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Phan tich dong Iyc hoc khung dan - deo CO ke den SLP pha hoai luy tien cua cac phan tip ket cau
Ngay nhan bai: 20/8/2014 Ngay su:a bai: 5/9/2014 Ngay chap nhan dang: 10/9/2014
Nguyen Van Hdi, Vu Van Hoang, Nguyen Van Tu
T O M TAT:
C^c bai toan phan tich dong Iflc hoc ket c5u khung dan - deo theo phfldi^ phap truyen thoi^ dfloc xem xet vdi quan diem, theo do cac khdp deo sau khi xuat hien se ton tai trong h§ (khfing bj pha hoai). Khac vdi cac bai todn tren, bai todn d^t ra trong bai bao ndy co xet d^n si^ phd hoai cua cac khcfp d^o khi vilcft qua gidi han chuyen vi cua no, dan den sii phd hoai luy tien ciia cac phan td trong ket cau theo thdi gian.
Bi gidi bai toan dat ra sd dyng mo hinh thanh DDLT vdi bien d a i ^ dec tap trimg, ti^u chuIn phd hoai deo theo chuydn vi xoay ciia tiet dien, phuong phap PTHH va phtiOng phap tich phan tnic tiep Newmark ket hop vdi phiioi^ phap lap Newton - Raphson bien dieu.
Cdc Mt qua tinh toan b ^ g so nh^n diiOc cho thay, viec k^ den phd hoai luy tien ciia cdc phan td Idm gidm khd nang chiu lilc cua k h t u ^ dan - deo, dan den qua trinh phd hoai cua ket cau nhanh htfn so vdi cdc k£t qua nhan diiOc theo phuong phap tinh todn truyen thong.
ABSTRACT
The traditional problem of dynamics analysis elastic-plastic frame structures has deal with the existence of the plastic hinges after their appearances in the system. In this paper, however, problem solved considering the failure of plastic hinges when their displacement exceeds a ultimate plastic displacement which leads to a progressive failure of the elements of structures with the time.
In this study are adopt elastic-perfectly plastic beam model with a concentrated plastic deformation, plasic-damage criteria accorchng to rotate displacement of the section, are use Newmark direct integral method in combination with Newton - Raphson approach. The results show that progressive failure of elements of structures impairs the load supporting ability of elastic-plastic frames, consequendy faster the fame's damage process in comparison with those estimated using traditional analysis approaches.
GS.TSKH. Nguyin ^^n Hoi Ih.SVu Van Hoang TS Nguyen Van Td Hpc vien Ky thudt Quan sif Mail: [email protected]
1 .Dcit b a i t o ^ n va cac g i i t h i e t t f n h t o a n Khdo sat ket cau khung phSng 6hn - deo chju t i c dung cua tcii t r g n g dpng nhU tren hinh ve l a . C5n xac djnh trang thai chiu lUC cCia he theo thdi gian Ichi l<^ den sU pha hoai luy tien ciia cdc phan t d ket cau.
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H'inh 1. Mo liinh xuat pliat va mfi hinli tinli cua ket cau (a) - mo hinh tfiuc cda he Itet cau va t3i trong;
(b) - mo hinli tinh cHa liet cau theo phuang phap PTHH.
Thda nhan cdc gid thiet sau 1 / Cdc phan t i l dam vi cot (goi chung ta phan t d dam - c6t hay phdn t d thanh) cua k^t cau khung lam viec theo m o hinh Euler - Bernoulli.
2/ Lien k^t dam - cot tai cac niit khung la cac li&n ket cdng.
3/ Vdt lieu ket cau bien dang theo m o hinh dan d§o ly t u d n g (ODLT), theo d o quan hk iing suat - bi^n dang cCia VEit lieu (o-£) co dang n h u tren hinh 2a, con quan he m d m e n udn - goc xoay (M-Q) cCia tiet dien thanh lam bdng vat lieu tren co dang n h u tren hinh 2b, trong do:
101
z.
E - m o dun dSn hdi ciia v^t lieu, a - dng su^t i l n g x d d a n hoi.
chSy d4o cua vdt li^u, k- d 6 cdng chSng udn cCia t i ^ t dien thanh, M^ - momen d§o cua tiet dien thanh.
