Laboratory report Thick Cylinder
Ujjwalraj Anitha (@00414004) Distribution, Transission systes syste s and desi!ns "sc# $etroleu and !as en!ineerin! "odule co%ordinator& $ro'#Ali kadir
ab e*ort on thick cylinder
Contents 1.
Objective......................................................................................................... 2
2.
Abstract........................................................................................................... 2
3.
Introduction..................................................................................................... 2
4.
Theory............................................................................................................. 2 4.1 LAME Theory...............................................................................................2
!.
Apparatus re"uired.......................................................................................... !
#.
$rocedure........................................................................................................ !
%.
&esu'ts ( co)putation....................................................................................#
*.
+iscussion.....................................................................................................1!
,.
Conc'usion..................................................................................................... 1!
&e-erences........................................................................................................... 1#
List o- tab'es Tab'e Tab'e Tab'e Tab'e Tab'e
1 2 3 4 !
+epth penetration /.#*............................................................................ # +epth penetration 1.!*............................................................................ * +epth penetration 2.2,..........................................................................1/ +epth penetration 3.#!..........................................................................12 0 error has been ca'cu'ated -or dierent penetrations.......................... 1!
List o- ure
iure 1 pressure in thic5 cy'inder........................................................................ 3 iure 2 Top vie6.................................................................................................. 3
List o- raph
7raph 1 8oop9 radia' stress :; 1;r2.....................................................................# 7raph 2 8oop and radia' strain :; 1;r 2................................................................% 7raph 3 8oop and radia' stress :; r....................................................................* 7raph 4 8oop9 radia' stress :; 1;r2.....................................................................* 7raph ! 8oop and radia' strain :; 1;r2............................................................... , 7raph # 8oop and radia' stress :;s r..................................................................1/ 7raph % 8oop9 radia' stress :; 1;r2...................................................................11 7raph * 8oop ( radia' strain :; 1; r2................................................................11 7raph , &adia' ( hoop stress :; r.....................................................................12 7raph 1/ 8oop9 radia' stress :; 1;r2.................................................................13 1
ab e*ort on thick cylinder 7raph 11 8oop strain ( radia' strain :; 1;r2.....................................................14 7raph 12 8oop stress ( radia' stress :; r.........................................................14
1. Objective To e
2. Abstract The e
3. Introduction $ressure vesse's are used in dierent industries 'i5e enery9 )echanica' industries and even in 6ater supp'y in the 6or'd to ho'd 'i"uids or ases at dierent pressures. These vesse's 6i'' be in dierent -or)s 'i5e pipes9 storae cy'inders9 heat e
4. Theory The Cy'indrica' vesse's 6i'' have 6a'' thic5ness to dia)eter not 'ess than 1 5
(
t 1 ) ≥ d 15
>ith hoop stress chanin a'' throuh the cy'inder thic5ness. Interna' radius has the )a
4.1 LAME Theory 8is theory is based on t6o assu)ptions 1@ Cy'inder )ateria' shou'd be ho)oeneous and isotropic.
2
ab e*ort on thick cylinder 2@ Lonitudina' a
ith this 6e can nd the hoop stress and radia' stress. The e"uations
8oop stress
&adia' stress
σ θ
σ r
9
9
E ?ouns Modu'us@ B 2/* 5;))2 D? $oisson=s ratio@ B /.3 The ure sho6s thic5 cy'inder 6ith pressure and stresses
+i!ure 1& *ressure in thick cylinder
3
ab e*ort on thick cylinder
+i!ure & To* -iew
ro) ure inner radius B &1 and outer radius B&29 Cy'inder havin pressure p?interna'@ at distance r?radia'@9 The LAME= e"uation is iven by9 8oop stress
σ θ
9
σ θ
Radial stress
=
A
+
B r
2
σ r
σ r
=
A
−
B r
2
Using boundary conditions,
Then the equations become,
4
ab e*ort on thick cylinder
!. Apparatus re"uired
1@ A thic5 cy'inder 6ith outer dia)eter B 3/4)) ( inner dia)eterB 1!!)) attached 6ith ten strain aues at respective positions. 2@ 8ydrau'ic ra) 3@ Tapered p'u inside the bore 4@ 8ydrau'ic cy'inder to supp'y 'ubricatin oi' !@ train aue #@ +ata 'oer ?Solatron Schlumber 3!3/@
#. $rocedure •
•
•
To attain ero re-erence 'ubricatin oi' is pu)ped to the tapered p'u. The p'u is rotated at the sa)e ti)e durin the 'ubrication to ensure uni-or)ity ti' the spira' roove is ''ed 6ith the oi'. In case o- hydrau'ic ra)9 an initia' 6eiht 6as app'ied and /.2#)) 6as obtained as initia' readin. o6 it is eroed and 1/ dierent channe's o- strain are initia'ised to the 'oin syste). The operator pushed the hand pu)p do6n to -orce the hydrau'ic ra) head ti'' the tapered p'u. i)u'taneous'y 'ubrication is done 6ith the he'p o- pu)p to ensure the uni-or)ity o- the ra) and the oi' pu)p. ote do6n the readins o- strain.
