Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

PUBLICATION NO. FHWA-HRT-23-077 OCTOBER 2023 

Structural Design with Ultra-High 

Performance Concrete
FOREWORD

Structural design of bridge components is a well-established field that is built on the fundamentals 

of engineering mechanics, the behaviors of the engaged materials, and the engineering judgment 

of the associated experts. Ultra-high performance concrete (UHPC) is a class of concrete with 

compelling properties that lie outside of those for which existing structural design guidance was 

developed. The Federal Highway Administration’s efforts to advance the state of the practice 

related to UHPC have recently focused on facilitating the development of formal U.S.-based 

structural design provisions for this class of materials. The information presented in this document 

provides background, context, and foundational knowledge to bridge owners, designers, and 

associated individuals interested in engaging this innovative solution in the design of our Nation’s 

highway bridges.

Jean A. Nehme, Ph.D., P.E.,

Director, Office of Infrastructure

Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

(USDOT) in the interest of information exchange. The U.S. Government assumes no liability for 

the use of the information contained in this document.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

manufacturers’ names appear in this report only because they are considered essential to the 

objective of the document.

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement.
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-23-077

2. Government Accession No. 3. Recipient’s Catalog No.

4. Title and Subtitle

Structural Design with Ultra-High Performance Concrete

5. Report Date

October 2023

6. Performing Organization Code:

7. Author(s)

Benjamin A. Graybeal (ORCID: 0000-0002-3694-1369) and 

Rafic G. Helou (ORCID: 0000-0003-0061-9439)

8. Performing Organization Report No.

9. Performing Organization Name and Address

Office of Infrastructure Research and Development 

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

10. Work Unit No.

11. Contract or Grant No.

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development 

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Final Report; 2017–2023

14. Sponsoring Agency Code

HRDI-40

15. Supplementary Notes

16. Abstract

Ultra-high performance concrete (UHPC) is a class of concrete that has emerged as a compelling material for use 

in the design, construction, and preservation of structures. UHPC provides mechanical and durability properties 

that are enhanced and distinct relative to conventional concrete, thus necessitating the use of design methodologies 

that facilitate the appropriate use of the material. With recent research and development-based advancements 

related to UHPC, and with a growing interest from bridge owners in the possibilities presented by this material, the 

U.S. bridge engineering community has an opportunity to consider and potentially adopt formal structural design 

guidance for UHPC structural elements. This document aims to support the development of formal UHPC 

structural design guidance by delivering a potential design framework for the community to consider. The report 

also provides two examples that will assist readers in better understanding key concepts included in the design 

framework.

17. Key Words

UHPC, structural design, material models, flexure, 

shear, design examples, analysis, pretensioned 

girder, fiber reinforced concrete

18. Distribution Statement

No restrictions. This document is available to the public 

through the National Technical Information Service, 

Springfield, VA 22161.

https://www.ntis.gov

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

290

22. Price

N/A

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized.
ii
iii

TABLE OF CONTENTS

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

Objective and Use ................................................................................................................... 1

Report Content........................................................................................................................ 1

CHAPTER 2. UHPC STRUCTURAL DESIGN........................................................................ 3

UHPC ....................................................................................................................................... 3

Material Properties................................................................................................................. 3

Availability............................................................................................................................... 3

UHPC Usage in Bridge Construction.................................................................................... 3

FHWA Research on Material Test Methods for UHPC...................................................... 4

FHWA Research on Structural Behavior of UHPC ............................................................ 5

Conceptual Draft UHPC Structural Design Guidance........................................................ 5

Analysis and Design Examples .............................................................................................. 7

CHAPTER 3. CONCLUDING REMARKS............................................................................... 9

APPENDIX A. GUIDE SPECIFICATION FOR STRUCTURAL DESIGN WITH 

ULTRA-HIGH PERFORMANCE CONCRETE.................................................................... 11

SECTION 1. STRUCTURAL DESIGN WITH ULTRA-HIGH PERFORMANCE 

CONCRETE................................................................................................................................ 13

1.1. Scope................................................................................................................................ 13

1.1.1. General.................................................................................................................... 13

1.1.2. Design Philosophy .................................................................................................. 14

