MODELING AND ANALYSIS OF STEEL GUSSET PLATES IN TRUSS

BRIDGES UNDER LIVE LOAD

by

MEGHAN M. MYERS

A thesis submitted to the

Graduate School - New Brunswick

Rutgers, The State University of New Jersey

in partial fulfillment of the requirements

for the degree of

Masters of Science

Graduate Program in Civil and Environmental Engineering

written under the direction of

Dr. Hani Nassif

and approved by

_________________________________________

_________________________________________

_________________________________________

New Brunswick, New Jersey

October, 2011

ii

ABSTRACT OF THE THESIS

MODELING AND ANALYSIS OF STEEL GUSSET PLATES IN TRUSS

BRIDGES UNDER LIVE LOAD

By MEGHAN M. MYERS

Thesis Director:

Dr. Hani Nassif

In the aftermath of the collapse of the I-35W over Mississippi River Bridge in Minnesota,

the Federal Highway Administration (FHWA) issued a technical advisory to bridge

owners to check the status of similarly-designed bridges. It was determined that under-

designed gusset plates contributed to the collapse. This sparked a nationwide effort to

investigate the design of these connection members and to develop more detailed

specifications for future gusset plate design. In order to thoroughly study complicated

bridge elements such as gusset plates, sophisticated analysis techniques are required.

One such technique is finite element modeling (FEM), which is used here to identify

critical loading cases for typical Warren truss gusset plates.

The specific gusset plates studied here are located on two bridges, herein referred to as

Bridge A and Bridge B, that are similar in design to the I-35W Bridge. Following the I-

35W collapse, independent investigations, which included finite element analysis, were

initiated on both bridges. In this thesis, information from these investigations is used to

develop a comprehensive FEM, which facilitates more in-depth analysis of such gusset

iii

plates. The analysis focuses on the investigation of stresses created in the gusset plates

by various types of live loading. The results are compared to the Method of Sections

approach recommended by FHWA following the I-35W Bridge collapse to determine if

better analysis specifications are needed. Although the results of the finite element

analysis and the Method of Sections approach are similar, the authors conclude that the

value of the Method of Sections approach is strongly dependent on the accuracy of the

load data input. Therefore, more detailed specifications are needed to ensure the

accuracy of future gusset plate analysis and design.

iv

ACKNOWLEDGMENTS

I would first like to thank my thesis advisor, Dr. Hani Nassif for giving me the

opportunity to conduct this research under him and for his guidance and support during

this time. I would also like to thank my committee members, Dr. Kaan Ozbay and Dr.

Perumalsamy Balaguru for their useful comments and input.

I would like to acknowledge Arora and Associates, P.C. for affording me the

opportunity to work on various fatigue-sensitive, fracture-critical steel bridges, which

introduced me to the field of gusset plate modeling and analysis. Special thanks are also

given to Dr. Nakin Suksawang and Mr. Dan Su for introducing me to other truss bridge

evaluations and assisting me in some of my finite element model development.

Much of this thesis was supported by previous publications that I would like to

acknowledge. My publications submitted to Safety and Reliability of Bridge Structures

for the New York City Bridge Conference (Myers 2009a), the NSBA World Steel Bridge

Symposium (Myers 2009b), and the NDE/NDT for Highways and Bridges: Structural

Materials Technology Conference were largely influential to the development of this

thesis.

Lastly, I would like to thank my family and friends, and especially my husband

Clayton, for their continuous support throughout my time completing the Rutgers

University Masters Program. Without their understanding and encouragement during this

busy time, my success in this endeavor would not have been possible.

v

TABLE OF CONTENTS

ABSTRACT OF THE THESIS .......................................................................................... ii

ACKNOWLEDGMENTS ................................................................................................. iv

TABLE OF CONTENTS .................................................................................................... v

LIST OF FIGURES .......................................................................................................... vii

LIST OF TABLES ............................................................................................................. xi

CHAPTER 1. INTRODUCTION ...................................................................................... 1

1.1 Motivation ................................................................................................................. 1

1.2 Justification ............................................................................................................... 3

CHAPTER 2. LITERATURE REVIEW ........................................................................... 5

2.1 Early Research .......................................................................................................... 5

2.2 Current Research ..................................................................................................... 10

CHAPTER 3. INITIAL ANALYSIS ............................................................................... 13

3.1 Bridge A Project Introduction ................................................................................. 13

3.2 Method of Sections Analysis .................................................................................. 14

3.3 Bridge A Finite Element Model .............................................................................. 22

3.4 Bridge A Instrumentation ....................................................................................... 28

3.5 Bridge A Conclusions ............................................................................................. 36

CHAPTER 4. MODEL DEVELOPMENT...................................................................... 38

4.1 Creating a Model in Abaqus ................................................................................... 38

vi

4.2 Bridge B Research and Model Development.......................................................... 42

4.3 Model Integration.................................................................................................... 57

CHAPTER 5. PARAMETRIC STUDY .......................................................................... 65

5.1 Varying Plate Thickness ......................................................................................... 65

5.2 Varying Live Load .................................................................................................. 67

5.3 Validating Integrated Plate Model .......................................................................... 71

5.4 Comparing In-Depth FEM to Method of Sections ................................................. 73

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS ................................... 76

REFERENCES ................................................................................................................. 80

vii

LIST OF FIGURES

Figure 1: Method of Sections notations and section locations .......................................... 2

Figure 2: Whitmore Section (Whitmore 1952) .................................................................. 6

Figure 3: Block Shear sections (Higgins et al. 2010) ...................................................... 11

Figure 4: Bridge A truss geometry for BAR7 model ....................................................... 14

Figure 5: Gusset plate shop drawings for the gusset plates in each category experiencing

the highest loads in the BAR7 analysis and used in the hand calculations .............. 16

Figure 6: Typical lower, odd numbered gusset plate type on Bridge A .......................... 20

Figure 7: Geometry of Gusset Plate L16 for STAAD model .......................................... 22

Figure 8: The STAAD finite element model for Gusset Plate L16 depicting nodes and

triangular plate elements .......................................................................................... 23

Figure 9: Stress contours on Gusset Plate L16 from STAAD finite element model ....... 26

Figure 10: Typical section loss in Bridge A gusset plate ................................................. 27

Figure 11: Sensor locations on Bridge A Gusset Plate L16 ............................................. 29

Figure 12: Bridge responses recorded by sensors on Bridge A Gusset Plate L16 ........... 30

Figure 13: Sensor readings before live loading event for four of the five truss members 31

Figure 14: Sensor readings before live loading event for fifth truss member ................. 31

Figure 15: Sensor readings at peak strain of vertical truss member ................................ 32

Figure 16: Sensor readings at peak strain of south diagonal truss member ..................... 33

Figure 17: Sensor readings at peak strain of south chord truss member ......................... 34

Figure 18: Sensor readings at peak strain of north diagonal truss member ..................... 34

Figure 19: Sensor readings at peak strain of north chord truss member .......................... 35

viii

Figure 20: Example gusset plate for detailed Abaqus finite element model ................... 39

Figure 21: Finite element model of Truss Spans 25, 26, and 27 in Bridge B (Nassif et al.

2007) ........................................................................................................................ 43

Figure 22: Integration point of (a) two-node, linear beam (B31) and (b) three-node,

quadratic beam (B32) elements along the length of the beam (Abaqus 2010) ........ 44

Figure 23: Four-node (S4) shell element (Abaqus 2010) ................................................ 45

Figure 24: Typical stress-strain curve of structural steel (Salmon and Johnson 1996) ... 47

Figure 25: Comparison of stresses in S5-S10 using the FE model and Static Load Test 1

(Nassif et al. 2007) ................................................................................................... 50

Figure 26: Comparison of stresses in S5-S10 using the FE model and Static Load Test 3

(Nassif et al. 2007) ................................................................................................... 51

Figure 27: Standard truck configurations used in the calibration of the finite element

model (Nassif et al. 2007) ........................................................................................ 52

Figure 28: Comparison of stresses in S5 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

................................................................................................................................... 53

Figure 29: Comparison of stresses in S6 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

................................................................................................................................... 54

Figure 30: Comparison of stresses in S7 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

................................................................................................................................... 54

ix

Figure 31: Comparison of stresses in S8 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

................................................................................................................................... 55

Figure 32: Comparison of stresses in S9 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

................................................................................................................................... 55

Figure 33: Comparison of stresses in S10 using the FE model and the actual field-test

data of a 5-axle truck with a GVW of 65 kips traveling WB on Lane 1 (Nassif et al.

2007) ........................................................................................................................ 56

Figure 34: Bridge B full truss model with gusset plate integration (red circles) in Abaqus

................................................................................................................................... 59

Figure 35: Test-truck configuration (top) and information (bottom) for controlled load

tests (Nassif et al. 2007) ........................................................................................... 60

Figure 36: Under-deck view of Span 26 Bay 1 between FB11 and FB12 (Nassif et al.

2007) Note: rectangles represent strain transducers ................................................ 61

Figure 37: Span 26 sensor layout involving 16 strain gauges and 4 LVDTs (Nassif et al.

2007) ........................................................................................................................ 61

Figure 38: Comparison of Abaqus FEM results at Sensor 6 location with and without

gusset plate integration ............................................................................................ 62

Figure 39: Comparison of FEM results at adjacent truss members with and without gusset

plate integration ....................................................................................................... 63

Figure 40: Chart graphing plate stress vs. plate thickness ............................................... 66

Figure 41: Results of varying live load on a gusset plate on Bridge B ............................ 69

x

Figure 42: Gusset plate in full bridge model stress contours from Abaqus ..................... 71

Figure 43: Individual 3D gusset plate model stress contours from Abaqus .................... 72

xi

LIST OF TABLES

Table 1: Gusset plate categories and worst-case plates for Bridge A .............................. 15

Table 2: Summary of D/C ratios calculated under HS20 live loading for all analyzed

gusset plates ............................................................................................................. 20

Table 3: Summary of D/C ratios calculated under Permit live loading for all analyzed

gusset plates ............................................................................................................. 21

Table 4: Comparing stresses for Horizontal Section A-A of gusset plate L16 ................ 25

Table 5: Comparing stresses for Vertical Section B-B of gusset plate L16 .................... 25

Table 6: Comparison between hand calculation results using the sensor data and the

BAR7 data ................................................................................................................ 36

Table 7: Truck configuration (Nassif et al. 2007) ............................................................ 53

Table 8: Configuration for various trucks used for live loading in model (italics indicate

estimations) .............................................................................................................. 68

Table 9: Stress comparisons between plate models ......................................................... 73

Table 10: Plate stresses using Method of Sections .......................................................... 74

1

CHAPTER 1. INTRODUCTION

1.1 Motivation

At 6:05 P.M. EST on Wednesday, August 1, 2007, the bridge over the Mississippi River

between University Avenue and Washington Avenue on highway I-35W in Minneapolis,

MN, collapsed. Numerous vehicles were on the bridge at the time, and, as is well known,

there was a tragic loss of life and a vital transportation link was severed. At the time, in

light of the uncertainty surrounding the cause of the collapse, the Federal Highway

Administration (FHWA) advised all State Transportation Agencies and other bridge

owners to immediately re-inspect all steel deck truss bridges with fracture critical

members or at a minimum to review inspection reports, including those for routine, in-

depth, fracture critical, and underwater, to determine whether more detailed inspections

were warranted. It was later discovered that under designed gusset plates were a

contributing cause of the collapse. After this discovery, interest in analyzing existing

gusset plates on other similar bridges was generated. This sparked a nation-wide interest

in looking deeper into the design of these connection members and the possibility of

developing new procedures for future use in such design.

