation r!1r!s!ntin& t#! tota$ 1ot!ntia$ !n!r&) o s)st!mL (1-cos)

L P

L cos

k Lsin

LkP#$erefore

LPksdeflectionsmallFor

LPk#$erefored

dme%&ilibri&For

LPkd

d

LPk

LP'

k

'

cr

e

e

=

==

=

=

=

=

=

=

,

0;

0sin,

0;

sin

)cos1(2

1

)cos1(2

1

2

2


38/192


T#! !n!r&) m!t#o( 1r!(i"ts t#at 8'";$in&


39/192


In stat!4/@ sta8$!


40/192

ENERGY 3ETH2D LARGE DE6LECTI2NS

Eam1$! / Lar&! (!$!"tion ana$)sis *ri&i( 8ar


41/192


Lar&! (!$!"tion ana$)sis

S!! t#! 1ost48'";$in& $oa(4(is1$a"!m!nt 1at# s#o


42/192


Lar&! (!$!"tion ana$)sis Eamin! t#! sta8i$it) o !?'i$i8ri'm 'sin&

#i!r or(!r (!ri%ati%!s o

00,

).,.(0

)tan

1(

cossin

sin,

cos

sin

)cos1(2

1

2

2

2

2

2

2

2

2

2

2

2

==

>

=

=

=

=

=

=

forddB&t

*#ABLEAl.a/s

ofal&esalleiAl.a/sd

d

kd

d

LL

kk

d

d

L

kPB&t

LPkd

d

LPkd

d

LPk


43/192


At @ t#! s!"on( (!ri%ati%! o . T#!r!or!@ in"on"$'si%!.

Consi(!r t#! Ta)$or s!ri!s !1ansion o at

D!t!rmin! t#! irst non4!ro t!rm o @

Sin"! t#! irst non4!ro t!rm is @ t#! stat! is sta8$! at P=Pcran(

n

n

n

d

d

nd

d

d

d

d

d

d

d

0

4

0

4

43

0

3

32

0

2

2

00 !

1.....

!4

1

!3

1

!2

1

======

++

+

+

+

+=

cos

sin

cos

sin

)cos1(2

1

4

4

3

3

2

2

2

LPd

d

LPd

d

LPkd

d

LPkd

d

LPk

=

=

=

=

=

kPLLPd

d

LPd

dd

d

d

d

===

==

=

=

=

=

=

=

=

=

cos

0sin

0

0

0

0

4

4

0

3

3

0

2

2

0

00

24

1

!4

1 44

0

4

4

>=

=

kd

d


44/192


STABLE

STABLESTABL

E


45/192

ENERGY 3ETH2D I3PER6ECT SYSTE3S

Consi(!r !am1$! / 8't as a s)st!m


46/192


aboe,ienisi-relations$Pme%&ilibri$e

me%&ilibri&forL

kP#$erefore

LPk#$erefore

ddme%&ilibri&For

LPkd

d

LPk

LP'

k

'

e

e

=

=

=

=

=

=

=

=

sin

)(,

0sin)(,

0;

sin)(

)cos(cos)(2

1

)cos(cos

)(2

1

0

0

0

0

2

0

0

2

0L (cos0-cos)

L P

L cos

k(0) Lsin

0


47/192


sinsin

)(

0

00

belo.s$o.nofal&esdifferentfori-srelations$PP

P

L

kP

cr

=

=


48/192

ENERGY 3ETH2DS I3PER6ECT SYSTE3S

As s#o


49/192

ENERGY 3ETH2DS I3PER6ECT SYSTE3S

Eamin! t#! sta8i$it) o t#! im1!r!"t s)st!m 'sin& #i!r or(!r

(!ri%ati%!s o

#i"# is a$


50/192


P

k

L

PL

L (1-cos)

L cos

Lsin

k L sinO

Eam1$! 4 Ri&i( 8ar s'11ort!( 8) trans$ationa$ s1rin& at !n(

Ass'm! (!orm!( stat! t#at a"ti%at!s a$$ 1ossi8$! (.o..

Dra< 6BD in t#! (!orm!( stat!


51/192


rit! t#! !?'ation r!1r!s!ntin& t#! tota$ 1ot!ntia$ !n!r&) o s)st!m

PL

L (1-cos)

L cos

Lsin

k L sinO

LkP#$erefore

LPLksdeflectionsmallFor

LPLk#$erefore

d

d

me%&ilibri&For

LPLkd

d

LPLk

LP'

LkLk

'

cr

e

e

==

==

=

=

=

==

=

,

0;

0sin,

0;

sin

)cos1(2

1

)cos1(

2

1)sin(

2

1

2

2

2

22

222


52/192


53/192


aboe,ienisi-relations$Pb&cklin,-ost#$e

me%&ilibri&forLkP#$erefore

LPLk#$erefore

d

d

me%&ilibri&For

LPLkd

d

LPLk

LP'