1.1.3. Loads and Load Combinations ............................................................................... 15

1.1.4. Limitations.............................................................................................................. 16

1.2. Definitions....................................................................................................................... 16

1.3. Notation........................................................................................................................... 21

1.4. Material Properties........................................................................................................ 26

1.4.1. General.................................................................................................................... 26

1.4.2. Ultra-High Performance Concrete .......................................................................... 26

1.4.2.1. General........................................................................................................... 26

1.4.2.2. Unit Weight.................................................................................................... 27

1.4.2.3. Modulus of Elasticity..................................................................................... 27

1.4.2.4. Compression Behavior................................................................................... 28

1.4.2.5. Tension Behavior........................................................................................... 29

1.4.2.6. Poisson’s Ratio............................................................................................... 32

1.4.2.7. Coefficient of Thermal Expansion................................................................. 32

1.4.2.8. Creep and Shrinkage ...................................................................................... 33

1.4.3. Reinforcing Steel .................................................................................................... 36

1.4.4. Prestressing Steel .................................................................................................... 36

1.5. Limit States and Design Methodologies....................................................................... 37

1.5.1. General.................................................................................................................... 37

1.5.2. Service Limit State.................................................................................................. 37

1.5.3. Fatigue Limit State.................................................................................................. 39

1.5.4. Strength Limit State ................................................................................................ 40
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1.5.4.1. General........................................................................................................... 40

1.5.4.2. Resistance Factors.......................................................................................... 40

1.5.4.3. Stability.......................................................................................................... 42

1.5.5. Extreme Event Limit State...................................................................................... 42

1.6. Design for Flexural and Axial Force Effects—B Regions.......................................... 42

1.6.1. Assumptions for Service and Fatigue Limit States................................................. 42

1.6.2. Assumptions for Strength and Extreme Event Limit States ................................... 43

1.6.3. Flexural Members................................................................................................... 45

1.6.3.1. Strain Compatibility Approach ...................................................................... 45

1.6.3.2. Flexural Resistance ........................................................................................ 46

1.6.3.3. Minimum Reinforcement............................................................................... 50

1.6.3.4. Moment Redistribution .................................................................................. 51

1.6.3.5. Deformations.................................................................................................. 51

1.6.4. Compression Members ........................................................................................... 51

1.6.5. Bearing.................................................................................................................... 52

1.6.6. Tension Members.................................................................................................... 52

1.6.6.1. Resistance to Tension .................................................................................... 52

1.6.6.2. Resistance to Combined Tension and Flexure............................................... 53

1.7. Design for Shear and Torsion—B-Regions.................................................................. 54

1.7.1. Design Procedures .................................................................................................. 54

1.7.2. General Requirements............................................................................................. 54

1.7.2.1. General........................................................................................................... 54

1.7.2.2. Transfer and Development Length ................................................................ 57

1.7.2.3. Regions Requiring Transverse Reinforcement .............................................. 57

1.7.2.4. Types of Transverse Reinforcement .............................................................. 57

1.7.2.5. Minimum Transverse Reinforcement ............................................................ 58

1.7.2.6. Maximum Spacing of Transverse Reinforcement ......................................... 58

1.7.2.7. Design and Detailing Requirements .............................................................. 58

1.7.2.8. Shear Stress on UHPC ................................................................................... 59

1.7.3. Sectional Design Model.......................................................................................... 61

1.7.3.1. General........................................................................................................... 61

1.7.3.2. Sections Near Supports.................................................................................. 61

1.7.3.3. Nominal Shear Resistance ............................................................................. 62

1.7.3.4. Procedures for Determining Shear Resistance Parameters θ and fv,α............. 63

1.7.3.5. Longitudinal Reinforcement .......................................................................... 68

1.7.3.6. Sections Subjected to Combined Shear and Torsion ..................................... 70

1.7.4. Interface Shear Transfer–Shear Friction................................................................. 72

1.7.4.1. General........................................................................................................... 72

1.7.4.2. Minimum Area of Interface Shear Reinforcement ........................................ 72

1.7.4.3. Interface Shear Resistance ............................................................................. 73

1.7.4.4. Cohesion and Friction Factors....................................................................... 75

1.7.4.5. Computation of the Factored Interface Shear Force for Girder/Slab 

Bridges........................................................................................................................ 77

