Solutions Manual for A First Course in the Finite Element Method (6th Edition, 2023) – Daryl L. Logan – Complete Step-by-Step Solutions

0
00 0
0
0



0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0



00 0

, 0
1234567080 80
1234567090 0
123456700 90
123456700 89 0
123456700 8 0
123456700 980
123456700 80
123456700 0
123456700 80
1234567080 80
12345670880 0
12345670890 80
123456708 0 90
123456708 0 0
123456708 0 90
1234567080 80
44600 0
44600 0
44600 80

, !
"#"#$%!&'(')*!*+*,*()!'-!.!-,.++!/012!03!4(')!'()*350((*5)*1!)0!0)6*3!4(')-!)0!,01*+!.!
+.37*3!-)345)43*!03!-2-)*,8!
"#9#$:'-53*)';.)'0(!,*.(-!1'<'1'(7!)6*!/012!=-2-)*,>!'()0!.(!*?4'<.+*()!-2-)*,!0&!
&'(')*!*+*,*()-!@')6!.--05'.)*1!(01*-!.(1!*+*,*()-8!
"#A#$B6*!,01*3(!1*<*+0C,*()!0&!)6*!&'(')*!*+*,*()!,*)601!/*7.(!'(!DEFD!@')6!)6*!
@03G!0&!H3*(('G0&&!'(!)6*!&'*+1!0&!-)345)43.+!*(7'(**3'(78!
"#I#$B6*!1'3*5)!-)'&&(*--!,*)601!@.-!'()30145*1!'(!DEFD!/2!H3*(('G0&&8!H0@*<*3J!')!
@.-!(0)!50,,0(+2!G(0@(!.-!)6*!1'3*5)!-)'&&(*--!,*)601!4()'+!DEKL8!
"#M#$%!,.)3'N!'-!.!3*5).(74+.3!.33.2!0&!?4.()')'*-!.33.(7*1!'(!30@-!.(1!50+4,(-!)6.)!
'-!0&)*(!4-*1!)0!.'1!'(!*NC3*--'(7!.(1!-0+<'(7!.!-2-)*,!0&!.+7*/3.'5!*?4.)'0(-8!
"#O#$%-!50,C4)*3!1*<*+0C*1!')!,.1*!C0--'/+*!)0!-0+<*!)604-.(1-!0&!*?4.)'0(-!'(!.!
,.))*3!0&!,'(4)*-8!
"#P#$B6*!&0++0@'(7!.3*!)6*!7*(*3.+!-)*C-!0&!)6*!&'(')*!*+*,*()!
,*)6018!Q)*C!D!
! :'<'1*!)6*!/012!'()0!.(!*?4'<.+*()!-2-)*,!0&!&'(')*!*+*,*()-!@')6!
! Q)*C!R!!
.--05'.)*1!(01*-!.(1!5600-*!)6*!,0-)!.CC30C3'.)*!*+*,*()!)2C*8!
! Q)*C!S!!
V600-*!.!1'-C+.5*,*()!&4(5)'0(!@')6'(!*.56!*+*,*()8!
! W*+.)*!)6*!-)3*--*-!)0!)6*!-)3.'(-!)630476!)6*!-)3*--X-)3.'(!+.@Y
! Q)*C!F!!
7*(*3.++2!5.++*1!)6*!50(-)')4)'<*!+.@8!
! :*3'<*!)6*!*+*,*()!-)'&&(*--!,.)3'N!.(1!*?4.)'0(-8!Z-*!)6*!1'3*5)!
! *?4'+'/3'4,!,*)601J!.!@03G!03!*(*372!,*)601J!03!.!,*)601!0&!
! Q)*C!K!!
@*'76)*1!3*-'14.+-!)0!3*+.)*!)6*!(01.+!&035*-!)0!(01.+!1'-C+.5*,*()-8!
! %--*,/+*!)6*!*+*,*()!*?4.)'0(-!)0!0/).'(!)6*!7+0/.+!03!)0).+!
! Q)*C!L!!
*?4.)'0(-!.(1!'()30145*!/04(1.32!50(1')'0(-8!
! Q)*C!T!
Q0+<*!&03!)6*!4(G(0@(!1*73**-!0&!&3**10,!=03!7*(*3.+';*1!
! Q)*C!U!
1'-C+.5*,*()->8!Q0+<*!&03!)6*!*+*,*()!-)3.'(-!.(1!-)3*--*-8!
[()*3C3*)!.(1!.(.+2;*!)6*!3*-4+)-!&03!4-*!'(!)6*!1*-'7(X.(.+2-'-!C305*--8!
"#\#$B6*!1'-C+.5*,*()!,*)601!.--4,*-!1'-C+.5*,*()-!0&!)6*!(01*-!.-!)6*!
4(G(0@(-!0&!)6*!C30/+*,8!B6*!C30/+*,!'-!&03,4+.)*1!-456!)6.)!.!-*)!0&!
-',4+).(*04-!*?4.)'0(-!'-!-0+<*1!&03!(01.+!1'-C+.5*,*()-8!
"#]#$^043!50,,0(!)2C*-!0&!*+*,*()-!.3*_!-',C+*!+'(*!*+*,*()-J!-',C+*!)@0`
1',*(-'0(.+! *+*,*()-J!-',C+*!)63**`1',*(-'0(.+!*+*,*()-J!.(1!-',C+*!
.N'-2,,*)3'5!*+*,*()-8!
"#"a$B63**!50,,0(!,*)601-!4-*1!)0!1*3'<*!)6*!*+*,*()!-)'&&(*--!,.)3'N!.(1!*?4.)'0(-!.3*!
bcd1'3*5)!*?4'+'/3'4,!,*)601!
bed@03G!03!*(*372!,*)601-!
bfd,*)601-!0&!@*'76)*1!3*-'14.+-!
"#""#$B6*!)*3,!ghijkiil!0&!&kiihmno!3*&*3-!)0!30).)'0(-!.(1!1'-C+.5*,*()-!)6.)!.3*!
.--05'.)*1!@')6!*.56!(01*8!





