K^T cAu - C 6 N G NGHE XAY Pg'NG
CAC PHU'QNG PHAP PHAN TICH DONG PHI TUY^N KET CAU THEO UCH Sir THOI GIAN TRONG SAP2000 {PHAN 1)
ThS. T R A N N G O C Cl/CTNG Vien KHCN Xay di>ng
Tom tit: Bai bao nay gom cd hai phin, gi&i thi$u nhimg phieang phap phan tich dong phi tuyen theo Ijch si> thai gian dieac tich hi?p sin trong phin mim SAP2000 nhim giOp nhOng nguxfi dung pho thong co du^yc nhung hieu biet ca ban ve d$c diem ciia tCmg phuv>ng phap cung nhir ph^m vi ap dyng cua ehung. Vai moi phuxyng phap, bai bao dwa ra cac tinh chat ca ban nhw:
Thuoc hg noi an th(rc hay ngoai hiin thu-c, dieu kien on dinh (niu co), do chinh xac trong kit qua tinh toan, do can nhat s6, tu- do dwa ra pham vi ap dyng di xuit. Bai bao cung dwa ra mot so vi du tinh to^n di minh /jpa va lam ro tinh chat cua timg phuang phap.
1. Dat van de
Van de phan tich <50ng lyc hoc phi tuy§n (goi tit la dgng phi tuyen) ket ciu cong trinh chju tai trgng dgng dit theo phu-ang phap Ijch sCr tho"!
gian {nonlinear time-history analyses) du'g'c quan tam gan day vi nhOng u'u diem cOa no nhu: Khae phuc du'g'c nhiing nhup'c diem cua phuong phap phan ti'ch dan hdi tuyIn tinh khi thi^t k4 nhung cong trinh co ket ciu dgc biet, phuc tgp (phuong phap nay yeu cau kit cau phai thoa man nhieu yeu cau mang tinh djnh lugng theo cac dieu khoan trong tieu chuan ap dyng), phan tich chi'nh xac hen ung xu cua ket ciu co the giCip giam bot kich thuoc tiet dien, iam giam chi phi dau tu ma v i n dam bao an toan khi su dung [1].
So sanh voi phuong phap phan tich pho phan i>ng dang dao dgng (myc 4.3.3.3, [2]), phuang phap phan tich dgng phi tuyIn cho phep xet den kha nang tieu tan nang lugng mgt each chinh xac va day du hon la chi thong qua mpt hg so ung xu q, dac bigt vo'i nhung he ket cau nha cao tang hoac cac hg ket ciu phuc tap ma tieu chuan chua de cap het Hon nua, phan tich dgng phi tuyen cd the cho biet chinh xac vj tri va thoi diem
hinh thanh kh6p deo trong kit cau, trong khi phuong phap phan tich pho phan ung dang dao dgng khong cho biet dugc dilu nay. Mgt uu diem nua cua vigc phan tich dgng, do la co the diJng no de hoan thien cac phuong phap phan tich khae, VI dy nhu xay dyng cac pho phan ung thiet k l hoac dieu chinh gia trj he so ung xu q.
Tieu chuan xay dyng thilt k l chong dgng dit cua Vigt Nam cung khuyin khfch su dung phuong phap dong phi tuyIn trong phan tich kit ciu cong trinh (muc 4.3.3., [2]). Tuy vay trong thuc hanh, viec thyc hign phan tich dgng phi tuyen hoac ke ca don gian hon la phuong phap day dan con khd khan va khong phai iuc nao cung thuc hien dugc, xet theo ca khia canh ky thuat lan kinh te, do yeu cau Ion ve nguon nhan lyc trinh dg cao, nang lyc may tinh va thoi gian phan tich [1].
