concrete filled rectangular tubular flange girders with corrugated and flat webs
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Lehigh UniversityLehigh Preserve
Theses and Dissertations
2004
Concrete filled rectangular tubular flange girderswith corrugated and flat websMark R. WimerLehigh University
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Recommended CitationWimer, Mark R., "Concrete filled rectangular tubular flange girders with corrugated and flat webs" (2004). Theses and Dissertations.Paper 862.
Wimer, Mark R.
Concrete FilledRectangularTubular FlangeGirders withCorrugated andFlat Webs
September 2004
Concrete Filled Rectangular Tubular Flange Girders
with Corrugated and Flat Webs
by
Mark R. Wimer
A Thesis
Presented to the Graduate and Research Committee
of Lehigh University
in Candidacy for the Degree of
Master of Science
In
Department of Civil Engineering
Lehigh University
August 2004
Acknowledgements
First of all, I wish to express my sincere thanks to Dr. Richard Sause for the
guidance he has given me throughout this project. I would also like to thank High
Steel Structures, Inc. for donating the test girders. In addition, Bong-Gyun Kim has
provided me with a great deal of assistance, and his help is greatly appreciated.
I would also like to thank all of the ATLSS laboratory technicians for their
help in preparing and testing my specimen. Most importantly, I want to express my
gratitude to my wife, Brandi, for her daily support and encouragement.
III
Table of Contents
List of Tables
List of Figures
Abstract
1. Introduction1.1 Background1.2 Objectives1.3 Approach1.4 Thesis Outline
2. Background2.1 Previous Research2.2 Tubular Flange Girders vs. Conventional I-Shaped Girders2.3 Additional Consideration for Tubular Flanges2.4 Corrugated Web Girders vs. Conventional Flat Web Girders2.5 Additional Considerations for Corrugated Web Girders2.6 AASHTO LRFD Bridge Design Specifications
3. Prototype Bridge Design Study3.1 Introduction3.2 Prototype Bridge3.3 Limit State Ratios3.4 Design Process3.5 Types of Designs3.6 Selection of Corrugated Web Geometric Parameters3.7 Discussion of Designs3.8 Efficiency of Corrugated Web
4. Test Specimen and Test Procedure4.1 Introduction4.2 Choice of Test Girders4.3 Scaling Process4.4 New Corrugated Web for Test Specimen4.5 Design Details
4.5.1 Stiffener Designs4.5.2 Fillet Weld Designs4.5.3 Selection of Deck4.5.4 Deck Construction
IV
VI
Vll
1
3445
71215152024
3131323336394151
626364666969727677
4.5.5 Shear Stud Design 784.6 Test Procedures 804.7 Test Instrumentation and Data Acquisition System 844.8 Stress-Strain Properties of Test Specimen Materials 864.9 Measured Girder Dimensions and Initial Tube Imperfection 90
5. Discussion of Experimental Results and Comparison with Analytical Results5.1 Introduction 1225.2 Test Stages 1225.3 Coordinate Axes and Instrumentation Identification 1255.4 Strain Gage Data 1265.5 Vertical Deflection Results 1345.6 Lateral Displacement Results 1385.7 Web Distortion 1455.8 Tube and Tension Flange Lateral Curvature of Scaled Design 19 1475.9 Plate Bending in Tension Flange of Scaled Design 19 148
6. Summary, Conclusions, and Recommendations6.1 Summary 2176.2 Conclusions 2186.3 Recommendations for Future Work 222
References 224
V~a 225
List of Tables
Table
3.1 Prototype Corrugated Web Girder Designs (Neglecting FlangeTransverse Bending Moments) 56
3.2 Prototype Conventional Flat Web Girder Designs 573.3 Prototype Corrugated Web Girder Designs (Incorporating Flange
Transverse Bending Moments) 58
4.1 Scaled Girder Designs 13 and 7 924.2 Scaled Girder Designs 19 and 7 (6 in. (152.4 mm) thick deck) 934.3 CW-T (48 ft.) Stress-Strain Properties 944.4 CW-T (12 ft.) Stress-Strain Properties 944.5 CW-W Stress-Strain Properties 944.6 CW-F Stress-Strain Properties 954.7 FW-T Stress-Strain Properties 954.8 FW-W Stress-Strain Properties 954.9 FW-F Stress-Strain Properties 964.10 Average Measured Girder Dimensions 964.11 Initial Imperfection (Sweep) of Tubes 96
5.1 Stage Identification Subscripts 1515.2 Analytical Values for Stiffness and Neutral Axis Location 1515.3 Comparison of Experimental Results and Analytical Results for
Stiffness and Neutral Axis Location 1515.4 Comparison of Experimental Results and Analytical Results for
Stiffness, Including Only Bending Deformation in AnalyticalCalculation 152
5.5 Comparison of Experimental Results and Analytical Results forStiffness, Including Bending and Shear Deformations inAnalytical Calculation 152
5.6 Description of FEM Models 1535.7 Curvatures Observed to Study Plate Bending in Tension Flange
of Scaled Design 19 153
List of Figures
Figure
3.1 Prototype Bridge 593.2 Shear Strength of 50 ksi (345 MPa) and 70 ksi (485 MPa) Flat
Webs 593.3 Prototype Bridge Corrugated Webs 603.4 Comparison of Corrugated Web Shear Strength to Unstiffened Flat
Web Shear Strength for 50 ksi (345 MPa) and 70 ksi (485 MPa)Webs 60
3.5 Comparison of Corrugated Web Shear Strength to Stiffened FlatWeb Shear Strength for 50 ksi (345 MPa) and 70 ksi (485 MPa)Webs 61
4.1 Prototype and Scaled Versions of Designs 13 and 7 974.2 Scaled Corrugated Webs 984.3 Trapezoidal Corrugated Web 984.4 Fatigue Crack 994.5 Web cuts and Splicing Arrangement 1004.6 Stiffener Geometry for Scaled Design 19 1014.7 Stiffener Geometry for Scaled Design 7 1024.8 Scaled Designs 19 and 7 1034.9 Illustrations for Tube Flange-to-Web Fillet Welds 1044.10 Example off(ax) versus ax (Used for Tube-to-Web Fillet Weld
Design) 1054.11 Illustrations for Tube Flange-to-Stiffener Fillet Welds 1064.12 Core Hole in Pre-Cast Deck 1074.13 Shear Studs Mounted in Core Hole 1074.14 Shear Stud Arrangement 1084.15 Core Hole Pattern 1084.16 Loading Arrangements 1094.17 Wood Cribbing 1104.18 Wood Shim 1104.19 Rollers III4.20 Haunch III4.21 Profile View of Scaled Design 7 Instrumentation 1124.22 Details of Scaled Design 7 Instrumentation 1134.23 Profile View of Scaled Design 19 Instrumentation 1144.24 Details of Scaled Design 19 Instrumentation 1154.25 Details of Lateral Displacement Instrumentation 1164.26 I\faterial Test Identifiers 117
VIl
4.27 CW-T (48 ft.) Coupon 1 1184.28 CW-T (12 ft.) Coupon 5 1184.29 CW-W Coupon 1 1194.30 CW-F Coupon 4 1194.31 FW-T Coupon 1 1204.32 FW-W Coupon 2 1204.33 FW-F Coupon 4 1214.34 CW-C and FW-C 121
5.1 Coordinate Axes for Test Girders 1545.2 Scaled Design 19 Instrumentation Identifiers 1555.3 Scaled Design 7 Instrumentation Identifiers 1565.4 Lateral Displacement Instrumentation Identifiers 1575.5 Moment at East Elastic Section versus Strain for Stage 1
(Scaled Design 7) 1585.6 Moment at West Elastic Section versus Strain for Stage 1
(Scaled Design 7) 1585.7 Moment at East Elastic Section versus Strain for Stage 1
(Scaled Design 19) 1595.8 Moment at West Elastic Section versus Strain for Stage 1
(Scaled Design 19) 1595.9 Moment at East Elastic Section versus Strain for Stage 2
(Scaled Design 7) 1605.10 Moment at West Elastic Section versus Strain for Stage 2
(Scaled Design 7) 1605.11 Moment at East Elastic Section versus Strain for Stage 2
(Scaled Design 19) 1615.12 Moment at West Elastic Section versus Strain for Stage 2
(Scaled Design 19) 1615.13 Moment at East Elastic Section versus Strain for Stage 3
(Scaled Design 7) 1625.14 Moment at West Elastic Section versus Strain for Stage 3
(Scaled Design 7) 1625.15 Moment at East Elastic Section versus Strain for Stage 3
(Scaled Design 19) 1635.16 Moment at West Elastic Section versus Strain for Stage 3
(Scaled Design 19) 1635.17 Moment at Midspan Section versus Strain for Stage 1
(Scaled Design 7) 1645.18 Moment at Midspan Section versus Strain for Stage 1
(Scaled Design 19) 1645.19 Moment at Midspan Section versus Strain for Stage 2
(Scaled Design 7) 165
VIII
5.20 Moment at Midspan Section versus Strain for Stage 2(Scaled Design 19) 165
5.21 Moment at Midspan Section versus Strain for Stage 3(Scaled Design 7) 166
5.22 Moment at Midspan Section versus Strain for Stage 3(Scaled Design 19) 166
5.23 Neutral Axis During Unloading of Scaled Design 7 in Stage 1 1675.24 Neutral Axis During Unloading of Scaled Design 19 in Stage 1 1675.25 Neutral Axis During Unloading of Scaled Design 7 in Stage 2 1685.26 Neutral Axis During Unloading of Scaled Design 19 in Stage 2 1685.27 Neutral Axis During Unloading of Scaled Design 7 in Stage 2-2 1695.28 Neutral Axis During Unloading of Scaled Design 19 in Stage 2-2 1695.29 Neutral Axis During Unloading of Scaled Design 7 in Stage 3 1705.30 Neutral Axis During Unloading of Scaled Design 19 in Stage 3 1705.31 Midspan Moment versus Vertical Deflection at Sections A and E
for Stage 1 (Scaled Design 7) 1715.32 Comparisqn of Experimental and Analytical Results at Section E
for Stage 1 (Scaled Design 7) 1715.33 Midspan Moment versus Vertical Deflection at Sections B and D
for Stage 1 (Scaled Design 7) 1725.34 Comparison of Experimental and Analytical Results at Section D
for Stage 1 (Scaled Design 7) 1725.35 Midspan Moment versus Vertical Deflection at Section C
for Stage 1 (Scaled Design 7) 1735.36 Comparison of Experimental and Analytical Results at Section C
for Stage 1 (Scaled Design 7) 1735.37 Midspan Moment versus Vertical Deflection at Sections A and E
for Stage 1 (Scaled Design 19) 1745.38 Comparison of Experimental and Analytical Results at Section E
for Stage 1 (Scaled Design 19) 1745.39 Midspan Moment versus Vertical Deflection at Sections B and D
for Stage 1 (Scaled Design 19) 1755.40 Comparison of Experimental and Analytical Results at Section D
for Stage 1 (Scaled Design 19) 1755.41 Midspan Moment versus Vertical Deflection at Section C
for Stage 1 (Scaled Design 19) 1765.42 Comparison of Experimental and Analytical Results at Section C
for Stage 1 (Scalcd Design 19) 1765.43 Midspan Moment versus Vertical Deflection at Sections A and E
for Stage 2 (Scaled Design 7) 1775.44 Comparison of Experimental and Analytical Results at Section E
for Stage 2 (Scaled Design 7) 1775.45 Midspan Moment versus Vertical Deflection at Sections Band D
for Stage 2 (Scalcd Design 7) 178
IX
5.46 Comparison of Experimental and Analytical Results at Section Dfor Stage 2 (Scaled Design 7) 178
5.47 Midspan Moment versus Vertical Deflection at Section Cfor Stage 2 (Scaled Design 7) 179
5.48 Comparison of Experimental and Analytical Results at Section Cfor Stage 2 (Scaled Design 7) 179
5.49 Midspan Moment versus Vertical Deflection at Sections A and Efor Stage 2 (Scaled Design 19) 180
5.50 Comparison of Experimental and Analytical Results at Section Efor Stage 2 (Scaled Design 19) 180
5.51 Midspan Moment versus Vertical Deflection at Sections B and Dfor Stage 2 (Scaled Design 19) 181
5.52 Comparison of Experimental and Analytical Results at Section Dfor Stage 2 (Scaled Design 19) 181
5.53 Midspan Moment versus Vertical Deflection at Section Cfor Stage 2 (Scaled Design 19) 182
5.54 Comparison of Experimental and Analytical Results at Section Cfor Stage 2 (Scaled Design 19) 182
5.55 Midspan Moment versus Vertical Deflection at Sections A and Efor Stage 3 (Scaled Design 7) 183
5.56 Comparison of Experimental and Analytical Results at Section Efor Stage 3 (Scaled Design 7) 183
5.57 Midspan Moment versus Vertical Deflection at Sections B and Dfor Stage 3 (Scaled Design 7) 184
5.58 Comparison of Experimental and Analytical Results at Section Dfor Stage 3 (Scaled Design 7) 184
5.59 Midspan Moment versus Vertical Deflection at Section Cfor Stage 3 (Scaled Design 7) 185
5.60 Comparison of Experimental and Analytical Results at Section Cfor Stage 3 (Scaled Design 7) 185
5.61 Midspan Moment versus Vertical Deflection at Sections A and Efor Stage 3 (Scaled Design 19) 186
5.62 Comparison of Experimental and Analytical Results at Section Efor Stage 3 (Scaled Design 19) 186
5.63 Midspan Moment versus Vertical Deflection at Sections B and Dfor Stage 3 (Scaled Design 19) 187
5.64 Comparison of Experimental and Analytical Results at Section Dfor Stage 3 (Scaled Design 19) 187
5.65 Midspan Moment versus Vertical Deflection at Section Cfor Stage 3 (Scaled Design 19) 188
5.66 Comparison of Experimental and Analytical Results at Section Cfor Stage 3 (Scaled Design 19) 188
5.67 Midspan Moment versus Vertical Deflection at Sections A and E(Scaled Design 7) 189
x
5.68 Midspan Moment versus Vertical Deflection at Sections B and D(Scaled Design 7) 189
5.69 Midspan Moment versus Vertical Deflection at Section C(Scaled Design 7) 190
5.70 Midspan Moment versus Vertical Deflection at Sections A and E(Scaled Design 19) 190
5.71 Midspan Moment versus Vertical Deflection at Sections B and D(Scaled Design 19) 191
5.72 Midspan Moment versus Vertical Deflection at Section C(Scaled Design 19) 191
5.73 Initial Imperfections at Midspan 1925.74 FEM Simulation Results for Model SD7-1 1935.75 FEM Simulation Results for Model SD7-2 1935.76 FEM Simulation Results for Model SD7-3 1945.77 FEM Simulation Results for Model SD7-4 1945.78 FEM Simulation Results for Model SD7-5 1955.79 FEM Simulation Results for Model SD7-6 1955.80 FEM Simulation Results for Model SD7-7 1965.81 FEM Simulation Results for Model SD7-8 1965.82 FEM Simulation Results for Model SD7-9 1975.83 FEM Simulation Results for Model SD7-10 1975.84 FEM Simulation Results for Model SD7-ll 1985.85 FEM Simulation Results for Model SD7-12 1985.86 FEM Simulation Results for Model SD19-1 1995.87 FEM Simulation Results for Model SD19-2 1995.88 FEM Simulation Results for Model SD19-3 2005.89 FEM Simulation Results for Model SD19-4 2005.90 FEM Simulation Results for Model SD19-5 2015.91 FEM Simulation Results for Model SD19-6 2015.92 FEM Simulation Results for Model SD 19-7 2025.93 FEM Simulation Results for Model SD19-8 2025.94 FEM Simulation Results for Model SD19-9 2035.95 FEM Simulation Results for Model SD19-7 (Including Post-Peak) 2035.96 FEM Simulation Results for Model SD 19-8 (Including Post-Peak) 2045.97 FEM Simulation Results for Model SD 19-9 (Including Post-Peak) 2045.98 FEM Simulation Results for Model SD19-8
(Including Initial Imperfections) 2055.99 FEM Simulation Results for Model SD 19-9
(Including Initial Imperfections) 2055.100 Schematic of FEM Simulation Results for Model SD19-8 2065.101 Schematic ofFEM Simulation Results for Model SD19-9 2065.102 Comparison of Experimental and Analytical Midspan Moment
versus Lateral Displacements (Scaled Design 7. Tube) 207
XI
5.103 Comparison of Experimental and Analytical Midspan Momentversus Lateral Displacements (Scaled Design 7, Tension Flange) 207
5.104 Comparison of Experimental and Analytical Midspan Momentversus Lateral Displacements (Scaled Design 19, Tube) 208
5.105 Comparison of Experimental and Analytical Midspan Momentversus Lateral Displacements (Scaled Design 19, Tension Flange) 208
5.106 Lateral Displacements of Scaled Design 7 (Tube, Stage 2) 2095.107 Lateral Displacements of Scaled Design 7
(Tension Flange, Stage 2) 2095.108 Lateral Displacements of Scaled Design 19 (Tube, Stage 2) 2105.109 Lateral Displacements of Scaled Design 19
(Tension Flange, Stage 2) 2105.110 Lateral Displacements of Scaled Design 7 (Tube, Stage 2-2) 2115.111 Lateral Displacements of Scaled Design 7
(Tension Flange, Stage 2-2) 2115.112 Lateral Displacements of Scaled Design 19 (Tube, Stage 2-2) 2125.113 Lateral Displacements of Scaled Design 19
(Tension Flange, Stage 2-2) 2125.114 Curvature throughout Web Depth for Stage 2 2135.115 Curvature throughout Web Depth for Stage 2-2 2135.116 Web Distortion 2145.117 Transverse Curvature Comparison (Tension Flange, Stage 2) 2145.118 Transverse Curvature Comparison (Tension Flange, Stage 2-2) 2155.119 Strain Gages Used to Study Plate Bending in Tension Flange of
Scaled Design 19 216
Xll
Abstract
Two different innovations to steel I-shaped highway bridge girders were
investigated in this thesis: (1) concrete filled tubular flanges and (2) corrugated webs.
Concrete filled tubular flanges make the section stiffer and stronger in bending
than a plate flange with the same amount of steel. The concrete filled tubular flange
increases the lateral torsional buckling capacity of the girder, and allows for a reduced
number of interior diaphragms over the length of the bridge.
Corrugated webs can be thinner than unstiffened flat webs, and therefore
lighter in weight. If a flat web were to be designed with the same thickness as a
corrugated web, then transverse stiffeners would be required. By eliminating the
transverse stiffeners, stiffener fabrication effort and Category C' fatigue details are
eliminated.
A design study was performed for tubular flange girders with corrugated webs
and with flat webs for a four girder, 131.23 ft. (40000 mm) prototype bridge. Two
girders were scaled down by a 0.45 factor, fabricated, and tested to investigate their
ability to carry their design loads. Also, experimental results were compared to
analytical results to verify the adequacy of the analytical models and tools.
TIle design study showed that tubular flanges allow for the use of large girder
unbraced lengths by increasing the torsional stiffness of the girder. The corrugated
web designs were only slightly lighter than their flat web counterparts for the 131.23
ft. (40000 mm) prototype bridge, with a girder length-to-depth ratio of approximately
22. Corrugated webs would be more efficient for deeper girders.
Experimental results showed that the test girders could effectively carry their
design loads, even for conditions with no interior diaphragms within the span.
Experimental results showed nonlinearity in the moment versus strain and moment
versus vertical deflection curves due to the presence of residual stresses in the steel.
After adjustments were made for the presence of the residual stresses, experimental
results compared quite well with analytical results.
Experimental lateral displacement results were generally smaller than those
predicted by Finite Element Method (FEM) simulations. Friction during testing and
uncertainty in actual test girder initial imperfections are possible reasons for this
result.
1. Introduction
1.1 Background
Two different innovations to steel I-shaped highway bridge girders are
investigated in this thesis: (I) concrete filled tubular flanges and (2) corrugated webs.
The behavior of I-shaped girders with tubular flanges and corrugated webs is
investigated with emphasis on the flexural behavior under highway bridge
construction and service conditions. A design study was conducted, and 0.45 scale
girders were fabricated and tested. The motivation for the study is as follows.
Concrete Filled Tubular Flanges
Concrete filled tubular flanges provide several advantages over traditional
plate flanges. Owing to the concrete within the tube, concrete filled tubular flanges
make the section stiffer and stronger in bending than a plate flange with the same
amount of steel. Also, the web depth is reduced when compared to an I-shaped girder
of the same total depth, which reduces web slenderness effects. Finally, the concrete
filled tubular flange increases the torsional stiffness, and therefore the lateral torsional
buckling capacity of the girder. Lateral torsional buckling is a flexural limit state for
non-composite bridges, as well as composite bridges during construction. before the
deck is composite with the girders. The increased lateral torsional buckling capacity
of tubular flange girders allows for an increased spacing of diaphragms. and therefore
a reduced number of diaphragms between girders.
Corrugated Webs
Corrugated webs have several advantages over traditional flat webs.
Corrugated webs can be designed to be thinner than unstiffened flat webs, and
therefore lighter in weight. If a flat web were to be designed with the same thickness
as a corrugated web, then transverse stiffeners would be required. In this case,
corrugated webs reduce the fabrication cost and effort involved with cutting and
welding numerous transverse stiffeners. Also, by eliminating the transverse stiffeners,
Category C' fatigue details are eliminated.
1.2 Objectives
The objectives of this research are: (1) to conduct a design study of tubular
flange girders with corrugated webs and with flat webs for a four girder, 131.23 ft.
(40000 mm) prototype bridge, (2) to design 0.45 scale test girders based on the results
of this design study, (3) to test the scaled girders to investigate their ability to carry
their design loads, and (4) to compare experimental and analytical results to verify the
adequacy of the analytical models and tools.
1.3 Approach
Rectangular tubular flange girder designs were studied because it is much
easier to attach a corrugated web to a rectangular tube than a round tube. A design
study of various combinations of rectangular tubular flange girders with corrugated
4
webs and flat webs was performed. This included composite and non-composite
designs, hybrid and homogeneous designs, as well as braced and unbraced designs.
The designs were generated based on modified AASHTO LRFD Bridge Design
Specifications (1999). Elastic section calculations were performed using equivalent
transformed sections to include the concrete in the tube and deck with the steel in the
girder cross-section properties.
One corrugated web girder design and one flat web girder design were scaled
down by a 0.45 factor and fabricated for use in a two-girder test specimen. The test
specimen was loaded to simulate various design loading conditions, and data was
recorded. Experimental results were compared with analytical results, and it was
determined that the use of modified AASHTO LRFD specifications and the use of
equivalent transformed sections to include the concrete in the cross-section properties
were adequate for design and analysis of tubular flange girders.
1.4 Thesis Outline
Chapter 2 discusses the design methodology used to design the tubular flange
girders in this research. Chapter 3 presents and discusses the results of the design
study. Chapter 4 discusses the selection of the girders to be scaled into test girders.
the scaling process, and design details for the test girders. It also describes the test
procedure and instrumentation used for the test specimen. Chapter 5 presents the
experimental results from the testing and compares these results with analytical
5
results. Chapter 6 summarizes the work and presents conclusions and
recommendations for future work.
6
2. Background
2.1 Previous Research
Previous research performed on tubular flange girders and corrugated web
girders is discussed here. Previous work by Smith (2001) and Kim (2004a) on tubular
flange girders, and by Easley (1975), Elgaaly et al. (1996), and Abbas (2003) on
corrugated web girders is reviewed. Although this section is not comprehensive in
discussing previous research on corrugated web girders, the summary provides
sufficient background for this thesis.
Smith (2001)
Smith (2001) performed design studies of four prototype bridges: (1) a four
girder prototype bridge with conventional composite I-girders, (2) a four-girder
prototype bridge with composite tubular flange girders, (3) a four-girder prototype
bridge with non-composite tubular flange girders, and (4) a through-girder prototype
bridge with two tubular flange girders. All prototype bridges were simple span
bridges with a span of 131.23 ft. (40000 mm). The designs were generated using High
Performance Steel (HPS) girders, including HPS-70W and HPS-I OOW steel. All
tubular flanges were round.
The design studies were perfonned according to AASHTO LRFD Bridge
Design Specifications (1998). Modifications. which \\ill be discussed later. were
made to the AASHTO LRFD specifications in order to account for the use of tubular
7
flanges. Constructability, Service II, Strength I, and Fatigue load combinations and
corresponding limit states were considered when generating the designs.
The results of the design studies showed that tubular flange girders are lighter
and need fewer diaphragms than conventional composite I-girders with flat plate
flanges. Also, as the number of diaphragms and/or transverse stiffeners is increased,
the girder weight will decrease. However, increasing the number ofdiaphragms
and/or transverse stiffeners increases the cost and effort involved in fabrication.
Kim (2004a)
Kim (2004a) is currently finishing a Ph.D. dissertation on tubular flange
girders for bridges. Preliminary design criteria were developed for tubular flange
girders. These criteria are compatible with AASHTO LRFD specifications. These
design criteria were used in the design study by Smith (2001), as well as the design
study presented in this thesis. Kim (2004a) performed design studies of three
prototype bridges: (I) a four-girder prototype bridge with conventional composite 1
girders, (2) a four-girder prototype bridge with composite tubular flange girders, and
(3) a four-girder prototype bridge with non-composite tubular flange girders. All
prototype bridges were simple span bridges with a span of 131.23 ft. (40000 mm).
Constructability, Service II, Strength L and Fatigue load combinations and
corresponding limit states were considered when generating the designs. The designs
used High Performance Steel (HPS) girders. including HPS-70W and HPS-l OOW
s
steel. Designs were optimized to minimize weight. All tubes used in this study were
round.
The Finite Element Method (FEM) package ABAQUS was used to perform a
parameter study of tubular flange girders. This parameter study varied the diameter
to-thickness ratio of the tube and the depth-to-thickness ratio of the web in order to
observe how the cross-section geometry affects the behavior of tubular flange girders.
This study was primarily used to investigate lateral torsional buckling capacity and
ultimate flexural strength of the tubular flange girders.
Tests were performed to verify the results of the FEM analyses. A prototype
bridge girder design was scaled by 0.45 for testing. Upon completion of this research,
tubular flange girder design criteria will be recommended that are compatible with the
AASHTO LRFD specifications.
Easley (1975)
Easley (1975) investigated the elastic shear buckling of light-gage corrugated
metal shear diaphragms. This work has application to corrugated webs for bridge
girders, as discussed later. Three different equations for elastic shear buckling
strength are discussed: (1) the Easley-McFarland equation, (2) the Bergmann-Reissner
equation. and (3) the Hlavacek equation. These equations provide global shear
buckling strength for buckling modes that occur over several folds of the corrugation
shape. Easley studied the theoretical derivation of each equation. and found that the
Easley-McFarland and Bergmann-Reissner equations are essentially the same for most
9
practical applications of light-gage corrugated metal shear diaphragms. The Hlavacek
equation provides results that differ by about 20%. Experimental results support the
theoretical results obtained by the Easley-McFarland and Bergmann-Reissner
equations.
Elgaaly et al. (1996)
Elgaaly et al. (1996) reported the results of shear tests on corrugated web
girders. It was observed that girders with corse corrugations would fail locally in a
single fold, whereas girders with dense corrugations would fail globally over several
corrugation folds. Finite Element Method (FEM) analyses were performed, which
support the results of the experiments. Some of the FEM analytical results provided
shear strengths higher than observed in the experiments, but this difference was due to
initial imperfections in the web. When these web imperfections were included in the
FEM analyses, the analytical and experimental results were very close.
Elgaaly et al. (1996) provided equations for calculating the elastic global and
local shear buckling strengths. When the elastic shear buckling stress is above 80% of
the shear yield stress, additional equations are provided for calculating inelastic global
and local shear buckling strengths. Abbas (2003) has ShO\\'l1 these equations are
unconservative for stocky webs, as discussed below.
IO
Abbas (2003)
Abbas perfonned a rigorous equilibrium analysis of corrugated web girders
under bending and shear forces. It was detennined that in-plane loading will cause
corrugated web girders to twist out of plane. This occurs because the corrugated web
is non-prismatic, and the shear is not always acting through the shear center. This
twisting is resisted by flange transverse bending. Finite Element Method (FEM)
analyses and large scale testing were perfonned to verify the results of the equilibrium
analysis.
Large scale fatigue tests were also perfonned. It was detennined that the
fatigue strength of the corrugated web girders that were studied is greater than that of a
Category C' fatigue detail, but less than that of a Category B fatigue detail. In other
words, the fatigue strength is greater than that of a conventional flat web girder with
transverse stiffeners, but less than that of a flat web girder with no transverse
stiffeners.
Theoretical shear strength equations for corrugated webs were developed by
Elgaaly et al. (1996), based on work by Easley (1975) and on experiments and FEM
analyses. However, using existing test data, Abbas showed that these equations were
unconservative for stocky web sections. This result was supported by additional large
scale shear tests and FEM analyses.
Based on this research. recommendations were made regarding the design of
corrugated webs for flexure, fatigue. and shear. These design criteria were used in this
thesis. and will be discussed later.
11
2.2 Tubular Flange Girders vs. Conventional I-Shaped Girders
Advantages of Tubular Flange Girders
As mentioned in Section 1.1, girders with concrete filled tubular flanges have
several advantages over conventional I-girders with flat plate flanges. The concrete
filled tubular flanges make the section stiffer and stronger in bending than a plate
flange with the same amount of steel. Also, the web depth is reduced when compared
to an I-shaped girder of the same total depth, which helps to reduce the web
slenderness. Finally, the concrete filled tubular flange increases the torsional stiffness,
and therefore increases the lateral torsional buckling capacity of the girder. This
increase in lateral torsional buckling capacity is the topic of the next section.
Lateral Torsional Buckling Strength on-Shaped Girders and Tubular Flange Girders
AASHTO LRFD Bridge Design Specifications (1999) currently provide
equations for the lateral torsional buckling strength of I-shaped girders. Lateral
torsional buckling strength depends on the unbraced length. For girders with slender
webs, the lateral torsional buckling strength is described by the AASHTO LRFD
specifications for three different ranges of unbraced length: (I) an elastic lateral
torsional buckling range. (2) an inelastic lateral torsional buckling range, and (3) a
yield range.
To be controlled by elastic lateral torsional buckling. the unbraced length of a
slender web girder. Lt>. must be greater than Lr :
12
Iycd EL, = 4.44 ----
S xc Fyc(Eq.2.1)
where, lye is the moment of inertia of the compression flange of the steel section about
a vertical axis in the plane of the web, d is the depth of the steel section, Sxe is the
section modulus about the horizontal axis of the section to the compression flange, E
is the modulus of elasticity of steel, and Fye is the yield stress of the compression
flange.
The elastic lateral torsional buckling strength of a girder with a stocky web is:
(Eq.2.2)
where, eb is a correction factor that accounts for moment gradient over the unbraced
length, Rh is a hybrid factor that accounts for a nonlinear variation of stresses when the
web has a lower yield stress than the flanges, Kr is the St. Venant torsional stiffness of
the girder, and My is the yield moment for the compression flange.
For slender web I-shaped girders, the S1. Venant torsional stiffness is taken as
zero. A slender web will distort, and therefore a slender web girder cross-section is
assumed to have little torsional stiffness. The elastic lateral torsional buckling
strength of a girder with a slender web is:
(Eq.2.3)
where. Rh is a factor that accounts for nonlinear variation of stresses caused by local
buckling of slender webs in flexure.
13
For a slender web girder controlled by inelastic lateral torsional buckling (Lp :S
Lb:S Lr), the AASHTO LRFD specifications define the strength by a linear transition
(Eq.2.4)
where, Lp is the unbraced length limit, below which the girder can yield in bending
(i.e., reach My) without lateral torsional buckling:
(Eq.2.5)
where, rt is the radius of gyration of the compression flange taken about the vertical
axIS.
To take advantage of the St. Venant torsional stiffness of the tubular flanges,
tubular flange girders should be designed with stocky webs. The web slenderness
limit for stocky webs is:
(Eq.2.6)
where, Dc is the depth of the web in compression in the elastic range and tw is the
thickness of the web. Ab is equal to 5.76 when Dc is less than half the web depth or
4.64 when Dc is greater than half the web depth.
The AASHTO LRFD specifications consider only elastic latcral torsional
buckling or yielding for stocky web girders. By neglccting inelastic buckling. thc
lateral torsional buckling strcngth. howcvcr. can bc seriously overestimated when Lb is
between Ll1 and Lr. Therefore. Kim (2004a) proposed that the inelastic straight line14
transition of Equation 2.4 be used for stocky web girders as well as slender web
girders. In this equation, Rb is set equal to 1.0 for stocky webs. Lr, for stocky web
girders is redefined as:
(Eq.2.7)
In summary, tubular flange girders are designed for lateral torsional buckling
using Equation 2.2, Equation 2.4 (with Rb=l.O), Equation 2.5, Equation 2.6, and
Equation 2.7.
2.3 Additional Consideration for Tubular Flanges
A tubular flange should not buckle locally before yielding in compression. The
following tube slenderness limit, provided by the AASHTO LRFD specifications for
rectangular tube compression members, was used:
(Eq.2.8)
where, b is the width of a tube wall, t is the wall thickness, E is the modulus of
elasticity. and Fy is the yield stress of the tube steel.
2.4 Corrugated Web Girders vs. Conventional Flat Web Girders
Advantages of Corrugated Web Girders
Corrugated webs have several advantages over traditional flat webs. They can
be designed to be thinner than unstitTened flat webs. and therefore lighter in weight. If
15
a flat web were to be designed with the same thickness as a corrugated web, then
transverse stiffeners would be required. In this case, corrugated webs reduce the
fabrication cost and effort involved with cutting and welding numerous transverse
stiffeners. Also, by eliminating the transverse stiffeners, Category C' fatigue details
are eliminated.
Shear Strength of Girders with Unstiffened Flat Webs
The shear design criteria for girders with unstiffened flat webs are outlined in
the AASHTO LRFD specifications. The unstiffened flat web shear strength equations
summarized here were used in this research. The nominal shear resistance, Vn, of an
unstiffened flat web is:
(Eq.2.9)
where, C is the ratio of the shear buckling stress to the shear yield stress, and V p is the
shear yield force, given by:
v = F'.I..... DtI' .fj " (Eq. 2.10)
where, Fyw is the yield stress of the web steel, D is the web depth, and tw is the web
thickness. The web depth-to-thickness ratio is used to determine the value of C. This
determines whether the web will yield in shear, buckle in the inelastic range. or buckle
in the elastic range.
If then
C =1.016
(Eq.l.ll)
and the shear strength is controlled by yielding.
If then
(Eq.2.12)
and the shear strength is controlled by inelastic buckling.
If then
(Eq.2.13)
and the shear strength is controlled by elastic buckling. In these equations, E is the
modulus of elasticity of steel and k is a shear buckling constant controlled by the web
boundary conditions.
