The elastic modulus and deadweight of carbon fiber-reinforced polymer (CFRP) cables are different from those of steel cables. Thus, the static and dynamic behaviors of cable-stayed bridges using CFRP cables are different from those of cable-stayed bridges using steel cables. The static and dynamic performances of the two kinds of bridges with a span of 1000 m were studied using the numerical method. The effects of geometric nonlinear factors on static performance of the two kinds of cable-stayed bridges were analyzed. The live load effects and temperature effects of the two cable-stayed bridges were also analyzed. The influences of design parameters, including different structural systems, the numbers of auxiliary piers, and the space arrangement types of cable, on the dynamic performance of the cable-stayed bridge using CFRP cables were also studied. Results demonstrate that sag effect of the CFRP cable is much smaller than that of steel cable. The temperature effects of CFRP cable-stayed bridge are less than those of steel cable-stayed bridge. The vertical bending natural vibration frequency of the CFRP cable-stayed bridge is generally lower than that of steel cable-stayed bridge, whereas the torsional natural vibration frequency of the former is higher than that of the latter.
Given their large spanning ability, elegant appearance, and good aerodynamic stability, cable-stayed bridges have experienced a prosperous development in the last 30 years and are still regarded as the first choice to cross rivers, valleys, and even straits. However, traditional steel cables are prone to corrosion and fatigue, which would cause premature failure of the cable. In addition, the substantial weight of steel cables results in a pronounced sag effect, thereby decreasing the effective stiffness of the cable in a superspan cable-stayed bridge. Carbon fiber-reinforced polymers (CFRP) possess many advantages, including light weight, high tensile strength, and excellent corrosion and fatigue resistance. These properties of CFRP make them very attractive in superspan cable-stayed bridges. In fact, as early as 1987, U. Meier and H. Meier (e.g., [
At present, a few scholars have studied the static and dynamic performances, as well as wind and seismic performances, of cable-stayed bridges using CFRP cables. Khalifa et al. [
With increasing span of the cable-stayed bridge, the geometric nonlinear issues become more and more acute, especially the sag effect of steel cable, which even becomes a control factor of designing the cable-stayed bridge. However, very little research on the effects of the geometric nonlinearity on the cable-stayed bridge using CFRP cables was conducted. In this study, the cable-stayed bridges using CFRP and steel cables, respectively, with the same span arrangement were designed. The effects of geometric nonlinearity on the static performance of two cable-stayed bridges were studied. Through comparing the geometric nonlinear effects of the two cable-stayed bridges, the advantages and disadvantages of the cable-stayed bridges using CFRP cables were explored. Although several scholars studied the dynamic characteristics of cable-stayed bridges using CFRP cables before, their research only focused on a special example. In this study, the dynamic characteristics of cable-stayed bridges with different design parameters, including structure system, the numbers of auxiliary piers, and arrangement pattern of cables, were studied. Besides, rational design parameters are also optimized, thus providing reference for the design of cable-stayed bridges using CFRP cables.
Two cable-stayed bridges using CFRP cables and steel cables, respectively, are preliminarily designed, with a 410 m + 1000 m + 410 m span arrangement, as shown in Figure
Parameters of main components.
Component name | Material | Sectional area (m2) | Bending moments of inertia (m4) | Transformed unit weight (kN/m3) | Modulus of elasticity (MPa) | Coefficient of linear expansion |
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Upper part of pylon | Concrete |
42~63 | 396~633 | 26 |
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|
Middle part of pylon | Concrete |
63~102 | 938~2314 | 26 |
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|
Lower part of pylon | Concrete |
102~108 | 2314~2542 | 26 |
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|
Main girder | 16 Mn steel |
1.47~2.31 | 4.0~6.5 | 109.9 |
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|
CFRP cable (Per) | CFRP tendons ( |
(0.7~1.25) |
0 | 16 |
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|
Steel cable (Per) | High-strength steel ( |
(0.7~1.25) |
0 | 80 |
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|
General layout of cable-stayed bridges with a 1000 m span (unit: m).
Section of the girder (unit: mm).
Section of girder A
Section of girder B
Layout of the pylon (unit: m).
Sectional area of the cables.
Finished-stage cable forces.
As the bridges are not symmetrical about the main pylon, counterbalance weights are set in the intensive-cable region and auxiliary piers region. The specific distribution of the counterbalance weights is as follows: In the region of 0~12 m from the end of the girder, the average weight is 230 kN/m; in the region of 12~92 m, the average weight is 330 kN/m; in the region of 92~102 m, the average weight is 280 kN/m; in the region of 102~122 m, the average weight is 180 kN/m.
