Dynamic performance of a concrete-filled steel tube high-pier curved continuous truss girder bridge due to moving vehicles

Abstract

The dynamic performance of a new type of concrete-filled steel tube high-pier curved continuous truss girder bridges under moving vehicles is studied combining field testing and numerical simulation method by an actual bridge example. The dynamic response data were obtained before opening to traffic for the bridge under moving testing vehicles. A three-dimensional vehicle and bridge coupled vibration analysis model for curved bridges is proposed and validated. The dynamic behavior characteristics and vehicular ride comfort for this new type bridge are studied considering variable road surface and other conditions. The results indicate that the dynamic impacts of this bridge from vehicles are underestimated compared with those defined in the design code. In addition, the dynamic impact factors vary significantly for the local components and some of them could exceed the design value. Finally, the riding comfort of this bridge is evaluated, which suggests a fairly uncomfortable condition.

Keywords

curved bridges dynamic analysis high piers loading test vehicle and bridge coupled vibration

Introduction

In recent years, the curved highway bridges with high piers are widely used to fit mountainous natural condition, such as in the southwest region of China. Considering the potential high earthquake intensity, large temperature difference, and strong wind environment in these areas, a new type of truss girder bridges with concrete-filled steel tubes (CFSTs) is proposed to be used together with high piers. The expected unique advantages, such as good geometric planar and vertical alignment adaptability, good seismic performances, light weight, low cost, easy construction, and short construction time, have made this new type of bridge obtain wide attentions. Finally, the Ganhaizi Bridge as the first one of this type of bridges, was constructed as a technology demonstration project in Sichuan Province of China, and was completed in April 2012 with 1811 m length and the maximal pier height of 110 m (shown as Figure 1).

Figure 1.

Layout of Ganghaizi Bridge: (a) elevation view of second unit of the bridge (units: cm), (b) cross section of the bridge at mid-span and vehicle loading position (units: cm), and (c) overview of the bridge.

As the literature shows, the CFST usually is used widely as arch ribs in arch bridges in the last several decades. However, CFST is rarely applied in girder bridges, especially for those with high piers. Compared with other types of arch bridges, such as steel, reinforced concrete, and so on, CFST arch bridges are found to be more sensitive for moving vehicles (Wu et al., 2008; Yoshimura et al., 2006). Similarly, high-pier girder bridges also show increased dynamic behavior compared with medium or short-pier girder bridges (Yin et al., 2011, 2013). In addition, due to the unique bend-twist coupling mechanical characteristics and driving centrifugal force action, curved bridges usually have more complex dynamic features than straight ones under moving vehicles (Wang et al., 1994). As for the aforementioned high-pier curved continuous truss girder bridge with CFST, which include all of the above features, strong dynamic effects from moving vehicles can be expected. As a result, it is necessary to carefully evaluate the dynamic effects to ensure the safety and reliability of the bridge, either experimentally, numerically, or by both. Mitigation measures, if necessary, should also be evaluated for their effectiveness.

In this article, the dynamic behavior characteristics of the high-pier curved continuous truss girder bridge with CFST is studied by combining the numerical analysis scheme and field tests. This article is organized as follows: after the introduction of the engineering background of the new bridge type, a three-dimensional (3D) finite element method (FEM) for the bridge is set up, followed by the establishment and verification of the vehicle and bridge coupled vibration (VBCV) model for curved bridge. Second, the VBCV analysis is performed to study the dynamic behavior characteristic of the bridge under moving vehicles. The global and local dynamic impacts and riding comfort are also discussed. Finally, some major findings are summarized with some brief discussions.

Description of the Ganhazi Bridge

As part of Yalu highway, the bridge deck of Ganhaizi Bridge is 24.5 m wide, carrying four vehicle lanes with a design vehicle speed of 80 km/h. The entire bridge includes three unit continuous girder bridges. The first bridge has 11 spans (40.7 m+9×44.5m+40.7 m=481m), the second bridge has 19 spans as shown in Figure 1(a) (45.1 m+3×44.5 m +11×62.5m+ 3×44.5m+45.1 m = 1044.7 m), and the third bridge has six spans (45.1 m+4×44.5m+45.1m= 268.2m).

As shown in Figure 1(b), the CFST space truss includes top chords, diagonal webs, lower chords, and lateral braces. The chords are made up of steel tubes filled with C50 micro-expansive concrete and hollow steel tubes are used for the web members. Over the CFST space truss, the bridge deck is built with pre-stressed concrete. Under the CFST space truss, most of the piers are CFST lattice piers, with a height ranging from 36 to 110 m. Some other short piers are reinforced concrete column piers (Nos 11, 12, and 13 piers) and are applied in side spans as shown in Figure 1(b). In this study, the second bridge is chosen for demonstration and the plane curvature radius for the bridge is 1130 m. For this unit of bridge, the longitudinal shifting bearing is installed between the 11th and 14th as well as 27th and 30th pier top and girder, and consolidations are set up between 15th and 26th piers top and girder.

VBCV numerical model for curved bridge

Compared with field test, numerical analysis on VBCV could be used to explore various loading scenarios and bridge conditions with a lower cost (Chen and Wu, 2009; Deng and Cai, 2010). In the past researches, numerical simulations on VBCV have been performed by many researchers (Han et al., 2018; Yau and Yang, 2008; Zhang and Cai, 2013). In this study, the modal superposition and coupled method is employed to establish the VBCV analysis model for curved bridge.

