Time-domain responses of immersing tunnel element under wave actions

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2009,21(6):739-749 DOI: 10.1016/S1001-6058(08)60208-5

TIME-DOMAIN RESPONSES OF IMMERSING TUNNEL ELEMENT UNDER WAVE ACTIONS* CHEN Zhi-jie, WANG Yong-xue, WANG Guo-yu State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: [email protected] HOU Yong CCCC-FHDI Engineering Co., Ltd, Guangzhou 510230, China

(Received December 28, 2008, Revised March 25, 2009)

Abstract: The time domain responses of the tunnel element under wave actions during its immersion are investigated based on the linear wave diffraction theory. The integral equation is derived by using the time-domain Green function that satisfies the free water surface condition in the finite water depth, and is solved by the boundary element method. The motion equations of the tunnel element are solved by the fourth order Runge-Kutta method. A comparison between the computed and measured results reveals that the numerical model can effectively simulate the motion responses of the tunnel element and the cable tensions when the motions of the tunnel element are within some limit. Taking the tunnel element of 100 m in length, 15 m in width and 10 m in height as an example, the computational results of the motion responses of the tunnel element and the cable tensions in different immersing depths are obtained under different incident wave conditions. Key words: Immersed tunnel, time-domain response, cable tension, boundary element method

1. Introduction  An immersed tunnel[1] is a kind of underwater transportion passage crossing a river, a canal, a gulf or a strait. It is built by dredging a trench on the river or sea bottom, immersing prefabricated tunnel elements one by one to the trench, connecting the elements, backfilling the trench and installing equipments inside it. Compared with a bridge, an immersed tunnel has advantages of being little influenced by big smog and typhoon, stable operation and good resistance against earthquakes. Due to the special economical and technological advantages of the immersed tunnel, more and more underwater immersed tunnel are built or are being built in the world.  * Project supported by the National Natural Science Foundation of China (Grant No. 50439010), the Key Project of the Ministry of Education of China (Grant No. 305003). Biography: CHEN Zhi-jie (1980-), Male, Ph. D.

Building an undersea immersed tunnel is generally a super-large and challenging project that involves many key engineering techniques[2,3], such as transporting and immersing, underwater linking, waterproofing and protecting against earthquakes. The immersed tunnel was widely studied with respect to transportation, in situ stability and seismic response [4-8] . The immersion of tunnel elements was also studied. Zhan et al.[9] carried out an experimental investigation on the hydrodynamics of the immersed tunnel. Zhan et al.[10] performed a simplified numerical simulation of the two-dimensional motion of the tunnel element, considering only the current action on the tunnel element. However, the results did not agree with the actual experimental results[11]. Zhou[12] computed the frequency responses of the tunnel element, starting from its initial state of immersion, regarding it as a floating body and ignoring the actions of the hanging lines over the tunnel element. It seems to be an over-simplified

740

model and one should consider more details of the motion dynamics of the tunnel element in the process of immersion. The immersion of a large-scale tunnel element is one of the most important procedures in the immersed tunnel construction, and its techniques involve barges immersing, pontoons immersing, platform immersing and lift immersing[11]. A study on the dynamic characteristics of the tunnel element in the immersion is desirable for tunnel construction safety. The aim of the present study is to investigate the motion dynamics of the tunnel element in the process of immersion, focusing on the time-domain motion responses of the tunnel element based on the twin-barge immersing method through numerical modeling and experiments. The numerical results of the motion responses of the tunnel element and the cable tensions in different immersing depths are obtained under different incident wave conditions.

reference coordinate system o0 - x0 y0 z0 and the moving coordinate system o - xyz , are employed, as shown in Fig.1. The origin of the reference coordinate system o0 - x0 y0 z0 is at the undisturbed water surface. The y0 -axis is directed to the right and the z0 coordinate is measured upwards as the positive direction. The coordinate system o-xyz is fixed on the tunnel element, moving with the tunnel element. The x-axis is directed along the length of the tunnel element. When the tunnel element is in the static equilibrium position, two coordinate systems coincide. The excursion of the tunnel element due to the external forces is described by the movement of the moving coordinate system o-xyz relative to the reference coordinate system o0 - x0 y0 z0 . 2.1 Governing equations With the linear assumption, the total velocity potential I is the sum of the incident wave potential II and the scattered potential IS , i.e.,

