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The Pennsylvania State University
The Graduate School
Department of Civil and Environmental Engineering
COMPUTATIONAL ASSESSMENT OF STEEL‐JACKETED
BRIDGE PIER COLUMN PERFORMANCE UNDER BLAST LOADS
A Thesis in
Civil Engineering
by
Edward V. O’Hare
© 2011 Edward V. O’Hare
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2011
ii
The thesis of Edward V. O’Hare was reviewed and approved* by the following:
Daniel G. Linzell Shaw Professor of Civil Engineering Thesis Advisor
Farshad Rajabipour Assistant Professor of Civil and Environmental Engineering
Eric S. Musselman Assistant Professor of Civil Engineering Special Signatory
Peggy Johnson Professor of Civil and Environmental Engineering Head of Department of Civil and Environmental Engineering
*Signatures are on file in the Graduate School.
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Abstract
Due to recent malicious acts on civil structures, blast resistant design of bridges has
become an important area of research. Bridge engineers have a unique challenge to
consider security in the design process due to the valuable assets that our country’s
infrastructure facilitates and its inherent vulnerability. This thesis summarizes the process
of investigating a steel jacket retrofit for circular reinforced concrete bridge pier columns
to increase their resistance against blast loads. A parametric study utilizing computational
models that are validated against available theoretical and experimental data was used to
accomplish this objective. Although limited experiments have been completed on steel‐
jacketed columns subjected to blast, a parametric study investigating the variation in
critical parameters has not been completed to date. Therefore, a 2k factorial design was
used to evaluate four critical steel‐jacketed bridge column parameters at minimum and
maximum values. Relevant results from this study were used to observe the sensitivity of
parameter variation on three critical failure modes, which were: direct shear, flexure, and
transverse shear. From these observations parameter characteristic recommendations
were offered which were shown to be advantageous in resisting undesirable brittle failure
modes and encouraging more ductile failure modes. Results from this study may also be
used as a basis for validation of future physical blast testing of steel‐jacketed bridge
columns.
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Table of Contents List of Figures ......................................................................................................................... vii List of Tables .......................................................................................................................... xii Acknowledgments ................................................................................................................. xiii Chapter 1 Introduction ....................................................................................................... 1
1.1 Background ............................................................................................................................................... 1 1.1.1 General Information ..................................................................................................................... 1 1.1.2 Bridge Column Retrofit ............................................................................................................... 2
1.2 Problem Statement ................................................................................................................................ 5 1.3 Objectives ................................................................................................................................................... 5 1.4 Scope ............................................................................................................................................................ 5 1.5 Summary .................................................................................................................................................... 8
Chapter 2 Literature Review ............................................................................................... 9
2.1 Overview .................................................................................................................................................... 9 2.2 Blast Analysis Methods ........................................................................................................................ 9 2.3 Blast Loading ......................................................................................................................................... 15 2.3.1 Air Blast Characteristics .......................................................................................................... 15 2.3.2 Blast Wave Propagation Programs ..................................................................................... 18
2.4 Relevant Model Assumptions, Parameters, and Materials ................................................ 20 2.4.1 Common Blast Modeling Assumptions ............................................................................. 21 2.4.2 Blast and Seismic Model Parameters ................................................................................. 24 2.4.2.1 Common Blast Parameters .............................................................................................................. 24
2.4.2.2 Common Steel‐Jacketed Pier Seismic Parameters ................................................................. 26
2.4.3 LS‐DYNA Material Models ....................................................................................................... 27 2.5 Relevant Research Findings ............................................................................................................ 32 2.5.1 Blast Resistant Results ............................................................................................................. 32 2.5.2 Steel‐Jacketed Column Seismic Resistance Results ..................................................... 37
2.6 Summary ................................................................................................................................................. 39 Chapter 3 Parametric Studies............................................................................................ 41
3.1 Overview ................................................................................................................................................. 41 3.2 Test Variables ........................................................................................................................................ 41 3.3 Constant Parameters .......................................................................................................................... 42 3.4 Varied Parameters .............................................................................................................................. 47 3.4.1 Column Aspect Ratio (L/D) .................................................................................................... 48 3.4.2 Transverse Reinforcement Ratio (ρs) ................................................................................ 49 3.4.3 Steel Jacket Diameter‐to‐Thickness Ratio (tj/D) ........................................................... 50 3.4.4 Steel Jacket Gap Distance‐to‐Diameter Ratio (Lg/D) ................................................... 51
3.5 Parameter Limits ................................................................................................................................. 52
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3.6 Parametric Matrix ............................................................................................................................... 53 3.7 Additional Parametric Considerations ....................................................................................... 57 3.8 Summary ................................................................................................................................................. 58
Chapter 4 Finite Element Modeling ................................................................................... 60
4.1 Overview ................................................................................................................................................. 60 4.2 Finite Elements ..................................................................................................................................... 60 4.3 Constraints and Boundary Conditions ....................................................................................... 61 4.4 Material Models .................................................................................................................................... 67 4.5 Summary ................................................................................................................................................. 69
Chapter 5 Model Validation .............................................................................................. 71
5.1 Overview ................................................................................................................................................. 71 5.2 Static Validation ................................................................................................................................... 71 5.2.1 Concentric Axial Quasi‐Static Load Validation ............................................................... 71 5.2.2 Combined Axial‐Flexural Quasi‐Static Load Validation ............................................. 77
5.3 Dynamic Blast Validation ................................................................................................................. 85 5.4 Summary ................................................................................................................................................. 95
Chapter 6 Parametric Study Results .................................................................................. 97
6.1 Overview ................................................................................................................................................. 97 6.2 Direct Shear Results ........................................................................................................................... 97 6.2.1 Axial Cross‐section Strain Profiles .................................................................................... 101 6.2.2 Normalized Direct Shear ....................................................................................................... 111
6.3 Parameter Variation Effects on Direct Shear ......................................................................... 113 6.3.1 Aspect Ratio Effects on Axial Cross‐Section Strain .................................................... 114 6.3.2 Transverse Reinforcement Ratio Effects on Axial Cross‐Section Strain ........... 116 6.3.3 Jacket Thickness Ratio Effects on Axial Cross‐Section Strain ................................ 118 6.3.4 Jacket Gap Ratio Effects on Axial Cross‐Section Strain ............................................ 121 6.3.5 Aspect Ratio Effects on Normalized Direct Shear ....................................................... 123 6.3.6 Transverse Reinforcement Ratio Effects on Normalized Direct Shear ............. 125 6.3.7 Jacket Thickness Ratio Effects on Normalized Direct Shear .................................. 126 6.3.8 Jacket Gap Ratio Effects on Normalized Direct Shear ............................................... 129
6.4 Flexure Results ................................................................................................................................... 130 6.4.1 Moment and Rotation Diagrams ........................................................................................ 132 6.4.2 Normalized Moment‐Rotation ............................................................................................ 137
6.5 Parameter Variation Effects on Flexure................................................................................... 142 6.5.1 Aspect Ratio Effects on Moment and Rotation ............................................................ 142 6.5.2 Transverse Reinforcement Ratio Effects on Moment and Rotation ................... 145 6.5.3 Jacket Thickness Ratio Effects on Moment and Rotation ........................................ 146 6.5.4 Jacket Gap Ratio Effects on Moment and Rotation ..................................................... 149 6.5.5 Aspect Ratio Effects on Normalized Moment‐Rotation ........................................... 152 6.5.6 Transverse Reinforcement Ratio Effects on Normalize Moment‐Rotation ..... 155
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6.5.7 Jacket Thickness Ratio Effects on Normalized Moment‐Rotation ....................... 157 6.5.8 Jacket Gap Ratio Effects on Normalized Moment‐Rotation .................................... 159
6.6 Transverse Shear ............................................................................................................................... 160 6.6.1 Transverse Strain Profiles .................................................................................................... 161
6.7 Parameter Variation Effects on Transverse Shear .............................................................. 167 6.7.1 Aspect Ratio Effects on Average Transverse Strain ................................................... 167 6.7.2 Transverse Reinforcement Ratio Effects on Average Transverse Strain ......... 169 6.7.3 Jacket Thickness Ratio Effects on Average Transverse Strain .............................. 171 6.7.4 Jacket Gap Ratio Effects on Average Transverse Strain ........................................... 173
6.8 Additional Parametric Results ..................................................................................................... 175 6.8.1 Steel Jacket Base Fixity Effects on Blast Resistance .................................................. 175 6.8.2 Steel Jacket Retrofitting Effects on Blast Resistance ................................................. 183
Chapter 7 Conclusions .....................................................................................................190
7.1 Overview ............................................................................................................................................... 190 7.1.1 Recommended Steel‐Jacketed Bridge Column Details to Resist Specific Failure Modes 190 7.1.2 Recommended Steel‐Jacketed Bridge Column Details to Resist Blast Loads . 197 7.1.3 Recommendations for Future Research ......................................................................... 199
References ........................................................................................................................201 Appendix A Axial Cross‐section Strain Profiles ....................................................................207 Appendix B Direct Shear Comparisons ................................................................................304 Appendix C Moment and Rotational Diagrams ...................................................................312 Appendix D Moment‐Rotation Capacity Curves ..................................................................328 Appendix E Transverse Strain Profiles ................................................................................337
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List of Figures
Figure Page Number
Figure 1.1: Typical Steel Jacket Retrofit (Washington State DOT, 2010) .................................... 2 Figure 2.1: Mach Front Region (Winget et al., 2005) ......................................................................... 16 Figure 2.2: Typical Blast Pressure‐Time Graph (Sriram et al., 2006) ........................................ 17 Figure 2.3: Propped‐Cantilever ................................................................................................................... 21 Figure 2.4: Moment‐Axial Load Interaction Diagram ........................................................................ 22 Figure 2.5: Test Set‐Up (Fujikura et al., 2008) ..................................................................................... 26 Figure 2.6: Material 72 Failure Surfaces (Malavar et al., 1997) .................................................... 28 Figure 2.7: CSCM Yield Surface (LSTC, 2007) ....................................................................................... 29 Figure 2.8: LS‐DYNA Material Type 3 (LSTC, 2007) .......................................................................... 30 Figure 2.9: Proposed DIF for ASTM A615 Steel Reinforcing Bar (Malvar, 1998) ................. 32 Figure 2.10: Direct Shear Model (Fujikura & Bruneau, 2008) ...................................................... 35 Figure 2.11: Direct Shear Model Comparison (ACI, 2008) .............................................................. 36 Figure 3.1: Moment Diagram of Plastic Hinge Analysis for Blast‐loaded Column (Williamson et al., 2010) .................................................................................................................................... 44 Figure 3.2: Blast Damage Threshold (FHWA, 2003) .......................................................................... 46 Figure 3.3: Parametric Model Label Definition .................................................................................... 54 Figure 4.1: Model Parts: a.) Steel Jacket, b.) Cover Concrete, c.) Core Concrete, d.) Reinforcement, e.) Entire Model, f.) Cross‐Section ................................................................................. 62 Figure 4.2: Nodal Compatibility Constraint .............................................................................................. 63 Figure 4.3: *CONSTRAINED _LAGRANGE_IN_SOLID Constraint ................................................... 64 Figure 4.4: Column to Foundation Constraint ......................................................................................... 66 Figure 4.5: Construction Joint Boundary Condition .............................................................................. 67 Figure 5.1: Concentric Load .......................................................................................................................... 72 Figure 5.2: Todeschini Curve (Wight & MacGregor, 2009) ............................................................. 73 Figure 5.3: Theoretical Force‐Displacement Determination Process ........................................ 75 Figure 5.4: Force‐Displacement Curve Comparison .......................................................................... 76 Figure 5.5: a.) Column normal vertical stress contours, b.) Reinforcing steel axial force contours ..................................................................................................................................................................... 77 Figure 5.6: Combined Axial and Flexural Load .................................................................................... 78
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Figure 5.7: Typical Axial‐Moment Interaction Diagram for Circular Columns (Wight & MacGregor, 2009) .................................................................................................................................................. 78 Figure 5.8: Modified Compression Field Theory Layered Model (Vecchio & Collins, 1988) ....................................................................................................................................................................................... 79 Figure 5.9: Modified Compression Field Theory Concrete and Steel Material Models (Vecchio & Collins, 1988) ................................................................................................................................... 80 Figure 5.10: Theoretical Axial‐Moment Diagram ................................................................................ 81 Figure 5.11: Model Data Points ................................................................................................................... 82 Figure 5.12: Axial‐Moment Interaction Comparison ......................................................................... 84 Figure 5.13: MCEER Report 08‐0028 Column SJ2 Blast Damage (Fujikura & Bruneau, 2008) ........................................................................................................................................................................... 86 Figure 5.14: Initial Blast Validation Model ............................................................................................. 87 Figure 5.15: Initial Blast Validation Damage ......................................................................................... 88 Figure 5.16: Modified Blast Validation Model ...................................................................................... 88 Figure 5.17: Blast Validation Model with Foundation Base Deflection Graph ....................... 90 Figure 5.18: Experimental Base Damage (Fujikura & Bruneau, 2008) ..................................... 90 Figure 5.19: Blast Validation Model with Foundation Pier Cap Gap Opening Graph .......... 91 Figure 5.20: Experimental Bent Damage (Fujikura & Bruneau, 2008) ..................................... 92 Figure 5.21: Final Blast Validation Model .............................................................................................. 92 Figure 5.22: Base Deflection Comparison .............................................................................................. 93 Figure 5.23: Pier Cap Gap Opening Comparison ................................................................................. 94 Figure 6.1: Cross‐section Locations .......................................................................................................... 99 Figure 6.2: Direct Shear Resistance ......................................................................................................... 100 Figure 6.3: 8_a0v1j1g1 Direct Shear at Base ....................................................................................... 100 Figure 6.4: Reported Base Steel Jacket Strains ................................................................................... 101 Figure 6.5: Cross‐sectional Strain Profile Definition ....................................................................... 102 Figure 6.6: 8_a0v1j1g1 Axial Cross‐section Strain Profiles .......................................................... 104 Figure 6.7: 3_a0v0j1g0 Axial Cross‐section Strain Profiles .......................................................... 106 Figure 6.8: Longitudinal Reinforcement Axial Strain Comparison at Base ............................ 109 Figure 6.9: Steel Jacket Axial Strain Comparison at Base .............................................................. 110 Figure 6.10: Direct Shear Capacity Model Load Definition ........................................................... 111 Figure 6.11: 8_a0v1j1g1 Direct Shear Comparison at Base .......................................................... 112 Figure 6.12: Normalized Direct Shear Comparison ......................................................................... 113 Figure 6.13: Aspect Ratio Effects on Longitudinal Bar Axial Cross‐Section Strain Profiles at Base ...................................................................................................................................................................... 115
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Figure 6.14: Aspect Ratio Effects on Steel Jacket Axial Cross‐Section Strain Profiles at Base ........................................................................................................................................................................... 116 Figure 6.15: Transverse Reinforcement Ratio Effects on Longitudinal Bar Axial Cross‐Section Strain Profiles at Base ....................................................................................................................... 117 Figure 6.16: Transverse Reinforcement Ratio Effects on Steel Jacket Axial Cross‐Section Strain Profiles at Base ........................................................................................................................................ 118 Figure 6.17: Jacket Thickness Ratio Effects on Long Bar Cross‐Sectional Strain Profiles for Columns without Gap ........................................................................................................................................ 119 Figure 6.18: Jacket Thickness Ratio Effects on Long Bar Cross‐Sectional Strain Profiles for Columns with Gap ............................................................................................................................................... 120 Figure 6.19: Jacket Thickness Ratio Effects on Steel Jacket Cross‐Sectional Strain Profiles ..................................................................................................................................................................................... 121 Figure 6.20: Jacket Gap Ratio Effects on Long Bar Cross‐Sectional Strain Profiles ............ 122 Figure 6.21: Jacket Gap Ratio Effects on Steel Jacket Cross‐Sectional Strain Profiles ....... 123 Figure 6.22: Aspect Ratio Effects on Normalized Direct Shear for Column without Gap 124 Figure 6.23: Aspect Ratio Effects on Normalized Direct Shear for Column with Gap ....... 125 Figure 6.24: Transverse Reinforcement Ratio Effects on Normalized Direct Shear .......... 126 Figure 6.25: Jacket Thickness Ratio Effects on Normalized Direct Shear............................... 127 Figure 6.26: 2_a0v0j0g1 and 4_a0v0j1g1 Normalized Direct Shear Comparison ............... 128 Figure 6.27: Jacket Gap Ratio Effects on Normalized Direct Shear ........................................... 130 Figure 6.28: 4_a0v0j1g1 and 11_a1v0j1g1 Moment Diagrams ................................................... 133 Figure 6.29: 4_a0v0j1g1 and 11_a1v0j1g1Rotation Diagrams .................................................... 133 Figure 6.30: Moment Diagram Comparison ........................................................................................ 135 Figure 6.31: Rotation Diagram Comparison ........................................................................................ 136 Figure 6.32: Moment‐Rotation Capacity Model Load Definition ................................................ 137 Figure 6.33: 4_a0v0j1g1 Static Moment‐Rotation Capacity Curve ............................................ 138 Figure 6.34: Normalized Moment‐Rotation Comparison .............................................................. 139 Figure 6.35: 4_a0v0j1g1 Flexure Capacity Analysis Axial Force Time History for Longitudinal Bars at Base ................................................................................................................................ 140 Figure 6.36: 4_a0v0j1g1 Direct Shear Capacity Analysis Axial Force Time History for Longitudinal Bars at Base ................................................................................................................................ 141 Figure 6.37: 14_a1v1j0g1 Moment‐Rotation Capacity Analysis Axial Force Time History for Longitudinal Bars at Base ......................................................................................................................... 142 Figure 6.38: Aspect Ratio Effects on Moment ..................................................................................... 143 Figure 6.39: Blast Effects on Double Curvature ................................................................................. 143 Figure 6.40: Aspect Ratio Effects on Rotation .................................................................................... 144
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Figure 6.41: Transverse Reinforcement Ratio Effects on Moment ........................................... 145 Figure 6.42: Transverse Reinforcement Ratio Effects on Rotation ........................................... 146 Figure 6.43: Jacket Thickness Ratio Effects on Moment for Columns without Gap ........... 147 Figure 6.44: Jacket Thickness Ratio Effects on Moment for Columns with Gap .................. 147 Figure 6.45: Jacket Thickness Ratio Effects on Rotation ................................................................ 149 Figure 6.46: Jacket Gap Ratio Effects on Moment ............................................................................. 150 Figure 6.47: Column 8_a0v1j1g1 Base Hinge Formation ............................................................... 150 Figure 6.48: Jacket Gap Ratio Effects on Rotation for Columns with Low Aspect Ratios 151 Figure 6.49: Jacket Gap Ratio Effects on Rotation for Columns with High Aspect Ratio.. 152 Figure 6.50: Aspect Ratio Effects on Normalized Moment‐Rotation ........................................ 153 Figure 6.51: Aspect Ratio Effects for Outliers on Normalized Moment‐Rotation ............... 154 Figure 6.52: Transverse Reinforcement Ratio Effects on Normalized Moment‐Rotation ..................................................................................................................................................................................... 155 Figure 6.53: Transverse Reinforcement Ratio Effects for Outliers on Normalized Moment‐Rotation ................................................................................................................................................................... 156 Figure 6.54: Moment‐Rotation Capacity Comparison for Outliers ............................................ 157 Figure 6.55: Jacket Thickness Ratio Effects on Normalized Moment‐Rotation for Columns without Gap ............................................................................................................................................................ 158 Figure 6.56: Jacket Thickness Ratio Effects on Normalized Moment‐Rotation for Columns with Gap ................................................................................................................................................................... 159 Figure 6.57: Jacket Gap Ratio Effects on Normalized Moment‐Rotation ................................ 160 Figure 6.58: Transverse Strain Profile Definition ............................................................................. 161 Figure 6.59: 9_a1v0j0g1 Average Transverse Strain Time History .......................................... 162 Figure 6.60: 2_a0v0j0g1 Transverse Strain Profile .......................................................................... 163 Figure 6.61: 9_a1v0j0g1 Transverse Strain Profile .......................................................................... 164 Figure 6.62: Hoop Average Transverse Strain Comparison ......................................................... 165 Figure 6.63: Steel Jacket Average Transverse Strain Comparison ............................................ 166 Figure 6.64: Aspect Ratio Effects on Hoop Average Transverse Strain ................................... 168 Figure 6.65: Aspect Ratio Effects on Steel Jacket Transverse Strain ........................................ 169 Figure 6.66: Transverse Reinforcement Ratio Effects on Hoop Average Transverse Strain ..................................................................................................................................................................................... 170 Figure 6.67: Transverse Reinforcement Ratio Effects on Steel Jacket Average Transverse Strain ......................................................................................................................................................................... 170 Figure 6.68: Jacket Thickness Ratio Effects on Hoop Average Transverse Strain .............. 172 Figure 6.69: Jacket Thickness Ratio Effects on Steel Jacket Average Transverse Strain .. 172
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Figure 6.70: Jacket Gap Ratio Effects on Hoop Average Transverse Strain ........................... 174 Figure 6.71: Jacket Gap Ratio Effects on Steel Jacket Average Transverse Strain .............. 174 Figure 6.72: Steel Jacket Fixity Effects on Longitudinal Bar Axial Cross‐section Strain at Base ........................................................................................................................................................................... 176 Figure 6.73: Steel Jacket Bearing and Prying ...................................................................................... 177 Figure 6.74: Steel Jacket Fixity Effects of on Steel Jacket Axial Cross‐section Strain at Base ..................................................................................................................................................................................... 178 Figure 6.75: Steel Jacket Fixity Effects on Normalized Direct Shear at Base ........................ 179 Figure 6.76: Steel Jacket Fixity Effects on Moment .......................................................................... 180 Figure 6.77: Steel Jacket Fixity Effects on Rotation .......................................................................... 181 Figure 6.78: Steel Jacket Fixity Effects on Normalized Moment‐Rotation ............................. 182 Figure 6.79: Plastic Strain Damage Contours – a.) Model 10 steel jacket, b.) Model 10 concrete inside jacket, c.) Model 10_Unjacketed ................................................................................... 184 Figure 6.80: Steel Jacket Retrofitting Effects on Longitudinal Bar Axial Cross‐section Strain at Base ......................................................................................................................................................... 185 Figure 6.81: Steel Jacket Retrofitting Effects on Normalized Direct Shear ............................ 186 Figure 6.82: Overall Steel Jacket Retrofitting Effects on Moment .............................................. 187 Figure 6.83: Overall Steel Jacket Retrofitting Effects on Normalized Moment .................... 188
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List of Tables
Table Page Number
Table 2.1: Comparison of Analysis Methods (Winget et al., 2005) .............................................. 13 Table 2.2: Column Parameters (Williamson et al., 2010) ................................................................ 25 Table 3.1: Bridge Column Parameters ..................................................................................................... 41 Table 3.2: Parameter Limits ......................................................................................................................... 52 Table 3.3: Constant Column Parameters ................................................................................................ 53 Table 3.4: Parametric Matrix ‐ Column and Jacket Details.............................................................. 55 Table 3.5: Parametric Matrix ‐ Transverse Reinforcement Details ............................................. 56 Table 3.6: Parametric Matrix ‐ Longitudinal Reinforcement Details .......................................... 57 Table 7.1: Desired Result Effects .............................................................................................................. 190 Table 7.2: Parameter Variation Recommendations to Resist Direct Shear ........................... 192 Table 7.3: Steel Jacket Bearing and Retrofitting Recommendations to Resist Direct Shear ..................................................................................................................................................................................... 193 Table 7.4: Recommendations to Enhance Flexure Resistance .................................................... 194 Table 7.5: Steel Jacket Bearing and Retrofitting Recommendations to Enhance Flexure Resistance ............................................................................................................................................................... 195 Table 7.6: Recommendations to Resist Transverse Shear Failure ............................................ 196
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Acknowledgments
First and foremost I would like to thank my family for their love, support and
understanding throughout this research process. My family is truly an inspiration to me
and gives me strength to persevere. My wife Paola’s encouragement and support have been
essential in my success. Her unwavering commitment to me and the children have allowed
us to endured long days, weeks and months as I have pursued this graduate degree. I am
also grateful my family and friends back home for their support.
I would also like to thank my advisor Dr. Daniel G. Linzell for his patience, support
and encouragement throughout this research process. I am grateful for his feedback which
encouraged me to think critically about research and allowed me to grow as a student and
engineer. I would also like to thank other faculty in the department, including Dr. Jeffery
Laman, Dr. Andrew Scanlon and Dr. Gordon Warn, who offered their advice and time
throughout my educational experience. I would like to thank my advisor and these faculty
members for their continued support and for giving me the tools to pursue a graduate
degree. Finally, I am grateful for the support of my fellow graduate students, including
Zachary Gabay, Gautham Ganesh Prasad, Shane Murphy, Shi Liu and Todd Rasey.
1
Chapter 1 Introduction
1.1 Background
1.1.1 General Information
Recent attacks on civil structures and the threat of similar attacks on existing
infrastructure provoke a need to better understand design processes and retrofit options
that will help resist these events. Much of the current knowledge on this subject involves
building performance against blast loads. Therefore, there is a need for research and
guidance related to bridge performance and design against such loads.
The threat on our country’s bridges is real and demonstrated through many arrests
that have been made in recent years. In July 2002, three men were arrested in Spain who
had close ties with al‐Qaeda. In their possession were video tapes of San Francisco’s
Golden Gate Bridge and the World Trade Center towers (CNN‐News, 2002). In June 2003,
an Ohio truck driver was arrested for conspiring with al‐Qaeda’s top leaders to cut the
Brooklyn Bridge’s suspension cables (Arena, 2003). This trend toward targeting surface
transportation systems is examined in Mineta Transportation Institute Report #97‐04
(Jenkins, 1997). This report compiles in chronological order terrorist attacks from 1920 –
1997. The report states that bombing is the most common terrorist tactic and that bridges
make up 6% of the targets.
Pier systems are possibly the most critical and vulnerable component of bridges
according to Williamson, et al (2006). Loss of a pier could result in collapse of multiple
spans and possibly the entire bridge. Their inherent exposure, dictated by traditional
2
bridge site layout, which typically locates them close to travel lanes and public access,
increases their vulnerability to these malicious acts. Therefore, experimental and
computational validation of retrofit effectiveness and resulting flexural and shear,
response, along with assessment of predicted damage levels and their mitigation after any
retrofitting is pursued, should be conducted.
This thesis discusses the retrofit option of using a steel jacket to improve the blast
resistance of circular reinforced concrete bridge pier columns. Jacketing consists of
surrounding a column with steel plates formed and welded along a vertical seam into a
single cylinder. The plates are oversized leaving a gap between the steel jacket and the
column which is then filled with grout to develop a jacket‐to‐column bond, thereby forming
a composite system (Chai et al., 1994). Steel jackets are typically terminated 50 mm (2”)
before both the top of the footing and pier cap to prevent the jacket from bearing on the
footing and producing increased moment demand on these adjoining members. Figure 1.1
demonstrates a typical steel jacket retrofit for a pier column.
Figure 1.1: Typical Steel Jacket Retrofit (Washington State DOT, 2010)
1.1.2 Bridge Column Retrofit
3
Traditionally steel jackets have been used in seismic zones as a retrofit for in‐situ
bridge pier columns that do not meet conventional seismic design standards. Steel jackets
have been used to retrofit circular, square, and rectangular columns. Many state
Departments of Transportation (DOTs) have completed research in this area, with
California on the forefront of conventional seismic design standards.
In the 1980’s the California Department of Transportation (Caltrans) implemented
an assessment and retrofit program for bridges that did not meet conventional seismic
design standards (Chai et al., 1994). Most of these inadequately detailed bridges were
constructed prior to the 1971 San Fernando Earthquake and commonly used No. 4 (12.7‐
mm‐diameter) hoops at 305 mm (12”) center‐to‐center spacing regardless of column
height. This percentage of transverse reinforcement results in inadequate column ductility
with respect to conventional seismic design load criteria. Lack of ductility in pre‐1971
detailing was also caused by inadequate lap‐splice length of the longitudinal steel (Chai et
al., 1994). The typical lap length of 20 times the bar diameter was not sufficient to develop
the full yield strength of the reinforcement. The Caltrans retrofit program initiated steel‐
jacket research which focused on the ductility and stiffness enhancement of bridge columns
under base excitation.
Steel jackets were used extensively for retrofitting inadequate columns to increase
the percentage of transverse reinforcement and to provide adequate confinement of the
column concrete, which increased column ductility. Steel jackets were also used to combat
inadequate lap‐splice length by increasing ductility and flexural capacity of the columns.
Although traditional steel jacket research has assessed the enhancement of bridge pier
4
columns against base excitation, their effectiveness against impulsive loading usually
associated with blast loads is still not well understood.
The behavior of steel‐jacketed bridge columns subjected to blast loads was predicted
in computational research conducted by Winget et al. (2005). As shown in traditional
seismic research, steel jackets increase the flexural stiffness of the column. As the flexural
stiffness of the column increases the shear at the supports also increases. Although not
modeled in the research, Winget et al. (2005) predicted that under blast steel‐jacketed
columns would not provide the necessary shear resistance needed to overcome the higher
shear forces produced by the increase in flexural stiffness. Similar inadequate steel‐
jacketed column shear resistance was calculated in research evaluating bridge columns
under multihazard events (Fujikura et al., 2008).
The possible shear failure predictions of Winget et al. (2005) were substantiated by
experimental research conducted by the University of Buffalo (Fujikura & Bruneau, 2008).
This case study examined the blast resistance of bridge piers that were designed according
to conventional seismic design standards. As series of experiments were conducted on ¼
scale typical reinforced concrete bridge columns, half of which were retrofitted with steel
jacketing. The retrofitted reinforced concrete columns did not exhibit ductile behavior and
failed in direct shear at the base of the column by fracturing all the longitudinal bars. The
gap between the steel jacket and the footing at the base of the column that intended to
prevent bearing of the jacket on the footing produced a discontinuity of shear resistance
and contributed to the direct shear failure mode. These experiments have shown that
adhering to seismic design criteria for blast resistant steel jackets produces a brittle failure
failure mode, rather than the preferred ductile failure mode of flexural yielding produced
5
under seismic loads. However, no work has been completed to study this topic
parametrically to observe if the steel jackets could be detailed to improve retrofitted
column shear performance.
1.2 Problem Statement
Pier columns are an extremely essential and vulnerable component in a bridge’s
structural system, and if one fails complete collapse of multiple spans and possibly the
entire structure can result. There exists a need to further examine the effectiveness of
retrofit options such as steel jackets for providing adequate protection to bridge pier
columns subjected to the increasingly realistic threat of blast loads. Studies completed to
date have examined bridge components under blast loads, including experiments on steel‐
jacketed bridge columns, but have not parametrically shown if steel jackets could be
detailed to increase bridge column blast performance.
1.3 Objectives
The primary objective of this study was to computationally investigate the
effectiveness of a steel jacket retrofit on circular reinforced concrete bridge pier columns
against blast loads through a parametric study. The secondary objective was to provide a
basis of validation for future physical testing of this retrofit option.
