ENGI6705-ClassNoteshandout2

8/20/2019 ENGI6705-ClassNoteshandout2

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2.5 INDETERMINATE ARCHES

An arch may be defined as a plane-cr!ed bar "r rib spp"r#edand l"aded in a $ay #ha# ma%es i# ac# in direc# c"mpressi"n.

&"r e'ample( #he s#rc#re in &i). 2.*a is a parab"lic

symme#rical arch #ha# is l"aded $i#h a dis#rib#ed l"ad #ha#

!aries linearly "!er #he span + "f #he arch. I# is fi'ed a# end A

and spp"r#ed by an imm"!able hin)e a# end ,. The h"ri"n#al

dis#ance be#$een #he end spp"r#s "f #he arch is #he span "f #he

arch( and #he line A, "inin) i#s p"in#s "f spp"r#s is #he

sprin)in) line.



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&i). 2.* /a0 1arab"lic arch "f !ariable #hic%ness l"aded as

sh"$n. /b0 1arab"lic arch l"aded $i#h redndan# f"rce A(

redndan# m"men# MA( and applied dis#rib#ed l"adin). /c0

&ree-b"dy dia)ram "f a se)men# "f #he arch.

&"r #he arch in &i). 2.*a( #he span and sprin)in) lines "f #he

arch are #he same becase p"in#s A and , are a# #he same le!el

$i#h respec# #" #he y a'is. The hi)hes# p"in# C "f #he arch is #he

cr"$n( and an arch may be a symme#rical arch "r an

nsymme#rical arch. If( f"r e'ample( "ne end "f #he arch is

l"$er #han #he "#her( #hen #he arch is nsymme#rical. 3ari"s

#ypes "f arches are sh"$n in &i)s. 2.4 and 2.6.

&i). 2.4 Circlar arches7 /a0 &i'ed a# "ne end and hin)ed a# #he

"#her. /b0 &i'ed a# b"#h ends. /c0 Three-hin)e arch. /d0 T$"-

hin)e arch $i#h spp"r#s a# differen# ele!a#i"n. /e0 T$"-hin)e

arch.



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I# is assmed here #ha# #he plane "f cr!a#re "f #he arch rib is

als" a plane "f symme#ry f"r each cr"ss sec#i"n "f #he arch( and

#he e'#ernally applied l"ads are assmed #" ac# "nly in #his

plane. 8n #his basis( $e ha!e a #$"-dimensi"nal pr"blem and#he def"rma#i"n "f #he arch $ill #a%e place in #he plane "f

symme#ry. The ma'imm !er#ical dis#ance fr"m #he sprin)in)

line #" #he arch a'is( den"#ed as H in &i). 2.*a( is #he rise "f #he

arch.

2.9. 1arab"lic Arch "f 3ariable Thic%ness

:e c"nsider #he linearly elas#ic !ariable-#hic%ness parab"lic

arch in &i). 2.*a #ha# is l"aded by a dis#rib#ed l"ad "f

ma'imm in#ensi#y $6 and !aryin) linearly "!er #he span "f

#he arch. A# any arc len)#h s fr"m spp"r# A( #he m"men# "f

iner#ia Is is assmed #" !ary as f"ll"$s7

c

s

s

cc s

I

I

dx

ds

ds

dx

I

I I I ==∴=== θ θ θ seccos,sec /2.6*0

$here Ic is #he m"men# "f iner#ia a# #he cr"$n C "f #he arch(

and

= −

dx

dy1tanθ /2.640



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&i). 2.6 /a0 T$"-hin)e parab"lic arch. /b0 T$"-hin)e ellip#icalarch. /c0 T$"-hin)e h"ll"$ arch. /d0 T$"-hin)e $i#h h"ri"n#al

#ie. /e0 T$in circlar arch

Since #he parab"lic arch in &i). 2.*a is s#a#ically inde#ermina#e

#" #he sec"nd de)ree( #he h"ri"n#al f"rce A and bendin)

m"men# MA a# spp"r# A are #a%en as #he redndan#s. 8n #his

basis( #he arch is redced #" "ne #ha# is hin)ed a# end ,(

spp"r#ed by r"ller a# end A( and l"aded as sh"$n in &i). 2.*b.,y c"nsiderin) a se)men# A6 "f #he arch as sh"$n in &i).

