CE2351 SA 2 Lecture Notes

7/25/2019 CE2351 SA 2 Lecture Notes

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UNIT-1

FLEXIBILITY MATRIX METHODS

Since twentieth century, ineter!in"te #tructure# "re $ein% wie&y u#e '(r it# ($)i(u#

!erit#* It !"y $e rec"&&e th"t, in the c"#e (' ineter!in"te #tructure# either the re"cti(n# (r the

intern"& '(rce# c"nn(t $e eter!ine 'r(! e+u"ti(n# (' #t"tic# "&(ne* In #uch #tructure#, thenu!$er (' re"cti(n# (r the nu!$er (' intern"& '(rce# ecee# the nu!$er (' #t"tic e+ui&i$riu!

e+u"ti(n#* In "iti(n t( e+ui&i$riu! e+u"ti(n#, c(!"ti$i&ity e+u"ti(n# "re u#e t( e)"&u"te the

un.n(wn re"cti(n# "n intern"& '(rce# in #t"tic"&&y ineter!in"te #tructure* In the "n"&y#i# ('

ineter!in"te #tructure it i# nece##"ry t( #"ti#'y the e+ui&i$riu! e+u"ti(n# /i!&yin% th"t the

#tructure i# in e+ui&i$riu!0 c(!"ti$i&ity e+u"ti(n# /re+uire!ent i' '(r "##urin% the c(ntinuity ('

the #tructure with(ut "ny $re".#0 "n '(rce i#&"ce!ent e+u"ti(n# /the w"y in which

i#&"ce!ent "re re&"te t( '(rce#0* e h")e tw( i#tinct !eth( (' "n"&y#i# '(r #t"tic"&&y

ineter!in"te #tructure eenin% u(n h(w the "$()e e+u"ti(n# "re #"ti#'ie2

1* F(rce !eth( (' "n"&y#i# /"( .n(wn "# '&ei$i&ity !eth( (' "n"&y#i#, !eth( (' c(n#i#tent

e'(r!"ti(n, '&ei$i&ity !"tri !eth(0

3* Di#&"ce!ent !eth( (' "n"&y#i# /"( .n(wn "# #ti''ne## !"tri !eth(0*

In the '(rce !eth( (' "n"&y#i#, ri!"ry un.n(wn "re '(rce#* In thi# !eth( c(!"ti$i&ity

e+u"ti(n# "re written '(r i#&"ce!ent "n r(t"ti(n# /which "re c"&cu&"te $y '(rce i#&"ce!ent

e+u"ti(n#0* S(&)in% the#e e+u"ti(n#, reun"nt '(rce# "re c"&cu&"te* Once the reun"nt '(rce#

"re c"&cu&"te, the re!"inin% re"cti(n# "re e)"&u"te $y e+u"ti(n# (' e+ui&i$riu!*

In the i#&"ce!ent !eth( (' "n"&y#i#, the ri!"ry un.n(wn# "re the i#&"ce!ent#* In thi#

!eth(, 'ir#t '(rce -i#&"ce!ent re&"ti(n# "re c(!ute "n #u$#e+uent&y e+u"ti(n# "re written

#"ti#'yin% the e+ui&i$riu! c(niti(n# (' the #tructure* A'ter eter!inin% the un.n(wn

i#&"ce!ent#, the (ther '(rce# "re c"&cu&"te #"ti#'yin% the c(!"ti$i&ity c(niti(n# "n '(rce

i#&"ce!ent re&"ti(n#* The i#&"ce!ent-$"#e !eth( i# "!en"$&e t( c(!uter r(%r"!!in%

"n hence the !eth( i# $ein% wie&y u#e in the !(ern "y #tructur"& "n"&y#i#* In %ener"&, the

!"i!u! e'&ecti(n "n the !"i!u! #tre##e# "re #!"&& "# c(!"re t( #t"tic"&&y eter!in"te

#tructure* F(r e"!&e, c(n#ier tw( $e"!# (' ientic"& cr(## #ecti(n "n #"n c"rryin%

uni'(r!&y i#tri$ute &(" "# #h(wn in Fi%* 4*1" "n Fi%* 4*1$*


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The &("# "re "( the #"!e in $(th c"#e#* In the 'ir#t c"#e, the $e"! i# 'ie "t $(th en# "n

thu# i# #t"tic"&&y ineter!in"te* The #i!&y #u(rte $e"! in Fi%* 4*1$ i# " #t"tic"&&y

eter!in"te #tructure* The !"i!u! $enin% !(!ent in c"#e (' 'ie- 'ie $e"! i# wL2/12

/which (ccur# "t the #u(rt#0 "# c(!"re t( w&356 /"t the centre0 in c"#e (' #i!&y #u(rte

$e"!* A( in the re#ent c"#e, the e'&ecti(n in the c"#e (' 'ie- 'ie $e"!

w&75867EI i# 'i)e ti!e# #!"&&er th"n th"t (' #i!&y #u(rte $e"! 9w&

75867EI * A(, there i#

rei#tri$uti(n (' #tre##e# in the c"#e (' reun"nt #tructure* Hence i' (ne !e!$er '"i, #tructure

(e# n(t c(&&"#e #uen&y* The re!"inin% !e!$er# c"rry the &("* The eter!in"te #tructur"&

#y#te! c(&&"#e# i' (ne !e!$er '"i* H(we)er, there "re i#")"nt"%e# in u#in% ineter!in"te

#tructure#* Due t( #u(rt #ett&e!ent, there wi&& $e "iti(n"& #tre##e# in the c"#e (' reun"nt

#tructure# where "# eter!in"te #tructure# "re n(t "''ecte $y #u(rt #ett&e!ent*

The "n"&y#i# (' ineter!in"te #tructure i''er# !"in&y in tw( "#ect# "# c(!"re t( eter!in"te

#tructure*

"0 T( e)"&u"te #tre##e# in ineter!in"te #tructure#, ""rt 'r(! #ecti(n"& r(ertie# /"re" (' cr(## #ecti(n"n !(!ent (' inerti"0, e&"#tic r(ertie# "re "( re+uire*