Vehicle model

In this study, the dynamic loading test for the bridge is performed with a standard three-axle truck (as shown in Figure 5(a)). The vehicular weight is 35,000 kg including front axle weight 7000 kg, center axle weight 14,000 kg, and rear axle weight 14,000 kg based on the measured results. The longitudinal axle distance for front and middle axle is 4.00 m, and the distance for middle and rear axle is 1.40 m. The horizontal wheel distance is 1.80 m. This type of truck is usually taken as a standard vehicle used in loading tests on highway bridges in China. The vehicle model corresponding to the actual used truck is set up and shown in Figure 2, and the detailed parameters are shown in Table 1. The dimensions, axle loads, and total weight of the vehicle were actually measured and can then be treated as reliable information. The values of suspension stiffness and damping might not be the same as the values for the actual trucks used in the test, which were obtained from references (Deng and Cai, 2010; Xu and Guo, 2003; Zhang and Cai, 2013) and generally can be accepted based on previous studies (Gim and Nikravesh, 1991; Livingston and Brown, 1969; Wang et al., 2016; Han et al., 2014; Yin et al., 2010).

Figure 2.

Vehicle model: (a) vehicle model and vehicle-bridge contact condition and (b) sketch of vehicle and bridge interaction.

Table 1.

Vehicular model parameters.

Mass	Truck body	31,980 kg
	First axle	1420 kg
	Second and third axle	800 kg
Momentof inertia	Pitching	123,704 kg m²
Momentof inertia	Rolling	63,524 kg m²
Springstiffness	Upper vertical, first axle	242.6 kN/m
	Upper vertical, second and third axle	951.6 kN/m
	Lower vertical, first axle	263.6 kN/m
	Lower vertical, second and third axle	533.3 kN/m
	Upper lateral, first axle	102.3 kN/m
	Upper lateral, second and third axle	501.5 kN/m
	Lower lateral	120 kN/m
Dampingcoefficient	Upper vertical, first axle	2.19 kN s/m
	Upper vertical, second and third axle	3.94 kN s/m
	Lower vertical, first axle	1.90 kN s/m
	Lower vertical, second and third axle	3.84 kN s/m
	Upper lateral, first axle	1.69 kN s/m
	Upper lateral, second and third axle	2.93 kN s/m
	Lower lateral	1.00 kN s/m
Length	L₁	3.76 m
	L₂	0.24 m
	L₃	1.64 m
	L₄	4.06 m
	h₁	1.00 m
	b₁	0.80 m

Bridge model

The FEM of the bridge is built by ANSYS as shown in Figure 3. The trusses, inclined chord and web members, as well as the piers are modeled using space beam elements (Beam 188). The concrete bridge deck is modeled using solid elements (Solid 65). The CFST components include chord members of the girder and main limbs of the pier for the bridge. The stiffness of CFST member is modeled by the equivalent stiffness theory and can be expressed as

E_{0} A_{0} = E_{s} A_{s} + E_{c} A_{c}

(1)

E_{0} I_{0} = E_{s} I_{s} + E_{c} I_{c}

(2)

where E, A, and I are the elasticity modulus, section area, and bending moment of inertia, respectively. The subscripts 0, s, and c represent the equivalent value, steel, and concrete.

Figure 3.

FEM of the bridge: (a) the overview of the finite element model, (b) bridge deck, and (c) truss girder.

In the bridge model, the consolidation connection between bridge deck and top chords of the girder is modeled by coupling corresponding nodal degrees. In addition, the fine and same size element dividing for the bridge deck and top chords also is used to guarantee their deformation consistency.

The boundary conditions of the model can be described as follows: the fixed constrains are applied on the bottom of all piers; the shifting bearings are simulated by releasing longitudinal coupled degree of freedoms between the 11th and 14th as well as 27th and 30th pier tops and corresponding girder; the rigid connection is modeled by coupling all the degree of freedoms between 15th and 26th pier tops and corresponding girder. The FE model will be used in the following VBCV numerical analysis.

Vehicle and bridge coupled analysis model

The dynamic equation of vehicle is expressed as

[M_{v}] {{\overset{\cdot\cdot}{X}}_{v}} + [C_{v}] {{\overset{\cdot}{X}}_{v}} + [K_{v}] {X_{v}} = {F_{v}}

(3)

where $[M_{v}]$ , $[K_{v}]$ ,and $[C_{v}]$ are vehicular mass, damping and stiffness matrix; ${X_{v}}$ is vehicular deflection vector, and ${F_{v}}$ is the vector of the wheel-road contact forces acting on the vehicle.

The dynamic equation of the bridge can be expressed as

[M_{b}] {{\overset{\cdot\cdot}{X}}_{b}} + [C_{b}] {{\overset{\cdot}{X}}_{b}} + [K_{b}] {X_{b}} = {F_{b}}

(4)

where $[M_{b}]$ is the mass matrix; $[C_{b}]$ is the damping matrix; $[K_{b}]$ is the stiffness matrix; ${X_{b}}$ is the node displacement vector for all degrees of freedom (DOFs) of the bridge; and ${F_{b}}$ the force vector acting on the bridge through the vehicle’s wheels.

The mode superposition method is employed to model the dynamic system of the bridge to cut down the calculation cost. The dynamic response ${X_{b}}$ of the bridge can be expressed as

\begin{matrix} {X_{b}} = [{Φ_{1}} \begin{matrix} {Φ_{2}} & \dots & {Φ_{N_{b}}} \end{matrix}] {\begin{matrix} q_{1} & q_{2} & \dots & q_{N_{b}} \end{matrix}}^{T} \\ = [Φ_{B}] {q_{B}} \end{matrix}

(5)

where $N_{b}$ is the involved mode number of the bridge; ${Φ_{i}}$ and $q_{i}$ are the ith mode of vibration and corresponding response. Equation (4) can be rewritten as

[M_{B}] {{\overset{\cdot\cdot}{q}}_{B}} + [C_{B}] {{\overset{\cdot}{q}}_{B}} + [K_{B}] {q_{B}} = {F_{B}}

(6)

where the subscript “B” is modal terms of the bridge. More details can be found in Li et al. (2016).