2
45625789 88 99888 98
998


8

8

8



8888
989


, !"!#"$%&'()*+,&-./).0(.1)23(0()*3()4&5&*()(/(6(5*)&1).,,/&(7).0().1)48//8219)
:;<=*0>-*>0./?1*0(11)).5./+1&1)
:@<A(.*)*0.514(0).5./+1&1)
:B<%/>&7)4/82).5./+1&1)
:C<D/(-*0&-)80)6.E5(*&-),8*(5*&./)7&1*0&F>*&85).5./+1&1)
:G<H&86(-3.5&-./))(5E&5((0&5E)
!"!I"$%&'().7'.5*.E(1)84)*3()4&5&*()(/(6(5*)6(*387).0()*3().F&/&*+)*8)
:;<J87(/)&00(E>/.0/+)13.,(7)F87&(1)K>&*()(.1&/+)
:@<A.57/()E(5(0./)/8.7)-857&*&851)2&*38>*)7&44&->/*+)
:B<J87(/)F87&(1)-86,81(7)84)1('(0./)7&44(0(5*)6.*(0&./1)F(-.>1()(/(6(5*)
(K>.*&851).0()('./>.*(7)&57&'&7>.//+)
:C<A.57/()>5/&6&*(7)5>6F(01).57)L&571)84)F8>57.0+)-857&*&851)
:G<M.0+)*3()1&N()84)*3()(/(6(5*1)*8)6.L()&*),811&F/()*8)>1()16.//)(/(6(5*1)
23(0()5(-(11.0+)






2
42562789 88 99888 98
998


8

8

8



8888
989


, "
"
#$%"
"

