Mechanical performance analysis and stiffness test of a new type of suspension bridge

Abstract

A new type of suspension bridge is proposed based on the gravity stiffness principle. Compared with a conventional suspension bridge, the proposed bridge adds rigid webs and cross braces. The rigid webs connect the main cable and main girder to form a truss that can improve the bending stiffness of the bridge. The cross braces connect the main cables to form a closed space truss structure that can improve the torsional stiffness of the bridge. The rigid webs and cross braces are installed after the construction of a conventional suspension bridge is completed to resist different loads with different structural forms. A new type of railway suspension bridge with a span of 340 m and a highway suspension bridge with a span of 1020 m were designed and analysed using the finite element method. The stress, deflection of the girders, unbalanced forces of the main towers, and natural frequencies were compared with those of conventional suspension bridges. A stiffness test was carried out on the new type of suspension bridge with a small span, and the results were compared with those for a conventional bridge. The results showed that the new suspension bridge had a better performance than the conventional suspension bridge.

Graphical abstract

Keywords

new type of suspension bridge / stiffness test / mechanical performance / railway bridge / space truss

1 Introduction

Suspension bridges have a wide range of applications in long-span bridge engineering because of their strong spanning ability, efficient material utilisation, and aesthetic values [1,2]. In recent decades, suspension bridges with spans greater than 1000 m have been built worldwide [3], with the Akashi Kaikyo Bridge [4] in Japan being the most famous. This bridge has three spans with a main span of 1991 m. The 1915 Çanakkale Suspension Bridge [5] with a main span of 2023 m is under construction in Turkey. The Strait of Messina Bridge in Italy [6] has been designed as a 3300 m suspension bridge connecting Sicily and the Apennine Peninsula. The Taizhou Yangtze River Highway Bridge [7] in China, which has the world’s longest span of any multi-tower suspension bridge, with a main span of 1080 m, has led to a new era of multi-tower suspension bridges. Currently, suspension bridges are still a good option for ultra-long continuous crossings [8,9].

A conventional suspension bridge is a flexible structure supported by main cables. The girders have large deflections under live loads, which results in large angles of rotation at the ends of the bridge, especially for railway bridges [10]. Additionally, a conventional suspension bridge may have a low fundamental frequency [11]. Out-of-plane swinging quickly occurs under a dynamic load, especially for high-speed train loads [12–14]. The application of the conventional suspension bridge for railway bridges is restricted because of this flexibility. In a long-span suspension bridge or railway suspension bridge, a truss girder is typically adopted to improve the structural stiffness [15]. For example, double-deck steel truss girders [16] were used in the Wufengshan Changjiang River Bridge and Akashi Kaikyo Bridge.

For a multi-tower suspension bridge, such as the three-tower suspension bridge, the main cable's unbalanced horizontal force at the top of the middle tower is large under live loads [17,18], especially for railway suspension bridges and super long-span suspension bridges. If the force exceeds the frictional resistance between the saddle and cable, the main cable will slip in the saddle, destroying the bridge. Therefore, limiting the unbalanced force becomes a key goal when designing a multi-tower suspension bridge [19]. Various measures have been shown to reduce the unbalanced force, including the following [20,21]. Firstly, design the tower with reasonable stiffness, such as by adopting a steel–concrete composite structure for the middle tower. Secondly, install a longitudinal restraint on the girder at the intersection of the middle tower and girder. Thirdly, break the main cables at the tower top and anchor them to the middle tower. This method was adopted for the Xiaonaruto Bridge constructed in Japan in 1961. It is worth noting that solving the excessive unbalanced force problem is an especially important issue when designing multi-tower suspension bridges subjected to harsh ocean conditions.

Another problem that has been faced for a long-span suspension bridge is its flutter stability [22,23]. The reason for the decrease in the flutter velocity with an increase in the span length is apparent. As the span length increases, both the bending and torsional frequencies decrease. They tend to become similar because the corresponding mode shapes consist of in-phase or out-of-phase cable vibrations. The deck’s contribution to the torsional stiffness of the bridge continuously decreases with an increasing span length. As a result, a rather poor flutter performance results for all the deck cross sections involving the so-called super-long-span range. For example, the Tacoma Narrows Bridge [24] experienced torsional vibration under a continuous wind load (winds of 26.1 kph and a frequency of 0.2 Hz) because of insufficient torsional stiffness. Flutter-vibration then developed, which eventually caused the bridge to collapse. Domestic and foreign scholars have proposed various structural systems to improve the torsional stiffness. In particular, methods for modifying the main cable system have been proposed, such as by using a crossed-hanger system [25], three-dimensional cable system [26], or cable-stayed-suspension hybrid bridge [27]. The Yavuz Sultan Selim hybrid cable-stayed and suspension bridge [28] is an important case. The bridge has three spans with a main span of 1408 m. In addition, the total weight of the stay-cables and main cable can be reduced by more than 40% through cable size optimization [29]. Another method to improve aerodynamic stability is to decrease the aerodynamic force by optimising the shape of the girder’s cross-section [30–32], such as by using a wind fairing structure, slotted box section (Tsing Ma Bridge), split-type box girder, or the addition of vertical and horizontal stabilising boards. Additionally, installing auxiliary devices on the girder to increase the damping of the structure and decrease the aerodynamic force acting on the structure can also improve aerodynamic stability [33,34]. There are two main types of such devices. One is a damper, and the other is a control surface installed on the windward and leeward edges of the cross-section of the girder. The above studies [25–34] mainly reduced the wind-induced vibration via structural optimisation, aerodynamic measures, and mechanical measures. Although these certainly provided results, some of these techniques are difficult to achieve and are still being discussed.

