Progressive collapse of multi-span bridges – A case study
Progressive collapse of multi-span bridges – A case study
Uwe STAROSSEK Uwe STAROSSEK Dr.-Ing., P.E. VSL Korea Co., Ltd Seoul, Korea
Uwe Starossek, born 1956, received his civil engineering degree from RWTH Aachen, Germany in 1982 and his doctoral degree from the University of Stuttgart, Germany in 1991. He is a director of VSL Korea being responsible for the design of prestressed concrete and composite bridges. He is also a lecturer at
Korea University, Seoul.
Summary
Summary
The significance of overall structural response to accidental local failure and the possibility of a failure progression throughout the structure are discussed. A progressive collapse study of a multi-span prestressed concrete bridge is presented. An analysis strategy is developed. Analytical results and the ensuing impact on the design of this bridge are discussed.
Keywords:
Keywords: Accidental load; local failure; progressive collapse; robustness; risk theory; response spectrum; dynamic amplification; plastic hinge; conceptual design; detailed design; design code
1.
Introduction
1.
Introduction
Local failure of a structural element may cause failure of another element of the same structure. In such a way, failure may progress throughout a major part or all of the structure. Different structural systems exhibit different degrees of sensitivity toward progressive collapse. These different degrees of sensitivity are neglected when using conventional design approaches, which typically focus on the safety of the structural elements but not directly on the safety of the entire structure.
Today’s design codes and technical textbooks give little guidance on how to prevent progressive collapse or, more precisely, how to provide a homogeneous level of global safety to different kinds of structures. Consequently, designers have to address the problem on a case-by-case basis by applying first principles and by using engineering judgement.
A progressive collapse study for a recently completed multi-span prestressed concrete bridge is presented as a practical example. Analysis strategy and working hypotheses are outlined. Some
details and results of the analysis as well as the ensuing impact on the final design of the bridge are discussed. It is shown that the progressive collapse criterion can have a strong impact on both conceptual design, including choice of structural system, and detailed design.
2.
2.
Northumberland
Northumberland Strait
Strait Crossing
Crossing Project
Project
The Northumberland Strait Crossing Project or, as now called, the Confederation Bridge is a prestressed concrete bridge between Prince Edward Island and the mainland of New Brunswick,
Canada. The bridge is 12.9 km long. It consists of the main bridge of 43 continuous 250-m spans and approach viaducts on both sides of the main bridge ( Fig. 1).
2.1
2.1 Final Final designdesign
The cross section of the superstructure is a mono-cellular box with deck-slab cantilevers ( Fig. 2). Girder depth changes continuously between 14.0 m at the piers and 4.5 m at midspan. The deck rises up to +59 m CGD (mean sea level). The piers are supported on ring foundations down to –38 m CGD [3].
Fig. 1 The Northumberland Strait Crossing Project
The entire main bridge, including its substructure, is made of large-scale components pre-fabricated on shore. The principal components are the pier bases which rise to +4.0 m CGD, the pier shafts which include massive conical ice shields extending down to –4.0 m CGD, the 192.5 m long cantilever main girders, and drop-in girders of 52 m or 60 m length ( Fig. 2). Moment-resistant connections are provided between cantilever main girders and pier shafts through the use of post-tensioning. Every second main span is closed with a drop-in girder made continuous with both cantilevers thus creating a series of two-column portal frames. Continuity of these drop-in girders to the cantilever ends is accomplished by cast-in-place closure joints and external post-tensioning. The remaining spans are completed with drop-in girders that are simply supported on the cantilever ends. Construction started in spring 1994. The bridge opened to traffic in summer 1997.
2.2
2.2 PrelimPreliminary inary designdesign
The final design described in the previous section deviates from a preliminary design in several regards. The lengths of the cantilever main girders and of the drop-in girders were 150 m and 100 m, respectively. The cantilever depth was to change linearly; the depth of the drop-in girders was to be constant. Instead of simply supported drop-in girders in every other span, continuity at one end, and a hinge at the other end, of these girders was called for.
This concept seemed advantageous in terms of constructibility, construction costs and maintenance. It proved inadequate, however, to impart enough robustness to the structure to prevent progressive collapse in case of the accidental loss of one span.
