Feasibility simulation of aseismic structure design for long-span bridges

2.1 Seismic isolation design schemes for long-span bridge structures

The traditional aseismic structural design scheme of long-span bridges is that No. 3 pier of the bridge adopts double thin-walled pier rigid joint to form T-type rigid frame system, and the other five piers adopt basin rubber bearings. This design results in low seismic resistance of long-span bridges [11]. In order to reduce the damage degree of long-span bridges under strong earthquake load and reduce the difficulty of pier design, five seismic isolation design schemes are proposed in this paper.

• Scheme 1: Double thin-walled pier rigid junction is adopted for the No. 3 pier of the bridge, and the remaining 5 piers are equipped with lock-up devices. One lock-up device is used on both sides of each thin-walled pier.

• Scheme 2: Double thin-walled pier rigid junction is adopted for the No. 3 pier of the bridge, while the other 5 piers adopt liquid viscous dampers. 1 liquid viscous damper is used on both sides of each thin-walled pier. Damping coefficient of liquid damper is c=1e6N/m⋅s2and each thin-walled pier is c = 2e6N/m⋅s2.

• Scheme 3: Double thin-walled pier rigid junction is adopted for the no. 3 pier of the bridge, while the other 5 piers adopt liquid viscous dampers. 1 liquid viscous damper is used on both sides of each thin-walled pier. Damping coefficient of liquid damper is c=2e6N/m⋅s2and each thin-walled pier is c = 4e6N/m⋅s2.

• Scheme 4: Double thin-walled pier rigid junction is adopted for the no. 3 pier of the bridge, lock-up device is installed for no. 4 pier, and liquid viscous damper is used for the other four piers. 1 liquid viscous damper is used on both sides of each thin-walled pier. Damping coefficient of liquid damper is c=1e6N/m⋅s2and each thin-walled pier is c=2e6N/m⋅s2.

• Scheme 5: Double thin-walled pier rigid junction is adopted for the no. 3 pier of the bridge, lock-up device is installed for no. 4 pier, and liquid viscous damper is used for the other four piers. 1 liquid viscous damper is used on both sides of each thin-walled pier. Damping coefficient of liquid damper is c=2e6N/m⋅s2and each thin-walled pier is c=4e6N/m⋅s2.

2.2 Numerical simulation method

In order to evaluate the merits and demerits of the five schemes, the response of long-span bridges under seismic loads is analyzed by numerical simulation and ANSYS software. According to the magnitude of earthquake response, we can judge the advantages and disadvantages of aseismic schemes.

2.3 Vibration analysis of long-span bridge structures

Finite element dynamic calculation model is used to analyze long-span bridge structures when liquid viscous dampers are used. Vibration equation of bridge structure (including foundation) in coordinate system is given by

where [M], [C], [K], and [R] are the mass matrix, damping matrix and stiffness matrix of the structure, {δ¨},{δ˙}and {δ} are the acceleration, velocity and displacement vector of the structure,{δ¨g}is the seismic acceleration, F_x and F_y are the bearing pressure of bridge structure on x axis and y axis.

Relationship between motion and damping force of liquid viscous damper is given by

where [C_d] is the damping stiffness matrix of liquid damper, {δ˙a}Deis the relative velocity between two ends of damper, α is the velocity index, M is the amplitude of damping force. Using 0.3-1.0 as the design parameter, the damper function is taken into account in the structural vibration. Then

When the lock-up device and liquid viscous damper are used in the long-span bridge structure, the lock-up device can be regarded as a simple speed switch in analyzing the finite element structural dynamic calculation model. When the lock-up device is used on the bridge structure, one end is located on the beam body and the other end is on the pier [12]. When the seismic load arrives, the lock-up device can constrain the horizontal displacement between the piers. The schematic diagram of the lock-up device is shown in Figure 1.