4/ B6 qua Snh hUdng cila luc doc N vd lUc cdt Q den giS trj momen d^o tai ceic tiet di#n, theo do, t i l t di&n dat den trang theii gi6i han
k h i : M = M ^ = Mp = OpWp (1) trong do. W - mdmen chong udn d^o CLia
tiet dien.
(a) i a (b) i M
PT thanh dAn hoi
Eli
Hinh2 M5 hinh DDLTtOavIt lieu vi kit cau (a)' quan he ihig suit- bien dang ci)a v3I llfu [b)' quan hi mflmen ufin - giit xoay ciJa tiet diln thanh 5/ Bien dang d^o trong cdc tiet d i f n ciia dam vi c^t cCia ket c l u 1^ c^c bi^n dang d^o tap trung xuat hien tai c5c t i l t di^n c6 mdmen uon Idn nhat.
2 . PhirtAtg t r i n h c h u y i n d d n g cua kit cau M i u n g DDLT k h i ki din p h d h o a i cda cac khoTpdto
2.1.Md hinh phdn tirthanh ODLT Bi tinh toan k^t cau theo nhiem vu dat ra ciia bai todn se sir dung phuang phap (PP) phan tif hOu han (PTHH). So do rdi rac hoa PTHH ciia ket cdu duoc the hi6n tren hinh 1 b. Viec rdi rac h6a PTHH doi vdi ket cau duoc thuc hien sao cho cac mdmen uon I6n nhat trong tat ca cac PTHH chi xuat hien tai cac ddu cu6i ciia PT. De thda man yeu cau nSy khi rdi rac hda PTHH ta cd t h i lUa chgn cdc PTHH nhU sau' tren PT khong cd tdi trong tac dung; n l u tren phdn tCr cd tai trong tap trung thi dau cu6i cOa phan tif dUOc chon triing vdi vi tri ciia tdi trgng; n l u phan tif chiu tdi trong phan bd thi phdn tCr d d can duoc chia ra thSnh n h i l u phan tir nhd hon sao cho mdmen udn Idn nhat chl xuat hien d cac tiet di&n dau CUOI cua cdc phdn tir con.
Vdi cdch rdi rac hda PTHH nhu tren va vdi gid thiet 3 (vSt li^u BDLT) vh gid t h i l t 5 ( b i l n dang d^o tdp trung) da thtfa nhdn 6 tren, co the md hinh hda PT thanh BDLT ciia k i t cau khung n h u t h e hi^n hinh ve 3, theo d d PT thanh gdm 2 thanh phdn co bdn: 1) c^c li^n k i t ODLT dat tai cdc dau cuoi PT; 2) PT thanh dan hoi nam gifla cac lien k i t DDLT. (Lien k i t dan - deo tai moi dau cuoi PT gom. 2 lien k i t thdng dan hoi theo phUdng cdc chuyen vi thdng vd 1 lien ket xoay ddn - d^o theo phuong chuyen vj xoay, nhUng de don gidn each trinh bdy tren hinh ve chi the biin lien k i t 6hn - d^o theo phUdng chuyen vi xoay). Vdi m o hinh hda n h u tren thi trong qua trinh chiu luc, b i l n dang d l o chl xuat hien trong cdc lien k i t xoay dcdc ddu cu6i PT, con PT thanh va 2 lien ket t h i n g d cdc d i u cudi PT ludn ludn
^ hen kit DDLT
Hinh3.M6 hinh PT thanh flSn - d*o ciia ket cSu khung 2.2.Md hinh phd hogi cda cdc khdp dio Cac lien ket dan - d^o cu6i PT thanh sau khi bien thanh khdp deo cdn cd the tiep tuc t d n tai d i n khi c h u y i n vi tai tiet di^n d d dat den mdt gidi han nao dd rdi mdi bi phd hoai. Mo hinh pha hoai (Damage model) ciia lien k i t dein - d l o (hay khdp deo) dUdc dac trung bdng chi so pha hoai (Damage Index) hoac tieu chuan phd hoai (Damage Cntena). Cd n h i l u tieu chuan phd hoai ciia lien ket 6in - d&o dUoc de nghj [3,8]
phu thudc vao loai vit lieu lam k i t cau (vdt \'\6u ddn, d l o , th^p, be tdng c6t thep, composite) vd tinh chat tac dung ciia tdi trong (tdi trong ddn dieu, tSi trgng lap lai cd chu ky). Cdc tieu chuSn pha hoai cd the xac djnh theo lUc (hay irng suat) hoac theo chuyen vl. Dudi ddy sir dung mdt trong cac tieu chuan noi tren theo c h u y i n vj xoay cua t i l t dien PT - Id ti^u chuan c6 the dp dung chung cho cac vat lieu tudn theo cdc gid t h i l t da thira nhan d tren va p h i i h o p vdi loai tai trgng don dieu. Theo tieu chuan ndy lifin k i t dan - deo (hoac tiet dien thanh ddn - d l o ) se bi pha hoai khi chuyen vi xoay ciia tiet di^n tai dd, 9, dat den gia trj gidi han [6]: 6 = e„ = p9 (2)
trong dd: 6 - gdc xoay tiet dien khioSt dau chay deo,
G^ - gdc xoay gidi han ciia tiet di^n sau khi chdy deo (tUcmg ung vdi t h d i diem lien k i t dan - deo bi pha hoai), p - chi so pha hoai d^o (hay cdn ggi la do deo) ciia t i l t dien (hinh 4).