2.3
2.3 Progressive Progressive collapse collapse studystudy
The engineers responsible for the final design, J. Muller International
-
Stanley Joint Venture, San Diego, studied possible mechanisms of, and means of design against, progressive collapse. This study has been performed during the author’s collaboration at J. Muller International. An outline is presented in the following.2.3.1 Philosophy of investigation and design
The conceivable triggers of collapse are manifold. A ship could go astray or an airplane might crash into the bridge; unexpectedly strong ice formations might collide with a pier; a fire caused by a traffic accident might damage the cantilever tendons in the top slab; a terrorist bomb placed at a vulnerable location might explode.
In view of the accidental nature of imaginable and unimaginable circumstances, and of the large dimensions of this structure, it would be unrealistic to design against progressive collapse just by preventing local failure at any expense. Instead, the possibility of a local failure must be accepted to
the extent that it becomes the starting-point of further investigation. The necessity of a progressive collapse analysis can be demonstrated by using the stochastic concepts of risk and reliability theory [4]. Application of such concepts to project-related design work, however, is not only difficult but, in the context of accidental loading and progressive collapse, also controversial: The statistical data is not sufficient to reliably establish the probability of triggering events; the magnitude of potential losses is not acceptable to society [1]. Thus, instead of a stochastic risk analysis, a deterministic analysis has been performed. It is based on certain premises regarding failure mechanisms and maximum failure progression.
Depending on the triggering accidental event, initial failure might occur in the vertical plane through the bridge axis or in transverse direction. Because of the joints in the bridge deck, a transverse failure would not give rise to substantial horizontal forces in the adjacent bridge sections (which are separated by joints). It would produce, however, large vertical forces and could eventually continue as a failure in the vertical plane. Only the latter case has therefore been investigated further, i.e., only effects in that two-dimensional plane are considered.
The following approach has been developed for the preliminary structural system ( Fig. 3). A collapse triggered by the failure of pier B or pier C should come to a halt, at the latest, at hinge H1 and at pier D. It is assumed that the drop-in girders slide off their respective bearings at hinges H1 and H2 so that the vertical supports, at these locations, are suddenly lost. The response of the remaining structures, to the right of H1 and to the left of H2, to these dynamic loads is investigated.
Fig. 3 Working hypothesis on progressive collapse onset
2.3.2 Loss of hinge H2
The response of the remaining structure to the left of H2 (see Fig. 3) after a sudden loss of this hinge has been investigated. The sequence of collapse, according to static and dynamic analyses, is marked by several distinct events ( Fig. 4):
The girder fails in bending under its own weight at the cast-in-place joint between cantilever and drop-in girder. The drop-in girder rotates around this point remaining connected to the cantilever through the continuity tendons. The free end of the drop-in girder hits the water, and the drop-in girder ruptures in bending under the inertia forces induced by its own mass [5].
Large forces are transmitted to the cantilever during this very violent event. Shear failure will occur at the cantilever end. The tendons cut through the bottom slab thus crippling the cantilever’s bending resistance. Rupture will progress throughout the cantilever toward the pier.
Further analytical prediction was deemed beyond credibility. Failure of the adjacent span (to the left of D in Fig. 3) and, thus, progressive collapse seemed possible. The only way to arrive at a predictable response was to provide for an early
separation of the falling drop-in girder from the remaining system. It was attempted to design a structural fuse within the cast-in-place joint between cantilever and drop-in girder. However,
no secure way to automatically cut (after collapse onset) the continuity tendons at that location was found, and the idea was abandoned.
An early separation seemed to be guaranteed only by inserting an additional hinge. The structural
system of the preliminary design has thus been modified ( Fig. 5): The drop-in girders in every other span are simply supported on the cantilever ends.
Fig. 4 Collapse after loss of hinge H2
Furthermore, the length of these drop-in girders has been reduced from an otherwise more advantageous 100 m to 60 m in order to assure final separation before the drop-in girder’s free end hits the water. (This change is also beneficial regarding the forces generated by the rotating drop-in girder and for the progressive failure resistance of the adjacent span.)
With these modifications, load case “loss of hinge H2” is equivalent to load case “loss of hinge H1” which still remains to be investigated.
2.3.3 Comment on the modification of the structural system
As stated in the previous section, insertion of additional hinges was the only way to arrive at a predictable response. When assuming that not only predictability but also robustness is improved by
this measure, we are in the intriguing situation to have to explain how reducing the system’s degree of static indeterminacy (which in turn is considered a measure of redundancy) can increase robustness.
