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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 14 "Question .3.32" }}{PARA 
0 "" 0 "" {TEXT -1 228 "\011\011A thin walled circular cylinder vessel
 of diameter d and wall thickness t is subjected to internal pressure \+
p .Given a small circular hole in the vessel wall, show that the maxim
um tangential and axial stresses at the hole are" }{OLE 1 4100 1 "[xm]
Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::
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KgU==:cK>JSnGQY==:WC:K:_KjDJy`j:JJF;;JSJhXj;>Z:Ffh_FK:_[<>O?J:<JJ<L>JS
^KAE;=:c;N@WMhPj:JQ:_;wB;=B:n?JSJiHj:jT<JBB:qAB:>l;Z<>Z<>ZJVdscRYEU^Z:
jPN:C:[q:>;N`D>Few<=:uK>JSJIDj:>:uKBB:qAB:>L=J:DJ:DZJVDvG:Uk:^:>X?J>JS
dJBEJWaj:B:aM>JSJHF:VH[Z:VY;;<:[>;b:DZJ:Y=jPN:C:[q:>;N`D>OSS:=J:<jrN;<
JMJ?vYxy:J:><<jw?<IZ:>:sg:B:=J;Dlc`qsLqlp`h_:f?=J<JrOZ:V:<J:::::::::::
::::::::::lk>@5:" }}{PARA 0 "" 0 "" {TEXT -1 19 "     , respectively" 
}}{PARA 0 "" 0 "" {TEXT 258 9 "Solution:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "The \+
stresses in a thin walled pressure vessel of diameter \"d\" and thickn
ess\" t\" subjected to a uniform pressure \"p\" is.:\n \n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sigmax:=p*d/(4*t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'sigmaxG,$*&*&%\"pG\"\"\"%\"dGF)\"\"\"%\"tG!\"\"
#F)\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sigmay:=p*d/(2*
t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigmayG,$*&*&%\"pG\"\"\"%\"d
GF)\"\"\"%\"tG!\"\"#F)\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "An
d, also, an infinite plate with a hole of radius\" " }{TEXT 256 2 "a\"
" }{TEXT -1 39 " subjected to a uniform axial tension\" " }{XPPEDIT 
18 0 "sigma;" "6#%&sigmaG" }{TEXT -1 50 "\"\ngives the tangential stre
ss at the hole edge as\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"sigmatheta:=proc(sigma,theta)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s
igma*(1-2*cos(2*theta));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+sigmathetaGR6$%&sigmaG%&thetaG6\"F)F)*&9$\"\"\",&F,F,-%$cosG6
#,$9%\"\"#!\"#F,F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Since the body is being subject
ed to both " }{XPPEDIT 18 0 "sigmax and sigmay;" "6#3%'sigmaxG%'sigmay
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "the resulatant tangential s
tress will be the superpostion of the tangential stres due to each com
ponent" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "sigmathetatotal:=sigmatheta(sigmax,theta1)+sigmatheta
(sigmay,theta2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0sigmathetatotal
G,&*&*(%\"pG\"\"\"%\"dGF),&F)F)-%$cosG6#,$%'theta1G\"\"#!\"#F)\"\"\"%
\"tG!\"\"#F)\"\"%*&*(F(F3F*F3,&F)F)-F-6#,$%'theta2GF1F2F)F3F4F5#F)F1" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "theta1:=theta;" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
162 "But, since theta2 is being measured from a different reference ax
is, we need to shift it to the same reference axis as theta1, so that \+
we can apply superposition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'theta1G%&thetaG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "theta2:=theta1-(Pi/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'theta2G,&%&thetaG\"\"\"%#PiG#!\"\"\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sigmathetatotal;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*&*(%\"pG\"\"\"%\"dGF',&F'F'-%$cosG6#,$%&thetaG\"\"
#!\"#F'\"\"\"%\"tG!\"\"#F'\"\"%*&*(F&F1F(F1,&F'F'F*F/F'F1F2F3#F'F/" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(sigmathetatotal);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*(%\"pG\"\"\"%\"dGF',&\"\"$F'-
%$cosG6#,$%&thetaG\"\"#F0F'\"\"\"%\"tG!\"\"#F'\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 95 "We conclude that this quantity can be maximum w
hen the cosine term in it becomes maximum i.e 1." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "subs(cos(2*theta)=1,sigmathetatotal);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"pG\"\"\"%\"dGF'\"\"\"%\"tG!\"
\"#\"\"&\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "Now, we have to
 find the \"axial stress in the hole\", which will be obtained by tran
sforming the polar coordinate stresses at the hole surface into cartes
ian stresses.so, let us first find the sigmar and taurtheta." }}{PARA 
0 "" 0 "" {TEXT -1 122 "Since the inner surface of the hole is a \"fre
e\" surface..in that no external loads are acting on it, we can conclu
de that:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "sigmar:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'si
gmarG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "taurtheta:=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*taurthetaG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
56 "Hence,the conversion from polar to cartesian reduces to:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sigmax:
=sigmathetatotal*(sin(theta)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'sigmaxG*&,&*&*(%\"pG\"\"\"%\"dGF*,&F*F*-%$cosG6#,$%&thetaG\"\"#!\"#F*
\"\"\"%\"tG!\"\"#F*\"\"%*&*(F)F4F+F4,&F*F*F-F2F*F4F5F6#F*F2F*)-%$sinG6
#F1F2F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "We note that this sig
max will be maximum at theta= 90 degrees, so we substitute that value \+
in radians in the formulae to get the maximum axial stress ( sigmax).