2.*c( #he n"rmal f"rce Ns( shear f"rce 3s( and bendin) m"men#

Ms a# #he end 6 "f #he se)men# may be de#ermined by sin) #he

#hree s#a#ic e;ilibrim e;a#i"ns. &"r e'ample( by #a%in)

m"men#s ab"# #he end 6 "f #he se)men# and assmin)

c"n#ercl"c%$ise m"men#s as p"si#i!e( $e ha!e

066

3

00

0 =+++−

−−=∑

L

xw M y X M x

L

M Lw M s A A

A/2.60

N"#e #ha# in &i). 2.*c is #he !er#ical reac#i"n a# end A in &i).2.*b( and i# may be de#ermined fr"m #his fi)re by sin)

s#a#ics. ,y s"l!in) E;. /2.60( $e find

)1(6

1 20 x

Lxw y X

L

x M M A A s −+−

−= /2.0



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,y se##in) e;al #" er" #he sm "f #he f"rces in #he

!er#ical direc#i"n( &i). 2.*c( and assmin) p$ard f"rces as

p"si#i!e( $e ha!e

0sincos

26

2

00 =+−−−=

∑ θ θ s s

A y N V

L

M

L

xw Lw F

/2.20,y se##in) e;al #" er" #he sm "f #he f"rces in #he h"ri"n#al

direc#i"n "f #he se)men#( $e find∑ =++= 0cossin θ θ s s A x N V X F /*.<0

The siml#ane"s s"l#i"n "f E;s. /2.20 and /2.<0 f"r 3s and

Ns yields

θ θ sin26

cos

2

00

−−−−=

L

M

L

xw Lw X N A

A s /2.90

θ θ cos26sin

2

00

−−+−= L

M

L

xw Lw X V A A s /2.50

&r"m E'ample .< "f Sec#i"n .5( an e;a#i"n anal")"s

#" E;. /60 may be $ri##en f"r #he c"mplemen#ary s#rain

ener)y = "f #he parab"lic arch. This e;a#i"n is

dsGA

KV ds

EI

M ds

AE

N U

s s s s

s

s s∫ ∫ ∫ ++=0 0 0

222

222/2.>0

$here S is #he arc len)#h "f #he parab"lic arch( A is i#s cr"ss-

sec#i"nal area a# any c""rdina#e s( E is #he m"dls "f

elas#ici#y( ? is #he shear m"dls( and @ is #he shear fac#"r.The shear fac#"r @ may be de#ermined as sh"$n in E'ample

.<. &"r rec#an)lar cr"ss sec#i"ns @ is .2.

:hen #he rise "f #he arch is lar)e c"mpared #" i#s

#hic%ness( say a ra#i" "f 6 "r lar)er( #hen #he c"mplemen#ary

s#rain ener)y de #" Ns and 3s $"ld be small c"mpared #" #he

"ne pr"dced by Ms and i# can be ne)lec#ed. 8n #his basis( E;.

/*.>0 yields

ds EI

M U s

s

s

∫ =0

2

2/2.0

The in#e)ra#i"ns in E;. /2.>0( "r E;. /2.0( may be

simplified by sin) #he e'pressi"n

dx I

I ds

c

s= /2.*0



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Ths( by sbs#i##in) E;s. /2.0 and /2.*0 in#" E;. /2.0(

$e find

dx x Lxw

y X L

x M

EI U

L

A A

c

2

0

20 )1(6

12

1∫

−+−

−= /2.40

2.9.2 Addi#i"nal Sbec#s and Me#h"ds

The !ales "f #he redndan# f"rce A and redndan#

m"men# MA may be "b#ained fr"m #he minimiin) c"ndi#i"ns

0=∂

∂

A X

U

/2.260

0=∂

∂

A M

U

/2.20

Applica#i"n "f E;s. /2.260 and /2.20 yields

0)()1(6

12

1

0

20 =−

−+−

−∫ dx y x

Lxw y X

L

x M

EI

L

A A

c

/2.220

0)1()1(6

12

1

0

20 =−

−+−

−∫ dx

L

x x

Lxw y X

L

x M

EI

L

A A

c

/2.2<0,y c"nsiderin) #he )e"me#ry "f #he arch in &i). 2.*a( $e

find

)(4

2 x L

L

Hx y −= /2.290

,y sbs#i##in) E;. /2.290 in#" E;s. /2.220 and /2.2<0 and

perf"rmin) #he re;ired in#e)ra#i"ns( $e "b#ain #he f"ll"$in)

