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A nonlinear model for gusset plate connections
Engineering Structures 62–63 (2014) 135–147
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
A nonlinear model for gusset plate connections Chiara Crosti a, Dat Duthinh b,⇑ a b
Sapienza University of Rome, Rome, Italy National Institute of Standards and Technology, 100 Bureau Dr. Gaithersburg, MD 20899-8611, United States
a r t i c l e
i n f o
Article history: Received 13 June 2012 Revised 4 November 2013 Accepted 19 January 2014 Available online 16 February 2014 Keywords: Bridges Connections Finite-element analysis Gusset plate Nonlinear springs Steel truss
a b s t r a c t The investigation of the 2007 collapse of the I-35 W Highway Bridge in Minneapolis, Minnesota, used very detailed nonlinear finite-element (FE) analysis. On the other hand, the Federal Highway Administration (FHWA) provided simple guidelines for the load rating of gusset plates, but load rating was never intended to capture the actual behavior of gusset plates. The approach proposed here combines the accuracy of the first method with the simplicity of the second. From the detailed FE analysis of a single joint, the stiffness matrix of semi-rigid equivalent springs (linear in a simple model, nonlinear in a more advanced model) was derived by applying forces and moments to the free end of each portion of member (hereafter called stub member) that framed into the joint, one action at a time, while keeping the ends of the other stub members fixed. The equivalent springs were then placed in a global model, which was in turn verified against a global, detailed FE analysis of the I-35 W Highway Bridge. The nonlinear equivalent spring model was able to predict the correct failure mode. The approach was applied to a Howe truss bridge as an example of performance prediction of bridges with semi-rigid connections, most of them of one type. As the simplified spring model was developed from a detailed FE analysis of the joint considered, this approach would not be justified if all joints had to be modeled in detail. Examples where the approach can be used include: structures where only specific joints need to be investigated (e.g., joints subjected to concentrated loads), and structures where the same joint model can be used repeatedly at multiple locations. In some cases, the effort required in performing detailed FE analyses of many joints in order to develop simplified models can be justified if the simplified models can be used in subsequent multiple load cases, thus leading to overall computational savings. Under these circumstances, the nonlinear connection model proposed here provides a simple and affordable way to account for connection performance in global analysis. Published by Elsevier Ltd.
1. Introduction The 2007 catastrophic collapse of the I-35 W Highway Bridge in Minneapolis, Minnesota (I-35 W Bridge for short), under ordinary traffic and construction loads, was triggered by the buckling of an undersized gusset plate [1]. Gusset plates are complicated structural components used to connect frame members such as beams, columns and braces. Their use in buildings and bridges goes back many decades and certainly predates the use of computers in structural analysis and design. Practical design methods ensure safety by providing a load path that satisfies equilibrium, boundary conditions and does not exceed material yield limits. The resulting stress field is by definition a statically possible yield state of stress. Safety against plastic failure is assured because, according to the ⇑ Corresponding author. Tel.: +1 301 975 4357; fax: +1 301 869 6275. E-mail addresses: [email protected] (C. Crosti), [email protected] (D. Duthinh). http://dx.doi.org/10.1016/j.engstruct.2014.01.026 0141-0296/Published by Elsevier Ltd.
Lower Bound Theorem of the Limiting Load [2], a statically possible yield state of stress is less than or equal to the limiting load, which is characterized by unrestricted plastic flow. There is, however, no guarantee that the design load path is the actual one, and thus the design methods provide no information on the load–displacement behavior or stiffness of the connection, even in the elastic range. Current procedures [3] for the design and load rating of multimember gusset plates consist in checking axial, bending and shear stresses along various sections deemed critical, using elastic beam theory. These procedures are intended to ensure a safe and conservative design, but produce results that can be quite different from those derived from more representative finite-element (FE) models, and cannot predict either stiffness or actual behavior. To do so would require highly sophisticated and detailed FE models, such as the ones used in the investigation of the I-35 W Bridge collapse [1]. The approach presented here combines the simplicity of one method with the accuracy of the other: a simplified connection in the form of semi-rigid springs is proposed to improve on current
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design analysis, which typically is linear and assumes rigid connections. Semi-rigid here means that the connection has a finite, defined rotational stiffness, as opposed to zero (hinged) or infinite (rigid) stiffness. The paper starts with a literature survey that focuses on approximate design and analysis methods, then follows with a brief account of the I-35 W Bridge collapse and its investigation. Next, a simple linear equivalent spring model and a more detailed nonlinear one are developed from detailed FE analyses of a joint, and used in a global model that, in the nonlinear case, correctly predicts the failure mode of the I-35 W Bridge. Finally, the spring model is applied to a Howe truss bridge to show how it can be used to predict the performance of bridges with connections mostly of the same type.
2. Review of the literature on approximate methods for gusset plates Extensive reviews of the literature were performed by Chambers and Ernst [4] and Astaneh-Asl [5]. In keeping with the theme of this paper, the focus here is on approximate methods of analysis and design and on buckling. Whitmore [6] tested 1:4 scale specimens of gusset plates for Warren trusses made of aluminum, masonite and bakelite using wire-bonded strain gages (a novelty in 1952), brittle lacquer and photoelastic techniques. Based on these experiments, he developed the effective width that now bears his name (Fig. 1), ‘‘by constructing lines making 30° with the axis of the member which originate at the outside rivets in the first row and continue until they intersect a line perpendicular to the member through the bottom row of rivets.’’ The maximum tensile and compressive stresses may be approximated by assuming the force in each diagonal is uniformly distributed over this width. Astaneh [7] proposed modeling gusset plates as wedges under a point load. In the elastic range, closed-form solutions exist for infinite wedges, but cannot account for the actual boundary conditions. Furthermore, actual loads are transferred to gusset plates by rows of rivets or bolts, rather than at a single point. The author also indicated that wedge models could not predict buckling, for which he proposed a fin truss model, where the cross section of each bar was the average of the cross section of the triangle it bisected. Astaneh [7] suggested an effective length factor of 0.7 for the one-dimensional struts to account for the restraint provided against buckling by the transverse direction in a two-dimensional plate. The use of multiple fin trusses to model gusset plates that connect multiple members rapidly becomes cumbersome, and makes the FE method very attractive for its accuracy and automation, when compared to Astaneh’s approach. An elegant, statically possible load path with no moment (i.e., concentric) in the gusset-to-beam and gusset-to-column connections was developed and called the Uniform Force Method by Thornton [8], who obtained the following connection forces (symbols are defined in Fig. 2):
a
eB b P; V B ¼ P; V C ¼ P; r r r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ ða þ eC Þ þ ðb þ eB Þ
HB ¼
HC ¼
eC P; r
r ð1Þ
The Uniform Force Method, with modifications to account for geometries that induce moments, is the method currently recommended by the American Institute of Steel Construction [9]. Dowswell and Barber [10] provided a quantitative definition of compact gusset plates and guidance on the required plate thickness tb to prevent their buckling:
sffiffiffiffiffiffiffiffiffiffi ry c 3 tb ¼ 1:5 EL2
ð2Þ
where c = the shorter of the distances from the corner bolt or rivet to the adjacent beam or column, E = modulus of elasticity, L2 = equivalent column length from the middle of the Whitmore width (Fig. 1) and ry = yield stress. The gusset plate is compact if its thickness t P tb and non-compact if t < tb. Dowswell and Barber [10] compared their theoretical buckling capacities Pth with experimental and FE calculations in the literature Plit. They used the average of the lengths from the middle and the ends of the Whitmore width for the equivalent column length. For compact gusset plates, using an effective length factor of 0.5, they found the ratio Plit/Pth = 1.47; and for non-compact gusset plates, using an effective length factor of 1.0, Plit/ Pth = 3.08. Thus, the separation of compact from non-compact gusset plates and the subsequent different effective length factors resulted in inconsistent and problematic factors of safety for design against buckling. Brown [11,12] developed analytical expressions for the edge buckling of gusset plates, based on the elastic buckling stress of a plate supported on its loaded edges, but otherwise unrestrained. For edge buckling, the critical section bisected the long free edge of length a and was perpendicular to the brace (Fig. 3). Its width b was different from the Whitmore width. Only a fraction of the total brace load contributed to the edge buckling of the gusset, and the rest was transferred directly to the steel frame. Brown [11,12] compared her predictions with 18 experiments and produced 16 ratios of experimental to predicted values ranging from 1.01 to 1.38, and 2 values below 1 (0.99 and 0.81). Yamamoto et al. [13] tested eight gusset plates (each connecting two horizontal chord members and two diagonal braces) and performed FE analysis to establish stability design criteria for the joints of the Warren truss designed to stiffen the deck of the Honshu–Shikoku suspension bridge. They focused on the development of plastic zones, local buckling and ultimate strength, and found that local yielding and local buckling preceded global buckling of the gusset plates. The load at which local buckling started depended on the extent of yielding, which covered the inner portions of the gusset plate, whose in-plane stiffness was constrained by the surrounding elastic region. Under the assumption that buckling occurred when the stress reached the allowable stress in the material, ra = 0.58 ry, and with l1 = length of the vertical free edge, Yamamoto et al. [13] proposed the following design thickness for local buckling:
rffiffiffiffiffiffi tcr ¼ 1:10l1
Fig. 1. Equivalent column.
ra E
ð3Þ
There have been numerous other FE studies of gusset plates over the last three decades, and they have been reviewed extensively by Chambers and Ernst [4] and Astaneh-Asl [5]. Among the earlier studies, for example, is the work of Cheng et al. [14]. They used ANSYS and performed a parametric study to calculate the elastic buckling strength of concentrically loaded gusset plates. In
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Fig. 2. Uniform Force Method.
Fig. 3. Critical section for gusset edge buckling. Fig. 4. Rotational spring four-parameter stiffness.
particular, they focused on the effects of the splice plate and the rotational restraint provided by the bracing member, but did not formulate design equations or derive equivalent springs. A widely used equivalent spring for beam–column connections of braced frames was formulated by Richard and Abbott [15]. Their nonlinear spring had one degree of freedom, and its moment–rotation stiffness was represented by four parameters that modeled a linear elastic part, a linear post-yielding, strain hardening part, and a curved transitional part (Fig. 4 and Eq. (4)).
2
3
6 7 K Kp M¼6 n 1=n þ K p 7 4 5h ¼ K CONN h ðKK Þ h 1 þ M0p
ð4Þ
where h = connection rotation angle; K = initial or elastic stiffness; Kp = plastic stiffness; KCONN = connection stiffness; M = connection moment; M0 = reference moment; and n = shape parameter. Williams [16] used computer program INELAS to analyze steel gusset plate connections and find values that fit the four-parameter model. For example, one of his model of a gusset plate and beam was comprised of 77 finite elements, a number not atypical of the computer technology of the time. He also used NASTRAN to calculate the capacity of gusset plates subjected to compressive brace forces. Gusset plate research, highlighted above, had been sporadic in the last 60 years, but would receive a jolt of interest in 2007 with the catastrophic failure of the I-35 W Highway Bridge in Minneapolis, Minnesota.
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3. I-35 W Bridge collapse The I-35 W Highway Bridge – Number 9340 in the National Bridge Inventory – spanned the Mississippi River in the city of Minneapolis, Minnesota. Construction started in 1964 and the bridge was opened to traffic in 1967. The major superstructure components consisted of welded plate girders and truss members with riveted and bolted connections. The deck truss portion of the bridge is shown schematically in Fig. 5. In the terminology of the National Transportation Safety Board (NTSB) Accident Report [1], the deck truss portion of the bridge excludes the north and south approach spans. The deck truss comprised two parallel Warren trusses (east and west) with vertical members. The upper and lower chords, the compression diagonal and vertical members were welded box sections, whereas the tension vertical and diagonal members were Hsections. Steel gusset plates, riveted to the side plates of the box members and the flanges of the H-members, were used on all the 112 connections of the two main trusses. All joints had at least two gusset plates on either side of the connection. In the late 1970s, the bridge was classified as ‘‘non-load-path-redundant’’, that is, if designated ‘‘fracture critical’’ members of the main trusses failed, the bridge would collapse. Unfortunately, that is what happened on August 1, 2007. On that day, roadway work was underway and four of the eight travel lanes were closed to traffic (two outside lanes northbound and two inside lanes southbound). In the early afternoon, construction equipment and material were positioned in the two closed inside southbound lanes. At 6:05 pm, a motion-activated surveillance camera showed the bridge center span separating from the rest and falling into the river. A total of 111 vehicles, including 25 associated with the construction, were on the bridge at the time, and 13 lives were lost. Examination of the bridge debris showed that all four gusset plates at the U10 nodes (U refers to the upper chord, L to the lower chord) had fractured into multiple pieces. More fractures in the lower chord members between the L9 and L10 (Fig. 5) nodes precipitated complete separation of the main truss, thereby causing the center span to drop. As a consequence of the collapse of the I-35 W Bridge, the Federal Highway Administration (FHWA) produced a guidance document [3] with simple, hand-calculable formulas for the load rating of gusset plates, with the intent that bridge owners may use the guidance to evaluate the connecting plates and fasteners of truss bridges. The gusset plates are rated for their resistance to tension, compression and shear. In tension, the modes of failure that need to be evaluated are yielding of gross section, fracture of net section, and rupture by block shear. For shear resistance, several sections must be investigated to find the governing one. Finally, the compressive strength of a gusset plate is that of an equivalent column determined as follows: 1. The thickness of the column is that of the gusset plate. 2. The width is the Whitmore effective width [6] presented Fig. 1.
3. The length of the column is the average of three lengths extending in the direction of the framing member from the middle and the ends of the last row of bolts to the edges of the gusset plate or adjacent groups of bolts (L1, L2 and L3 in Fig. 1). This approximation is adapted from Thornton [17], who used the middle length L2, but then went on to propose the average of L1, L2 and L3 as ‘‘a more reasonable approximation’’ of the critical length of the column strip, with an effective length factor of 0.65, corresponding to a column with both ends fixed. The examples in the Guidance ignore any lateral constraint to the gusset and use an effective length factor of 1.2, which corresponds to a column with one end fixed and the other restrained against rotation but free to translate. Brown [11,12] had previously proposed the factor of 1.2 based on experimental observation and calculations. 4. Detailed finite-element investigation From the literature review, it is seen that there are simple design methods based on equilibrium and elastic behavior and proven safe by experiments. There is, however, no simple way of calculating the actual behavior of a gusset plate, even in the elastic range. Designers ensure that the connections are stronger than the members, then proceed with a structural analysis that assumes rigid connections. Such a structural analysis is incapable of predicting connection failure, or account for the flexibility of the connection in the global behavior of the structure. On the other hand, there exist detailed models such as the ones analyzed by the NTSB. As part of the investigation of the collapse of the I-35 W Bridge, the NTSB commissioned the FHWA, the State University of New York (SUNY) at Stony Brook and the software company Simulia to develop an FE model of the bridge. FHWA constructed a three-dimensional global model of the entire deck truss portion of the bridge with (two-node, linear or cubic) beam and (four-node) shell elements. The model, including the boundary conditions at the piers, was calibrated with strain gage data from a fatigue assessment conducted by the University of Minnesota in 1999 [18]. In addition, as field evidence pointed to gusset plates as the trigger of the collapse, SUNY/Simulia developed detailed models of the U10 (Fig. 6) and L11 nodes and incorporated them into the global model. More details about the highway accident and its analysis can be found in [1,19,20]. In [19] Hao provided a possible explanation for why some of the gusset plates were under-designed: the main frame gusset plates and the upper chords may have been designed from a one-dimensional model of the bridge, i.e., a uniformly loaded beam supported at four points. Each gusset plate of the detailed model was composed of four layers of eight-node, linear, solid brick elements, whose largest dimension in the plane of the gusset plate was 5 mm in highly stressed regions and less than 15 mm elsewhere [20]. The 289,000 solid elements used reduced integration and hourglass control to alleviate spurious displacement modes. The connection model also included the five main truss members, cut at 2/5 of
Fig. 5. Schematics of the deck truss portion of I-35 W Bridge.
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Fig. 6. Detailed FE model 1 of gusset plate (ABAQUS).
their lengths in the bridge (the so-called stub members), and modeled with solid and shell elements (Fig. 6). In their largest dimension, the shell elements ranged from 25 mm to 50 mm, and the solid elements, which were in the transition zones between the truss members and the gusset plates, from 8 mm to 25 mm. Transition between shell and solid elements was performed by surfacebased coupling constraints. All members in the connection model other than the main truss members and gusset plate had maximum element dimension of about 15 mm. The sophistication of these models, compared with Williams’s [16] for example, reflects the extraordinary advance of computer and finite-element technologies in the last quarter of a century. Even with such a detailed connection model, the rivets could not be modeled. The shanks of the rivets had a radius of 13 mm and the hemispherical heads a radius of 20 mm. The rivets were modeled by coupling nodes of the fastened components normal to the rivet axis within a radius of influence of 13 mm. Contact pairs, with a Coulomb friction coefficient of 0.1, were also used with the fastener elements. Such a detailed analysis required advanced skills and powerful computers and was clearly beyond routine design.
5. Simplified linear spring model 5.1. Stiffness matrix of linear springs A simplified model of a gusset plate connection of the bridge was developed as part of the present work. First, the linear model 2 (Table 1) is described. It started with the analysis of the NTSB detailed FE model 1 [1,20], formulated in software ABAQUS [21], of gusset plate U10 (Fig. 6). The results of the analysis of model 1 established the equivalent stiffness of springs that completely modeled the elastic behavior of the connection model 2 [22,23]. This simplified connection model 2 was placed in a global 2D model of the West main truss (model 5), using overall strategies for integration of substructures proposed in [24]. Table 1 lists all the
computer models run in the present work. They are described in more detail when they are first used. The ABAQUS detailed FE model of U10 (model 1) had five stub elements connected to a pair of gusset plates (Fig. 6). In the NTSB global analysis, the stub members were reattached to their corresponding truss members. To develop the simplified linear connection model, the stub elements and gusset plates were replaced by five user-defined springs, that each had a full 6 6 stiffness matrix for all 6 degrees of freedom (DoFs). To establish the flexibility of the spring that is equivalent to element 1 in the detailed model (Fig. 6) for example, the ends of elements 2–5 in the detailed model were fixed and an arbitrary concentrated force or moment was applied to obtain the displacements and rotations at the end of element 1. In general, the concentrated force or moment was applied at a node at the end of a stub member, as close to the center of the cross section as possible. For tubular members, two concentrated forces or moments were applied at the middle of the long sides of the end cross section, and displacements and rotations were calculated at these same points. The process was performed using ABAQUS and repeated by applying arbitrary forces and moments corresponding to all 6 DoFs, and thus, after normalization, the 6 6 flexibility matrix for element 1 in ABAQUS coordinates was obtained. This flexibility matrix was inverted to obtain the stiffness matrix, which was then transformed to the coordinate system specified by STRAND for user-defined elements and applied to the simplified spring model (Fig. 7, model 2). Fig. 8 is a flowchart of the procedure. This process was repeated for all five elements of the connection. In calculating the equivalent spring stiffness, software STRAND7/STRAUS7 [25] was used, and care had to be exercised in defining the local coordinates of the user-defined element [STRAND7/STRAUS7 required the local axis 3 to be in the longitudinal direction and pointing away from the end of the stub (Fig. 7)].
5.2. Verification of the stiffness matrix After the equivalent springs were assembled, the response of the simplified connection in STRAND (model 2) was verified against the detailed FE analysis in ABAQUS (model 1), for the same set of applied forces and moments. As done previously, forces and moments were applied one at a time, for example at end node 1 of element 1, while the other end nodes were fixed. Agreement in the displacements and rotations calculated was good, especially for the diagonal terms (maximum relative error 0.5%), with acceptable errors in the off-diagonal terms, which were several orders of magnitude smaller (Tables 2 and 3). Similar results were obtained for the other elements of the connection [22], but were not presented here to save space. As stated in the introduction, current design analysis is typically linear and assumes rigid connections. The next part of this work investigates the validity of the rigid connection assumption. For this purpose, ABAQUS model 3 was developed by changing the
Table 1 Computer models run. Model
Software
Analysis
Purpose
1. 2. 3. 4. 5. 6. 7. 8. 9.
ABAQUS STRAND ABAQUS STRAND STRAND ABAQUS STRAND STRAND STRAND
Linear Linear Linear Linear Linear Nonlinear Nonlinear Nonlinear Nonlinear
Stiffness of 5 members Stiffness of 5 springs Stiffness of 5 members Validity of rigid connection assumption Behavior with various connection models Stiffness of 5 members Stiffness of 5 springs Behavior with various connection models Behavior with various connection models
Detailed connection Spring connection Detailed connection, various gusset thickness Rigid connection West main truss (2D) Detailed connection Spring connection I-35 W (3D) Howe truss (2D)
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Fig. 7. Equivalent spring model 2 (STRAND).
thickness of the gusset plates in the NTSB gusset connection model 1. Also, STRAND7/STRAUS7 [25] model 4 was developed with beams that had the same stiffness as the members framing into U10, but were rigidly connected to each other. Table 4 compares the displacements d and rotations q predicted by models 2 (semi-rigid connection) and 4 (rigid connection) with the results from the detailed ABAQUS model 3 with various gusset plate thicknesses. As expected, the thicker the gusset plates, the smaller the deflections and rotations. Also, as expected, the rigid joint model 4 produced results that were not as accurate as the more complicated spring model 2, which corresponded to a gusset thickness of 12.7 mm (base case, also diagonal terms of Tables 2 and 3). The displacements and rotations predicted by the rigid connection model 4 (Table 4, column 2) were not always smaller than the base case (column 4), which included the stiffness of the gusset plates. 6. Planar linear global analysis The next phase of the analysis consisted in using the spring elements in a global analysis of the bridge. The first global analysis was linear, with the various connection models placed in a 2D model of the I-35 W Bridge, at a location corresponding to the U10 gusset plate where the bridge collapse initiated. For this 2D analysis, the simplified connection model only had three degrees of freedom corresponding to planar displacements and rotation. The analysis was performed with software STRAND7/STRAUS7 [25], using a model consisting of 496 beam elements (which have axial stiffness) and loaded with the gravity loads (dead weight, vehicle and construction loads, Fig. 9) present at collapse, and divided into the East and West trusses [26]. Liao et al. [27] showed by influence lines that the temporary construction loads placed near U10 significantly affected the forces imposed on the joint and contributed to the collapse. The 2D analysis was performed for the more heavily loaded West main truss (model 5). Although the model was 2D, the finite elements used were formulated for the general 3D case, and therefore some of the support points had to be restrained in 3D (Fig. 9). Four cases were run, resulting in the following midspan vertical deflections d: (1) all joints rigid, d = 281 mm; (2) U10 modeled with linear springs, all other joints rigid, d = 285 mm; (3) U10 modeled with detailed FE, all other joints rigid, d = 285 mm; and (4) all five-element joints hinged, all other joints rigid, d = 286 mm. Results showed that modifying the stiffness of one connection within the elastic range produced no noticeable effect on the maximum vertical deflection of the bridge (at midspan). It was concluded that, in the elastic range, the connections could be assumed rigid to a good approximation. 7. Simplified nonlinear spring model
Fig. 8. Flowchart to calculate equivalent spring stiffness.
In order to investigate the failure of the bridge, it was necessary to progress beyond the linear elastic range. To capture the nonlinear behaviour of the gusset connection up to failure, the detailed
Table 2 Element 1, node 1, displacements d and rotations q obtained in ABAQUS for applied forces and moments (detailed model 1). Node 1
dx (mm)
dy (mm)
dz (mm)
qx (mrad)
qy (mrad)
qz (mrad)
Fx = 444.8 (kN) Fy = 444.8 (kN) Fz = 444.8 (kN) Mx = 228.8 (kN m) My = 113.0 (kN m) Mz = 161.0 (kN m)
1.84E1 8.19E2 1.88E5 4.19E6 7.83E7 7.43E3
8.19E2 3.50E0 3.39E4 1.22E5 1.94E5 7.09E1
1.88E5 3.42E4 2.10E+1 9.30E1 1.29E0 5.10E5
6.72E6 4.51E5 1.51E0 5.15E0 3.71E2 4.41E6
7.72E5 7.66E5 5.08E0 8.98E2 7.04E1 1.11E5
2.05E2 1.96E0 1.37E4 8.80E6 8.01E6 6.16E1
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dx (mm)
dy (mm)
dz (mm)
qx (mrad)
qy (mrad)
qz (mrad)
Fx = 444.8 (kN) Fy = 444.8 (kN) Fz = 444.8 (kN) Mx = 228.8 (kN m) My = 113.0 (kN m) Mz = 161.0 (kN m)
1.85E1 8.27E2 2.51E4 1.60E5 3.63E5 7.60E3
8.27E2 3.50E0 3.29E4 2.83E5 1.86E5 7.07E1
2.51E4 3.29E4 2.10E + 1 9.30E1 1.29E0 4.75E5
2.61E5 4.60E5 1.51E0 5.15E0 3.71E2 4.66E6
1.43E1 7.31E5 5.09E0 9.00E2 7.06E1 1.02E5
2.10E2 1.95E0 1.31E4 7.92E6 7.14E6 6.15E1
Table 4 Comparison of displacements d and rotations q of STRAND7/STRAUS7 model 2 with semi-rigid connection, model 4 with rigid connection and ABAQUS FE model 3 with various gusset plate thicknesses [numbers in brackets are ratios to base case with thickness of 12.7 mm]. STRAND mod. 4 rigid joints
dx (mm) dy (mm) dz (mm) qx (mrad) qy (mrad) qz (mrad)
0.184 [1.00] 7.42 [2.12] 12.8 [0.61] 3.12 [0.61] 0.759 [1.08] 0.802 [1.30]
STRAND mod. 2 semi-rigid joints
0.185 [1.00] 3.50 [1.00] 21.0 [1.00] 5.15 [1.00] 0.706 [1.00] 0.615 [1.00]
ABAQUS FE model 3 with gusset thickness: 12.7 mm
15.87 mm
19.05 mm
25.4 mm
0.184 [1] 3.50 [1] 21.0 [1] 5.15 [1] 0.704 [1] 0.616 [1]
0.1748 [0.95] 3.276 [0.94] 18.54 [0.88] 4.688 [0.91] 0.6618 [0.94] 0.5998 [0.97]
0.1677 [0.91] 3.105 [0.89] 16.90 [0.80] 4.393 [0.85] 0.6302 [0.90] 0.5872 [0.95]
0.1575 [0.86] 2.860 [0.82] 14.83 [0.71] 4.021 [0.78] 0.5862 [0.83] 0.5682 [0.92]
Fig. 9. 2D model of I-35 W Bridge showing loads and support conditions. Circles and squares indicate support conditions (restrained displacements d and rotations q) for both main trusses.
ABAQUS FE model 1 of the five-element connection was loaded as described in Section 6, but into the nonlinear range. Both geometric and material nonlinearities are taken into account (model 6). Nonlinear material behaviour was modelled using the Von Mises yield criterion and the isotropic hardening rule (Fig. 10, from [1]). A large strain–large displacement formulation (ABAQUS default option) was used to carry out the nonlinear analysis. The results were force–displacement and moment–rotation curves, which were then transformed into the STRAND coordinates of the equivalent spring model. In the spirit of simplification, to make the problem tractable and to use what was available in STRAND7/STRAUS7 [25], the stiffness matrix of each of the five elements of the connection was diagonal only and derived from an initially perfect gusset plate. The diagonal terms were in the form of load–displacement or moment–rotation curves (model 7). Notice that this nonlinear connection model included all six DOF.
Fig. 10. Stress–strain curve [1].
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Richard’s four-parameter nonlinear connection model (Fig. 4, Eq. (4)) only had one DOF and would not have captured the buckling of the gusset plate under axial load. Fig. 11 presents the load–displacement and moment–rotation curves of element 3 of the connection, where x, y and z refers to the global coordinate system. For clarity, only the positive values are shown. Similar curves model the behavior of the other elements of the connection and can be found in [22].
8. 3D nonlinear global analysis As the global nonlinear analysis might involve buckling and out-of-plane deformation, it was necessary to use a full 3D model of the I-35 W Bridge, as described in Section 3. The FE model 8 was composed of 1894 beam elements representing all the primary steel members, including all main lateral braces (Figs. 12 and 13), but the deck was not included in the model. The dead load was always present and not factored, but the construction loads were applied gradually, with a load factor of 1.0 corresponding to the actual loads at collapse ([26] and Fig. 9). The support conditions for both main trusses were the same as described in Fig. 9. Nonlinear analyses (for material and geometry) were run on two different models: 1. Model 8A: All joints were rigid, the five structural members intersecting at U10-W were elasto-plastic (Fig. 10) and all the other members were elastic; 2. Model 8B: U10-W joint was semi-rigid (modeled by nonlinear connection elements with diagonal stiffness matrix as described above) and the five members around the U10-W joint were elasto-plastic. All the other joints were rigid and all the other members were elastic.
9. Results
Fig. 11. Force–displacement and moment rotation curves for element 3 of connection (KFXX, KFYY, KFZZ and KMXX, KMYY, KMZZ are the elastic stiffness for force– displacement and moment–rotation for x, y and z axis respectively).
For model 8A (with rigid joints), failure was initiated at a load factor of 6.36 by yielding of the end of element 1 that connected with U10 W. Model 8B (nonlinear semi-rigid connection) provided a much better simulation of the actual behavior. The semi-rigid joint model predicted that U10 W began to fail at a load factor of 0.92, and completely failed at load factor 1.7 (Fig. 14), leading to the collapse of the bridge. Fig. 15 and Table 5 present the axial forces and load ratios (ratio of force carried to capacity) in the five elements of the connection. Collapse initiated when connection element 3 reached axial capacity in compression (buckling, Fig. 16a), which agreed with the NTSB analysis. Fig. 16a shows the deflection of the detailed ABAQUS model of U10-W under the
Fig. 12. Modeling of lateral bracing.
Fig. 13. 3D beam model of I-35 W (model 8).
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ultimate axial compressive force of 12.1 MN applied at the end of stub member 3 (Fig. 6), with the ends of the other stubs fixed. It was thus seen that the equivalent spring model produced a good approximation of the behavior of a gusset plate connection in the elastic and post-elastic range. Clearly, the model was not as accurate as the detailed FE model used by NTSB. The discrepancy was measured by a load factor of 0.92, instead of 1.0, applied to the construction loads at failure initiation. Moreover, the simplified connection model assumed an initially perfect gusset plate. Had the model accounted for initial out-of-plane displacements, the load factor would have been less than 0.92. (The NTSB connection model [20] accounted for an initial out-of-plane displacement of one thickness or 12.7 mm at the free edge between connection elements 2 and 3, Fig. 6. After dead weight and traffic load application, the maximum out-of-plane deformation was 17.3 mm, in agreement with the values 15.2 ± 3.8 mm estimated from inspection photographs, Fig. 16b.) Nevertheless, the results of the global 3D analysis using the simplified nonlinear connection model would have attracted the attention of the analyst to joint U10 and the location and magnitude of the construction loads, something that the rigid connection model, which resulted in a load factor of 6.36, would have failed to do. A major difficulty in the development of the simplified connection model was that it was based on a prior detailed FE
Fig. 14. Load–deflection curve (deflection at midspan as indicated in Fig. 9).
Table 5 Ratio of calculated force to axial capacity for connection U10-W at load factor 0.92. Connection element
Load ratio
Tension or compression
1 2 3 4 5
0.28 0.56 1.00 0.02 0.41
Compression Tension Compression Tension Tension
model. Various attempts to circumvent this difficulty were unsuccessful and the literature review confirmed that previous simplified models were very limited in accuracy and use. This simplified connection model is then most useful for structures that use the same connection repeatedly, or only with changes in plate thickness or material properties that can be accommodated easily in the same detailed FE model; or for loadings that would cause the analyst to focus on a few specific connections only; or for multiple load cases, where a detailed model can be used efficiently multiple times. An example follows.
10. Application to a bridge with identical connections The general procedure of using a nonlinear simplified gusset connection was applied to a Howe truss bridge to show the importance of properly accounting for connection behavior. The truss was 12.14 m high, spanned 93.91 m, and consisted of repeated panels of length 11.58 m or 9.00 m defined by vertical members, horizontal chords and diagonal braces (Fig. 17). For simplicity, the structure considered was planar and the three-member joints at the ends were assumed rigid. All five-element connections and the members they attach were the same as for U10-W analyzed above (elasto-plastic, Fig. 10, dimensions shown in Fig. 17). Besides its dead load, the bridge was subjected to traffic load uniformly distributed on the top chord, and construction loads modeled as three point loads on the top chord (Fig. 17). The bridge was restrained against horizontal and vertical translations at one end and against vertical translation at the other end (model 9).
Fig. 15. Axial forces in the five elements of semi-rigid connection U10-W.
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Fig. 16. (a) ABAQUS model showing failure of U10 under axial compression of diagonal member 3 and (b) inspection photograph of U10 [1].
Fig. 17. Geometry of the Howe truss bridge (model 9).
Four models with large displacement formulations, elasto-plastic members and different connection models were considered: 9A: All joints rigid; 9B: one critical joint (at the concentrated loads) modeled by the nonlinear, five-element connection previously described, all other joints rigid; 9C: all five-element joints modeled by the nonlinear connection elements previously described; and 9D: all five-element joints hinged. Nonlinear analyses (that accounted for both material and geometric nonlinearity) were run by applying an increasing load factor
to the concentrated loads only, with the uniform loads held constant. Fig. 18 presents the load factor versus midspan vertical displacement for the four different models considered. As expected, model 9D (pinned joints), had the lowest load capacity, model 9A (rigid joints) had the highest, and the others (semi-rigid joints) fell in between. 10.1. Model 9A: Rigid joints Model 9A reached the ultimate load factor of 3.26. Fig. 19 presents the ratio of bending moment to flexural capacity in the members at the ultimate load factor. Since the joints were rigid, collapse occurred because of flexural yielding of the members.
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the critical joint under the concentrated loads, and the other joints remained elastic as the critical joint failed by flexural yielding. Section 6 showed that the assumption of rigid behavior was a good approximation for elastic joints, and the difference between models 9B and 9C was in the treatment of joints that turned out to remain elastic. If there were instead several heavily loaded joints, then model 9C (with all joints semi-rigid) would predict a different load distribution than model 9B (with only one semi-rigid joint) as some of the heavily loaded joints begin to plastify. In this case, the failure loads predicted by models 9B and 9C would be expected to be different. 10.4. Model 9D: All five-member joints pinned
Fig. 18. Load factor versus midspan vertical displacement.
10.2. Model 9B: Semi-rigid connection (five element-connection at concentrated load) Fig. 20 illustrates the details of the joint under study, and Fig. 21 and Table 6 the axial capacity of each connection element. The failure of the truss bridge could be attributed to the achievement of axial capacity in the connection elements. At a load factor of 1.98, connection element 2 reached axial capacity (Table 6), followed by connection element 1 and finally connection elements 3 and 4.
10.3. Model 9C: Semi-rigid connections (all five-member joints modeled with five element-connections) Model 9C predicted the same failure load as model 9B because in this example the highest stresses were heavily concentrated at
Fig. 22 presents the ratio of bending moment to flexural capacity in the members at the ultimate load factor of 1.46. For this model, failure was reached as soon as a plastic hinge was developed in member 1 of the joint under concentrated loads. Table 7 summarizes the results. Models that did not account for realistic connection behavior could be quite erroneous, especially when concentrated loads made a joint critical. When structural members were assumed rigidly connected, the predicted capacity could be quite erroneous if the assumption that joints were stronger than members turned out to be incorrect. The methodology proposed here made global analysis with proper accounting of connection performance efficient and affordable. 11. Conclusion Conclusion 15 of the NTSB investigation of the I-35 W Bridge collapse stated [1]: ‘‘Because bridge owners generally consider gusset plates to be designed more conservatively than the other
Fig. 19. Model 9A, rigid joints: bending moment ratio at load factor of 3.26.
Fig. 20. Model 9B, semi-rigid, five-element connection at concentrated loads.
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Fig. 21. Model 9B, axial forces in the five-element connection at concentrated load.
Table 6 Ratio of calculated force to axial capacity for critical connection at load factor 1.98. Connection member
Load ratio
Tension or compression
1 2 3 4 5
0.71 1.00 0.24 0.11 0.38
Compression Compression Compression Compression Compression
members of a truss, because the American Association of State Highway and Transportation Officials (AASHTO) provides no specific guidance for the inspection of gusset plates, and because computer programs for load rating analysis do not include gusset plates, bridge owners typically ignore gusset plates when performing load ratings, and the resulting load ratings might not accurately
reflect the actual capacity of the structure.’’ The work presented herein illustrates the problem. In the linear elastic range the assumption of rigidly connected members produced good results, and that was why FE analysis coupled with load tests [18] did not find the design flaw that turned out to be fatal. In the nonlinear range, however, structural capacity might not be well predicted if connection behavior and strength were not properly accounted for. If a particular joint was under highly concentrated loads, that node needed to be scrutinized, possibly by the method explained in this paper. The example in the previous section showed the method to be especially economical when only specific nodes needed to be analyzed in detail, or all nodes were similar. Future work, which will include comparison with experiments, for example full scale tests at FHWA [28], will delineate more precisely the level of detail required for the modeling of connections, and how the approach can be used in structural health monitoring [29].
Fig. 22. Model 9D, pinned joints. Bending moment ratio at load factor of 1.46.
Table 7 Analysis of Howe truss with various connection models. Model
Description
Ultimate load factor
Failure mode
9A 9B 9C 9D
All joints rigid Semi-rigid joint at point loads All 5-member joints semi-rigid All 5-member joints pinned
3.26 1.98 1.98 1.46
Flexural yielding of members Connection element 2 reached axial capacity Connection element 2 reached axial capacity Plastic hinge in member 1 of joint
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Accurate prediction of bridge failure should include proper modeling of connections. This can be done economically and accurately by the method presented herein if only specific joints need to be investigated in detail, or the same joint model can be used repeatedly at multiple locations, or multiple load cases must be run. The nonlinear connection model proposed here provides a simple and affordable way to account for connection performance in global analysis. Disclaimer The full description of the procedures used in this paper requires the identification of certain commercial software. The inclusion of such information should in no way be construed as indicating that such products are endorsed or recommended by NIST or that they are necessarily the best software for the purposes described. Acknowledgments The authors are grateful to Joey Hartmann of FHWA TurnerFairbank Highway Research Center for granting access to the FE model of the I-35 W Bridge and to the research team of www.francobontempi.org for support and advice. References [1] National Transportation Safety Board. Collapse of I-35 W Highway Bridge, Minneapolis, Minnesota, August 1, 2007. Accident Report, NTSB/HAR 08/03 PB 2008-916213. Washington D.C. 20594; 2008. [2] Kachanov LM. Fundamentals of the theory of plasticity. Moscow, USSR: Mir Pub.; 1974. [3] Federal Highway Administration. Load rating guidance and examples for bolted and riveted gusset plates in truss bridges. FHWA-IF-09-014, Washington, D.C. 20590; 2009. [4] Chambers JJ, Ernst CJ. Brace frame gusset plate research – Phase 1 literature review, vols. 1 and 2. Dept. of Civil Engg. University of Utah, Salt Lake City, UT 84112–0561; 2005. [5] Astaneh-Asl A. Gusset plates in steel bridges – design and evaluation. Steel TIPS report, structural steel educational council technical information & product services. Moraga, CA; April, 2010. 156 pp. [6] Whitmore RE. Experimental investigation of stresses in gusset plates. University of Tennessee Engineering Experiment Station Bulletin no. 16, May; 1952. [7] Astaneh A. Simple methods for design of steel gusset plates. In: Proc. ASCE Structures Congress ’89, San Francisco, CA; 1989. p 345–54.
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[8] Thornton WA. On the analysis and design of bracing connections. In: Proc. National Steel Construction Conf. American Institute of Steel Construction, Washington D.C.; June, 1991. p. 26–33. [9] American Institute of Steel Construction. Manual of Steel Construction. 14 th ed. AISC, Chicago, IL; 2011. [10] Dowswell B, Barber S. Buckling of gusset plates: a comparison of design equations to test data. In: Proc. Annual Stability Conf. Structural Stability Research Council, Rolla, MO; 2004. [11] Brown VL. Stability of gusseted connections in steel structures. Ph.D. thesis. University of Delaware, Newark, DE; 1988. [12] Brown VL. Stability design criteria for gusseted connections in steel-framed structures. In: Proc. An. Tech. Sess. on Stability of Bridges. Structural Stability Research Council, St. Louis, MO; 1990. p. 357–64. [13] Yamamoto K, Akiyama N, Okumura T. Buckling strength of gusseted truss joints. ASCE JSE 1988;114(3):575–90. [14] Cheng JJR, Yam MCH, Hu SZ. Elastic buckling strength of gusset plate connections. ASCE JSE 1994;120(2):538–59. [15] Richard RM, Abbott BJ. Versatile elastic-plastic stress-strain formula. ASCE J Eng Mech Div 1975;101(4):511–5. [16] Williams GC. Steel connection designs based on inelastic finite-element analysis. Ph.D. dissertation, Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721; 1986. [17] Thornton WA. Bracing connections for heavy construction. Eng J AISC 1984;21(3):139–48. [18] O’Connell HM, Dexter RJ, Bergson P. Fatigue evaluation of the deck truss of bridge 9340. Minnesota Dept. of Transportation Final Report; 2001. [19] Hao S. I-35W bridge collapse. ASCE J Bridge Eng 2010;15(5):608–14. [20] National Transportation Safety Board. Structural and local failure study of gusset plate in Minneapolis bridge collapse. Modeling group chairman final report 08–119, NTSBC070010, Nov. Washington D.C. 20594; 2008. [21] ABAQUS Analysis User’s Manual v. 6.5. West Lafayette, IN; 2004.. [27] Liao M, Okazaki T, Ballarini R, Schultz A, Galambos T. Nonlinear finite-element analysis of critical gusset plates in the I-35W Bridge in Minnesota. ASCE JSE 2011;137(1):59–68. [28] Ocel JM, Zobel RS, White D. Prediction of gusset plate limit states. In: 29th Int. Bridge Conf., Pittsburgh, PA; June 2012. [29] Bontempi F, Gkoumas K, Arangio S. Systemic approach for the maintenance of complex structural systems. Struct Infrastruct Eng 2008;4:77–94. http:// dx.doi.org/10.1080/1573247060115 523. ISSN: 1573-2479.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
A nonlinear model for gusset plate connections Chiara Crosti a, Dat Duthinh b,⇑ a b
Sapienza University of Rome, Rome, Italy National Institute of Standards and Technology, 100 Bureau Dr. Gaithersburg, MD 20899-8611, United States
a r t i c l e
i n f o
Article history: Received 13 June 2012 Revised 4 November 2013 Accepted 19 January 2014 Available online 16 February 2014 Keywords: Bridges Connections Finite-element analysis Gusset plate Nonlinear springs Steel truss
a b s t r a c t The investigation of the 2007 collapse of the I-35 W Highway Bridge in Minneapolis, Minnesota, used very detailed nonlinear finite-element (FE) analysis. On the other hand, the Federal Highway Administration (FHWA) provided simple guidelines for the load rating of gusset plates, but load rating was never intended to capture the actual behavior of gusset plates. The approach proposed here combines the accuracy of the first method with the simplicity of the second. From the detailed FE analysis of a single joint, the stiffness matrix of semi-rigid equivalent springs (linear in a simple model, nonlinear in a more advanced model) was derived by applying forces and moments to the free end of each portion of member (hereafter called stub member) that framed into the joint, one action at a time, while keeping the ends of the other stub members fixed. The equivalent springs were then placed in a global model, which was in turn verified against a global, detailed FE analysis of the I-35 W Highway Bridge. The nonlinear equivalent spring model was able to predict the correct failure mode. The approach was applied to a Howe truss bridge as an example of performance prediction of bridges with semi-rigid connections, most of them of one type. As the simplified spring model was developed from a detailed FE analysis of the joint considered, this approach would not be justified if all joints had to be modeled in detail. Examples where the approach can be used include: structures where only specific joints need to be investigated (e.g., joints subjected to concentrated loads), and structures where the same joint model can be used repeatedly at multiple locations. In some cases, the effort required in performing detailed FE analyses of many joints in order to develop simplified models can be justified if the simplified models can be used in subsequent multiple load cases, thus leading to overall computational savings. Under these circumstances, the nonlinear connection model proposed here provides a simple and affordable way to account for connection performance in global analysis. Published by Elsevier Ltd.
1. Introduction The 2007 catastrophic collapse of the I-35 W Highway Bridge in Minneapolis, Minnesota (I-35 W Bridge for short), under ordinary traffic and construction loads, was triggered by the buckling of an undersized gusset plate [1]. Gusset plates are complicated structural components used to connect frame members such as beams, columns and braces. Their use in buildings and bridges goes back many decades and certainly predates the use of computers in structural analysis and design. Practical design methods ensure safety by providing a load path that satisfies equilibrium, boundary conditions and does not exceed material yield limits. The resulting stress field is by definition a statically possible yield state of stress. Safety against plastic failure is assured because, according to the ⇑ Corresponding author. Tel.: +1 301 975 4357; fax: +1 301 869 6275. E-mail addresses: [email protected] (C. Crosti), [email protected] (D. Duthinh). http://dx.doi.org/10.1016/j.engstruct.2014.01.026 0141-0296/Published by Elsevier Ltd.
Lower Bound Theorem of the Limiting Load [2], a statically possible yield state of stress is less than or equal to the limiting load, which is characterized by unrestricted plastic flow. There is, however, no guarantee that the design load path is the actual one, and thus the design methods provide no information on the load–displacement behavior or stiffness of the connection, even in the elastic range. Current procedures [3] for the design and load rating of multimember gusset plates consist in checking axial, bending and shear stresses along various sections deemed critical, using elastic beam theory. These procedures are intended to ensure a safe and conservative design, but produce results that can be quite different from those derived from more representative finite-element (FE) models, and cannot predict either stiffness or actual behavior. To do so would require highly sophisticated and detailed FE models, such as the ones used in the investigation of the I-35 W Bridge collapse [1]. The approach presented here combines the simplicity of one method with the accuracy of the other: a simplified connection in the form of semi-rigid springs is proposed to improve on current
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design analysis, which typically is linear and assumes rigid connections. Semi-rigid here means that the connection has a finite, defined rotational stiffness, as opposed to zero (hinged) or infinite (rigid) stiffness. The paper starts with a literature survey that focuses on approximate design and analysis methods, then follows with a brief account of the I-35 W Bridge collapse and its investigation. Next, a simple linear equivalent spring model and a more detailed nonlinear one are developed from detailed FE analyses of a joint, and used in a global model that, in the nonlinear case, correctly predicts the failure mode of the I-35 W Bridge. Finally, the spring model is applied to a Howe truss bridge to show how it can be used to predict the performance of bridges with connections mostly of the same type.
2. Review of the literature on approximate methods for gusset plates Extensive reviews of the literature were performed by Chambers and Ernst [4] and Astaneh-Asl [5]. In keeping with the theme of this paper, the focus here is on approximate methods of analysis and design and on buckling. Whitmore [6] tested 1:4 scale specimens of gusset plates for Warren trusses made of aluminum, masonite and bakelite using wire-bonded strain gages (a novelty in 1952), brittle lacquer and photoelastic techniques. Based on these experiments, he developed the effective width that now bears his name (Fig. 1), ‘‘by constructing lines making 30° with the axis of the member which originate at the outside rivets in the first row and continue until they intersect a line perpendicular to the member through the bottom row of rivets.’’ The maximum tensile and compressive stresses may be approximated by assuming the force in each diagonal is uniformly distributed over this width. Astaneh [7] proposed modeling gusset plates as wedges under a point load. In the elastic range, closed-form solutions exist for infinite wedges, but cannot account for the actual boundary conditions. Furthermore, actual loads are transferred to gusset plates by rows of rivets or bolts, rather than at a single point. The author also indicated that wedge models could not predict buckling, for which he proposed a fin truss model, where the cross section of each bar was the average of the cross section of the triangle it bisected. Astaneh [7] suggested an effective length factor of 0.7 for the one-dimensional struts to account for the restraint provided against buckling by the transverse direction in a two-dimensional plate. The use of multiple fin trusses to model gusset plates that connect multiple members rapidly becomes cumbersome, and makes the FE method very attractive for its accuracy and automation, when compared to Astaneh’s approach. An elegant, statically possible load path with no moment (i.e., concentric) in the gusset-to-beam and gusset-to-column connections was developed and called the Uniform Force Method by Thornton [8], who obtained the following connection forces (symbols are defined in Fig. 2):
a
eB b P; V B ¼ P; V C ¼ P; r r r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ ða þ eC Þ þ ðb þ eB Þ
HB ¼
HC ¼
eC P; r
r ð1Þ
The Uniform Force Method, with modifications to account for geometries that induce moments, is the method currently recommended by the American Institute of Steel Construction [9]. Dowswell and Barber [10] provided a quantitative definition of compact gusset plates and guidance on the required plate thickness tb to prevent their buckling:
sffiffiffiffiffiffiffiffiffiffi ry c 3 tb ¼ 1:5 EL2
ð2Þ
where c = the shorter of the distances from the corner bolt or rivet to the adjacent beam or column, E = modulus of elasticity, L2 = equivalent column length from the middle of the Whitmore width (Fig. 1) and ry = yield stress. The gusset plate is compact if its thickness t P tb and non-compact if t < tb. Dowswell and Barber [10] compared their theoretical buckling capacities Pth with experimental and FE calculations in the literature Plit. They used the average of the lengths from the middle and the ends of the Whitmore width for the equivalent column length. For compact gusset plates, using an effective length factor of 0.5, they found the ratio Plit/Pth = 1.47; and for non-compact gusset plates, using an effective length factor of 1.0, Plit/ Pth = 3.08. Thus, the separation of compact from non-compact gusset plates and the subsequent different effective length factors resulted in inconsistent and problematic factors of safety for design against buckling. Brown [11,12] developed analytical expressions for the edge buckling of gusset plates, based on the elastic buckling stress of a plate supported on its loaded edges, but otherwise unrestrained. For edge buckling, the critical section bisected the long free edge of length a and was perpendicular to the brace (Fig. 3). Its width b was different from the Whitmore width. Only a fraction of the total brace load contributed to the edge buckling of the gusset, and the rest was transferred directly to the steel frame. Brown [11,12] compared her predictions with 18 experiments and produced 16 ratios of experimental to predicted values ranging from 1.01 to 1.38, and 2 values below 1 (0.99 and 0.81). Yamamoto et al. [13] tested eight gusset plates (each connecting two horizontal chord members and two diagonal braces) and performed FE analysis to establish stability design criteria for the joints of the Warren truss designed to stiffen the deck of the Honshu–Shikoku suspension bridge. They focused on the development of plastic zones, local buckling and ultimate strength, and found that local yielding and local buckling preceded global buckling of the gusset plates. The load at which local buckling started depended on the extent of yielding, which covered the inner portions of the gusset plate, whose in-plane stiffness was constrained by the surrounding elastic region. Under the assumption that buckling occurred when the stress reached the allowable stress in the material, ra = 0.58 ry, and with l1 = length of the vertical free edge, Yamamoto et al. [13] proposed the following design thickness for local buckling:
rffiffiffiffiffiffi tcr ¼ 1:10l1
Fig. 1. Equivalent column.
ra E
ð3Þ
There have been numerous other FE studies of gusset plates over the last three decades, and they have been reviewed extensively by Chambers and Ernst [4] and Astaneh-Asl [5]. Among the earlier studies, for example, is the work of Cheng et al. [14]. They used ANSYS and performed a parametric study to calculate the elastic buckling strength of concentrically loaded gusset plates. In
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Fig. 2. Uniform Force Method.
Fig. 3. Critical section for gusset edge buckling. Fig. 4. Rotational spring four-parameter stiffness.
particular, they focused on the effects of the splice plate and the rotational restraint provided by the bracing member, but did not formulate design equations or derive equivalent springs. A widely used equivalent spring for beam–column connections of braced frames was formulated by Richard and Abbott [15]. Their nonlinear spring had one degree of freedom, and its moment–rotation stiffness was represented by four parameters that modeled a linear elastic part, a linear post-yielding, strain hardening part, and a curved transitional part (Fig. 4 and Eq. (4)).
2
3
6 7 K Kp M¼6 n 1=n þ K p 7 4 5h ¼ K CONN h ðKK Þ h 1 þ M0p
ð4Þ
where h = connection rotation angle; K = initial or elastic stiffness; Kp = plastic stiffness; KCONN = connection stiffness; M = connection moment; M0 = reference moment; and n = shape parameter. Williams [16] used computer program INELAS to analyze steel gusset plate connections and find values that fit the four-parameter model. For example, one of his model of a gusset plate and beam was comprised of 77 finite elements, a number not atypical of the computer technology of the time. He also used NASTRAN to calculate the capacity of gusset plates subjected to compressive brace forces. Gusset plate research, highlighted above, had been sporadic in the last 60 years, but would receive a jolt of interest in 2007 with the catastrophic failure of the I-35 W Highway Bridge in Minneapolis, Minnesota.
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3. I-35 W Bridge collapse The I-35 W Highway Bridge – Number 9340 in the National Bridge Inventory – spanned the Mississippi River in the city of Minneapolis, Minnesota. Construction started in 1964 and the bridge was opened to traffic in 1967. The major superstructure components consisted of welded plate girders and truss members with riveted and bolted connections. The deck truss portion of the bridge is shown schematically in Fig. 5. In the terminology of the National Transportation Safety Board (NTSB) Accident Report [1], the deck truss portion of the bridge excludes the north and south approach spans. The deck truss comprised two parallel Warren trusses (east and west) with vertical members. The upper and lower chords, the compression diagonal and vertical members were welded box sections, whereas the tension vertical and diagonal members were Hsections. Steel gusset plates, riveted to the side plates of the box members and the flanges of the H-members, were used on all the 112 connections of the two main trusses. All joints had at least two gusset plates on either side of the connection. In the late 1970s, the bridge was classified as ‘‘non-load-path-redundant’’, that is, if designated ‘‘fracture critical’’ members of the main trusses failed, the bridge would collapse. Unfortunately, that is what happened on August 1, 2007. On that day, roadway work was underway and four of the eight travel lanes were closed to traffic (two outside lanes northbound and two inside lanes southbound). In the early afternoon, construction equipment and material were positioned in the two closed inside southbound lanes. At 6:05 pm, a motion-activated surveillance camera showed the bridge center span separating from the rest and falling into the river. A total of 111 vehicles, including 25 associated with the construction, were on the bridge at the time, and 13 lives were lost. Examination of the bridge debris showed that all four gusset plates at the U10 nodes (U refers to the upper chord, L to the lower chord) had fractured into multiple pieces. More fractures in the lower chord members between the L9 and L10 (Fig. 5) nodes precipitated complete separation of the main truss, thereby causing the center span to drop. As a consequence of the collapse of the I-35 W Bridge, the Federal Highway Administration (FHWA) produced a guidance document [3] with simple, hand-calculable formulas for the load rating of gusset plates, with the intent that bridge owners may use the guidance to evaluate the connecting plates and fasteners of truss bridges. The gusset plates are rated for their resistance to tension, compression and shear. In tension, the modes of failure that need to be evaluated are yielding of gross section, fracture of net section, and rupture by block shear. For shear resistance, several sections must be investigated to find the governing one. Finally, the compressive strength of a gusset plate is that of an equivalent column determined as follows: 1. The thickness of the column is that of the gusset plate. 2. The width is the Whitmore effective width [6] presented Fig. 1.
3. The length of the column is the average of three lengths extending in the direction of the framing member from the middle and the ends of the last row of bolts to the edges of the gusset plate or adjacent groups of bolts (L1, L2 and L3 in Fig. 1). This approximation is adapted from Thornton [17], who used the middle length L2, but then went on to propose the average of L1, L2 and L3 as ‘‘a more reasonable approximation’’ of the critical length of the column strip, with an effective length factor of 0.65, corresponding to a column with both ends fixed. The examples in the Guidance ignore any lateral constraint to the gusset and use an effective length factor of 1.2, which corresponds to a column with one end fixed and the other restrained against rotation but free to translate. Brown [11,12] had previously proposed the factor of 1.2 based on experimental observation and calculations. 4. Detailed finite-element investigation From the literature review, it is seen that there are simple design methods based on equilibrium and elastic behavior and proven safe by experiments. There is, however, no simple way of calculating the actual behavior of a gusset plate, even in the elastic range. Designers ensure that the connections are stronger than the members, then proceed with a structural analysis that assumes rigid connections. Such a structural analysis is incapable of predicting connection failure, or account for the flexibility of the connection in the global behavior of the structure. On the other hand, there exist detailed models such as the ones analyzed by the NTSB. As part of the investigation of the collapse of the I-35 W Bridge, the NTSB commissioned the FHWA, the State University of New York (SUNY) at Stony Brook and the software company Simulia to develop an FE model of the bridge. FHWA constructed a three-dimensional global model of the entire deck truss portion of the bridge with (two-node, linear or cubic) beam and (four-node) shell elements. The model, including the boundary conditions at the piers, was calibrated with strain gage data from a fatigue assessment conducted by the University of Minnesota in 1999 [18]. In addition, as field evidence pointed to gusset plates as the trigger of the collapse, SUNY/Simulia developed detailed models of the U10 (Fig. 6) and L11 nodes and incorporated them into the global model. More details about the highway accident and its analysis can be found in [1,19,20]. In [19] Hao provided a possible explanation for why some of the gusset plates were under-designed: the main frame gusset plates and the upper chords may have been designed from a one-dimensional model of the bridge, i.e., a uniformly loaded beam supported at four points. Each gusset plate of the detailed model was composed of four layers of eight-node, linear, solid brick elements, whose largest dimension in the plane of the gusset plate was 5 mm in highly stressed regions and less than 15 mm elsewhere [20]. The 289,000 solid elements used reduced integration and hourglass control to alleviate spurious displacement modes. The connection model also included the five main truss members, cut at 2/5 of
Fig. 5. Schematics of the deck truss portion of I-35 W Bridge.
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Fig. 6. Detailed FE model 1 of gusset plate (ABAQUS).
their lengths in the bridge (the so-called stub members), and modeled with solid and shell elements (Fig. 6). In their largest dimension, the shell elements ranged from 25 mm to 50 mm, and the solid elements, which were in the transition zones between the truss members and the gusset plates, from 8 mm to 25 mm. Transition between shell and solid elements was performed by surfacebased coupling constraints. All members in the connection model other than the main truss members and gusset plate had maximum element dimension of about 15 mm. The sophistication of these models, compared with Williams’s [16] for example, reflects the extraordinary advance of computer and finite-element technologies in the last quarter of a century. Even with such a detailed connection model, the rivets could not be modeled. The shanks of the rivets had a radius of 13 mm and the hemispherical heads a radius of 20 mm. The rivets were modeled by coupling nodes of the fastened components normal to the rivet axis within a radius of influence of 13 mm. Contact pairs, with a Coulomb friction coefficient of 0.1, were also used with the fastener elements. Such a detailed analysis required advanced skills and powerful computers and was clearly beyond routine design.
5. Simplified linear spring model 5.1. Stiffness matrix of linear springs A simplified model of a gusset plate connection of the bridge was developed as part of the present work. First, the linear model 2 (Table 1) is described. It started with the analysis of the NTSB detailed FE model 1 [1,20], formulated in software ABAQUS [21], of gusset plate U10 (Fig. 6). The results of the analysis of model 1 established the equivalent stiffness of springs that completely modeled the elastic behavior of the connection model 2 [22,23]. This simplified connection model 2 was placed in a global 2D model of the West main truss (model 5), using overall strategies for integration of substructures proposed in [24]. Table 1 lists all the
computer models run in the present work. They are described in more detail when they are first used. The ABAQUS detailed FE model of U10 (model 1) had five stub elements connected to a pair of gusset plates (Fig. 6). In the NTSB global analysis, the stub members were reattached to their corresponding truss members. To develop the simplified linear connection model, the stub elements and gusset plates were replaced by five user-defined springs, that each had a full 6 6 stiffness matrix for all 6 degrees of freedom (DoFs). To establish the flexibility of the spring that is equivalent to element 1 in the detailed model (Fig. 6) for example, the ends of elements 2–5 in the detailed model were fixed and an arbitrary concentrated force or moment was applied to obtain the displacements and rotations at the end of element 1. In general, the concentrated force or moment was applied at a node at the end of a stub member, as close to the center of the cross section as possible. For tubular members, two concentrated forces or moments were applied at the middle of the long sides of the end cross section, and displacements and rotations were calculated at these same points. The process was performed using ABAQUS and repeated by applying arbitrary forces and moments corresponding to all 6 DoFs, and thus, after normalization, the 6 6 flexibility matrix for element 1 in ABAQUS coordinates was obtained. This flexibility matrix was inverted to obtain the stiffness matrix, which was then transformed to the coordinate system specified by STRAND for user-defined elements and applied to the simplified spring model (Fig. 7, model 2). Fig. 8 is a flowchart of the procedure. This process was repeated for all five elements of the connection. In calculating the equivalent spring stiffness, software STRAND7/STRAUS7 [25] was used, and care had to be exercised in defining the local coordinates of the user-defined element [STRAND7/STRAUS7 required the local axis 3 to be in the longitudinal direction and pointing away from the end of the stub (Fig. 7)].
5.2. Verification of the stiffness matrix After the equivalent springs were assembled, the response of the simplified connection in STRAND (model 2) was verified against the detailed FE analysis in ABAQUS (model 1), for the same set of applied forces and moments. As done previously, forces and moments were applied one at a time, for example at end node 1 of element 1, while the other end nodes were fixed. Agreement in the displacements and rotations calculated was good, especially for the diagonal terms (maximum relative error 0.5%), with acceptable errors in the off-diagonal terms, which were several orders of magnitude smaller (Tables 2 and 3). Similar results were obtained for the other elements of the connection [22], but were not presented here to save space. As stated in the introduction, current design analysis is typically linear and assumes rigid connections. The next part of this work investigates the validity of the rigid connection assumption. For this purpose, ABAQUS model 3 was developed by changing the
Table 1 Computer models run. Model
Software
Analysis
Purpose
1. 2. 3. 4. 5. 6. 7. 8. 9.
ABAQUS STRAND ABAQUS STRAND STRAND ABAQUS STRAND STRAND STRAND
Linear Linear Linear Linear Linear Nonlinear Nonlinear Nonlinear Nonlinear
Stiffness of 5 members Stiffness of 5 springs Stiffness of 5 members Validity of rigid connection assumption Behavior with various connection models Stiffness of 5 members Stiffness of 5 springs Behavior with various connection models Behavior with various connection models
Detailed connection Spring connection Detailed connection, various gusset thickness Rigid connection West main truss (2D) Detailed connection Spring connection I-35 W (3D) Howe truss (2D)
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Fig. 7. Equivalent spring model 2 (STRAND).
thickness of the gusset plates in the NTSB gusset connection model 1. Also, STRAND7/STRAUS7 [25] model 4 was developed with beams that had the same stiffness as the members framing into U10, but were rigidly connected to each other. Table 4 compares the displacements d and rotations q predicted by models 2 (semi-rigid connection) and 4 (rigid connection) with the results from the detailed ABAQUS model 3 with various gusset plate thicknesses. As expected, the thicker the gusset plates, the smaller the deflections and rotations. Also, as expected, the rigid joint model 4 produced results that were not as accurate as the more complicated spring model 2, which corresponded to a gusset thickness of 12.7 mm (base case, also diagonal terms of Tables 2 and 3). The displacements and rotations predicted by the rigid connection model 4 (Table 4, column 2) were not always smaller than the base case (column 4), which included the stiffness of the gusset plates. 6. Planar linear global analysis The next phase of the analysis consisted in using the spring elements in a global analysis of the bridge. The first global analysis was linear, with the various connection models placed in a 2D model of the I-35 W Bridge, at a location corresponding to the U10 gusset plate where the bridge collapse initiated. For this 2D analysis, the simplified connection model only had three degrees of freedom corresponding to planar displacements and rotation. The analysis was performed with software STRAND7/STRAUS7 [25], using a model consisting of 496 beam elements (which have axial stiffness) and loaded with the gravity loads (dead weight, vehicle and construction loads, Fig. 9) present at collapse, and divided into the East and West trusses [26]. Liao et al. [27] showed by influence lines that the temporary construction loads placed near U10 significantly affected the forces imposed on the joint and contributed to the collapse. The 2D analysis was performed for the more heavily loaded West main truss (model 5). Although the model was 2D, the finite elements used were formulated for the general 3D case, and therefore some of the support points had to be restrained in 3D (Fig. 9). Four cases were run, resulting in the following midspan vertical deflections d: (1) all joints rigid, d = 281 mm; (2) U10 modeled with linear springs, all other joints rigid, d = 285 mm; (3) U10 modeled with detailed FE, all other joints rigid, d = 285 mm; and (4) all five-element joints hinged, all other joints rigid, d = 286 mm. Results showed that modifying the stiffness of one connection within the elastic range produced no noticeable effect on the maximum vertical deflection of the bridge (at midspan). It was concluded that, in the elastic range, the connections could be assumed rigid to a good approximation. 7. Simplified nonlinear spring model
Fig. 8. Flowchart to calculate equivalent spring stiffness.
In order to investigate the failure of the bridge, it was necessary to progress beyond the linear elastic range. To capture the nonlinear behaviour of the gusset connection up to failure, the detailed
Table 2 Element 1, node 1, displacements d and rotations q obtained in ABAQUS for applied forces and moments (detailed model 1). Node 1
dx (mm)
dy (mm)
dz (mm)
qx (mrad)
qy (mrad)
qz (mrad)
Fx = 444.8 (kN) Fy = 444.8 (kN) Fz = 444.8 (kN) Mx = 228.8 (kN m) My = 113.0 (kN m) Mz = 161.0 (kN m)
1.84E1 8.19E2 1.88E5 4.19E6 7.83E7 7.43E3
8.19E2 3.50E0 3.39E4 1.22E5 1.94E5 7.09E1
1.88E5 3.42E4 2.10E+1 9.30E1 1.29E0 5.10E5
6.72E6 4.51E5 1.51E0 5.15E0 3.71E2 4.41E6
7.72E5 7.66E5 5.08E0 8.98E2 7.04E1 1.11E5
2.05E2 1.96E0 1.37E4 8.80E6 8.01E6 6.16E1
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dx (mm)
dy (mm)
dz (mm)
qx (mrad)
qy (mrad)
qz (mrad)
Fx = 444.8 (kN) Fy = 444.8 (kN) Fz = 444.8 (kN) Mx = 228.8 (kN m) My = 113.0 (kN m) Mz = 161.0 (kN m)
1.85E1 8.27E2 2.51E4 1.60E5 3.63E5 7.60E3
8.27E2 3.50E0 3.29E4 2.83E5 1.86E5 7.07E1
2.51E4 3.29E4 2.10E + 1 9.30E1 1.29E0 4.75E5
2.61E5 4.60E5 1.51E0 5.15E0 3.71E2 4.66E6
1.43E1 7.31E5 5.09E0 9.00E2 7.06E1 1.02E5
2.10E2 1.95E0 1.31E4 7.92E6 7.14E6 6.15E1
Table 4 Comparison of displacements d and rotations q of STRAND7/STRAUS7 model 2 with semi-rigid connection, model 4 with rigid connection and ABAQUS FE model 3 with various gusset plate thicknesses [numbers in brackets are ratios to base case with thickness of 12.7 mm]. STRAND mod. 4 rigid joints
dx (mm) dy (mm) dz (mm) qx (mrad) qy (mrad) qz (mrad)
0.184 [1.00] 7.42 [2.12] 12.8 [0.61] 3.12 [0.61] 0.759 [1.08] 0.802 [1.30]
STRAND mod. 2 semi-rigid joints
0.185 [1.00] 3.50 [1.00] 21.0 [1.00] 5.15 [1.00] 0.706 [1.00] 0.615 [1.00]
ABAQUS FE model 3 with gusset thickness: 12.7 mm
15.87 mm
19.05 mm
25.4 mm
0.184 [1] 3.50 [1] 21.0 [1] 5.15 [1] 0.704 [1] 0.616 [1]
0.1748 [0.95] 3.276 [0.94] 18.54 [0.88] 4.688 [0.91] 0.6618 [0.94] 0.5998 [0.97]
0.1677 [0.91] 3.105 [0.89] 16.90 [0.80] 4.393 [0.85] 0.6302 [0.90] 0.5872 [0.95]
0.1575 [0.86] 2.860 [0.82] 14.83 [0.71] 4.021 [0.78] 0.5862 [0.83] 0.5682 [0.92]
Fig. 9. 2D model of I-35 W Bridge showing loads and support conditions. Circles and squares indicate support conditions (restrained displacements d and rotations q) for both main trusses.
ABAQUS FE model 1 of the five-element connection was loaded as described in Section 6, but into the nonlinear range. Both geometric and material nonlinearities are taken into account (model 6). Nonlinear material behaviour was modelled using the Von Mises yield criterion and the isotropic hardening rule (Fig. 10, from [1]). A large strain–large displacement formulation (ABAQUS default option) was used to carry out the nonlinear analysis. The results were force–displacement and moment–rotation curves, which were then transformed into the STRAND coordinates of the equivalent spring model. In the spirit of simplification, to make the problem tractable and to use what was available in STRAND7/STRAUS7 [25], the stiffness matrix of each of the five elements of the connection was diagonal only and derived from an initially perfect gusset plate. The diagonal terms were in the form of load–displacement or moment–rotation curves (model 7). Notice that this nonlinear connection model included all six DOF.
Fig. 10. Stress–strain curve [1].
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Richard’s four-parameter nonlinear connection model (Fig. 4, Eq. (4)) only had one DOF and would not have captured the buckling of the gusset plate under axial load. Fig. 11 presents the load–displacement and moment–rotation curves of element 3 of the connection, where x, y and z refers to the global coordinate system. For clarity, only the positive values are shown. Similar curves model the behavior of the other elements of the connection and can be found in [22].
8. 3D nonlinear global analysis As the global nonlinear analysis might involve buckling and out-of-plane deformation, it was necessary to use a full 3D model of the I-35 W Bridge, as described in Section 3. The FE model 8 was composed of 1894 beam elements representing all the primary steel members, including all main lateral braces (Figs. 12 and 13), but the deck was not included in the model. The dead load was always present and not factored, but the construction loads were applied gradually, with a load factor of 1.0 corresponding to the actual loads at collapse ([26] and Fig. 9). The support conditions for both main trusses were the same as described in Fig. 9. Nonlinear analyses (for material and geometry) were run on two different models: 1. Model 8A: All joints were rigid, the five structural members intersecting at U10-W were elasto-plastic (Fig. 10) and all the other members were elastic; 2. Model 8B: U10-W joint was semi-rigid (modeled by nonlinear connection elements with diagonal stiffness matrix as described above) and the five members around the U10-W joint were elasto-plastic. All the other joints were rigid and all the other members were elastic.
9. Results
Fig. 11. Force–displacement and moment rotation curves for element 3 of connection (KFXX, KFYY, KFZZ and KMXX, KMYY, KMZZ are the elastic stiffness for force– displacement and moment–rotation for x, y and z axis respectively).
For model 8A (with rigid joints), failure was initiated at a load factor of 6.36 by yielding of the end of element 1 that connected with U10 W. Model 8B (nonlinear semi-rigid connection) provided a much better simulation of the actual behavior. The semi-rigid joint model predicted that U10 W began to fail at a load factor of 0.92, and completely failed at load factor 1.7 (Fig. 14), leading to the collapse of the bridge. Fig. 15 and Table 5 present the axial forces and load ratios (ratio of force carried to capacity) in the five elements of the connection. Collapse initiated when connection element 3 reached axial capacity in compression (buckling, Fig. 16a), which agreed with the NTSB analysis. Fig. 16a shows the deflection of the detailed ABAQUS model of U10-W under the
Fig. 12. Modeling of lateral bracing.
Fig. 13. 3D beam model of I-35 W (model 8).
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ultimate axial compressive force of 12.1 MN applied at the end of stub member 3 (Fig. 6), with the ends of the other stubs fixed. It was thus seen that the equivalent spring model produced a good approximation of the behavior of a gusset plate connection in the elastic and post-elastic range. Clearly, the model was not as accurate as the detailed FE model used by NTSB. The discrepancy was measured by a load factor of 0.92, instead of 1.0, applied to the construction loads at failure initiation. Moreover, the simplified connection model assumed an initially perfect gusset plate. Had the model accounted for initial out-of-plane displacements, the load factor would have been less than 0.92. (The NTSB connection model [20] accounted for an initial out-of-plane displacement of one thickness or 12.7 mm at the free edge between connection elements 2 and 3, Fig. 6. After dead weight and traffic load application, the maximum out-of-plane deformation was 17.3 mm, in agreement with the values 15.2 ± 3.8 mm estimated from inspection photographs, Fig. 16b.) Nevertheless, the results of the global 3D analysis using the simplified nonlinear connection model would have attracted the attention of the analyst to joint U10 and the location and magnitude of the construction loads, something that the rigid connection model, which resulted in a load factor of 6.36, would have failed to do. A major difficulty in the development of the simplified connection model was that it was based on a prior detailed FE
Fig. 14. Load–deflection curve (deflection at midspan as indicated in Fig. 9).
Table 5 Ratio of calculated force to axial capacity for connection U10-W at load factor 0.92. Connection element
Load ratio
Tension or compression
1 2 3 4 5
0.28 0.56 1.00 0.02 0.41
Compression Tension Compression Tension Tension
model. Various attempts to circumvent this difficulty were unsuccessful and the literature review confirmed that previous simplified models were very limited in accuracy and use. This simplified connection model is then most useful for structures that use the same connection repeatedly, or only with changes in plate thickness or material properties that can be accommodated easily in the same detailed FE model; or for loadings that would cause the analyst to focus on a few specific connections only; or for multiple load cases, where a detailed model can be used efficiently multiple times. An example follows.
10. Application to a bridge with identical connections The general procedure of using a nonlinear simplified gusset connection was applied to a Howe truss bridge to show the importance of properly accounting for connection behavior. The truss was 12.14 m high, spanned 93.91 m, and consisted of repeated panels of length 11.58 m or 9.00 m defined by vertical members, horizontal chords and diagonal braces (Fig. 17). For simplicity, the structure considered was planar and the three-member joints at the ends were assumed rigid. All five-element connections and the members they attach were the same as for U10-W analyzed above (elasto-plastic, Fig. 10, dimensions shown in Fig. 17). Besides its dead load, the bridge was subjected to traffic load uniformly distributed on the top chord, and construction loads modeled as three point loads on the top chord (Fig. 17). The bridge was restrained against horizontal and vertical translations at one end and against vertical translation at the other end (model 9).
Fig. 15. Axial forces in the five elements of semi-rigid connection U10-W.
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Fig. 16. (a) ABAQUS model showing failure of U10 under axial compression of diagonal member 3 and (b) inspection photograph of U10 [1].
Fig. 17. Geometry of the Howe truss bridge (model 9).
Four models with large displacement formulations, elasto-plastic members and different connection models were considered: 9A: All joints rigid; 9B: one critical joint (at the concentrated loads) modeled by the nonlinear, five-element connection previously described, all other joints rigid; 9C: all five-element joints modeled by the nonlinear connection elements previously described; and 9D: all five-element joints hinged. Nonlinear analyses (that accounted for both material and geometric nonlinearity) were run by applying an increasing load factor
to the concentrated loads only, with the uniform loads held constant. Fig. 18 presents the load factor versus midspan vertical displacement for the four different models considered. As expected, model 9D (pinned joints), had the lowest load capacity, model 9A (rigid joints) had the highest, and the others (semi-rigid joints) fell in between. 10.1. Model 9A: Rigid joints Model 9A reached the ultimate load factor of 3.26. Fig. 19 presents the ratio of bending moment to flexural capacity in the members at the ultimate load factor. Since the joints were rigid, collapse occurred because of flexural yielding of the members.
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the critical joint under the concentrated loads, and the other joints remained elastic as the critical joint failed by flexural yielding. Section 6 showed that the assumption of rigid behavior was a good approximation for elastic joints, and the difference between models 9B and 9C was in the treatment of joints that turned out to remain elastic. If there were instead several heavily loaded joints, then model 9C (with all joints semi-rigid) would predict a different load distribution than model 9B (with only one semi-rigid joint) as some of the heavily loaded joints begin to plastify. In this case, the failure loads predicted by models 9B and 9C would be expected to be different. 10.4. Model 9D: All five-member joints pinned
Fig. 18. Load factor versus midspan vertical displacement.
10.2. Model 9B: Semi-rigid connection (five element-connection at concentrated load) Fig. 20 illustrates the details of the joint under study, and Fig. 21 and Table 6 the axial capacity of each connection element. The failure of the truss bridge could be attributed to the achievement of axial capacity in the connection elements. At a load factor of 1.98, connection element 2 reached axial capacity (Table 6), followed by connection element 1 and finally connection elements 3 and 4.
10.3. Model 9C: Semi-rigid connections (all five-member joints modeled with five element-connections) Model 9C predicted the same failure load as model 9B because in this example the highest stresses were heavily concentrated at
Fig. 22 presents the ratio of bending moment to flexural capacity in the members at the ultimate load factor of 1.46. For this model, failure was reached as soon as a plastic hinge was developed in member 1 of the joint under concentrated loads. Table 7 summarizes the results. Models that did not account for realistic connection behavior could be quite erroneous, especially when concentrated loads made a joint critical. When structural members were assumed rigidly connected, the predicted capacity could be quite erroneous if the assumption that joints were stronger than members turned out to be incorrect. The methodology proposed here made global analysis with proper accounting of connection performance efficient and affordable. 11. Conclusion Conclusion 15 of the NTSB investigation of the I-35 W Bridge collapse stated [1]: ‘‘Because bridge owners generally consider gusset plates to be designed more conservatively than the other
Fig. 19. Model 9A, rigid joints: bending moment ratio at load factor of 3.26.
Fig. 20. Model 9B, semi-rigid, five-element connection at concentrated loads.
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Fig. 21. Model 9B, axial forces in the five-element connection at concentrated load.
Table 6 Ratio of calculated force to axial capacity for critical connection at load factor 1.98. Connection member
Load ratio
Tension or compression
1 2 3 4 5
0.71 1.00 0.24 0.11 0.38
Compression Compression Compression Compression Compression
members of a truss, because the American Association of State Highway and Transportation Officials (AASHTO) provides no specific guidance for the inspection of gusset plates, and because computer programs for load rating analysis do not include gusset plates, bridge owners typically ignore gusset plates when performing load ratings, and the resulting load ratings might not accurately
reflect the actual capacity of the structure.’’ The work presented herein illustrates the problem. In the linear elastic range the assumption of rigidly connected members produced good results, and that was why FE analysis coupled with load tests [18] did not find the design flaw that turned out to be fatal. In the nonlinear range, however, structural capacity might not be well predicted if connection behavior and strength were not properly accounted for. If a particular joint was under highly concentrated loads, that node needed to be scrutinized, possibly by the method explained in this paper. The example in the previous section showed the method to be especially economical when only specific nodes needed to be analyzed in detail, or all nodes were similar. Future work, which will include comparison with experiments, for example full scale tests at FHWA [28], will delineate more precisely the level of detail required for the modeling of connections, and how the approach can be used in structural health monitoring [29].
Fig. 22. Model 9D, pinned joints. Bending moment ratio at load factor of 1.46.
Table 7 Analysis of Howe truss with various connection models. Model
Description
Ultimate load factor
Failure mode
9A 9B 9C 9D
All joints rigid Semi-rigid joint at point loads All 5-member joints semi-rigid All 5-member joints pinned
3.26 1.98 1.98 1.46
Flexural yielding of members Connection element 2 reached axial capacity Connection element 2 reached axial capacity Plastic hinge in member 1 of joint
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Accurate prediction of bridge failure should include proper modeling of connections. This can be done economically and accurately by the method presented herein if only specific joints need to be investigated in detail, or the same joint model can be used repeatedly at multiple locations, or multiple load cases must be run. The nonlinear connection model proposed here provides a simple and affordable way to account for connection performance in global analysis. Disclaimer The full description of the procedures used in this paper requires the identification of certain commercial software. The inclusion of such information should in no way be construed as indicating that such products are endorsed or recommended by NIST or that they are necessarily the best software for the purposes described. Acknowledgments The authors are grateful to Joey Hartmann of FHWA TurnerFairbank Highway Research Center for granting access to the FE model of the I-35 W Bridge and to the research team of www.francobontempi.org for support and advice. References [1] National Transportation Safety Board. Collapse of I-35 W Highway Bridge, Minneapolis, Minnesota, August 1, 2007. Accident Report, NTSB/HAR 08/03 PB 2008-916213. Washington D.C. 20594; 2008. [2] Kachanov LM. Fundamentals of the theory of plasticity. Moscow, USSR: Mir Pub.; 1974. [3] Federal Highway Administration. Load rating guidance and examples for bolted and riveted gusset plates in truss bridges. FHWA-IF-09-014, Washington, D.C. 20590; 2009. [4] Chambers JJ, Ernst CJ. Brace frame gusset plate research – Phase 1 literature review, vols. 1 and 2. Dept. of Civil Engg. University of Utah, Salt Lake City, UT 84112–0561; 2005. [5] Astaneh-Asl A. Gusset plates in steel bridges – design and evaluation. Steel TIPS report, structural steel educational council technical information & product services. Moraga, CA; April, 2010. 156 pp. [6] Whitmore RE. Experimental investigation of stresses in gusset plates. University of Tennessee Engineering Experiment Station Bulletin no. 16, May; 1952. [7] Astaneh A. Simple methods for design of steel gusset plates. In: Proc. ASCE Structures Congress ’89, San Francisco, CA; 1989. p 345–54.
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