Lk

'

e

e

=

==

=

=

=

=

=

cos,

0sincossin,

0;

sincossin

)cos1(sin2

1

)cos1(

)sin(2

1

2

2

22

2

PL

L (1-cos)

L cos

Lsin

O

rit! t#! !?'ation r!1r!s!ntin& t#! tota$ 1ot!ntia$ !n!r&) o s)st!m


54/192


Lar&! (!$!"tion ana$)sis S!! t#! 1ost48'";$in& $oa(4(is1$a"!m!nt 1at# s#o


55/192


Lar&! (!$!"tion ana$)sis Eamin! t#! sta8i$it) o !?'i$i8ri'm 'sin&

#i!r or(!r (!ri%ati%!s o

)*#ABLEE)CEAL'A*d

d

Lkd

d

LkLkd

d

LkLkd

d

LkPme%&ilibri&For

LPLkdd

LPLkd

d

LPLk

.0

sin

cos)sin(cos

cos2cos

cos

cos2cos

sincossin

)cos1(sin2

1

2

2

22

2

2

22222

2

2

222

2

2

2

2

2

2

22

alculate t*tal strain ,*r each ,iber

t*t=r

;ssume a material stress&strain

cure ,*r each ,iber>alculate stress in each ,iber ,ib

>alculate ;6ial *rce = P

Sum ,ib;,ib

>alculate aerage stress = = P8;

>alculate the tangent '($3and '($4,*r the

'($3= sum'$&,ibIy,ib2;,ib(6&,ibJ

'($y= sum'$&,ibI6,ib2;,ib (y&,ibJ

3 4#93&cr= srt K'($68PL

#9y&cr= srt K'($y8PL

D

C

N10

11

12

1?

1@

Tan&!nt mo('$'s 8'";$in& 4 n'm!ri"a$


177/192

'ection i*ension

12 fier no. ,fi %fi &fi rfi rfi -%fi -&fi

d 4 1 2.4 5." 0 22.5 "."59E04 !.2 "8.05& 50 2 2.4 5.1 0 1".5 6.0!4E04 !.2 62.50

! 2.4 4.5 0 12.5 4.!10E04 !.2 48.6"

o. of fiers 20 4 2.4 !.9 0 ".5 2.586E04 !.2 !6.58

5 2.4 !.! 0 2.5 8.621E05 !.2 26.21

6 2.4 2." 0 2.5 8.621E05 !.2 1".5"

, 48 " 2.4 2.1 0 ".5 2.586E04 !.2 10.66

-% 64 8 2.4 1.5 0 12.5 4.!10E04 !.2 5.4"

-& 5"6.00 9 2.4 0.9 0 1".5 6.0!4E04 !.2 2.02

10 2.4 0.! 0 22.5 "."59E04 !.2 0.29

11 2.4 0.! 0 22.5 "."59E04 !.2 0.29

12 2.4 0.9 0 1".5 6.0!4E04 !.2 2.02

1! 2.4 1.5 0 12.5 4.!10E04 !.2 5.4"

14 2.4 2.1 0 ".5 2.586E04 !.2 10.66

15 2.4 2." 0 2.5 8.621E05 !.2 1".5"

16 2.4 !.! 0 2.5 8.621E05 !.2 26.21

1" 2.4 !.9 0 ".5 2.586E04 !.2 !6.58

18 2.4 4.5 0 12.5 4.!10E04 !.2 48.6"