The stiffened flat web shear equations include the effects of tension field
action, provided by the stiffeners. These equations are not presented here because
stiffened flat web designs were not generated for the design study presented in this
thesis.
Shear Strength Equations for Girders with Corrugated Webs
Shear design criteria for corrugated web girders have been developed by Sause
et al. (2003). The shear strength of a corrugated web may be controlled by yielding.
local buckling. or global buckling. Local and global buckling can be elastic or17
inelastic buckling. Local buckling is concentrated in a single corrugation fold with
deformation in the adjacent folds, whereas global buckling spans many corrugation
folds.
The elastic local buckling stress can be determined using classical plate
buckling theory. The elastic global buckling stress can be determined by treating the
corrugated web as an orthotropic plate, based on the work of Easley (1975). An
empirical equation was presented by Elgaaly et al. (1996) to calculate the inelastic
buckling stress for both local and global buckling. This equation is to be used if the
elastic buckling stress is greater than 80% of the shear yield stress. However, Abbas
(2003) showed that these equations do not provide an adequate lower bound to
existing test results. It appears that when the failure is governed by inelastic local
buckling or yield, the test results are lower than calculated by the empirical equation
presented by Elgaaly et al. (1996). Therefore, an interaction equation has been
proposed to better model the test results (Abbas 2003).
In the design criteria developed by Sause et al. (2003), a web slenderness
criterion is imposed to make the calculated global elastic shear buckling stress equal to
1.25 times the shear yield stress. Therefore, corrugated webs of bridge girders are not
permitted to buckle globally. The web slenderness criterion developed by Sause et al.
(2003) was derived for a trapezoidal corrugation. whereas most of the designs
generated in this research used a triangular corrugation. Therefore. the following.
more general equation. was derived for any corrugation shape:
18
(Eq.2.14)
where, Dx and Dy are the flexural rigidities of an orthotropic plate model of the
corrugated web about the weak and strong axes, respectively. Dx and Dy can be
calculated using the method presented by Easley (1975).
If Equation 2.14 is satisfied, then the shear strength is governed by elastic
local buckling, inelastic local buckling, or yield. The dimensionless parameter shown
below determines which of these shear failure modes is used to calculate the shear
strength:
(Eq.2.15)
where, w is the corrugation fold width for a triangular corrugation or the maximum
fold width for a trapezoidal corrugation. If AL ~ 2.586, then the corrugated web will
yield in shear. The shear strength is:
(F,.... )
V" =0.707 jj DI.. (Eq.2.16)
If 2.586 ~ AL ~ 3.233, then inelastic local shear buckling controls, and the shear
strength is:
V = I (F\"')D" 2 I"I + 0.1 50}./. .J3
(Eq.2.17)
If I'L::: 3.233. then elastic local shear buckling controls. and the shear strength is:
I (F,... )V = 4 (;; DI"" I+ 0.01 43)./. v3
19
(Eq.2.18)
As shown in these shear strength equations, the corrugated web shear strength
does not decrease as the web depth-to-thickness ratio increases. Thus, corrugated web
designs can be thinner than flat web designs. There are issues, however, that must be
considered for a corrugated web girder to be more efficient than a flat web girder.
These issues are discussed in Section 3.6.
2.5 Additional Considerations for Corrugated Web Girders
There are additional design considerations for corrugated web girders. These
considerations are flexural strength of corrugated web girders under overall bending,
the fatigue strength of corrugated web girders, and flange transverse bending moments
created by corrugated webs.
Overall Bending of Corrugated Web Girders
It is often assumed that corrugated webs do not carry flexural stresses due to
overall bending. The corrugated web behavior under axial stresses is similar to an
accordion. Therefore, flexural stresses do not develop in the corrugated web. This
assumption was verified by Abbas (2003).
Fatigue Strength of Corrugated Web Girders
Abbas (2003) determined that the fatigue strength of corrugated web girders is
greater than that of a Category C' fatigue detail. but less than that of a Category B
fatigue detail. In other words. the fatigue strength is greater than that of a flat web
20
with stiffeners, but less than that of an unstiffened flat web. The AASHTO LRFD
specifications provide the following equation for fatigue resistance:
(Eq.2.19)
For corrugated web girders, Abbas (2003) recommended using A equal to 6lx108 ksi3
(20xI0 11 MPa\ based on Category B' of the AASHTO LRFD specifications. N is the
number of fatigue cycles to be applied in the design life of the bridge. (i1F)TH is the
constant amplitude fatigue threshold value, which is recommended as 14 ksi (96.5
MPa) by Abbas (2003).
Flange Transverse Bending Moments Created by Corrugated Webs
Abbas (2003) showed that in-plane loading acting on a corrugated web will
cause a corrugated web girder to twist out of plane. This is because the corrugated
web girder is non-prismatic, and the shear does not always act through the shear
center. This resulting twisting moment is carried by flange transverse bending (Abbas
2003).
The corrugated web design criteria developed by Sause et al. (2003) provide an
equation for calculating flange transverse bending moments for a trapezoidal
corrugated web. Most of the designs generated in this thesis have triangular
corrugated webs. so the equation is presented here in its general form (Abbas 2003):
21v IIJI,I = ~d Ao
21
(Eq.2.20)
where, M, is the flange transverse bending moment, Vref is a reference vertical shear
associated with overall bending in the span, 0 is the web depth, and Ao is the
accumulated area under a half wave of the corrugation shape. The flange transverse
bending moment actually varies as the accumulated area varies along the corrugation,
but Ao provides the maximum effect. Sause et al. (2003) recommended using either
25% of the maximum shear in the span or the shear design envelope value at a given
cross-section for Vref. Abbas (2003) proposed that the maximum shear in the span be
used as Vref because of the numerous factors that can influence the value of M,. These
factors include things such as web misalignment and non-uniform web geometry. The
maximum shear in the span was used for this research. Equation 2.20 assumes the
following: (1) shear is constant over the length of a corrugation; (2) there are a large
number ofcorrugations in the span; (3) the girder is braced with diaphragms at the
ends of the span; and (4) the girder bearings are located at the center of inclined folds.
Equation 2.20 is applicable for the following: (1) the girder span contains an even
number of half corrugation wavelengths, regardless of the existence of interior
diaphragms; or (2) the girder span contains an odd number of half corrugation
wavelengths, but is braced by at least one interior diaphragm (if only one or two
interior diaphragms are provided. then they must be equally spaced).
The flange transverse bending moments must be amplified (to account for
second order effects) in the following manner for compression flanges:
()M,
,\I, AW = \I1__'_"_
STtFcr
..,..,
(Eq. 2.21)
where, (Mt)AMP is the amplified flange transverse bending moment, Mu is the factored
overall bending moment, and Sxe is the section modulus to the compression flange. Fer
is calculated using the equation below:
(Eq.2.22)
where, E is the modulus of elasticity of steel, Lb is the girder unbraced length, and rt is
the radius of gyration of the flange about a vertical axis through its midpoint.
The design criteria developed by Sause et a1. (2003) describe, in detail, the
method in which to incorporate the effects of flange transverse bending moments into
corrugated web girder designs. The key concepts will be summarized here. The
stresses created by flange transverse bending moments are to be superimposed with
the stresses from overall bending moments. This is to be done in both compression
and tension flanges. When the compression flange is composite with the deck, the
flange transverse bending moments in the compression flange can be neglected. For
situations where the girder is designed to be linear elastic, the stress superposition is
straight forward. When the plastic strength of a girder is considered, flange transverse
bending moments created by the corrugated web are treated similarly to flange
transverse bending moments from wind loads (AASHTO LRFD 1999). Under plastic
conditions. the flange transverse bending moments will create fully yielded regions at
the flange tips. The flange force due to overall bending must then be placed on the
remaining section. discounting the yielded regions. The \\idth of the yielded region.'- -.,'..... -' '-
b". at each edge of the flange is:
(Eq.2.23)
where, br is the width of the flange, tf is the thickness of the flange, and Fyr is the yield
stress of the flange.
2.6 AASHTO LRFD Bridge Design Specifications
The designs developed in this research were based on the AASHTO LRFD
Bridge Design Specifications (1999) for I-sections in flexure, with the modifications
discussed in Sections 2.2 through 2.5. This section presents an overview of the
general design equation, limit states, loads, load combinations, and important
calculations involved in the girder designs presented in this thesis.
General Design Equation
The general design equation of the AASHTO LRFD specifications is:
(Eq.2.24)
where Qi refers to force effects from various loads, and Rn refers to the resistance of
the specific bridge component. Yi is a statistically based load factor that generally
increases the value on the left side of Equation 2.24. and ~ is a statistically based
resistance factor that generally reduces the value on the right side of Equation 2.24. 11
is a load modifier based on the ductility. redundancy, and importance. Equation 2.24
states that the factored loads must be less than or equal to the factored resistance.
24
Limit States and Load Combinations
The AASHTO LRFD specifications require four limit states be considered for
steel I-girders. These are the Strength limit state for flexural resistance, the Service
limit state, the Fatigue and Fracture limit state, and the Strength limit state for shear
resistance. The investigation of Constructability is also required, though this is not
specifically identified as a limit state. These limit states can be reached under
different loading conditions, and each loading condition is identified by a Roman
numeral after the limit state name. The limit states that were investigated for the
designs generated in this thesis are Strength I, Strength III, Strength V, Service II,
Fatigue, and Constructability.
Strength I is the set ofloading conditions that relate to the normal use of the
bridge without wind. Flexural strength and shear strength are investigated under
Strength I. Strength III is the set of loading conditions for a bridge exposed to winds
exceeding 55 mph (90 km/hr), with a reduced live load. Strength V is the set of
loading conditions for normal use of a bridge with 55 mph (90 km/hr) winds. Flexural
strength is investigated under Strength III and Strength V. Service II is the set of
loading conditions under which yielding and permanent deformation of the steel
structure is prevented. Fatigue is the set ofloading conditions investigated to prevent
failure from repetitive load cycles. Constructability, which is termed "Construction"
in this thesis. relates to the loads to be considered in investigating the incomplete
bridge under construction.
25
The AASHTO LRFD specifications use 2-letter symbols to refer to different
loads. The loads considered in this thesis are DC, OW, LL, 1M, and WS. DC is the
dead load of the structural components and attachments. OW is the superimposed
dead load of wearing surfaces and utilities. LL is the live load created by specified
combinations of a Design Truck, Design Tandem, and Design Lane loads. 1M is a
dynamic load allowance applied to LL. WS is the wind load on the bridge.
The girder designs presented in this thesis are based on the four-girder
prototype bridges with composite tubular flange girders developed by Smith (200 I).
The bending moments and shears used to design girders in the present research are
those calculated by Smith (200 I) for girders with circular tubular flanges and flat
webs. Due to the different girder geometry, these load effects are only approximately
accurate (but sufficiently accurate) for the designs developed in this thesis.
The AASHTO LRFD specifications describe a set of loads and load factors to
create the load combination that should be considered for each limit state. These load
combinations are listed below.
Strength I
1.25DC + 1.50DW + 1.75(LL + 1M)
Strength III
I.25DC + I.50DW+ I.40WS
Strength V
I.25DC + I.50DW + 1.35(LL + 1M) + OAOWS
26
(Eq.2.25)
(Eq.2.26)
(Eq.2.27)
Service II
I.OODC + I.OODW + I.30(LL + 1M)
Fatigue
0.75(LL + 1M)
(Eq.2.28)
(Eq.2.29)
In addition, AASHTO LRFD specifications require the bridge design engineer to
investigate the Constructability of a design. For this purpose, a load combination was
established called the "Construction" load combination, given below.
I.5DC (Eq.2.30)
Strength I Limit State - Flexure
As discussed further in Chapter 3, composite and non-composite designs were
investigated in this thesis. For composite designs, the girders and an effective width
of the deck contribute to the moment carrying capacity. For non-composite designs,
the girders alone carry the moment, but they are assumed to be laterally braced by the
deck. In either case, lateral torsional buckling is not a flexural limit state to be
considered under the Strength I loading conditions. The compression flange (concrete
filled tube) was deemed compact by satisfying the tube slenderness limit given in
Section 2.3. The flat web was deemed compact by satisfying the AASHTO LRFD
specifications for web compactness. This web compactness specification was not
applied to corrugated web girders because the corrugated web will not experience web
buckling under bending stresses due to the "accordion effect" mentioned in Section
2.5.
27
A compact section should be able to develop the full plastic moment.
However, for composite sections the AASHTO LRFD specifications require a
concrete ductility check that reduces the ultimate flexural strength below the plastic
moment, based on empirical equations. For tubular flange girders, the ultimate
flexural strength is calculated using a strain compatibility analysis (Smith 2001). The
flexural strength from this analysis is less than the full plastic moment. The concrete
ductility check of the AASHTO LRFD specifications is not required. In the strain
compatibility analysis, the concrete is represented by an equivalent rectangular stress
block and the steel exhibits elastic-perfectly plastic behavior. Ultimate strength is
reached when the strain is 0.003 at the top of the deck for composite sections, or at the
top of the tube concrete for non-composite sections.
Strength III and Strength V Limit States - Flexure
The Strength III and Strength V limit states include wind load. The wind load
is applied laterally to the bridge, and it is assumed that the load on the lower half of
the exterior girder is carried by the bottom flange, creating a transverse bending
moment in the flange between bearings or benveen diaphragms. Wind load on the
upper half of the girder is carried by the deck to the end diaphragms and bearings. For
compact sections. the AASHTO LRFD specifications allow the transverse flange
moment to be carried by yielded portions at the bottom flange tips. The equation to
calculate the \\idth of these yielded portions was presented in Section 2.5. The flange
28
force due to overall bending must then be placed on the remaining section, discounting
the yielded regions.
Strength I Limit State - Shear
The Strength I limit state for shear was investigated using either the flat web or
corrugated web shear strength design criteria discussed in Section 2.4.
Constructability
For Constructability, lateral torsional buckling of a girder before the girder was
composite with the deck was investigated using design equations discussed in Section
2.2. The Construction load combination (1.5 DC) was used in this investigation. In
all cases investigated in this thesis, the unbraced length was greater than Lp, and
therefore, lateral torsional buckling controlled. Equations presented in Section 2.2
were used to calculate the lateral torsional buckling strength.
Service II
Calculations of the elastic section properties were perfonned using a
transfom1ed section for concrete filled tubular flange girders, in the composite and
non-composite conditions. In these calculations, the concrete was transfom1ed to an
equivalent area of steel using the modular ratio. For the Service II limit state check.
stresses on a composite section due to the Service II load combination were calculated
using a three-step process. The factored DC moment was applied to the steel girder
29
and tube concrete. The factored DW moment was applied to the long term composite
section and the factored LL moment was applied to the short term composite section.
The long term composite section -is based on an increased modular ratio (by a factor of
3) to account for creep that will occur over time in the concrete. The tube concrete is
neglected for both the short and long term composite section calculations because it is
assumed that the tube concrete is fully stressed by the DC moment (Smith 2001). The
purpose of this limit state check is to confirm that yielding will not occur in the
flanges under the Service II load combination.
Fatigue
AASHTO LRFD specifies two types of Fatigue limit states: load induced
fatigue and distortion induced fatigue. The load induced fatigue is investigated using
Equation 2.19. The constants used in the equation were based on whether the girder
is a corrugated web girder or a flat web girder. The values to be used for a corrugated
web girder were discussed in Section 2.5, and those for a flat web girder with
stiffeners are provided in the AASHTO LRFD specifications. The corrugated and flat
web girders with interior diaphragms have diaphragm connection plates at midspan
only, as discussed in Chapter 4. Distortion induced fatigue was not directly
considered after it was detemlined that it would not control the designs.
30
3. Prototype Bridge Design Study
3.1 Introduction
Chapter 2 introduced previous research on tubular flange girders and
corrugated web girders. Lateral torsional buckling of conventional I-shaped girders
and tubular flange girders was discussed. Shear strength of conventional I-shaped
girders and corrugated web girders was discussed. Corrugated webs create transverse
bending moments in the flanges of corrugated web girders, which must be taken into
account in design. The relevant AASHTO LRFD Bridge Design Specifications (1999)
were briefly discussed, as well as the modifications that must be applied to the
AASHTO LRFD specifications in order to design rectangular tubular flange girders
with corrugated webs. In this chapter, a prototype bridge is introduced, and the design
process used in conjunction with the modified AASHTO LRFD specifications is
discussed. To facilitate this design process, MathCAD files were developed to
evaluate a girder design according to these modified AASHTO LRFD specifications.
All of the designs generated for the study are described and compared in detail. Also,
the efficiencies of corrugated webs are investigated.
3.2 Prototype Bridge
The prototype bridge is the full scale bridge for which all designs in this study
are generated. The prototype bridge is 131.23 ft. (40000 mm) long, simply supported,
and a single span. It has a 50 ft. (15240 mm) wide, 10 in. (254 mm) thick concrete
31
deck. There are four girders, equally spaced at 12.5 ft (3810 mm). The deck
overhangs are 6.25 ft. (1905 mm) wide. The bridge carries two 12 ft. (3658 mm)
traffic lanes, and has 13 ft. (3962 mm) on either side for a shoulder and parapet. The
loads, however, were generated assuming four 11.5 ft. (3505 mm) design lanes with 2
ft. (610 mm) on either side for a shoulder and parapet. This resulted in more
conservative designs, which would be adequate for a future change in use. Figure 3.1
shows the prototype bridge. The geometry of this prototype bridge has been used in
other recent research at Lehigh University (Smith 2001, Kim 2004a). Therefore, the
results of the present study can be compared with those of the other studies. More
importantly, testing has been perfonned on 0.45 scale girders designed for this
prototype bridge (Kim 2004a). Thus, for testing purposes, two prototype designs from
the present research were scaled to 0.45 offull size so that the same footings, deck,
and loading procedure could be used, as discussed in Chapter 4. The prototype girders
have concrete filled rectangular tubular compression flanges. The tube and deck
concrete have an ultimate compressive strength of 6 ksi (40 MPa).
3.3 Limit State Ratios
As mentioned in the chapter introduction (Section 3.1), MathCAD files were
developed and used to generate the designs in this study. The MathCAD files were
based on the modified AASHTO LRFD specifications (Chapter 2). In the files, limit
state ratios were used as indicators of whether certain design criteria were satisfied. A
limit state ratio is calculated by dividing factored loads by factored resistance for a
32
specific limit state. Thus, a ratio less than 1.0 indicates that the design criterion for
this limit state is satisfied. The limit state ratios used in this design study, and the
design criteria that they refer to, are listed below.
Ratiolnexure - Strength I limit state for flexure
RatioIIInexure - Strength III limit state for flexure
RatioVflexure - Strength V limit state for flexure
Ratioshear - Strength I limit state for shear
Ratiowebslendemess - AASHTO LRFD proportion limit for web slenderness
Ratiotensionflange - AASHTO LRFD proportion limit for tension flange
Ratioflangebracing - ratio of Lbto Lp (a ratio over 1.0 requires a check oflateral
torsional buckling strength)
Ratioltbresistance - Constructability considering lateral torsional buckling under
the Construction load combination
RatioserviccIl - Service II limit state for prevention of yield
RatiofatigucCW - Fatigue limit state for corrugated web-to-tension flange detail
Ratiofatigucconnplatc - Fatigue limit state for transverse stiffener-to-tension flange
detail
Ratiotubcthickncss - compactness check for tube
3.4 Design Process
The design process involved choosing a section depth. choosing a web depth
and thickness. choosing the tube size. and choosing a bottom flange width and
thickness. The design decisions were made in that order. These design decisions are
discussed below.
Section Depth
Generally, bridge girder length-to-depth ratios are kept between twenty and
thirty. Making the corrugated web deep and thin made it more economically efficient
(Sect. 3.8). With this in mind, a length-to-depth ratio close to twenty was appealing.
It was decided to use a combined tube and web depth of70 in. (1778 mm), which gave
a length-to-depth ratio for the steel section of approximately 22. This fixed depth
reduced the extensive number of designs that could have been generated.
Web Depth and Thickness
The tube depth dictated the web depth. The three tube depths considered are 4
in. (101.6 mm), 6 in. (152.4 mm), and 8 in. (203.2 mm). It was desired to obtain a
minimum weight design, so the design process was iterated three times, once using
each tube depth. The iteration that provided the minimum weight was selected as the
final design. The web depth changed for each iteration so that the combined tube and
web depth remained consistent at 70 in. (1778 mm). Once the web depth was decided,
a web thickness was chosen in order to satisfy the necessary shear strength criteria for
the Strength I limit state. The web thickness was varied by 1/16 in. (1.6 mm).
All web designs generated in the design study were unstiffened flat webs. This
allowed for a direct comparison of weight between corrugated web and flat web
34
designs. Also, the webs of tubular flange girders are required to be stocky so that the
girders benefit from the torsional stiffness of the tube, as discussed in Chapter 2.
Thus, flat webs could not have the large depth-to-thickness ratio that is necessary for
the inclusion of transverse stiffeners to significantly increase the shear strength.
Tube Size
Before selecting the tube size, an approximate tension flange size was
specified so that the tube could be designed. With the tube depth already selected,
only the tube width and thickness had to be selected. The tube width was chosen so
that the girder would satisfy the criteria for lateral torsional buckling. The tube
thickness was chosen to satisfy Equation 2.8, which made the tube compact. Tube
thicknesses of3/8 in. (9.5 mm), 1/2 in. (12.7 mm), and 5/8 in. (15.9 mm) were
investigated. The tube sizes used in the designs were selected from a list of tubes
suggested by industry advisors to the project, and will be referred to as "suggested"
tube sizes.
Tension Flange Width and Thickness
The tension flange was designed last. The tension flange was likely to be
governed by the Fatigue or Service II limit states. The thickness was varied by 1/4 in.
(6.4 mm). and the width was varied by I in. (25.4 mm). A maximum of2 in. (50.8
mm) was set for the flange thickness. and the maximum width was governed by the
AASHTO LRFD tension flange proportion limit. In most cases. several combinations
35
of width and thickness satisfied the limit states. Thus, the minimum weight/minimum
area tension flange was chosen.
Other Checks
After all of the cross-section dimensions were selected, the final design was
then checked to make sure that it satisfied all remaining limit states. This included a
check to make sure that Strength III and Strength V were satisfied.
It was of interest to determine how much the initial approximation of the
tension flange size would affect the selected tube size. For several cases, a large initial
flange size and a small initial flange size were tried in order to observe the effect.
This never caused a change in the tube size, and therefore the initial approximation of
the tension flange size had little effect on the design.
3.5 Types of Designs
At the earliest stages of the design study, certain important design conditions
were uncertain. These design conditions governed the types of girders that were
designed. These design conditions are discussed here, followed by a detailed
discussion of the hybrid girder considerations.
Design Conditions
One design condition is the use of composite or non-composite conditions for
designing the bridge for service conditions. A composite design uses shear connectors
36
attached to the top flange that allow the concrete deck to act compositely with the
girders. An effective width of the concrete deck thus contributes to the load carrying
capacity of the cross-section. For a non-composite design, the loads are carried by the
girders alone. For tubular flange girders, shear connectors must be attached to the
tube.
A second design condition is the use of homogeneous or hybrid girder sections.
A homogeneous cross-section employs a single strength steel in the section whereas a
hybrid section uses different strength steels.
A third design condition is the diaphragm arrangement. Diaphragms ·provide
torsional bracing to the girders. As mentioned in Section 1.1, a primary advantage of
the tubular compression flange is the added torsional stiffness, which increases the
lateral torsional buckling strength. By increasing the lateral torsional buckling
strength, the number of necessary interior diaphragms is decreased.
Given these uncertain design conditions, a range of different designs were
developed. This included composite and non-composite, and homogeneous and
hybrid designs. Designs with and without interior diaphragms at midspan were
developed. Also, as mentioned in Chapter 1, both corrugated web and unstiffened flat
web designs were generated. Considering several important combinations of the
design conditions discussed above, twelve total types of designs were studied.
37
Homogeneous and Hybrid Designs
The homogeneous designs use ASTM A709 Grade 50W steel with a minimum
yield stress of 50 ksi (345 MPa), which is the most commonly used steel strength in
bridge design. The hybrid arrangement that was considered incorporates a tube made
of ASTM A588 steel, with a minimum yield stress of 50 ksi (345 MPa), coupled with
a web and bottom flange made from ASTM A709 Grade HPS 70W steel with a yield
stress of 70 ksi (485 MPa). The higher strength steel allowed the bottom flange to be
smaller and the web to be thinner, thus resulting in a lower weight design. It is
difficult to obtain tubes with 70 ksi (485 MPa) strength, so the complete 70 ksi (485
MPa) design was not considered. In addition, the tube design was controlled by lateral
torsional buckling. For the comparatively large unbraced lengths (i.e., 65.62 ft.
(20000 mm) and 131.23 ft. (40000 mm» used in this study, lateral torsional buckling
is more affected by the tube geometry than the steel strength.
The 70 ksi (485 MPa) web was not efficient for the hybrid unstiffened flat web
designs because the depth-to-thickness ratio for the flat webs caused them to be
governed by elastic shear buckling. In the elastic shear buckling range, the buckling
strengths of a 70 ksi (485 MPa) flat web and a 50 ksi (345 MPa) flat web are the same,
because the elastic buckling strength depends only on the web dimensions. Figure 3.2
illustrates the unstiffened shear strength ('tnu) of 50 ksi (345 MPa) and 70 ksi (485
MPa) flat webs versus the web depth-to-thickness ratio (D/t\\). Note that the curves
converge in the clastic shear buckling range.
38
3.6 Selection of Corrugated Web Geometric Parameters
Before corrugated web designs were generated, the corrugated web geometric
parameters for the 50 ksi (345 MPa) and 70 ksi (485 MPa) webs were established.
The parameters are the corrugation shape, corrugation angle (a), and corrugation fold
width (w). Figure 3.3 illustrates the shape and parameters chosen for this study. The
parameters are discussed below.
Corrugation Shape
The decision to use a triangular shape was based on work performed by Abbas
(2003). In deriving a C-Factor Correction Method for calculating flange transverse
bending moments caused by vertical shear acting on the corrugated web, C was
defined as the ratio of the area under one-half wave of a corrugation shape to the area
under one-half wave of an equivalent sinusoidal corrugation shape. Equivalent
corrugation shapes have the same wavelength (q) and corrugation depth (hr). Abbas
(2003) detennined that flange transverse bending moments for a specific corrugation
shape could be calculated by multiplying the C factor by the flange transverse bending
moments calculated for the equivalent sinusoidal corrugation. It was evident that a
triangular shape would minimize the flange transverse bending moments, thus
prompting the choice for this design study.
39
Corrugation Angle
It was observed during preliminary studies of corrugated webs that the global
buckling capacity of a corrugated web is increased by increasing the angle change
between two successive corrugation folds. However, the fatigue life of a girder may
be shortened when the angle is large (Abbas 2003). A value between 30 and 45
degrees is often used for the angle change between two successive corrugation folds,
and a value of 40 degrees was selected for this study. a, defined here for a triangular
shape as halfthe angle change between the successive folds, is 20 degrees.
Corrugation Fold Width
Given a prescribed tube and web depth of 70 in. (1778 mm), and a minimum
tube depth of 4 in. (101.6 mm), the maximum depth of the web is 66 in. (1676 mm). It
was assumed that the web will be proportioned such that the global and local shear
strength will be shear yielding. In this case, 0.707 times the shear yield stress and web
area must be greater than or equal to the factored shear force for the Strength I limit
state. This means that the minimum web thicknesses for the 50 ksi (345 MPa) and 70
ksi (485 MPa) webs are 7/16 in. (11.1 mm) and 5/16 in. (7.9 mm), respectively. Using
Equation 2.15 with AL equal to 2.586, the maximum corrugation fold width for the 50
ksi (345 MPa) and 70 ksi (485 MPa) webs are 27.25 in. (692.2 mm) and 16.45 in.
(417.8 mm). respectively.
To reduce fabrication effort. the maximum allowable fold width should be
used. In order to have an even number of half wavelengths (Section 2.5) in the
40
predetermined specimen of 131.23 ft. (40000 nun), the actual maximum fold width
was not used. However, the corrugation folds were sized very close to maximum
width, such that an even number of half wavelengths could be used. The web shape
and dimensions are shown in Figure 3.3. One corrugation geometry was used for all
50 ksi (345 MPa) webs, and one geometry was used for all 70 ksi (485 MPa) webs.
Calculations were performed, and Equation 2.14 was checked to confirm that
the global shear strength of these webs was shear yielding. The use of Equation 2.15
to calculate maximum fold width mandated that the local shear strength of the webs
was also shear yielding. The resulting corrugated webs are as efficient as possible.
3.7 Discussion of Designs
Tables 3.1 through 3.3 provide the geometry and limit state ratios for each
type of girder designed for the prototype bridge. Three types of designs were
generated: corrugated web girders designed without considering flange transverse
bending (Designs 1 through 6 shown in Table 3.1), conventional flat web girders
(Designs 7 through 12 shown in Table 3.2), and corrugated web girders designed
considering flange transverse bending (Designs 13 through 18 shown in Table 3.3).
These designs are discussed in the following sections.
Designs 1 through 6
Design 1 is a composite. corrugated web design. and the girder is
homogeneous. There are two end diaphragms. and no interior diaphragm. The first
41
two design iterations were performed with a 4 in. (101.6 mm) deep tube and a 6 in.
(152.4 rom) deep tube, respectively. It was determined that designs using the
suggested tubes with these depths could not provide sufficient lateral torsional
buckling strength. The third iteration was performed with an 8 in. (203.2 rom) deep
tube, which results in a 62 in. (1575 rom) deep web. A 7/16 in. (11.1 rom) thick web
was required to satisfy the shear strength criteria. A 20x8x5/8 in. (508x203.2x15.9
rom) tube satisfies the lateral torsional buckling criteria and a 27xl-3/4 in. (685.8x44.5
mm) tension flange satisfies the remaining limit states.
Design 2 is a composite, corrugated web design, and the girder is
homogeneous. There are three diaphragms, one at each end and one at midspan. The
interior diaphragm provides torsional bracing to the girder, and reduces the unbraced
length of the girders, therefore a smaller tube can be used. In this design, therefore,
either a 4 in. (101.6 mm) deep tube or a 6 in. (152.4 mm) deep tube can satisfy the
lateral torsional buckling criteria. These were the first and second iterations,
respectively. An 8 in. (203.2 mm) deep tube, however, led to the minimum weight
design. Once again, the 8 in. (203.2 mm) deep tube led to a 62 in. (1575 mm) deep
web, and the required web thickness of 7/16 in. (11.1 mm) is the same as for Design 1.
This time, however, a smaller tube satisfies the lateral torsional buckling criteria. The
required tube size is 16x8x3/8 in. (406.4x203.2x9.5 mm). A 27x 1-3/4 in. (685.8x44.5
mm) tension flange satisfies the rest of the limit states. It is interesting to note that the
only change to the design conditions from Design 1 to Design 2 was the addition of an
interior diaphragm. and this change affects only the size of the tubular flange.
42
Design 3 is a composite, corrugated web design, and the girder is hybrid. The
tube has a yield stress of 50 ksi (345 MPa), and the web and bottom flange both have a
yield stress of70 ksi (485 MPa). There are two end diaphragms, and no interior
diaphragm. The suggested 4 in. (101.6 mm) and 6 in. (152.4 mm) deep tubes could
not satisfy the lateral torsional buckling criteria. An 8 in. (203.2 mm) deep tube was
investigated. The web depth is 62 in. (1575 mm), and a 5/16 in. (7.9 mm) web
thickness is required to satisfy the shear criteria. A 20x8x5/8 in. (508x203.2x15.9
mm) tube satisfies the lateral torsional buckling criteria and a 22xl-l/2 in. (558.8x38.1
mm) tension flange satisfies the rest of the limit states. Notice that the use of a 70 ksi
(485 MPa) web allowed the thickness to decrease from 7/16 in. (11.1 mm) to 5/16 in.
(7.9 mm) when comparing Designs 1 and 3. Also notice that the use of a 70 ksi (485
MPa) tension flange allowed the cross-sectional area of the tension flange to decrease
from 47.25 in.2 (30484 mm2) to 33 in? (21290 mm2
).
Design 4 is similar to Design 3, the only difference being the addition of an
interior diaphragm. Design 4 is a composite, corrugated web design, and the girder is
hybrid. As seen previously, when an intermediate diaphragm was added, either a 4 in.
(101.6 mm) deep tube or a 6 in. (152.4 mm) deep tube can satisfy the lateral torsional
buckling criteria in the first and second iterations, respectively. An 8 in. (203.2 mm)
deep tube, however, led to the minimum weight design. The 62 in. (1575 mm) web
depth requires a 5/16 in. (7.9 mm) thickness to satisfy the shear criteria. The tube
required to satisfy the lateral torsional buckling criteria is smaller than in Design 3
because of the addition of the interior diaphragm. The required tube size is l6x8x3/8
43
in. (406.4x203.2x9.5 mm). A 25xl-l/2 in. (635x38.1 mm) tension flange satisfies the
rest of the limit states. Design 4 can be compared with both Designs 3 and 2. The
introduction of the interior diaphragm from Design 3 to 4 allows for the smaller tube.
The use of a hybrid girder rather than homogeneous girder when going from Design 2
to 4 allows for a thinner web and a smaller tension flange.
One of the possible design permutations is a homogeneous non-composite
design. However, the non-composite design must satisfy the Strength limit states
using only the girder and concrete within the tube. Preliminary studies showed the
size of the girder tension flange would be extremely large to satisfy all limit states.
Therefore, only the hybrid non-composite case was considered.
Design 5 is a non-composite, corrugated web design, and the girder is hybrid.
Just as in the composite hybrid designs, the tube has a yield stress of 50 ksi (345 MPa)
and the web and tension flange both have a yield stress of 70 ksi (485 MPa). There
are two end diaphragms, and no interior diaphragm. The suggested 4 in. (101.6 mm),
6 in. (152.4 mm), and 8 in. (203.2 mm) deep tubes could not satisfy the lateral
torsional buckling criteria. It seems that an 8 in. (203.2 mm) deep tube should be able
to satisfy the lateral torsional buckling criteria, just as in the previous designs. The
lateral torsional buckling strength under the Construction loading conditions is
independent of whether the bridge will be composite or non-composite in service.
However. the suggested 8 in. (203.2 mm) tubes do not satisfy the lateral torsional
buckling criteria because the estimated girder dead loads are higher for non-composite
designs since non-composite girders are generally heavier than composite girders.
44
Thus, it was necessary to use a deeper tube for Design 5. However, deeper
tubes were not on the list of suggested tubes provided by industry advisors to the
project. Therefore, the decision was made to use a tube size that, when scaled down
for testing, would be a tube on the list of suggested tubes (so that it could be used in
the test specimen). As mentioned earlier, the scale factor to be used for testing was
predetermined to be 0.45 to take advantage of an existing test setup. Three design
iterations were performed with 4 in. (101.6 mm), 6 in. (152.4 mm), and 8 in. (203.2
mm) tubes each scaled by 1/0.45. The 8.89 in. (225.8 mm) deep tube yielded the
lowest weight. The web was, therefore, 61.11 in. (1552 mm) deep, and a web
thickness of 3/8 in. (9.5 mm) was needed to satisfy the shear criteria. A tube size of
31.11 x8.89xO.83 in. (790.2x225.8x21.1 mm) satisfies the lateral torsional buckling
criteria and a 26xl-3/4 in. (660.4x44.5 mm) tension flange satisfies the rest of the limit
states.
