This is an open access article under the CC BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/.. Peer-review under responsibility of the organizing committee of SCESCM
Trang 1Procedia Engineering 171 ( 2017 ) 147 – 158
1877-7058 © 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
Peer-review under responsibility of the organizing committee of SCESCM 2016.
doi: 10.1016/j.proeng.2017.01.321
ScienceDirect
6XVWDLQDEOH&LYLO(QJLQHHULQJ6WUXFWXUHVDQG&RQVWUXFWLRQ0DWHULDOV6&(6&0 3RVWEXFNOLQJEHKDYLRXURID[LDOO\)*0SODQDUEHDPVDQGIUDPHV
a
Department of Architecture, College of Engineering, Nihon University, 1 Nakagawara, Koriyama, Fukushima 963-8642, Japan
b Department of Solid Mechanics, Institute of Mechanics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
$EVWUDFW
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
7KH$XWKRUV3XEOLVKHGE\(OVHYLHU/WG
3HHUUHYLHZXQGHUUHVSRQVLELOLW\RIWKHRUJDQL]LQJFRPPLWWHHRI6&(6&0
Keywords:$[LDOO\)*0 /DUJHGLVSODFHPHQWDQDO\VLV 3ODQDUEHDPDQGIUDPHVWUXFWXUHV )LQLWHHOHPHQWPHWKRG
,QWURGXFWLRQ
7KH SRVWEXFNOLQJ EHKDYLRXU RI SODQDU EHDPV DQG IUDPHV LV YHU\ LPSRUWDQW LQIRUPDWLRQ IRU GHVLJQ HQJLQHHUV ([SHULPHQWVRQEXFNOLQJRIORQJDQGVKRUWF\OLQGHUEHDPFDQEHIRXQGLQ>@(VWDEOLVKLQJHTXLOLEULXPSDWKVLVWKH PRVWFRPPRQZD\RIXQGHUVWDQGLQJWKHVWUXFWXUHV¶EHKDYLRXULQWKHSRVWEXFNOLQJVWDWH+RZHYHUDPDMRUGLIILFXOW\ LVWKHJHRPHWULFQRQOLQHDULW\ZKHQWKHVWUXFWXUHVXQGHUJRODUJHGLVSODFHPHQW'XHWRWKLVFKDOOHQJHLWLVGLIILFXOWWR DQDO\VHWKHSRVWEXFNOLQJSUREOHPXVLQJDQDO\WLFDOPHWKRGVDQGDQXPHULFDOPHWKRGHVSHFLDOO\WKHILQLWHHOHPHQW PHWKRGLVRIWHQHPSOR\HGLQVWHDG,QRUGHUWRDQDO\VHWKHODUJHGLVSODFHPHQWVRIWKHVWUXFWXUHVE\WKHILQLWHHOHPHQW PHWKRG D QRQOLQHDU ILQLWH HOHPHQW ZKLFK PDNHV LW SRVVLEOH WR PRGHO WKH QRQOLQHDU EHKDYLRXU RI WKH VWUXFWXUHV
E-mail address:EXQWDUD#DUFKFHQLKRQXDFMS
© 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
Peer-review under responsibility of the organizing committee of SCESCM 2016.
