Abstract
Although the application of the Alternate Path method has well proven its efficiency towards increasing the structural robustness of bare frame structures against progressive collapse, it should be further developed for bridge structures, and especially cable stayed bridges, as they are the only type routinely designed for cable loss phenomenon. The main objective of this paper is to analyse a cable stayed bridge for multiple types of cable loss phenomenon to develop a profound knowledge regarding the overall structural response to local failure and the possibility of a failure progression throughout the structure. This paper demonstrates numerical modelling and analysis of a typical cable stayed bridge through a nonlinear static procedure using SAP 2000. The response of the structural model is discussed for multiple types of cable loss cases to identify a definite progressive collapse pattern. Furthermore, a categorical investigation of the impact ratios was also done for different structural properties through a dynamic analysis to recognize the lack of robustness in the structure. The results indicate different design considerations for different elements of the structure and also a variation in the possibility of failure progression through the cable stayed model depending upon the location of the failed cables. The prevailing collapse type is discussed with some suggestions for a more robust design.
Keywords
Introduction
Every collapse in a manner can be regarded as a progressive collapse if it is inconsistent with its original cause. The disproportionality refers to the circumstances in which failure of any one affiliate causes a key collapse of a larger magnitude compared to the initial event. It is alike to the fall of cycles in a cycle stand when the first one is pushed. These disproportionate failures occur due to the small initial local failures caused by unforeseen events. These lead to collapse of the entire structure, showing its inadequacy in offering resistance against the development of damage due to insufficient load carrying capacity. For example, that first cycle might get pushed by a storm, or get hit by someone and initiate the fall of the other cycles. This is an initial local failure and the storm or collision is a rare unpredicted incident. This results in a fall of hundred or more cycles standing right next to the first one to describe the inconsistent magnification of the damage level. Vulnerability of buildings and bridges, to a disproportionate collapse has often been highlighted by the tragic incidents like Ronan Point Apartment collapse in 1968, Skyline Plaza collapse in 1970, Civic Arena Roof collapse in 1978, Haeng Ju Grand Bridge failure in 1992, Xiaonanmen Bridge collapse in 2001, Guangdong Jiujiang Bridge collapse in 2007 and more importantly the collapse of the twin towers of World Trade Centre in 2001. Although, a lot of studies had been carried out over the past few years to determine various methods to increase the resistance of building structures against collapse progression, not many investigations have focussed on Bridge Structures. This present piece of study is nothing but a rational structural analysis of a cable stayed bridge to the cognition of robustness and collapse resistance, aiming for the development of a clear concept of progressive collapse mechanism of bridges. As bridges are horizontally aligned structures with one main axis of extension and are sensitive to dynamic excitation, offering less redundancy, their possible mechanisms of collapse are different compared to buildings. The reasons behind a bridge progressive collapse could be (i) unexpected events, such as collision with heavy vehicles, and earthquake, (ii) degradation of structure performance due to corrosion and creep effect, or (iii) improper design or wrong construction methods.
According to unique description, two types of bridge progressive collapse are proposed by Liu et al. [1]; the Bearing failure type and the Partial Failure type. The first consists of (a) Reduction of degree of static indeterminacy, and (b) Internal force distribution and the second involves failure of a load bearing element, which causes a major collapse. Since, the cross sections of cables in a Cable-stayed bridge are small and hence provide low resistance to accidental lateral loads, they should be considered as a possible local failure. Wolff and Starossek [2] examined the dynamic response of a 3D cable-stayed bridge with the loss of one or more cables. The collapse progression of the bridge was analysed considering geometric and material nonlinearities. A quasi-static analysis using dynamic amplification factors was conducted to study the dynamic effects. The robustness of the bridge could be increased by increasing the stiffness of the bridge girder or by reducing the unsupported length, i.e. closer cable spacing. The former was recommended due to higher chance of failure in the latter case.
Cai et al. [3] used four analytical procedures, i.e. static analysis for linear and nonlinear cases and dynamic analysis for linear and nonlinear cases, for a 2D cable stayed bridge. The authors found that the loss of a single cable did not cause a progressive collapse as the cable tensions remain small. Plastic hinges formed in the girder when cables started to yield for simultaneous failure of two adjacent cables and deformations increased rapidly without collapse of the bridge. The study concluded that when live load was increased by a factor greater than or equal to two, the cables adjacent to the lost cables would fail causing complete collapse.