Design 6 is a non-composite, corrugated web design, and the girder is hybrid.
The difference from Design 5 is the addition of an interior diaphragm. Suggested 4 in.
(101.6 mm), 6 in. (152.4 mm), and 8 in. (203.2 mm) tubes satisfied the lateral
torsional buckling criteria. However, none of them could satisfy the Strength I and
Service II flexural limit states. In non-composite design. increasing the tension flange
size decreases the demand on the compression flange only to a certain extent because
the corrugated web does not contribute to overall bending. Therefore. the size of the
compression and tension flanges need to be balanced for the Strength I limit state.
45
Once again, it was necessary to use tube depths other than those originally
suggested by the industry advisors to the project. The 8.89 in. (225.8 mm) deep tube
provided the minimum weight design. Interestingly, both Designs 5 and 6 have low
lateral torsional buckling limit state ratios because Strength I and·Service II control the
compression flange design. Since lateral torsional buckling was not a critical limit
state, the fact that one design has an interior diaphragm was of no consequence, and
the two designs were the same. For Design 6, the web depth and thickness are 61.11
in. (1552 mm) and 3/8 in. (9.5 mm), respectively. The tube size is 31.11x8.89xO.83
in. (790.2x225.8x21.1 mm) and the tension flange is 26x1 3/4 in. (660Ax44.5 mm).
Designs 7 through 12
Design 7 is a composite, unstiffened flat web design, and the girder is
homogeneous. There are two end diaphragms. The first two design iterations were
performed with a 4 in. (101.6 mm) deep tube and a 6 in. (15204 mm) deep tube,
respectively. Similar to the corrugated web designs, it was determined that none of
the suggested tubes with these depths could provide sufficient lateral torsional
buckling strength. The third iteration was successful, using an 8 in. (203.2 mm) deep
tube. The web depth of 62 in. (1575 mm) requires a thickness of 11/16 in. (17.5 mm)
to satisfy the shear strength criteria. A tube size of 20x8x5/8 in. (508x203.2x15.9
mm) satisfies the lateral torsional buckling criteria. A 24x1-1/2 in. (609.6x38.1 mm)
tension flange satisfies the rest of the limit states. This design can be directly
compared to Design 1. the only difference being the switch from corrugated web to
46
flat web. The flat web must be thicker than the corrugated web to carry the shear.
The thickness of the corrugated web is 7/16 in. (II.! mm), whereas the thickness of
the flat web is 11/16 in. (17.5 mm). On the other hand, the corrugated web girder
needed a bigger tension flange than the flat web girder because the corrugated web
does not carry overall bending stresses. The tension flange size for the corrugated web
girder is 47.25 in.2 (30483.8 mm2), whereas the tension flange size for the flat web
girder is 36 in.2 (23225.8 mm2).
Design 8 is also a composite, unstiffened flat web design, and the girder is
homogeneous. This design has three diaphragms, one at each end and one at midspan.
An interior diaphragm allowed successful designs to be generated using 4 in. (101.6),
6 in. (152.4), and 8 in. (203.2) deep tubes. The 8 in. (203.2 mm) deep tube provided
the minimum weight design. An 11/16 in. (17.5 mm) thick web is required to satisfy
the shear strength criteria. However, the 16x8x3/8 in. (406.4x203.2x9.5 mm) tube
required to satisfy the lateral torsional buckling criteria is smaller than used in Design
7 because of the shorter unbraced length. A 25xl-l/2 in. (635x38.1 mm) tension
flange is required to satisfy the rest of the limit states. Design 8 can be directly
compared to Design 2, to illustrate the differences between a corrugated web design
and a flat web design with all other design conditions held constant. The web
thickness is larger for the flat web design than for the corrugated web design, and the
tension flange area is larger for the corrugated web design than for the flat web design.
Design 9 is a composite, unstiffened flat web design, and is hybrid. Design 9
has two end diaphragms. The only successful design iteration was for a tube depth of
47
8 in. (203.2 mm). The web depth and thickness are 62 in. (1575 mm) and 11/16 in.
(17.5 nun), respectively. Note that the web is the same thickness as in the
homogeneous Design 7, because the shear strength was governed by elastic buckling.
The elastic buckling strengths of a 70 ksi (485 MPa) web and a 50 ksi (345 MPa) web
are the same. A 20x8x5/8 in. (508x203.2xI5.9 nun) tube satisfies the lateral torsional
buckling criteria, and a 21xl in. (533.4x25.4 mm) tension flange satisfies the rest of
the limit state ratios. The transition to hybrid did allow Design 9 to have a smaller
tension flange than Design 7. The differences between the corrugated web and flat
web designs are illustrated once more by comparing Designs 3 and 9. Design 9
requires a thicker web to carry the shear, but Design 3 requires a larger tension flange
to carry bending.
Design 10 is also a composite, unstiffened flat web design, and is hybrid. It is
different from Design 9 in that an interior diaphragm has been introduced. The web
depth and thickness are 62 in. (1575 mm) and 11/16 in. (17.5 mm), respectively. A
16x8x3/8 in. (406.4x203.2x9.5 mm) tube satisfies the lateral torsional buckling
criteria. This tube is smaller than the tube used in Design 9. The tension flange that
satisfies the rest of the limit states is 18x1-1 /4 in. (457.2x31.8 mm). Once again, the
web thickness of the hybrid Design 10 is the same as that of its homogeneous
counterpart, Design 8. The flange is smaller than that of Design 8, as would be
expected. Designs 4 and 10 again illustrate the differences between corrugated web
designs and flat web designs.
48
Design 11 is a non-composite, unstiffened flat web design, and is hybrid.
There are two end diaphragms. The first three design iterations were performed with 4
in. (101.6 mm), 6 in. (152.4 mm), and 8 in. (203.2 mm) deep tubes, respectively.
None of the suggested tubes with these depths could provide sufficient lateral torsional
buckling strength. Tube sizes that would scale by 0.45 to suggested tube sizes were
considered. The minimum weight design used an 8.89 in. (225.8 mm) deep tube. The
web depth and thickness are 61.11 in. (1552 mm) and 11/16 in. (17.5 mm)
respectively. The tube that satisfies the lateral torsional buckling criteria is
31.1 Ix8.89xO.83 in. (790.2x225.8x21.1 mm). A 28xI-l/4 in. (7I1.2x31.75 mm)
tension flange satisfies the rest of the limit states.
Design 12 is exactly the same as Design 11. This situation occurred before
with the non-composite corrugated web designs, Designs 5 and 6.
Designs 13 through 18
For the first set of twelve designs, the corrugated web flange transverse
bending moments were neglected. At this point, however, a second set of six
corrugated web designs that includes the effects of flange transverse bending moments
will be briefly discussed (see Table 3.3). The detailed geometry of the new designs
will not be discussed because there is very little change from the previous set of
corrugated web girder designs. The only change between Designs 1 through 6 and
Designs 13 through 18 is the size of the tension flange. The increased stresses in the
compression flange due to the Construction loading conditions were not significant
49
enough to require a larger tube. However, the increased stresses in the tension flange
due to Service II loading did require an increase in size.
Discussion of the Weights of the Designs
It is important to discuss the weights of the designs in more detail. For this
discussion, refer again to Tables 3.1 through 3.3. The girder designs are lighter when
an interior diaphragm is added, because the tube required to satisfy the lateral torsional
buckling criteria is smaller. For example, Design 14 is approximately 13% lighter
than Design 13. However, although the girder is lighter, the fabrication effort and cost
are increased by the necessary interior diaphragm.
In addition, the hybrid girders are lighter than the homogeneous girders. For
example, Design 15 is approximately 19% lighter than Design 13. However, the
weight savings must offset the cost of more expensive steel if the hybrid girders are to
be more economical. The non-composite designs are generally much heavier than the
composite designs because the girders alone carry the Strength I, Service II, and
Fatigue load combinations. The weight savings from using composite girders must
offset the cost and effort involved in making the girders composite with the deck if the
composite girders are to be more economical.
All of the corrugated web girders are lighter than their flat web counterparts.
This \vas one of the goals in setting the length-to-depth ratio close to 20. However.
some of the corrugated web girders are only slightly lighter than their flat web
counterparts. This is particularly evident when comparing Designs 13 and 7 or
50
Designs 14 and 8. The difference between Designs 13 and 7 is only 0.2% and the
difference between Designs 14 and 8 is only 0.5%. The savings in weight for these
designs do not seem to justify the added expense of forming and welding a corrugated
web. However, it is well known that corrugated web girders become more efficient
for deeper girders because the web depth-to-thickness ratio (D/tw) is not a controlling
factor as it is with flat web girders. This is discussed in more detail in Section 3.8.
Therefore, the corrugated web should prove to be more efficient for deeper designs.
The length-to-depth ratio of bridge girders is between 20 and 30, 20 being the deepest
practical design. Thus, to generate deeper designs, the prototype bridge must be
longer. The 131.23 ft. (40000 mm) prototype bridge was chosen because it had been
used for previous studies. The findings of this study, however, support a future design
study using a longer (e.g., 196.85 ft. (60000 mm)) prototype bridge.
3.8 Efficiency of Corrugated Web
As mentioned in Section 3.7, efficient design of corrugated web girders
(relative to flat web girders) was investigated. When the girder web is not deep
enough, a flat web of the same thickness as a corrugated web may have equal or
greater shear strength. It would be inefficient to corrugate a web in this situation. In
addition. since a corrugated web does not contribute to overall bending, corrugated
web girders require more steel in the tension flange than a similar flat web girder.
These issues are discussed below.
51
Efficiency of Corrugated Webs in Shear
The maximum design shear stress resistance of a corrugated web is 0.707 times
the shear yield stress of the web steel (Section 2.4). However, the maximum shear
stress capacity for an unstiffened flat web is the full shear yield stress. Therefore,
there is a clear range over which an unstiffened flat web has equal or greater strength
than a corrugated web. Though stated in Section 2.4, the AASHTO LRFD shear
strength specifications for flat webs are briefly reviewed here for convenience. The
nominal shear resistance of an unstiffened flat web is:
(Eq.3.1)
where, C is the ratio of the shear buckling stress to the shear yield stress. Vp is the
plastic shear force, or shear yield force, given by:
(Eq.3.2)
where Fyw is the yield stress of the web, D is the web depth, and tw is the web
thickness. The web depth-to-thickness ratio (Dltw) is used to calculate C, which in
tum, will detennine whether the web will yield in shear, buckle in the inelastic range,
or buckle in the elastic range. As mentioned earlier, given the appropriate web depth-
to-thickness ratio, C can have a value of one, thus giving the unstiffened flat web the
full shear yield strength.
Given the strength of the web steel. and the modulus of elasticity. the shear
strength can be plotted as a function of the web depth-to-thickness ratio. At a specific
depth-to-thickness ratio. the shear strength of an unstiffened flat web \\ill fall below
0.707 multiplied by the shear yield stress of the web. This is illustrated in Figure 3.4.52
If it is assumed that the corrugated web is designed as discussed in Section 3.6, with
AL less than or equal to 2.586, then for any web with a depth-to-thickness ratio larger
than the specific ratio mentioned above, the corrugated web will be stronger than the
unstiffened flat web. As stated earlier, two web steels with different yield stresses
were investigated in this research. These yield stresses were 50 ksi (345 MPa) and 70
ksi (485 MPa). For a 50 ksi (345 MPa) web, the depth-to-thickness ratio at which a
corrugated web becomes stronger than an unstiffened flat web is approximately 79.
For a 70 ksi (485 MPa) web, the depth-to-thickness ratio is approximately 67. Thus,
all corrugated web designs were checked to confirm that their depth-to-thickness ratios
were above the appropriate value.
Note that the results of this simple analysis do not imply that the web depth-to
thickness ratio of corrugated web girders can increase without consequence.
Depending on the corrugated web geometry, there is a maximum web depth-to
thickness ratio that can not be exceeded without the shear resistance being limited by
global web buckling. This ratio is quite large, however, and is not considered in
Figure 3.4. When considering only local buckling, as in Figure 3.4, the corrugated
web can have any depth-to-thickness ratio without a loss of strength.
Although the flat webs designed for the tubular flange girders described in
Section 3.7 were designed as unstiffened webs. a flat web which uses the minimum
amount of stiffeners is relatively easy to fabricate. The small number of stiffeners is
often considered negligible in tenns of cost. Often. stiffeners must be placed on webs
simply for the use of diaphragm connection plates. Thus. it should be detennined
53
when a corrugated web design is stronger than a flat web with a minimum number of
stiffeners. The AASHTO LRFD specifications provide the shear resistance of a
stiffened flat web. The nominal shear resistance is:
(Eq. 3.3)
where, Vp, C, and D are as defined earlier, and do is the stiffener spacing. The first
term in parentheses addresses shear buckling or yield, whereas the second term
addresses tension-field-action developed after shear buckling. Equation 3.3 may also
include a reduction factor based on the maximum moment within the shear panel,
however, for the present discussion, this factor is neglected.
The minimum number of stiffeners occurs when the stiffeners are spaced at the
AASHTO LRFD specified maximum stiffener spacing of three times the depth of the
web (i.e., dofD equal to 3). For this case, the shear strength can be plotted as a
function of the web depth-to-thickness ratio, and the depth-to-thickness ratio at which
the strength of a flat web with minimum stiffeners falls below the strength of a
corrugated web can be determined. For a 50 ksi (345 MPa) web, this depth-to-
thickness ratio is approximately 91. The 70 ksi (485 MPa) web is part of a hybrid
girder design. and the AASHTO LRFD specifications do not permit the use of tension-
field-action in the design of hybrid girders. and thus the shear strength of hybrid
girders is limited to that of an unstiffened flat web. Therefore. the results for an
54
unstiffened flat web and a flat web with stiffeners are the same. Figure 3.5 illustrates
these results.
Efficiency of Corrugated Webs, Considering the Tension Flange
The above discussions of corrugated web efficiency are not comprehensive in
making corrugated web girders efficient. They are simply a first check to eliminate
inefficient designs. Another step in the check of efficiency is to consider that
corrugated webs do not contribute to bending strength of a girder. Due to this lack of
bending strength contribution, a corrugated web girder must have a larger tension
flange than a girder with a flat web. Efficient corrugated web girder designs, relative
to flat web girder designs, will trade off steel from the web to the tension flange.
However, the reduction of steel in the web must be greater than the steel put back into
the tension flange. Otherwise, the corrugated web girder will be heavier and therefore
more expensive. Specific criteria have not been developed to address this issue.
55
Table 3.1 Prototype Corrugated Web Girder Designs (Neglecting Flange TransverseBending Moments)
DesiQn 1 2 3 4 5 6
Corruaated Web yes yes yes yes yes yes
Composite yes yes yes yes no no
Hybrid no no yes yes yes yes
Interior Diaphraam no yes no yes no yes
Dweb (in.l 62 62 62 62 61.11 61.11
Tweb (in.) 7/16 7/16 5/16 5/16 3/8 3/8
Tube Size (in.) 20x8x5/8 16x8x3/8 20x8x5/8 16x8x3/8 31.11 x8.89xO.83 31.11 x8.89xO.83
Bbf (in.) 27 27 22 25 26 26
Tbf(in.l 1-3/4 1-3/4 1-1/2 1-1/2 1-3/4 1-3/4
Ratioll\exuJl! 0.95 0.99 0.97 0.90 0.98 0.98
RatiolilnexuJl! 0.52 0.55 0.57 0.50 0.54 0.54
RatioVl\exuJl! 0.87 0.91 0.90 0.83 0.89 0.89
Ratio'he., 0.95 0.95 0.97 0.97 0.84 0.84
Ratio...b'Ie""eme.. NA NA NA NA NA NA
Ratio,en,oonflanne 0.64 0.64 0.61 0.69 0.62 0.62
Ratioflanaeb<3c>n" 5.71 3.60 5.71 3.60 3.76 1.88
RatioftbJ1!'~I.nce 0.93 0.88 0.95 0.89 0.49 0.46
Ratio,eMOOIl 0.99 0.99 1.00 0.95 1.00 1.00
Ratio,.,nueCw 0.68 0.67 0.96 0.84 0.82 0.82
Ratio,"",uP'COnnol.'e NA 0.78 NA 0.98 NA 0.95
Ratio,ube<hd.ne.. 0.72 0.98 0.72 0.98 0.85 0.85
WeiQhl (kips) 48.92 41.78 38.88 33.74 59.63 59.63NA =Not Applicable
56
Table 3.2 Prototype Conventional Flat Web Girder Designs
DeslQn 7 8 9 10 11 12
Corrugated Web no no no no no no
Composite yes yes yes yes no no
Hvbrid no no yes yes yes yes
Interior Diaphraqm no yes no yes no yes
Dv-Ieb (in.) 62 62 62 62 61.11 61.11
Tweb (in.) 11/16 11/16 11/16 11/16 11/16 11/16
Tube Size (in.) 20x8x5/8 16x8x3/8 20x8x5/8 16x8x3/8 31.11x8.89xO.83 31.11x8.89x0.83
Bbf (in.) 24 25 21 18 28 28
Tbf(in.) 1-1/2 1-1/2 1 1-1/4 1-1/4 1-1/4
Ratioln.xure 0.76 0.77 0.73 0.73 0.84 0.84
Ratiollln.xure 0.42 0.42 0.41 0.43 0.45 0.45
RatioVn.xure 0.70 0.70 0.67 0.67 0.76 0.76
Ratioshe., 0.79 0.79 0.79 0.79 0.81 0.81
Ratio...'bslendeme.. 0.35 0.50 0.30 0.44 0.20 0.20
Ratio,ensionnanne 0.67 0.69 0.88 0.60 0.93 0.93
Ration'noebn>ano 5.71 3.55 5.71 3.55 3.76 1.88
Ratio't><eSislance 0.94 0.88 0.95 0.91 0.48 0.46
RatioseMcell 0.99 0.99 0.98 0.98 1.00 1.00
Ratiof"",,ueCw NA NA NA NA NA NA
Ratio,at"",econnpl.te NA 0.72 NA 0.99 NA 0.96
Ratio,ube1hidn... 0.72 0.98 0.72 0.98 0.85 0.85
Weiqht (kips) 50.05 43.57 43.35 36.87 62.82 62.82NA = Not Applicable
57
Table 3.3 Prototype Corrugated Web Girder Designs (Incorporating FlangeTransverse Bending Moments)
Design 13 14 15 16 17 18
Corruqated Web yes yes yes yes yes yes
Comoosite yes yes yes yes no no
Hybrid no no yes yes yes yes
Interior Diaphraqm no yes no yes no yes
Dweb (in.) 62 62 62 62 61.11 61.11
Tweb (in.) 7/16 7/16 5/16 5/16 3/8 3/8
Tube Size (in.) 20x8x5/8 16x8x3/8 20x8x5/8 16x8x3/8 31.11 x8.89xO.83 31.11 x8.89xO.83
Bbf (in.) 33 29 24 19 32 32
Tbf (in.) 1-1/2 1-3/4 1-1/2 2 1-112 1-1/2
Ratiol tlexure 0.93 0.95 0.92 0.92 0.95 0.95
Ratioili tlerure 0.49 0.51 0.51 0.54 0.50 0.50
RatioVtlerure 0.84 0.87 0.85 0.86 0.86 0.86
Ratio'h... 0.95 0.95 0.97 0.97 0.84 0.84
RatiOw.bslond.m... NA NA NA NA NA NA
Ratio..n'ion',no. 0.92 0.69 0.67 0.40 0.89 0.89
Ratianann.tncono 5.77 3.60 5.77 3.60 3.79 1.89
Ratio'tn,"'.n", 0.98 0.99 0.98 0.99 0.49 0.46
Ratio,.",,,," 1.00 0.99 0.99 0.96 1.00 1.00
Ratio'a'"uoCW 0.64 0.63 0.88 0.83 0.78 0.78
Ratio'''"u.connol'''. NA 0.73 NA 0.97 NA 0.91
Ratio.UtlO'\hidnm 0.72 0.98 0.72 0.98 0.85 0.85
Weight (kips) 49.93 43.34 40.22 33.97 60.75 60.75NA =Not Applicable
58
50' (15240 rll'1)
2' (610 l'1l'1) 46' (14021 MM) 2' (610 rlrl)
hi 1rt;:J t:;:l l:';:l t;:l
-... -... -... -...6'-3' (1905 l'1l'1) 12'-6' (3810 l'1l'1) 12'-6' (3810 l'1rl) 12'-6' (3810 rlrl) 6'-3' 0905 l'1l'1)
Figure 3.1 Prototype Bridge
I
- ---J
--_._-----
- FW-unstiffened 50 ksi (345 MPa) I;I'I'
- FW-unstiffened 70 ksi (485 MPa) I!I
15.0 i~--------- ----,----------- ------~-------~~
10.0 ~--~--------- .......
5.0 ~
O.O:-~-"--~--·-- .~-~.---.--~--.~~-.-- ------"~-----.-. -
o 50 100 150 200 250
30.0 L -- ----
45.0
40.0 ~L====\
35.0 ~f- \ _
,~ 25.0 L .~ _~ ~-- ,~ 20.0:----------\
D/tw
Figure 3.2 Shear Strength of 50 ksi (345 MPa) and 70 ksi (485 MPa) Flat Webs
59
q=50.80' 0290 I'll'l)
(a) 50 ksi (345 MPa) Corrugated Web
q=30.88' 084.4 MI'l)
(b) 70 ksi (485 MPa) Corrugated Web
Figure 3.3 Prototype Bridge Corrugated Webs
_FW-unstiffened 50 ksi (345 MPa) i
--FW-unstiffened 70 ksi (485 MPa) .
35.0 -~------\----------i _CW 50 ksi (345 MPa),t _ CW 70 ksi (485 MPa)
30.0 -;'~;;;;;;;;;;~~::;::;===:::::::::::::~~:::::::~~-.
45.0 -;:-------- .------ ..-~.~===========,
f40.0 t:
f~'::!::!:::!::!::"~,
~ 25.0 -{-----------''''\------~ .
15.0
10.0 L __
5.0 ~------~----------~----~~---
0.0 .:- .--..--.-~--.-.--~.-~.-~--~-.~~-.-.~.. _.. ~~.~.__ L __ .--~.--.~
o 50 100 150 200 250D/tw
Figure 3.4 Comparison of Corrugated Web Shear Strength to Unstiffened Flat WebShear Strength for 50 ksi (345 MPa) and 70 ksi (485 MPa) Webs
60
25020015010050
__ FW-stiffened 50 ksi (345 MPa) I:
~~~'::!:::!~.-------J __ FW-unstiffened 70 ksi (485 MPa) 11
+------\-------1 _CW50ksi(345MPa) i~
- CW 70 ksi (485 MPa) iiII
45.0
40.0
35.0
30.0
~ 25.0:.
c 20.0l"
15.0
10.0
5.0
0.0
0D/tw
Figure 3.5 Comparison of Corrugated Web Shear Strength to Stiffened Flat WebShear Strength for 50 ksi (345 MPa) and 70 ksi (485 MPa) Webs
61
4. Test Specimen and Test Procedure
4.1 Introduction
In the previous chapter, the prototype bridge was described, and the results of a
design study were discussed. This design study investigated twelve different tubular
flange girder designs, which used various combinations of corrugated or flat web,
composite or non-composite, homogeneous or hybrid, and braced or unbraced
conditions. The tubular flange girder design process and the method used to establish
the corrugation geometry of the designs with corrugated webs were also discussed.
Also, design considerations for an economically efficient corrugated web were
discussed.
This chapter discusses the development of the test specimen and the testing
procedures used in this research. The first topic is the choice of prototype girder
designs to be scaled down into the test girders. A new corrugated web design is
introduced in order to make fabrication of the test specimen easier. The detailed
design of the test specimen is described, including stiffeners, welds, and shear studs.
The diaphragms used for the test specimen are those used previously by Kim (2004a).
These diaphragm designs were checked to make sure that they were adequate for the
test specimen. The loading conditions and instrumentation used in the tests are
discussed so that the reader can more fully understand the results provided in Chapter
5. Finally. the stress-strain properties of the materials used in the test girders. the
62
measured cross-section dimensions, and selected geometric imperfections of the test
girders are presented.
4.2 Choice of Test Girders
As mentioned in Section 4.1, the first task was to decide which prototype
girder designs to scale down into test girders. Composite designs were selected
because they are viewed as more economically efficient in practice. The deck
contributes to the load carrying capacity, and therefore composite girders are lighter
than non-composite girders. Tests that are valuable from an engineering practice
standpoint will encourage the future use of these types of girders. It was also decided
that a prototype girder design without an interior diaphragm should be used to provide
a good representation of the advantages of tubular flange girders.
For the composite designs with no interior diaphragm, estimated costs of the
homogeneous and hybrid designs were compared. It was determined that
homogeneous designs would be less expensive to fabricate because of the lower price
of the 50 ksi (345 MPa) steel. Fabrication of the test girders was performed by High
Steel Structures, Inc., located in Lancaster, Pennsylvania.
Industry advisors to the project from High Steel Structures and other
companies and agencies have some reservations about the economic efficiency of
corrugated web girders. The initial scope of this research project, however, was to
design and test concrete filled rectangular tubular flange girders with corrugated webs.
Thus. a compromise was reached in which the two test girders would consist of one
63
corrugated web girder and one flat web girder. Given all of the considerations
discussed above, it was decided that Designs 13 and 7 would be scaled down into test
girders, and these two test girders would be used in the test specimen.
4.3 Scaling Process
Scaling the Moment and Shear
Bending moment and shear were scaled to create the same stress levels in the
test girders as in the prototype girders. Flexural stress is calculated as follows:
MU=-
S(Eq.4.1)
where, cr is the flexural stress, M is the moment in the cross-section, and S is the
section modulus of the cross-section. The section modulus incorporates the
dimensional scale factor cubed, so the moment is multiplied by the dimensional scale
factor cubed. The shear stress is calculated as follows:
VQT=-
/1(Eq.4.2)
where, 't is the shear stress, V is the shear in the cross-section, and Q is the first
moment of the area, above or below the point in question, about the neutral axis. I is
the moment of inertia of the cross-section and t is the thickness of the cross-section at
the point in question. It is evident that using the scale factor on all girder dimensions
will create a net factor equivalent to the dimensional scale factor squared in the
denominator. TIms, the shear is multiplied by the dimensional scale factor squared.
64
Scaling the Prototype Girders
Initially, each girder dimension for prototype girder Designs 13 and 7 was
scaled by 0.45, and input along with the scaled moment and shear into the MathCAD
files which were used for design, in order to verify the scaling process. It was
observed that all limit state ratios were the same as they were for the corresponding
prototype girders. Simply scaling all girder dimensions by 0.45, however, does not
provide available tube sizes and plate thicknesses. It was therefore decided to choose
available tube sizes and plate thicknesses so that fabrication of the test specimen was
feasible. Some scaled dimensions were rounded up and others were rounded down in
order to keep limit state ratios similar to those of the prototype designs. Depths and
widths were chosen in 1/2 in. (12.7 mm) increments. Thicknesses were chosen in 1/16
in. (1.6 mm) increments. Also, the tube dimensions were chosen using input from
High Steel Structures regarding tube costs. A low cost tube was chosen over one that
provided limit state ratios closer to those ofthe prototype designs. This low cost tube
was slightly larger than necessary, and led to lower limit state ratios for lateral
torsional buckling. The scaled dimensions and limit state ratios are shown in Table
4.1. The scaled test girder cross-sections are shown with the corresponding prototype
girders in Figure 4.1.
When scaling the corrugated web, a (20 degrees) was not changed. but the
plate \\'idth (w) was multiplied by the 0.45 scale factor. The scaled webs are shO\\'TI in
Figure 4.2. Note that the shear limit state ratio for the 3/16 in. (4.8 mm) thick web of
scaled Design 13 is 1.02. If the web is incrementally increased in size by 1/16 in. (1.6
65
nun), then the shear limit state ratio falls to 0.75. The value of 1.02, though too large,
more closely matches the shear limit state ratio of the prototype Design 13, and since
the shear limit state would not be approached during testing, the 3/16 in. (4.8 mm)
web was maintained.
4.4 New Corrugated Web for Test Specimen
Due to the limited funding available for the project, it was desired to mitigate
fabrication effort and expense for the corrugated web test girder, and alternative ways
to obtain a corrugated web were considered. Six trapezoidal corrugated web girders
had been tested in fatigue at Lehigh University by Abbas (2003), and webs from these
girders were available for re-use. The corrugated web shape of these girders is shown
in Figure 4.3. a is the corrugation angle, b is the width of the longitudinal fold, c is
the width of the inclined fold, pis the ratio of the longitudinal fold width to the
inclined fold width, d is the projection of the inclined fold in the longitudinal
direction, hr is the corrugation centerline depth, q is the corrugation wavelength, and tw
is the thickness of the web. These webs were fabricated using steel with a 70 ksi (485
MPa) yield stress.
Several important design criteria were considered before these webs were re
used. As stated earlier, the corrugated web used in the test girder should have global
and local shear strength equal to the shear yield stress. In other words. neither global
nor local buckling could occur before the web yielded in shear. Shear strength design
criteria developed by Sause et al. (2003) show the follo\\ing modified version of
66
Equation 2.13, which guarantees that the global buckling shear strength of a
trapezoidal corrugated web is shear yielding:
(Eq.4.3)
where, D is the depth of the web, E is the modulus of elasticity of steel, Fyw is the
yield stress of the web material, and F(a,~) is:
F(a,[3)= (1+[3)sin3a{ 3[3+1 }3/4
[3 + cosa ,82 ([J +1)(Eq.4.4)
It was determined that the inequality of Equation 4.3 was satisfied. Also, it was
determined that AL:S 2.586 (Sect. 2.4), which guarantees that the local buckling shear
strength of a corrugated web is shear yielding. The test girder with the re-used web
was treated as a homogeneous design, even though the web material has a 70 ksi (485
MPa) yield stress. When assuming that the web has a yield stress of 50 ksi (345
MPa), the previous two inequalities are still satisfied.
The re-used trapezoidal corrugated web was incorporated into the MathCAD
files used for girder design, and new limit state ratios were obtained. These limit state
ratios are given in Table 4.2. The scaled Design 13 with the trapezoidal corrugated
web is hence forth called scaled Design 19. The deck thickness used in obtaining
these limit state ratios was 6 in. (152.4 mm) rather than the lOin. (254 mm) scaled by
0.45. for reasons that \..ill be discussed later. Therefore. new limit state ratios are also
provided in Table 4.2 for scaled Design 7 \\ith a 6 in. (254 mm) thick deck.
67
RatioIIIflexure and RatioVflexure were not reevaluated because the extra effort was not
justified.
The re-used corrugated web of scaled Design 19 was overstrength for shear, as
illustrated by the shear limit state ratio of 0.79. Note that the lateral torsional buckling
limit state ratio for scaled Design 19 was 0.93 with the re-used corrugated web. This
is because the flange transverse bending moments were increased by using a
trapezoidal web. The trapezoidal shape has an accumulated area under one half
corrugation (See Sect. 3.6) that is 3.9 times as large as that of the triangular shape.
Note that the Service II limit state ratio is 1.11 for scaled Design 19. Obviously, this
should be a concern for a girder that will be put in service, but is of little consequence
for the scaled Design 19. As described later in this chapter, the Service II load was not
a specific load condition applied during testing. As mentioned above, scaled Design
19 was treated as homogeneous, with the web yield stress assumed to be 50 ksi (345
MPa). No benefit was obtained from the actually greater web strength during testing.
Under the Strength I loading conditions, the girder was uniformly loaded to create a
moment of approximately 18500 kip-in (2.1 x109 N-mm) at midspan. This
corresponds to a maximum shear of approximately 104 kip (4.63x I05 N). The shear
strength of the corrugated \veb is approximately 135 kip (6.00xI05 N) when a 50 ksi
(345 MPa) yield stress is assumed for the web.
To re-use the trapezoidal webs from the previously tested girders. web pieces
were cut from three of the fatigue test girders and spliced together to provide the
necessary length of web. Webs from girders G3A. GSA. and G6A (Abbas. 2003)
68
were re-used for the scaled Design 19 test girder. Girders G3A and GSA had been
fatigue tested until cracking occurred. They were repaired and then tested without
further failure. G6A had failed early, and was never repaired. One of the girders had
a crack in the web at the end of a partial stiffener, and it can be viewed in Figure 4.4.
This crack was repaired by the fabricator. Figure 4.5 illustrates how the web pieces
were cut from the three girders and spliced together to provide a continuous
corrugated web. The dashed lines in the figure represent the cuts that were made.
Each corrugation wavelength is numbered and the N and S refer to North and South
longitudinal folds, respectively.
4.5 Design Details
4.5.1 Stiffener Designs
For scaled Designs 19 and 7, bearing stiffeners were required at the ends, and
diaphragm connection plates were required at midspan. Even though the prototype
girder Designs 13 and 7 were designed under the Construction loading conditions
without interior diaphragms, tests of scaled Designs 19 and 7 were planned both with
and without a midspan diaphragm present. Recall that scaled Design 19 has a
trapezoidal corrugated web and scaled Design 7 utilizes a flat web that was designed
without stiffeners (for shear). The flat web. however, was provided with additional
stiffeners at the midspan and quarter points. These were to prevent web distortion
(Kim 2004a). not to develop tension field action after shear buckling. Web distortion
69
is not considered to be an issue with the corrugated web. It was decided to design
these intermediate stiffeners to be the same as the bearing stiffeners.
The AASHTO LRFD specifications require that bearing stiffeners are
connected to both sides of the web, extend to the full depth of the web, and extend as
closely as practical to the outer edges of the flanges. In order to prevent local buckling
of the bearing stiffener plates, the following inequality must be satisfied:
b, S 0.48/pJ EF"'j
(Eq.4.5)
where, bl is the width of the stiffener, tp is the thickness of the stiffener, E is the
modulus of elasticity of steel, and Fys is the yield stress of the stiffener steel. It was
decided to use 50 ksi (345 MPa) steel for the stiffeners. Using Equation 4.5, with bl
chosen so that the stiffeners extend to the outer edge of the tubular flange, a stiffener
thickness of 0.50 in. (12.7 mm) was found to be satisfactory for both scaled Designs
19 and 7.
It is also necessary to verify that bearing stiffeners have the bearing resistance
to carry the reaction force at the bearing. In these calculations, the scaled maximum
shear in the test girders under the Strength I loading conditions was used as the
maximum value of the end reaction. The factored bearing resistance, Br• is:
(Eq.4.6)
where. Qb is the resistance factor for bearing (1.00) and Apn is the net area of the
stiffeners. taking into account the portion of stiffener that must be clipped to fit around
the flange-to-web fillet welds. This clip length was assumed to be 1 in. (25.4 mm).
70
Using Equation 4.6, it was determined that the stiffeners had satisfactory bearing
resistance.
Finally, the pair of bearing stiffeners was evaluated as part of an axial
compression member. This axial compression member consists of the stiffeners and a
portion of the web that extends nine times the thickness of the web on each side of the
stiffeners. The effective length of the axial compression member is 0.75 times the
web depth, due to buckling restraint provided by the flanges. The factored axial
compression resistance, Phis:
(Eq.4.7)
where, ~c is the resistance factor for compression (0.9), and Pn is the nominal axial
compression resistance. Note that the corrugated web bearing stiffeners were treated
as part of an asymmetric axial compression member because the stiffeners are attached
to an inclined fold. The stiffener geometry for scaled Designs 19 and 7 is presented in
Figures 4.6 and 4.7, respectively. St1 is a set of stiffeners located at a quarterpoint,
St2 is a set of midspan stiffeners, and St3 is a set of bearing stiffeners. St2 and St3
have bolt holes because they function as diaphragm connection plates. St I is located
on the flat web of scaled Design 7 to prevent web distortion. The stiffener locations,
as well as other information about scaled Designs 19 and 7. are illustrated in Figure
4.8.
71
4.5.2 Fillet Weld Designs
Tube Flange-to-Web Fillet Welds
The first welds considered were the tube flange-to-web and the bottom flange-
to-web fillet welds. Figure 4.9 provides illustrations for the discussion of these welds.
The design process is essentially the same for both flanges, but only the tube flange-
to-web fillet welds are shown in the figure. A unit longitudinal dimension of the
girder was considered for design.
The fillet welds were designed to resist web out-of plane bending along with
the horizontal shear transferred between the flange and web, and overall bending
stresses. The web was assumed fully plastified in plate bending, as shown in the
figure. The dimension, a, is the size of the fillet weld, and yielding of the weld is
assumed to occur along the throat. A tensile force, T, is developed on the throat of
one fillet weld, whereas a compressive force, C, is developed on the throat of the
other. Shear forces, Vz, also develop along the throat of the two fillet welds. The
shear and normal forces provide a vertical force resultant, F. The plastic moment of
the web, Mp, is a known quantity. The distance between forces, d, can be expressed as
a function of a. F is calculated using the following expression:
M"F=-d
(Eq.4.8)
Considering the tension side fillet weld. T and Vz can be expressed in terms of F. The
stresses, cry and 'tyz, are calculated by dividing force by the area.
The horizontal shear force carried by one fillet weld. Vx. is the lesser of one
half the fully yielded concrete filled tubular flange force or one half the fully yielded
72
web and bottom flange force. In the case of the corrugated web girder, the web was
not included in this calculation for reasons discussed previously. 'txy is calculated by
dividing Vx by the area. The stress calculations are:
T T(j =--=----
y area 0.707a(1)
v. V.T - • - •
>= - area - 0.707a(1)
Vx VxT ------.::.......,..-
" - area - O.707a(~J
(Eq.4.9)
(Eq. 4.1 0)
(Eq.4.11)
where, L is the span length of the girder. The stresses are substituted into Von Mises
yield criterion:
2 ( \2 ( )2 (2 2 2) 2((jx - (j y ) + (j y - (Y: J + (j: - (y x + 6 T X)' + T y: + Tr. = 2(j>p (Eq.4.12)
az and 'txz are assumed zero, so the only unknown is the contribution from overall
bending, ax. The yield criterion can be written as a function of ax, equal to the left
side minus the right side of Equation 4.12. For values of ax for which the function
remains negative, the distortion energy per unit volume in the state of combined stress
is less than that associated with yielding in a simple tension test, and therefore the
weld does not yield. An example of the function plotted versus ax is shown in Figure
4.10. The weld size was chosen so that the function ofax is negative over the overall
bending stress range that will be experienced by the girder at the weld location.
Figure 4.10 shows the function being negative for ax ranging from approximately -50
ksi (-345 :MPa) to 50 ksi (345 MPa). so the girder steel will yield in bending before the
73
weld metal does. The compression side of the tube flange-to-web fillet weld was
investigated in the same manner discussed above, but the tension side was determined
to be more critical. The calculations required the tube flange-to-web fillet welds to be
3/16 in. (4.8 mm), and the bottom flange-to-web fillet welds to be 1/4 in. (6.4 mm).
Tube Flange-to-Stiffener Fillet Welds
Next, the tube flange-to-stiffener fillet welds were investigated. Figure 4.11
provides illustration for the discussion of these welds. These welds are designed to
allow the stiffeners to develop their in-plane plastic moment, as shown in the figure.
This moment can develop as the stiffener helps the tube develop torsional moments to
restrain lateral torsional buckling of the girder (Kim 2004a). Based on the plastic
moment of the stiffeners, the force per unit length along the stiffener on the welds,
Fys·tp, was determined. Fys is the yield stress of the stiffener steel and tp is the
thickness of the stiffener. Three possible failure modes are shovm in Figure 4.11. In
Case I, there is tensile yielding due to normal stress on the horizontal faces of the
welds. The force per unit length required along the stiffener to cause this yielding, R1,
IS:
(Eq.4.13)
where, CJywcld is the yield stress of the weld and a is the size of the weld. Case 3
illustrates the situation where shear yielding occurs on the vertical faces of the welds.
The force per unit length required along the stiffener to cause this yielding. R3• is:
74
(Eq.4.14)
where, tyweld is the shear yield stress of the weld. Case 2 is combined shear and tensile
yielding on the weld throat, and can be analyzed using virtial work. A diagram of the
virtual displacements is also shown in Figure 4.11. Observing the virtual
displacements, it is evident that the force per unit length, R2, is:
R2 ·Li= 2(a(O.707a)Li a +r(O.707a)Lib)
After substitution for Lia and Lib, the following is obtained:
(Eq.4.15)
(Eq.4.16)
Using the Von Mises yield criterion, the following relationship between cr and t is
obtained:
(Eq.4.17)
Substitution of Equation 4.17 into Equation 4.16 yields:
(Eq.4.18)
R2 is a function of t, so the derivative is taken to find the minimum value of R2. This
is determined to be:
(Eq.4.19)
Therefore, Cases 2 and 3 provide the same resistance. and are the critical cases. A
weld size was chosen so that the resistance. R, was greater than the load. Fys·tp• The
calculations required the tube flange-to-stiffener welds to be 9/16 in. (14.3 mm). The
75
remainder of the weld designs were not complicated, and do not require extensive
explanation.
4.5.3 Selection of Deck
Initially a composite cast-in-place deck was considered for the test specimen.
Shear studs could be mounted to the tubular flanges, and then the deck could be
poured. However, this created some difficulties in testing the test specimen under
Construction loading conditions. As will be discussed in Section 4.6, the tests
performed under the Construction loading conditions required the girders to be loaded
in the non-composite state. If the deck was cast-in-place, the cast-in-place deck would
have to be thick enough for the concrete to provide the full Construction loading
conditions. After investigating this possibility, it was detennined that, due to
dimensional scaling and other factors, the deck would have to be 20 in. (508 mm)
thick. The prototype bridge deck thickness is only lOin. (254 mm) and the scaled
deck thickness should only be 4.50 in. (114.3 mm). Obviously, the excessive
thickness required did not seem practical.
Therefore, it was decided to re-use the pre-cast deck previously used by Kim
(2004a) for similar sized test specimens. Similar to the present research, the previous
research by Kim (2004a) used a 0.45 scale test specimen based on a 131.23 ft. (40000
mm) bridge, so the pre-cast deck was the correct length. The deck concrete had an
ultimate strength of 6 ksi (41.4 MPa). as assumed for the designs in the present
76
research. In addition, the pre-cast deck design was determined to be adequate for the
loading applied in the present research.
The existing deck consisted of six panels, 6 in. (152.4 rnm) thick by 156 in.
(3962 rnm) wide by 120 in. (3048 rnm) long. The panels are post-tensioned
longitudinally. The post-tensioning strands used by Kim (2004a) were 0.60 in. (15.2
rom) diameter seven-wire strand, with an ultimate tensile strength of 270 ksi (1862
MPa). A post-tensioning stress of 187 ksi (1289 MPa) was applied to each tendon.
Kim (2004a) used nine post-tensioning strands, but for reasons discussed in Section
4.5.5, only seven strands were used in the present study. Calculations were made to
verify that the lower number of post-tensioning strands would be adequate. The steps
taken to make this pre-cast deck composite with the girders are discussed in Section
4.5.4.
4.5.4 Deck Construction
As mentioned in the previous section, the deck was pre-cast, and therefore
steps were taken to make it composite with the girders. The first step was to core
holes in the deck panels so that shear studs could be placed through the holes and
welded to the girders. The hole size and location were determined based on the shear
stud design discussed in the next section. The second step was to make the deck level.
A laser level was used to establish a level plane. and the deck was shimmed to parallel
with this level plane. The third step was to post-tension the deck. The fourth step was
to place and weld the shear studs to the top of the tubular flange through the core
77
holes. Then, wood forms were built along the sides of the test girder tubes to form a
haunch, and grout was poured through the core holes to fill the haunch and the core
holes up to the top surface of the deck. Photographs ofa core hole and shear studs are
provided in Figures 4.12 and 4.13, respectively.
4.5.5 Shear Stud Design
The AASHTO LRFD specifications outline the design requirements for shear
studs. The ratio of the height to the diameter of the shear stud must not be less than
4.0. Also, the clear depth of concrete cover over the top of the shear stud must be at
least 2 in. (50.8 mm) and the shear stud must penetrate into the deck at least 2 in. (50.8
mm). Both 5 in. (127 mm) and 6 in. (152.4 mm) long shear studs were used, based on
the girder deflections under the weight of the pre-cast deck panels and requirements
for clearance and penetration. Shear studs must not be closer than four stud diameters
center-to-center transverse to the longitudinal axis of the girder, and the clear distance
between the edge of the top flange and the edge of the nearest shear stud must not be
less than I in. (25.4 mm). Given these requirements, as well as the fact that the studs
had to fit within a core hole, it was decided to use the four stud diamond shaped shear
stud arrangement illustrated in Figure 4.14. A 0.75 in. (19.1 mm) stud diameter was
chosen.
After the shear stud diameter and arrangement were chosen. the pitch of the
shear studs. p. was determined based on the fatigue limit state:
J!"Q < ll RZr
I - p
78
(Eq.4.20)
where, Vsr is the shear force range under live load plus impact for the fatigue limit
state, Q is the first moment of the transformed area of the slab about the neutral axis of
the short-term composite section, I is the moment of inertia of the short term
composite section, ng is the number of shear studs in a group (ng=4), and Zr is the
shear fatigue resistance of an individual shear stud determined from the AASHTO
LRFD specifications. All of the values used in Equation 4.20 were scaled values.
The inequality expressed in Equation 4.20 states that the shear flow at the de~k-to-
girder interface under the fatigue loading must be less than or equal to the resistance of
a stud group divided by the pitch. In addition to Equation 4.20, the center-to-center
pitch of the shear studs is not allowed to exceed 24 in. (609.6 mm) by the AASHTO
LRFD specifications. It was determined from Equation 4.20 that the pitch must be
less than 22.08 in. (560.8 mm).
The number of shear studs required by the strength limit state was investigated.
This number, n, is:
Vn=_hQ,
(Eq.4.21)
where, Vh is the nominal horizontal shear force, equal to the lesser of the plastic
strength of the deck or the girder, and Qr is the factored shear resistance of one stud
from the AASHTO LRFD specifications. Thus, Equation 4.21 states that the number
of studs required is the total shear divided by the shear strength of one stud. These
calculations showed that 39 shear studs were required between the points of zero and
ma..ximum moment. The pitch detemlined from the fatigue limit state was determined
to be more critical.
79
The pitch of the shear studs was not a flexible design choice because the core
hole locations were limited. There were various reinforcement bars in the pre-cast
deck panels that needed to be avoided. A core hole pattern was established, however,
that satisfies the strength limit state requirement (44 shear studs were used). The core
hole spacing is not constant, and does not always satisfy the fatigue limit state pitch
requirement. However, fatigue loading was not part of the test program (see Section
4.6). The core hole pattern is illustrated in Figure 4.15. The transverse reinforcing is
the prestressed strands and the longitudinal reinforcing is the post-tensioning strands.
The core holes interfered with two of the post-tensioning strands used previously by
Kim (2004a), so only seven strands were used, as noted earlier. Note that the shear
stud design was generated for the scaled Design 19 test girder, and the web was not
included in calculating the plastic strength of the girder. The shear stud design is not
conservative for scaled Design 7, which requires 52 shear studs between the points of
zero and maximum moment. However, scaled Design 7 was taken to only 67% of its
plastic moment during the testing.
4.6 Test Procedures
The third objective stated in Section 1.2 was to test the scaled girders to
investigate their ability to carry their design loads. The tests were performed outdoors.
due to the size of the test specimen and lack of space on the lab floor. The load was
applied by placing 24 x 24 x 72 in. (609.6 x 609.6 x 1829 mm) concrete blocks.
having an average weight of 3.3 7 kip (15.0 kN). on the deck of the test specimen
80
(Figure 4.16). The blocks were placed two across on the deck so that the weight of
one block could be assumed to act on each girder of the test specimen. The blocks
were placed onto the deck by a crane.
Two loading conditions are of particular interest in this research. The first was
the Construction loading condition. The test loading condition which simulates this
loading condition will be referred to as the Simulated Construction loading condition.
The tests under this loading condition investigated the lateral torsional buckling
strength of the tubular flange girders before they were composite with the deck. The
second loading condition of importance was the Strength I loading condition, which
investigated the flexural strength of the composite girder and deck. The test loading
condition which simulates this loading condition will be referred to as the Simulated
Strength I loading condition.
An additional consideration was the stress in the test girders when the girders
were made composite with the deck. The test specimen was loaded with blocks at the
time the test girders were made composite with the deck to create the scaled moment
at midspan that produced the correct stresses in the test girders at midspan for the
beginning of the composite condition. This loading condition is called the Simulated
Mdc loading condition. After the girders were made composite with the deck, blocks
from the Simulated Mdc loading condition were augmented with the additional blocks
to reach the Simulated Strength I loading condition.
The number and spacing of concrete blocks was calculated such that the
moment at midspan was the same as the moment under the corresponding loading
81
condition in the AASHTO LRFD specifications. The moment diagram across the
simply supported span simulated the moment diagram from a uniform distributed load.
The moment created by the girder steel, concrete within the rectangular tube
compression flange, deck, and details such as stiffeners and instrumentation cables
was considered. A number and spacing of blocks was determined for the Simulated
Construction loading condition and the Simulated Strength I loading condition. The
scaled midspan moment levels that were to be reached in the Simulated Construction,
Simulated Mdc, and Simulated Strength I loading condition tests were 7946 kip-in
(9.0x108 N-mm), 5297 kip-in (6.0x108 N-mm), and 18500 kip-in (2.1 X 109 N-mm),
respectively. Figure 4.16 shows the corresponding block arrangements.
The order that the blocks were placed on the test specimen was also important.
The loading was intended to be similar to an increasing uniform load, even though the
loading was applied by a number of concentrated loads. This was achieved by
determining the new moment diagram after each block was placed. This moment
diagram was compared to the moment diagram from a uniform distributed load equal
to the weight of the blocks divided by the length of the test specimen. Block
placement sequences were determined to make the two diagrams very similar. The
midspan moments calculated from the block arrangements for the Simulated
Construction. Simulated Mdc • and Simulated Strength I loading condition tests were
7962 kip-in (9.0x108 N-mm). 5870 kip-in (6.6xl08 N-mm). and 18711 kip-in (2.1x109
N-mm). respectively.
82
The block arrangements are illustrated in Figure 4.16. The Simulated Mdc
loading condition arrangement illustrated in Figure 4.16 (b) was adjusted slightly
before the blocks of the Simulated Strength I loading condition were added. The
Simulated Mdc loading condition blocks are shaded in the Simulated Strength I loading
condition arrangement (Figure 4.16 (c». Note that each block illustrated in the figure
represents two blocks, the one shown and one directly behind it.
Blocks were placed on wood cribbing that in turn transferred the load down to
the deck. Figure 4.17 shows the wood cribbing. The longitudinal members were
parallel to the girders. Two longitudinal members transferred load to a single girder,
and were spaced equidistantly from the girder. The cribbing included transverse wood
pieces which created the proper block spacing. During the non-composite stages
(Simulated Construction and Simulated Mdc loading conditions), each block applied
two loads to each girder through the shims that the deck panels sat on. The bottom of
the wood shims had a layer ofTeflon. An additional layer ofTeflon was placed on the
top flange of the test girders, at the shim locations, creating a Teflon-on-Teflon
interface. Figure 4.18 shows a picture of a wood shim. For some of the non
composite testing stages, rollers oriented to roll in the transverse direction of the test
specimen were placed between the deck and scaled Design 7 (see Sect. 5.2). For
stability purposes, the deck still sat on wood shims on scaled Design 19 during this
time. Figure 4.19 shows a picture of a roller. During the composite stage (Simulated
Strength I loading condition). each block applied a single load to the composite
83
section because the deck sat on a continuous haunch. Figure 4.20 shows a picture of
the haunch.
4.7 Test Instrumentation and Data Acquisition System
The instrumentation used during the tests included 127, 120 ohm, uniaxial
strain gages, sixteen +/- 2 in. linear variable differential transformers (LVDT), twelve
+/- 3 in. LVDT, and eight string potentiometers with various ranges from lOin. to 25
in. Thus, 163 total channels were monitored during testing. The strain gages were
conditioned by Vishay signal conditioners. Each channel was run to one of four
analog-to-digital boards, where the signals were converted and read by a PC. The PC
was equipped with the TestPoint data acquisition software. The TestPoint program
was written to save data to an output file, and to plot selected vertical deflections and
lateral displacements of the test girders during the tests.
Figures 4.21 through 4.25 illustrate the locations of the instrumentation. The
small rectangles represent strain gages, whereas the small circles indicate locations
where a displacement transducer is attached. Figure 4.21 shows the profile view of
the instrumentation located on scaled Design 7, and Figure 4.22 provides the details
of the instrumentation. The purpose of the instrumentation shown in each drawing
detail is discussed in this paragraph. Detail A shows a set of vertically oriented web
strain gages used to measure web distortion in the flat web. Detail B shows a set of
longitudinally oriented web strain gages used to detem1ine the location of the neutral
a.xis. Detail C shows a set of longitudinally oriented flange strain gages used along
84
with those in Detail B to locate the neutral axis, and observe bending behavior. It was
determined through calculation that the cross-section labeled by G in Figure 4.21 and
depicted in Detail G in Figure 4.22 would remain elastic throughout testing, and thus
could be used as an "Elastic" section for monitoring the load level. Flange strain
gages shown in Detail C were also located at these Elastic sections. Detail D shows
the flange strain gages used to check first yield due to bending and the transducers
used to measure vertical deflection at midspan. Detail E shows the transducers used to
measure vertical deflection at other cross-sections, and Detail F shows the transducers
used to measure longitudinal displacement and twist of the tension flange.
Figure 4.23 shows the profile view of the instrumentation located on scaled
Design 19, and Figure 4.24 provides the details of the instrumentation. Details A and
J show the vertically oriented web strain gages used to measure web distortion in the
corrugated web. Detail D shows the transducers used to measure vertical deflection at
midspan, and Detail E shows the transducers used to measure vertical deflection at
other cross-sections. Detail F shows the transducers used to measure longitudinal
displacement and twist of the tension flange. Detail I shows the strain gages used to
differentiate between overall, plate, and flange transverse bending, as well as to
measure strain on an Elastic section.
Detail K shows strain gages used to measure flange transverse bending in the
tube and bottom flange. The C-Factor Correction Method (Abbas 2003) was used to
deternline the flange transverse bending moments. and it was determined that the
ma.ximum flange transverse bending moment would occur in the first inclined fold.
85
followed by a zero flange transverse bending moment in the second inclined fold. The
strain gages shown in Detail K were used to measure strains from these bending
moments. The C-Factor Correction Method also revealed that flange transverse
bending moment stresses would combine with overall bending moment stresses to
create a more critical point than midspan. Hence, the strain gages shown in Detail L
were used to observe first yield. Figure 4.25 shows Detail H, which shows the
transducers used to measure lateral displacements of the two test girders.
4.8 Stress-Strain Properties of Test Specimen Materials
Steel tension coupon tests and concrete cylinder compression tests were
conducted on the test specimen materials. The tension coupon tests were performed
according to ASTM E8-00. The coupons were standard 8 in. (203.2 mm) gage length
coupons. Coupons were tested using a bracket with linear displacement
potentiometers mounted to each side. The average displacement from the two linear
displacement potentiometers was divided by the gage length to provide the strain data.
Also, some of the coupons were instrumented with uniaxial strain gages on both sides
at the midpoint of the gage length. The tensile force ofthe test machine was divided
by an average of three cross-sectional area measurements, within the gage length, to
provide the stress value. The yield stress was determined using the 0.2% offset
method.
Figure 4.26 illustrates the identifiers used for the material. The letters before
the hyphen (CW of FW) refer to whether the material is from the corrugated web
86
girder (scaled Design 19) or the flat web girder (scaled Design 7). The letters after the
hyphen represent the tube (T), concrete (C), web (W), or flange (F). Scaled Design 19
was made using two tubes spliced together from different heats. One is approximately
48 ft. (14630 mm) long, whereas·the other is approximately 12 ft. (3658 mm) long.
Coupons were taken from both tubes. Whenever coupons were taken from a tube,
they were taken at six places around the cross-section, including two from each of the
long tube sides, and one from each of the short tube sides.
Scaled Design 7 was also made using two tubes, but both were from the same
heat. Therefore, only one set of coupons was tested. For scaled Design 7, the coupons
were not flat, and had a large curvature after being cut from the tube. The curvature
created approximately I in. (25.4 mm) of out-of-flatness from the endpoints to the
midpoint of the coupons, which had a length of26 in. (660.4 mm). During the tension
tests, the linear displacement potentiometers did not provide accurate results for small
values of strain, because of the interaction of the transducer bracket and the curved
coupons. For this reason, only the results from the coupon with strain gages were used
in determining the yield stress and strain. The strain gage data showed early softening
of the steel. Calculations supported this by showing that an out-of-flatness of
approximately I in. (25.4 mm) at the midpoint of the coupon will cause one side of the
coupon plate to reach the yield strain as the coupon becomes straight under the tensile
load.
The web for scaled Design 19 is the web used by Abbas (2003). Recall that
the web was created from three of the girders tested by Abbas. Tension coupon data
87
for the web material of Abbas' girder G6A, which provided the web plate for the
midspan area of scaled Design 19 was used in the present study. Scaled Design 7 has
several splices in the web, but the web came from a single plate. The tension flange
steel for both test girders came from the same plate, so a single set of four coupons
was taken from this plate.
Tables 4.3 through 4.9 provide tension test data from the coupons. alp and Elp
are the proportional limits of stress and strain, ayand Ey are the yield stress and strain,
and au and Eu are the ultimate stress and strain. The tables also include a stress-strain
point which corresponds to a stress equal to the average of the yield stress and ultimate
stress. Data is provided for each coupon, and the average of the set of coupons, for
each type of material. In most cases where strain gages and linear displacement
potentiometers were used in the test, the data was nearly identical, and only the linear
displacement potentiometer data is presented. For the FW-T coupons, it was noted
earlier that the linear displacement potentiometer data was not accurate for small
values of strain, so the data for small values of strain is from the strain gages on one
coupon.
A "Material Model" row is included for each set of coupons. The Material
Model row was used as input for finite element analyses discussed in Section 5.6. In
most cases. the Material Model is a quadra-linear curve based on the data sho"t'TI in
this row of the tables. The data sho"t'TI in this row is the average data for CW-T (48
ft.). CW-T (12 ft.). CW-W. CW-F. FW-W. and FW-F materials. For the FW-T
material. the Material Model row uses the data from the single coupon tested \\;th
88
strain gages for the linear proportional limit and yield properties. The linear
displacement potentiometers provided accurate results for large values of strain, so
averages were used for the ultimate strain. The strain corresponding to the average of
the yield stress and ultimate stress was beyond the range of the strain gages, but less
than the point when the linear displacement potentiometers started providing accurate
results. Therefore, this strain had to be approximated. Figures 4.27 through 4.33
provide illustrations of the stress-strain curves from the tests, as well as the
corresponding quadra-linear curves. It is evident that the quadra-linear curves provide
a good representation of the test data. For the FW-T case shown in Figure 4.31,
several additional points were added to the Material Model curve so that it would
better represent the early softening of the material.
The concrete within the two tubes came from the same pour. At the time ofthe
pour, fifteen 6 in. (152.4 mm) by 12 in. (304.8 mm) cylinders were made. The
concrete cylinder compression tests were performed according to ASTM C 39/C 39M
01. The cylinders were tested at 7, 14,21, and 28 days for the ultimate strength. At a
time close to testing of the test specimen, a stress-strain test was performed on the
remaining three cylinders. A bracket was placed on the cylinders, and an LVOT was
mounted to each side. Strain was calculated by dividing the average displacement of
the LVOTs by the gage length. Stress was calculated by dividing the compressive
force in the test machine by the cross-sectional area of the cylinder. All three
cylinders provided similar results. and one of the stress-strain curves is illustrated in
89
Figure 4.34. The average ultimate strength of the three cylinders was 7.8 ksi (53.8
MPa).
4.9 Measured Girder Cross-Section Dimensions and Initial Tube
Imperfection
Measured Girder Cross-Section Dimensions
The actual cross-section dimensions of the test girders were measured and the
averages are presented in Table 4.10. The tension flange width and thickness was
measured at 10 places along the test girder spans. The tube was assumed to have its
nominal outer dimensions of lOin. x 4 in. The tube thickness was measured from the
tension test coupons. The web thickness was measured at three places on each end
and the web depth was measured once at each end.
Test Girder Initial Imperfection
The initial imperfection (out-of-straightness or sweep) of the tube was
measured for both test girders. Table 4.11 provides the initial tube imperfection of
scaled Designs 7 and 19. In Table 4.11, x represents the longitudinal distance along
the test girder measured from the west bearing. A more detailed coordinate system is
provided in Section 5.3. The maximum amplitude of the initial tube imperfections
was equal to Ll945 and Ll1512 for scaled Designs 7 and 19. respectively. where Lis
the test girder span. Initial imperfection was considered positive in the south
direction. Thus. it is evident that the scaled Design 7 initial tube imperfections were to
90
the south, whereas the scaled Design 13 initial tube imperfections were to the north.
At the time, it was thought that the tube out-of-straightness was the critical
imperfection. As discussed in Chapter 5, however, it was observed that experimental
results did not agree well with Finite Element Method (FEM) simulation results, and
various tension flange imperfections were studied using FEM simulations.
91
Table 4.1 Scaled Girder Designs 13 and 7
Scaled Girder Design 13 7CorruQated Web yes noComposite yes yesHybrid no noInterior DiaphraQm no noDweb (in.) 28 28Tweb (in.) 3/16 5/16Tube Size (in.) 10x4x1/4 10x4x1/4Bbf (in.) 14 10Tbf (in.) 3/4 3/4
Ratiolnexure 0.88 0.74
Ratioilinexure 0.46 0.41
RatioVnexure 0.80 0.67
Ratioshear 1.02 0.77
RatiOwebslendemess NA 0.30
RatiOtensionnanoe 0.78 0.56
Rationanoebracinn 5.22 5.22
Rationbresistance 0.75 0.73
Ratioservicell 0.95 0.94
RatiOtatioueCW 0.61 NA
Ratiotatioueconnniate NA NA
RatiOtubethickness 0.91 0.91NA=Not Applicable
92
Table 4.2 Scaled Girder Designs 19 and 7 (6 in. (152.4 nun) thick deck)
Scaled Girder DesiQn 19 7Corrugated Web yes noComposite yes yesHybrid no noInterior Diaphragm no noDweb (in.) 28 28Tweb (in.) 1/4 5/16Tube Size (in.) 10x4x1/4 10x4x1/4Bbf (in.) 14 10Tbf (in.) 3/4 3/4
Ratiolnexure 0.88 0.67
Ratioilinexure NA NA
RatioVnexure NA NA
Ratioshear 0.79 0.77
Ratiowebslendemess NA 0.30
RatiOtensionnanne 0.78 0.56
RatiOnanaebracina 5.22 5.22
Ratio"bresistance 0.93 0.73
Ratioservicell 1.11 0.91
RatiofatiaueCW 0.58 NA
Ratiofatiaueconnolate NA NA
RatiOtubethicl<ness 0.91 0.91NA=Not Applicable
93
Table 4.3 CW-T (48 ft.) Stress-Strain Properties
alp a y au (ay+au)/2 Eat (ay+au)/2CW-T (48 tt) (ksi) EJD (ksi) (ksi) Ev (ksi) (ksl) &u (ksi) (ksi) (ksll
Coupon 1 40 0.001379 54.7 0.003927 65.8 0.143637 60.3 0.018800
Coupon 2 40 0.001379 55.1 0.004231 65.0 0.152312 60.1 0.019601
Coupon 3 35 0.001207 54.6 0.003882 64.9 0.140590 59.8 0.017889
Coupon 4 40 0.001379 56.9 0.003973 66.8 0.110591 61.9 0.015905
Coupon 5 40 0.001379 60.7 0.004040 68.7 0.047456 64.7 0.011781
Coupon 6 40 0.001379 52.7 0.003937 63.5 0.153025 58.1 0.025269
Average 39.2 0.001351 55.8 0.003998 65.8 0.124602 60.8 0.018208MaterialModel 39.2 0.001351 55.8 0.003998 65.8 0.124602 60.8 0.018208
Table 4.4 CW-T (12 ft.) Stress-Strain Properties
alp ay au (ay+au)/2 Eat (ay+au)/2CW-T (12 tt) (ksi) EJD(ksl) (ksi) Ev (ksi) (ksi) &u (ksi) (ksi) (ksi)
Coupon 1 40.0 0.001379 59.3 0.004024 62.5 0.019689 60.9 0.006577
Coupon 2 45.0 0.001552 64.9 0.004410 67.6 0.020228 66.3 0.006902
Coupon 3 40.0 0.001379 61.3 0.004238 64.0 0.025915 62.7 0.007472
Coupon 4 45.0 0.001552 60.1 0.004175 63.3 0.023300 61.7 0.010360
Coupon 5 50.0 0.001724 61.1 0.003998 64.5 0.028545 62.8 0.010743
Coupon 6 45.0 0.001552 62.1 0.003916 64.4 0.020650 63.3 0.010386
Averaqe 44.2 0.001523 61.5 0.004127 64.4 0.023055 62.9 0.008740MaterialModel 44.2 0.001523 61.5 0.004127 64.4 0.023055 62.9 0.008740
Table 4.5 CW-W Stress-Strain Properties
alp cry au (ay+au)/2 Eat (ay+au)/2CW-W (ksi) EJD (ksi) (ksi) Ev (ksi) Iksi) &u Iksi) (ksl) (ksil
Coupon 1 40.0 0.001379 69.1 0.004488 97.8 0.118143 83.5 0.021833MaterialModel 40.0 0.001379 69.1 0.004488 97.8 0.118143 83.5 0.021833
94
Table 4.6 CW-F Stress-Strain Properties
alp a, au (a,+a.)/2 Eat (a,+a.)/2CW·F (ksil E!D(ksil (ksil Ev (ksil (ksil Eu lksil iksil (ksi)
Coupon 1 61.3 0.002114 61.4 0.010560 85.0 0.124452 73.2 0.030316
Coupon 2 60.8 0.002097 60.9 0.010547 84.6 0.124977 72.8 0.030637
Coupon 3 60.2 0.002076 60.3 0.011512 83.6 0.129828 72.0 0.031206
Coupon 4 60.4 0.002083 60.5 0.011922 83.7 0.123372 72.1 0.031626
Average 60.7 0.002092 60.8 0.011135 84.2 0.125657 72.5 0.030946MaterialModel 60.7 0.002092 60.8 0.011135 84.2 0.125657 72.5 0.030946
Table 4.7 FW-T Stress-Strain Properties
alp a, au (a,+a.)/2 E at (a,+a.)/2FW·T (ksi) E!D(ksl) (ksi) Ev (ksl) (ksi) Eu (ksi) (ksl) (ksl)
Coupon 1 10.0 0.000345 54.4 0.003881 68.9 0.151539 61.7 NA
Coupon 2 NA NA NA NA 65.3 0.174318 59.9 NA
Coupon 3 NA NA NA NA 68.0 0.134961 61.2 NA
Coupon 4 NA NA NA NA 69.1 0.154870 61.8 NA
Coupon 5 NA NA NA NA 74.0 0.153341 64.2 NA
Average NA NA NA NA 69.1 0.153806 61.7 NAMaterialModel 10.0 0.000345 54.4 0.003881 69.1 0.153806 61.7 0.007000
NA=Not Applicable
Table 4.8 FW-W Stress-Strain Properties
alp a, a. (a,+a.)/2 Eat (a,+a.)/2FW·W (ksi) CtD (ksl) (ksl) Ev (ksi) (ksl) Eu (ksl) (ksl) (ksl)
Coupon 1 35.0 0.001207 57.9 0.004371 71.5 0.106899 64.7 0.027839
Coupon 2 40.0 0.001379 59.9 0.004329 73.9 0.129492 66.9 0.029496
Coupon 3 45.0 0.001552 60.6 0.004627 74.3 0.118111 67.5 0.030119
Coupon 4 40.0 0.001379 62.1 0.004911 75.7 0.130339 68.9 0.029743
Averaae 40.0 0.001379 60.1 0.004560 73.9 0.121210 67.0 0.029299MaterialModel 40.0 0.001379 60.1 0.004560 73.9 0.121210 67.0 0.029299
95
Table 4.9 FW-F Stress-Strain Properties
alp a y au (ay+au)/2 Eat (ay+au)/2FW-F (ksil £JD (ksil (ksil &, (ksl) (ksl) Eu (ksil (ksi) (ksi)
Coupon 1 61.3 0.002114 61.4 0.010560 85.0 0.124452 73.2 0.030316
Coupon 2 60.8 0.002097 60.9 0.010547 84.6 0.124977 72.8 0.030637
Coupon 3 60.2 0.002076 60.3 0.011512 83.6 0.129828 72.0 0.031206
Coupon 4 60.4 0.002083 60.5 0.011922 83.7 0.123372 72.1 0.031626
Average 60.7 0.002092 60.8 0.011135 84.2 0.125657 72.5 0.030946MaterialModel 60.7 0.002092 60.8 0.011135 84.2 0.125657 72.5 0.030946
Table 4.10 Average Measured Girder Dimensions
TensionTube Tension Flange Web Web
Thickness Flange Thickness Depth ThicknessTest Girder (in.) Width (in.) (in.) (in.) (in.)
Scaled Design 19 0.23 13.99 0.76 28.09 0.25Scaled Design 7 0.24 9.95 0.75 28.00 0.32
Table 4.11 Initial Imperfection (Sweep) of Tubes
Scaled Design 7 Scaled Design 19x location (in.) Imperfection (in.) x location (in.) Imperfection (in.)
708.66 0.00000 708.66 0.00000648.66 0.18750 648.66 -0.15625588.66 0.56250 588.66 -0.21875528.66 0.65625 528.66 -0.28125468.66 0.75000 468.66 -0.31250408.66 0.71875 408.66 -0.25000354.33 0.75000 354.33 -0.31250348.66 0.71875 348.66 -0.37500288.66 0.62500 288.66 -0.46875228.66 0.53125 228.66 -0.43750168.66 0.50000 168.66 -0.37500108.66 0.28125 108.66 -0.2187548.66 0.09375 48.66 0.03125
0 0.00000 0 0.00000
96
20x8x~ in.
10x4x! In.
52' (1 75 rUT)
28' Oil. MM)
33' (838.2 fTlfTl)3, <19.1 MM)4 14' (355.5 MfTl)
(a) Design 13
2Dx8xa In.
JOx4x~ In.
62' 0575 fTlfTl)
ft,' <17.5 nM)
28' 011.2 MM)
5, (7.9 MM)fb
1~' (38.1 nn) 34
, 09.124' (609.6 rlrI)
(b) Design 7
Figure 4.1 Prototype and Scaled Versions of Designs 13 and 7
97
q=22.8£>' (580.£> PH'))
(a) 50 ksi (345 MPa) Corrugated Web
q=13.90' (353.1 MM)
(b) 70 ksi (485 MPa) Corrugated Web
Figure 4.2 Scaled Corrugated Webs
8.66' (220 I'll'l)
hr=5.91' <150 ",,) R=4.72' (120 ,.,,.,)
d=7.87' (200 ,.,,.,) d=7.87' 200 I'll'l)b=Il.81' (300 I'l")
q=39.37· <1000 I'l")
Figure 4.3 Trapezoidal Corrugated Web
98
Figure 4.4 Fatigue Crack
99
INTENTIONAL SECOND EXPOSURE
Figure ..L-t Fatiguc Crack
llll
G3A
1'-9~' (547.7 1'1I'I) 1'-5it;· (449.3 /1/1)
2B' OIL
I I
I - I'- 1- - l- I- . - ~ - I- I- ... 1- r1/1/1) I 1 S I N 2 S 2 N 3 S 3 N 4 S 4 N 5 S 5 N 6 S 6 N 71 5
- '- I- . - ~ ~ . - -I- "'" - I-- I- I- - Hr
t I
South [jJ,.de,., No,.th f ltee ~lt,.tiltl Stirfene,.~
G6A
IB'-O~' (5501 /I,.,)
2B' OIL
I lr - - - - ~ - ..- ,- r- - p.- I- . 11
. ,.,,.,) 7Js 7 f~ 8 S 8 N 9 S 9 N 10 S 10 f 11 S 11 N 12 l~ ~
I- "'" - - -I- - - - I- I- - 1-0 - -I- 1r
I t
South Gir"de,., fjo,.th fltce ~Q,.tiQIStiffene,.~G5A
21'-O~' (6402 nn) 1'-9~' (547.7 nn)
2B' (711
II
,- . - - ~ 1- - -t- - - .. --- ... -If• /1/1) I~t 13 13 t 14 < 14 r 15 15 r 16 16 r 17 17 r IB 18 , I
L I-- . - - i- I- .1- - .. ~ - - . ... - 100- -~,I II
South Glr"der, North fltce ~ortlolStlrrener~
Noteo The end beltrlng stlHeners have been rel'loved ;rol'l G6A.The p:lrtlol stltrenl'rS sha~n In the dl':lwlng h:l ve been
rel'loved to below the cut
Figure 4.5 Web cuts and Splicing Arrangement
100
28.00' (
~ IIn
V
~ ~00
00
o 0
11.2 r'lr'I)00
00
00
00
00
----
.50' <12.7 r'lr'I) thickness
4.19' Cl06.4 MM)
Cross section with InterMedlo. te stiffener (St2)and beo.rlng stiffener (St3)
Plan View
1'- -+-"-'0.""50"--'-.:<12.7 Mn)
Figure 4.6 Stiffener Geometry for Scaled Design 19
101
28.00' (
II n
n
~/
~
11.2 1'11'1)
..,j~
"---
50' <12.7 1'11'1) thicKness
4.20' <106.7 1'11'1)
Cross section with Interl'ledlate stiffener CStD
28,00' C
II n
n
I~V
~o 0g g
o g
11.2 1'11'1)o 0o 0o g
o 0o 0
-
.50' 02,7 1'11'1) thickness
4.20' Cl06J nn)\
Cross section with internedia te stiffener CSt2)and bearing stiffener CSt3)
Figure 4.7 Stiffener Geometry for Scaled Design 7
102
Scoled Design 19 (CorrugQtlons not shown for dorlty)
6' U52A 1'1II)
xl' <254xIOl.6x6.4 m) TUb~ Veb Spl1c~a' (4,8 Ml
Veb Spl1c~ E70 I r1' (4.8 "I'll v \
GJA G6A 14'x3/4' (3~~ K!9.1 rn) Botto" r
\:.t 3 28',1' <711.2><6,4 ""I veb~ ,d' (6A "ft) f\~t 2 E70 /,' (6A nn) v \ St
-6' (~2.4 I'tll
10'x4'
South Girdl1r, North rOell
Scoled Dl'Slgn 7
14'-94' (4500 Ml G' <152.4 on)14'-94' (4500 M)
'xl' <254xIOL6x6,4 m) TUb~ .a' (4.8 Ml f\
E70/ d' (4,8 I'IIIl v \
2B'xa C71L2x7,9 I'JI) \leb~ .,1' (6,4 nnl f\St 3 St 1 St 2 E70 /1, (1;,4 III'Il \ St 3
10'x3/4' (2 .19.1 ""I Bottoft n~g\ St 1
.....6' <152.4 m) 14'-94' (4500 on)
10'x4
tJorth Girder, North race
Figure 4.8 Scaled Designs 19 and 7
103
x--- ~---<
a.
z
I..F
web
ax
d
F
a.
0.707a. (throQ t of weld)
Figure 4.9 Illustrations for Tube Flange-to-Web Fillet Welds
104
20 40 60 80 100o
10000 --r---'-~~~-~-~-i
8000 -'-:-~~~~-~--
I6000 -T----~_4.--~-~-~-------~-----1I
4000 -r--2000 -:--~-----'l.~-~-~~~-~~~--J'-~---j
!
- Ib 0 _I------~~-~~~~~-~ I
1
-2000 ~--~---'"I
-4000 ~II
-6000 .,'~-~---~-~------------
-8000 ~I-----_----~I
-10000 -1------~~---,---'-~~-~~-~
-100 -80 -60 -40 -20
Figure 4.10 Example of f(ax) versus ax (Used for Tube-to-Web Fillet Weld Design)
105
<J
"\ V.J
Case 1
II1III1 Fys1111111~
Mp
Case 2 Case 3
O"ywcld O"ywcld 0" 't
R2
't 0"
'tywcld'a 1 stlrr. I 'tywcld'a
RJ
A:iO'
Figure 4.11 Illustrations for Tube Flange-to-Stiffener Fillet Welds
106
Figure 4.12 Core Hole in Pre-Cast Deck
.. -' ,,
.,';
...
Figure 4.13 Shear Studs Mounted in Core Hole
107
INTENTIONAL SECOND EXPOSURE
Figure 4.12 Core Hole in Pre-Cast Deck
Figure 4.13 Shear Studs Mounted in Core Hole
107
9" (228.6 M[Y))
0.75" C19.l [Y)[Y) DIA. TYP.
Figure 4.14 Shear Stud Arrangement
Figure 4.15 Core Hole Pattern
108
3 15 II 7 1 13 9 5 b 10 14 2 8 12 16 4
:--r-30.33' 070.4 11M) 11 spes @ 60' USe4 I'lIll = 660' (16764 "1'1) 30.33' 070.4 nl'l):r-
(a) Simulated Construction Loading Condition
120' (3048 "'" 120' (3048 I'IMl
(b) Simulated Mdc Loading Condition
19.38' ( 923.38' .3n 19 (492 l'1l'i) 25 spes @ 27.25' (692.2 M) = 681.25' (J7~5 MI'I) Ll'II'Ilr
51 45 39 35 29 49 43 37 33 27 47 41 31 32 42 49 29 34 39 44 50 30 3. 40 46 52
25 19 13 9 ~ 23 17 II I ~ 21 15 I 8 16 22 I ~ 12 18 24 I 10 14 20 26~. ';Wij ~
I I
(c) Simulated Strength I Loading Condition
Figure 4.16 Loading Arrangements
109
Figure 4.17 Wood Cribbing
Figure 4.18 Wood Shim
110
Figure 4.19 Rollers
Figure 4.20 Haunch
111
6(152.4 I'm) (2249 m)
1-;-;---;4'-~'-----'3'
(4501 1'lII) (2249 n~) (152.4 I'ln)
'-------i4·-9a·'----ol.----i4.-9~··---"'-----l4·-9a··---ol.--;
(4501 0") (4501 "0) (45l1 0.,)
EottoM of bOttg~ OgoeF
Figure 4.21 Profile View of Scaled Design 7 Instrumentation
112
l' (25.4 I'lM)
l' <25.4 I'lr'l)
2' <50.8 r'lf,) • 2' (50.8 I'lM)
6" 052.4 MM) 6' (152.4 riM)
•6' <152.4 MI'l) 6' (152.4 I'lM)
•5" <152.4 MI'l) 5' <152.4 MM)
• •6" 052.4 f'lf'l) 6' (152.4 I'lf'l)
2' (50.8 MM) • 2" (50.8 r'lM) •
Detail A: strain oat;Jes Detail B: strain t;Jat;Jes
Detail DI strain gagesood strimJ pots
l' (25.4 I'IM)
•4' (101.5 MI'l)
• l' (25.4 I'll'l)4' <101.6 I'lr'l)
•
Detail C, strain gages
2' (50.8 r'lM).-J-+--I2' (50.8 I'lM)2' (50.8 MM)
2' (50.8 I'lM)
l' (25.4 I'lr'l)
l' (25.4 MM)
l' (25.4 r'lr'l)
5' (127 MMlEtj5' Cl27 I'lM) 1' (25.4 I'IM)
Detail [I string potsand LVDrS
-4' ClO1.6 I'lM)
>-- •10' <254 MM)
i) i)
w • <10' <254 MM)
I-- •4' Cl01.6 nn)
~
De-tail G' strain oaoes
Figure ....22 Details of Scaled Design 7 Instrumentation
113
!'-71f<-l.......-..;.
(50~! Ml........-.........>-i··-7M·
CSO~I M)
!'-7U........l.--......r'-I0G;' l'-7/t'-l.o......"-i'-7/t(50~1 "I'll (874.7 m) (500.1 m) (50~1 "N
. .l....-,..L.----\4·-9aL-----'-~·
(4~01 N'll
E.llcn of bgUQ'! fltJ';.
Figure 4.23 Profile View of Scaled Design 19 Instrumentation
114
-4' (101.6 I'll'l)
I-- •
I) [)
20' (508 rll'!) ( <2' (50.8 MM)
10' (254 MI'l)
2' (50.8 MM)
o
o
DE'toil AI strain oooes Detail P, strin';! pots
7' 077.8
7' 077.8
Detail (, string potsand LVPIS
I' <25.4 MM)
l' (25.4 MM) l' <25.4 MM)
Detail F'
l' <25.4 I'lM)
I' (25.4 /'1M)
l' (25.4 1'11'1)...-........_ ...
6' (152.4 1'11'1) o' 052.4 MM)
De tail II stl'aln Qooes
2' <50.8 MM)
~' (9.5 1'1,,)
I
Dotall ~;, strain <;0<;:;0;l'
Detail J' strain caoes
I' (25.4 Mr-I) l' (25.4 1'10)
'--........~~1· (25.4 M)
r,n) 6' 052.4 1'1,,)
Detail LI st·'o'o oo.ors
Figure 4.24 Details of Scaled Design 19 Instrumentation
115
7~' 0461 M) between tube sj' (14l0 ~~) between botto~ fl sides
7'-lO~"~'"----l4'-9~"-----'-
(24ll2 ml (4~1 n~)
.-"-----i4·-9~··---'""'---7'-l~'
(4~01 ml C24ll2 ml
Top Vi•• of Grdrr' forI gtecp,t PSMrenen1 I YDTs
Figure 4.25 Details of Lateral Displacement Instrumentation
116
r'W-T ~'W-C,
\ .r -C IW -~
I~'<
VJ-F
VJ- T 'W-C
r'W-\JVJ-F
Figure 4.26 Material Test Identifiers
117
70 ~f----------
60 h~~:::::======:::=:======-=-=-==--C>..----JI 1
50jf~---I
:::- !! 40 ~
UlUl
~ 30 -1------------------------------'en
20 -1------------------------
0.2 0.25
"
'"I',I-.....Material Modelil
,I
0.150.1
o ~--------~----,- ----o 0.05
--Test Data10 ------------------------~
Strain (in./in.)
Figure 4.27 CW-T (48 ft.) Coupon 1
70 --,---- ------~----~~----------~--------------.,
601-1---- ----~----------
50 -1---------------\----- --~---~-----.
-~ 40 -- --- - --------------------------~---------------
--Test Data
. -..... Material Model
UlUlCl)
30...-en
20
10 -
00 0.05 0.1 0.15
Strain (in.lin.)0.2 0.25
Figure 4.28 CW-T (12 ft.) Coupon 5118
-
~~-
----- ::::--- r ~r--- ,
/-
~
-
,
-Test Data
i-.-Material Model
120
110
100
90
80-'iii 70:.lZ 60een 50
40
30
20
10
oo 0.05 0.1 0.15
Strain (in.lin.)
Figure 4.29 CW-W Coupon 1
0.2 0.25
90----
80-'-----~
60 '
----- ~---------~-1
!
-- -~ ~-- -~ ------
'iii .:. 50 -: --~~--------~- ~----~----- -~-~~----~---~~~-~----
CIlCIle 40 - --------- --- - -- ----~-------~~ ~-~----~--------en
30 ~ -- --------
20
10 ___~ ~_~ __~_____ _ --Test Data
-r-Material Modelo
o 0.05 0.1 0.15Strain (in.lin.)
Figure 4.30 CW-F Coupon-+119
0.2 0.25
80 -,--------------------------____,
0.30.250.20.1 0.15Strain (in.lin.)
0.05o
~ .---.~ I-f r
~ 'I~
IrrLI
i
~~,~
rL i--Test Data (Lin. Disp. Pot.)
t i --Test Data (Strain Gages)1- I, I-+--Material Modelfo
-0.05
10
60
20
70
&l40e....
CJ) 30
__ 50
~--
Figure 4.31 FW-T Coupon 1
80 -,-----
60 -'r----- ------~
-- ----~----------------------------~----
__ 50"iii~
&l 40e....
CJ) 30 ~ ~ ~ ~ .
20 ------- -------- -~--~-------
10 t~---------~-------~----------~------~. --Test Data
oo 0.05 0.1 0.15
Strain (in.lin.)
-+-- Material Model
0.2 0.25
Figure 4.32 FW-W Coupon 2120
90 ~------~-~---------
80 -----~-=---____=_~~------= ......
70
60 ..;po..------------------
-~ 50-enen~ 40 -1-------------------en
30 -I-------~ ----------------~
20-1-----
_______--.;' --Test Data
-.-Material Model
10 -1----
o ...-~o 0.05 0.1 0.15
Strain (in.lin.)0.2 0.25
Figure 4.33 FW-F Coupon 4
9000 -:----~~~---- ---- -------------
8000 i---~----- ---.-----------
7000 ~----~,,
6000 .:-~- .·iii i.e: 5000 -:-------~----- --------------en •en '~ 4000 ~------~---- -----------------------. ---. -----.-- --------- .-en
3000 ~-- ------- -- --
2000:- --- ---~--
1000 :~--.
o '-.---~~---------.-.. -.---. ----.--~-.---.--.-- -. -. .-.------- ---.--.-.-o 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
Strain (in.lin.)
Figure 4.34 CW-C and FW-C121
5. Experimental Results and Comparison with Analytical Results
5.1 Introduction
In the previous chapter, the development of the test specimen was discussed.
The choice of the prototype girders to scale and test, and detail designs, such as
stiffeners, welds, and shear studs were discussed. The load conditions to be developed
during testing were described in detail. The data acquisition system and
instrumentation were presented. Finally, the stress-strain properties of the materials
used in the test girders, the measured cross-section dimensions, and selected geometric
imperfections of the test girders were presented.
This chapter presents the experimental results. The specific stages of testing,
and a set of coordinate axes to facilitate understanding of the experimental results are
presented first. Then, data from strain gages is presented. Next, the vertical deflection
of the test girders, and lateral displacements of the test girders are presented.
Comparisons are made between experimental results and analytical results. Other
results that are only summarized are the web distortion for both test girders, and tube
and tension flange lateral curvatures, and tension flange plate bending for scaled
Design 19.
5.2 Test Stages
Each stage of the testing progranl is explained in this section. A stage of
testing is considered to be any period of time when data was continuously obtained
122
from the instrumentation. An identification scheme is provided so the reader will
know the test stage corresponding to given experimental results.
Deck Placement
The first stage of testing was the Deck Placement stage. This stage involves
the placement of the six deck panels on the test girders. The panels sat on wood
shims. The bottom of the wood shims had a layer ofTeflon. An additional layer of
Teflon was placed on the top flange of the test girders, at the shim locations, creating a
Teflon-on-Teflon interface. This was an attempt to minimize friction between the
deck panels and girders for the Simulated Construction loading condition.
Stages 1, 2, and Mdc
The next stage of testing was referred to as Stage 1. This stage used the
Simulated Construction loading condition with a single interior diaphragm located at
midspan (and end diaphragms at the bearings). The test girders were non-composite
with the deck panels, and the deck panels were supported by shims with a Teflon-on
Teflon interface to the girders. Stage 2 was a repeat of Stage 1, with the exception that
the interior diaphragm was removed. After Stage 1 and Stage 2, the Simulated Mdc
loading condition was applied to the non-composite test girders. As mentioned in
Chapter 4, the purpose of the Simulated Mdc loading condition was to create the san1e
stress levels in the non-composite test girders as in the non-composite prototype
123
girders, at the time the girders are made composite with the deck. This stage is
referred to as the Mdc stage.
Roller Placement
As discussed later in Section 5.6, the lateral displacements of the test girders
during Stage 2 were much smaller than expected from analytical results. Friction was
thought to have caused the test girders to be braced by the deck and/or the adjacent
test girder. Therefore, to reduce this friction, rollers oriented to roll in the transverse
direction of the test specimen were placed between the deck and scaled Design 7. For
stability purposes, the deck still sat on wood shims with Teflon-on-Teflon interface to
scaled Design 19. The stage in which the wood shims on scaled Design 7 were
replaced with rollers is referred to as the Roller stage.
Stage 2-2, Mdc-2, and Stage 3
Stage 2 was repeated, and then the Simulated Mdc loading condition was again
placed on the test specimen. These stages were referred to as Stage 2-2 and Mdc-2,
respectively. With the Simulated Mdc loading condition in place, the work was
performed to make the test girders composite with the deck. Then, the Simulated
Strength I loading condition was applied to the test specimen in a final test stage
referred to as Stage 3.
124
Stage Identification
The data recorded from the instrumentation was balanced (set to zero) at the
beginning of each stage. For some of the stages, it is convenient to present
experimental results from only that stage. For other stages, it makes more sense to
present the total results. This can be explained by considering moment versus vertical
deflection for Stage 3. If the data zeroed at the beginning of Stage 3 is presented,
moment versus vertical deflection will include only the effects of the blocks placed in
Stage 3. If the total results are presented, then the moment versus vertical deflection
plot will include the girder self-weight, Deck Placement, Simulated Mdc loading
condition, and the effects of the other test stages. It is thus important to be able to
differentiate between the test stage results and total results, and the subscripts of
Table 5.1 are placed on symbols to indicate that these results are for an individual test
stage. If none of the subscripts of Table 5.1 are used, then the results are total results.
5.3 Coordinate Axes and Instrumentation Identification
Coordinate axes for the test girders and instrumentation identifiers are given in
this section to aid in interpreting experimental results. Figure 5.1 illustrates the
coordinate axes of the test girders. The origin is at the west bearing, at the center of
the bottom of the tension flange. x is positive in the eastward direction, y is positive in
the vertical direction, and z is positive in the southward direction.
Identifiers for the test instrumentation on scaled Designs 19 and 7 are sho\'m in
Figures 5.2 through 5.4. The letter before the hyphen (C or F) refers to whether the
125
instrumentation was located on the corrugated web girder (scaled Design 19) or the
flat web girder (scaled Design 7). The letter after the hyphen (S or D) refers to
whether the instrumentation was a strain gage or a displacement transducer. The letter
after the hyphen is followed by a number. Scaled Design 19 had 69 strain gages and
18 displacement transducers, whereas scaled Design 7 had 58 strain gages and 18
displacement transducers. Note also that each cross-section, at which strain gages
have been placed, has been given an identifier. The identifier is located at the top of
the list of strain gages for that cross-section. Each cross-section at which a vertical
deflection transducer is located is also given an identifier.
5.4 Strain Gage Data
Moment versus strain graphs were generated for each stage at the "Elastic"
sections shown in Figures 5.2 and 5.3. According to beam theory, strain through the
height of a cross-section varies linearly from zero at the neutral axis. In addition, the
strain at a specific point in a cross-section increases linearly with moment in the cross-
section according to the following equation:
(Eq.5.1)
where M is the moment in the cross-section, E is the modulus of elasticity of steel, I is
the moment of inertia of the cross-section, y' is the distance from the neutral axis to
the point in question, and E is the strain at the point in question. However, strain at a
point \\;thin the cross-section may not increase linearly with moment on the cross-
section if a nonlinearity. such as the effect of residual stresses. is present. Eq. 5.1 is
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only true while Hooke's Law is valid on the entire cross-section. The moment will not
be linearly related to strain after residual stresses have caused partial yielding of the
material in the cross-section. After some initial study, the strain data was thought to
include nonlinearity, even at sections expected to remain linear elastic. To verify this,
the bending moment was calculated from static equilibrium, using the measured
weights of the individual loading blocks (Kim 2004b), and the bending moment was
compared to the strain data.
Figures 5.5 through 5.16 illustrate moment versus strain graphs for Stage 1,
Stage 2, and Stage 3 at the Elastic sections. The strain value graphed for scaled
Design 19 is the average of the four comer strain gages of the tension flange at each
Elastic section. This average eliminates lateral curvature or plate bending of the
tension flange (see Sect. 5.9). The strain value graphed for scaled Design 7 is the
average of the three gages on the tension flange of the Elastic section. This average
eliminates lateral curvature of the tension flange. The strain gage identifiers are
provided on the graphs, and can be compared against Figures 5.2 through 5.4.
Stage 1 Elastic Sections
The Stage 1 graphs (Figures 5.5-5.8) show nonlinearity on the loading branch
of the curves. Linear regression lines were developed using the unloading branches of
each curve, because cross-sections that yield will unload elastically. These regression
lines were plotted along the loading branches of the curves to further illustrate the
nonlinearity in the data. Note that the nonlinearity creates pemlanent strain at the end
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of Stage 1. The nonlinearity and resulting pennanent strain are believed to be from
local yielding on the cross-section due to residual stresses.
Stage 2 Elastic Sections
In the case where residual stresses exist in a steel girder, loading and unloading
within a selected load range will eventually nearly eliminate the residual stresses.
This effect is often referred to as the "shakedown" of residual stresses. The
elimination of residual stresses is illustrated in the Stage 2 graphs (Figures 5.9-5.12).
Notice that the loading and unloading portions of the curves are identical, and do not
show serious nonlinearity. The same load level is used in Stages 1 and 2, so in Stage 2
the test girders are being loaded along the unloading branch from Stage 1. The graphs
for Stage 2-2 are not shown because they are similar to Stage 2, and do not provide
any additional infonnation.
Stage 3 Elastic Sections
The Stage 3 graphs (Figures 5.13-5.16) appear to be linear in the initial portion
of the moment versus strain curves. After the stresses become larger than those in
Stages 1 and 2, the curves become nonlinear.
Summarv of Results from Elastic Sections
From the graphs discussed above, it is obvious that Equation 5.1 cannot be
used to calculate moment from strain on the Elastic sections. Therefore. the bending
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moments for the various test stages were determined from static equilibrium. Only
tension flange strain gage results were presented in the above discussion, but
nonlinearity was observed in all strain gages at the Elastic sections. The residual
stresses that cause local yielding and the related nonlinearity are not observed in the
tension tests of the web and tension flange material. The residual stresses are likely
introduced into the flanges by web-to-flange welding. Residual stresses appear to be
present in the tubes, especially FW-T, as discussed in Section 4.8. These stresses
would contribute to the observed nonlinearity.
Midspan Section
Moment versus strain graphs for Stages 1, 2, and 3 were also generated at the
"Midspan" sections. These results are presented in Figures 5.17 through 5.22. The
strain gage identifiers are provided on the graphs, and can be compared against
Figures 5.2 through 5.4. These graphs display similar behavior to that shown in
Figures 5.5 through 5.16. As expected, the nonlinearity is larger at midspan because
of the larger moment levels.
Investigation of Stresses
The stresses in the steel and concrete during the various loading stages were
estimated assuming linear elastic behavior of the cross-section. For scaled Designs 19
and 7. the steel cross-section and the concrete in the tube as well as the deck were
modeled as an equivalent transfonned section. The stresses were estimated and
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compared to the material strengths to verify that the observed nonlinearity in the
moment versus strain curves was not from gross yielding of the steel in the cross
section, or stresses exceeding the linear elastic range of the concrete. The midspan
stresses were studied. During the tests under the Simulated Construction loading
condition (Stage 1, Stage 2, and Stage 2-2), the stress in the top of the tube concrete
reached 3.8 ksi (26 MPa) and 3.6 ksi (25 MPa) for scaled Designs 19 and 7,
respectively. The stress at the top of the tube reached 25 ksi (173 MPa) and 24 ksi
(168 MPa) for scaled Designs 19 and 7, respectively. The stress at the bottom of the
tension flange reached 25 ksi (175 MPa) and 28 ksi (196 MPa) for scaled Designs 19
and 7, respectively.
During the Simulated Strength I loading condition test (Stage 3), the stress in
the top of the tube concrete reached 2.6 ksi (18 MPa) for both test girders. The stress
in the top of the deck concrete reached 1.3 ksi (9 MPa). The stress at the top of the
tube reached 17 ksi (120 MPa) for both test girders. The stress at the bottom of the
tension flange reached 51 ksi (350 MPa) and 52 ksi (357 MPa) for scaled Designs 19
and 7, respectively.
As mentioned in Section 4.8, the ultimate strength of the concrete was 7.8 ksi
(54 MPa), and the linear elastic limit can be estimated as 3.9 ksi (27 MPa). The
stresses in the tube concrete and in the deck did not exceed this value, so concrete
material nonlinearity does not appear to contribute to the nonlinearity in the moment
versus strain curves. The steel used in the test specimen, except for the FW-T
material, had a proportional limit of at least 40 ksi (276 MPa). as shO\\11 in Section
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4.8. Some of the steel stresses presented above exceed 40 ksi (276 MPa), however,
the moment at which the Midspan sections started to exhibit nonlinearity was
estimated from Figures 5.17 through 5.22, and the stress levels corresponding to the
onset of nonlinearity were calculated assuming linear elastic behavior. The stress
levels at which the two test girders started to show nonlinearity in the midspan
moment-strain curves was approximately 18 ksi (124 MPa) for scaled Design 7 and 10
ksi (69 MPa) for scaled Design 19. Therefore, since the midspan moment-strain
graphs exhibit nonlinearity of stresses well below 40 ksi (276 MPa), the nonlinearity is
most likely due to residual stresses in the steel.
Strain Jumps
Two jumps in strain at the "East Elastic" sections of the test girders, caused by
placing block number four of the Simulated Construction loading condition
arrangement are observed in Figures 5.5 and 5.7. This is observed as ajump forward
in the strains of scaled Design 7 (Figure 5.5), and a jump backward in the strains of
scaled Design 19 (Figure 5.7). It is speculated that the non-composite deck panels
were not in contact with the scaled Design 7 test girder near the East Elastic section
until after block number four was placed. In Figure 5.5, the slope ofthe data for
scaled Design 7, before block number four is placed, is approximately 7% steeper than
the linear regression line. In Figure 5.7, the slope of the data for scaled Design 19,
before block number four is placed. is approximately 9% flatter than the linear
regression line. In other words, scaled Design 7 was not being fully loaded and scaled
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Design 19 was being overloaded, before block number four was placed. This strain
jump is not observed for the Stage 3 test after the girders are composite with the deck.
In the composite case, the deck is supported by a continuous haunch, and therefore
makes contact everywhere along the test girders.
Neutral Axis Location
Plots were made to determine the experimental location of the neutral axis in
the test girders during the unloading phase of each test stage. This was done by
plotting strains at the Elastic sections. Longitudinal strains were plotted at the
different heights over the cross-section, and a linear regression line was fit to the data.
The intersection of the linear regression line with the zero strain axis was the location
of the neutral axis. This was performed after each block was removed. The fully
loaded condition was taken as the initial point for the longitudinal strain values. Thus,
the neutral axis location could be plotted throughout the unloading phase ofeach test
stage.
Figures 5.23 through 5.30 display the location of the neutral axis for each test
stage. The neutral axis location does not change in Stages I, 2, and 2-2. However, in
Stage 3, the neutral axis dropped over the course of unloading from slightly above the
analytically calculated location (discussed later) to slightly below. The reason may be
as follows. When the deck panels were post-tensioned, a compressive force in the test
girders was introduced by friction between the wood shims and the girders. This
condition existed when the instrumentation was balanced for Stage 3. As the test
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specimen was loaded, the deck panels were compressed more closely together, and
some of the compression in the girders was lost. This caused the neutral axis to be
above predicted at the beginning of unloading. As the test specimen was unloaded,
gaps may have opened at the deck panel joints. This would cause the compression
area to become less effective, and the neutral axis to drop.
Comparison of Experimental and Analytical Results
Table 5.2 presents analytical results for the slope of the moment-strain plots
(stiffness) and the neutral axis location, measured from the bottom ofthe tension
flange. The actual, measured girder dimensions were used along with transformed
sections to calculate these values of the stiffness and neutral axis location. The
analytical calculations are shown for both the East Elastic sections and West Elastic
sections of the test girders because the haunch thickness was slightly different at these
two locations.
Table 5.3 presents a comparison of the experimental results and analytical
results. The experimental stiffnesses range from approximately the same as the
analytical, to 11% more stiff. No comprehensive reason for the discrepancy has been
detemlined. However, the two possible causes for difference in stiffness are suggested
here. During the test stages using the Simulated Construction loading condition when
the deck is non-composite. the deck may assist in carrying load. A net tension may
develop in the girder and a net compression in the deck. This \\ill cause the test
girders to appear stiffer. and also push the neutral a,is position upward. The
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corrugated web may also contribute slightly to the moment carrying capacity of scaled
Design 19, even though it was neglected in the analytical calculations. The
combination of the corrugated web and deck contributions could cause the neutral axis
to move upward or downward, depending on the size of the individual contributions.
5.5 Vertical Deflection Results
Experimental Measurements
Vertical deflections were monitored at five cross-sections along the bottom of
the tension flange of each test girder. These cross-sections were Section A, Section B,
Section C, Section D, and Section E. Refer to Figures 5.2 and 5.3 for details about
these section locations. Midspan moment versus vertical deflection graphs were
generated for each of these sections. Sections A and E were plotted on the same graph
to show the symmetry of the test girder deflections. Likewise, Sections Band D were
plotted on the same graph. The two vertical deflection transducers at Section C were
plotted on the same graph to show twisting of the tension flange at midspan of the test
girders.
Analytical Calculations
Graphs were generated to compare the experimental results \vith analytical
results. The analytical results were calculated using transfomled sections for the
girders. Vertical deflections due to bending defonnations and shear deformations~ ~
were considered in the analytical calculations. In calculating vertical deflections due
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to shear, an effective shear area must be determined. In the non-composite case, the
effective shear area is assumed to be the area of the web. In the composite case, the
effective shear area contributed by the haunch and the deck are included. The total
effective shear area ofa composite section is calculated by dividing the transformed
section cross-sectional area by a form factor, fs, calculated as follows:
(Eq.5.2)
where, A is the cross-sectional area, I is the cross-sectional moment of inertia, Qis the
first moment of the area above or below the point in question about the neutral axis,
and b is the width of the cross-section at the point in question. One important note
concerning the shear deformation calculations is that the shear stiffness of a
corrugated web must be multiplied by the following term (Abbas, 2003):
f3 + cosa
f3 + 1(Eq.5.3)
where Pis the ratio of the longitudinal fold width, b, to the inclined fold width, c, and
a is the corrugation angle for a trapezoidal corrugation shape. This factor decreases
the shear stiffness of a corrugated web, when compared to a flat web with the same
effective shear area.
Comparison of Experimental Results and Anah1ical Results
Figures 5.31 through 5.66 show the midspan moment versus vertical
deflection graphs. as well as the comparisons between experimental results and
analytical results. The experimental results for midspan moment versus vertical
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deflection are similar to the results for moment versus strain. The results are nonlinear
for Stage 1, and linear for Stage 2. The results of Stage 3 are linear until the stress
levels of the previous stages are exceeded, and then the results become nonlinear. In
Stages 1 and 3, the analytical results are compared to the unloading branch of the
experimental data because the loading branch is nonlinear. In Stage 2, the analytical
results are compared to the loading branch of the test data. In the non-composite
stages (Stages 1 and 2), the comparison between experimental and analytical is shown
after block number four was placed, so that the contact issue discussed in Section 5.4
does not influence the comparison.
It is important to note that the experimental and analytical data are not linear
for moment versus vertical deflection, even after the shakedown of residual stresses.
If the loading were proportional, then the moment and curvature of a linear elastic
girder would increase linearly, and the vertical deflections along the span would also
increase linearly. However, the loading used in the experimental program only
approximates a uniformly distributed loading (i.e., a proportional loading), as
discussed in Section 4.6, and therefore the graphs are only approximately linear.
Tables 5.4 and 5.5 present comparisons of experimental results and analytical
results for stiffness of the midspan moment versus vertical deflection plots. Table 5.4
includes only bending deformations in the analytical vertical deflection calculations.
whereas Table 5.5 includes bending and shear defom1ations. TI1e inclusion of shear
defom1ations increases the ratios by approximately 5% for scaled Design 19 and 3%
for scaled Design 7. TI1e difference is due to the fact that corrugated webs are not as
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stiff in shear, as discussed above. The results are similar to those obtained for the
moment versus strain graphs, in that the experimental results are stiffer than the
analytical results. The reader may refer to Section 5.4, where possible reasons for the
increased stiffness are discussed.
Total Vertical Deflections
The vertical deflection results presented thus far have been for individual test
stages only. It is equally important to look at total vertical deflection throughout all
testing. Figures 5.67 through 5.72 present total midspan moment versus total vertical
deflection at the five sections for each girder. These results are presented for all test
stages through the end of Stage 3. Horizontal offsets have been added between stages
to make viewing easier. The figures include calculated girder self-weight effects,
measured Deck Placement effects, measured Stage 1 effects, measured Stage 2 effects,
measured Roller Placement effects, measured Stage 2-2 effects, measured Mdc-2
effects, calculated haunch weight effects, and measured Stage 3 effects. The girder
self-weight and haunch weight effects had to be calculated because no data was taken
to measure these effects. The Mdc stage was assumed to unload elastically before the
Roller Placement stage, and is not shown in Figures 5.67 through 5.72.
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5.6 Lateral Displacement Results
Finite Element Simulations
Numerous simulations were generated by Kim (2004b) using the Finite
Element Method (FEM). The FEM models used by Kim for these simulations were
similar to those used in his Ph.D. dissertation (Kim 2004a), but the models used in this
study had rectangular tubular flanges. The FEM simulations were performed to
understand the lateral displacement results obtained from testing. As discussed in
Chapter 4, only the tube flange imperfections (sweep) were measured before testing.
FEM simulation results were used to investigate how the imperfection in the tension
flange could affect the lateral displacements of the test girders. Numerous
imperfection shapes were investigated in the FEM models. Six initial imperfection
shapes, illustrated in Figure 5.73, were considered in the simulations. The
displacements shown in the figure are the midspan displacements. The value of Ux is
an amplitude that was defined for each FEM model. The shapes of the imperfections
along the lengths of the tube and tension flange are half sine waves. The midspan
displacements are positive in the south direction. Table 5.6 provides descriptions for
the different FEM models that were investigated. SD7 and SO 19 stand for scaled
Designs 7 and 19, respectively. The shape column refers to the shape number in
Figure 5.73. The amplitude column indicates the value ofux, where L is the span of
the test girders. The material column refers to the stress-strain models that were used
in the FEM simulations.
138
The material stress-strain models presented in Section 4.8 were used
consistently in the FEM simulations except for the model used to represent the tube
steel in scaled Design 7. Recall that the accuracy of the stress-strain data for the tube
steel (FW-T) for scaled Design 7 was questionable (see Sect. 4.8). The original
quadra-linear curve was initially used for the stress-strain model in the FEM
simulations. However, the early softening observed in the tension tests led to a poor
fit between the stress-strain model and the actual stress-strain data, and the simulation
results showed a lateral torsional buckling strength less than the Simulated
Construction loading condition applied during the Stage 1 and Stage 2 tests. The use
of this stress-strain model for the scaled Design 7 tube steel, along with all other
stress-strain models from Section 4.8, is referred to as MaW in Table 5.6. More points
were added to the FW-T stress-strain model, as shown in Figure 4.27. This led to a
better fit between the stress-strain model and the actual stress-strain data, but the FEM
simulation results still showed that the lateral torsional buckling strength of scaled
Design 7 was less than that required for the Simulated Construction loading condition.
The use of this stress-strain model, along with all other stress-strain models from
Section 4.8, is referred to as MatI in Table 5.6. Finally, the stress-strain model for the
tube of scaled Design 19 (CW-T) was used for that of scaled Design 7. The use of this
stress-strain model, along with all other stress-strain models from Section 4.8, is
referred to as Mat2.
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Scaled Design 7 FEM Simulation Results
For all the FEM simulations, it was assumed that the girders had their specified
initial imperfections at zero load. Figures 5.74 through 5.94 illustrate the midspan
moment versus midspan lateral displacement results obtained from the FEM models
described in Table 5.6. The midspan lateral displacement shown in these figures does
not include the initial displacement due to the initial imperfection. Figures 5.74
through 5.79 (for models SD7-1 through SD7-6) provide good qualitative information.
The second three of these models have larger initial imperfections than the first three.
It is observed that larger initial imperfections cause larger lateral displacements during
loading and a lower lateral torsional buckling strength. Also, it is observed that initial
lateral displacement of the girder, with no twist (Figures 5.74 and 5.77), will lead to
twist in the same direction (i.e., with the top flange displacement greater than the
bottom flange displacement). Initial lateral displacement accompanied by initial twist
in the same direction, however, is more critical. For instance, compare Figures 5.74
and 5.75. Model SD7-1 has the same tube imperfection as model SD7-2, but model
SD7-2 has initial twist. The initial twist causes the lateral displacements of model
SD7-2 to be larger, and lateral torsional buckling strength to be smaller.
Figures 5.76 and 5.79 (for models SD7-3 and SD7-6) introduce an interesting
situation where the initial twist is in the opposite direction of the initial lateral
displacements (i.e., with the top flange displacement less than the bottom flange
displacement). It is observed in these figures that the lateral displacement occurs in
the direction of the initial twist. not the initial lateral displacement. For these cases.
140
the initial twist dominates the movement during loading. It should be noted for these
girders that the tension flange initially displaces more than the tube, thus reducing the
twist. This issue was investigated further using the FEM models generated for scaled
Design 19. The remaining figures for scaled Design 7 (Figures 5.80 through 5.85)
show the results of changing the tube steel properties, as discussed above.
Scaled Design 19 FEM Simulation Results
A discussion similar to the one above could be presented for Figures 5.86
through 5.91 (for models SD19-1 through SD19-6), but is not necessary. Models
SD 19-3 and SD19-6 were generated to investigate the situation where initial twist is in
the opposite direction of initial lateral displacements. This was done with the tension
flange being initially displaced 1.5 times the initial displacement of the tube. Models
SD 19-7 through SD19-9 investigated additional initial displacement factors of 1.25,
1.375, and 1.3125 (see Figure 5.73), and Figures 5.92 through 5.94 show the results.
While Figures 5.92 through 5.94 provide a good view of the early behavior ofFEM
models SD19-7 through SD19-9, Figures 5.95 through 5.97 illustrate the post-peak
behavior of FEM models SD19-7 through SD19-9. Consider the post-peak behavior
first. It can be seen that at a certain point, when the initial displacement factor is
between 1.3125 and 1.375, scaled Design 19 will switch from failing in the same
direction as the initial lateral displacements to failing in the opposite direction of the
initial lateral displacements. If the initial 1\\ist is large enough (in the opposite
141
direction of the initial lateral displacements), then it dominates the motion, causing the
girder to fail in the opposite direction of the initial lateral displacements.
The early behavior is similar for all magnitudes of initial displacement used in
the tension flange. The girder displaces in the direction of the twist, and the twist is
reduced. At some point, the girder becomes vertical. The rest of the behavior depends
on the size of the initial twist. The complete behavior is best explained using two
graphs and two schematics. Figures 5.98 and 5.99 are the same as Figures 5.93 and
5.94, except that the initial imperfections have been added in. Figures 5.100 and
5.101 show schematics of the initial imperfection (1), the vertical position (2), and the
behavior after the vertical position is reached (3). SD19-8 reaches vertical with tube
and flange lateral displacements to the north, and then displaces and twists to the
north. SD19-9 reaches vertical with tube and flange lateral displacements to the south,
and then displaces and twists to the south.
Comparison of Experimental Results with Analytical Results for Scaled Design 7
Figures 5.102 and 5.103 compare the scaled Design 7 experimental results for
the tube and tension flange lateral displacement during Stages 2 and 2-2 with selected
FEM simulation results. In the figures, ST2 and ST22 represent the experimental
results from Stage 2 and 2-2, respectively. The remaining curves are FEM simulation
results identified by the model nanles from Table 5.6. For both stages, the
experimental results show the tube displacing more than the tension flange. It was
therefore speculated that the imperfection shape did not include an initial t"ist in the
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opposite direction of initial lateral displacements. The experimental results also show
much smaller lateral displacements than analytical results. The analytical results
chosen for comparison were from FEM models that had initial imperfections with
lateral displacement, but without twist, because they compared the best with the
experimental results. The figures show that the material models MatO and MatI for
the scaled Design 7 tube steel (used in models SD-l, SD-4, and SD-7) are probably
inaccurate. Even the results from model SD7-lOusing the Mat2 material model do
not compare well with the experimental results.
Recall that the Stage 2-2 test was conducted after rollers were placed between
the scaled Design 7 girder and the precast deck to eliminate friction that could restrain
lateral displacement of the girder. A further attempt to allow more lateral
displacement in the test girders was made during Stage 2-2, shown by the plateau at
the end of the curve ST22. At this point, the test specimen was fully loaded with the
Simulated Construction loading condition. The deck panels were unconnected during
this stage, but it was suspected that wood spacers between panels were causing a
certain amount of friction between panels, allowing the deck to act to some extent as a
large lateral beam, which inhibited lateral displacements of the girders. Cuts were
made through the wood shims to try to separate the deck panels. This process allowed
the movements shO\m in the plateau, and these movements suggest that the
unconnected deck panels were inhibiting lateral displacement of the test girders.
It is possible that the imperfection shape was quite different than those used in
the FEl\'1 models. As stated earlier. the initial imperfection shapes in the FEM models
143
were assumed to be at zero load. The tube imperfection in the test girders was
measured under the self weight of the girders. Also, placement of the panels caused
some lateral displacement of the test girders, but these lateral displacements do not
necessarily correspond to the FEM simulation results for the corresponding added
load, because the girders may be pushed laterally unintentionally during placement of
the panels. It is also possible that there is some lateral bracing being applied to scaled
Design 7 by the deck and/or scaled Design 19. Any of these factors could be the
source of the discrepancy in the comparison.
Comparison of Experimental with Analytical Results for Scaled Design 19
Figures 5.104 and 5.105 compare the scaled Design 19 experimental results
for the tube and tension flange lateral displacement during Stages 2 and 2-2 with
selected FEM simulation results. In the figures, ST2 and ST22 represent the
experimental results from Stages 2 and 2-2, respectively. The remaining curves are
FEM simulation results identified by the model names from Table 5.6. Note that
these names are preceded by a negative sign, because the initial imperfections and the
lateral displacement results are in the opposite direction to that used in the FEM
model. This change in direction was for comparison purposes because the initial
imperfection of the tube of scaled Design 19 was to the north.
For both stages. the experimental results show the tension flange moving more
than the tube. but both in the south direction. It is therefore believed that the initial
imperfection shape of scaled Design 19 included a twist in the opposite direction to
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the tube flange displacement. Therefore, the analytical results chosen for comparison
were from FEM models with initial imperfections of this type. The comparison
between experimental and analytical results is acceptable.
Once again, it is possible that the imperfection shape is something other than
what was used in the FEM models. It could be that the initial imperfection
incorporates lateral displacement and twist in the same direction. The scaled Design
19 girder could have been pushed southward by the scaled Design 7 girder. The
possibilities for discrepancy discussed for scaled Design 7 are also possible.
Lateral Displacement Shape Plots
Figures 5.106 through 5.113 show the lateral displacements of the tube and
tension flange for Stages 2 and 2-2, along the entire length ofscaled Designs 7 and 19.
The individual curves in each figure represent different load levels during testing
(refer to Figure 4.16). The curve entitled "Wood Spacers Cut" refers to data taken
after the cutting of wood spacers between deck panels in Stage 2-2.
5.7 Web Distortion
As stated in Section 4.7, gages were placed on the \vebs of the test girders to
measure the distortion. As the web distorts, the tubular flange will become less
effective in supplying torsional stiffness to the girder (Kim 2004a). The test girders
were the most susceptible to web distortion during Stage 2 and Stage 2-2. and results
for these stages are discussed here.
145
Web Distortion of Scaled Design 7
Sections 1 and 2 were used to measure web distortion on scaled Design 7, and
Section 2 was observed to be the most critical. The gages on either side of the web
allowed curvature to be calculated at the gage locations over the depth of the web.
Positive curvature corresponds to a radius of curvature in the positive z direction.
Figures 5.114 and 5.115 show the curvatures at Section 2 plotted over the depth of the
web for Stages 2 and 2-2, respectively. These are the curvatures when the test
specimen is fully loaded with the Simulated Construction loading condition. The
figures show that the curvature is somewhat linear over the depth of the web, with
positive curvature at the bottom and negative curvature at the top. Figure 5.116
shows a schematic view of the web distortion as the girder twists and displaces
laterally. This distortion correlates well with the measured lateral displacements.
Take note, however, that the curvature is very small. The largest curvature (0.000244
in:') corresponds to a radius of curvature of approximately 4098 in. (104089 mm).
Web Distortion of Scaled Design 19
Before testing. it was thought that web distortion of a corrugated \veb girder
would occur as vertical tension only or vertical compression only in adjacent
longitudinal folds. depending which side of the girder they were on. However. this
was not observed in Sections 3. 4. 6. 7. 9 and 10 of scaled Design 19. There was web
plate distortion \\ithin the individual folds. but no clear trends were observed.
146
However, curvatures were quite small. The maximum observed value was 0.000352
in.-I, corresponding to a radius of curvature of 2841 in. (72161 mm).
5.8 Tension Flange Transverse Curvature of Scaled Design 19
The C-Factor Correction Method (Abbas 2003) was used to calculate the
flange transverse bending moments due to vertical shear acting on the corrugated web
of scaled Design 19, under the Simulated Construction loading condition. These
transverse bending moments were then converted to transverse curvature in the
flanges. These analytical calculations were compared to experimental results.
However, the flanges experienced additional transverse bending during loading, due to
initial imperfections in the form of flange out-of-straightness. This resulted in
combined effects of transverse curvature from transverse bending due to the
corrugated web and transverse bending due to initial imperfections. Figures 5.117
and 5.118 show the experimental transverse curvature in the tension flange (using
Sections 1, 2, West Elastic, 5, Midspan, 8, East Elastic, 11, and 12) from Stage 2 and
Stage 2-2, compared to the analytically calculated transverse curvature that would be
present due to the corrugated web. The curvature is taken as positive when the radius
of curvature is measured in the positive z direction. The figures show curvature in the
tension flange only, because two failed gages created incomplete results for the tube.
Close to the ends of the simple span, the vertical shear is high. which causes
the flange transverse bending moments due to the corrugated web to be large as well.
Also. the transverse curvature in the flanges caused by initial imperfections is small at
147
the ends. Therefore, in Figures 5.117 and 5.118, the experimental results closely
match the analytical results near the ends. Out near midspan, the vertical shear is low
and the flange transverse bending moments due to the corrugated web are also small.
The transverse curvature is dominated by the effects of the initial imperfections. Thus,
the experimental results do not match the analytical results in this region.
5.9 Plate Bending in Tension Flange of Scaled Design 19
From beam theory, strains are expected to vary linearly from the neutral axis,
and to increase with distance from the neutral axis when a cross-section is subjected to
bending. Behavior contradictory to this was observed in the tension flange of scaled
Design 19 during testing. The Elastic sections and the Midspan section were used to
look at this behavior more closely. As shown in Figure 4.20, these sections have four
strain gages on the tube and five strain gages on the tension flange. Results from the
four gages on the tube and the four comer gages on the tension flange were studied.
Curvature Calculations using Experimental Data
The strain gages were used to calculate the overall curvature of the cross
section and the local curvature of the tension flange. The curvatures that were studied
are vertical curvatures. Figure 5.119 illustrates the eight gages that were studied. The
upper and lower tube gages on the north side are referred to as UTGN and LTGN.
respectively. The UTGN and LTGN are equidistant from the mid-surface of the tube.
In a similar manner. the upper and lower tension flange gages on the north side are
148
referred to as UFGN and LFGN, respectively. The UFGN and LFGN are equidistant
from the mid-surface of the flange. Similar strain gages are on the south side of the
girder. The strains at the mid-surface of the tube and mid-surface of the tension flange
were obtained by averaging the upper and lower values. In order to eliminate the
effects of lateral bending, averages obtained on the north side were averaged with
those obtained on the south side. Using these average strains (at the centroids of the
tube and flange), the overall curvature of the cross-section was determined by taking
the difference in strains divided by the distance between the two centroids. The local
(plate bending) curvature of the north and south tips of the tension flange was
determined by taking the difference between the upper and lower tension flange
strains divided by the thickness of the flange.
Comparison of Overall Cross-Section Curvature to Tension Flange Curvature
Table 5.7 shows comparisons between the overall cross-section curvature and
the local curvature of the tension flange. The specific sections are identified using
their section names. The table was generated using data from Stages 2 and 2-2 under
the full Simulated Construction loading condition. The local curvature in the tension
flange is given for both the north and south flange tips. When the flange local
curvature is opposite to the overall cross-section curvature, it will be referred to as
reverse curvature. The relationship between the section locations and the corrugation
folds can be observed in Figure 5.2. It can be seen that all of the studied sections are
located close to bend regions at the ends of longitudinal folds. The curvature in the
149
tension flange at these sections is reverse curvature, and is somewhat consistent for the
flange tip closest to the longitudinal fold. The curvature on the flange tip opposite to
the side with the longitudinal fold is not consistent. In all but two cases, the flange
local curvatures are larger than the overall cross-section curvatures. These local
curvatures are due to flange plate bending. This plate bending was also observed by
Abbas (2003) in the tension flanges of corrugated web test girders.
150
Table 5.1 Stage Identification Subscripts
Stage SubscriptDeck Placement DP
Stage 1 ST1Stage 2 ST2
Mdc MDCRoller Placement RP
Stage 2-2 ST22
Mdc-2 MDC2Stage 3 ST3
Table 5.2 Analytical Values for Stiffness and Neutral Axis Location
Girder and ConditionAnalytical Results
Stiffness (kip-in.) NA (in.)
Scaled Design 19, Noncomposite 9.41 16.43Scaled Design 7, Noncomposite 8.12 17.66
Scaled Design 19, Composite, East Elastic Section 11.99 33.01Scaled Design 19, Composite, West Elastic Section 12.04 33.11Scaled Design 7, Composite, East Elastic Section 12.24 32.37Scaled Design 7, Composite, West Elastic Section 12.39 32.60
Table 5.3 Comparison of Experimental Results and Analytical Results for Stiffnessand Neutral Axis Location
Experimental Results I Analytical Results
Test Scaled Design 19 Scaled Design 7
Stage East Elastic West Elastic East Elastic West ElasticSection Section Section Section
Stiff. NA Stiff. NA Stiff. NA Stiff. NAStage 1 1.11 1.00 1.08 0.99 1.05 1.03 1.07 1.03Stage 2 1.10 1.00 1.08 0.99 1.02 1.01 1.04 1.01
Stage 2-2 1.10 1.00 1.05 1.00 1.01 1.00 1.05 1.00Stage 3 1.09 0.96 1.10 0.99 0.99 1.00 0.99 1.02
151
Table 5.4 Comparison of Experimental Results and Analytical Results for Stiffness,Including Only Bending Deformation in Analytical Calculation
Test Experimental I AnalyticalStage C-D5 C-D8 C-D11 F-D5 F-D8 F-D11
Staqe 1 1.06 1.02 1.02 1.04 1.00 1.00Staqe 2 1.02 1.05 1.05 1.01 1.04 1.06
Stage 2-2 1.01 1.00 1.02 1.02 1.04 1.04Staqe 3 1.03 1.07 1.09 1.00 1.04 1.02
Table 5.5 Comparison of Experimental Results and Analytical Results for Stiffness,Including Bending and Shear Deformations in Analytical Calculation
Test Experimental I AnalvticalStage C-D5 C-D8 C-D11 F-D5 F-D8 F-D11
Staqe 1 1.10 1.06 1.06 1.07 1.03 1.02Staqe 2 1.07 1.10 1.09 1.04 1.07 1.09
Stage 2-2 1.06 1.04 1.06 1.06 1.07 1.06Stage 3 1.09 1.12 1.14 1.04 1.08 1.06
152
Table 5.6 Description of FEM Models
FEM Model Id. Shape Amplitude Materials807-1 1 U1000 MatO
807-2 2 U1000 MatO
807-3 3 U1000 MatO
807-4 1 U500 MatO
807-5 2 U500 MatO
807-6 3 U500 MatO
807-7 1 U1000 Mat1
807-8 2 U1000 Mat1
807-9 3 U1000 Mat1
807-10 1 U1000 Mat2
807-11 2 U1000 Mat2
807-12 3 U1000 Mat28019-1 1 U1500 MatO8019-2 2 U1500 MatO8019-3 3 U1500 Mato
8019-4 1 U1000 MatO8019-5 2 U1000 MatO
8019-6 3 U1000 Mato8019-7 4 U1000 Mato
8D19-8 5 U1000 MatO
8D19-9 6 U1000 MatO
Table 5.7 Curvatures Observed to Study Plate Bending in Tension Flange of ScaledDesign 19
Flange, North Flange, SouthSection Id. Stage Total ell (in:1
) ell (in:1) ell (in:1
)
EastElasticSection Stage 2 1.71x10·s -2.64x10-s -4.88x10·s
Midspan3.09x1O·s -4.75x10·s 2.98x10-4Section StaQe 2
WestElasticSection Stage 2 1.81x1O·s -3.70x10·s 5.54x10·s
EastElasticSection 8tage 2-2 1.73x10·s -1.32x1O-s -5.81x10·s
Midspan2.97x10·s -3.83x10·s 1.21x10-4Section Stage 2-2
WestElasticSection 8tage 2-2 1.84x10·s -3.17x10·s 6.34x10·s
153
\Jest
y Vertico.l Direction
~---- x Eo.stwo.rd Direction
2 Southwo.rd Direction
Figure 5.1 Coordinate Axes for Test Girders
154
Sectl2 Sect.1!C-Sl C-S5C-S2 C-S6(C-S3) (C-S7l(C-S4) (C-S8)
K K
EllstElQstlc MidspilnSection SectionC-S9 C-S31C-SIO Sect.IO C-S32C-SI1 C-SI8 " t 9 Sect.7 C-S33 Sect.6(C-SI2) C-:19 ~=~22 C-S29 (C=S34) C-S:O(C-SI3) (C S20) (C-S23) (C-S30) (C S35) (C-.AD(C-SI4) (C-S2D (C-S36)
I A J Sect.8C-S24( -
................-4·-7M·(~OOJ "I'll
VestEillsticSt'ctlonC-S53
Sed.3 C-S54 Sed.2 Sect.1SectA C-S49 C-S55 C-S62 C-S66C-S47 C-S5O (C-S56) C-S63 C-S67(C-S48) (C-S51)(C-S57) (C-S64) (C-S68)
(C-S52)(C-S58) (C-S65) (C-S69)J A I K K
/T'-'-........"-!.-7M'(~OOJ "I'll
C-S59C-S60 Sect.AC-S61 C-Dl8
I
+....J..----l4·-stt'----J..--:
(4~0l ""I
C-SI5Sect.E C-SI6C-D5 C-SI7
E I
7·-10Ji<--..J-----i4·-Srt'----J--l-eo(2402 "") (4~C1 N'll
C-DlC-D2
Botten of betic., Qg,npg
SCllled Design 19South Girder, North Side( ) Denotes Opposite Sidt'
Figure 5.2 Scaled Design 19 Instrumentation Identifiers
155
MidspanSectionF-S20F-S2lF-S22
Sect.2 F-S23 Sect.!F-S!O F-S24 F-S49
East F-SIl F-S25 Vest F-S50Elastic F-S!2 F-S26 Elastic F-S51S!C'etion F-S!3 (F-S27) Section F-S52F-S! F-SI4 (F-S28) F-S40 F-S53F-S2 (F-SIS) (F-S29) F-S4! (F-S54)F-S3 (F-SI6) (F-S30) F-S42 (F-S55)(F-S4) (F-S17) (F-S3D (F-S43) (F-S56)(F-S5) (F-SIB) (F-S32) (F-S44) (F-S571(F-S6) (F-SI9) (F-S33) (F-S45) (F -S58)
G G
'-10 """"".----i4·-9a:'---....[2lJ97 M) (J~2.4 IVI) (4~0l N'Il
EleYJl1kln
Sect.CF-S37F-DlI
F-57 F-S34 F-S38 F-S46F-Dl SectE F-58 SectD F-S35 r -Dl2 Sect.B F-S47 Seci.A
F-~ F-~ F~_9 'f__F-_sr_s3_9__F_I_15 F_-~8 F~_8 _
=t=7'-IOIt' p 14'-9tt:....·------i-----J14.-9tt··---i.......----114.-9f.. g 7'-lojt;](2402 ,,") (4SH "I'll (4501 N'Il (4501 MJ (2402 ""J
!ct1on cf bQ1100 £IMPR
Scaled Design 7North Girder. North Side( ) Denotl?s Opposite Side
Figure 5.3 Scaled Design 7 Instrumentation Identifiers
156
Scaled Design 7
C-Dl6(C-D17)
C-Dl3 C-D9(C-D14) (C-DlO)
C-D(j(C-D7)
C-D3(C-D4)
<2402 ",,)
No ( ) DE'notE's Top rlangE'( ) DE'notE's Boti:on rlangE'
<--.......-----14·-9i/;-------'--'(~501 rnJ
IQQ VIp ... of Gl"'d§'f'5 ferLot!'es! D:mtpcrcrnt Lyor,
Figure 5.4 Lateral Displacement Instrumentation Identifiers
157
3500 -------
----I
--+-A\9. (F-S7, F-S8, F-S9) II
--- Linear Regression of Unload IiI!
i -1- ~-
.-1'-'- '
o~~~-.~~--~--o 50 100 150 200 250 300 350 400 450 500
~SsT1 (in,/in. x106)
Figure 5.5 Moment at East Elastic Section versus Strain for Stage 1(Scaled Design 7)
2500 +--------
rr
3000 +------ ---
-C i" 2000 +------------ ""-=~/----------~-~----,~ !
~I-rn'" 1500 -,-_~Placement of __~__::!: BlOCK#4----."rf"---r------
1000 +------fl---~-"J
3500 ---~------- -~--._-----~~~-------------- ---
_____-J
--1
_ --+-A\9. (F-S46, F-S47, F-S48) II
--- Linear Regression of Unload II
-.... -_._.~~ - ~-_ .. - ....- ._--_.__.~-- ,_ .. ~-~_.. --.--._.. "'~..,.--.-..,-:-::-:--~~~:--,---:- -~ -.,.---:;:---.,.- -..,.- .,,_ .. ..,.--:;-~._..,.-~.-'.-::._--~
50
oo 100 150 200 250 300 350 400 450 500
~SsT1 (in.lin. x106)
Figure 5.6 t\loment at West Elastic Section ycrsus Strain for Stage 1(Scaled Design 7)
158
500
2500'------------- ---~- --
1000
3000 l ~,
~ 1500:'- -- --::!:
- 'C '" 2000 -- -.-- -~ ,
~
3500 -,-----------
i-----JlI------- .....---,
----------_._--I
I
I--.'v.,.t~-------------____,
I
100 150 200 250 300 350 400 450 500~SsT1 (in./in. x106
)
50
o~~~~~.~.o ",,-0_-
o
~------------,i
i -+-A\g. (C-511, C-514, C-515, C-517)i!500 +--+--'......-'111.---- _.--~ - I'
1 Linear Regression of Unload !:il
3000 -'--------
1000 +---
2500 +--------
~ 1500 +---=-o--.-,-;-;---..yrc/·
~
-c"j 2000 +------Co:i-
Figure 5.7 Moment at East Elastic Section versus Strain for Stage I(Scaled Design 19)
3500 ,---' - .----. --.- ---_._.-_._---_._._._----_... _ .._~-._-_._--- ---_.._--_._-,
____ Linear Regression of Unload
-+-A\g_ (C-5SS, C-5S8, C-5S9. C-561)
2500 ~---_.-
500 :.---
1000 ..-.
3000 .:--...--
oo SO 100 1S0 200 2S0 300 3S0 400 4S0 SOO
~SsTl (in.lin. x106)
Figure 5.8 l\Ioment at West Elastic Section yersus Strain for Stage 1(Scaled Design 19)
159
- .c .OJ" 2000 ..---.Co •
;g.~ 1500;'-·~
3500 ~------
3000 -T----------
- --I------~-----~-_l
250 300 350 400 450 500
I/------------------------4
I
-+-A'll. (FoS7. FoSS, FoS9) I
--Linear Regression of Unload I~ ~ ==r=:
150 20010050
+----f --l-------lII
o ~-'---"-L-+-"-~---~~
o
i
2500 +---------- -------------~------~
,-c"j 2000 +------C.~Nlll)
:E
J1SsT2 (in./in. x106)
Figure 5.9 Moment at East Elastic Section versus Strain for Stage 2(Scaled Design 7)
3500 ------- ----------- ------- ----III
_ Linear Regression of Unload
--+--A\g. (F-S46, F-S47, F-S48)
50o
o 100 150 200 250 300 350 400 450 500
J1SsT2 (in.lin. x106)
Figure 5.10 Moment at 'West Elastic Section yersus Strain for Stage 2(Scaled Design 7)
160
.1000 ---------
f2500 L~ _
l
3000 -'---------
N •
t; 1500,-------:E
-C lI 2000 ~'----------C. L
~
3S00 ---------
SOO4S04003S0
I
I: Linear Regression of Unload
Iu---------------------j
II
Ii
100soo __.L-.L-'--t--~--~~~---.~l.-=- =:;====;========;====1
o 1S0 200 2S0 300
IlSsn (in.lin. x106)
Figure 5.11 Moment at East Elastic Section versus Strain for Stage 2(Scaled Design 19)
SOO +-.-----+-41L:--f--------f -+-A\9. (C-511, C-514, C-51S, C-S17)
1000 +-.------
2S00 +---------
3000 +-.-----------
3S00 ------_._------ -----
___ Linear Regression of Unload
-+-A\9. (C-5SS, C-SS8, C-5S9, C-5S1)
o ~-.-'---- ..--.-------------o 50 100 1S0 200 250 300 3S0 400 450 SOO
IlSsn (in.lin. x106)
Figure 5.12 Moment at West Elastic Section yersus Strain for Stage 2(Scaled Design 19)
161
1000 ~---
SOO
3000 -'------
2S00:- ----------- --- --
--c ." 2000 ~- ~-----Co .
~N ~
~ 1S00 ~-------- - -- --:!:
9000
!~
I,
~------------~-i
300200100
a --~-¥-~--+--~------.
a 400 500 600 700 800 900 1000
~SsT3 (in./in. x106)
Figure 5.13 Moment at East Elastic Section versus Strain for Stage 3(Scaled Design 7)
,,!
2000 -+----.r"--~----_,-----------~--I-+-A\9. (F-57, F-S8, F-59) Ii
ii--- Linear Regression of Unload Ii
1000 -r---c*--~------ I
6000 ~'--------- -~"'.r--
7000
8000 ~-------~,
t:! 4000 ~------~III
:!:3000 -,---------=~-_T'_----------------~--'
--c"6. 5000 -,------------,..---.-------~~----___,~ !--
800 900 1000700
___ Linear Regression of Unload
-+-A\9. (F-546, F-547, F-548)
a '--a 100 200 300 400 500 600
~SsT3 (in.lin. x106)
Figure 5.14 Moment at West Elastic Section yersus Strain for Stage 3(Scaled Design 7)
162
3000:----- - --
2000~-~------
1000 ~-
7000 -'~---------------~----l
9000 -,~-------~-------------- --------~----~------~
8000 -'----~---~---~-----~--------~ ~---.-----------'
__ 6000 ~-------.--c"6. :~ 5000 ------------ ---~------ ....I- •
:E 4000 ~-- - ----
9000 .~---~---
8000 ----~-------.-----.~,..~~~~-
,JP------~~----~-_____,
100 200 300 400 500 SOO 700 800 900 1000
J1SsT3 (in./in. x106)
Figure 5.15 Moment at East Elastic Section versus Strain for Stage 3(Scaled Design 19)
2000 -'--~~-/'-~-f'-------,---------------------i
-+-A\9. (C-S11, e-S14, C-S1S, C-S17) I____ Linear Regression of Unload j1000 -i
r
o ...t~'-t-~--~o
7000 r
3000 +---------:cr--=-=---------------
SOOO ~_._---------r.~ -..---------'2 .•- rC. 5000 ,--------------- w--*----------
g ~~ 4000 -j-----------;r.7FIn
:!:
9000 ---- .-._-------------- --.------------- ------..---------.--
8000 -:--~--------- --~---------------- .....--------- --~-..
-+-A\9. (C-SSS, C-SS8, C-SS9, C-SS1) I
--- Linear Regression of Unload Io
o 100 200 300 400 500 SOO 700 800 900 1000J1SsT3 (in.lin. x106)
Figure 5.16 Moment at West Elastic Section versus Strain for Stage 3(Scaled Design 19)
163
1000~-
SOOO~----- ------------/-M'
7000 .:...----------------r-
.....l: .
'6. 5000 .:-~-------.g .~ 4000:---- ---CIl •
:!:3000 ~--~-----
2000 .:- -----
6000 ~---------- ------ ---------
___ Linear Regression of Unload
-+-A\9. (F-534, F-535, F-536)
---/-------------
I/--------------11000 +----/-
4000 ~--------
5000
o -~~-~
o 100 200 300 400 500 600 700 800 900 1000
~SsT1 (in.lin, x106)
Figure 5.17 Moment at Midspan Section versus Strain for Stage 1(Scaled Design 7)
j:III
:!:2000 +------,-F--::/--------------------'
-c:'"iCo
;g, 3000 +------------,£-/'-----,-,1'--------------------- --.
6000------
/----------_.
___ Linear Regression of Unload
-+-A\9. (C-533, C-536, C-537, C-539)
-/----/--/--------------
300200100o
o 400 500 600 700 800 900 1000
~SsT1 (in.lin, x106)
Figure 5.18 Moment at Midspan Section yersus Strain for Stage 1(Scaled Design 19)
164
1000 --
4000 +----_.-------
5000
j:III
:!:2000-------- . -- --
'"iCo ,
;g, 3000 -"--------- -
-c:
6000 ----------~--- -- --
___ Linear Regression of Unload
-+-A\g. (F-534, F-535, F-536)
.---~:========-=---:..:~ ..----:-=:....
100 200 300 400 500 600 700 800 900 1000
~SsT2 (in.lin. x106)
Figure 5.19 Moment at Midspan Section versus Strain for Stage 2(Scaled Design 7)
rrI
2000 -~-------/".=------
1000 +---h'-----
4000 ~--------- -~------------------~
5000 +---------
NIUI
:!:
-c"Q.
g 3000 -'---------h"----
6000
___ Linear Regression of Unload
--+-A\g. (C-533, C-536, G-s37, C-539)
oo 100 200 300 400 500 600 700 800 900 1000
~SsT2 (in.lin, x106)
Figure 5.20 Moment at Midspan Section versus Strain for Stage 2(Scaled Design 19)
165
1000 .----
2000 ~-
,4000· --- -- -----------
5000
"
.~ .:. 3000 ~-------g
NI<Il
:!:
-c
14001200
4f-----------
400
~---------~I
~ ~ -+-A\g. (F-S34, F-535, F-536) I~
, Linear Regression of Unload i!"
__----++---'"~-+____~---~~----.,....._-----,..__....L
600 800 1000
J.1SsT3 (in.lin. x106)
Figure 5.21 Moment at Midspan Section versus Strain for Stage 3(Scaled Design 7)
14000
12000
10000 ~
- rl:
8000 -t-----"iCog I
M[
I- 6000 ~III
:!: ,I
r4000 ~----
2000
00 200
1200 1400
___ Linear Regression of Unload
________ .-+-A\g. (C-533. C-536. C-537. C-539).
400200o
o 600 800 1000
J.1SsT3 (in./in. x106)
Figure 5.22 Moment at Midspan Section yersus Strain for Stage 3(Scaled Design 19)
166
2000
4000 ---
10000 ----------------
12000 -- ------------- ---------- -- -- ---
14000
-.!i= 8000'---------Cog
M
lii 6000:!:
, ,I'
9
___ West Elastic Section
,-----------1-+- East Elastic Section !i
-----------------
5 7Block Removal
3
,o -:
1
"30,-Ee... --"C C "Q) ._ 25 +------------------------... ...... ,;::, Q) "
CIl Cl 'met~ ~ 20 +l-----------------------.~ c ••---I.---~.------I•.-----I•._-__tl.l__-__tl.l__- .I---_tt.~.~ f- :g 15 ,f _~ Q).... 1-;::, ...~ 0... E 10 ,-----------o 0c:t:lo 0... m~ 5o.J
Figure 5.23 Neutral Axis During Unloading of Scaled Design 7 in Stage 1
--------- -------- -------
9
--- -------- -
___ West Elastic Section
-+- East Elastic Section
------------------
5 7Block Removal
3
5 ----
o --1
E 30e... .-."C. ,Q) .: 25 -----... ...... .;::, ellCIl ClIII CQ) III ':!: - 20 --------~--------------------------CIl U. •'- c)( a<'iii· • • • , • " • •ia c 15 ~-------------------- --------- --------------.. Q) •.... 1-;::, ...~ a... E 10o 0c:t:lo 0... 0)IIIoo.J
Figure 5.24 Neutral Axis During Unloading of Scaled Design 19 in Stage 1
167
~ East Elastic Section .[I
____ West Elastic Section : i
30Ee.... --g .~ 25 -y---------------... -:::J Q)1/1 ClCIS C
~ ~ 20 +--------------------------
.~ ~ .~~==lat==--lI.....-===*~=-_t.t_-~~~---.---l,t_-__•~ .* t~ ; 15 --t----------------------------.... 1:::J ....~ 0.... E 10 -t---- ----------
o 0c~o 0~a:l
~ 5--1--.-------------o
...J
3 5 7Block Removal
9
Figure 5.25 Neutral Axis During Unloading of Scaled Design 7 in Stage 2
r--- ----------------_._--- _._-
----------- -.
~ East Elastic Section
___ West Elastic Section
-----------
5
o
,30 +------------
rEo....... 't:J •Q) _: 25 --I--.-------.---.-.--.-,,-~--~----------... -:::J Q)1/1 C)CIS CQ) CIS:E iL 20.~ c)( 0ct u; ••---t......- ........,---t.t----4I.f--....~==~~~==1t~===*~~ ; 15-"--.... 1:::J ...~ 0... E 10 ~--- ------o 0c~o 0~a:lCIS(.)
o...J
3 5 7Block Removal
9
Figure 5.26 Neutral Axis During Unloading of Scaled Design 19 in Stage 2
168
,I
--4,
95 7Block Removal
3
_30 +----------'0 .11l .=.. -~ ~25 +------------Q) C
:!:~ ,.~ C 20 f _
~ .~ 1;:::==~l:=:::="-.---..,--..,1--~~,F===l,t===t,===4,!'O C ~ • == ,; ~ 15 +r---------------Q) .... rZ 0 t I
'0 ~ 10 t iC :::: +[--------- I
o 0 ~ 1,1;Om L
~ E +t ----:-------;=-::-:-:-;::::-;-:-::::-:--;::~~IIo 0 5 I -+- East Elastic Section Ii..J .:: t
r
!I___ West Elastic SectionIIo -t------+----
1
Figure 5.27 Neutral Axis During Unloading of Scaled Design 7 in Stage 2-2
i--------------------_._-_ ..__ . -
---, -+- East Elastic Section-
---West Elastic Section
5 ------- ------
Ee... " .CD .: 25 ~----------... -~ CDIII C)
~ C ,-= ~ 20 ----II: U. ~
.~ c~ .~ ~~---t.I-----.,I-----j,IJ---4,l---.......---i.....- .......I-----.E; 15 -------I~ ...~ 0 .... E 10 -----------------o 0 -c:S:::o 0~mcooo-J
o -- ---- -- ----- ---.-----1 3 579
Block Removal
Figure 5.28 Neutral Axis During Unloading of Scaled Design 19 in Stage 2-2
169
40 ------------------------
-----,I
-+- East Elastic Section
___ West Elastic Section
o
5+-----------
~ 35~:: :: ~r-~=:::::::j,t::::::::::t,====l,t::::::::::::t.==~~:::::!"' ... ~ • = ::::~ ::. 30 -r------------------------:::s CIlen C)III c::E ~ 25 -r------------
u. "o~ c: ~>< 0 r< °iii 20.,--- c: rE Q) t.... I- ':::s It- 15 "~ 0It- E~ ~ 10 -t------------o 0.. mIIICJo...J
3 5 7Block Removal
9
Figure 5.29 Neutral Axis During Unloading of Scaled Design 7 in Stage 3
40 -----------------------
-+- East Elastic Section
___ West Elastic Section5
,E 35 1 _
i ~ 30 t~=-=--~-:=~==~=:=~:==:;:==::::;::::;::::-=:::---- ---,:::s CIl ~
~ C) ~Q) ; 25 ~------------:!:u: .o~ c: .~ ~ 20-:en •iii c:... CIl •:; I- .Q) It- 15---- -- - ----z 0It- Eo 0c: ~ 10 ~--- --. ---- ---o 0.. mIIICJo
...J
o -3 5 7
Block Removal9
Figure 5.30 Neutral Axis During Unloading of Scaled Design 19 in Stage 3
170
~~~'Wt-~~---~~
5000 ,~r~
4500 rr
;:-:E 2000 +-~~~~~~~~~~~~~~~~~~~-"\~~--
rr
1500 -,-~~~~~~~----,~~~~
ti1000 +-1 -+- F-D5
ri500 f _F-D18 ,-----.-------\-\,-;
,I Io -~---·=---==~=f==~===-==+T~-'--.,--~~~
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1Vertical Deflection 5T1 (in,)
Figure 5.31 Midspan Moment versus Vertical Deflection at Sections A and E forStage 1 (Scaled Design 7)
-c: 3000 -i-----~--------------~:L-\\::___-
r4000 -'-~--~~~~~~~~~~-------'l,-\\-----____.
f3500 -:------------------".~----
r
"Co;g. 2500 +--~----------~------~c_\\::__-___,
--------- ~---- ._---_ ..
;:-:E 2000 ~-------------------. --.------
------_ .._._----~
--,._-~~._.__._-----------_.__ . -Deflections after--~--- .
. ~__ J:@cement~ .Block #4
1500:-------
1000 ~. -+- F-D5
5000 r--.------4500 ~~-------
4000 l--_
3S00 ~ ---,
-:- 3000 .c ,-. ~Co ';g, 2500 ~----
SOO ~- --.-Analytical F-DS---·------
O:..······~-···~-·-·c ~.~. -.- --.------ .. -..~ -' ..-..-.. - -.-~. ~ -- ..--- .
-2.5 -2.3 -2.1 -1.9 -1.7 -1.S -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection 5T1 (in.)
Figure 5.32 Comparison of Experimental and Anal)tical Results at Section E forStage I (Scaled Design 7)
171
5000 -r~--- ------------------~f
4500 -,--'-------"111!,,~!.----------------
4000 -~--~------~------""..--------------r
3500 +------------~~..----------~
-c: 3000 +--------------'~-__y...______-------",c.:i: 2500 -+-------------~---''rt___------___i- !~ I:E 2000 !
1500 Tr=,======~-------____\.\.____"\->, __--~, I
1000 lJ -+-F-D8r I
t I,
500 ~: -If- F-D15r IO~~-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.33 Midspan Moment versus Vertical Deflection at Sections B and D forStage 1 (Scaled Design 7)
5000 -~----~
Deflections after __~: Elacemenl of ~
Block #4500 ~~; -If- Analytical F-D8-~---~~ --~------ ------- ---- ----
o -:----------~-~~ -----~--.- ---.-~ .-~--- --.-~- .. --~ .. -~.~ --" ._.-. --- -.-.--~--
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
1500:-~-~-------
1000 ~. -+- F-D8
-c: 3000 .---",c.~ 2500 ~:----------------
,4500 -:--------------'".
t4000 -',---------------~
3500
Vertical Deflection ST1 (in.)
Figure 5.34 Comparison of Experimental and Analytical Results at Section D forStage 1 (Scaled Design 7)
172
5000 ,-----------------------~
4500 ~-------~-~---------------__1
4000 +---------""""",,------',--------------
3500 +----------~___"\;;,_____---------____i
-C 3000 -'--------------"-------'~--------_______i''jCo;g. 2500 ~-----------____"\:--,-----~:___------
~CIl 2000 -+-r--------------....~-,,~----_____1~ ~
1500 t--~ .~u.u •• -. __ u-. I
1000] --+-F-Dl1 i
500 T F-D12 I·----------~,-----"
oL-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.35 Midspan Moment versus Vertical Deflection at Section C forStage 1 (Scaled Design 7)
5000 -,---------
4500 .4-- _
4000 ,--------------~,
3500 ~-
~ 3000 .:I ,
Co •;g. 2500 ~--_.
~ tCIl 2000 ,--~--~--~--~- -.--~-----.-. ---- -.-~ l
1500!--·-~---··..-·~··~---·-~--~ ..-Defieaionsafter·- ~
1OOO~- -+- F-D11 Pla_c~eme-'ltof_~_. __.: Block #4
500 .~ _Analytical F-D11 .-------- .. ---~--~-----.
0:------··---·---- ~-~~~-..~....~--.-.-.-,-.-.-,- ...... -- ...-...~.--- ..
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.36 Comparison of Experimental and Analytical Results at Section C forStage 1 (Scaled Design 7)
173
5000
4500
4000 ct
3500 ~~- f
l: 3000~'j" r
Co >-
;g 2500 rf..
I-:E 2000 +--------------------\\--\\
~1500 If---~ ---1000 i -+- C-D5
500 ~~ -ll-C-D18
o -j".'-"---,-----r---,L~-~~~"~>-----..-.~~~--'-+-~---,--'---'-'-.~.
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.37 Midspan Moment versus Vertical Deflection at Sections A and E forStage 1 (Scaled Design 19)
------ -----------=\'\
5000 --,---f
4500 ~f__
4000 --'-------------"
t3500:-~--------
~ 3000 ~----~-~-------'j"Co;g 2500:-----------
t=:E 2000 -:---------
1500>-~--------------- ----Deflections after-------~ "
~ -+---- C-DS Placement of1000 .:: --- ---------~----~-------Block #4
500 .:~ __ Analytical C-DS---- - ------~-- -------~------
--o ~-"-~~~~=~.~c,~-~~=_._.__~ • • • ._._. __ L • - -
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.38 Comparison of Experimental and Ana1)tical Results at Section E forStage 1 (Scaled Design 19)
174
___ G-015
-+-G-08
5000 -~
4500 +-------~"'"--------------1
4000 +--------~_A:__~c__---------______!
3500 -t------------'~~=--------__1
-C 3000 +------------~----'~-------'j"c.:i: 2500 +--------------~:--~._____----____1.......~ 2000 +----------------".,-"<--"'-,,----------i
1500 -t-F~~~~~~,___---------'\'\_-----\."':____---j
1000
1500 1o -.jL----,-----,-----r---f---'-'-'--+~__+~'----t--"-~_r"_~. ~--<--L~---'-+_.-_'___+_••
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.39 Midspan Moment versus Vertical Deflection at Sections Band D forStage 1 (Scaled Design 19)
----Deflections after__--.'.Placement of
----~-------~.-_._._-------
Block #4
5000 -,-------------.-----------------c
4500 +------------------.;:"<-----------'
4000 -t---------------,'\.-------
3500 +--------
~ 3000 i _.- ,I ,
C. f:i: 2500 -!------------ t;: t~ 2000
1500 .
1000 ~. -+-- G-08
500 + Analytical C-08 ,------ - ----- --- -- -------- ~---
o ~-.~__ ~-'~-~----'---... -_..._.- ... -.~ 4 " ....• ---- .• ------.-.-- ..-- ..-.-~.-------- .... -
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.40 Comparison of Experimental and AnaI}1ical Results at Section D forStage I (Scaled Design 19)
175
5000r----
4500 +-------~.....__"',--------------__i
4000 +f-------~o--~-----------__i
3500 +----------=~~c_=_---------_1
'';Q.:i: 2500 +--------------~~---"\o;o_-------i=.;. 2000 -j----------------""------"....----------i::::r: ~D-11-
500 1 -II- G-D12
o +---, _ .-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5.41 Midspan Moment versus Vertical Deflection at Section C forStage 1 (Scaled Design 19)
-C 3000 +-----------~;_o_____'\.;;.____-------_1
5000 -rl--------------------------,rr
4500 {--------------..,,------------'
4000 -j--------------,,-------_____,
3500 -;-----------------~,_,__----_____,
Deflections after _---.._________elacementof ~
Block #4
--------------------"'-0------'
.1500:---
1OOO~- -+- G-D11
i=.;. 2000
-C 3000 -----------------'u....----~'';Q.
;g, 2500 ~---
500 ~. _Analytical C-D11 -------
o-'------------0·--·-- .. --.-"-.. ---"-.-"-----~-"--~.---"-.. "-~-.
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST1 (in.)
Figure 5042 Comparison of Experimental and Ana1)tical Results at Section C forStage 1 (Scaled Design 19)
176
5000 14500 i-------------------'IIIll'------!
4000 i-----.------------~~------J
3500 +------
-C 3000 +------------------~~----!"Q,
;g. 2500 +--------------------\lI~----iN
~ 2000 t1500 ,-------~-------------___\~\-_.j
~ -+-F-D5 I
1::: +.'_---~ F-D18 1=,============~==========:===: ...-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.43 Midspan Moment versus Vertical Deflection at Sections A and E forStage 2 (Scaled Design 7)
--------------------~-C 3000 ~-~"Q,
;g. 2500 ~------ ..----------------l.
NI-
.;, 2000 ~-----~--
1500~·-_------_.---------- .. --.- ----------;;;-----:-;------:;:-----y
Deflections after ~~ -+--- F-DS
1000 ~- elacemenLof _Block #4
5000 t4500 _f__.__. . ~-----i,
~
t4000 -~---.--------------,3500 +-[-------------.--------~---
500 ~- Analytical F-DS-----------
o -~ .---.- ._---- .-....-.-...-.-.---.-------- -....-....--2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.44 Comparison of Experimental and Anal)tical Results at Section E forStage 2 (Scaled Design 7)
177
5000 -,------
4500 +--------- ~t..--------~--------'
4000 +------------',,"'''<----------------.,
3500 +----------~~----------______;
c: 3000 +----------------'\''\\'\-------------.,'"jCo;g, 2500 +--------NI-~ 2000 +---------------.>.".",----------i
1500 +ri,======-
1000 j -+- F-D8q
500 ~ F-D15
~o -;---- =~.=.=='-"--'--"_T___~>---------"-~~----.-'~----;--~+__ • .._~,__'____'__1
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5
Vertcal Deflection ST2 (in.)
Figure 5.45 Midspan Moment versus Vertical Deflection at Sections B and D forStage 2 (Scaled Design 7)
--~--~-----------
Deflections after ---1'-_________Placementot ~
Block #4
------------------l
. -+-F-D8
5000 -r--------------~--~~------,
NI-
~ 2000
1500
1000
3500 -:------~------~---------~---------',~---------<
.-.C 3000·------- -- -------------"j
c.;g, 2500~-------
4500 -:--~------------ - -------
4000 -:-----~~--~---------------"'\.------,
500 - ___ Analytical F-D8
o .~~ -- -~-- ---~--- -- ---- --- -- --~-~- -. ~-------------.~~-~-- --- ~- -- --- --2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertcal Deflection ST2 (in.)
Figure 5,46 Comparison of Experimental and Analytical Results at Section D forStage 2 (Scaled Design 7)
178
5000 14500 -j--------~------
~4000 +-r ------'-'"~-__
3500 -j----------~~---
-C 3000 +-----------~",____-
___ F-D12500
1000
1500 ~~~~~~-------~ \.'\..~------~
-+-F-D11
o~ I -~.~ ~ ..•~~
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5
Vertical Deflection ST2 (in.)
Figure 5.47 Midspan Moment versus Vertical Deflection at Section C forStage 2 (Scaled Design 7)
"'j"Q,
~ 2500 +-----------------""N...~ 2000 +---------------'\.'\c
Deflections after ----J'*_.. EJacemenloL _
Block #4
---- -------------.5000 t
4500 +f---L
4000 ~:-----f,
3500 ~~----~---------
N...~ 2000:------------ - - --~-.-.. -.-.---.
1500 :
1ooo~. -+- F-D11
,- ..~ 3000 t------------------~-Q, ,
~ 2500 +-----
500 ~. Analytical F-D11 .----- -.-. -
O~---------- - .--~---- - - ... '- -. ------"-"------.....--.---- ..
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.48 Comparison of Experimental and Anal)1ical Results at Section C forStage 2 (Scaled Design 7)
179
--+-C-D5
-ll-C-D18
5000 -,-------------------~------~
4500 +---------------------'~----__i
4000 +----~-----------
3500 +----------------~---=m:---__i
-C 3000 +-----------------------,ill.---__i"j'Co;g. 2500 +-------------- -----.:-----j
N....:E 2000 ~--------------------~\_-_it
1500 !~_-~~~~~~"_"=;-------- -------\\r
1000 ~1500 i
o }I-I-------,----~~~~~~____,-~~~---r--~'1-1~-'----'-i~.
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.49 Midspan Moment versus Vertical Deflection at Sections A and E forStage 2 (Scaled Design 19)
5000 --,---~---~-
,4500 {------------------~------------\\---~
4000 -"----
3500 ----
-C 3000 ,----------------.j"
Co;g. 2500 r--- -----------------------~---- --_.-
N.... .:E 2000 -:------
1500~,--------·~----------------·--------nelle-dions afte--r------ ,
1000~. --+-C-D5 PlacementJ2L ~: Block #4
500 -~ Analytical C-D5-- -
o -~ ,~-.-~'.'~ -~-._'~-o_~~,,~=_"_..__• •__ .__ ,-.----,-.-0-.-.. ---.--
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.50 Comparison of Experimental and Anal}tical Results at Section E forStage 2 (Scaled Design 19)
180
-------------"'\.'.\----------;
NI-
~ 2000 +-----
1500 -f~I-~~[:
1000] -+- C-D8
500 ti C-D15ilflo +---L.-..-.--=-~:-::-"1'1=~---+--~ ~.......---~t---'-"--'-t
f4500 -f--
f
4000 -'-~--
5000 ~---
3500 +-----_ c
c: 3000 +-----~----------~~---------_i-. ~Co f~ 2500 -,-----
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5
Vertical Deflection ST2 (in.)
Figure 5.51 Midspan Moment versus Vertical Deflection at Sections B and D forStage 2 (Scaled Design 19)
Deflections after---J~, ~Iacemenlof , __ ~ .
Block #4
NI-
~ 2000
1500
1000 ~- -+- C-D8
3500 ~------ ---------
; 3000 ~;---.- .I •
Co •
~ 2500 ~
5000 ~,----
4500 i -....'"r
4000 L-__ ~ " ,----------r
500 .:. ---------------------------~- -- ~---------~--"
: Analytical C-D8
o .:. --- ---2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.52 Comparison of Experimental and Anal)tical Results at Section D forStage 2 (Scaled Design 19)
181
5000 -,--------~-----------------_____,
4500 -t---------""'li~-----F
4000 -[----~---_'..,,"----------____i
~ :::: t----------~--------------1Co ,;g. 2500 +'---------------"'__----------1
NI-~ 2000 +--------------~,-------------1
1500 +r=~~~--~--~---~--~--,--------~~-----__I
___ C-D12
-+-C-D111000
500~
o r' , I '
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5
Vertical Deflection ST2 (in.)
Figure 5.53 Midspan Moment versus Vertical Deflection at Section C forStage 2 (Scaled Design 19)
=-2'------~--J
-~--------"
Deflections after ----f•••__________E18_cemeotof, _
Block #4
1500: _-= -_~-_-- _
1000 ~ -+-C-D11
5000 -,--~----,4500 {------------~- .....--------__1
~4000
3500---
NI-~ 2000 -:------- -----------------
-C 3000 -:-----------------------~~--.. ~Co ,
;g. 2500 -,-.:----------- ---------~----~--------"lo""'----
500 ...:: Analytical C-D11
O~---------------~·--_·-·-·--·---·~~
-2.5 -2.3 -2.1 -1.9 -1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1
Vertical Deflection ST2 (in.)
Figure 5.54 Comparison of Experimental and Anal)1ical Results at Section C forStage 2 (Scaled Design 19)
182
___ F-D18
-+-F-D5
-2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
14000 -~
13000 I
12000 -~11000 -~
10000
- 9000c
8000'j'c.;g. 7000..,
6000I-CIl
:!: 50004000
3000
20001000
0-3
Figure 5.55 Midspan Moment versus Vertical Deflection at Sections A and E forStage 3 (Scaled Design 7)
-----.- -.---~---.-----.lI:
14000 }13000 +,~~~~~~-
12000 -r-~-----
11000 +L---10000 ~ .~--- ~~~~--~--~-.
- 9000 -~~,~ 8000 4--~~~~~~ __~~~~~_C.;g. 7000 -c--------
~ 6000 -~~~~~:!: 5000~
4000 ~------------- - - -------- ----- ------------
3000 ~ -+- F-D5 -- --2000 ,-------------- ------
1000, Analytical F-D5 --.- -----o -----.-----.-----.. -..-- -'" -- -.- -.- -.-.. - .
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in,)
Figure 5.56 Comparison of Experimental and Analytical Results at Section E forStage 3 (Scaled Design 7)
183
14000 -,-------------------------,
13000 +--------__,.-----------------
12000 111000 +---------------"'1..=__-------------1
10000 +----------~-~------------I
_ 9000 +---------------"It:=---~,.___------_____1.~ 8000 +------------ - -------_____1
Co;g, 7000 +1-------------.~-I.__----------1~ 6000 +------------------'__-~----_____1
::E 5000 +--------------------'''I<--......r----_____1
4000 ~~~~~~~~-----------"Il,.--~---___l
3000 -+- F-D8
20001000 F-D15
o _so=;===t==p==r===\~'__j__"__'__"_t~'_t_'_"_"_+~~~"_'_+~-+-'-"_'_+"....__;__'_"___
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.57 Midspan Moment versus Vertical Deflection at Sections B and D forStage 3 (Scaled Design 7)
14000 -,--------------------------,
13000 +-j---------w ~------------O
12000 +-r-----------...-)r,,.---------------~
11000 -t-----------""II"i.._------------'
10000 +-1---------------l~----------
; 9000 {----------'6. 8000 1--------·------.----·;g, 70001---~-------------; ..-------.., ,Iii 6000 +-~-----
:!: '5000 1---------- .-4000i===~~==- 0
3000 -+- F-D82000
1000 _ Analytical F-D8 --o --~~- -- -__ - ...-~......--.-....~~-~.-_. ~.---- ~~ -~ -~- .----~ .0- - ...... _ ..... ~ • --
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.58 Comparison of Experimental and Anal)1ical Results at Section D forStage 3 (Scaled Design 7)
184
14000 113000 +!---~----.. t----------------J
12000 t----~--~.---------------~11000 +--------~
10000 +1------------'~ .._411(1...~---------~
_ 9000 +--------------=.
.~ 8000 +-----------~t_~I-------Co;g. 7000 +------------__l-~II___------
~ 6000 +-------------~,-~.-----
:!: 5000 +--------------~r__---!II .....---
4000 TFr--~-~,-'~~~~~~-------_"'l~-_..----
3000 i -+- F-D11
2000 l1000 1 --- F-D12
o ....,--,----,--,----,----l.t--.-p---'--1f-'-L-+-"-'-'-1'~'__t__-----4~'-o--L_'_'_t......"+_'_~.-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.59 Midspan Moment versus Vertical Deflection at Section C forStage 3 (Scaled Design 7)
"-~"~----------~---~
14000 -,----------------------------.
13000 -L'---~-----'_.--------------i
12000 J-- ..... ~ ____',11000 ~-------------.:~r------------'
10000 -L--------------.l--------------'
_ 9000 ------
.~ 8000 -----------------'11~-----,--~Co;g. 7000~'----------------~.
~ 6000·~------------------.
:!: 5000 --~----
4000 .3000 . -+- F-D11
2000 ,1000 . Analytical F-D11----~---~--------,--~----~,-----
o - .--.~.,~~~--.-----~~~-.--~~~,., ...-~.. ~~--3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.60 Comparison of Experimental and Anal)tical Results at Section C forStage 3 (Scaled Design 7)
185
14000 113000 1--12000 -
1100010000 +1---
_ 9000~r -.~ 8000 -------------~--------jCo:i: 7000 ---------------4.-..---1-~ 6000 -~
:i!: 5000 +---------------------......-----1
4000 +-------l-----------~~'________i
3000 1 ,: C-DS2000
1OO~ --- C_-_D,-18_---,----lt-~~..Lf_'_____'____'____'~'_t_'_'___'____r~t__"_'_-Y--"-_'_'__i~'+tIf--J-.-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.61 Midspan Moment versus Vertical Deflection at Sections A and E forStage 3 (Scaled Design 19)
----.---------------........
14000 --,---- !
13000 +----- -------J11
12000 .:.---------------~~------!If ,
11000 1---10000 ~--------------------I~----~!
_ 9000 -:-------------- -------------.:~--~
.~ 8000 --,----Co ";g, 7000.,;..'---------------------a
~ 6000 -;-:i!: 5000 ~----------- -----
4000 =-_c,_=cccc~c=c,==-_==__-------------------"
3000 . -+- C-D52000 - ~--------------
1000 . Analytical C-D5--------------------------- ;o ~- ---- --- ------------, -~.-"~--~~-.-.-.-.-.------.--.-~-~~"-.-.---.--~-.-.~
-3 -2.8-2.6-2.4-2.2 -2 -1.8-1.6-1.4-1.2 -1 -0.8-0.6-0.4-0.2 0
Vertical Deflection ST3 (in.)
Figure 5.62 Comparison of Experimental and Analytical Results at Section E forStage 3 (Scaled Design 19)
186
___ C-D15
-+-C-D8
+------~~-- ----------
14000
13000
12000
11000
10000
- 9000c
8000"c.g 7000...
6000 Il-ll)
:!i: 5000
4000
3000
2000
1000
o -;-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.63 Midspan Moment versus Vertical Deflection at Sections B and D forStage 3 (Scaled Design 19)
-----='"11 ,.--.. -----
-----_.__.~----~_._----,
._--_.._----..
4000 -"-------------~--.
I3000 J -+- C-D8,,2000 ~
1000 1 Analytical C-D8-o 1 .-- -~ - -0·· , ·-0 • ,_._o~~~__ •__ .~. __ .__ .o ,_. ._•••_~_ •••••_0" 0_
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
14000 ~
13000 +!,-----f
12000 i----11000 -"--!---
10000 -"-----l
_ 9000 ~---~---
.~ 8000 -:----c. ,g 7000 ~.~~-----~ 6000 ~-----_._._-~-_._._---_.~ ---------lI)
:!i: 5000------ ---~----~---------~--
Vertical Deflection ST3 (in.)
Figure 5.64 Comparison of Experimental and Analytical Results at Section D forStage 3 (Scaled Design 19)
187
14000 -~
13000 1----12000 -~
11000 t-----------9Il;:-.~---------1
10000 +----------......---.~---------______1
_ 9000 +-----------..-__..--------______1
.~ 8000 +------------ -'._------______1Co;g. 7000 +-------------It----._-----______1
~ 6000 +----------------"'1It---~ __----______1
::E 5000 +-------------------lII..----'~---___j
4000I3000 -+-C-D11
2000
100~ ] C-D12
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Vertical Deflection ST3 (in.)
Figure 5.65 Midspan Moment versus Vertical Deflection at Section C forStage 3 (Scaled Design 19)
14000 -,--------
13000 +----------_--..-----------~
12000 -'-----------'.
11000 -'-----------'CJ1
10000 -'-------------......
_ 9000 -'--------------..~,-------__i
.~ 8000 -;.----Co ,
;g. 7000 ~
~ 6000 J
::E 5000~----~.-.-
4000 ---
3000 . -+-C-D11
2000 .1000 . Analytical C-D11-·-·--···_··--··· ----.--------.--- -.
o --_. __~_.~-- -- _-. -- ---0---- ..-.-----~------.-----
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Vertical Deflection ST3 (in.)
Figure 5.66 Comparison of Experimental and Analytical Results at Section C forStage 3 (Scaled Design 19)
188
o-1-2
20000
18000
16000
14000
-:- 12000c:
~" r.90 10000:. r~ 8000 ~
~6000 ]
~. -+-F-D54000 II
2000 ]_F-D18rlr!
o+i-9 -8 -7 -B -5 -4 -3
Vertical Deflection (in.)
Figure 5.67 Midspan Moment versus Vertical Deflection at Sections A and E(Scaled Design 7)
o-1-2
_F-D15
-+-F-D8
-7-8
-,------~ '!t.------------~---------
-9
2000
o .:_~_.~.~_"_~~_~._~~ __."_~._~~_u_~~. __...-.--~••~ ... '-' .-... -. -..._- _.- .~-
4000 ----
6000 -.-------~--- • .._ ___
~ 8000~---------- -.-.---~ ...
20000
18000
16000,
14000 L
- 12000c:
"Co 10000~
-B -5 -4 -3Vertical Deflection (in.)
Figure 5.68 Midspan Moment yersus Vertical Deflection at Sections B and D(Scaled Design 7)
189
o-1-2
___ F-D12
-6 -5 -4 -3Vertical Deflection (in.)
Figure 5.69 Midspan Moment versus Vertical Deflection at Section C(Scaled Design 7)
20000 -f18000
16000
14000
-:- 12000c:.;-
.~ 10000:.:E 8000
6000
4000
2000
o I
-9 -8 -7
o-1-2-7
___ C-D18
o ..::--~-~-... --------..- ... -.-.. ~-.. ,.-..~.-.._~-,.~. '-."~"'~-_.'--'---"'-".'--
-9 -8
2000 ..::
14000 ~------------
20000 J18000 +-,----~---------;K:
16000 +-~------------l..-------_
:E 8000 .~--
6000 ~.-.------~
-+-C-D54000 ~
-6 -5 -4 -3
Vertical Deflection (in.)
Figure 5.70 Midspan Moment versus Vertical Deflection at Sections A and E(Scaled Design 19)
190
-:- 12000 -L- ~ _
c:...~ 10000 ~--:.
20000 ~------
o-1-2
____ C-D15
--+-C-D8
-8
16000 +----....--14000 +I---~'a_-~
18000 +---~---
-7 -6 -5 -4 -3Vertical Deflection (in.)
Figure 5.71 Midspan Moment versus Vertical Deflection at Sections B and D(Scaled Design 19)
'"":' 12000 -i------...=._-c
"'j
"~ 10000 +--------------~-------------.:.:-
11,
IIIiiJiiIr
!~
____ C-D12
--+-C-D11
._- ----- ----------
~ 8000 .:------~tL
6000 --'----
4000 l: 1 " offset typo
2000 -~---~-----------
O:,~~,~~~--~~-- ---- ---- .-~-.- ....-.. ,-,-., ~~.~~".--- '-' .~.---.-~.'--.-~'"~~ .... ----
-9 -8 -7 -6 -5 -4 -3 -2 -1 0Vertical Deflection (in.)
Figure 5.72 Midspan Moment \"ersus Vertical Deflection at Section C(Scaled Design 19)
191
20000 ~
180001--oz.---
16000 {---It...------------- ",L
14000 L!---'1.......---
f'"":' 12000 1----C i-, t.~ 10000 -1'----..:.: '- '
Ux
Ux
Shupe
Ux
1.25 Ux
Shope 4
Ux
0.67 Ux
Shape 2
Ux
1.375 Ux
ShupE' 5
Ux
1.5 Ux
Shope 3
1.3125 Ux
Shope 6
Note: Ux is positive in the South direction.
Figure 5.73 Initial Imperfections at Midspan
192
32.50.5 1 1.5 2Midspan Lateral Displ. (in.)
o
-,k----.----.- ...........-..-. .tr-.~
-~~___Ir
.,--
~/'//'
1/i(IJi
Maximum Moment=6955 kip-in.I4i,
--ux(top)
1--..- ux(bot.),o
-0.5
1000
2000
7000
8000
.j"
~
~ 5000
1 6000
c:CI>E 4000o~
; 3000~Ul'C
~
Figure 5.74 FEM Simulation Results for Model SD7-1
8000 -.------------------------
7000 j------------------------i
2.5 321.5
Maximum Moment=6594 kip-in.
.-----------------._----' --ux(top)
--.,\- ux(bot.)
0.5o
./-----/.. r-------/
o -l---_-t'- .-_-_---_--_--_-_--_---.J--
-0.5
1000
16000 -j ~---.. .... ---,t-.--.----•..-- ... --, _...'~-,-_..a.-----~--~ -~~ 5000 -1---------"'-·--~r-----------1
./-c:CI>g 4000~
; 3000~Ul'C~ 2000
Midspan Lateral DispJ. (in.)
Figure 5.75 FEM Simulation Results for Model SD7-2193
0.5o-2 -1.5 -1 ~.5
Midspan Lateral Displ. (in.)-2.5
-.A-----..- __A-__
~'~'J._,.~
'"",-
~
""~'~\~
Maximum Moment=7288 kip-in. \\--ux(top)
\-,0- ux(bot.)o
-3
1000
7000
8000
~ 6000''jc.;g. 5000....c:CIl
E 4000o~
lij 3000c.III'C~ 2000
Figure 5.76 FEM Simulation Results for Model SD7-3
8000 ,--------------------------,
7000 -I----------~--------
----~- --ux(top)
- .. - ux(bot.)
Maximum Moment=6536 kip-in.