Trang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
&RQVLVWHQWVKDSHIXQFWLRQVIRUD[LDOO\)*0EHDPHOHPHQW
2.1 Problem definition
,QWKH%HUQRXOOL(XOHUEHDPWKHFURVVVHFWLRQRIWKHEHDPLVDVVXPHGWRUHPDLQVWUDLJKWDJDLQVWWKHD[LVDIWHULW KDVGHIRUPHG KHQFHQRVKHDUGHIRUPDWLRQLVWDNHQLQWRDFFRXQW+HQFHWKHUHVXOWLQJVKDSHIXQFWLRQVFRQWDLQRQO\WKH OHQJWKLQIRUPDWLRQRIWKHEHDPLQVLGHWKHSRO\QRPLDOIRUPV)RUDVKRUWEHDPZKHUHWKHVKHDULQJGHIRUPDWLRQLVD GRPLQDQW IDFWRU IRU JHWWLQJ DQ DFFXUDWH UHVXOW WKH %HUQRXOOL(XOHU EHDP LV QRW UHFRPPHQGHG 7KHUHIRUH WKH 7LPRVKHQNREHDPKDVWREHFRQVLGHUHG
5HIHUHQFH>@GHULYHGWKHVKDSHIXQFWLRQVIRUWKH7LPRVKHQNREHDP )LJ E\XVLQJ+DPLOWRQ¶VSULQFLSOHWR VDWLVI\WKHKRPRJHQHRXVIRUPRIWKHVWDWLFHTXDWLRQVRIHTXLOLEULXPDV
u
EA
w AG
w
T
ZKHUHADQG IDUHWKHEHDPFURVVVHFWLRQRIWKHDUHDDQGWKHDUHDPRPHQWRILQHUWLDUHVSHFWLYHO\EDQGGDUHWKH
<RXQJ¶V DQG VKHDU PRGXOL RI WKH EHDP PDWHULDO UHVSHFWLYHO\ DQGNLV WKH VKHDU FRHIILFLHQW RI WKH FURVVVHFWLRQ
Trang 3JHRPHWU\ 7KH UHVXOWLQJ FRPPRQ DQG ZLGHO\ XVHG VKDSH IXQFWLRQV IRU D 7LPRVKHQNR EHDP FRQVLGHULQJ VKHDU GHIRUPDWLRQFDQEHREWDLQHGDV
u
u
l x
N
l
x
N
l
E I
G A
l
I
N
w
w
w
w
N
N
N
N
I
I
I I
I I
x l
I
°
°
°
°
°
°
®
°
°
°
§ ·
¯
N
N
N
N
T
T
T
T
I
I
I
I I
°
®
°
¯
ZKHUH"LVWKHHOHPHQWOHQJWK I LVWKHUDWLRRIEHDPEHQGLQJVWLIIQHVVWRVKHDUVWLIIQHVVADQGI DUHWKHFURVVVHFWLRQ DUHDDQGPRPHQWRILQHUWLDRIWKHXQLIRUPEHDPEDQGGDUHWKH<RXQJ¶VDQGVKHDUPRGXOLRIWKHPDWHULDODWWKHOHIW HQGRIWKH)*0EHDPZKLOH>N u N u@>N w N w N w N w@DQG>NT NT NT NT@DUHWKHQRGDOD[LDO WUDQVYHUVHGLVSODFHPHQWDQGURWDWLRQDOVKDSHIXQFWLRQVRIWKHEHDP
)LJ7ZRQRGHVL[GHJUHHVRIIUHHGRPEHDPHOHPHQW
,WFDQEHREVHUYHGWKDW(T LVWKHSUHGHILQHGVKDSHIXQFWLRQVIRUKRPRJHQHRXVPDWHULDO FRQVWDQWE DQGXQLIRUP FURVVVHFWLRQ FRQVWDQWA RID7LPRVKHQNREHDP+RZHYHUXSWRQRZWKHVKDSHIXQFWLRQVKDYHEHHQZLGHO\XVHGWR
