Condition evaluation for through and half-through arch bridges considering robustness of suspended deck systems

Abstract

Through and half-through arch bridges is one of the main bridge types in China, having arch ribs and simple suspended deck systems, which are connected by high-strength wire suspenders. However, due to the lack of robustness, the suspended deck systems of these types of bridges caused progressive collapse after the suspender corrosion damage. In this study, the internal structural factor of progressive collapse induced by suspender failure was revealed by comparison of five cases of suspender failure, and the robustness requirement of the suspended deck systems was proposed as redundancy to avoid transverse beam collapse when two supporting suspenders on both ends were broken at the same time. Meanwhile, a typical rigid-framed tied through concrete-filled steel tube arch bridge was investigated, and structural condition evaluation was carried out based on the Chinese standard (JTG/T H21-2011). It was found that the evaluation results of bridges with different deck systems were unreasonable, as the better the robustness was, the worse the condition of the deck system was rated, which would mislead the bridge maintenance decisions. A robustness-based evaluation method was then proposed by adopting different evaluation method according to the robustness of the suspended deck system, which is determined by qualitative classification or quantitative calculation. In addition, weight-adjusting method considering robustness was introduced in the evaluation process. The result showed that the new method could objectively reflect the condition of the bridge and thus fits the needs of bridge maintenance and management better.

Keywords

arch bridge condition evaluation failure progressive collapse robustness suspender

Introduction

China involves wide mountainous areas covering 69% of the country’s land area. Many arch bridges have been built in Chinese traffic infrastructure since ancient times to date (Tan and Yao, 2019). According to the relative positions of the deck and the arch rib, arch bridges can be classified into deck bridges, through arch bridges, and half-through arch bridges (Chen, 2013). In some through or half-through arch bridges, the suspended deck systems are local structures that transfer the load to the arch ribs, different from the deck systems in the tied-arch bridges which are fixed with the arch and take part in the load resistance of the whole structure (Chen et al., 2017). The suspended deck systems are used commonly in half-through arch bridges (Figure 1) and rigid-framed tied-arch bridges (Figure 2) (deck and suspenders are not shown). The former is a traditional bridge type, while the latter is a new one which first appeared in concrete-filled steel tube (CFST) arch bridges in China (Chen and Wang, 2009).

Figure 1.

Half-through arch bridges.

Figure 2.

Rigid-framed tied-arch bridge: (a) through arch bridge and (b) half-through arch bridge.

Nevertheless, defects increase with the extent of service duration due to harsh environmental conditions and increasing traffic. In particular, high-strength strands are mostly used as suspenders with small size; however, they are prone to corrosion and rupture (Chen et al., 2016). To study the effects of suspender rupture, Wu et al. (2014) analysed the dynamic hanger breaking of a through tied-arch bridge with ANSYS/LS-DYNA software using the contact–collision method. The results indicated that the hanger next to the ruptured hanger suffered the most significant impact. Moreover, the overall influence on the longitudinal girders could result in the local collapse of the bridge deck. Li and He (2014) investigated the robustness of a swallow-type arch bridge in which the suspenders in different positions were removed successively to determine the most adverse damage model of the structure. The results showed that the partial collapse of the bridge deck caused by the failure of a single suspender could be controlled within an allowable range, though a disproportionate collapse would be inevitable in the case of a simultaneous rupture of two suspenders. Therefore, bridge damage degree greatly varied due to the differences in the structural robustness of suspended deck systems (Chen et al., 2016).

Structural robustness mainly refers to the ability of a structure to withstand fire, explosion, impact or human error without being disproportionately damaged, as defined in UNI EN 1991-1-7-2014 (2014) of the Accidental Actions Eurocode. Although the structural robustness of buildings has been widely investigated (Eskew and Jang, 2020; Nav et al., 2018; Russell et al., 2018; Zhang et al., 2018), research on the structural robustness of bridges is relatively lacking. A new double-hanger system was designed by Wu et al. (2018) for tied-arch bridges to enhance its robustness based on Miner’s linear cumulative damage law, and its fatigue life analysis as well as the time–history simulation of the transitory fracture impact due to the fracture of one or more hangers were conducted, through which its reliability of resistance to progressive collapse was verified. Shoghijavan and Starossek (2018) carried out a structural robustness analysis for long-span cable-supported bridges considering a cable-loss scenario based on an approximation function for a simplified bridge as well as the least-squares method. As a result, the stress increase ratio of the cable adjacent to the failed cable was decreased, suggesting that the structural robustness of this bridge type could be improved by increasing the ratio of the bending stiffness of the girder to the axial stiffness of the cables.

Meanwhile, by comparing U.S. specification (LRFD-8, 2017) and Chinese standard (JTG D60-2015, 2015), it could be found that the current design rules for highway bridges were mainly written on probability-based limit states, including ultimate limit state and serviceability limit state. Although the extreme event limit state has been involved in the specification LRFD-8 (2017) during the recent years, it is still related with accidental actions, such as earthquakes, ice load and vehicle or vessel collision, with return periods in excess of the design life of the bridges (JTG D60-2015, 2015), rather than the extreme events like suspender rupture induced by corrosion or damage by human activities, and relevant conceptual and quantitative design requirements for robustness are still relatively lacking in the specification system.