Investigated bridge system and modelling
An 822.96 m long cable stayed bridge supported by four 122.3 m high pylons is considered for this present investigation. Figure 1 shows the 2D cable nomenclature and the 3D geometrical configuration of the model whereas Fig. 2 represents the dimensional details. Each pair of pylons has two portals between them and rest on two 13 m high cellular box type concrete piers. The deck is supported by 136 stay cables with 68 on each half of the bridge, having diameters rising linearly from 0.14 m to 0.2 m as either end or the centre of the bridge is approached from each of the pylons. The cables are prestressed accordingly with a convergence study under gravity load. The numerical investigation is done using the finite analysis software SAP 2000. Material properties and sectional details used for the different components of the bridge model have been considered as per the M.Tech. Dissertation report of the corresponding author, Das [4]. Roller-rocker support condition is adopted as the boundary conditions of the bridge model. To obtain a more realistic behaviour of the cable stayed bridges; nonlinearity should be incorporated in terms of material and geometrical nonlinearity. The modelling and analysis is done using plastic hinges to represent material nonlinearity for different elements. Since two prominent modes of failure are observed for each case of cable loss viz. (i) Overloading of adjacent cables due to local redistribution and (ii) Flexural failure of the steel girders and crack occurrence in the deck; material nonlinearity is considered for the cables, girders and the deck.
As the cables are capable of withstanding tensile forces alone, a compression limit of zero is assigned to them and an axial plastic hinge is introduced in the middle of each cable element. The yield stress has been taken as 0.6 GPA. The strain at onset of strain hardening is taken to be 2% ; then the strain at rupture is taken to be 5%. Since the girders are subjected to both axial forces and bending moment, thus P-M2-M3 hinges are introduced into these elements. Hinge properties have been adopted based on ASCE-365. Nonlinear behaviour of the deck has been considered only for in-plane behaviour. The out of plane behaviour has been taken as linear. Since large deflections are involved, the effect of geometrical nonlinearity is considered in the nonlinear static and dynamic analysis. All materials used are assumed to be isotropic and homogeneous in nature. The bridge girder, piers, portals and pylons are modelled as equivalent frame elements. For investigations with nonlinear material behaviour, the deck is modelled using thin shell elements. The cables are modelled with a series of frame elements with distributed mass and self-weight. Connections between the cables and the pylon and deck are considered to be pinned whereas the base of the piers is taken as fixed ignoring the soil behaviour to avoid complications.
Progressive collapse analytical procedure
Cable stayed bridge is the only type of bridge structure routinely designed for cable loss. One or multiple cables are assumed to lose their load carrying capacity primarily due to failure of anchorage joints, to trigger the collapse progression and initiate the analytical investigation. Then the cable stayed bridge model is analysed by a nonlinear static procedure for different cable loss cases. This analysis can be performed in two ways as described in Fig. 3. After assuming the specific cables that are to be eliminated (dotted cables), (a) the damaged cable(s) can be removed and the analysis is carried out on the remaining structure as shown in Fig. 3a, or, alternatively (b) an equivalent and opposite yield force can be introduced to the damaged cables to nullify their contribution to the structure as shown in Fig. 3b. The first method is considered for the static analysis. The axial force carried by the eliminated cable gets redistributed to the supporting elements after its elimination from the structure. Stresses increase and after their exceedance over the yield limit, corresponding elements fail. So a collapse propagates through the structure. Failure of different elements is indicated by hinges.
The initial elimination method is used in this study to analyse the model for subsequent cable losses as described below. A nonlinear gravity load analysis is carried out on the undamaged original model to determine the deflected and stressed condition of the structure. Initial axial forces (P) in cables are noted down from the nonlinear gravity load analysis of the undamaged model. The cables which were primarily assumed to lose their load carrying capacity are eliminated from iv the structure as shown by the dotted lines representing them in Fig. 3(a). Again a nonlinear gravity load analysis is carried out on the modified structure to investigate the change in its behaviour. Hinges are found in relevant cables and girders after their corresponding yield points were reached. Stresses are noted to increase in the decks and cracks are expected to occur subsequently.