1.4 Scope
The scope of this study was to investigate the use of steel jackets for circular bridge
pier columns against blast loads. To investigate the steel jacket retrofit option a parametric
study was conducted using finite element (FE) models. These computational models were
6
created and validated against theoretical column performance and published physical blast
test data. After model validation they were subjected to simulated blast loads. The results
of the steel jacket retrofit were then accessed through examination of the produced data
(summarized below) and recommendations on retrofit effectiveness were offered. The
following list of tasks were followed to fulfill the objectives of this investigation.
1. Literature Review:
This task served three purposes. The first was to investigate and compare
current practices of computational analyses to choose the most effective method
for this study. The second purpose was to investigate the current state‐of‐the art
in bridge blast and seismic resistant design to validate the need for investigating
the steel jacket retrofit option and identify important parameters. The last
purpose was to find and apply current steel jacket proportioning limits to initially
size the specimens that were examined against blast loads.
2. Computational Modeling:
Coupled FE models of steel‐jacketed bridge pier columns, that included
influential reflective surfaces (i.e. the ground), were created in LS‐DYNA with
appropriate blast loads being generated using CONWEP, a blast load function that
is integrated into LS‐DYNA. Model parameters include:
• Commonly used DOT bridge column aspect ratios;
• Explicit transverse reinforcement ratios;
• Variations in steel jacket thickness‐to‐diameter ratios; and
• Variations in steel jacket gap distance‐to‐diameter ratios.
3. Model Validation:
7
To validate the computational models, bare and steel‐jacketed concrete
column models were generated and compared to theoretical column performance
and published experimental blast testing research. Experimental blast
information came from MCEER technical report 08‐0028 (Fujikura & Bruneau,
2008). This work, which investigated the blast performance of seismically
detailed bridge columns including steel‐jacketed columns, was used to validate
the computational models through comparison of deformed shape and
deflections.
4. SteelJacketed Columns Computational Study:
Once the models were validated, the models were subjected to simulated blast
loads. The analyses’ output data was then be compiled and organized. Relevant
results used to assess the sensitivity of steel‐jacketed column parameter variation
in resisting brittle shear failure modes and encouraging a ductile flexural failure
mode include:
• Axial cross‐section strain profiles for longitudinal reinforcement and steel
jackets;
• Normalized direct shear;
• Moment and rotation diagrams;
• Normalized moment‐rotation plots; and
• Transverse strain profiles for hoop reinforcement and steel jackets.
5. Computational Result Comparisons and Recommendations:
After the analyses’ data was compiled, the results were compared to assess the
sensitivity of column parameter variation on increasing or decreasing their
8
resistance to respective failure modes. Finally, utilizing the result comparisons,
recommendations were offered pertaining to the critical column parameter
characteristics that were observed to resist brittle shear failure modes and
encourage a ductile flexural failure mode.
1.5 Summary
Due to recent attacks on civil structures and arrests that have occurred, there exists
an urgent need to protect our country’s infrastructure. The response of bridge pier
columns to blast loads was an identified area of needed research related to the resilience of
these structures against possible terrorist threats. Hardening of circular reinforced
concrete bridge pier columns was the focus of this research due to their prevalence and
vulnerability. Since steel jackets are commonly used in seismic zones to harden bridge
columns against base excitation, this retrofit was studied to assess its effectiveness against
blast loads. Computational models were used in a parametric study to obtain results for
various steel jacketing scenarios and those results were analyzed and compared against
published physical data and one another to offer retrofit recommendations.
9
Chapter 2 Literature Review
2.1 Overview
This literature review focused on three areas. The first was to investigate and
compare studied approaches for computational analysis of pier columns under dynamic
loads, including blast. Since the effectiveness of the retrofit option was determined entirely
through computational means, analysis accuracy greatly influences the study results. The
second focus of this literature review was to investigate the current state‐of‐the art in
bridge blast and seismic resistance design to substantiate the need for investigating the
steel jacket retrofit option and identify important model and result parameters. The final
focus was to investigate the current use of steel jackets for seismic retrofits to assist with
developing initial jacket proportions and to assess possible blast strengthening
contributions.
To accomplish these objectives, literature related to blast resistance and seismic steel
jacket retrofitting was examined and discussed. Results and findings that were relevant to
developing reliable computational models to examine steel jacket retrofit effectiveness
under blast loads were also presented and were organized as follows:
• Blast Analysis Methods;
• Blast Loading;
• Relevant Model Assumptions, Parameters, and Materials; and
• Relevant Research Findings.
2.2 Blast Analysis Methods
10
The first purpose of this literature review was to explore possible analysis methods to
correctly represent circular steel‐jacketed reinforced concrete pier columns and their
response to blast loads. This section discusses the various analysis options and compares
their advantages and disadvantages to select an appropriate method for this case study.
There were three common analysis categories that could be utilized to achieve a
spectrum of result accuracy for a range of problem types.
• Uncoupled vs. Coupled
• Static vs. Dynamic
• Single‐degree‐of‐freedom (SDOF) vs. Multiple‐degree‐of‐freedom (MDOF)
Uncoupled methods require two analyses. One analysis calculates the blast loads and
one calculates the structural response. In contrast, coupled analyses account for the effects
of structural response in conjunction with the fluid dynamics behavior of an explosion.
Static analysis is independent of the rate of loading, while dynamic analysis is dependent
on the rate of loading. Degrees of freedom (DOF) are defined as the number of
independent displacements and rotations required to define the displacement positions of
all the masses relative to their original position (Chopra, 2007). As the names suggest,
SDOF systems have only one required displacement (usually lateral displacement) to
define the model’s deformed position, whereas MDOF systems have multiple required
displacements to define the model’s deformed positions.
There were several analysis options for bridge blast research. They ranged from
SDOF dynamic analysis to finite element models (FEM) that couple the effects of blast loads
and structural response. The most common analysis method was a SDOF or MDOF
11
nonlinear dynamic uncoupled analysis. Loads are determined using a blast wave
propagation program and then applied to the structure through pressure‐time histories to
obtain the structural response from a dynamic analysis. However, if the blast environment
is extremely altered during the loading period a more advanced, coupled analysis may be
necessary.analysis was an accurate tool to predict the intensity and location of deformation
in the columns given a specific standoff distance, charge, and height (Fujikura et al.,
2008)(Fujikura et al., 2008). Table 2.1 was reproduced from a case study by Winget et al
(2005) and compares the advantages and disadvantages of each category of analysis,
ranging from simple uncoupled static SDOF problems to more complex coupled dynamic FE
problems. In general, when comparing levels of analysis there is a tradeoff between the
faster simplified analyses, which tend to produce overly conservative results, and the
slower complex analyses, which produce more accurate results but are characterized by
cumbersome input and output files. In the case study by Winget et al, (2005), dynamic
uncoupled SDOF models were selected to examine the response of bridge pier columns
under blast loads. This level of analysis was chosen because the goal of this research was
to perform numerous fast running analyses using simplified, but still accurate, models to
examine the accuracy of this type of tool. SDOF models also represent the current state of
practice in blast design and provided reasonable estimates of blast effects. The results of
the case study indicated that the simplified dynamic uncoupled SDOF models provided an
accurate prediction of maximum column deflection and support rotations. A study which
examined the response of concrete filled steel tube (CFST) bridge columns founded in fiber
reinforced concrete against blast also indicated that simplified SDOF analysis was an
12
accurate tool to predict the intensity and location of deformation in the columns given a
specific standoff distance, charge, and height (Fujikura et al., 2008).
13
Table 2.1: Comparison of Analysis Methods (Winget et al., 2005)
Uncoupled Analysis Coupled analysis
Static analysis/ equivalent static
analysis Dynamic analysis SDOF analysisMDOF analysis (including FEM)
Advantages Much simpler to Accounts for Simplified analysis Increased accuracy Reasonable Increased accuracyperform coupled effects of procedures because it accuracy for for more complex
structural response accounts for simple structures componentswith fluid inertial effects, because because it accountsdynamicsbehavior which can fundamental mode for higher‐orderof an explosion contribute dominates modes or responseload, considering significantly to displacementtime and spatial stresses and responsecoupling strains
Provides Provides better Simplifiedreasonably predictions of calculationsaccurate, ultimate behaviorconservativepredictions
Disadvantages Overestimates Complex models Inaccurate results More difficult to Difficult to Difficult to performblast loads may require in most cases perform analysis correctly model a without abecause it does extensive because it only complex structure computer, complexnot account for computational accounts for one models may requiremembers yielding resources deformation mode several hours ofand failing and does not computational time
include inertialeffects
Programs often Difficult to As more modes Estimates may belack conservatively are added beyond very conservative,user‐friendliness determine static a certain limit, depending ondue to large design load accuracy geometry andnumber of input because improvements will contribution fromparameters, magnitudes and be insignificant higher modesexperience locations of blast while analysisnecessary to create can vary, time increasesan accurate model producingaccounting for the significantlycorrect failure different responsesmodes, and
14
Another consideration for choosing the proper analysis method was the availability of
public domain analysis programs. Although blast wave propagation programs such as
BlastX (BlastX, 2001) are advantageous for this type of research, many of them are licensed
by the Department of Defense and are not available to the general public. LS‐DYNA is a
combined implicit/explicit solver and integrated into LS‐DYNA is an coupled empirically
based blast load function code called CONWEP, as well as constitutive material models
developed for blast loads (LSTC, 2007).
The CONWEP blast load function was implemented into LS‐DYNA based on a report
by Randers‐Pehrson and Bannister (1997). The report summarized a study conducted to
verify the empirical equations that are use in the loading function against physical testing
of a vehicle’s response to a land mine. The study concluded that the blast load functions
allowed for the application of blast loads on structures created in LS‐DYNA without having
to run CONWEP explicitly and were adequate for modeling similar types of blast loading
problems.
Recently the blast load function has been modified to account for the reflection of the
blast waves off the ground surface or any rigid body. This modified function provides the
user with the option to choose between a spherical air blast load producing only one initial
peak pressure, initial and reflected pressures, or a combined mach front pressure that
merges the effects of both incident and reflected pressures (LSTC, 2007).
Therefore, due to the ability and reliability of LS‐DYNA to use the mechanics and
characteristics of fluids and flows to calculate variations in pressures as a function of time
and position (coupled) it was used in this research to model steel‐jacketed columns under
blast loads. FE models were created using embedded constitutive material models and the
15
CONWEP loading functions were used to perform nonlinear coupled dynamic
computational analyses of representative pier columns.
2.3 Blast Loading
2.3.1 Air Blast Characteristics
Air blast loads on structures are affected by the weight, shape, location, and
orientation of the explosive charge. Commonly assumed charge shapes include spherical
air charges, hemispherical contact charges, or cylindrical vehicular charges. In the process
of an explosion an incident pressure wave travels from the center of a charge until it strikes
an object. Once striking the object, it is reflected back towards the point of explosion
(Sriram et al., 2006).
If a blast initiates near an abutment or between girders, the confined space allows
reflection of the pressure waves off the structure or ground. These reflected waves
continue to load the structure until they are vented from the space. Reflections of the
pressure waves may also increase damage to pier columns by creating a mach front region
(Winget et al., 2005; Williamson et al., 2010). This region, demonstrated in Figure 2.1, is
the area under the bridge where the incident and reflected waves combine to form a wave
front with almost twice the original pressure.
16
Figure 2.1: Mach Front Region (Winget et al., 2005)
This merge occurs because the incident wave heats and compresses the air behind it
allowing the reflected wave to travel faster through the superheated compressed air. The
standoff distance plays an important role in allowing this mach region to form. If the
charge is detonated too close to the deck or piers it will not allow enough time for the
reflected wave to combine with the initial wave (Winget et al., 2005). When modeling pier
columns for blast loads it is important to account for confinement of the blast pressures by
correctly representing the surrounding environment and its effects on peak and reflected
pressures and the time for pressure relief to occur.
Two possible confinement areas affecting pier columns located away from
abutments are reflections off the ground surface and reflections off the superstructure
(Williamson et al., 2010). Confinement and reflections caused by the ground surface are an
unavoidable consideration for this research due to the most likely threat scenario entailing
detonation close to or at ground level. These reflected pressures and possible mach front
pressures can be accounted for using CONWEP. LS‐DYNA can account for the reflected
pressure off of rigid surfaces and is considered a coupled dynamic analysis because the
blast pressures are dependent on pressure arrival time, distance to the loaded surface, and
angel of incidence, all of which are modified depending on the non‐rigid structural
17
response (LSTC, 2007). Experimental tests by Williamson et al. (2010) of blast loaded
bridge columns confirmed that the pressures and impulses experienced at the bottom of
the column from blast and ground reflected pressure waves were significantly higher and
arrived much sooner than those at the top of the column. This difference was also shown
to increase as the physical standoff distance decreases. Therefore, the second confinement
concern of reflected blast waves off the superstructure will not likely control the response
of bridge columns because they will reach their peak response much earlier in time due to
the ground reflections (Williamson et al., 2010).
Air pressures generated on a structure from blast loads are characterized as an
impulse excitation (Chopra, 2007). A typical pressure verses time graph for a blast wave is
shown in Figure 2.2.
Figure 2.2: Typical Blast PressureTime Graph (Sriram et al., 2006)
The curve is defined by a peak over‐pressure (Po) which decays to atmospheric pressure
(Ps) in time (to) and continues to drop, creating a partial vacuum of small magnitude before
returning to atmospheric pressure. This curve shape creates two regions in the graph, a
“positive phase” and a “negative or suction phase”. In most blast resistant studies the
18
effects of the “negative phase” are ignored and only the “positive phase” and its parameters
are considered because damage to the structure is caused by this phase (Sriram et al.,
2006). To evaluate values of Po, to, and the positive region impulse, I+ (area under the
positive curve), the blast waves are assumed to obey the widely accepted modified
Friedlander’s equation (Sriram et al., 2006).
1
(2.1)
Where P is the instantaneous over‐pressure at time t, and α is the dimensionless wave form
parameter. This equation can then be integrated between 0 and to to obtain I+ (Sriram et
al., 2006).
1 11 exp
(2.2)
When modeling blast over‐pressures on a structure, individual pressure‐time history
graphs can be generated and applied specifically to each element in a mesh by wave
propagation programs. These computer programs are widely used for uncoupled analysis
of blast loads on structures.
2.3.2 Blast Wave Propagation Programs
The two most commonly used blast wave propagation programs for blast resistant
research are discussed in this section. The first program, BlastX (2001), was created and
licensed by the Department of Defense to analyze blast loads on military structures. The
second program, CONWEP, uses empirically based blast load models created by Kingery
19
and Bulmash to generate pressure‐time histories for individual model elements (Sriram et
al., 2006).
BlastX has been utilized to examine the blast resistance of various bridge
components. It uses methods based on first principles of wave reflection to track blast
pressure magnitudes as they radiate from the center of an explosion and reflect off
surfaces. Studies include components such as girders (Baylot et al., 2003), decks, pier
columns and bridge systems (Winget et al., 2005). In the study by Winget et al. (2005)
circular pier columns, created for computational models, were approximated using a
square cross‐section due to solid modeling limitations in BlastX. The dimension for the
cross‐section was selected so that one side was equal to the projected diameter of a
circular column, which allowed the blast waves to encounter an object with the same
relative size as the actual dimension of the column. Because the blast pressures, reflective
shock waves and time of pressure relief are different for a square column verses a circular
column, the computed BlastX pressures were reduced by a factor of 0.80.
The empirical model used in CONWEP, presented in studies by Kingery and Bulmash,
produces a pressure load (P) in terms of a reflected (normal‐incident) pressure (P1),
incident (side‐on incident) pressure (P2) and angle of incidence (θ) (Sriram et al., 2006):
1 2.3
This code also includes the use of the modified Friedlander’s equation and the Hopkinson’s
cubic root scaling law. This law states that a given peak pressure will occur at a distance
from an explosion that is proportional to the cube root of the energy yield:
20
1 3⁄ (2.4)
where Z is the scaled distance, R is the distance from the center of explosion, and W is the
weight of the explosive. In addition to distance, CONWEP uses the cubic root scaling law to
scale the time (tsc) in terms of the given time (t) with the following equation.
⁄ (2.5)
With the use of this empirical model CONWEP calculates the pressure loads on a structure
for a given amount of Tri‐Nitro‐Toluene (TNT) at a given distance.
Both blast wave propagation programs can consider the effects of incident and
reflected blast waves and provide an accurate solution to the loads produced by an
explosion. However, due to solid modeling limitations and unavailability to the general
public of the BlastX program, the CONWEP blast load function incorporated into LS‐DYNA
was a logical choice for this research and was used to generate the loads for the
computational bridge column models.
2.4 Relevant Model Assumptions, Parameters, and Materials
This section discusses common model assumptions, parameters, and material models
that were implemented in the blast and seismic case studies reviewed for this research.
The discussed modeling assumptions allowed for more simplistic, but still accurate, models
to be created. The discussed parameters are separated into blast and seismic categories.
Within each of these categories, commonly used variables that attempted to encompass a
wide range of geometric properties are presented. Relevant LS‐DYNA constitutive models
21
for concrete and steel are also discussed to assist with selection of the most appropriate
method for modeling the bridge pier column’s response to blast loads.
2.4.1 Common Blast Modeling Assumptions
The following assumptions have been identified in the literature to produce reliable
results for computational models of bridge components subjected to air blast loads. The
majority of assumptions are related to support conditions and failure modes.
One typically assumed support condition relates to the transfer of uplift forces
produced by below deck explosions. An explosion occurring below the deck would create
tremendous uplift forces. The only resistance to these uplift forces is the deck‐shear stud
interface. Some of the forces could be transferred through the shear studs to the girders.
However, girder uplift is assumed to provide no contribution to axial loads experienced by
pier columns due to the simple bearing pad support conditions that are common to many
bridges designed in the United States (Winget et al., 2005). Therefore, support conditions
would be that of a propped cantilever, where the top of the column is modeled as a roller
support that prevents lateral translation, as shown in Figure 2.3.
Figure 2.3: ProppedCantilever
Pier Column
22
A propped cantilever is a conservative assumption because it produces the greatest shear
demand at the base of the column (apart from a pure cantilever), which was expected to be
the controlling mode of failure (Williamson et al., 2010).
Another commonly used assumption in blast resistant bridge column research
neglects traffic loads and superstructure dead load reactions from the analyses. This
assumption is conservative for flexural failure modes due to the increase in flexural
capacity associated with axial loads up to the balanced point on a concrete pier moment‐
axial load interaction diagram (Winget et al., 2005). Figure 2.4 demonstrates this increase
in flexural capacity on a representative moment‐axial load interaction diagram.
Figure 2.4: MomentAxial Load Interaction Diagram
However, because direct shear failures rather than flexural failures are commonly
observed at the base of blast loaded steel‐jacketed columns, assuming no axial loads would
not necessarily be conservative. The interaction between the normal forces on the column
produced by the axial load and the shear forces produced by the blast load may result in
failure of the steel‐jacketed column earlier than anticipated if axial loads are neglected.
Increase in flexural capacity
Increase in axial load
Balanced Point
Mn
Pn
23
Therefore, typical column axial loads must be represented in steel‐jacketed blast loaded
bridge columns.
Practical axial loads were included in seismic research that investigated the shear
enhancement of steel‐jacketed bridge columns (Priestley et al., 1994). This research
considered axial load ratios of the form / , where P = axial load, f’c = 28 day
concrete compressive strength, and Ag = gross cross‐sectional area of the column. The
ratios were varied from 0.06 to 0.18. Research results showed shear bond failure,
involving the bond degradation of bars on the sides of the column, for columns with higher
axial load.
The ground surface surrounding the pier columns must also be considered to
produce desired effects, such as reflection of blast waves. Research involving blast
performance on fiber reinforced concrete barriers assumed that the ground was infinitely
rigid to account for the reflection of the blast wave on the structure (Coughlin, 2008).
Similar assumptions are made in other bridge component blast performance studies
(Baylot et al., 2003; Winget et al., 2005).
Blast loads can cause shear or membrane failure of pier columns (Winget et al.,
2005). Membrane failure includes spalling or cratering of the concrete. Spalling is a
tension failure that occurs when a shock wave travels through a member and reflects off
the back of the object causing tension as it travels back towards the center of the member.
Cratering is defined as a compression failure through crushing of the concrete subjected to
blast loads. These types of membrane failures can cause loss of cover and affect the bond
between the reinforcement and the concrete, greatly reducing the column’s capacity.
However, previous research has indicated that, for pier columns retrofitted with steel
24
jackets, membrane failure can be mitigated and ignored during an analysis by selecting an
appropriate jacket thickness (Winget et al., 2005; Fujikura et al., 2008).
2.4.2 Blast and Seismic Model Parameters
This section summarizes commonly examined reinforced concrete pier column
parameters that have been to shown to predominantly influence column behavior and
resistance when subjected to either blast or seismic loads. In addition, parameters that
have been considered for steel jacket seismic retrofitting are also summarized to assist
with evaluating their possible contributions to blast load resistance.
2.4.2.1 Common Blast Parameters
Bridge pier columns are designed with a variety of different shapes, dimensions,
reinforcement types, ratios, and reinforcement splice locations. This provides a unique
challenge for modeling their response to blast loads. Representative columns must be
justified to encompass a logical range of possibilities. In a recent study, ten representative
reinforced concrete pier columns were examined under blast loads (Williamson et al.,
2010). This research was completed to study the influence of reinforcement splice
location, cross‐section shape and size, and transverse reinforcement type and spacing on
the behavior of blast‐loaded bridge columns. The tested half‐scale specimens represented
the most commonly used bridge column design parameters according to consultations of
state DOT guidelines and representatives. Table 2.2 presents the column specimen
parameters.
25
Table 2.2: Column Parameters (Williamson et al., 2010)
Column Label
ShapeDiameter mm(ft)
Longitudinal Reinforcement
Ratio
Transverse Steel Type
Transverse Steel Design
1A1 Circular 457(1.5) 1.04 Hoops Typical1A2 Circular 457(1.5) 1.04 Hoops Typical1B Circular 457(1.5) 1.04 Spiral Typical2A1 Circular 762(2.5) 1.13 Hoops Typical2A2 Circular 762(2.5) 1.13 Hoops Typical2B Circular 762(2.5) 1.13 Spiral Typical
2‐seismic Circular 762(2.5) 1.13 Spiral Seismic2‐blast Circular 762(2.5) 1.13 Spiral Blast3A Square 762(2.5) 1.18 Ties Typical
3‐blast Square 762(2.5) 1.18 Ties Blast
The work by Fujikura el al., (2008) discussed in Section 2.2, which computationally
and experimentally evaluated concrete filled steel tube (CFST) columns under blast loads,
considered the diameter to thickness ratio of the steel tube (D/t) as the controlling
parameter for the initial column sizing. The tests varied the loading parameters while the
column parameters remained constant. The six tested specimens consisted of two bents
having ¼ scale columns with diameters of 101.60 mm (4”), 127.00 mm (5”), and 152.40
mm (6”) each having a steel tube thicknesses of 3.18 mm (0.125”). A photograph of the test
set‐up is shown in Figure 2.5.
The reseach investigated the tube section’s performance for the blast load
considered and showed that D/t ratio is an important parameter with respect to column
blast resistance. The D/t ratio has also a been shown to be a commonly used parameter for
studies assessing the performance of steel‐jacketed bridge columns under seismic loads
(Chai et al., 1994).
26
Figure 2.5: Test SetUp (Fujikura et al., 2008)
2.4.2.2 Common Steel‐Jacketed Pier Seismic Parameters
A number of parameters have been studied related to the performance of steel
jackets under seismic loads. Steel jackets are oversized allowing the gap between the
jacket and the column to be filled with a cement‐based grout to ensure composite action.
This bond strength, along with the steel jacket height and jacket thickness to column
diameter ratio (tj/D), and column aspect ratio (L/D) have been shown to affect the lateral
stiffness of the columns (Chai et al., 1994; Chai, 1996). The studies have also shown that
longitudinal steel lap‐splice length and location also influenced the column’s resistance.
The design jacket thickness shown in Equation 2.6 is currently specified by the
California Department of Transportation for circular columns and has been proven
experimentally and in field applications to perform well under seismic loads (Chai, 1996)
200 6.4 (2.6)
where D = diameter of the column, and coefficient depending on the details of the
longitudinal reinforcement at the base of the column defined by:
27
1.0 for continuous longitudinal reinforcemen1.2 for lapped longitudinal reinforcement
The minimum thickness of the jacket is specified to primarily provide for sufficient rigidity
to ease handling and field installation. The maximum thickness is usually taken at 25 mm.
Equation 2.6 implies that the thickness to diameter ratio (tj/D) is either 0.005 for
continuous reinforcement or 0.006 for spliced reinforcement.
2.4.3 LS‐DYNA Material Models
As stated earlier, LS‐DYNA, a Lagrangian finite element code with explicit and implicit
time integration, has been proven to be a reliable tool for modeling structures subjected to
blast loads. Extensive material libraries have been developed and integrated into the code
and can be utilized to realistically represent reinforced concrete and reinforcing steel
behavior under a variety of conditions (LSTC, 2007). LS‐DYNA material models that are
able to represent the high strain rate effects of blast loading on concrete and steel are
discussed in this section. Magallanes (2008) studied various LS‐DYNA concrete models
with autogenerated parameters using tri‐axial and blast test data. It was stated that only
models using an equation of state (EOS) approach to represent the encountered tri‐axial
stress states can accurately predict the response of concrete flexural members under blast.
Computational tests employing LS‐DYNA Material Types 72, 72 Release 3, and 159 were
shown to produce reliable results compared to lab tri‐axial and blast tests and will be
examined in this section (Magallanes, 2008). For reinforcing steel models a bilinear elasto‐
plastic Material Type 3 constitutive relationship with kinematic and isotropic hardening is
discussed for quasi‐static loading cases. Material Type 24 is also discussed which allows
user defined strain rate effects for dynamic load cases.
28
Material 72 (keyword *MAT_COCNCRETE_DAMAGE), created by Malavar et al. (1997),
was a modification of Material Type 16. The original Material Type 16 constitutive model
decoupled the volumetric and deviatoric material responses to account for confinement
effects on concrete. The EOS, which produces current pressures in terms of volumetric
strain and defines the movable yield surface limiting the second invariant of the deviatoric
stress tensor, was deemed reliable for the original model. However, the original deviatoric
response, which has the limitation of not incorporating shear dilation of the concrete, will
result in softer than expected behavior. This original deviatoric response is a linear
combination of two fixed three‐parameter functions of pressure (maximum and residual),
whereas, the modified response incorporates a third independent fixed surface that
represents yield. Figure 2.6 represents the failure surfaces associated with Material 72.
Figure 2.6: Material 72 Failure Surfaces (Malavar et al., 1997)
Material Type 72 Release 3 (keyword *MAT_CONCRETE_DAMAGE_REL3) is the latest
revision of the original Material Type 16. This model, created by Karagozian and Case, Inc.,
utilizes the same shear failure surfaces shown in Figure 2.6. The most significant user
29
improvement is a model parameter generation component that is specified by the
unconfined concrete compressive strength, which allows default parameters to be defined
for normal weight concrete. Strain rates affect both Material 72 versions through radial
enhancement factors that are defined for values of pressure calculated from the EOS.
Another recent addition to the LS‐DYNA material library is Material Type 159
(keyword *MAT_CSCM_{OPTION}). It is a smooth continuous surface cap model (CSCM)
containing a smooth intersection between the yield surface and the hardening cap, as
shown in Figure 2.7 (LSTC, 2007).
Figure 2.7: CSCM Yield Surface (LSTC, 2007)
The user has the option of inputting unique material properties or requesting default
material properties for normal weight concrete. It was developed for the Federal Highway
Administration and used to model concrete impact for transportation applications
(Coughlin, 2008). Material 159 is modeled with a three invariant yield surface and contains
a damage model to simulate the softening of concrete in tensile and moderate compressive
regimes.
The damage accumulation is base on two distinct formulations representing brittle
and ductile damage. A softening function then applies a damage parameter to the six
30
elemental stress components that is equal to the current maximum of the brittle or ductile
parameters. Although this damage softening is mesh sensitive, the model maintains a
constant user defined fracture energy that can be used to calculate the damage parameter
regardless of element size. Strain rate effects are modeled using a viscoplasticity approach.
Material 159 also allows the user to enhance the fracture energy as a function of the
effective strain rate.
There are a variety of LS‐DYNA material models that allow for definition of static
and dynamic reinforcing steel material properties. Material Type 3
(*MAT_PLASTIC_KINEMATIC), a bilinear elasto‐plastic model, is a useful material model for
quasi‐static and cyclic loading due to its ability to define either kinematic or isotropic strain
hardening (LSTC, 2007). This model utilizes a bilinear representation of the stress‐strain
curve, as seen in Figure 2.8, with modulus of elasticity (E) defining the first line and a
tangent modulus (Et) defining the second strain hardening line.
Figure 2.8: LSDYNA Material Type 3 (LSTC, 2007)
31
Kinematic hardening defines the difference in ultimate tension and compression stresses
as 2 , whereas isotropic hardening defines the difference
as 2 . Expected yield and ultimate strengths of ASTM A615 Grade 60
reinforcing bar can be assumed as 475 and 750 MPa (69 and 109 ksi) respectively (Malvar,
1998). The modulus of elasticity can be assumed 200 GPa (29,200 ksi) with failure
occurring at 12% strain (Malvar, 1998). Material Type 3 also accommodates strain rate
effects through the Cowper and Symonds model which scales the yield stress with the
factor 1 ⁄ ⁄ where is the strain rate.
It is important to consider the effects of high strain rates associated with blast loads
for reinforcing steel. Reinforcing bars subjected to high strain rates can experience a yield
stress increase of 60 percent or more, depending on the grade of steel (Malvar, 1998). LS‐
DYNA contains constitutive material models that realistically represent reinforcing steel
response at very high strain rates for structures subjected to blast loads.
Another such material model that can account for the increase in capacity caused by
high strain rates is Material Type 24 (*MAT_PIECEWISE_LINEAR_PLASTICITY) (LSTC,
2007). This material model offers three separate options for defining strain rate effects.
The first option uses the Cowper and Symonds model mentioned above. The second option
allows for generality by requiring the user to input a curve to scale the yield stress. This
curve is defined as a scale factor which is a function of strain rate. These yield stress scale
factors are usually referred to as dynamic influence factors (DIF). A yield stress DIF curve
that has been shown to correctly represent strain rate effects in the range of 10‐4 to 10 s‐1 is
defined by Equation 2.7 (Malvar, 1998):
32
10 2.7
where 0.074 0.040 60⁄ . Figure 2.9 represents this proposed DIF curve for ASTM
A615 reinforcing steel Grade 40, 60, and 75. The third model strain rate option allows the
user to input stress‐strain curves for various strain rates. These curves can also be defined
in LS‐DYNA with reference to a table.