GHULYHWKHVWLIIQHVVDQGPDVVPDWULFHVIRU7LPRVKHQNREHDPVZLWKLQKHUHQWLQKRPRJHQHLW\DQGQRQXQLIRUPLW\+HQFH LWLVREYLRXVWKDWWKHEHKDYLRXURIEHDPVLVLQDFFXUDWHO\SUHGLFWHGLQWKHDQDO\VHVDQGWRVROYHWKLVSUREOHPTXLWHD ODUJH QXPEHU RI EHDP GLYLVLRQV DUH UHTXLUHG 7KXV XQQHFHVVDU\DQG XQUHDVRQDEOH HIIRUWV DUH RIWHQ PDGH ZLWKRXW XQGHUVWDQGLQJ WKH QDWXUH RI WKH SUREOHP 7KLV SUREOHP FDQ RQO\ EH VROYHG E\ XVLQJ FRQVLVWHQW DQG FRUUHFW VKDSH IXQFWLRQVIRUWKHLQKRPRJHQHRXV7LPRVKHQNREHDPHOHPHQW
2.2 Consistent shape functions
$EHDPZLWKOHQJWK"DQGDUHFWDQJXODUFURVVVHFWLRQDQGPDGHRIWZRGLIIHUHQWPDWHULDOVIXQFWLRQDOO\JUDGHG
DORQJWKHEHDPD[LVLVFRQVLGHUHG7KHHIIHFWLYHSURSHUW\P UHSUHVHQWLQJWKH<RXQJ¶VPRGXOXVRUVKHDUPRGXOXV RI
WKHEHDPPDWHULDOVLVDVVXPHGWRYDU\OLQHDUO\LQWKHEHDPD[LVDV
ZKHUHPUHSUHVHQWVWKHHIIHFWLYHSURSHUWLHVDWx RIWKHEHDPDQGnLVWKHUDWLREHWZHHQWKHHIIHFWLYHSURSHUWLHVDW
ERWKHQGV
1RQKRPRJHQHRXVD[LDOO\)*0EHDPVZLWK<RXQJ¶VPRGXOXVRIWKHEHDPVYDU\DVIROORZV
Trang 4ZKHUH E x LVWKHILUVWRUGHU n ZLWKUHVSHFWWRx,QWKLVSDSHU(T ZLOOEHXVHGIRUQXPHULFDOYHULILFDWLRQVDQG
H[DPSOHV
,QWHJUDWLQJWKHILUVWWHUPRI(T DQGVROYLQJIRU u x UHVXOWVLQ
k
ZKHUH k 1LVDQLQWHJUDWLRQFRQVWDQW
%\DSSO\LQJWKHERXQGDU\FRQGLWLRQVDWERWKHQGQRGHVRIWKHEHDPZKHUHu x uDQGu x " uVROYLQJ IRUkDQGkDQGWKHQVXEVWLWXWLQJWKHUHVXOWEDFNLQWR(T WKHVKDSHIXQFWLRQVIRUWKHD[LDOGLVSODFHPHQWRIWKH EHDPFDQEHREWDLQHGDV
u
u
½
® ¾
¯ ¿
1
,WFDQEHREVHUYHGWKDWWKHLQWHJUDWLRQRIWKHVHFRQGWHUPLQ(T LVHQFDSVXODWHGLQWKHWKLUGWHUPRI(T +HQFH WKH WUDQVYHUVH GLVSODFHPHQW w x VKRXOG EH RQH RUGHU KLJKHU WKDQ WKH URWDWLRQ T x RI WKH EHDP +HUH WKH WUDQVYHUVHGLVSODFHPHQWDQGURWDWLRQILHOGVFDQEHH[SUHVVHGLQSRO\QRPLDOHTXDWLRQVDVIROORZV
x b b x b x
w x a a x a x a x