In 2008, an investigation was initiated on a bridge henceforth referred to as Bridge A due

to its similarity to the I-35W Bridge. The gusset plates on the bridge were to be analyzed

to determine whether they were adequate to carry the current loads that were being seen

on the bridge. This investigation initiated the development of a finite element model to

2

predict the stresses in typical gusset plates on such a bridge. This bridge consists of a

continuous haunched Warren deck truss that makes up the three main spans of the bridge.

To begin this investigation, documents regarding the bridge geometry and history were

collected and reviewed. Using information found in these documents, the bridge was

analyzed and loads were generated that represent the current state of stress on the truss.

The gusset plates were then investigated using the analysis techniques presented in the

FHWA Turner-Fairbank Highway Research Center report on the I-35W Bridge dated

January 11, 2008. This analysis consisted of a method of sections calculation, where

equilibrating loads are calculated for a horizontal and vertical section through a gusset

plate to balance the applied loads from the connecting truss members. These sections can

be seen in Figure 1 below.

Figure 1: Method of Sections notations and section locations

After hand calculations were performed to analyze the stresses in the above defined

sections, a finite element model was created using STAAD PRO 2005 software to

calculate similar stresses. This was done to try to verify the results of the hand

calculations. As an extension of the Bridge A analysis, the authors of this paper used

3

some of the information found from the investigation of Bridge A as a starting point for a

more in-depth analysis. This more in-depth analysis focused on the integration of an

individual gusset plate finite element model (FEM) into a 3D full truss bridge FEM.

Once the integrated model was complete, variations of live loading that may be

experienced by a bridge of this type were applied to the model to determine the stress on

gusset plates under such loading. Comparisons were also made between the integrated

FEM and the Method of Sections calculations suggested by FHWA.

1.2 Justification

Gusset plate connections are very complicated bridge elements. For this reason, many

researchers have looked into using finite element modeling in order to really determine

how these connections behave. However, since these connections need to be investigated

and analyzed often by bridge owners, especially in the last few years, more simplified

methods are needed. Development of good, detailed finite element models is too time

consuming to be used frequently by these agencies. Though, for research purposes, FEM

is a useful tool in learning more about the behavior of these members. As such, the

research presented in this paper uses finite element models to find out critical loading

cases for typical Warren truss gusset plates.

As a focus, this research investigates various types of live loading and how the truss

member reactions from these trucks affect the stresses in the gusset plate. For instance,

the permit trucks used in typical analyses and ratings of bridges are normally heavy

trucks that are also longer than the typical HS20 live load case specified by the American

4

Association of State Highway and Transportation Officials (AASHTO). These permit

trucks are meant to represent the largest legal trucks allowed on roads. However, there

could be heavier trucks that make it onto roads unnoticed. Also, in some cases, trucks

that are slightly less heavy, or maybe just as heavy as permit trucks, but are much shorter

in length, such as full dump trucks, may end up creating a more critical reaction in trusses

since the heavy load would be carried by fewer truss members. These are the types of

concepts investigated in this study.

5

CHAPTER 2. LITERATURE REVIEW

2.1 Early Research

Some of the previous research done specifically to evaluate locations and magnitudes of

stress in gusset plates, and to derive a simple way to determine maximum stresses for

designing these structural members was performed by Whitmore (Whitmore 1952). In

his investigation, Whitmore mainly studied the joints in Warren type truss configurations,

one of the most common truss types. For the gusset plates tested, he determined that

maximum tension and compression forces were located around the ends of the diagonal

members and maximum shearing stresses were located near the chord member and

toward the center of the plate, with much lower stresses toward the edge of the plate. The

results of this analysis contradicted what had been the regularly assumed distribution of

stresses in such gusset plates. Whitmore then found that the maximum stresses at the

ends of the diagonal members could be approximated by dividing the force from the truss

member by an area equal to the width of the plate multiplied by a length measured

perpendicular to the truss member axis along the bottom row of bolts and between two

lines measured 30 degrees from the outside columns of bolts along the truss member axis.

This section, known now as the ―Whitmore Section,‖ can be seen in Figure 2 below.

6

Figure 2: Whitmore Section (Whitmore 1952)

The Whitmore Section analysis became the more widely used procedure to design and

check gusset plates until the development of the block shear analysis method.

In the 1980’s there were a number of analyses performed in order to further understand

the specific stress devices working in typical gusset plate configurations. As mentioned

above, Warren, along with Pratt, trusses are the most common truss shapes. Yamamoto,

Akiyama, and Okumura (Yamamoto et al. 1985) performed experiments and theoretical

analyses on elastic stress distributions in gusset plates of these common truss

configurations. The paper stated that at that time ―the design of gusseted joints is based

on rather simple methods of analyses and current design specifications hardly establish

definite rules about gusset plate thickness.‖ This is attributed to the general lack of

experimental research available on the design of gusset plates. This idea that current

design standards do not adequately deal with design of gusset plates is still somewhat

relevant today, as will be discussed later in this section. Yamamoto et al. proposes

formulae to calculate the design thickness of gusset plates. These formulae establish

7

rules for calculating required gusset plate thickness to transmit axial forces and bending

moments and transmit shear forces.

Another consideration within the study of gusset plates is the evaluation of fatigue and

fracture. A study performed under the National Cooperative Highway Research Program

(NCHRP) Project 12-25 (Fisher et al. 1987) researched these mechanisms in riveted

connections on bridges. It was confirmed through this study, which examined data from

a Department of Transportation (DOT)-sponsored research study, as well as other studies

and full-scale tests, that the type of riveted connection does not affect resistance to

fatigue stresses in any significant way. It also confirmed that primary members in these

riveted bridges are not expected to develop fatigue cracks, though the study did not cover

gusset plates specifically. Fatigue stresses were also studied by Kitzawa, Kanaji,

Ohminami, and Furukawa (Kitazawa et al. 1994), though with slightly different results.

Kitazawa et al. found that the fatigue strength of the gusset plates on the cable-stayed

Higashi-Kobe Bridge were not sufficient to resist the largest stresses from live loading.

These differing results may very well be attributed to the design standards in Japan versus

the United States and the difference between stresses in the types of bridges (cable-stayed

versus riveted connections in other kinds of bridges).

An aspect of analyzing bridges that has not been considered in many truss analyses is the

specific differences that may result from various types of live loads passing over the

bridge, specifically, heavier trucks than the standard permit trucks used for rating bridges.

One such research project that was conducted on this topic was by Laman, Pechar, and

8

Boothby (Laman, et al. 1999). This study looked at the affect of bridge component type,

component peak static stress, live load type, and live load speed on the dynamic stresses

in steel through-truss bridges. The results do not seem to conclude anything specific

about the live load type, and the gusset plates on the bridge were not one of the

components analyzed. This leaves some gaps in the topic of truss live load analysis for

consideration in future research.

In 2006, a Rutgers University team performed field tests, analysis, simulations, and

laboratory tests in order to identify if cracking in the deck of a bridge (herein referred to

as Bridge B) is caused by shrinkage or by live load vibrations (Nassif 2007). To

accurately measure the loads that were crossing the bridge, multiple sensors were

installed on the bridge, including a portable Weigh-in-Motion (WIM) system and two

piezo-axle sensors connected to a main data collection unit. Using these sensors to

measure live load on the bridge, as well as other sensors to measure bridge response, it

was possible to create accurate, calibrated finite element models to determine the overall

impact of the live load on the bridge, though the gusset plates in particular were not

studied at this time.

In 2006 Huns et al. conducted research with the objective of developing a finite element

model that could predict the tension and shear block failure of gusset plates and

conducting a reliability analysis of existing test results to evaluate current design

equations and propose new limit state design equations. The current practices in North

America, Europe, and Japan for tension and shear block design (Kulak and Grondin

9

2000, 2001) were reviewed and it was discovered that the equations give a satisfactory

prediction of capacity of gusset plates, but do not predict the failure mode as well. Some

other studies were conducted in search of similar results. Chakrabarti and Bjorhovde

(1983) and Hardash and Bjorhovde (1984) studied inelastic behavior of gusset plates in

tension, Hu and Cheng (1987) and Yam and Cheng (1993) studied gusset plates in

compression, and Rabinovitch and Cheng (1993), Walbridge et al. (1998), and Nast et al.

(1999) studied gusset plates under cyclic loading. The conclusions of the Huns et al.

research state that the existing literature as well as the finite element analysis performed

in this study indicate that tension fracture always occurs before shear rupture. It also

indicates that most of the connections showed the full capacity of the gusset plate being

reached before the rupture occurs. It was determined through the reliability analysis that

equations posed by Hardash and Bjorhovde (1984) and Driver et al. (2004) provide a

good prediction of the test results of the gusset plates, where the equations in the design

standards were overly conservative and in some cases did not predict the failure mode

accurately.

Also in 2006, Li, Zhou, Chan, and Yu (Li et al. 2006), published a paper on their research

in multi-scale numerical analysis on long-span bridges. This research focused on local

damage and dynamic responses in such bridges and used the stiffening truss of a

suspension bridge in China as a case study. Li et al. believe that the most effective way

to analyze such a complicated structure is through models that account for damage and

deterioration and the behavior of the connections between main structural elements. This

study concluded that such multi-scale modeling was necessary for the evaluation of long-

10

span bridges and the effects of damage on them. A discrete evaluation of the structural

elements in such a bridge cannot adequately evaluate more complex responses such as

fatigue in the trusses of these bridges, although this simpler form of analysis can be

useful in that it can be easily applied for quicker analyses.

