Finite element analysis of the aseismicity of a large aqueduct

Soil Dynamics and Earthquake Engineering 94 (2017) 102–108

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Finite element analysis of the aseismicity of a large aqueduct

MARK

⁎

Yunhe Liu, Kangning Dang , Jing Dong School of Water Resources & Hydro-Electrical Engineering, Xi’an University of Technology, Xi’an 710048, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Aqueduct Fluid-structure dynamic coupling Bilinear deformation characteristics LRB Numerical calculation

Finite element analysis has been applied to a large aqueduct to investigate the efficiency of lead rubber bearings (LRBs) in reducing the effects of earthquake shocks. Theoretical relationships have been derived for dynamic interaction of a coupled fluid-solid system such as an aqueduct. A numerical program has been developed by considering the coupled fluid-structure dynamics between the water in the aqueduct and the side wall of the aqueduct and the bilinear deformation characteristics of LRB. The numerical analysis shows that the incorporation of LRBs is a valuable technique in creating aseismicity for a variety of conditions. Since aseismic efficiency varies with LRB type, it is feasible to have an optimal design. The analyses show that LRBs work best with large earthquakes, however, the displacement of the trough sat on top of the LRBs is enhanced so the design of joints within the aqueduct needs special attention.

1. Introduction Aqueducts are hydraulic structures that transport water across valleys, rivers, traffic lines and other obstructions, played an important role in utilization and allocation of water resources. A large number of major aqueduct projects are to be undertaken in China as part of the South-North Water Transfer Project, including an aqueduct with a total length of 3.5 km and maximum flow rate of 500 m3/s across the Yellow River [1], the Diao River Aqueduct [2] with a design flow capacity of 610 m3/s. These projects are so large that their equivalents are rarely found in the world. Most of these large aqueducts are located in areas of intense earthquake activity and ensuring that the aqueducts are safe when subjected to seismic load is an issue that must be addressed. To date little research work has been undertaken into the seismic design of large aqueducts. However, it is readily apparent that there are a lot of similarities between the structural form of an aqueduct and a bridge and there has been a lot of research conducted into bridges in seismic areas [3–5] and some of the findings may be applicable to aqueducts. The specific difference between bridges and aqueducts is that the latter carries a large body of water at the top of the structure. This causes two significant problems, i.e., how to account for the behaviour of the water when subjected to seismic load, how to reduce the effect of the large water body which makes the aqueduct structure ‘top-heavy’ by comparison to a normal bridge. The presence of the large quantify of water at the top of the aqueduct is very detrimental under the action of an earthquake. The use of a passive "resistance" strategy is not economic, it is very difficult to guarantee the safety of the structure.

⁎

At present, the effect of the water in an aqueduct which is subjected to seismic excitation can be accounted for in four ways: 1) the added mass approach [6], 2) the spring-mass system approach [7] as put forward by Housner [8–10], 3) the fluid-solid coupling approach with the trough of the aqueduct being treated as a rigid body [11], 4) the fluid-solid coupling approach with the trough being considered as an elastic body [12]. The added mass and spring-mass system approaches are relatively simple and practical to apply. In Chinese code NB 35047-2015 [10], based on the general Housner model, the improved formula of the dynamic water pressure and the added mass of the trough wall and the trough bottom are given. Analytical results show that the fluid-solid interaction is more intense when the trough of the aqueduct is comparatively thin and flexible compared to when the trough is rigid. When the trough of the aqueduct is treated as an elastic body the analytical results have better correlation with actual behaviour. Disaster investigation of earthquake shows that the steel storage tank in the earthquake occurred a serious damage, especially round thinwalled steel tanks, prone to nonlinear local buckling [13–15]. Aqueducts are concrete structure, its materials and shapes are different from steel tanks, the existing literature shows that the main failure occurs in the supporting part [12], and so far, there has been no instance of trough failure during earthquakes, so elastic body was considered in this paper.