Figure 2.9: Proposed DIF for ASTM A615 Steel Reinforcing Bar (Malvar, 1998)
2.5 Relevant Research Findings
This section discusses relevant case study results that focused on the response of
bridge pier columns against both blast and seismic loads. Resulting failure modes and
parameters that contribute to preventing these failure modes are considered. Parameters
that were shown to improve the response of the pier columns against their respective loads
are also discussed.
2.5.1 Blast Resistant Results
33
Computational and physical tests show that bridge columns have greater blast capacity
than initially thought (Williamson et al., 2010). Two parameter changes that showed the
most improvements with respect to shear and flexural capacity were transverse
reinforcement steel ratio, which defines the ductility and shear capacity of the column, and
the elimination of longitudinal reinforcement splices in vulnerable regions, such as near the
foundation level. It was also demonstrated that spiral shear reinforcement performed
better than hoops or ties against blast loads with respect to adequately confining the
concrete.
Breaching failure of the concrete, which represented spalling and cratering , was
shown to govern pier design (Winget et al., 2005; Williamson et al., 2010). However, if
proper protection against this failure mode was provided, such as via the use of fiber
reinforced polymer (FRP) jackets, steel jackets, or steel tubes, direct shear failure at the
supports became a dominant failure (Winget et al., 2005; Williamson et al., 2010). This
failure mode was also produced in experimental tests performed by Fujikura et al., (2008)
on steel‐jacketed bridge pier columns that were designed according to conventional
seismic design standards and subjected to blast loads. These ¼ scale typical reinforced
steel‐jacketed concrete bridge columns did not exhibit ductile behavior and failed in direct
shear at the base of the column by fracturing all the longitudinal bars. These results
indicated that the increase in flexural stiffness associated with adhering to seismic design
criteria for blast resistant steel jacketing encourages a brittle shear failure rather than the
preferred ductile failure mode of flexural yielding. However, these experiments did not
consider variations in column and jacket detailing that could improve retrofitted column
shear performance.
34
Direct shear failure in a column can occur between different members, or member
parts that can slide along a common interface, such as the column‐foundation interface.
The failure is due to high inertial shear forces associated with high strain rate dynamic
loading (Fujikura & Bruneau, 2008). It especially needs attention when considering
concrete members that were not cast monolithically or between different material
interfaces, such as, concrete and steel interfaces. Direct shear is also referred to as “shear
friction” in ACI 2008 and equations are discussed to prevent this type of failure (ACI, 2008).
Direct shear capacity in a reinforced concrete column is a function of friction and
cohesion resisting forces within the shear plane. Factors influencing capacity are global
and local interface roughness, applied compressive stress normal to the interface, levels of
reinforcement that cross the interface, and concrete strength (Fujikura & Bruneau, 2008).
Global and local roughness determines the coefficient of friction between the two
interfaces. Combining these items with the applied compressive stress normal to the
interface and the total column cross‐sectional area produces a resisting frictional force to
shearing. Steel reinforcement, such as longitudinal bars in bridge columns that cross the
interface will experience tension as the two surfaces move laterally relative to each other.
This bar tensile force is equilibrated by the applied normal compressive stress as seen in
Figure 2.10.
35
Figure 2.10: Direct Shear Model (Fujikura & Bruneau, 2008)
This reinforcement will also produce dowel action to resist the shear forces (Fujikura &
Bruneau, 2008). Steel jackets that are properly developed across the interface would also
increase direct shear resistance by providing resisting doweling forces and possibly
shifting the blast load resistance at the base of the column to a more flexural resistance.
Concrete strength influences the resisting cohesion forces.
ACI 2008 provides and equation for nominal direct shear strength under static
loading in Section 11.7, which assumes that direct shear resistance is a function of friction
force only as shown in the following equation (ACI, 2008):
2.8
where Avf is the area of shear‐friction reinforcement across the shear plane, fy is the yield
strength of the reinforcement, and µ is the coefficient of friction. ACI 2008 specifies a
coefficient of friction of 0.6 for normal weight concrete placed against hardened concrete
not roughened intentionally. A “modified shear‐friction method” equation is also discussed
in the commentary of ACI 2008 Section 11.7 (ACI, 2008). This form of the equation
36
considers the direct shear resistance as a function of both friction (1st term) and cohesion
(2nd term):
0.8 2.9
where 0.8 represents the coefficient of friction, Ac is the area of the concrete section shear
plane, and K1 = 1.7MPa (400psi) for normal weight concrete.
As seen if Figure 2.11, the ACI shear‐friction model gives a conservative prediction
of the direct shear capacity while the modified shear‐friction model matches well with
experimental data.
Figure 2.11: Direct Shear Model Comparison (ACI, 2008)
Some of the possible failure modes discussed by Fujikura et al. (2008), which
involved the CFST blast research, included buckling of the steel tube at the mid‐height
hinge and rupture of the steel tube at the base. The columns were also examined for the
second order moment effects caused by local member displacement and found to be stable
against these P‐δ failures for the considered axial forces.
37
2.5.2 Steel‐Jacketed Column Seismic Resistance Results
Steel jackets have been shown to extend the length of the effective plastic hinge in
columns subjected to base excitation resulting in a larger increase in ductility compared to
columns without jackets (Chai et al., 1994). This increase in ductility and confinement of
the dilated concrete in compression allows for an increase in the ultimate compressive
strain of the concrete, thereby increasing the column’s capacity and blast resistance. The
capacity increase may put additional strain on the extreme longitudinal steel fiber in the
column and result in an ultimate failure mode similar to low‐cycle fatigue fracture.
Steel jackets were also shown to greatly affect the shear and moment‐curvature
response of columns (Chai et al., 1994; Chai, 1996). Full‐height jackets showed a lateral
stiffness increase of 23‐41%. This increase was dependent on the jacket‐to‐column bond
strength, thickness ratio of the jackets (tj/D), and aspect ratio of the column (L/D).
Computational tests by Chai et al. (1994) demonstrated that initial moment‐curvature
response was largely unaffected because the jacket had not yet been engaged. However,
once enough lateral expansion of the columns occurred, spalling of the concrete was
delayed by confinement provided by the jacket allowing for greater column curvatures to
be developed. Ductility factors, which consist of the ratio of ultimate curvature to yield
curvature (φu/φy), increased from 10 for unconfined columns to 41 for columns with
thickness ratios of 0.005. Columns with jacket thicknesses between 5 mm and 15 mm (0.2”
and 0.59”) were shown to fail at a failure mode corresponding to the ultimate compressive
strain of the concrete. Columns with jacket thicknesses between 20 mm and 25 mm (0.79”
and 1.0”) were shown to fail at an ultimate failure mode corresponding to the ultimate
tensile strength of the longitudinal steel.
38
In computational tests, structural damage to the columns at the ultimate failure
mode was assessed with a modified Park and Ang damage model (1985) which assumes
damage to be a linear combination of normalized peak response displacement and
normalized total hysteretic energy dissipated by the structure during the response.
Damage above 1.0 is considered failure. All unretrofitted columns analyzed with the
damage model were shown to have values greater than 1.0 and, therefore, failed. However,
the retrofitted columns resulted in damage values that were extremely low, showing that
current design thicknesses were sufficient against the assumed seismic loads.
Experimental correlation between the damage models and physical testing was
accomplished through damage indices that were computed at various stages of cyclic
loading for the steel‐jacketed columns. The test models were jacketed up from the footing
with a jacket length of 2 times the diameter of the column. These tests showed that
columns experiencing damage indices up to 0.61 were repairable with the steel jacket
retrofit. However, those that experienced damage indices greater than 0.61 were
irreparable due to the expense of repairing severe spalling of the concrete at the pier base
and the subsequent exposure and buckling of longitudinal steel at this location.
These studies show that the current guidelines for steel jacketing of circular bridge
columns in Caltrans Bridge Design Aid 14‐2 (2008) which specifies steel jacket thickness
ratios of 0.005 and 0.006 for columns with continuous and lap‐spliced longitudinal
reinforcement, respectively, provide significant enhancement in the ultimate compressive
strain of the concrete. This increase in concrete compressive strain is accompanied by
large increases in curvature ductility of the columns. Specified Caltrans jacket thickness
ratios also are shown to increase column lateral stiffness in these studies. These retrofits
39
provide adequate protection against damage from ground motions that had spectral
accelerations comparable to those specified in current design spectra (Chai, 1996). As a
result, these designs were utilized to assist with initial sizing of steel jackets for this
research.
2.6 Summary
After evaluation of current blast analysis methods, a nonlinear coupled dynamic FE
analysis utilizing LS‐DYNA was chosen to evaluate the performance of circular steel‐
jacketed reinforced concrete bridge pier columns under blast loads. Investigation of
general blast loading theory and blast wave propagation programs further validated the
use of the empirically based blast load function, CONWEP, in LS‐DYNA as the load
application method. Commonly used blast modeling assumptions such as neglecting the
effects of girder uplift, traffic and girder gravity loads, and spalling and cratering on steel‐
jacketed columns were identified through case study investigations and incorporated into
this research. Important modeling parameters to evaluate the columns were also extracted
from these studies. Some of the parameters included: steel jacket thickness ratio (tj/D);
column aspect ratio (L/D); jacket‐to‐column bond strength; longitudinal reinforcement
splice location; and transverse reinforcement type and spacing. Constitutive material
models available in LS‐DYNA were investigated to obtain a list of models that realistically
represent the response of reinforced concrete under blast loads. LS‐DYNA Material Types
72, 72 Release 3, and 159 were shown to produce acceptable computational response for
the concrete. Finally, results of recent case studies evaluating the performance of pier
columns under blast and seismic loads have shown that the dominant failure mode for
40
steel‐jacketed columns was direct shear failure at the pier column base. Although
experimental research on steel‐jacketed bridge columns under blast produces this brittle
failure mode, no research to date has evaluated the effects of variations in column and
jacket parameters that may improve jacketed column shear performance under blast.
Parameters that were shown to contribute most to resisting this type of failure were
transverse steel ratio and type, steel jacket thickness ratio (tj/D), column aspect ratio
(L/D), jacket‐to‐column bond strength and eliminating longitudinal reinforcement splices
in vulnerable regions.
41
Chapter 3 Parametric Studies
3.1 Overview
This chapter examines the method used to determine the effectiveness of steel‐
jacketed bridge columns against blast loads. Essential parameters needed to fully describe
bridge columns according to current code criteria are presented. These variables are then
examined to extract commonly used values that can be held constant throughout this
research. The blast loading and corresponding scaled stand‐off distances are also
discussed. Finally, a discussion of possible limits for remaining parameters obtained using
factorial design is presented.
3.2 Test Variables
At a minimum, reinforced concrete bridge columns are designed to meet American
Association of State Highway and Transportation Officials (AASHTO) LRFD Bridge Design
Specifications (2007). However, state Departments of Transportation (DOTs) commonly
have more stringent design requirements depending on local design issues such as
earthquakes and other environmental conditions. Table 3.1 compiles a list of the essential
parameters needed to design a steel‐jacketed reinforced concrete bridge column specified
in the AASHTO LRFD Bridge Design Specifications and in Caltrans Bridge Design Aid 14‐2
for steel jacket sizing (AASHTO, 2007; Caltrans, 2008).
Table 3.1: Bridge Column Parameters
Column Geometry Column Connection Concrete Stds.
Shape Footing f'C
42
Aspect ratio Bent Class Cover
Long. Reinf. Stds. Tran. Reinf. Stds. Steel Jacket Stds.
Grade Grade Grade Min. size Min. size Height Min. No. Min. No. Diameter‐to‐thickness ratio
ρL ρS Jacket‐to‐column bond Splice location Type Gap distance
3.3 Constant Parameters
In NCHRP Report 645 (Williamson et al., 2010) a state DOT design survey was
conducted to represent current national bridge column design standards. This survey
highlighted consistent design standards throughout the nation that could realistically be
held constant for this study. Parameters identified from this survey that were not varied
are:
• Column geometry – circular;
• f’c = 27.57 MPa (4000 psi);
• 50.8 mm (2”) concrete cover;
• A615 Gr. 60 steel reinforcing bars (std. uncoated, fully developed);
• 1% long. reinf. ratio (ρL);
• Propped‐cantilever boundary conditions; and
• Full height Gr. 36 steel jackets;
Additional parameters that were held constant for this parametric included:
• Longitudinal splice location at 0.25L;
• Discrete hoop transverse reinforcement;
43
• Perfectly bonded steel jacket‐to‐column; and
• Scaled distance = 0.8 for representative blast load.
Longitudinal splices have been proven to be detrimental to the ductile response of
bridge columns under blast and seismic demands (Williamson et al., 2010; Chai et al.,
1994). However, elimination of longitudinal splices is not an option when considering in‐
situ bridge columns and is not consistent with the objective of bridge column retrofit.
Research has also shown that, although longitudinal splices exist, the presence of a steel
jacket will restrain the dilation on the flexural tension side of the column and provide
effective constraint against failure at the longitudinal bar splices (Priestley et al., 1994).
Traditionally longitudinal splices are located near the base of the column due to the
ease of construction. The ideal splice, however, should be located where the bending
moment in the column is zero or very small. These zero moment locations correspond to
inflection points of the column’s deflected shape. The location of inflection points depends
on the assumed column boundary and loading conditions.
In NCHRP Report 645 Williamson et al., (2010) also assumed a propped cantilever
column boundary condition and a linearly varying blast load distribution, which is typical
for small physical stand‐off distances. To reduce the chances of shear failure at small
stand‐off distances the study designed the blast‐loaded columns using plastic analysis. This
analysis forms two plastic hinges in the column, as seen in Figure 3.1, both of which reach
the full plastic capacity of the column.
44
Figure 3.1: Moment Diagram of Plastic Hinge Analysis for Blastloaded Column (Williamson et al., 2010)
Figure 3.1 shows that the location of zero moment from the plastic hinge analysis is at
around ¼th of the height of the column from the base. Therefore the splice location was
held constant at ¼th of the height from the base of the column (0.25L) for this research.
Splices were represented as internal hinges in the longitudinal reinforcement of the FE
models. These hinges allowed rotation of the reinforcement and produced a reduction in
flexural capacity and overall ductility, which is representative of the effects of longitudinal
splices.
Discrete hoops and continuous spiral reinforcement are typically used as transverse
reinforcement in bridge column construction. Although spiral reinforcement is a common
method in seismic zones to increase concrete confinement and column ductility, many
existing bridge columns are not equipped with this type of reinforcement (Williamson et
al., 2010). Discrete hoops, however, are commonly used in bridge column detailing and are
preferred by most DOTs due to their ease of construction (Williamson et al., 2010). One
~ 0.25L
45
concern for these types of hoops is development length and pullout resistance against blast
and seismic loads (Williamson et al., 2010). This research focused on the effects of discrete
hoops, assuming a perfect bond between the concrete and bars, due to their prevalence in
existing bridge columns and favorability amongst a majority of state DOTs for new
construction.
Increase in the lateral stiffness of steel‐jacketed bridge columns is fairly sensitive to
the increase in bond strength between the jacket and the column ( ) although the rate of
increase in stiffness diminishes with increased bond strength (Chai, 1996). This increase in
lateral stiffness will reduce the effective plastic hinge length of the column, resulting in a
large increase in ductility at the critical section of the column (Chai et al., 1994). Variation
in can be expected in practice with values ranging from 150 kPa – 2100 kPa (Chai,
1996). However, steel jacketing seismic research has already defined increases in as
advantageous for achieving the required lateral stiffness to resist the considered ground
accelerations. Therefore, this research assumed a perfectly bonded composite system
between the steel jacket and the column.
As mentioned in the literature review chapter, CONWEP was used to generate blast
loads for this research. Although charge weight, shape, and location are unpredictable
when considering terrorist threats on bridge columns, column accessibility and
vulnerability to direct attack are high due to traditional site layout which places columns
close to travel lanes and public access. Easy accessibility to bridge columns likely suggests
small stand‐off distances. However, there are limitations to the load prediction techniques
involving close‐in detonations due to the variability of temperature and pressures in the
vicinity of an explosion (Williamson et al., 2010; LSTC, 2007).
46
Classic scaling distances were used to represent charge weight and stand‐off
distances for this research. Loads were held constant by specifying a constant scaled
stand‐off distance, thus allowing the variation in column parameters rather than loading
parameters to be assessed. According to the LS‐DYNA keyword manual the empirical
equations underlying the blast load function are valid between scaled distances, Z, of 0.136
ft/lbm1/3 and 100 ft/lbm1/3,where Z = R/W1/3 (LSTC, 2007), R is the physical standoff
distance, and W is the equivalent weight of TNT in lbm. This research considered an
airblast with ground reflection. This option in the LS‐DYNA blast function considers an
incident shock wave impinging on the ground’s surface which is reinforced by a reflected
wave producing a mach stem region (LSTC, 2007). The equivalent weight of TNT was set to
W = 400 lbm based on weights predicted by the Federal Highway Administration (2003)
for terrorist actions using sedans as shown in Figure 3.2.
Figure 3.2: Blast Damage Threshold (FHWA, 2003)
47
Physical stand‐off distance was close‐in to conservatively consider the vulnerability
of most bridge columns. This also allows blast waves to reach the face of the column
quickly, reducing model run time. Close‐in physical stand‐off distances were explained in
the literature review to maximize pressures and impulses at the base of the column and
cause the reflected pressure waves to arrive sooner with higher amplitude than those at
the top of the column. Therefore, reflection of the blast waves off of the superstructure will
not likely control the response of the bridge columns because the columns will likely reach
their peak response much earlier in time at their base due to the ground reflections.
Physical standoff was set at 1795 mm (5.89’) for this research. Therefore, utilizing the
predicted weight and physical standoff distance the scaled distance for this research of Z =
0.8 fell within the valid range for load prediction.
3.4 Varied Parameters
Bridge column parameters that were varied include design details that are
anticipated to be the most critical with respect to the performance of steel‐jacketed
reinforced concrete bridge columns under blast loads. These parameters include:
• Column aspect ratio (L/D);
• Transverse reinforcement ratio (ρS);
• Steel jacket diameter‐to‐thickness ratio (tj/D); and
• Steel jacket gap distance‐to‐diameter ratio (Lg/D).
Although blast and seismic loading produce somewhat similar dynamic responses,
seismic detailing cannot be assumed to provide adequate protection for bridge columns
against blast loads. However, research conducted on the seismic performance of steel‐
48
jacketed bridge columns has focused on shear and ductility enhancement, which is also of
concern for blast resistance. Therefore, seismic standards provide a good starting point for
blast resistant detailing and provide a means of further refining parameters considered for
this research.
3.4.1 Column Aspect Ratio (L/D)
The length to diameter (L/D) ratio is an important parameter for blast loaded
columns because of its effects on shear capacity, lateral stiffness, and steel jacket
confinement pressure. Throughout this research L/D was varied by holding the diameter
of the column fixed at 1524 mm (60”) while adjusting the length. The column’s diameter
can be defined in terms of d, the effective diameter, which is essentially the diameter of the
core concrete encompassed within the longitudinal bars of the column. This effective
diameter is directly related to its shear capacity as shown in Equation 3.1 from the AASHTO
LRFD Bridge Design Specifications (2007).
2 (3.1)
The L/D parameter has also been shown to affect the lateral stiffness of steel‐
jacketed columns under seismic events (Chai, 1996). In this study the column’s lateral
stiffness, defined as Δ⁄ , where = the lateral force necessary to cause first‐yielding of
the longitudinal reinforcement, and Δ = the lateral displacement at first yield of the
longitudinal reinforcement, increased from 46 percent for L/D = 3 to 59 percent for L/D=9
for steel‐jacketed columns. These results demonstrated that, as L/D increased, the
column’s lateral stiffness also increased. Finally, the L/D parameter was also shown to
49
affect the required minimum confinement pressure of 2.0 MPa (300 psi) specified by steel
jacket design thickness Equation 2.6. To maintain this confinement pressure it was
determined that the steel jacket design thickness must increase as L/D decreases. Common
values of L/D from this study ranged from 1.5, referred to as “shear columns,” to 9, referred
to as “flexural” columns (Chai, 1996).
3.4.2 Transverse Reinforcement Ratio (ρs)
Another important parameter associated with blast loaded columns is the ratio of
transverse reinforcement to the effective cross‐sectional area of the column. Transverse
reinforcement ratio not only dictates shear capacity, but also ductility and column concrete
confinement pressure. AASHTO LRFD Bridge Design Specifications (2007) requires a
minimum transverse reinforcement ratio for typically designed circular columns which is
given by Equation 3.2
0.45 1
(3.2)
where = the gross area of the column (in2), = the area of the concrete core (in2), =
specified compressive strength of the concrete at 28 days (psi), and = yield strength of
reinforcing bars (psi). For columns designed to AASHTO LRFD Bridge Design
Specification’s (2007) seismic standards the minimum transverse reinforcement ratio is
defined in Equation 3.3.
0.12
(3.3)
50
Williamson et al. (2010), suggested an even more stringent transverse reinforcement ratio
for blast design shown in Equation 3.4.
0.18
(3.4)
This recommended minimum ratio will essentially provide 50% more confinement over
current seismic standards to improve ductility, energy absorption, and dissipation capacity
of the cross‐section (Williamson et al., 2010). This thesis research focused on the limits of
transverse reinforcement ratio encompassed by Equation 3.2 and 3.4.
3.4.3 Steel Jacket Diameter‐to‐Thickness Ratio (tj/D)
Similar to L/D, the ratio of steel jacket thickness to column diameter (tj/D)
parameter is directly related to column shear capacity, lateral stiffness, and concrete
confinement pressure. In seismic research tj/D has also contributed to an increase in
ductility and in preventing lap‐splice failure (Chai, 1996). Work by Priestley et al. (1994),
on the shear enhancement of steel‐jacketed bridge columns related both tj, and Dj which is
the steel jacket interior diameter, to the added column shear capacity provided by the steel
jacket. This added capacity can be superimposed onto the shear capacity given in Equation
3.1 for circular reinforced concrete bridge columns specified in the AASHTO LRFD Bridge
Design Specifications (2007). This equation is reproduced as Equation 3.5, which is in
terms of tj and Dj.
2 0.865 (3.5)
51
Here = shear capacity contribution of the steel jacket, = steel jacket thickness, =
steel jacket yield strength, and = steel jacket interior diameter. As with L/D, the tj/D
parameter is directly proportional to column lateral stiffness through the steel jacket’s
shear capacity enhancement (Chai, 1996). Finally, to obtain the minimum concrete
confinement pressure of 2.0 MPa (300 psi) as specified by Caltrans (2008), tj/D must equal
0.005 for continuous reinforcement and 0.006 for spliced reinforcement. These minimum
values, along with the maximum value 0.0164 used by Chai (1996) in his steel‐jacketed
column seismic research, were considered for this research. It should be noted though that
Caltrans Bridge Design Aid 14‐2 specifies a 25 mm (1”) maximum steel jacket thickness.
3.4.4 Steel Jacket Gap Distance‐to‐Diameter Ratio (Lg/D)
Steel jackets are typically terminated 50 mm (2”) before both the footing and pier
cap ends of bridge columns. The purpose of this gap distance, defined as Lg, is to prevent
the jacket from bearing on the adjoining members avoiding an increase in moment demand
on both the pier cap and footing due to the steel jacket’s ability to increase the column’s
moment capacity in these adjoining areas (Chai et al., 1994; Fujikura & Bruneau, 2008).
However, this gap contributes to the discontinuity of shear resistance at the column base
which is proven to be a critical area to resist the commonly observed direct shear failure
for steel‐jacketed columns subject to blast loads (Fujikura & Bruneau, 2008). Steel jacket
seismic research also has shown that spalling of the concrete is commonly observed at the
base of the column where the steel jacket was terminated (Chai, 1996; Priestley et al.,
1994). Therefore, in an attempt to prevent the discontinuity of shear resistance at the base
of the column, reduction of the steel jacket gap, Lg, may prove advantageous in resisting
52
direct shear failures. To nondimensionalize the Lg parameter, this research considered a
ratio of Lg/D, where Lg = steel jacket gap distance, and D = diameter of the column.
3.5 Parameter Limits
Factorial design is an efficient method to study the effects of the four parameters
mentioned above. Factorial design is defined as an investigation of all possible
combinations of parameter levels (Montgomery, 2009). The symbol ‘Nk ’ represents a
factorial design where k = the number of parameters, and N = equals the number of
parameter levels. Therefore, the total number of experiments required is Nk. To further
refine the factorial design method, a 2k factorial design method can be implemented. This
type of factorial design considers only two levels of each parameter. These levels can
either be quantitative, such as the L/D value, or qualitative, such as using discrete hoops or
continuous spiral reinforcement. These two levels are commonly considered as minimum
and maximum value limits. With respect to this study, k represents the number of
parameters that are being examined and equals four. 2k factorial design is an intelligent
choice for this study because most of the quantitative parameters can be bound to
minimum and maximum limits specified in current codes and relevant blast and seismic
research literature. Table 3.2 shows a list of the parameters along with their anticipated
minimum and maximum levels and sources.
Table 3.2: Parameter Limits
Parameters Minimum (Source) Maximum (Source)
L/D 1.5 (Chai, 1996) 9 (Chai, 1996)
53
ρs 0.45 1 (ASSHTO, 2007) 0.18 (Williamson et al., 2010)
tj/D 0.005 (Chai, 1996) 0.0164 (Chai, 1996)
Lg/D 50 mm(2”)/D (n/a) Fixed to footing (n/a)
When associated with 2k factorial design these minimum and maximum limits produce 24 =
16 possible combinations of the four considered blast loaded steel‐jacketed bridge column
parameters.
3.6 Parametric Matrix
Using the constant and variable column parameters in conjunction with the varied
parameter limits discussed in the previous section, sixteen bridge pier columns were
designed for use in the parametric study. This study used a constant column diameter
method to assign the parameters to each column. This simplified the FE modeling by
maintaining a constant size and number of longitudinal reinforcement in every parametric
column. A constant diameter of 1524 mm (60”) was chosen, which was used for the
columns subject to blast loads in the investigation conducted by Williamson, et al. (2010).
Table 3.3 lists the values of the constant parameters used to develop the parametric matrix
of the sixteen FE model columns.
Table 3.3: Constant Column Parameters
Constant Parameters U.S. Units LS‐DYNA Units Diameter 60 in 1524.00 mm f'c 4 ksi 27.58 MPa fy 69 ksi 475.74 MPa
54
Cover 2 in 50.80 mm ρL 0.01 0.01 P/f'cAg 0.06 0.06 WTNT 400 lb 1.81E‐01 Mg Z 0.8 ft/lb⅓ 537575.18 mm/Mg⅓ RBlast 5.89 ft 1796.63 mm
Each model was labeled with a quick reference number from 1 to 16 along with codes that
represented each of the four varied parameters and their limits. The codes consisted of a
for aspect ratio, v for transverse reinforcement ratio, j for jacket thickness ratio and g for
jacket gap ratio. Parameter limits were defined with a 0 for minimum values and 1 for
maximum values. An example parametric model label is defined in Figure 3.3.
Figure 3.3: Parametric Model Label Definition
Table 3.4 to Table 3.6 present the parametric matrix for the sixteen FE models listing the
parameters and their limits, along with details on the column and jacket, and the transverse
and longitudinal reinforcement. The color shades help differentiate between minimum and
maximum parameter limits and combinations.
Min(0)/Max(1) Limit
Quick Reference #
Aspect Ratio
Trans. Reinf. Ratio
JacketThick. Ratio
Min(0)/ Max(1) Limit
Min(0)/Max(1) Limit
Min(0)/Max(1) Limit
Jacket Gap Ratio
55
Table 3.4: Parametric Matrix Column and Jacket Details
Parametric Matrix Parameter Limits Column and Jacket Details
Model # L/D ρS tj/D Lg/D Axial
Load (N) Ag
(mm2) Ac (mm2)L
(mm) tj
(mm)Lg
(mm)1_a0v0j0g0 3 0.001830 0.005 0.000 3.02e+06 1824147 1704564 4572 7.62 0.02_a0v0j0g1 0.033 3.02e+06 1824147 1704564 4572 7.62 50.83_a0v0j1g0 0.0164 0.000 3.02e+06 1824147 1704564 4572 24.99 0.04_a0v0j1g1 0.033 3.02e+06 1824147 1704564 4572 24.99 50.85_a0v1j0g0 0.010435 0.005 0.000 3.02e+06 1824147 1704564 4572 7.62 0.06_a0v1j0g1 0.033 3.02e+06 1824147 1704564 4572 7.62 50.87_a0v1j1g0 0.0164 0.000 3.02e+06 1824147 1704564 4572 24.99 0.08_a0v1j1g1 0.033 3.02e+06 1824147 1704564 4572 24.99 50.89_a1v0j0g0 9 0.001830 0.005 0.000 3.02e+06 1824147 1704564 13716 7.62 0.010_a1v0j0g1 0.033 3.02e+06 1824147 1704564 13716 7.62 50.811_a1v0j1g0 0.0164 0.000 3.02e+06 1824147 1704564 13716 24.99 0.012_a1v0j1g1 0.033 3.02e+06 1824147 1704564 13716 24.99 50.813_a1v1j0g0 0.010435 0.005 0.000 3.02e+06 1824147 1704564 13716 7.62 0.014_a1v1j0g1 0.033 3.02e+06 1824147 1704564 13716 7.62 50.815_a1v1j1g0 0.0164 0.000 3.02e+06 1824147 1704564 13716 24.99 0.016_a1v1j1g1 0.033 3.02e+06 1824147 1704564 13716 24.99 50.8
Ag = Column gross cross‐sectional area Ac = Column concrete core cross‐sectional area L = Column height tj = Steel jacket thickness Lg = steel jacket gap length
56
Table 3.5: Parametric Matrix Transverse Reinforcement Details
Parametric Matrix Parameter Limits Trans Reinforcement Details
Model # L/D ρS tj/D Lg/D Trans Bar Av (mm2) sv (mm) 1_a0v0j0g0 3 0.001830 0.005 0.000 #3 70.97 109.05 2_a0v0j0g1 0.033 #3 70.97 109.05 3_a0v0j1g0 0.0164 0.000 #3 70.97 109.05 4_a0v0j1g1 0.033 #3 70.97 109.05 5_a0v1j0g0 0.010435 0.005 0.000 #5 200.00 53.90 6_a0v1j0g1 0.033 #5 200.00 53.90 7_a0v1j1g0 0.0164 0.000 #5 200.00 53.90 8_a0v1j1g1 0.033 #5 200.00 53.90 9_a1v0j0g0 9 0.001830 0.005 0.000 #3 70.97 109.05 10_a1v0j0g1 0.033 #3 70.97 109.05 11_a1v0j1g0 0.0164 0.000 #3 70.97 109.05 12_a1v0j1g1 0.033 #3 70.97 109.05 13_a1v1j0g0 0.010435 0.005 0.000 #5 200.00 53.90 14_a1v1j0g1 0.033 #5 200.00 53.90 15_a1v1j1g0 0.0164 0.000 #5 200.00 53.90 16_a1v1j1g1 0.033 #5 200.00 53.90
Av = Transverse reinforcement (volumetric) area sv = Required transverse reinforcement spacing
57
Table 3.6: Parametric Matrix Longitudinal Reinforcement Details
Parametric Matrix Parameter Limits Long. Reinforcement Details
Model # L/D ρS tj/D Lg/D Long Bar
No. Bars
Splice Loc (mm)
As, PROVIDED
(mm2) As, REQ'D (mm2)
1_a0v0j0g0 3 0.001830 0.005 0.000 #8 36 1143 18348 18241 2_a0v0j0g1 0.033 #8 36 1143 18348 18241 3_a0v0j1g0 0.0164 0.000 #8 36 1143 18348 18241 4_a0v0j1g1 0.033 #8 36 1143 18348 18241 5_a0v1j0g0 0.010435 0.005 0.000 #8 36 1143 18348 18241 6_a0v1j0g1 0.033 #8 36 1143 18348 18241 7_a0v1j1g0 0.0164 0.000 #8 36 1143 18348 18241 8_a0v1j1g1 0.033 #8 36 1143 18348 18241 9_a1v0j0g0 9 0.001830 0.005 0.000 #8 36 3429 18348 18241 10_a1v0j0g1 0.033 #8 36 3429 18348 18241 11_a1v0j1g0 0.0164 0.000 #8 36 3429 18348 18241 12_a1v0j1g1 0.033 #8 36 3429 18348 18241 13_a1v1j0g0 0.010435 0.005 0.000 #8 36 3429 18348 18241 14_a1v1j0g1 0.033 #8 36 3429 18348 18241 15_a1v1j1g0 0.0164 0.000 #8 36 3429 18348 18241 16_a1v1j1g1 0.033 #8 36 3429 18348 18241 As, PROVIDED = Area of longitudinal steel provided As, REQ’D = Area of longitudinal steel required
3.7 Additional Parametric Considerations
In addition to the sixteen parametric models discussed above, two FE models were
created to examine the effects of steel jacket fixity and bearing on the foundation and to
assess the advantages of steel jacket retrofitting compared to unjacketed columns. The
first of the two models was a modification of parametric Model 9_a1v0j0g0 that compared
blast response for steel‐jacketed columns without a gap and the jacket completely fixed to
the foundation (as in model 9_a1v0j0g0) to those without a gap and the jacket not fixed and
bearing on the foundation (9_a1v0j0g0_Not Fixed). The second additional model was a
modification of parametric model 10_a1v0j0g1, which showed the best blast resistance for
columns containing a steel jacket gap in Chapter 6. The modified model (10_a1v0) was
58
subjected to the same axial and blast loading, however, the steel jacket was removed to
compare its response to further examine advantageous of steel jacket retrofitting against
blast.