A key to the answer might be found in the fact that a failure progression requires a certain degree of connectivity and interaction between neighboring structural elements
-
properties usuallyassociated with the system’s degree of static indeterminacy. A further ingredient to a proper explanation might be that failure progression involves violent dynamic effects.
2.3.4 Loss of hinge H1
The response of the remaining structure to the right of H1 (see Fig. 3) after a loss of this hinge is to be investigated. Because of the modification of the structural system, however, this loss does no
longer need to be a sudden event, associated with a step-impulse type of loading. Instead, sudden loss of the hinge at the opposite end of the drop-in girder might occur leaving this girder connected, for some time during its fall, to hinge H1. Final separation from H1 will take place at a certain angle of rotation defined by the geometry of the hinge corbel.
The vertical hinge force at H1, during this more gradual event, has been analyzed on the basis of simplifying assumptions. During the fall, the displacements of the drop-in girder will be much larger than the displacements of the remaining structure. A fixed bearing at H1 has therefore been assumed in a first analysis approach ( Fig. 6a). This will result in both vertical and horizontal forces at H1. No major horizontal forces can be resisted, however, and the drop-in girder will soon enter into a sliding motion at H1. In a second analysis approach, a sliding bearing at H1 has been assumed ( Fig. 6b). The drop-in girder’s center of gravity, in this case, moves in vertical instead of circular direction. The girder’s actual trajectory is expected to lie somewhere in between these two extreme cases. Both cases have been analyzed as outlined in the following.
The equations of motion (1a) and (1b) follow from the conditions of dynamic equilibrium of the moments around the point of support. In both cases, second-order differential equations are obtained for the angle of rotation Θ as function of timet . Both equations of motion are nonlinear and cannot be solved in closed form. Numerical solutions have thus been pursued.
For this purpose, eqs. (1a) and (1b) have been re-arranged. Separating variables and integrating twice leads to the equivalent formulae (2a) and (2b), respectively, which are formal expressions for the timet as integral function ofΘ. These integrals can be evaluated numerically after
l g m x y x F F y H1 P Θ Θ l m F x F y x g y H1
Fig. 6a Motion of hinged drop-in girder Fig. 6b Motion of sliding drop-in girder
0 cos 2 3 Θ= − Θ l g
(1a) Θ(1+3cos2Θ0)−3Θ2cosΘsinΘ−6 cosΘ=
l g
(1b) Θ Θ Θ = Θ 0 sin 3 ) ( d g l t (2a) Θ Θ Θ Θ + = Θ 0 2 sin cos 3 1 12 ) ( d g l t (2b) ) 2 sin( 8 9 Θ = mg F x (3a) 0 = x F (3b)(
+ Θ)
= 2 sin 9 1 4 mg F y (4a)(
2)
2 2 cos 3 1 sin 3 4 Θ + Θ + =mg F y (4b)Furthermore, invoking the dynamic equilibrium conditions in horizontal and vertical directions, and making use of eqs. (1a), (1b), expressions (3a), (3b), (4a), (4b) have been found which are closed-form solutions for the hinge forces F x and F y as functions of the rotation angleΘ. When using
these expressions in conjunction with eqs. (2a), (2b), parametric-numerical relationships between hinge forces F x, F y and timet can be established.
Further analysis details are given in [4].
It was found that, during the relevant time period, the vertical hinge force F y obtained from the first
analysis approach is larger than F y obtained from
the second analysis approach. Only the force according to eq. (4a) has therefore been used further. As it can be seen in Fig. 7 , this force suddenly drops to one half its static value when the drop-in girder starts to fall. During the fall, the force gradually increases and eventually exceeds the static value. When reaching the angle of disengagement, the force suddenly disappears. Fig. 7 Vertical force at cantilever tip during
The vertical hinge force corresponds to the vertical force acting at the tip of the cantilever of the remaining structure to the right of H1. The response of the remaining structure to this force has first been investigated in a linear time-history space-frame analysis. It was found that large moments –
difficult to design for – develop at the top of the first adjacent pier and in the cast-in-place joints of the first adjacent span.