" "
*&#'%+","" "&'" ("""")"&'"""""(" "
(" (" (" (" "
" )"&'"""""(" &'" (" "
"
(" (" (" (" "
*&#-%+","" "("""(" (" ("
(""""(" (" ("
"("""(" &-" )"&-"" "
"("""("""")"&-" &-"
"(" (" (" (""
*&".#.%+","(" &"." (""")"&".""" ""
"(" (" (" ("
"("""")"&." (" &."""" "
*/+","*&#'%+"0"*&#-%+"0"*&#.%+"
""&'" (" )"&'" ("
*/+","" (" &." (" )"&."
)"&'" (" &'""""" 1&-"
&-"
(" )"&." 1& -" -"""""".""
& &"
#2%""34567"'"$85"-"$96":;<65"74"='","("$85"=-","("$85"*/+"26>4?67"
" */+",""
&'"""
&-"
" 1&-"
)"&-" &-""""""&."" "
" @A B"
A.<"", "*/
""+ "@5
&'""""B"
" )"&-" " """ =."""
""A" , &
"" )"&" & & =
-
" "C<
" "
(""
" """&"""" "" C""
" -)"""&""""" .="""""""""
-
"." "
,""& " """ """ "
"'"" --"
"D" " )"&-" )&'-""""""&.""""""")='C"""
@AB","*/+"@5B"*/ +"@A)'B","*/"+"*/+"@5B"
/"+"@AB","@5B"
E7;8F"GH6"$5I4;8G"?6GH45"G4":;85"*/")'+"
J''","&-"0"&." J-'",""#)"'%."#)"&-%"
J'-","#)"'%'"0"-"#)"&-%","&-" J--",""&'"0"&-"




1
3567589 99  999 
9
9

9 
9
9



9
9!9
9
!9 

,% &'(%)%% *+%%% % *+%% % %-./%%%'0% % *+%%% %*,% *+%
*,% ) %
*+% *1%%%%%%*+%% % *+% *1%%%%%%*+%% %
/23%&4(%)%5%&4(%5%)%6*1%7%*+8%6*+%7%*,8%9%6%9+%*+8%69%*+8%
% 5%&4(%5%)%60*%1%7%*+8%6*+%7%*,8%9%*"% %
&4%91(%)%%&'%%(%
*+%%%%%%*,% *+%% %
/23%4%
% * * *%
&4%91(%)% *+% *1%%%%%%*+% % +%*%%+%%%%,%*1%%%%+%%*+%% %
6*1%%%%%%*+%8%6*+%%%%%%*,%8%9%*+"%) *1%*+%%%%%%*1%*,%%%%%%*+%*,%
%*+%%%%%%*,% *+% :%% %
% % ;
%%%;%%
, %%%
%)% * +% * 1% % %% %*
% + %
%%
%%
%%<% %
%=%% % *1%*+%%%*%%%%%*<1%* % ,%%%%%%*+%*,%
;,%) % + %
*1%*+%6%*%%%%%%*%%1%%**%,%8%%%%<%%*%+%*,%
;=%)%*1%*+%%%%%1%*%1%*+,%%%%%%%*+%*,%
6>8%%?.%@A/2A%3@%BC./%3D2%A2->3C@.%B@A>2E%F2%G@%H->*%3@%3D2%GI@H-I%J-3ACK%L%)%&4(%M/N%
% L1K%%% % *%1 % :% ;1%% %
*1%
L % :%
: ;%%"%%%%%
%
%%%+%K %% % *%,
) %*,%  %
% %
%%
:% ;,%%
L
%,K%% %%
%% 1* % *1%%%%% % %% %
*+% *+%
% :%
% L=%K%% %% :% *,% % * %%*+%%%%%%*,%%%%%%;=%%
< %
* + %
L1K%)%9%*1%;,%)%9%*1% +%
*1%*+%%%%%%*1%*,%%%%%%*+%
*,%
* * < %
L%#%%$)%*1%*+%%%%%1%*%1%+*%%,%%%%%%*+%*,%
L+K%)%9%*,%;=%)%9%*,% 6*%1%%%%%*+%%8%<%
*1%*+%%%%%%*1%*,%%%%%*+%
%* % % 6 * % % %*% % 8%< % *,%
% L+K%)%*1%*+,%%%%%%%%%%%*11%%*,%+%%%%%%%*+%*%,%
% %
%
% %
% %
% %
% %
% %
%
*1%)%*+%)%*,%)%1:::%%IH%
C.O%
&*618(%)% 61*8%*
%%%6%+%681%8% 6+8 *6+%8%6,8%
P%%&* (%)%% *%%
% 6+8%
%
%
%
 *
  % * %% 6
+8% %
*%
  *%%
,8%
6
QR%3D2%J23D@/%@B%E;S2AS@EC3C@.%3D2%GI@H-I%E3CBB.2EE%J-3ACK%CE%>@.E3A;>32/O%
%




