ce579 half course summary
TRANSCRIPT
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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$: a#%arma01'r('!.!('
2i"! #o'rs: 3446 9:4//: a.m.
mailto:[email protected]:[email protected] -
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C#a1t!r /. Intro('"tion to Str'"t'ra$ 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$$ (!$!"tion ana$)s!s
Eam1$!s $ar&! (!$!"tion ana$)s!s
Eam1$!s im1!r!"t s)st!ms
D!si&n o st!!$ str'"t'r!s
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STABILITY DE6INITI2N
C#an&! in &!om!tr) o a str'"t'r! or str'"t'ra$ "om1on!nt'n(!r "om1r!ssion r!s'$tin& in $oss o a8i$it) to r!sist $oa(in&
is (!in!( as instability in t#! 8oo;.
Insta8i$it) "an $!a( to "atastro1#i" ai$'r!m'st 8! a""o'nt!(
in (!si&n. Insta8i$it) is a str!n&t#4r!$at!( $imit stat!.
#) (i(
Sta8i$it) is not !as) to (!in!. E%!r) str'"t'r! is in !?'i$i8ri'm stati" or ()nami". I it is not in
!?'i$i8ri'm@ t#! 8o()
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STABILITY DE6INITI2N
Str'"t'r! is in sta8$! !?'i$i8ri'm
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STABILITY DE6INITI2N
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BUCLING Vs. STABILITY
C#an&! in &!om!tr) o str'"t'r! 'n(!r "om1r!ssion t#atr!s'$ts in its a8i$it) to r!sist $oa(s "a$$!( instability.
Not tr'! t#is is "a$$!( buckling.
Buckling is a 1#!nom!non t#at "an o""'r or str'"t'r!s 'n(!r
"om1r!ssi%! $oa(s. T#! str'"t'r! (!orms an( is in sta8$! !?'i$i8ri'm in stat!4/.
As t#! $oa( in"r!as!s@ t#! str'"t'r! s'((!n$) "#an&!s to
(!ormation stat!4 at som! "riti"a$ $oa( P"r.
T#! str'"t'r! 8'";$!s rom stat!4/ to stat!4@
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BUCLING Vs. STABILITY
T#'s@ t#!r! ar! t
Sta8i$it) o !?'i$i8ri'm As t#! $oa(s a"tin& on t#! str'"t'r! ar!
in"r!as!(@
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TYPES 26 INSTABILITY
Str'"t'r! s'8F!"t!( to "om1r!ssi%! or"!s "an 'n(!r&o:/. B'";$in& 8i'r"ation o !?'i$i8ri'm rom (!ormation stat!4/ to
stat!4. Bi'r"ation 8'";$in& o""'rs or "o$'mns@ 8!ams@ an( s)mm!tri"
ram!s 'n(!r &ra%it) $oa(s on$)
. 6ai$'r! ('! to insta8i$it) o !?'i$i8ri'm stat!4/ ('! to $ar&!
(!ormations or mat!ria$ in!$asti"it) E$asti" insta8i$it) o""'rs or 8!am4"o$'mns@ an( ram!s s'8F!"t!(
to &ra%it) an( $at!ra$ $oa(s.
In!$asti" insta8i$it) "an o""'r or a$$ m!m8!rs an( t#! ram!.
!
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TYPES 26 INSTABILITY
BI6URCATI2N BUCLING 3!m8!r or str'"t'r! s'8F!"t!( to $oa(s. As t#! $oa( is
in"r!as!(@ it r!a"#!s a critical%a$'!
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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$!
s)mm!tri" 8i'r"ation.
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ASY33ETRIC BI6URCATI2N
Post48'";$in& 8!#a%ior t#at is as)mm!tri" a8o't $oa( ais.
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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#! $oa( "a1a"it) is r!a"#!(
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INSTABILITY 6AILURE
6AILURE 26 BEA34C2LU3NSP
P
3
3
K
K=0
K
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INSTABILITY 6AILURE
Sna14t#ro' 8'";$in&P
Snap-through
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INSTABILITY 6AILURE
S#!$$ B'";$in& ai$'r! %!r) s!nsiti%! to im1!r!"tions
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C#a1t!r /. Intro('"tion to Str'"t'ra$ Sta8i$it)
2UTLINE D!inition o sta8i$it)
T)1!s o insta8i$it)
3!t#o(s o sta8i$it) ana$)s!s Eam1$!s sma$$ (!$!"tion ana$)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
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3ETH2DS 26 STABILITY ANALYSES
Bi'r"ation a11roa"# "onsists o
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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& $oa(4
(!ormation 1at#.
T#! !n!r&) a11roa"# is t#! 8!st
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STABILITY ANALYSIS
T#! ()nami" m!t#o( is %!r) 1o
stin!ss. T#is is si&nii"ant )o' #a%! not s!!n it 8!or!.