$0 Stre##e# "re e)e&(e in ineter!in"te #tructure ue t( #u(rt #ett&e!ent#, te!er"ture ch"n%e "n

'"$ric"ti(n err(r# etc*


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UNIT-3 >&"#tic An"&y#i#

1. Introduction

1.1 Background

U t( n(w we h")e c(ncentr"te (n the e&"#tic "n"&y#i# (' #tructure#* In the#e "n"&y#e# we u#e

#uer(#iti(n ('ten, .n(win% th"t '(r " &ine"r&y e&"#tic #tructure it w"# )"&i* H(we)er, "n e&"#tic

"n"&y#i# (e# n(t %i)e in'(r!"ti(n "$(ut the &("# th"t wi&& "ctu"&&y c(&&"#e " #tructure* An

ineter!in"te #tructure !"y #u#t"in &("# %re"ter th"n the &(" th"t 'ir#t c"u#e# " yie& t( (ccur "t

"ny (int in the #tructure* In '"ct, " #tructure wi&& #t"n "# &(n% "# it i# "$&e t( 'in reun"ncie#

t( yie&* It i# (n&y when " #tructure h"# eh"u#te "&& (' it# reun"ncie# wi&& etr" &(" c"u#e# it

t( '"i&* >&"#tic "n"&y#i# i# the !eth( thr(u%h which the "ctu"& '"i&ure &(" (' " #tructure i#

c"&cu&"te, "n "# wi&& $e #een, thi# '"i&ure &(" c"n $e #i%ni'ic"nt&y %re"ter th"n the e&"#tic &("

c""city*

T( #u!!"ri?e thi#, >r('* Se"n e =(urcy /U=D0 u#e t( #"y2

a structure only collapses when it has exhausted all means of standing*

Be'(re "n"&y?in% c(!&ete #tructure#, we re)iew !"teri"& "n cr(## #ecti(n $eh")i(ur

$ey(n the e&"#tic &i!it*

Basis of Plastic Design

2.1 Material Behaviour

A uni"i"& ten#i&e #tre## (n " ucti&e !"teri"& #uch "# !i& #tee& tyic"&&y r()ie# the

'(&&(win% %r"h (' #tre## )er#u# #tr"in2


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A# c"n $e #een, the !"teri"& c"n #u#t"in #tr"in# '"r in ece## (' the #tr"in "t which yie&

(ccur# $e'(re '"i&ure* Thi# r(erty (' the !"teri"& i# c"&&e it# ductility*

Th(u%h c(!&e !(e ( ei#t t( "ccur"te&y re'&ect the "$()e re"& $eh")i(ur (' the

!"teri"&, the !(#t c(!!(n, "n #i!&e#t, !(e& i# the idealized stress-strain curve* Thi# i#

the cur)e '(r "n ie"& e&"#tic-&"#tic !"teri"& /which (e#n@t ei#t0, "n the %r"h i#2

A# c"n $e #een, (nce the yie& h"# $een re"che it i# t".en th"t "n ine'inite "!(unt (' #tr"in c"n

(ccur* Since #( !uch (#t-yie& #tr"in i# !(e&e, the "ctu"& !"teri"& /(r cr(## #ecti(n0 !u#t "(

$e c""$&e (' "&&(win% #uch #tr"in#* Th"t i#, it !u#t $e #u''icient&y ucti&e '(r the ie"&i?e

#tre##-#tr"in cur)e t( $e )"&i* Net we c(n#ier the $eh")i(ur (' " cr(## #ecti(n (' "n ie"&

e&"#tic-&"#tic !"teri"& #u$:ect t( $enin%* In (in% #(, we #ee. the re&"ti(n#hi $etween "&ie

!(!ent "n the r(t"ti(n /(r !(re "ccur"te&y, the cur)"ture0 (' " cr(## #ecti(n*

Cross Section Behaviour

Moment-Rotation Characteristics of General Cross Section

e c(n#ier "n "r$itr"ry cr(##-#ecti(n with " )ertic"& &"ne (' #y!!etry, which i# "( the &"ne

(' &("in%* e c(n#ier the cr(## #ecti(n #u$:ect t( "n incre"#in% $enin% !(!ent, "n "##e##

the #tre##e# "t e"ch #t"%e*

=r(## #ecti(n "n Stre##e#


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M(!ent-R(t"ti(n =ur)e

Stage 1 Elastic Behaiour

The "&ie !(!ent c"u#e# #tre##e# ()er the cr(##-#ecti(n th"t "re "&& &e## th"n the yie& #tre## ('

the !"teri"&*

Stage ! "ield Moment

The "&ie !(!ent i# :u#t #u''icient th"t the yie& #tre## (' the !"teri"& i# re"che "t the

(uter!(#t 'i$re/#0 (' the cr(##-#ecti(n* A&& (ther #tre##e# in the cr(## #ecti(n "re &e## th"n the

yie& #tre##* Thi# i# &i!it (' "&ic"$i&ity (' "n e&"#tic "n"&y#i# "n (' e&"#tic e#i%n* Since "&&

'i$re# "re e&"#tic, the r"ti( (' the eth (' the e&"#tic t( &"#tic re%i(n#,

Stage # Elasto-Plastic Bending

The !(!ent "&ie t( the cr(## #ecti(n h"# $een incre"#e $ey(n the yie& !(!ent* Since $y

the ie"&i?e #tre##-#tr"in cur)e the !"teri"& c"nn(t #u#t"in " #tre## %re"ter th"n yie& #tre##, the

'i$re# "t the yie& #tre## h")e r(%re##e inw"r# t(w"r# the centre (' the $e"!* Thu# ()er the

cr(## #ecti(n there i# "n e&"#tic c(re "n " &"#tic re%i(n* The r"ti( (' the eth (' the e&"#tic c(re

t( the &"#tic re%i(n i# * Since etr" !(!ent i# $ein% "&ie "n n( #tre## i# $i%%er th"n the yie&