$IWHUVXEVWLWXWLQJ(T LQWRWKHVHFRQGDQGWKLUGWHUPVRI(T WKHSRO\QRPLDOFRHIILFLHQWVb b b a a a DQGaLQ(T DUHWKHQUHSUHVHQWHGE\WKHLQWHJUDWLRQFRQVWDQWVc c c c UHVXOWLQJLQWKHURWDWLRQDODQG
WUDQVYHUVHGLVSODFHPHQWHTXDWLRQVDVIROORZV
I E x I E x
c
A G x
T
N
7KHERXQGDU\FRQGLWLRQVw x w T x T w x " w T x " TDUHDSSOLHGWR(T VROYLQJ IRUc c c c DQGWKHQVXEVWLWXWLQJWKHUHVXOWEDFNLQWR(T +HUHWKLVUHVXOWHGLQVRPHQDWXUDOORJDULWKPLF WHUPVLQ(T ZKLFKOHDGVWRLQILQLW\ZKHQDKRPRJHQHRXVFRQGLWLRQLVLPSRVHG7RHOLPLQDWHWKLVSUREOHPD FXELFRUGHU7D\ORUVHULHVH[SDQVLRQVFKHPHLVXVHGIRUUHSUHVHQWLQJWKHQDWXUDOORJDULWKPLFWHUPVLQ(T ZKLFK UHVXOWVLQ
w
w
T T
w
w
T T
T
Trang 5
w
w
w
N
N
n
n N
n n
I I
I I
"
"
w
I I
I
°
°
°
°
°
®
°
°
°
°
°
"
"
N
N
N
n n x n N
T
T
T
T
I
I I
I
"
"
"
"
"
"
x nx
I I
°
°
°
°
°
®
°
°
°
°
°
¯
"
"
E I
G A l
I
N ZLWKEDQGGEHLQJWKH<RXQJ¶VDQGVKHDUPRGXOLRIWKHPDWHULDODWx RIWKH)*0EHDP D[LV,WFDQEHVKRZQWKDWIRUKRPRJHQHRXVPDWHULDO n ERWKWKHWUDQVYHUVHDQGURWDWLRQDOVKDSHIXQFWLRQVLQ(T EHFRPH(T ,WVKRXOGEHQRWHGWKDWLQWKHVHFRQVLVWHQWVKDSHIXQFWLRQVJLYHQE\(T WKHLQIOXHQFHRIWKH
HIIHFWLYHSURSHUWLHVUDWLRnLVUHIOHFWHG
)LQLWHHOHPHQWIRUPXODWLRQ
3.1 Global formulation
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
)LJXUHVKRZVDWZRQRGHSODQDUEHDPHOHPHQWDQGLWVNLQHPDWLFVLQDJOREDOFRRUGLQDWHV\VWHP(X,Z)$ORFDO V\VWHP(x,z)DWWDFKHGWRWKHHOHPHQWLVFKRVHQ LWVRULJLQLVDOZD\VDWQRGHDQGLWVxD[LVLVGLUHFWHGWRQRGH%\
FKRRVLQJVXFKDORFDOV\VWHPWKHD[LDOGLVSODFHPHQWDWQRGHDQGWKHWUDQVYHUVHGLVSODFHPHQWDWERWKRIWKHWZR QRGHVLQWKHORFDOV\VWHPDUHDOZD\V]HUR
Trang 6)LJ$SODQDUFRURWDWLRQDOEHDPHOHPHQWDQGLWVNLQHPDWLFV
7KHORFDODQGJOREDOQRGDOGLVSODFHPHQWYHFWRUVDUHUHVSHFWLYHO\JLYHQE\
T
u T T
T
,Q(T DQGKHUHDIWHUWKHVXSHUVFULSWTGHQRWHVWKHWUDQVSRVHRIDYHFWRURUDPDWUL[DQGWKHEDULVXVHGWR
LQGLFDWHWKDWWKHYDULDEOHLVGHILQHGLQWKHORFDOV\VWHP
7KHYHFWRUVRIQRGDOIRUFHVDQGPRPHQWVDVVRFLDWHGZLWKWKHQRGDOGLVSODFHPHQWVRI(T DUHDVIROORZV