However, bridge strengthening or retrofitting decisions should be determined according to evaluation results. Evaluation methods concerned with structural safety can mainly be classified into two types: structural condition ratings and load-bearing capacity assessment (FHWA NHI 12-049, 2012; JTG/T H21-2011, 2011; JTG/T J21-2011, 2011; MCEB-2-2011, 2013). Structural condition ratings (named as technical condition evaluation in China) mainly focuses on current structural damage and deterioration compared with the initial condition of the bridge, and the evaluation result ranged from poor to good based on visual inspection or non-destructive testing. Load-bearing capacity and further analysis will be carried out only if the bridge condition is complicated or it is severe damaged or accurate carrying capacity data are needed. For structural safety evaluation, it is widely used as a preliminary bridge assessment method, because its data source is relatively widely available, and the working cycle is shorter, the cost is lower, and the method is more convenient to use.

However, compared with bridge design, structural robustness is rarely involved in condition evaluation. The current standards for the condition evaluation of bridges have no criteria on robustness; and there is hardly any relevant research about robustness based on condition evaluation. Indeed, suspender failure induced disasters with the progressive collapse of the deck structures took place in bridges with weak or no robustness of deck systems, while no progressive collapse happens in other bridges with good robustness. Therefore, the existing bridges with weak robustness should be strengthened to improve their robustness, and the robustness of the suspended deck systems shall be considered within the condition evaluation for providing an accurate basis for the maintenance, to ensure their safety.

In this study, the robustness of the suspended deck systems in through and half-through arch bridges was analysed based on the suspender failure accidents, and the robustness requirement was proposed; then condition evaluation for a real bridge by standard JTG/T H21-2011 was conducted to investigate the problems brought by ignorance of robustness. Finally, a condition evaluation method considering robustness of suspended deck systems was proposed and was verified with a case study.

Robustness analysis on suspended deck systems

Cases of suspender failure and deck collapse

In through and half-through arch bridges, the dead load of the deck and live loads on it are transferred to the arch rib by suspenders. For high-strength steel strand suspenders, many defects were detected in the existing bridges because of suspender corrosion, fatigue failure, as well as insufficient protection (Figure 3). Therefore, suspenders are vulnerable components and key elements for the safety of bridges.

Figure 3.

Photos of some defects of the suspenders: (a) corrosion in anchorage and (b) wire breaking on the suspender.

In recent years, many suspender failure induced accidents reported in China and five selected cases were investigated in the chronological order, as follows:

On 7 November 2001, the South Gate Bridge located in Yibin City, Sichuan Province, is a half-through reinforced concrete arch bridge built in 1990. Some suspenders close to the arch springing were ruptured, causing the failure of transverse beams and deck slabs they carried, as shown in Figure 4. A passenger car on it fell into the river, and four people were killed in the accident (Wei, 2003).

On 14 July 2011, the Gongguan Bridge in Wuyishan City, Fujian Province, is a three-span half-through reinforced concrete arch bridge built in 1999. The two shortest suspenders in one span were ruptured by an overloaded traffic accident, resulting in the rupture of seven pairs of suspenders, followed by the falling of the transverse beams and deck slabs to the river, as shown in Figure 5. One passenger was killed and 22 were injured (Tian et al., 2012).

On 12 April 2011, the Peacock River Bridge located in Korla City, Xinjiang Province, is a half-through CFST arch bridge built in 1998. A pair of the second-shortest suspenders was ruptured, causing the transverse beams as well as the deck slabs on it fell down to the river, as shown in Figure 6. No one was killed or injured (Chen et al., 2016).

On 11 January 2010, the Yuping Mountain Bridge in Nanping City, Fujian Province, is a half-through reinforced concrete arch bridge built in 1995. A long suspender close to the mid-span was ruptured, and the cross beam hung by the ruptured suspender did not fall but was interrupted by a water pipe with a diameter of 500 mm, as shown in Figure 7 (Wu et al., 2018).

On 10 December 2012, the Luoguo Bridge in Panzhihua City, Sichuan Province, is a half-through CFST arch bridge built in 1995. One pair of suspenders slipped out from the bottom anchors due to the anchorage head failure, causing support loss of the transverse beam, and the continuous deck slabs tied to the transverse beam deformed in V-shaped, but prevented them from falling into the river, as shown in Figure 8 (Zhao and Pu, 2014).

To reveal the internal structural factors by comparison, the five accidents are divided into three levels, according to the seriousness of the consequence, as follows:

Level I: South Gate Bridge and Gongguan Bridge. After the rupture of the suspenders of these two bridges caused by corrosion and overload, the transverse beams and deck slabs without stiffening longitudinal girders fell down to the river after suspender failure. The sudden breakage of suspenders of the South Gate Bridge caused violent vibration and suspender rupture on the other side. While for the Gongguan Bridge, the failure of the suspender by the running truck resulted in a zipper-type failure of other suspenders as well as the deck system. Accidents did cause not only damage to the bridge structure but also many casualties.

Level II: Peacock River Bridge. The deck system was composed of transverse beams and deck slabs without stiffening longitudinal girders. Although there was no heavy traffic on bridge when the suspenders were ruptured, the transverse beams and the deck slabs supported on them collapsed in progressive.

Level III: Luoguo Bridge and Yuping Mountain Bridge. The transverse beams lose support when the suspenders were damaged by corrosion and anchor failure; however, the transverse beams and the deck slabs on them did not fall into the river, though both of them were not built with stiffening girders in the longitudinal direction. The loss of support seen in the transverse beams in the Luoguo Bridge was tied to the continuous slabs and that in the Yuping Mountain Bridge was temporarily supported by the water pipe.

Figure 4.

Deck system collapse of South Gate Bridge.

Figure 5.

Deck system collapse of Gongguan Bridge.

Figure 6.

Deck system collapse of Peacock River Bridge.

Figure 7.

Deck system collapse of Yuping Mountain Bridge: (a) over deck and (b) under deck.