This study had the goal of solving the above issues through structural system innovation, improving a suspension bridge’s mechanical performance, and expanding its railway applications. A new suspension bridge was proposed in this study, based on the gravity stiffness of the main cables. In this new system, rigid webs are added between the main cable and girder, forming a truss structure with the main cable as the upper chord and the girder as the lower chord (Fig. 1). Cross braces are added between the main cables forming a closed space truss structure, to improve the torsional stiffness of the suspension bridge. The rigid webs and cross braces are installed after the suspension bridge (including the main cables, towers, girders, etc.) is completed so that the main cables bear the dead load and the space truss bears the live loads. Additionally, to reduce the free length of the rigid webs in super-long-span bridges, another improved method can be used for the new bridge (Fig. 2). The main cable alignment control [35,36] of the proposed bridge is similar to that of a traditional suspension bridge. The difference is that the dead weight of rigid webs and cross brace need to be considered in the proposed bridge. With the new design proposed in this paper, the bridge retains the advantage of the strong bearing capacity of the traditional structure and has the added advantage of the large rigidity of a truss structure. The bridge’s overall stiffness is improved, and the main cables’ unbalanced horizontal forces at the tops of the towers are reduced. This paper details the structural form and mechanical principles of this new bridge design. It shows how the mechanical properties were investigated using the finite element method for a two-line railway bridge with a 340 m main span and a highway bridge with a 1020 m main span. The new system was also investigated using a three-tower a physical model with a small span.

2 Structural form and methodology

Under dead loads, a conventional suspension bridge has a strong carrying capacity because of the main cables’ high tensile strength. However, under live loads, it has a weak performance because of its low bending stiffness and torsional stiffness. This paper proposes a new type of suspension bridge (Fig. 3) to solve this issue.

In the proposed bridge, rigid webs connect the main girder and main cable to form a truss, which can provide better bending rigidity. Cross braces connect the cables, so that the main cables, main girders, rigid webs, and cross braces form a closed structure, providing better torsional rigidity. Additionally, to reduce the free length of the webs in super-long-span bridges, an improved method can be used for the proposed bridge (Fig. 3(b)). The mechanical principles of the proposed bridge are explained as follows.

2.1 Compressive stiffness of main cable based on gravity stiffness

For convenience, the following analysis is carried out with a flexible horizontal cable without gravity (without bending rigidity) as an example.

1) Tensile stiffness of main cable

As shown in Fig. 4, the unstressed length of a cable is l, and the ends of the cable are hinged. The cable is subjected to a concentrated force at point C, forming a new balanced state (AC′B). It can be assumed that the pull of the cable is T and the angle between AC′ and AB is t in the new balanced state. The vertical force P following can be obtained from the vertical balance condition:

(1)P=2T\sin t.

Based on Hooke’s law, the following is obtained:

(2)\frac{Tl}{2EA}=\frac{l}{2\cos t}-\frac{l}{2},

where E and A are the elastic modulus and cross-sectional area, respectively. According to Eq. (2), the pull of the cable, T, can be obtained as follows:

(3)T=EA\left(\frac{1}{\cos t}-1\right).

Substituting Eq. (3) into Eq. (1) gives the following:

(4)P=2EA(\tan t-\sin t).

Eqation (4) shows the nonlinear relationship between P and t, from which the tensile stiffness of the cable can be obtained:

(5)K_{\text{t}}=\frac{\text{d}P}{\text{d}t}\text{=}2EA({\sec }^{2}t-\cos t).

If P is force due to gravity, then the tensile stiffness, K_t, that it causes is the gravity stiffness.

2) Compression stiffness of main cable

A force Q opposite to force P is added at point C′ (Fig. 5), resulting in rotation angle r (0 ≤ r ≤ t) and tensile force T^{\prime }. The following can be obtained from the vertical balance condition:

(6)P-Q=2T^{\prime }\sin (t-r).

The following can be obtained from Eq. (6):

(7)Q=P-2T^{\prime }\sin (t-r).

Based on Hooke’s law, the following can be stated:

(8)\frac{T^{\prime }l}{2EA}=\frac{l}{2\cos (t-r)}-\frac{l}{2}.

The following can be obtained from Eq. (8):

(9)T^{\prime }\text{=}2EA\left[\frac{1}{\cos (t-r)}-1\right].

Substituting Eqs. (4) and (9) into Eq. (7) gives the following:

(10)Q=2EA\left[\tan t-\sin t-\tan (t-r)+\sin (t-r)\right].

