The past century can be remembered as the most successful period for bridge engineering, which has also brought up great challenges to bridge engineers to digest rapidly-developed ideas, new tools, and novel methodologies in related engineering sciences while to assure new designs and existing bridges’ operation being safe, economic, and technologically advancing. While enjoying great success for decades, we did have suffered several significant setbacks. One example is the collapse of the Minneapolis I35W highway bridge.
August 1st of 2007, 6:05PM, the interstate I35W Bridge in the City Minneapolis, Minnesota, collapsed suddenly. The 8 traffic-lane, 1000-foot-long deck of the 1907-foot-long bridge fell into Mississippi River within seconds, resulted in 13 fatalities and 145 injuries see Figs. 1, 2.
Designed in 1964 and opened to traffic at 1967, the I35W Bridge stretches from north to south over the Mississippi River in the City Minneapolis in Hennepin County, Minnesota. The major part of the bridge is a 1000 foot (304.8 meters) long, 108 foot (32.92 meters) wide, three-span steel deck-truss superstructure, which collapsed completely. The bridge originally had six traffic lanes. After two major rehabs in last century, it has been widened into eight lanes with two auxiliary lanes and its concrete deck has been reinforced from 6.5 inches to 8.5 inches thick. NTSB’s official investigation [1, 2] indicates that, at the moment of the collapse, two lanes of the bridge were occupied by piled construction materials and heavy trucks; four lanes were opened to traffic; and the other two lanes were empty.
This article summarizes an independent investigation of this disaster based on material evidences and advanced computations. The theoretical background and the analytical methodologies applied can be found, for examples, those in the references list [5-26, 30-33, 43-46, 50-56]. A part of preliminary results of this analysis have been submitted to the related investigation and administrative agencies since the second week after the bridge’s collapse [27, 28].
This article is organized as follows: the next section introduces the structural and material’s models applied in the analysis. The third section analyzes the results obtained and compares the data with materials’ evidences. The following section discusses the reasons that had caused this bridge’s undersized design. The fifth section explores further the underlying structural issues that triggered the collapse, from which the lessons learned do have certain significance for the procedure of gusset plates’ assessment currently applied. A model based on two-dimensional plate buckling theory has been introduced. The last section summarizes the conclusions.
2. METHODOLOGY AND MODEL APPLIED
2.1 Structural Model: As depicted in Fig. 2, the steel superstructure of I35W Bridge contains three parts: (i) two main truss-frames, west and east; (ii) lateral bracings and 27 floor trusses, (iii) reinforced concrete-slab deck. Fig. 3 is a two-dimensional model of the bridge’s main frame and the locations of major gusset plates, i.e. the nodes in the truss-network. Additional subscript “E” or “W” may be used in the following text to denote a node at eastern or western main frame. As illustrated in Fig. 4a, a two-level computational model has been developed, i.e. using 3D finite element to obtain force-deflection behavior of gusset plate, which is implemented into truss network of the bridge superstructure.
2.2 Materials Model: A structural failure is often started by material’s failure in one or several key-structural components, which introduces extra complexities to an analysis of a structural collapse like I35W Bridge. In a new sustainable structure’s design, it is generally required to keep the stress level in all components within material’s yield limit. By contrast, a failure means applied stress reaches a material’s strength limit while the material’s strain surpasses yield strain. As explained by the example in Fig. 4b, for the defect-free plate under uniaxial tension on left-upper corner, the corresponding average stress-strain curve is the solid line that linearly increases from the origin and connects the thick dash line that ascends nonlinearly, plotted in middle. A material’s failure is the result of evolution of one or many defects. When a defect in the form of crack exists in the plate, as plotted in the lower-left, the average stress-strain relations can be represented by the linearly solid line connected with these thin short-dashed lines corresponding for different crack lengths. When the material elements at crack tip fails, the crack grows, for example, from a0 to a2, the plate’s elongation will increase continuously while applied stress decreases, as illustrated by the solid line connected through the dots. For common 30-50 grade mild structural steels at room temperature, such a crack growth is an accumulation of micro-scaled damages in the form of voids nucleation, growth and coalescence [9-13,30-33]. The “spring model” or “cohesive model”, developed in [12,13], have been implemented into finite element analysis to count such a damage-induced material’s failure.
RESEULTS AND DISCUSSION
(1) Forces and Bending Moments Distributions: Plotted in Fig. 6(a) are the uniaxial forces in the four groups of truss members in main frame, which are upper chords, lower chords, truss diagonals, and truss verticals. The force in upper chords, the solid line connecting small solid dots, reaches its positive maximum above the piers of center span while the negative maximum at the center. By contrast, the force in lower chords, the light dash-dot line, has a positive peak at the center and two negative maximums above the piers. The distribution of the force in truss diagonals (solid line) is characterized by the peaks around the spots where the upper chord’s force decreases from positive to negative while the lower chord’s force inversely uprises. As plotted in Fig. 6(b), the bending moment in truss-verticals is ignorable whereas that in other reaches positive maximum in the centre of middle span and negative maximum just above two piers. In this case the bending moment in the diagonal members is generally lower than that in upper and lower chords.
