seismic damage and failure analysis of arcch dam …authors should be written like a.mehmet and...
TRANSCRIPT
SEISM
ABSTRA
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1 INTRO
Variouresponsecompresconcretethrough DruckerChina. Hresponse
One oconcreteparametedevelopmapplied t
In thisseismic 1 Doctor, 2 Professo3 Professo4 Professo
MIC DAMDIFFE
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or, China Instior, China Instior, China Insti
MAGE ANERENT M
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and rock con damping dund solving m
am-foundatioomputation b
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LURE ANAL MOD
eyu Li2, Jin
linear dynammethod in thontraction joam and foun
y dissipationling contact psystem anddomain decodation are u
plastic modeken as an exmodels of fouamage mode
mic response
uring the recan isotropic ge of an arcture of the cd the extendmpare the seial rotating s
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Hydropower nd Hydropownd Hydropownd Hydropow
NALYSISELS OF F
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mic analysis he time domoint openingndation, andto the far fieproblem andd enormousomposition msed for coml. The studyample. The rundation, anl for dam an
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Research, Beier Research, Ber Research, Ber Research, B
OF ARCFOUNDA
un Chen4
model of amain is prese
and closingd viscous-spreld foundatiod nonlinear m computatio
method is intmparison, incl
y of Shapai result showsd the damagd foundation
orce model;
or modeling del which alet al2. studieterial and thement, plasticage of Dagack approach
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mass, to expl models fo
ijing, guoss@Beijing, lideyuBeijing, tujin@Beijing, chenh
CH DAM WATION
arch dam-fouented. In thig, nonlinear ring boundaron are includmaterial modon work is troduced accluding elastii arch dam s that differege of dam-fon is close to t
foundation
the seismicllows for tened seismic fe model wasc damage mangshan arch
to study the
i-brittle matroscopic mend stiffness refore, dama
plore the effeor rock mas
@iwhr.com [email protected] @iwhr.com [email protected]
WITH
undation-is model,
material ry model
ded. Since del is very
needed, cordingly. ic model, in China nt failure
oundation the actual
radiation
c damage nsion and failure of s verified odel, and h dam in e seismic
terial like echanical with the
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ect on the ss, some
m
2
comparative investigations are taken. The seismic damage process of Shapai dam-foundation system is investigated by using damage model applied to dam-foundation system. As comparative studies, the seismic damage process of dam-foundation system is investigated by using damage model for dam, using Drucker-Prager elastoplastic and elastic model for rock mass. Since the computation is enormous, parallel computation is introduced in this paper and all computing work is completed by the parallel finite element program developed by China Institute of Water Resources and Hydropower Research.
2 Time domain dynamic equation
Dynamic equation after finite element discretization:
MU CU KU F (1)
Where, M , C and K are the mass, damping and stiffness matrixes respectively,and F , U , U andU are the load, acceleration, velocity and displacement vectors respectively. For large scale nonlinear dynamical calculations, the decoupled explicit integration algorithm is more efficient. Thus, this paper adopts the decoupled explicit numerical integration format which combines the center differential and unilateral differential method as:
n n 1n dt
U UU
,
n 1 n n 1n 2
2
dt
U U U
U. (2)
The discrete dynamic equation at time step n+1 is written as:
2 2
n 1 n n 1 n n n 1 n2 MU M U U KU C U U Fdt dt dt (3)
If the diagonal lumped mass matrix M is used, the equation is solved explicitly.
3 Foundation radiation damping
For the seismic response of the dam, since foundation is infinite with respect to the dam, the seismic analysis of dam is actually to simulate the wave propagation of the open system consisting of infinite foundation and dam, which includes both the vibration of dam generated by the input wave and the scattering wave to the foundation generated by the dam as vibration source. The energy gradually escapes as the scattering wave transmits to the foundation due to the geometric spreading and damping dissipation. In the actual calculation, it is impossible to simulate the scattered wave dissipation process of infinite foundation, and the foundation can be only taken a limited range. Theoretically, as long as the scope of foundation meets the need of 2L CT , where C is the velocity of foundation
and T is the seismic wave duration,the dissipation effect can be included. In static analysis, the foundation far away from the dam could take a larger size grid, but in dynamic analysis oversized grid could not be taken due to the limit of the wave length. Thus, if the scope of foundation is taken by
/L CT 2,considerable computing would be brought and difficulty to apply to engineering practice. If a smaller range of foundation is taken,the scattered wave which was supposed to spread to foundation would reflect back to dam and seismic response of dam would be amplified. The effect that energy spreads to foundation is equivalent to damping, so called ‘foundation radiation damping’. In order to simulate the effect of ‘foundation radiation damping’, in the context of a limited range of foundation, the concept of artificial boundary is proposed to simulate the effect of infinite foundation to near –field wave.