1.7.4.6. Interface Shear in Box Girder Bridges........................................................... 77

1.8. Design of D-Regions....................................................................................................... 77

1.9. Prestressing..................................................................................................................... 78
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1.9.1. General Design Consideration ................................................................................ 78

1.9.1.1. General........................................................................................................... 78

1.9.1.2. Design UHPC Strengths ................................................................................ 78

1.9.1.3. Section Properties .......................................................................................... 79

1.9.1.4. Crack Control................................................................................................. 79

1.9.1.5. Buckling......................................................................................................... 79

1.9.2. Stress Limitations.................................................................................................... 79

1.9.2.1. Stresses Due to Imposed Deformation........................................................... 79

1.9.2.2. Stress Limitations for Prestressing Steel........................................................ 79

1.9.2.3. Stress Limits for UHPC ................................................................................. 79

1.9.3. Prestress Losses ...................................................................................................... 81

1.9.4. Details for Pretensioning......................................................................................... 81

1.9.4.1. Minimum Spacing of Pretensioning Strand................................................... 81

1.9.4.2. Maximum Spacing of Pretensioning Strand in Slabs .................................... 82

1.9.4.3. Development of Pretensioning Strand ........................................................... 82

1.9.4.4. Pretensioned Anchorage Zones...................................................................... 84

1.9.4.5. Temporary Strands......................................................................................... 85

1.10. Reinforcement .............................................................................................................. 85

1.10.1. UHPC Cover......................................................................................................... 85

1.10.2. Hooks and Bends .................................................................................................. 86

1.10.3. Spacing of Reinforcement..................................................................................... 87

1.10.4. Transverse Reinforcement for Compression Members ........................................ 87

1.10.5. Transverse Reinforcement for Flexural Member.................................................. 88

1.10.6. Shrinkage and Temperature Reinforcement ......................................................... 88

1.10.7. Reinforcement for Hollow Rectangular Compression Members.......................... 88

1.10.8. Development and Splices of Reinforcement......................................................... 88

1.10.8.1. General......................................................................................................... 88

1.10.8.2. Development Length of Reinforcement....................................................... 89

1.10.8.3. Development by Mechanical Anchorages................................................... 93

1.10.8.4. Splices of Bar Reinforcement ...................................................................... 93

1.10.8.5. Splices of Welded Wire Reinforcement ...................................................... 94

SECTION 2. MATERIAL CONFORMANCE GUIDANCE FOR ULTRA-HIGH 

PERFORMANCE CONCRETE ............................................................................................... 95

2.1. Scope................................................................................................................................ 95

2.1.1. General.................................................................................................................... 95

2.1.2. Limitations.............................................................................................................. 95

2.2. Definitions....................................................................................................................... 96

2.3. Notation........................................................................................................................... 96

2.4. Referenced Standards, Test Methods, and Reports ................................................... 99

2.5. Material Constituents.................................................................................................. 101

2.5.1. Granular Constituents ........................................................................................... 101

2.5.1.1. Hydraulic Cement ........................................................................................ 101

2.5.1.2. Fine Aggregates........................................................................................... 101

2.5.1.3. Supplementary Cementitious Materials, Mineral Fillers, and Other 

Granular Constituents ............................................................................................... 101

2.5.2. Liquid and Frozen Water ...................................................................................... 102
vi

2.5.3. Chemical Admixtures ........................................................................................... 102

2.5.4. Fiber Reinforcement ............................................................................................. 103

2.5.4.1. Steel Fiber Reinforcement ........................................................................... 103

2.5.4.2. Non-steel Fiber Reinforcement.................................................................... 103

2.6. Material Qualification ................................................................................................. 104

2.6.1. General.................................................................................................................. 104

2.6.2. Qualification Frequency........................................................................................ 105

2.6.3. Plastic Properties................................................................................................... 105

2.6.3.1. General......................................................................................................... 105

2.6.3.2. Flow ............................................................................................................. 105

2.6.3.3. Fiber Segregation......................................................................................... 105

2.6.3.4. Unit Weight.................................................................................................. 106