2.2 Current Research

A variety of topics in the realm of truss or gusset plate analysis were discussed in the

previous section. These connections have always been a somewhat lesser understood

element, in that their responses and reactions are complex and are many times designed

to well exceed necessary capacity in order to assure that they are not the first thing to fail

on the bridge. However, in 2007, the collapse of the I-35W Bridge over the Mississippi

River in Minneapolis, Minnesota brought the topic of truss analysis to the forefront of

structural engineering research. Many studies were performed and papers were written

on the analysis of the collapse of the I-35W Bridge (e.g. Holt & Hartmann 2008, Minmao

et al. 2009, Hao 2010, Liao et al. 2011). Through the forensic structural analyses

performed, it was determined that underdesigned gusset plates were a contributing cause

of the collapse of the bridge. The inadequate gusset plates were yielded at the time of the

collapse, and increased weight on the bridge due to construction on the deck (materials,

equipment, etc.) caused the failure of the plate and the collapse of the bridge.

In July of 2009, the FHWA issued a publication entitled Load Rating Guidance and

Examples for Bolted and Riveted Gusset Plates in Truss Bridges. This publication was

produced in order to give guidance to bridge owners on how to analyze and load rate

11

gusset plates. This publication and its guidelines were based on the existing practices and

knowledge in the field. However, these guidelines could be updated with information

received from a current study that is underway. This study is being conducted by FHWA

and is being sponsored jointly by FHWA and AASHTO through the NCHRP.

In 2010, Higgins et al. performed a study to compare the methods of block-shear

(depicted in Figure 3 below) and Whitmore Section stress evaluation on truss gusset

plates. As noted earlier, these kinds of studies became more prevalent after the collapse

of the I-35W Bridge. Block shear and Whitmore section analyses are two of the most

prevalent methods of gusset plate analysis, which is why these were the two compared in

this study. The specific failure mode in a gusset plate will depend on the geometry,

distribution of loads, and material properties for that specific gusset plate. This is why

there are many failure planes in the block shear method. Higgins et al. found that using

the block-shear method on plates that were originally designed with the Whitmore section

methods and allowable stress design will produce rating factors below 1.0. They also

determined that gusset plates made from higher yield strength steel and those with more

bolts produce lower rating factors with the block-shear method.

Figure 3: Block Shear sections (Higgins et al. 2010)

12

From the current AASHTO bridge specifications, Article 6.14.2.8 states,

―Gusset or connection plates should be used for connecting main members, except

where the members are pin-connected. The fasteners connecting each member shall

be symmetrical with the axis of the member, so far as practicable, and the full

development of the elements of the member should be given consideration.‖

The comments to this article state that,

―Following the 2007 collapse of the I-35W Bridge in Minneapolis, the traditional

procedures for designing gusset plates, including the provisions of this Article, have

been under extensive review. As of Spring 2008, new design procedures have not

been codified. Guidance from FHWA is expected shortly. Designers are advised to

obtain the latest approved recommendations from Owners.‖

This comment coincides with the statement made by FHWA on their website that was

mentioned in the above. When the joint study between FHWA and AASHTO has been

completed, it is expected that new guidelines will be given in AASHTO as to how to go

about designing future truss bridge gusset plates. This same article also describes the

design equations as follows,

―The maximum stress from combined factored flexural and axial loads shall not

exceed φfFy based on the gross area. The maximum shear stress on a section due to

the factored loads shall be φvFy/√3 for uniform shear and φv 0.74 Fy/√3 for flexural

shear computed as the factored shear force divided by the shear area. If the length of

the unsupported edge of a gusset plate exceeds 2.06(E/Fy)1/2

times its thickness, the

edge shall be stiffened. Stiffened and unstiffened gusset edges shall be investigated

as idealized column sections.‖

Besides the reference that is given to the more general Sections 6.13.4 and 6.13.5 in

AASHTO (Block Shear Rupture Resistance and Connection Elements), these are the only

guidelines for specific design of gusset plates. This shows the relative uncertain nature of

gusset plate design to this point.

Obviously the topic of gusset plate analysis is a broad and complex one. This paper

focuses on the topic of varying live load cases and their impact on truss gusset plates.

13

CHAPTER 3. INITIAL ANALYSIS

3.1 Bridge A Project Introduction

As mentioned before, many authorities began investigations of their bridges with similar

details to the I-35W Bridge after its collapse. In particular, Bridge A was the subject of

such gusset plate analyses. Bridge A is a five-span bridge consisting of a continuous

haunched Warren deck truss, which makes up the three main spans, and two deck girder

side spans. The overall bridge span is approximately 740 ft, and its width is 61 ft 6 in.

The longest of the five spans is the center span, which is 280 ft long.

To initiate this analysis, a review of the bridge inspection reports, design calculations,

photograph logs, and plans was performed, and a detailed history of the bridge was

compiled considering its structural configuration, modifications made through

rehabilitation and maintenance contracts, and the condition of the primary members.

This was helpful in determining what changes had been made to the structure since the

original design and construction and the current state of stress in each of the primary

members and, hence, the gusset plates. Using the information found in these documents,

the most recent computer analysis run performed on this structure was reviewed and

updated. The rating program BAR7 was used to reanalyze the bridge and generate new

loads that represent the current state of stress on the truss. Based on the results and the

gusset plate geometry detailed in the original shop drawings, the gusset plates were

grouped into categories for further analysis. The gusset plates were then investigated

14

using the analysis techniques presented in the FHWA Turner-Fairbank Highway

Research Center report on the I-35W Bridge dated January 11, 2008 (FHWA 2008).

3.2 Method of Sections Analysis

The model that was used to analyze the three-span, continuous deck truss unit is shown in

Figure 4.

Figure 4: Bridge A truss geometry for BAR7 model

With revised input and an updated model, BAR7 was run and new member forces were

generated. The live load cases considered were HS20 and Permit loadings.

For the Permit live load run, it was assumed that there were three lanes loaded with

100%, 55%, and 55% of the truck load respectively. The two 55% lanes approximate the

force of an HS20 live load, while the 100% represents the one lane of Permit truck

loading.

There are 58 gusset plates in each of the deck trusses. Rather than analyze every gusset

plate, a representative sample of gusset plates with worst-case scenario loads and

geometry was selected for analysis. The first step was to split the plates into categories

15

based on their thickness and their location on the truss. This resulted in 9 different

categories. Within each of the categories, the worst-case location was determined using

the forces calculated in the new BAR7 run. Table 1 lists the various categories by which

each gusset plate could be identified and the gusset plate(s) chosen for analysis for each

category.

Gusset Plate Worst-Case Gusset Plates

Categories High Loads Other

5/8‖ Upper Chord U7

11/16‖ Upper Chord U9

3/4‖ Upper Chord U13

1/2‖ Upper Chord U14 U8

3/4‖ Lower Chord L0 L28

11/16‖ Lower Chord L14

1/2‖ Lower Chord L15 L7

5/8‖ Lower Chord L16

Multiple Gussets L8

Table 1: Gusset plate categories and worst-case plates for Bridge A

One of these categories (―Multiple Gussets‖) consisted solely of the panel points at the

pier supports. These panel points contain 4 gusset plates stacked together and represent a

unique and important situation. In some categories, two gusset plates were analyzed due

to the large differences in force direction (tension vs. compression) and geometry of the

connecting truss members (right angles vs. more extreme angles). In the end, the 58

gusset plates in each truss were narrowed down to only 12 that were analyzed. The shop

drawing for the high-load gusset plates in each category is shown in Figure 5 below.

16

Figure 5: Gusset plate shop drawings for the gusset plates in each category experiencing

the highest loads in the BAR7 analysis and used in the hand calculations

17

Calculations were performed to analyze the representative gusset plates to determine the

forces and stresses acting on them. The calculations and methods used were based on the

reconstructed design calculations for the I-35W over Mississippi River Bridge illustrated

in the FHWA report (FHWA 2008). The procedure that was used makes two cuts in the

plate, one horizontal, just above or below the horizontal member (section A-A) and one

vertical, just next to the vertical member (section B-B), as shown in Figure 1 in the

introduction section of this paper.

Based on the truss member forces found from BAR7, equilibrating axial, shear, and

moment forces are calculated to balance the truss member forces from one side of the cut

plane. Then the axial (fa = P/A), flexural (fb = M/S), and shear (fv-avg = V/A) stresses

along the section and the principal tension and compression stresses are calculated, taking

into account any splice plates that contribute to the gusset plate section properties. For

this analysis, the principal stresses were taken at either the section neutral axis or at the

edge of the plate depending on which was greater. The neutral axis principal stresses

were calculated by

22

/2

3

22

avgv

aacompten f

fff (1)

and the principal stresses at the edge of the gusset plate were calculated by

bacompten fff / (2)

The plus or minus was applied in each of these equations to create the biggest tension and

compression stresses for each case. For the final step in this procedure, a

18

Demand/Capacity (D/C) ratio is produced using AASHTO allowable stresses as the

capacities.

The review of the available bridge plans and documents did not reveal which edition of

AASHTO was used to perform the original bridge design or what grade of steel was used.

Therefore, the AASHTO Manual for Condition Evaluation of Bridges, Second Edition

was used to determine the allowable stresses. In the Manual, there is a category for

―unknown steel‖ constructed between 1936 and 1963, which specifies Grade 33 steel.

Since the bridge was designed in 1954 and constructed in 1957, Grade 33 steel was

assumed in order to determine the allowable stresses for these calculations.

For Inventory Rating (IR) compression allowable stress, the AASHTO Manual for

Condition Evaluation of Bridges formula for compression in concentrically loaded

columns for bridges built between 1936 and 1963,

2

45.0570,15

r

KLFcomp (3)

was used, since the FHWA procedure models the edge of the gusset plate as a column. It

is noted that the 1949 version of AASHTO uses

2

2

4

1000,15

r

LFcomp (4)

for compression in concentrically loaded columns having values of L/r not greater than

140 and with riveted ends. The differences between Equation 3 and Equation 4 would

not have significantly changed the results of the analysis. Therefore, the AASHTO

19

Manual equation (Eq. 3) was the one chosen, since it is a more general guideline. The

Operating Rating (OR) equation for compression allowable stress that was used is

2

56.0410,19

r

KLFcomp (5)

The AASHTO Manual was also consulted to determine the allowable stresses for tension

(Ften = 18,000 psi for IR and 24,500 psi for OR) and shear (Fv-avg = 11,000 psi for IR and

15,000 psi for OR). The same allowable stresses used for tension were also used for the

bending allowable stresses, Fb.