For the curved bridges, the conversion of coordinates is an important issue in the VBCV model. The relationship of global and local coordinate system is shown in Figure 4(a). The subscript “B” represents overall bridge coordinate. The Z axis of local coordinates is the same as overall coordinate, while the stiffness and damping of vehicle axles are projected to local coordinates considering the inclined angle of cross section $θ$ .

Figure 4.

Schematic diagram interaction for vehicle and curved bridge: (a) relationship between local and global coordinate and (b) the driving centrifugal force.

The left vehicle wheel of the ith axle is taken as an example to interpret the interaction between the vehicle and bridge (shown as in Figure 4(b)). The mass and rotation centers are the center of vehicle body mass and rolling moment caused by centrifugal force, respectively. Since the rolling moment will drive increasing loads at outer wheels and decreasing loads at inner wheels, the effect of rolling moment is reflected by the redistributed wheel load $Δ F_{zi}$ in this study. The interaction forces on the left wheel from the bridge can be written as

\begin{matrix} {\begin{matrix} F_{wyL}^{i} \\ F_{wzL}^{i} \end{matrix}} \\ = - [\begin{matrix} \cos (θ) k_{ylL}^{i} + \sin (θ) k_{vlL}^{i} \\ \cos (θ) k_{vlL}^{i} + \sin (θ) k_{ylL}^{i} \end{matrix}] \\ {\begin{matrix} Δ_{ylL}^{i} \\ Δ_{vlL}^{i} \end{matrix}} \\ - [\begin{matrix} \cos (θ) c_{ylL}^{i} + \sin (θ) c_{vlL}^{i} \\ \cos (θ) c_{vlL}^{i} + \sin (θ) c_{ylL}^{i} \end{matrix}] \\ {\begin{matrix} {\overset{\cdot}{Δ}}_{ylL}^{i} \\ {\overset{\cdot}{Δ}}_{vlL}^{i} \end{matrix}} + {\begin{matrix} 0 \\ Δ F_{zi} \end{matrix}} \end{matrix}

(7)

where $Δ F_{z i} = m_{v} (V^{2} / 2 R) (h_{0} R_{i} / \sum R_{i} D_{2}) \times \cos θ$ ; $vertical force$ , $F_{wyL}^{i} = horizontal force$ horizontal forces on the left wheel; $Δ_{vlL}^{i}$ and $Δ_{ylL}^{i}$ are vertical and horizontal relative displacement of the lower springs for the wheel in local coordinates. m_v is the overall weight of vehicle. $h_{0}$ is the distance from barycenter to rotation center, which is 1.0 m given by Rill (2007), R is the radius of the curvature, and $R_{i}$ is the rotation stiffness of ith axis, which equals to the stiffness of upper vertical spring times the distance D₂/2.

The compatible deflection matrix on the contact point of the tire and bridge deck can be written as

{\begin{matrix} Δ_{ylL}^{i} \\ Δ_{vlL}^{i} \end{matrix}} = {\begin{matrix} Y_{sL}^{i} \\ Z_{sL}^{i} \end{matrix}} - {\begin{matrix} Y_{bL}^{i} \\ Z_{bL}^{i} \end{matrix}} - {\begin{matrix} 0 \\ r_{Li} (x) \end{matrix}}

(8)

{\begin{matrix} {\overset{\cdot}{Δ}}_{ylL}^{i} \\ {\overset{\cdot}{Δ}}_{vlL}^{i} \end{matrix}} = {\begin{matrix} {\overset{\cdot}{Y}}_{sL}^{i} \\ {\overset{\cdot}{Z}}_{sL}^{i} \end{matrix}} - {\begin{matrix} {\overset{\cdot}{Y}}_{bL}^{i} \\ {\overset{\cdot}{Z}}_{bL}^{i} \end{matrix}} - {\begin{matrix} 0 \\ {\overset{\cdot}{r}}_{Li} (x) \end{matrix}}

(9)

where ${\overset{\cdot}{r}}_{Li} (x) = (d r_{Li} (x) / dx) (dx / dt) = (d r_{Li} (x) / dx) V (t)$ , $Z_{sL}^{i}$ and $Y_{sL}^{i}$ are the vertical and horizontal displacement of the i axle suspension; r is the road surface profile, and V is vehicular speed. $Z_{bL}^{i}$ and $Y_{bL}^{i}$ are the vertical and horizontal displacements at the touch points of bridge as shown in Figure 2 and can be written as

{\begin{matrix} Y_{bL}^{i} \\ Z_{bL}^{i} \end{matrix}} = \sum_{n = 1}^{N_{b}} {\begin{matrix} q_{n} ϕ_{h}^{n} (x_{iL}) \\ q_{n} ϕ_{v}^{n} (x_{iL}) \end{matrix}}

(10)

where $x_{iL}$ represents the wheel position on the bridge and the q_n represents generalized modal coordinates; $ϕ_{v}^{n} (x_{iL})$ and $ϕ_{h}^{n} (x_{iL})$ are the vertical and horizontal vectors of the nth mode of vibration and can be expressed with vehicular local coordinate as follows

\begin{matrix} ϕ_{h}^{n} (x_{iL}) = \sin (ψ (x_{i})) ϕ_{x}^{n} (x_{iL}) + \cos (ψ (x_{i})) ϕ_{y}^{n} (x_{iL}) \\ ϕ_{v}^{n} (x_{iL}) = ϕ_{z}^{n} (x_{iL}) \end{matrix}}

(11)

where $ϕ_{x}^{n} (x_{iL})$ , $ϕ_{y}^{n} (x_{iL})$ , and $ϕ_{z}^{n} (x_{iL})$ are the X-direction, Y-direction, and Z-direction components of the nth mode of vibration under bridge global coordinate system, respectively. $ψ (x_{i})$ is the intersection angle between X-direction of global coordinates and X-direction of local ones at the location of wheel.