3.8 Summary
Essential parameters needed to design a steel‐jacketed circular reinforcement
concrete bridge column were defined through current code criteria which included
AASHTO LRFD Bridge Design Specifications (2007) and Caltrans Bridge Design Aid 14‐02
(2008). These design parameters were then further defined to extract commonly used
values that were considered constant throughout this research. This constant parameters
consisted of:
• Column geometry – circular;
• f’c = 27.57 MPa (4000 psi);
• 50.8 mm (2”) concrete cover;
• A615 Gr. 60 steel reinforcing bars (std. uncoated, fully developed);
• 1% long. reinf. ratio (ρL);
• Propped‐cantilever boundary conditions; and
• Full height Gr. 36 steel jackets;
• Longitudinal splice location at 0.25L;
• Discrete hoop transverse reinforcement;
• Perfectly bonded steel jacket‐to‐column; and
• Scaled distance = 0.8 for representative blast load.
59
The remaining essential bridge column parameters of aspect ratio, transverse
reinforcement ratio, jacket thickness ratio and jacket gap ratio were varied through a 2k
factorial design. This parametric study method considers only two levels of each
parameter. These two levels were chosen as minimum and maximum values that were
identified in current codes and seismic and blast research literature. The variation of four
critical parameters at two levels each produced 16 parametric combinations (24 = 16).
Finally, a parametric matrix was produced for this research from the 16 parameter
combinations. This matrix defined column and jacket detailing, as well as, longitudinal and
transverse detailing which was used to create sixteen parametric column models for this
research. Two additional parametric columns were considered to examine the importance
of steel jacket fixity in the columns without a steel jacket gap and the overall advantages of
steel jacket retrofitting for columns subjected to blast. This corresponds to eighteen total
parametric models.
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Chapter 4 Finite Element Modeling
4.1 Overview
This chapter describes the FE modeling techniques used to create validation and
parametric study bridge column models in LS‐DYNA. The validation models were initially
used for quasi‐static validation and agreement with predicted response from theory. Once
the models were deemed acceptable under these loading cases, they were modified,
jacketed and validated under blast loads. Finally, additional steel‐jacketed models were
parameterized following information presented in the previous chapter and used to
examine effective detailing methods.
4.2 Finite Elements
Three finite element types were utilized in the FE bridge column. Constant stress
solid brick elements were used to represent the concrete, shell elements represented the
steel jacket and beam elements represented concrete reinforcement. LS‐DYNA material
models Type 72, Type 72 Release 3, and Type 159, which have been shown to realistically
represent concrete under blast loads, require the use of solid brick elements (LSTC, 2007).
Therefore, the constant stress brick elements were chosen to represent the concrete and
can efficiently provide complete stress and strain results.
LS‐DYNA offers two desirable shell element types: the Hughes‐Liu, and Belytschko‐
Tsay elements (LSTC, 2007). Although both have been shown to perform well, this
research used the Hughes‐Liu type shell element. It is a computationally efficient and
robust element that allows for the treatment of finite strains with reduced computational
61
time (LSTC, 2006). Furthermore, this element type is compatible with the brick elements
used to represent the concrete and is versatile enough to be used in both implicit and
explicit problem solutions.
For beam elements, LS‐DYNA offers two similar types: the Hughes‐Liu, and the
Belytschko‐Schwer elements. For similar reasons as the shell element, the Hughes‐Liu
beam element will be utilized.
4.3 Constraints and Boundary Conditions
Elements were constrained or coupled using three different techniques: nodal
compatibility, the *CONSTRAINED_LAGRANGE_IN_SOLID LS‐DYNA function, and the
*CONTACT_NODES_TO_SURFACE LS‐DYNA function. Boundary conditions differed
between validation models depending on the theoretical or experimental setup and are
discussed in detail in Chapter 5. Calibration of the blast validation models required
boundary conditions that simulated the restraints of a bridge column at typical
construction joints located at base and pier cap ends. Parametric models utilized this type
of boundary condition and it is discussed later in this section.
Nodal compatibility is often used in computational research to couple different
elements that are in close proximity to each other and assumes a perfect bond between the
connected elements. Although nodal compatibility is widely accepted, it can cause meshing
difficulties and undesirable element sizes and angles in certain modeling situations. To
utilize the nodal compatibility constraint it was necessary to locate common nodes
throughout a column that were coincident with neighboring element nodes. This technique
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resulted in five basic element groups as shown in Figure 4.1a‐e: the steel jacket shell, cover
and core concrete zones, and transverse and longitudinal reinforcement zones.
a.) b.) c.) d.) e.)
f.)
Figure 4.1: Model Parts: a.) Steel Jacket, b.) Cover Concrete, c.) Core Concrete, d.) Reinforcement, e.) Entire Model, f.) CrossSection
63
Using nodal compatibility for each model required that the element density around
the column perimeter, shown in Figure 4.1f, be chosen to accommodate locations of
longitudinal reinforcement throughout the height of the column. It was necessary to
specify a perimeter density that produced nodal locations at the correct longitudinal
reinforcement spacing. The density of elements across the diameter was then chosen to
formulate element aspect ratios as close to a 1:1 ratio as possible. This cross‐section
discretization level remains constant throughout the height of the column, thereby
permitting connection of every element via nodal compatibility as shown in Figure 4.2
where both solid bricks and beam elements share common nodes. This approach directly
coupled elements and avoided the use of surface‐to‐surface contact or other methods to tie
elements together that would increase computational time and could produce erroneous
results.
Figure 4.2: Nodal Compatibility Constraint
Element heights were chosen as a function of the transverse reinforcement spacing. Mesh
refinement occurred near the base and top of the column since these areas were where
Common nodes
64
inelastic response, large deformations and failure were anticipated to occur. Transition
techniques matched those from previous research that assessed the response of reinforced
concrete vehicle barriers to blast loads (Coughlin, 2008) and involved coarse to fine brick
element transitions from 101.6 mm to 25 mm (4” to 1”) .
The second technique used to constrain elements was the
*CONSTRAINED_LAGRANGE_IN_SOLID LS‐DYNA function. This constraint was used to
embed reinforcement beam elements in the concrete brick elements without the need of
coincident nodal locations. Evaluation of this constraint has been performed both in the
literature (Murray et al., 2007) and during model validation for this research and is shown
to produce acceptable results. This function requires the definition of a set of Lagrangian
beam “slave” parts that are constrained at their nodes through velocities and accelerations
to a defined set of “master” brick parts. As seen in Figure 4.3, this type of constraint is
useful when modeling spiral reinforced columns which would produce meshing difficulties
and elemental distortions if nodal compatibility was used.
Figure 4.3: *CONSTRAINED _LAGRANGE_IN_SOLID Constraint
Embeddednodes
65
The final type of element contact was the *CONTACT_NODES_TO_SURFACE LS‐DYAN
function. This type of constraint was used in initial blast validation models containing
foundations which were eventually simplified to a rigid surface rendering this contact type
obsolete for parametric models. However, this contact modeled the interaction between
the column base with the surface segments of the column’s foundation. This contact
requires the definition of a set of “slave” nodes that are not permitted to penetrate the set
of “master” segments in compression, but are allowed to pull away from the segments in
tension. This contact, shown in Figure 4.4, models the construction joint typically found at
the base of the column and assumes that the interface concrete has no capacity in tension
but has capacity in compression (Murray et al., 2007). This constraint also requires the
definition of a friction coefficient to further refine interaction between the column and
adjoining foundation surfaces. This column base interface contact conservatively resists
direct shear failure as it does not model concrete cohesion resistance but only models
friction resistance. Additionally, rebar was modeled between the column and footing at
this contact to provide tension capacity.
66
Figure 4.4: Column to Foundation Constraint
Although boundary conditions varied for the validation models, blast validation
models were explicitly used to evaluate boundary conditions planned for the parametric
models at the base. The boundary condition necessary for these models is similar to the
elemental constraint, *CONTACT_NODES_TO_SURFACE, as it models the construction joint
at the base of the column. To define the boundary a rigid surface was created at the base
and was assigned an appropriate coefficient of friction to define the interaction between
the adjoining column and footing surfaces. Next, a set of “slave” nodes was defined
corresponding to the concrete brick elements. These nodes were not permitted to
penetrate the rigid surface in compression but were allowed to pull away in tension, as
seen in Figure 4.5. This boundary condition is also conservative against resisting direct
shear failure as it does not model concrete cohesion resistance but only models friction
resistance. The nodes corresponding to the longitudinal beam elements were also
restrained in all translational and rotational directions representing a completely fixed
situation of the rebar penetrating into the footing. This boundary condition is a
Compression capacity Tension
pull away
67
simplification of the blast validation models containing foundations and was shown to
produce similar conservative results in the blast validation section.
Figure 4.5: Construction Joint Boundary Condition
Parametric models were also translationally restrained in the plane of the cross‐
section at the top of the columns representing the roller in a propped cantilever system as
discussed in Section 2.4.1. Here the longitudinal beam elements were also restrained in all
translational and rotational directions representing a completely fixed situation of the
rebar penetrating into the pier cap.
4.4 Material Models
The material model selected to represent the concrete brick elements was LS‐DYNA
normal weight concrete option Material Type 159 (*MAT_CSCM_CONCRETE). As
mentioned Chapter 2, this material has been shown to produce realistic results for concrete
subjected to impact and blast loads due to its viscoplastic strain rate effects and inclusion
of three invariant failure surface with integrated brittle and ductile damage regions
68
(Murray et al., 2007; Murray, 2007; Coughlin, 2008). The normal weight concrete option
requires input of the unconfined concrete compressive strength and maximum concrete
aggregate size to generate default material properties. These values were assumed 28 MPa
(4 ksi), and 19.0 mm ¾" respectively which were taken from the national DOT survey
conducted by Williamson et al. (2010).
Two additional defined material parameters for Material Type 159 were the Erode
and Recov items in the material parameter cards. Erode defines at what damage value and
maximum principle strain concrete elements will fail and, subsequently be deleted from the
problem solution. If the value is set to zero, the element will not erode. A value of unity
prescribes element erosion when that element reaches 99 percent damage based on the
model’s defined material softening function and is irrespective of principle strain. Previous
research recommends specifying erosion at 5 to 10 percent of the maximum principle
strain (1.05 ≤ ERODE ≤ 1.10) (Murray et al., 2007). A value of 10 percent maximum strain
was conservatively selected for this research. The Recov parameter defines the recovery
material modulus (stiffness) when switching between compression and tension within a
given element and attempts to model crack closing in concrete. If the value is not specified
the tensile modulus is completely recovered in compression. A value between 0 and 1
models partial recovery and the model remains at the brittle damage level if set to unity.
Research recommends that the value be set at 10 where modulus recovery is based on the
sign of both the pressure and the volumetric strain (Murray et al., 2007). A value of 10 was
specified for this research.
Beam elements used to represent the reinforcement were modeled using LS‐DYNA
Material Type 3 (*MAT_PLASTIC_KINEMATIC). This model utilizes a bilinear stress‐strain
69
curve representation, as shown in Figure 2.8, and was defined with a yield and ultimate
stress of 475 and 750 MPa (69 and 109 ksi) respectively (Malvar, 1998), a modulus of
elasticity of 200 GPa (29,200 ksi) (Malvar, 1998), and a tangent modulus of 2110 MPa (306
ksi) (Coughlin, 2008). Ultimate strain for eroding beam elements was defined as 12%
(Malvar, 1998). Strain rate effects for this model were defined with the Cowper Symonds
model which was fit to the DIF curves for steel reinforcement discussed in Chapter 2.
4.5 Summary
Finite elements representing the concrete, steel jacket, and reinforcement were
defined in LS‐DYNA as constant stress solids, and Hughes‐Liu shells and beams respectively
(LSTC, 2007). These elements are computationally efficient and robust enough to produce
the desired results for this research in a timely manner. Three contact methods were
utilized to combination of these elements forming steel‐jacketed bridge column validation
and parametric models. Contact types included: nodal compatibility, the
*CONSTRAINED_LAGRANGE_IN_SOLID LS‐DYNA function and the
*CONTACT_NODES_TO_SURFACE LS‐DYNA function. Boundary conditions at the column
bases were representative of the construction joint typically found in this base location. A
rigid surface was created at the base to represent the top surface of the footing. This
surface was defined with a coefficient of friction and resisted column nodes in compression
while allowing column nodes to pull away in tension. However, the longitudinal
reinforcement was completely fixed in all directions of translation and rotation to model its
development into the footing. Column boundary conditions at the top were representative
of the roller in a propped cantilever system. This corresponded to all nodes at the top of
70
the columns being restrained in all translational directions in the plane of the cross‐section.
Longitudinal reinforcement at the top of the column was also restrained in all translational
and rotational directions to model its development into the pier cap. Finally, LS‐DYNA
Material Type 159 and 3 were defined for concrete and steel reinforcement model
materials.
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Chapter 5 Model Validation
5.1 Overview
This Chapter discusses the model validation methodology that was used to
demonstrate that acceptable bridge column response could be obtained under various
loads using the LS‐DYNA models. Model validation was conducted statically and
dynamically. Static validation was conducted with two loading scenarios. The first static
scenario consisted of a concentric quasi‐static load validation and the second a combined
axial‐flexural quasi‐static load validation. The dynamic scenario focused on blast load
validation through comparison to experimental results obtained from MCEER Report 08‐
0028 (Fujikura & Bruneau, 2008). The purpose of both static and dynamic validation was
to incrementally subject the model to more complex loadings to systematically ensure
acceptable predictive capabilities prior to use in the parametric study.
5.2 Static Validation
5.2.1 Concentric Axial Quasi‐Static Load Validation
The purpose of this static load validation is to compare the model’s axial response
under concentric quasi‐static load at the top of the column to theoretically predicted
reinforced concrete column axial behavior. Concentric loading is defined by a purely axial
load with the resultant applied through the center of the column at the top, as shown in
Figure 5.1.
72
Figure 5.1: Concentric Load
Model accuracy was assessed by comparing theoretical force‐displacement relationships in
both the concrete and reinforcement to force‐displacement relationship from the model.
There are a variety of theoretical elastic stress‐strain relationships for normal
weight concrete; all of which provide somewhat similar results. These relationships differ
slightly with respect to assumed compressive concrete strength and strain as well as
ultimate strain and the shape of the softening portion of the curve. Two common stress‐
strain relationships are the Modified Hognestad Curve and the Todeschini Curve (Wight &
MacGregor, 2009). Both of these methods have been shown to produce comparable results
to experimental testing, although, the Todeschini Curve has the advantage of being defined
with a single formula (Wight & MacGregor, 2009) and was selected for comparison with FE
models in this research.
The theoretical Todeschini non‐linear stress‐strain curve represents concrete
softening, as seen in Figure 5.2, and is defined by a reduced concrete strength, " 0.9
occurring at a strain, 1.71 ⁄ .
73
Figure 5.2: Todeschini Curve (Wight & MacGregor, 2009)
These values define a stress‐strain curve, represented by the following formula:
2 " ⁄1 ⁄
(5.1)
with an ultimate concrete compressive strain, 0.003. The reduction in concrete
compressive strength accounts for the difference between cylinder and member strengths
resulting from different curing and placing techniques, which gives rise to different
concrete compressive strengths due to vertical migration of bleed‐water (Wight &
MacGregor, 2009). This concrete strength reduction becomes unnecessary when
comparing theory and modeling predictions since curing and placing techniques are not
considered in the material models. Therefore, " 28 4 for this scenario.
The theoretical steel stress‐strain curve for the reinforcement was developed from
the LS‐DYNA material model definition as discussed in Section 4.4. Figure 2.8 represents
74
this bilinear stress‐strain curve with 200 29,200
and 2110 306 ) (Malvar, 1998; Coughlin, 2008).
Using the theoretical models discussed above, both concrete and steel stress‐strain
curves were derived for strains, ε = 0 to εult = 0.003, which correspond to the ultimate
strain of concrete in compression. Theoretical strains were then converted to
corresponding theoretical deflections (Δ), which were assumed to act at the top of the
column, by using the following relationship:
Δ 3429 135" (5.2)
Individual theoretical concrete and steel forces were obtained by multiplying the stresses
obtained from their respective material models by their respective gross cross‐sectional
areas. Due to the interaction between the concrete and steel reinforcement and the
assumption of perfect bond between the two, the individual force‐displacement
relationships were superimposed and plotted creating the theoretical total force‐
displacement curve for the top of the column. This process of theoretical total force‐
displacement determination for the column is demonstrated in Figure 5.3. This figure
summaries the method to obtain the theoretical force‐displacement curve for the column.
Validated material models were chosen from literature as the theoretical bases for the
behavior of the concrete and steel components. Column detailing, such as number and size
of longitudinal reinforcement, was then inputted into the material models to obtain
theoretical stress‐strain relationships for a predisposed strain range of ε = 0 to εult = 0.003,
which corresponds to the ultimate strain capacity of concrete in compression. These
relationships were then converted to force‐displacement relationships. Concrete and steel
75
cross‐sectional areas were used to convert the theoretical stress to force, while Equation
(5.2) was used to convert theoretical strain to deflection. Finally, concrete and steel force‐
displacement relationships were superimposed to form a theoretical column force‐
displacement curve that was used for comparison to FE model results. This process
required only one iteration.
Figure 5.3: Theoretical ForceDisplacement Determination Process
The theoretical elastic ultimate deflection at the top of the column was obtained by
substituting εult = 0.003 into Equation 5.2. The resulting maximum theoretical
displacement of 10.29 mm (0.405”) was then used as the maximum prescribed top nodal
displacement in the quasi‐static FE model. Nodal displacements at the top of the column
model were prescribed rather than an ultimate load because this method allowed for a
non‐linear resultant load to be obtained that softens as yielding occurs throughout the
column, a behavioral aspect that is not possible in LS‐DYNA when the load itself is
prescribed. To ensure a quasi‐static loading, prescribed displacement was defined to reach
76
its maximum in 2 seconds to avoid wave propagation and reduce strain rate effects in the
material models. Resulting model force‐displacement curves obtained at the top of the
column are plotted against theoretical values in Figure 5.4 for the total column system and
the reinforcement.
Figure 5.4: ForceDisplacement Curve Comparison
The model curves show good comparison to the theoretical curves in the elastic
range to the maximum concrete compressive strength. After this point the total model
curve deviates from the theoretical curve and shows a decrease in total resultant force
capacity. This point of deviation occurs when the longitudinal reinforcement reaches its
yield strength and is most likely due to excessive deformation and yielding of the
longitudinal and the transverse reinforcement. Figure 5.5a shows normal stress contours
(MPa) in the vertical direction that show softening near the top of the column at the fully
prescribed displacement.
0 0.1 0.2 0.3 0.4
0
500
1000
1500
2000
2500
3000
3500
0
2000000
4000000
6000000
8000000
10000000
12000000
14000000
16000000
0 2 4 6 8 10
Displacement (in)
Axial Load (kips)
Axial Load (N)
Displacement (mm)
Theo. Concrete
Theo. Steel
Theo. Total
Model Steel
Model Total
77
a.) b.)
Figure 5.5: a.) Column normal vertical stress contours, b.) Reinforcing steel axial force contours
The reinforcing steel is shown in Figure 5.5b to contain areas of high axial tension in the
transverse reinforcement and areas of high axial compression in the longitudinal
reinforcement near the column top. It can be deduced that failure of the model column and
deviation of its force‐displacement curve from theoretical values was due to dilation of the
core concrete and subsequent yielding of transverse and vertical reinforcement near the
column’s softening zone at its top. Effects of transverse hoop yielding due to concrete
dilation was not considered in the formulation of the theoretical force‐displacement curve
and therefore showed deviation from the modeled results.
5.2.2 Combined Axial‐Flexural Quasi‐Static Load Validation
The purpose of the second static model validation step was to validate the model’s
combined axial and flexural prediction accuracy under a combination of vertical and
horizontal loads (see Figure 5.6). Validation occurred via the use of axial‐moment
78
interaction diagrams that plotted axial verses flexural capacity, as shown in Figure 5.7 for a
representative circular column. Interaction diagrams are typically derived for assumed
column cross‐sectional dimensions and details and are a function of concrete compressive
strength, column diameter, longitudinal reinforcement yield strength, and longitudinal
reinforcement ratio and spacing.
Figure 5.6: Combined Axial and Flexural Load
Figure 5.7: Typical AxialMoment Interaction Diagram for Circular Columns (Wight
& MacGregor, 2009)
FE model results were validated against both simplistic and advanced theoretical
methods to provide a range of acceptable agreement. The first was a simplified method
79
based on the Whitney stress block assumption, which replaces the nonlinear stress
distribution for concrete in compression using a rectangular stress distribution (Wight &
MacGregor, 2009). In addition, concrete in tension is assumed to have no contribution to
the column’s capacity. The second more advanced theoretical method was based on the
modified compression field theory developed by Vecchio and Collins (1988). This method
discretizes a given reinforced concrete section into concrete layers and longitudinal
reinforcement elements depending on their respective properties (see Figure 5.8) and
assumed concrete and steel material models (see Figure 5.9) in an attempt to more
accurately reflect the nonlinear behavior.
Figure 5.8: Modified Compression Field Theory Layered Model (Vecchio & Collins, 1988)
80
Figure 5.9: Modified Compression Field Theory Concrete and Steel Material Models (Vecchio & Collins, 1988)
Through equilibrium and compatibility of the individual sections and elements, this theory
can accurately predicted response of reinforced concrete members subjected to shear,
moment and axial load.
To create the Whitney stress block theoretical interaction diagram, the strain
distribution across the cross‐section of the column was assumed to be linear with a
concrete ultimate strain εcu = 0.003. An iterative process was then used to create the
interaction diagram. The iterations were controlled by prescribing an extreme tension
fiber strain and calculating the strains in the remaining bars and the concrete in
compression. From these strains, stress and force values were then computed and
converted to nominal axial and moment effects using internal equilibrium. As the derived
points of the theoretical interaction diagram reached the purely axial tension point, the
iteration control was switched to incrementing the neutral axis depth. This control switch,
seen in Figure 5.10, requires less iteration to be completed as curve generation terminates.
81
Figure 5.10: Theoretical AxialMoment Diagram
Response 2000 software developed by Evans and Collins (1998) was used to
develop the axial‐moment diagram based on modified compression field theory. As
discussed above, the program requires the user to input concrete and reinforcement
properties, as well as, overall concrete member characteristics and external loading. A
457.2 mm (18”) diameter circular column with a height of 3429 mm (135”) was used for
validation. The column contained six #6 longitudinal bars and #4 bars for transverse
reinforcement at 152.4 mm (6”). The concrete cover was 12.7 mm (0.5”).
In the LS‐DYNA model seven data points, shown in Figure 5.11, were collected from
model results to define its interaction diagram.
82
Figure 5.11: Model Data Points
These points defined separate loading conditions for the model and consisted of five
significant points on the curve: the purely axial condition (A); the “balanced” condition (E),
which relates to the loading condition where the extreme steel tension fiber and the
concrete reach yield simultaneously; the purely flexural condition (F); and purely axial
tension condition (G). Three intermediate points were also evaluated to better define the
curve and corresponded to axial forces equal to 4.45 x 103 kN (1000 kips) (B), 3.56 x 103
kN (800 kips) (C), and 2.67 x 103 kN (600 kips) (D).
In general, to simulate static effects, quasi‐static axial and lateral loads were applied
to the column using the implicit nonlinear LS‐DYNA time dependent solver. These loads
were slowly applied over a thirty second period to avoid wave propagation and reduce
strain rate effects in the model and simulated a static loading situation. Two stages of
loading were also used in the analysis to simulate the static application of axial and lateral
loads. The first loading stage consisted of application of the axial load, which was applied
monotonically to its maximum value over a nine second period and remained constant
Pn
A B C
D
E
G
F Mn
83
until analysis termination at thirty seconds. A period of one second was then used to allow
LS‐DYNA to solve for the effects of maximum axial load without considering the next
loading stage. Next, a lateral load was also applied at mid‐height of the column at ten
seconds into the analysis, and increased monotonically to its maximum value at twenty‐five
seconds where it remained constant until analysis termination. Conditions that did not
require combined vertical and axial loads, such as the purely axial condition, the purely
tension condition, and the purely flexural condition, were applied to the column in a single
monotonic stage over a twenty‐five second period where they reached their maximum
value and then remained constant until analysis termination.
The vertical and lateral loads were applied in this manner to produce a critical
cross‐sectional moment capacity from the application of the lateral load at a given axial
load. Boundary conditions for the column being used for validation were a propped
cantilevered column, and, as a result, the critical cross‐section was at the base. Resulting
capacity due to the combined axial and bending effects was defined as the cross‐sectional
moment, at the prescribed axial load, that produced either the ultimate strain in the
concrete (assumed to be εcu = 0.003) or yield at the extreme tensile steel (assumed to be εt
= 0.00207).
Results for models including and excluding transverse reinforcement were
developed to obtain axial‐moment curves for comparison to theoretical values. Theoretical
results were obtained using both the Whitney and modified compression field approaches.
As shown in Figure 5.12, initial model results, which included transverse reinforcement,
indicated an increase in predicted column capacity in the compression controlled regions
of the interaction diagram.
84
Figure 5.12: AxialMoment Interaction Comparison
This increase was attributed to an increase in concrete confinement produced by the
reinforcement which is not considered in either theoretical method. Therefore, to more
closely correlate the results of the model to theoretical values, transverse reinforcement
was excluded to eliminate its effects on concrete confinement and capacity.
Results showed good correlation between model and modified compression field
theory values after elimination of the transverse reinforcement, as shown in Figure 5.12.
Due to the conservative assumptions built into the Whitney stress block method these
values where shown to be lower than both the LS‐DYNA model values and the modified
compression field theory values in the compression controlled region of the interaction
diagram. However, in the tension controlled region of the diagram, the Whitney stress
block values more closely compared to the LS‐DYNA model values. This was due to the
method by which the FE model results were obtained which correlated better with the
0.0E+00 1.0E+08 2.0E+08 3.0E+08
‐1.9E+06
‐8.8E+05
1.2E+05
1.1E+06
2.1E+06
3.1E+06
4.1E+06
5.1E+06
6.1E+06
7.1E+06
‐400
‐200
0
200
400
600
800
1000
1200
1400
1600
0 500 1000 1500 2000 2500 3000
Moment(Nmm)
Axial Force (N)
Axial Force (kips)
Moment (kin)
Whitney Stress Block Theory
Modified Compression Field Theory
LS‐DYNA Model with Hoops
LS‐DYNA Model without Hoops
85
manner of tension capacity prediction associated with the Whitney stress block method.
This method assumes the capacity force of the column in tension is purely contributed by
the resisting summation of longitudinal bar forces at first yield. This is similar to the FE
model where the tension capacity was obtained in the column at the point in the analysis
when all longitudinal bars had reached yield.
5.3 Dynamic Blast Validation
The blast load validation component was used to assess realistic steel‐jacketed
column response to blast loads. To accomplish this objective a FE model was constructed
with the same attributes as ¼ scale steel‐jacketed bridge column specimen SJ2 from
MCEER Technical Report 08‐0028 (Fujikura & Bruneau, 2008). This specific column was
chosen because it demonstrated the most visual damage for the tested circular steel‐
jacketed columns and therefore allowed for more robust comparison.
The column had a length of 1499 mm (59”) and a diameter of 213 mm (8.39”). The
steel jacket was 1.2 mm (0.05”) thick and was terminated 13 mm (0.5”) before both the
footing and pier cap ends of the column, which corresponded to a full scale 2” steel jacket
gap. Longitudinal reinforcement consisted of sixteen D3 bars, while transverse
reinforcement consisted of a spiraled D1 bar with a pitch of 41 mm (1.625”). Concrete
cover was 13 mm (0.5”), which also corresponded to a full scale 2” concrete cover. D1 and
D3 reinforcing bars correspond to scaled‐down version of #4 and #6 bars respectfully.
Further details of the bar properties are found in MCEER Report 08‐28 (Fujikura &
Bruneau, 2008). Similar blast loads to those used for the tests were applied to the FE
model. After the actual test, it was stated that the column incurred a permanent lateral
86
deflection of 78 mm (3.0625”) at its base. It exhibited a brittle direct shear failure at the
base with all longitudinal bars fracturing. A 6 mm gap at the top of the columns was also
observed along with crushing of the concrete inside the steel jacket and steel jacket bulging
at the base (Fujikura & Bruneau, 2008). These items were used for comparison to the finite
element models (Figure 5.13).