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Hinh 4. Mo hinh pha hoai tiet dien PT thanh d i n - dto theo chuyen VI xoay
2.3, Cdc ma trdn cua phdn tiJf thanh DDLT khi ki din phd hogi cda cdc khdp dio
Trong qua trinh chiu liic, li^n k i t d ddu cudi cua PT thanh DDLT khi k l d i n phd hoai ciia cdc khdp deo cd t h i trdi qua cac trang thdi: bien dang dan hoi, b i l n dang d^o hoac bi phd hoai deo (lien k i t bi 6ilt vi taeh ra khdi nut ket cau), theo do cac phan tif thanh ddn - d l o ndy se lam viec n h u cac phan tif thanh dan hoi nhung vdi lien ket tai cac dau cudi tUdng ifng ciia PT Id:
hoac ngdm, hodc khdp (khdp d l o ) , hoac t u d o (khdng lien k i t ) ; dan den cac ma trdn ciia phdn tir thanh khdo sat se khac nhau phu thugc vdo
qud trinh chat tdi. Gia trj cac ma tran ndy ciia phan tif thanh thiJ " m " trong hi toa dd cue b$
(ma tran d d ciJng [ K l ^ ma tran khoi lUdng [M]
m, ma tran cdn [Q^ vec t d tdi trong quy nijt (R) ) CO t h i true t i l p nhan dugc tif cac ma trjn ciia vat lieu dan - deo song tuyen tinh (DDSTT) t u a n g iing da t h i l t lap trong cac cdng trinh [1,2].
2.4. Phuang trinh ehuyin dgng eda kit ciu khung DDLT khi keden phd hogi cua cdc khdp dio
Sit dung cdc ma t r | n ciia phan tif thanh ODLT trong he toa d g cue b d da ndi d muc 2.3, ma trdn bien d6i chuyen vj vd lUc niit cua phdn tif, phUcmg phap "dd cutig trifc tiep" trong PP PTHH [3,7] vd cdc dieu kien bien ciJa k i t cSu, CO t h i nhSn dUdc phuong trinh c h u y i n ddng trong he tga d d chung tai thdi diem bat ky ciia k i t cau khung ODLT khi ke den pha hoai cua cic khdp deo dudi dang sau d a y
[ M ] f j ) + [ C ] { u } t [ K ] { U } . ( R } . (3) trong d d : f U} - vec t o c h u y i n vj niit ciia Icit cau trong he tga d d chung (trong dd v6c to chuyen vi mdi niit i cd dang; {U}^ = {U,, V,, e,)^, gdm c h u y i n vi t h a n g theo phuong true x, tryc y vd chuyen vj xoay quanh true z; [M], [C], [K], {R} - t u a n g iJng la ma tran khoi Iflong, ma trdn edn, ma tran do c i l n g vd vec t d tdi trgng quy niit cOa todn he.
Trong cac ma tran tren, ngodi sU phu thudc vdo trang thai lien ket tai cac dau cudi FT, ma tran d d eifng k i t cau [K] cdn phu thude vdo chuyen vi niit (U):
[ K ] - [ K { U ! 1 . (4) Tinh d i n (4), phUong trinh (3) cd the v i l t l^i
dudi dang:
[ M ] ^ } ^ [ C ] { j } + f , ( ( U } > { R } , (5) vdi:f,({U)) = [K(U}]{U) (6) f^({U}) - duoc g g i Id vec t o luc phuc hdi (U)
eiia ket cau.