#tre##, etr" r(t"ti(n (' the #ecti(n (ccur#2 the !(!ent-r(t"ti(n cur)e &(##e# it# &ine"rity "n

cur)e#, %i)in% !(re r(t"ti(n er unit !(!ent /i*e* &((#e# #ti''ne##0*

Stage $ Plastic Bending


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The "&ie !(!ent t( the cr(## #ecti(n i# #uch th"t "&& 'i$re# in the cr(## #ecti(n "re "t yie&

#tre##* Thi# i# ter!e the >&"#tic M(!ent =""city (' the #ecti(n #ince there "re n( 'i$re# "t "n

e&"#tic #tre##, A( n(te th"t the 'u&& &"#tic !(!ent re+uire# "n in'inite #tr"in "t the neutr"& "i#

"n #( i# hy#ic"&&y i!(##i$&e t( "chie)e* H(we)er, it i# c&(#e&y "r(i!"te in r"ctice* Any

"tte!t "t incre"#in% the !(!ent "t thi# (int #i!&y re#u&t# in !(re r(t"ti(n, (nce the cr(##-#ecti(n h"# #u''icient ucti&ity* There'(re in #tee& !e!$er# the cr(## #ecti(n c&"##i'ic"ti(n !u#t

$e &"#tic "n in c(ncrete !e!$er# the #ecti(n !u#t $e uner-rein'(rce*

Stage % Strain &ardening

Due t( #tr"in h"renin% (' the !"teri"&, " #!"&& "!(unt (' etr" !(!ent c"n $e #u#t"ine*

The "$()e !(!ent-r(t"ti(n cur)e rere#ent# the $eh")i(ur (' " cr(## #ecti(n (' " re%u&"r e&"#tic-

&"#tic !"teri"&* H(we)er, it i# u#u"&&y 'urther #i!&i'ie "# '(&&(w#2

ith thi# ie"&i?e !(!ent-r(t"ti(n cur)e, the cr(## #ecti(n &ine"r&y #u#t"in# !(!ent u t( the

&"#tic !(!ent c""city (' the #ecti(n "n then yie in r(t"ti(n "n ineter!in"te "!(unt*

A%"in, t( u#e thi# ie"&i?"ti(n, the "ctu"& #ecti(n !u#t $e c""$&e (' #u#t"inin% &"r%e r(t"ti(n#

th"t i# it !u#t $e ucti&e*

Plastic &ingeN(te th"t (nce the &"#tic !(!ent c""city i# re"che, the #ecti(n c"n r(t"te 'ree&y th"t i#, it

$eh")e# &i.e " hin%e, ecet with !(!ent (' M "t the hin%e* Thi# i# ter!e "plastic hinge, "n


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i# the $"#i# '(r &"#tic "n"&y#i#* At the &"#tic hin%e #tre##e# re!"in c(n#t"nt, $ut #tr"in# "n

hence r(t"ti(n# c"n incre"#e*

'nal(sis of Rectangular Cross Section

Since we n(w .n(w th"t " cr(## #ecti(n c"n #u#t"in !(re &(" th"n :u#t the yie& !(!ent, we "re

intere#te in h(w !uch !(re* In (ther w(r# we w"nt t( 'in the yie& !(!ent "n &"#tic

!(!ent, "n we ( #( '(r " rect"n%u&"r #ecti(n* T".in% the #tre## i"%r"!# 'r(! th(#e (' the

!(!ent-r(t"ti(n cur)e e"!ine re)i(u#&y, we h")e2


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Sha)e *actor

Thu# the r"ti( (' e&"#tic t( &"#tic !(!ent c""city i#2

Thi# r"ti( i# ter!e theshape factor,f, "n i# " r(erty (' " cr(## #ecti(n "&(ne* F(r "rect"n%u&"r cr(##-#ecti(n, we h")e2


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An #( " rect"n%u&"r #ecti(n c"n #u#t"in 9C !(re !(!ent th"n the yie& !(!ent,

$e'(re " &"#tic hin%e i# '(r!e* There'(re the #h"e '"ct(r i# " %(( !e"#ure (' the e''iciency

(' " cr(## #ecti(n in $enin%* Sh"e '"ct(r# '(r #(!e (ther cr(## #ecti(n# "re

Methods of Plastic 'nal(sis

3.1 IntroductionThere "re three !"in "r("che# '(r er'(r!in% " &"#tic "n"&y#i#2

+he Incremental Method

Thi# i# r($"$&y the !(#t ($)i(u# "r("ch2 the &("# (n the #tructure "re incre!ente unti& the

'ir#t &"#tic hin%e '(r!#* Thi# c(ntinue# unti& #u''icient hin%e# h")e '(r!e t( c(&&"#e the

#tructure* Thi# i# " &"$(ur-inten#i)e, $rute-'(rce@, "r("ch, $ut (ne th"t i# !(#t re"i&y #uite

'(r c(!uter i!&e!ent"ti(n*

+he E,uilirium or Statical/ Method


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In thi# !eth(, 'ree "n re"ct"nt $enin% !(!ent i"%r"!# "re r"wn* The#e i"%r"!# "re

()er&"i t( ienti'y the &i.e&y &(c"ti(n# (' &"#tic hin%e#* Thi# !eth( there'(re #"ti#'ie# the

e+ui&i$riu! criteri(n 'ir#t &e")in% the tw( re!"inin% criteri(n t( eri)e there 'r(!*

+he 0inematic or Mechanism/ Method

In thi# !eth(, " c(&&"#e !ech"ni#! i# 'ir#t (#tu&"te* irtu"& w(r. e+u"ti(n# "re then written'(r thi# c(&&"#e #t"te, "&&(win% the c"&cu&"ti(n# (' the c(&&"#e $enin% !(!ent i"%r"!* Thi#

!eth( #"ti#'ie# the !ech"ni#! c(niti(n 'ir#t, &e")in% the re!"inin% tw( criteri" t( $e eri)e

there 'r(!*

e wi&& c(ncentr"te !"in&y (n the ine!"tic Meth(, $ut intr(uce n(w the Incre!ent"&