I ( p , t ) = I I ( p , t ) + IS ( p , t )

(1)

where p is the field point, t is the time and

II =

gA cosh k ( z + h) sin(kx cos E + Z cosh kh

ky sin E  Zt ) in which A is the wave amplitude, E wave angle, g the gravity acceleration, depth, k the wave number and Z the the incident wave. In the fluid domain, the velocity satisfies the Laplace equation

(2) the incident h the water frequency of potential I

’ 2I = 0 (: , t > 0) Fig.1 Schematic diagram of a twin-barge immersing tunnel

2. Mathematical model The mathematical model of the twin-barge immersing tunnel shown in Fig.1 is developed based on the linear potential theory for the wave diffraction and radiation of an immersed tunnel element. The fluid is assumed inviscid, incompressible and the motion irrotational and periodic. In the simplification, it is assumed that the movements of the two barges are small and can be ignored. Two Cartesian coordinate systems, i. e., the

(3)

On the quiescent free water surface, the velocity potential I satisfies the following linearized free surface boundary condition

w 2I wI +g =0 2 wt wz ( t ! 0 on the water surface S F (t ) )

(4)

The boundary condition on the surface of the body can be expressed as

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G f ( p, t ; q,W ) = 2 ³

wIS wI = Vn  I wn wn

0

( t ! 0 on the surface of the body S B (t ) )

wIS = 0 ( t ! 0 on the sea bottom S D ) wn

(6)

J ( z , ] , R )dk in which t and W denote the time, p is the field point, q the source point, J the distance between p , q and R the horizontal distance between them, and r2 the distance between p and the image of q with respect to the sea bottom,

J ( z , ] , R ) = cosh[ k ( z + h)]cosh[ k (] + h)]<

The far boundary condition can be expressed as

wIS , ’I o 0 ( t ! 0 at infinity Sf ) wt

(7)

The initial condition is

IS

t =0

= 0,

wIS wt

t =0

=0

( t ! 0 on the water surface S F (t ) )

(9) where G and G are the transient term and the memory term, respectively, defined by f 1 1 e  kh G ( p; q ) = +  2 ³ J ( z , ] , R)dk , 0 cosh kh r r2 0

where J0 is the Bessel function of the zero-th order. By applying Green’s theorem, the problem for the scattered potential IS may be expressed by the following integral equation[14,15]:

³³

SB (t )

(8)

G ( p, t ; q,W ) = G (t  W )G 0 ( p, q) + G f ( p, t; q,W ) f

J 0 (kR)

2SIS ( p, t ) +

Equations (3) through (8) form a problem with initial and boundary conditions for a definite solution. 2.2 Solution method There are some ready-made methods available to solve the above problem expressed as Laplace equation. In this article, the problem is solved by the Green’s function method to obtain the velocity potential. For convenience, the sea bottom is assumed to be flat, which means that the influence of the trench on the flow field is ignored. So the time-domain Green function in the finite water depth obtained by Wehausen[13] is used:

0

1 < cosh kh sinh kh

[1  cos(t  W ) gk tanh kh ]<

(5)

where n is the unit normal vector pointing outwards from the body surface and Vn the normal velocity on the surface of the body. The boundary condition on the sea bottom of a finite water depth is expressed as

IS ,

f

IS

wI wG 0 ds = ³³ G 0 S ds + wn wn SB (t )

wGWf f wI ³0 dW [S³³(W ) (IS wn  GW wnS )ds + B t

1 (Is GWWf  IsW GWf )VN dl ] v ³ g * (W )

(10)

where p  S B (t ) , * (W ) are the waterline contour and VN is the two-dimensional normal velocity on * (W ) . It is assumed that the motions of the body are small. Then the body surface condition can be considered to be satisfied in the mean position of the body surface. Thus, S B (t ) and S B (W ) in Eq.(10) can be replaced by the fixed mean boundary S B . In addition, because the motions of the immersed tunnel element are under the water, the part of the integral on the waterline at the right hand of Eq.(10) disappears. Then Eq.(10) can be re-written as