T
in N M M
T
7KHFRPSRQHQWVRIWKHORFDOYHFWRULQ(T FDQEHREWDLQHGIURPu "n" T TT Tr TTr ,Q(T " "nDQGT DUHWKHLQLWLDOOHQJWKFXUUHQWOHQJWKDQGULJLGURWDWLRQRIWKHHOHPHQWUHVSHFWLYHO\DQGFDQr
EHFRPSXWHGIURPWKHFRRUGLQDWHVRIWKHQRGHV$VVXPLQJWKDWWKHVWUDLQHQHUJ\URIWKHHOHPHQWKDVEHHQGHULYHG WKHHOHPHQWQRGDOIRUFHYHFWRUIinDQGWKHWDQJHQWVWLIIQHVVPDWUL[NtLQWKHJOREDOV\VWHPFDQEHREWDLQHGE\VXFFHVVLYH GLIIHUHQWLDWLRQRIWKHVWUDLQHQHUJ\ZLWKUHVSHFWWRWKHQRGDOGLVSODFHPHQWV
T
G
T
G
,Q(T IinDQGNtDUHWKHORFDOQRGDOIRUFHYHFWRUDQGWDQJHQWVWLIIQHVVPDWUL[UHVSHFWLYHO\ 7 71 DQG7DUH WKHWUDQVSRVHPDWULFHV>@ZKLFKFDQEHFRPSXWHGIURP(T DV
w
w
G
7
G
u
w w
7
G
r
T w w
7
(TXDWLRQV DQG FRPSOHWHO\GHILQHWKHHOHPHQWIRUPXODWLRQSURYLGHGWKDWWKHORFDOQRGDOIRUFHYHFWRUDQG WDQJHQWVWLIIQHVVPDWUL[DUHNQRZQ
Trang 73.2 Local formulation
&RQVLGHU D WZRQRGH EHDP HOHPHQW LQ D ORFDO V\VWHP x z ZKHUH WKH xD[LV FRLQFLGHV ZLWK WKH QHXWUDO D[LV
$GRSWLQJ7LPRVKHQNREHDPWKHRU\WKHD[LDODQGWUDQVYHUVDOGLVSODFHPHQWVRIDQ\DUELWUDU\SRLQWDUHJLYHQE\
ZKHUHuLVWKHD[LDOGLVSODFHPHQWRIDSRLQWDWWKHQHXWUDOD[LV zLVWKHGLVWDQFHIURPWKHFRQVLGHUHGSRLQWWRWKH
x D[LVDQG T x LVWKHURWDWLRQRIWKHFURVVVHFWLRQ7KHD[LDODQGVKHDUVWUDLQVDUHGHGXFHGIURP(T DV
ZKHUH D FRPPD GHQRWHV WKHGHULYDWLYHZLWK UHVSHFW WR WKH x YDULDEOH H F J DUH WKH PHPEUDQHVWUDLQ FXUYDWXUHm VWUDLQ DQG VKHDU VWUDLQ UHVSHFWLYHO\ $VVXPLQJ HODVWLF EHKDYLRXU WKH D[LDO DQG VKHDU VWUHVVHV DVVRFLDWHG ZLWK WKH VWUDLQVRI(T DUHJLYHQE\
ZKHUH E x DQG G x DUH WKH HIIHFWLYH <RXQJ¶V PRGXOXV DQG VKHDU PRGXOXV UHVSHFWLYHO\ N GHQRWHV WKH VKHDU FRUUHFWLRQIDFWRUZKLFKLVHTXDOWRIRUDUHFWDQJXODUVHFWLRQ,QRUGHUWRH[SUHVVWKHGLVSODFHPHQWDWDQDUELWUDU\ SRLQW LQ WHUPV RI WKH QRGDO GLVSODFHPHQWV LW LV QHFHVVDU\ WR LQWURGXFH VKDSH IXQFWLRQV IRUu w T 1RWLQJ WKDW
ZKHUH ^ `T