19 2.4 5.1 0 1".5 6.0!4E04 !.2 62.50

20 2.4 5." 0 22.5 "."59E04 !.2 "8.05

Tan&!nt 3o('$'s B'";$in& 4 n'm!ri"a$

'train -ncre*ent


178/192

'train -ncre*ent

ier no. tot fi Efi T4i8 T)4i8 Pi8

0.000! 1 1.0"6E0! !1.2 29000 92800 2.26E+06 "4.882 9.0!4E04 26.2 29000 92800 1.81E+06 62.88

! ".!10E04 21.2 29000 92800 1.41E+06 50.88

4 5.586E04 16.2 29000 92800 1.06E+06 !8.88

5 !.862E04 11.2 29000 92800 ".60E+05 26.88

6 2.1!8E04 6.2 29000 92800 5.09E+05 14.88

" 4.1!8E05 1.2 29000 92800 !.09E+05 2.88

8 1.!10E04 !.8 29000 92800 1.59E+05 9.12

9 !.0!4E04 8.8 29000 92800 5.85E+04 21.12

10 4."59E04 1!.8 29000 92800 8.!5E+0! !!.12

11 4."59E04 1!.8 29000 92800 8.!5E+0! !!.1212 !.0!4E04 8.8 29000 92800 5.85E+04 21.12

1! 1.!10E04 !.8 29000 92800 1.59E+05 9.12

14 4.1!8E05 1.2 29000 92800 !.09E+05 2.88

15 2.1!8E04 6.2 29000 92800 5.09E+05 14.88

16 !.862E04 11.2 29000 92800 ".60E+05 26.88

1" 5.586E04 16.2 29000 92800 1.06E+06 !8.88

18 ".!10E04 21.2 29000 92800 1.41E+06 50.88

19 9.0!4E04 26.2 29000 92800 1.81E+06 62.88

20 1.0"6E0! !1.2 29000 92800 2.26E+06 "4.88

Tan&!nt 3o('$'s B'";$in& 4 N'm!ri"a$

# L L / *Lr, *Lr,


179/192

# T T) L4"r L)4"r T/$ *Lr, *Lr,)0.0003 41".6 1856000 16704000 9.-95/ +./57 0.1"4 181.!"99!64 181.!"99!64

0.0004 556.8

1856000 16704000//.799+- 5--./99 0.2!2 15".0"96!2" 15".0"96!2"

0.0005 696 1856000 16"04000 162.2!114" 486.69!4411 0.29 140.4962946 140.4962946

0.0006 8!5.2 1856000 16"04000 148.09609"9 444.28829!8 0.!48 128.25498! 128.25498!

0.000" 9"4.4 1856000 16"04000 1!".110!442 411.!!10!25 0.406 118."410412 118."410412

0.0008 111!.6 1856000 16"04000 128.25498! !84."64949 0.464 111.0"20"!5 111.0"20"!5

0.0009 1252.8 1856000 16"04000 120.91995"6 !62."598"28 0.522 104."19"551 104."19"551

0.001 1!84.8 16"0400 121""216 109.11051 294.598!""1 0.5"" 94.4924"!52 85.04!2261"

0.0011 1510.08 16"0400 121""216 104.4864889 282.11!5199 0.6292 90.48"95!"1 81.4!9158!4

0.0012 1624.!2 1484800 8552448 94.98!4"542 22".960!41 0.6"68 82.25810265 65.80648212

0.001! 1"!4."2 1299200 5"294"2 85.9"51982! 180.54"916! 0."228 "4.456"05"6 52.1196940!

0.0014 18!2.16 1299200 5"294"2 8!.65""5001 1"5.6812"5 0."6!4 "2.449"!6"! 50."14815"10.0015 1924.8 111!600 !608064 "5.5651"26! 1!6.01"!10" 0.802 65.441!5914 !9.26481548

0.0016 2008.!2 111!600 !608064 "!.9""22!46 1!!.1590022 0.8!68 64.06615482 !8.4!969289

0.001" 208!.2 928000 2088000 66.!0684"06 99.4602"059 0.868 5".42!414 28."11"0"

0.0018 2152.8 928000 2088000 65.22619108 9".8!92866! 0.89" 56.48"5!84" 28.24!"6924

0.0019 2209.92 "42400 1069056 5".581182!! 69.09"4188 0.9208 49.866"6668 19.946"066"

0.002 226!.2 556800 451008 49.2"629185 44.!486626" 0.94! 42.6"452055 12.802!5616

0.0021 2!04.96 556800 451008 48.82"8"11 4!.94508!99 0.9604 42.2861"6"9 12.68585!04

0.0022 2!40.48 !"1200 1!!6!2 !9.5641089" 2!."!8465!8 0.9"52 !4.26!52!44 6.852"04688

0.002! 2!68.!2 !"1200 1!!6!2 !9.!!088015 2!.59852809 0.9868 !4.061541!6 6.812!082"!

0.0024 2!86.08 185600 16"04 2"."0"4!"25 8.!122!11"6 0.9942 2!.995!445! 2.!995!445!0.00249 2!98.608 185600 16"04 2".6!498414 8.29049524! 0.99942 2!.9!2598! 2.!9!2598!

Tan&!nt 3o('$'s B'";$in& 4 N'm!ri"a$


180/192

Inelastic Column Buckling

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140 160 180 200

KL/r ratio

(KL/r)% (KL/r)&

Column Inelastic Buckling

1 2 1 2


181/192

0

0.2

0.4

0.6

0.8

1

1.2

0.0 0.5 1.0 1.5 2.0

Lam#da

0.0

0.2

0.4

0.6

0.8

1.0

1.2

u*% u*& ,na&tica%

Eastic ,-'esin ,na&tica&


182/192


183/192

ELASTIC BUCLING 26 BEA3S

Goin& 8a"; to t#! ori&ina$ t#r!! s!"on(4or(!r (i!r!ntia$


184/192

& &

!?'ations:

( ) ( )

( ) ( )

0

0

0

0

,

( ) ( ( ) )

( ( ) ) ( ) ( ) 0

3 B # B B< #< B


185/192

Consi(!r t#! "as! o a 8!am s'8F!"t!( to 'niaia$ 8!n(in& on$):