Corresponding author. E-mail address: [email protected] (K. Dang).

http://dx.doi.org/10.1016/j.soildyn.2017.01.018 Received 4 September 2015; Received in revised form 3 November 2016; Accepted 15 January 2017 Available online 20 January 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 94 (2017) 102–108

Y. Liu et al.

2. Seismic resistance using lead rubber bearings Seismic isolation and absorption technology for bridges, which has been the subject of much research and is in an advanced state of development [3–5,16,17], would seem to be worth applying to aqueduct structures. There are three kinds of common used isolation bearings: laminated rubber bearings, sliding friction isolation bearings, viscous damping bearings [18,19]. The bridge isolation device which is composed of laminated steel plate and rubber bearing is the main bridge seismic isolation system. The plate type rubber bearing is made of thin rubber sheet and thin steel plate, and is formed by bonding with each other. In the middle or centre of the plate type rubber bearing, the lead core is pressed into the lead core with a purity of 99.9%, and the lead rubber bearing (LRB) is formed. Its function of dissipation vibration energy through the shear deformation of the lead core. The principle of sliding friction isolation bearing is that when the structure is subjected to a small ground vibration, the static friction force can prevent the structure sliding and keep the structure stable, when the ground excitation exceeds a certain limit that is more than the maximum static friction, the sliding surface begins to slip, at this time, even the ground excitation is increasing the seismic force to the upper structure will not be increased. Now there are resilient-friction base isolator (R-FBI) and friction pendulum system (FPS). The viscous damping bearing is a viscous damping energy dissipation device. The vibration isolation principle is to achieve the purpose of absorbing and dissipating the vibration energy through the viscous shear of the viscous material. Its damping effect is more ideal, but its structure and processing complex. The lead rubber bearing is a well-established seismic isolation system for use in buildings and bridges. It consists of a laminated rubber bearing with a lead core pressed into its centre as depicted in Fig. 2-1. The rubber/steel laminated part of the bearing is designed to carry the weight of the structure and provide post-yield elasticity. The lead core deforms plastically under shear deformations and its size can be chosen to provide the required amount of damping. This type of bearing has the advantage of being a simple structure and has been used widely throughout the world. [3]. A large number of quasi-static experiments have shown that the load-deformation behaviour of an LRB is in the form of a hysteresis curve [4,16] which can be represented by the bi-linear model shown in Fig. 2-2, where Q and Δ are horizontal force and displacement, respectively. In this Figure Qy represents the characteristic strength which is elastically related to the yield stress of the lead, K1 and K2 are the elastic and post-elastic shear stiffness, respectively. The small amount of research which has been undertaken on aqueducts [20,21], has utilised the added mass or spring-mass system approaches wherein the water is treated as part of the aqueduct. This paper concerns numerical study of the aseismic behaviour of a large aqueduct structure to assess the feasibility and effectiveness of using LRBs within such a structure.

Fig. 2-2. LRB Load-deformation model.

3. Fluid-structure dynamic interaction A coupled fluid-solid dynamic model, which considers the water container as an elastic body, was used to simulate the behaviour of the water in the aqueduct. This fluid-structure system must satisfy the controlling equations and the general boundary conditions in the fluid and structural fields, it must also satisfy the constraint conditions at the fluid and structure interface. The generalised variable energy function for the coupled fluid-structure dynamic system has been given by Liu et al. [22] as:

Π = Π0 + QC + QL

(3-1)

where, Π0 is the generalised variable energy function of the fluid-solid system; QC and QL represent the internal work done during non-slip and slip conditions, respectively. The specific form of Π0 is: Π0 = Π0f + Π0s −

∬V

(ρFif ui̇ + p f

∂ui̇ ∂u ̇ ∂u ̇ − μ i ⋅ i )dV − Tif ui̇ dΓ − ∂xj ∂xj ∂xi Γfσ

∫

∬V Fisui̇ dV − ∫Γ s

Tisui̇ dΓ

sσ

(3-2) wherein,

δΠ0f =

δΠ0s =

∬V

ρ( f

∂ui̇ ∂u ̇ + uk̇ i )δui̇ dV ∂xk ∂t

(3-3)

∬V (ρs uï δui̇ + σijδεij̇ )dV

(3-4)

s

In this paper, the superscript and subscript f and s represent the fluid domain and solid domain, respectively.