Figure 8.

Deck system collapse of Luoguo Bridge.

Therefore, the decrease in the resistance of suspender failure due to corrosion was the main cause of accidents. The stress on the corroded suspenders increases due to the loss of effective area of the suspender sections; when an overloaded truck passes over the bridge, the stress is further increased, eventually exceeding the carrying capacity of the suspenders. Furthermore, the deck systems of these five bridges were simply supported slabs, as shown in Figure 9(a). The non-robustness of the deck system due to the inherent deficiency of the structure caused the disproportionate collapse. If the deck systems are integral structures or the slabs were continuous structures, as shown in Figure 9(b), the longitudinal elements with enough stiffness and strengths would have provided with a tie connection between the transverse beams, which lost the support from the ruptured suspenders and thus prevented the transverse beams from falling into the river.

Figure 9.

Structure diagram of suspended deck system: (a) simply supported slabs in suspended deck system and (b) slabs with continuous structures in suspended deck system.

Robustness requirement of suspended deck systems based on the statistical analysis of suspender failure

The suspender failure of through and half-through arch bridges is accidental, which is induced by complex reasons, although with a very small probability. The possible causes are mainly the coupling effect of cable corrosion and fatigue loading, the defects of design and construction, insufficient maintenance, vehicle overload, the conflagration, the vehicle impact, and other man-made sabotages. Moreover, most suspender failure events were influenced by many factors, and thus it is relatively difficult to theoretically analyse the probability of suspender failure.

To evaluate the frequency of suspender failure, an investigation on the existing 535 through and half-through arch bridges with a span larger than 50 m in China was conducted. The statistics of the number of bridges by year is shown in Figure 10.

Figure 10.

Annual statistics of through and half-through arch bridges.

According to the completion time, the average service time of each bridge is 13.37 years, and a total of five suspender failure events are collected, as shown in Table 1. The annual frequency of suspender failure induced accidents in through and half-through arch bridges can be calculated as 0.7 × 10⁻³. As for the annual frequency of accidental action, according to the International Organization for Standardization (ISO) DP10252 (1994), the central value of ‘Accidental effects of human activity’ which meets the requirements of high-energy impact acting on the structure should be lower than 1.0 × 10⁻⁴. Therefore, it can be taken as the design annual frequency of suspender failure induced accidents.

Table 1.

Annual suspender failure induced accidents in through and half-through arch bridges.

Statistical parameter	Statistical magnitude	Average service time (year)	Total service time (year)	Total number of accidents	Annual frequency
Statistical result	535	13.37	7154	5	0.7 × 10⁻³

Within the five suspender failure induced accidents investigated, the number of ruptured suspenders was more than or equal to one pair; among which there were three accidents with more than one pair of ruptured suspenders. Even though they are considered as accidents induced by multi-pairs of suspender failure, the annual frequency was calculated as 4.2 × 10⁻⁴, lower than the annual frequency of Grade E2 earthquake (with a recurrence interval of 2000 years) corresponding to the second fortification level of bridge seismic design in China, which is 5.0 × 10⁻⁴ (JTG/T B02-01-2008, 2008). Therefore, by comprehensively considering reliability and economy, the probability of the accident with more than one pair of ruptured suspenders should not be taken into account. Moreover, these three accidents were brought by preceding failure of one pair of suspenders, which were caused by the poor robustness of the deck system and the effect of overload. Therefore, the suspended deck systems should be designed to resist the failure of one pair of suspenders, and the remaining suspenders should resist the progressive failure.

The specific requirements of robustness design for simply suspended deck system can be proposed as follows:

As key elements, suspenders must not be ruptured;

In the case of a rupture of one pair of suspenders, the most serious situation is limited to accident Level III. Level II accident should not occur, while Level I accident must be prohibited.

As for the second point, for through and half-through CFST arch bridges, there is a mandatory provision (Clause 7.5.1) in the standard GB 50923-2013 (2013), which is stipulated as ‘deck systems shall adopt the scheme of integrated structures. Deck systems that are mainly supported by transverse beams must be provided with stiffening longitudinal girders and shall have the redundancy to avoid the collapse of the transverse beam due to the failure of the supporting suspenders at both ends. This requirement can also be applied to suspended decks in through and half-through concrete arch bridges.

Condition evaluation by the current standard

Evaluation method in JTG/T H21-2011

CFST arch bridges are not specifically listed in the U.S. specifications (FHWA NHI 12-049, 2012; MCEB-2-2011, 2013); while in the Chinese standard JTG/T H21-2011 (2011), they are classified further into three subtypes, and CFST arch bridge should be classified as the steel–concrete composite arch bridge. Herein, the condition of a perfect bridge is rated as 100. A bridge structure is discretized into three types of segments (deck, superstructure and substructure). The bridge condition rating D_r (in percentage) is evaluated by equation (1)

D_{r} = BDCI \times W_{D} + SPCI \times W_{SP} + SBCI \times W_{SB}

(1)

where $D_{r}$ stands for bridge condition index (BCI for short) ranging from 0 to 100; BDCI, SPCI and SBCI stand for BCI of deck system, superstructure and substructure, respectively; and W_D, W_SP and W_SB stand for weights of deck system, superstructure and substructure, respectively.

The segments (deck system, superstructure and substructure) of each type of bridge are divided into various components with different weights. The structural condition of a highway bridge is rated from CS I to CS V, as listed in Table 2. CS I indicates that the bridge is completely new or in good condition, while CS V means that the bridge is unqualified or in danger and should not be used.

Table 2.

Bridge condition assessment standard in JTG/T H21-2011.

$D_{r}$	[95,100]	[80,95]	[60,80]	[40,60]	[0,40]
Condition state (CS)	I	II	III	IV	V

As discussed in section ‘Introduction’, structural robustness is seldom considered in current standards for condition evaluation. As a result, the structural failure risk due to extreme or accidental events is ignored, leading to a potential external cause of the progressive collapse of the bridge.

Engineering background

The analytic engineering background is a rigid-frame tied through CFST arch bridge, with a span of 80 m. The overall length of the bridge is 200 m, and the clear width is 12 m with two sidewalks of 1.5 m wide. The bridge was completed in 1999 (Figure 11). Each suspender is composed of 110 high strength and low relaxation steel wire whose diameter is 5 mm, with PE protection and a spacing of 5 m. The members of the suspended deck system are all prefabricated by normal concrete of 30 MPa. The transverse beam is designed with an I-shaped cross section. Two stiffening longitudinal girders are fixed at the suspender anchorage of two neighbouring transverse beams with a T-shaped cross section (Figure 12(a)). Moreover, to strengthen the stiffness of the deck system, an extra steel longitudinal stiffening girder with an I-shaped cross section was installed outside the initial small longitudinal stiffening girder (Figure 12(b)). The deck is composed of reinforced concrete solid slabs with a width of 1.5 m and thickness of 0.25 m, which are simply supported by transverse beams connected by wet joints and the deck pavement with a thickness of 8 cm, the deck forms a structure of multi-span elastic supported continuous beam. The reinforced concrete double cylindrical pier of the main bridge is designed with a diameter of 2 m; there are four cast-in-place piles with a diameter of 1.50 m under each pier.

Figure 11.

Photo of the bridge.

Figure 12.

Design of stiffening girder before and after modification (unit: cm): (a) section of stiffening longitudinal girder in original structure and (b) section of stiffening longitudinal girder in strengthened structure.

To evaluate the structural condition of the main bridge, a visual inspection was conducted. Only the defects concerned with the suspended deck system are described as follows:

For the suspender, corrosion and concrete cracks were found on the protection box of the lower anchor end, and the short suspender was installed with deviation. Long concrete cracks with widths of 0.08–0.47 mm were found on the lower anchor end of 1# suspender, as shown in Figure 13.

For the longitudinal stiffening girder, most girders were installed with displacement; most of the joints of the longitudinal and transverse beams cracked with external concrete peeling off; and there were many voids and pits on the surface, accompanied with exposed reinforcement, as shown in Figure 14.

For the suspender transverse beam, many orifices were found on the concrete surface as well as exposed reinforcement; dense vertical cracks were discovered on the lower flange slab of I-shaped cross section, along with some diagonal cracks on the web.

For the deck, through cracks on the transverse direction of the bridge were detected on the mid-span slab undersurface, as shown in Figure 15.

Figure 13.

Defects of suspender: (a) deviation of short suspender and (b) corrosion and concrete cracks on the protection box.

Figure 14.

Defects of longitudinal stiffening girder: (a) installation displacement and (b) exposed reinforcement.

Figure 15.

Defects of deck: (a) cracks on the transverse beam and (b) cracks on the deck slab.

Condition evaluation by JTG/T H21-2011

For comparison, on the basis of the bridge with the initial deck structure (shown in Figure 12(a), referred to as structure B), two other structures were added. The initial longitudinal stiffening girder was too small and have insufficient strength and stiffness, which cannot meet the requirements for robustness (shown in Figure 12(a)); thus, Q345c steel stiffening longitudinal girders with I-shaped cross section was added (shown in Figure 12(b)), referred to as Structure C. The steel girders were found with defects as corrosion and bolt loosening after several years in service. In addition, a structure without longitudinal stiffening girder is added, corresponding with the bridges in accidents of Level I, which is obviously insufficient of robustness, referred to as structure A. For the suspended deck system, there is no member listed as ‘deck, transverse beam, longitudinal stiffening girder’ in the standard; thus, the member ‘deck (girder)’ was applied to evaluate the Structure A, Structure B and Structure C, as shown in Table 3.

Table 3.

Evaluation result of superstructure (by standard JTG/T H21-2011).

Member	Weight	Structure A		Structure B		Structure C
		Member score	Synthetic score	Member score	Synthetic score	Member score	Synthetic score
Rib	0.32	49.39	15.80	49.39	15.80	49.39	15.80
Transverse coupling system	0.06	49.39	2.96	49.39	2.96	49.39	2.96
Suspender	0.15	25.86	3.88	25.86	3.88	25.86	3.88
Tie bar	0.32	30.13	9.64	30.13	9.64	30.13	9.64
Deck (girder)	0.09	55.00	4.95	37.50	3.38	30.13	2.71
Support	0.06	75.00	4.50	75.00	4.50	75.00	4.50
Summation	–	–	41.74	–	40.16	–	39.50

Although the suspenders and other members are in the same condition, compared with Structure A, additional defects can be seen on the small stiffening longitudinal girders on Structure B; furthermore, there are additional defects on the big stiffening longitudinal girders on Structure C. When evaluated by Standard JTG/T H21-2011, the deduct marks of the members increase with the number, which are adapted by the coefficient $t$ ; in this way, the score of the member ‘deck (girder)’ in Structure B and Structure C decrease, resulting in the lower total score than Structure A, which has no longitudinal stiffening girder. Through the analysis in the next section, the robustness of the three structures is ranked as C, B and A from strong to weak. Nevertheless, on the contrary, the condition by the evaluation result is ranked as A, B and C from good to poor, and Structure C with the best robustness is yet rated as CS V (unqualified).

Indeed, if there are defects on suspenders in through and half-through arch bridges, managers would be most worried about Structure A, instead of Structure C, when making maintenance decisions with concerning of structural operation safety. The evaluation result received should be ranked as C, B and A from good to poor correspondingly, which would be encouraged to conduct reinforcement or repair on bridges with poor structural robustness preferentially. Therefore, the current condition evaluation method is unreasonable for through and half-through arch bridges, which influence the consequences of structural failure regardless of robustness.

Condition evaluation considering robustness

Evaluation process

Distinct evaluation methods should be applied to bridges with different robustness. The suspenders and the deck system should be seen as connected members to consider their interaction in suspender failure induced accidents. The improved condition evaluation considering robustness is shown in Figure 16.

Figure 16.

Process of condition evaluation considering robustness.

Robustness assessment

First, the robustness of the suspended deck system can be divided into three classes. The first class is the integral structure, such as the integral deck, gridwork deck and U-ribbed stiffener steel box girder, which are apparently with strong bending and shear resistance, and there has been no suspender failure induced accident reported on bridges with this type of suspended deck system. For the structure of the second class, the traffic load is mainly carried by transverse beams, and stiffening longitudinal girders are set between transverse beams; this class is further divided into two subclasses: when the stiffening longitudinal girders are strong or are with good continuity, the structure is assigned to subclass I of the second class; otherwise, the structure is assigned to subclass II of the second class. For the structure of the third class, the traffic load is also carried by transverse beams, but there are no stiffening longitudinal girders within the deck systems, which are apparently with week bending and shear resistance, and the investigated five accidents were all related to structure of this type.

Therefore, the robustness of the suspended deck system can be judged preliminarily and qualitatively: the first class is with the best robustness obviously, while the third class is with the worst robustness. However, for the second class, the method for the quantitative calculation of robustness as suggested by Yu (2015) should be introduced to calculate the dynamic effect of the remaining structure after suspender failure, to which is calculated as follows:

The equivalent force of suspender breaking F_i is calculated by load combination under the original normal operating condition, considering the equivalent load multiplier.

One pair of the equivalent forces is applied on the two ends of the breaking suspender whose directions are opposite to the initial suspender force. According to the alternate path method, the shortest and the second-shortest suspender are removed, respectively (Figure 17), and the forces of the remaining suspenders, the bending moment as well as the shearing force of the stiffening longitudinal girders are calculated.

Figure 17.

Layout diagram for equivalent force of suspender breaking.

If the suspended deck system can resist the rupture of one pair of suspenders (which has the redundancy to avoid the collapse of the transverse beam due to the failure of the supporting suspenders at its two ends), it would be determined as subclass I of the second class with good robustness; otherwise, it would be determined as subclass II of the second class with poor robustness.

Condition evaluation

Different methods are adopted to evaluate structures with different robustness:

1. Structures with good robustness

(a) When the structural condition of suspenders and stiffening longitudinal girders are assessed as CS I ∼ CS III, which means that they are all in good condition or with slight defects, the method in standard JTG/T H21-2011 (hereinafter referred to as ‘standard method’) can still be adopted.

(b) When one pair of suspenders or some longitudinal stiffening girders are assessed as CS IV–CS V, which means there are some severe defects detected, the weights of the suspender and the deck system should be adjusted to modify the standard method.

(c) When many pairs of suspenders are assessed as CS IV–CS V, which means that suspenders are at high risk of fracture, the bridge would be directly determined as CS V (unqualified).

(d) The score of the longitudinal stiffening girder should be equal to that of the girder with the worst condition, because when the suspender breaks, the possibility of progressive collapse depends upon the longitudinal stiffening girder with the weakest bearing capacity. Thus, the longitudinal stiffening girder should be taken as an important member, and its score should be equal to the lowest score of the longitudinal stiffening girder, according to Clause 4.1 in standard JTG/T H21-2011 (2011), rather than simply decreasing as the number and defects of the girders increase.

2. Structures with poor robustness

(a) When the structural conditions of suspenders are assessed as CS I–CS III, the weights of the suspender and the deck system should be adjusted to modify the standard method.

(b) The score of the suspended system should be rated as zero, because when the suspender breaks, the structure with poor robustness will inevitably be subjected to progressive collapse, no matter how the condition of the longitudinal stiffening girder is, it is unable to complete the designed function, which is equivalent to complete failure; thus, the score of the suspended system should be equal to the longitudinal stiffening girder, which is zero, according to Clause 4.1 in standard JTG/T H21-2011.

Variable weight considering robustness

Quantitative indicators of robustness

The dynamic analysis index of demand-capacity ratios (DCR) is introduced in the Standard GSA2013 (2013) to reflect the carrying capacity reserve; on this basis, a safety factor of carrying capacity I_i is put forward as equation (2)

I_{i} = 1 / DC R_{i} = Q_{CE} / Q_{UDLim} (i = 1, 2)

(2)

where I₁ and I₂ stand for the safety factor of suspender and the longitudinal stiffening girder; $Q_{UDLim}$ stands for the maximum dynamic load effect of the member, which can be calculated by the method presented in section ‘Robustness assessment’; and $Q_{CE}$ stands for the ultimate bearing capacity of the member, which can be calculated by the method presented in standards GB 50923-2013 (2013) and JTG 3362-2018 (2018). The value of I_i increases with the redundancy of member-carrying capacity.