Figure 5.13: MCEER Report 080028 Column SJ2 Blast Damage (Fujikura & Bruneau, 2008)
An LS‐DYNA model of column SJ2 was created following Chapter 4 modeling
techniques (Figure 5.14).
87
Figure 5.14: Initial Blast Validation Model
Boundary conditions were modeled a propped cantilever as mentioned in Sections 2.4.1.
All nodes at the base were restrained in all translational and rotational directions. All
nodes at the top were restrained in translational directions in the plan of the column cross‐
section only.
When compared against qualitative information from the actual test, these
boundary conditions were shown to overestimate the strength of the column at its base.
Initial models results were not able to recreate fracture of the longitudinal bars at the base
as shown in Figure 5.15.
88
Figure 5.15: Initial Blast Validation Damage
Boundary conditions were revised to include a tributary volume of foundation at the
column base and a rigid surface at the column top representing the pier cap. At the base
the boundary conditions were modified to explicitly model the construction joint in this
location that is typically created from casting the footing and column at different times
(Figure 5.16).
Figure 5.16: Modified Blast Validation Model
89
As discussed in Section 4.3 the *CONTACT_NODES_TO_SURFACE LS‐DYAN function was
used to model contact between the column and footing surfaces. Longitudinal
reinforcement was completely fixed at the base in all translational and rotational directions
to model its development into the footing. Boundary conditions at the top were modeled
similarly and allowed nodes to lift off the top surface in tension while bearing normally in
compression, which again mimicked a construction joint. Similar to base conditions,
longitudinal reinforcement at the top was also completely fixed in all translational and
rotational directions to model its development into the pier cap.
Both end constraints required the definition of the coefficient of friction to further
refine the interaction between the column and its adjoining surfaces. To obtain the proper
coefficient of friction, model iterations were performed that varied the interface coefficient
of friction until experimental column base deflections were reproduced in the model
results. Figure 5.17 shows the final model deflection at the base was 80.8 mm with μ = 0.3.
The model results show complete fracture of the longitudinal bars and a bulge in the steel
jacket, just as observed in Figure 5.18, which shows experimental damage and a base
deflection of 78 mm.
90
Figure 5.17: Blast Validation Model with Foundation Base Deflection Graph
Figure 5.18: Experimental Base Damage (Fujikura & Bruneau, 2008)
‐83.59 ‐80.78‐4
‐3
‐3
‐2
‐2
‐1
‐1
0
‐90
‐80
‐70
‐60
‐50
‐40
‐30
‐20
‐10
0
0.E+00 2.E‐02 4.E‐02 6.E‐02 8.E‐02 1.E‐01 1.E‐01 1.E‐01 2.E‐01
Deflection (in)
Deflection (mm)
Time (s)
Blast Validation Model with Foundation Base Deflections
91
Figure 5.17 shows a dynamic deflection of 83.59 mm (3.29”) and a static deflection at the
termination of the analysis of 80.78 mm (3.18”) which was deemed an acceptable
correlation to experimental base deflection results.
Similarly, Figure 5.19 shows a dynamic crack opening width of 7.4 mm (0.29”) and a
static crack opening of 5 mm (0.20”) at the end of the analysis for the top of the modeled
column
Figure 5.19: Blast Validation Model with Foundation Pier Cap Gap Opening Graph
This was deemed an acceptable correlation to the experimental crack opening of 6 mm
(0.24”) shown in Figure 5.20 .
‐7.36
‐5.02
‐0.315
‐0.265
‐0.215
‐0.165
‐0.115
‐0.065
‐0.015
‐8
‐7
‐6
‐5
‐4
‐3
‐2
‐1
0
0.E+00 2.E‐02 4.E‐02 6.E‐02 8.E‐02 1.E‐01 1.E‐01 1.E‐01 2.E‐01
Deflection (in)
Deflection (mm)
Time (s)
Blast Validation Model w/ Foundation Crack Opening Widths at Pier Cap
92
Figure 5.20: Experimental Bent Damage (Fujikura & Bruneau, 2008)
To further improve the results and reduce computational time the next step in blast
model validation was to replace the foundation with a similar rigid surface as that used at
the top end of the column as shown in Figure 5.21.
Figure 5.21: Final Blast Validation Model
93
For this model iteration, Figure 5.22 and Figure 5.23 compare the base deflection
and pier cap crack opening results, respectively, between the model that included the
foundation, the model that included a rigid surface substitution and the experimental
result.
Figure 5.22: Base Deflection Comparison
‐86.78 ‐86.60
‐3.94
‐3.44
‐2.94
‐2.44
‐1.94
‐1.44
‐0.94
‐0.44
‐100
‐90
‐80
‐70
‐60
‐50
‐40
‐30
‐20
‐10
0
0.E+00 5.E‐02 1.E‐01 2.E‐01Deflection (in)
Deflection (mm)
Time (s)
Rigid Surface ModelExperimental ResultFoundation Model
94
Figure 5.23: Pier Cap Gap Opening Comparison
Although the static deflection values for both the base deflection and the crack opening
width increased slightly they were deemed an efficient conservative correlation to both the
experimental results and the models including a modeled foundation.
Since blast validation models containing the rigid surface at the base of the column
showed acceptable correlation to experimental results, the modeling techniques used to
model the construction joint boundary conditions were utilized in all parametric models.
The base rigid surface boundary condition was defined with a coefficient of friction of μ =
0.3 and with all longitudinal reinforcement completely fixed to properly model the
construction joint.
‐6.39‐6.14
‐0.31
‐0.26
‐0.21
‐0.16
‐0.11
‐0.06
‐0.01
‐8
‐7
‐6
‐5
‐4
‐3
‐2
‐1
0
0.E+00 5.E‐02 1.E‐01 2.E‐01
Deflection (in)
Deflection (mm)
Time (s)
Rigid Surface ModelExperimental ResultFoundation Model
95
5.4 Summary
Both static and dynamic validation was completed for the LS‐DYNA models. Static
validation consisted of two loading scenarios. The first was a concentric axial load and the
second was a combined axial flexural load. Force‐displacement curves obtained from the
models were compared to theoretical curves for the concentric axial loading scenario.
Model results showed acceptable correlation to the theoretical values with slight
differences observed in the later portions of the curve due to yielding of the transverse
reinforcement in the column models that was not accounted for in theoretical
determination. Theoretical axial‐moment diagrams were used to compare models results
for the second static loading scenario. Once again, FE models showed acceptable
correlation to the theoretical values. However, it was necessary to exclude the transverse
reinforcement in the FE models to decrease the capacity to more closely resemble
theoretical values. This was most likely due to the increase in confinement, and therefore
capacity, associated with the in inclusion of transverse reinforcement. Finally, dynamic
blast validation was completed by comparing FE model results to experimental blast data
from MCEER Report 08‐28 (Fujikura & Bruneau, 2008). FE models showed acceptable
correlation to experimental base deflection and pier cap crack opening width results. They
were also able to replicate the direct shear failure of the columns at their base which was
observed in the experiments and was accompanied by complete fracture of the longitudinal
bars. This blast validation calibrated the construction joint boundary conditions for the FE
columns resulting in a rigid surface at the column base representing the top surface of the
footing which was defined with a coefficient of friction of µ = 0.3. Longitudinal
96
reinforcement was modeled as fixed for this boundary condition modeling its development
into the footing.
97
Chapter 6 Parametric Study Results
6.1 Overview
This chapter discusses results used to assess the effects of the variation of specific
design parameters on the behavior and performance of steel jacketed bridge pier columns
under blast loads. Failure modes that focused on shear and flexural behavior were used to
assess jacketed column performance. The failure modes and the methods by which column
performance was measured in association with each included:
• Direct Shear
o Longitudinal reinforcement and steel jacket axial cross‐section strain profiles
at 10% column height intervals;
o Normalized direct shear plots at the column base;
• Flexure
o Moment and rotation diagrams throughout the height of the columns;
o Normalized moment‐rotation plots at critical locations and times; and
• Transverse Shear
o Transverse reinforcement and steel jacket strain profiles throughout the
height of the column.
The following sections include detailed descriptions of each failure mode and subsequent
performance of the jacketed columns in association with that failure mode over the range
of parameters that were examined.
6.2 Direct Shear Results
98
As mentioned in Section 2.5.1, direct shear at the column base is the most common
mode of failure for steel‐jacketed bridge columns. Friction and cohesion forces across the
construction joint interface have the biggest influence on a column’s resistance to this
failure mode. Conservatively this research ignores the resistance of concrete cohesion, as
suggested ACI 318‐08 (2008), and only relies on the friction force at the column base
construction joint interface and the longitudinal reinforcement and steel jacket doweling
forces which cross the interface. Both resisting force effects on parameter variation were
monitored in this research. The doweling resisting force was examined through axial
cross‐sectional strain profiles for the longitudinal reinforcement and the steel jacket. The
interface resisting friction force was examined through normalized direct shear values
taken from the base cross‐section.
Axial cross‐section strain profiles for the longitudinal reinforcement and the steel
jacket were used to assess the column’s resistance to direct shear. These profiles were
evaluated at intervals of 0.1h, where h = column height, throughout the height of the
column starting at the base. This produced eleven shear planes throughout the height of a
given column for shear capacity examination (Figure 6.1).
99
Figure 6.1: Crosssection Locations
In addition to examining axial cross‐section strain profiles at a number of sections
along the column height , critical cross‐sectional direct shear resisting forces at the base of
the columns were obtained from each model with LS‐DYNA’s
*DATABASE_CROSS_SECTION_PLANE function. This function was defined by a circular
plane at the base and allows forces and moments to be calculated from any element types
that cross the defined plane. The direct shear resisting force corresponds to a resultant
cross‐sectional force that is parallel to and opposing the blast wave, as shown in Figure 6.2.
Resulting total direct shear capacity values were then normalized for each model with
respect to an ultimate static capacity obtained from a separate, quasi‐static analysis. The
quasi‐static capacity analysis provided ultimate values of direct shear resisting force in the
base cross‐section for each model. These capacity values were used to normalize the
Base
0.1h
0.2h
0.3h
0.4h
0.5h
0.6h
0.7h
0.8h
0.9h
Top
100
corresponding blast model direct shear resistance results and permitted the evaluation of
parameter variation on direct shear resistance.
Figure 6.2: Direct Shear Resistance
All models were shown to experience the highest direct shear demand throughout
the analysis at a time of 7.5 ms after evaluation of all axial strains and direct shear time
histories. Figure 6.3 shows that 7.5 ms was the critical time step by plotting a
representative direct shear time history for Model 8 at the column base.
Figure 6.3: 8_a0v1j1g1 Direct Shear at Base
7.50E03, ‐8.21E+06 ‐2.E+03
‐2.E+03
‐1.E+03
‐7.E+02
‐2.E+02
3.E+02
8.E+02
1.E+03
‐1.E+07
‐8.E+06
‐6.E+06
‐4.E+06
‐2.E+06
0.E+00
2.E+06
4.E+06
6.E+06
0.00E+00 5.00E‐02 1.00E‐01 1.50E‐01 2.00E‐01 2.50E‐01
Base Direct Shear Force (kips)
Base Direct Shear Force (N)
Time (sec)
101
Also, all even numbered parametric model steel jacket strain values are not taken
directly at the bottom of the column due to the jacket gap in this region (see Table 3.4).
Steel jacket strains for these models are reported at the first row of steel jacket shell
elements, located 50.8 mm (2”) from the base of the column as shown in Figure 6.4.
Figure 6.4: Reported Base Steel Jacket Strains
This technique was used for all parametric models that had a gap, which corresponds to the
models having the maximum jacket gap ratio values in Table 3.4.
6.2.1 Axial Cross‐section Strain Profiles
Axial cross‐section strain profiles where obtained from the parametric models by
separately reporting either an average axial strain at a distance from the column’s centroid
corresponding to a specific pair of longitudinal reinforcement bar’s or the steel jacket
elements in the same cross‐sectional location. They were used to demonstrate the dowel
action in the cross‐section and evaluations occurred at each of the eleven direct shear
planes along the column height. Figure 6.5 demonstrates this technique by showing that
102
the two longitudinal bar axial strain values at section A‐A are averaged and plotted at the
distance on the y‐axis corresponding to section A‐A. Likewise two steel jacket shell
element axial strain values at section A‐A are averaged and plotted separately at the
distance on the y‐axis corresponding to section A‐A. This was done for all levels of
longitudinal reinforcement and steel jacket elements through a single cross‐section.
Figure 6.5: Crosssectional Strain Profile Definition
Representative axial cross‐section strain profiles obtained from parametric Model
8_a0v1j1g1 (see Table 3.4), which is characterized with all maximum parameter limits
Blast side
Average axial strain in steel jacket at section A‐A
Compression strain Tension strain
‐ Distance from centroid
Average axial strain in long. reinf. at section A‐A
+ Distance from centroid
A A
103
except aspect ratio, are shown Figure 6.6. Note the plot indicates axial strain levels at each
of the eleven evaluated direct shear planes along the height of the column for both
longitudinal reinforcement and steel jackets. This column experienced the some of the
highest tensile axial strains in the longitudinal reinforcement, and therefore, high direct
shear demand.
104
Figure 6.6: 8_a0v1j1g1 Axial Crosssection Strain Profiles
‐30.00
‐20.00
‐10.00
0.00
10.00
20.00
30.00
‐762
‐562
‐362
‐162
38
238
438
638
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
Base Bar
Base Jacket
0.1h Bars
0.1h Jacket
0.2h Bars
0.2h Jacket
0.3h Bars
0.3h Jacket
0.4h Bars
0.4h Jacket
0.5h Bars
0.5h Jacket
0.6h Bars
0.6h Jacket
0.7h Bars
0.7h Jacket
0.8h Bars
0.8h Jacket
0.9h Bars
0.9h Jacket
Top BarsBack Side
Blast Side
Base Jacket
Base Bars
Top Bars
105
This plot indicates that the column base was the critical cross‐section for direct
shear. All longitudinal reinforcement at the base was acting in tension, which
demonstrates dowel action across the shear plane. Remaining cross‐sections along the
column (0.1h – Top) had low strain profiles and included tension and compression in the
reinforcement and steel jackets, indicative of bending rather than direct shear behavior
Longitudinal reinforcement at the top of the column also experienced only tension
throughout the cross‐section due to its fixed boundary condition in this location. Note that
the steel jacket experienced virtually no strain at the base since this particular column had
a 50 mm (2”) jacket gap at the base.
Figure 6.7 shows representative axial cross‐section strain profiles obtained from
parametric Model 3_a0v0j1g0 (see Table 3.4), which is characterized with only a maximum
jacket thickness ratio parameter limit and all other at minimum values. Note the plot
indicates axial strain levels at each of the eleven evaluated direct shear planes along the
height of the column for both longitudinal reinforcement and steel jackets. This column
experienced the some of the lowest tensile axial strains in the longitudinal reinforcement,
and therefore, low direct shear demand.
106
Figure 6.7: 3_a0v0j1g0 Axial Crosssection Strain Profiles
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
Base Bar
Base Jacket
0.1h Bars
0.1h Jacket
0.2h Bars
0.2h Jacket
0.3h Bars
0.3h Jacket
0.4h Bars
0.4h Jacket
0.5h Bars
0.5h Jacket
0.6h Bars
0.6h Jacket
0.7h Bars
0.7h Jacket
0.8h Bars
0.8h Jacket
0.9h Bars
0.9h Jacket
Top Bars
Top JacketBack Side
Blast Side
Base Jacket
Base Bars
Top Bars
107
This plot also indicates that the column base was the critical cross‐section for direct
shear, although Model 3 experienced much lower longitudinal reinforcement strain values
compared to Model 8. All longitudinal reinforcement at the base was acting in tension,
which demonstrates dowel action across the shear plane. Remaining cross‐sections along
the column (0.1h – Top) had low strain profiles and included tension and compression in
the reinforcement and steel jackets, indicative of bending rather than direct shear behavior
Longitudinal reinforcement at the top of the column also experienced only tension
throughout the cross‐section due to its fixed boundary condition in this location. Note that
for this model the steel jacket experienced a linearly varying strain profile encompassing
tension and compression indicative of bending in the base cross‐section due to its fixity in
this location. This is reduction in longitudinal reinforcement axial strain and shift to
flexural resistance in the steel jacket is advantageous in resisting direct shear and
encouraging a more ductile flexural failure. Remaining parametric column axial cross‐
section strain profiles are located in Appendix A.
Figure 6.8 and Figure 6.9 plot critical cross‐sectional longitudinal reinforcement and
steel jacket strain profiles respectively at the base of the column for all parametric models.
Figure 6.8 shows that for every model the longitudinal reinforcement is in complete
tension throughout the base cross‐section indicative of doweling action associated with
direct shear. Models 6 and 8 (see Table 3.4), which are characterized as stub columns
having a steel jacket gap, are shown to have had the highest longitudinal reinforcement
strain values, while Models 3, 11 and 15, which are mostly characterized by being slender
and not having a gap, had some of the lowest strain values. Figure 6.9 shows that the odd
numbered models, which do not have steel jacket gaps (see Table 3.4), had almost linear
108
strain profiles encompassing both tension and compression which is indicative of steel
jacket bending in the base cross‐section. Insignificant axial cross‐section strains were
observed in the even numbered models, which had steel jacket gaps at the base (see Table
3.4), because the steel jacket had not yet developed its capacity at this first reported row of
elements as indicated in Figure 6.4. Steel jacket strain values further up the columns also
indicated low values for all models as demonstrated in Figure 6.6. The large tensile strains,
for these models, observed in some of the steel jackets on the blast side were attributed to
local damage effects due to the fact that this was the location closest to the blast.
109
Figure 6.8: Longitudinal Reinforcement Axial Strain Comparison at Base
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.005 0.01 0.015 0.02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
Blast Side
Back Side
110
Figure 6.9: Steel Jacket Axial Strain Comparison at Base
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐0.015 ‐0.01 ‐0.005 0 0.005 0.01 0.015
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
Blast Side
Back Side
111
6.2.2 Normalized Direct Shear
Direct shear (V) at the base of the column in the parametric models was normalized
against a capacity value (Vult) obtained from separate quasi‐static analyses on equivalent
columns to evaluate the level of base direct shear in the blast loaded columns and offer a
means of parametric model comparison. The quasi‐static capacity analysis models match
corresponding parametric model columns with respect to boundary conditions and applied
axial loads, however, they were subjected to a linearly increasing quasi‐static lateral load
concentrated at the base. This load was increased until direct shear failure was observed
(see Figure 6.10).
Figure 6.10: Direct Shear Capacity Model Load Definition
A graph of the direct shear time histories for both blast and quasi‐static capacity
models is shown in Figure 6.11 for Model 8 (see Table 3.4), which is identified in Figure 6.8
as exhibiting some of the highest longitudinal reinforcement axial cross‐section strain.
Remaining parametric model direct shear time history comparisons are located in
Appendix B. These graphs demonstrate the process of acquiring both the maximum direct
112
shear force (V) obtained from the blast models and the ultimate capacity value (Vult)
obtained from the quasi‐static analysis models to formulate the normalized direct shear
values of V/Vult for each parametric model.
Figure 6.11: 8_a0v1j1g1 Direct Shear Comparison at Base
A compilation of all parametric model normalized values is shown in Figure 6.12.
This graph compares the values of normalized direct shear obtained from the process
described above for the sixteen parametric models. Parametric columns with lower
normalized direct shear values showed better resistance to this failure mode. This graph
agrees with the observations from the axial cross‐section strain profiles where Models 3,
11 and 15 showed good resistance to direct shear, while Models 6 and 8 showed poor
resistance. Overall observations showed that the even numbered models, which have a
steel jacket gaps (see Table 3.4), achieved higher values of normalized direct shear than the
‐8.21E+06
8.02E+06
‐2.2E+03
‐1.7E+03
‐1.2E+03
‐7.5E+02
‐2.5E+02
2.5E+02
7.5E+02
1.3E+03
1.8E+03
‐1.E+07
‐8.E+06
‐6.E+06
‐4.E+06
‐2.E+06
0.E+00
2.E+06
4.E+06
6.E+06
8.E+06
0.00E+00 5.00E‐02 1.00E‐01 1.50E‐01 2.00E‐01 2.50E‐01
Base Direct Shear Force (kips)
Base Direct Shear Force (N)
Time (sec)
Blast
Capacity
113
models without gaps. Figure 6.12 also shows that some columns achieved values higher
than unity meaning that the blast models, in these cases, reached larger values of direct
shear than the quasi‐static capacity models achieved.
Figure 6.12: Normalized Direct Shear Comparison
This behavior can likely be attributed to the increase in column base friction and doweling
capacities at higher loading rates associated with blast. The effects of higher strain rates on
both concrete and steel capacities were discussed in Sections 2.4.3 and 4.4. Although direct
shear failure, which is defined as complete fracture of the longitudinal bars at the base
cross‐section, was not observed in any of the models, normalized direct shear values for
columns having a steel jacket gap (see Table 3.4) suggest that the columns were quite close
to failure.
6.3 Parameter Variation Effects on Direct Shear
0.000
0.200
0.400
0.600
0.800
1.000
1.200
V/V
ult
Parametric Model
114
This section discusses the sensitivity of direct shear performance to the variation in
parameters presented in Chapter 3. Performance sensitivity is again evaluated using axial
cross‐section strain profiles and normalized direct shear plots for the base cross‐section
which clearly shows the influences of blast resistance to changes in important parameters
identified in Section 3.4. Those parameters are listed again here for clarity:
• Column aspect ratio (L/D);
• Transverse reinforcement ratio (ρS);
• Steel jacket diameter‐to‐thickness ratio (tj/D); and
• Steel jacket gap distance‐to‐diameter ratio (Lg/D).
Each plot contains a select few representative model results. Each result is paired with
their corresponding parametrically similar model with the only difference in models being
the examined parameter minimum and maximum values. To help facilitate this
comparison paired results are plotted with the same color. The darker color shade
represents the maximum limit of the examined parameter, while the lighter color shade
represents the minimum limit.
6.3.1 Aspect Ratio Effects on Axial Cross‐Section Strain
Figure 6.13 demonstrates that increase in aspect ratio produced a decrease in
longitudinal reinforcement axial cross‐section strain.
115
Figure 6.13: Aspect Ratio Effects on Longitudinal Bar Axial CrossSection Strain Profiles at Base
Overall reductions in longitudinal bar strains signifies reduced dowel action at the critical
base section, and therefore reduced direct shear demand, which is advantageous for blast
resistance.
Figure 6.14 demonstrates that increase in aspect ratio produced little effect in the
axial cross‐section strain values for the steel jackets.
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.005 0.01 0.015 0.02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
3_a0v0j1g0
11_a1v0j1g0
5_a0v1j0g0
13_a1v1j0g0
8_a0v1j1g1
16_a1v1j1g1
Blast Side
Back Side
116
Figure 6.14: Aspect Ratio Effects on Steel Jacket Axial CrossSection Strain Profiles at Base
6.3.2 Transverse Reinforcement Ratio Effects on Axial Cross‐Section Strain
Figure 6.15 demonstrates that decrease in transverse reinforcement ratio produced
a decrease in longitudinal reinforcement axial cross‐section strain.
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐0.015 ‐0.01 ‐0.005 0 0.005 0.01 0.015
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
3_a0v0j1g0
11_a1v0j1g0
5_a0v1j0g0
13_a1v1j0g0
8_a0v1j1g1
16_a1v1j1g1
Blast Side
Back Side
117
Figure 6.15: Transverse Reinforcement Ratio Effects on Longitudinal Bar Axial CrossSection Strain Profiles at Base
As the amount of transverse reinforcement decreased the lateral stiffness also decreased
reducing the direct shear demand at the base, which is advantageous to blast resistance.
Figure 6.16 demonstrates that decrease in transverse reinforcement ratio produced
an insignificant effect in the cross‐sectional strain values for the steel jackets.
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.005 0.01 0.015 0.02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
3_a0v0j1g0
7_a0v1j1g0
9_a1v0j0g0
13_a1v1j0g0
4_a0v0j1g1
8_a0v1j1g1
Blast Side
Back Side
118
Figure 6.16: Transverse Reinforcement Ratio Effects on Steel Jacket Axial CrossSection Strain Profiles at Base
6.3.3 Jacket Thickness Ratio Effects on Axial Cross‐Section Strain
Figure 6.17 demonstrates that increase in jacket thickness ratio produced a
decrease in longitudinal reinforcement axial cross‐section strain for models without a steel
jacket gap.
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐0.015 ‐0.01 ‐0.005 0 0.005 0.01 0.015
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
3_a0v0j1g0
7_a0v1j1g0
9_a1v0j0g0
13_a1v1j0g0
4_a0v0j1g1
8_a0v1j1g1
Blast Side
Back Side
119
Figure 6.17: Jacket Thickness Ratio Effects on Long Bar CrossSectional Strain Profiles for Columns without Gap
This observed reduction in longitudinal reinforcement axial cross‐section strain for
columns without a gap as jacket thickness increases is due to the shift in resistance from
direct shear to flexure. Flexure results discussed in Sections 6.5.3 and 6.5.7 support this
argument by demonstrating increase in jacket thickness produced increase in the flexural
resistance at the base of the columns without gaps, thereby producing the decrease in
longitudinal bar strain shown here. This is advantageous to blast resistance.
However, models containing a gap showed different behavior (Figure 6.18). These
models showed an increase in strain on the side of the section facing the blast and a
decrease on the side away from the blast.
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.002 0.004 0.006 0.008 0.01 0.012
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
3_a0v0j1g0
5_a0v1j0g0
7_a0v1j1g0
9_a1v0j0g0
11_a1v0j1g0
13_a1v1j0g0
15_a1v1j1g0
Blast Side
Back Side
Blast Side
Back Side
120
Figure 6.18: Jacket Thickness Ratio Effects on Long Bar CrossSectional Strain Profiles for Columns with Gap
Just as the effects of transverse reinforcement ratio discussed in the previous section,
increase in jacket thickness for columns with a gap also increases lateral stiffness, which
caused greater direct shear demand in the critical base cross‐section on the blast side. The
reasons for the shift to a decreasing strain on the back side of the cross‐section are
inconclusive. However, considering the lateral stiffness reasoning and the results from
Section 6.3.2, it seems that decrease in jacket thickness ratio assists is resisting direct shear
in columns with a gap, and is advantageous in blast resistance.
Figure 6.19 demonstrates that increase in jacket thickness ratio produced decrease
in the axial cross‐section strain values for the steel jackets.
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.005 0.01 0.015 0.02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
2_a0v0j0g1
4_a0v0j1g1
6_a0v1j0g1
8_a0v1j1g1
10_a1v0j0g1
12_a1v0j1g1
14_a1v1j0g1
16_a1v1j1g1
Blast Side
Back Side
Blast Side
Back Side
121
Figure 6.19: Jacket Thickness Ratio Effects on Steel Jacket CrossSectional Strain Profiles
This is due to the increase in steel jacket cross‐sectional area, which reduces strains in the
material at equivalent loading.
6.3.4 Jacket Gap Ratio Effects on Axial Cross‐Section Strain
Figure 6.20 demonstrates that decrease in jacket gap ratio produced a decrease in
longitudinal reinforcement axial cross‐section strain values.
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐0.015 ‐0.01 ‐0.005 0 0.005 0.01 0.015
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
3_a0v0j1g0
5_a0v1j0g0
7_a0v1j1g0
Blast Side
Back Side
122
Figure 6.20: Jacket Gap Ratio Effects on Long Bar CrossSectional Strain Profiles
Just as in Section 6.3.3 which discusses the advantages of increasing jacket thickness for
columns without a gap, the reductions in longitudinal reinforcement axial cross‐section
strain for columns having low jacket gap ratios are due to the shift in resistance from direct
shear to flexure. Flexure results discussed in Sections 6.5.4 and 6.5.8 support this
argument by demonstrating decrease in jacket gap ratio produced increase in flexural
resistance at the base of the columns, thereby decreasing the longitudinal bar strain shown
here. This is advantageous to blast resistance.
Figure 6.21 demonstrates that increase in jacket gap ratio produced decrease in the
axial cross‐section strain values for the steel jackets.
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.005 0.01 0.015 0.02
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
2_a0v0j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
Blast Side
Back Side
Blast Side
Back Side
123
Figure 6.21: Jacket Gap Ratio Effects on Steel Jacket CrossSectional Strain Profiles
This observation is a result of the reporting technique discussed in Section 6.2 which
defines the reported axial strains for the steel jacket in columns having a gap as the first
available row of shell elements at the base which are located 50 mm (2”) from the base of
the column. The jacket in this location has not yet been able to develop its resistance and
therefore shows low values of strain. The decreased jacket strains are not necessarily
advantageous, however, increase in jacket strain for columns without a gap allowed to
columns to resist the loading through flexure rather than direct shear which is
advantageous to blast resistance.
6.3.5 Aspect Ratio Effects on Normalized Direct Shear
Figure 6.22 demonstrates that increase in aspect ratio produced a decrease in
normalized direct shear values for models without a steel jacket gap.
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐0.015 ‐0.01 ‐0.005 0 0.005 0.01 0.015
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
2_a0v0j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
Blast Side
Back Side
124
Figure 6.22: Aspect Ratio Effects on Normalized Direct Shear for Column without Gap
This observation is once again dependent on flexural results from Sections 6.5.1 and
6.5.5which discuss the advantages of increasing aspect ratio on the increase in flexural
resistance at the column base. This shift in resistance to flexure reduces the direct shear, as
shown here for columns without a gap, and is advantageous in blast resistance.
However, Figure 6.23, conversely demonstrates that decrease in aspect ratio
produced a decrease in normalized direct shear values for models containing a gap.
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
V/V
ult
Parametric Model
125
Figure 6.23: Aspect Ratio Effects on Normalized Direct Shear for Column with Gap
This observation is contradictory to the results from Section 6.3.1 which discusses the
advantages of increasing aspect ratio on axial cross‐section strain values and direct shear
resistance of all columns with and without gaps. Also, flexure results in Sections 6.5.1 and
6.5.5 showed increased flexural resistance and therefore reduced direct shear demand at
the column base for columns with high aspect. In this research, as aspect ratio increased
the column diameter remained constant and the column height increased. It is speculated
that due to this increase in column height the resultant blast load also increased. Columns
with higher aspect ratio (taller) had more surface area loaded by the blast, and therefore, a
higher resultant lateral force. This may have lead to the increase in normalized direct
shear observed for these columns, although results are inconclusive.
6.3.6 Transverse Reinforcement Ratio Effects on Normalized Direct Shear
Figure 6.24 demonstrates that decrease in transverse reinforcement ratio produced
a decrease in normalized direct shear values.
0.000
0.200
0.400
0.600
0.800
1.000
1.200V/V
ult
Parametric Model
126
Figure 6.24: Transverse Reinforcement Ratio Effects on Normalized Direct Shear
Just as in Section 6.3.2, which discusses the advantages of decreasing transverse
reinforcement ratio on axial cross‐section strain, it is also advantageous in reducing
normalized direct shear to decrease the transverse reinforcement ratio. Once again, the
decrease in transverse reinforcement caused a decrease in lateral stiffness which reduced
the direct shear demand at the column base, and is advantageous in blast resistance.