!
ab e*ort on thick cylinder •
&eadins 6ere noted -or penetrations o- 4)) 6ith a step dierence o- /.%!)). Lubrication is done si)u'taneous'y. Load shou'd be re)oved 6hen you are ta5in the readins.
%. &esu'ts ( co)putation
a@ +epth o- penetration B /.#*
Table 1& De*th *enetration 0#./
#
ab e*ort on thick cylinder
ra*h 1& oo*, radial stress 23 13r
From graph, A= 5.4N/mm2 B= (0!"#$0#0!%&2 ' 0) = 57050N *= !!+&mm2 By ame-s equation (theoretically%, 4.4∗77.5
A=
2
152
− 77.5
= 1.545N/mm
2
2
∗77.5 ∗152 152 − 4.4
4.4
B=
2
.n graph
2
σ θ
2
2
,
2
σ r
= 35711.18N /& r and
eθ,
er
/& &r 2 are plotted
%
ab e*ort on thick cylinder
ra*h & oo* and radial strain 23 13r
ra*h 5& oo* and radial stress 23 r
b@ +epth o- penetration B 1.!*
Table & De*th *enetration 1#6/
*
ab e*ort on thick cylinder
ra*h 4& oo*, radial stress 23 13r
From graph, 2 A= 9.9N/mm B= (1)$#%&2'0) = 184600N 2
*= 10 N/mm ame-s eqn, 2
∗ A= 152 − 77.5 10 77.5 2
10∗77.5
B=
2
152
3i4en graph
= 3.51N/mm
2
2
2
2
∗152
−4.4
σ θ
2
,
= 81161.79N σ r
/& r and
eθ ,
er
/& &r 2 5or penetration )6 has
been plotted
,
ab e*ort on thick cylinder
ra*h 6& oo* and radial strain 23 13r
ra*h .& oo* and radial stress 23s r
c@ +epth o- penetration B 2.2,
1/
ab e*ort on thick cylinder
Table 5& De*th *enetration #7
ra*h 8& oo*, radial stress 23 13r
From graph, A= 10.9N/mm2 B= (2012$")0%&2 ' 0 ) = 300600N * = 20.2N (3raph% By ame-s 5ormula, 20.2∗77.5
A=
2
152
− 77.5
20.2∗77.5
B=
152
.n graph
2
2
2
2
∗152
−4.4
σ θ
= 7.09N/mm
,
2
σ r
2
2
= 163946.81N /& r and
eθ,
er
/& &r 2 ( penetration 22% has been
dra7n belo7
11
ab e*ort on thick cylinder
ra*h /& oo* 9 radial strain 23 13 r
ra*h 7& adial 9 hoo* stress 23 r
d@ +epth o- penetration B 3.#!
Table 4& De*th *enetration 5#.6
12
ab e*ort on thick cylinder
ra*h 10& oo*, radial stress 23 13r
From graph, A= 22N/mm 2 B= (!1$!!62%&2'0) = 472150N *= 30N By ame-s theory, 30∗77.5
A=
2
152
− 77.5
30∗77.5
B=
152
3raph o5
2
2
2
2
2
∗152
−4.4
σ θ
= 10.53N/mm
2
,
2
σ r
= 243485.37N /& r and
eθ,
er
/& &r 2 ( penetration "1)% has been
dra7n belo7
13
ab e*ort on thick cylinder
ra*h 11& oo* strain 9 radial strain 23 13r
ra*h 1& oo* stress 9 radial stress 23 r
$ercentae error ca'cu'ations
14
ab e*ort on thick cylinder
Table 6& : error has been calculated 'or di;erent *enetrations
*. +iscussion •
•
•
>hen the p'u penetration increases9 the pressure inside the cy'inder a'so increases. >hen the depth o- penetration is /.#*))9 the interna' pressure beco)es 4.4;))29 and by increasin the deepness to 1.!*9 2.2,9 ( 3.#!))9 the re'ated pressure va'ues are 1/9 2/.2 ( 3/;))2 respective'y. The -o''o6in strain readins -or dierent heihts 6ou'd va'idate the increase in va'ues o- interna' pressure. Fy co)parin the va'ues o- theoretica' ( e
,. Conc'usion The test to nd the stress distribution in a thic5 cy'inder based on La)e=s theory 6as investiated and ana'ytica' resu'ts 6ere co)pared 6ith ca'cu'ated ones. The dierence in va'ues can be accepted as it co)es under acceptab'e rane and these 6i'' be due to e
1!
ab e*ort on thick cylinder &e-erences •
•
•
Fhavi5atti9 . . Mechanics Of Solids. e6 +e'hi e6 Ae Internationa' Ltd.9 2/1/. $rint. 8earn9 E. ?1,,%@. Mechanics of materials. O<-ord Futter6orthH 8eine)ann. C'ar5e9 . ?1,#,@. An investigation of stress redistribution caused by creep in a thick-walled circular cylinder subjected to axial and thermal loading. London 8.M..O.
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