Global artificial boundary and local artificial boundary are two main boundaries. Global artificial boundary5,6 is based on the frequency domain and coupled equation is formed in space domain. To solve the coupled equation, cumbersome and enormous computation is needed. The nonlinearity of foundation can’t be considered due to the frequency domain. Thus, the local artificial boundary which is a decoupled method in space and time domain is proposed. The local artificial boundary includes
Authors should be written like A.Mehmet and M.Ahmet 3
displacement artificial boundary and stress artificial boundary. Transmitting boundary7 belongs to displacement artificial boundary which has second –order accuracy, but the phenomenon of numerical instability may occur and usually requires repeatedly trial. Viscous-spring artificial boundary8 belongs to stress artificial boundary, which needs to impose external force and spring-damping system on the boundary. The boundary nodes and internal nodes use a uniform format to solve and the algorithm has first order accuracy and good stability.
The viscous-spring artificial boundary is used in the present paper. This boundary condition is applied at the far –end boudanry of the foundation in 3D space given as :
n nK u C u , 1 s sK v C v , 2 s sK w C w (4)
Where and are the normal and shear tractions; u , v , and w are the displacement of normal and two shear components; u , v , and w are the velocity of normal and two shear components; nK
and sK are the spring coefficient of normal and shear components; nC and sC are the damping
coefficient of normal and shear components; K and C are given as:
n
EK
r2 ,
s
GK
r2 (5)
n pC c, s sC c (6)
Where r is the distance from wave source to boundary; E is modulus of elasticity; G is shear modulus of elasticity; is density ; pc is pressure wave velocity; sc is shear wave velocity; pc and
sc are given as:
p
(1 )
(1 )(1 2 )
Ec
,
s 2(1 )
Ec
(7)
Where is Poisson’s ratio.
4 The contact nonlinearity of contraction joints
Contraction joints are an important aspect of seismic response of arch dam. During the construction process, the placement of arch dam concrete is divided into several dam sections as a section of 20m width. After concrete cools to the steady temperature and contraction joints grouting between dam sections, dam sections combines into a whole model to resist water load. Under the static load, contraction joints work in compression. During earthquake, the arch tensile stress generated overcomes the arch compressive stress under static load and then contraction joints open. Pacoima arch dam in America has experienced the Richter 6.6 earthquake and dam itself is not damaged, but the opening and closing of contraction joints has obviously occurred. The contact nonlinearity caused by contraction joints attracts the attention of scholars.
Contact problem has clearly physical concept, mainly including opening, closing and slipping of contact surface, and easy to determine the constraint condition. The point is to get appropriate numerical solution method aiming to the discontinuous nonlinear problem.
The contact model can be divided into two categories aiming to the different treatment of constraint condition. One is contact force model based on the nonlinear boundary condition, taking Lagrange multiplier method as representative, and Lagrange multiplier represents the unknown contact force. Another is represented as penalty function. The introduction of contact stiffness into model is to meet the contact boundary constraint condition. To meet the condition of no embedding between contact surface, penalty function method needs to introduce greater stiffness. Theoretically, there still exists embedded distance, and there is a certain sensitivity to the stiffness value. In the contact force model, applying the unknown force as the contact force to the contact interface is to meet the constraint
4
condition, and there is no problem of artificially assumed stiffness. The Lagrange multiplier method is used in this paper.
The contact surface constraint condition:
a. Opening condition B A n
s
0 0
0
u u n
b. Closing condition
B A n
n 1 n 1 n 1B A B A s n
0 0
u u n
u u t u u t
c. Slipping condition
B A n
n 1 n 1 n 1B A B A s n
0 0
u u n
u u t u u t
Where, the location of point A and B is Au and Bu ; n is the direction vector from A to B; s is
shear contact force; n is normal contact force; is friction coefficient; t , n 1t is the current step and
previous step.
From (3), A M , 2 2n n 1 n n n 1 n2 F M U U KU C U U Fdt dt dt , n 1U U ,the
equation can be written as:
AU F (8)
The equation including contact force can be given as:
AU F Bλ (9)
Where, B is contact constraint matrix, λ is contact force vector.
Contact constraint equation can be given as:
T B U γ (10)
Where, γ is displacement constraint vector.
From (9) and (10), contact force equation is given as:
l Cλ DU (11)
Where lλ is contact force vector in the local coordinate system, C is flexibility matrix , DU is displace vector .C and DU are given as:
T 1 T TC TB A B T , T 1 l DU TB A F γ (12)
Where lγ is displace constraint vector in the local coordinate system, T is coordinate system transformation matrix.
The flexibility matrix of contact force equation C is full matrix due to added mass considering the effect of dynamic water pressure, normal contact force and shear contact force are coupled each other. The modified Gauss-Seidel iterative method solving normal contact force and shear contact force alternatively is used to solve the equation.