The remaining suspenders and the longitudinal stiffening girder were the most sensitive members when some suspender was ruptured, while the other members of the superstructure were less affected (Wu et al., 2014). Among the suspended system, the longitudinal stiffening girder is vulnerable; the progressive collapse of the suspended deck system after suspender failure would perform as an implicated rigid body motion of the transverse beams and the decks due to the loss of support. Therefore, the failure risk of the suspended deck system can be represented by the carrying capacity redundancy of the longitudinal stiffening girder. Hence, the robustness index of the superstructure can be defined as equation (3)

I_{rob} = min {I_{1}, I_{2}}

(3)

The robustness of the suspended deck system is poor when $I_{rob}$ < 1, while the robustness is good if $I_{rob}$ ≥ 1.

However, it is unreasonable that the member weights in current standard JTG/T H21-2011 (2011) are fixed when adopted to evaluate bridges with different robustness (Fan et al., 2019); instead, the member weights should be adjusted according to robustness. Indeed, the progressive collapse risk for the failure path is mainly concerned with the actual carrying capacity of those members, which is comprised of the remaining suspenders, the stiffening longitudinal girders, the transverse beams and the decks. Whereupon the robustness based on member weight factors and the robustness weight are proposed as equation (4)

W_{rob} = {[r_{1}, r_{2}]}^{T}

(4)

where $W_{rob}$ stands for the robustness weight vector group obtained by the member safety factor I₁ and I₂, which are normalized. It can reflect the comparative value of the actual carrying capacity of the two members, as well as their comparative influence degree on the progressive collapse risk induced by suspender failure.

When $I_{rob}$ ≥ 1, if the carrying capacity redundancy of the stiffening longitudinal girders is low, the corresponding value of r₂ would be low too; then, the progressive collapse risk depends mainly upon the carrying capacity redundancy of the remaining suspenders, which become more important with a higher value of r₁ and vice versa. However, when $I_{rob}$ < 1, the robustness of the suspended deck system is poor, no matter how the robustness of the stiffening longitudinal girders is, as long as some suspender breaks, progressive collapse of the suspended deck system would be inevitable. The stiffening longitudinal girders have little impact on the whole condition of the superstructure. In this case, the value of r₂ should be equal to zero, and the normalized robustness weight vector group is $W_{rob} = {[1, 0]}^{T}$ .

Weight-adjusting method

For the weight set obtained by different calculation methods or decision-making methods, the integration method should be adopted to smooth their diversity with comprehensive consideration. Hence, the robustness weight set $W_{rob}$ should be integrated with the initial weight set $W_{0}$ , to achieve the purpose of adjusting the member weight according to robustness. For this purpose, a convex linear combination v consists of m vectors is proposed as equation (5) (Yi and Chen, 2014)

w_{i} = {[w_{i 1}, {w_{i}}_{2}, \dots, w_{in}]}^{T} (i = 1, 2, \dots, m)

(5)

where $w_{i}$ is the weight set obtained and n is the number of the structural members.

The dispersion between some weight set and other weight sets is calculated by equation (6)

D (d) = \sum_{j = 1}^{n} {\sum_{d' = 1}^{m} [w_{ij} (d) - w_{ij} (d')]}^{2} (d' = 1, 2, \dots, m)

(6)

The dispersion summation of all the weight sets should be minimized, and thus, a weighting model is established based on non-linear programming approach, to search the optimized weight coefficient vector $α_{i}$ (equation (7)), which is calculated by equation (8)

α_{i} = {[α_{1}, α_{2}, \dots, α_{m}]}^{T} (i = 1, 2, \dots, m)

(7)

min_{(α_{1}, α_{2}, \dots, α_{m})} \sum_{d = 1}^{m} {α_{d}}^{2} D (d), s . t . \sum_{d = 1}^{m} α_{d} = 1, α_{d} ⩾ 0

(8)

where $α_{d}$ is the weight of some weight set of structural members.

Then, a Lagrangian function is established to solve the weighting model, which is calculated by equation (9)

L (α_{d}, β) = \sum_{d = 1}^{m} {α_{d}}^{2} D (d) + 2 β [\sum_{d = 1}^{m} α_{d} - 1]

(9)

If the extreme point exists, the necessary condition should be that the partial derivative of the Lagrange function $L (α_{d}, β)$ is equal to zero; the optimum solution $α_{i}^{*}$ for $α_{d}$ is calculated and normalized therefrom, and the final compositive member weight coefficient set $w^{*}$ is calculated by equation (10)

w^{*} = \sum_{d = 1}^{m} α_{i}^{*} w_{di} (i = 1, 2, \dots, n)

(10)

Initial weight

As described above, the ignorance of robustness and the lack of dedicated clauses for the suspended deck system of through and half-through arch bridges caused inaccurate condition evaluation. Meanwhile, the weight of the suspender is 0.13, which is much smaller than that of the rib and the tie bar; when there are severe defects on the suspender, the impact on the total score of the superstructure would still be relatively limited. Thus, based on expert surveys, Fan et al. (2019) set up the member weight analysis model, and the group decision assembling method and the analytic network process method were introduced to calculate the initial member weights. As shown in Table 4, with targeted consideration on through and half-through arch bridge, the suspended deck system took a more important place; thus, the weight highly increased, so does the suspender; also, the rib and the tie bar were still members with the highest weight. Therefore, the initial weight proposed by Fan et al. (2019) tended to be more reasonable for through or half-through arch bridge, which is adopted in the condition evaluation of the case study.

Table 4.

Superstructure member weights comparison.