T T
ș 1u 1w1TDUHWKHPDWULFHVRILQWHUSRODWLQJIXQFWLRQVIRU wDQGT UHVSHFWLYHO\,QWKHSUHVHQW
ZRUNWKHFRQVLVWHQWVKDSHIXQFWLRQVRI(TV DQG DUHHPSOR\HGWRLQWHUSRODWHIRU u wTUHVSHFWLYHO\
7KHHOHPHQWIRUPXODWHGIURPWKHDERYHXQEDODQFHGLQWHUSRODWLQJIXQFWLRQVGRHVKRZHYHUVXIIHUIURPPHPEUDQH ORFNLQJ>@,QRUGHUWRDYRLGWKLVSUREOHPWKHPHPEUDQHVWUDLQLQ(T LVUHSODFHGE\WKHHIIHFWLYHVWUDLQDQGE\ VXEVWLWXWLQJ(TV DQG LQWR(T RQHJHWV
T T
u
x
w
w
1
w x
w w
1
8VLQJWKHFRQFHSWRIHIIHFWLYHVWUDLQRQHFDQH[SUHVVWKHYLUWXDOZRUNLQWKHIRUP
V
ZKHUHN E x A Heff M E x I F Q \G x A J b u
" T eff
H T
w w
x
T T
w w
1
(TXDWLRQ JLYHVWKHFRPSRQHQWVRIWKHORFDOLQWHUQDOIRUFHYHFWRUDV
Trang 8
f ³Nb dx IT ³ NHT METQE 1T dx
" "
7KHWDQJHQWVWLIIQHVVPDWUL[LQWKHORFDOV\VWHPFDQEHH[SUHVVHGE\VXEPDWULFHVDVIROORZV
uu u
t T
u
T
T TT
N
ZKHUH
T
°
°
®
°
³
"
"
"
DQG T
T
w w
H
7RJHWKHUWKHORFDOLQWHUQDOIRUFHYHFWRU (T WKHVWLIIQHVVPDWULFHV (T DQGWKHWUDQVIRUPDWLRQPDWULFHV (TV DQG FRPSOHWHO\GHVFULEHWKHHOHPHQWIRUPXODWLRQLQWKHJOREDOFRRUGLQDWHV\VWHP
1XPHULFDOSURFHGXUH
7KHIRUPXODWHGHOHPHQWLQWHUQDOIRUFHYHFWRUDQGWDQJHQWVWLIIQHVVPDWUL[DUHDVVHPEOHGWRFRQVWUXFWWKHVWUXFWXUDO HTXLOLEULXPHTXDWLRQV>@ZKLFKFDQEHZULWWHQLQWKHIRUP
M in M ext
ZKHUHWKHRXWRIEDODQFHIRUFHYHFWRUJLVDIXQFWLRQRIWKHFXUUHQWVWUXFWXUDOQRGDOGLVSODFHPHQWVGDQGWKHORDGLQJ SDUDPHWHUM TinLVWKHVWUXFWXUDOQRGDOIRUFHYHFWRUDVVHPEOHGIURPWKHHOHPHQWIRUFHYHFWRUIin IextLVWKHFRQVWDQW H[WHUQDOORDGLQJYHFWRU
7KHQRQOLQHDUHTXDWLRQV FDQEHVROYHGE\XVLQJDQLQFUHPHQWDOLWHUDWLYHSURFHGXUHDQGLQRUGHUWRWUDFHD FRPSOHWHHTXLOLEULXPSDWKWKHDUFOHQJWKFRQWUROPHWKRGGHVFULEHGE\&ULVILHOG>@LVHPSOR\HG
1XPHULFDOH[DPSOHV
5.1 Formulation verification
7R YHULI\ WKH DFFXUDF\ RI WKH SUHVHQW IRUPXODWLRQ DQG WKH QXPHULFDO SURFHGXUH RI WKH XQLIRUP KRPRJHQRXV 7LPRVKHQNREHDPDQGXQLIRUPD[LDOO\)*0%HUQRXOOL(XOHUEHDPXQGHUYDULRXVERXQGDU\FRQGLWLRQV FODPSHG±