6.3.7 Jacket Thickness Ratio Effects on Normalized Direct Shear
Increase in jacket thickness ratio produced a decrease in normalized direct shear for
all parametric columns, as shown in Figure 6.25, with the exception of Columns 2 and 4.
0.000
0.200
0.400
0.600
0.800
1.000
1.200
3_a0v0j1g0 7_a0v1j1g0 9_a1v0j0g0 13_a1v1j0g0 4_a0v0j1g1 8_a0v1j1g1
V/V
ult
Parametric Model
127
Figure 6.25: Jacket Thickness Ratio Effects on Normalized Direct Shear
This observation for models without a steel jacket gap agree with Section 6.3.3, which
discusses the advantages of increasing jacket thickness on axial cross‐section strains for
columns without a gap. Reduction in normalized direct shear for columns without a gap as
jacket thickness increases is due to the shift in resistance from direct shear to flexure.
Flexure results discussed in Sections 6.5.3 and 6.5.7 support this argument by
demonstrating increase in jacket thickness produced increase in the flexural resistance at
the base of the columns without gaps, thereby producing the decrease in normalized direct
shear shown here. This is advantageous to blast resistance.
However, the same Section 6.3.3, also demonstrated that it was more advantageous
to decrease jacket thickness to reduce axial cross‐section strains for columns with a gap.
This disagrees with these normalized direct shear results for columns with a gap, except for
Model 2 and 4. Given the large number of cases where exceptions to decreased normalized
direct shear due to decreased jacket thickness exist, no conclusive observations could be
0.000
0.200
0.400
0.600
0.800
1.000
1.200V/V
ult
Parametric Model
128
made on effects of jacket thickness ratio variation on the normalized direct shear response
of blast loaded columns with gaps.
Models 2 and 4 are consistent with axial cross‐section strain results for columns
with a gap discussed in Section 6.3.3, which showed that the advantages of decreasing
jacket thickness on axial cross‐section strains. Figure 6.26 is used to establish the cause for
differences in response between Models 2 and 4 and the remaining parametric models by
evaluating both the blast and quasi‐static capacity model direct shear time histories to
demonstrate which value is behaving differently than other parametric models with
increased jacket thickness ratio.
Figure 6.26: 2_a0v0j0g1 and 4_a0v0j1g1 Normalized Direct Shear Comparison
This figure shows that the static capacity value increased with increased jacket thickness
ratio, as with all other parametric models. However, the blast model demonstrated a
higher increase in direct shear (V), which subsequently dominated the normalized direct
shear value and caused it to increase. This increase in available direct shear within the
V4 = ‐7.95E+06
Vult, 4 = 8.50E+06
V2 = ‐7.36E+06
Vult, 2 = 8.11E+06
‐2.2E+03
‐1.7E+03
‐1.2E+03
‐7.5E+02
‐2.5E+02
2.5E+02
7.5E+02
1.3E+03
1.8E+03
‐1.E+07
‐8.E+06
‐6.E+06
‐4.E+06
‐2.E+06
0.E+00
2.E+06
4.E+06
6.E+06
8.E+06
1.E+07
0.0E+00 1.0E‐01 2.0E‐01 3.0E‐01 4.0E‐01 5.0E‐01
Base Crosssectional Direct Shear Force
(kips)
Base Crosssectional Direct Shear Force
(N)
Time (sec)
4_a0v0j1g1 Blast4_a0v0j1g1 Capacity2_a0v0j0g1 Blast2_a0v0j0g1 Capacity
129
blast model was likely caused by an increase in capacity of both the concrete and steel at
the higher strain rates associated with the blast. Note that both normalized direct shear
values and cross‐sectional strains from the previous sections show contraditory results
when considering the effects of jacket thickness ratio on columns containing steel jacket
gaps.
Jacket thickness ratio effects on normalized direct shear are clear for columns
without a gap and showed that increase in jacket thickness is advantageous in reducing
normalized direct shear and subsequently blast resistance. However, due to the
contradictory reports no conclusive observations could be made on effects of jacket
thickness ratio variation on the normalized direct shear response of blast loaded columns
with gaps.
6.3.8 Jacket Gap Ratio Effects on Normalized Direct Shear
Figure 6.27 demonstrates that decrease in jacket gap ratio produced a decrease in
normalized direct shear values.
130
Figure 6.27: Jacket Gap Ratio Effects on Normalized Direct Shear
Normalized direct shear is reduced from the decrease in jacket gap ratio due to the shift in
resistance from direct shear to flexure, just as in Section 6.3.4, which discusses the
advantages of decreasing jacket gap ratio on the reductions in longitudinal reinforcement
axial cross‐section strain. Flexure results discussed in Sections 6.5.4 and 6.5.8 support this
argument by demonstrating decrease in jacket gap ratio produced increase in flexural
resistance at the base of the columns, thereby decreasing the longitudinal bar strain shown
here. This is advantageous to blast resistance.
6.4 Flexure Results
The evaluation of moment‐rotation throughout the column is an informative
measure of the influence of steel‐jacketed column parameters on column flexural capacity
and ductility under blast. As discussed in Section 2.4.2.2, steel jackets have been used in
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1_a0v0j0g0 2_a0v0j0g1 7_a0v1j1g0 8_a0v1j1g1 9_a1v0j0g0 10_a1v0j0g1
V/V
ult
Parametric Model
131
seismic retrofits to increase insufficiently detailed column ductility, which can be evaluated
by examining their deformations via rotational response for a given load. The same
research has not shown an increase in flexural capacity under seismic loads. However, due
to the variation in jacket gap ratio in this research, which fixes the steel jacket at the base
for the lower parameter limit, flexural capacity was considered throughout the column
height to observe it variation and contribution to resisting direct shear.
To assess parametric column flexural behavior under blast, cross‐sections were
once again defined at intervals of 0.1h along the height of the columns, including the base,
using the LS‐DYNA *DATABASE_CROSS_SECTION_PLANE function as shown in Figure 6.1.
Cross‐sectional moment and average nodal rotation were calculated at each of these eleven
locations for the duration of the analysis. Maximum absolute values of moment were then
obtained from each model and the corresponding analysis time step was noted.
Unlike the critical analysis time step for maximum direct shear result values that
was consistent for all models, the critical analysis time step for flexure results differed
between models. Most of the columns with low aspect ratios (see Table 3.4) reached peak
moment around 10 ms into the analysis, while columns with high aspect ratios reached
peak moment around 30 ms. This difference in critical analysis time is due to the formation
of double curvature in the columns with high aspect ratios and it effects on maximum
positive and negative moment and rotation during the blast. Aspect ratio effects on double
curvature experienced by parametric models are discussed in Section 6.5.1 of this research.
Average nodal rotations were then obtained at the eleven cross‐sections
corresponding to the critical analysis time step and critical moment and rotational
diagrams were plotted along the height of each parametric column and compared.
132
Furthermore, in similar fashion to direct shear examinations completed in the previous
section, maximum moment and rotation values were normalized with respect to their
ultimate static capacities obtained from a separate quasi‐static analysis and plotted
parameter variation effects on column flexural demand were compared.
6.4.1 Moment and Rotation Diagrams
Moment and rotation diagrams were developed from the parametric models by
plotting the moments and rotations in each of the cross‐sections along the column height at
the critical analysis time step, defined as that time when the given column experienced
maximum moment. Representative moment and rotational diagrams are presented in
Figure 6.28 and Figure 6.29 for Models 4_a0v0j1g1 and 11_a1v0j1g0 (see Table 3.4) which
experienced some of the highest moments, and therefore, highest flexural demands of all
the parametric columns that were examined. Model 4 is characterized by minimum values
of aspect and transverse reinforcement ratios, and maximum values of both jacket
thickness and jacket gap ratios. Model 11 is characterized by minimum values of
transverse reinforcement and jacket gap ratios, and maximum values of aspect and jacket
thickness ratios. Both aspect ratio and jacket gap ratio differ between the two models.
133
Figure 6.28: 4_a0v0j1g1 and 11_a1v0j1g1 Moment Diagrams
Figure 6.29: 4_a0v0j1g1 and 11_a1v0j1g1Rotation Diagrams
Figure 6.28 shows that Model 4 experienced its maximum moment near the mid‐height of
the column, while Model 11 experienced its maximum moment at the base. Model 4
M4 = ‐1.07E+10
M11 = 1.07+10
‐8851 ‐3851 1149 6149
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.20E+10 ‐2.00E+09 8.00E+09
Moment (kft)
Column Height (in)
Column Height (mm)
Moment, M (Nmm)
4_a0v0j1g1
11_a1v0j1g0
θ4 = 1.46E‐02θ11 = 1.31E‐02
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02
Column Height (in)
Column Height (mm)
Rotation, θ (rads)
4_a0v0j1g111_a1v0j1g1
134
exhibited low fixity at its base most notably due to the steel jacket gap in this location.
Although the maximum moment for both columns were close in magnitude, it is more
advantageous to form the maximum moment at the base of the column as opposed to the
mid‐height. When compared to direct shear results, the columns that experienced
maximum moment at the base resisted direct shear at the base better than those that
experience maximum moment at mid‐height. Therefore, evaluation of moment diagrams
considered an increase in base moment to be advantageous in resisting blast loads, as it
signifies more flexural resistance and subsequent reduction in direct shear demand at the
column base. Figure 6.29 shows that rotation values at the base and top of the column for
Model 4 are higher than those of Model 11. This is also due to the absence of the steel
jacket in these locations. The steel jacket in Model 11 provides higher rigidity at the base
and top reducing the rotation experienced by the column, which is also advantageous in
resisting direct shear and shifting to a ductile flexural resistance. Remaining parametric
column moment and rotation diagrams are found in Appendix C.
Figure 6.30 and Figure 6.31 plot the critical moment and rotation diagrams over the
height of the columns for all sixteen parametric models.
135
Figure 6.30: Moment Diagram Comparison
‐11,063 ‐6,063 ‐1,063 3,937 8,937
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.5E+10 ‐1E+10 ‐5E+09 0 5E+09 1E+10 1.5E+10
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
136
Figure 6.31: Rotation Diagram Comparison
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.01 ‐0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Column Height (in)
Column Height (mm)
Rotation (rads)
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
137
Both moment and rotation diagram comparisons show the change in fixity between
parametric models at the base. These diagrams also demonstrate that an increase in aspect
ratio (increase in column height) produced double curvature in the columns. Both moment
and rotation diagrams for the taller (high aspect ratio) columns fluctuate between positive
and negative values along the height of the column substantiating double curvature effects.
6.4.2 Normalized Moment‐Rotation
Normalized moment‐rotation plots were once again used to further evaluate column
performance. Here, maximum absolute values for resisting cross‐sectional moment (M)
and rotation (θ) at critical locations and analysis time steps for parametric columns were
normalized with respect to capacity values (Mult and θult) obtained from separate quasi‐
static analyses of on equivalent columns. The quasi‐static capacity models were subjected
to a linearly increasing quasi‐static lateral load at column mid‐height until flexural failure
was observed, as seen in Figure 6.32Figure 6.32.
Figure 6.32: MomentRotation Capacity Model Load Definition
138
A graph of each parametric model’s static moment‐rotation capacity curve (Figure 6.33) is
found in Appendix D.
Figure 6.33: 4_a0v0j1g1 Static MomentRotation Capacity Curve
A summary of all parametric model normalized moment rotation values is shown in Figure
6.34.
θult = 3.29E‐02, Mult = 3.61E+09
0
500
1000
1500
2000
2500
0.E+00
5.E+08
1.E+09
2.E+09
2.E+09
3.E+09
3.E+09
4.E+09
4.E+09
3.E‐03 8.E‐03 1.E‐02 2.E‐02 2.E‐02 3.E‐02 3.E‐02 4.E‐02
Mom
ent (kft)
Mom
ent (Nmm)
Rotation (rads)
139
Figure 6.34: Normalized MomentRotation Comparison
The majority of the parametric models experienced levels of normalized moment‐
rotation below unity, which suggests that the blast models did not reach their static flexural
capacity. However, as shown in Figure 6.34, Models 2, 4, 6 and 8 achieved levels of
normalized moment higher than unity, while Models 1, 12 and 16 achieved levels of
normalized rotation higher than unity. This behavior is due to the difference in loading
rate between the static capacity values used in the denominator of the normalized values
and the values in the numerator that were obtained from the blast models. The steel jacket
provided better resistance to flexure at the higher strain rates associated with blast.
Furthermore, the static capacity analyses used to determine flexural capacity for the
columns above unity showed a possible failure mode not of flexure, but rather direct shear
0
0.5
1
1.5
2
2.5
3
3.5
0.1 1 10
M/M
ult
θ/θult (log10)
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
140
at their base, which indicates the susceptibility of these columns to this type of failure. This
coupled with the increase in flexural capacity at higher strain rates could account for the
normalized values exceeding unity.
Figure 6.35 plots the axial force time history for the 36 longitudinal bars from the
static capacity analysis for parametric Column 4, which experienced normalized moment
above unity and is most notably characterized as having a steel jacket gap (see Table 3.4).
Figure 6.35: 4_a0v0j1g1 Flexure Capacity Analysis Axial Force Time History for Longitudinal Bars at Base
This plot is used to demonstrate that the longitudinal bars all failed in tension, indicative of
direct shear failure, rather than a combination of tension and compression which is
indicative of flexural failure. This is shown in the graph as every bar’s axial force peaks on
the positive (tension) side of the y‐axis before failure around the analysis time of 0.35 secs.
When the longitudinal bar axial forces at the base (Figure 6.35) are compared to a graph of
their base axial forces from the direct shear capacity analysis (Figure 6.36), similarities in
are noted.
Failure of bars
141
Figure 6.36: 4_a0v0j1g1 Direct Shear Capacity Analysis Axial Force Time History for Longitudinal Bars at Base
In both cases all 36 longitudinal bars were shown to fail in tension which points to a direct
shear failure at the base. This supports the statement that some of the flexure capacity
analysis models failed in direct shear and were not able to develop their maximum flexure
capacity.
These direct shear effects at the base of the column were not seen in the models
that showed normalized moment‐rotation values less than unity, as shown in Figure 6.37
for Column 14, which is most notably characterized by not having a steel jacket gap (see
Table 3.4). This figure shows that the longitudinal bars in the base section experienced
both compression and tension at their time of failure, which suggests a flexural failure.
Therefore, the increased loading rate experienced by columns in the blast analyses allowed
parametric columns to develop higher moments and rotation at critical sections compared
to their static moment‐rotation capacity analysis counterparts producing normalized
values above unity.
Failure of bars
142
Figure 6.37: 14_a1v1j0g1 MomentRotation Capacity Analysis Axial Force Time History for Longitudinal Bars at Base
6.5 Parameter Variation Effects on Flexure
This section discusses the effects of varying the four critical parameters on flexure.
First, the effects of the individual parameters on moment and rotation diagrams for the
critical analysis time steps are discussed for representative model results. After which, the
effects of the individual parameters on normalized moment‐rotation plots are discussed.
6.5.1 Aspect Ratio Effects on Moment and Rotation
Figure 6.38 demonstrates that increase in aspect ratio produced a decrease in
moment along the height of the column and an increase in moment at the base of the
column.
Failure of bars
143
Figure 6.38: Aspect Ratio Effects on Moment
The converse effects on moment between locations along the height of the column and at
the base are due the formation of double curvature in parametric models with high aspect
ratios, as seen in Figure 6.39.
Figure 6.39: Blast Effects on Double Curvature
‐11,063 ‐6,063 ‐1,063 3,937 8,937
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.5E+10 ‐5E+09 5E+09 1.5E+10
Moment (kft)
Column height (in)
Column Height (mm)
Moment (Nmm)
3_a0v0j1g0
11_a1v0j1g0
5_a0v1j0g0
13_a1v1j0g0
8_a0v1j1g1
16_a1v1j1g1
144
The formation of double curvature allowed columns with high aspect ratios to experience
reducing in moment along the height of the column and increase moment at the base.
Increase in moment at the column base is advantageous in resisting blast loads as it
signifies and increase or shirt to flexural response in this location where direct shear needs
to be avoided.
Figure 6.40 demonstrates that increase in aspect ratio produced an increase in
rotation at the base of the column and a decrease in rotation at the top of the column.
Figure 6.40: Aspect Ratio Effects on Rotation
Similar double curvature effects for columns with high aspect ratios are observed in
rotation plots just as in moment. Increased aspect ratio produced fluctuation of rotation
between positive and negative values along the height of the column. Ultimately, increased
aspect ratio produced high rotations at the column base which, once again, signifies
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.01 0 0.01 0.02 0.03 0.04
Column Height (in)
Column Height (mm)
Rotation (rads)
3_a0v0j1g0
11_a1v0j1g0
5_a0v1j0g0
13_a1v1j0g0
8_a0v1j1g1
16_a1v1j1g1
145
increased flexural response at the base which is advantageous in resisting blast loads and
avoiding direct shear.
6.5.2 Transverse Reinforcement Ratio Effects on Moment and Rotation
Figure 6.41 demonstrates that increase in transverse reinforcement ratio produced
virtually no effect in moment throughout the column.
Figure 6.41: Transverse Reinforcement Ratio Effects on Moment
These results agree with steel‐jacketed column seismic research which shows no effect in
moment capacity with variation in transverse reinforcement ratio, as discussed in Section
2.5.2.
Increased rotation is commonly associated with increased transverse reinforcement
ratio due to the increase in concrete confinement that it produces and corresponding
greater concrete core ductility. Figure 6.42 demonstrates that increased transverse
reinforcement produces an increase in rotation at the base of the column under blast and
‐11,063 ‐6,063 ‐1,063 3,937 8,937
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.5E+10 ‐5E+09 5E+09 1.5E+10
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
3_a0v0j1g0
7_a0v1j1g0
9_a1v0j0g0
13_a1v1j0g0
4_a0v0j1g1
8_a0v1j1g1
146
has little effect at the top of the column for all parametric columns with the exception of
Columns 1 and 5, 4 and 8 and 10 and 14 (see Table 3.4).
Figure 6.42: Transverse Reinforcement Ratio Effects on Rotation
Reduction in rotation capacity of these models is speculated to be caused by an increase in
lateral stiffness associated with increased transverse reinforcement ratio. Lateral stiffness
increase could cause the columns to perform less ductile at the base and increase direct
shear demand increasing deflections rather than rotations. However, given the large
number of cases where exceptions to increased rotation exist, no conclusive observations
could be made on effects of transverse reinforcement ratio variation on the rotational
response of blast loaded columns.
6.5.3 Jacket Thickness Ratio Effects on Moment and Rotation
Figure 6.43 demonstrates that increase in jacket thickness ratio produced increase
in moment throughout the height of the column and at the base for models without a steel
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.01 ‐0.005 0 0.005 0.01 0.015 0.02 0.025 0.03
Column Height (in)
Column Height (mm)
Rotation (rads)
3_a0v0j1g0
7_a0v1j1g0
9_a1v0j0g0
13_a1v1j0g0
4_a0v0j1g1
8_a0v1j1g1
147
jacket gap. Figure 6.44 demonstrates that increase in jacket thickness ratio produces
increase in moment throughout the height of the column and a decrease in moment at the
base for models containing a steel jacket gap.
Figure 6.43: Jacket Thickness Ratio Effects on Moment for Columns without Gap
Figure 6.44: Jacket Thickness Ratio Effects on Moment for Columns with Gap
‐11,063 ‐6,063 ‐1,063 3,937 8,937
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.5E+10 ‐5E+09 5E+09 1.5E+10
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
1_a0v0j0g0
3_a0v0j1g0
5_a0v1j0g0
7_a0v1j1g0
9_a1v0j0g0
11_a1v0j1g0
13_a1v1j0g0
15_a1v1j1g0
‐11,063 ‐6,063 ‐1,063 3,937 8,937
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.5E+10 ‐5E+09 5E+09 1.5E+10
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
2_a0v0j0g1
4_a0v0j1g1
6_a0v1j0g1
8_a0v1j1g1
10_a1v0j0g1
12_a1v0j1g1
14_a1v1j0g1
16_a1v1j1g1
148
The increases in moment in all cases are due to the increase in steel cross‐sectional area,
and therefore, an increase in flexural resistance due to a stiffer steel jacket and due to the
jacket being placed at the critical flexural location. The dramatic reduction in moment
capacity at the base for columns having a gap is not advantageous because they are not
developing the additional flexural resistance provided by the jacket at the critical base
section. This increased resistance is largely shifted to the next critical section, located at
mid‐height of the column. This observation may also be a clue to the contradictory
observations in both normalized direct shear, and cross‐sectional strain values for columns
containing steel jacket gaps and high jacket thickness ratios. As stated in Section 6.3.7,
parametric Models 2 and 4 show a disadvantageous increase in normalized direct shear
when the jacket thickness ratio is increased, which is contradictory to all other parametric
models. This may be caused by a shift in resistance at the base of the column from flexural
to direct shear due to the increased flexural capacity and stiffness at mid‐height of the
column from the thicker steel jacket. Figure 6.44 shows this shift in flexural capacity at the
base of the column between Models 2 and 4, while Figure 6.25 shows the increase in
normalized direct shear due to this shift in resistance caused by the increased jacket
thickness ratio.
Figure 6.45 demonstrates that increase in jacket thickness ratio produced a
decrease in rotation at both the base and top of the column.
149
Figure 6.45: Jacket Thickness Ratio Effects on Rotation
Once again, this observation supports the above argument that increased jacket thickness
ratio may be detrimental to increased rotational resistance at the column base, just as with
moment resistance, although not all parametric models are in agreement. Observed
decrease in both moment and rotation for columns having high jacket thickness ratios and
steel jacket gaps, although not completely conclusive, agrees with increases in direct shear
demand discussed in Sections 6.3.3 and 6.3.7 and may be detrimental to blast resistance.
6.5.4 Jacket Gap Ratio Effects on Moment and Rotation
Figure 6.46 demonstrates that increase in jacket gap ratio produced an increase in
moment along the height of the column and a decrease in moment at the base of the
column.
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.01 0 0.01 0.02 0.03 0.04
Column Height (in)
Column Height (mm)
Rotation (rads)
1_a0v0j0g0
3_a0v0j1g0
6_a0v1j0g1
8_a0v1j1g1
10_a1v0j0g1
12_a1v0j1g1
150
Figure 6.46: Jacket Gap Ratio Effects on Moment
For columns containing a steel jacket gap, moment resistance is relocated from the critical
base section to locations further up the column. These critical section relocation patterns
mimic the hinging patterns within the columns as shown in Figure 6.47 for Column 8,
which has a steel jacket gap.
Figure 6.47: Column 8_a0v1j1g1 Base Hinge Formation
‐11,063 ‐6,063 ‐1,063 3,937 8,937
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1.5E+10 ‐5E+09 5E+09 1.5E+10
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
1_a0v0j0g0
2_a0v0j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
151
This figure shows the formation of a hinge at the base of Column 8 in terms of plastic strain.
Due to the hinge formation at its base, Column 8 shifts moment resistance to the mid‐height
of the column, as shown in Figure 6.46. Therefore, increase in the gap reduces the
likelihood of ductile flexural failures by allowing hinges to be formed at the base and
shifting the resistance to the mid‐height of the column which increases the likelihood of
more brittle failures, such as direct shear which is detrimental to blast resistance.
Figure 6.48 demonstrates that increase in jacket gap ratio produces increase in
rotation at the top of columns and decrease in rotation at the base of columns with low
aspect ratios. Figure 6.49 demonstrates that increase in jacket gap ratio produces increase
in rotation both at the base and top of columns with high aspect ratios.
Figure 6.48: Jacket Gap Ratio Effects on Rotation for Columns with Low Aspect Ratios
0
20
40
60
80
100
120
140
160
180
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐0.01 0 0.01 0.02 0.03 0.04
Column Height (in)
Column Height (mm)
Rotation (rads)
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
7_a0v1j1g0
8_a0v1j1g1
152
Figure 6.49: Jacket Gap Ratio Effects on Rotation for Columns with High Aspect Ratio
Increase in the jacket gap produced a decrease in the column’s rotational capacity at the
critical base section, which is exaggerated in columns with low aspect ratios that are more
susceptible to brittle failures at the base and is detrimental to blast resistance.
Observations showing an increase in rotation for columns having both high jacket gap and
aspect ratios are substantiated by the increase in rotational capacity for columns
containing a high aspect ratio, as discussed in Section 6.5.1, combined with the increase in
rotation observed for columns containing low jacket thickness ratio (such as in the areas
containing a gap), as discussed in Section 6.5.3.
6.5.5 Aspect Ratio Effects on Normalized Moment‐Rotation
Figure 6.50 demonstrates that for the majority of the parametric models increased
aspect ratio decreased both normalized moment and rotation.
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.01 0 0.01 0.02 0.03 0.04
Column Height (in)
Column Height (mm)
Rotation (rads)
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
15_a1v1j1g0
16_a1v1j1g1
153
Figure 6.50: Aspect Ratio Effects on Normalized MomentRotation
This decrease in both normalized moment and rotation is due to the increase in the flexural
capacity associated with higher aspect ratios, as discussed in Section 6.5.1, which is
advantageous to blast resistance.
Outliers to this trend are shown in Figure 6.51 where increase in aspect ratio
produced the same decrease in normalized moment, however increased normalized
rotation.
0
0.5
1
1.5
2
2.5
3
3.5
0.1 1 10
M/M
ult
θ/θult (log10)
1_a0v0j0g0
9_a1v0j0g0
2_a0v0j0g1
10_a1v0j0g1
3_a0v0j1g0
11_a1v0j1g0
5_a0v1j0g0
13_a1v1j0g0
154
Figure 6.51: Aspect Ratio Effects for Outliers on Normalized MomentRotation
As discussed in Section 6.4.2, this can be attributed to two reasons. The first is the
observation of a direct shear failure for the static moment‐rotation capacity analysis
models which experienced reduced rotational capacity at this slower loading rate for the
outlier models. The second contributor is that the increased loading rate associated with
the blast models allowed the columns to experience higher rotations than that of their
static capacity analysis model counterparts. Once again, as discussed for general
observations on normalized moment‐rotation values above unity in Section 6.4.2, it
appears that the steel jacket offers more benefits to the rotation capacities, for certain blast
models, as the strain rate increases. Therefore, the outliers showed larger normalized
rotation values not due to of a reduction in capacity, but rather due to an increase in the
rotation capacity at higher strain rates, which is also advantageous to blast resistance.
0
0.5
1
1.5
2
2.5
3
3.5
0.1 1 10
M/M
ult
θ/θult (log10)
4_a0v0j1g1
12_a1v0j1g1
6_a0v1j0g1
14_a1v1j0g1
8_a0v1j1g1
16_a1v1j1g1
155
6.5.6 Transverse Reinforcement Ratio Effects on Normalize Moment‐Rotation
As discussed in Section 2.5.2 transverse reinforcement ratio has the greatest effect
on column rotation capacity and virtually no effect on column moment capacity, and
therefore, the focus of this section is on the variation of normalized rotation only. Figure
6.52 shows that for the majority of the parametric models an increase in transverse
reinforcement ratio produced a decrease in normalized rotation.
Figure 6.52: Transverse Reinforcement Ratio Effects on Normalized MomentRotation
0
0.5
1
1.5
2
2.5
3
3.5
0.1 1 10
M/M
ult
θ/θult (log10)
1_a0v0j0g0
5_a0v1j0g0
2_a0v0j0g1
6_a0v1j0g1
3_a0v0j1g0
7_a0v1j1g0
10_a1v0j0g1
14_a1v1j0g1
156
This effect can be associated to the increase in confinement within the column, and is
advantageous in blast resistance. However, Figure 6.53 conversely demonstrates that
increase in transverse reinforcement produced an increase in normalized rotation.
Figure 6.53: Transverse Reinforcement Ratio Effects for Outliers on Normalized MomentRotation
The reason behind this effect may once again be due to the column’s susceptibility to direct
shear failure, even in flexure capacity analysis models, which would decrease the
denominator values and the increases in column capacity at high strain rates, as discussed
in Sections 6.4.2 and 6.5.5, which would increase the numerator value resulting in
increased normalized rotation. As seen in Figure 6.54, the comparison of moment‐rotation
capacity curves shows a decrease in the rotational capacity as transverse reinforcement
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 1 10
M/M
ult
θ/θult (log10)
9_a1v0j0g0
13_a1v1j0g0
11_a1v0j1g0
15_a1v1j1g0
12_a1v0j1g1
16_a1v1j1g1
157
ratio increases supporting the argument of load resistance shifting from flexure to direct
shear, which is detrimental in blast situations.
Figure 6.54: MomentRotation Capacity Comparison for Outliers
Given the large number of cases where exceptions to decreased normalized rotation exist,
no conclusive observations could be made on effects of transverse reinforcement ratio
variation on the normalized rotation response of blast loaded columns.
6.5.7 Jacket Thickness Ratio Effects on Normalized Moment‐Rotation
The effects of variation in jacket thickness ratio on normalized moment‐rotation are
shown to be dependent on whether the columns contain a steel jacket gap or not. Figure
6.55 shows that for columns without a gap increase in jacket thickness ratio produced
decrease in normalized moment and rotation with the exception of the Models 9 and 11
(see Table 3.4).
0
5000
10000
15000
20000
25000
30000
0.00E+00
5.00E+09
1.00E+10
1.50E+10
2.00E+10
2.50E+10
3.00E+10
3.50E+10
4.00E+10
4.50E+10
0.0E+00 5.0E‐02 1.0E‐01 1.5E‐01 2.0E‐01
Mom
ent, M (kft)
Mom
ent, M (Nmm)
Rotation, θ (rads)
9_a1v0j0g0
13_a1v1j0g0
11_a1v0j1g0
15_a1v1j1g0
12_a1v0j1g1
16_a1v1j1g1
158
Figure 6.55: Jacket Thickness Ratio Effects on Normalized MomentRotation for Columns without Gap
Models 9 and 11 showed increase in jacket thickness ratio produced the same decrease in
normalized moment while slightly increasing normalized rotation. However, the majority
of the models without gaps agreed that increasing jacket thickness ratio is beneficial for
decreasing both normalized moment and rotation and subsequently beneficial to resisting
blast loads.
Figure 6.56 demonstrates that decrease in jacket thickness ratio produced a
decrease in normalized moment and rotation for columns with a gap in all cases but one.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.1 1 10
M/M
ult
θ/θult (log10)
1_a0v0j0g0
3_a0v0j1g0
5_a0v1j0g0
7_a0v1j1g0
9_a1v0j0g0
11_a1v0j1g0
13_a1v1j0g0
15_a1v1j1g0
159
Figure 6.56: Jacket Thickness Ratio Effects on Normalized MomentRotation for Columns with Gap
Models 2 and 4 showed the same decrease in rotation; however, they showed that decrease
in jacket thickness ratio produced increased normalized moment. However, the majority of
the models with gaps agreed that decreasing jacket thickness ratio is beneficial for
decreasing both normalized moment and rotation and subsequently beneficial to resisting
blast loads.