The VBCV system with global coordinates needs to be changed into local coordinates during solving process. During post-processing, the dynamic response under local coordinates is transformed back to the global coordinates. The acting force on the bridge from the left wheel of the vehicular ith axle in the local coordinates can be written as

{F_{bL}^{i}} = {\begin{matrix} F_{byL}^{i} \\ F_{bzL}^{i} \end{matrix}} = {\begin{matrix} \frac{m_{i} V^{2}}{2 R} \\ \frac{m_{i} g}{2} \end{matrix}} - {\begin{matrix} F_{wyL}^{i} \\ F_{wzL}^{i} \end{matrix}}

(12)

where $m_{i}$ is the distributed mass to the wheels of the ith axle due to the total mass of the truck. $F_{bzL}^{i}$ and $F_{byL}^{i}$ are the vertical and horizontal forces on the bridge in local coordinates. Then the mode generalized force on the bridge can be obtained and related details can be found in Li et al., 2016.

In addition, the road surface condition (RSC) is also an important factor affecting the dynamic response of VBCV system (Yin et al., 2013). The RSC usually is simulated by Fourier inversion of the power density spectrum of bridge deck surface roughness. More details can be obtained in the reference (Huang and Wang, 1992). The ISO road roughness classification from A (very good) to H (very bad) is considered in this article based on the ISO 8068:1995 (1995).

The dynamic analysis model of VBCV system is established by combining the motion equations of vehicle and bridge considering the deflection coordination and force’s equilibrium condition (Li et al., 2016)

\begin{matrix} [\begin{matrix} M_{v} & 0 \\ 0 & M_{B} \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{X}}_{v} \\ {\overset{\cdot\cdot}{q}}_{B} \end{matrix}} + [\begin{matrix} C_{v} & C_{vB} \\ C_{Bv} & C_{B} + C_{B}^{v} \end{matrix}] {\begin{matrix} {\overset{\cdot}{X}}_{v} \\ {\overset{\cdot}{q}}_{B} \end{matrix}} \\ + [\begin{matrix} K_{v} & K_{vB} \\ K_{Bv} & K_{B} + K_{B}^{v} \end{matrix}] {\begin{matrix} X_{v} \\ q_{B} \end{matrix}} = {\begin{matrix} F_{v}^{r} \\ F_{B}^{rG} \end{matrix}} \end{matrix}

(13)

where $M, C, K$ and F are the mass, damping, stiffness, and force terms, respectively; the subscripts “Bv” and “vB” express vehicle and bridge coupled terms; the parameters in details can be referred in Li et al. (2016).

A calculation program for setting up and solving the VBCV system is developed in Matlab 2012 and Newmark-β method based on above algorithm. More details for the model can be obtained in Li et al (2016).

Field dynamic test

Before the opening to traffic of the Ganhaizi Bridge in Apirl of 2012, a dynamic test including moving vehicle loading and ambient excitation testing was performed to check and verify the dynamic performance of the bridge (Jiang, 2006).

Measurement instrument

During the dynamic testing for the bridge, the displacement, strain, and acceleration responses are recorded at selected representative points and sections (as shown in Figure 5) based on theoretical analysis results. A data collection system with 32 channels (IMC CL1032, German) is used to record the time history response for all acceleration sensors. The dynamic strain response is collected by the DH3817 data recording system (DH, China). The displacement response of the bridge under loading by moving trucks is measured by IBIS-S, which is a new type of deformation monitoring and measurement instrumentation with 0.01 mm measurement accuracy and maximum sample rate 200 Hz, produced by IDS Company in Italy. The acceleration sensors were installed on the bridge deck of all the spans of the second unit continuous bridge and the lower chord member at the top of 20th pier only. The arrangement form of accelerometers on the bridge deck of every span is same with that of the 20th span as shown in Figure 5(a). The dynamic displacement measuring points were arranged at the lower chord of the girder in middle of the 20th span and the 20th pier top as shown in Figure 5(b) and (c). The dynamic stress measuring points were arranged at the bottom of the bridge deck, web members, top and lower chord members of the girder as well as diagonal bracing and main limb of the pier for the 19th, 20^th, and 21st span. Because the arrangement of strain sensors is the same for the above spans, the 20th span are taken as an example to show the detail as Figure 5.

Figure 5.

Measured points layout for field testing and numerical analysis: (a) elevation view of the measure points arrangement in the 20th span, (b) cross section A, (c) cross section B, (d) cross section C, (e) cross section D, (f) cross section E, (g) cross section F.

The AV represents the sensor location on the bridge deck that is used to obtain the vertical acceleration; and AH represents the location of the sensor for horizontal acceleration. DVⁱ and DHⁱ represent vertical and horizontal displacement sensors. The superscript suggests the location of the sensors along the bridge, namely “1” represents the mid-span and “2” represents the pier top. S_i represents the ith strain sensor. In addition to these sensors discussed above, CD represents vertical displacement, and CS denotes strain. Subscripts “c” and “w” represent the chord member and web member; and subscripts “u” and “l” represent upper and lower. The subscript “b” represents bridge deck.

