[1] The I35W Bridge is numbered as 9340 in national bridge inventory.
Figure 14 An illustration of the main “load path” in central span of I35W, where the blue solid-line arrows represent compression and the red dotted line arrows represent tension. The compression is mainly corresponding to the resistance from the pier and the global bending moment; whereas the tension is mainly corresponding to span’s weight and deck load. The gusset plates U10, L11, U12 are the pivots to balance load and resistance forces transferred by diagonal members. The major function of vertical trusses is to provide redundancy to the truss cells except the member L8-U8.
5. “TRUSS APPROXIMATION” AND GUSSET PLATE
A practical question is: how to determine a gusset plate’s capacity in practical applications? Conventional consideration treats gusset plates as additional stiffeners that do not have significant effect to a truss network load capacity. The author of this paper had also been convinced for while that “truss-approximation”, i.e. omitting bending moment and shear in slender structural members of a truss network, was good enough for applications. Under this approximation, a gusset, which fastens connected truss-members, becomes a “hinge” that allows free rotation between adjacent members because of no bending moment exists. This approximation generally provides acceptable accuracy for long beams/trusses while significantly simplifies analysis, which has been successfully applied over centuries. However, it should be noticed that the resulted “hinge approximation” implies the ignorance of the secondary stress caused by bending moment at a gusset plate and the corresponding stress concentration. These stresses may not have significant effect to truss members but can be crucial for a gusset plate. To demonstrate this point, two simple examples are given in Fig. 15 to explain the differences.
Fig. 15 (a) The truss-cell is an unstable structure when hinges are at the corner. By contrast, gusset plates keep the adjacent trusses perpendicular to each other, resulted in the bending moments at the corners; (b) An additional diagonal bracing stabilizes the frame; the bending moments become more remarkable at the upper-right node gusset and its amplitude increases when the ratio of the widths of diagonal bracing (
h_{D}) and vertical trusses (
h) increases; when this ratio is large, the upper horizontal truss is like a cantilever fixed at right end by the gusset plate.
Fig. 15(a) shows a square truss-cell that is unstable when the nodes at the corners are hinges. Adding gusset plates to keep the adjacent trusses perpendicular to each other, which makes the cell becomes a stable frame but results in bending moment at the gusset. When an additional diagonal truss is added to the cell in (a), as illustrated in Fig.15 (b), the cell becomes a stable structure despite hinges at the nodes. The corresponding bending moment in upper chord is plotted when a concentrated force is imposed. As a comparison, the lower part of this figure is an analytical solution when the nodes are stiffened by gusset plates; stronger diagonal member results in higher bending moment at the node, hence, the gusset plate.
It is no doubt that “truss approximation” can provide satisfied solution for the uniaxial forces of slender members in many cases. However, when force flows in a truss-network are not uniform, instead, varying drastically, the corresponding bending moments, particularly, at gusset plates, can be remarkable. This may occur at least under the following three conditions: (i)concentrated load presents; (ii) inhomogeneous internal stress, for example, thermal stress; (iii) residual stresses caused by, for example, in-appropriated welding or cambering. For a structural member like a gusset plate with complicated geometry, a bending moment-induced stresses and corresponding stress concentration may have substantial effects on its load capacity and fatigue life. The effect of bending moment can be viewed directly by the numerical analysis presented in Fig.9(c), which shows significant relative rotation between adjacent trusses around those pivotal nodes such as U10, L11 when the superstructure starts to fail. This result can be verified by the bowing of the actual U’10 gusset plate observed four years before the bridge’s collapse, see Fig. 16(a), which could be caused by a past deck’s overload or the result of cambering during erection. In either case, such a bowing implies reduction of the gusset’s load capacity and, thus, its vulnerability to this kind of loads applied later.
Fig. 16 Three patterns of gussets’ failures:(a) bowing, I35W Bridge’s gusset plate U10’ [2]; (b) a cut of the U10’ plate’s lower part; (c) buckling due to the large ratio of unbraced length /gusset thickness, which occurred in the highway I90 Grand River Bridge, Ohio, 1997 [47];
6. GUSSET PLATES’ LOAD RATING
(1 )Failure patterns of gusset plates: The gusset plates’ failure in I35W Bridge is not a unique case. Fig. 16(c) shows a gusset’s buckling that occurred 1997 at the Grand River Bridge that carried highway I90 of Ohio [47, 48]. Therefore, it can be assure that gusset plate may fail at least by three different patterns: (i) bending and associated rotation-induced bowing, on the part of a gusset plate with free-edge between two adjacent trusses, see Fig. 16(a); (ii) tension-dominated ductile failure between the edges of two adjacent trusses, which can be a cause or the result of a bowing presented at another edge of one truss, or occurs independently; (iii) buckling in the unbraced area in the front of an attached truss, which may be accompanied by bowing nearby, as demonstrated in Fig. 16(c).