6.5.8 Jacket Gap Ratio Effects on Normalized Moment‐Rotation
Just as in the direct shear discussion, the reduction of jacket gap ratio produced an
overall flexural effect of decrease in both normalized moment and rotation, as seen in
Figure 6.57.
0
0.5
1
1.5
2
2.5
3
3.5
0.1 1 10
M/M
ult
θ/θult (log10)
2_a0v0j0g1
4_a0v0j1g1
6_a0v1j0g1
8_a0v1j1g1
10_a1v0j0g1
12_a1v0j1g1
14_a1v1j0g1
16_a1v1j1g1
160
Figure 6.57: Jacket Gap Ratio Effects on Normalized MomentRotation
This observation is contributed to the increase in flexural capacity associated with the
decrease in jacket gap ratio which is advantageous in resisting blast loads.
6.6 Transverse Shear
The final failure mode discussed in this study is transverse shear. Transverse shear
is defined as the shear experienced along the height of the column due to the lateral blast
load. Although research has not shown this failure mode to be of concern in steel‐jacketed
columns under blast loads (Fujikura & Bruneau, 2008), it is typically considered in the
design of bridge columns (AASHTO, 2007), and therefore, was examined in this study.
Variation in the vertical distribution of axial strains in hoop reinforcement and
0
0.5
1
1.5
2
2.5
3
3.5
0.1 1 10
M/M
ult
θ/θult (log10)
3_a0v0j1g0
4_a0v0j1g1
7_a0v1j1g0
8_a0v1j1g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
161
Cross‐sectional cuts at 10% column height intervals
corresponding strains in steel jackets were used to assess the resistance of the columns to
transverse shear.
6.6.1 Transverse Strain Profiles
Transverse strain profiles were obtained from the parametric models by reporting a
maximum, minimum, and average axial strain value for the hoop reinforcement and the
corresponding strains for the steel jacket at each of the eleven transverse shear planes
throughout the height of the column. This is demonstrated in Figure 6.58.
Figure 6.58: Transverse Strain Profile Definition
Maximum, minimum, and average values of all 72 hoop elements
Maximum, minimum, and average values of all 96 steel jacket elements
162
Maximum, minimum and average values were used to evaluate performance and
ranges of transverse shear distribution along the height of the columns to determine an
acceptable limit at which to report the distribution of transverse strain within one
particular hoop or corresponding steel jacket cross‐section. The critical analysis time for all
parametric models was 7.5ms (see Figure 6.59) and will be used for all transverse strain
profiles.
Figure 6.59: 9_a1v0j0g1 Average Transverse Strain Time History
A representative transverse strain profile is shown in Figure 6.60 for parametric
Model 2_a0v0j0g1 (see Table 3.4). This column experienced some of the highest transverse
shear strains in the hoop reinforcement and the steel jacket, and therefore, a high
transverse shear demand. All parametric model transverse strain profiles are located in
Appendix E.
7.50E‐03
0.0E+002.0E‐044.0E‐046.0E‐048.0E‐041.0E‐031.2E‐031.4E‐031.6E‐031.8E‐032.0E‐03
0.0E+00 5.0E‐02 1.0E‐01 1.5E‐01 2.0E‐01 2.5E‐01 3.0E‐01
Average Transverse Strain
[(mm/m
m) or (in/in)]
Time (sec)
163
Figure 6.60: 2_a0v0j0g1 Transverse Strain Profile
The variation in transverse strain between minimum and maximum values can be
associated with localized hoop and jacket deformation caused by the blast load. All
parametric column transverse strain plots showed the range between the minimum and
maximum strains in the hoops and jacket decreasing as you moved away from the column
base and blast load, with slight increases in the range occurring at the top due to restraint
conditions. Columns containing high aspect ratios (see Table 3.4) showed smaller
transverse strain ranges along their height (see Figure 6.61).
0
20
40
60
80
100
120
140
160
180
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐0.004 ‐0.002 0 0.002 0.004 0.006 0.008 0.01
Column Height (in)
Column Height (mm)
Trans Strain [(mm/mm) or (in/in)]
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
164
Figure 6.61: 9_a1v0j0g1 Transverse Strain Profile
Since high aspect ratio correspond to taller columns, upper portions of these columns
experienced less local deformation due to the blast which contributed to the transverse
strain range reduction. This demonstrates that the local deformation due to the blast is a
major contributor to the range in transverse strain plotted in these graphs. To account for
more global or cumulative behavior in the hoops and jacket the average value was used as
a method of transverse strain comparison with the local blast effects. Figure 6.62 and
Figure 6.63 plot all parametric column average transverse shear strain profiles for the hoop
reinforcement and steel jackets, respectively.
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐2.E‐03 ‐1.E‐03 ‐5.E‐04 0.E+00 5.E‐04 1.E‐03 2.E‐03 2.E‐03 3.E‐03
Column Height (in)
Column Height (mm)
Transverse Strain [(mm/mm) or (in/in)]
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
165
Figure 6.62: Hoop Average Transverse Strain Comparison
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015 0.002
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
166
Figure 6.63: Steel Jacket Average Transverse Strain Comparison
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015 0.002 0.0025
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in /in )]
1_a0v0j0g0
2_a0v0j0g1
3_a0v0j1g0
4_a0v0j1g1
5_a0v1j0g0
6_a0v1j0g1
7_a0v1j1g0
8_a0v1j1g1
9_a1v0j0g0
10_a1v0j0g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
15_a1v1j1g0
16_a1v1j1g1
167
Comparison of the hoop and steel jacket average transverse strains shows relatively low
strains magnitudes in all the parametric columns. These findings agree with previous
steel‐jacketed column blast effectiveness research, which did not identify transverse shear
as a critical condition. Furthermore, almost all average transverse strain values are in
tension showing resistance to transverse shear cracking. The exception to this is at the
base of parametric columns that had a steel jacket gap. These models show low values of
compression strain in the hoops and jacket at the base, which can be attributed to local
damage from the blast load in the unprotected gap region in similar fashion to the findings
discussed in above for average transverse strain.
6.7 Parameter Variation Effects on Transverse Shear
This section discusses the effects of varying the four critical parameters on the
transverse shear. The effects of the individual parameters on average transverse strain
profiles are discussed for representative model results.
6.7.1 Aspect Ratio Effects on Average Transverse Strain
Figure 6.64 and Figure 6.65 demonstrate that increase in aspect ratio produced
decrease in hoop reinforcement and steel jacket average transverse strain respectively.
168
Figure 6.64: Aspect Ratio Effects on Hoop Average Transverse Strain
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015 0.002
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
9_a1v0j0g0
2_a0v0j0g1
10_a1v0j0g1
7_a0v1j1g0
15_a1v1j1g0
8_a0v1j1g1
16_a1v1j1g1
169
Figure 6.65: Aspect Ratio Effects on Steel Jacket Transverse Strain
As shown in Sections 6.5.1 and 6.5.5 increased aspect ratio encourages flexural resistance,
especially at the base of the columns, and therefore transverse shear along the height of the
column, but most notably at the base, is reduced which is advantageous to both transverse
shear and blast resistance.
6.7.2 Transverse Reinforcement Ratio Effects on Average Transverse Strain
Figure 6.66 and Figure 6.67 demonstrate that increase in transverse reinforcement
ratio produced decrease in hoop reinforcement and steel jacket average transverse strain
respectively.
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015 0.002 0.0025
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in /in )]
1_a0v0j0g0
9_a1v0j0g0
2_a0v0j0g1
10_a1v0j0g1
7_a0v1j1g0
15_a1v1j1g0
8_a0v1j1g1
16_a1v1j1g1
170
Figure 6.66: Transverse Reinforcement Ratio Effects on Hoop Average Transverse Strain
Figure 6.67: Transverse Reinforcement Ratio Effects on Steel Jacket Average Transverse Strain
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in/in)]
3_a0v0j1g0
7_a0v1j1g0
9_a1v0j0g0
13_a1v1j0g0
4_a0v0j1g1
8_a0v1j1g1
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in /in )]
3_a0v0j1g0
7_a0v1j1g0
9_a1v0j0g0
13_a1v1j0g0
4_a0v0j1g1
8_a0v1j1g1
171
This agrees with seismic steel‐jacketed column research and is due to the increase in
transverse steel area which reduces strains within the material itself and contributes more
toward transverse shear resistance reliving demand on the steel jacket. However, increase
in transverse reinforcement ratio has been shown to be detrimental to direct shear
resistance, as discussed in Sections 6.3.2 and 6.3.6, and its effects on the flexural resistance
of blast loaded columns are unclear (see Sections 6.5.2 and 6.5.6). Direct shear results
coupled with the knowledge that transverse shear is not a critical condition denotes that
decrease in transverse reinforcement ratio is more advantageous to resisting the more
critical direct shear failure and blast.
6.7.3 Jacket Thickness Ratio Effects on Average Transverse Strain
Figure 6.68 and Figure 6.69 demonstrate that increase in jacket thickness ratio
produces decrease in hoop reinforcement and steel jacket average transverse strain
respectively.
172
Figure 6.68: Jacket Thickness Ratio Effects on Hoop Average Transverse Strain
Figure 6.69: Jacket Thickness Ratio Effects on Steel Jacket Average Transverse Strain
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015 0.002
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in/in)]
1_a0v0j0g0
3_a0v0j1g0
6_a0v1j0g1
8_a0v1j1g1
10_a1v0j0g1
12_a1v0j1g1
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015 0.002
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in /in )]
1_a0v0j0g0
3_a0v0j1g0
6_a0v1j0g1
8_a0v1j1g1
10_a1v0j0g1
12_a1v0j1g1
173
As in the previous section this is due to the increase in steel jacket area which reduces
strains within the material itself and contributes more toward transverse shear resistance
reliving demand on hoop reinforcement. Increase in jacket thickness ratio agrees with
direct shear and flexure results for column without a steel jacket gap. However, increase in
jacket thickness is in disagreement with flexure and direct shear results for columns with a
gap, although normalized direct shear results for columns with a gap are unclear.
Therefore it is most likely advantageous to decrease jacket thickness ratio for columns with
a gap to increases blast resistance, although normalized direct shear results are unclear on
this matter.
6.7.4 Jacket Gap Ratio Effects on Average Transverse Strain
Figure 6.70 and Figure 6.71 demonstrate that increase in jacket gap ratio produces a
slight decrease in hoop reinforcement and steel jacket average transverse strain
respectively throughout the height of the column and an increase in hoop reinforcement
and steel jacket average transverse strain at the column base.
174
Figure 6.70: Jacket Gap Ratio Effects on Hoop Average Transverse Strain
Figure 6.71: Jacket Gap Ratio Effects on Steel Jacket Average Transverse Strain
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in/in)]
7_a0v1j1g0
8_a0v1j1g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.0005 0 0.0005 0.001 0.0015
Column Height (in)
Column Height (mm)
Axial Strain [(mm/mm) or (in /in )]
7_a0v1j1g0
8_a0v1j1g1
11_a1v0j1g0
12_a1v0j1g1
13_a1v1j0g0
14_a1v1j0g1
175
The sharp decrease in transverse strain at the base of the columns show that columns with
a steel jacket gap are not developing both the hoop’s and jacket’s resistance to shear at this
location subjecting them to a possible brittle failure mode.
6.8 Additional Parametric Results
Results summarized herein were obtained from two additional parametric models
discussed in Section 3.7: 9_a1v0j0g0_Not Fixed and 10_a1v0. These models were used to
examine the importance of steel jacket base fixity for retrofitted columns without a steel
jacket gap and the overall advantages of steel jacket retrofitting for columns subjected to
blast loads. Both direct shear and flexure failure parameters were used to assess these
columns for blast resistance. Since transverse shear was not shown to be a critical failure
mode in the previous sections, it was not considered for these additional columns.
6.8.1 Steel Jacket Base Fixity Effects on Blast Resistance
Direct shear and flexure resistance for similarly proportioned steel‐jacketed
columns having varying levels of jacket fixity at the base were evaluated by comparing
three columns: Model 9_a1v0j0g0, Model 9_a1v0j0g0_Not Fixed (9_Not Fixed) and Model
10_a1v0j0g1. These models represent slender columns with low transverse reinforcement
ratios and low jacket thickness ratios. The jacket gaps for these models vary from not
having a steel jacket gap with the jacket fixed to the foundation, to columns with jackets
that bear on the foundation, to a 50 mm (2”) jacket gap. The importance of steel jacket
fixity on blast resistance is evaluated through comparisons of: axial cross‐section strain
profiles in the longitudinal reinforcing bars and the steel jackets; normalized direct shear,
moment and rotation diagrams; and normalized moment‐rotation plots.
176
Figure 6.72 shows the axial cross‐section strain profiles for longitudinal
reinforcement at the column base as defined in the previous sections.
Figure 6.72: Steel Jacket Fixity Effects on Longitudinal Bar Axial Crosssection Strain at Base
As discussed in Section 6.3.4, strains in the longitudinal bars were shown to decrease as the
jacket gap ratio decreased. This is shown in Figure 6.72 where Model 9 bar strains are
lower that Model’s 10. This suggests less direct shear demand at the base and better blast
resistance. However, as shown through Model 9_Not Fixed results, if fixity of the steel
jacket is removed only relying on the bearing of the jacket on the foundation surface,
longitudinal bar strains increased beyond the level of Model’s 10 results, which has a steel
jacket gap. This signifies that bearing of the jacket on the foundation has increased direct
shear demand. It is suspected that the steel jacket in Model 9_Not Fixed may essentially be
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.002 0.004 0.006 0.008 0.01 0.012
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
9_a1v0j0g0
10_a1v0j0g1
9_a1v0j0g0_Not Fixed
Blast Side
Back Side
177
causing prying of the longitudinal bars due to its bearing on the foundation surface, as
shown in Figure 6.73.
Figure 6.73: Steel Jacket Bearing and Prying
This prying action may be leading to the higher longitudinal bar strains observed in Figure
6.72.
Figure 6.74 shows the axial cross‐section strain profiles for steel jackets at the
column base. Note that steel jacket strain values for Model 9_Not Fixed were not only
reported directly at the base of the column, but also were reported at the same location as
Model 10, which is 50 mm (2”) from the base of the column (see Figure 6.4).
Blast Side Back Side
Increased Uplift due to Prying Increased Rotation
due to Prying
Steel Jacket Bearing
Increased dowel action due to Prying
178
Figure 6.74: Steel Jacket Fixity Effects of on Steel Jacket Axial Crosssection Strain at Base
This profile comparison shows that steel jackets contribute little to the resistance of direct
shear when they are not fixed to the foundation. This is shown by the low values of jacket
strain, and therefore direct shear resistance contribution, in Models 9_Not Fixed and 10.
Whereas, Model 9 shows higher jacket strains that are distributed between tension and
compression signifying a flexural resistance to the blast load at the base. Jacket strains for
Models 9_Not Fixed and 10, which are reported at the same location 50 mm (2”) up from
the base of the column are comparable. However, Model 9_Not Fixed shows slightly higher
jacket contribution and strain in the back side of the cross‐section.
Figure 6.75 compares the normalized direct shear values for the three columns.
‐30
‐20
‐10
0
10
20
30
‐762
‐562
‐362
‐162
38
238
438
638
‐0.015 ‐0.01 ‐0.005 0 0.005 0.01 0.015
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
9_a1v0j0g0
9_a1v0j0g0_Not Fixed reported at base
9_a1v0j0g0_Not Fixed reported at same location as Model 1010_a1v0j0g1
Blast Side
Back Side
179
Figure 6.75: Steel Jacket Fixity Effects on Normalized Direct Shear at Base
This plot demonstrates that Model 9_Not Fixed behaves similarly to Model 10, which has a
gap of 50 mm (2”). Column 9_Not Fixed showed high normalized direct shear, and
therefore, direct shear demand. However, its normalized value was slightly lower than
Model 10. This decrease in normalized direct shear is due to the increased frictional force
at the base cross‐section due to the jacket bearing at this location.
Figure 6.76 considers flexure through moment diagram comparisons between the
three columns. Diagrams are plotted along the height of the column.
0.000
0.200
0.400
0.600
0.800
1.000
1.200
9_a1v0j0g0 9_a1v0j0g0_Not Fixed 10_a1v0j0g1
V/Vult
Parametric Model
180
Figure 6.76: Steel Jacket Fixity Effects on Moment
This figure shows that Model 9_Not Fixed behaves very similarly to Model 10 both at the
base and along the column height. Therefore, insignificant moment capacity at any location
in the column is gained from extending the steel jacket to the foundation of the column
without developing its fixity.
Figure 6.77 compares rotation along the height of the three columns.
‐7,376 ‐5,376 ‐3,376 ‐1,376 624 2,624 4,624 6,624
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐1E+10 ‐5E+09 0 5E+09 1E+10
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
9_a1v0j0g0
10_a1v0j0g1
9_a1v0j0g0_Not Fixed
181
Figure 6.77: Steel Jacket Fixity Effects on Rotation
Although, moment capacity effects were insignificant for these columns, the rotation at the
base of Model 9_Not Fixed more than double in value compared to Model 10. This
increased rotation showed decreased flexural resistance and may once again be due prying
or increased uplift on the blast side of the cross‐section due to the increased stiffness and
bearing of the steel jacket on the back side of the column (see Figure 6.73).
The final flexure consideration for comparison of the three columns is normalized
moment‐rotation, as shown in Figure 6.78.
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
Column Height (in)
Column Height (mm)
Rotation (rads)
9_a1v0j0g0
10_a1v0j0g1
9_a1v0j0g0_Not Fixed
182
Figure 6.78: Steel Jacket Fixity Effects on Normalized MomentRotation
This plot agrees with the moment and rotation diagrams by showing the slight decrease in
normalized moment for Model 9_Not Fixed signifying increased moment resistance, while
the normalized rotation value increased dramatically and was above unity signifying a drop
in rotation resistance. As was discussed earlier, the increase in normalized rotation
beyond unity can be attributed an increase in material capacity associated with increased
loading rate during a blast event.
The model that had the jacket bearing on the foundation demonstrated a slight
decrease in normalized direct shear when compared to the column with a steel jacket gap.
However, it showed little effect in moment resistance at the base and an increase in
longitudinal reinforcement axial cross‐section strain, base rotation and normalized
rotation due to prying of the steel jacket. Therefore, due to the inconsistencies between
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1 1 10
M/M
ult
θ/θult (log10)
9_a1v0j0g0
9_a1v0j0g0_Not Fixed
10_a1v0j0g1
183
results the blast resistance benefits of extending the steel jacket so that it bears on the
foundation is inconclusive and may suggest that this level of base fixity is detrimental to
blast resistance.
6.8.2 Steel Jacket Retrofitting Effects on Blast Resistance
The effects of including a steel jacket on direct shear and flexural blast resistance
were examined by comparing the behavior of steel‐jacketed Model 10_a1v0j0g1 to a
similar unjacketed Model 10_a1v0 (10_Unjacketed). Steel‐jacketed Model 10 showed some
of the highest blast resistance among the columns having a steel jacket gap. Both are
slender columns with low transverse reinforcement ratios. In addition, Model 10_a1v0j0g1
has low jacket thickness ratio and high jacket gap ratio. The importance of steel jacket
retrofitting on blast resistance was evaluated through comparisons of observed damage,
longitudinal reinforcing bar axial cross‐section strain profiles, normalized direct shear
comparisons, moment diagrams, and normalized moment comparisons.
One of the most notable differences between jacketed and unjacketed models was
erosion and damage of the concrete at blast level for Model 10_Unjacketed that was not
shown in Model 10. Figure 6.79a‐c compares levels of damage between the unjacketed and
jacketed models via plastic strain contours at the same analysis time step. Figure 6.79a and
Figure 6.79b show Model 10 plastic strain contours for the steel jacket and the concrete on
the inside of the jacket, respectively. Figure 6.79c shows Model 10_Unjacketed plastic
strain contours.
184
a.) b.) c.)
Figure 6.79: Plastic Strain Damage Contours – a.) Model 10 steel jacket, b.) Model 10 concrete inside jacket, c.) Model 10_Unjacketed
These figures show extensive damage that occurred in the unjacketed column at the blast
level. Complete erosion of the cross‐section was observed in the column at the blast level
that subsequently contributed to complete failure and collapse of the column. The jacketed
column, however, showed no concrete erosion under the steel jacket. Small areas of
damage were observed at the base of the column where the gap existed but these did not
lead to column failure.
In consideration of direct shear, Figure 6.80 shows axial cross‐section strain profiles
for longitudinal reinforcement of the jacketed and unjacketed columns.
Blast Direction
185
Figure 6.80: Steel Jacket Retrofitting Effects on Longitudinal Bar Axial Crosssection Strain at Base
Although reductions in longitudinal bar strains were observed on the blast side of the
Column 10_Unjacketed, large increases in strain beyond Model 10 values on the back side
of the cross‐section were also evident. A larger variation in strains was observed in the
unjacketed model, whereas the jacketed model was able to distribute the demand more
evenly through all longitudinal bars in the cross‐section. Overall, the jacketed model
showed higher direct shear resistance.
Figure 6.81 compares the normalized direct shear values for the two columns.
‐28
‐18
‐8
2
12
22
‐711.2
‐511.2
‐311.2
‐111.2
88.8
288.8
488.8
688.8
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Distance fm
Centroid (in)
Distance fm
Centroid (mm)
Axial Strain [(mm/mm) or (in/in)]
10_a1v0j0g1 Jacketed
10_a1v0 Unjacketed
Blast Side
Back Side
186
Figure 6.81: Steel Jacket Retrofitting Effects on Normalized Direct Shear
This figure demonstrates that normalized direct shear was reduced in the unjacketed
model. However, the direct shear value for Model 10 reached higher levels due to the
presence of the jacket and its enhanced contribution to resisting direct shear at high
loading rates.
Figure 6.82 compares moment diagrams along the height of the two columns.
0.000
0.200
0.400
0.600
0.800
1.000
1.200
10_a1v0j0g1 Jacketed 10_a1v0 Unjacketed
V/Vult
Parametric Model
187
Figure 6.82: Overall Steel Jacket Retrofitting Effects on Moment
This plot shows that both columns reach similar values of maximum moment at the base.
The jacketed column was able to more evenly distribute the moment along the height of the
column while Model 10_Unjacketed showed high moments towards the base of the column
and virtually no moment at the top. This lack of moment distribution along the unjacketed
column prevented proper flexural resistance and contributed to failure at blast level.
The final flexure consideration for comparison of the two columns is normalized
moment in Figure 6.83.
‐3000 ‐2000 ‐1000 0 1000 2000 3000 4000 5000 6000
0
100
200
300
400
500
0
2000
4000
6000
8000
10000
12000
‐2E+09 0 2E+09 4E+09 6E+09
Moment (kft)
Column Height (in)
Column Height (mm)
Moment (Nmm)
10_a1v0 Unjacketed
10_a1v0j0g1 Jacketed
188
Figure 6.83: Overall Steel Jacket Retrofitting Effects on Normalized Moment
This plot demonstrates a large increase in normalized moment for the unjacketed column.
This increase, compared to Model 10, is due to larger moment demand in the column with
respect to a decreased capacity associated with the removal of the steel jacket. This allows
Model 10_Unjacketed to achieve values of moment closer to capacity.
Results in this section clearly demonstrate advantages of steel jacket retrofitting for
resisting close in blasts. Local damage and concrete erosion was prominent in the
unjacketed model case and were not observed in the corresponding jacketed case. Direct
shear resistance was also decreased and less evenly distributed throughout the base cross‐
section for the unjacketed model. Moment was also less evenly distributed along the
column height and more concentrated at the column base. As a result, the unjacketed
model was shown to reach its moment capacity near the base. Based on these observations
0.6
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
10_a1v0j0g1 Jacketed 10_a1v0 Unjacketed
M/M
ult
Parametric Model
189
it can be concluded that the jacketed model allowed for better blast resistance by
increasing direct shear resistance and enhancing flexure resistance along the column
height.
190
Chapter 7 Conclusions
7.1 Overview
This Chapter provides recommendations for detailing steel jacket bridge column
retrofits to resist blast loads. These recommendations were based on the result
comparisons from the previous chapter that were governed by the sensitivity of failure
mode results to the variation of critical parameters. Finally, recommendations for future
research are discussed.
7.1.1 Recommended Steel‐Jacketed Bridge Column Details to Resist Specific Failure Modes
This section presents a summary of the recommended steel jacket column details
from Chapter 6. They are obtained by examining desired effects on the studied failure
modes (direct shear, flexure and transverse shear) and matching appropriate parameter
variations those desired effects. Table 7.1 lists the desired effects in order of failure mode
importance. It also describes the advantages of achieving these desired effects with respect
to blast resistance. Listed sources used to establish the order of importance are also
provided.
Table 7.1: Desired Result Effects
Failure Mode
Importance (greatest to
least)
Effect Importance
(greatest to least)
Failure Mode
Desired Effect Advantages Source
1
Direct Shear
1 Decreased Normalized Direct Shear
Signifies reserve direct shear capacity, offers comparison
(Fujikura & Bruneau, 2008)
2 Decreased Long Bar Strain Signifies decrease in doweling (ACI, 2008)
191
action
3 Increased Steel Jacket Strain Signifies contribution of retrofit Sect. 6.2 & 6.3
2
Flexure
1 Decreased Normalized Moment
Signifies reserve moment capacity, offers comparison Sect. 6.4 & 6.5
2 Decreased Normalized Rotation
Signifies reserve rotation capacity, offers comparison Sect. 6.4 & 6.5
3 Increased Moment at Base Signifies increase in base flexural resistance Sect. 6.4 & 6.5
4 Increased Rotation at Base Signifies increase in base flexural resistance Sect. 6.4 & 6.5
5 Decreased Moment throughout Height
Signifies increase in base flexural resistance (no shift in critical location to mid‐height fm hinge formation at base) Sect. 6.4 & 6.5
3
Transverse Shear
1 Decreased Hoop Avg Transverse Strain Signifies decrease in demand
(Priestley et al., 1994)
2 Decreased Jacket Avg Transverse Strain Signifies decrease in demand
(Priestley et al., 1994)
Table 7.2 to Table 7.6 summarize recommended variations in parameters to obtain
the desired effects in relation to each failure mode. The entries below each critical
parameter column heading indicate whether an increase or decrease in the respective
parameter will achieve the desired effect. Finally, each table contains summary column
detailing recommendations. These recommendations are separated for columns having
steel jacket gaps and for those that have steel jackets developed into the foundation, which
was an important distinction observed in Chapter 6.
Table 7.2 lists the recommended variation in parameters to achieve the desired
effects that contribute most to direct shear resistance.
192
Table 7.2: Parameter Variation Recommendations to Resist Direct Shear
Desired Effects in order of importance (least to greatest)
Aspect Ratio (L/D)
Trans Reinf Ratio (ρs)
Jacket Thickness Ratio (tj/D)
Jacket Gap Ratio (Lg/D)
w/o Gap w/ Gap
w/o Gap w/ Gap
Increased Steel Jacket Strain no effect no effect no effect decrease decrease decrease Decreased Long Bar Strain increase increase decrease increase decrease decrease Decreased Normalized Direct Shear increase decrease decrease increase inconclusive decrease
Recommendations for Columns w/o Gap increase decrease increase decrease
Recommendations for Columns w/ Gap Inconclusive (increase) decrease inconclusive (decrease) decrease
Results for direct shear resistance offer insight for detailing steel‐jacket columns
with the jackets developed into the foundation. As summarized in Table 7.2, these columns
should have high aspect ratios, meaning that they are tall compared to their width. They
should have a low transverse reinforcement ratio, meaning that the number of transverse
hoops should be keep to a minimum, while spacing should be at its maximum possible
value. Steel jackets, for these columns, should be as thick as possible and adequate
development into the foundation is important.
Detailing to improve direct shear resistance for steel‐jacketed columns having a gap
is not as conclusive. Both aspect ratio and jacket thickness ratio results are unclear as to
the best variation. However, it is clear that these columns should contain low transverse
reinforcement ratios and transverse reinforcement spacing should be at the maximum
193
possible distance. The jacket gap ratio should also be kept to a minimal value to increase
direct shear resistance.
Table 7.3 addresses the influence of jacket base fixity on blast performance and if
using steel jackets as a blast retrofitting technique was advantageous to achieving the
desired direct shear effects. Steel jacket fixity effects on performance were studied relative
to Model 10_a1v0j0g1, which was shown to have the highest resistance to blast loads
among columns with a steel jacket gap. Steel jacket retrofitting was examined relative to
an unjacketed column.
Table 7.3: Steel Jacket Bearing and Retrofitting Recommendations to Resist Direct Shear
Desired Effects in order of importance (least to greatest)
Steel Jacket Bearing on the Foundation
Overall Steel Jacket Retrofitting
Increased Steel Jacket Strain no effect n/a
Decreased Long Bar Strain disadvantage advantage Decreased Normalized Direct Shear advantage disadvantage
Damage and Concrete Erosion n/a advantage
Recommendations for all Columns
Complete jacket base fixity is most advantageous, however extension to and
bearing on the foundation helps. advantage
Direct shear resistance is affected by both the presence of a steel jacket retrofit and
its base fixity or bearing on the foundation as summarized in Table 7.3. Steel jacket
retrofitting avoided excessive concrete damage and erosion and contributed to reduced
longitudinal bar strains. Complete steel jacket base fixity was shown to be most
advantageous in resisting direct shear, however, extension to and bearing on the
194
foundation was also shown to lower normalized direct shear values and enhanced blast
resistance.
Table 7.4 lists the recommended parameter variations to achieve desired effects
that heavily influence flexure resistance.
Table 7.4: Recommendations to Enhance Flexure Resistance
Desired Effects in order of importance (least to greatest)
Aspect Ratio (L/D)
Trans Reinf
Ratio (ρs)Jacket Thickness
Ratio (tj/D)Jacket Gap Ratio
(Lg/D)
w/o Gap w/ Gap
Low Aspect Ratios
High Aspect Ratios
Decreased Moment throughout Height increase no effect decrease decrease decrease decrease
Decreased Rotation at Base decrease inconclusive increase increase increase decrease
Increased Moment at Base increase no effect increase decrease decrease decrease
Decreased Normalized Rotation inconclusive inconclusive increase decrease decrease decrease
Decreased Normalized Moment increase no effect increase decrease decrease decrease
Recommendations for Columns w/o Gap increase inconclusive increase decrease
Recommendations for Columns w/ Gap increase inconclusive decrease decrease
Details which enhance flexural resistance for steel‐jacketed columns without gaps include
designing columns with a high aspect ratio, meaning that they are tall compared to their
width. Steel jackets should be as thick as possible and their development into the
foundation is critical.
195
Steel‐jacketed columns with a gap should also be tall compared to their width.
However, it is more advantageous for flexural resistance to decrease the thickness of the
steel jacket. The jacket gap ratio should also be keep to a minimal value to increase flexure
resistance.