Meth( t( i&&u#tr"te the !"in c(ncet#*

Incremental Analysis

Illustratie Eam)le Pro))ed Cantileer

e n(w "##e## the $eh")i(ur (' " #i!&e #t"tic"&&y ineter!in"te #tructure uner

incre"#in% &("* e c(n#ier " r(e c"nti&e)er with !i-#"n (int &("2


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Since the e". !(!ent# "re &e## th"n the yie& !(!ent#, we .n(w th"t yie& #tre## h"# n(t $een

re"che "t "ny (int in the $e"!* A(, the !"i!u! !(!ent (ccur# "t A"n #( thi# (int wi&&

'ir#t re"ch the yie& !(!ent*


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Equilirium Method

Introduction

T( er'(r! thi# "n"&y#i# we %ener"&&y '(&&(w the '(&&(win% #te#2

1* Fin " ri!"ry #tructure $y re!()in% reun"nt unti& the #tructure i# #t"tic"&&y eter!in"teG

3* Dr"w the ri!"ry /(r 'ree0 $enin% !(!ent i"%r"!G

8* Dr"w the re"ct"nt BMD '(r e"ch reun"nt, "# "&ie t( the ri!"ry #tructureG

7* =(n#truct " c(!(#ite BMD $y c(!$in% the ri!"ry "n re"ct"nt BMD#G

9* Deter!ine the e+ui&i$riu! e+u"ti(n# 'r(! the c(!(#ite BMDG


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F(r Ste 4, we #(&)e thi# e+u"ti(n '(r the c(&&"#e &("2


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!inematic Method "sing #irtual $ork

Introduction

>r($"$&y the e"#ie#t w"y t( c"rry (ut " &"#tic "n"&y#i# i# thr(u%h the ine!"tic Meth( u#in%

)irtu"& w(r.* T( ( thi# we "&&(w the re#u!e #h"e "t c(&&"#e t( $e the c(!"ti$&e

i#&"ce!ent #et, "n the etern"& &("in% "n intern"& $enin% !(!ent# t( $e the e+ui&i$riu!

#et* e c"n then e+u"te etern"& "n intern"& )irtu"& w(r., "n #(&)e '(r the c(&&"#e &(" '"ct(r

'(r th"t #u(#e !ech"ni#!*

Re!e!$er2

E+ui&i$riu! #et2 the intern"& $enin% !(!ent# "t c(&&"#eG

=(!"ti$&e #et2 the )irtu"& c(&&"#e c(n'i%ur"ti(n /#ee $e&(w0*


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N(te th"t in the "ctu"& c(&&"#e c(n'i%ur"ti(n the !e!$er# wi&& h")e e&"#tic e'(r!"ti(n in

$etween the &"#tic hin%e#* H(we)er, #ince " )irtu"& i#&"ce!ent (e# n(t h")e t( $e re"&, (n&y

c(!"ti$&e, we wi&& ch((#e t( i%n(re the e&"#tic e'(r!"ti(n# $etween &"#tic hin%e#, "n t".e

the !e!$er# t( $e #tr"i%ht $etween the!*

'ctual Colla)se Mechanism

S( '(r (ur re)i(u# $e"!, we .n(w th"t we re+uire tw( hin%e# '(r c(&&"#e /(ne !(re th"n it#

e%ree (' reun"ncy0, "n we thin. th"t the hin%e# wi&& (ccur uner the (int# (' e". !(!ent,

A "nC* There'(re i!(#e " unit )irtu"& i#&"ce!ent "tC "n re&"te the c(rre#(nin% )irtu"&

r(t"ti(n# (' the hin%e# u#in% ,


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2ther Colla)se Mechanisms

F(r the c(&&"#e !ech"ni#! &((.e "t re)i(u#&y, it #ee!e ($)i(u# th"t the &"#tic hin%e in the

#"n #h(u& $e $ene"th the &("* But why U#in% )irtu"& w(r. we c"n e"!ine "ny (##i$&e

c(&&"#e !ech"ni#!* S( &et@# c(n#ier the '(&&(win% c(&&"#e !ech"ni#! "n #ee why the &"#tic

hin%e h"# t( $e &(c"te $ene"th the &("*

Plastic &inge et3eenAand C4

I!(#in% " unit )irtu"& e'&ecti(n "t, we %et the '(&&(win% c(&&"#e !ech"ni#!2


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An #( we #ee th"t the c(&&"#e &(" '"ct(r '(r thi# !ech"ni#! een# (n the (#iti(n (' the

&"#tic hin%e in the #"n*

+heorems of Plastic 'nal(sis

%.1 Criteria

In >&"#tic An"&y#i# t( ienti'y the c(rrect &(" '"ct(r, there "re three criteri" (' i!(rt"nce2

1* E,uilirium2 the intern"& $enin% !(!ent# !u#t $e in e+ui&i$riu! with the etern"& &("in%*

3* Mechanism2 "t c(&&"#e the #tructure, (r " "rt (', c"n e'(r! "# " !ech"ni#!*

8* "ield2 n( (int in the #tructure c"n h")e " !(!ent %re"ter th"n the &"#tic !(!ent c""city ('

the #ecti(n it i# "&ie t(*

B"#e (n the#e criteri", we h")e the '(&&(win% the(re!#*


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&he "''er ound ("nsa)e* &heorem

Thi# c"n $e #t"te "#2

If a !ending moment diagram is found which satisfies the conditions of e"uili!rium

and mechanism #!ut not necessarily yield$% then the corresponding load factor is

either greater than or e"ual to the true load factor at collapse&

Thi# i# c"&&e the un#"'e the(re! $ec"u#e '(r "n "r$itr"ri&y "##u!e !ech"ni#! the &(" '"ct(r i#

either e"ct&y ri%ht /when the yie& criteri(n i# !et0 (r i# wr(n% "n i# t(( &"r%e, leading a

designer to thin' that the frame can carry more load than is actually possi!le*

The Lower bound (Safe) Theorem

If a !ending moment diagram is found which satisfies the conditions of e"uili!rium and yield#!ut not necessarily that of mechanism$% then the corresponding load factor is either less than or

e"ual to the true load factor at collapse&

Plastic Analysis of Beams

Example 1 Fixed-Fixed Beam with Point Load


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To start the problem, we examine the usual elastic BMD to seewhere the plastic

hinges are likely to form:

e "( nee t( .n(w h(w !"ny hin%e# "re re+uire* Thi# #tructure i# 8 #t"tic"&&y ineter!in"te

