Phan tich tuyen tinh coc tiet dien tron chju tai trong diFng trong nen dat nhieu lop
Linear analysis of circular pile under vertical load in layered soil
Ngay nhan bai: 10/12/2015 Ngay sCfa bai: 9/02/2016 Ngay chap nhan (Jang: 22/03/2016
Nguyen Van Vien, Nghiem Manh Hien,Tr|nh Viet Cu'dng
TOM TAT
Bai bdo t r i n h bay m o t philOng p h a p m<5i d d n gian trong phdn tich ling xif tuyen tinh ciia coc dan tiet diSn t r o n chiu tai t r o n g diJng t r o n g n l n nhiSu ldp. Phdcfng p h a p nay d d a tren nguyen ly nang liidng va p h d d n g p h a p bien p h a n do Vallabhan va Mustafa (1996) de xuat vdi IcJi giai tdcJng m i n h theo p h d d n g p h a p p h a n tt£ hQu h a n trong viec xap xi chuyen VI c i a coc. D o tin c^y cua p h d o n g p h a p d d d c k i l m chdng bang each so sanh ket qua tinh toan vdi ldi giai giai tich va p h a n tich phSn tii hi3u h a n ba chilu.
TU khda: phdOng phdp phan td hiiu han; coc; bi^n phan nang IdOng; tai trpng ddng.
ABSTRACT
This paper presents a new simple method for analyzing linear response of smgle pile with circular cross section under vertical load in layered soils. The m e t h o d is based on energy principles and variational approach proposed by Vallabhan and Mustafa (1996) with explicit solution by using finite element m e t h o d in pile displacement approximation.
Effectiveness of the proposed method is verified by comparing its results to analytical solutions and 3D finite element analyses.
N g u y i n Van V i l n
Vien Khoa hoc Cong nghe Xay ddng NghiSm Manh Hien
Khoa Xay ddng, trddng Dai hoc Kien true Ha Noi Trinh Viet Cdtfng
Vien Khoa hpc Cong nghe Xay ddng
1 . Gidi thieu
Phan tich tuyen t i n h coc don d u d i tac dung ciia t ^ i t r o n g dilng duoc sd d u n g t r o n g viec xac d i n h d o cdng cOa he cpc-nen dat khi phgn tich t u o n g tac dat-nen mong-ket c^u Da co nhieu tac gia nghien cdu giai bai toan coc chiu tSi t r p n g d d n g t h e o phUOng phap g\h'\ tich va phUOng phap so.
Pouios va Davis (1968,1980) [16][T7] va Butterfield va Banerjee{1971) [2] phan tich chuyen vi cua coc don chiu tai trpng ddng trong nen dat dan h6i l ^ t u d n g s d d y n g phUdng trinh ciia Mindlin (1936) [9].
Ldi gi^i ddi vdi cpc trong nen Gibson (Gibson 1967) [5] trong do rno dun dan hoi truot bien doi tuyen tfnh vdi dp sau dUpc de xuat bdi nhieu tac gia nho Randolph vh Wroth (1978) [19], Poulos (1979) [15], Rajapakse (1990) [18], GuD va Randolph (1997) [7] va Goo (2000) [6].
Lee va cong sU (1987) [13], Lee (1991) [10] va Lee and Small (1991) [11] de xuat cac ldi giai t u y e n t i n h d d i v d i cpc ddng trong nen d a t nhieu Idp. Ai va Yue (2009) [1] t r i n h bay p h ^ n tich tuyen t i n h d d i vdr coc d a n t r o n g n^n d a t nhieu Idp sir d u n g bien S6'i Hankel va ldi giai nay du'pc so s^nh v d i cac ldi giai cCia Poulos vi Davis (1980) [17], Randolph va W r o t h (1978) 119], Poulos va Davis (1980) [17], Lee va Small (1991) [ 1 1 ] , Chin va cQng sU (1990) [3], va Guo v a c p n g s U { 1 9 8 7 ) [8].
Vallabhan va Mustafa (1996) [24] de xuat phuong phap ti'nh to^n gan diing dUa tren nguyen ly nang iUong vdi gia thiet ve tru'dng chuydn vi.