Mgt trong nhung each d l ap dung cac phuong phap phan tich d6ng phi tuyIn theo ijch su thcyi gian la su dyng cac phin mem may tinh dugc lap trinh san. SAP2000 la mgt phin mim phan tich kit cau dugc su dyng tuong doi rgng rai 6 Viet Nam. SAP2000 co the tfnh toan vo'i ca dang phi tuyen vat ligu va phi tuyIn hinh hgc (co xet din higu ung P-A). SAP2000 phien ban Advance va Ultimate cho phep phan tich dgng phi tuyen theo lich su thai gian voi mot so phuong phap dugc tich hgp san, gom co: Newmark, Wilson, Collocation, Hilber - Hughes - Taylor, Chung&Hulbert. Viec thyc hien tuan ty cac buo'c de ap dyng cac phuong phap nay ve co ban khong qua phuc tap, co the tham khao nhung tai lieu chi dan viec phan tich nay trong phin Tra giup (Help) cua chuong trinh, hogc tham khao cac video chi dan tinh toan tren trang web chinh thuc cua hang CSI [3]. Tuy nhien, nguoi tinh toan CO the gap kho khan trong viec lua chpn Tap chl KHCN Xay dimg - s6 1/2016
K£T CAU - CONG NGHE XAV DUNG
phuomg phap va cac tham s6 tinh toan. Do vay, bai bao hu6ng den nh&ng nguoi su dung thong thuo-ng bing each tong hgp va giiip ngucyi dung nam dugc mgt so nhOng dac diem co ban nhit cua cac phuong phap phan tich dgng phi tuyen theo Ijch SLP tho'i gian, nham tang hieu qua su dung SAP2000. Tren co so do, bai bao co mgt so npi dung chinh nhu sau:
Phan 1:
- Tdm luoc mpt so kiln thuc co ban trong phan tich dpng phi tuyen theo Ijch SLP thoi gian;
- Gio'i thieu cac phuong phap phan tich dugc tich hgp san trong SAP2000 va cac tinh chit cua ehung;
- Dua ra mpt so khuyin nghj su dung cho mot s6 truong hgp tinh toan.
Phan 2: (dy kien dang trong s6 tilp theo cua tap chi)
- Gioi thigu mgt so vi dy tfnh toan bang SAP2000 nham lam ro cae dgc dilm cua cac phuong phap tinh.
2. Cac khai niem c c ban
De danh gia mgt phuong phap phan tich, ngud'i ta dya vao mgt vai tieu chi khae nhau.
Theo Hilber, Hughes va Taylor [4], nhCrng yeu cau CO ban eua mgt phuong phap phan tich phi tuyen theo ijch su thoi gian bao gom:
(1) Phai khong cd dieu kign on djnh voi hg ket elu tuyIn tfnh;
(2) He so tieu tan so hgc {numerical dissipation) - khai niem nay tuong duong vdi he so can nhdt so {numerical damping ratio) se dugc nhic den trong bai bao nay - cd the dugc kiem soat bang cac tham s6 thay vi kiem soat bing bude thdi gian. Ngoai ra, phai cd kha nang tinh toan vdi trudng hgp khdng cd he so tieu tan;
(3) Hg s6 tieu tan sd hpc khdng dugc anh hudng qua Idn din cac dgng dao dpng bgc thap.
Ngoai ra, cdn mdt tieu ehf khae dugc ehip nhgn tuong doi rpng rai trong nhieu tai lieu, vi dy nhu [5], dd la:
(4) Phai cd dp chinh xac cap 2 (sai so liy den Bai bao se bam theo bdn tidu ehi nay d l danh^
gia cac phuong phap tinh.
2.1 Ngoai hiin thOv, ngi an thvpc
Theo [6], nhung phuong phap thupe hp ngoai hiln thuc {explicit) la nhung phuong phap co dang: { D V I =f({D}i,{b}i,{b}i,{D}i-i...) ^ (1)
trong dd, gia tn (chuyin vj, van tde, gia toe...) eOa budc thu (/+1) cd the dugc tinh true tiep tii cac gia tri eua cac bude trude dd. Nhung phuong phap thupc hp ndi an thuc {Implicit) co dang:
{D}i+i=f({b}i+i,{D}i+i,{D}i....) (2) trong dd, gia trj cua budc thi> (/ +1) khong chl lien
quan din cac gia trj cCia cae budc trude dd ma cdn lien quan din gia tn cua ehinh budc hign tgi, do vay, d l tinh can giai phuong trinh d l tim ra nghiem. Phuong phap thudng dung de giai phuong trinh dugc ap dung trong SAP2000 la phuong phap tinh lap Newton Raphson va Newton Raphson cai tien {Modified Newton Raphson).
Nhiing uu diem va nhugc diem cua hai hp phuong phap nay cd the chi ra nhu sau [7]:
Vdi hp phuong phap ngoai hiln thiic:
Lfu diem:
- it mit cdng tfnh toan hon trong mpt budc;
Thuat toan don gian, logic, de ap dung vdi kit elu phi tuyen;
- Can it dung bp nhd may tinh hon khi tfnh toan (so sanh vdi hg ngi an thiic tinh bang vong lap Newton Raphson);
- Thfch hgp de thu cac thuat toan mdi vl viet cac doan ma chuong trinh may tfnh don gian hon;
- Do tin cay va chinh xac cao hon.
Nhugc dilm:
- Cd dilu kign on djnh, do vgy nhilu trudng hgp so budc tinh toan yeu cau rit ldn.
Ben eanh dd, khi so sanh vdi hp phuong phap npi an thuc, nhung uu diem cua hg phuong phap ngoai hien thCrc la nhugc diem cua hp npi an thiie va ngugc lai.