Furthermore, the equivalent and opposite yield force method (shown in Fig. 3(b)) was used along with an instantaneous unloading function as shown in Fig. 4 to get a better understanding of the local and global behaviour of the structure due to the sudden snap of the cables.
This behaviour might be referred in terms of impact ratios discussed in Section 4.5. When this function is applied on the model, the assigned axial forces are automatically doubled. The first P cancels out the load carrying capacity of the cable that was assumed to fail and the second P introduces the impact force to the pylon and the deck on the anchorage points. So, the corresponding cable collapses and subsequently generates an impact force on the anchorage points with the unloading function. The unloading duration is taken to be 10 ms in this study.
Progressive collapse analysis results
The original, undamaged structural model and the models modified accordingly for different cable loss cases are analysed to visualize the load distribution, variation of different response parameters and to determine the path of collapse progression throughout the structure using a nonlinear static procedure. Three types of cable loss scenarios were investigated as follows.
1. Failure of an End Cable; 2. Failure of a Middle Cable (Cables joining the centre region of the deck with the pylons); 3. Failure of a Near Cable (Cables positioned just beside pylons).
Loss of Cable 1-1 (Case-1)
The comparison study of structural response of the cable stayed bridge to the sudden loss of cable 1-1 is premeditated in this section. The vertical deflection of the centre (1) and the quarter points (2 & 3) of the deck and the longitudinal displacement of the pylon tops (5, 6, 7 & 8) are plotted in Fig. 5 respectively. The deflection values of the undamaged (main) structure are also given for a better understanding. In both of these graphs it can easily be observed that there is no significant variation in the vertical and longitudinal displacements obtained from the nonlinear static analysis of the modified model. Hence, the bridge remains intact during the first cable loss case.
Figure 6 shows the cable forces in four most vulnerable cables after the mentioned cable failure. Cable 1-1 is the leftmost end cable. It is observed that, considerable increase in demand occurs in one of the end cables (Cable 2-1) at the opposite end of the bridge. For this particular cable loss case, as shown in Fig. 6, the increase in the axial force is observed to stay within the limit. Therefore failure hasn’t occurred.
Loss of Cable 1-1 and Cable 1-2 (Case-2)
The variation in the vertical deflection of the centre (1) and quarter points (2 &3) of the bridge deck and the lateral displacement of the pylon tops (5, 6, 7 & 8) due to the sudden loss of cable 1-1 and 1-2 are described in Fig. 7 along with the deflection values of the undamaged structure. In Fig. 8, the increase in axial force and resulting failure of the first four vulnerable cables are explained by the nonlinear static procedure.
Cable 1-1 and 1-2 are the two leftmost end cables. So, significant changes are observed in the axial forces in the end cable at the opposite end of the bridge and also the nearby cables. The graphical figures can be explained in a similar manner as done in the previous section. In Fig. 7, it can be observed that there is a considerable variation in the vertical and longitudinal displacements obtained from the nonlinear static analysis of the modified model. The vertical displacement of the centre of the deck resulted to be more than 4.5 meters, which defines that the bridge deck has collapsed. Figure 7 also shows that the top of the first and forth pylon have undergone a longitudinal displacement of 3 meters due to increased flexure and vertical deflection of the longitudinal girders and the concrete deck. This reflects upon the failure of pylon 1 and pylon 4.