1
3567589 99  999 
9
9

9 
9
9



9
9!9
9
!9 

, $%&' $(&' $)&'
*'*' +' *'*' +' '
$
%&'
,-.'/'' '*' *'' '' *''$(&'',-.'/''' ' *' (*'*''
'
*'
+'*' *''$)&' +'*' *'''
0123'%'45'64732' 8%'/'+'9:2'8)'/''
;<='/',-.';2='
'<%7''''''>'' ' *' *' +' ''8%''''+'' '
'<'
(' +
'7'''''' ' ''' /
''' ' ( * '* ' ' '8
' '('' ' ' '
> '
'''
*'
' <)7''' ''' +>'''
''' ''''' ''*'*''8'
+'''' *'''
'''8'
)''''''' '(*8'''''*'
'* + ' '
' '
' ' '''
' /'(' ''' ' ''' (
'''' '' ' (
' <)7'' '' *' *'' ' '''''''' <)7'''''''' *'8'(''''*' '
''
' *' '/'' ' %' '
' 8('/' (* /4:'?' 8('/'+?@'
' (' ('
<)7'/'A'*'$+?@&CD'B''*'$%&' CD'&'$%&'
<)7'/'$A'%+++''4:?'$&'%$+++?@& +''B ' '
4:?'
'
' '




<)7'/'@++'
CD5'E:F3G:9C'61GH35''''''''
IC3J3:F'$%&'
''6%7'$&%' *'*''''8%'''''+'
/'
''6('7'$&'( *' *'''' ' '+?@' ''
8(' CD'
' ' ''
" #$%&'/'$A'%+++'' &'$+?@&''6$% /'A'@++'CD'
%7' 4:?' %
7'&'
' ' CD' ' ''
6('7'$&%' /'$%+++'4':?'&'$+?@&''6('$&%' /'@++'CD'
IC3J3:F'
'$'6((&'7''' *'*''''8' '
'+?@'
'
7'
' 6''' 'A'@++'CD'
$( $(&'
&'
'
'''''




/'' (' ('7'
'6)7'$&'(
' '*' *''' 8)''' ' 6)7'$&'( @++'CD'
'%' '
''''



'
'
*9'&'',*$%&.'/',*$(&.'/',*$)&.'/',*$K&.'/''''' '
$ *'''
' '*' *''
'
LM'FN3'J3FN12'16'58O3GO154F41:'P3'H1:5FG8HF'FN3'QC1D9C',-.'9:2'*:1P4:Q';<='/'

,-.';2='P3'N9R3'
' <%7''''''>''' *'*' +' +' +''''8%'''''+' '
' <
' ' (7
'''''''+''
'' ' ( *'* ' +' + '
''
'8
'('
*'
')7'''''' ' +*''''''' (*'*' +'''''8)'
< S' ''/
' ' +*'' +'''''' (*'*'''''8K'
<K'7''''''+''
' <@7''''''>''
' +' + ' *' *''''8@'''
'+'
'
' +'
1
3567589 99  999 
9
9