Similar to the tensile stiffness of the cable, the compression stiffness can be calculated as follows:

(11)K_{\text{p}}=\frac{\text{d}Q}{\text{d}r}\text{=}2EA\left[{\sec }^2(t-r)-\cos (t-r)\right].

3) Comparison of tensile stiffness and compression stiffness

For convenience, it is assumed that function F(x) is as follows:

(12)F(x)\text{= 2}EA\text{(}{\sec }^{2}x-\cos x).

We have K_t = F(t) and K_p = F(t − r). The derivative of function F(x) is written as follows:

(13)f(x)\text{= 2}EA({\sec }^{2}x\tan x\text{+}\sin x).

It can be seen that f(x) > 0, x ∈ [0,π/2], so that F(x) is an increasing function in the domain. Therefore, the following can be obtained:

(14)

that is

(15)

4) Effect of vertical force P on compression stiffness

The following can be obtained from Eq. (11):

(16)\frac{\text{d}{K}_{\text{p}}}{\text{d}t}\text{=}2EA\left[{\sec }^2(t-r)\tan (t-r)+\sin (t-r)\right].

It can be seen that {\displaystyle \frac{\text{d}{K}_{\text{p}}}{\text{d}t}} > 0. Thus, K_p is an increasing function. From Eq. (5), it can be seen that t is proportional to P, which indicates that K_p is proportional to P. Therefore, a greater vertical force P, can cause greater compressive stiffness.

The research suggests that the gravity stiffness of a cable is an indication of its compressive stiffness. From the above analysis, the following conclusions can be drawn.

a) K_p increases with P (the greater the gravity stiffness, the greater the compressive stiffness);

b) K_{\text{p}} < K_{\text{t}};

c) when the gravity stiffness is zero, the compressive stiffness of the cable is also zero.

2.2 Forming truss structure to bear live load

The analysis in Section 1.1 suggests that the main cable bears the weight of the structure and second-stage dead load to obtain the gravity stiffness, thereby obtaining the compressive stiffness. Based on this, a truss structure with the main cables as the upper chords and the main girders as the lower chords can be formed to bear the live load. This achieves the goals of reducing the unbalanced force of the towers of a multi-tower suspension bridge and improving the structure’s mechanical properties (stiffness, strength, dynamic characteristics, etc.).

2.3 Adding rigid webs and cross braces in time to resist different loads with different structural forms

If the rigid webs and cross braces are added arbitrarily, the main cable shape of the suspension bridge may be destroyed, causing the rigid webs to bear a more massive dead load, and resulting significant compressive stresses and instability of the members. Therefore, the proposed suspension bridge introduces the concept of ‘using different structures to bear different loads’ (Fig. 6). The method is to install the added components and complete the system transformation after the conventional suspension bridge is built (During construction, first install the main cables, with an unloaded cable shape, then the hangers and main girder, and finally the rigid webs and cross braces. As a result, the main cable is tensioned to the cable shape under the dead loads). Thus, it is possible to achieve the purpose of allowing the conventional bridge to carry the dead loads (including the first and second stages), while the space truss takes the moving load. Therefore, the new structure can prevent the compressive stresses of the rigid webs from being too large and improve their resistance to deformation.

2.4 Connection of main cable and rigid web

In the proposed bridge, the rigid webs are connected to the main cable using cable clamps with gusset plates, as shown in Fig. 7. These cable clamps can be prefabricated in a factory. The rigid webs can be hinged or welded to the gusset plates. The length of the cable clamp is determined by the anti-skid friction of the truss joint. In terms of the spatial treatment of the rigid web and hanger, the rigid web adopts a double-limb lattice structure, which can prevent a space conflict between the rigid web and hanger.

3 Finite element analysis

3.1 Comparison of conventional and proposed bridges for two-line railway

A two-line railway bridge was selected as an example to study the stiffness, strength, and dynamic characteristics of the proposed bridge. A finite element analysis (FEA) was performed, and the results were compared with those for a conventional two-line railway suspension bridge with the same span arrangement.

3.1.1 Calculation models for proposed bridge and conventional bridge

As shown in Fig. 8, the main span was 340 m, and the sag ratio of the mid-span was 1/9. H-shaped concrete towers with a height of 50 m were used for two bridges, the proposed new bridge and a conventional bridge, and a 4.5 m-deep main girder was adopted. The main girder was made of steel and covered by a concrete layer with a thickness of 25 cm. The rigid webs of the proposed bridge adopted latticed steel structures, which prevented a space conflict with the hangers. The cross braces adopted steel box structures. The detailed parameters of the two bridges were summarised in Tables 1 and 2. The amount of Q345 steel used for the proposed bridge was 16.4% more than that of the conventional bridge. The amount of Q345 steel used for the former was 8121.2 t (1035.9 kg/m²), of which the amount of steel used for the truss structure was 1147.2 t, accounting for 14.1% of the total amount of steel used.

The mechanical properties and unbalanced horizontal forces at the tops of the towers of the two bridges were tested and analysed using the MIDAS^TM program. In the element analysis, firstly, considering the dead loads, the cable shape of the completed bridge was obtained by the main cable shape of suspension system in MIDAS. Secondly, the unloaded cable shape was obtained through a reverse analysis. The stressless cable length and the completed bridge cable length of the proposed bridge were 662.691 and 664.157 m, while those of the conventional bridge were 662.765 and 664.155 m, respectively.