Phuang trinh (5) la phuang trinh phi tuyin ddi vdi chuyen vj nut ciia khung.
3. PhUdng p h a p g i d i b a i t o d n dong cOa k h u n g DDLT k h i ke d e n p h d h o a i luy tl6n cita c^c p h S n t A k e t c K u
3.1.Khdi niim viphd hogi Idy tiin cua kit cSu khung BDLT
Khdi mem "pha hoai" de cap d i n dudi ddy duoc h i l u la pha hoai deo (pha hoai sau khi vlt lieu da chay deo). Vi chi cd edc Win k i t cudi PT thanh m d i cd bien dang d l o , n i n md hinh phS hoai PT thanh DDLT thUc chat Id md hinh phd hoai cdc k h d p deo d cudi PT.
Cae bdi toan phdn tieh d d n g life hoc k i t cau khung dan - d^o t r u y i n t h d n g dfldc xem xdt vdi quan diem, theo d d cae k h d ^ d^o sau khi xult hien t r o n g k i t cau se t d n tai theo thcri gian ma khdng bj phd hoai neu tai t h d i d i l m khdo sdt tiep theo tai trpng vdn edn tiep tuc tang (hoac gidm) d o n di6u n h u t a i t h d i diem khdo sdt tnidc d d dil cho ehuyen vi xoay tai edc khdp dd sau
108
ehdy d^o t i e p tue phat t n l n d i n gid tri bao nhifeu t u y y vd ket edu chi bi sap do (phd hoai) khi s6 khdp ddo trong khung cd sd Ifldng dCi d l b i l n k i t cdu da cho thanh cd cau dSo. Khdc vdi cdc bdi toan t r ^ n , bdi toan dat ra d day (ed xet den Slf phd hoai ciia cac khdp 6io), theo d d cae khdp d l o sau khi x u l t hien se bj pha hoal khi ehuyin vj xoay tai cae khdp d d dat d i n mgt gia tri gidi han theo m d t tieu ehuan pha hoai nao dd, chdng han theo he thflc (2). Khi khdp d^o bi pha hoai thi t i l t d i f n thanh tai dd bi dirt vd taeh ra khdi k i t ciu. Neu mdt PT cd cd 2 ddu bi dirt vd tdch ra khdi khung (cd the khdng ddng thdi) thi PT do duoc coi Id bi pha hoai hoan toan va bi loai ra khdi ket edu.
Sy pha hoai ciia PT ddu ti^n Idm eho dd eflng ciia he k i t eau xuat phdt bj gidm y l u , dan tdi h? qua lam cho PT t h f l 2 bi pha hoai. Su pha hoai eiia PT t h f l 2 Idm cho do cflng ciia he ket cdu cdng bj suy y l u hon, ddn tdi sfl phd hoai eiia PT thif 3, v.v...Hien tflOng tren xdy ra n h u mdt phdn flng day chuyen va duac goi Id pha hoai luy tien (Progressive Failure; Progressive Damage). SU phd hoai luy t i l n (PHLT) cd t h i tiep dien cho den khi k i t edu bj sdp do hodn todn - goi Id sdp do luy t i l n (Progressive Collapse).