Table 7.5 addresses the influence of jacket base fixity on blast performance and if
using steel jackets as a blast retrofitting technique was advantageous to enhance flexural
resistance. Steel jacket fixity effects on performance were studied relative to Model
10_a1v0j0g1, which was shown to have the highest resistance to blast loads among
columns with a steel jacket gap. Steel jacket retrofitting was examined relative to an
unjacketed column.
Table 7.5: Steel Jacket Bearing and Retrofitting Recommendations to Enhance Flexure Resistance
Desired Effects in order of importance (least to greatest)
Steel Jacket Bearing on the Foundation
Overall Steel Jacket Retrofitting
Decreased Moment throughout Height no effect advantage
Decreased Rotation at Base disadvantage n/a
Increased Moment at Base no effect no effect
Decreased Normalized Rotation disadvantage n/a
Decreased Normalized Moment advantage advantage
Recommendations for all Columns inconclusive advantage
Flexural enhancement was shown to be affected by both the presence of a steel
jacket and its level of base fixity. Including a steel jacket retrofit enhanced moment
resistance both along the height of the column and at its base. The jacket bearing on the
196
foundation showed an increase in normalized moment value, which is advantageous to
flexural resistance. However, little effect on moment resistance at the base was shown and
base rotation and normalized rotation values were shown to increase due to prying caused
by the jacket bearing on the foundation. These inconsistencies between flexural results
lead to an inconclusive determination of the advantageous of the level of base fixity on
enhanced flexural resistance.
Table 7.6 lists the recommended variation in parameters to achieve the desired
effects which contribute most to transverse shear resistance.
Table 7.6: Recommendations to Resist Transverse Shear Failure
Desired Effects in order of
importance (least to greatest)
Aspect Ratio (L/D)
Trans Reinf Ratio (ρs)
Jacket Thickness Ratio (tj/D)
Jacket Gap Ratio (Lg/D)
Along Height Base
Decreased Jacket Avg Transverse Strain increase increase increase increase decrease
Decreased Hoop Avg Transverse Strain increase increase increase increase decrease
Recommendations for all Columns increase increase increase decrease
Results are not dependent on the steel jacket gap for transverse shear resistance. Column
should be slender and should contain the highest amount of transverse reinforcement as
possible with transverse spacing at a minimum distance. They should also be detailed with
steel jackets as thick as possible. The jacket gap should be keep to a minimal distance.
197
However, as mentioned in Section 6.6, this failure mode is not critical and column detailing
recommendations for direct shear and flexure supersede these recommendations.
7.1.2 Recommended Steel‐Jacketed Bridge Column Details to Resist Blast Loads
This section recommends variations in critical parameters for overall resistance to
blast loads when considering all three examined failure modes. Discussion of the critical
parameters begins by presenting overall recommendations for jacket gap and jacket
thickness ratios. Recommendations related to including a steel jacket and its level of base
fixity are also discussed. The discussion concludes with recommendations for in the
column transverse reinforcement and column aspect ratios.
Decreasing the jacket gap ratio most significantly influenced direct shear resistance,
which is the most common failure mode of for steel‐jacketed bridge columns under blast.
In this research, a jacket gap ratio equaling zero, which implied that the jacket was
developed at the base of the column into the footing, was observed to shift the failure from
a brittle direct shear mode to a more ductile flexural mode. This was observed in Table 7.4,
where decrease in jacket gap ratio produced an increase in moment at the base of the
column and implied that the column was resisting the load through more largely flexural
behavior. It was also shown that extending the steel jacket so that it was bearing on the
foundation was inconclusive to direct shear resistance and flexural enhancement (see
Table 7.3 and Table 7.5) and may suggest that this level of base fixity is detrimental to blast
resistance. Therefore, the jacket gap should be kept to a minimal distance if developed into
the foundation to resist blast loads.
198
The presence of the steel jacket was shown to be advantageous for both direct shear
and flexural resistance compared to unjacketed columns, and therefore, is advantageous to
blast resistance. The effects of jacket thickness ratio were shown to be dependent on
whether the steel‐jacketed column has a gap for both direct shear and flexure. An increase
in jacket thickness ratio was advantageous in resisting blast loads for columns that do not
have a steel jacket gap. For columns not having a steel jacket gap, steel jackets should be
detailed at maximum thickness to resist blast loads.
However, for columns that having a steel jacket gap results were inconclusive
whether an increased jacket thickness ratio was advantageous. When considering both
direct shear failure and flexure, it seemed that a decrease in jacket thickness ratio for
columns having a gap may be more advantageous. As mentioned in Section 6.5.3, an
increase in jacket thickness ratio provided higher flexural capacity and stiffness at column
mid‐height, but may shift the column’s resistance to blast loads at its base from flexural to
direct shear. Therefore, for steeljacketed columns having a gap, the jackets should be
detailed with a minimum thickness to control lateral stiffness and enhance flexural resistance
at the column base, thereby increasing blast resistance.
A decrease in transverse reinforcement ratio proved to be advantageous for the
resistance of direct shear. This is not an option in retrofit schemes; however, it is helpful
for the design of new bridges having steel‐jacketed columns to resist blast loads. Although,
a decrease in transverse reinforcement ratio was shown to decrease resistance to
transverse shear, transverse shear was not shown to be a critical failure mode, and
therefore, the advantage of increasing direct shear resistance outweighed the disadvantage
of decreasing transverse shear resistance. The variation in transverse reinforcement ratio
199
produced inconclusive effects on flexural response for the blast loaded columns. Once
again, this may be due to the susceptibility of the columns to failing in direct shear even in
flexural loading situations. Therefore, the number of transverse hoops should be keep to a
minimum, while their spacing should be detailed at maximum possible distance to resist blast
loads.
An increase in column aspect ratio was shown to be advantageous for resisting all
failure mode results except normalized direct shear in columns having steel jacket gaps.
This is not an option in retrofit schemes; however, it is helpful in the design of new steel‐
jacketed bridge columns to resist blast loads. The majority of the results, however,
concluded that columns with high aspect ratios were able to reach higher direct shear
resistance and increased flexural resistance at the column base. Therefore, steeljacketed
columns not having a steel jacket gap should be detailed with a high aspect ratio to resist
blast loads.
The influence of aspect ratio on direct shear was inconclusive for columns having a
gap. All results for all failure modes showed that increased aspect ratio is advantageous in
resisting or promoting their respective desired result effects except the normalized direct
shear failure mode. Results from this failure mode conversely showed that decrease in
aspect ratio was advantageous for preventing direct shear failure. Since normalized direct
shear values were considered to be the most important determination of direct shear, and
therefore, blast resistance (see Table 7.1), no conclusive column aspect ratio
recommendation can be made for detailing steeljacketed columns with a gap.
7.1.3 Recommendations for Future Research
200
Since reduction of jacket gap ratio was the single most beneficial variation of the
four critical parameters examined in this study, further investigation into methods to
achieve this are warranted. However, due to the development of the steel jacket at the
column base, as in this research, the flexural demand on the footing and possibly the pier
cap could increase. Therefore, steel jacket detailing and placement methods that effectively
development and distribute this demand would be advantageous and may prevent excess
footing or pier cap damage. Further investigation into varying steel jacket gap distance
would also be beneficial as this research only considered two extreme limits with limited
examination of jacket base fixity effects. Also, investigating the effects of column cross‐
section geometries would be beneficial to further understanding blast response.
201
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206
207
Appendix A Axial Cross‐section Strain Profiles
The appendix contains plots of all parametric model axial cross‐section strain
profiles throughout the eleven cross‐sections taken along the height of the column.
Longitudinal bar strain and steel jacket strain are plotted on the same graph at each
location.
208
Model 1_a0v0j0g1:
1.28E‐02
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
1_a0v0j0g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
209
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
210
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
211
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
212
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
213
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
1_a0v0j0g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
214
Model 2_a0v0j0g1:
‐762
‐562
‐362
‐162
38
238
438
638
‐2.50E‐02 ‐1.50E‐02 ‐5.00E‐03 5.00E‐03 1.50E‐02 2.50E‐02
Distance fm
Centriod (mm)
Long Strain
2_a0v0j0g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
215
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
216
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
217
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
218
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
219
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
2_a0v0j0g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
220
Model 3_a0v0j1g0:
1.03E‐02
‐762
‐562
‐362
‐162
38
238
438
638
‐1.5E‐02 ‐1.0E‐02 ‐5.0E‐03 0.0E+00 5.0E‐03 1.0E‐02 1.5E‐02
Distance fm
Centriod (mm)
Long Strain
3_a0v0j1g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
221
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
222
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
223
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
224
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
225
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
3_a0v0j1g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
226
Model 4_a0v0j1g1:
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
4_a0v0j1g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
227
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
228
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
229
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
230
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
231
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
4_a0v0j1g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
232
Model 5_a0v1j0g0:
1.35E‐02
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
5_a0v1j0g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
233
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
234
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
235
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
236
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
237
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
5_a0v1j0g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
238
Model 6_a0v1j0g1:
1.98E‐02
1.17E‐04
‐762
‐562
‐362
‐162
38
238
438
638
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02 2.50E‐02
Distance fm
Centriod (mm)
Long Strain
6_a0v1j0g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
239
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
240
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
241
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
242
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
243
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
6_a0v1j0g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
244
Model 7_a0v1j1g0:
1.02E‐02
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
7_a0v1j1g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
245
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
246
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
247
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
248
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
249
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
7_a0v1j1g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
250
Model 8_a0v1j1g1:
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐5.00E‐03 5.00E‐03 1.50E‐02 2.50E‐02
Distance fm
Centriod (mm)
Long Strain
8_a0v1j1g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
251
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
252
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
253
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
254
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
255
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
8_a0v1j1g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
256
Model 9_a1v0j0g0:
1.25E‐02
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
9_a1v0j0g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
257
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
258
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
259
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
260
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
261
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
9_a1v0j0g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
262
Model 10_a1v0j0g1:
4.32E‐04
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
10_a1v0j0g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
263
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
264
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
265
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
266
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
267
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
10_a1v0j0g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
268
Model 11_a1v0j1g0:
7.82E‐03
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
11_a1v0j1g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
269
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
270
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
271
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
272
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
273
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
11_a1v0j1g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
274
Model 12_a1v0j1g1:
5.54E‐05
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
12_a1v0j1g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
275
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
276
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
277
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
278
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
279
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
12_a1v0j1g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
280
Model 13_a1v1j0g0:
1.25E‐02
‐762
‐562
‐362
‐162
38
238
438
638
‐2.E‐02 ‐1.E‐02 ‐5.E‐03 0.E+00 5.E‐03 1.E‐02 2.E‐02
Distance fm
Centriod (mm)
Long Strain
13_a1v1j0g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
281
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
282
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
283
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
284
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
285
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
13_a1v1j0g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
286
Model 14_a1v1j0g1:
1.04E‐03
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
14_a1v1j0g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
287
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
288
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
289
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
290
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
291
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
14_a1v1j0g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
292
Model 15_a1v1j1g0:
8.82E‐03
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
15_a1v1j1g0 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
293
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
294
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
295
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
296
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
297
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
15_a1v1j1g0 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
298
Model 16_a1v1j1g1:
‐3.00E‐04
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centriod (mm)
Long Strain
16_a1v1j1g1 Base Crosssectional Strain Profile
Long Bar
Jacket
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.10h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
299
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.20h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.30h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
300
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.40h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.50h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
301
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.60h Crosssectional Strain Profile
Long Bars
Jacket
Back Side
Blast Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.70h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
302
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.80h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 0.90h Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
303
‐762
‐562
‐362
‐162
38
238
438
638
‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02
Distance fm
Centroid (mm)
Long Strain
16_a1v1j1g1 Top Crosssectional Strain Profile
Long Bars
Jacket
Blast Side
Back Side
304
Appendix B Direct Shear Comparisons
‐1.50E+07
‐1.00E+07
‐5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
1_a0v0j0g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.00E+07
‐8.00E+06
‐6.00E+06
‐4.00E+06
‐2.00E+06
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
2_a0v0j0g1 Direct Shear Comparison at Column Base
Blast
Capacity
305
‐2.00E+07
‐1.00E+07
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
3_a0v0j1g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.00E+07
‐8.00E+06
‐6.00E+06
‐4.00E+06
‐2.00E+06
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
4_a0v0j1g1 Direct Shear Comparison at Column Base
Blast
Capacity
306
‐1.50E+07
‐1.00E+07
‐5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
5_a0v1j0g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.00E+07
‐8.00E+06
‐6.00E+06
‐4.00E+06
‐2.00E+06
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
6_a0v1j0g1 Direct Shear Comparison at Column Base
Blast
Capacity
307
‐2.00E+07
‐1.00E+07
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
7_a0v1j1g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.00E+07
‐8.00E+06
‐6.00E+06
‐4.00E+06
‐2.00E+06
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
0.00E+00 5.00E‐02 1.00E‐01 1.50E‐01 2.00E‐01 2.50E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
8_a0v1j1g1 Direct Shear Comparison at Column Base
Blast
Capacity
308
‐1.50E+07
‐1.00E+07
‐5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
9_a1v0j0g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.00E+07
‐8.00E+06
‐6.00E+06
‐4.00E+06
‐2.00E+06
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
10_a1v0j0g1 Direct Shear Comparison at Column Base
Blast
Capacity
309
‐2.00E+07
‐1.00E+07
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
5.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
11_a1v0j1g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.50E+07
‐1.00E+07
‐5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
12_a0v0j1g1 Direct Shear Comparison at Column Base
Blast
Capacity
310
‐1.50E+07
‐1.00E+07
‐5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
2.50E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
13_a1v1j0g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.20E+07
‐1.00E+07
‐8.00E+06
‐6.00E+06
‐4.00E+06
‐2.00E+06
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
14_a1v1j0g1 Direct Shear Comparison at Column Base
Blast
Capacity
311
‐2.00E+07
‐1.00E+07
0.00E+00
1.00E+07
2.00E+07
3.00E+07
4.00E+07
5.00E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
15_a1v1j1g0 Direct Shear Comparison at Column Base
Blast
Capacity
‐1.50E+07
‐1.00E+07
‐5.00E+06
0.00E+00
5.00E+06
1.00E+07
1.50E+07
0.00E+00 1.00E‐01 2.00E‐01 3.00E‐01 4.00E‐01 5.00E‐01
Base Crosssectional Direct Shear Force
(N)
Time (sec)
16_a1v1j1g1 Direct Shear Comparison at Column Base
Blast
Capacity
312
Appendix C Moment and Rotational Diagrams
This appendix contains the moment and rotational diagrams for each parametric
model.
Model 1_a0v0j0g0:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10
Column Height (mm)
Moment (Nmm)
1_a0v0j0g0 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.E‐02 ‐5.E‐03 0.E+00 5.E‐03 1.E‐02 2.E‐02 2.E‐02 3.E‐02 3.E‐02 4.E‐02 4.E‐02
Column Height (mm)
Rotation (rads)
1_a0v0j0g0 Rotation Diagram
313
Model 2_a0v0j0g1:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09
Column Height (mm)
Moment (Nmm)
2_a0v0j0g1 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02 2.50E‐02
Column Height (mm)
Rotation (rads)
2_a0v0j0g1 Rotation Diagram
314
Model 3_a0v0j1g0:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.50E+10 ‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10 1.50E+10
Column Height (mm)
Moment (Nmm)
3_a0v0j1g0 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐4.E‐03 ‐2.E‐03 0.E+00 2.E‐03 4.E‐03 6.E‐03 8.E‐03 1.E‐02 1.E‐02 1.E‐02 2.E‐02 2.E‐02
Column Height (mm)
Rotation (rads)
3_a0v0j1g0 Rotation Diagram
315
Model 4_a0v0j1g1:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.50E+10 ‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09
Column Height (mm)
Moment (Nmm)
4_a0v0j1g1 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02
Column Height (mm)
Rotation (rads)
4_a0v0j1g1 Rotation Diagram
316
Model 5_a0v1j0g0:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10
Column Height (mm)
Moment (Nmm)
5_a0v1j0g0 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02 2.50E‐02
Column Height (mm)
Rotation (rads)
5_a0v1j0g0 Rotation Diagram
317
Model 6_a0v1j0g1:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10
Column Height (mm)
Moment (Nmm)
6_a0v1j0g1 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.00E‐02 ‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02 2.50E‐02
Column Height (mm)
Rotation (rads)
6_a0v1j0g1 Rotation Diagram
318
Model 7_a0v1j1g0:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.50E+10 ‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10 1.50E+10
Column Height (mm)
Moment (Nmm)
7_a0v1j1g0 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02
Column Height (mm)
Rotation (rads)
7_a0v1j1g0 Rotation Diagram
319
Model 8_a0v1j1g1:
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐1.50E+10 ‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09
Column Height (mm)
Moment (Nmm)
8_a0v1j1g1 Moment Diagram
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐6.E‐03 ‐4.E‐03 ‐2.E‐03 0.E+00 2.E‐03 4.E‐03 6.E‐03 8.E‐03 1.E‐02 1.E‐02 1.E‐02
Column Height (mm)
Rotation (rads)
8_a0v1j1g1 Rotation Diagram
320
Model 9_a1v0j0g0:
‐284
1716
3716
5716
7716
9716
11716
13716
‐5.00E+09 0.00E+00 5.00E+09 1.00E+10
Column Height (mm)
Moment (Nmm)
9_a1v0j0g1 Moment Diagram
‐284
1716
3716
5716
7716
9716
11716
13716
‐4.E‐03 ‐2.E‐03 0.E+00 2.E‐03 4.E‐03 6.E‐03 8.E‐03 1.E‐02 1.E‐02 1.E‐02 2.E‐02 2.E‐02
Column Height (mm)
Rotation (rads)
9_a1v0j0g0 Rotation Diagram
321
0
2000
4000
6000
8000
10000
12000
‐4.00E+09 ‐2.00E+09 0.00E+00 2.00E+09 4.00E+09 6.00E+09 8.00E+09
Column Height (mm)
Moment (Nmm)
10_a1v0j0g1 Moment Diagram
0
2000
4000
6000
8000
10000
12000
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02 2.50E‐02 3.00E‐02
Column Height (mm)
Rotation (rads)
10_a1v0j0g1 Rotation Diagram
322
Model 11_a1v0j1g0:
‐284
1716
3716
5716
7716
9716
11716
13716
‐5.00E+09 0.00E+00 5.00E+09 1.00E+10 1.50E+10
Column Height (mm)
Moment (Nmm)
11_a1v0j1g0 Moment Diagram
‐284
1716
3716
5716
7716
9716
11716
13716
‐4.E‐03 ‐2.E‐03 0.E+00 2.E‐03 4.E‐03 6.E‐03 8.E‐03 1.E‐02 1.E‐02 1.E‐02
Column Height (mm)
Rotation (rads)
11_a1v0j1g0 Rotation Diagram
323
Model 12_a1v0j1g1:
‐284
1716
3716
5716
7716
9716
11716
13716
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09
Column Height (mm)
Moment (Nmm)
12_a1v0j1g1 Moment Diagram
‐284
1716
3716
5716
7716
9716
11716
13716
‐4.E‐03 ‐2.E‐03 0.E+00 2.E‐03 4.E‐03 6.E‐03 8.E‐03 1.E‐02 1.E‐02 1.E‐02 2.E‐02
Column Height (mm)
Rotation (rads)
12_a1v0j1g1 Rotation Diagram
324
Model 13_a1v1j0g0:
0
2000
4000
6000
8000
10000
12000
‐5.00E+09 0.00E+00 5.00E+09 1.00E+10
Column Height (mm)
Moment (Nmm)
13_a1v1j0g0 Moment Diagram
0
2000
4000
6000
8000
10000
12000
‐5.00E‐03 0.00E+00 5.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02 2.50E‐02 3.00E‐02
Column Height (mm)
Rotation (rads)
13_a1v1j0g0 Rotation Diagram
325
Model 14_a1v1j0g1:
0
2000
4000
6000
8000
10000
12000
‐4.00E+09 ‐2.00E+09 0.00E+00 2.00E+09 4.00E+09 6.00E+09 8.00E+09
Column Height (mm)
Moment (nmm)
14_a1v1j0g1 Moment Diagram
0
2000
4000
6000
8000
10000
12000
‐6.00E‐03 ‐5.00E‐03 ‐4.00E‐03 ‐3.00E‐03 ‐2.00E‐03 ‐1.00E‐03 0.00E+00 1.00E‐03 2.00E‐03 3.00E‐03
Column Height (mm)
Rotation (rads)
14_a1v1j0g1 Rotation Diagram
326
Model 15_a1v1j1g0:
0
2000
4000
6000
8000
10000
12000
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10 1.50E+10
Column Height (mm)
Moment (Nmm)
15_a1v1j1g0 Moment Digaram
0
2000
4000
6000
8000
10000
12000
‐2.E‐03 0.E+00 2.E‐03 4.E‐03 6.E‐03 8.E‐03 1.E‐02 1.E‐02 1.E‐02 2.E‐02 2.E‐02
Column Height (mm)
Rotation (rads)
15_a1v1j1g0 Rotation Diagram
327
Model 16_a1v1j1g1:
0
2000
4000
6000
8000
10000
12000
‐1.00E+10 ‐5.00E+09 0.00E+00 5.00E+09 1.00E+10
Column Height (mm)
Moment (Nmm)
16_a1v1j1g1 Moment Diagram
0
2000
4000
6000
8000
10000
12000
‐1.E‐02 ‐5.E‐03 0.E+00 5.E‐03 1.E‐02 2.E‐02 2.E‐02 3.E‐02 3.E‐02 4.E‐02 4.E‐02
Column Height (mm)
Rotation (rads)
16_a1v1j1g1 Rotation Diagram
328
Appendix D Moment‐Rotation Capacity Curves
This appendix contains the moment‐rotation capacity curves for all parametric
columns that were used to determine normalized moment‐rotation denominator values.
‐2.0E+10
‐1.5E+10
‐1.0E+10
‐5.0E+09
0.0E+00
5.0E+09
1.0E+10
1.5E+10
2.0E+10
‐1.5E‐02 ‐1.0E‐02 ‐5.0E‐03 0.0E+00 5.0E‐03 1.0E‐02 1.5E‐02
Mom
ent (Nmm)
Rotation (rads)
1_a0v0j0g0 MomentRotation Comparison
Static Mom‐Rot Capacity Curve
329
‐3.0E+09
‐2.0E+09
‐1.0E+09
0.0E+00
1.0E+09
2.0E+09
3.0E+09
‐4.0E‐02 ‐2.0E‐02 0.0E+00 2.0E‐02 4.0E‐02
Mom
ent (Nmm)
Rotation (rads)
2_a0v0j0g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
330
‐4.0E+10
‐3.0E+10
‐2.0E+10
‐1.0E+10
0.0E+00
1.0E+10
2.0E+10
3.0E+10
4.0E+10
‐8.0E‐02 ‐6.0E‐02 ‐4.0E‐02 ‐2.0E‐02 0.0E+00 2.0E‐02 4.0E‐02 6.0E‐02 8.0E‐02
Mom
ent (Nmm)
Rotation (rads)
3_a0v0j1g0 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
0.00E+00
5.00E+08
1.00E+09
1.50E+09
2.00E+09
2.50E+09
3.00E+09
3.50E+09
4.00E+09
2.54E‐03 7.54E‐03 1.25E‐02 1.75E‐02 2.25E‐02 2.75E‐02 3.25E‐02 3.75E‐02
Mom
ent (Nmm)
Rotation (rads)
4_a0v0j1g1 Static MomentRotation Capacity Curve
Static Mot‐Rot Capacity Curve
331
‐2.5E+10
‐2E+10
‐1.5E+10
‐1E+10
‐5E+09
0
5E+09
1E+10
1.5E+10
2E+10
2.5E+10
‐0.06 ‐0.04 ‐0.02 0 0.02 0.04 0.06
Mom
ent (Nmm)
Rotation (rads)
5_a0v1j0g0 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
‐4E+09
‐3E+09
‐2E+09
‐1E+09
0
1E+09
2E+09
3E+09
4E+09
‐0.15 ‐0.1 ‐0.05 0 0.05 0.1 0.15
Mom
ent (Nmm)
Rotation (rads)
6_a0v1j0g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
332
‐4.00E+10
‐3.00E+10
‐2.00E+10
‐1.00E+10
0.00E+00
1.00E+10
2.00E+10
3.00E+10
4.00E+10
‐1.00E‐01 ‐5.00E‐02 0.00E+00 5.00E‐02 1.00E‐01
Mom
ent (Nmm)
Rotation (rads)
7_a0v1j1g0 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
0.00E+00
5.00E+08
1.00E+09
1.50E+09
2.00E+09
2.50E+09
3.00E+09
3.50E+09
4.00E+09
3.53E‐03 8.53E‐03 1.35E‐02 1.85E‐02 2.35E‐02 2.85E‐02
Mom
ent (Nmm)
Rotation (rads)
8_a0v1j1g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
333
‐2.5E+10
‐2E+10
‐1.5E+10
‐1E+10
‐5E+09
0
5E+09
1E+10
1.5E+10
2E+10
2.5E+10
‐0.2 ‐0.15 ‐0.1 ‐0.05 0 0.05 0.1 0.15 0.2
Mom
ent (Nmm)
Rotation (rads)
9_a10j0g0 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
‐1.5E+10
‐1E+10
‐5E+09
0
5E+09
1E+10
1.5E+10
‐0.08 ‐0.06 ‐0.04 ‐0.02 0 0.02 0.04 0.06 0.08
Mom
ent (Nmm)
Rotation (rads)
10_a1v0j0g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
334
‐5E+10
‐4E+10
‐3E+10
‐2E+10
‐1E+10
0
1E+10
2E+10
3E+10
4E+10
5E+10
‐0.15 ‐0.1 ‐0.05 0 0.05 0.1 0.15
Mom
ent (Nmm)
Rotation (rads)
11_a1v0j1g0 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
‐1.5E+10
‐1E+10
‐5E+09
0
5E+09
1E+10
1.5E+10
‐0.008 ‐0.006 ‐0.004 ‐0.002 0 0.002 0.004 0.006 0.008
Mom
ent (Nmm)
Rotation (rads)
12_a1v0j1g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
335
‐2.5E+10
‐2.0E+10
‐1.5E+10
‐1.0E+10
‐5.0E+09
0.0E+00
5.0E+09
1.0E+10
1.5E+10
2.0E+10
2.5E+10
‐1.0E‐01 ‐5.0E‐02 0.0E+00 5.0E‐02 1.0E‐01
Mom
ent (Nmm)
Rotation (rads)
13_a1v1j0g0 MomentRotation Comparison
Static Mot‐Rot …
‐1.5E+10
‐1.0E+10
‐5.0E+09
0.0E+00
5.0E+09
1.0E+10
1.5E+10
‐2.0E‐02 ‐1.5E‐02 ‐1.0E‐02 ‐5.0E‐03 0.0E+00 5.0E‐03 1.0E‐02 1.5E‐02 2.0E‐02
Mom
ent (Nmm)
Rotation (rads)
14_a1v1j0g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
336
‐5.0E+10
‐4.0E+10
‐3.0E+10
‐2.0E+10
‐1.0E+10
0.0E+00
1.0E+10
2.0E+10
3.0E+10
4.0E+10
5.0E+10
‐1.0E‐01 ‐5.0E‐02 0.0E+00 5.0E‐02 1.0E‐01
Mom
ent (Nmm)
Rotation (rads)
15_a1v1j1g0 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
‐1E+10
‐8E+09
‐6E+09
‐4E+09
‐2E+09
0
2E+09
4E+09
6E+09
8E+09
1E+10
‐0.01 ‐0.005 0 0.005 0.01
Mom
ent (Nmm)
Rotation (rads)
16_a1v1j1g1 MomentRotation Comparison
Static Mot‐Rot Capacity Curve
337
Appendix E Transverse Strain Profiles
This appendix contains the transverse strain profiles along the height of the column
for each parametric model. These graphs plot minimum, maximum and average transverse
strains for both the hoop reinforcement and the steel jacket.
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐0.0015 ‐0.001 ‐0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003
Height (mm)
Trans Strain
1_a0v0j0g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐0.004 ‐0.002 0 0.002 0.004 0.006 0.008 0.01
Column Height (mm)
Trans Strain
2_a0v0j0g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
338
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐0.001 ‐0.0005 0 0.0005 0.001 0.0015 0.002
Height (mm)
Trans Strain
3_a0v0j1g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0
500
1000
1500
2000
2500
3000
3500
4000
4500
‐0.006 ‐0.004 ‐0.002 0 0.002 0.004 0.006 0.008 0.01
Height (mm)
Trans Strain
4_a0v0j1g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
339
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
3500.00
4000.00
4500.00
‐0.002 ‐0.0015 ‐0.001 ‐0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003
Height (mm)
Trans Strain
5_a0v1j0g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
3500.00
4000.00
4500.00
‐0.004 ‐0.002 0 0.002 0.004 0.006 0.008
Height (mm)
Trans Strain
6_a0v1j0g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
340
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
3500.00
4000.00
4500.00
‐0.001 ‐0.0005 0 0.0005 0.001 0.0015
Height (mm)
Trans Strain
7_a0v1j1g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0.00
500.00
1000.00
1500.00
2000.00
2500.00
3000.00
3500.00
4000.00
4500.00
‐0.004 ‐0.002 0 0.002 0.004 0.006
Height (mm)
Trans Strain
8_a0v1j1g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
341
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐1.50E‐03 ‐1.00E‐03 ‐5.00E‐04 0.00E+00 5.00E‐04 1.00E‐03 1.50E‐03 2.00E‐03 2.50E‐03
Height (mm)
Trans Strain
9_a1v0j0g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐0.002 0 0.002 0.004 0.006 0.008 0.01
Height (mm)
Trans Strain
10_a1v0j0g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
342
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐0.001 ‐0.0005 0 0.0005 0.001 0.0015 0.002 0.0025
Height (mm)
Trans Strain
11_a1v0j1g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐4.00E‐03 ‐2.00E‐03 0.00E+00 2.00E‐03 4.00E‐03 6.00E‐03 8.00E‐03
Height (mm)
Trans Strain
12_a1v0j1g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
343
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐2.00E‐03‐1.50E‐03‐1.00E‐03‐5.00E‐040.00E+005.00E‐041.00E‐031.50E‐032.00E‐032.50E‐03
Height (mm)
Trans Strain
13_a1v1j0g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐3.00E‐03‐2.00E‐03‐1.00E‐030.00E+001.00E‐032.00E‐033.00E‐034.00E‐035.00E‐036.00E‐03
Height (mm)
Trans Strain
14_a1v1j0g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
344
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐8.00E‐04‐6.00E‐04‐4.00E‐04‐2.00E‐040.00E+002.00E‐044.00E‐046.00E‐048.00E‐041.00E‐031.20E‐03
Height (mm)
Trans Strain
15_a1v1j1g0 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
‐3.00E‐03 ‐2.00E‐03 ‐1.00E‐03 0.00E+00 1.00E‐03 2.00E‐03 3.00E‐03 4.00E‐03 5.00E‐03
Height (mm)
Trans Strain
16_a1v1j1g1 Transverse Strain Along Column Height
Bar Max
Bar Ave
Bar Min
Jacket Max
Jacket Ave
Jacket Min