In the finite element models (Fig. 9), beam elements were used to simulate the girders, towers, rigid webs, and cross braces, and cable elements were used to simulate the cables. For boundary conditions, all the degrees-of-freedom of the tower foot nodes were fixed. The main girder had rigid vertical support at the piers and was coupled to the closest node in each tower to restrain the vertical movement, transverse movement and rotation. The following conditions and assumptions were used in the modelling process. The railway load grade used followed code [37]. The railway load was applied to the most unfavourable position based on the influence line. The loads specified and their combinations are described below.

1) The live load was composed of a uniformly distributed load of 64 kN/m and four concentrated forces of 200 kN. In the middle position, four concentrated forces are arranged at an interval of 1.6 m. In the two side positions, the uniform distribution forces are arranged.

2) The secondary dead load was 170 kN/m.

3) To determine the effect of temperature, the initial temperature was 15°C, which was increased by 25°C and then decreased by 20°C.

4) The following load combinations were used:

-combination I: dead load;

-combination II: dead load + train load;

-combination III: dead load + train load + action due to the highest temperatures;

-combination IV: dead load + train load + action due to the lowest temperatures.

3.1.2 FEA results

3.1.2.1 Analysis of structural internal force and stresses

(1) Structural internal force

An FEA was performed. The axial force values of the main girder and main cable under load combination I and load combination II (the combination of the dead load and static live load) were obtained for the two different bridge systems. Figures 10 and 11 show the results, respectively. Table 3 lists the percentages of change for axial force distributions of main cables and main girders.

The obtained results suggest that under load combination I, the axial force distributions of the main cable and main girder for the two systems are similar. Under load combination II, for the proposed bridge, the axial force of the main girder gradually increases along the mid-span direction, and the axial force of the main cable gradually decreases along the mid-span direction. In contrast, for the conventional bridge, the axial force of the main girder is smaller and remains the same in every position, and the axial force of the main cable is the same in every position. The axial force distribution of the main girder for the proposed bridge is 7388.5% larger than that of the conventional bridge. Therefore, the suspension bridge in this paper brings the tensile capacity into full play for the main girder. The axial force distribution of the main cable for the proposed bridge is 16.64% larger than that of the conventional bridge.

In the proposed bridge, the main girder, the main cables and the rigid webs are connected to form a truss structure that bears the live load and the self-weight of the rigid webs. (System transformation has been described in Section 1.3). The main girder is tensioned as the lower chord of the truss. Therefore, the axial force increases in the main girder of the proposed bridge. The results indicate that the space truss formed by the main cable and girder plays the role of a truss under the live load.

(2) Structural stresses

Table 4 lists the stresses in the towers, main cables, girders, hangers, and rigid webs of the bridges. It shows that under load combination I, the maximum stresses of the main cables, girders, and main towers of the proposed bridge are similar to those of the conventional bridge. This is because the force transmissions in the two systems are the same under a dead load. In other words, the dead load of each system is mainly borne by the main cable. In the proposed bridge, the maximum tensile and compressive stresses of the rigid webs are 32.4 and 35.9 MPa, respectively, mainly because of their self-weight. Under the action of load combination II, the maximum stress of each component of the proposed bridge shows a significant reduction or remains the same as that in the conventional bridge. The results indicate that the proposed bridge adopts a truss structure to bear the moving load, which has better strength.

Table 5 lists the stresses in the different components of the bridges under load combinations III and IV. It can be seen that the stress increment (compared with load combination II) in the proposed bridge is larger than that of the conventional bridge because of the higher statically indeterminate internal structure. However, the stresses of the main cables and girders of the proposed bridge, under load conditions I and II, are smaller than those of the conventional bridge. Thus, under these load combinations (load conditions III and IV), the stresses in the main cables and girders of the proposed bridge are still smaller than those of the conventional bridge.

3.1.2.2 Analysis of structural stiffness

Table 6 reports the deflections of the main girders under the action of moving loads. Figure 12 presents the envelopes for the two bridges. Figure 13 illustrates the deflections of the girders for the two bridges under the most unfavourable vehicle-loading scenario. The maximum deflection of the main girder occurs at the middle of the main span for the proposed bridge, but at 1/4 of the main span for the conventional bridge. The maximum negative deflection of the girder is 975.2 mm for the conventional bridge and 325.9 mm for the proposed bridge, which is 66.6% smaller. Figure 13 suggests that the maximum positive deflection of the main girder is 375.9 mm for the conventional bridge and 89.1 mm for the proposed bridge, which is 76.3% smaller. The results indicates that the method proposed in this article can significantly improve the stiffness of a conventional suspension bridge. Because of the stiff truss, the horizontal displacement of the top of the main towers of the proposed bridge is also significantly reduced, with a value that is 66.53% smaller than that of the conventional bridge.