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "thet
a:=Pi/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG,$%#PiG#\"\"\"\"
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(sigmax);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"pG\"\"\"%\"dGF'\"\"\"%\"tG!\"
\"$\"+++++D!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sigmathet
atotal;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"pG\"\"\"%\"dGF'\"
\"\"%\"tG!\"\"#F'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "si
gmax;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"pG\"\"\"%\"dGF'\"\"
\"%\"tG!\"\"#F'\"\"%" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 13 "Final A
nswer:" }{TEXT 261 0 "" }{TEXT 262 72 "  The maximum values of the tan
gential and axial stress are  5pd/4t  at " }{OLE 1 2564 1 "[xm]Br=WfoR
rB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::fyyyyyqyyyY:vY::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyj;J;tV;B:;JZC:bKB::>r\\`]?O^lxq;V:>
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::::::J;>:[U:B:nYnYV:^:f:;B:G:IJ:nYvY:::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::JH>:?JfIWBB:SIy;t[seT]YS
UTKYtO=J:VZ;FZ=FZDFZ<^jrX:>::::F:DZ:B::::::::::vYxy;J<j:B:yayQ:>:>f;jp
A:gS:<j:F;HjGK;V<>:E:Mb:B:C:?R:=j<j:Djyyyy=J<JQ@JFf:F`<F:<j<j?D:<j<J@D
JZAJZ@JCJMTjAN=yyyxY:<JB<Z:FgOjPjNU;UAQB:n>^;UTRcETcTX[US<JR><;B:qi:;f
yB:>l;F:;b:;b:<kc`qpDpql@C:US:>;N`D>f;>\\:FZ:B:]mC<Z:n>^=U<RSeTO[j;@j:
F::FCOJ:n>N;yayI:>:[Z:VY[j=J:^q<J:<j:DJ;Dlc`qsLqlp`h_:f?=J<JrA:AB:<J::
:::::5:" }{TEXT 263 27 " = 0 degrees and  pd/4t at " }{OLE 1 2564 1 "[
xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyyqyyyY:vY::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyj;J;tV;B:;JZC:bKB::>r\\`]
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::::::::::::::::::::::::::::::::::::::::::::::::::::::::JH>:?JfIWBB:CD
OAt[seT]YSUTKYtO=J:VZ;FZ=FZDFZ<^jrX:>::::F:DZ:B::::::::::vYxy;J<j:B:ya
yQ:>:>f;jpA:gS:<j:F;HjGK;V<>:E:Mb:B:C:?R:=j<j:Djyyyy=J<JQ@JFf:F`<F:<j<
j?D:<j<J@DJZAJZ@JCJMTjAN=yyyxY:<JB<Z:FgOjPjNU;UAQB:n>^;UTRcETcTX[US<JR
><;B:qi:;fyB:>l;F:;b:;b:<kc`qpDpql@C:US:>;N`D>f;>\\:FZ:B:]mC<Z:n>^=U<R
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rA:AB:<J:::::::5:" }{TEXT 264 25 "=90 degrees  respectively" }{TEXT 
259 0 "" }{TEXT 260 0 "" }}}}{MARK "21 0 0" 7 }{VIEWOPTS 1 1 0 1 1 
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