Handling the exceptions: robustness assessment of a complex structural system Franco Bontempi, Luisa Giuliani, Konstantinos Gkoumas School of Engineering, University of Rome “La Sapienza
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Keywords: complex structural systems, suspension bridge, robustness, nonlinear dynamic analysis, initial failure, collapse propagation. ABSTRACT: The robustness of a structure, intended as its ability not to suffer disproportionate damages as a result of a limited initial failure, is an intrinsic requirement, inherent to the structural system organization. A robust design cannot therefore be limited to design the structure just for additional accidental load cases, but should take into account the response of the structure to different damage conditions. Complex structural systems like the long suspended bridge considered in this paper, in case some failures involving the suspension system occur, can be inclined to the propagation of the damage, due to the high continuity of the long central span. Some specific investigations aimed to robustness assessment have therefore to be carried out on the structure: following a bottom-up approach the critical event modeling can be neglected and an initial failure has to be a-priori assumed on the structure, in order to evaluate if and how much the overall response of the structure is influenced from a local failure. The investigations presented in this paper start from assuming an abrupt rupture of some hanger ropes of the suspension system and investigate the response of the system to different damage levels and locations. This kind of investigations require quite complex nonlinear dynamic analyses to be performed, in order to account for the dynamic amplification caused from such abrupt initial failures and for the plastic reserves of the materials that the system can exploit during the collapse. 1
ROBUSTNESS ASSESSMENT
Within the achievement of structural safety, most recent codes and regulations (Model Code 1990, ASCE 7-95, TU 2005) agree in considering the importance of assuring a satisfying behavior of the structure not only in its nominal integer configuration, but also in an imperfect one, where element failures or structural damages have occurred, either in consequences of human errors (in the construction design or execution) or in consequence of malevolent attack or natural disasters. It’s not to be expected that the structure will resist all these possible occurrences without any damage, but a disproportionate collapse consequent to a localized triggering failure should be avoided, i.e. the structural system should prove a robust behavior. If methods to satisfy the service and ultimate limit states are wide studied and documented, regulations for the practical achievement of an acceptable level of structural integrity after a failure are not as much codified. Most recommendations deal with the problem of structural integrity in a general way, providing just for indirect design criteria.
With the term indirect design all those measures are meant, that, without requiring any specific investigation, are aimed to provide the structure with an high level of continuity and ductility: the continuity is supposed to allow for a redistribution of loads on the elements adjoining the failed ones, while the ductility should favor the dissipation of the dynamic energy consequent to an abrupt failure. This kind of generic prescriptions and recommendations let them to be easily contemplated by codes and regulations, but don’t allow for a clear identification of performance design objectives (MMCPPC 2002). For this purpose it’s necessary to perform instead a direct design, with this term meaning an investigation aimed to highlight the structure behavior after the occurrence of failure or a critical event. In this regard, it is possible to follow two different approaches: a) in a top-down investigation, the critical event has to be modeled (a collision between a ship and a bridge pier, the explosion of a bomb in a building, the detachment of some bridge hangers from the collars that tie them to the main cables) and
the following structural response has to be evaluated; b) in a bottom-up investigation, the particular triggering cause is set aside and the investigation proceeds from the lowest hierarchical structural level to the higher ones, e.g. the consequences due to the failure of an element on the rest structure are identified. Contrary to a top-down investigation, which can provide information on the structure insensitivity just limited to a particular accidental circumstance (the term collapse resistance is in this regards proposed by Starossek & Wolff., 2005), this approach provides information on the robustness degree of the structure. In a direct design however (either when performing a top-down or a bottom up investigation), quite complex and onerous analyses are required (full geometric and material non linearity, dynamic actions, triggering of local mechanisms, possible collisions within elements, as well as many different scenarios), but the response of the structure consequent to a critical event is known and in case of a top-down analysis an intrinsic robust behavior could be achieved (Fig.1) Failure is prevented
Failure is presumed
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Figure 1: Strategies and methods for robustness achievement.