3.1.2.3 Analysis of structural frequency

The dynamic responses of the structures were analysed using the linear elasticity principle. Figure 14 displays the first five natural vibration modes. It indicates that the first out-of-plane vibration frequency of the conventional bridge is 0.2656 Hz, and that of the proposed bridge is 0.3006 Hz, which is 13.2% greater. The first in-plane vibration frequency of the former is 0.2851 Hz, and that of the latter is 0.4911 Hz, which is 72.3% greater. These results indicates that the proposed bridge exhibits better dynamic characteristics, especially in-plane dynamic characteristics.

3.2 Comparison of conventional and proposed bridges with three-tower system

A three-tower suspension bridge with large spans was used to study the stiffness, strength, dynamic characteristics, and unbalanced forces of the second type of proposed bridge. An FEA was performed, and the calculation results were compared with those for a conventional suspension bridge with the same span arrangement. The total material consumption of the proposed bridge was equivalent to that of the conventional bridge.

3.2.1 Calculation models for proposed bridge and conventional bridge

The span arrangement of the proposed and conventional bridges was 315 + 1020 + 1020 + 315 m, as shown in Fig. 15. The sag ratio of the mid-span was approximately 1/10. The main tower was 195 m high. The main girder was a steel–concrete composite girder with a width of 28 m covered by a concrete layer with a thickness of 15 cm. A box-shaped concrete structure was adopted for the side towers, and a steel box structure was adopted for the middle tower. The rigid webs of the proposed bridge adopted a latticed steel structure, which could prevent a space conflict with the hangers. The oblique chord and cross braces adopted steel box structures. Tables 7 and 8 detail the component parameters and material consumption of the proposed bridge and conventional bridge, respectively. For the sake of comparison, the total material consumption of the proposed bridge was designed to be equivalent to that of the conventional bridge. It can be seen from Tables 8 and 9 that the amount of Q345 steel used for the proposed bridge was 37960.6 t (506.3 kg/m²), of which the amount of steel used for the truss structure was 10377.6 t, accounting for 27.3% of the total amount of steel used. The steel used for the conventional bridge was 37853.0 t, which was approximately equal to that of the proposed bridge. The stressless cable length and the completed bridge cable length of the proposed bridge are 2899.06 and 2907.10 m, while those of the conventional bridge are 2898.89 and 2907.11 m, respectively. In the finite element models of the bridges (Fig. 16), the elements of each component and the boundary conditions were similar to those in Section 2.1.1. The following conditions and assumptions were used in the modelling process. The traffic load grade used followed the Chinese code ‘General Specifications for Design of Highway Bridges and Culverts’ (Ministry of Transport of P.R. China 2015) [38]. The traffic load was applied to the most unfavourable position based on the influence line. The loads specified and their combinations are described below.

1) The following design loads and load combinations were considered in the calculations.

2) The secondary dead load was 70 kN/m.

3) To determine the effect of temperature, the initial temperature was 15°C, which was increased by 25°C (to 40°C) and decreased by 20°C (to –5°C).

4) The following load combinations were used:

-combination I: dead load;

-combination II: dead load + moving load;

-combination III: dead load + moving load + action due to the highest temperature;

-combination IV: dead load + moving load + action due to the lowest temperature.

3.2.2 FEA results

3.2.2.1 Analysis of structural stresses and internal force

1) Structural internal force

The axial force values of the main girder and main cable under load combinations I and II for the two different bridge systems are shown in Figs. 17 and 18, respectively. The percentages of change for axial force distributions of main cables and main girders are listed in Table 9.

Under load combination I, the axial force distributions of the main girder and main cable for the two bridges are similar. Under load combination II, the axial force of the main girder gradually increases along the mid-span direction, and the axial force of the main cable gradually decreases along the mid-span direction. The axial force distribution of the main girder for the proposed bridge is 3638.36% larger than that of the conventional bridge. The axial force distribution of the main cable for the proposed bridge is 10.18% larger than that of the conventional bridge. Compared with the example in Section 2.1, the change in the axial force is not obvious. This is because the live load of the highway bridge is smaller than that of the railway bridge.

2) Structural stresses

Tables 10 and 11 present the stresses of the proposed bridge and conventional bridge. Table 10 indicates that under the dead load, the maximum stresses of the main cables, main girder and main towers of the proposed bridge are similar to those of the conventional bridge. The maximum tensile and compressive stresses of the newly added components are 54.3 and 52.1 MPa, respectively. Under the action of load combination II (dead load + moving load), the maximum stress of each component of the proposed bridge shows a significant reduction from or remains the same as that of the conventional bridge. The results indicate that the proposed bridge is stronger.

Table 11 illustrates that under load combination III, the maximum tensile stresses of the main girders of the proposed bridge and conventional bridge are 55.6 and 103.7 MPa, respectively. Under the action of load combination IV, the maximum tensile stresses of the main girders of the two are 110.3 and 103.7 MPa, respectively. The results show that under the actions of different temperatures, the maximum stresses for the main girder of the conventional bridge are unchanged, whereas the stresses of the proposed bridge change significantly, indicating that the latter’s temperature response is more pronounced than that of the former. Under the effect of load combination III or IV, all the maximum stresses of the proposed bridge’s members (except the main girder) are also reduced or unchanged compared with those of the conventional suspension bridge, and the stress of each component meets the strength requirements.