Member	Half-through arch bridge		Through arch bridge
	Fan et al. (2019)	Current standard	Fan et al. (2019)	Current standard
Rib	0.23	0.28	0.25	0.32
Transverse coupling system	0.03	0.05	0.03	0.06
Suspender	0.19	0.13	0.21	0.15
Tie bar	0.23	0.28	0.25	0.32
Deck system	0.20	0.08	0.22	0.09
Support	0.04	0.05	0.04	0.06
Upright column	0.08	0.13	–	–

Case study

The bridge case shown in section ‘Engineering background’ is re-evaluated by the improved method presented above.

Robustness calculation of suspended deck system

The deck system of Structure A is designed without stiffening longitudinal girder, and thus it can be rated qualitatively and directly as the third class of structure with poor robustness.

Structure B and Structure C could be initially determined as the structure of the second class, the quantitative method is introduced to calculate the dynamic response after suspender failure, and the results are shown in Table 5. For Structure B, the safety factor of axial force of the suspender is I₁ = 2.33; the safety factor of the bending moment of the stiffening longitudinal girder is 0 (as there is no negative bending moment reinforcement on the support; M stands for the positive bending moment of the mid-span section and M′ stands for the negative bending moment on the support section); thus, the safety factor of the stiffening longitudinal girder is I₂ = 0; eventually, the safety factor of the suspended system of Structure B is I_rob = 0 < 1, indicating that the robustness is poor. For Structure C, the safety factor of the suspender is I₁ = 2.08; the safety factor of the bending moment of the stiffening longitudinal girder is 1.35, and the safety factor of the shearing force of the stiffening longitudinal girder is 2.91; thus, the safety factor of the stiffening longitudinal girder is I₂ = 1.35; eventually, the safety factor of the suspended system of Structure C is I_rob = 1.35 > 1, indicating that the robustness is good.

Table 5.

Analysis result of suspender breaking robustness.

Structure	$F$			$M$			$M'$			$V$
	$Q_{UDLim}$ (kN)	$Q_{CE}$ (kN)	$I_{i}$	$Q_{UDLim}$ (kN m)	$Q_{CE}$ (kN m)	$I_{i}$	$Q_{UDLim}$ (kN m)	$Q_{CE}$ (kN m)	$I_{i}$	$Q_{UDLim}$ (kN)	$Q_{CE}$ (kN)	$I_{i}$
B	1525.42	3560.45	2.33				–949.91	0	0	596.99	4772.09	7.99
C	1710.55	3560.45	2.08	2097.21	2838.55	1.35	–995.87	–2411.48	2.42	629.93	1835.31	2.91

Weight-adjusting

Structure B is with poor robustness, and the suspender is rated as CS IV; then, Structure B is rated as CS V. So there is no need to adjust the member weights.

For Structure C, the robustness is poor, and the suspender is rated as CS IV; thus, the weights of the suspender and the deck system should be adjusted. The safety factors I₁ and I₂ are taken and normalized to establish the robustness weight vector group

W_{rob} = {[0.61, 0.39]}^{T}

The weights can be adjusted by the weight-adjusting method, as shown in Table 6.

Table 6.

Robustness-based weight modification (Structure C).

Member	$W_{0}$	$W_{rob}$	$W^{*}$
Suspender	0.21	0.61	0.23
Deck system	0.22	0.39	0.19

Condition evaluation

Structure A has poor robustness, and the suspender is rated as CS IV and Structure A as CS V. For Structure C, with adjusted weights, the condition evaluation considering robustness is shown in Table 7. The score of the longitudinal stiffening girder is equal to that of the girder with the worst condition; thus, the score of the deck should be 55.0, rather than 30.13.

Table 7.

Robustness based on the condition evaluation (Structure C).

Member	$W^{*}$	Member score	Synthetic score
Rib	0.25	49.39	12.35
Transverse coupling system	0.03	49.39	1.61
Suspender	0.23	25.86	5.99
Tie bar	0.25	30.13	7.53
Deck	0.19	55.00	10.57
Support	0.04	75.00	3.26
Summation	–	–	41.31

Discussion

For Structure A and Structure B having poor robustness, the re-evaluation results indicate that the deck system would suffer from progressive collapse as soon as the suspender breaks when the suspender is in poor condition, which can be directly rated as CS V (unqualified). However, for Structure C, which has the best robustness, the total score is 41.31, and it is rated as CS IV, meaning that although it cannot ensure normal operation, it is not yet in danger; even if some suspender breaks, the remaining structure is still capable of carrying the load instead of suffering from progressive collapse. Consequently, traffic restriction, reinforcement or rehabilitation should be arranged with priority to Structures A and B as soon as possible; for Structure C, monitoring and maintenance should be conducted when necessary. With the constraints of limited resources, such bridge maintenance decisions can be reasonably made.

Conclusion

In this study, the robustness of the suspended deck systems within a through or half-through arch bridge was involved in developing an accurate evaluation of bridge structure. An evaluation of three structures with different robustness on an existing bridge was conducted by the current standard, and the results revealed unreliable conclusions resulting from the negligence of robustness. Furthermore, a robustness-based condition evaluation method was established, and the three structures were re-evaluated. The following conclusions were obtained:

For the suspended simply supported deck systems with poor robustness, the failure of the vulnerable suspender with high strength may lead to the progressive collapse of the deck system. Hence, this kind of deck systems should be designed by stiffening longitudinal girders, which have sufficient stiffness and strength. The strengthened and retrofitted bridge structures shall have the redundancy to avoid the collapse of the transverse beam due to the failure of the supporting suspenders at its two ends, in the case of one pair of corresponding suspenders ruptured.

The evaluation result by the standard JTG/T H21-2011 shows that the structure with better robustness gets lower score due to the negligence of structural robustness. Therefore, structural robustness should be involved in the condition evaluation of through or half-through arch bridges.

