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Influence of earthquake input angle on seismic response of curved girder bridge
Accepted Manuscript Influence of earthquake input angle on seismic response of curved girder bridge Yongjun Ni, Jiayu Chen, Hualiang Teng, Hui Jiang PII:
S2095-7564(15)00051-3
DOI:
10.1016/j.jtte.2015.05.003
Reference:
JTTE 23
To appear in:
Journal of Traffic and Transportation Engineering (English Edition)
Please cite this article as: Ni, Y., Chen, J., Teng, H., Jiang, H., Influence of earthquake input angle on seismic response of curved girder bridge, Journal of Traffic and Transportation Engineering (English Edition) (2015), doi: 10.1016/j.jtte.2015.05.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Influence of earthquake input angle on
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seismic response of curved girder bridge
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Yongjun Nia,b,*, Jiayu Chenb,c, Hualiang Tenga, Hui Jiangb
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Department of Civil and Environmental Engineering, University of Nevada-Las Vegas, Las Vegas
a
89119, USA b
School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
c
Central Research Institute of Building and Construction Co., Ltd., Beijing 100088, China
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Abstract
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The maximum seismic response of curved bridge is significantly related to the input angle of
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designated earthquake. Owing to structure irregularities, bridge reactions result from the interaction
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between the moment and torsion forces. Based on the solving of the seismic response of structure
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excited by a one-way earthquake input, a uniform expression of the unfavorable angle of the
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earthquake input was derived, and the corresponding maximum response of structure was determined.
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Considering the orthotropic and skewed dual-directional earthquake input manners, the most
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unfavorable angles for the two cases were also derived, respectively. Furthermore, a series finite
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element models were built to analyze the multi-component seismic responses by examining an
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example of curved girder bridge considering the variation of curvature radius and the bearings
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arrangement. The seismic responses of the case bridges, were excited by earthquakes at different
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input angles, and were calculated and analyzed using a response spectrum method. The input angles
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of earthquake excitation were progressively increased. From the analysis and comparison based on
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the calculation results mentioned above, the most unfavorable angle of earthquake excitation
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corresponding to the maximum seismic response of the curved bridge could be determined. It was
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shown that the most unfavorable angles of earthquake input resulted from the different response
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combination methods were essentially coherent.
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Keywords:
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curved girder bridge; earthquake input; seismic response; unfavorable angle
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* Corresponding author. Tel.: +86 10 51687252, fax: +86 10 51683764. E-mail: [email protected] (Y. Ni).
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1 Introduction
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Curved bridges have been extensively applied to the construction of roads and railways. As an
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important variation of the girder bridge, the curved bridge is playing a significant role in bridge
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engineering. Since the 1970s, engineers have observed numerous devastating earthquakes globally
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that have severely impacted a large number of curved bridges. One of the most destructive cases
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known by engineering researchers was the San Fernando earthquake in 1971, which caused serious
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damage to a multi-span girder bridge spanning between two large grade-separated interchanges
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projects. Since then, the research on the seismic responses of curved bridges has been given
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increased attention in the field of bridge engineering. Tseng and Penzien (1975a, 1975b) published two
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articles on the analysis results of the nonlinear seismic response of continuous curved bridges under
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severe earthquake. Williams and Godden (1979) published their experimental results derived from
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shake table model of curved girder bridge that had collapsed in the San Fernando earthquake and the
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corresponding theoretical results of their finite element analysis. Kawashima and Penzien (1979)
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established the mechanical model of expansion joints by considering collision and yielding
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phenomena, and studied the influences of expansion joints on the seismic response of curved bridges.
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Wilson and Button (1982) discussed the stress direction of the structure response excited by the
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multi-directional earthquake input, but their results proved to be in the later documents applied only to
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the situation of single-degree-of-freedom (SDOF) structure as of one-way input. Li et al. (1984)
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developed a curved coordinate system to study the seismic response of the curved bridge. Yuan et al.
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(1996) analyzed the linear and nonlinear response of a 9-span continuous curved bridge considering
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the wave passage effect. Based on the analysis of two curved bridge newly constructed, Qin et al.
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(1996) discussed the seismic performance respecting on the yielding and deformation of piers and the
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sliding and collision of the expansion joints. Zhu et al. (2000, 2002) discussed the principle input angle
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of the irregular bridges based on the SRSS combination method, pointed out that the maximum
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response of irregular bridge could be obtained by using response spectrum analysis inputted along the
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arbitrary two directions in plane. The factors such as the curvature and the type of the connecting of the
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pier and the beam were also analyzed. Zhang et al. (1999) and Fan et al. (2003) determined the most
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unfavorable input direction based on the yielding surface function theory. Applying the fiber element in
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piers, the sliding elements to simulate the bearings and the contacting elements to simulate the
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collision of the adjacent upper structures, Nie et al. (2004) evaluated the seismic performance of a
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curved bridge. Based on the detailed comparison analysis of the multiple calculating methods on the
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curved bridge, Gao and Zhou (2005) verified that the CQC3 (complete quadratic combination 3)
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method was the proper method for obtaining the maximum seismic response under the
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multiple-direction earthquake inputs. Relying on the multiple shaking table array tests of small scale
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curved girder bridge, Saad et al. (2012) and Wieser et al. (2012) analyzed the influence of indices such
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as beam curvatures, seat types, foundation areas, and isolation measures on the curved girder bridges,
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it was shown that the seismic response of curved bridges was significantly affected by the parameters
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mentioned above.
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The newly issued Caltrans Seismic Design Criteria (2013) presents two methods to calculate the
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elastic earthquake response of curved bridges, namely that the maximum of the two cases is used for
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the bridge design; the response combination of the longitudinal direction and the transverse direction
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by CQC3 method. Whether the Caltrans method is suitable for the curved girder bridge is still a
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question worthy of verification because of the detailed construction difference in California, USA and
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China.
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Owing to the difference in determining the most unfavorable input direction of curved bridge by different
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methods, the unfavorable input angle of earthquake ground motion is much needed in order to
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evaluate the seismic behavior of curved bridge more coherently.
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2.1
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Suppose an x-y coordinate system is to be adopted by the structure, as shown in Fig. 1(a). The
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single-direction earthquake acceleration
Theory of computation
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Single-direction earthquake acceleration input along a random direction in plane
a&&1(t ) is inputted along a random direction (0°< α <180°) in the
x-y plane. (a)
(b)
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(c)
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Fig. 1 Earthquake acceleration input angle. (a) Single-direction input.