2
A CASE OF STUDY
The previously exposed concept of structural robustness finds a practical exemplification in the further described investigations of a long suspended bridge: the bridge considered is based on the preliminary design of the Messina Strait Bridge that should connect the Italian peninsula with the Sicily Island. This bridge was deeply studied by the authors in the years 2002-2005 during the reexamination of the design of the bridge: for its unique features and mainly because of the long main central span (3300 m over a total length of 3666 m) that distinguishes the structural system (Fig.2), this bridge is particularly interesting from the point of
view of a robust investigation (Bontempi et al. 2005; Bontempi 2006, Giuliani et al. 2007) For a detailed description of the bridge refers to that one presented in “Handling the exceptions: dependability of systems and structural robustness” (Bontempi et al., 2007), whose this paper is meant to represent an applicative exemplification. The investigations concern the bridge behavior consequent to an initial damage, intended here as a sudden removal of some hangers. The approach followed is a bottom-up method: it is irrelevant to the robustness assessment of the bridge if such an abrupt failure is caused from explosive charges at the bottom of the ropes, or to an unexpected break off of the collars that tie the hangers to the cables, or to an erroneous evaluation of the material resistance of those hangers. The analyses are aimed to identify the minimum number of removed hangers needed to cause a subsequent damage in the adjoining one, this being a meaningful parameter that depends on the particular structural system and on the damage location. 2.1 Contingency and load scenarios The investigations performed considered four different locations for the initial damage within the main span length (zone A, B, C, D of Fig.2) in order to identify the most sensitive position of the bridge to the failure and two different damage conditions of symmetrical and asymmetrical failure (respectively on one side only or on both sides of the bridge). Among all the investigations that have been carried out, some significant results are presented in the following, that refer to these contingency scenarios: 1. Asymmetrical and symmetrical failure of hangers in zone A (damage centered at 345 m of distance from the left tower). 2. Asymmetrical failure of hangers in zone B (damage centered at 900 m of distance from the left tower). 3. Symmetrical failure of hangers in zone C (damage centered 450 m left from the mid-span). 4. Symmetrical and asymmetrical failures of hangers in zone C (damage centered at mid-span). TOWER 345 m
MID-SPAN
A 900 m
B C
450 m
D
section type 1
section type 2
section type 3
Figure 2: Different damage locations.
As regards the load scenarios, the hanger failures are always considered to occur on the unloaded structure, i.e. neither the train nor the car loads are
assumed on the bridge, which results therefore deformed only under the self weight. 2.2 Bridge modeling The model of the bridge (Fig.3) has been implemented in a current commercial code using onedimensional finite elements: particularly, the hangers have been modeled through tension-only frame elements with moment releases at both ends.
Figure 3: Bridge structural scheme and hangers modeling.
Different starting schemes were considered for the investigations, each referring to the above mentioned damaged configurations of the bridge. In each structural model the hangers that are intended to be removed during the analysis were modeled by means of equivalent forces: so it’s possible to perform the element killing needed during the analysis just by activating a time-history function on those forces, as in the following section better explained. A preliminary static analysis has therefore been performed on the integer structure, in order to compute the reactions exerted by the hangers on the rest of the structure. Then, for each considered contingency scenarios, the interesting hangers have been substituted in the static scheme with its previously computed reactions: a static analysis on the model that consider the load case assigned to these reaction should provide the same results of the integer bridge analysis, ensuring that the nominal configuration is correctly restored by the equivalent forces; a static analysis where this load case is neglected would instead be able to show the final equilibrated configuration for the damaged bridge only if no subsequent plasticizations occurred due to dynamic amplification after the equilibrium point. 2.3 Analysis type Investigating the response of a complex system after a structural failure requires to take into account in the performed analyses many different aspects that will be briefly remarked in the following. 2.3.1 Geometric nonlinearities The responses of a long span suspended bridge like the considered one is strongly nonlinear, due to
the presence of the cables. Therefore, geometrical nonlinearities have to be considered even within the elastic response, since it’s fundamental for the equilibrium equations to be written with respect to the deformed geometry. Being this deformed configuration not known in advance, the solution cannot be obtained in a single analysis step but requires developing through several steps, updating the tentative solution after each step until a convergence test is satisfied (Catallo, 2005). The geometrical nonlinearities considered in the performed analyses are limited to the 2nd order Theory (the so called P-Delta effects), whereas the large displacement formulation is not actually used. 2.3.2 Material nonlinearities Since the “tension only” frame elements used to model the hangers have a nonlinear behavior, material nonlinearities have been considered starting from the first static analysis. Moreover, a nonlinear behavior in the axial hangers response is also considered, by means of plastic hinges with a tributary equivalent length equal to the length of the element on which the hinge is assigned. The considered hinges have no hardening branch and drop load when the ultimate displacement is reached. The definition of the hinge properties parameters is summarized in the image below (Fig.4). It’s worth noting that in the performed analyses the deck is considered to behave elastically and only some qualitative considerations about a possible yielding of the continuous box-girders of the deck are provided. fy
B
C plastic range
rigid branch
hinge drops load
residual deformation
D 0,2 fy
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y
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Figure 4: Definition of the axial hinge properties for hangers.