Lee va Xiao (1999) [12], Seo va Prezzi (2006) [21], Fidel (2014) [4], Seo va cdng su (2009) [22] va Salgado va cOng sU (2013) [20] phat trien phUOng phap ti'nh toan do Vallabhan va Mustafa (1996) [24] de xuat ddi v6i nen dat nhi^u Idp Lee va Xiao (1999) [12] xay dUng moi quan he giffa chuyen • VI diJng va noi lyc trong coc bang hang so tich phan xac dinh bang viec dUa vao cac dieu kien bien.
Trong bai bao nay, cac tac gia trinh bay m6t phuong phap dOn ket hop gida giai tich va phuong phap so trong phan tich coc don tiet dien tron chiu th'i trong ddng trong nen nhieu Idp (Nghiem, 2009) [14].
Phuong tr'inh can bang dua tren phUOng phap de xuat bdi Vallabhan va Mustafa (1996) [24] va phUdng phap phan t d hdu han.
2. Mo hinh tinh to^n
Coc tiet dien tron cd chieu dai L^, ban kfnh r^ n h u trgn hinh 1, t^l trong P dat tai trong tam cila tiet d i ^ n , d^t nen c6 n Idp nam ngang Coc xuyen qua m Idp d a t v a mui cpc dUpc dat vao day ciia ldp dat t h d m . Do vay CO n-m Idp dat nam dUdi mui cpc. Oac trUng cQa Idp dat t h d 1 bao gom mo dun flan hoi, E,, he so Poisson, v , , va mo dun dan hoi trdpt, G, Coc dupc chia thanh nhieu phan t d vdi phan tff coc thff j co chieu dai L| nam trong Idp dat thff i . He tpa d o tru dUpc sff dung trong tfnh toan vdi goc tpa do dat tai trong tam t i ^ t dien d dinh cpc va true z cb' chieu duong hiidng t d tren xudng dUdi. Vat lieu cua coc va dat n^n xung quanh dUoc gia thiet la dang hUdng, tuyen tinh va chuyen vj tai vi tritiep^
xuc gida coc va dat nen la lien tuc.
78|0S*niEi»9i 04.2016
^ + , . G „ ^ „ v „ + ,
Hlnh1:Hecoc-dat
3. Quan h f gida chuyen vj-iimg suat-bi^n dang
Dudi tac dung cua tai trpng d d n g , bien dang theo phUOng tiep tuyen va theo phuong ban kinh la rSt nho so vdi bien dang theo phuong d d n g ngn CO the bfi qua. Do chuyen vi cua cpc theo phuong ban kinh gISm dan khi khodng each theo phuong ban kinh tang len ngn trddng chuydn vj trong dat nen d u o c x a p x i theo bieu thffc sau:
u . ( r , z ) = w ( z ) * ( , ) (1) Trong d o M ' ( Z ) la chuyen vi theo phuong ddng ciia cpc tai d p sau
z ; i]i(r) la ham khong t h d nguyen mo t ^ s u suy gl^m chuyen vj cilia coc theo phuong ban kfnh ke t d t S m cpc. GiS thiet \h iti(r) = l tai r = rp va (j)(r) = 0 tai r = 0 vdi r^ la ban kinh ciia cpc. Quan he gida bien dang va chuyen vj cff a cpc dupc xac djnh n h u sau:
Trong d o E. la m o d u n dan hdi ciia phan tff coc thff j ,neu j < m t h i E- = E , n^u j > m t h i Ej = E | ; A\h dien tich t i ^ t dien cua cpc; w , , va w J t u o n g ung la chuyen vj cffa nut dau va nff t cuoi cua phan tff cpc t h d j ; P va w^^ t u o n g dng la tai trpng vh chuyen vj tai z = z,,.
Nang lupng bien dang dupc xac i^nh theo bieu thde:
Trong do CT^, va e^ la cac ten xo dng suat vh bi^n dang.
The bleu thffc (5) vao bieu thffc (4), vh tfch phhn d d i vdi 9 . t h e nang bien dang dupc xac dinh nhusau:
dw."l
2£UJ__5UB_ L r 59 3r
Su, flu,
d w ( z )
-w(r).
1 5 u , dUg
~'r~m~~dz
Mdi quan he gida Ong suat va bien dang cffa dat nen dUoc viet dUdi dang tong quat theo djnh luat Hooke n h u sau:
•?i+2G \ X X+2Q X
X X + 2 G
0 G O O (3)
n = Z ^ j E , A [ ^ T d z + 2 : l J j f w d r d O d z - P w „ . (4)
•.^ 1 / f d w , n = T - f E A ll.—^
triJ ^ L%z,
.I.jJ(,.2C,)(,5j.G{w,
rdrdz-Pw^_^Phuong t r m h can bSng cffa he coc nen t h u dUpc t d viec t d i t h i ^ u the nang hay bien phan bac nh^t cffa t h e nang phSi bSng khdng { 8 n = 0 ) .
Phuang trinh vi phan sau day ddi vdi phan t d c o c thu dUpc tdviec lay bien phan theo bien w :
Phuong trinh (7) dupc viet gon lai \k
Trong do k^ va t j l a c a c h f s o n e n c S t v S n ^ n d U O C x S c d j n h t h e o :
t , = E , A + 2 ! t j { ^ , + 2G,)i|iVdr (10)
5. Xap xi chuydn vi
Theo phuong phap ph^n t d hffu han, chuyen vj dffng trong phan tff cpc dupc xap xi bang cac chuyen vj tai hai d i u cpc (hlnh 2):
w , = N „ W j , + N,jW,j (11) Trong d o w^, va Wj^ tUOng d n g la chuyen vi dffng cua coc tai n i i t
dau va n u t cuoi cua phan t d coc t h d j ; Nj, vh Nj^ Ici cac ham dang. Cac ham dang dUpc xac dinh nhU sau:
c o s h ( a j z ] s i n h ( a j L j ] - c o s h ( a j L j ) s i n h ( a | z ) sinhfa,?) trong do G va A va cac hSng so Lame cCia dat nen.
4. Ph dtfng trinh can b^ng
Ham t h e nang EI ciia he coc-d^t n4n dUpc djnh nghTa la tong cffa nang lUpng cffa npi lUcva cua ngoai lUC.
'•' sinh(ajL) ' '^ s i n h ^ a j l j ) Trong do z tpa dp dta phuong va a^ dUOc t i n h theo bieu thffc:
(12)
•4
Toi thieu ham the nang b§ng c^ch lay bien phan ham the nang theo bien ^ , phUdng trinh vi phan can bSng ciia dat nen xung quanh cpc thu duoc la:
l|79
d'<t> 1 d ^ p ' t = o
E . A
2
Hinh 2: Ptian tiy Hianti
-I
• - Z ^ J w J d z (16)
b . | ; ( l , + 2 G , ) J f ^ l 6z (17)
DUa tren bleu thffc xap xi c h u y i n vj theo bieu thffc (11), gia tri cffa a a b dupc tfnh toan nhU sau:
.'f-'-^j.
> = X 6 , (N,,>
^ w , , w , j S i n h ( L ] a , ) + ( w ^ , + w ' j ) [ - 2 L , a j + sinh(2L|a
,_, cosh(L^a.)
^-Zft«=oJ[^'
= Y , / \ M L , a . w , . w , , c o s h f L , a , ) (,qi t f 4 s i n h ' ( L j a j ) l ' ' *•' '-= ^ > " ('9) -4Wj|W|jSinh(Ljaj)
+(w^, + w j j )[2L|a, + sinh(2L,a,)]}
Phuong trinh vi phan (14) cd dang cffa phuong trinh vi phan Bessel cai tien va Idi giai cho phuong trinh vi p h i n nay la.
4i=C,l„(pr)+C,K„(pr) (20) Trong d d I,, la ham Bessel cai t i ^ n dang mpt va bac khdng, va K^, ia
ham Bessel cai tien dang hai va bac khdng. Ap dung dieu kien bien <|> = 1 tai r^Tp.va ii^O tai r = co vao bieu thffc (20), Idi giai cho phUong trinh (14) la:
H e s o n e n theo bieu thffc (9) va (10) ddOc viet nhusau:
K,.2.jG{*J,d,=.|g[K.(pr,)K,(p,,)-K;(p,,)] ,22,
t j = E , A + 25tJ(>., + 2G,)tti^rdr
= E , A . ^ & g ^ [ K ! K ) - K ; ( p r . ) ] 6. G i i i b a i t o a n
The bieu thde (10) va bieu thde (7) t h u 6\iaz bieu thde n h u sau:
"dz-
-K ^u] w'r'^'C'^''' "'-C"r°
(24)Tfch phan bieu thde (24) theo p h u o n g phdp Galerkin va ly thuyet Green (Smith va Griffiths, 2004) [23] t h u dUoe bieu thffc ma tr^n d p cilng cua phan t d c o c nen nhUsau:
cosh^OjLjJ
n-'f
sinh(tijLj) s i n h { a | L j c o s h f a L , )(251 sinh(a,L,) sinhfajL,]
Trong do [ K ] la ma tran d p cdng cffa phan t d cgc n l n t h d j . Quy trinh tinh toan don gian sau day duac ap dung eho t n f c ^ g hpp tai trpng p dat tai dtnh coc. PhifOng trinh can bSng d o i vcfi phan tdcoc t h d j duuc viet nhusau:
I K ] , { " } , - { F ) , E6) Phuong trinh (26) duac viet lai dudS dang ma tran la:
H\
(27)T r o n g d 6 P | | l a t a i t r o n g d a t t a i d l n h c u a p h a n t f f c o c ; Kj^, ladocdng tUOng dUcmg cffa phan tff coc (j +1). GiSi phudng trinh (27), ehuyin w ddng tai niit d^u vh niit cuoi eiia phan tff epc la:
w*r=d^ (28)
(29) '^ t,a,cosh(ajLj) + K , . , s l n h ( a , L j ' '
Trong do K^ la dp cuTng tUong duang cua phan tff cpc thff j trong bieu thffc (28) co dang n h u sau:
^^t,a,cosh(a,L,) [ t ^ ' sinh{a,L,) s i n h { a , L j [ t j a , e o s h ( a , L , ) + K ^ , 5 i n h ( a j L j ) ]
Xet phan t d cpc cudi cung ngam d nOt t h d 2, dp cdng tUong dUdng cffa phan t d nay dUpc xac ^ n h theo bieu thde sau vdi K^, = c o :
^ _ t ^ a ^ c o s h ( a ^ } s l n h ( a ^ )
Lap lai qua trinh tfnh toan d p cung tUOng dUong cila phan tdcoc t i i N den 1, dp cung tuong duong cila toan b p he coc nen b§ng d p cdng, t u o n g duong cffa phan t d c o c t h d nhSt. Chuyen vi tai ^ n h coc dupe tfnh toan nhusau:
W z ^ = w , , , = ^ (321 Vi ehuyen vi cffa nut t h d hai cOa phan t d t h d j la ehuyen vi cua niit
thff nhat cffa phan tff thff (j+1) nen ehuyen vj dUpc t i n h toan t d phan tiJ t h d nhat den phan t d t h d N t h e o b i e u thde (31). Npi lUe t r o n g phSntd, cling dupe tinh toan dong t h d i vcfi ehuyen vj theo bieu tiidc sau: ^
(301
(31)
80!
Pi., = Wi.,K, (331 7 . V f d u t r n h t o a n
Seo vh Prezzi (2006) [21 ] phan tich cpe chju tai trpng d u n g trong nen n h i l u ldp n h u hinh 4 vdi d d lieu trinh bay trong bang t . K l t q u c i nghien cilu tuOng t u dupc thuc h i i n theo ly t h u y e t da trinh bay d tren va theo phSn tfch phan t d h f f u han ba chieu bang phan m e m SS13D.
P Trudng hpp 1 Trudng hpp 2 Trudng hop 3
/ .
DCP ll#U , . G , , G i . V i , H | . P
1 Pn™=1,P.IJ=l|
1 K ^ . 1 - "
1 1^1-1
[mil 3: Quy trinh tinh loan khi t^i trong dat tai dinh rac Bang 1. Dff lieu phan tfeh (Seo va Prezzi, 2006) E21]
Ldp dat
1 2 3
E,/E,., Case!
1.0 2.0 4.0
Case 2 4.0 2 d 1.0
Case 3 2.0 1.0 4.0 Ket quh tinh toan he so ^nh hudng trinh bay trong hhng 2. Cae gia trj tfnh toan cho thay he so anh hUdng tinh theo phuong phap de xuat bang vdi he sd anh hUdng tfnh toan bcfi Seo va Prezzi (2006) [21 ] .
I 0.3L Es I
Mr
0.4L |2E,
2E, V .
Hlnh 4: Vf du tfnh toan coc trong nen ba lOp (Seo va Prezzi, 2006) [21]
B^nq 2. Ket qua t m h toan he so anh hudnq Trudng
hop
1 2 3
H e s d a n h h U d n g 1 ^^^—i^L p Lee
(1991) 0.0361 0.0372 0.0358
Seova Prezzi (2006)
0.0336 0.0309 0.0323
3D FEM
0.0367 0.0365 0.0372
O l xuat 0.0336 0.0309 0.0323
Vf du tfnh toan tiep theo tff nghien cdu cffa Salgado vh cfing sU(2013) [21 Uc6 ke d i n bien dang ngang va so shnh vdi ket quh phSn tich cffa Seo v^ Prezzi (2006) [20] khong ke d i n b i l n dang ngang. Cpc co Lp= IS m, rp=0.5 m, ED=25 GPa. €)at n l n co mo dun dan hoi lOOMPa, hg SO Poisson 0.15. Tai trpng tac dung tai dinh cpc la 1000 kN. Cpc duoc phan tfch lai theo phuong phap de xuat d trgn va theo phuong phap p h i n t d hOXi han ba chieu sff dung phan mem SSi3D.
K i t qu5 tfnh toan trinh bay trong hinh 5 va bang 3 cho thay sai so chuyen vi dinh cpc theo Seo va Prezzi (2006) [21] la 31.2%, theo Salgado v^ cdng su (2013) [20] % 23.5% so vdi phuong phap phan tff hdu han d o Salgado va cong sff (2013) [20] thUc hien. PhUOng phap de xuat co gia tri ehuyen vj dinh cpc va mui cpc sat vdi ket qu3 tfnh toan cua Seo va Prezzi (2006) [21]. Phan tich phan tff hffu han ba ehieu bang SSI3D cho thay chuyen vi dinh coc va mui epc nho hOn phan tich theo phUOng phap phan tff hffu han do Salgado va cpng sU (2013) [20] thUc hien la 14.7%.
N l u lay k i t qua phan tich theo phuong ph^p bang phan mem SSI3D lam eO sd danh gia t h i sai so cua phuong phap de xuat la 17.3%.
B^ng 3: Chuyen vj cua ei Nguon
Seo va Prezzi (2006) Salgado va cdng sU (2013) FEM (Salgado va eong sU, 2013) De xuat
FEM 3D (SSI3D)
Chuyen vj dinh eoe (m) 0.00160 0.00171 0.0021 0.00156 0.00183
Chuyen vj mui coc (m) 0.00044 0.00048 0.00075 0.00043 0.00054
/ /
ll //
1 / •
*
r ^
0.0000 0.0005 0.0010 0.0015 0.0020 Chuygn vi(m)
Be xuat — — F E M 3D Hinh 4: Chuyen ^ nia coc
9. Ket l u a n
Phucmg phap dem gian tinh tcjan ung x d cua cpc dtm chiu tai trang dung dupc xay dung dua tren phucmg phap giSi tieh va phuong phap phan t d huU han. Trong phuong phap nay, dp edng tucmg duong cffa coc duoc tinh toan t d cac phan doan cpc t d d a y den dinh cpc va chuyen vj va npi lUc trang cpe duoc tinh toan tff dinh coc den day c p c Cac ket q u i so sanh cho thay.phucmg phap ti'nh toan CO d p tin cay cao phil hop wSeacgia tiiiet dua ra. Tuy nhien, ket qua tfnh toan cdn sai khae vdi phucmg phap phap phan t d hdu han ba chieu do gia thiet bo qua bien dang theo phuong ngang.
Van de nay se dugc ke den trong die nghien cffxi t i l p theo.
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