Tap Chi KHCN Xay dimg - so 1/2016
K i r c A U - CONG NGHE XAY DU-NG U'u diem:
- Khong cd dieu kign on dinh nen gia tn buo'c thdi gian cd the Idn hon nhilu lan so vdi hg phuong phap ngogi hien thiie.
Nhugc dilm:
- Chuong trinh tinh toan thudng Idn va phiic tap, vi dg nhu khi SLP dyng phuong phap tinh Igp Newton Raphson;
- Dp lin cay kem hon;
- T i n nhieu dung lugng xii ly hon.
2.2 Pham vi ap dung cua moi phiFO^g phap Theo [8], cae bai toan dpng lyc hpc cong trinh dugc ehia lam hai dgng chinh:
Dang 1 ia cac bai toan dang truyen sdng (ivave propagation problems), vf du nhu khi cdng trinh chiu tac dpng va cham hoac cac vu no.
Trong dgng nay, anh hudng eua cac dang dao dpng t i n so cao din tong the eong trinh la dang ke va ta cin phai quan tam den higu ipng cua cac sdng ling suit. Thdi gian bj anh hudng eua cdng trinh thudng la ngan.
Dgng 2 la eae bai toan dpng lyc hgc {structural dynamics problems), dugc dinh nghTa la nhung bai toan khong nam trong dang 1, vi du nhu khi tac ddng cCia dOng dit. Trong dang nay, iyc quan tfnh ddng vai trd quan trpng trong iing xu tong the cua cdng trinh, cdng trinh chiu anh hudng cua cac dgng dao dpng bac thap la chu ylu.
Do cd dieu kien on dinh nen phuong phap ngogi hien thiic thudng dugc dung de giai cac bai toan thuge dang 1. Thdng thudng gia tn budc thdi gian dugc chpn thoa man dilu kign on djnh thi dilu kien v l do chinh xac cung ty dgng dugc thda man. Vdi cac bai toan dpng lyc hpe, thudng chiem da so trong cac bai toan v l xay dung, phuong phap npi an thiic thudng dugc ehgn do khdng ed dilu kien on djnh. Trong trudng hgp nay. bude thdi gian tfnh toan khong ehgn theo dieu kign on djnh ma ehgn theo yeu cau ve dp chinh xac trong ket qua tinh.
2.3 Sai so tu^ng doi cua chu ky
Be danh gia dp chinh xac trong kit qua cua moi phuong phap phan tich dgng phi tuyIn, khai nigm sai so tuong ddi cua chu ky {relative period error) dugc sii dung va dieQfc tfnh bang:
T trong dd:
PE ia ky hieu cua sai s i tuong doi eua chu ky (Period Error).
T la chu ky dao dpng eua hg kit ciu tinh bang phuong phap phan tich phi tuyen theo lich si> thdi gian.
7" la ehu ky dao dgng thyc.
Sai so tuong doi eua chu ky dugc md ta nhu trong hinh 1 [9]. Vige tinh toan cac thong sd nay thudng phiic tgp va can sii dyng din may tinh.
Sai so tuong ddi cua chu ky cang nhd thi kit qua tinh se cang chinh xac.
(3)
Hinh 1. Cac thong so danh gia dg chinh xac cua phwong phap tinh 2.4 He so can nh&t so
Cac phuong phap phan tfch phi tuyIn theo lich sii thdi gian hign tgi thudng cho kit qua khdng chinh xac vdi cac dgng dao dpng bac cao, Tap chi KHCN Xay dung - so 1/2016
them vao dd, vdi cae bai toan dpng lye thupc dgng 2 thi anh hudng cua cac dang dao ddng bgc cao vdi tong t h i kit elu la khdng dang k l , do vay he so can nhdt so {algorithmic damping ratio
KET CAu - CONG NGHE XAY DlfNG hogc numerical damping ratio) cd tac dyng lam tat nhanh chdng anh hudng cua eae dang dao dgng bac cao trong khi khdng lam anh hudng din dp ehinh xac eua cac dgng dao dgng bae thap. Mgt phuong phap tinh dugc danh gia la tdt nlu nd cd kha nang kiem soat va dilu ehinh he so can nhdt so bang each thay doi eae tham so tinh toan. Trong mdi phuong phap dugc gidi thigu d muc 3, eae gia tri nay deu dugc dua ra lam eo sd so sanh.