2.3.3 Dynamic analysis The abrupt failure of one or more hangers is a dynamic event, that can hardly be considered through a quasi-static approach by means of an amplification factor: the estimation of this coefficient would indeed not be easy for such a complex structural system, since it depends on many factors, like the ratio between the time taken by the failure event and the period of the vibration mode solicited from that failure (i.e. how much the failure event is felt as “dynamic” or as “quasi-static” from the structure)
3
RESULTS
The most significant outcomes resulting from the previously described investigations are presented and briefly discussed in the following sections. 3.1 Initial failures in zone A The bridge response to hanger abrupt failures localized in this zone results to be a quite robust one. With reference to an asymmetrical initial damage, if up to 8 hangers together abruptly fail, nothing significant occurs to the other elements of the bridge, that bear the overloading due to stress redistribution and dynamic amplification without exceeding the elastic range. When 9 hangers together fail the hanger adjoining the failed ones to the right (RH) reaches the plastic range but don’t actually break, since the dynamic peak is too short to exhaust the hanger ductility. So the plastic hinge unloads elastically and preserves the following hangers from yielding. The result is just an increment in the vertical displacement of the unsupported deck segment with respect to the static equilibrium of the elastic analysis. It’s worth noting that the same localized ef-
fect on the right adjoining hanger occurs also when 10 or 11 or all the 12 hangers of this zone (section type 2) are abruptly removed. With respect to an initial damage that affects symmetrically both sides of the bridge, nothing significant happen up to the abrupt failure of 5 hangers per side and just a localized effect regarding the plasticization of the hangers adjoining the damaged zone (RH & LH) occurs when 6 hangers per side together fail. In order to trigger a collapse progression, an initial damage of 7 hangers per side is needed. It’s interesting to notice that in this case the collapse propagates faster and faster toward the centre, while the hangers toward the tower resist the propagation of the collapse much longer, since the one to the left of the damaged zone (LH) eventually breaks: its rupture occurs 6 seconds after the right one (RH), i.e. when 8 right hangers have already failed (Fig.5). Axial force [10^3 Ton]
and the energy absorbed from possibly occurring plasticization of elements, thing that cannot be known in advance. Moreover, if it is still true that in a first evaluation (aimed to exclude the possibility of damage propagation), only the first peak of the dynamic wave is relevant (if the structure survive it without any subsequent damage, it will resist all the more the following dumped waves), not the same can be said when other elements collapse as result of the triggering events. In this case, as better shown in the following, a full dynamic analysis is required to correctly investigate the propagation of the collapse. The hanger failure has been therefore considered through a sharp ramp function (the time needed for the failure is taken equal to 1/100 second) assigned to the equivalent forces that simulated the presence of the considered hangers in the structure. The dynamic analysis performed consists in a directintegration time-history analysis where no damping is assigned and all the previous exposed nonlinearities are considered. The time integration method used for the time-history analysis was the HilbertHughes-Taylor (HHT) method, that use a single parameter alpha, that can assume value in [-1/3, 0] and that was set to zero in the performed analyses, letting the method to be equivalent to the so called trapezoidal rule method (i.e. method of the average acceleration). In order to let the hanger failure occurs in a selfweight equilibrated configuration, the dynamic analysis starts at the end of a previous static case that provides for the initial conditions of the subsequent time-history case.
LH
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LH unloads elastically
3 2,5 2 1,5 1
RH & LH yield
6 seconds
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Figure 5: LH and RH behavior in symmetrical damage, zone A.
It has to be emphasize that such long unsupported span segments like that ones shown by these analyses would probably not actually develop. In fact the rise of high negative moment on the extreme section of the unsupported deck length would not be bearable by standard box girder sections that would break when the ultimate moment value is reached. If in case of a symmetrical initial damage a proper design of joint details could provide for an early detachment of the deck after the yielding of the extreme sections, not the same can be easily said for the asymmetrical failure where the moment distribution results to be non symmetric too. 3.2 Initial failures in zone B When the initial damage happen to occur in a zone farer from the tower, an higher number of hangers is needed to be abruptly removed in order to cause the plasticization of the adjoining ones or the triggering of the chain ruptures. With regards to an asymmetrical failure, the abrupt removal of 7 hangers causes the hangers to the left and right of the damage zone to yield, versus the 9 needed nearby the tower (zone A). Also here the two hangers don’t break, preserving the following hangers from yielding.
In order to cause the rupture of the two hangers that trigger the chain ruptures to the left and to the right of the damaged zone, 9 hangers have to abruptly fail together. Also in this case the collapse propagation triggers in direction of the bridge centre first (after 2 second since the removal) and the hanger to the left of the damaged zone (LH) surmount the rupture of 12 hangers to the right and resist up to 4 seconds after the rupture of the first right hanger before breaking and triggering the collapse also toward the tower (Fig.6). Tension [10^3 Ton]
9
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Figure 7: Moment propagation in case of a high resistant deck.
3.4 Initial failures in zone D
0
Chain ruptures propagate
If the deck has enough resistance to withstand the negative moments that develop on the girders just before the first hanger rupture, it will then let the collapse propagate, since, due to the velocity of collapse propagation, no higher negative value of the moment will be reached from the deck (see Fig.7).
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Figure 6: Rupture propagation to the right and to the left of the removed hangers, in case of an asymmetric damage in zone B.
The centre of the bridge proves to be the most sensible zone to a damage involving the hanger system: in this zone the asymmetrical abrupt failure of just 7 hangers and the symmetrical failure of 5 hangers are sufficient for the triggering of the collapse. Furthermore the failure propagations result to be very fast: referring to the propagation toward the centre, 7 hangers break in just three second (Fig.8), while in the previous exposed case of asymmetrical failure in zone B (Fig.6) one second more was needed to let the same number of hangers break in the same direction. Tension [10^3 Ton]
It can be also observed that in the direction of the bridge center the hangers yielding and rupture speed up with the time: in the first second after the first rupture just 2 hangers break, while in the next second the chain rupture involve the following 4 hangers (Fig.6). The velocity of the collapse propagation, even if not very high at first becomes rapidly hasty. 3.3 Initial failures in zone C In case a symmetrical failure is assumed, the propagation of the collapse is shown by removing just 5 hangers per side, versus the 7 needed to trigger the collapse nearby the tower (zone A). Again, the collapse propagates faster toward the centre of the bridge and only some seconds later the two adjoining left hangers of both sides also break, triggering the chain rupture also toward the tower. As above already mentioned and observable in the image sequence below, the peculiarity of a symmetrical rupture is the fact that it could lead to an early separation of the unsupported deck segment, which could allow for a collapse standstill. The resistance and the ductility of the deck result therefore to be fundamentals parameter in this kind of failure.
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Figure 8: Rupture propagation to the right and to the left of the removed hangers, in case of an asymmetric damage in zone D.