"n #( we !i%ht eect the nu!$er (' &"#tic hin%e# re+uire t( $e 7* H(we)er, #ince (ne (' the

ineter!in"cie# i# h(ri?(nt"& re#tr"int, re!()in% it w(u& n(t ch"n%e the $enin% $eh")i(ur ('

the $e"!* Thu# '(r " $enin% c(&&"#e (n&y 3 ineter!in"cie# "&y "n #( it wi&& (n&y t".e 8

&"#tic hin%e# t( c"u#e c(&&"#e*

S( &((.in% "t the e&"#tic BMD, we@&& "##u!e " c(&&"#e !ech"ni#! with the 8 &"#tic hin%e# "t

the e". !(!ent &(c"ti(n#2A,, "n C*


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We need to check that this is the correct solution using theUniueness Theorem:

An #( the "&ie &(" i# in e+ui&i$riu! with the 'ree BMD (' the c(&&"#e BMD*

(& )echanism*

Fr(! the r((#e c(&&"#e !ech"ni#! it i# ""rent th"t the $e"! i# " !ech"ni#!*

+& ,ield*

Fr(! the c(&&"#e BMD it c"n $e #een th"t n(where i# eceee* > M

Thu# the #(&uti(n !eet# the three c(niti(n# "n #(, $y the Uni+uene## The(re!, i# the c(rrect

#(&uti(n*

Example 2 Propped Cantilever with Two Point Loads


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F(r the '(&&(win% $e"!, '(r " &(" '"ct(r (' 3*, 'in the re+uire &"#tic !(!ent

c""city2

!llowing for the load factor, we need to design the beam for thefollowing loads:

"nce again we try to picture possible failure mechanisms# $ince

maximum moments occur underneath point loads, there are two

real possibilities:


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Mech"ni#!-1

Mech"ni#!-3

Therefore, we analyse both and apply the Upperbound Theorem to%nd the

design plastic moment capacity#

Mechanism 1: Plastic Hinge at C:


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Mechanism 2: Plastic Hinge at :


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& E"uili!rium*


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U#in% the BMD "t c(&&"#e, we@&& chec. th"t the hei%ht (' the 'ree BMD i# th"t ('

the e+ui)"&ent #i!&y-#u(rte $e"!* Fir#t&y the c(&&"#e BMD 'r(! Mech"ni#!

1 i#2

&ence, the total heights of the free BMD are:

'hecking these using a simply(supported beam analysis:


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Thu#, u#in% "r(ri"te 'ree $(y i"%r"!# ('AC"n.2

An #( the "&ie &(" i# in e+ui&i$riu! with the 'ree BMD (' the c(&&"#e BMD*

(& )echanism*

Fr(! the r((#e c(&&"#e !ech"ni#! it i# ""rent th"t the $e"! i# " !ech"ni#!* A(, #ince

it i# " r(e c"nti&e)er "n thu# (ne e%ree ineter!in"te, we re+uire tw( &"#tic hin%e# '(r

c(&&"#e, "n the#e we h")e*

+& ,ield*

Fr(! the c(&&"#e BMD it c"n $e #een th"t n(where i# the e#i%n eceee* 177

.N! Thu# $y the Uni+uene## The(re! we h")e the c(rrect #(&uti(n*

L"#t&y, we@&& e"!ine why the Mech"ni#! 3 c(&&"#e i# n(t the c(rrect #(&uti(n* Since the )irtu"&

w(r. !eth( r()ie# "n uer$(un, then, $y the Uni+uene## The(re!, it !u#t n(t $e the

c(rrect #(&uti(n $ec"u#e it !u#t )i(&"te the yie& c(niti(n*

U#in% the c(&&"#e Mech"ni#! 3 t( eter!ine re"cti(n#, we c"n r"w the '(&&(win% BMD '(r

c(&&"#e Mech"ni#! 32


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)rom this it is apparent that Mechanism * is not the uniue solution, and so

the design plastic moment capacity must be + k-m as implied pre.iously

from the Upperbound Theorem#

Basic Collapse Mechanisms

In 'r"!e#, the $"#ic !ech"ni#!# (' c(&&"#e "re2

Beam-type collapse

Sway Collapse

Combination Collapse


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Combination of Mechanisms

One (' the !(#t (wer'u& t(( in &"#tic "n"&y#i# i# =(!$in"ti(n (' Mech"ni#!#* Thi# "&&(w# u#

t( w(r. (ut the )irtu"& w(r. e+u"ti(n# '(r the $e"! "n #w"y c(&&"#e# #e"r"te&y "n then

c(!$ine the! t( 'in the c(&&"#e &(" '"ct(r '(r " c(!$in"ti(n c(&&"#e !ech"ni#!*

=(!$in"ti(n (' !ech"ni#!# i# $"#e (n the ie" th"t there "re (n&y " cert"in nu!$er ('

ineenent e+ui&i$riu! e+u"ti(n# '(r " #tructure* Any 'urther e+u"ti(n# "re ($t"ine 'r(! "

c(!$in"ti(n (' the#e ineenent e+u"ti(n#* Since e+ui&i$riu! e+u"ti(n# c"n $e ($t"ine u#in%

)irtu"& w(r. "&ie t( " (##i$&e c(&&"#e !ech"ni#!, it '(&&(w# th"t there "re ineenent

c(&&"#e !ech"ni#!#, "n (ther c(&&"#e !ech"ni#!# th"t !"y $e ($t"ine '(r! " c(!$in"ti(n

(' the ineenent c(&&"#e !ech"ni#!#*

Simple Portal Frame

/n this example we will consider a basic prismatic 0so all membersha.e the same

plastic moment capacity1 rectangular portal frame with pinnedfeet:

e wi&& c(n#ier thi# %ener"& c"#e #( th"t we c"n in'er the r(ertie# "n $eh")i(ur (' "&& #uch