3. Cac phiro'ng phap phan tich 3.1 Phuxyng phap Nevifmark
Phuong phap Newmark [10] la phuong phap dugc biet den rpng rai nhat trong tat ca cac phuong phap phan tfch dong phi tuyIn theo lich sii thdi gian. Phuong trinh md ta phuong ph^p.
nay dugc vilt nhu sau:
= d.+(At)Vj + ^ [ ( l - 2 p ) a ^2P^i.lJ (4)
Cd rat nhieu each d l lya chpn cac he so ^ va Y cho phuang phap nay, tuy nhien ed 4 each lya chpn dugc biet den rpng rai nhit nhu sau:
Bang 1. Cac phwong phap thwang dung trong hg phwong phap Newmark Ten thuo-ng goi bang tieng Anh
Average Acceleration Method (AAM) Linear Acceleration Method Fox-Goodwin Method Newmark Explicit Method (NEM)
Kieu
Npi an fhCrc Ngi an thiic Npi an thiic Ngoai hien thiic
)3
1/4 1/6 1/12 0
V
K Vi
%
Vz
Dieu kien on dinh {*) Khong n ^ , - 2 - , ^ ^3,464 n „ , - ^ / 6 ^ 2,449 n „ , = 2
W% so can nho'tsS Kh6ng Khong Khong Khong (*) Dieu kien on dinh cua phuang phap duo-c tinh theo cong thiic: CL = ai{^t)={27zlT){^t)<.Q.^, hay
^tlT< Ii„„ / [ i n ) , vai 7" la chu ky dao dgng tdn nhat cua he k i t cau. Dieu kign on djnh con phu thugc vao dp can nhdt vgt ly cua he ket cau, d day xet vdi trudng hap khong co can nhdt vat ly (f = 0).
03
\
02% ^ • '
^
•0 2
•03
• f ) 1
o fli^m m^l fin djnh
1 ' . 1 • 1 ..-•••^=1,'4
" • ' * . - .
.-,
•••^
\ ^
= ol ;/(= i;j2
, r . !,
Fiijiirej. V'anj[ioni>ffe!diin:[Krnid(.-rrorwilh'\(i riiirPFM
Hinh 2. Sais6 twcmg d^i cua chu k^ Cmg v&itruung /7pp Y=1/2 Trong 4 phuang phap nay thi hai phuong
phap AAM va NEM thudng duge dung nhilu hon. Tit ea cae phuang phap nay deu khdng cd hg so can nhdt so (thye t l la phuong phap 6
Newmark cd t h i cd he s6 can nhdt so vdi y > 1/2, tuy nhien nd se lam giam dp chinh xac cCia ket qua tfnh tii bac 2 xuong bgc 1 nen trudng hgp nay thudng it dugc quan tam). Sau nay, chiing Tap chl KHCN Xay dung - s6 1/2016
KET cAU - CONG NGHE XAY Pg^lG dugc dieu chinh cai tien de dua them he so can nhdt so vao nhu phuang phap HHT se not d myc 0. Sai sd tuong doi cua chu ky ciJa ca 4 phuong phap nay dugc trinh bay trong hinh 2. Hinh 2 cho thiy neu xet v l dp chinh xac trong kit qua tinh toan thi trudng hgp ^ = 1/12 cho kit qua ehinh xac hon ea, tuy nhidn, nd Igi cd dilu kign on djnh.
Trudng hgp j8 = % thud-ng dugc dung do nd khong cd dteu kien on dinh. Vdi /3 = 0, du cd dieu kign on dinh nhung lai hay dugc dung de doi chieu kit qua vi nd la phuong phap ngoai hiln thiic.
3.2 Phuxmgphap Wilson Theta
Gia thilt co ban cDa phuang phap Wilson Theta [11] dd la gia tie cua hg kit eau thay doi tuyen tfnh trong khoang thdi gian tir thdi diem t din thdi diem ( + e(AO vdi 9 > 1 va 0 dugc xac dinh dya vao vtec toi uu hda dp on dinh va dp chinh xac cua kit qua tinh toan. Ggi r la khoang thdi gian tinh thdi diem t den thdi dilm dang xet, vdi 0 s z-s 0(AO, 0 S 1; nhu vay trong khoang thdi gian tu t din Uidi diem t+9{At), ta ed:
Tich phan edng thiie (5) de cd vec-to van toe nhu sau:
'''^'""'"'t^irah-iiti-'t]
Tfch phan mot Ian nOa tir (6) de thu du-pc vec-to" chuyen vj:
^ = d , + r v , ^ - - a
% { A t ) h * I A i r d tho'i diem (+(A0, fa co:
, ( i O r
•Witl " t '
tr
(7)
(8)
* ( i t ) v (9)
(6)
(Air r- 't-H4t) "t • > " " • ! • • 5 rtMHn*
Thay cac cong thuc (5), (6) va (7) vao phuong trinh chuyen dong ca ban voi r = 6{Mj, ta co:
•^^^lAt) * "^mm * %B(Ati = ''mM} <'°) Giai phuang trinh (10) vdi mgt an duy nhat chua bilt la ajtsjat). sau dd t h i vao cong thiic (8) va (9) ta thu duge cac gia tri chuyen vi. vgn tde, gia toe d thdi diem f+(AO- Khao sat phuong phap nay, ta thu dugc khoang tdi uu eho gia tri 0 la 1,37 £ 9 ^ 1,4, trong khoang gia tn nay phuang phap nay khdng co dilu kien on djnh. Luu y rang, vdi 9 = 1, phuang phap nay se trd thanh phuong phap Newmark tuong iing vdi trudng hgp j3 = 1/6 va y = 14, khi do nd se cd dieu kien on djnh.
