INVESTIGATION OF DYNAMIC CHARACTERISTICS OF FRP CABLES IN THOUSAND-METER SCALE CABLE-STAYED BRIDGE

Xin Wang and Zhishen Wu Department of Urban and Civil Engineering, Ibaraki University,

Nakanarusawa-cho 4-12-1, Hitachi, 316-8511, Japan Email: zswu@mx.ibaraki.ac.jp ABSTRACT This paper investigates a new type of hybrid fiber-reinforced polymer (FRP) cable applied in thousand-meter scale cable-stayed bridge with a special emphasis on the dynamic characteristics of the bridge with these hybrid FRP cables. The limitations of conventional steel stay cables and the advantages of hybrid basalt and carbon FRP (B/CFRP) cables were first clarified. The dynamic behavior of cable-stayed bridges in two main spans (1088m and 2176m) with conventional steel cables, CFRP, BFRP and hybrid B/CFRP cables were analyzed by means of finite element method, respectively. Results show that the hybrid B/CFRP cable processes much higher natural frequencies than steel cables, which could lower the possibility of resonance induced by interaction between stay cables and the bridge. Furthermore, the aerodynamic stability of hybrid B/CFRP cables may be superior to that of the other cables due to its designable internal damping. KEYWORDS FRP cable, hybrid, cable-stayed bridge, dynamic characteristics, aerodynamic stability INTRODUCTION Cable-stayed bridges have become a widely used type of long-span bridges, due to their superior self-balancing structural system, higher overall stiffness and better aerodynamic behavior in comparison to suspension bridges (Gimsing 1997). Currently, cable-stayed bridges have reached thousand-meter scale, such as Stonecutters Bridge (1,018 m) and Sutong Bridge (1,088 m), which was previously the economical span only for suspension bridges. Moreover, even longer spans have been planned (Meier 1987, Nagai 2004). However, traditional materials, such as steel cables, and concrete or steel main girders, may restrict the advantages of cable-stayed bridge with increasing span, because of their significantly increased weight and relatively poor durability. Among these problems, corrosion and fatigue are mostly common seen in steel cables. To essentially solve above problems, fiber-reinforced polymer (FRP) materials have been recommended to replace steel cables and achieve both short-term and long-term superior performance, owing to their superior mechanical and chemical properties. Applications of FRPs in super-long span bridges have been discussed worldwide for many years, the most famous being the 8,400 m span cable-stayed bridge with CFRP cables across the Strait of Gibraltar proposed by Meier (Meier 1987). After that, many investigations and experiments have been conducted regarding FRP cables (Keller 2003), especially for anchorage (Meier 1996). Meanwhile, some pedestrian or vehicular bridges have been constructed for the purpose of testing the actual performance of FRP cables (Scalea 2000). In previous studies of composite cable-stayed bridges, carbon fiber composites are usually applied as stay cables to overcome the shortcomings of steel cables (Lu and Mei 2007, Chen and Lau 2006). However, the high and continually increasing cost of carbon fibers restricts their applications in large-scale constructions like bridges. Moreover, the sensitivity of CFRP cables to the wind load is difficult to control due to their extremely light weight and high strength. In these cases, basalt fibers may be a superior choice for long stay cables. Basalt fibers not only have similar costs and mechanical properties as glass fibers, but they are much more chemically stable when exposed in severe environment (Wu et al. 2009a). The only problem for basalt fibers, their low modulus, could be ameliorated by hybridizing with a small proportion of carbon fibers (Wu 2004). In a previous study (Wu and Wang 2008, Wang and Wu 2009), it was suggested that hybridization of basalt and carbon fiber cables with a proportion of 2.57:1 can achieve better integrated performance in terms of static analysis. Based on this study, the dynamic characteristics of cable-stayed bridge with hybrid B/CFRP cables will be discussed; meanwhile the bridges with steel, CFRP and BFRP cables are also analyzed as references. FUNDAMENTAL PROPERTIES OF STAY CABLES

381

Mechanical properties

0

500

1000

1500

2000

2500

3000

3500

4000

0 0.005 0.01 0.015 0.02 0.025 0.03Strain (ε)

Stre

ss (M

Pa)

PC steel Carbon FRPAramid FRPE-Glass FRPBasalt FRPHybrid B/CFRP

40%

50%

60%

70%

80%

90%

100%

110%

0 1 2 3 4 5 6

LogN

S=Pm

ax/P

avg

CFRP

GFRP

BFRP

C1G2

C1B1

S=1.001-0.026logN

S=1.011-0.042logN

S=1.004-0.062logN

S=1.015-0.069logN

S=0.997-0.071logN

Fig. 1 Idealized Stress-Strain relationship Fig. 2 Fatigue behavior of different FRP composites

The mechanical properties of the materials of stay cables are listed in Fig. 1. It is shown that hybrid B/C fibers with a proportion of 2.57:1 (B:C) exhibit higher stiffness than basalt fibers and linearity before the rupture of carbon fibers, after which the capacity of basalt fibers could continue to be exerted as ductility stock, although this would not be considered in the design of cable. On one hand, this hybrid method improved the material modulus to satisfy the requirement of stiffness; on the other hand, the small amount of carbon fibers would not greatly increase the total cost.

Fatigue behavior of FRP

Besides the short-term mechanical properties described above, the properties related to their long-term behavior of fatigue resistance are usually considered strictly in the design of stay cables subjected to the dynamic tensile loading. Fatigue tests on the sheets of carbon fibers, basalt fibers, and hybrid C1B1 (Ratio of C/B=1:1) were conducted (Wu et al. 2009b). Test results of S-N curves (Fig.2) show the fatigue-resistance strength of hybrid C1B1 sheet was increased significantly close to the level of carbon fiber sheet. This hybrid effect on improving fatigue resistance makes hybrid B/C fibers more suitable as stay cables for carrying dynamic load. Design of cable section For hybrid B/CFRP cables, a special arrangement of wires or stands could be operated to consider aerodynamic stability. The hybrid cable section is shown in Fig 3(c), where each hybrid B/CFRP wire consists of hybridizing carbon fibers and basalt fibers in a certain proportion. This design is different from common cable wires, as shown in Fig 3 (a) and (b), BFRP wires are arrayed in the center of cable and hybrid B/CFRP wires are arrayed in outer layers, where the inner sleeve is preset for fixing the shape. A gap between the two layers would be left, which is filled with a viscoelastic material. With this design, relative movement between inner BFRP wires and outer hybrid wires may occur when the cable is excited by dynamic load due to their inherent differences in sag. This relative movement can consume vibrational energy through compressing the filled viscoelastic material, which may also be regarded as improving the internal damping. Based on this design, the hybrid C/BFRP cables are expected to have superior performance over steel, CFRP, and BFRP in terms of aerodynamic stability.

Figure 3 The arrangement of wires or stands in the cable Figure 4 Elevation of the bridge model (unit:m)

100 100 300 544

230

70

YX

Lr

11@12 22@16 33@16100100300544

11@1222@1633@16

Lr Lr Lr

(600) (1088)(200)(200)

(11@24) (22@32) (33@32)

(600)(1088)(200) (200)

(11@24)(22@32)(33@32)

(460

)(1

40)

Middle span line

Steel box girderPylon

Single wire

Single 7-wire strand

(a) Parallel wires (b) Parallel strands

BFRP wires(strands)

Hybrid B/CFRP wires(strands)

Viscoelatic material

Inner sleeve

(c) Hybrid cable section

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BRIDGE MODEL AND PRINCIPLES Since the present longest cable-stayed bridge spans 1,088 m, the models are built according to this size; meanwhile, the models with a main span of 2,176m (twice the size of the existing longest one) are also built to verify the influence of FRP cables with the elongation of span. The shape and dimensions of the bridge model are shown in Fig 4, which are the dimensions of the Sutong Bridge with a main span of 1,088 m (Zhang 2003). Detailed parameters of every portion of the bridge are listed can be found in (Wang and Wu 2009). Stiffness equivalent principle is adopted to design FRP cables due to the deficiency of cable stiffness obtained by strength equivalent principle, which was described in detail in Wang and Wu 2009. With stiffness equivalent principle, the design parameters are listed in Table 2.