Most gusset plates in steel bridges are made of 30-70 grade mild steels. This class of steels often fails in the form of ductile fracture except under some particular circumstances such as temperature below ductile-brittle transition point or within the area near or at a welded joint. For a thin gusset plate, buckling or bowing is when compression force reaches instability limit when applied stress may be much lower than yield strength. By contrast, a tension-dominated failure is an accumulation of shear deformation along slip-planes of polycrystalline matrix at yield limit, which may present in two modes: the out-plane shear that induces necking; and the in-plane shear that often starts at a free edge with stress concentration, for example, a rivet’s hole or a crack, and then leads to a Lude’s band. Both modes present in Fig. 16(b) but within different areas.
(2)Load-rating gusset plates: The “bowing” of the U’10 gusset plate of I35W in Fig. 16a is considered as an early warning sign. That tragedy proofs a failure of such a gusset can lead to entire bridge’s fall. Therefore, in a new truss-bridge’s design or an aged bridge’s load-rating, a practical issue is to identify its load capacity based on each structural components’ capacities including all gusset plates, whereby the following two steps are essential:
To this end, a procedure and associated computer programs have developed to investigate the failure process of I35W Bridge based on the theories and methodologies introduced in [5-26, 30-33, 43-46, 50-51]. A series of in-depth three-dimensional computations of I35W Bridge under the actual live loads at the instance of collapse, see Fig. 17, as well as the bridge under various design conditions, have been performed [28,29] according to original design drawings and in the light of the material evidences of NTSB’s investigation[1,2] and FHWA’s technical advisory [3].
In addition to the discussions of more undersized-components disclosed from original design drawings and the evidence of the one-dimensional influence line solution-biased original design, another conclusion obtained in [29] is that the nodes connecting floor trusses and main truss-frame, see Fig. 17b, was inappropriately designed. This is because it actually divides deck load to two high-amplitude force flows input into a weak gusset plate, i.e. compression on the plate’s top and downward tension on its bottom edge. The particular structural features presented in this part and the implications to load capacity are to be discussed in follows because they can be useful for other gusset plates’ load rating in general.
Figure 17. (a) A fully-scaled computation of I35W Bridge [29] based on design drawings and live load at the time of collapse, where the trucks, cars, and construction materials are represented by the brick elements with the densities corresponding to their weights; (b) a finite element model for the connection between floor truss and main flame at the node U10; (c) computed stress contour on an inside gusset plate[27b], which indicates the stress on the inside horizontal lateral bracing introduces lateral force that affects the deformation of the gusset plate.
Fig. 18 Stress distribution in floor trusses connect to U10’ and U11’, which result in the out-of-plane force
Q_{M} and bending moment
M onto the gusset plate; see the discussion in the text.
Plotted in Fig. 18 are the computed uniaxial stress in the floor trusses that connected the south U10’ and U11’ gusset plates of I35W. Here we focus on the stress distribution in the diagonal members of the floor truss attached to U10’, which is plotted as the empty diamonds linked by dash lines. One may notice that the two highest stress peaks occur in the diagonal floor truss members just inside the gusset plate on each side, by which the amplitude of the peak near western U10’ is about 13% higher than that near the eastern U10’. This difference is denoted as
Qp in the figure, which, obviously, is caused by the eccentric deck live load at the moment of the collapse, see Fig. 19a. Another notable phenomenon in Fig. 18 is the difference in stress level between the diagonal floor trusses member inside and outside a gusset. On the western side, it is denoted as
Q_{D}. Because all floor truss diagonals are made of the similar sized H-beams, therefore, the magnitude of
Q_{D} means the force that inside diagonal input to the node below the gusset plate is about 2.6 times higher than the sum of the forces from two outside diagonal members.
Obviously, the difference
Q_{D} reflects the unbalance of the forces at the node that connects main flame’s vertical truss, the diagonals and lower (bottom) chord in floor truss. Its horizontal resultant must be balanced by the tensions shared by the bottom chord, the diagonal lateral bracing, and the lateral shear in the main flame vertical truss. According Fig. 18 there is no significant stress in the bottom chord; whereas the inside diagonal lateral bracing is a very slender beam that has very limited capacity to carry horizontal force. Therefore, the majority of the horizontal force induced by the inside floor truss diagonal is balanced by the lateral shear of the main flame’s truss vertical, which is denoted as
Q_{M} in the figure.
Q_{M} produces a bending moment to the attached gusset plate, denoted as
M.