Even if the first rupture trigger very soon and propagates also very rapidly, though it does not ac-
celerate, contrarily to what happened in the previous exposed case: by carefully comparing the graphics above (Fig.8) it can be observed that two hangers break in each half second after the first rupture. Nevertheless the collapse doesn’t come to a stop, not even in correspondence of the higher resistant hangers (section type 2). These hangers are very far from the centre though, so when the ruptures get so far, the unsupported deck involves the major of the central span, making very difficult (and also not very significant) a collapse standstill. Further investigation should be carried on in order to test a possible effectiveness of an earlier change of sections in the hangers for the collapse arrest. 4
CONCLUSIONS
The review of the results provided from the performed investigations permit to identify some characteristics in the behavior of the bridge after a damage in the hanger suspension system, that are intrinsic to the structural system. The bridge results to be more sensible to the damage at mid-span, where just 5 hanger abrupt removals on both sides (asymmetrical rupture) or 7 hanger removals on one side only (asymmetrical rupture) are needed to trigger the collapse propagation. Shifting the initial damage location aside (about at 1/3 of the span) the asymmetrical rupture of 9 hangers is required for the collapse propagation, while moving the initial damage near the tower even the asymmetrical removal of 12 hangers has no global effects on the structure and very 7 hangers must be symmetrically removed on both sides in order to trigger the propagation of the ruptures on the adjoining hangers. This higher damage sensibility of the bridge central zone counterpoises a lower acceleration of the collapse progression triggered by central ruptures, with respect to that one triggered by lateral ruptures. This effect is due to the particular configuration of the structural system that requires a growing hanger length from the centre to the sides of the bridge: when a chain rupture trigger, the ultimate elongation required to the hangers adjoining the failed ones increases as the collapse propagates (because the unsupported deck length also increases). If the initial damage occurs at mid-span, it involves the shortest hangers and the collapse propagation is partially slowed down from the growing element ductility of sideward hangers. On the contrary, a more intense initial damage is required sideways to trigger chain ruptures, but then the hanger breakdowns speeds up when moving toward the centre, where the hanger length decreases. If in the first case a closer increment in the section of the hangers (that remain instead the same for about 5/6 of the span length) could possibly provide
for a collapse standstill, in the second case the progressive collapse shows a preferential direction, making probably less effective such a measure. Another consideration about the possible collapse standstill concerns the higher susceptibility of the bridge to an unsymmetrical hanger failure than to a symmetrical one: in the last case the symmetrical hinge formations provide for a symmetrical moment increment on the deck box-girders, thus possibly allowing for an early deck segment detachment that would arrest the collapse. ACKNOWLEDGMENTS The authors wish to thank Professors R. Calzona, F. Casciati and U. Starossek for discussions related to this study. The financial supports of University of Rome “La Sapienza” and COFIN2004 are acknowledged. The opinions and the results presented here are responsibility of the authors and cannot be assumed to reflect the ones of University of Rome “La Sapienza”. REFERENCES Model Code 1990. Buletin d’information No. 203, Lausanne, 1991. American Society of Civil Engineers. Minimum design loads for buildings and other structures, ASCE 7-02, Reston, VA, 2002. Testo Unitario - Norme tecniche per le costruzioni, D.M. 14/09/05, Italy 2005 (in Italian). Multihazard Mitigation Council Prevention on Progressive Collapse: Report on the July 2002 National Workshop and recommendation for future efforts, Washington D.C., 2003. U. Starossek, M. Wolff, Design of collapse-resistant structures, JCSS and IABSE Workshop on robustness of structures, Building Research Establishment, Garston, Waterford, UK, November 2005. F. Bontempi, L.Giuliani, K.Gkoumas, Handling the exceptions: dependability of systems and structural robustness., Proceedings of the 3rd international conference on structural engineering, mechanics and computation (SEMC07), Cape Town, South Africa, September 10-12, 2007 F. Bontempi, K. Gkoumas, G. Righetti, Conceptual aspects and considerations for the Risk Analysis of Complex Structural Systems such as long suspension bridges., Proceedings of the 9th International Conference on Structural Safety and Reliability (ICOSSAR2005), Rome, Italy, June 19-23, 2005 F. Bontempi, 2006. Basis of Design and expected Performances for the Messina Strait Bridge. Proc. of the International Conference on Bridge Engineering – Challenges in the 21st Century, Hong Kong, 1-3 November, 2006. L. Giuliani, K. Gkoumas, F. Bontempi, Nonlinear dynamic analysis for the dependability of complex structural systems, Proceeding of the international conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN2007), Rethymno, Crete, Greece, June 13-15, 2007. L. Catallo, The robust design of long span suspension bridge., Proceedings of the 10th International Conference on Civil, Structural and Environmental Engineering Computing, Rome, Italy, 30 August - 2 September 2005.