Based on the calculated stresses and D/C ratios, it was evident which gusset plates would

require further investigation. There is one D/C ratio over 1.00 for the HS20 live load

case, which occurs for fcomp. The ratio occurs in gusset plate U13 (1.03 at Section B-B).

The gusset plates in the table that have NOT APPLICABLE written in the Section B-B

columns are placed at the intersection of a vertical web truss member and a chord truss

member. These are located at the centers of continuous chord members and are therefore

not carrying much, if any, of the horizontal load. Also, as the gusset plates at these

locations are only slightly wider than the vertical truss member, the location of the

vertical section would effectively be at the edge of the plate and would not reveal internal

gusset plate forces and stresses. This is why the vertical cut was not considered

applicable for the analysis. An example of one of these thinner gusset plates can be seen

in Figure 6 below.

20

Figure 6: Typical lower, odd numbered gusset plate type on Bridge A

A summary of the analysis results is shown in Table 2 and Table 3 below for HS20 and

Permit loading, respectively.

HS20 Section A-A Section B-B

fb fv-avg ften fcomp fb fv-avg ften fcomp

U7 0.32 0.57 0.48 0.72 0.70 0.38 0.99 0.34

U8 0.00 0.00 0.00 0.68 NOT APPLICABLE

U9 0.39 0.77 0.67 1.00 0.57 0.69 0.75 0.66

U13 0.22 0.36 0.29 0.45 0.60 0.25 0.12 1.03*

U14 0.00 0.00 0.00 0.82 NOT APPLICABLE

L0 0.26 0.39 0.22 0.77 0.08 0.40 0.34 0.48

L7 0.00 0.01 0.03 0.03 NOT APPLICABLE

L8 0.04 0.05 0.01 0.36 0.05 0.20 0.14 0.32

L14 0.00 0.00 0.05 0.00 0.36 0.13 0.81 0.13

L15 0.00 0.00 0.06 0.00 NOT APPLICABLE

L16 0.29 0.58 0.51 0.69 0.38 0.40 0.78 0.26

L28 0.30 0.39 0.21 0.84 0.10 0.40 0.34 0.48

*Case where D/C ratio is greater than 1.0.

Table 2: Summary of D/C ratios calculated under HS20 live loading for all analyzed

gusset plates

21

Permit Section A-A Section B-B

fb fv-avg ften fcomp fb fv-avg ften fcomp

U7 0.28 0.50 0.43 0.70 0.69 0.34 0.91 0.46

U8 0.00 0.00 0.00 0.66 NOT APPLICABLE

U9 0.31 0.61 0.53 0.87 0.36 0.54 0.56 0.60

U13 0.19 0.31 0.26 0.42 0.50 0.21 0.10 0.94

U14 0.00 0.00 0.00 0.79 NOT APPLICABLE

L0 0.24 0.36 0.20 0.77 0.08 0.37 0.32 0.48

L7 0.00 0.01 0.02 0.03 NOT APPLICABLE

L8 0.03 0.03 0.00 0.32 0.04 0.16 0.11 0.28

L14 0.00 0.00 0.06 0.00 0.29 0.12 0.67 0.11

L15 0.00 0.00 0.04 0.00 NOT APPLICABLE

L16 0.24 0.48 0.43 0.64 0.32 0.33 0.66 0.23

L28 0.27 0.36 0.20 0.84 0.10 0.37 0.32 0.48

Table 3: Summary of D/C ratios calculated under Permit live loading for all analyzed

gusset plates

Even though the stresses under Permit live loading are larger, the D/C ratios that were

calculated were lower than those under HS20 live loading. This was a result of the fact

that the Permit live load stresses were compared to Operating Rating allowable stresses.

Unsupported edge length adequacy was also studied. Only one of the applicable gusset

plates (gusset plates connecting diagonal members) investigated did not comply with the

unsupported edge limit, expressed by

thicknessplate 000,11

Limit Edge dUnsupporte

yF (6)

This equation is for the unstiffened unsupported edge limit from the AASHTO Standard

Specifications for Highway Bridges, Ninth Edition. The gusset plate that did not comply

with this limit is located at panel point U9. Since the long unsupported edge of this

gusset plate is in tension, it was concluded that all the gusset plates on the bridge were

adequate to meet the criteria for unsupported edge length limit.

22

3.3 Bridge A Finite Element Model

For this investigation of Bridge A, analyses were taken a step beyond the FHWA

suggested procedure. As a verification of the hand calculations, a 2D finite element

model using STAADPRO 2005 software was developed. Interior gusset plate L16 was

chosen as a representative example for this model. The gusset plate dimensions,

connection configuration and the limits of splice plates were confirmed after a review of

the original shop drawings and were input into the FE model. A copy of the original shop

drawing for gusset plate L16 depicting its geometry can be seen in Figure 7 below.

Figure 7: Geometry of Gusset Plate L16 for STAAD model

Nodes (joints) were generated to define boundaries for the gusset plate, limits of splice

plates, and locations of the fasteners in each of the five connections. For simplicity, the

23

connection holes in the gusset plate were not modeled. After these fixed nodes were

established, a 4‖ x 4‖ grid of nodes was placed over the model to fill in the areas between

these key points. The node geometry was reviewed and joints from the 4‖ x 4‖ grid that

conflicted with any of the initially determined ―fixed‖ nodes were deleted. Locations of

conflict were determined by engineering judgment. Triangular plate elements were then

generated between the remaining nodes. Figure 8 shows the layout of the nodes and

triangular plate elements.

Figure 8: The STAAD finite element model for Gusset Plate L16 depicting nodes and

triangular plate elements

The triangular plate elements within the limits of the splice plate were assigned the

thickness of the combined gusset and splice plates (2‖), while elements outside this area

were assigned the thickness of the gusset plate alone (5/8‖). The basic geometry model

was analyzed for the two sections evaluated with the hand calculations. For the

horizontal Section A-A, truss member forces from members B, D, and E were applied to

24

the plate. For vertical Section B-B, the member forces from members A and C were

applied.

Support conditions for the finite element model were given careful consideration. Axial

forces taken from the BAR7 output are envelope forces; they represent the maximum

values in a given member, but do not necessarily occur under the same loading conditions

or at the same time. As a result, the gusset plate as an individual free body is not in

equilibrium when these maximum loads are applied simultaneously. Therefore, separate

support conditions were established to develop equilibrating forces in the truss members

that did not have their maximum forces applied to the model. This was accomplished by

placing additional nodes in line with the fasteners of the existing connection and just

beyond the gusset plate boundary. These nodes were assigned the properties of a pinned

support. From these supports, truss elements were generated in the model to connect

each of the fastener nodes in the adjacent connection to the new virtual pinned support.

These additional truss elements were assigned a proportion of the area of the

corresponding truss member. Loads were applied to the remaining connector nodes

depending upon which section of the gusset plate was being evaluated. For example:

When evaluating vertical Section B-B, forces from truss members A and C were applied

to the model and supports were generated at additional nodes located in line with the

existing connections of truss members B, D and E.

Unit (1 kip) axial forces, resolved into components along the global X- (horizontal) and

Y- (vertical) axes, were applied to each of the fastener nodes. Component values were

25

calculated based upon the geometry of the individual connection. The forces from each

truss member were considered as a separate load case. The live and dead load axial force

values obtained from the BAR7 analysis were divided equally amongst the number of

connectors in each member. This ―per connector‖ value was then applied as a factor to

the unit load cases in the Load Combination command.

Stresses were checked using the feature in STAAD that allows the user to define a cutting

plane. Using this STAAD command, horizontal and vertical cutting planes were

established at the same locations in the model that were evaluated in the hand

calculations. The stresses are summarized in Table 4 and Table 5 below. Positive values

represent tensile stresses; negative values represent compressive stresses.

Section A-A Left Side of Plate Right Side of Plate

ksi ksi

Hand Calculations -4.7 5.7

STAAD Model -2.8 0.4

Table 4: Comparing stresses for Horizontal Section A-A of gusset plate L16

Section B-B Top of Plate Bottom of Plate

ksi ksi

Hand Calculations -3.1 14.03

STAAD Model -4.9 14.0

Table 5: Comparing stresses for Vertical Section B-B of gusset plate L16

Qualitatively, the STAAD model yields results that match those of the hand calculations.

However quantitatively there are differences, especially in section A-A. The difference

in the stress values for the Section A-A model is attributed to local effects due to the

26

support conditions. It is also noted that when the forces in members B and D are in

tension, as they are in Figure 9, they have a tendency to want to become parallel, creating

a lower tensile area in the right side of the gusset plate along Section A-A (between

members B and D). This secondary effect is not accounted for in the hand calculations

and would be an area for consideration in future study.

Figure 9: Stress contours on Gusset Plate L16 from STAAD finite element model

The model for evaluating Section B-B provides stresses that are in general agreement

with the hand calculations. Also of note is the distribution of the stresses in the gusset

plate. The larger stresses are found to occur at the edges of the cut section and reduce

substantially towards the center of the plate. See Figure 9 for stress contours on the

gusset plate.

27

After the initial FE analysis was done on the gusset plate, the model was modified to

account for the section loss in the gusset plate that was documented in the latest available

bridge inspection report. Inspection photos reveal that the typical corrosion on the plate

occurs just above the bottom chord members, as can be seen in Figure 10. Based upon a

review of this information, the gusset plate elements just above the limit of the splice

plates were reduced in thickness from 5/8‖ to 1/2‖. The model was reanalyzed and it is

found that the section loss raises the stresses in the gusset plate by approximately 20% in

the area of the section loss.

Figure 10: Typical section loss in Bridge A gusset plate

28

3.4 Bridge A Instrumentation

Recognizing that the recommended analysis procedure includes a number of assumptions

that may be overly conservative, it was decided that continuous monitoring would offer

data necessary to quantitatively evaluate the actual behavior of the bridge, and allow for a

more informed decision making process regarding the bridge operation and any

rehabilitation needs. In April 2009, a continuous monitoring program was implemented

to gather additional information on the behavior of the gusset plates through sensors

installed on the representative gusset plate L16 used in the finite element analysis. 12

strain gauges were installed on various locations on and around the gusset plate. Five

extensometers were installed on the truss members, one on each member just beyond the

limits of the plate, and seven extensometers were installed on the plate itself at various

locations of high stress, as determined by the STAAD FEA. The sensors were installed

and monitored continuously for one year by Osmos USA. The yearlong monitoring cycle

allowed for all seasonal variations and traffic cycles to be observed. See Figure 11 for

sensor locations.