Sai so tuang doi cua ehu ky, hg so can nhdt so ling vdi cac trudng hgp 9= 1,37 va 1,4 dugc t h i hign trong hinh 3. Sai s6 tuong ddi cua chu ky eua phuang phap Wilson Theta va hinh 4, trong khi cac tinh chit cua phuong phap dugc the hign trong bang 2.
Bang 2. Cac gia tn tham so 6 va cac tinh chit cua phwong phap Wilson Theta Gia trj B
1,37 1,4 1,0 (•)
Kieu
Ngi I n thiic Ngi I n thiie Ndi an thiic
Dieu kien o n djnh
Khong Khong Cd
Hg so can n h d t so
Cd Cd Cd (*): Vdi 9 = 1 , phuong phap nay se trd thanh mot trudng hgp ciia phuong phap NevflTiark, ligt ke d day ctil d l tham khao
Tap chi KHCN Xay dung - s6 1/2016
K£T CAU - CdNG NGHE XAY DU'NG
0 5
| 0 4
-n
£ 0 - 3
| o ^ 01
•
•
•
y^
1 • — 1
1 . 1 K"^
Hinh 3. Sai sS tirang 661 cua chu ky cua phuvng phip Wilson Theta
Hinh 4. Dp can nhat so cua 3.3 Phtmng phap collocation
Phuong phcip collocation ia su i<et hap cua hai phuong phap Newton va Wilson Theta voi phuong trinh mo ta duoc vilt nhu sau [12]:
= (l-e)Ma,+eMa.
=(i-e)F+eF|., e(At)v
'H-1
^(At)^
(11)
[(l-2p)a| + 2(ia,^
V e = V|*e(At)[(i-v)a|+va„e]
Vdi 6=1, phuong phap nay trd thanh phuong phap Newmark, eon vdi j3 = 1/6 va y = 14, phuang phap nay trd thanh phuang phap Wilson Theta.
Dilu kien de phuang phap nay cd dp chinh xac cap hai la y = Vz. De cd dp chfnh xac trong kit qua len din cap 3 thi can them mot dilu kign
phwong phap Wilson Theta
nua, dd la p^^-^9{9-\) , tuy nhien trong trudng hgp nay Igi yeu ciu cd dilu kign on djnh.
Khoang gia tri ^ 1 phuang phap nay thda man khdng ed dieu kien on dinh va eo dp ehinh xac cap 2 la (xem hinh 5):
2(9
(12)
Cac gia trj ^ va 0 dugc lua chpn lya ehgn theo dieu kign tdi uu hda do chfnh xac cua kit qua va gia tri dp can nhdt so, each lya ehgn nay thyc hien nhu sau: gia tn y co dinh bang Yz d l phuang phap cd dp chinh xac cap hai, tuong ling vdi mdi gia trj J3 se tim lay mgt gia tn 0 sao eho sai s i cua kit qua la nhd nhit va dg can nhdt sd la ldn nhat. Mpt sd cap gia tri (/3, y, 9) dugc khuyin nghi lya chpn trong bang 3. Vdi cac gia tri tham so khae cd the dugc lya chpn bang each ngi suy tuyen tinh giiia cac gia tn da cho.
Tap chi KHCN Xay dung - s6 1/2016
K^T CAU - CONG NGHE X A Y D I / N G
Hinh 5. Viing gia tri khong co diiu ki^n on djnh cua phwong phap collocation (y = 14}
Bang 3. Cac cap gia tn tham so khuyen nghi su dung cho phwong phap collocation /3
y . 0.24 0.22 0.20 0.18 1/6
V
0.25 0.25 0.25 0.25 0.25 0.25
8 1 1,021712 1,077933 1,159772 1,287301 1,420315
Kieu Noi an thCfc NQI an thl>c Ngi an thuc Npi an thi>c Ngi an thuc Ngi an thuc
He so can n h o t so Khong
Co Co Co Co Co
Ghi chu T r d thanh phuong phap AAM
Tro thanh phuong phap Wilson Theta Nhin ehung, vdi nhiing yeu cau sii dyng thdng
thudng thi phuang phap nay khong d l sii dyng, vigc lya ehgn cac cgp gia tn tham so khd hon so vdi cac phuang phap khae, do dd khong khuyin nghi lya ehgn phuang phap nay trong tinh toan.