FODPSHGFODPSHG±IUHHDQGKLQJHG±KLQJHG VXEMHFWHGWRD[LDOORDGP 1DEHDPZLWKDVKHDUFRUUHFWLRQIDFWRUȥ 3RLVVRQ¶VUDWLRȣ E *3DLVFRQVLGHUHG,QDQDO\VLQJWKH%HUQRXOOL(XOHUEHDPSUREOHPVWKHVKDSH IXQFWLRQVGHULYHGLQWKLVSDSHUFDQEHXVHGE\XVLQJDODUJHYDOXHRIN7KH<RXQJ¶VPRGXOXVRIWKHD[LDOO\)*0
EHDPIROORZVDOLQHDUYDULDWLRQDORQJWKHxD[LVDVJLYHQLQ(T
7RIDFLOLWDWHWKHSUHVHQWDWLRQRIUHVXOWVWKHGLPHQVLRQOHVVFULWLFDOEXFNOLQJSDUDPHWHULVLQWURGXFHGDV
cr
P L
P
7DEOHVDQGVKRZYHU\JRRGDJUHHPHQWVEHWZHHQWKHFULWLFDOORDGVREWDLQHGLQWKHSUHVHQWZRUNDQGWKRVHRI UHIHUHQFHV
Trang 9
%RXQGDU\&RQGLWLRQV &ODPSHG±&ODPSHG +LQJHG±+LQJHG &ODPSHG±)UHH
5HIHUHQFH>@
7DEOH'LPHQVLRQOHVVFULWLFDOORDGRIDXQLIRUPD[LDOO\)*0%HUQRXOOL(XOHUEHDP
%RXQGDU\&RQGLWLRQV &ODPSHG±&ODPSHG +LQJHG±+LQJHG &ODPSHG±)UHH
5HIHUHQFH>@
5HIHUHQFH>@
7KHFULWLFDOORDGVRIWKHVLPSO\VXSSRUWHGEHDPPDGHIURPVWHHODQGDOXPLQDZLWKGLIIHUHQWQXPEHUVRIHOHPHQWV DUHOLVWHGLQ7DEOH7KHILUVWFROXPQLQFOXGHVWKHQXPEHURIHOHPHQWVWKHVHFRQGFROXPQVKRZVWKHFULWLFDOORDGVRI WKHEHDPXVLQJWKHXQLIRUPVKDSHIXQFWLRQVRI(T IRULQWHUSRODWLRQDQGWKHODVWFROXPQGLVSOD\VWKHFULWLFDOORDGV RIWKHEHDPXVLQJWKHQHZO\GHULYHGVKDSHIXQFWLRQVRI(TV DQG
$VFDQEHVHHQIURPWKH7DEOHWKHEHDPHOHPHQWEDVHGRQWKHQHZO\GHULYHGVKDSHIXQFWLRQVFRQYHUJHVIDVWHU WKDQWKHRQHEDVHGRQWKHKRPRJHQHRXVVKDSHIXQFWLRQV7RDFKLHYHWKHFRQYHUJHQFHRIEXFNOLQJFRHIILFLHQWWKH VROXWLRQE\XVLQJ(T ZKLFKLVWKHKRPRJHQHRXVVKDSHIXQFWLRQVUHTXLULQJPRUHWKDQHOHPHQWGLYLVLRQZKLOH E\XVLQJ(TV DQG RQO\UHTXLULQJRQHRUWZRHOHPHQWGLYLVLRQVFDQDFKLHYHWKHFRQYHUJHQFH
7DEOH'LPHQVLRQOHVVFULWLFDOORDGRIDXQLIRUPVWHHO±DOXPLQDEHDPHOHPHQW
(OHPHQW
GLYLVLRQ (T (TV DQG
7DEOH%XFNOLQJFRHIILFLHQWIRUFDQWLOHYHUEHDPPDGHRIVWHHO±DOXPLQD EXFNOLQJFRHIILFLHQW
f B 3 FU / (,EHDPZLWKZLGWKE PKHLJKWK PDQGWRWDOOHQJWK/ P