Fig.7 (a) shows a comparison of the upper chords’ uniaxial forces for the cases 0, 1, 2, and 5, which indicates that the deck’s slope and the horizontal constraint/forces from approaching spans do not have remarkable effects on this force’s distribution. By contrast, Fig.7(b) are the bending moments in upper chords when temperature changes while supporting bearings are locked, i.e. the cases 3 and case 4 in Fig. 5, respectively; compared with the case 2 that is the normal design condition. It demonstrates a significant combined effect of temperature change and locked bearing. In practice, severe corrosion may drastically increase the friction resistance of a bridge’s roller bearing; an extreme case is that a roller bearing eventually is fully locked so it becomes a pin point that does not allow horizontal movement. This induces considerable high thermal stress when temperature changes, as demonstrated by this plot.
Fig. 6 (a) Uniaxial force in the four groups of major truss members in main frame, where Nx is the uniaxial force and NYield is that when material yields; (b) the bending moment distributions, where Mx is in-plane moment and MYield is the moment at yield. No bearing-lock or temperature change (case 0).
Fig. 7: The same plots as that in Fig. 6 but considering temperature change and locked bearings, where RT refers to room temperature; (a) Uniaxial forces in upper chords and (b) the corresponding bending moment.
(2) Scenario of the Collapse: Now we discuss the following two issues: the sequence of the bridge’s superstructure’s collapse and what we can learn from it. One may notice that the force and bending moment distributions in Figs.6 and 7 are nearly symmetric to the central span of the bridge. Just after the disaster, there was an argument regarding the sequence of the failure, i.e., which approach of the bridge failed first. An opinion gained many echoes was that all structural components between the two piers over the river failed almost simultaneously, i.e. a horizontal fall of the central span. To clarify this issue, Fig.8 are the photographic pictures recorded by Army Engr. Corp. from south side of the bridge, which was reported by medium (CNN). The water splashes at the moment that middle span was fallen, i.e. Fig. 8(d), reveals the collapse initiated from south side [27a]. The north approach span failed subsequently without showing any capacity to resist the collapse.
In bridges’ design, the three-span like I35W is a common structure; the underlying idea is to gain long central span with minimized cost. This can be explained by the influence-line solutions in Fig. 9. By comparing the bending moments in the three cases with the same middle span length “b” and under the same distributed load “q”, one can find that the maximum bending moment of the case (a) is about 80% higher than that in case (c); similarly, the moment of case (b) is about 20% higher than case (c).
Fig. 8 The scenario of the progressive collapse presented by I35W (Army Corps’ Video)
However, from the viewpoint of structural integrity, the advantage of the three-span design (c) is traded-off by lower redundancy. This is because, when one of three spans is severely damaged or fails, the force distributions in other two shifts back to the situation similar to the case (a). Such a scenario is illustrated in Fig. 10. This, the author believes, was a reason that caused the progressive collapse of I35W Bridge. This leads to a conclusion that states as following: to avoid a similar progressive collapse that occurred in I35W Bridge, an additional safety factor may be necessary for those key single-load path structural components in a multi-span bridge if its design of load capacity is merely based on the influence-line analysis described in Fig. 9(c).
(3) Where the Collapse Started: The early study [27a] suggested that damage-induced material’s failure caused the structural failure. Such a material’s damage can be, for example, fatigue-induced cracks or significant corrosion-resulted section loss. Obviously, an undersized design has the same effect as such damage. Three possible failure patterns were focused, see Fig. 11, and the case (c) was considered as mostly close to reality [27a]. Although this simulation was done just a couple of weeks after the collapse (submitted to NTSB at August 17th, 2007), it does demonstrate significant deflection around the node U’9 and U’10. The “undersized” gusset plate U’10 actually initiated the bridge’s failure [1, 2, 41].
(4)NTSB’s Findings and Conclusions: after thorough examination of the wrecked pieces of the bridge, at January 15, 2008 and November of 2008, respectively, NTSB has released the results of the official investigation [1, 41], which disclosed that the gusset plates at the nodes U10, U’10, L11, L’11 in main frames of central span and U4, L3, U’4, L’3 of the approach-spans are undersized, which are half inch thick.
By contract, all other gusset plates in the nodes that connect four or five truss members were one inch in thickness or more see Fig. 12. Also, the NTSB’s analyses of the video recorded at the instance of the collapse indicates that the half-inched gusset plate at node U’10 failed first, which triggered the subsequent collapse of the bridge. The record of the design firm of the I35W Bridge reveals the bridge designed without appropriated calculation, which resulted in the undersized gusset plates.
(to be continued)