From Fig. 8 shown above, the failure of cables can easily be explained with increasing analysis steps. As Cable 1-1 fails, stresses get redistributed and the opposite end Cable 2-1attracts a large percentage of that. Since, being an end cable, it was already carrying a high axial force; this small increase in axial tension is noticed to trigger its failure. In the process, as the bridge tries to regain its symmetric configuration, Cable 2-2 fails after Cable 2-1. Again the failure path shifts to the other half of the bridge model due to stress redistribution and the whole process repeats itself until the entire structure falls apart. Failure of an element means its zero contribution to the structure. So, a cable carrying zero axial tension is considered to lose its load carrying ability. In Fig. 8, it is seen that Cable 2-1 fails after 90 steps and Cable 2-2 fails after 180 steps. Then Cable 1-3 and Cable 2-3 follows. Thus, in this cable loss case the nonlinear static procedure gives us a certain sequence of failure progression as described below. The final state of the structure before total collapse is shown in Fig. 9. As observed in the study, after an initial elimination of Cable 1-1 and Cable 1-2, axial force in cable 2-1 exceeds the yield point leading to its complete failure.Girders between cables 1-1, 1-2, 1-3, 2-1 and 2-2 anchorage joints fails consequently in tension. Cable 2-2 also fails in tension. After that, girder between cable 2-2 and 2-3 anchorage joints was observed to fail in tension with Cable 1-3 and 2-3 following it. One by one, the girders in the mid portion of the deck failed in flexure. Cable 1-35 and 2-35 failed in tension. Pylon 1 and Pylon 4 bent towards the centre of the deck and failed in flexure and thus the whole bridge structure collapsed.
Loss of Cable 1-30, 1-31, 1-32, 1-33 and Cable 1-34 (Case-3)
The change in force and variation of displacements due to the sudden removal of cables 1-30, 1-31, 1-32, 1-33 and 1-34 are studied in this section. The vertical deflection of the centre (1) and quarter points (2 & 3) of the bridge deck and the longitudinal displacement of the pylon tops (5, 6, 7 & 8) are plotted in Fig. 10.
Since, Cables 1-30, 1-31, 1-32, 1-33 and 1-34 are the cables holding the middle portion of the deck, significant changes are observed in the axial forces in cable 2-1 (right end cable) as shown in Fig. 11. The figure shows the variation in cable forces of the first four damage prone cables after the mentioned elimination. Progressive failure is seen to occur after the failure of cable 2-1 which is an important load carrying end cable. More hinges are found in different members. After the initial removal of the mentioned cables, Cable 2-1, the rightmost end cable failed with subsequent failure of girders between cable 1-32,1-33, 1-3 and 1-34 anchorage joints. Next, Cables 2-2, 2-3, 2-4, 2-5, 2-35 and 2-36 failed in the corresponding manner. Girders in the mid portion of the deck were observed to lose their load bearing ability and thus the whole structure collapsed. The final stressed state before entire collapse, with hinges in relevant elements is shown in Fig. 11.
Loss of Cable 1-11, 1-12, 1-13, 1-14 and Cable 1-15 (Case-4)
In this section the behaviour of the bridge model is studied for initial failure of cables 1-11, 1-12, 1-13, 1-14 and 1-15. It is clearly noticed from the nonlinear static analysis procedure, that the model remains in a safe situation after these five cables rupture initially due to some accidental cause. All the axial forces in the remaining cables are observed to stay within the yield limit. The girders are also seen to be safe and no cracks have occurred in the bridge deck. So, further investigation was done by rupturing more nearby cables and it was observed that after an initial failure of two more cables, cable 1-16 and 1-17, the model still remained safe. When cables 1-9 and 1-8 were ruptured, a collapse progression was observed to originate. So, at least seven cables on the same side of the pylon were required to fail to start a failure sequence. This is considered to be a very rare case and therefore is not included in this present study.
Impact ratio calculation
Once a singular or multiple numbers of cables snap because of some accidental loads, it generates a sudden impact force on the anchorage points which are composed of the inertia loads from the failed cable plus the inertia forces from the vibrating bridge deck. A rapid increase occurs in forces in the neighbouring elements (or some critical elements) for a few seconds which sometimes results in the failure of one of them. Thus, a collapse progresses through the whole structure damaging the other relevant cables, where the stresses had increased due to load distribution. Similar to the static analysis part, a set of dynamic analyses are carried out to observe the local and global response of the structure due to sudden cable failures. The variations in the displacement of the centre of the deck, axial force variation in a neighbour cable, and variation in bending moment throughout the longitudinal axis of the bridge for different cable loss cases are compared with the undisturbed model (main) to get a clear understanding of the impact ratios.