Figure 1

Schematic diagram of lock-up device

In Figures 1a and 1b there are bridge superstructures, c is pier structure, the node 2 is beam node, the node 4 is pier top node, and lock-up device is installed between 2 and 4. The degrees of freedom of nodes 2 and 4 are coupled with vertical degrees of freedom, and they can be freely rotated. Before the unit c participates in system integration, the finite element equation in its own coordinate system can be expressed as

where KaKabKbaKbeis the finite element coordinates, δ_a is the degree of freedom to be retained, δ_b is the degree of freedom to be condensed. P_a and P_b are the influence coefficient of the nodes a and b. According to the second equation in Eq. (4)

Substitute Eq. (5) into the first equation in Eq. (4)the unit equation after condensation is obtained as K∗δa=Pa∗,where K∗=Ka−KabKb−1Kba,Pa∗=Pa−KabKb−1Pb.The element stiffness matrix considering the effect of lockup is adopted to integrate K^* into system integration.

2.4 Numerical simulation of long-span bridge structure

The response of long-span bridges under seismic loads under five kinds of seismic isolation schemes is analyzed by establishing calculation model and improved layered Winkler spring model.

The long-span bridge group pile foundation can be regarded as a spatial rigid frame with rigid base plate. According to the actual force characteristics of the pier, the pile group foundation is usually simplified as a plane rigid frame along the symmetrical plane. The displacement (vertical displacement, horizontal displacement and rotation angle) of the origin of the cap coordinate is solved by the displacement method of structural mechanics, as shown in Eq. (6) The internal forces (bending moment, shear force and axial force) at the top of any pile are obtained.

where

where γbb,γaa,γββ,andγαβare the stiffness coefficient of bridge pile top, ₁ is the axial force at the top of the component when the displacement of the underside of the pile cap occurs along the axis of the member, ρ₂ is the transverse force at the top of the component when the lateral displacement of the underside of the cap is generated along the axis of the vertical component, ρ₃ is the bending moment at the top of the component when the lateral displacement of the underside of the cap is generated along the axis of the vertical component, or the lateral force at the top of the component when the unit angle is generated along the bending moment of the top surface of the pile cap, and ρ4is the bending moment at the top surface of the bottom of the pile cap when the unit angle is generated along the bending moment of the top part of the component. ρ1is given by

For bored friction piles, assume the lateral friction is evenly distributed, that is ξ = 0.5,the axial load on the top of a single pile is shared by the lateral friction and the soil resistance at the bottom of the pile. Therefore, the axial force from pile below the ground gradually decreases [13]. The value of ρ₁ is related to the side friction resistance and the resistance stiffness of the soil under the pile, and affects the translation displacement α and rotation angle β of the origin of the base coordinate of the pile cap, and then affects the force and deformation of the single pile group foundation. However, the layered Winkler spring model usually only focuses on the stiffness of horizontal soil spring on the side of pile, ignoring the effect of vertical stiffness ρ₁ on the top of piles. Therefore, an improved layered Kerr spring model is proposed in this paper.

The improved layered Winkler spring model indirectly considers the influence of the vertical friction and the resistance of the pile bottom soil through the equivalent area of the pile body. The section area of the pile body is adjusted according to the principle of equal axial stiffness of the pile body according to the size of ρ₁, but the moment of inertia and the length of the pile remain unchanged. The equivalent area of pile body is calculated by using Eq. (10)

Let

then

In Eq. (8) ~ Eq. (10)C₀ is the vertical foundation coefficient of pile bottom, C₀ = m₀L. If ,L = 10m. For friction bored pile, ξ is set to 0.5. A is the sectional area of pile body, A₁ is the equivalent section area of pile body, A₀ is the stress area of foundation at pile bottom. For frictional pile with scouring lines from the ground or part of the pile side, the area spread downward to the tip of the pile according to φ/4 (φ is the internal friction angle of the soil layer).