Dynamic testing process and results

The dynamic responses on the measured points is recorded under one moving truck on the lane 2 (as shown in Figure 6) at the speeds from 10 km/h to 60 km/h. In addition the environmental excitation testing also is performed for the bridge.

Figure 6.

Field test: (a) testing vehicles and (b) dynamic response test under one moving truck load.

The modal parameters of the bridge are identified based on the measured acceleration response data. The comparison of tested and numerical results is shown in Table 2, where f_t, f_c and ς are the tested frequency, computed frequency, and damping ratio, respectively.

Table 2.

Modal parameters from experimental and numerical methods.

Modal description	f_t (Hz)	f_c (Hz)	Relatively Error (%)	ς(%)	Measured mode shape	MAC
Pier: longitudinalbending	–	0.18	–	–	–	–
Girder: lateralbending	0.28	0.26	7.1	0.5		0.90
Girder: verticalbending	1.10	1.11	0.9	0.9		0.92
Girder: twisting	–	1.12	–	–	–	–

MAC: modal assurance criteria.

The first two measured mode shapes are lateral and vertical bending of girder. The first vertical bending modal with the 1.10 Hz appears after several lateral bending modes, and the torsional modes of the bridge deck appear continuously for several times after the 18th mode based on the numerical simulation results. The good match as shown in Table 2 between tested modal data and numerical ones with less than 7.1% errors and over 0.9 modal assurance criteria (MAC) values, indicates that the established FEM for the bridge is suitable to be used in the VBCV numerical simulation.

Furthermore, the influence of the boundary condition on dynamic characteristic is discussed for the bridge. The original continuous rigid bridge is redefined as continuous girder bridge by changing the support condition between piers and girder. The original model is named as Model 1 and the changed model is named as Model 2. The calculated longitudinal bending modal shape of piers, lateral, vertical and twisting bending modal shapes of girder are 0.17, 0.24, 1.10, and 1.11, respectively for the Model 2. Comparison of the dynamic characteristic for the Model 1 (as shown in Table 2) and Model 2 indicates that the change of the boundary condition has slightly effect on dynamic characteristic of the bridge.

Another important issue is the impact effect of the bridge due to moving vehicle. The impact effect usually is expressed with an impact factor as follows

IF = \frac{R_{d} (x)}{R_{s} (x)} - 1

(14)

where $R_{d} (x)$ and $R_{s} (x)$ are maximal dynamic and static response of the bridge due to passing trucks. The IF is defined with equation (15) in the design codes in China (JTG D60-2015, 2005) as follows

IF = {\begin{matrix} 0.05 & f < 1.5 Hz \\ 0.1767 \ln f - 0.0157 & 1.5 Hz ⩽ f ⩽ 14 Hz \\ 0.45 & f > 14 Hz \end{matrix}

(15)

where f represents the first vertical bending mode frequency. For the bridge the f = 1.10 Hz, and the corresponding designing IF is 0.05.

According to the calculated data the maximal vertical displacement happened in the mid of span of the bridge considering the passing trucks. As a result, the IF in mid-span can be taken as index of the bridge. The measured maximal IF is 0.3 at DV¹ as in Figure 5. The corresponding maximum vertical dynamic displacement is 6.3 mm, appeared at the mid-span for the 20th span due to one truck with the speed of 50 km/h at lane 2. The impact factor is larger than the designing value 0.05, which indicates that the designed code possibly underestimates the dynamic impact of vehicles on this bridge.

Besides the vertical response, the structural lateral response is also worthy of attention for this high-pier curved girder bridge due to moving vehicles. Taking velocity 50 km/h as an instance, the lateral displacement and acceleration time history of 20th pier top are plotted in Figure 7. The result shows that the maximal lateral response appears at the moment of vehicle passing the mid-span.

Figure 7.

The measured results of selected points: (a) lateral displacement time history of DH² and (b) lateral acceleration time history of AH².

Validation of numerical analysis model

The measured road surface profile for the bridge before opening traffic is verified to belong to good level RSC. A typical field dynamic loading case is simulated by the VBCV analysis program for bridge considering one truck passing with 60 km/h velocity and a measured road surface sample as shown in Figure 8. The time step is defined with 0.005 s in subsequent numerical analysis based on trial analysis and comparison of convergence condition under different time step length. The comparison of computed and tested response results on the point DV¹ and DH² as shown in Figure 8 indicates that the relative difference of the peak value for both lateral and vertical displacement is less than 5%. The displacement time history curves of tested and calculated also well matched with each other, respectively. The comparisons indicate that the proposed VBCV model work very well to describe dynamic behavior and can be used to simulate dynamic responses of the bridge under some other loading cases.

Figure 8.

Dynamic response comparison of tested and calculated results: (a) vertical displacement of measure point DV¹ and (b) lateral displacement of measure point DH².

Numerical study

The verified VBCV analysis model is usually applied as a supplement and extension of field experiment, and more influence factors and cases can be considered by this way. In this part, the dynamic performance of the bridge due to moving vehicles is studied considering several main effect parameters including vehicle number, velocity, and RSC with numerical method.