Case-1 (undamaged structure)
The original undamaged structure was first investigated to get an overall idea of the axial forces in the cables, as well as the deflections at the centre of the deck. The bending moments along the longitudinal axis of the bridge and on the cable rupture plane were also studied to understand the variations on the deck and the main girders. Following are some case studies with different cable loss considerations to highlight the impact ratios inherent in the sudden cable loss phenomenon.
Case-2 (Loss of Cable 1-1)
As Cable 1-1 ruptures in this case, a sudden impact is generated on its anchorage points on the pylon and the deck leading to a rapid increase in the displacement of the centre of the deck and also in the axial force in Cable 1-2, one of its neighbouring cables, as observed in Fig. 12 and Fig. 13 respectively. This rapid excitement stays for a few seconds and stabilises after 100-150 seconds to a rather increased value due to load redistribution throughout the structure.
Case-3 (Loss of Cable 1-4)
The prompt rise in the displacement of the centre of the deck and also in the axial force in Cable 1-2, one of the nearby cables, due to the sudden rupture of Cable 1-4 are plotted in Fig. 14 and Fig. 15 respectively.
Case-4 (Loss of Cable 1-7)
Figure 16 and Fig. 17 display the escalation in the displacement of the centre of the deck and also in the axial force in Cable 1-6, due to the sudden rupture of Cable 1-7.
Case-5 (Loss of Cable 1-10)
The displacement of the centre of the deck and also in the axial force in Cable 1-9 due to the sudden rupture of Cable 1-10 are plotted in Fig. 18 and Fig. 19 respectively.
Case-6 (Loss of Cable 1-13)
The apt rise in the displacement of the centre of the deck and also in the axial force in Cable 1-12, one of the immediate cables, due to the sudden rupture of Cable 1-13 are plotted in Fig. 20 and Fig. 21 respectively.
The bending moment also gets affected in each and every cable loss scenario due to this sudden impact phenomenon. Thus, the variation in maximum and minimum bending moments on the cable ruptureplane, along the longitudinal axis of the bridge are compared for every cable loss case as shown in Fig. 22 and Fig. 23.
Discussion of results
The mid part or the centre of the bridge model deck undergoes maximum vertical displacement and the top of the pylons are subjected to maximum transverse and longitudinal displacements resulting due to high rotation developed under its self-weight.
Important elements
The important elements of the structure are those elements whose failure cause the highest responses and leads to a disproportionate damage. For the investigated bridge, axial stresses in the end cables under self-weight of the structural model are very high compared to the intermediate cables. The nonlinear static cases demonstrate that the initiation of every collapse progression distinctly involves rupture of one of the end cables. Moreover, from Fig. 22 and Fig. 23, it is certainly observed that the loss of an end cable causes a much higher variation in the bending moments on the deck girder than the others. Thus, these are the most important load carrying members in the model and failure of these could always initiate the collapse progression throughout the structure.
If any random cable fails, a huge amount of stress gets redistributed to the remaining structure ensuring a significant amount of increase in the axial forces in the remaining cables and stresses in the girders and deck. This redistribution ratio also depends upon the position of the elements. The remaining end cables attract more axial force than the middle cables. Since, these end cables were already carrying high axial tensions; they are more vulnerable to damage as a slight increase in the axial force can trigger their failure. Therefore, their chance of failure is much greater than the others due to the load redistribution. As perceived in the last failure case discussed, if failure occurs in a cable near to the pylon, the possibility of progressive collapse becomes less as the small amount of axial force incorporated with that cable(s) before failure, easily gets redistributed among the other nearby cables.
The structure’s susceptibility towards twist might also be taken into account as it creates a significant amount of concern. As discussed in Section 4.2, after the assumed failure of Cable 1-1 and Cable 1-2, Cables 2-1, 2-2, 1-3, and 2-3 failed one after another on the same cable plane thus creating a rotational imbalance on the deck. Thus the deck experiences a large twisting action causing a substantial increase in axial forces in the remaining cables on the same cable plane. This leads to failure of all the cables and perhaps the whole structure.