2.5 Seismic performance test of long-span bridge structure

The seismic performance of long-span bridge structures is tested by multi-point excitation motion equation and response power spectrum. In order to analyze seismic performance of discrete long-span bridges subjected to inhomogeneous ground motions, assume the structure has N bearings. In the absolute coordinate system as the center of the earth, the equation of structural motion can be written as the following block form.

where x_m is the forced ground displacement of support, is the displacement of all non-support nodes, f_m is the force acting on N supports by the ground. Using the discretization model obtained by lumped mass method, M_sm and M_ms are set to zero. The absolute displacement in Eq. (11) is decomposed into the sum of quasi-static displacement {y_s} and dynamic relative displacement y_r. Then

Assume the damping force of the structure is directly proportional to the dynamic relative displacement. According to Eq. (12) the equation of motion given by Eq. (11) is expanded to

which is the general expression of the motion equation of long-span bridge structure considering the ground motion of the space change.

For long-span bridges subjected to multi-point zero-mean normal stationary random earthquake excitation, the acceleration power spectrum matrix of ground motion has the following form

where

where Sx¨K(ω)is the ground acceleration power spectral density function at the site K. Assume the seismic wave reaches the reference point at t = 0, T_N is the time when the earthquake reaches N point, ρ_KL is the coherence coefficient of the point K and the point L, e−iωTis the aseismic function of structure. Eq. (15) ~ (17) respectively reflects the local site effect, traveling wave effect and partial coherence effect of spatial variation of ground motion of long-span bridges under earthquakes.

The power spectrum matrix Sx¨mx¨m(ω)is a nonnegative definite Hermitian matrix and the coherence coefficient matrix [R] is a nonnegative definite real symmetric matrix. If the rank of the [R] matrix is r(r ≤ N), the following decomposition can be performed.

where [Q] is the real N × r matrix. Substitute Eq. (18) into Eq. (14)

where

A virtual excitation acceleration vector is constructed by using every column vector {b_j}(j = 1, 2, . . , r) in [B].

The corresponding virtual displacement vector is given by

By substituting {x~¨mj}in Eq. (22) into Eq. (13) the dynamic relative displacement y~rjof long-span bridge structure under the j-th virtual excitation is obtained. The pseudo-static displacement {y~sj}under the j-th virtual excitation can be obtained by ignoring all the dynamic terms in the equation of motion Eq. (11) and using Eq. (22) According to Eq. (12)the absolute displacement {x~sj}under the j-th virtual excitation is obtained. Then, the response power spectrum matrix of long-span bridge structures under arbitrary coherent stationary excitation is obtained. The absolute response power spectrum matrix of long-span bridge structures under r independent excitations is given by

If all the elements in the coherence function matrix [R] of Eq. (17) are 1, the case is degenerated to one in which only traveling wave effect is considered. Further, if the time difference of each support movement is T_K = 0(k = 1, 2, . . , N), it will become a uniform ground motion condition.

2.6 Dynamic reliability analysis of bridge structures based on first exceeding failure criterion

The first transcendental failure criterion is widely applied in detecting the seismic behavior of long-span bridges. It is defined that: If the dynamic response of a long-span bridge exceeds the critical value or safety limit for the first time (such as stress at the control point, strain, displacement and elongation of the control layer, etc.),the structure is considered to be damaged [14]. In the dynamic reliability analysis of bridge structures based on the first transcendental failure criterion, there are two commonly used methods for the dynamic reliability analysis of bridge structures, according to the probability distribution of the bridge structural response and the number of crossing boundaries.