Dynamic impact effect analysis

First, the effect of RSC, vehicular velocity, and transversal position on dynamic impact due to moving trucks is studied with the numerical method for the bridge. To consider effect of the randomness of RSC generation on the results, for each specific case the vehicle and bridge interaction analysis was set to run 12 times with 12 sets of randomly generated road surface profiles under the given RSC. The pilot calculation results indicated that the coefficient of variation (COV) of the dynamic response indexes are less than 10% and the number of 12 was thus considered to be sufficient. The vehicle speed ranging from 40 km/h to 100 km/h, RSCs from “good” to “very bad” levels and two loading cases (loading case I and II as shown in Figure 5(b)) are considered during the VBCV analysis. The displacement IFs on the point DV¹ in mid-span are shown in Figure 9. The results indicate that the IF magnifies with the RSC deteriorating, but for the vehicular speed the effect trend is not clear. Generally, the extreme values of IFs under every RSC levels occur within the velocity scope of 90–100 km/h. In addition, the maximal IFs under loading Case I are bigger than that of loading Case II, and the former are about 1.5 times of the latter under various RSCs.

Figure 9.

Predicted displacement impact factor of point DV¹: (a) loading case I with one truck on lane 1 and (b) loading case II with two trucks.

The analysis results of the stress impact factor at the point ${CS}_{uc}^{1}$ in the section above the top of pier are demonstrated in Figure 10. The contrast of Figures 9 and 10 shows that the influence of RSC on impact effect for the section at top of pier is weaker than that of mid-span, while the general increasing trend of the impact factor for the two sections is consistent. Moreover, the influence of truck number on the impact factor at the point ${CS}_{uc}^{1}$ is not such obvious and clear as that of point DV¹ under “Good,”“Average,” and “Bad” RSCs. Particularly, the IF peak value under loading case II is bigger than that of loading case I for the “average” RSC level.

Figure 10.

Predicted impact factor of point ${CS}_{uc}^{1}$ : (a) loading case I with one truck on lane 1 and (b) loading case II with two truck on lane 1 and 2.

As discussed earlier, the computed displacement impact factor 0.31 in the mid-span under good RSC and loading case I agrees very well with the corresponding dynamic impact factor of 0.3 from the tests. The maximal impact factors 0.51 and 0.25 in mid-pan under good RSC with loading case I and case II exceed the design value 0.05 with about 10 and 5 times, respectively. It indicates that the present design code underestimates the impact effect of moving vehicles for this new type of bridge.

In addition, it is necessary to pay attention to the lateral dynamic response for this bridge due to its curved and high-pier feature. The influence of RSC and velocity on lateral dynamic response at mid-span and top pier for the bridge under loading case I is demonstrated in Figure 11. The result indicates that the lateral dynamic displacement peak values generally increase with the deterioration of RSC and enlargement of equivalent vertical load, and the effect trend is similar to the vertical displacement response. The extreme lateral displacement is 1.2 and 1.5 mm at the point DV¹ and DH², respectively, under an average RSC, and the ratio of the extreme lateral displacements to the corresponding vertical response is about 25%. However, different with theoretical predictions, the effect of vehicle speed on lateral dynamic impact does not have a clear trend. The lateral displacement caused by vertical load is relatively high, contributing a large proportion of displacement to the overall lateral displacement. Generally, the torsional and vertical deflection would occur simultaneously for the girder section under uneven and even symmetrical vertical load due to the particular bending-torsional coupling mechanic feature of the curved girder bridges. For the high-pier curved continuous girder bridge in this study, the torsion deflection of girder under uneven vehicles load leads to the radial bending of the high pier, and the lateral displacement of the girder occurs due to the radial bending of the high pier with relative weak bending stiffness. Further analysis is needed to further explain this phenomena. The lateral vibration issue for this new bridge type deserves more attention.

Figure 11.

Maximum lateral displacement in key section of the bridge under loading case I: (a) extreme value in mid-span and (b) extreme value in top pier.

Finally, the effect of transversal lane position of moving truck on the dynamic impact for the bridge is also evaluated by considering one truck passing on various lanes under 90 km/h and “average” RSC level. The vertical displacement responses in the mid-span at the point DV¹ and CD¹ with one truck on either lane 1, 2, 3, or 4 are shown in Figure 12. The extreme displacement on the point DV¹ with 8.172 and 7.150 mm under loading lane 1 and 2 is higher than the point CD¹ with 7.708 and 6.112 under loading lane 3 and 4 due to the different equivalent radius of curvature. However, the transversal lane position of vehicles has insignificantly influence on the impact effect of the bridge.

Figure 12.

Predicted response and IF of the mid-span under one moving truck on various lanes.

The stress responses on the top of pier at the point ${CS}_{uc}^{2}$ and ${CS}_{uc}^{2'}$ under truck on the lane 1–4 are listed in Table 3. The stress under loading the outside part of the bridge with 0.161 and 0.140 MPa is larger than the inside part with 0.091 and 0.110 MPa. Conversely, the impact factor under loading lane 1 and 2 is lower than the value under loading lane 3 and 4. It indicates that the impact factor of pier top section decreases with the lane position closing to the outside, that is, the equivalent increase of radius of curvature.

Table 3.

Predicted response and IF of the pier top under one moving truck on various lanes.

Lane position	Lane 1	Lane 2	Lane 3	Lane 4
Tensile stress (MPa)	0.161	0.140	0.091	0.110
Impact factor	0.278	0.094	1.528	1.918

In addition to the global dynamic effect, local dynamic impact effect for bridge components could also be critical for the bridge design and condition assessment in their life cycle. To this end, several key local components are selected to calculate the local dynamic stress response. The influence of the vehicle speed, RSC, and vehicle number is discussed. The selected bridge components include web members (the point ${CS}_{w}^{1}$ ), chord members (the point ${CS}_{uc}^{1}$ and ${CS}_{lc}^{1}$ ), and bridge deck (the point ${CS}_{b}^{1}$ ) as shown in Figure 5.