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(b) Orthotropic dual-direction inputs. (c) Skewed dual-direction inputs
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The dynamic equilibrium equation of the structure is given as Eq. (1)
Mv&&(t ) + Cv&(t ) + Kv(t ) = − M I x cos (α ) + I y sin (α ) a&&1(t )
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(1)
where M, C and K are the mass matrix, damping matrix and stiffness matrix of the system, respectively, &&1( t ) and Ix and Iy are the unit column vectors along the coordinates x and y, respectively, − MI x a
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− Μ I y a&&1(t ) are the earthquake inertia forces excited by the single-direction earthquake acceleration
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a&&1(t ) inputted along the x axis and y axis, respectively, α is the input angle anticlockwise from the x axis. Solving Eq. (1), we can get
vijα = γ αj Sd (Ti , ξi ) φij
(2)
where φij is the element of the vibration mode vector for the j-th mode of the i-th mass point of the system, Sd(Ti, ξi) is the displacement response spectrum calculated by Duhamel integral method for
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modal mass of the system excited by the single-direction earthquake along angle α. It can be then calculated using Eq. (3)
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the period Ti and the damping ratio of the structure ξi,
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Φj
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where
is the vibration mode matrix, and
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is the modal participation coefficient of the j-th
Φ jT M I x cos (α ) + I y sin (α ) γj = = γ xj cos (α ) + γ jy sin (α ) T Φ j MΦ j α
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γ αj
Φ jT
is its transport matrix,
γ xj
and
γ jy
(3) are the modal
participation coefficients of the j-th modal mass of the system along the global coordinate of x and y axes, respectively. Substituting the Eq. (3) into Eq. (2), we can get
vijα = γ jx cos (α ) + γ jy sin (α ) Sd (T j , ξ j ) φij = vijx cos (α ) + vijy sin (α ) (4) x y where vij and vij are the relative displacements for the j-th mode and the i-th mass point of the system under the single-direction earthquake acceleration respectively. Similarly, we can get
where
a&&1(t ) along the global coordinate of x and y axes,
Rijα = Rijx cos (α ) + Rijy sin (α )
(5)
Rijα is the earthquake response (such as the bending moment, shearing force, deformation etc.)
for the j-th mode and the i-th mass point under the single-direction earthquake along a random included
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angle α with x axis,
Rijx and Rijy are the responses for the j-th mode and the i-th mass point along the
global coordinate of x and y axes, respectively. When the natural vibration property of the structure is controlled by the predominant mode, then the earthquake response
Riα as the input angle α is
Riα = Rix cos (α ) + Riy sin (α )
R and R are the responses for the i-th mass point of the system under the single-direction
where
earthquake along x and y axes, respectively. From Eq. (6), taking
we can get the maximum value
Riαmax
R1y α = arctan x R1 α of Ri by
(R ) + (R ) x 2 i
Riαmax =
∑(R ) n
α 2
=
ij
j =1
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Rix =
where
∑(R ) n
x 2 ij
j =1
and
(R )
Riy =
x 2 i
(8)
cos2 α + ( Riy ) sin 2 α + ∑Rijx Rijy sin ( 2α ) n
2
(9)
j =1
∑(R ) n
y 2 ij
j =1
are the responses for the i-th mass point under the
single-direction earthquake along the global coordinate x and y axes, respectively, and combined using the SRSS method. Both sides of Eq. (9) can been squared, then its uniform formula can be described as
y 2 = A cos 2 α + B sin 2 α + Csin ( 2α )
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(7)
When the related coefficient of each mode of the structure is smaller, we can adopt the SRSS method to combine the response of each mode. Then, we can get
Riα =
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y 2 i
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(6)
y i
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x i
(10)
A = ( Rix ) 2 , B = ( Riy ) and C = ∑ Rijx Rijy . n
2
where
j =1
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The condition for y to reach its maximum
2 ymax is that the derivative of the right part of Eq. (10) is equal
to zero, we can get the uniform calculating formulas of the most unfavorable angle of the earthquake input and the maximum response Eqs. (11) and (12)
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2
1
2C
α = arctan 2 A− B A+ B A− B 2 ± +C 2 2 2
2 ymax =
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Taking
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n 2 Rijx Rijy ∑ 1 j =1 α = arctan 2 2 x 2 ( Ri ) − ( Riy ) α α we can get the maximum value Rimax of Ri by
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(11) (12)
(13)
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(R ) α
2
2
imax
=
2
2
4 ( R x )2 − ( R y )2 n i i x y ± + ∑Rij Rij 2 j =1 2
(14)
When the related coefficient of each mode of the structure cannot be neglected, we can adopt the CQC method to combine the response of each mode, and arrive at n
(R )
n
∑∑ρ jk Rijα Rikα =
Riα =
cos2 α + ( Riy ) sin 2 α + ∑∑ρ jk Rijx Riky sin ( 2α )
x 2 i
j =1 k =1
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( Rix ) + ( Riy )
n
2
n
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where ρjk is the related coefficient between the j-th mode and the k-th mode,
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R =
n
n
∑∑ρ
jk
x ij
x ik
R R
and
R = y i
j =1 k =1
n
n
∑∑ρ
jk
Rijy Riky are the responses for the i-th mass point of the
j =1 k =1
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x i
(15)
j =1 k =1
system under the single-direction earthquake along the global coordinate of x and y axes, respectively, and combined using the CQC method. Taking
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n 2∑ ρ jk Rijx Rijy 1 α = arctan j =1 2 2 2 ( Rix ) − ( Riy ) α α we can get the maximum value Rimax of Ri by
(R ) α
2
2
=
2
2
(16)
4 ( R x )2 − ( R y )2 n i i x y ± + ∑ ρ jk Rij Rij 2 j =1 2
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imax
( Rix ) + ( Riy )
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(17)
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2.2 Orthotropic dual direction earthquakes input along a random direction in plane
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As shown in Fig. 1(b), the orthotropic double-direction earthquake accelerations
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along a random direction α (0°< α <180°) in plane are perpendicular to each other.
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Supposing that the structure response is controlled by the predominant mode, we can adopt the SRSS
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method to combine the seismic responses of two directions, obtaining
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Riα =
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a&&1(t ) , a&&2(t ) , inputted
(R ) + (R ) 1α 2 i
2α 2 i
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2 2 2 2 (18) = ( Ri1 x ) + ( Ri2 y ) cos 2 α + ( Ri1 y ) + ( Ri2 x ) sin 2 α + ( Ri1 x Ri1 y − Ri2 x Ri2 y ) sin ( 2α ) α where Ri is the response of the i-th mass point of the system under the orthotropic double-direction
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earthquakes along the directions shown in Fig. 1(b), each of
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orthotropic double-direction earthquake accelerations
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1(b). Likewise, the meanings of
Ri1α , Ri1x and Ri1 y stands for one of the
a&&1(t ) inputted along the direction shown in Fig.
Ri2α , Ri2 x , and Ri2 y are similar to that of Ri1α , Ri1x and Ri1 y .
Supposing that the displacement response spectrum of dual-direction has the same conversion, and
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their relation is as Eq. (19)
S2 (Ti , ξi ) = λ S1 (Ti , ξi )
(19)
where λ is defined as the percentage coefficient to adjust the peak ground acceleration (PGA) of the designated earthquake along the different input direction, λ <1. Eq. (18) can be derived as Eq. (20)
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2 2 2 2 (20 Riα = ( Ri1 x ) + λ 2 ( Ri1 y ) cos2 α + ( Ri1 y ) + λ 2 ( Ri1 x ) sin 2 α + (1 − λ 2 ) Ri1 x Ri1 y sin ( 2α )
, we can get the maximum value Riαmax of Riα as Eq. (21) 2 2 1 x 1 y ( Ri ) − ( Ri )
1 2
2 Ri1 x Ri1 y
α = arctan
(R ) α
2
imax
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(R ) + (R ) = x 2 i
y 2 i
2
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Taking
( R x )2 − ( R y )2 4 i i + ( Rix Riy ) ± 2 2
(21)
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When the related coefficient of each mode of the structure is smaller, and Eq. (19) is satisfied, we can adopt the SRSS method to combine the seismic responses of each mode for two directions, in a similar process to reach
n n 2 2 2 2 Riα = ( Ri1 x ) + λ 2 ( Ri1 y ) cos 2 α + ( Ri1 y ) + λ 2 ( Ri1 x ) sin 2 α + (1 − λ 2 ) ∑∑Rij1 x Rik1 y sin(222 j =1 k =1 )
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(R )
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which has the same expression as Eq. (21). When the related coefficient of each mode of the structure cannot be neglected, we can adopt the CQC method to combine the seismic response of each mode and use the SRSS method to combine the two directions to get:
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(R ) + (R ) =
y 2 i
2
( R x )2 − ( R y )2 4 i i + ( Rix Riy ) ± 2 2
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x 2 i
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α
imax
2
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n n 1x 1 y 2∑∑ Rij Rij 1 j =1 k =1 , we can get the maximum value Riamax of Riα as Taking α = arctan 1x 2 1y 2 2 ( Ri ) − ( Ri )
n n 2 2 2 2 Riα = ( Ri1 x ) + λ 2 ( Ri1 y ) cos2 α + ( Ri1 y ) + λ 2 ( Ri1x ) sin 2 α + (1 − λ 2 ) ∑∑ρ jk Rij1 x Rik1 y sin ( 2α ) j =1 k =1
(23)
2∑ ρ jk Rijx Rijy 1 j =1 , we can get the maximum value R a Taking α = arctan x 2 y 2 2 ( Ri ) − ( Ri ) n
of
i
x a
8
i m
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R a as
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( Rix ) + ( Riy ) 2
2
( R x )2 − ( R y )2 4 i i + ( Rix Riy ) ± 2 2
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(R )
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which has the same expression as Eq. (21).
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2.3 Skewed dual-direction earthquakes inputted along a random direction in plane
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For some special mountainous sites, the earthquakes maybe inputted as the skewed angle as shown in Fig. 1(c), it could be realized and verified by the investigation of 2008 Wenchuan Earthquake in China &&1(t ) , a&&2(t ) , are inputted along a (Zhou, 2010). The skewed dual-direction earthquake accelerations a
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random direction α , β (0°< α, β <180°) in the same plane, respectively. Adopting the SRSS method to combine the seismic responses of the two directions, we can get
α
2
=
2
(R ) + (R )
Riαβ =
β 2
i
(24)
Adopting the percentage method to combine the two directions, we can get
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i
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imax
Riαβ = Riα + λ Riβ αβ
Ri
(25)
180
where
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earthquakes inputted along the direction as shown in Fig. 1(c),
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3.1 Introduction of case bridge
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We now examine a three-span continuous curved bridge (Fig. 2) with the span of 73 m +130 m +73 m. The piers of the bridge with piles foundation are single columns with rectangular sections (13 m×3 m), with pier heights varied as 9, 10, 11 and 12 m. The main beam is a single-box double-cell girder, and the heights of the top beam and mid-span beam are 8.0 and 3.5 m, respectively. The height of the beam changes along the span direction in accordance with a second-degree parabola. All bearings are pot rubber bearings, allocated as Fig. 3, in which, DX means the bearing can only be moved along longitudinal direction or transverse directions; SX indicates the bearing can be moved both along the longitudinal and transverse directions; GD implies that the bearing is fixed. The stiffness of seating is 6 taken as 12.5×10 N/m. The concrete strength for the box girder and the piers are C50 and C40, respectively. The bridge is located on soft site (classified as Site IV). Its design fortification intensity of earthquake is 8 (namely the peak ground acceleration of the horizontal design earthquake is 0.2g). Two types of bearing arrangement were considered as seen in Fig. 4. Fig. 4(a) means that the bearings are considered as the ideal sliding or fixed bearings. Fig. 4(b) indicates that all the bearings are considered as their actual stiffness. The soil-structure interaction was neglected. The modal damping ratio of 5% was adopted.
is the response of the i-th mass point of the system excited by the skewed dual-direction
Riα , Riβ are the responses for the i-th
mass point of the system under the skewed earthquakes as shown in Fig. 1(c). Using the similar procedures like Eqs. (10)-(12), the most unfavorable angle of the earthquake input α
α
Rimax of Ri , a similar expression can be obtained.
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α
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Numeric example of the curved bridge
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Fig. 2 Elevation of curved girder bridge
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Fig. 3 Plane allocation of curved girder bridge
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(a)
Fig. 4 Two types of bearing arrangement. (a) Idealized bearing arrangement of continuous bridge.
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(b) Actual bearing arrangement.
Modal analysis was carried out to choose the top 20 modes for the response spectrum analysis. The maximum of the response spectrum was determined as Eq. (26) according to Guidelines for Seismic Design of Highway Bridges (2008). S max = 2.25CiCsCd A (26) where Ci is the importance coefficient, 0.5 for E1 earthquake and 1.7 for E2 earthquake, Cs is the site coefficient, 1.0 for site IV, Cd is the damping adjustment coefficient, 1.0 for damping ratio of 5%, A is the peak ground acceleration of the horizontal design earthquake, 0.2g for the design fortification intensity of earthquake. In this paper, only E1 earthquake was considered.
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3.2 Single-direction earthquake inputted along any direction in plane
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The case curved bridges (curvature radius are varied as 75, 175, 275 m and infinity (namely a direct-line layout bridge)) were excited by a series of one-way earthquake spectra along a varying angle (the angle increment is 3°) of the x axis within the range of 0° to 180°. The global co ordinate of x and y axes were defined as the direction along the chord connection of the center abutments and its perpendicular direction shown. The seismic responses were combined using the CQC method. The base shearing forces for all piers along the longitudinal and transverse direction were calculated. For conciseness, only the responses for P5 were listed from Figs. 5-8. The symbols R75, R175, R275 and R representing the curvature radius of the bridge are 75, 175, 275 m and infinity, respectively. F2 and F3 are the base shearing forces of piers along the radial and tangential direction in kN, respectively. M2 and M3 indicates the bending moments of piers around the radial and tangential directions in kN·m, respectively.
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Fig. 5 Base shearing forces of P5 for bearing arrangement as Fig. 4(a). (a) Base shearing forces along radial direction. (b)
(a)
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Base shearing forces along tangential direction. (b)
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Fig. 6 Base shearing forces of P5 for bearing arrangement as Fig. 4(b). (a) Base shearing forces along radial direction. (b) Base shearing forces along tangential direction.
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(b)
Fig. 7 Bending moments of P5 for bearing arrangement as Fig. 4(a). (a) Bending moments around radial direction. (b)
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(a)
Bending moments around tangential direction. (a)
(b)
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Fig. 8 Bending moments of P5 for bearing arrangement as Fig. 4(b). (a) Bending moments around radial direction. (b)
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Bending moments around tangential direction.
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Under the condition of the bearing arranged as Fig. 4(a), it can be seen from Figs. 5-7 that the base shearing forces and bending moments are varied as a trigonometric function rule with the incremental input angle. The phase angles of the trigonometric function are gradually increased while their wave lengths decreased with the reduction of the curvature radius of the bridge. Most of the response parameters but F2 show the gradually reduced tendency with the reduction of the curvature radius of the bridge. It can be seen from Figs. 5-8 that all the responses in case of the bearings arranged as Fig. 4(b) are much less than those from Fig. 4(a). The amplitudes of the parameters F3 and M2 are greatly influenced by the bearings arrangement. For the parameters F2 and M3, not only their amplitudes are greatly influenced by the bearings arrangement, but also their varying rules with the curvature radius variation. Considering the influences of the curvature radius of the bridge and the bearings arrangement on the seismic response of the curved bridge, the general varying tendency of the pier responses is shown in Table 1, in which the symbols "↑", "↓" and "≈" are used to describe the progressive increase, the progressive decrease and almost no change, respectively. The symbols ①, ②, ③ and ④ designate the amplitude, phase angle, wave length and functional value of the triangular function, respectively. And the symbol "★" means the maximum of the response for the pier in the cases mentioned above. R designates the curvature radius, “(a)” and “(b)” mean the bearing arrangements as Figs. 4(a) and 4(b), respectively.
AC C
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Table 1 Summary of varying tendencies of pier responses for curved bridge
(a)→(b) P5
R↓ (a)→(b)
P6
R↓ (a)→(b)
P7
R↓ (a)→(b)
① ② ③ ④ ① ② ③ ④ ① ② ③ ④ ① ② ③ ④
F2 (a) ≈ ↑ ↓
F3 (b) ↓ ↓ ↓
(a) ≈ ↑ ≈
↑ ≈ ↑ ↓
≈ ≈ ↓
↓ ↑ ↓
(a) ≈ ↑ ≈
≈★ ↓ ↓
↑ ↓ ↓
↓ ↑ ↓
(a) ↓ ↑ ↓
↑ ↓ ↓
≈ ↓ ≈
↓ ↑ ↓
≈ ↑ ↓
≈
↓ ↓ ↓
↑
↓★ ↓ ↓ ↑
≈ ↓ ≈ ≈
≈ ≈ ↓
↓
↓★ ↓ ↓
≈ ↓ ≈
(b) ↓ ↓ ↓
↑
↓
↓★ ↓ ↓
≈ ↓ ≈
↑
(b) ≈ ↑ ≈
↓ ↑ ↓
↑
↓ ↓ ↓
M3
≈
↓
↑
↓ ↓ ↓
(b) ≈ ↑ ≈
≈
↓ ↓ ↓ ↓
M2
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R↓
Variable
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P4
Influential factor
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↓ ≈ ↓
↓ ↓ ↓ ↑
3.3 Dual-directional earthquakes inputted along two directions in plane
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For brevity, the curvature radius of the curved bridge was fixed to 500 m, and the bearings were arranged as Fig. 4(b). According to the calculating procedures above, considering the assumption of Eq. (19), the seismic responses under the dual-directional earthquakes were obtained using the SRSS and the percentage combination methods according to Eqs. (24) and (25), respectively. The orthogonal and skewed bi-direction earthquake inputs were included in the combination procedures. In this paper, only the base shearing forces for P4 are illustrated in Fig. 9. (b)
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(a)
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Fig. 9 Base shearing forces of P4 excited by bi-directional response spectra along radial direction based on bearing arrangement as Fig. 4(b). (a) SRSS combination method. (b) Percentage combination method.
It can be seen from Fig. 9 that the results from the two combination methods are significantly differed.
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The results from the SRSS combination method are much greatly than those from the percentage method. The rationality of the calculating results should be verified by the results from the time history analysis. But the varying tendency from the two methods are essentially same. The former response can provide a conservative result for the seismic design of a curved bridge. For further understand of the varying rule from the combination of the bi-directional responses, the results of Fig. 9(a) was changed to drawn as a three dimension chart as Fig. 10. In Fig. 10, the coordinate axis with X18, …, represents the input angle of the response spectrum; the horizontal axis indicates the intersection angle from the former axis. The vertical axis means the base shearing forces of P4 along the radial direction (in kN). The unfavorable angle for the designated response parameter under the condition of the dual-directional earthquake input can be easily determined.
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Fig. 10 Radial shearing forces of P4 excited by bi-directional response spectra
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(1) Under the reaction of single response spectrum, the seismic responses of the piers vary as a trigonometric tendency with respect to changes of the input angle, and the maximum response of an individual pier is far higher than the rest of the piers with respect to the reaction amount. (2) The maximum response of each pier comes out at different angles, even for the same pier the maximum of differency appears from different angle. The application of the theoretical formula reduces the workload of statistics to a large extent, and makes it is more convenient to evaluate the most unfavorable input angles. (3) The varying tendency results from the SRSS or percentage combination method tend to be similar. The results from the SRSS combination method can provide a conservative evaluation for the seismic design of curved bridge.
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Acknowledgments
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This research was supported by the National Natural Science Foundation of China (No. 51378050), China Scholarship Council (No. 201307095008). The authors also acknowledge the assiduous work for the check and verification of the English compilation completed by Henry Long Teng in University of California, Berkeley.
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Fan, L. C., Nie L. Y., Li, J. Z., 2003. Discussion on standard of critical angle of seismic wave in seismic
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Zhang, J. J., Lin, D. J., Hu, S. D., 1999. determine the most unfavorable input direction by seismic response spectrum method. World Earthquake Engineering, 15(4): 38-40.
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Zhu, D. S., Liu, S. Z.,Yu, L. S., 2002. Research on seismic response of curved girder
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S2095-7564(15)00051-3
DOI:
10.1016/j.jtte.2015.05.003
Reference:
JTTE 23
To appear in:
Journal of Traffic and Transportation Engineering (English Edition)
Please cite this article as: Ni, Y., Chen, J., Teng, H., Jiang, H., Influence of earthquake input angle on seismic response of curved girder bridge, Journal of Traffic and Transportation Engineering (English Edition) (2015), doi: 10.1016/j.jtte.2015.05.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Influence of earthquake input angle on
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seismic response of curved girder bridge
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Yongjun Nia,b,*, Jiayu Chenb,c, Hualiang Tenga, Hui Jiangb
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Department of Civil and Environmental Engineering, University of Nevada-Las Vegas, Las Vegas
a
89119, USA b
School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
c
Central Research Institute of Building and Construction Co., Ltd., Beijing 100088, China
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Abstract
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The maximum seismic response of curved bridge is significantly related to the input angle of
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designated earthquake. Owing to structure irregularities, bridge reactions result from the interaction
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between the moment and torsion forces. Based on the solving of the seismic response of structure
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excited by a one-way earthquake input, a uniform expression of the unfavorable angle of the
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earthquake input was derived, and the corresponding maximum response of structure was determined.
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Considering the orthotropic and skewed dual-directional earthquake input manners, the most
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unfavorable angles for the two cases were also derived, respectively. Furthermore, a series finite
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element models were built to analyze the multi-component seismic responses by examining an
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example of curved girder bridge considering the variation of curvature radius and the bearings
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arrangement. The seismic responses of the case bridges, were excited by earthquakes at different
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input angles, and were calculated and analyzed using a response spectrum method. The input angles
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of earthquake excitation were progressively increased. From the analysis and comparison based on
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the calculation results mentioned above, the most unfavorable angle of earthquake excitation
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corresponding to the maximum seismic response of the curved bridge could be determined. It was
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shown that the most unfavorable angles of earthquake input resulted from the different response
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combination methods were essentially coherent.
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Keywords:
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curved girder bridge; earthquake input; seismic response; unfavorable angle
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* Corresponding author. Tel.: +86 10 51687252, fax: +86 10 51683764. E-mail: [email protected] (Y. Ni).
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1 Introduction
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Curved bridges have been extensively applied to the construction of roads and railways. As an
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important variation of the girder bridge, the curved bridge is playing a significant role in bridge
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engineering. Since the 1970s, engineers have observed numerous devastating earthquakes globally
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that have severely impacted a large number of curved bridges. One of the most destructive cases
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known by engineering researchers was the San Fernando earthquake in 1971, which caused serious
41
damage to a multi-span girder bridge spanning between two large grade-separated interchanges
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projects. Since then, the research on the seismic responses of curved bridges has been given
43
increased attention in the field of bridge engineering. Tseng and Penzien (1975a, 1975b) published two
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articles on the analysis results of the nonlinear seismic response of continuous curved bridges under
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severe earthquake. Williams and Godden (1979) published their experimental results derived from
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shake table model of curved girder bridge that had collapsed in the San Fernando earthquake and the
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corresponding theoretical results of their finite element analysis. Kawashima and Penzien (1979)
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established the mechanical model of expansion joints by considering collision and yielding
49
phenomena, and studied the influences of expansion joints on the seismic response of curved bridges.
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Wilson and Button (1982) discussed the stress direction of the structure response excited by the
51
multi-directional earthquake input, but their results proved to be in the later documents applied only to
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the situation of single-degree-of-freedom (SDOF) structure as of one-way input. Li et al. (1984)
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developed a curved coordinate system to study the seismic response of the curved bridge. Yuan et al.
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(1996) analyzed the linear and nonlinear response of a 9-span continuous curved bridge considering
55
the wave passage effect. Based on the analysis of two curved bridge newly constructed, Qin et al.
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(1996) discussed the seismic performance respecting on the yielding and deformation of piers and the
57
sliding and collision of the expansion joints. Zhu et al. (2000, 2002) discussed the principle input angle
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of the irregular bridges based on the SRSS combination method, pointed out that the maximum
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response of irregular bridge could be obtained by using response spectrum analysis inputted along the
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arbitrary two directions in plane. The factors such as the curvature and the type of the connecting of the
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pier and the beam were also analyzed. Zhang et al. (1999) and Fan et al. (2003) determined the most
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unfavorable input direction based on the yielding surface function theory. Applying the fiber element in
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piers, the sliding elements to simulate the bearings and the contacting elements to simulate the
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collision of the adjacent upper structures, Nie et al. (2004) evaluated the seismic performance of a
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curved bridge. Based on the detailed comparison analysis of the multiple calculating methods on the
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curved bridge, Gao and Zhou (2005) verified that the CQC3 (complete quadratic combination 3)
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method was the proper method for obtaining the maximum seismic response under the
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multiple-direction earthquake inputs. Relying on the multiple shaking table array tests of small scale
69
curved girder bridge, Saad et al. (2012) and Wieser et al. (2012) analyzed the influence of indices such
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as beam curvatures, seat types, foundation areas, and isolation measures on the curved girder bridges,
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it was shown that the seismic response of curved bridges was significantly affected by the parameters
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mentioned above.
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The newly issued Caltrans Seismic Design Criteria (2013) presents two methods to calculate the
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elastic earthquake response of curved bridges, namely that the maximum of the two cases is used for
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the bridge design; the response combination of the longitudinal direction and the transverse direction
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by CQC3 method. Whether the Caltrans method is suitable for the curved girder bridge is still a
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question worthy of verification because of the detailed construction difference in California, USA and
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China.
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Owing to the difference in determining the most unfavorable input direction of curved bridge by different
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methods, the unfavorable input angle of earthquake ground motion is much needed in order to
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evaluate the seismic behavior of curved bridge more coherently.
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2
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2.1
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Suppose an x-y coordinate system is to be adopted by the structure, as shown in Fig. 1(a). The
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single-direction earthquake acceleration
Theory of computation
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Single-direction earthquake acceleration input along a random direction in plane
a&&1(t ) is inputted along a random direction (0°< α <180°) in the
x-y plane. (a)
(b)
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(c)
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Fig. 1 Earthquake acceleration input angle. (a) Single-direction input.
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(b) Orthotropic dual-direction inputs. (c) Skewed dual-direction inputs
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The dynamic equilibrium equation of the structure is given as Eq. (1)
Mv&&(t ) + Cv&(t ) + Kv(t ) = − M I x cos (α ) + I y sin (α ) a&&1(t )
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(1)
where M, C and K are the mass matrix, damping matrix and stiffness matrix of the system, respectively, &&1( t ) and Ix and Iy are the unit column vectors along the coordinates x and y, respectively, − MI x a
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− Μ I y a&&1(t ) are the earthquake inertia forces excited by the single-direction earthquake acceleration
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a&&1(t ) inputted along the x axis and y axis, respectively, α is the input angle anticlockwise from the x axis. Solving Eq. (1), we can get
vijα = γ αj Sd (Ti , ξi ) φij
(2)
where φij is the element of the vibration mode vector for the j-th mode of the i-th mass point of the system, Sd(Ti, ξi) is the displacement response spectrum calculated by Duhamel integral method for
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modal mass of the system excited by the single-direction earthquake along angle α. It can be then calculated using Eq. (3)
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the period Ti and the damping ratio of the structure ξi,
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Φj
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where
is the vibration mode matrix, and
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is the modal participation coefficient of the j-th
Φ jT M I x cos (α ) + I y sin (α ) γj = = γ xj cos (α ) + γ jy sin (α ) T Φ j MΦ j α
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γ αj
Φ jT
is its transport matrix,
γ xj
and
γ jy
(3) are the modal
participation coefficients of the j-th modal mass of the system along the global coordinate of x and y axes, respectively. Substituting the Eq. (3) into Eq. (2), we can get
vijα = γ jx cos (α ) + γ jy sin (α ) Sd (T j , ξ j ) φij = vijx cos (α ) + vijy sin (α ) (4) x y where vij and vij are the relative displacements for the j-th mode and the i-th mass point of the system under the single-direction earthquake acceleration respectively. Similarly, we can get
where
a&&1(t ) along the global coordinate of x and y axes,
Rijα = Rijx cos (α ) + Rijy sin (α )
(5)
Rijα is the earthquake response (such as the bending moment, shearing force, deformation etc.)
for the j-th mode and the i-th mass point under the single-direction earthquake along a random included
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angle α with x axis,
Rijx and Rijy are the responses for the j-th mode and the i-th mass point along the
global coordinate of x and y axes, respectively. When the natural vibration property of the structure is controlled by the predominant mode, then the earthquake response
Riα as the input angle α is
Riα = Rix cos (α ) + Riy sin (α )
R and R are the responses for the i-th mass point of the system under the single-direction
where
earthquake along x and y axes, respectively. From Eq. (6), taking
we can get the maximum value
Riαmax
R1y α = arctan x R1 α of Ri by
(R ) + (R ) x 2 i
Riαmax =
∑(R ) n
α 2
=
ij
j =1
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Rix =
where
∑(R ) n
x 2 ij
j =1
and
(R )
Riy =
x 2 i
(8)
cos2 α + ( Riy ) sin 2 α + ∑Rijx Rijy sin ( 2α ) n
2
(9)
j =1
∑(R ) n
y 2 ij
j =1
are the responses for the i-th mass point under the
single-direction earthquake along the global coordinate x and y axes, respectively, and combined using the SRSS method. Both sides of Eq. (9) can been squared, then its uniform formula can be described as
y 2 = A cos 2 α + B sin 2 α + Csin ( 2α )
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(7)
When the related coefficient of each mode of the structure is smaller, we can adopt the SRSS method to combine the response of each mode. Then, we can get
Riα =
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y 2 i
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(6)
y i
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x i
(10)
A = ( Rix ) 2 , B = ( Riy ) and C = ∑ Rijx Rijy . n
2
where
j =1
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The condition for y to reach its maximum
2 ymax is that the derivative of the right part of Eq. (10) is equal
to zero, we can get the uniform calculating formulas of the most unfavorable angle of the earthquake input and the maximum response Eqs. (11) and (12)
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2
1
2C
α = arctan 2 A− B A+ B A− B 2 ± +C 2 2 2
2 ymax =
134
Taking
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n 2 Rijx Rijy ∑ 1 j =1 α = arctan 2 2 x 2 ( Ri ) − ( Riy ) α α we can get the maximum value Rimax of Ri by
6
(11) (12)
(13)
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(R ) α
2
2
imax
=
2
2
4 ( R x )2 − ( R y )2 n i i x y ± + ∑Rij Rij 2 j =1 2
(14)
When the related coefficient of each mode of the structure cannot be neglected, we can adopt the CQC method to combine the response of each mode, and arrive at n
(R )
n
∑∑ρ jk Rijα Rikα =
Riα =
cos2 α + ( Riy ) sin 2 α + ∑∑ρ jk Rijx Riky sin ( 2α )
x 2 i
j =1 k =1
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( Rix ) + ( Riy )
n
2
n
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where ρjk is the related coefficient between the j-th mode and the k-th mode,
139
R =
n
n
∑∑ρ
jk
x ij
x ik
R R
and
R = y i
j =1 k =1
n
n
∑∑ρ
jk
Rijy Riky are the responses for the i-th mass point of the
j =1 k =1
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x i
(15)
j =1 k =1
system under the single-direction earthquake along the global coordinate of x and y axes, respectively, and combined using the CQC method. Taking
142
n 2∑ ρ jk Rijx Rijy 1 α = arctan j =1 2 2 2 ( Rix ) − ( Riy ) α α we can get the maximum value Rimax of Ri by
(R ) α
2
2
=
2
2
(16)
4 ( R x )2 − ( R y )2 n i i x y ± + ∑ ρ jk Rij Rij 2 j =1 2
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imax
( Rix ) + ( Riy )
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(17)
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2.2 Orthotropic dual direction earthquakes input along a random direction in plane
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As shown in Fig. 1(b), the orthotropic double-direction earthquake accelerations
145
along a random direction α (0°< α <180°) in plane are perpendicular to each other.
146
Supposing that the structure response is controlled by the predominant mode, we can adopt the SRSS
147
method to combine the seismic responses of two directions, obtaining
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Riα =
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a&&1(t ) , a&&2(t ) , inputted
(R ) + (R ) 1α 2 i
2α 2 i
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2 2 2 2 (18) = ( Ri1 x ) + ( Ri2 y ) cos 2 α + ( Ri1 y ) + ( Ri2 x ) sin 2 α + ( Ri1 x Ri1 y − Ri2 x Ri2 y ) sin ( 2α ) α where Ri is the response of the i-th mass point of the system under the orthotropic double-direction
150
earthquakes along the directions shown in Fig. 1(b), each of
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orthotropic double-direction earthquake accelerations
152 153
1(b). Likewise, the meanings of
Ri1α , Ri1x and Ri1 y stands for one of the
a&&1(t ) inputted along the direction shown in Fig.
Ri2α , Ri2 x , and Ri2 y are similar to that of Ri1α , Ri1x and Ri1 y .
Supposing that the displacement response spectrum of dual-direction has the same conversion, and
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their relation is as Eq. (19)
S2 (Ti , ξi ) = λ S1 (Ti , ξi )
(19)
where λ is defined as the percentage coefficient to adjust the peak ground acceleration (PGA) of the designated earthquake along the different input direction, λ <1. Eq. (18) can be derived as Eq. (20)
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2 2 2 2 (20 Riα = ( Ri1 x ) + λ 2 ( Ri1 y ) cos2 α + ( Ri1 y ) + λ 2 ( Ri1 x ) sin 2 α + (1 − λ 2 ) Ri1 x Ri1 y sin ( 2α )
, we can get the maximum value Riαmax of Riα as Eq. (21) 2 2 1 x 1 y ( Ri ) − ( Ri )
1 2
2 Ri1 x Ri1 y
α = arctan
(R ) α
2
imax
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(R ) + (R ) = x 2 i
y 2 i
2
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Taking
( R x )2 − ( R y )2 4 i i + ( Rix Riy ) ± 2 2
(21)
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When the related coefficient of each mode of the structure is smaller, and Eq. (19) is satisfied, we can adopt the SRSS method to combine the seismic responses of each mode for two directions, in a similar process to reach
n n 2 2 2 2 Riα = ( Ri1 x ) + λ 2 ( Ri1 y ) cos 2 α + ( Ri1 y ) + λ 2 ( Ri1 x ) sin 2 α + (1 − λ 2 ) ∑∑Rij1 x Rik1 y sin(222 j =1 k =1 )
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(R )
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which has the same expression as Eq. (21). When the related coefficient of each mode of the structure cannot be neglected, we can adopt the CQC method to combine the seismic response of each mode and use the SRSS method to combine the two directions to get:
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(R ) + (R ) =
y 2 i
2
( R x )2 − ( R y )2 4 i i + ( Rix Riy ) ± 2 2
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x 2 i
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α
imax
2
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n n 1x 1 y 2∑∑ Rij Rij 1 j =1 k =1 , we can get the maximum value Riamax of Riα as Taking α = arctan 1x 2 1y 2 2 ( Ri ) − ( Ri )
n n 2 2 2 2 Riα = ( Ri1 x ) + λ 2 ( Ri1 y ) cos2 α + ( Ri1 y ) + λ 2 ( Ri1x ) sin 2 α + (1 − λ 2 ) ∑∑ρ jk Rij1 x Rik1 y sin ( 2α ) j =1 k =1
(23)
2∑ ρ jk Rijx Rijy 1 j =1 , we can get the maximum value R a Taking α = arctan x 2 y 2 2 ( Ri ) − ( Ri ) n
of
i
x a
8
i m
170
R a as
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( Rix ) + ( Riy ) 2
2
( R x )2 − ( R y )2 4 i i + ( Rix Riy ) ± 2 2
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(R )
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which has the same expression as Eq. (21).
173
2.3 Skewed dual-direction earthquakes inputted along a random direction in plane
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For some special mountainous sites, the earthquakes maybe inputted as the skewed angle as shown in Fig. 1(c), it could be realized and verified by the investigation of 2008 Wenchuan Earthquake in China &&1(t ) , a&&2(t ) , are inputted along a (Zhou, 2010). The skewed dual-direction earthquake accelerations a
177 178
random direction α , β (0°< α, β <180°) in the same plane, respectively. Adopting the SRSS method to combine the seismic responses of the two directions, we can get
α
2
=
2
(R ) + (R )
Riαβ =
β 2
i
(24)
Adopting the percentage method to combine the two directions, we can get
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i
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imax
Riαβ = Riα + λ Riβ αβ
Ri
(25)
180
where
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earthquakes inputted along the direction as shown in Fig. 1(c),
185
3
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3.1 Introduction of case bridge
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We now examine a three-span continuous curved bridge (Fig. 2) with the span of 73 m +130 m +73 m. The piers of the bridge with piles foundation are single columns with rectangular sections (13 m×3 m), with pier heights varied as 9, 10, 11 and 12 m. The main beam is a single-box double-cell girder, and the heights of the top beam and mid-span beam are 8.0 and 3.5 m, respectively. The height of the beam changes along the span direction in accordance with a second-degree parabola. All bearings are pot rubber bearings, allocated as Fig. 3, in which, DX means the bearing can only be moved along longitudinal direction or transverse directions; SX indicates the bearing can be moved both along the longitudinal and transverse directions; GD implies that the bearing is fixed. The stiffness of seating is 6 taken as 12.5×10 N/m. The concrete strength for the box girder and the piers are C50 and C40, respectively. The bridge is located on soft site (classified as Site IV). Its design fortification intensity of earthquake is 8 (namely the peak ground acceleration of the horizontal design earthquake is 0.2g). Two types of bearing arrangement were considered as seen in Fig. 4. Fig. 4(a) means that the bearings are considered as the ideal sliding or fixed bearings. Fig. 4(b) indicates that all the bearings are considered as their actual stiffness. The soil-structure interaction was neglected. The modal damping ratio of 5% was adopted.
is the response of the i-th mass point of the system excited by the skewed dual-direction
Riα , Riβ are the responses for the i-th
mass point of the system under the skewed earthquakes as shown in Fig. 1(c). Using the similar procedures like Eqs. (10)-(12), the most unfavorable angle of the earthquake input α
α
Rimax of Ri , a similar expression can be obtained.
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α
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Fig. 2 Elevation of curved girder bridge
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Fig. 3 Plane allocation of curved girder bridge
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Fig. 4 Two types of bearing arrangement. (a) Idealized bearing arrangement of continuous bridge.
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203 204
(b) Actual bearing arrangement.
Modal analysis was carried out to choose the top 20 modes for the response spectrum analysis. The maximum of the response spectrum was determined as Eq. (26) according to Guidelines for Seismic Design of Highway Bridges (2008). S max = 2.25CiCsCd A (26) where Ci is the importance coefficient, 0.5 for E1 earthquake and 1.7 for E2 earthquake, Cs is the site coefficient, 1.0 for site IV, Cd is the damping adjustment coefficient, 1.0 for damping ratio of 5%, A is the peak ground acceleration of the horizontal design earthquake, 0.2g for the design fortification intensity of earthquake. In this paper, only E1 earthquake was considered.
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3.2 Single-direction earthquake inputted along any direction in plane
221 222 223 224 225 226 227 228 229 230 231 232 233
The case curved bridges (curvature radius are varied as 75, 175, 275 m and infinity (namely a direct-line layout bridge)) were excited by a series of one-way earthquake spectra along a varying angle (the angle increment is 3°) of the x axis within the range of 0° to 180°. The global co ordinate of x and y axes were defined as the direction along the chord connection of the center abutments and its perpendicular direction shown. The seismic responses were combined using the CQC method. The base shearing forces for all piers along the longitudinal and transverse direction were calculated. For conciseness, only the responses for P5 were listed from Figs. 5-8. The symbols R75, R175, R275 and R representing the curvature radius of the bridge are 75, 175, 275 m and infinity, respectively. F2 and F3 are the base shearing forces of piers along the radial and tangential direction in kN, respectively. M2 and M3 indicates the bending moments of piers around the radial and tangential directions in kN·m, respectively.
SC
(b)
M AN U
(a)
Fig. 5 Base shearing forces of P5 for bearing arrangement as Fig. 4(a). (a) Base shearing forces along radial direction. (b)
(a)
TE D
Base shearing forces along tangential direction. (b)
239 240 241 242
AC C
EP
234 235 236 237 238
RI PT
220
Fig. 6 Base shearing forces of P5 for bearing arrangement as Fig. 4(b). (a) Base shearing forces along radial direction. (b) Base shearing forces along tangential direction.
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RI PT
(b)
Fig. 7 Bending moments of P5 for bearing arrangement as Fig. 4(a). (a) Bending moments around radial direction. (b)
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245 246 247 248 249
(a)
Bending moments around tangential direction. (a)
(b)
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Fig. 8 Bending moments of P5 for bearing arrangement as Fig. 4(b). (a) Bending moments around radial direction. (b)
252
Bending moments around tangential direction.
EP
Under the condition of the bearing arranged as Fig. 4(a), it can be seen from Figs. 5-7 that the base shearing forces and bending moments are varied as a trigonometric function rule with the incremental input angle. The phase angles of the trigonometric function are gradually increased while their wave lengths decreased with the reduction of the curvature radius of the bridge. Most of the response parameters but F2 show the gradually reduced tendency with the reduction of the curvature radius of the bridge. It can be seen from Figs. 5-8 that all the responses in case of the bearings arranged as Fig. 4(b) are much less than those from Fig. 4(a). The amplitudes of the parameters F3 and M2 are greatly influenced by the bearings arrangement. For the parameters F2 and M3, not only their amplitudes are greatly influenced by the bearings arrangement, but also their varying rules with the curvature radius variation. Considering the influences of the curvature radius of the bridge and the bearings arrangement on the seismic response of the curved bridge, the general varying tendency of the pier responses is shown in Table 1, in which the symbols "↑", "↓" and "≈" are used to describe the progressive increase, the progressive decrease and almost no change, respectively. The symbols ①, ②, ③ and ④ designate the amplitude, phase angle, wave length and functional value of the triangular function, respectively. And the symbol "★" means the maximum of the response for the pier in the cases mentioned above. R designates the curvature radius, “(a)” and “(b)” mean the bearing arrangements as Figs. 4(a) and 4(b), respectively.
AC C
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272
TE D
250 251
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273 274 275 276
Table 1 Summary of varying tendencies of pier responses for curved bridge
(a)→(b) P5
R↓ (a)→(b)
P6
R↓ (a)→(b)
P7
R↓ (a)→(b)
① ② ③ ④ ① ② ③ ④ ① ② ③ ④ ① ② ③ ④
F2 (a) ≈ ↑ ↓
F3 (b) ↓ ↓ ↓
(a) ≈ ↑ ≈
↑ ≈ ↑ ↓
≈ ≈ ↓
↓ ↑ ↓
(a) ≈ ↑ ≈
≈★ ↓ ↓
↑ ↓ ↓
↓ ↑ ↓
(a) ↓ ↑ ↓
↑ ↓ ↓
≈ ↓ ≈
↓ ↑ ↓
≈ ↑ ↓
≈
↓ ↓ ↓
↑
↓★ ↓ ↓ ↑
≈ ↓ ≈ ≈
≈ ≈ ↓
↓
↓★ ↓ ↓
≈ ↓ ≈
(b) ↓ ↓ ↓
↑
↓
↓★ ↓ ↓
≈ ↓ ≈
↑
(b) ≈ ↑ ≈
↓ ↑ ↓
↑
↓ ↓ ↓
M3
≈
↓
↑
↓ ↓ ↓
(b) ≈ ↑ ≈
≈
↓ ↓ ↓ ↓
M2
RI PT
R↓
Variable
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P4
Influential factor
M AN U
Pier no.
↓ ≈ ↓
↓ ↓ ↓ ↑
3.3 Dual-directional earthquakes inputted along two directions in plane
278 279 280 281 282 283 284 285
For brevity, the curvature radius of the curved bridge was fixed to 500 m, and the bearings were arranged as Fig. 4(b). According to the calculating procedures above, considering the assumption of Eq. (19), the seismic responses under the dual-directional earthquakes were obtained using the SRSS and the percentage combination methods according to Eqs. (24) and (25), respectively. The orthogonal and skewed bi-direction earthquake inputs were included in the combination procedures. In this paper, only the base shearing forces for P4 are illustrated in Fig. 9. (b)
AC C
(a)
EP
TE D
277
286 287 288 289 290 291
Fig. 9 Base shearing forces of P4 excited by bi-directional response spectra along radial direction based on bearing arrangement as Fig. 4(b). (a) SRSS combination method. (b) Percentage combination method.
It can be seen from Fig. 9 that the results from the two combination methods are significantly differed.
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The results from the SRSS combination method are much greatly than those from the percentage method. The rationality of the calculating results should be verified by the results from the time history analysis. But the varying tendency from the two methods are essentially same. The former response can provide a conservative result for the seismic design of a curved bridge. For further understand of the varying rule from the combination of the bi-directional responses, the results of Fig. 9(a) was changed to drawn as a three dimension chart as Fig. 10. In Fig. 10, the coordinate axis with X18, …, represents the input angle of the response spectrum; the horizontal axis indicates the intersection angle from the former axis. The vertical axis means the base shearing forces of P4 along the radial direction (in kN). The unfavorable angle for the designated response parameter under the condition of the dual-directional earthquake input can be easily determined.
302 303 304
Fig. 10 Radial shearing forces of P4 excited by bi-directional response spectra
M AN U
SC
RI PT
292 293 294 295 296 297 298 299 300 301
305
4
306 307 308 309 310 311 312 313 314 315
(1) Under the reaction of single response spectrum, the seismic responses of the piers vary as a trigonometric tendency with respect to changes of the input angle, and the maximum response of an individual pier is far higher than the rest of the piers with respect to the reaction amount. (2) The maximum response of each pier comes out at different angles, even for the same pier the maximum of differency appears from different angle. The application of the theoretical formula reduces the workload of statistics to a large extent, and makes it is more convenient to evaluate the most unfavorable input angles. (3) The varying tendency results from the SRSS or percentage combination method tend to be similar. The results from the SRSS combination method can provide a conservative evaluation for the seismic design of curved bridge.
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Acknowledgments
317 318 319 320 321
This research was supported by the National Natural Science Foundation of China (No. 51378050), China Scholarship Council (No. 201307095008). The authors also acknowledge the assiduous work for the check and verification of the English compilation completed by Henry Long Teng in University of California, Berkeley.
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