#at #a11!ns
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STABILITY ANALYSIS
FOR ANY KIND OF BUCKLING OR STABILITY ANALYSIS
NEED TO DRAW THE FREE BODY DIAGRAM OF THE DEFORMED
STRUCTURE.
WRITE THE EQUATION OF STATIC EQUILIBRIUM IN THE
DEFORMED STATE
WRITE THE ENERGY EQUATION IN THE DEFORMED STATE TOO.
THIS IS CENTRAL TO THE TOPIC OF STABILITY ANALYSIS
NO STABILITY ANALYSIS CAN BE PERFORMED IF THE FREE
BODY DIAGRAM IS IN THE UNDEFORMED STATE
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BI6URCATI2N ANALYSIS
A$
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BI6URCATI2N ANALYSIS
rit! t#! !?'ation o stati" !?'i$i8ri'm in t#! (!orm!( stat!
T#'s@ t#! str'"t'r!
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BI6URCATI2N ANALYSIS
P
k
L
PL
L (1-cos)
L cos
Lsin
k L sinO
Eam1$! 4 Ri&i( 8ar s'11ort!( 8) trans$ationa$ s1rin& at !n(
Ass'm! (!orm!( stat! t#at a"ti%at!s a$$ 1ossi8$! (.o..
Dra< 6BD in t#! (!orm!( stat!
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BI6URCATI2N ANALYSIS
rit! !?'ations o stati" !?'i$i8ri'm in (!orm!( stat!
PL
L (1-cos)
L cos
Lsin
k L sinO
LkL
LkP
nsdeformatiosmallFor
L
LkP
LPLLkM
cr
o
===
=
=+=
2
2
sin
sin
sin
0sin)sin(0
JT#'s@ t#! str'"t'r!
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BI6URCATI2N ANALYSIS
Eam1$! T#r!! ri&i( 8ar s)st!m
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BI6URCATI2N ANALYSIS
rit! !?'ations o stati" !?'i$i8ri'm in (!orm!( stat!PP
A D
kk
LL
1
2
L sin 1L sin 2
BC(1 2)
(1 2)
P
A
L
1
B
L sin 1
1+(12!
0)2(0sin)2(0 121121 === LPkLPkMB
k(212!
P
D
k
L2L sin
2
C
2(1 2)
k(221!
0)2(0sin)2(0 212212 =+=+= LPkLPkMC
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BI6URCATI2N ANALYSIS
E?'ations o Stati" E?'i$i8ri'm
T#!r!or! !it#!r /an( ar! !?'a$ to !ro or t#! (!t!rminant o t#!
"o!i"i!nt matri is !?'a$ to !ro. #!n /an( ar! not !?'a$ to !ro t#at is
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BI6URCATI2N ANALYSIS
T#is is t#! "$assi"a$ !i&!n%a$'! 1ro8$!m. ([K]-[I]){x}={0}.
! ar! s!ar"#in& or t#! !i&!n%a$'!s *, o t#! stin!ss matri K.T#!s! !i&!n%a$'!s "a's! t#! stin!ss matri to 8!"om! sin&'$ar Sin&'$ar stin!ss matri m!ans t#at it #as a !ro %a$'!@
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BI6URCATI2N ANALYSIS
Ea"# !i&!n%a$'! or "riti"a$ 8'";$in& $oa( *P"r, "orr!s1on(s to a 8'";$in& s#a1!
t#at "an 8! (!t!rmin!( as o$$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
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BI6URCATI2N ANALYSIS
S!"on( !i&!n%a$'!
R!m!m8!r t#at t#is is a sma$$ (!$!"tion ana$)sis. So@ t#! %a$'!s ar!n!&$i&i8$!. #at
B
C
21
21
212
212
0
03)2(
3
0)2(
==+
=+
==
=+
kk
kk
LkPPLet
LPk
cr
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BI6URCATI2N ANALYSIS
Hom!
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C#a1t!r /. Intro('"tion to Str'"t'ra$ Sta8i$it)
2UTLINE D!inition o sta8i$it)
T)1!s o insta8i$it)
3!t#o(s o sta8i$it) ana$)s!s
Bi'r"ation ana$)sis !am1$!s sma$$ (!$!"tion ana$)s!s
En!r&) m!t#o( Eam1$!s sma$$ (!$!"tion ana$)s!s
Eam1$!s $ar&! (!$!"tion ana$)s!s Eam1$!s im1!r!"t s)st!ms
D!si&n o st!!$ str'"t'r!s
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ENERGY 3ETH2D
!
