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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 13 "Question.2.39" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 238 "\011A solid bron
ze sphere ( E = 110 GPa, n=1/3  , r = 150 mm) is subjected to hydrosta
tic pressure p so that its volume is reduced by 0.5 %. Determine(a) th
e pressure p, and(b) the strain energy U in the sphere.( Note: Volume \+
of a sphere    " }{OLE 1 3076 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:
F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::fyyyyyqyyyYJ<Jyky;::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::_lqvGcMJ<j\\nGcMsFI>:::
::::JEf:yyyxIV:NZHQ:<J:>R<:T><::;GcEDh=cfyW:A:;B;B:fB]mtFFcmnvGWMJnC==
nHE=;:::::JJNZ:vyyuy:>:<::::::JjB:j:vCSmlJ::::::::::OJ;Zy=J:B::::::N:;
B:c:;:?ja:[LsfFaMR>`:J:<:::::::>=?R:?:=J:vYxY:B::::::V:^a=Z:j:JyKyK<j<
>Z:n:v:>;F;N;;j?J@j@>:W:YJ:nYvY:::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::^=N:kxnCL:>m<IbA`idGx_gc?wcQG:;j;@j
:Hj:dj:Dk;U_<njwr:WJ:@J;\\RTR:V@NZ@xT?B[;x:?b[ip;d<]r<]R:?B\\R==b>AfVB
::::::j:<Z<>Z:B::::::::::vYxy;B:CZ:F:<jysy?:;:EW<fb>J;H:=j>r:=V:VZ:V@<
J:f:F[<Z:^:<J;@j:f:FZ<vyyyy:^:n_;>=E:]c:=Z:f:V[<Z:f:^[<>v=>rBN<n^@v;oj
yyiyAZ:>\\:B:;xyyQVyyyyY:]:kv:V[:B:G;Sj`@Pt\\Pd`QrP`:>r:F[:B:oi:><<:N>
C:US:f:<JDDZ<^v<f:V\\<b:Sx:eJ:Vy<>jxM:<:[V:=B:;B:Cb:;B>cTTUUSaEBWTSiEB
_tUUURWmJ^Z:jPF:KZ:N`DVl<nfAF:NHK:_KjDJEF:nDe:qQJxI;B:>L;Z:^:^D::f?C:U
K;^:>x;F:JSVs;F\\BF:V@e:qQ:[:JB[:^::::=K<jPF:C:[Q;<J>JSfk=ny<F:;jS>;N@
WU:wi:=:cKBB:qAB:>Lb:;b:;B>aTXDpql@CB:f??J<JrG:K:_c<;MaTj:>:cO>JSJCLj:
JT;_c<Fn;F:;jX><<jw;<:[n:b:;b:<;vG:Uk:^:>X?J>JSd:SR;=Z:>HaK:<JMJhfCZ_g
dO?DAR:=:bORIZ;B\\cK:@:a;?b[iP;r:?b[V:F\\EJ;ZGZlK:<ZiywiyWJXtp^N;<JMJ?
vYyyy=:;JBB:qQBv:<J:^q<Z:F:?bBaTXaEWEUUtP<B:UK;^:>X=B:AJ:<J:::::Bv1:" 
}{TEXT -1 4 "   )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 9 "Solution:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The \+
\"dilatation\" e, or the change in volume per unit volume," }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "e:=
0.5e-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG$\"\"&!\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The Bulk modulus of elasticity," }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "k:=E/(3*(1-2*nu));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
131 "Now, we input the given values of the engineeering constants, E a
nd \"nu\" and find the value of ''k'' by using the \"evalf\" function.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"kG*&%\"EG\"\"\",&\"\"$\"\"\"%#nuG!\"'!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "nu:=1/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nu
G#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "E:=110e9;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG$\"$5\"\"\"*" }}}{EXCHG {PARA 
11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e
valf(k);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "This evaluates
 the floating point value of \"k\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$5\"\"\"*" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 35 "Also, by defintion of \"e\", we have:" }{MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eqn:=e =-p/k;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/$\"\"&!\"$,$%\"pG$!+\"4444*!#@
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(eqn,p);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$!+++++b!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 93 "The negative sign indicates that the pressure applied wou
ld have to be compressive in nature." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 203 "Now, in order to find the strain energy, we first write the fo
rmula for strain energy density, and since it is constant over the ent
ire volume, multiply it with the volume to get strain energy in Joules
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "
For the sphere which is being subjected to an external hydrostatic pre
ssure of \"p\", the state of stress is:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tauxy:=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&tauxyG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "tauyz:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tauyzG\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tauxz:=0;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&tauxzG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "sigmax:=-p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigmaxG,$%\"
pG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sigmay:=-p;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigmayG,$%\"pG!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sigmaz:=-p;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'sigmazG,$%\"pG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 39 "The defintion of strain energy density;" }{MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "uo:=(1/(2*E))*(sigmax^
2+sigmay^2+sigmaz^2)-(nu/E)*(sigmax*sigmay+sigmay*sigmaz+sigmax*sigmaz
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#uoG,$*$)%\"pG\"\"#\"\"\"$\"+]
XXXX!#@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Now, we find the volum
e of the sphere." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "r:=150e
-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"$]\"!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:=(4*Pi/3)*(r^3);" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$%#PiG$\"+++++X!#7" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "As we said earlier, the strain energy;" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "U:=uo*V;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG,$*&)%\"pG\"\"#\"\"\"%#PiG\"
\"\"$\"+[XXX?!#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(U)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"pG\"\"#\"\"\"$\"+#)\\)fU'
!#B" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Now, we substitute the val
ues of p and Pi into U, so as to get the exact flaoting point value of
 U." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "s
ubs(p=0.550e9,Pi=3.142,U);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.D6W
>!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 13 "Final Answer:" }{TEXT 
258 0 "" }{TEXT -1 83 "         The hydrostatic pressure, p= 550MPa an
d Starin energy,U = 19441.12 Joules " }}}}{MARK "28 0 0" 38 }
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