(OHPHQW
Trang 105.2 Cantilever beam subjected to eccentric axial load
&RQVLGHUWKHFDQWLOHYHUEHDPVXEMHFWHGWRDQHFFHQWULFD[LDOORDGVKRZQLQ)LJD7KLVSUREOHPZDVSUHYLRXVO\
DQDO\VHGE\:RRGDQG=LHQNLHZLF]>@DQGE\1JX\HQ>@7KHEHDPGDWDDUH L = Pb P, h P, E =
12, DQGG = E/27KHORDGGLVSODFHPHQWFXUYHVREWDLQHGE\XVLQJRQO\WKUHHEHDPHOHPHQWVDUHVKRZQLQ)LJE
)LJ/RDG±GLVSODFHPHQWUHODWLRQIRUFDQWLOHYHUEHDPXQGHUHFFHQWULFD[LDOORDG ,WFDQEHREVHUYHGIURPWKHILJXUHWKDWWKHSXUHVWHHOPDWHULDOKDVWKHORZHVWEXFNOLQJORDGFRPSDUHGZLWKWKH OLQHDUO\JUDGHGVWHHO±DOXPLQDPDWHULDO,WLVYHU\LQWHUHVWLQJWKDWHYHQZKHQWKHVDPHOLQHDUO\JUDGHGVWHHO±DOXPLQD PDWHULDOLVXVHGWKHEXFNOLQJORDGVDUHVLJQLILFDQWO\GLIIHUHQWGHSHQGLQJRQWKHVLGHRIWKHPDWHULDOEHLQJORDGHGRU IL[HG7KHEXFNOLQJORDGFDQEHLQFUHDVHGE\DUUDQJLQJRQO\VWHHORQWKHORDGLQJVLGHDQGDOXPLQDRQWKHIL[HGVLGH EHFDXVHWKHDOXPLQDKDVJUHDWHUVWLIIQHVV
5.3 Williams' toggle FGM frame
7KH:LOOLDPV¶ WRJJOH IUDPH )LJD ZKLFKKDVEHHQ HPSOR\HGE\ PDQ\ DXWKRUV WR WHVW WKHLUQRQOLQHDUEHDP
HOHPHQWVLVFRQVLGHUHGKHUH7KHGDWDIRUFRPSXWDWLRQDUHDVIROORZV L = LQ P H LQ
P b = LQ P h = LQ P DQGER= îOE î03D ,Q)LJEWKHORDG GLVSODFHPHQWFXUYHVIRUWKHIUDPHDUHREWDLQHGE\XVLQJRQO\WZREHDPHOHPHQWV7KHJUHHQFXUYHFRUUHVSRQGLQJWR WKHKRPRJHQRXVPDWHULDOZKLFKLVLQJRRGDJUHHPHQWZLWKWKHUHVXOWVREWDLQHGE\1JX\HQ>@LVVKRZQLQURXQGEOXH FLUFOHV7KHHIIHFWRIWKHPDWHULDOGLVWULEXWLRQRQWKHOLPLWORDGRIWKH)*0IUDPHLVFOHDUO\VHHQIURPWKHILJXUH7KH SXUHVWHHOPDWHULDOKDVWKHORZHVWOLPLWORDGFRPSDUHGZLWKWKHJUDGHGVWHHO±DOXPLQDPDWHULDO3XWWLQJWKHJUDGHG GLVWULEXWLRQRIDOXPLQDDWWKHIL[HGHQGVDQGWKHVWHHODWWKHSHDNRIWKHIUDPHZLOOJLYHWKHKLJKHVWOLPLWORDGRIWKH
IUDPHVXEMHFWHGWRDYHUWLFDOORDGLQJP
... http://creativecommons.org/licenses/by-nc-nd/4.0/ ).Peer-review under responsibility of the organizing committee of SCESCM 2016.
Trang 2