As shown in Figure 13, the stress impact factors for all the above selected components increase significantly when RSC becomes worse. The maximum dynamic impact factor 8.237 under very bad RSC is over 8 times of the value 0.941 under good RSC as shown in Figure 13(a). Within the velocity scope of 90–100 km/h, the IF under each RSC usually reaches its maximum value. The extreme IFs for the point ${CS}_{lc}^{1}$ appear at the velocity of 90 km/h under different RSCs, and the value is lower than that of the point ${CS}_{uc}^{1}$ (shown in Figure 13(b)). The extreme IFs for point ${CS}_{w}^{1}$ are 0.977, 1.360, 2.686, and 5.338 for RSC being from good to very bad as shown in Figure 13(c). Similar pattern could also be found for point ${CS}_{b}^{1}$ as shown in Figure 13(d).

Figure 13.

Predicted impact factor of local components under one truck: (a) the point ${CS}_{uc}^{1}$ for upper chord, (b) the point ${CS}_{lc}^{1}$ for lower chord, (c) the point ${CS}_{w}^{1}$ for web member, and (d) the point ${CS}_{b}^{1}$ for bridge deck.

As discussed earlier, the dynamic impact effect of structural components increases remarkably with the deterioration status of RSC for the bridge. The maximum impact factors for these bridge components are larger than the design value of 0.3. Taking the average RSC as an example, the impact factors for the points ${CS}_{uc}^{1}$ , ${CS}_{lc}^{1}$ , ${CS}_{w}^{1}$ , and ${CS}_{b}^{1}$ are 1.915, 1.181, 1.360, and 1.045, respectively. In addition the dynamic impact effect could also vary for different bridge components even under the same RSC and the same velocity. The upper chord usually experiences larger dynamic impact effects than the other structural members.

Evaluation for riding comfort

Besides structural safety, the dynamic interaction of vehicle and bridge could also introduce potential concerns for riding comfort of moving vehicles. The method from ISO 2631-1:1997 (1997) has been used widely to evaluate the ride comfort for many bridges (Yin et al., 2013). Specifically, the weighted root-mean-square (RMS) magnitudes for the accelerations are used as the criterion of discomfort rating and listed in Table 4.

Table 4.

Discomfort rating criterion in ISO 2631-1.

Discomfortrating	a_w (m/s²)range	ISO 2631-1 discomfortlevel
1	<0.315	Not uncomfortable
2	0.315–0.63	A little uncomfortable
3	0.50–1.0	Fairly uncomfortable
4	0.8–1.6	Uncomfortable
5	1.25–2.5	Very uncomfortable
6	>2.0	Extremely uncomfortable

For vibrations in more than one direction, the weighted RMS acceleration $a_{w}$ determined from vibrations in orthogonal coordinates is calculated as

a_{w} = {[{(k_{x} a_{wx})}^{2} + {(k_{y} a_{wy})}^{2} + {(k_{z} a_{wz})}^{2}]}^{1 / 2}

(16)

where $a_{wx}$ , $a_{wy}$ , and $a_{wz}$ are the RMS of longitudinal, lateral, and vertical acceleration from the driving seat position, respectively, and the corresponding weighting factors are k_x = 1.4, k_y = 1.4, and k_z = 1.0.

For the bridge, the driver seat position is chosen to evaluate the riding comfort referring to some existing related researches (Li et al., 2016; Yin et al., 2013). The riding comfort is analyzed and evaluated considering one truck passing on the lane 1 of the bridge under various velocities and RSCs. The results shown as Table 5 demonstrate that the RSC has a significant effect on the ride comfort, which decreases from “fairly uncomfortable” to “uncomfortable” when the RSC changes from “good” to “bad.” For the bridge even the RSC maintains a good level, the riding comfort will change from “fairly uncomfortable” to “uncomfortable” as the vehicle speed exceeds 90 km/h; when the RSC degrades to the bad level, the riding comfort reaches the “uncomfortable” level at the velocity of 40 km/h. Overall, the riding comfort of the Ganhaizi Bridge is not satisfactory, and it is worse than most of the regular girder bridge (Yin et al., 2013).

Table 5.

Comfort rating result for the bridge.

Speed(km/h)	a_w (m/s²)good	Discomfortlevel	a_w (m/s²)average	Discomfortlevel	a_w (m/s²)bad	Discomfortlevel
40	0.633	Fairly uncomfortable	0.777	Fairly uncomfortable	1.066	Uncomfortable
50	0.697	Fairly uncomfortable	0.833	Fairly uncomfortable	1.122	Uncomfortable
60	0.751	Fairly uncomfortable	0.861	Fairly uncomfortable	1.132	Uncomfortable
70	0.813	Fairly uncomfortable	0.895	Fairly uncomfortable	1.138	Uncomfortable
80	0.881	Fairly uncomfortable	0.929	Fairly uncomfortable	1.138	Uncomfortable
90	0.958	Fairly uncomfortable	0.978	Fairly uncomfortable	1.153	Uncomfortable
100	1.045	Uncomfortable	1.050	Uncomfortable	1.198	Uncomfortable

To evaluate the contribution of mode shapes to the riding comfort for the bridge further, the contribution factor for riding comfort induced by different direction vibration is defined as following

C_{j | j = x, y, z} = \frac{{(k_{j} a_{wj})}^{2}}{{(k_{x} a_{wx})}^{2} + {(k_{y} a_{wy})}^{2} + {(k_{z} a_{wz})}^{2}}

(17)

Generally, the vehicular z direction vibration acceleration caused by the vertical and pitching motion mainly is affected by vertical bending mode shapes of the bridge; the x and y direction vibration accelerations principally are controlled by the longitudinal and lateral bending mode shapes, respectively. Taking the case with 60 km/h vehicle speed and average RSC as an example, the calculated contribution factors C_x, C_y, and C_z are 3.2%, 6.4%, and 90.4%, respectively. The result indicates that the riding comfort of the bridge is mainly affected by the vertical mode shapes with over 90% contribution factor.