9 
9
9



9
9!9
9
!9 

, # $#### %&## $#####'% %&'%#####&'(#
&# #
$#
)*+## ,# #-####& # &### #'##(###
## #,### &'%###
##### #%&'(###
#&'.#
# % #
&######
# $###
## # # &#%&## ## '#.## #$##########&'(#####%&'.#
$# '# '
'%#-#%(##/#'.#-###%#(##
0'*1232'2345#34#267#78'923:4#34#267#;3<<=7#
,#-#>#&#'%#?#%&#'(#>#&#'.#
# ,#-#>#&##'(## ##?#%&# >#&##"""
# '(##
"
""%# " ' %## "
# ,#-#&,#'#(# ( #
'(#-## &#
#
#




'%#-#%,# /#'
&# %&# .#-#
,#
)@+##A4#:B<7B#2:#C34<#267#B79@23:41#92#267#C3D7<#4:<71#E#94<#F#G7#5:#*9@&#2:#267#
5=:*9=#78'923:4#HIJ#-#KLM#H<J#
#
# IED#-#>#&#'%#-##>&#%,&## IED#-## ,%##
# IFD#-#>#&#'.#-##>&#%,&## IFD#-## ,%##
#
#




#
#
#
#




#
N67@&#
ID#-#$# IED#,# ?#IFD#?#,# ,##-##$#
##
#####?#,#-#$#
#
#
# %##?##%#
#
########################




# $#-#$#
#
#
# #
&9#+##K&)E+M#-#K&)%+M#-#K&)(+M#-#K&).+M#-##### #
) &## #
# # &# &## #
OP#267#;726:<#:C#1'Q7BQ:13 23:4#267#5=:*9=#KLM#31#
@:412B'@27<R#S=1:# HIJ#-#KLM#H<J#94<#'E#-#$#94<#'F#-##
# IED#####T## # &#&# $# $# $####'E#####$## #
#I $ #  UVW UVU V X  $ ### #' T##
I(D######$#### U $####'(######T##
# %D
# ####
## # %###
##
#
# -# XUWU #
#
#
#
I.#D######$#### X$# X$
IFD######T## UW #U&## &##
&######
'.###
#'F###
#T##
##
#
##
#
# $#



1
3567589 99  999 
9
9

9 
9
9



9
9!9
9
!9 

, $%&''(')'*+',*'-'+',.' $/&'
(')'-'+,*'0'*+',.'-'+',1' $*&'
(')'-'+',.'0'*+',1'-'+'' $.&'
2345'$*&'
,.')'*',*'
2345'$.&'
,"'')''''''''**'',*'
6,%7898,89:;'9:'<=,>894:'$*&' *''
-'+'$,'#&'0'*+'$*',#'&'-'+''''''''' *'?''
'
*'
-',*'0'1',*'-',*'-' )'(' ,*')'
'

' ''
*' 1'
,.')'*'1' ,.')'' ' '
*'
' ' ''''*' '
''
'
,1')' '" ,1')'1'
.''
'

*' '




$@&''A49:;'%>@+'84'8BC';D4%>D'C=,>894:'
E2F')'GHI'EJF'
' ' 2/?')''+'
2/?')'-'+'',*')''+'1'
'
'
'' '.'' '
'' 1' '
'




2K?')'-'+',1'0'+' )'-'+'''' 0'+''
1'+''
' 2K?')'1''' ' '
' ' '
'
'
'' '
'
'
'
' *'''+9:9LM'' .'''+9:9LM''
' /' *' '
/' *' . K'
' 1' .' ?'
/'''+9:9LM'' 1' K''''+9:9LM'''
'
1'''+9:9LM''
G+'$/&I')' J//''''J''*/'N''G+'$*&I')'*J*'' 'J1'
'*''
'' /' /' ' '' '*' *' '
G+'$.&I')' J.*''''''J'.1''N'G+'$1&I')'1'J*' '
J1'
'1'
'
'' '.' .' ' '' '1' 1' '
G+'$K&I')' JK1''''''J'K.'''
'
'
 '
K ' K '
'




1
3567589 99  999 
9
9

9 
9
9



9
9!9
9
!9 

, THOSE WERE PREVIEW PAGES

TO DOWNLOAD THE FULL PDF

CLICK ON THE L.I.N.K

ON THE NEXT PAGE