Usually, two methods are applied to replace steel cables with CFRP cables. They are equal-strength and equal-stiffness principles, respectively. In terms of equal-strength principle, the design strength of CFRP is usually greater than that of steel due to its much higher tensile strength, and the safety factor is similar to steel, which results in a much smaller sectional area of the CFRP cable than that of the steel one and decreases the stiffness of the whole bridge. But with equal-stiffness principle, larger sectional area of CFRP cable would be required, which is uneconomic. In our paper, a compromise plan for designing the CFRP cable with the same sectional area as the steel cable is adopted. A safety coefficient of 2.5 is selected for steel cables, and then the safety coefficient for CFRP cables is about 3.4 according to the parameters shown in Table
Before discussing the static and dynamic properties of CFRP cable-stayed bridges, the material weight and cost of CFRP cables versus steel cables are first investigated. The material dosages for the designed CFRP cable-stayed bridge and steel cable-stayed bridge in this paper are calculated and listed in Table
Material dosages for CFRP and steel cable-stayed bridges.
Cable type | Weight of each component (104 kN) | ①/② | ①/(① + ② + ③) | ||||
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① Cable | ② Main girder | ③ Second dead load and counter weights | ④ Pylon | ①/2 + ② + ③ | |||
CFRP | 1.072 | 38.41 | 19.85 | 129.6 | 58.80 | 2.8% | 1.8% |
Steel | 5.360 | 38.41 | 19.85 | 129.6 | 60.94 | 14.0% | 8.4% |
Table
Cable-stayed bridge is a complex structure in which lateral bending and torsion are often coupled together. The entire bridges are modeled by fishbone beam using spatial finite models, by which the girders, pylons, and piers are simulated by three-dimensional beam elements and the cables are simulated by link elements. The sectional properties of the girders, pylons, and piers are modeled in terms of equivalent sectional area and moments of inertia. The analysis model is shown in Figure
Model for analysis.
The influencing factors of geometric nonlinearity can be classified into three types: sag effect, large deformation effect, and beam-column effect. Compared with beam bridges, the effects of geometric nonlinearity on the static performance of cable-stayed bridges are greater. The effects of geometric nonlinearity on the static performance of cable-stayed bridges were studied in this paper. Commonly, the sag effect of a cable is considered by an equivalent elastic modulus
The large deformation effect is solved by theoretical formulation of TL or UL methods and the beam-column effect is considered by introducing a geometric stiffness matrix.
According to the Chinese Highway Engineering Technique Standard (JTGB01-2014), the live load (vehicle load) is calculated to be 42 kN/m, with an additional concentrated load of 1440 kN at the midspan.
Analyses were described as follows:
According to the six above-mentioned situations, the effects of living load arranged at midspan of the two cable-stayed bridges are calculated. The results are presented in Table
Comparison of geometric nonlinear static analysis.
Analysis content | Displacement (m) | Bending moment (104 kN⋅m) | Axial force of girder (104 kN) | ||
---|---|---|---|---|---|
At midspan of main span (vertical) | At midspan of main span | At bottom of the pylon | At the pylon | ||
CFRP cables | L | −1.406 | 7.57 | 74.2 | −2.58 |
IS | −1.367 | 7.42 | 71.6 | −2.51 | |
SE | −1.364 | 7.41 | 71.4 | −2.49 | |
BE | −1.429 | 7.60 | 73.4 | −2.62 | |
LDE | −1.369 | 7.40 | 72.1 | −2.51 | |
CE | −1.432 | 7.62 | 74.2 | −2.96 | |
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Steel cables | L | −1.175 | 6.92 | 63.6 | −2.62 |
IS | −1.154 | 6.81 | 61.8 | −2.56 | |
SE | −1.240 | 7.11 | 63.8 | −2.54 | |
BE | −1.196 | 6.97 | 63.0 | −2.66 | |
LDE | −1.156 | 6.81 | 62.1 | −2.56 | |
CE | −1.331 | 7.39 | 66.4 | −3.01 |
According to Table
Given the geometry nonlinearity effect, the linear superposition principle is not applicable for long-span cable-stayed bridges. Therefore, the traditional influence line-loading algorithm cannot be used for the analysis of live load effect. However, the live load accounts for only 15% of the dead load of superstructures for long-span bridges, so the dead load can be used as its initial state to calculate the live load effect. This method is called the second-order theory for live load effect calculation.
The CFRP cable-stayed bridge was calculated by the second-order theory analysis and linear analysis methods. Figures
Envelope diagram of girder’s bending moments.