Based on the present study, the specific dynamic characteristics of this new type of bridge need to take main responsibility for the unsatisfactory riding comfort. In addition, the results indicate that performing the riding comfort evaluation is necessary during designing process for some new type of bridges.

Conclusion

In this article, the dynamic performance of a new type CFST high-pier curved continuous truss girder bridge due to moving vehicles is studied combing field test and numerical methods. Some conclusions are drawn as follows:

A VBCV analysis model for curved bridge is proposed based on the modal synthesis method and verified by the field test data. A suitable local and global coordinate’s transformation strategy also is introduced for the method.

The experimental and numerical results indicate that the existing design code in China underestimates the vertical impact effect for this type of bridges. In addition, the impact factors for the local key components are also far larger than that defined in the codes, and these impact factors could differ significantly with each other for different components. The peak lateral displacement is about 25% of vertical displacement, and the vertical load also contributes significantly to the lateral displacement.

The riding comfort varies from “fairly uncomfortable” to “uncomfortable” under various vehicle speeds and RSCs for the bridge, which are worse than that of the regular high-pier girder bridge. The riding comfort is mainly affected by the vertical mode shapes of the bridge.

It is noted that above conclusions are obtained just according to the dynamic study for the Ganhaizi Bridge, and general conclusions could only be made after more studies are performed.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the financial support provided by National Natural Science Foundation of China (Grant No. 51108132).

References

Chen

(2009) Dynamic performance simulation of long-span bridge under combined loads of stochastic traffic and wind. Journal of Bridge Engineering; 15(3): 219–230.

Deng

Cai

(2010) Development of dynamic impact factor for performance evaluation of existing multi-girder concrete bridges. Engineering Structures 32(1): 21–31.

Gim

Nikravesh

(1991) An analytical model of pneumatic types for vehicle dynamic simulations. Part 2: comprehensive slips. International Journal of Vehicle Design 12(1): 19–39.

Han

Cai

Zhang

et al . (2014) Effect of aerodynamic parameters on the dynamic responses of road vehicles and bridges under cross winds. Journal of Wind Engineering and Industrial Aerodynamics 134: 78–95.

Han

et al . (2018) Stress analysis of a long-span steel-truss suspension bridge under combined action of random traffic and wind loads. Journal of Aerospace Engineering 31(3): 04018021.

Huang

Wang

(1992) Impact analysis of cable-stayed bridges. Computers & Structures 43(5): 897–908.

ISO 8068:1995 (1995) Mechanical vibration—road surface profiles—reporting of measured data.

ISO 2631-1:1997 (1997) Mechanical vibration and shock—evaluation of human exposure to whole body vibration—part I: general requirements.

Jiang

(2006) Static and dynamic analysis of composited truss beam bridge with steel tube and conctrete-filled tube. Masters’ Thesis, Southwest Jiaotong University, Leshan, China.

10.

JTG D60-2015 (2005) General Code for Design of Highway Bridges and Culverts. Beijing, China: China’s Ministry of Transport.

11.

Cai

Liu

et al . (2016) Dynamic analysis of a large span specially shaped hybrid girder bridge with concrete-filled steel tube arches. Engineering Structures 106: 243–260.

12.

Livingston

Brown

(1969) Physics of the slipping wheel-force and torque calculations for various pressure distributions. Rubber Chemistry and Technology 43(2): 45–56.

13.

Rill

(2007) Vehicle Dynamics-Fundamentals and Modeling Aspects. Regensbg: Regensbg Fachhochschule.

14.

Wang

Huang

Shahawy

(1994) Dynamic behavior of slant-legged rigid-frame highway bridge. Journal of Structural Engineering 120(3): 885–902.

15.

Chen

Takahashi

et al . (2008) Vehicle-bridge dynamic analysis and riding comfort evaluation of new Saikai bridge. Journal of Highway and Transportation Research and Development 25(5): 61–67.

16.

Wang

Deng

Shao

(2016) Number of stress cycles for fatigue design of simply-supported steel I-girder bridges considering the dynamic effect of vehicle loading. Engineering Structures 110: 70–78.

17.

Guo

(2003) Dynamic analysis of coupled road vehicle and cable-stayed bridge system under turbulent wind. Engineering Structures 25: 473–86.

18.

Yoshimura

Takahashi

et al . (2006) Vibration analysis of the second Saikai bridge—a concrete filled tubular (CFT) arch bridge. Journal of Sound and Vibration 290: 388–409.

19.

Yau

Yang

(2008) Vibration of a suspension bridge installed with a water pipeline and subjected to moving trains. Engineering Structures 30(3): 632–642.

20.

Yin

Cai

Fang

et al . (2010) Bridge vibration under vehicular loads-tire patch contact versus point contact. International Journal of Structural Stability and Dynamics 10(3): 529–554.

21.

Yin

Cai

Liu

et al . (2013) Experimental and numerical studies of non-stationary random vibrations for a high-pier bridge under vehicular loads. Journal of Bridge Engineering 18(10): 1005–1020.

22.

Yin

Fang

Cai

(2011) Lateral vibration of high-pier bridge under moving vehicular loads. Journal of Bridge Engineering 16(3): 400–412.

23.

Zhang

Cai

(2013) Reliability-based dynamic amplification factor on stress ranges for fatigue design of existing bridges. Journal of Bridge Engineering 18(6): 538–552.