29

Figure 11: Sensor locations on Bridge A Gusset Plate L16

In order to manage the sheer volume of available data, the software which was used to

record, view, and download the information from the bridge was programmed to save

measurements that exceeded a certain threshold. The real-time data could be viewed

through a web browser to allow the user to see the bridge’s response while the structure

was subjected to live load. Alternatively, the historical data that has been downloaded

and archived can be reviewed through a program installed on the user’s personal

computer. The historical data can be organized and viewed in various formats, useful for

comparing readings from different sensors at concurrent times. The following screen

30

shots depict data that was collected to compare against the results from the hand

calculations and finite element model. Figure 12 shows approximately four minutes of

monitoring in dynamic mode for the bridge. This information represents the results for

four of the truss member sensors.

Figure 12: Bridge responses recorded by sensors on Bridge A Gusset Plate L16

Figure 13 and Figure 14 below show a zoomed-in view of the five truss member sensors

(four sensors in Figure 13, the fifth sensor in Figure 14) over a time period of less than

one minute. The black vertical line represents the cursor function in the data program.

The numbers displayed in the bottom left corner of each screen represent the strain

recorded by each sensor at the time indicated by the cursor. The various colors in the

bottom left corner correspond to the sensor color in the graph. The sensor name is

displayed in line with the readings below the graph at the center of the screen.

31

Figure 13: Sensor readings before live loading event for four of the five truss members

Figure 14: Sensor readings before live loading event for fifth truss member

The strain values in the above figures provide a base reading against which to compare

the readings from the following set of figures. It can be seen that the responses depicted

32

in the graphs above (Figure 13 and Figure 14) occur just before the live loading event and

therefore represent the approximate sensor readings when the truss members are

subjected only to dead load.

Figure 15 below depicts the sensor strain readings of all five truss members at the time

when the vertical truss member (green) is experiencing a peak reading.

Figure 15: Sensor readings at peak strain of vertical truss member

As can be seen, the peak values for each truss member do not occur at the same time,

which is consistent for a truss bridge subjected to a moving load.

Below are four other screen shots (Figure 16 through Figure 19) depicting the same live

load event as Figure 12. In each graph, the cursor has been located at the peak value of a

33

different truss member sensor. Therefore, the strains recorded by all five sensors are

displayed for the time period corresponding to each of these peak sensor values.

Figure 16: Sensor readings at peak strain of south diagonal truss member

34

Figure 17: Sensor readings at peak strain of south chord truss member

Figure 18: Sensor readings at peak strain of north diagonal truss member

35

Figure 19: Sensor readings at peak strain of north chord truss member

The data from Figure 18 represented by the pink line (from the sensor affixed to the 45

degree member extending from the gusset plate to the north) suggests that there may have

been loosening of the bolted connection that attaches the sensor mounting plate to the

member based on the excessive vibrations. The results may not be 100% accurate, but

do show the increased strain value during the live load event.

The actual strains in each member were found by taking the readings of the peak values

on these graphs, subtracting the value at the flat line area of the graph occurring just

before the live load event, and dividing the difference by 2000 mm, which is the length of

the sensors on the truss members. These strains were then converted to stresses and

forces in order to be substituted into the spreadsheets created for the hand calculations.

Table 6 compares the final total stresses resulting from using the sensor live load data

36

with the total stresses obtained from BAR7. Since it is assumed that the sensor data is

only detecting live load strains, the live load stress values obtained from the sensors are

added to the dead load stresses calculated in BAR7.

Sensor Data BAR7

Section Location on Plate

SC

Peak

S45

Peak

VM

Peak

N45

Peak

NC

Peak HS20 P82

A-A Left Side of Plate -3.41 -3.56 -2.98 -3.66 -3.02 -4.7 -5.11

Right Side of Plate 4.23 4.06 4.16 4.02 3.46 5.7 6.75

B-B Top of Plate -1.72 -1.67 -1.87 -1.67 -1.99 -3.18 -3.56

Bottom of Plate 9.13 8.95 9.34 9.04 9.47 14.03 16.22

Table 6: Comparison between hand calculation results using the sensor data and the

BAR7 data

From the data that has been obtained, it appears that the actual loads and strains

experienced by the bridge are significantly less than the hand calculations would predict.

The stresses calculated from the sensor data appear to be at most about 75% of the

stresses calculated using the envelope forces in BAR7. When comparing the live load

forces alone from the sensors to the live load forces from BAR7, the live load forces seen

by the sensors are much smaller than those calculated from BAR7.

3.5 Bridge A Conclusions

The hand calculation analysis of the Bridge A gusset plates reveal only one location with

a Demand/Capacity Ratio greater than 1.0, which is at Gusset Plate U13 where the D/C

Ratio is 1.03. A finite element analysis of the Gusset Plate L16 was performed, and the

resultant maximum principal stress is found to be 3% above the allowable value. The

finite element analysis of the L16 plate indicates that a 20% loss in plate thickness

37

occurring just above the bottom chord, which is typically where the gusset plate section

loss occurs on the Bridge A structure, results in a 20% increase in stress at the reduced

area. However, these stresses are still below the yield strength of the material. The

results of the analysis indicate that the gusset plates that do not exhibit significant section

loss have adequate capacity to support the current dead load and design live loads on the

bridge. Instrumentation of the bridge provides more data regarding the actual state of

stress of the gusset plates on Bridge A. Since the actual strains and resulting

stresses/forces are far below those obtained from both the hand calculations and FE

model, it can be concluded that the structure has adequate capacity.

38

CHAPTER 4. MODEL DEVELOPMENT

4.1 Creating a Model in Abaqus

In order to perform a more refined analysis of gusset plates in a typical Warren deck

truss, multiple steps must be taken beyond the above discussed hand calculations and

simple FEM. As a first step, it is decided that a more comprehensive finite element

model program should be used. Abaqus is the program chosen to advance the gusset

plate analysis. A similar gusset plate to the one that was used in the previous FE model is

used for the Abaqus model. However, since the Abaqus model uses live loads measured

on Bridge B (discussed later in this paper), a high-loaded gusset plate from the truss on

that bridge (Panel Point L10) is chosen for the Abaqus FEM. In order to improve upon

the simple FEM developed in the initial analyses, the Abaqus model is created as a 3D

model.

Given the complicated shape of gusset plates, it is decided that the Abaqus CAE will be

used to develop the model. As stated on the Simulia website,

―With Abaqus/CAE you can quickly and efficiently create, edit, monitor, diagnose,

and visualize advanced Abaqus analyses. The intuitive interface integrates modeling,

analysis, job management, and results visualization in a consistent, easy-to-use

environment that is highly productive. Abaqus/CAE supports familiar interactive

computer-aided engineering concepts such as feature-based, parametric modeling,

interactive and scripted operation, and GUI customization. Users can create

geometry, import CAD models for meshing, or integrate geometry-based meshes

that do not have associated CAD geometry.‖

With the visualizations and multiple manipulation tools, it is easy to create a more

complex model.

39

Figure 20: Example gusset plate for detailed Abaqus finite element model

Abaqus defines its models beginning with ―parts.‖ The gusset plate model being created

in this step of the analysis is based off of the shop drawing shown in Figure 20 above and

consists of just one part: the gusset plate. The gusset plate is approximately 73.5‖ wide

by approximately 45.1‖ tall and is 5/8‖ thick. At one point, it was considered to refine

this model compared to the previous one by modeling the rivet holes in the gusset plate.

There are a total of 125 rivets that connect the gusset plate to the truss members. Each of

the rivet holes are 1 1/6‖ in diameter for 1‖ rivets. Each one of the holes for these rivets

could have been included in the gusset plate. However, since in the actual field condition

the rivets effectively fill out the voided area of the plates, it was decided that the rivet

holes would not be included.

40

After each of these parts are modeled and input into Abaqus, the next step is to create a

material for the plates, the type of material being steel. In Abaqus, the density is input as

0.000284 kips per cubic inch. The grade of steel for Bridge B (as well as many other

similar truss bridges) is Grade 36 (36ksi yield strength). Therefore, this yield strength is

input into the Abaqus model. Elastic material properties are then input into the program

for the material. A Young’s Modulus of 29,000 ksi is used, as well as a Poisson’s Ratio

of 0.3. This material is then applied to the parts of the model through the use of

―sections.‖ A section is created as a solid, homogeneous type with the steel material

previously defined.

The next step to the model development is to create an assembly. The assembly function

in Abaqus uses ―instances‖ of the parts created in order to create a completed model.

There can be more than one instance of a part, and any changes made to the part are made

to the instance automatically. In the model under discussion here, only one instance of

the plate part is used in the assembly. In actuality, these kinds of gusset plate connections

are typically double gusset plates. This means that there are symmetrical gusset plates on

either side of the truss members. However, since it is assumed that these gusset plates

would carry the same stresses as each other on either side of the truss, it is decided that

only one gusset plate will be modeled and simply half of the loads felt by the truss

members will be transferred to the gusset plate model.

After the assembly is positioned correctly, the steps for the model run are created. For

this model, three steps are created. The first step, called the ―Initial‖ step, is created

41

automatically for every model developed in the Abaqus CAE. This step applies the

boundary conditions, which will be discussed below. The second step created is called

the ―Contact‖ step. This step establishes the contact for the loads carried by the truss

members under truck live loading. The third and final step is called the ―Load‖ step.

This step applies the loading itself in iterative steps as defined in the input.

Next, the boundary conditions are applied to the model assembly. It is decided that the

nodes at the bottom of the vertical truss member will function as the pinned supports for

the plate with rotation allowed in all directions, but no translation allowed.

After boundary conditions are established, the individual part is meshed to create the

finite elements that are to be analyzed during the model run. The first step to the meshing

function is to seed the part. For the gusset plate, a maximum deviation factor of 0.1 is

tried to limit the size of the finite elements. An approximate global size of 5 is input to

accompany the deviation factor. Then, the mesh controls are established. The element

shape tested is a Wedge, or a six-node linear triangular prism. This is chosen since the

2D STADD model had used a triangular plate element. This results in an appropriately

sized mesh with similarly sized elements in a sweep across the plate that will both give

good resolution to the model, but will not take an unreasonably long time to run.