Bai bao eung khdng trinh bay ky hon v l phuong phap nay, neu bgn dpc quan tam ed t h i tham khao tai lieu [12, pp. 114-119].
3.4 PhiFang phap Hilber - Hughes - Taylor (HHV
Hilber, Hughes va Taylor da de xuat mpt phuang phap tfnh mdi, trong dd dua them vao he so a d l dilu chinh he s i can nhdt so tinh toan, dieu ma hp phuang phap Newmark khong lam dugc [4]. Phuang phap nay dugc viet nhu sau:
J.^, = d + ( A t ) v , * ^ [ ( l - 2 | 3 ) a , t 2 P a . ^ , ] v.^,=V|+(At)[(l-v)a|+vai+i]
Gia trj a khuyen nghj nim trong l<hoang [-1/3, 0], trong do voi ot = 0 thi phuong phap nay tro thanh phuong phap AAM (thuoc hg phuong phap Newmarl<). Cac gia tri jS, y duoc lua chon can cu theo a nhu sau:
(13)
(14) 1-2a ^ (l-a)
Gia tri a cang giam thi he so can nhdt so cang tang. Vdi trudng hgp a = 0, nd khdng cd he so
Tap chi KHCN Xay dung - s6 1/2016
K^T c A U • CONG NGHE XAY D l / N G
can nhdt so, tuong ty nhu vdi phuang phap trudng hgp nay, day chinh la phuang phap pji,lj\_ AAM nhu da ndi d" mue 0. Ivlpt so cgp gia tri a, 13, Y va cac tinh chat cDa chCing dugc trinh bay Phuang phap nay dugc ehgn la phuong ^ihu trong bang 4 de thugn tign eho ngudi sii' phap tinh mac dinh trong SAP2000 vdi he sd dung,
mac dinh a = 0, f3 = 0,25 va y = 0,5. Trong
Bang 4. Cac gia tri thong s6 diu vao thong dung cho phwong phap HHT Gia tri a
-1/3 -1/6 0
Kilu Ngi an thCrc Ngi an thuc Ngi an thuc
P 0,444 0,340 0,25
Y
0,833 0,667 0,5
Dieu kien on djnh Khong Khong Khong
He so can nh6t s6 Co Co Khong Ghi chu: Trgng SAP2000, ngudi dung chi can nhap gia tri a, gia tri p, y se do chuong trinh tu dgng tinh toan.
Sai so tuong doi cua chu ky, h$ so can nhot so i>ng voi cac truong h{?p a khae nhau duoc the hien trong hinh 6 va hinh 7. Ta th§y rang, muon co dup-c hp so can nhdt s6 thi ta phai giam dp chinh xac, dg can nho-t so cang tang thi dp chfnh xac trong ket qua giam xuong.
Hinh 6. Sa; so tuxrng do; cua chu ky cua phuung phap HHT
Hinh 7. He so can nhat s6 cua phuang phap HHT
10 Tap cfli KHCN Xay dvng - s6 1/2018
K £ T C A U - C O N G N G H E X A Y DU'NG
3.5 Phuvng phap Chung&Hulbert
Phuong phap Chung & Hulbert [13], doi khi c6n duoc gpi la phuong phap he so a (a-method), la phuong phap tong quat bao ham ca phuong phap Newmark va HHT, dugc viet nhu sau:
(•'-°m)'^=it1 + ".'''3i *(1-<')CV.^1 *aCV|+(l-ci)Kd.^.|+aKd. =(l-a)F|^.|+aF.
= d , t ( A t ) v ( A t ) '
[(l-2p)a|+2Pa.^J (15)
(16) v.^,=V| + ( A t ) [ ( l - v ) a , + v a | + l ]
Hai gia trj Om , or, /3 va y duo-c xac dinh dua vao dieu kien on djnh, do chinh xac va sal so cua ket qua tinh. De thay rang, voi a^ = 0, phuong phap nay se tro thanh phuong phap HHT, va voi ca am= 0 va a = 0, phuong phap nay tro thanh phuong phap Newmark. De phuong phap nay khong co dieu kign on djnh, cac gia trj On,, a, jS va y dugc kien nghj lua chon nhu sau:
_2p-1 p „ 1
'•--J^ °=p^ '-^-f
Gia tri p nam trong khoang [0, 1]. Vdi p = 1, phuang phap nay khdng cd h§ sd can nhdt so (trd thanh phuang phap Newmark), p c^ng nhd se cho he s i can nhdt so cang tang, p = 0,5 se trd thanh phuang phap HHT tuong iing vdi a = -1/3.