'r"!e#* e wi&& c(n#ier e"ch (' the (##i$&e !ech"ni#!# (ut&ine "$()e*

Beam colla)se4

The (##i$&e $e"! c(&&"#e &((.# "# '(&&(w#2


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S3a( Colla)se

The )irtu"& e'&ecti(n '(r the #w"y c(&&"#e i#2


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Comined Colla)se

The )irtu"& e'&ecti(n '(r thi# '(r! (' c(&&"#e i#2


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Colla)se Mode

Since we (n@t .n(w the re&"ti)e )"&ue# (' / "n 0, we c"nn(t eter!ine the

c(rrect c(&&"#e !(e* H(we)er, we c"n ienti'y the#e c(&&"#e !(e# i' we &(t

the three &(" '"ct(r e+u"ti(n# eri)e "$()e (n the '(&&(win% inter"cti(n ch"rt2


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-otice that each mechanism de%nes a boundary and that it isonly the region inside all of these boundaries that is safe# -ow, for

a gi.en ration of Vto H, we will be able to determine the critical

collapse mechanism# -ote also that the beam collapse

mechanism is only critical for this frame at point Pon the chart 2

this point is also included in the 'ombined mechanism#

The bending moment diagrams corresponding to each of themechanisms are approximately:

eam 1way Com!ined


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!n interesting phenomenon is obser.ed at point Qon the chart,

where the $way and 'ombined mechanisms gi.e the same result#

3ooking at the bending moment diagrams, we can see that this

occurs as the moment at the top of the left column becomes

eual to the mid(span moment of the beam:

!mportant Problem"


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56I+-#

S+I**6ESS M'+RI7 ME+&2DS

The %i)en ineter!in"te #tructure i# 'ir#t !"e .ine!"tic"&&y eter!in"te $y intr(ucin%c(n#tr"int# "t the n(e#* The re+uire nu!$er (' c(n#tr"int# i# e+u"& t( e%ree# (' 'ree(! "t then(e# th"t i# .ine!"tic ineter!in"cy 4.* The .ine!"tic"&&y eter!in"te #tructure c(!ri#e# (''ie ene !e!$er#, hence, "&& n("& i#&"ce!ent# "re ?er(* The#e re#u&t# in #tre## re#u&t"nti#c(ntinuitie# "t the#e n(e# uner the "cti(n (' "&ie &("# (r in (ther w(r# the c&"!e:(int# "re n(t in e+ui&i$riu!* In (rer t( re#t(re the e+ui&i$riu! (' #tre## re#u&t"nt# "t the n(e#the n(e# "re i!"rte #uit"$&e un.n(wn i#&"ce!ent#* The nu!$er (' #i!u&t"ne(u# e+u"ti(n#rere#entin% :(int e+ui&i$riu! (' '(rce# i# e+u"& t( .ine!"tic ineter!in"cy 4.* S(&uti(n ('the#e e+u"ti(n# %i)e# un.n(wn n("& i#&"ce!ent#* U#in% #ti''ne## r(ertie# (' !e!$er# the!e!$er en '(rce# "re c(!ute "n hence the intern"& '(rce# thr(u%h(ut the #tructure* Sincen("& i#&"ce!ent# "re un.n(wn#, the !eth( i# "( c"&&e i#&"ce!ent !eth(* Sincee+ui&i$riu! c(niti(n# "re "&ie "t the :(int# the !eth( i# "( c"&&e e+ui&i$riu! !eth(*Since #ti''ne## r(ertie# (' !e!$er# "re u#e the !eth( i# "( c"&&e #ti''ne## !eth(*

Intr(ucti(n

A&& .n(wn !eth(# (' #tructur"& "n"&y#i# "re c&"##i'ie int( tw( i#tinct %r(u#2-

/i0 '(rce !eth( (' "n"&y#i# "n

/ii0 i#&"ce!ent !eth( (' "n"&y#i#*


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In !(u&e 8, the '(rce !eth( (' "n"&y#i# (r the !eth( (' c(n#i#tent e'(r!"ti(n i# i#cu##e* An

intr(ucti(n t( the i#&"ce!ent !eth( (' "n"&y#i# i# %i)en in !(u&e 8, where in #&(e-e'&ecti(n

!eth( "n !(!ent- i#tri$uti(n !eth( "re i#cu##e* In thi# !(u&e the irect #ti''ne## !eth( i#

i#cu##e* In the i#&"ce!ent !eth( (' "n"&y#i# the e+ui&i$riu! e+u"ti(n# "re written $y

ere##in% the un.n(wn :(int i#&"ce!ent# in ter!# (' &("# $y u#in% &("-i#&"ce!ent re&"ti(n#*

The un.n(wn :(int i#&"ce!ent# /the e%ree# (' 'ree(! (' the #tructure0 "re c"&cu&"te $y #(&)in%

e+ui&i$riu! e+u"ti(n#* The #&(e-e'&ecti(n "n !(!ent-i#tri$uti(n !eth(# were eten#i)e&y u#e

$e'(re the hi%h #ee c(!utin% er"* A'ter the re)(&uti(n in c(!uter inu#try, (n&y irect #ti''ne##

!eth( i# u#e*

The i#&"ce!ent !eth( '(&&(w# e##enti"&&y the #"!e #te# '(r $(th #t"tic"&&y eter!in"te "n

ineter!in"te #tructure#* In i#&"ce!ent 5#ti''ne## !eth( (' "n"&y#i#, (nce the #tructur"& !(e& i#

e'ine, the un.n(wn# /:(int r(t"ti(n# "n tr"n#&"ti(n#0 "re "ut(!"tic"&&y ch(#en un&i.e the '(rce

!eth( (' "n"&y#i#* Hence, i#&"ce!ent !eth( (' "n"&y#i# i# re'erre t( c(!uter

i!&e!ent"ti(n* The !eth( '(&&(w# " r"ther " #et r(ceure* The irect #ti''ne## !eth( i# c&(#e&y

re&"te t( #&(e-e'&ecti(n e+u"ti(n#*

The %ener"& !eth( (' "n"&y?in% ineter!in"te #tructure# $y i#&"ce!ent !eth( !"y $e tr"ce t(

N")ier /1469-168


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Let "n e&"#tic $(y i# "cte $y " '(rce F "n the c(rre#(nin% i#&"ce!ent $e uin the irecti(n

(' '(rce* In !(u&e 1, we h")e i#cu##e '(rce- i#&"ce!ent re&"ti(n#hi* The '(rce /30 i#

re&"te t( the i#&"ce!ent /u0 '(r the &ine"r e&"#tic !"teri"& $y the re&"ti(n

where the c(n#t"nt (' r((rti(n"&ity 'i# e'ine "# the #ti''ne## (' the #tructure "n it h"# unit#

(' '(rce er unit e&(n%"ti(n* The "$()e e+u"ti(n !"y "( $e written "#


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=h"ter-3 O$:ecti)e#

A'ter re"in% thi# ch"ter the #tuent wi&& $e "$&e t(

1* Deri)e !e!$er #ti''ne## !"tri (' " tru## !e!$er*

3* De'ine &(c"& "n %&($"& c(-(rin"te #y#te!*

8* Tr"n#'(r! i#&"ce!ent# 'r(! &(c"& c(-(rin"te #y#te! t( %&($"& c(-(rin"te #y#te!*

7* Tr"n#'(r! '(rce# 'r(! &(c"& t( %&($"& c(-(rin"te #y#te!*

9* Tr"n#'(r! !e!$er #ti''ne## !"tri 'r(! &(c"& t( %&($"& c(-(rin"te #y#te!*


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>&"ne tru##e# "re !"e u (' #h(rt thin !e!$er# interc(nnecte "t hin%e# t( '(r! tri"n%u&"te

"ttern#* A hin%e c(nnecti(n c"n (n&y tr"n#!it '(rce# 'r(! (ne !e!$er t( "n(ther !e!$er $ut n(t

the !(!ent* F(r "n"&y#i# ur(#e, the tru## i# &("e "t the :(int#* Hence, " tru## !e!$er i#

#u$:ecte t( (n&y "i"& '(rce# "n the '(rce# re!"in c(n#t"nt "&(n% the &en%th (' the !e!$er* The

'(rce# in the !e!$er "t it# tw( en# !u#t $e (' the #"!e !"%nitue $ut "ct in the ((#ite irecti(n#

'(r e+ui&i$riu! "# #h(wn in Fi%* 37*1*

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56I+-$- C'B8ES '6D S5SPE6SI26 BRIDGES

In#tructi(n"& O$:ecti)e#2

A'ter re"in% thi# ch"ter the #tuent wi&& $e "$&e t(

1* Di''erenti"te $etween ri%i "n e'(r!"$&e #tructure#*

3* De'ine 'unicu&"r #tructure*

8* St"te the tye #tre## in " c"$&e*

7* An"&y#e c"$&e# #u$:ecte t( uni'(r!&y i#tri$ute &("*

9* An"&y#e c"$&e# #u$:ecte t( c(ncentr"te &("#*

81*1 Intr(ucti(n

="$&e# "n "rche# "re c&(#e&y re&"te t( e"ch (ther "n hence they "re %r(ue in thi# c(ur#e in

the #"!e !(u&e* F(r &(n% #"n #tructure# /'(r e*%* in c"#e $ri%e#0 en%ineer# c(!!(n&y u#e

c"$&e (r "rch c(n#tructi(n ue t( their e''iciency* In the 'ir#t &e##(n (' thi# !(u&e, c"$&e#

#u$:ecte t( uni'(r! "n c(ncentr"te &("# "re i#cu##e* In the #ec(n &e##(n, "rche# in

%ener"& "n three hin%e "rche# in "rticu&"r "&(n% with i&&u#tr"ti)e e"!&e# "re e&"ine* In

the &"#t tw( &e##(n# (' thi# !(u&e, tw( hin%e "rch "n hin%e&e## "rche# "re c(n#iere*

Structure !"y $e c&"##i'ie int( ri%i "n e'(r!"$&e #tructure# eenin% (n ch"n%e in

%e(!etry (' the #tructure whi&e #u(rtin% the &("* Ri%i #tructure# #u(rt etern"&&y "&ie

&("# with(ut "reci"$&e ch"n%e in their #h"e /%e(!etry0* Be"!# tru##e# "n 'r"!e# "re

e"!&e# (' ri%i #tructure#* Un&i.e ri%i #tructure#, e'(r!"$&e #tructure# uner%( ch"n%e# in

their #h"e "cc(rin% t( etern"&&y "&ie &("#* H(we)er, it #h(u& $e n(te th"t e'(r!"ti(n#

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"re #ti&& #!"&&* ="$&e# "n '"$ric #tructure# "re e'(r!"$&e #tructure#* ="$&e# "re !"in&y u#e t(

#u(rt #u#en#i(n r(('#, $ri%e# "n c"$&e c"r #y#te!* They "re "( u#e in e&ectric"&

tr"n#!i##i(n &ine# "n '(r #tructure# #u(rtin% r"i( "ntenn"#* In the '(&&(win% #ecti(n#, c"$&e#

#u$:ecte t( c(ncentr"te &(" "n c"$&e# #u$:ecte t( uni'(r! &("# "re c(n#iere*

The shape assumed by a rope or a chain (with no stiffness) under the action of external

loads when hung from two supports is known as a funicular shape. Cable is a funicular

structure. It is easy to visualize that a cable hung from two supports subected to external

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load must be in tension (vide !ig. "#.$a and "#.$b). %ow let us modify our definition of

cable. & cable may be defined as the structure in pure tension having the funicular shape of

the load.