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ENERGY 3ETHD
T#! !n!r&) m!t#o( is t#! 8!st or !sta8$is#in& t#! !?'i$i8ri'm
!?'ation an( !aminin& its sta8i$it) T#! (!ormations "an 8! sma$$ 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$$ 1ossi8$! (!&r!!s o
r!!(om.
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ENERGY 3ETH2D
Eam1$! / Ri&i( 8ar s'11ort!( 8) rotationa$ s1rin&
Ass'm! sma$$ (!$!"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
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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
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ENERGY 3ETH2D S3ALL DE6LECTI2NS
T#! !n!r&) m!t#o( 1r!(i"ts t#at 8'";$in&
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ENERGY 3ETH2D S3ALL DE6LECTI2NS
In stat!4/@ sta8$!
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ENERGY 3ETH2D LARGE DE6LECTI2NS
Eam1$! / Lar&! (!$!"tion ana$)sis *ri&i( 8ar
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ENERGY 3ETH2D LARGE DE6LECTI2NS
Lar&! (!$!"tion ana$)sis
S!! t#! 1ost48'";$in& $oa(4(is1$a"!m!nt 1at# s#o
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ENERGY 3ETH2D LARGE DE6LECTI2NS
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
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ENERGY 3ETH2D LARGE DE6LECTI2NS
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$! at P=Pcran(
n
n
n
d
d
nd
d
d
d
d
d
d
d
0
4
0
4
43
0
3
32
0
2
2
00 !
1.....
!4
1
!3
1
!2
1
======
++
+
+
+
+=
cos
sin
cos
sin
)cos1(2
1
4
4
3
3
2
2
2
LPd
d
LPd
d
LPkd
d
LPkd
d
LPk
=
=
=
=
=
kPLLPd
d
LPd
dd
d
d
d
===
==
=
=
=
=
=
=
=
=
cos
0sin
0
0
0
0
4
4
0
3
3
0
2
2
0
00
24
1
!4
1 44
0
4
4
>=
=
kd
d
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ENERGY 3ETH2D LARGE DE6LECTI2NS
STABLE
STABLESTABL
E
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ENERGY 3ETH2D I3PER6ECT SYSTE3S
Consi(!r !am1$! / 8't as a s)st!m
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ENERGY 3ETH2D I3PER6ECT SYSTE3S
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
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ENERGY 3ETH2D I3PER6ECT SYSTE3S
sinsin
)(
0
00
belo.s$o.nofal&esdifferentfori-srelations$PP
P
L
kP
cr
=
=
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ENERGY 3ETH2DS I3PER6ECT SYSTE3S
As s#o
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ENERGY 3ETH2DS I3PER6ECT SYSTE3S
Eamin! t#! sta8i$it) o t#! im1!r!"t s)st!m 'sin& #i!r or(!r
(!ri%ati%!s o
#i"# is a$
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ENERGY 3ETH2D S3ALL DE6LECTI2NS
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!
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ENERGY 3ETH2D S3ALL DE6LECTI2NS
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
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ENERGY 3ETH2D LARGE DE6LECTI2NS
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
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ENERGY 3ETH2D LARGE DE6LECTI2NS
Lar&! (!$!"tion ana$)sis S!! t#! 1ost48'";$in& $oa(4(is1$a"!m!nt 1at# s#o
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ENERGY 3ETH2D LARGE DE6LECTI2NS
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$
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'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
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'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,
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# 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!
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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
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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&
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ELASTIC BUCLING 26 BEA3S
Goin& 8a"; to t#! ori&ina$ t#r!! s!"on(4or(!r (i!r!ntia$
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& &
!?'ations:
( ) ( )
( ) ( )
0
0
0
0
,
( ) ( ( ) )
( ( ) ) ( ) ( ) 0
3 B # B B< #< B
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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
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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$an! 8!n(in&
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( )
2
2 2
2 2 2 2
0 0
0
( ) 0
,
( ) ( )
2 2
/ o
. # o
A
o
3
oo o
3A
oo o
3
E I & M
E I 9 8 8 & M
.$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
= + + +
ELASTIC BUCLING 26 BEA3S
2 22o
M8 / 3 / dA / I
= +
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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
+ =
+ + =
ELASTIC BUCLING 26 BEA3S
(2) oM
E%&ation ,ies &E I
=
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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$e combined d e of L#B
ELASTIC BUCLING 26 BEA3S
( )4 2 0
6
6
Ass&me sol&tion is of t$e form e
=
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( )
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$( ) sin( ) cos( )
in, real and ima,inar/ terms
9 6 9 6 9 6 9 6 = + + +
ELASTIC BUCLING 26 BEA3S
Ass'm! sim1$) s'11ort!( 8o'n(ar) "on(itions or t#! 8!am:
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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
=
+ =
= =
ELASTIC BUCLING 26 BEA3S
2
n
L
=
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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
+ =
+ =
+ + = =
= +