#$" e;a#i"ns( $hich are in #erms "f #he redndan#s A and

MA7

01016 2

0 =−− Lw M HX A A /2.25007120120 2

0 =++− Lw M HX A A /2.2>0

Siml#ane"s s"l#i"n "f E;s. /2.250 and /2.2>0 yields



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H

Lw X A

72

5 2

0= /2.20

90

2

0 Lw M

A = /2.2*0

:i#h %n"$n A and MA( #he !ales "f Ms( Ns( and 3s may

be de#ermined fr"m E;s. /2.0( /2.90( and /2.50(

respec#i!ely. They are as f"ll"$s7

)1(672

51

90

20

2

0

2

0 x Lxw

H

y Lw

L

x Lw M s −+−

−= /2.240

θ θ sin9026

cos72

5 2

0

2

00

2

0

−−−−=

L

Lw

L

xw Lw

H

Lw N s /2.<60

θ θ cos9026

sin72

5 20

200

20

−−+−=

L

Lw

L

xw Lw

H

LwV s /2.<0

2.9.< Semicirclar Arch $i#h Hin)ed Ends

:e c"nsider n"$ #he nif"rm semicirclar arch in &i). 2.a

#ha# is hin)ed a# #he spp"r# p"in#s A and , and l"aded by a

nif"rmly dis#rib#ed l"ad $ as sh"$n. The m"men# "f iner#iaI is nif"rm #hr")h"# #he arch. Since #he arch is s#a#ically

inde#ermina#e #" #he firs# de)ree( #he h"ri"n#al reac#i"n A a#

#he end A is #a%en as #he redndan#. 8n #his basis( $e ha!e an

arch #ha# is hin)ed a# end ,( spp"r#ed by r"ller a# end A( and

l"aded by #he reac#i!e f"rce A and applied l"ad $ as sh"$n in

&i). 2.b.

,y c"nsiderin) #he free-b"dy dia)ram "f #he arch se)men#

AC in &i). 2.c and applyin) #he #hree s#a#ic e;ilibrim

e;a#i"ns( #he e'pressi"ns f"r #he n"rmal f"rce



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&i). 2. /a0 Semicirclar arch hin)ed a# #he end spp"r#s. /b0

Semicirclar arch l"aded $i#h redndan# f"rce A and applied

dis#rib#ed l"ad $. /c0 &ree-b"dy dia)ram "f a se)men# "f #he

arch.

2.9.9 Addi#i"nal Sbec#s and Me#h"ds



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Ns( shear f"rce 3s( and bendin) m"men# Ms( may be

de#ermined. &"r e'ample( by se##in) e;al #" #he sm "f #he

m"men#s ab"# p"in# C "f #he se)men#( $e find

222 )cos1(2

sin)cos1( θ θ θ −−−−= wr r X wr M A s /2.<20

The s#a#ic e;ilibrim e;a#i"ns in #he h"ri"n#al and !er#ical

direc#i"ns "f #he arch se)men# yield

0cossin)cos1( =+−−− θ θ θ s s N V wr wr /2.<<00sincos =++ θ θ s s A N V X /2.<90

Siml#ane"s s"l#i"n "f E;s. /2.<<0 and /2.<90 yields

θ θ 2

cossin wr X N A s −−= /2.<50

θ θ θ cossincos A s X wr V −= /2.<>0

,y c"nsiderin) "nly #he c"mplemen#ary s#rain ener)y de #"

bendin) and sin) E;. /2.<20( $e find

θ θ θ θ

θ

π

π

d wr

r X wr EI

d EI

M U

A

s

22/

0

22

2/

0

2

)cos1(2

sin)cos1(1

22

∫

∫

−−−−=

=

The redndan# reac#i"n A may be de#ermined by sin) #he

e;a#i"n

0=

∂

∂

A X

U

/2.<0E;a#i"n /2.<0 yields

0)sin()cos1(2

sin)cos1(2 2/

0

22

2 =−

−−−−∫ θ θ θ θ θ

π

d r wr

r X wr EI

A /2.<*0



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,y in#e)ra#in) E;. /2.<*0 and s"l!in) f"r A( $e find

AB6.9644$r /2.<*a0

,y sbs#i##in) E;. /2.<*a0 in#" E;s. /2.<20( /2.<50( and/*.<>0( $e find

θ θ θ sin4099.0)cos1(2

)cos1( 222

2 wr wr

wr M s −−−−= /2.<40

θ θ 2cossin4099.0 wr wr N s −−= /2.960

θ θ θ cos4099.0sincos wr wr V s

−= /2.90

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