A# #t"te e"r&ier, the c"$&e# "re c(n#iere t( $e er'ect&y '&ei$&e /n( '&eur"& #ti''ne##0 "n

ineten#i$&e* A# they "re '&ei$&e they ( n(t re#i#t #he"r '(rce "n $enin% !(!ent* It i# #u$:ecte t(

"i"& ten#i(n (n&y "n it i# "&w"y# "ctin% t"n%enti"& t( the c"$&e "t "ny (int "&(n% the &en%th* I' the

wei%ht (' the c"$&e i# ne%&i%i$&e "# c(!"re with the etern"&&y "&ie &("# then it# #e&' wei%ht i#

ne%&ecte in the "n"&y#i#* In the re#ent "n"&y#i# #e&' wei%ht i# n(t c(n#iere*

=(n#ier " c"$&e "# &("e in Fi%* 81*3* Let u# "##u!e th"t the c"$&e &en%th# "n #"% "t /0 "re .n(wn*

The '(ur re"cti(n c(!(nent# "tAC.E"n, c"$&e ten#i(n# in e"ch (' the '(ur #e%!ent# "n three

#"% )"&ue#2 " t(t"& (' e&e)en un.n(wn +u"ntitie# "re t( $e eter!ine* Fr(! the %e(!etry, (ne c(u&

write tw( '(rce e+ui&i$riu! e+u"ti(n# /,KKyx330 "t e"ch (' the (int "n.CA,,,Ei*e* " t(t"&(' ten e+u"ti(n# "n the re+uire (ne !(re e+u"ti(n !"y $e written 'r(! the %e(!etry (' the c"$&e*

F(r e"!&e, i' (ne (' the #"% i# %i)en then the r($&e! c"n $e #(&)e e"#i&y* Otherwi#e i' the t(t"&

&en%th (' the c"$&e i# %i)en then the re+uire e+u"ti(n !"y $e written "#

Cale su9ected to uniform load.

="$&e# "re u#e t( #u(rt the e" wei%ht "n &i)e &("# (' the $ri%e ec.# h")in% &(n% #"n#*

The $ri%e ec.# "re #u#ene 'r(! the c"$&e u#in% the h"n%er#* The #ti''ene ec. re)ent#

the #u(rtin% c"$&e 'r(! ch"n%in% it# #h"e $y i#tri$utin% the &i)e &(" !()in% ()er it, '(r "

&(n%er &en%th (' c"$&e* In #uch c"#e# c"$&e i# "##u!e t( $e uni'(r!&y &("e*

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Consider a cable which is uniformly loaded as shown in !ig "#."a

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'ue to uniformly distributed load the cable takes a parabolic shape. owever due to its

own dead weight it takes a shape of a catenary. owever dead weight of the cable is

neglected in the present analysis.

Example 31.1

'etermine reaction components at & and * tension in the cable and the sag of the cableshown in !ig. "#.+a. %eglect the self weight of the cable in the analysis

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,ince there are no horizontal loads horizontal reactions at & and * should be the same.Taking moment about E yields

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T;!M


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UNIT-5 SPCE T!USSES

Tetr"her(n2 #i!&e#t e&e!ent (' #t"$&e #"ce tru## /#i !e!$er#, '(ur :(int#0 e"n $y "in%

8 !e!$er# "n 1 :(int e"ch ti!e

Deter!in"cy "n St"$i&ity

$ r 8: un#t"$&e

$ r K 8: #t"tic"&&y eter!in"te /chec. #t"$i&ity0 $

r 8: #t"tic"&&y ineter!in"te /chec. #t"$i&ity0

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>r(ceure '(r An"&y#i#

Meth( (' Secti(n#

Meth( (' ;(int#

6umerical eam)le

In the '(&&(win% e"!&e we #h"&& c(n#truct the intern"& '(rce# i"%r"!# '(r the %i)en in Fi%* 6

#"ce 'r"!e #tructure* The intr(uce %&($"& c((rin"te #y#te! i# #h(wn in the #"!e 'i%ure*

The intr(uce &(c"& c((rin"te #y#te!# (' the i''erent e&e!ent# (' the #"ce 'r"!e "re

re#ente in Fi%* P* The tyic"& #ecti(n# where the intern"& '(rce# !u#t $e c"&cu&"te, in (rer t(

c(n#truct the re&e)"nt i"%r"!#, "re nu!$ere 'r(! 1 t( 6 in the #"!e 'i%ure* The tyic"&

#ecti(n# "re &"ce "t &e"#t "t the $e%innin% "n "t the en (' e"ch e&e!ent /#e%!ent0 (' the

'r"!e* The intern"& '(rce# i"%r"!#, in the &i!it# (' e"ch e&e!ent, c(u& $e eri)e $y u#in% the

c(rre#(nin% re'erence "n $"#e i"%r"!#*

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Internal *orces

In (rer t( ($t"in the intern"& '(rce# "t " #eci'ie (int, we #h(u& !".e #ecti(n cut

erenicu&"r t( the "i# (' the !e!$er "t thi# (int* Thi# #ecti(n cut i)ie# the #tructure in tw(

"rt#* The (rti(n (' the #tructure re!()e 'r(! the "rt int( c(n#ier"ti(n #h(u& $e re&"ce

$y the intern"& '(rce#* The intern"& '(rce# en#ure the e+ui&i$riu! (' the i#(&"te "rt #u$:ecte t(

the "cti(n (' etern"& &("# "n #u(rt re"cti(n#* A 'ree $(y i"%r"! (' either #e%!ent (' the

cut !e!$er i# i#(&"te "n the intern"& &("# c(u& $e eri)e $y the #i e+u"ti(n# (' e+ui&i$riu!

"&ie t( the #e%!ent int( c(n#ier"ti(n*

e #h"&& #.i the eri)"ti(n (' intern"& '(rce# in #ecti(n 1 'r(! Fi%* 6, $ec"u#e they c"n $e eri)e

with(ut "ny tr(u$&e#* Let u# %( irect t( the intern"& '(rce# in #ecti(n 3* I' we "## " #ecti(n cut "t

(int 3 the #"ce 'r"!e wi&& $e #e"r"te "# #h(wn in Fi%* 1* The (#iti)e irecti(n# (' intern"&

'(rce#, in "cc(r"nce with the intr(uce &(c"& c((rin"te #y#te! '(r the !e!$er#

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E+ui&i$riu! (' the :(int#

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Nu!eric"& E"!&e-3

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CE2351 SA 2 Lecture Notes

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