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2SIS ( p, t ) + ³³ IS SB

wI wG ds = ³³ G 0 S ds + wn wn SB

wGWf f wI ³0 dW ³³S (IS wn  GW wnS )ds B t

the body, which will be discussed in the next sub-section. The cable tensions are calculated by Wilson formula and is expressed as

0

(11)

By applying the boundary element method[16], Eq.(11) is discretized to obtain a set of algebraic equations as follows

[ A]mum {Is } = {B}

(12)

in which m is the total number of the discrete panel elements on the body surface, [ A] is the coefficient matrix, of which the computation is relatively easy for its simple relation with the transient term of the Green function. For the computation of the right hand matrix {B} , the memory term of the Green function and its normal derivative must be recalculated at every new time instant, so must their convolutions with the velocity potential IS and its normal derivative, since they are all time-dependent variables. For saving the computational time, the values of the memory term of Green function and its normal derivative at the previous time are stored and recalled at every new time instant, although this will require a comparatively large computer memory. In solving Eq.(11), the calculations of the Green function and its spatial derivatives are rather complicated, for which it is important to select appropriate and effective algorithms. They are discussed in Refs.[17] and [18]. 2.3 Computation for wave forces and cable tensions The hydrodynamic pressure at any point in the fluid can be expressed as

p = U

wI wt

(13)

where U is the mass density of the fluid. The wave forces can be determined by integrating the pressure over the body surface, S B , i.e.,

F =  ³³ pndS

(14)

SB

After obtaining the velocity potentials in the fluid by solving the integral equation, the wave forces may be computed from Eq.(14). In solving the integral equation, the motion velocity of the body is required, which is obtained by solving the motion equations of

§ 'S · T = Ce
n

2

(15)

where T denotes the cable tension, Ce is the elasticity coefficient, d the cable diameter, S the original length, 'S the deformation of the cable and n a constant. It is assumed that the steel cables are used in the immersing system of the tunnel element and Ce = 2.75 u 1010 kg / m 2 , n = 3 / 2 . 2.4 Motion equations of immersed tunnel element In this article, the scattered potential is not separated into the diffracted potential and the radiated potential, so the motion equations of the body can be written as 6

¦[M

kj

[j (t ) + Bk [ j (t ) + Ckj[ j (t )] =

j =1

Fk (t ) + Gk (t ) ( k = 1, 2, " , 6)

(16)

where Fk is the total generalized wave force, Gk the total generalized force that acts on the tunnel element by the cables of the immersing system, [ M ] the inertia matrix of mass, [ B ] the viscous damping matrix and [C ] the hydro-restoring matrix. As a potential flow problem, the viscous damping is not taken into account in this article. Furthermore, the hydro-restoring force is equal to zero because the tunnel element moves under the water. The above six coupled differential equations of motion can be reduced to the following second order differential equation

[ = F [t , [ , [ ]

(17)

Equation (17) is solved by applying the fourth order Runge-Kutta method, and the displacement and velocity of the body can be expressed as

[ (t + 't ) = [ (t ) + 't[(t ) +

't ( M 1 + M 2 + M 3 ) 6 (18)

[ (t + 't ) = [(t ) +

( M 1 + 2M 2 + 2M 3 + M 4 ) 6 (19)

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where

M 1 = 't < F [t , [ (t ), [(t )] , M 2 = 't < F [t +

M 't 't[ (t )  , [ (t ) + , [ (t ) + 1 ] , 2 2 2

M 3 = 't < F [t +

't 't[ (t ) , [ (t ) + + 2 2

M 'tM 1  , [ (t ) + 2 ] , 4 2

M 3 = 't < F [t + 't , [ (t ) + 't[(t ) + 'tM 2  , [ (t ) + M 3 ] 2 After the displacement and the velocity of the immersed tunnel element at the time t are obtained, the cable tensions are computed according to Wilson formula, and the wave forces are determined by hydrodynamic analysis, then the function  F [t , [ (t ), [ (t )] is obtained. Using Eqs.(18)-(19) the displacement and the velocity of the tunnel element at the time t + 't are obtained. Repeat the computation at the next time step as above until the desired termination time is reached. Based on the above procedures, a time-domain computer code is developed.