The results show that the temperature response of the proposed bridge is larger than that of the conventional bridge, but the stress caused by the moving load is small. Under the action of the combined loads, the maximum stresses of most members decrease. Moreover, the stresses of most hangers of the proposed bridge are smaller than those of the conventional bridge under the various load combinations.

3.2.2.2 Analysis of unbalanced forces of main towers

Table 12 and Fig. 19 present the unbalanced forces of the main towers of the proposed and conventional bridges under the action of a moving load. The maximum unbalanced force of the side tower of the conventional bridge is 3418.7 kN, and that of the proposed bridge is 7426.3, which is 117.2% larger. The maximum unbalanced force of the middle tower of the conventional bridge is 17174.3 kN, and that of the proposed bridge is 10938.7 kN, which is 36.3% smaller. The results show that the unbalanced force of the side tower of the proposed bridge is more significant than that of the conventional bridge, but that of the middle tower is significantly reduced. This is because the newly formed space truss has better integrity, and the longitudinal deformation of the main towers is more coordinated. However, there only exists slight difference between the side towers and the middle towers of the new bridge for the maximum unbalanced force. That is, the peak value of the unbalanced force of the bridge’s main towers is effectively reduced. Therefore, the new design can significantly reduce the unbalanced force of the main towers.

3.2.2.3 Analysis of structural stiffness

Table 13 lists the deflections for the two main girders of the two bridges under the action of moving loads. Figure 20 shows the envelopes. Figure 21 presents the deflections of the main girders under the most unfavourable vehicle-loading scenario.

The maximum positive deflection, minimum negative deflection, and sum of the maximum absolute values of positive and negative deflection of the conventional bridge are 1598.7, 2539.9, and 4138.9 mm, respectively. Those of the proposed bridge are 638.5 mm, 1076.8 mm, and 1715.3 mm, respectively, or 57.6%, 60.1%, and 58.6% smaller, respectively. The maximum displacements of the side tower and middle tower of the conventional bridge under a moving load are 233.5 and 758.4 mm, respectively, while those for the proposed bridge are 133.1 and 346.6 mm, respectively, or 43.0% and 54.2% smaller, respectively.

The results indicate that the proposed bridge exhibits apparent stiffness advantages due to the high stiffness of the formed truss structure. Therefore, the new design can significantly improve the stiffness of a multi-tower suspension bridge and reduce the displacement of the main tower.

3.2.2.4 Analysis of structural fundamental frequency

The dynamic responses of the structures were analysed based on the linear elasticity theory. The first five natural vibration modes are shown in Fig. 22. It can be seen from the calculation results that the first out-of-plane vibration frequency of the conventional bridge is 0.0800 Hz, and that of the proposed bridge is 0.0821 Hz, which is 2.6% greater. The first in-plane vibration frequency of the conventional bridge is 0.1023 Hz, and that of the proposed bridge is 0.1424 Hz, which is 39.2% greater. The calculation results show that the proposed suspension bridge’s overall stiffness is significantly improved even when similar amounts of materials are used. The proposed bridge thus exhibits better dynamic characteristics, especially in-plane dynamic characteristics.

4 Stiffness test of proposed bridge with small span

4.1 Test bridges designed with span of 10 m

Two test bridges were designed to study the stiffness of the new type of suspension bridge. One was a new type of three-tower suspension bride (model l) with a span arrangement of 3 + 10.2 + 10.2 + 3 = 26.4 m, and a vertical span ratio of 1/9.27, as shown in Fig. 23(a). Another was a comparative test bridge (model 2) with the same span arrangement, as shown in Fig. 23(b). The main parameters of the two models are listed in Table 14, and photos are shown in Fig. 24(a). In order to save test materials, the stiffness tests of model 1 and model 2 were performed through the system conversion of one bridge, by loosening the cable clamps to form model 1 (Fig. 25(a)), and tightening the cable clamps to form model 2 (Fig. 25(b)).

4.2 Load test setup of test bridges and finite element simulation

Figure 24 shows a sketch of the experimental setup. Vertical load q in Fig. 26 was loaded on the test bridges in stages: 2.8, 5.6, 8.4, and 11.2 kN/m. A water tank was used to simulate the load. Dial indicators were used to test the deflection. The arrangement of the measurement points is shown in the sketch. The experimental process is shown in Fig. 27.

The stiffness of the test bridge was analysed using the MIDAS^TM program. In the finite element models of the bridges (Fig. 28), beam elements were used to simulate the girders, towers, and rigid webs, and cable elements were used to simulate the cables and hangers. For boundary conditions, the tower foot nodes and ends of the main cable were fixed. The main girder had rigid vertical supports at the piers.

4.3 Results and discussion

Figures 29–32 show the deflections of the girders of the two models given by MIDAS^TM and the experiments under loads of 2.8, 5.6, 8.4, and 11.2 kN/m. The following results are found under the load at each stage.