Mpt so trudng hgp tinh eua phuong phap Chung&Hulbert dugc the hien trong, sai so tuang doi cua chu ky va he so can nhdt sd dugc t h i hien trong hinh 8 va hinh 9.
Bang 5. Gia tn diu vao ciJa mgt so trwang hap tinh cua phwang phap Chung&Hulbert
Gia tri p 1 0,5 0
Kieu Ngi an thuc Ngi an thUc Ngi an thuc
Um
0,5 0 -1,0
a 0,5 1/3 0
/5 0,25 0,444 1,0
V
0,5 0,833 1,5
DiSu kien on dinh Khcng Khong Khgng
H$ so can n h o t s o Khong Co Co
^ / . • •
00
0 0 0.1 02 0 3 0.4 0.5 .VI
Hinh 8. Sai so twang ddi cua chu ky cua phwang phap Chung&Hulbed
Tap chi KHCN Xay dung - s6 1/2016
KET CAU - CONG NGHE XAY DU'NG
5
1
%
?
(H
.,
)<i2
lIUI
-
• , ( •
tl
I — 1 ^ , -
/ v' / /
1 , 1 i — t
•
•
•
Figiirc 4. Vmiilion of numi;rif!il damping rjiio wilh I/. T li-r aLlTtrtnl j> \ aliif
Hinh 9. He so can nh6t s6 cOa phuvng phap Chung&Hultiert 4. Lg'a chon cac phu>cng phap tinh
fle de hinh dung va so sanh cac phuong phap tinh, ti> do ii,fa chpn mgt phuang phap tinh thicti hgp, sal so tuong d6i cua chu ky va dp can nhot so cua cac phuong phap dugc in ehung trong hinh 10 va hinh 11. Cac phuong phap dugc lua chpn so sanh dugc ky higu nhu sau:
- /\AM: phuong phap Newmark vdi )3 = 0,25, / = 0,5;
- HHT: phuong phap HIiber - Hughes - Taylor voi a = -1/3;
- WiL: phuong phap Wilson Theta voi 0 = 1,4;
- C&H: phuong phap Chung&Hulbert vol p = 0.
u.ii
u: eo4
^ 0 . 3
•^
. 0 . 2 U.1, , , , . ., / /
/ <•
/ , ' . - •
/ ' f'.'' /
' ' '••'' y
<' //'
' . •
'^
^*"
•
•
,l;r
Hinh 10. So sanh do sai s6 tuvng ddi cua chu ky giOa cac phuxmg phap tinh
12
Tap cfli KHCN Xay dung - so 1/2016KiT CAU - C 6 N G NGHE XAY DU'NG
0.04
= O.DI
3 0.02
0 01
nnn
•
"
/
1 1
i
1 1 * 1 /
' ''
•!..-••• 1
'!
! j
,
1
#
.
.\\\i
. /.
Figure 4. Vanaiion of numencal damping ratio wilh J/^7'fordiftercni/)\aluc Hinh 11. So sanh he so cin nhcAsoglwa cac phwang phap tinh TO hinh 10 va hinh 11, ta ed the dua ra mdt sd
kit luan sau:
- Sai so cua phuong phap C&H la cao nhat, do vgy phuong phap nay thud'ng khong duge lya chpn de tinh toan. Vdi phuang phap Chung&Hulbert, khoang gia tri 0 < p < 0.5 thudng it dugc Iga chpn do sai so Idn.
- Dp chinh xac cua phuang phap AAM la cao nhit, do vgy trong nhung trudng hgp tinh khdng yeu cau cd dp can nhdt sd, phuang phap nay thudng duge lua chpn de tinh toan. De d l nhd, CO the ehgn phuong phap HHT va gan he so a = 0, trudng hgp nay phuang phap HHT se tuong ung vdi phuang phap AAM. Day chinh la trudng hgp tinh mac dinh cua SAP2000.
Trong trudng hgp can cd dp can nhdt sd, nen su dung phuong phap HHT ung vdi a < 0 do de nhd (chi can nhap mgt gia tri a, SAP2000 se tg tinh vdi cac gia tri p, y con lai).
- Khi tinh toan vdi hg kit cau phuc tap, ed nhilu bac tu do va cd tfnh phi tuyen eao, nen ehgn phuang phap HHT vdi he so a <0 (vi du: a
= -0,1. -0.2 hoac -0.33), vl vigc logi bd anh hudng cua cac dang dao dgng ed tan so eao se lam tang kha nang hpi tu eua kit qua tinh toan khi chuang trinh dung thuat toan tinh lap.
5. Ket luan
Bai bao (phan 1) da tdm luge mdt so dae diem cua eae phuang phap phan tieh phi tuyIn theo Ijch su thdi gian trong SAP2000. Vdi moi phuong phap, cd the danh gia tinh chat cua nd qua cac tinh chat nhu: npi an thue hay ngoai hiln thuc, dieu kign on dinh, dp chinh xac trong kit qua tfnh, he so can nhdt s6. Bai bao da dua ra nhung trudng hgp tfnh toan co ban nhit vdi mdi phuong phap lam co sd d l si> dung tfnh toan vdi SAP2000. Trong cac phuang phap dugc thilt lap san trong SAP2000, phuang phap Chung&Hulbert cd thi bao quat gan nhu du hit cac trudng hgp cdn lai (tru phuang phap Wilson Theta). Tuy nhien, phuong phap nen Iga chpn su dung la phuang phap HHT do cac tfnh ehit cua nd dugc kiem soat duy nhit vdi mdt thdng so.
Tat ea cac trudng hgp tfnh vdi cac phuang phap deu thupe ho ngi an thuc, duy nhat mdt trudng hgp tfnh cua phuong phap Newmark thuoc hp ngoai hiln thuc, nhung phuang phap nay khdng dugc khuyin nghi su dung do ed dilu kien on djnh, chi dung d l so sanh kit qua hoac d l giai cae bai toan dang truyen sdng, dang nay it gap trong xay dung.
Phin 2 tilp theo eua bai bao se gidi thigu mot s i vi du tfnh toan de lam ro cac tinh chit cua phuong phap nay.
Tap chi KHCN Xay dung - s6 1/2016 13
K^T CAu - CONG NGHE XAY DU'NG
T A I L I E U THAM KHAO
[1] Nguyin Hong Hai (2015), "Nghien cuu s u lam vigc cua nha cao tang be tdng cot thep cd tang cCeng chju tac dgng cua dpng dat d Viet Nam", Vi$n Khoa hgc Cong nghe Xay dyng. Ha Ndi, 2015.
[2] TCVN9386:2012, "Thiit ki cong trinh chju dgng dif, BO Khoa hgc Cdng nghg, Ha Ngi, 2012.
[3] I. Computers and Structures, "SAP2000 Watch and Learn," |T"VC tuyen]. Available:
http://www.csiamerica.eom/products/sap2000/w atch-and-learn. [Da truy cap 1 1 2016].
[4] Hilber, H.M., Hughes, T.J.R. and Taylor, R.L.,
"Improved numerical dissipation for time integration algorithms In structural dynamics.,"
Earthquake Engineering and Structural Dynamics, tgp 5, pp. 283-292,1977.
[5] X. Zhou va K. Tamma, "Algorithms by design with illustrations to solid and structural mechanics/dynamics," International Journal for Numerical Methods In Engineering, tap 66, pp.
1738-1790, 2006.
[6] Robert D. Cook, David S. MalKus, Michael E.
Plesha, "Finite Elements in Dynamics and Vibrations," trong Concepts and Applications of Finite Element Analysis, Madison, Wisconsin, John Wiley & Sons, 1988, p. 396.
[7] T Belytschko, Semidiscretization
"An Oven/iew of and Time Integration
Procedures," trong Computational Methods for Transient Analysis, North Holland, Elsevier Science Publisher B.V. 1983, p. 55.
[8] R. D. Cook, D. S. Malkus va M. E. Plesha, Concepts and applications of finite eiemeni analysis, Madison, Wisconsin: John Wiley &
Son, 1988.
[9] T J. Hughes, The Finite Element Method, Toronto, Canada: General Publishing Company Ltd., 1987.
[10] Newmark, N.M., "A method of computation for strutural dynamics," Journal of Engineering Mechanics Division, ASCE, tgp 85, pp. 67-94, 1959.
[IIJBathe, K.J. and Wilson, E.L. "Stability and accuracy analysis of direct integrator) methods.," Earthquake Engineering and Structural Dynamics. Vol.1, pp. 283-291,1973.
[12] Belytschko, T. and Hughes, T.J.R., Computational methods for transient analysis., North-Holland. 1983.
[13] J. Chung and G.M. Hulbert., "A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-a method," Journal of Applied Mechanics, pp.
60:371-375, 1993.
Ngaynh$n bai: 21/11/2015.
Ngay nhan bai sira lan cu6i: 01/01/2016.
14 T^p chi KHCN Xay dung - so 1/2016