Table 1 Design parameters by stiffness equivalent principle Type of cable Stiffness Modulus (GPa) Sectional area Design stress(MPa)

Steel Ks 195 As 708 Carbon Ks 230 0.85As 835 Basalt Ks 91 2.14As 330

Hybrid B/C Ks 130 1.50As 472 Since the sag effect existing in stay cables, the equivalent modulus is introduced to simplify the nonlinear stress-strain property of stay cable. The tangent equivalent modulus is a common method used in finite element analysis, which use the initial design stress to describe stress-strain relation, as shown in Eq (4). Another method is secant equivalent modulus also in Eq(4), which present stress-strain relation within two stresses’ states. The accurate modulus Ecat can be derived through the catenary curve. However, it is too complex to apply it in finite element analysis. The comparison of these three methods are listed in Table 2

tan 2

31 .12cb

EEL Eγσ

=

+

and sec 2

321124

cb

EEL E k

kγ

σ

=+ + +

(4)

Table 2 Comparison of modulus expressions

Type 1088m span 2176m span Steel Carbon Basalt Hybrid B/C Steel Carbon Basalt Hybrid B/C

(Etan- Ecat) /Ecat 4.93% 0.16% 2.72% 1.08% 12.32% 0.49% 7.43% 3.17% (Esec-Ecat) /Ecat 1.78% 0.06% 1.00% 0.40% 5.02% 0.22% 3.12% 1.37%

For the span of 1088m, Etan could achieve sufficient precision (<5%) regardless the types of cables. However, when span increases to 2176m, Etan is obviously not suitable for basalt and steel cables. Thus, Esec would be used in the analysis of bridges with 2176m span. DYNAMIC PERFORMANCES OF CABLE-STAYED BRIDGE Parameters of analysis model

Figure 5 Dynamic model of cable-stayed bridge Figure 6 Natural frequencies of the entire bridge

(with and without element shapes)

Main girder Rigid beam

Cables Transversal constrains

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Order of natural frequency

Nat

ural

freq

uenc

y (H

z

steel cables-1088 CFRP cables-1088BFRP cables-1088 B/C FRP cables-1088steel cables-2176 CFRP cables-2176BFRP cables-2176 B/C FRP cables-2176

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The entire bridge was modeled by the single-beam method, by which the bridge girders, pylon and piers are simulated by three-dimensional beam elements, and the bridge deck was simulated by rigid beams, through which the cables simulated by the two-dimensional link elements are connected (Fig. 5). The deformations at the connection of piers and main girders are constrained except for longitudinal displacement, and the transverse displacement between the main girder and pylon at the location of the pylon are constrained. The ends of the pylons and piers are fixed. The model of a single cable was divided into thirty sections to achieve sufficient accuracy, where each section was simulated by two-dimensional beam elements that can automatically calculate sags. The initial strain in the cable was also taken into account. The two ends of cable were fixed. Results of analysis 1) Dynamic characteristics of the entire bridge From Fig.6, it can be seen that the mode shapes of different bridges at both spans (1088m and 2176m) remain invariant despite differences of cables, which indicates that the influences of cables to the entire bridge is very small. The natural frequencies of entire bridge decrease with the increasing of span and the corresponding mode shapes are partially changed which torsion mode shapes appear at 20th order in comparison to 18th of 1088 span bridge (Table 3). The modal characteristic is only determined by the main girder and pylon of the bridges.

Table 3 The first eighteenth natural frequencies of bridges with different cables Order 1 2 3 4 6 8 9 10 11 18 20

Mode shape

1088m span Floating

Sway sym

.

Heave sym.

Heave anti-sym.

Sway anti-sym.

Tower bending

sym.

Tower bending

anti-sym. / / Torsion /

2176m span Floating Sway

sym. Heave sym.

Heave anti-sym.

Sway anti-sym.

/ / Tower

bending sym.