The weight-adjusting method, in which robustness indicators characterized by the redundancy of actual carrying capacity redundancy, can be utilized to reasonably reflect the risk of progressive collapse of the deck system after suspender failure.

The condition evaluation with the consideration of robustness can reflect the actual condition of the bridge structure. Furthermore, it is conducive to encouraging the reinforcement and reconstruction of the existing structures with poor robustness to leverage both economy and safety.

Footnotes

Acknowledgements

The authors gratefully acknowledge the support from Fujian Yongzheng Construction Quality Inspection CO., LTD.

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: This work was supported by the Natural Science Foundation of Fujian Province (Grant No. 2020J01480 and 2020J01481).

ORCID iD

Jia-Zhan Su

References

Chen

(2013) Arch bridges. In: Chen

Duan

(eds) Handbook of International Bridge Engineering. 2nd ed. Boca Raton, FL: CRC Press, pp. 907–950.

Chen

Wang

(2009) Overview of concrete filled steel tube arch bridges in China. Practice Periodical on Structural Design and Construction 14(2): 70–80.

Chen

Fan

, et al. (2016) Robustness design of concrete-filled steel tube arch bridges. Bridge Construction 46(6): 88–93 (in Chinese).

Chen

Wei

Zhou

, et al. (2017) Application of concrete-filled steel tube arch bridges in China: current status and prospects. China Civil Engineering Journal 50(6): 50–61 (in Chinese).

Eskew

Jang

(2020) Remaining capacity estimation for buildings after an explosion using the adaptive alternate path analysis. Advances in Structural Engineering 23(4): 630–641.

Fan

Chen

(2019) Robustness based technical condition evaluation method with variable weight of through and half-through arch bridge. Journal of Fuzhou University (Natural Science Edition) 47(2): 223–229 (in Chinese).

FHWA NHI 12-049 (2012) Bridge inspector’s reference manual.

GB 50923-2013 (2013) Technical code for concrete-filled steel tube arch bridges.

GSA2013 (2013) Progressive collapse analysis and design guidelines for new federal office buildings and major modernization projects.

10.

ISO DP10252 (1994) Accidental actions due to human activities.

11.

JTG 3362–2018 (2018) Specifications for design of highway reinforced concrete and pre-stressed concrete bridges and culverts.

12.

JTG D60-2015 (2015) General specifications for design of highway bridges and culvert.

13.

JTG/T B02-01-2008 (2008) Guidelines for seismic design of highway bridges.

14.

JTG/T H21-2011 (2011) Standard for technical condition evaluation of highway bridges.

15.

JTG/T J21-2011 (2011) Specification for Inspection and Evaluation of Load-bearing Capacity of Highway Bridges.

16.

(2014) Research on robustness performance of swallow type arch bridge based on suspender damage. Journal of Hefei University of Technology (Natural Science) 37(10): 1254–1258 (in Chinese).

17.

LRFD-8 (2017) AASHTO LRFD Bridge Design Specifications. 8th ed. Washington, DC: American Association of State Highway and Transportation Officials.

18.

MCEB-2-2011 (2013) The Manual for Bridge Evaluation. 2nd ed. Washington, DC: American Association of State Highway and Transportation Officials.

19.

Nav

Usefi

Abbasnia

(2018) Analytical investigation of reinforced concrete frames under middle column removal scenario. Advances in Structural Engineering 21(9): 1388–1401.

20.

Russell

Owen

Hajirasouliha

(2018) Nonlinear behaviour of reinforced concrete flat slabs after a column loss event. Advances in Structural Engineering 21(14): 2169–2183.

21.

Shoghijavan

Starossek

(2018) Structural robustness of long-span cable-supported bridges in a cable-loss scenario. Journal of Bridge Engineering 23(2): 4017133

22.

Tan

Yao

(2019) Optimization of hanger arrangement in pedestrian tied arch bridge with sparse hanger system. Advances in Structural Engineering 22(12): 2594–2604.

23.

Tian

Zou

Tan

, et al. (2012) Explosive demolition of half-supporting suspension arch bridge. Blasting 29(3): 78–81 (in Chinese).

24.

UNI EN 1991-1-7-2014 (2014) Eurocode 1 – actions on structures, part 1-7: general actions-accidental actions.

25.

Wei

(2003) Urgent reinforcement and restoration of Xiaonanmen bridge in Yibin City. Highway 48(4): 34–38 (in Chinese).

26.

Chen

(2014) Dynamic analysis for cable loss of a rigid-frame tied through concrete-filled steel tubular arch bridge. Journal of Vibration and Shock 33(15): 144–149 (in Chinese).

27.

Wang

Zhu

, et al. (2018) New hanger design approach of tied-arch bridge to enhance its robustness. KSCE Journal of Civil Engineering 22: 4547–4554.

28.

Chen

(2014) Research on structural damage assessment based on integrated weight extension theory. Earthquake Engineering and Engineering Vibration 34(2): 137–145 (in Chinese).

29.

(2015) Study on Robustness of Suspended Floor System in Half-Through and Through Arch Bridges. Fuzhou, China: Fuzhou University (in Chinese).

30.

Zhang

Jiang

(2018) Modeling structural behavior of reinforced concrete beam-slab substructures subject to side-column loss at large deflections. Advances in Structural Engineering 21(7): 1051–1071.

31.

Zhao

(2014) Mechanical characteristics of tied-arch bridge under structural defects and traffic increase. Journal of Chongqing University (Natural Science Edition) 37(6): 25–32 (in Chinese).