Defining the mesh is the last main step in the development of the gusset plate finite

element model. The next step is to analyze the truss model developed by the research

team at Rutgers, The State University of New Jersey, for Bridge B.

42

4.2 Bridge B Research and Model Development

As discussed previously in the Literature Review, a team of researchers at Rutgers

University performed tests and analysis to determine the cause of cracking in the deck of

what is here referred to as Bridge B. As part of this effort, a detailed 3-D finite element

model was created to help process the results and conclusions. The model used beam and

shell elements. The model was validated using results from field tests that were

performed on the bridge. These tests were performed using a test-truck of known axle

weights and consisted of taking measurements through sensors installed on the bridge,

which included a portable Weigh-in-Motion (WIM) system and two piezo-axle sensors

connected to a data collection unit. The finite element model was developed to examine

the behavior of the bridge structure at various loading stages.

Just as was the decision for the gusset plate model described in the previous section,

Abaqus was used for developing the model of Bridge B. The program was chosen here

because of its vast materials and elements library that is suited for civil engineering

applications. Figure 21 shows part of this full truss model.

43

Figure 21: Finite element model of Truss Spans 25, 26, and 27 in Bridge B (Nassif et al.

2007)

Various element types were used to model the bridge. The various element types are

described in detail. Moreover, the description of the boundary conditions, loads, and

constraints are also detailed in the following paragraphs.

The beam element is used to assemble the trusses, floor beams, and stringers. It is a one-

dimensional line element that cannot deform in its own plane; under bending the plane

sections remain plane. Two types of beam elements were chosen for the analysis: two-

node, linear beam (B31) and three-node, quadratic beam (B32) elements (Figure 22a&b

respectively). Both beam elements were modeled in spaces with six degrees of freedom

at each node. Abaqus also includes an I-beam section in the beam element cross-section

library. The advantage of using the cross-section library is that the moment of inertia and

44

torsional rigidity are automatically calculated. The user only needs to input the

dimensions of the I-beam.

(a) (b)

Figure 22: Integration point of (a) two-node, linear beam (B31) and (b) three-node,

quadratic beam (B32) elements along the length of the beam (Abaqus 2010)

The shell element is used to model the concrete slab on the bridge. It was used because

the concrete slab has one dimension that is significantly smaller than the others (i.e.,

thickness of the slabs is smaller than its width and length). Abaqus contains a vast library

of shell elements, but the most common and general type of shell element is the four-

node shell element (S4). This element is a fully integrated, general purpose, finite-

membrane-strain shell element that allows in-plane bending (Abaqus 2010). The S4

element has six degrees of freedom at each node. Figure 23 shows S4 element.

45

Figure 23: Four-node (S4) shell element (Abaqus 2010)

Bridge piers and abutments were idealized using boundary conditions to represent the

actual bearings used in the field. Piers and abutment were assumed not to be affected by

the live load (i.e., no settlement or side-sway) in the FEM model.

The bridge model consists of multiple parts that needed to be joined together to construct

the entire bridge structure. This is achieved using constraint elements, specifically a

multi-point constraint (MPC). In Abaqus, there are predefined MPCs, including BEAM

and PIN. The BEAM MPC provides a rigid beam between two nodes to constrain the

displacement and rotation at the first node to the displacement and rotation at the second

node (Abaqus 2010). It is mainly used for constraining the slab nodes to the stringer

nodes for composite action. For non-composite or zero moment connection, such as the

connection between the stringers and floor beams, PIN MPC is used. PIN MPC provides

a pin connection between two nodes.

46

In addition to the constraints, there were some members that shared the same nodes (e.g.

the diaphragms and the stringers). Abaqus assumes a rigid connection if the same initial

or terminal node of two elements is used. Thus, to model the diaphragm connections, the

rotation of the starting and ending nodes of the diaphragms (that are connected to the

stringer) needed to be released. This was done using the RELEASE commands specified

by Abaqus.

Three different types of steel properties were used in the finite element model: structural

steel, reinforcing steel, and prestressing steel. The structural steel (I-girder and

diaphragms) is subdivided into two grades: A36 carbon steel and A572 high-strength,

low-alloy carbon steel. The A36 carbon steel has a minimum yield strength of 36,000

lb/in2, where the ultimate strength varies between 58,000 lb/in

2 to 80,000 lb/in

2. The

A572 high-strength, low-alloy carbon steel has a minimum yield strength of 50,000 psi,

where the ultimate strength varies between 70,000 lb/in2 to 100,000 lb/in

2. A36 carbon

steel was used in most older bridges. A572 high-strength, low-alloy carbon steel is used

for newly constructed bridges. Figure 24 shows a typical stress-strain curve of the two

grades of steel used in the FE model.

47

Figure 24: Typical stress-strain curve of structural steel (Salmon and Johnson 1996)

Depending on the age of concrete, the deck slab typically has a design compressive

strength ranging from 4,000 lb/in2 to 6,000 lb/in

2. The compressive strength of the deck

slab was assumed to be 5,000 lb/in.2.

As mentioned earlier, the modulus of elasticity and Poison’s ratio need to be specified in

the model for elastic analysis. Unlike steel, the modulus of elasticity of concrete varies

significantly with compressive strength, types of aggregates, paste content, and

admixture. For simplicity, a relationship between the modulus of elasticity and

compressive strength has been established. The American Concrete Institute (ACI)

Building Code (ACI 318 Article 8.5.1, 2005) gives the modulus of elasticity, Ec, as

follows:

48

Ec 33wc1.5 f c for

90 wc 155 lb/ft3

(7)

or for normal-strength concrete:

Ec 57,000 f c

(8)

where, wc and cf are the unit weight (lb/ft3) and compressive strength (lb/in.

2) of

concrete, respectively.

The tensile strength of concrete was also considered in the FEM model. A good

approximation of the tensile strength of concrete is 10% to 20% of the compressive

strength (Nawy, 2005). However, if subjected to bending, the modulus of rupture rather

than the tensile strength should be used. ACI 318 Article 9.5.2.3 specifies the modulus of

rupture of concrete, fr, for normal-weight concrete as follows:

cr ff 5.7 (9)

The FE model was validated by comparing it to both the static and dynamic field load

tests. The FE model was validated with the static load test results by doing the following:

A 3-axle dump truck with a gross vehicle weight (GVW) of 67 kips was positioned at the

center of S8 (Stringer 8) (i.e., the left and right wheels of the truck were evenly

distributed to the north and south of S8). A diagram depicting the location of S8 on the

bridge can be seen in Figure 37 below. The test was controlled and isolated from other

trucks traveling over the bridge. It was also unaffected by the dynamic impact factor

since the test-truck was moving at a relative low speed (<10 mph).

49

Figure 25 and Figure 26 show the comparison of stresses in S5 through S10 from the

field test results and the FE model for static load tests 1 and 3, respectively. In the

figures, the field test data is denoted by ―EXP‖ and represented with a red solid line, and

the FE model is denoted as ―FEM‖ with a blue dashed line. Overall, the FE model

correlated well with the field test results with variations within 15% of the field test

results. The FE model does provide an accurate stress-strain calculation and represents

the actual bridge very well.

50

Figure 25: Comparison of stresses in S5-S10 using the FE model and Static Load Test 1

(Nassif et al. 2007)

51

Figure 26: Comparison of stresses in S5-S10 using the FE model and Static Load Test 3

(Nassif et al. 2007)

52

The dynamic load tests were also used for validating the FE model. The dynamic field

tests were conducted using the actual truck traffic traveling on the bridge. Three dynamic

load cases were used for the comparison. These cases consisted of three 5-axle trucks

with gross vehicle weights of 65, 55, and 42 kips. Figure 27 and Table 7 show the

configuration of the most common truck used for the FE model.

Figure 28 through Figure 33 illustrate the comparison of the dynamic load test of the 5-

axle trucks with GVW of 65 kips for S5 through S10, respectively. Overall, the FE

model correlated well with the field test results for the maximum stresses, having only a

2% variation. At lower stresses, especially over S8, S9, and S10, the variation is

significantly high, which could have been for multiple reasons.

A B C D

1 2 3 4 5

Figure 27: Standard truck configurations used in the calibration of the finite element

model (Nassif et al. 2007)

53

Truck GWV (kips) 65.2

Axle 1 (kip) 9.2

Axle 2 (kip) 13.3

Axle 3 (kip) 12.4

Axle 4 (kip) 15.1

Axle 5 (kip) 15.2

Spacing A (ft) 17.4

Spacing B (ft) 4.3

Spacing C (ft) 35.9

Spacing D (ft) 4.1

Table 7: Truck configuration (Nassif et al. 2007)

Figure 28: Comparison of stresses in S5 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

54

Figure 29: Comparison of stresses in S6 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

Figure 30: Comparison of stresses in S7 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

55

Figure 31: Comparison of stresses in S8 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

Figure 32: Comparison of stresses in S9 using the FE model and the actual field-test data

of a 5-axle truck with a GVW of 65 kips traveling WB in Lane 1 (Nassif et al. 2007)

-1

-0.5

0

0.5

1

1.5

2

2.5

0 500 1000 1500 2000

S9-EXPS9-FEM

Stresses (ksi)

Time (1/100 s)

56

Figure 33: Comparison of stresses in S10 using the FE model and the actual field-test

data of a 5-axle truck with a GVW of 65 kips traveling WB on Lane 1 (Nassif et al. 2007)

Multiple simulations representing various load cases were made using the FE model.

Two actual trucks obtained from the portable WIM system were used for the simulations:

78.7 kip, 4-axle and 50 kip, 3-axle dump trucks. These types of trucks cause the highest

stress range and also represent approximately 20% of the average daily traffic.

Multiple simulations were performed with these two truck types:

Loading the WB left lane with a 78.7 kip, 4-axle dump truck

Loading the EB left lane with a 50 kip, 3-axle dump truck

Loading the WB right lane with a 78.7 kip, 4-axle dump truck

Loading the WB lanes with two 78.7 kip, 4-axle dump trucks side-by-side

-1

-0.5

0

0.5

1

1.5

2

2.5

0 500 1000 1500 2000

S10-EXPS10-FEM

Stresses (ksi)

Time (1/100 s)

57

Loading the WB left lane with a 50 kip, 3-axle dump truck

Loading the EB left lane with a 50 kip, 3-axle dump truck

Two impact factors, 1.33 and 1.5, are used in the analyses for trucks traveling WB and

EB, respectively. The 1.33 impact factor is based on dynamic test results as well as the

recommendation of the AASHTO LRFD Standard Specification. It was noted that the

observed impact factor of dynamic test-trucks traveling EB was as high as 2.0.