5 Damag
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ai dam is a thhighest RCCarthquake ofearthquake r
ake. Accordinent announceand PGA ran
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trength w
90 120Time/s
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e distance frovice at the dantour map of earthquake, tn 177gal-286exceeded de
dam site. Since reestablish
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ed artificial acwn in Figure 6and 1276704,ough the heigof dam, the dis taken as 2
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foundation sys
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-0.1
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atio
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structure, of 1On May 12, 2om Shapai aram site, the ef seismic intethe seismic in6gal. The seiesign values. ce no ground
hed by using ecorded at 7 ccelerogram. The seismic
ccelerograms a6, the total nu, respectivelyght directiondam is divide2m. The grid ear material
des and elem contracting
m.
stem
n as: density
namic tensile
he adjusted
e damage var
30 60 90Time/s
132m high, c2008, Wenchurch dam to thearthquake reensity and bentensity of thsmic intensit After the ea
d motion hasthe “stochasstations duri
ms for the Shac wave is sho
at Shapai damumber of noy. The dam of dam, the
ed into 11 palength of neelements is1ents is 67326joints and in
Figure 7 FE
2400 k e strength tf
function of c
riable w D
120 150-
-Acc
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atio
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completed inuan in China
he epicenter iecording faildrock peak ahe Shapai damty of the damarthquake, nobeen recordtic finite fauing the earthqapai dam siteown as Figure
m site (from Zhdes, elementand near fielgrid length o
arts. Accordinear foundatio82867, whic
6 and 58757 nduced joints
EM mesh of no
3kg/ m , origin
0 4.3MPacracking disp
.
0 30 6-0.2
-0.1
0.0
0.1
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n 2006, was ta was attackeis 36 km. Sinled to obtain acceleration am site rangem site and theo damage wa
ded at the damult method” aquake. Fig. 1
e with PGA oe 5.
hang Cuiran) ts, and degreld is modeledof dam is takng the cross
on is about 3mch is 45% of respectively
s. The elemen
onlinear elem
nal elastic m
a , fracture en
placement to
60 90 120
Time/s
the ed by the nce there during PGA the s 8 - 9 e as found m site and 13 is the of 0.262
es of d by ken as 2m. river
m. As the total
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modulus
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o the
150
Authors should be written like A.Mehmet and M.Ahmet 9
Figure 8 tw f Figure 9 w D
The rock material parameters are taken as: density 32600kg/ m , original elastic modulus
0 11GP aE , Poisson ratio 0.23 , friction angel 47.73 , cohesive stress 2.0MPac ,
dynamic tensile strength t 0 2 cos 1 sin 1.55MPaf c , F 107 N mG . The parameters of
viscous damping boundaries of the foundation are taken as: 7 3n 1.61 10 N mK ,
6 3n 5.76 10 N s mC , 7 3
s 0.65 10 N mK , 6 3n 3.41 10 N s mC . Figure 10 and Figure 11
show the adjusted function of cracking displacement to the degradation tensile strength tw f and to
the damage variable w D .
Figure 10 tw f Figure 11 w D
The shear strengths of contraction and induced joints are taken as: f=1.1, c=1.1MPa. The tensile
strength of contraction joints and induced joints are zero and half of the tensile strength of concrete, respectively.
Damage model for concrete and rock: Figure 12 shows that the dam is basically not damaged even near the bottom pedestal, but the
jointed foundation rock body is damaged, as the tensile strength and fracture energy of the cracked foundation rock body are less than those of the dam concrete.
0 50 100 150 200 2500
1
2
3
4
5
f t/M
Pa
w/m
GF=296N/m
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
D
w/m
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
f t/M
Pa
w/m
GF=107N/m
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
D
w/m
‐Dt
Damadamaged
age model for near the botto
Figure 13
Upstream da
Downstream
Upstream
Downstre
Figuconcrete and
om pedestal an
Damageable
m surface
m dam surface
m dam surface
eam dam surfac
ure 12 DamagDrucker-Pragnd the jointed
model for dam
can
Fou
e
10
ge of jointed fger elastoplastd foundation ro
m with Druker
Crown ntilever section
undation
Crown cantilever
Foundation
foundation roctic model for rock is damage
r-Prager mode
ck masses rock: Figure 1ed in the plasti
el for foundati
El. 1842 arsection
El. 1798 assessss
El. 1754 ar
El
El. 1
El
13 shows thattic form.
ion rock mass
rch section
arch section sssecsection
rch section
l. 1798 arch secti
1842 arch sectio
l. 1754 arch secti
t dam is
ses
ion
n
ion
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By c
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Conclusi
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REFERE
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ions
s study, a relbased on finitai dam-foundAs comparat
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