8!"a's! most st!!$ str'"t'r!s #a%! 8!ams in 'niaia$ 8!n(in& B!ams 'n(!r 8iaia$ 8!n(in& (o not 'n(!r&o !$asti" 8'";$in&

P 3TY3BY

T#! t#r!! !?'ations sim1$i) to:

E?'ation */, is an 'n"o'1$!( (i!r!ntia$ !?'ation (!s"ri8in& in4

1$an! 8!n(in& 8!#a%ior "a's!( 8) 3TXan( 3BX

( )

( )

( ) ( ) ( ) 0

3 B< #< B


186/192

E?'ations *, an( *, ar! "o'1$!( !?'ations in ' an( t#at

(!s"ri8! t#! $at!ra$ 8!n(in& an( torsiona$ 8!#a%ior o t#! 8!am.In a"t t#!) (!in! t#! $at!ra$ torsiona$ 8'";$in& o t#! 8!am.

T#! 8!am m'st satis) a$$ t#r!! !?'ations */@ @ an( ,. H!n"!@

8!am in41$an! 8!n(in&


187/192

( )

2

2 2

2 2 2 2

0 0

0

( ) 0

,

( ) ( )

2 2

/ o

. # o

A

o

3

oo o

3A

oo o

3

E I & M

E I 9 8 8 & M

.$ere

8 'a,ner s effect d&e to .ar-in, ca&sed b/ torsion

8 a dA

MB&t / ne,lectin, $i,$er order terms

I

M8 / 3 3 / / dA

IM

8 / 3 3 33 / / // dAI

+ =

+ + =

=

=

=

= +

= + + +

2 2 2 2 2

0 2 2

A

oo o o

3 A A A A A

M8 3 / dA / 3 / dA 3 3/ dA / / dA / / dA

I

= + + +


2 22o

M8 / 3 / dA / I

= +


188/192

2 2

2 2

2

2

, 2

s#c

o 3

3 A

Ao o

3

Ao 3 3 o

3

3

8 / 3 / dA / II

/ 3 / dA

8 M /I

/ 3 / dA

8 M .$ere /I

is a ne. tional -ro-ert/

+

+ =

+ = =

( )

(2) 0

(3) ( ) 0

/ o

. # o 3 o

#$e beam b&cklin, differential e%&ations become

E I & M

E I 9 8 M & M

+ =

+ + =


(2) oM

E%&ation ,ies &E I

=


189/192

2

2

2

2

1 2 2

1 2

(2) (3)

( ) 0

s#c 0

0

,

0

/

i o. # o 3

/

3

i o#

. / .

o#

. / .

i

E I

*&bstit&tin, & from E%&ation in ,ies

ME I 9 8 M

E I

For do&bl/ s/mmetric tion

M9 8

E I E I I

M9 8Let and

E I E I I

+ =

= =

= =

= . .becomes t$e combined d e of L#B


( )4 2 0

6

6

Ass&me sol&tion is of t$e form e

=


190/192

( )

1 1 2 2

4 2

1 2

4 2

1 2

2 2

1 1 2 1 2 12

2 2

1 1 2 1 1 2

1 2

1 2 3 4

0

0

4 4,

2 2

4 4

,2 2

, ,

6

6 6 i 6 i 6

e

i

Let and i

Aboe are t$e fo&r roots for

C e C e C e C e

collect

=

=

+ + + =

+ + + + =

=

= + + +

1 1 2 1 3 2 4 2cos$( ) sin$( ) sin( ) cos( )

in, real and ima,inar/ terms

9 6 9 6 9 6 9 6 = + + +


Ass'm! sim1$) s'11ort!( 8o'n(ar) "on(itions or t#! 8!am:


191/192

1 ) 11 )

12 2

1 2 2

1 1 2 2 3

2 2 2 2 41 1 1 1 2 2 2 2

(0) (0) ( ) ( ) 0. .

1 0 0 1

0 00

cos$( ) sin$( ) sin( ) cos( )

cos$( ) sin$( ) sin( ) cos( )

L L*ol&tion for m&st satisf/ all fo&r b c

9

9

L L L L 9

9L L L L

For b&ckl

= = = =

=

( )2 21 2 1 2

2

2

sin

#t in 0

sin$( ) sin$( ) 0

sin$( ) 0

in, coefficient matri3 m&st be ,&lar

er ant of matri3

L L

4f t$eseonl/ L

L n

=

+ =

= =


2

n

L

=


192/192

21 2 1

22

1 2 1 2

22 2 2

21 1 12 2 2

2

2 2

2 12 2

4

2

24

2 2 22

4 4

L

L

L L L

L L

+ =

+ =

+ + = =

= +

CE579 Half Course Summary

Documents

Transcript of CE579 Half Course Summary