3.2. Phuang phdp gidi bdi todn dpng cua khung ODLT khi ki den phd hogi luy tUn cda cdc fdtdn tdkit cdu
D l gidi phUCfng trinh c h u y i n ddng cua khung DDLT kh! ke d i n phd hoai luy tien eiia cac phdn tCr k i t cau ed dang phi tuyen (3) se sfldung phuong phap tich phdn trflc tiep Newmark ket hop vdi p h u a n g phdp lap Newton - Raphson b i l n di&u quen b i l t vdi thuat giai ciia nd da dfldc trinh bay cu t h i trong [2,4]. Khi gidi bai todn ndy, sau mdi bflde t h d i gian can phdi kiem tra vd cap nhat lai cac t h d n g sd sau day phuc vu cho viee t i n h todn tai budc t h d i gian t i l p theo eac khdp deo mdi xuat hien, cae khdp d l o bi pha hoai, cdc PT bi pha hoai, sa do k i t cdu mdi, cae ma tran phan t f l (ma trdn do cflng, ma tran khdi Ifldng, ma trdn can nhdt, vec t o tdi trong nut) t f l d n g flng vdi trang thai li^n ket mdi tai cdc ddu thanh (ngdm, khdp, t f l do), chieu chuyen ddng xoay tiep theo cua t i l t dien vd sU ddng - md cOa cdc khdp deo Can liAi y Id, khi khdp d^o tai dau thanh ndo bi phd hoai thi dau thanh dd bi 6ilt vi taeh ra khdi he, nghia la dau thanh dd t r d thdnh t u do khdng lien k i t , ddng t h d i he k i t cau si phat sinh mdt niit mdi, tflong flng se phdt smh ba bde t f l do mdi; edn khi PT nao cd khdp d l o d ed 2 dau thanh deu bi phd hoai thi phdn t f l d d dfloc loai ra khdi he. Do do, Idi gidi cCia bdi todn dang khdo sdt s& phflc tap hon nhieu so vdi bai todn phan tieh ddng Iflc hgc khung ddn d e o t r u y i n t h d n g [khdng ke den sU phd hoai luy tien ciia cdc phan t f l k i t c l u ) vd do dd, bde p h i t u y i n cung se cao hdn.
D i l u kien phd hoai ciia khdp d l o dfldc kiem tra theo cdng thifc: B s e__ = u6, (7)
Jiin ca sd cdc thuat todn t h i l t lap d trgn.
cdc tae gid da lap trinh d l giai bdi toan phdn tfch d d n g life hoc khung dan - d^o cd ke d i n sfl phd hoai luy t i l n ciia cdc phan t f l k i t eau. Phan mem tinh toan mang ten FRADYPROCOL duge viet bang ngdn n g f l lap trinh MATLAB.
Thi d u s d DUcH day trinh bdy k i t qua phan tieh ddng Iflc hgc mdt k i t cau khung DDLT cu the cd ke d i n sU pha hoai luy t i l n ciia cdc phan tOr bang phan mem FRADYPROCOL de minh hga cho do tin cdy cua phuong phdp, thudt todn vd chuong trinh da lap.
• Khdp dio
H'inh 6. Su 66 ink cau va vi In cac Hidp deo tai thiii dienn t = 0,08{s)
Hinh 7. So do k^t cSu va vi tri cic khdp deo tai thtii dilm t=0,08(s) khi ke'de'n PHLTviJin = 2
a
Hinh S, So do ket cau (a) vi dang tii trong dong ngan han (b) K i t cau khung cd dang n h u h i n h 5 lam bang vdt li^u t h e p hinh chCr I vdi t i l t di&n cd kich thudc 1(200x100x5,5x8mm), m d dun dan hdi E
= 2.1e+5 MPa, ifng suat chdy d^o =210Mpa.Tdi trong ddng tde dung tr4n k i t cau gdm cac tdi trgng tap trung: Pl(t) = 25f(t)kN, P2(t) = 30f(t) kN va cac tdi trong phdn bd d i u : q l ( t ) = 15f(t) kN, q2(t) = SOf(t) kN, q3(t) = 25f(t) kN, trong d d f(t) CO dang ngan han hinh tam giac (hinh 5b), thdi gian duy tri tac dung ciia tdi trgng T = 0.05(s). Ty sd cdn duoc chon i., = ^^ = 0.05.
Cac ket qua tfnh toan bang sd duge the hien tren hinh 6 den hinh 13.
- T r e n h i n h 7 , 8 v d 9 l d e a c s a d d k l t c a u v d v j trf ede khdp deo (ky hieu bSng cham trdn den) trong khung dflOc cap nhat sau cac thdi diem t i n h todn khac nhau khi k l den PHLT vdi ehi sd pha hoai deo |J = 2. Trinh t f l pha hoai ciia cae PT trong k i t cau la- (3), (1), (5), (4), {2),...va cudi cung la (7), tflong flng vdi eae thdi diem bj phd hoai: t3=0.075(s), t1=0.0818(s), tS=0.0956(s), t4=0.0960{s), t2=0.0977(s), ,t7=0,1019(s). D l so sanh cac k i t qua nhan dflge vdi eac k i t qud tfnh todn k i t edu dan - deo theo phfldng phap tfnh todn truyen thong, tren hinh 6 dan ra so d d ket eau vd vi trf khdp deo khi khdng ke den PHLT (tuang flng vdi ehi sd pha hoai d§o |i = " ) .
Hinh 8 So do ket du vi vi tri cac khdp deo tai thOi diem
^0,0S18(s) khi ke de'n PHLT vdi p=2
I
• KMpdeo
Hinh 9. So do ket cau vi vi tri cac khi^ deo tai tiidi di^m t=0,1019(s)khiked^nPHLTvffl p=2
-Tren hinh 10 vd 11 Id do thi mdmen udn vd gdc xoay theo thdi gian tai nut 9 thude phan tir (12) ciia khung, tren dd dfldng net dam t f l d n g flng vdi tinh toan ket eau dan d^o cd ke den pha hoai luy tien ciia k i t cdu vdi ehi sd phd hoai d l o M = 2, edn dfldng net dflt tUOng flng vdi t i n h toan k i t eau dan d l o t r u y i n t h o n g (khdng ke den phd hoai ciia khdp deo, nghia la t f l d n g flng vdi n = « ) . TCr k i t qud tren hinh 10 cd the thay rang, mdmen udn t i i t i l t di§n cu t h i ndi tren khi ke d i n pha hoai luy tien eOa k i t eau cd gid t n tuyet ddi Idn hon ddng ke so vdi t i n h todn k i t cdu ddn d l o truyen t h d n g (khdng k l den pha hoai ciia khdp d§o).
Hinh 10. flo thi momen uon theo thai gian PT (12) cfla ke't cau khi ke den phi hoai luy tien chisophihoaidJop=2
tai nut 9 thuoc ket cau vdi
4fiinii:[|QIVI -IQg
•ma Bim I , (s)
Hinh 11. fi6 thj g6c xoay theo thdi gian tai mit 9 thudc PT (12) cfla ket cau khi k4 den phi hoai liiy tien vdi chl s6 phi hoai
- Tren hinh 12 Id dd t h j gdc xoay theo thdi gian tai nut 11 thudc phdn t f l (7) khj ke den phd hoai luy t i l n eua ede phdn t f l k i t eau vdi chi sd pha hoai d^o p ed edc gid t n khae nhau. Tfl k i t qua nhdn dflOC cd t h i thay rang, \i dnh hfldng rat Idn d i n gdc xoay tai niit khung.
4. Chopra, A.K. (2007), Dynamics of Structures: Theory and Applications to Earthquake Engineering [3nJ Edition)/ Prentice Hall Intemational Inc
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&Technohx|v,Vol34
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Stmctural Analysis and Steel Structures Institute Hamburg UniversityofTechnology,Geimany
8 Park Y.J. (1987).Inelastic damage Analysis of reinforced concrete frame-seat-wall structures /Technicai Report NCEER-87-00D8,NewVoik
9. Zienkiewicz O.C,, Taylor J i , fliu J Z (200S),The Finite Element Method: Its Basis and Fundamentals/NEW YORK
Hinh 12. Eld thi goc xoay theo thdi gian tai nflt 11 thudc phantfl'(7)
!,
i
Kinh 13. Bi Ih! W ^^ aiil^ c^ii ^° P^i^ ^°i' <i^o vdi th6i gian phi hoai cua phin tflsd(3)
- Trdn hinh 13 Id d o t h j quan he gifla chi sd phd hoai d l o vdi thdi gian pha hoai eua phdn t f l s d {3).Tfl k i t qud nhdn duge ed t h i thay rdng, chi sd phd hoai ddo cdng n h d t h i thdi gian phd hoai ciia phdn t f l cdng nhanh, dieu ndy hodn todn p h d hop vdi cdc quy luat ca hgc.
TAI LlEU THAM KHAO
1 Nguyin Vin Hoi. Nguyen Van TO (20J0), Phin iJng fifing cfla ket cau khung phing bien dang theo mo hinh din -dio song tuyen tfnh chiu tac dung cfla tii trong ngin han /Tap chlKH&ICT, Hoc vien KTQS, sd 132 (02-2010), trang 63 - 72.
2. Nguy^ Van Tu {2010),Tinh toin khung phing din - dio diju tic dung ciia tii trong ddng ngdn han/Uiin i n TS,Ha npi
3. Bath^ KJ. (1996), Finite Element Procedures/ Prentice Hall International Inc, Englewood Qifh.