2.6.4. Hardened Properties.............................................................................................. 106

2.6.4.1. General......................................................................................................... 106

2.6.4.2. Compressive Strength .................................................................................. 108

2.6.4.3. Tensile Response ......................................................................................... 110

2.6.4.4. Modulus of Elasticity................................................................................... 114

2.6.5. Durability .............................................................................................................. 115

2.6.5.1. Electrical Resistivity .................................................................................... 115

2.6.5.2. Alternative Testing....................................................................................... 118

2.6.6. Supplemental Flexural Testing Supporting Alternative Tensile Response 

Acceptance Methods....................................................................................................... 118

2.7. Material Acceptance .................................................................................................... 121

2.7.1. General.................................................................................................................. 121

2.7.2. Plastic Properties................................................................................................... 122

2.7.2.1. Flow ............................................................................................................. 122

2.7.2.2. Fiber Segregation......................................................................................... 123

2.7.2.3. Temperature ................................................................................................. 123

2.7.3. Hardened Properties.............................................................................................. 123

2.7.3.1. General......................................................................................................... 123

2.7.3.2. Compressive Strength .................................................................................. 124

2.7.3.3. Tensile Response ......................................................................................... 126

ADDENDUM A1. TYPICAL MATERIAL PROPERTIES OF ULTRA-HIGH 

PERFORMANCE CONCRETE ............................................................................................. 131

ADDENDUM A2. SHEAR DESIGN TABLES FOR θ AND fv,α .......................................... 133

A2.1. General....................................................................................................................... 133

A2.2. Members Without Transverse Steel Reinforcement.............................................. 133

A2.3. Members With Transverse Steel Reinforcement................................................... 134

APPENDIX B. ANALYSIS OF A RECTANGULAR, MILD STEEL REINFORCED

UHPC BEAM ............................................................................................................................ 141

CHAPTER B1. INTRODUCTION ......................................................................................... 147

CHAPTER B2. MATERIALS................................................................................................. 149

CHAPTER B3. FLEXURAL ANALYSIS.............................................................................. 151

B3.1. Strain Compatibility Analysis.................................................................................. 152
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B3.1.1. Material Models for UHPC and Reinforcing Steel ............................................ 152

B3.2. Moment Curvature Diagram ................................................................................... 153

B3.3. Factored Flexural Resistance ................................................................................... 157

B3.4. Flexure Demand vs. Capacity Check....................................................................... 158

CHAPTER B4. SHEAR ANALYSIS ...................................................................................... 159

B4.1. Contribution of UHPC to Shear Resistance ........................................................... 159

B4.2. Calculation of Inclination Angle and Stress in Transverse Steel.......................... 161

B4.2.1. General Approach .............................................................................................. 161

B4.2.2. Simplified Approach .......................................................................................... 163

B4.3. Total Shear Resistance.............................................................................................. 164

B4.4. Shear Demand vs. Capacity Check.......................................................................... 164

APPENDIX C. DESIGN EXAMPLE OF A PRETENSIONED UHPC I-BEAM 

BRIDGE WITH A CONVENTIONAL CONCRETE DECK .............................................. 167

CHAPTER C1. INTRODUCTION......................................................................................... 185

C1.1. Terminology............................................................................................................... 186

CHAPTER C2. MATERIALS................................................................................................. 187

CHAPTER C3. CROSS-SECTIONAL PROPERTIES FOR A TYPICAL INTERIOR 

GIRDER..................................................................................................................................... 189

C3.1. Noncomposite, Nontransformed Girder Section.................................................... 189

C3.2. Composite Girder Section ........................................................................................ 189

C3.2.1. Effective Flange Width ...................................................................................... 189

C3.2.2. Modular Ratio Between Deck and Girder UHPC.............................................. 189

C3.2.3. Section Properties............................................................................................... 189

CHAPTER C4. SHEAR FORCES AND BENDING MOMENTS....................................... 193

C4.1. Shear Forces and Bending Moments Due to Dead Loads..................................... 193

C4.1.1. Dead Loads......................................................................................................... 193

C4.1.2. Unfactored Shear Forces and Bending Moments............................................... 194

C4.2. Shear Forces and Bending Moments Due to Live Loads ...................................... 195