However, a conservative value of 1.5 was used since the dynamic impact factor from

heavier trucks will be lower and will vary from truck to truck.

4.3 Model Integration

In order to complete the final steps of the analysis for this thesis research, the individual

gusset plate finite element model that was developed based on the Bridge B as built

drawings is integrated into the truss model developed by the Rutgers team described in

the previous section. This allows the stresses in the gusset plate to develop under actual

loading conditions in a calibrated model.

As a first step, it is decided to convert the 3D model into a 2D shell element in order to

more closely transition to the 1D beam elements used for the truss members. The Abaqus

CAE model for the gusset plate is used to export an input file to define the coordinates of

all of the points in the gusset plate FE mesh. The coordinates from one side of the 3D

gusset plate are then added as new nodes into the truss model with the appropriate offsets

to the base coordinate system in the truss bridge model. The gusset plate coordinates are

58

integrated into the truss model at 4 symmetric locations, two on each truss, where this

type of gusset plate is located on the bridge.

The elements are then created between each node by copying the element definitions

from one side of the gusset plate model. The carbon steel material used for some of the

members in the truss is applied to the integrated plate. The thickness of each element in

the plate is then defined. In the areas connected to truss members, the thickness is

increased to account for the two gusset plates as well as the truss member. In the areas in

between, the thickness is simply the two gusset plates.

In order to insert the gusset plate into the truss, some sections of the truss member are

removed around the associated node. Then the connection between the remaining truss

members and the nodes at the corresponding edge of the plate are established by using

the same node for both the end of the member and the center of the plate edge. A

kinematic coupling restraint is also established between the connector node and the

adjacent node on the plate edge to establish that the whole edge of the plate at the truss

member would move as one. Figure 34 below shows the full bridge model with gusset

plate integration. This figure also displays the stress results of the bridge being loaded

with a heavy live load, the results of which will be discussed in the next section.

59

Figure 34: Bridge B full truss model with gusset plate integration (red circles) in Abaqus

Now the model is ready to be loaded. The test live load case shown in Figure 35 below is

used on the new combined model as well as the original truss bridge model without the

gusset plates. Both models are run with the live load and the results are compared to

validate that the gusset plates in the new model do not significantly affect the outcome

compared to the original model.

60

August/Sept Test-truck

3-axle single body dump

truck

Gross Vehicle Weight =

53.7k

Axles 1 2 3

Weights 15.5k 19.1k 19.1k

Spacing --- 12.5 ft 5.0 ft

Figure 35: Test-truck configuration (top) and information (bottom) for controlled load

tests (Nassif et al. 2007)

After consulting the results from the Rutgers study, the node corresponding to the

location of the sensor on S6 (as shown in Figure 36 and Figure 37) in both models is

chosen to generate the validation data, since this has the highest responses from the

dynamic tests.

61

Figure 36: Under-deck view of Span 26 Bay 1 between FB11 and FB12 (Nassif et al.

2007) Note: rectangles represent strain transducers

Toward River 6534

S11

6138

S10

6117 5979 6348 6538 5122 Top

6489 Bot.

S9

6139 Top

6536 Bot.

S8

6111 5088 5116 6258 S7

6612

S6

S5

6535

S4

S3

S2

Pier 25

FB9’

FB10’ FB11’ FB12’ S1

―Bay 4‖ ―Bay 3‖ ―Bay 2‖ ―Bay 1‖ End of 4 span

stringer

Bottom flange

only

Top & Bot

Flange

LVDT vert. or

horizontal

Figure 37: Span 26 sensor layout involving 16 strain gauges and 4 LVDTs (Nassif et al.

2007)

62

The results from the integration points at the bottom of the stringer S6 in both models are

shown in Figure 38 below.

Figure 38: Comparison of Abaqus FEM results at Sensor 6 location with and without

gusset plate integration

As can be seen, the results are nearly identical. The models are also compared with

respect to the response of the truss members immediately around the gusset plate

location. The following images (Figure 39) show the responses of these truss members

with and without the plate integrated into the model.

63

Figure 39: Comparison of FEM results at adjacent truss members with and without gusset

plate integration

-210

-110

-10

90

190

0 20 40 60 80

Member A w/Plate

-210

-110

-10

90

190

0 20 40 60 80

Member A w/o Plate

-210

-110

-10

90

190

0 20 40 60 80

Member B w/Plate

-210

-110

-10

90

190

0 20 40 60 80

Member B w/o Plate

-210

-110

-10

90

190

0 20 40 60 80

Member C w/ Plate

-210

-110

-10

90

190

0 20 40 60 80

Member C w/o Plate

-210

-110

-10

90

190

0 20 40 60 80

Member D w/Plate

-210

-110

-10

90

190

0 20 40 60 80

Member D w/o Plate

-210

-110

-10

90

190

0 20 40 60 80

Member E w/Plate

-210

-110

-10

90

190

0 20 40 60 80

Member E w/o Plate

64

In general, the results of these comparisons show that the member stresses with or

without the gusset plates are similar. The various lines on each chart are the different

integration points within the section of the member. There is some difference in the

magnitude of the stresses for Member B (horizontal), though the general response shape

is the same and is deemed to be similar enough to continue with the model including the

plate. This allows us to follow through using the new model to start analyzing the gusset

plate stresses under various live loads.

65

CHAPTER 5. PARAMETRIC STUDY

5.1 Varying Plate Thickness

In this step of the research, a parametric study is conducted to compare the gusset plate

responses under various conditions. The first step is to simply see how the response of

the gusset plate will change under two lanes of the test dump truck live load that was

used in the static analysis of Bridge B with varying plate thickness. The graph also

shows the relationship of the gusset plate stress to thickness under two lanes of the

AASHTO defined HL-93 standard design live loading. According to as-build drawings

of Bridge B, the test gusset plates chosen for this analysis are each 5/8‖ thick. In each

separate finite element run, the thickness is decreased by an additional 1/16‖ down to

5/16‖ each, which is the minimum thickness for any steel members, as defined in the

AASHTO Bridge Design Specifications. The results of this parametric study are shown

in Figure 40 below.

66

Figure 40: Chart graphing plate stress vs. plate thickness

As would be expected, the stress in the plate increases as the plate thickness decreases

under both live loading cases. These graphs follow power trends, approaching infinity

when the gusset plate thickness approaches zero (0.0). It can be seen in the graph that the

HL-93 loading case is the far more critical case for this type of comparison. And

although the stresses for both live loads increase, they do not reach a critical level for

steel with a yield strength of 36ksi, even at the minimum plate thickness. This is

considered a good result, since it is undesirable for gusset plates to approach failure under

any live loading. Gusset plates are typically designed such that they are not the most

critical members on a bridge, as is the case with any connection point between main

bridge members.

67

5.2 Varying Live Load

The second parametric study that is performed for this research involves using the

original integrated FEM with the as-build gusset plate thickness and varying the live load

according to the WIM data that was collected by the Rutgers team during the Bridge B

Deck Evaluation. The test dump truck is the first and smallest load applied. However,

instead of running one lane of the load down the center of the bridge model, two lanes of

the live load are placed along the edge of the deck, with the first lane located

approximately 2’-0‖ from the edge of the parapet, and the second lane located 10 feet

from the first, as per AASHTO standard specifications. The next largest live load that is

tested is an HS20 AASHTO design live load. Again, two lanes of the load are placed on

the bridge at the same location as the dump truck case. The third live load that is used is

the 78.7-kip Short Heavy Vehicle recorded by the WIM data in the Rutgers study. This

truck is coupled with a second lane of the test dump truck. The fourth load is an

approximation of the heaviest load measured by the Rutgers WIM system. This is a

105.8-kip, 6-axle vehicle. The spacing of the wheels is assumed to be approximately 5

feet between each of the back 5 axles under a heavy load and an approximately 16’

between the front axle and the first of the back axles. The distribution of the weight is

assumed to be 15.8 kips on the front axle and 18 kips on each of the back axles. This

truck is then also paired with a second lane of the dump truck. The final and heaviest live

load used for this study is a Permit load, representing the largest legal live load allowed

on the bridge. This load is coupled with the AASHTO HS20 truck in the second lane to

represent the most probable configuration of permit live loading on a bridge per

68

AASHTO Bridge Design Specifications. The spacing between and load on each of the

axles for all of the above mentioned live loads are listed in the table below.

LL Case

Test

Dump

Truck

HS20 SHV Max

WIM

NJ

Permit

PA

Permit

Axle 1

Load (kips) 15.5 8 14.3 15.8 16 15

Space (ft) 12.5 14 13.5 16 12 11

Axle 2

Load (kips) 19.1 32 15.6 18 16 27

Space (ft) 5 14 4.5 5 4 4

Axle 3

Load (kips) 19.1 32 26.3 18 28 27

Space (ft) - - 4.5 5 4 4

Axle 4

Load (kips) - - 22.5 18 28 27

Space (ft) - - - 5 21 24

Axle 5

Load (kips) - - - 18 28 27

Space (ft) - - - 5 4 4

Axle 6

Load (kips) - - - 18 28 27

Space (ft) - - - - 4 4

Axle 7

Load (kips) - - - - 28 27

Space (ft) - - - - 4 4

Axle 8

Load (kips) - - - - 28 27

Table 8: Configuration for various trucks used for live loading in model (italics indicate

estimations)

After each of these live load cases are created and the FE models are run for each case,

the results are graphed on the chart in Figure 41 below. The results shown are the Tresca

69

stresses in the node with the maximum stress in the gusset plate, which happens to be at

the edge of the plate next to the vertical truss member.

Figure 41: Results of varying live load on a gusset plate on Bridge B

The results of this parametric study comparison are as would be expected for the

respected live load cases. The test dump truck, which is the lightest of the vehicles

produces the smallest stresses, with the maximum stress (when the centroid of the truck is

placed over the location of the gusset plate) being 3.41kips. The HS-20 live load case

and the 4-axle short heavy vehicle produce very similar results, which makes sense since

the loads of these trucks are less than 10% different from each other. However, the

70

second lanes for these live load cases are more significantly different. The HS20 second

lane is another HS-20 truck, whereas the SHV second lane is the test dump truck, which

is only 53.7 kips, compared to the 72 kip HS20. Despite this difference, the stress results

still come out similarly, with the maximum stress for the HS20 being slightly higher.