springs and nylon strings that deform elastically. The water depth ( h ) of the wave flume is 0.80 m. It is known that the tunnel element in engineering is actually immersed under the ballast water action, namely under a negative buoyancy, inside the tunnel element. The weight of the tunnel element used in this experiment is 1208.34 N. When the tunnel element is completely submerged in the water, the buoyancy force acting on it is 1176.0 N. So the negative buoyancy is equal to 32.34 N, which is 2.75 percent of the buoyancy force of the tunnel element. The negative buoyancy provides the cable with initial tensions. Regular waves are generated by a piston-type wave generator, with the wave height ( H ) of 0.03 m, 0.04 m and 0.05 m, and the wave period ( T ) of 0.7 s, 0.85 s, 1.1 s and 1.4 s, respectively. The experiments are conducted for the cases of three different immersing depths of the tunnel element, i.e., d = 0.1m, 0.3m and 0.5m, where d is defined as the distance from the water surface to the top surface of the tunnel element.

Fig.2 Schematic diagram of the experimental setup

3. Experimental investigation The experiments are carried out in a wave flume of 50 m long, 3.0 m wide and 1.0 m deep. The schematic diagram of the experimental setup is shown in Fig.2. The immersed tunnel element considered in this study is a hollow cuboid sealed at its two ends and is made of acrylic plate and concrete. The tunnel element is 2 m long, 0.3 m wide and 0.2 m high. The body is immersed into the flume under the control of cables. In the experiments, the cables are modeled by

Fig.3 The tunnel element in the wave flume

There are four vertical strings connected with springs to support the immersed tunnel element under wave actions, two are on the offshore side and the

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other two on the onshore side. To measure the tensions in the strings, four tensile force gages are installed on the four strings. Two lights with a certain distance are installed at the front surface of the tunnel element, as shown in Fig.3. When the tunnel element moves under wave actions, the positions of two lights are recorded with a Charge Coupled Device (CCD) camera. The sway, heave and roll motions of the tunnel element are obtained from CCD recorded images with an image analysis program. 4. Results and discussion 4.1 Verification of the numerical model The time domain method is validated by experimental results. In the following calculations, the number of elements used to model the tunnel surface is 15 in length, 3 in width and 2 in height, so the total number of the elements is 162. Time step is chosen as 1/40 times of the wave period and the gross computation time chosen as 10 times of the wave period. The numerical results of the displacements of the tunnel element against time for the typical cases of d = 0.3m , H = 0.04m , T = 1.1s and d = 0.5m , H = 0.05m , T = 1.4s are compared with the experimental results, as shown in Figs.4 and 5. It can be seen that the agreement between them is good, especially for the sway motion. The numerical results slightly overestimate the heave motion and slightly underestimate the roll motion as compared with the experimental results. From Figs.4 and 5, it can be seen that the displacement curves of sway and heave motions at some peaks and troughs are truncated in the experimental results, which may be due to the limitation of the image resolution of the CCD system. In the case of the immersing depth d = 0.3m , the motions of the tunnel element show a strong nonlinearity, especially the heave and roll motions. The heave motion is asymmetrical about the equilibrium position of the tunnel element, with the displacement along the upward direction being larger than that along the downward direction, and the amplitudes are uneven, so are the amplitudes of the roll motion. Furthermore, it is shown that the motion range of the sway motion in every period is even while its center position is moveable. Figures 4 and 5 reveal that the numerical model can well simulate the motion responses of the immersed tunnel element under wave actions. The numerical computation results and experimentally measured dada of the cable tensions against time are presented in Figs.6 and 7. It can be seen that they agree quite well , with the numerical

results slightly overestimating the cable tensions. In Fig.6, the zero cable tensions appear periodically with the increase of time, which shows that there is a slack state in the cable during the movement of the tunnel element. This is the main cause for the nonlinear feature of the motions of the tunnel element as shown in Fig.4. While in the case of Fig.7 there is not any slack state in the cable. Here, only the tension on the offshore side cable is given. The tension on the onshore side cable is very similar to that on the offshore side cable, with only a slight difference in the phase and value.

Fig.4 Comparison between numerical and experimental results of displacements of the tunnel element ( d = 0.3m , H = 0.04m , T = 1.1s )

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Fig.7 Comparison between numerical and experimental results of the tension acting on the offshore side cable ( d = 0.5m , H = 0.05m , T = 1.4s )

Fig.5 Comparison between numerical and experimental results of displacements of the tunnel element ( d = 0.5m , H = 0.05m , T = 1.4s )

Fig.6 Comparison between numerical and experimental results of the tension acting on the offshore side cable ( d = 0.3m , H = 0.04m , T = 1.1s )

The numerical and experimental results of motion displacements of the tunnel element and cable tensions for the cases of wave height H = 0.03m are, respectively, shown in Tables 1 and 2. The differences between the numerical results and the experimental ones are also given in the tables. It is shown that the numerical and experimental results of motion displacements are in fairly good agreement except for a slightly large difference in the roll motion. As for the cable tensions, the differences between the numerical and experimental results are generally quite small. The differences may result from both the precisions of the experimental measuring systems and limitation of numerical computation. Generally speaking, the numerical model can well simulate the dynamic displacements of the tunnel element and the cable tensions. 4.2 Numerical computation The motion responses of the tunnel element due to the wave actions are numerically calculated. The dimensions of the tunnel element are 100 m in length, 15 m in width and 10m in height. The water depth ( h ) is 40 m, the incident wave angle is 90o, and the tunnel element moves with three degrees of freedom, i.e., in sway, heave and roll motions. The numerical computations are performed for the waves of four different wave periods, i.e., T = 5.0s, 6.0s, 8.0s and 10.0s. The relative wave height ( H / h ) is 0.025, 0.0375 and 0.05. The negative buoyancy is taken to be equal to 2 percent of the buoyancy of the tunnel element. And two relative immersing depths are considered, i.e. d / h = 0.375, 0.625. Similarly, the total number of elements on the tunnel surface is 162, and the time step is 1/40 times of the wave period. Figure 8 shows the displacements of motion of the tunnel element as d / h = 0.375 . In the figure, ] denotes the displacement of the tunnel element, A the wave amplitude, L the wave length, k the

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Table 1 Numerical and experimental results of motion displacements Case

Motion

Results Exp.

Num.

Err.

Sway (10-2m)

0.55

0.48

12.7%

Heave (10-2m)

1.11

1.24

11.7%

Roll (o)

6.30

6.90

9.5%

Sway (10-2m)

0.93

0.95

2.2%

Heave (10-2m)

1.95

2.10

7.7%

Roll (10-2m)

11.40

13.10

14.9%

Sway (10-2m)

0.31

0.28

7.2%

Heave (10-2m)

0.55

0.51

6.4%

Roll (10-2m)

0.89

0.79

11.2%

Sway (10-2m)

0.93

0.87

6.5%

Heave (10-2m)

1.87

1.91

2.1%

Roll (o)

1.80

1.40

22.2%

Sway (10-2m)

0.53

0.48

9.4%

Heave (10-2m)

0.57

0.62

9.2%

Roll (o)

0.52

0.44

15.5%

Sway (10-2m)

1.32

1.20

8.7%

d = 0.5m , H = 0.03m , T = 1.4s

Heave (10-2m)

1.13

1.28

13.5%



Roll (o)

0.88

0.74

15.9%

d = 0.1m , H = 0.03m , T = 0.7s

d = 0.1m , H = 0.03m , T = 0.85s

d = 0.3m , H = 0.03m , T = 0.85s

d = 0.3m , H = 0.03m , T = 1.1s

d = 0.5m , H = 0.03m , T = 1.1s

wave number and B the width of the tunnel element. For the asymmetrical movement about the equilibrium position of the tunnel element and the uneven response amplitudes, the computational results are given by the maximum non-dimensional displacements of the tunnel element in the positive and negative directions, respectively. It is shown that in the case of small B / L , the motion responses of the tunnel element are comparatively large and decrease rapidly with the increase of B / L . The effects of the wave height on the non-dimensional displacements are also shown in the figure. It can be seen that at large B / L the effects of

the wave height are slight while at small B/L the effects are perceptible, especially for heave and roll motions. Moreover, the heave motion in the upward direction is seen to be larger than that in the downward direction at smaller B / L . This is mostly due to the constraint of the cables when the tunnel element moves downwards and the slack of the cables when the tunnel element moves upwards. Figure 9 shows the motion responses of the tunnel element as d / h = 0.625 . In this case, the motion amplitudes of the tunnel element are even. So in the figure, the results are given by the values of the motion amplitudes. It can be seen that the amplitudes

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Table 2 Numerical and experimental results of cable tensions Case

d = 0.1m , H = 0.03m , T = 0.7s 

d = 0.1m , H = 0.03m , T = 0.85s 

Results

Cable Exp.

Num.

Err.

C1 (N)

28.4

33.8

19.0%

C2 (N)

30.0

35.4

18.0%

C1 (N)

38.3

39.6

3.4%

C2 (N)

42.7

44.5

4.2%

C1 (N)

11.5

11.5

0.0%

C2 (N)

11.3

11.7

3.5%

C1 (N)

19.4

20.8

7.2%

C2(N)

19.5

21.3

9.2%

C1 (N)

9.8

10.6

8.2%

C2 (N)

9.9

10.8

9.1%

C1 (N)

12.1

13.0

7.4%

C2 (N)

12.2

13.7

12.3%

d = 0.3m , H = 0.03m , T = 0.85s 

d = 0.3m , H = 0.03m , T = 1.1s 

d = 0.5m , H = 0.03m , T = 1.1s 

d = 0.5m , H = 0.03m , T = 1.4s 

of sway and heave motions are close and they decrease with the immersing depth, and so is the amplitude of the roll motion. In the long wave case, the motion responses are seen to be much larger than that in the short wave case. Furthermore, the effects of the wave height on the motions of the tunnel element are worth noticing and it seems that they are linear in the case of small motion responses of the tunnel element. The results of the cable tensions are presented in Fig.10, where F denotes the cable tension, FNB the negative buoyancy acting on the tunnel element, C1 the onshore side cable and C 2 the offshore side cable. The maximum tensions of the cables are obtained during the interaction of the tunnel element with waves. From Fig.10, it is shown that the cable tensions are larger in the case of small B / L and they decrease with the increase of B / L . The tensions decrease as the wave height decreases and the immersing depth increases. The tensions of the cable at the onshore side are slightly small as compared to the tensions of the cable at the offshore side.

5. Conclusions In this study, a time-domain numerical model of the immersed tunnel element due to the wave actions is presented, and the dynamic displacements of the immersed tunnel element and the cable tensions are obtained both by the numerical simulations and experiments. The motion responses of the tunnel element and the cables tensions are analyzed. The motions of the tunnel element show a strong nonlinearity, when the immersing depth is relatively small, and the heave motion is asymmetrical about the equilibrium position of the tunnel element and its amplitudes are uneven. The relative width B / L has a significant effect on the motions of the tunnel element. The motion responses decrease rapidly with the increase of B / L . Moreover, the effects of wave height on the non-dimensional displacements are perceptible at small B / L and slight at large B/L in the case of small immersing depth, while the effects are slight at any B / L in the case of large immersing depth.

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Fig.10 Tensions of the cables

The zero cable tensions appear periodically with the increase of time in the case of small immersing depth and the cables remain in tension in the case of large immersing depth. The curves of tensions vs. time for the offshore and onshore side cables are very similar. The cable tensions increase with the decrease of B / L , and decrease as the wave height decreases and the immersing depth increases. In this article, the movements of the barges are ignored and only wave actions are considered. The influences of the movements of the barges and current actions on the motion responses of the tunnel element will be considered in our further researches. References [1] [2]

[3] Fig.8 Motion responses of the tunnel element ( d / h = 0.375 ) [4]

[5]

[6]

[7] Fig.9 Motion responses of the tunnel element ( d / h = 0.625 )

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