ρ and ρs represent the fluid and solid densities, respectively. u̇ and ü represent the system velocity and acceleration, respectively. Tif and Tis represent the known surface force at the fluid and solid interface, respectively. Fif and Fis represent the fluid unit mass volume vector and the solid volume force vector, respectively. ε̇ij represents the strain rate tensor for small deformations of the 1 ∂u ̇

∂uj̇

j

∂xi

solid and is equal to 2 ( ∂xi +

).

If the non-slip region of the internal contact boundary is taken as Γfsc and the slip contact region is denoted by Γfsl , then,

QC =

∫Γ

πi(ui̇ f − ui̇ s )dΓ

∫Γ

πn(uṅ f − uṅ s )dΓ +

(i = n , τ )

(3-5)

fsc

QL =

fsl

∫Γ

fsl

g(λτ − ast )dΓ +

∫Γ

λτ (uτ̇ f − uτ̇ s )dΓ

fsl

(3-6) where πi, πn and g are Lagrangian multipliers which import constraint conditions. The final term in Eq. (3-6) is the work generated by the internal contact force along the slip direction in unit time.

Fig. 2-1. Lead rubber bearing (LRB).

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Y. Liu et al.

of fluid; [Cij ] is the contact constraint matrix of the coupled fluidstructure system, which can be obtained from Eqs. (3-5) and (3-6). It is then possible to introduce the equivalent internal force of dotpairs on the contact surface when represented as a collection of nodes according to the finite element method. Variations of u̇ and the Lagrange multiplier πi, πn, g then show that, for the non-slip and slip states [26]:

4. Finite element representation of fluid-structure dynamic interaction The finite element method has been used to solve the coupled fluidsolid problem of an aqueduct containing water being subjected to seismic excitation [23]. To do this it was necessary to choose an interpolation function similar to that which expresses the field variables at any position inside an element, this interpolation function has to provide accuracy and consistency inside an element. In the coupled fluid-solid dynamic system fluid momentum transfer is expressed by a first order equation, because it relates to fluid pressure, but viscous momentum transfer is represented by a second order equation because of strain rate dependency. Hence the pressure and velocity require different interpolation functions. In addition, in order to achieve compatibility of deformation at the interface between the fluid and solid domains, the interpolation function for velocity and displacement within the solid domain has been assigned the same second-order interpolation function as the fluid velocity. Therefore the interpolation functions for displacement, velocity, pressure of any position inside an element can be expressed as: [24]

(4-4) where:

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ [T ] = ⎢ cos α sin α ⎥ , [T ]* = ⎢ cos α sin α ⎥ , [R] = ⎢ 0 0 ⎥ 0 ⎦ ⎣− sin α cos α ⎦ ⎣ 0 ⎣0 1⎦ [T] is the co-ordinate transformation matrix, α is the angle between the normal direction (the n-direction) of the local coordinates and the positive direction of the X-axis in the global coordinates. 5. Solution of the dynamic finite element equation

8

ui(x, y, t ) =

∑ Nα(x, y)uai(t )

(4-1a)

α =1

The coefficients in Eq. (4-3) are variable and nonlinear so this equation needs to be solved by an incremental method. In this paper, the Wilson-θ algorithm was used in a step-by-step integral. When the Wilson-θ algorithm is applied to the coupled fluid-structure system, it is still supposed that the acceleration is linear during the period [t, t +θ△t]. Velocity instead of displacement, the pressure and the internal contact force need to be taken as the unknown variables, the commonly adopted Wilson-θ algorithm need to be modified. By applying symmetric treatment to Eq. (4-3), as proposed by Liu [26], and using a time-increment method, the following incremental equation is obtained:

8

ui̇ (x, y, t ) =

∑ Nα(x, y)uai̇ (t )

(4-1b)

α =1 4

P (s , y , t ) =

∑ NβP(x, y)Pβ (t )

(4-1c)

β =1

where, α and β are the number of nodes inside an element, Nα (x, y ) are the shape functions for displacement and velocity; Nβp(x, y ) are the shape functions for pressure. Using Eq. (4-1) the fluid and solid domains are converted into a unified discrete domain and the generalised variable function (Eq. (31)) for this discrete domain becomes: Nf

Π=

Ns

Mc

i =1

i =1

⎡ Aαβ ⎢ ⎢0 ⎢0 ⎢⎣ 0

ML

∑ Π f(e)+ ∑ Πs(e)+ ∑ QC(e)+ ∑ QL(e) i =1

i =1

(4-2)

⎡0 ⎢ +⎢ 0 ⎢0 ⎣0

where N and N represent the number of elements in the fluid and solid domains, respectively. M c and M L represent the number of nonslip elements and slip elements at the fluid-solid interface, respectively. By combining Eqs. (4-1) and (4-2), the dynamic finite element equation can be obtained through variation of u̇i f and u̇is in Eq. (4-2). The constraint equation for the coupled fluid-solid system can be obtained through variation of the Lagrange multipliers π and g in Eqs. (3-5) and (3-6). When the dynamic and constraint equations are combined, the finite element equation for the whole coupled fluidsolid system is obtained as: f

⎡ Aαβ ⎢ ⎢ 0 ⎢ 0 ⎢⎣ 0

s

0 K 0 0

0 0 0 0

0 ⎤⎧ u f ⎫ ⎧ Ff ⎫ ⎪ ⎪F ⎪ u⎪ ⎪ 0 ⎥⎪ ⎥⎨ s ⎬ = ⎨ s ⎬ 0 ⎥⎪ 0 ⎪ ⎪ ατ ⎪ ⎪ 0 ⎦⎩ 0 ⎭ ⎪ ⎩H ⎭

⎡ D + C11 0 0 ⎤⎧ Δu f̈ ⎫ ⎢ αβ ⎥⎪ ⎪ Δu ̈ ⎪ ⎪ ⎢C21 M 0 0 ⎥⎨ s ⎬ + ⎢ 0 0 0 ⎥⎪ 0 ⎪ ⎢C31 ⎪ ⎪ ⎥ T 0 0 0 ⎦⎩ 0 ⎭ ⎢⎣Cαβ Δu f ⎫ ⎧ ΔGf ⎫ 0 0 0 ⎤⎧ ⎪ ⎪ ⎪ ⎪ K 0 0 ⎥⎪ Δu ⎪ ⎪ ΔG ⎪ ⎥⎨ s ⎬ = ⎨ s ⎬ 0 0 0 ⎥⎪ 0 ⎪ ⎪ ΔaI ⎪ ⎪ ⎪ 0 0 0 ⎦⎪ ⎩0 ⎭ ⎪ ⎩0 ⎭ 0

C12 C13 Cαβ ⎤⎧ Δu ̇ ⎫ ⎥⎪ f ⎪ C + C22 C23 0 ⎥⎪ ⎪ ⎨ Δuṡ ⎬ C32 C33 0 ⎥⎪ Δλ ⎪ ⎥⎪ ⎪ 0 0 0 ⎥⎦⎩ ΔP ⎭

(5-1)

where [Cij ] represents the contact constraint matrix (resulting from the symmetric treatment), {ΔGf } and {ΔGs} represent the load vectors after symmetric treatment [25]. According to the assumptions within the Wilson-θ method, there is a linear changes in the acceleration during a time step, so that the acceleration response at any time during the interval [t, t+θ△t] is given by [27]:

⎡ j C12 C13 Cαβ ⎤⎧ u ̇ ⎫ 0 0 0 ⎤⎧ u f̈ ⎫ ⎢ Bαβγ + Dαβ + C11 ⎥ f ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ u̇ ⎪ C21 C + C22 C23 0 ⎥⎪ M 0 0 ⎥⎨ us̈ ⎬ + ⎢⎢ ⎥⎨ s ⎬ 0 0 0 ⎥⎪ 0 ⎪ ⎢ C31 C23 C33 0 ⎥⎪ λ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 0 ⎥⎦⎩ 0 ⎭ ⎢ Gαβ 0 0 0 ⎥⎦⎩ P ⎭ ⎣ ⎡0 ⎢ + ⎢0 ⎢0 ⎣0

⎡ 0 0 [T ] ⎤ ⎥ ⎢ 0 −[T ]⎥ (For slip ) (For non − slip ), [Cij ] = ⎢ 0 ⎢⎣[T ]* −[T ]* [R] ⎥⎦

⎡ 0 0 [T ] ⎤ ⎢ ⎥ [Cij ] = ⎢ 0 0 −[T ]⎥ ⎣[T ] −[T ] 0 ⎦

Δuẗ + θΔt =

2Δuṫ + θΔt − 2uẗ θΔt 2

Δut + θΔt = θΔt⋅uṫ +

(5-2)

2

θ Δt θΔt uẗ + Δuṫ + θΔt 6 3

(5-3)

When Eqs. (5-2) and (5-3) are combined with the incremental Eq. (5-1), the equivalent static equation is obtained as;

⎡K ⎢ 11 ⎢ K21 ⎢K ⎢ 31 ⎢⎣ K41

(4-3)

where, [M], [C] and [K] are respectively the mass, damping and j stiffness matrices of the solid; [ Aαβ ], [Bαβγ ] and [Dαβ ] are the matrices of mass, convection motion and damping, [Cαβ ] and [Gαβ ] are pressure matrices for the fluid [25]; Ff and Fs are the vectors of the external force on the fluid and solid respectively; ατ is the vector of the internal force on the contact surface; H is the tensor of velocity on the boundary

K12 K13 K14 ⎤⎧ Δu ̇ ⎫ ⎧ ΔGf ⎫ f ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Δu ̇ ⎪ K22 K23 K24 ⎥⎪ s ⎬ = ⎨ ΔGs ⎬ ⎨ ⎥ K32 K33 K34 ⎥⎪ Δλ ⎪ ⎪ ΔaI ⎪ ⎪ ⎪ ⎪ ⎪ K42 K43 K44 ⎥⎦⎩ ΔP ⎭ ⎩ ΔH ⎭

(5-4)

In Eq. (5-4) the first term represents the equivalent stiffness matrix, and as shown by Liu [26]; 104

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Y. Liu et al.

⎡K ⎢ 11 ⎢ K21 ⎢K ⎢ 31 ⎣⎢ K41

K12 K22 K32 K42

K13 K23 K33 K43

⎡ α K14 ⎤ ⎢ θΔt Aαβ ⎥ K24 ⎥ ⎢ =⎢ K34 ⎥ ⎢ ⎥ K44 ⎦⎥ ⎢⎢ ⎣

+ Dαβ + C11

C12 Cs +

C21

2 M θΔt s

+ C22 +

C31

C32

T Cαβ

0

θΔt K 3 s

C13 Cαβ ⎤ ⎥ C23 0 ⎥ ⎥ C33 0 ⎥ ⎥ 0 0 ⎥⎦

Table 6-1 LRB physical parameters. Bearing

Elastic stiffness (K1)

Post-elastic stiffness (K2 )

Characteristic shear strength (Qy )

type

kN/m

kN/m

kN

S40 H40 S60

45680 36800 75840

7200 5200 7200

150 155 297

(5-5) j {ΔGf } = {ΔGf + 2uẗ f Aαβ − Bαβγ uαj̇ f Δu ḟ },

(5-6a)

⎧ θ 2Δt 2 s ⎫ uẗ )Ks⎬ {ΔGs} = ⎨ΔGs + 2uẗ sMs − (θΔt⋅uṫ s + 6 ⎩ ⎭

(5-6b)

5. Width of the pier at top and bottom is 4.5 m and 5.5 m, respectively.

The incremental value of the specific time t+θ△t can be obtained by solution of Eq. (5-4), and the acceleration, velocity, displacement, fluid pressure and internal contact forces are subsequently obtained using the following formulae:

Δuẗ + Δt =

Δuẗ + θΔt Δuṫ + θΔt 2 = − uẗ θ θ θ 2Δt

The depth H of water in the aqueduct was 3.0 m. the width D of the aqueduct trough was taken as 4.5 m. The density of the water is 1000 kg/m3, and its coefficient of kinematic viscosity is 1.435×10−3 Pa· s. The elastic modulus, density and Poisson's ratio of the material of the aqueduct was 2.1×1010 N/m2, 2.5×103 kg/m3 and 0.167, respectively. The LRBs were arranged in two parallel lines as shown in Fig. 6-1. According to the total weight of the trough and water there were eight LRBs set in each line. Three different types of LRB seismic damper were studied, the physical parameters of each bearing type are shown in Table 6-1. To test the aqueduct structure it was subjected to both El-Centro (NS) and Taft (EW) earthquake excitation in the transverse direction. In order to observe the effect of shock absorption under small and medium earthquake, the peak acceleration of each seismic wave was amplitude modulation to 1.0 m/s2 and 2.0 m/s2, respectively.

(5-7)

Δuṫ + Δt = uẗ Δt +

Δt (uẗ + Δt − uṫ ) 2

(5-8)

Δut + Δt = uṫ Δt +

Δt 2 Δt 2 uẗ Δt + (uẗ + Δt − uẗ ) 2 6

(5-9)

ΔP = ΔPt + θΔt / θ

(5-10)

Δλ = Δλt + θΔt / θ

(5-11)

Because Eq. (5-4) has an implicit form, it has to be solved by iteration.

6.2. Results

6. Numerical analysis

Tables 6-2, 6-3 show the maximum shear force and bending moment in the bottom of the trough wall and at the base of the pier (since these are the key parts of the aqueduct) when subjected to ElCentro (NS) and Taft (EW) seismic excitation, respectively. It is readily apparent that for each LRB type and category of seismic loading there was some degree of shock absorption and thus it is technically feasible to create an earthquake-resistant aqueduct using LRBs. Furthermore, the relative magnitudes of the values in Tables 6-2, 6-3 show that different LRBs exhibited different degrees of efficiency in damping seismic excitation, particularly in the case of a medium-sized earthquake. Figs. 6-2 to 6-4 showing the bending moments at the bottom of the trough side wall and at the base of the aqueduct, which set the S60 bearings and subjected El-Centro (NS) earthquake excitation. These Figures show that the incorporation of the S60 bearing reduces the seismic response of the aqueduct by varying degrees and the LRB certainly provided the system with enhanced shock absorption capability. Fig. 6-3 shows that the presence of the bearing produced very little reduction of the maximum bending moment in a pier when it was subjected to a small earthquake. In fact, the bearing had only a very small effect on any part of the bending moment-time profile over the whole duration of the seismic excitation (Fig. 6-3). On the other hand the incorporation of the bearing produced very significant reduction in the maximum bending moment in the pier when it was subjected to a medium earthquake (Fig. 6-4). From Figs. 6-2 and 6-4 it is clear that the isolation bearing was most effective in reducing the very large bending moments generated in the aqueduct without bearings in the time period immediately after the start of earthquake excitation, i.e. in the first 10 s. The presence of the bearing had very little effect on the magnitude of the much smaller bending moments induced subsequently (this is in keeping with the trend shown in Fig. 6-3). It is believed that this difference in performance of the LRB was due to the extent to which its stressstrain behaviour was mobilized. Under major seismic excitation the bearing undergoes large deformation and there is significant mobilisa-

6.1. System and test parameters To investigate the seismic response characteristics of an aqueduct containing bilinear LRBs, the model shown in Fig. 6-1 was analysed. It was treated as a two-dimensional structure with the following dimensions: 1. Aqueduct length - 30 m 2. Thickness of the trough side wall and base - 0.20 m and 0.50 m, respectively 3. Height of the aqueduct pier (Hd) - 20.0 m 4. Thickness of the pier - 2.5 m

D/2

D/2

y H/2 H/2

A

h(x,t)

x

Hd

Fig. 6-1. Calculation model of aseismic aqueduct.

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Table 6-2 Maximum shear force and bending moment under El-Centro excitation. Peak acceleration (m/s2)

LRB type

Shear force of wall side (kN)

Bending Moment of wall side (kN m)

Shear force of pier (kN)

Bending Moment of pier (kN m)

2.0

None S40 Bearing S60 Bearing H40 Bearing

24.76 7.38 10.45 6.911

45.09 16.54 20.37 15.12

1255.00 606.40 937.80 598.30

21.55 8.14 12.30 7.92

1.0

None S40 Bearing S60 Bearing H40 Bearing

12.26 4.84 7.71 4.72

22.34 9.96 14.39 9.02

621.7 385.80 583.20 366.4

10.67 5.27 8.44 5.22

Table 6-3 Maximum shear force and bending moment under Taft excitation. Peak acceleration (m/s2)

LRB type

Shear force of wall side (kN)

Bending moment of wall side (kN m)

Shear force of pier (kN)

Bending moment of pier (kN m)

2.0

None S40 Bearing S60 Bearing H40 Bearing

26.73 8.26 11.33 8.14

47.03 17.58 22.01 17.40

1236.75 580.30 711.50 539.30

22.28 7.44 10.06 7.31

1.0

None S40 Bearing S60 Bearing H40 Bearing

13.20 5.42 7.92 5.37

23.22 10.14 14.85 9.79

613.5 351.70 493.10 328.50

10.97 4.82 7.82 4.74

Bending moment (N·m)

50 䇮 ᭟ ᓗ S60㖞bearing ᑨ 䇮 䇑 No 㿴 bearing

25

0

-2 5

-5 0

0

5

10

15

20

Time (s) Fig. 6-2. Bending moment at bottom of trough wall (El-Centro excitation, medium).

Bending moment (MN·m)

1 3 .0

S60 bearing No bearing

6 .5

0 .0

-6 .5

-1 3 .0

0

5

10

15

20

Time (s) Fig. 6-3. Bending moment at bottom of pier (El-Centro excitation, small).

reduction of the maximum internal contact force induced by an earthquake. The same behaviour was observed for the shear forces and bending moments in the trough wall as shown in Fig. 6-2 and Tables 6-2, 6-3. The presence of the bearing significantly reduced the effect of seismic excitation on every internal parameter. The bearing reduces the rigidity of the connection between the trough and the pier so that seismic loads have less effect on the water-containing trough of

tion of the system's damping capability, so that it reduces to the largest values of seismic response. Thus the bearing was more aseismic under larger earthquake than smaller earthquakes. The internal contact forces at point A on the trough wall, the displacement of the pier top and the hysteresis curve of the LRB (type S60) in the aqueduct are shown in Figs. 6-5 to 6-7. It is readily apparent that the incorporation of the bearing leads to a large 106

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Y. Liu et al.

Bending moment (MN·m)

2 5 .0

No bearing S60 bearing

1 2 .5

0 .0

-1 2 .5

-2 5 .0

0

5

1 0

1 5

2 0

Time (s)

Contact internal force (kN)

Fig. 6-4. Bending moment at bottom of pier (El-Centro excitation, medium).

-8 0

S60 bearing No bearing

-1 0 0 -1 2 0 -1 4 0 -1 6 0 -1 8 0

0

5

1 0

15

2 0

Time (s) Fig. 6-5. Internal contact forces at A point on the trough wall (El-Centro excerption, Medium).

Top pier displacement (cm)

1 .5

No bearing S60 bearing

1 .0 0 .5 0 .0 -0 .5 -1 .0 -1 .5

0

5

10

15

20

Time (s) Fig. 6-6. Displacement at the pier top (El-Centro excitation, medium).

Resistant Force (kN)

600

to this aspect at the design stage if it is intended to use LRBs. The displacement at the top of the pier and the shear forces and bending moments at the bottom of the pier were all reduced significantly when the bearing was installed in the aqueduct. The numerical analysis results shows that the mass of the aqueduct trough and the LRB bearing constitute a tuned mass damping (TMD) nonlinear system. The seismic response values of the pier were decreased by the interaction of the trough and the pier. In addition, the LRB produces large hysteresis damping (as shown in Fig. 6-7) by its virtue of deformation, the seismic response of the entire aqueduct system is reduced.

300

0

-300

-600

-2.0

-1.0

0.0

1.0

2.0

3.0

7. Conclusions

Deformation of Bearing (cm)

The general conclusion from the study is that it is has a good degree of aseismicity for aqueduct when using LRBs. The varying values of seismic response that occurred within the aqueduct showed that different bearings produce different vibration attenuation when subjected to seismic loads. With regard to bending moments and shear forces it was possible to obtain good aseismic efficiency and for each aqueduct structure there would be a bearing which gave optimal aseismic efficiency.

Fig. 6-7. Hysteresis curve of the S60 bearing (El-Centro excitation, medium).

the aqueduct, i.e. the bearing acted as a seismic isolator. However, this isolation effect makes the bearing has larger deformation (as shown in Fig. 6-7), the trough of the aqueduct had a large rigid body displacement. This large displacement places major demands on the water stops within the joints of the trough. Special attention needs to be paid

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Whilst the presence of an LRB caused large reduction of the peak values of bending moment, shear force and internal contact force induced by an earthquake it was not efficient in reducing small values of these parameters. This behaviour was due to the large deformation of the selected LRB which was required to achieve significant mobilisation of its damping features, so the bearing was particularly effective with large earthquakes and provides significant opportunities for seismic design. However, it must be noted that the presence of the bearing causes large rigid-body displacement of the aqueduct trough and particular attention must be paid to the water seals within the system. Acknowledgments This study was supported by the National Natural Science Foundation of China (NSFC 51179154). References [1] Anon . The summary report of project design across the Yellow River in the Southto-North Water Convey projects, 7. The Planning Design, Survey and Research Institute of Yellow River Conservancy Commission of the Ministry of Water Resources; 2001. [2] Xing Guibi. The discussion of selection of the structure of Diao River Aqueduct in the South-to-North Water Convey projects, hydraulic engineering and project application. HuaZhong University of Technology Press; 1996. [3] Lin J, Williams FW. An introduction to seismic isolation. Eng Struct 1995;17(3):233–4. [4] Skinner RI, Robinson WH, Mcverry GH. An introduction to seismic isolation. John Wiley & Sons Ltd; 1993. [5] Priestley , Nigel MJ, Seible Frieder, Michele Calvi Gian. Seismic design and retrofit of bridges. John Wiley & Sons, Ltd.; 1996. [6] Wang Bo, Li Qingbin. A beam segment element for dynamic analysis of large aqueducts. Finite Elem Anal Des 2003;39:1249–58. [7] Ri-Chen JI, Xiao-Feng SU, Yan J. Transverse seismic performance of beam aqueduct considering water mass influence. China Earthq Eng J 2013. [8] Housner GW. Dynamic pressure on accelerated fluid containers. Bull Seismol Am 1957;47(1):15–35.

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