Impact ratios
It was observed that a unique impact ratio cannot be specified. Instead, the value is dependent on the location of the ruptured cable as well as the type and location of the state variable being considered. Very different impact rations result if the rupture of one cable is considered depending upon its location and load bearing capacity, i.e. its importance to the structure. Impact ratios in case of tensile forces, are always higher for the end cables, whereas, for bending moment or vertical displacement at a particular point, it depends upon the location of the ruptured cable. As observed from the figures discussed in Section 4.6, concerning the vertical deflections at the centre of the deck, an impact ratio between 1.1 and 1.8 results at these locations. The impact ratio for the positive bending moments in proximity to the ruptured cable lies between 1.2 and 1.6, while that for the negative bending moments between 1.5 and 2.2. Although failure of an end cable makes a distinct exception and the impact ratio becomes much higher in this scenario. The impact ratios for the design relevant cable forces – which develop in the cables adjacent to the lost cable – are between 1.9 and 2.8, depending on the lost cable being considered. Thus, a categorical calculation of the DAF can be beneficial.
Progressive collapse behaviour
A definite concept behind the progressive collapse may be derived from the static procedure. Vertical deformations occur in the longitudinal girder of the damaged cable plane due to the sudden loss of the mentioned cables for each case. The extra axial force initially carried by the failed cables gets redistributed throughout the structure. A larger percentage of this extra axial force is attracted by the nearby (or remaining) end cables. Moreover, normal forces acting on this bridge girder gets transferred to the longitudinal girder of the undamaged cable plane. These girders can’t resist the additional normal forces and commence to buckle in the vertical direction. So, vertical deformations grow strongly and can’t be stopped since the bridge deck is not restrained by fix supports in the longitudinal direction and thus, ultimate stresses exceeds in the bridge girder resulting in a relevant end cable failure. After losing this important load carrying member, the bridge structure tries to regain its symmetrical configuration. So failure occurs to the relevant cable to achieve symmetry. The failure path shifts to the other half of the bridge due to stress redistribution and a cable failure is observed. This continues till the girders in the middle of the decks fail due to flexure. Forces get transferred to the two pylons of the more damaged cable plane and they are pulled towards the centre of the deck, causing the final break down of the entire bridge structure.
Robustness
As discussed in the previous Sections (Sec 4.1 – Sec 4.4), if plastifications are allowed, the examined bridge acts robustly only for one end cable loss or for four intermediate (cables joining the Deck Centre with the Pylons) and adjacent cable losses. Alternate paths can be developed as loads are transferred through the bridge girder to the cables nearby to the ruptured one. Only local plastifications develop in the bridge girder and the cables remain elastic. The failure of any additional cable was observed to be impossible without total collapse. For the investigated bridge, the bridge girder proved to be the critical structural element as its resistance against instability was crucial for the structural robustness in case of two adjacent end cable failures. The resistance can either be increased by increasing the stiffness or by reducing the cable spacing i.e. increasing the cable density. Although, to avoid disproportionate collapse, the first alternative offers more advantage because as investigated by Shankar Nair [5]; if the cables are placed close together, the probability that more cables fail due to the same accidental event is higher, therefore the consequence to the girder remains the same.
Conclusions
The centre of the bridge deck and the top of the pylons are the most critical points of a cable stayed bridge. The end cables on either side of the bridge are the most vulnerable and important elements as failure of these increases the probability of a failure progression throughout the structure. Lesser the distance of the cable from the pylon, lesser will be the probability of total collapse. A conceptual understanding of the collapse progression throughout the structure is also gained as discussed in the previous section of this study. The results also highlighted the difficulty to specify a unique impact ratio. Instead it depends upon the location of the ruptured cable and also the considered type of state variable. Thus a more categorical and detailed calculation of the impact ratios through a time history analysis might be beneficial for an overall design of the cable stayed bridge. The robustness of the cable stayed model can be increased by preventing instability. This is possible by increasing the stiffness of the bridge girder or by reducing the unsupported length, i.e. the cable spacing. The former is suggested due to a higher failure probability of closely spaced cables.
Footnotes
Acknowledgments
Authors would like to thank the Department of Earthquake Engineering for providing the computational facilities required for this work.