This could indicate that a shorter vehicle which fits entirely between two stringers on a

truss bridge could have a similar or possibly even more critical reaction in gusset plates

than a longer vehicle with a larger load. The load case with the next largest stress is the

heavier vehicle with a longer assumed total length. When comparing this longer vehicle

with the short heavy vehicle (both with the test dump truck in the second lane), a 20%

increase in the total live load on the bridge results in an increase in stress of only 11%.

This again indicates that the length of the vehicle can have an inverse effect on the gusset

plate stress. However, it does appear that the effect is not so great as to cause concern

about using the HS20 vehicle vs. anything much shorter than that. Especially when all of

these loading cases are compared to the Permit load, also specified by AASHTO to be

used for design of bridges. The Permit truck produces a peak stress that is more than

50% larger than the next largest peak stress. Although in the AASHTO design

specifications these types of Permit loads are scaled down compared to a more typical

HS20 truck load, the fact that these trucks are used for the design of bridges indicates that

most of the largest stresses experienced by gusset plates due to live load are being

covered in current designs.

71

5.3 Validating Integrated Plate Model

One final comparison of the FEM results in order to validate the models used to analyze

gusset plates in the future is to compare the results of an individual plate model with the

results from the plate shell element integrated into a full truss model. To do this, the

stresses in the truss elements of the full bridge model without the gusset plates are found

and converted to forces. Then these forces are applied as pressure uniformly to the edges

of the plate within the limits of the truss member width. Figure 42 and Figure 43 below

show the gusset plate stress distribution on the gusset plate in the truss model and the 3D

individual plate model, respectively.

Figure 42: Gusset plate in full bridge model stress contours from Abaqus

72

Figure 43: Individual 3D gusset plate model stress contours from Abaqus

At first glance, these results look very different, but when analyzed more closely, they

seem to be much more similar than may be first assumed. The most obvious difference

between the two is the added thickness of the plate in the bridge model to account for the

truss members that were removed from the original Rutgers bridge model. These

locations of the gusset plate model will not experience nearly as large of a stress as the

individual model will. It is probably most accurate from a tension/compression stress

standpoint that the gusset plate would not fail where it is closely connected to the truss

member. Therefore, we can focus mostly on the other locations within the individual

model.

73

There is also obviously going to be a discrepancy in the area of the boundary condition

within the individual model, which is where the highest stress is occurring in our

individual model. When these discrepancies are set aside, the remaining plate stresses in

between the truss member locations actually reveal a similar pattern. If the location of

highest stress in the bridge model gusset plate is also looked at in the individual plate

model, a similar stress is revealed. The stress results are shown in Table 9 below.

Max

Principal

(ksi)

Min

Principal

(ksi)

Mises

(ksi)

Tresca

(ksi)

3D Plate 0.336023 -4.99244 4.36662 5.03196

Bridge 0.226118 -4.58197 4.131292 4.581972

% Diff 32.71% 8.22% 5.39% 8.94%

Table 9: Stress comparisons between plate models

The four types of stresses shown are the ones that are controlling in either negative or

positive stress in both cases. For 3 of the 4 of these stresses, the difference is below 10%,

which can be considered acceptable. The magnitude of the Maximum Principal stresses

are so small in this location that the large difference can be disregarded. These results

indicate that an individual model can be used to create accurate results if the correct loads

from the truss members are input into the model.

5.4 Comparing In-Depth FEM to Method of Sections

In our final step of this research, we compare the stresses in the finite element model with

the ones calculated from the FHWA suggested Method of Sections procedure used in our

74

previously-described Bridge A analysis. To do this, we used the spreadsheet created

during the Bridge A part of our work, though instead of inputting the envelope forces of

the truss members into the spreadsheet, we input the truss member loads calculated from

the full bridge FEM. These are the same forces used in our plate vs. full bridge FEM

comparison above. The table below shows the final calculation of principal stresses

using the Method of Sections.

Vertical Section B-B

Principal Stresses (at neutral axis)

Principal Stresses (at edge of

gusset plate) Maximum

R = 5.65 kip/in2

ften = 7.3 kip/in2 ften = 4.61 kip/in

2 7.3 kip/in

2

fcomp = -4.01 kip/in2 fcomp = 1.97 -4.01 kip/in

2

Horizontal Section A-A

Principal Stresses (at neutral axis)

Principal Stresses (at edge of

gusset plate) Maximum

R = 1.55 kip/in2

ften = -0.6 kip/in2 ften = 1.31 1.31 ksi

fcomp = 2.5 kip/in2 fcomp = 2.49 2.5 ksi

Table 10: Plate stresses using Method of Sections

Comparing to the values in Table 9, it can be seen that by an order of magnitude

comparison, these stresses actually appear to be close to what is being seen in the finite

element models. Especially when considering the tension at the edge of the gusset plate

in the Vertical Section B-B (4.61ksi) vs. the Mises and Tresca maximum stresses from

both FEM, which are measured at the top edge of the plate as well. Given these results, it

can be concluded that the Method of Sections is an appropriate way to quickly check a

gusset plate’s capacity. However, this is only the case if accurate loads are input into the

calculations, and in order to do this, a more in-depth approach needs to be taken to find

these loads. From the Bridge A analysis, it was learned that doing a simple computer

75

analysis model and using the envelope forces from the model in the Method of Sections

do not produce accurate results.

76

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

The analyses performed during the course of this research resulted in various different

conclusions on the topic of gusset plate stresses under live loading. The topics of

adequate plate thickness, varying degrees of heavy truck live loading, and recommended

analysis techniques were all studied and the results documented in this paper.

In the first stage of this research, the Bridge A gusset plates were looked at to determine

if they had adequate capacity to carry the loads on the structure. Using the FHWA

suggested method of sections calculations, it is concluded that there is only one location

on the truss with a Demand/Capacity Ratio greater than 1.0 (1.03 at U13), which is

considered to be an acceptable result. A finite element analysis of a representative gusset

plate was then performed, and aside from the typical analysis, the effect of section loss

was studied. The finite element analysis indicates that a 20% loss in plate thickness

occurring just above the bottom chord results in a 20% increase in stress at the reduced

area, though stresses are still below the yield strength of the material. The results of the

analysis indicate that the gusset plates that do not exhibit significant section loss have

adequate capacity to support the current dead load and design live loads on the bridge.

The second part to the Bridge A analysis was to compare readings from instrumentation

on the bridge to the hand calculations and FEM. From the data that was obtained, the

actual loads and strains experienced by the bridge are significantly less than the hand

calculations would predict. The stresses calculated from the sensor data appear to be at

most about 75% of the stresses calculated using the envelope forces in BAR7. Since the

77

actual strains and resulting stresses/forces were found to be far below those obtained

from both the hand calculations and FE model, it is again concluded that the structure has

adequate capacity.

The second stage in this research focused on more in depth finite element modeling on

the trusses and gusset plates in another similar bridge (Bridge B). One aspect of this

stage was to compare the stresses in gusset plates integrated into a full truss model to the

stresses in an individual gusset plate FEM. At first glance, the results look very different,

but when analyzed more closely, they seem to be much closer than may be first assumed.

The most significant difference between the two is obviously going to be a discrepancy in

the area of the boundary condition within the individual model, which is where the

highest stress is occurring in our individual model. It is difficult to model an individual

plate element properly without the truss elements to hold it in place. Another

discrepancy is the handling of the section properties within the area of the truss elements,

which causes large variations in stress in those locations. However, if it is assumed that

the plate will buckle or yield in a location that is not tightly connected to the truss

element, an analysis of the remaining plate stresses in between the truss member

locations can be focused on. The intermediate locations reveal a similar pattern in both

models, and when looking at the location of highest stress from the bridge model gusset

plate and comparing it to that location in the individual model, a similar stress is

revealed. This helps determine that an individual plate model can be used as a quicker

analysis tool for gusset plates, rather than creating a full scale truss model.

78

A parametric study was conducted on the full scale truss model to study the effect of

various live loads on gusset plates. The results of this parametric study comparison are in

general as would be expected for the respected live load cases. The test dump truck,

which is the lightest of the vehicles, produces the smallest stresses. The HS-20 live load

case and the 4-axle short heavy vehicle produce very similar results, which makes sense

since the loads of these trucks are less than 10% different from each other. However, the

effect of the second lanes for these live load cases is more interesting. The HS-20 second

lane was another HS-20 truck, whereas the SHV second lane was the test dump truck.

Given this difference, one might think the stress results of these two cases should be more

different. This suggests that a shorter vehicle which fits entirely between two stringers

on a truss bridge can have a similar or possibly even more critical reaction in gusset

plates than a longer vehicle with an even larger load. The load case with the next largest

stress is the heaviest vehicle with a longer assumed total length. When comparing this

longer vehicle with the short heavy vehicle (both with the test dump truck in the second

lane), a 20% increase in the total live load on the bridge results in an increase in stress of

only 11%. This again indicates that the length of the vehicle can have an opposite effect

on the gusset plate stress. However, it does appear that the effect is not so great as to

cause concern about using the HS20 vehicle vs. anything much shorter than that. The

largest stresses are a result of the heaviest load, the Permit load. These stresses are more

than 50% larger than the long heavy vehicle. This is a good result since Permit vehicles

are used as critical case designs for actual bridges. This indicates that in designs of

bridges, the most critical live loading cases are generally being used.

79

The last part of the full truss FEM analysis was to compare the results of the model to the

FHWA Method of Sections approach. It is found that by an order of magnitude

comparison, the stresses from the Method of Sections appear to be close to the finite

element model results. This is especially true when considering the location of highest

stress in the plate model, which are almost exactly the same as the stresses in that

location from the hand calculations. Given these results, it can be concluded that the

Method of Sections is an appropriate way to quickly check a gusset plate’s capacity.

However, this is only the case if accurate loads are input into the calculations. In order to

find accurate load inputs, an in-depth approach needs to be taken, rather than running a

simple computer analysis model and taking the envelope forces.

In the end, the authors on this paper suggest that, although the method of sections appears

to be an adequate approach to gusset plate analysis, more detailed guidelines are needed

in this topic to assure that whichever approach is used to analyze a truss bridge gusset

plate will be done using accurate live loading to what is seen on the subject bridge and

accurate corresponding forces in the truss members connected to the gusset plate.

80

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