1) The results of the FEA and test shows good agreement. The FEA results for the sum of the maximum and minimum deflections (absolute value) for model 1 are 21.72, 42.85, 63.53, and 83.72 mm. The test results are 24.18, 37.35, 68.42, and 92.05 mm, and their relative errors are 11.33%, 12.83%, 7.70%, and 9.95%, respectively. Moreover, the FEA results for model 2 are 4.52, 10.35, 16.26, and 22.50 mm. The test results are 5.10, 12.11, 19.36, and 22.92 mm, and their relative errors are 12.83%, 17.0%, 19.06%, and 1.87%, respectively.

2) Compared with model 2, the deflection of model 1 given by the FEA has decreased by 79.19%, 75.84%, 74.41%, and 73.66%, and those given by the tests has decreased by 78.91%, 67.76%, 71.70%, and 75.10%.

The comparisons of these two models make it obvious that the stiffness of the new type of suspension bridge is significantly improved compared with the conventional suspension bridge with the same span.

5 Conclusions

To improve upon the mechanical properties of a conventional suspension bridge, a new type of suspension bridge was proposed in the presented study. A new type of railway suspension bridge with a main span of 340 m and a new type of highway suspension bridge with a main span of 1020 m were considered and compared with conventional bridges. An experimental study was carried out to validate this new type of bridge. Based on the results, the following conclusions can be drawn.

1) The main cable that bears the dead load and obtains the gravity stiffness has compressive rigidity. Therefore, the main cable, rigid webs, cross braces, and girder can form a space truss structure, which significantly improves the overall stiffness of the suspension bridge. Compared with conventional suspension bridges, the deflection of the main girder of the proposed bridge is reduced by 69.3%, 58.6%, and 75.1% for the railway, highway and test bridges, respectively.

2) The unbalanced forces at the tops of the towers of the proposed bridge are reduced, especially in the multi-tower suspension bridge. Compared with a conventional suspension bridge, the distribution of the unbalanced forces on the side towers and middle towers of the new type of multi-tower suspension bridge is more even, and the peak values of these unbalanced forces are significantly reduced.

3) The frequency of the first out-of-plane vibration of the proposed bridge is equal to that of a conventional suspension bridge, but the frequency of the first in-plane vibration is significantly increased in the proposed bridge by 32.97% in the highway bridge and 55.4% in the railway bridge.

4) The new type of suspension bridge uses the suspension cable structure to bear the dead load and the space truss structure to bear the live loads through a timely system transformation. The newly added components mainly participate in the live loads. Thus, their axial forces are small, and its stresses and stabilities can easily meet the requirements of the bridge design code.

Based on the presented results, it can be concluded that the proposed method can significantly improve the overall stiffness and reduce the unbalanced forces of the main towers. Therefore, this new type of suspension bridge will be particularly suitable for high-speed railway bridges.

References

[1] Ren W X, Harik I E, Blandford G E, Lenett M, Baseheart T M. Roebling suspension bridge. II: Ambient testing and live-load response. Journal of Bridge Engineering, 2004, 9( 2): 119– 126

[2] Ge Y J. Bridge engineering: Science, technology and engineering. China Civil Engineering Journal, 2019, 52( 8): 1– 5

[3] Zhou S Z. State-of-the-art of suspension bridges in China. Bridge Construction, 2003, 5 : 30– 34

[4] Miyata T, Yamaguchi K. Aerodynamics of wind effects on the Akashi Kaikyo Bridge. Journal of Wind Engineering and Industrial Aerodynamics, 1993, 48( 2−3): 287– 315

[5] Gülka P, Caner A, Apaydin N M. Developments in International Bridge Engineering: Selected Papers from Istanbul Bridge Conference. Cham: Springer Nature Switzerland AG, 2018, 19 :

[6] Diana G, Fiammenghi G, Belloli M, Rocchi D. Wind tunnel tests and numerical approach for long span bridges: the Messina bridge. Journal of Wind Engineering and Industrial Aerodynamics, 2013, 122 : 38– 49

[7] Yan J K, Peng T B, Li J Z. Shake table test of Taizhou Changjiang Highway Bridge: Test design and result analysis of seismic structural system. Journal of Southwest University (Natural Science Edition), 2014, 44( 2): 357– 362

[8] Dai X R, Wang L, Wang C J, Wang X Y, Shen Y L. Anti-slip scheme of full-vertical friction plate for multi-pylon suspension bridge. Journal of Zhejiang University, 2019, 53( 9): 1697– 1703

[9] Wang H, Yang M, Tao T Y, Li A Q. Parameter sensitivity analysis on dynamic characteristics of long-span quadruple-tower suspension bridge. Journal of Southwest University (Natural Science Edition), 2016, 46( 3): 559– 564

[10] Tang H Q, Xu G Y, Liu H S. Feasibility analysis of suspension bridge type to railway bridges. Bridge Construction, 2017, 47( 2): 13– 18

[11] Capsoni A, Ardito R, Guerrieri A. Stability of dynamic response of suspension bridges. Journal of Sound and Vibration, 2004, 393 : 285– 307

[12] Zhai W M, Wang S L. Influence of bridge structure stiffness on dynamic performance of high-speed train-track-bridge coupled system. China Railway Science, 2012, 33( 1): 19– 26

[13] Cantero D, McGetrick P, Kim C, OBrien E. Experimental monitoring of bridge frequency evolution during the passage of vehicles with different suspension properties. Engineering Structures, 2019, 187 : 209– 219

[14] Zhang X, Du B, Xiang T Y. Parameter sensitivity analysis of dynamic characteristics of long-span road-rail suspension bridge. Railway Standard Design, 2018, 6 : 77– 82

[15] Zhang W M, Liu Z, Xu S. Jindong bridge: Suspension bridge with steel truss girder and prefabricated RC deck slabs in China. Structural Engineering International, 2019, 29( 2): 315– 318

[16] Feng C B. Control techniques for the superstructure construction of Wufengshan Changjiang River Bridge. Bridge Construction, 2020, 50( 1): 99– 103

[17] Wang C J, Wang L, Ye Y Q, Bai Y D. Test study of anti-slip scheme of horizontal friction plates for middle tower saddle of multi-tower suspension bridge. Bridge Construction, 2020, 48( 2): 13– 18

[18] Shen R L, Hou K, Wang L. Requirements of vertical stiffness and anti-slip safety for three-pylon suspension bridge. Journal of Southwest University (Natural Science Edition), 2019, 49( 3): 474– 480

[19] Cao H, Qian X, Zhou Y, Chen Z, Zhu H P. Feasible range for midtower lateral stiffness in three-tower suspension bridge. Journal of Bridge Engineering, 2018, 23( 3): 06017009–

[20] Chai S, Xiao R, Li X. Longitudinal restraint of a double-cable suspension bridge. Journal of Bridge Engineering, 2014, 19( 4): 06013002–

[21] Wang X, Chai S, Xu Y. Sliding resistance of main cables in double-cable multispan suspension bridges. Journal of Bridge Engineering, 2017, 22( 3): 06016011–

[22] Zhang X J, Ying L D. A summary of wind-resistant measures of long span suspension bridges. Highway, 2006, 11 : 73– 80

[23] Andersen M S, Johansson J, Brandt A, Hansen S O. Aerodynamic stability of long span suspension bridges with low torsional natural frequencies. Engineering Structures, 2016, 120 : 82– 91

[24] Arioli G, Gazzola F. Torsional instability in suspension bridges: The Tacoma Narrows Bridge case. Communications in Nonlinear Science and Numerical Simulation, 2017, 42 : 342– 357

[25] Li C J, Li Y L, Tang M L, Qiang S Z. Improvement of flutter stability of long span suspension bridge with CFRP cable by crossed hangers. China Civil Engineering Journal, 2017, 3 : 83– 90

[26] Qi D C, Wang H X. Model test of main cable torsion of spatial cable suspension bridge. Railway Engineering, 2016, ( 2): 14– 17

[27] Zhang X J, Shun B N, Chen A R, Xiang H F. Flutter stability of cable-stayed-suspension hybrid bridges. China Civil Engineering Journal, 2004, 7 : 106– 110

[28] de Ville de Goyet V, Duchêne Y, Propson A. 16.08: The dynamic behaviour of the third Bosporus Bridge. Special Issue: Proceedings of Eurosteel 2017, 2017, 1( 2−3): 4098– 4107

[29] Barbaros A, Tayfun D, Maksym G. Optimization of cables size and prestressing force for a single pylon cable-stayed bridge with Jaya algorithm. Steel and Composite Structures, 2020, 34( 6): 853– 862

[30] Xiang H F, Ge Y J. Modern theory for wind resistant bridge and its application. Mechanical Engineering (New York), 2007, 1 : 1– 13

[31] Liu Z W, Xie P R, Chen Z Q, Xu G P, Xu J. Aerodynamic optimization of flutter stability of long-span streamlined box girder suspension bridge. Journal of Hunan University, 2019, 3 : 1– 9

[32] Liu J, Liao H L, Li M S, Mei H Y. Effect of stabilizer on flutter stability of truss girder suspension bridges. Journal of Vibroengineering, 2017, 19( 3): 1915– 1929

[33] Abdel-Rohman M, John M J. Control of wind-induced nonlinear oscillations in suspension bridges using a semi-active tuned mass damper. Journal of Vibration and Control, 2006, 12( 10): 1049– 1080

[34] Guo Z W, Ge Y J, Zhao L, Li K. Flutter suppression of long-span suspension bridge based on active control surface. China Journal of Highway and Transport, 2017, 2 : 57– 68

[35] Xu H Z, Huang P M. Cable tension control in anchorage span of suspension bridge. Journal of Chang’an University, 2002, 5 : 32– 41

[36] Pan Y R, Du G H, Fan L C. A fine calculation of the geometry and internal force of suspension bridge under dead load. China Journal of Highway and Transport, 2000, 4 : 35– 38

[37] TB 10002-2017. Code for Design on Railway Bridge and Culvert. Beijing: China Railway Publishing House Co., Ltd., 2017

[38] JTG D60-2015. General Specifications for Design of Highway Bridges and Culverts. Beijing: China Communications Press Co. Ltd., 2015

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