Envelope diagram of girder’s axial forces.
Envelope diagram of girder’s vertical displacements.
Figures
Figures
The linear expansion coefficient of CFRP is approximately 1/14 of steel. When structures are made of the same material, the smaller the linear expansion coefficient, the smaller the temperature effect. To evaluate the temperature susceptibility of the CFRP cable-stayed bridge, the temperature effect of the two cable-stayed bridges mentioned before with different structure systems was examined. Restraint stiffness is 12,000 kN/m in the tower-beam elastic restraint system. According to Guidelines for Design of Highway Cables-Stayed Bridge (JTG/T D65-01-2007), the combination of temperatures is as follows: the temperature of girder rise 20°C + the temperature of pylon rise 15°C + the temperature of cables rise 30°C. Temperature effects of different structural systems are shown in Table
Temperature effects of different structural systems.
Cable type | Sectional position (force or displacement) | The floating system | The tower-beam elastic restraint system | The semifloating system | Rigid-frame system |
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CFRP cables | Girder’s bending moment at the pylon (kN⋅m) | 20387 | 20359 | 24125 | 12373 |
Bending moment at bottom of the pylon (kN⋅m) |
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Displacement at midspan (m) | 0.383 | 0.383 | 0.384 | 0.326 | |
Horizontal displacement of beam end (m) | −0.179 | −0.178 | −0.179 | −0.129 | |
Cable’s axial force at the pylon (kN) | 167 | 167 | 204 | 220 | |
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Steel cables | Girder’s bending moment at the tower (kN⋅m) | −35196 | −35227 | −38490 | −67794 |
Bending moment at bottom of the pylon (kN⋅m) |
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Displacement at midspan (m) | −0.474 | −0.479 | −0.474 | −0.542 | |
Horizontal displacement of beam end (m) | −0.182 | −0.181 | −0.182 | −0.126 | |
Cable’s axial force at the pylon (kN) | −359 | −359 | −395 | −349 |
A significant difference can be found by comparing the results of the two cable-stayed bridges. The deformations of CFRP cables are small because the linear expansion coefficient value of CFRP is much less than that of steel. Correspondingly, the tension forces increase rather than decrease because of the uncoordinated deformation. Moreover, the deflections of the girder are upward and the bending moments of the girder at the pylon are positive. On the contrary, the elongation of the steel cable is large, resulting in the decrease of cable tension, downward deflection of the girder, and a great negative bending moment at the pylon. To sum up, the girder’s bending moment at the pylon and displacement at midspan of CFRP cable-stayed bridge due to temperature are less than those of the steel cable-stayed bridge, which is beneficial for the service performance of the whole bridge.
To analyze the dynamic characteristics of cable-stayed bridge using different kinds of stay cables, the bridges with the same span arrangement as that in static analysis were adopted. Using FEM model, the first 16 orders of vibration mode and the corresponding frequency considering or not considering initial stress of the cable-stayed bridges are calculated, respectively, and listed in Table
Dynamic performance of the two cable-stayed bridges.
Frequency (Hz) | Modal characteristics | |||||||
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CFRP cable-stayed bridge | Steel cable-stayed bridge | |||||||
Order | With initial stress | Without initial stress | The difference (%) | Order | With initial stress | Without initial stress | The difference (%) | |
1 | 0.1012 | 0.1067 | 5.4 | 1 | 0.0988 | 0.1045 | 5.8 | Longitudinal floating + |
2 | 0.1290 | 0.1292 | 0.2 | 2 | 0.1248 | 0.1253 | 0.4 | First lateral lending |
3 | 0.2028 | 0.2046 | 0.9 | 3 | 0.2075 | 0.2095 | 1.0 | First vertical bending |
4 | 0.2493 | 0.2529 | 1.4 | 4 | 0.2575 | 0.2614 | 1.5 | Second vertical bending |
5 | 0.3403 | 0.3429 | 0.8 | 5 | 0.3338 | 0.3366 | 0.8 | Second lateral bending |
6 | 0.3463 | 0.3556 | 2.7 | 6 | 0.3546 | 0.3640 | 2.7 | Third vertical bending |
7 | 0.4227 | 0.4352 | 3.0 | 7 | 0.4346 | 0.4470 | 2.9 | Fourth vertical bending |
8 | 0.4493 | 0.4621 | 2.8 | 10 | 0.4612 | 0.4738 | 2.7 | Fifth vertical bending |
9 | 0.4776 | 0.4792 | 0.3 | 8 | 0.4568 | 0.4594 | 0.6 | Synchrony lateral bending of pylon |
10 | 0.4800 | 0.4816 | 0.3 | 9 | 0.4586 | 0.4612 | 0.6 | Reverse lateral bending of pylon |
11 | 0.5154 | 0.5325 | 3.3 | 11 | 0.5318 | 0.5486 | 3.2 | Sixth vertical bending |
12 | 0.5277 | 0.5412 | 2.6 | 13 | 0.5529 | 0.5675 | 2.6 | Vertical bending |
13 | 0.5533 | 0.5625 | 1.7 | 12 | 0.5432 | 0.5529 | 1.8 | First torsion + lateral bending |
14 | 0.5703 | 0.5855 | 2.7 | 14 | 0.5595 | 0.5743 | 2.6 | Reverse lateral bending of pylon + torsion |
15 | 0.5906 | 0.6081 | 3.0 | 16 | 0.6160 | 0.6338 | 2.9 | Vertical bending |
16 | 0.5912 | 0.5949 | 0.6 | 15 | 0.5816 | 0.5849 | 0.6 | First torsion |
First 20-order frequencies of CFRP cable-stayed bridge.
First 20-order frequencies of two cable-stayed bridges.
To verify the rationality of the preliminary design of the bridges and the reliability of calculation, the main fundamental frequencies of the two cable-stayed bridges with similar main span were listed for comparison. For a cable-stayed bridge with span arrangement of 76 m + 100 m + 298 m + 1008 m + 298 m + 100 m + 76 m [
(1) The frequencies decrease if initial stress is considered. This is because the main girder and pylon are mainly compressed, and their structural stiffnesses decrease after considering initial stress. However, the difference between the two results is unobvious.
(2) The vertical bending natural vibration frequencies of the cable-stayed bridge using CFRP cables are lower than those using steel cables because the elastic modulus of the CFRP cable is smaller than that of steel cable. This leads to smaller elastic support stiffness for the main girder, which is in accordance with the static characteristic. It is also found that, with increasing span of the bridge, the sag effect increases, and the equivalent elastic modulus decreases. However, if the span is greater than a certain value, the vertical bending natural vibration frequency of CFRP cable-stayed bridge will be higher than steel cable-stayed bridges.
(3) The torsional natural vibration frequency of CFRP cable-stayed bridge is a bit higher than that of steel cable-stayed bridge because the CFRP cables are lighter and provide a smaller mass moment of inertia. With increasing span, the equivalent elastic modulus of CFRP cables increases faster than that of steel cables. Hence, the torsional natural vibration frequency of CFRP cable-stayed bridge will be much higher than that of steel cable-stayed bridge, which is beneficial for improving the flutter critical wind speed.
The common structural systems of long-span cable-stayed bridges can be classified into four types: full-floating system, half-floating system, rigid-frame system, and tower-beam elastic restraint system. The dynamic properties of different kinds of structural systems of the CFRP cable-stayed bridge are compared in this section. The longitudinal elastic restraint stiffness is 12,000 kN/m for tower-beam elastic restraint system. Table
Dynamic characteristics of CFRP cable-stayed bridge with different structural systems.
Full-floating system | Half-floating system | Rigid-frame system | Tower-beam elastic restraint system | Modal characteristics | ||||
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Order | Frequency (Hz) | Order | Frequency (Hz) | Order | Frequency (Hz) | Order | Frequency (Hz) | |
1 | 0.0731 | 1 | 0.0731 | 1 | 0.1067 | Longitudinal floating + vertical bending of pylon | ||
2 | 0.1291 | 2 | 0.1295 | 1 | 0.1365 | 2 | 0.1292 | First lateral bending |
3 | 0.2046 | 3 | 0.2046 | 2 | 0.2050 | 3 | 0.2046 | First vertical bending |
13 | 0.5625 | 13 | 0.5636 | 15 | 0.6650 | 13 | 0.5625 | First torsion + lateral bending |
15 | 0.5949 | 15 | 0.5984 | 12 | 0.5853 | 15 | 0.5949 | First torsion |
Table
The influence of the numbers of side-span auxiliary piers on structure dynamic performance is analyzed in the following. Table
Dynamic characteristics of CFRP cable-stayed bridge with different numbers of auxiliary piers.
No auxiliary pier | 1 auxiliary pier | 2 auxiliary piers | 3 auxiliary piers | Modal characteristics | ||||
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Order | Frequency (Hz) | Order | Frequency (Hz) | Order | Frequency (Hz) | Order | Frequency (Hz) | |
1 | 0.1023 | 1 | 0.1067 | 1 | 0.1067 | 1 | 0.1076 | Longitudinal floating + vertical bending of pylon |
2 | 0.1121 | 2 | 0.1286 | 2 | 0.1292 | 2 | 0.1351 | First lateral bending |
3 | 0.1151 | 3 | 0.1673 | 3 | 0.2046 | 3 | 0.2094 | First vertical bending |
7 | 0.3578 | 7 | 0.3762 | Second vertical bending + side-span first vertical bending | ||||
9 | 0.4243 | Side-span synchrony first lateral bending | ||||||
12 | 0.4639 | Side-span reverse first lateral bending | ||||||
18 | 0.5628 | 15 | 0.5625 | 13 | 0.5625 | 13 | 0.5632 | First torsion + lateral bending |
21 | 0.5956 | 17 | 0.5946 | 15 | 0.5949 | 15 | 0.5980 | First torsion |
Table
Single-cable plane and double-cable plane are two general arrangements in cable-stayed bridges. The double cable plane includes parallel double cable plane and inclined double cable plane. The influences of arrangement forms on structure dynamic performance are analyzed in this section. Table
Dynamic characteristics of the two cable-stayed bridges with different types of cable plane.
Frequency (Hz) | Modal characteristics | |||||||
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CFRP cable-stayed bridge | Steel cable-stayed bridge | |||||||
Order | Inclined plane | Order | Parallel plane | Order | Inclined plane | Order | Parallel plane | |
1 | 0.1067 | 1 | 0.1053 | 1 | 0.1045 | 1 | 0.1032 | Longitudinal floating + vertical bending of pylon |
2 | 0.1292 | 2 | 0.1292 | 2 | 0.1253 | 2 | 0.1254 | First lateral bending |
3 | 0.2046 | 3 | 0.2031 | 3 | 0.2095 | 3 | 0.2080 | First vertical bending |
13 | 0.5625 | 12 | 0.5529 | First torsion + lateral bending | ||||
15 | 0.5949 | 11 | 0.4999 | 15 | 0.5849 | 11 | 0.4883 | First torsion |
Torsional vibration mode of the cable-stayed bridges with different cable planes.
Inclined cable plane
Parallel cable plane
Table
Two 1000 m span cable-stayed bridges using CFRP cable and steel cables, respectively, were preliminarily designed. The static and dynamic performances of the two kinds of bridges were analyzed using FEM. The effects of geometric nonlinear factors on static performance and the influences of design parameters, including different structural systems, the numbers of auxiliary piers, and the space arrangement types of cable, on the dynamic performance of the cable-stayed bridge using CFRP cables were also studied. The following specific conclusions can be drawn.
(1) The displacements of the cable-stayed bridges using CFRP cables exhibit a little difference, whereas the displacements at the midspan of steel cable-stayed bridge increase by 7.5%, when the sag effect is taken into account.. With increasing span, the sag effect of steel cable will be larger than that of CFRP cable. Besides, the beam-column effects of the two bridges are both great, whereas the large displacement effects are relatively small.
(2) The girder’s bending moment at the pylon and displacement at midspan of CFRP cable-stayed bridge due to temperature are less than those of steel cable-stayed bridge, which is beneficial for the service performance of the whole bridge.
(3) The vertical bending natural vibration frequency of the CFRP cable-stayed bridge is generally lower than that of steel cable-stayed bridge, whereas the torsion natural vibration frequency of the former is higher than that of the latter. A higher torsion-bend frequency ratio is beneficial for enhancing the critical flutter wind speed.
(4) Comparing the full-floating system and the half-floating system, only the longitudinal floating mode natural vibration frequency of the tower-beam elastic restraint system increases, whereas the other frequencies change slightly. Obviously, the tower-beam elastic restraint system is applicable in the super-long-span cable-stayed bridge.
(5) Setting up one or two auxiliary piers can improve the lateral and vertical fundamental frequencies. However, when three auxiliary piers are set up, only a slight increase is observed. Therefore, setting up 2-3 auxiliary piers is enough and the numbers of piers should not be more than 3.
(6) The first torsional natural vibration frequency of the main girder with inclined cable plane is approximately 20% higher than that with parallel cable plane. Considering that the higher torsional natural vibration frequency can improve wind resistance stability of cable-stayed bridge, the inclined cable plane is more suitable for long-span cable-stayed bridge.
The authors declare that they have no conflicts of interest.
The authors gratefully acknowledge the National Natural Science Foundation of China (no. 51778059) for their support.