Tower bending

anti-sym. / Torsion

2) Dynamic characteristics of stay cables From Table 4, obvious differences in the natural frequencies of cables can be found, CFRP cable exhibits the highest natural frequencies in every order due to its low density and sectional area determined by the equivalent stiffness principle, whereas the natural frequencies of BFRP cable are very close to those of steel cable because its weight per unit length is similar to steel cable. Although the natural frequencies of hybrid B/CFRP are lower than CFRP cables, an obvious improvement still could be seen in comparison with steel cable. The natural frequencies of different cables in 2176 m span occupy nearly half of the one with 1088 m span, which shows linearity with respect to the length.

Table 4 The first fifth natural frequencies of different stay cables Order

Freq. 1 2 3 4 5

Steel

Length of cable

575m 0.27 0.55 0.82 1.10 1.37 1150m 0.14 0.27 0.41 0.55 0.68

CFRP 575m 0.65 1.31 1.96 2.62 3.27 1150m 0.33 0.65 0.98 1.31 1.64

BFRP 575m 0.32 0.63 0.95 1.27 1.58 1150m 0.16 0.32 0.48 0.64 0.80

Hybrid B/CFRP

575m 0.41 0.81 1.22 1.63 2.03 1150m 0.20 0.41 0.61 0.81 1.02

3) Impact of stay cables to the vibration of bridge The possibility of resonance between stay cables and the bridge could be evaluated by comparing the natural

frequencies of the bridge and stay cables, which may be conducted through two aspects: one is by comparing the first natural frequencies of stay cables of different lengths with the first eighteen natural frequencies of bridges; the other is by comparing the first few natural frequencies of the longest cables with the first eighteen natural frequencies of the bridges.

0

0.5

1

1.5

2

2.5

3

150 200 250 300 350 400 450 500 550 600

Length of stay cables (m)

Firs

t nat

ural

freq

uenc

y (H

z)

Steel cable

CFRP cable

BFRP cable

Hybrid B/CFRPcablemin

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

300 500 700 900 1100

Length of stay cables (m)

Firs

t nat

ural

freq

uenc

y (H

z

Steel cable

CFRP cable

BFRP cable

Hybrid B/CFRPcablemin

(a) 1088m span (b) 2176m span

Figure 7 Possibility of resonance between bridges and cables with different lengths

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Regarding the first aspect, the actual lengths of the cables (form shortest to longest:161 m, 202m, 305, 407m, 500 m, and 575m for 1088m span; 322 m, 404m, 610, 814m, 1000 m, and 1150m) were chosen to calculate their first natural frequencies. The analysis results are shown in Fig. 7, where the dashed lines represent the range of the first eighteen (for 1088m) and twentieth (for 2176m) natural frequencies of the bridges. It is shown that the majority of steel cables are in the range of possible resonance between the cables and bridge. Obviously, the first natural frequencies of hybrid B/CFRP cables are improved compared with steel cables. Thus, for the cables shorter than 350m, which is 61% of the longest cable, resonance between the cables and bridge will not occur. And this advantage will become more apparent in 2176m span bridge, where resonance will not occur when the cable length below 814m (71% of the longest cable).

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

The order of natural frequencies

The

natu

ral f

requ

enci

es (H

z)

Steel cable

CFRP cable

BFRP cable

Hybrid B/CFRPcableEntire bridge

0.00

0.20

0.40

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0.80

1.00

1.20

1.40

1.60

1.80

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The order of natural frequenciesTh

e na

tura

l fre

quen

cies

(Hz

Steel cable

CFRP cable

BFRP cable

Hybrid B/CFRPcableEntire bridge

(a) 1088m span (b) 2176m span

Fig. 8 Possibility of resonance between bridges and cables From the other aspect, the first five natural frequencies of different cables with the length of 575m (1150m for 2176m span) are adopted to compare the first eighteen/twentieth natural frequencies of the bridge, which is plotted in Fig.8, where the dashed lines also represent the range of the first eighteen/twentieth natural frequencies of the bridge. It is shown that the first and second natural frequencies of the steel cable and BFRP cable are located in the range of the risk of resonance, whereas only the first natural frequencies of the CFRP cable and hybrid B/CFRP cable are at risk of resonance. No obvious difference could be found in comparison to the figures with two spans. Therefore, a lower possibility of resonance could be realized through applying CFRP and hybrid FRP cables. It is proven from above two aspects that the hybrid B/CFRP cables possess similarly low possibility of resonance between cables and bridge as CFRP cables, which would benefit stability of the entire bridge under the vibration. 4) Aerodynamic stability of stay cables The aerodynamic stability of stay cables is often evaluated by the Scruton number (Sc=2δme/ρD2) (Caetano 2007), which is a non-dimensional parameter that represents the occurrence of violent vortex-induced vibrations. The detailed parameters can be referred to Caetano 2007, and the comparisons of Sc are shown in Table 5.

Table 5 The Sc number of different stay cables Types Steel CFRP B/C FRP BFRP

575m 1150m 575m 1150m 575m 1150m 575m 1150m Sc 50 55 7 11 11 17 12 19

Decrease (%) 100% 100% 14% 20% 23% 31% 25% 35% Table 5 shows that all of the FRP cables exhibit much smaller Sc compared with the steel cable due to their low densities. It is reported that the risk of violent vortex-induced vibration can be avoided only when Sc is greater than 20. Moreover, the risk is very significant if Sc is less than 10 (Caetano 2007). For two spans of bridges, the CFRP cable only has Sc of 7 and 11, which is 14% and 20% of the steel cable and which could make it very sensitive to vortex-induced vibration. The B/CFRP cable behalves better than the CFRP cable and similar to the BFRP cable, but it is still insufficient to thoroughly avoid the risk of violent vortex-induced vibration. The only parameter that could be designed to improve the Sc is the logarithmic decrement of damping, which usually consists of two portions, internal damping of cable and the damping provided by external damper. Internal damping of the CFRP cable can not be improved sufficiently in comparison with steel cables due to their identical structural style. Thus, the cost would be increased by installing external dampers in order to improve the Sc. However, for the hybrid B/CFRP cable, not only is the Sc essentially higher than that of the CFRP cable, but also its internal damping could be designed to achieve a much higher value than the usual steel or CFRP cables by means of separating basalt wires and carbon wires in one cable, the design of which is shown in Fig. 3(c). Through this method the energy of vibration could be consumed by the relative movement of basalt and carbon fiber wires due to their inherent differences in sag. From this perspective, the hybrid FRP cable can exhibit potentially superior advantages in enhancing the aerodynamic stability.

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CONCLUSIONS The advantages of using hybrid FRP cables in thousand-meter scale cable-stayed bridge were investigated in this paper. The results show not only that hybrid C/BFRP cables exhibit better economy than CFRP cables but also that the bridge with hybrid C/BFRP cables exhibits similarly high performances to a bridge with CFRP cables. Major conclusions are drawn as follows: (1) Hybrid B/CFRP could integrate the advantages of both CFRP and BFRP composites, and simultaneously

eliminate their disadvantages, which could not only provide superior mechanical performances (relatively high stiffness, small sag, superior fatigue resistance) for stay cables, but also lower the cost and enable practical application.

(2) The 1088m span bridges with different FRP cables exhibit similar natural frequencies compared with steel cable bridge, while the mode shapes are constant among bridges with different cables. The similar rules are observed for 2176m span bridge. The small influence of different FRP cables on the entire bridge proves that vibrational properties are determined by bridge girders and pylons instead of stay cables.

(3) The vibrational properties of a single stay cable differ greatly for different materials. Higher natural frequencies of different FRP stay cables are all observed as compared to steel cables, while CFRP and hybrid B/CFRP cables possess much higher natural frequencies than steel and BFRP cables, which results in a low possibility of resonance between stay cables (CFRP or hybrid B/C FRP cables) and the entire bridge. This advantage manifests more apparently for the cables in 2176m span bridges.

(4) Hybrid B/CFRP cables exhibit potentially superior aerodynamic stability than the CFRP and BFRP cables due to their designable internal damping. This advantage would greatly enhance the design for controlling vibration under a dynamic load.

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