On the other hand, the
Q_{P} in Fig. 18 represents the difference of the forces that the floor truss input to eastern and western main flames, respectively. Again, because the stresses on the two ends of floor truss’ horizontal members attached to the two main flames are almost equal,
Q_{P} should be balanced by the lateral shear on main truss vertical and the horizontal lateral bracings of two main flames, i.e. the top horizontal lateral bracing attached to gusset plate, as illustrated in Fig. 17(c). Fig. 19(a) gives the deck load’s distribution of the I35W Bridge when it collapsed. Although majority of the live loads laid on the western side, it should be noticed that the stock-piled construction materials, which is the heaviest part of total live-load, were on the lane W1 that is just beside the central line. This is the reason that the difference
Q_{P} is less than 15% of the total force in the floor truss diagonal. Obviously, there is no need to address the fact that the deck load-induced vertical force on the western gusset plate is much higher than that on the eastern main flame. In Fig. 19(a) the live-load on the lane W1 equals the weight of several design trucks (80 kips each). In practical application, a worst scenario could be the case that multiple presence of heavy trucks line on one side of a bridge’s deck, for example, the lane W4 in Fig. 19(a). Under this situation,
Q_{P} can be significantly higher.
The forces
Q_{P} and
Q_{D} represent eccentric effect, caused by eccentric deck’s live load and original design. They introduce extra out-of-plane forces and bending moment to a gusset plate, which may have significant impact to the plate’s stability. For an engineering evaluation, conventional procedure often simplifies a structure into a one-dimensional model, then applying Eular’s instability criterion, as illustrated in Fig. 19b. This criterion is exact for long slender structures such as bars and columns. For a two-dimensional structure like a gusset plate, Eular’s criterion is generally over-conservative when all external forces lie in the same plane as the plate does. However, for either a one-dimensional bar or a two-dimensional plate, when lateral force or out-of-plane bending presents, see Fig. 19b, corresponding modifications are necessary. Theoretical solutions of lateral forces have already been obtained for one-dimensional bar, for examples, see [5,6]. Theoretical analyses in these literatures indicate that, when an out-of-plane force
Q or bending moment
M is proportional to in-plane compression force
P, for an elastic slender structure component the following relationship holds:
where
is the maximum bending moment in the component;
is the Eular’s prediction of the load at onset of instability. This relationship implies that a lateral load does not affect instability criterion under elastic condition. However, the maximum bending moment in the structure may increase faster because it is proportional to the square of in-plane load. This may cause material’s yielding earlier than the case without out-of-plane forces, resulted in reduction of system stiffness and subsequent loss of stability even when applied load is still lower than Eular’s prediction [50, 51].
A thin-plate theory-based model has been developed which partitions a gusset plate into several zones characterized by different failure modes [49a] and the model enables to obtain analytical or semi-analytical solution for each mode, as outlined in [49b]. For buckling, a group of semi-analytical solutions has been obtained based on Von Karman’s thin plate stability theory [38]. Introductions of this theory and related applications can be found, for examples, in [5,6,40]. Further theoretical developments of structural stability can be found, for examples, in [43,44,50-51]. According to this theory, the stability of this fan-shaped is governed by the stationary solution of the following equilibrium condition (in polar coordinate system):
A gusset plate can be treated as “thin plate” because its largest unbraced dimension is generally one order greater than its thickness. Such a thin plate may fail through two mechanisms:
tension-induced material failure or buckling under compression, as discussed early in this paper. Accordingly, the two cases in Fig. 16 demonstrate the three corresponding patterns: (i) buckling at unbraced area due to the large ratio of the clearance between attached beams and the plate’s thickness, Fig. 15c; (ii) compression-induced bowing of the triangle area of a gusset, Fig. 16a; (iii) tension induced necking in another triangle area of the gusset, together with shear failure around rivets, Fig. 16b. These patterns may occur concurrently or a dominant one triggers another. The challenges for design and load-rating a gusset plate are to identify the key dimension-parameters that are able to characterize geometric complexities and to compute the limit load at critical conditions considering lateral perturbation.
After some additional geometric modifications to the Whitemore model, a group of semi-analytical solutions of gusset plates’ buckling has been obtained based the theory in [5,6] and the Von Karman’s thin plate stability criterion [38]. Introductions of this theory and related applications can be found, for examples, in [5,6,40]. More advanced theoretical developments of structural stability, including elastoplastic buckling, post buckling, imperfection sensitivities, can be found, for examples, in [43,44,50-51]. According to this class of theories, the elastic stability of this fan-shaped plate is governed by the stationary solution of the following equilibrium condition (in polar coordinate system):
where
denote in turn the in-plane radial, hoop, and shear force densities;
w is out-of-plane displacement; and
K is plate stiffness: