A Nonlinear Model for Gusset Plate Connections Engineering Structures

8/9/2019 A Nonlinear Model for Gusset Plate Connections Engineering Structures

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A nonlinear model for gusset plate connections

Chiara Crosti a, Dat Duthinh b,

a Sapienza University of Rome, Rome, Italyb 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

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 Administra-

tion (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 accu-

racy 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 con-

sidered, 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 repeat-

edly 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 sub-sequent 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 con-

nection 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 asbeams, 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

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 loaddisplacement

behavior or stiffness of the connection, even in the elastic range.

Current procedures [3]for the design and load rating of multi-member 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 conser-

vative design, but produce results that can be quite different from

those derived from more representative finite-element (FE) mod-

els, 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

http://dx.doi.org/10.1016/j.engstruct.2014.01.026

0141-0296/Published by Elsevier Ltd.

Corresponding author. Tel.: +1 301 975 4357; fax: +1 301 869 6275.

E-mail addresses: [email protected] (C. Crosti), [email protected] (D.

Duthinh).

Engineering Structures 62-63 (2014) 135147

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design analysis, which typically is linear and assumes rigid connec-

tions. Semi-rigid here means that the connection has a finite, de-

fined rotational stiffness, as opposed to zero (hinged) or infinite

(rigid) stiffness. The paper starts with a literature survey that fo-

cuses on approximate design and analysis methods, then follows

with a brief account of the I-35 W Bridge collapse and its investiga-

tion. Next, a simple linear equivalent spring model and a more de-

tailed nonlinear one are developed from detailed FE analyses of a

joint, and used in a global model that, in the nonlinear case, cor-

rectly 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 Cham-

bers 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 speci-

mens 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 assum-

ing 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 infi-

nite wedges, but cannot account for the actual boundary condi-

tions. Furthermore, actual loads are transferred to gusset platesby 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 bi-

sected. 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 automa-

tion, when compared to Astanehs approach.

An elegant, statically possible load path with no moment (i.e.,

concentric) in the gusset-to-beam and gusset-to-column connec-

tions was developed and called the Uniform Force Method byThornton[8], who obtained the following connection forces (sym-

bols are defined inFig. 2):

HB arP; VB

eBr P; VC

b

rP; HC

eCr P; r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffia eC

2 b eB2

q 1

The Uniform Force Method, with modifications to account for

geometries that induce moments, is the method currently recom-

mended by the American Institute of Steel Construction [9].

Dowswell and Barber[10]provided a quantitative definition ofcompact gusset plates and guidance on the required plate thick-

nesstbto prevent their buckling:

tb 1:5

ffiffiffiffiffiffiffiffiffiffiryc3

EL2

s 2

wherec= 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) andry= yield stress.

The gusset plate is compact if its thicknesstP tband non-com-

pact ift< 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 themiddle and the ends of the Whitmore width for the equivalent col-

umn length. For compact gusset plates, using an effective length

factor of 0.5, they found the ratio Plit/Pth= 1.47; and for non-com-

pact gusset plates, using an effective length factor of 1.0, Plit/

Pth= 3.08. Thus, the separation of compact from non-compact gus-

set plates and the subsequent different effective length factors re-

sulted 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 lengtha and was perpendicular to the brace (Fig. 3). Its width

bwas different from the Whitmore width. Only a fraction of the to-tal 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 pro-

duced 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 connect-

ing 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 Hon-

shuShikoku 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 de-

pended 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 oc-

curred when the stress reached the allowable stress in the mate-

rial, 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:

tcr 1:10l1

ffiffiffiffiffiffiraE

r 3

There have been numerous other FE studies of gusset plates

over the last three decades, and they have been reviewed exten-

sively 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 theelastic buckling strength of concentrically loaded gusset plates. InFig. 1. Equivalent column.

136 C. Crosti, D. Duthinh / Engineering Structures 62-63 (2014) 135147


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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 beamcolumn connections

of braced frames was formulated by Richard and Abbott[15]. Their

nonlinear spring had one degree of freedom, and its momentrota-

tion 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. 4and Eq.(4)).

M K Kp

1 KKp h

M0 n

1=n

Kp

2

664

3

775h KCONNh 4

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-parame-

ter 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 Minneap-

olis, Minnesota.

Fig. 2. Uniform Force Method.

Fig. 3. Critical section for gusset edge buckling.

Fig. 4. Rotational spring four-parameter stiffness.

C. Crosti, D. Duthinh / Engineering Structures 62-63 (2014) 135147 137
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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 Min-

neapolis, Minnesota. Construction started in 1964 and the bridge

was opened to traffic in 1967. The major superstructure compo-

nents consisted of welded plate girders and truss members with

riveted and bolted connections. The deck truss portion of thebridge 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 sec-

tions, whereas the tension vertical and diagonal members were H-

sections. 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 asso-

ciated 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 frac-

tures 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 Fed-

eral Highway Administration (FHWA) produced a guidance docu-

ment [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. Fi-

nally, the compressive strength of a gusset plate is that of an equiv-

alent 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]presentedFig. 1.

3. The length of the column is the averageof three lengths extend-

ing 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,L2andL3inFig. 1). This approx-

imation is adapted from Thornton [17], who used the middle

length L2, but then went on to propose the average of L1, L2and 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 corre-

sponds to a column with one end fixed and the other restrained

against rotation but free to translate. Brown [11,12]had previ-

ously proposed the factor of 1.2 based on experimental observa-

tion and calculations.

4. Detailed finite-element investigation

From the literature review, it is seen that there are simple de-

sign methods based on equilibrium and elastic behavior and pro-

ven 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 ri-

gid connections. Such a structural analysis is incapable of predict-

ing 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 con-

structed 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 un-

der-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 hourglasscontrol 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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their lengths in the bridge (the so-called stub members), and mod-

eled with solid and shell elements (Fig. 6). In their largest dimen-

sion, 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. Tran-

sition between shell and solid elements was performed by surface-

based coupling constraints. All members in the connection model

other than the main truss members and gusset plate had maxi-

mum element dimension of about 15 mm. The sophistication of

these models, compared with Williamss[16]for example, reflects

the extraordinary advance of computer and finite-element tech-

nologies 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 ad-

vanced skills and powerful computers and was clearly beyond rou-

tine 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 de-

tailed 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 mod-el of the West main truss (model 5), using overall strategies for

integration of substructures proposed in[24].Table 1lists 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 corre-

sponding truss members. To develop the simplified linear connec-

tion 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 25 in the detailed model

were fixed and an arbitrary concentrated force or moment was ap-

plied to obtain the displacements and rotations at the end of ele-

ment 1. In general, the concentrated force or moment was

applied at a node at the end of a stub member, as close to the cen-

ter of the cross section as possible. For tubular members, two con-

centrated forces or moments were applied at the middle of the

long sides of the end cross section, and displacements and rota-

tions were calculated at these same points. The process was per-

formed using ABAQUS and repeated by applying arbitrary forces

and moments corresponding to all 6 DoFs, and thus, after normal-

ization, the 6 6 flexibility matrix for element 1 in ABAQUS coor-

dinates 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 connec-

tion. 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 longitu-

dinal 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 er-

rors in the off-diagonal terms, which were several orders of magni-

tude 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 workinvestigates the validity of the rigid connection assumption. For

this purpose, ABAQUS model 3 was developed by changing the

Fig. 6. Detailed FE model 1 of gusset plate (ABAQUS).

Table 1

Computer models run.

Model Software Analysis Purpose

1. Detailed connection ABAQUS Linear Stiffness of 5 members

2. Spring connection STRAND Linear Stiffness of 5 springs

3. Detailed connection, various gusset thickness ABAQUS Linear Stiffness of 5 members

4. Rigid connection STRAND Linear Validity of rigid connection assumption

5. West main truss (2D) STRAND Linear Behavior with various connection models

6. Detailed connection ABAQUS Nonlinear Stiffness of 5 members

7. Spring connection STRAND Nonlinear Stiffness of 5 springs

8. I-35 W (3D) STRAND Nonlinear Behavior with various connection models

9. Howe truss (2D) STRAND Nonlinear Behavior with various connection models

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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 thick-

nesses. 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 compli-

cated 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 ele-

ments 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 theU10 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 di-

vided 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 pointshad 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. Re-

sults 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

In order to investigate the failure of the bridge, it was necessary

to progress beyond the linear elastic range. To capture the nonlin-

ear behaviour of the gusset connection up to failure, the detailed

Fig. 7. Equivalent spring model 2 (STRAND).

Fig. 8. Flowchart to calculate equivalent spring stiffness.

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) 1.84E1 8.19E2 1.88E5 6.72E6 7.72E5 2.05E2

Fy= 444.8 (kN) 8.19E2 3.50E0 3.42E4 4.51E5 7.66E5 1.96E0

Fz= 444.8 (kN) 1.88E5 3.39E4 2.10E+1 1.51E0 5.08E0 1.37E4

Mx= 228.8 (kN m) 4.19E6 1.22E5 9.30E1 5.15E0 8.98E2 8.80E6

My= 113.0 (kN m) 7.83E7 1.94E5 1.29E0 3.71E2 7.04E1 8.01E6

Mz= 161.0 (kN m) 7.43E3 7.09E1 5.10E5 4.41E6 1.11E5 6.16E1



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ABAQUS FE model 1 of the five-element connection was loaded asdescribed in Section6, but into the nonlinear range. Both geomet-

ric 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 strainlarge displacement formulation (ABAQUS de-

fault option) was used to carry out the nonlinear analysis.

The results were forcedisplacement and momentrotation

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 ele-

ments of the connection was diagonal only and derived from an

initially perfect gusset plate. The diagonal terms were in the form

of loaddisplacement or momentrotation curves (model 7). No-tice that this nonlinear connection model included all six DOF.

Table 3

Element 1, node 1, displacements and rotations obtained in STRAND7/STRAUS7 [25]with user-defined elements for applied forces and moments (simplified model 2).

Node 1 dx (mm) dy (mm) dz (mm) qx (mrad) qy (mrad) qz (mrad)

Fx= 444.8 (kN) 1.85E1 8.27E2 2.51E4 2.61E5 1.43E1 2.10E2

Fy= 444.8 (kN) 8.27E2 3.50E0 3.29E4 4.60E5 7.31E5 1.95E0

Fz= 444.8 (kN) 2.51E4 3.29E4 2.10E + 1 1.51E0 5.09E0 1.31E4

Mx= 228.8 (kN m) 1.60E5 2.83E5 9.30E1 5.15E0 9.00E2 7.92E6

My= 113.0 (kN m) 3.63E5 1.86E5 1.29E0 3.71E2 7.06E1 7.14E6

Mz= 161.0 (kN m) 7.60E3 7.07E1 4.75E5 4.66E6 1.02E5 6.15E1

Table 4

Comparison of displacementsd and rotationsq 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 STRAND mod. 2 semi-rigid joints ABAQUS FE model 3 with gusset thickness:

12.7 mm 15.87 mm 19.05 mm 25.4 mm

dx (mm) 0.184 [1.00] 0.185 [1.00] 0.184 [1] 0.1748 [0.95] 0.1677 [0.91] 0.1575 [0.86]

dy (mm) 7.42 [2.12] 3.50 [1.00] 3.50 [1] 3.276 [0.94] 3.105 [0.89] 2.860 [0.82]

dz(mm) 12.8 [0.61] 21.0 [1.00] 21.0 [1] 18.54 [0.88] 16.90 [0.80] 14.83 [0.71]

qx (mrad) 3.12 [0.61] 5.15 [1.00] 5.15 [1] 4.688 [0.91] 4.393 [0.85] 4.021 [0.78]

qy (mrad) 0.759 [1.08] 0.706 [1.00] 0.704 [1] 0.6618 [0.94] 0.6302 [0.90] 0.5862 [0.83]

qz (mrad) 0.802 [1.30] 0.615 [1.00] 0.616 [1] 0.5998 [0.97] 0.5872 [0.95] 0.5682 [0.92]

Fig. 9. 2D model of I-35 W Bridge showingloads and support conditions. Circles and squares indicate support conditions (restrained displacementsd and rotationsq) for both

main trusses.

Fig. 10. Stressstrain curve[1].

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Richards 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 loaddisplacement and momentrotation

curves of element 3 of the connection, where x , y and zrefers to

the global coordinate system. For clarity, only the positive values

are shown. Similar curves model the behavior of the other ele-

ments 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 con-

struction 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 de-

scribed inFig. 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

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. 11. Forcedisplacement and moment rotation curves for element 3 of

connection (KFXX, KFYY, KFZZand KMXX, KMYY, KMZZ are the elastic stiffness for force

displacement and momentrotation forx,y andzaxis respectively).

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 ele-

ments 2 and 3, Fig. 6. After dead weight and traffic load application,

the maximum out-of-plane deformation was 17.3 mm, in agree-

ment 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

model. Various attempts to circumvent this difficulty were unsuc-

cessful and the literature review confirmed that previous simpli-

fied 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 impor-

tance 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-ele-

ment connections and the members they attach were the same

as for U10-W analyzed above (elasto-plastic, Fig. 10, dimensions

shown inFig. 17). Besides its dead load, the bridge was subjected

to traffic load uniformly distributed on the top chord, and con-

struction 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. 14. Loaddeflection curve (deflection at midspan as indicated inFig. 9).

Fig. 15. Axial forces in the five elements of semi-rigid connection U10-W.

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 0.28 Compression

2 0.56 Tension

3 1.00 Compression

4 0.02 Tension

5 0.41 Tension

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Four models with large displacement formulations, elasto-plas-

tic 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 geo-metric nonlinearity) were run by applying an increasing load factor

to the concentrated loads only, with the uniform loads held con-

stant.Fig. 18presents the load factor versus midspan vertical dis-

placement 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. 19pre-

sents the ratio of bending moment to flexural capacity in the mem-

bers at the ultimate load factor. Since the joints were rigid, collapseoccurred because of flexural yielding of the members.

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).



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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 fail-

ureof the truss bridge couldbe attributed to the achievement of ax-

ial 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

the critical joint under the concentrated loads, and the other joints

remained elastic as the critical joint failed by flexural yielding. Sec-

tion6 showed that the assumption of rigid behavior was a good

approximation for elastic joints, and the difference between mod-

els 9B and 9C was in the treatment of joints that turned out to re-

main 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. 22presents the ratio of bending moment to flexural capac-

ity in the members at the ultimate load factor of 1.46. For this mod-

el, 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, thepredicted 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. 18. Load factor versus midspan vertical displacement.

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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members of a truss, because the American Association of StateHighway and Transportation Officials (AASHTO) provides no spe-

cific guidance for the inspection of gusset plates, and because com-

puter programs for load rating analysis do not include gusset

plates, bridge owners typically ignore gusset plates when perform-

ing 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 nodesneeded to be analyzed in detail, or all nodes were similar. Future

work, which will include comparison with experiments, for exam-

ple 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. 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 0.71 Compression

2 1.00 Compression

3 0.24 Compression

4 0.11 Compression

5 0.38 Compression

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 All joints rigid 3.26 Flexural yielding of members

9B Semi-rigid joint at point loads 1.98 Connection element 2 reached axial capacity

9C All 5-member joints semi-rigid 1.98 Connection element 2 reached axial capacity

9D All 5-member joints pinned 1.46 Plastic hinge in member 1 of joint



13/13

Accurate prediction of bridge failure should include proper

modeling of connections. This can be done economically and accu-

rately 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 re-

quires the identification of certain commercial software. The inclu-

sion 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 Turner-

Fairbank 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.

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A Nonlinear Model for Gusset Plate Connections Engineering Structures

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