Davenport assumes that any two time difference events that respond to stochastic processes and boundaries are independent of each other. That is, the stochastic process in the time (0, τ] is bounded by the Poisson distribution. The dynamic reliability of symmetric bilateral D boundaries is given by

where is the standard deviation of stochastic process y(t), b is the symmetric bilateral threshold value, υ is the average zero crossing rate of stochastic process, and

where λ₀ and λ₂ are power spectrum spectral moments of the stochastic process,

Based on the assumption that the structural response and the number of bounded intersections occur in groups and obey the two-state Markov process, Vanmarcke proposed the formula for calculating the reliability of the bilateral D-bound of the symmetrical long-span bridge structure, which is given by

where q is the power spectrum bandwidth parameter

The above method of calculating reliability is only for single component under single failure mode. It is much more difficult to calculate the reliability of actual bridge structure, mainly because there may be more than one failure member, and the failure mode of the structure may be more than one form. The calculation of reliability of general structural system is to adopt an equivalent system to simulate the actual structure while considering all the corresponding failure modes [15, 16, 17, 18, 19, 20,21]. In aseismic engineering of long-span bridges, the bridge structure itself is a complex statically indeterminate structure system. There are many failure modes, and the failure modes may not be completely correlated or independent of each other. The lower structure of bridge (pier and tower) is an important subsystem of the system, and it is also the main resistance system of bridge against earthquake. Therefore, it can be used as the main research object of bridge dynamic reliability. Assume each pier (tower) has n dangerous sections (control cross section), and can be simulated by series system. The failure probability of structure is given by

where P_f,i is the failure probability of the component i. The failure probability is calculated by multiple integrals, and the joint probability distribution function of each random variable is difficult to determine. The upper and lower limit of the failure probability is usually calculated only, which is given by

The upper bound of failure probability P_f for a series system corresponds to the case where the failure of each component is independent of each other or there is no simultaneous failure component. The lower bound corresponds to the case where the failure of each component is completely correlated. The upper and lower limits of dynamic reliability P_s of long-span bridge structures can be further determined by Eq. (30)

3.1 3.1 Comparison of seismic response calculation results

Due to the time-lag effect of liquid viscous damper, there is a time lag between the appearance of the most disadvantageous stress state of rigid and non-rigid piers and the appearance of the most disadvantageous stress state when using lock-up device. In data processing, the forces acting on the lock-up device and the liquid viscous damper are shown in Table 1 ~ Table 3 according to the total forces acting on the device on each thin-walled pier.

3.2 Engineering example

The deck and bridge of a long-span suspension bridge are described by three-dimensional beam elements with rigid arms, while the main cable and suspension are described by one-dimensional cable elements. The total bridge consists of 769 nodes (29 ground nodes), 1010 units, and 2257 degrees of freedom. The damping ratio of each mode of bridge is set to 0.01. The first 200 order of the vibration mode are taken. The apparent wave velocity of longitudinal seismic P wave along the bridge deck is 3000 m/ s.

Series model simulation is applied to single bridge. The proposed algorithm is used to calculate the dynamic reliability of suspension bridge A. A total of 8 sections are calculated for each of the 4 sections at the bottom of each branch of the bridge. Three cases of fortification intensity of 7,8 and 9 are calculated respectively (actual site intensity is between 7 and 8 degrees). Table 4 shows the dynamic reliability of bridge structures with uniform ground excitation. Table 5 and Table 6 show the calculation results considering traveling wave effect and partial coherence effect respectively.

3.3 Analysis of factors affecting the equivalent area of pile in the improved layered Kerr spring model

In order to study the influence of pile length, pile diameter and vertical foundation coefficient of pile bottom soil on the equivalent area of pile body in the improved layered Winkler spring model for long-span bridges, parametric analysis is carried out in this paper. The length of pile L is set to 20 m, 40 m, and 60 m, respectively. The pile diameter D is set to 1.0 m, 1.25 m, and 1.5 m. The vertical foundation coefficient m0 of pile bottom soil is set to 1000 kN/m4, 2000 kN/m4, 4000 kN/m4 and 8000 kN/m4. The change of the equivalent area ratio η of pile section with the change of m0, pile diameter D and pile length L is as shown in Figure 2 up to Figure 4.