In view of the accidental nature of the considered loading, however, and for sake of an economical design, the formation of plastic hinges within the remaining structure was deemed acceptable. The plastic reserves of the structural system have thus been utilized in the design against progressive
collapse. The further investigation advanced into the realm of the theory of plasticity. To keep analysis manageable, a quasi-static approach has been used (instead of a nonlinear time-history space-frame analysis).
The key to that approach is the determination of a dynamic amplification factor. Fig. 8shows the dynamic part (defined in Fig. 7 ) of the load function and the response of a single-degree-of-freedom system, computed with a Duhamel integral. All quantities are made dimensionless by relating to the respective static values. The
load function’s initial step impulse of ½ excites the system to a maximum response of almost 1. The second loading step, to a final value of 1, causes the system to swing far beyond 2, which would be the maximum value of response in case of a simple unity-step impulse loading [2]. Hence a gradual separation of the drop-in girder can produce higher forces in the remaining structure than a sudden loss. The reason for this is the second step impulse, resulting from final disengagement, which might produce a resonance-like dynamic amplification. The actual maximum response depends on the ratio of the time of final disengagement to the system's period of vibration. Analysis has thus been repeated for a spectrum of many different periods of vibration. The resulting Fig. 9 shows
the spectra of extreme responses (to the same dynamic loading) of a single-degree-of-freedom system as functions of its period of vibration. Based on this investigation, it was concluded that the dynamic response could be up to 2.6 times the static response (i.e., when the same loading is applied very slowly). The remaining structure has thus been loaded with the static hinge force at H1 times a dynamic
amplification factor of 2.6. The detailed design was based on a plastic space-frame analysis performed for this quasi-static loading [4]. Fig. 8 Impulsive load and response of
single-degree-of-freedom system = dynamic load function = response
Fig. 9 Impulsive load and response spectra of single-degree-of-freedom system
= dynamic load function
= spectrum of extreme positive response = spectrum of extreme negative response
2.3.5 Additional design modifications
The final design of the Northumberland Strait Crossing Project was strongly influenced by the investigation on progressive collapse. Additionally to the design changes already mentioned, the following modifications have been found necessary to avoid progression of a local failure into the adjacent spans:
• Post-tensioning between superstructure and piers was increased to the practicably possible maximum to limit the moments to be redistributed into the superstructure after formation of a plastic hinge at pier top.
• The form of the superstructure’s soffit was changed from haunched to curved to increase section depth, and moment capacity, at the cast-in-place joints.
• Top and bottom reinforcement was added around the quarter points of the continuous spans to limit the number of plastic hinges in the superstructure to one.
• Transverse reinforcement was added in the regions of expected plastic hinges to provide sufficient rotational capacity.
A detailed account of the investigation outlined here and its impact on the final design of the Northumberland Strait Crossing Project can be found in [4].
3.
Conclusions
3.
Conclusions
The requirement to avoid progressive collapse in case of local failure is an important design criterion for multi-span bridges and other complex structures. It can have strong impact on both conceptual design, including choice of structural system, and detailed design.
Current design codes do not strictly require the prevention of progressive collapse. Recent disasters and theoretical considerations on the basis of risk theory indicate that codes should be improved to more clearly address this problem.
In the meantime, owners and engineers should be encouraged to use judgement and discretion to implement the necessary measures even if not yet specifically required by codes.
4.
References
4.
References
[1] Breugel K. van, “Storage System Criteria for Hazardous Products,”Structural Engineering International , IABSE, Zürich, Vol. 7, No. 1, 1997, pp. 53-55.
[2] Clough R. W., J. Penzien, Dynamics of Structures, McGraw-Hill, New York, 1975. [3] Sauvageot G., “Northumberland Strait Crossing, Canada,” Fourth International Bridge
Engineering Conference, Transportation Research Board, Vol. 1, August 1995, pp. 238-248. [4] Starossek U., “Zum progressiven Kollaps mehrfeldriger Brückentragwerke” Bautechnik ,
Ernst & Sohn, Berlin, Vol. 74, No. 7, 1997, pp. 443-453.
[5] Starossek U., G. Sauvageot, Discussion of “Bridge Progressive Collapse Vulnerability” by A. Ghali, G. Tadros, Journal of Structural Engineering , ASCE, Vol. 124, No. 12, 1998, pp. 1497-1498.