______-----Jo._-~,-----------
1000
....c:CIl
~ 4000:!:lij 3000c.III'C
~ 2000
-1~-----------~_._.-I<---~-'--.--~_" __ "'- --.--~6000 .''jc.;g.5000
2.5 30.5 1 1.5 2Midspan Lateral Displ. (in.)
oo l-__-A- ---'--=-=-=-..=..-=-.=--=::..J--
-0.5
Figure 5.77 FEM Simulation Results for I\lodel SD7-4194
32.50.5 1 1.5 2Midspan Lateral Displ. (in.)
o
,._~ .......___----t:;---
_--A---«-----~----.//J>---/~
/1>'/
/
,ft'
,1/II Maximum Moment=6115 kip-in.
--ux(top)
f -..- ux(bot.)o-0.5
1000
7000
8000
'''jCo
;g. 5000....I:Q)
~ 4000::i!:~ 3000CoUl'C~ 2000
1 6000
Figure 5.78 FEM Simulation Results for Model SD7-5
8000 -,----------------------------,
7000 ~---------~----------------------------j
0.5o
~_._-_._-~_ .. - ---------~-~-_ .._-
-2 -1.5 -1 -0.5Midspan Lateral Displ. (in.)
-2.5
1000 - --ux(top) -------------------------
--. -- ux(bot. )o -'---'--""==-----'---0..----'-'---'-'-.;;..:-- -"-__----'
-3
....~,
E 4000 -I~--~--------- --~-~------' ..-,- ~---------lo '-= --~.:: \
~ 3000~----------------------------'\- -------~-----CoUl'C~ 2000 -1--- ---
Maximum Moment=7065 kip-in.
1 6000 ~_----=-tI"""-"
'''jCo
;g. 5000 -j~----
Figure 5.79 FEM Simulation Results for Model S07-6195
32.50.5 1 1.5 2
Midspan Lateral Displ. (in.)
o
k_~--.-*----l:.··~ k-~·-'
~,..-g- "
)(/,/Lr'''-
--
////f(11,
Maximum Moment=7393 kip-in.
I. l-ux(toP)
I - .. -- ux(bot.)o-0.5
1000
7000
8000
~ 6000'j"Q.
;g. 5000...c:Q)
5 4000~
; 3000Q.en"0i 2000
Figure 5.80 FEM Simulation Results for Model SD7-7
3
- 1.----.1.- --.-
2.5
---------------
Maximum Moment=7026 kip-in.
--------- ---------------- . - ux(top)
-----..- ux(bot. )
0.5 1 1.5 2Midspan Lateral Displ. (in.)
j----------:.<-=-.....-------
8000
7000 -
~6000..Q.
;g.5000 -...c:CIl
54000~
; 3000Q.en
"0i 2000
1000
0-
-0.5 0
Figure 5.81 FEM Simulation Results for Model SD7-S196
0.5a-2 -1.5 -1 -0.5Midspan Lateral Displ. (in.)
-2.5
1::-"'-'.40---.- -a --&----.---:.;::...................'-
~.........'''"'i.,
"'~"'-,
".~~
\:~
\\\\
Maximum Moment=7749 kip-in. \--ux(top)
\.....,- ux(bot.)..o
-3
1000
7000
8000
;6000'j'Co
;g,5000...cQ)
54000~
; 3000CoIII
"C:1E 2000
Figure 5.82 FEM Simulation Results for Model SD7-9
10000 ,--------------------------,
9000 -t~--------------------------..., _ __=:=- __ - ·'k·-}r-
8000".-~j .
.~ 7000 l~ ;._k_·__
Co~~ 6000 l~ /_:/~A_/./------.~..--.-.-----------.-
i ::::.1
1-_-_----------,?-.(.-./----.-(-----~-.~~---....-.
!3000 !/-.. ------..-·~-M-a-x-im..·u·-m-M-o-m-e-n-t=--8-87·-3-k-ip-in.
2000 -t~---Icl-------·-·---·----- ----.---.-
30.5 1 1.5 2 2.5
Midspan Lateral Displ. (in.)
a
___ ....~_ --ux(top)
- •.- ux(bot.)a J.........__-4' -J
-0.5
1000 I~---I---"- '-"--"-
Figure 5.83 FEM Simulation Results for Model SD7-IO197
~--~---
-b-~:":'~-- ....- -,,-_-"----.1-~ ----..--.-'",r'
/~~~/A/--
///'//II
II Maximum Moment=8501 kip-in.
tl I~UX(IOP)i
/ ------T-- ux(bot.)l _.. _.. __._ •
10000
9000
8000-c7000'j"
Q.;g.
6000....cQ)
E 50000:!:c 4000nlQ.til 3000'0
:!:2000
1000
o-0.5 o 0.5 1 1.5 2
Midspan Lateral Displ. (in.)2.5 3
Figure 5.84 FEM Simulation Results for Model SD7-11
10000 ,------------------------,
9000 ~ .- +--'---..~~~
8000 ~t-------~'_--_-l_--_-~.._---_--_"'-'-~- ..,.~~_.---~-------i
0.5o-2 -1.5 -1 -0,5
Midspan Lateral Displ. (in.)
1000 :-- ux(top) .------------ ----------~------ --
: .- ux(bot. )o --~ -- - --3 -2.5
c", 7000 -f~----------~------------"-\,--------Q.:i:::6000 f~--cQ)
~ 5000:!:c 4000nlQ.~ 3000 -f-------~----------~--~------~-- --1, --~-~-- ---~-~~~-
:!: Maximum Moment=9203 kip-in.2000 -1-----------..----------.------ .-..---.-.--.--.---.-~._-~ ..-- -;~ .----------------
Figure 5.85 FEM Simulation Results for ~todel SD7-12198
._._ -Ix·· ...-..--j;
//'~
IiIIIIt~ Maximum Moment=12428 kip-in.
I i-- ux(top) I1--..-- ux(bot.) I
14000
12000
-.~ 10000Co
~..8000c
Q)
E0:E 6000c111CoIII
4000'tl
:E
2000
o-0.5 o 0.5 1.5 2 2.5 3
Midspan Lateral Displ. (in.)
Figure 5.86 FEM Simulation Results for Model SO19-1
14000
12000
.~ 10000Co
~..8000c
Q)
E0:E 6000c111CoIII
4000'tl
:E
2000
• io~il-·-.-·---
A._.-~'...l·_---"""---_· .------.---~.....::..--.:......- .-:---=-----1,..l ,.-
/4'--,-r>..~-_~--------------i
....
------f=---"r--------~--------------- -
ii.
._----~ ..~-_._--------~-----_._---~----_.._---
-------~---- --------------
Maximum Moment=12050 kip-in.
-------------------- -- ux(top)
-.--- ux(bot.)
2.5 30.5 1 1.5 2
Midspan Lateral Displ. (in.)
oo .L-__--"- .. _--_--_.---_-~.:::.__-_J
-0.5
Figure 5.87 FEM Simulation Results for Model S019-2199
- - ,_......'. -I.-
~-...~
'\\'\~
Maximum Moment=12796 kip-in. \1~UX(IOP) i
,
\I"· ux(bot. )
14000
12000
-.~ 10000Co
~.... 8000cQ)
E0~ 6000cI1lCoen
4000'0
~
2000
o-3 -2.5 -2 -1.5 -1 -0.5 o 0.5
Midspan Lateral DispJ. (in.)
Figure 5.88 FEM Simulation Results for Model SO 19-3
32.521.5
Maximum Moment=12209 kip-in.
----- n:_ ~:::\ 1
0.5
I~----,.-I--------~~--------------~--~-- --
,/.x/
i-------,.~?~---------
A"/
A
14000
12000
-.~ 10000Co
~....8000c
al
E0~ 6000 -CI1lCoen
4000 .'0
~
2000
0
-0.5 0
Midspan Lateral DispJ. (in.)
Figure 5.89 FEM Simulation Results for Model S019-4200
_K__-Ir--...---:a---·-'-'··- ir--"
.....--------------."'.41·'-·......--
--
~//'
///1
/I Maximum Moment=11760 kip-in.
j --ux(top)
~ux(bot.)
14000
12000
.~ 10000Q.:i:-... 8000cCI)
E0:E 6000cIIIQ.lJl
4000'C
~
2000
o-0.5 o 0.5 1 1.5 2
Midspan Lateral Displ. (in.)
2.5 3
Figure 5.90 FEM Simulation Results for Model SD19-5
14000 ,---------------------------,
0.5-2 -1.5 -1 -0.5 0
Midspan Lateral Displ. (in.)
-2.5
Maximum Moment=12645 kip-in.
------------_.._-._.... ~.~---_--'..
\L------- .__~ . ~.__•...._. ._
--ux(top)
--. .. ux(bot.)o
-3
2000
- ...--.-......---...'-'-I~=.:::;;;-±.-- -:::.,-··~i..>'+.-----1
12000 -1-----------=-<-....:
C 8000 -1-------~------~----_______'t\-----___1CI)
Eo:E 6000 j--------
CIIIQ.lJl'C 4000:E
.~ 10000 j----------------,'------Q.:i:-
Figure 5.91 FEM Simulation Results for Model SD19-6201
~'_'-A-" -....__'.-1-- -' .....-....._----,.,.--_.. -...---.0 ~
//I
ff111
t...~ Maximum Moment=12754 kip-in..~\~
l~ux(IOp)•.-,r,- ux(bot.)
14000
12000
--.~ 10000c.:i:--.. 8000cQ)
E0:!: 6000cI1lC.III
4000't:l
:!:
2000
o-D.5 o 0.5 1 1.5 2
Midspan Lateral Displ. (in.)
2.5 3
Figure 5.92 FEM Simulation Results for Model SD19-7
14000 -,....-----------------------,
0.5o-2 -1.5 -1 -D.5Midspan Lateral Displ. (in.)
-2.5
------------------._.~----._~ -------\-----~
4000 i-------------------------- -----------. ~~----1
2000~ax~::\anent=13183 ~i~~____ . ~\---Ol--~_~_~ __'A_______J
-3
.~--*.~.~
12000 1---------------- t~~
"
---------------------~_____.:-__----iI
~
~C 8000 -t----------------------:-~ -1-----Q) \
E •o i:!: 6000cI1lC.III
't:l
:!:
.~ 10000c.~
Figure 5.93 FEM Simulation Results for Model SD 19-8202
J<----*-,4t------ - - ... -. --iii···· ~
.".---~
(~(
t T
tt+~4I
!
~
.l.I Maximum Moment=12977 kip-in.~\t
~~UX(toP)J0,
:t -1>- ux(bot)
14000
12000
.~ 10000Q.
:i:-C 8000Q)
Eo~ 6000cIVQ.
~ 4000~
2000
o-0.5 o 0.5 1 1.5 2 2.5 3
Midspan Lateral Displ. (in.)
Figure 5.94 FEM Simulation Results for Model SO19-9
4 5 6 7 8 9 10 11 12 13 14 15Midspan Lateral Displ. (in.)
32o
-1------ ------------------------------ -----
:!-.t------------------------------- --
A Maximum Moment=12754 kip-in.
-1---------------- -------------------------~-;(t~p)-
1 ~=_:..u~(b_ot.)o_L-_------------------~-------J
-1
14000
12000
.~ 10000Q.
~.. 8000cQ)
E0~ 6000cIVQ.en
4000'C
~
2000
Figure 5.95 FEi\1 Simulation Results for Model SO 19-7 (Including Post-Peak)203
-.-..:~~1f;!}:.L!J1................" ..<JoU.....,
........tL".... ..........'
i,I
i.,,.
Maximum Moment=13183 kip-in.
\M-- ux(top) I
1---- ux(bot.) I r:Io
-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0Midspan Lateral Displ. (in.)
14000
12000
-.; 10000Co:i:-... 8000r::(I)
E0:!: 6000r::IIICoII)
4000'C
:!:
2000
Figure 5.96 FEM Simulation Results for Model SO 19-8 (Including Post-Peak)
--1------ ------ --- ..--.---------------
-.I-----.------.--------~---~-------_.-
Maximum Moment=12977 kip-in.
------ ------- -----~------ .. - . -- ux(top)
-*- ux(bot.)
:!.!!---.i------- --~.\
-~~------ ----
I--j----------
J
14000
12000
-~ 10000Cog... 8000r::(I)
E0:!: 6000r::IIICoII)
4000'C
:!:
2000
2 3 4 5 6 7 8 9 10 11 12 13 14 15
Midspan Lateral Displ. (in.)
o .L- -======_...-1 0
Figure 5.97 FEM Simulation Results for Model SO 19-9 (Including Post-Peak)204
0.5-2.5
~-lk-'.-Q .~ '" ""~-'-a-4-- -''- ._--~ -~
.,,~\
.-..+
-ofi..\..I
t\
~I
it~I
~lI
Maximum Moment=13183 kip-in. ~~\
--ux(top) \ ~
-a- ux(bot.) ~o
-3 -2 -1.5 -1 -0.5 0Midspan Lateral Displ. (in.)
Figure 5.98 FEM Simulation Results for Model SD19-8 (Including InitialImperfections)
14000
12000
-.~ 10000Q.~-- 8000cQ)
E0~ 6000cnsQ.Ul
4000't:l
~
2000
14000 -r-------------------------,.... -- •• 'f'-~--'-""-""'~r
120004---------~ .
Maximum Moment=12977 kip-in.
------------- --------- --------~------.•
•i.
.~ 10000 I-~~~~~~~~--.I,"i---~~~-~~--~~~-Q. Ig 4
•1: 8000 4-~-~~~----J_+'~~-~--~--------- -...._-Q) ..
E ..o 1~ 6000cnsQ.
~ 4000~
32.521.5
__ __ _~.4. .__ , ~ . .._..__----~ ---- ._-------ux(top)
• ux(bot.)
0.5o-0.5
\ .Ol....--------~--<~.....----------'='-'---.;..;;:"--'=....:.=....J
-1
2000
Midspan Lateral Displ. (in.)
Figure 5.99 FE~1 Simulation Results for Model SD19-9 (Including InitialImperfections)
205
3
N
2
o s
1
Figure 5.100 Schematic ofFEM Simulation Results for Model SD19-8
N o s
2 1 3
Figure 5.101 Schematic ofFEM Simulation Results for Model SD19-9
206
___ S07-1
-.-S07-4---:X- S07-7~S07-10
--+-S12___ S122[
o ~f~.-~~~~~~'_j__~--
5000 Tt---------------------------,f
4500 +----r-~hF=::::::~------~;;;;o-o~=-------_____j
4000 L--~--I--==--==~~-'--='=~~------------;. ~
.~ 3500 f---,..'---~----~'-----------__=__4
~ 1:: 3000 +-i--I
lii tE 2500 +f----JI---I---Y----"""7"""---'-----=::;~~-------~__:"o:E[ 2000 ~
~ 1500 r--f-f--'---r'--;T"'---------:;;,,~-----_I
~ ~1000 -r-- -lo~!Y'-----....--------------1
f
500 -t--ILI'-/"-------------------1
-0.10 0.10 0.30 0.50 0.70 0.90 1.10
Midspan Lateral Displ. (in.)
1.30 1.50
Figure 5.102 Comparison of Experimental and Analytical Midspan Moment versusLateral Displacements (Scaled Design 7, Tube)
1.501.30
___ S07-1
-.-S07-4-"i;- S07-7
~S07-10
--+-S12___ S122
1.100.300.10
0':
-0.10
1000 -~-
500 -:-
r
f4500 ~.----ir~F:::+t-----~,--:--------------j
t4000 ~---j~-~~-=-
-- fC :'6, 3500 -,---....----/-'------~.:i ::: 3000 ~:--,c
5000
0.50 0.70 0.90
Midspan Lateral Displ. (in.)
Figure 5.103 Comparison of Experimental and Anal)1ical Midspan Moment yersusLateral Displacements (Scaled Design 7. Tension Flange)
207
__-{SD19-3)
......... -(SD19-B)
-+-S12____ S122
5000 ~,-----.----1---[---------------
~4500 +--~-~
4000 -l-t-- --4/--J./-----L.Cl
; ,~"j 3500 +--..-------:H--,..-------------------1Co:i:;:; 3000 -;----1C
E2500 +f 1.•-.I. --;
o fi 200011~ 1500 ~
:E 1000 +----flT.l----------~~500 +,--/IL'----------------<!
f !o ~r---.-----,--.-----t-~-~_~_~JL__---_~"
-0.10 0.10 0.30 0.50 0.70 0.90
Midspan Lateral Displ. ST2 (in.)
Figure 5.104 Comparison of Experimental and Analytical Midspan Moment versusLateral Displacements (Scaled Design 19, Tube)
__-{SD19-3)
--.--{SD19-6)
-+-S12____ S122
0.70 0.90
---------------
~----- ---_.
0.30 0.50
500:--
O~--c--
-0.10 0.10
4500~-------T-- -- 94 .1_L-'~LDisQI. Fro"-'-m'-'- ----'
Cutting Wood Spacers
CfaCo~ 1500~-- ---- .
:E 1000~----IlTt
,4000 .L-----.l
- IC ~" 3500 -'---- - -1,-·------------------Co ':i: :;:; 3000 -:-' -.--.-- -J--I/------c .Cl) •
E 2500 -'-----•..ij.,------------.-o ::E
2000 ~------
Midspan Lateral Displ. (in.)
Figure 5.105 Comparison of Experimental and Analytical Midspan Moment \"ersusLateral Displacements (Scaled Design 19. Tension Flange)
208
....III
~E.g"CE::::IIll'"':"111 cQ) .-
~;... CQ)w
"EGcoco;
111(J
o...J
r-800~~~~~~~~~~~~1~~~~~~~_
I
' [-+- Before Load Block 1
r-70u---¥--------------------'I --- Before Load Block 6
i -.-Before Load Block 10
r O -..-After Load Block 16I
~Ou-Hr-\.----".,----------------i
hOO-t---t--\-----'r----------------II
~300-f---+--f----J--------------Ir-200-f-I-----I-----/--------------
~100-LJI-<;;/__-------------_iI~-El'-'-~~-r"-"-~~- .~~---'--'-t~~'---'--"--t~~~~~~+-L-L-~~
-0.1 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Lateral Displacement 512 (in.)
Figure 5.106 Lateral Displacements of Scaled Design 7 (Tube, Stage 2)
~'-600-
·----100-
:------400-·- - --- ------- --------.-------------------- -------------
,--800-,---------- -, ~ ! -+- Before Load Block 1 ii
f---700-¥----------------~--j --- Before Load Block 6 :;i Iii-.-Before Load Block 10(!
------------' -#-After Load Block 16 1i
coco;III(J
o...J
....III
~E.g"CE::::IIll'"':"111 CCll .-~;... CCllW"E
0.70.60.50.2 0.3 0.4
Lateral Displacement 512 (in.)
. ------. O· -------------.-------.-
-0.1 0 0.1
Figure 5.107 Lateral Displacements of Scaled Design 7 (Tension Flange. Stage 2)209
..III
~E,g'0~:::J1Il'-;'l'lI l:Q) .-:E-
'0... l:Q)W
'EC>l:ol:o;;l'lIUo.J
~800f----r--------------,,---------
I i~ Before Load Block 1
L70~~--------------i: --- Before Load Block 6 I
II 1-.-Before Load Block 10 i1 I
~OlJ-l\----""------------11 "",*-After Load Block 16 I
1
f--50Q--I--\--\.-----'.....----------1
i:---400-1--+--1,---\.------------------1
~ 30u-+~\--'r-----\.----\-------------II
~200iI-I-100-hU----o'----------------------iI
-0.1 a 0.1 0.2 0.3 0.4 0.5
Lateral Displacement ST2 (in.)
0.6 0.7
Figure 5.108 Lateral Displacements of Scaled Design 19 (Tube, Stage 2)
-100-1--..J-J-~'"
----200--'
I~ Before Load Block 1I i
'---700-~------------' --- Before Load Block 61
I-.-Before Load Block 101
-----~! ~After Load Block 16 I
--800-,r
., 0-- ..~. __"_.~""_~ c._c _L ----- •••_0 •••_. __._~.~.__• •• _ -.~ -----.__ ••••
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Lateral Displacement ST2 (in.)
..III
~E,g'0~:::J1Il'-;'l'lI l:(I) .-:E-
'0... l:(l)W'EC>l:ol:o;;l'lIUo.J
Figure 5.109 Lateral Displacements of Scaled Design 19 (Tension Flange. Stage 2)210
....l/l
~E,g"Ce~l/l-:11l c:Q) .-:a;... c:Q)w
'EC)
c:oc:o
+::11luo
...J
~OO---,-----------~· .~~~--"'::-~.~-~.--~-~--~-....~-~-~--~-Ii--+-Before Load Block 1
~t'Ou-'l~=----------------1! ---- Before Load Block 6i-.-Before Load Block 10i I
i I ->f-After Load Block 16~600---i ~Wood Spacers Cutit
~OU---+l--'\------"--~--------"""""""----------i
~OO----H----+-----------"-~L 300-t------I1
I
t 200
iL100~t
I
-0.1 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Lateral Displacement ST22 (in.)
Figure 5.110 Lateral Displacements of Scaled Design 7 (Tube, Stage 2-2)
0.70.5 0.60.40.3
-------_._------------,-=. - --_ .._...~=-=~'"'._' •
. --+- Before Load Block 1
I ---- Before Load Block 6. -.-Before Load Block 10 1
i ->f-After Load Block 16__J
-r-Wood Spacers Cut
0.2
,------------''''''---- ------ ----------- ----~ --
0.1o
o
100-
~-800------
·---200-- --
:-600~
~700-~-----------
!
1-500- - --'1.----,
-0.1
....1Il
~E,g"CCl)...::J1Il-:l1l c:
:E~... c:~w...ac:oc:o
+::l1luo
...J
Lateral Displacement ST22 (in.)
Figure 5.111 Lateral Displacements of Scaled Design 7 (Tension Flange. Stage 2-2)211
.----800-~-
! I -+- Before Load Block 1 Ir----+0u-'l~-----------------1; Before Load Block 6
1
I-Al.-Before Load Block 10I I """,*-After Load Block 16~O I! t -*-Wood Spacers Cut !I
~Ol}--l'-t----\------'\--------"'."-------':--""'~---------------I
Iroo ~r--300-!r 200- -- -I----/---~.._L
i~100-
-0. 1 a 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Lateral Displacement ST22 (in.)
Figure 5.112 Lateral Displacements of Scaled Design 19 (Tube, Stage 2-2)
0.70.3 0.4 0.5 0.60.20.1
[-800-r-----~--~------~----- i -+-Before Load Block 1 :
I ~ : Before Load Block 6 i~OO-""~---------------l -AI.- Before Load Block 101
•~After Load Block 16 j
~c-----~I -*-Wood Spacers Cut !
--100- - -
··· .. ·0
-0.1 0
...III
~E.g'Cl!!~--:-III CCIl .-:i::;... CCIlw
"EC)
CoCo;IIIoo
...J
Lateral Displacement ST22 (in.)
Figure 5.113 Lateral Displacements of Scaled Design 19 (Tension Flange. Stage 2-2)212
-----.--- .-- -------- -------------1
5.E-Q4
--~
3.E-Q4
+
+
+
+
-1.E-Q4 1.E-Q4
cPsT2 (in:1)
----+,-----------------0
-3.E-Q4
+---- ~~-------
30 -f
~
25 ftr- fc:
::;.. 20 r.0 ~Q) I
~,,Ic: 15 r
0 [c: I0
~..III0 100 t
-l ,I f>-
5
0
-5.E-Q4
Figure 5.114 Curvature throughout Web Depth for Stage 2
30 ---~~----
, +f
25 +-~-------
:§. 20 +---- ---- ----- -+---.0
~
+10 ~- - -- -- ------ -----
c: •o 15 -"------~- ----- ---------------------+--------- -- -.- .---- - -.---------.
c:o..IIIoo-l.>-
5
+
o ..-5.E-Q4 -3.E-Q4 -1.E-Q4 1.E-Q4
C>sm (in:1)
3.E-Q4 5.E-Q4
Figure 5.115 Cur\'3ture throughout Web Depth for Stage 2-2
213
-----I..~ S
Figure 5.116 Web Distortion
S.OE-QS ,-------------------------- -----
~4.0E-QS -t------
t3.0E-QS -t--------~
l,2.0E-QS +',-------
.-. 1.oE-Qs l-";' "c- O.OE+OO -
N >I- ,
~ -1.0E-QS ~--
-2.0E-QS +--------------------------3.0E-QS ~--------- ---- ---~~===--==~_:....==__ ==: _
: -+- Stage 2 Experimental-4.0E-QS ~-----~----
--Analytical-S.OE-QS -~----~----.- --- --<----.--- --- -~-.---~c~.-.-~c-~~c~--=~-- ------~- --- -----~
o 100 200 300 400 sao 600 700
x Location on scaled Design 19 (in.)
Figure 5.117 Trans\'erse Curvature Comparison (Tension Flange. Stage 2)
214
700600SOO400300200100
S f~ -1.0E-QS -~
-2.0E-QS +-,------------
~-3.0E-QS -f--------------------------i
l!
-+- Stage 2-2 Experimental I'
-4.0E-QS +c------------, 11
f __Analytical !it Ii
-S.OE-QS +-1-~--r-----;------:==r===========,===--,=i-
o
S.OE-QS
4.0E-QS
3.0E-QS
2.0E-QS -----
f"' 1.0E-QSt:- O.OE+OO
x Location on scaled Design 19 (in.)
Figure 5.118 Transverse Curvature Comparison (Tension Flange, Stage 2-2)
215
UTGN
LTGN
North Side ofSeo.led Design 19
UFGN
LFGN
Figure 5.119 Strain Gages Used to Study Plate Bending in Tension Flange of ScaledDesign 19
216
6. Summary, Conclusions, and Recommendations for Future Work
6.1 Summary
Two different innovations to steel I-shaped highway bridge girders were
investigated in this thesis: (1) concrete filled tubular flanges and (2) corrugated webs.
The objectives of this research were: (1) to conduct a design study of tubular flange
girders with corrugated webs and with flat webs for a four girder, 131.23 ft. (40000
nun) prototype bridge, (2) to design 0.45 scale test girders based on the results of this
design study, (3) to test the scaled girders to investigate their ability to carry their
design loads, and (4) to compare experimental and analytical results to verify the
adequacy of the analytical models and tools.
A design study investigated twelve different tubular flange girder designs,
which used various combinations of corrugated or flat web, composite or non
composite, homogeneous or hybrid, and braced or unbraced conditions. Six additional
designs were generated to incorporate flange transverse bending effects into the
original six corrugated web designs. The designs were generated based on modified
AASHTO LRFD Bridge Design Specifications (1999). Elastic section calculations
were perfomled using equivalent transfomled sections to include the concrete in the
tube and deck with the steel in the girder cross-section properties.
One corrugated web girder design and one flat web girder design were scaled
down by 0.45. re-designed and fabricated for use in a two-girder test specimen. The
re-dcsign involvcd small modifications of the dimensions to make fabrication of thc
217
scaled girders feasible, and design of the test girder welds, stiffeners, shear studs, and
deck construction. The test specimen was instrumented with 163 channels, consisting
of strain gages and displacement transducers, and connected to a data acquisition
system. The test specimen was then loaded to simulate various design loading
conditions while data was recorded. Numerous analytical calculations and FEM
simulations were performed so that experimental results could be compared with
analytical results.
6.2 Conclusions
Design Study
The eighteen design study combinations (Sect. 3.7) supported the following
conclusions: (l) tubular flanges allow for the use oflarge girder unbraced lengths by
increasing the torsional stiffness of the girder; (2) corrugated webs create lighter
weight designs than unstiffened flat webs because the corrugated web is thinner,
although the flanges are slightly larger; (3) composite designs are lighter weight than
non-composite designs because the deck contributes to the load carrying capacity in a
composite design; (4) hybrid designs create lighter weight designs than homogeneous
designs because of the increased steel yield stress; (5) the presence of interior
diaphragms provides torsional bracing to a girder, increases its lateral torsional
buckling strength, and allows for the use of a slightly smaller tubular compression
flange.
218
Although each of the conclusions above suggests that certain designs are more
advantageous, other issues should be considered. Composite designs require the
added cost and effort of making bridge girders composite with the deck. Hybrid
designs may cost more than homogeneous designs if the weight savings do not make
up for the cost of the higher strength steel. Interior diaphragms are costly to fabricate
and connect to the girders, and these costs may offset the benefits of the lighter weight
girders.
The design study showed corrugated web girders to be only slightly lighter
than their flat web counterparts for a 131.23 ft. (40000 mm) bridge, with a girder
length-to-depth ratio of approximately 22. As the depth-to-thickness ratio of the web
increases (for deeper girders), the corrugated web retains the same shear strength but
the flat web shear strength decreases (Sect. 3.8). Also, a corrugated web requires
more steel in the tension flange because the web does not contribute to overall bending
strength. A deeper section will help reduce the need for this excess tension flange
steel, and therefore, corrugated webs will be more advantageous for deeper girders.
Since the girder length-to-depth ratio should not be too small (say, less than 20),
deeper girders are practical only for bridges longer than the prototype bridge.
Transformed Section Calculations for Concrete Filled Tubular Flange Girders
The use of an elastic transformed section for the combined properties of the
steel tubular flange girder and the concrete in the tube and deck is a valid method for
analysis ofa tubular flange girder. As discussed in Section 5.4. the experimental
219
results compare quite well with the analytical results based on elastic transformed
sections. There was a small increased stiffness observed in experimental results
compared to the analytical results, but this increased stiffness was not necessarily due
to the use of elastic transformed sections.
Strain Nonlinearity
As discussed in Section 5.4, nonlinearity was observed in the loading branch of
Stage 1 and Stage 3 moment versus strain and moment versus vertical deflection
experimental results. It was shown that this nonlinearity was due to the existence of
residual stresses within the steel. The nonlinearity due to residual stresses prevented
the use of strains for determining the bending moment, even at sections expected to be
linear elastic during the tests. Stage 1 of the tests eliminated the residual stresses for
the Simulated Construction loading condition, and therefore Stage 2 and Stage 2-2
provided linear results.
Ability to Carry Design Loads
During the Simulated Construction loading condition, a moment equal to 0.51
times the yield moment was placed on scaled Design 19 and a moment equal to 0.57
times the yield moment was placed on scaled Design 7. Flange transverse bending
moments caused the Construction loading condition stress to equal 0.67 times the
yield stress for scaled Design 19. The increase in stress due to flange transverse
bending moments was calculated using maximum shear in the span, as assumed in the
220
design of the prototype corrugated web girders. Also during the Simulated
Construction loading condition, a moment equal to 0.70 times the lateral torsional
buckling moment capacity was placed on scaled Design 19 and a moment equal to
0.73 times the lateral torsional buckling moment capacity was placed on scaled Design
7. Flange transverse bending moment effects caused the Simulated Construction
loading condition stress to equal 0.93 times the lateral torsional buckling stress in the
compression flange for scaled Design 19.
During the Simulated Strength I loading condition, a moment equal to 1.01
times the yield moment and 0.88 times the plastic moment was placed on scaled
Design 19. A moment equal to 1.03 times the yield moment and 0.67 times the plastic
moment was placed on scaled Design 7.
It was shown through these tests that the test specimen could effectively carry
the loads for which it was designed. The loads placed on the test girders closely
simulated the Construction and Strength I loading conditions for the prototype bridge
and the test specimen generally behaved as expected. The presence of the residual
stresses increased the experimental vertical deflections, but once the residual stresses
were taken into account, very good comparisons existed between experimental and
analytical results.
The experimental lateral displacements were generally less than those
predicted by FEM simulations, so it is difficult to comment on the latcral torsional
buckling strength of the test girders. It is kno\\l1. howevcr, that the lateral torsional
221
buckling strength of the test girders was greater than the moment created by the
Simulated Construction loading condition.
6.3 Recommendations for Future Work
Longer Prototype Bridge
As discussed in Sections 3.7 and 3.8, a prototype bridge with a longer span
would require deeper girders that would more efficiently use a corrugated web. This
research showed that the 131.23 ft. (40000 rom) prototype bridge resulted in
corrugated web designs that were only slightly lighter than flat web designs. It is
suggested that a design study be performed for a longer (e.g., a 196.85 ft. (60000 rom)
span prototype bridge.
Lateral Torsional Buckling Test
As discussed in Section 5.6, the experimental lateral displacement results did
not compare well with the FEM simulation results. It is recommended that a
laboratory experiment be performed in order to study the lateral displacements and
lateral torsional buckling strength of tubular flange girders more thoroughly. The load
should be purely vertical, without accidental lateral loading, something that was not
necessarily achieved in the tests performed in this research. The recommended test
should use a single girder, with no unintentional bracing at the ends or within the span.
In addition, a detailed set of initial imperfection measurements of the tube. web. and
tension flange should be made.
222
Tube Compactness Requirement
The tube compactness requirement (Eq. 2.7) discussed in Section 2.3 only
prevents elastic buckling of the tube before yielding. It does not guarantee that the
section can be fully plastified before inelastic buckling occurs in the tube. This was
not an issue in this research because the stresses in the tube stayed below yield for the
tests. As discussed in Section 2.6, the plastic moment was calculated based on strain
compatibility, which does not assume the section to be fully plastified. In order to
calculate the plastic moment based on full plastification of the section, an appropriate
compactness limit is needed.
223
References
AASHTO, "AASHTO LRFD Bridge Design Specifications," Second Edition, 1999.
Abbas, H.H., "Analysis and Design of Corrugated Web I-Girders for Bridges usingHigh Perfonnance Steel," Ph.D. Dissertation, Department of Civil and EnvironmentalEngineering, Lehigh University, 2003.
Easley, IT., "Buckling Fonnulas for Corrugated Metal Shear Diaphragms," ASCEJournal of the Structural Division, Vol. 101, No. ST7 (July), pp. 1403-1417, 1975.
Elgaaly, M., R.W. Hamilton, A. Seshadri, "Shear Strength of Beams with CorrugatedWebs," ASCE Journal of Structural Engineering, Vol. 122, No.4 (April), pp. 390-398,1996.
Kim, B.G., "HPS Bridge Girders with Tubular Flanges," Ph.D. Dissertation,Department of Civil and Environmental Engineering, Lehigh University, 2004a.
Kim, B.G., Personal Communication, Department of Civil and EnvironmentalEngineering, Lehigh University, 2004b.
Sause, R., H.H. Abbas, W.G. Wassef, R.G. Driver, M. Elgaaly, "Corrugated WebGirder Shape and Strength Criteria," Report No. 03-18, pp. 51-58, Center forAdvanced Technology for Large Structural Systems, Lehigh University, 2003.
Smith, A. "Design ofHPS Bridge Girders with Tubular Flanges," M.S. Thesis,Department of Civil and Environmental Engineering, Lehigh University, 2001.
224
Vita
Mark Robert Wimer was born in Wooster, Ohio on November 22, 1978. His
parents' names are Robert and Betheny Wimer, and he has a younger sister, Stacy. He
graduated from Waynedale High School in 1997. He went on to study Civil
Engineering at Ohio Northern University. He was married to his wife, Brandi, in
2000. He graduated first in his class with a Bachelor of Science in 2002. He is
currently attending Lehigh University, and this thesis is partial fulfillment ofa Master
of Science.
225
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