Figure 2

The change curve of D = 1.0 m

Figure 3

The change curve of D = 1.25 m

Figure 3

The change curve of D = 1.5 m

Figure 4

The change curve of L = 40 m

4.1 Discussion on seismic response calculation results

From Table 1 ~ Table 3, it can be seen that, the maximum horizontal thrust at the top of no. 3 rigid pier and at the top of six continuous piers in three schemes are: Scheme 1: 4.97E+06, 3.97E+07, Scheme 2: 8.04E+06, 2.31E+07, Scheme 3: 7.18E+06, 2.59E+07, Scheme 4: 6.27E+ 062, 0.50E+07, Scheme 5: 6.00E+06,2.78E+07.

Lock-up device makes it possible to transfer horizontal thrust between non-rigid pier and main beam, and significantly reduces the force of rigid pier and displacement of pier top, but it has no energy dissipation effect.

The use of liquid viscous damper makes it possible to transfer horizontal thrust between non-rigid pier and main beam, but the effect of reducing the force and displacement of rigid pier is less obvious than that of lock-up device. From the maximum thrust sum, the liquid damper has a significant energy dissipation effect, and has a significant effect on improving the overall force of long-span bridge structure.

Due to the time-lag effect of liquid viscous dampers, the most unfavorable stress state of rigid-jointed piers and non-rigid-jointed piers of long-span bridges appears, and there is a time-lag when the most unfavorable stress state appears in the use of the lock-up device.

The effect of reducing the most disadvantageous force of pier structure by increasing the damping value of damper is not obvious.

The combination of lock-up device and liquid viscous damper can give full play to the advantages of both structural protection devices.

Because of the use of lock-up device and liquid damper, long-span beam bridges can reduce the number of rigid joints between main girders and piers, thereby reducing the adverse effects of temperature effects on long-span bridge structures.

The lock-up device reduces the internal force on the rigid pier of the long-span bridge, and also increases the internal force of the non-rigid pier. But the internal force of rigid pier and non-rigid pier tends to be uniform, which is conducive to the residual bearing capacity of non-rigid pier.

Based on the above analysis, the 5 seismic design schemes for long-span bridges are feasible. The combination of lock-up device and liquid viscous damper in scheme 4 and scheme 5 can give full play to the advantages of the two structural protective devices and have the strongest seismic effect.

4.2 Discussion on engineering example

From Table 4, it can be seen that, with the increase of fortification intensity from 7 to 9,the calculated bridge structural reliability decreases from 0.9999 9978 to 0.8055786548. In Table 5, with the increase of fortification intensity, the reliability of the calculated bridge structure decreases correspondingly. Comparing tables 4 and 5, the reliability of bridge structure is greater than that of uniform ground excitation when considering traveling wave effect. From Table 6, it can be seen that the traveling wave effect of earthquake ground motion causes the bridge structural response to become larger or smaller. The influence of reliability index on bridge structure is sometimes large and sometimes small. Comparing the results of Poisson-based Davenport method and Markov-based Vanmarcke method in Table 5 and Table 6, the reliability results of the two methods are not much different. The Poisson hypothesis has a slightly smaller result. It is concluded that for the proposed study method in most practical cases of seismic engineering research, it is not important to consider the Markov hypothesis. In most cases, this hypothesis may lead to non-conservative results.

4.3 Analysis and discussion on influential factors of equivalent area of pile in improved layered Kerr spring model

From Figure 2 ~ Figure 4, it can be seen that equivalent area ratio η of pile body in bridge structure is related to pile diameter D, pile length L and vertical foundation coefficient at the bottom of pile mO. The value may be less than 1.0 or greater than 1.0.

Under the condition of unchanged mO(or L),the equivalent area ratio η increases with the increase of pile length L (or mO), but with the increase of L (or mO), the change η becomes gentle.

When the pile length is constant, the equivalent area ratio of pile decreases with the increase of pile diameter.

The larger the equivalent area ratio of pile body, the greater the influence on the response of long-span bridge structure. Only when the equivalent area ratio of pile body is close to 1.0, the layered Winkler spring model can be used for seismic analysis, that is, it is not necessary to modify the section area of pile body of bridge structure.