C4.2.1. Live Loads.......................................................................................................... 195

C4.2.2. Live Load Distribution Factors for a Typical Interior Beam ............................. 195

C4.2.2.1. Distribution Factor for Bending Moment.................................................. 196

C4.2.2.2. Distribution Factor for Shear Force .......................................................... 197

C4.2.3. Dynamic Allowance........................................................................................... 198

C4.2.4. Unfactored Shear Forces and Bending Moments............................................... 198

C4.2.4.1. Due to Truck Load: VLT and MLT............................................................... 198

C4.2.4.2. Due to Design Lane Load: VLL and MLL .................................................... 199

C4.3. Load Combinations................................................................................................... 199

CHAPTER C5. REQUIRED PRESTRESSING.................................................................... 203

C5.1. Strand Pattern........................................................................................................... 203

C5.2. Steel Transformed Section Properties..................................................................... 204

C5.2.1. Noncomposite Transformed Section at Transfer at Midspan............................. 205

C5.2.2. Noncomposite Transformed Section at Girder Ends at Transfer (14 Strands 

Debonded)....................................................................................................................... 205
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C5.2.3. Noncomposite Transformed Section at Midspan at Final Time ........................ 206

C5.2.4. Composite Transformed Section at Midspan at Final Time .............................. 207

CHAPTER C6. PRESTRESS LOSSES.................................................................................. 209

C6.1. Elastic Shortening ..................................................................................................... 209

C6.2. Time-Dependent Losses Between Transfer and Deck Placement ........................ 210

C6.2.1. Creep of UHPC .................................................................................................. 210

C6.2.2. Shrinkage of UHPC............................................................................................ 212

C6.2.3. Relaxation of Prestressing Strands..................................................................... 212

C6.3. Time-Dependent Losses Between Deck Placement and Final Time..................... 213

C6.3.1. Creep of UHPC .................................................................................................. 213

C6.3.2. Shrinkage of UHPC............................................................................................ 215

C6.3.3. Relaxation of Prestressing Strands..................................................................... 216

C6.3.4. Shrinkage of Deck Concrete .............................................................................. 216

C6.4. Total Time-Dependent Losses.................................................................................. 218

C6.5. Total Losses at Prestress Transfer .......................................................................... 218

C6.6. Total Losses at Service Loads.................................................................................. 218

CHAPTER C7. UHPC STRESSES AT PRESTRESS TRANSFER.................................... 221

C7.1. Stress Limits for UHPC at Prestress Transfer, Before Losses ............................. 221

C7.2. Stresses at Transfer Length Section........................................................................ 221

C7.3. Recomputed Stresses at Transfer Length Section with 14 Strands Debonded ... 222

C7.4. Stresses at Midspan................................................................................................... 222

C7.5. Summary of Stresses at Transfer ............................................................................ 223

C7.6. Principal Tensile Stress in the Web of the Girder ................................................. 223

C7.6.1. Principal Stress at the Centroid of the UHPC Gross Section............................. 224

C7.6.2. Principal Stress at Top of the Web..................................................................... 224

CHAPTER C8. CONCRETE STRESSES AT SERVICE LOADS ..................................... 227

C8.1. Stress Limits for UHPC Girder and Conventional Concrete Deck at Final 

Time, After Losses .............................................................................................................. 227

C8.2. Stresses at Midspan................................................................................................... 227

C8.2.1. Net UHPC Girder Stresses................................................................................. 231

C8.2.2. Net Conventional Concrete Deck Stresses......................................................... 231

C8.3. Principal Tensile Stress in Webs.............................................................................. 232

C8.3.1. Principal Stress at the Centroid of the Noncomposite Gross Section ................ 232

C8.3.2. Principal Stress at the Centroid of the Composite Nontransformed Section ..... 234

CHAPTER C9. STEEL STRESS AT SERVICE LOADS .................................................... 237

C9.1. Service Limit State .................................................................................................... 237

C9.2. Fatigue Limit State.................................................................................................... 237

C9.2.1. UHPC Compression Stress Check ..................................................................... 237

C9.2.2. UHPC Tension Stress Check.............................................................................. 238

C9.2.3. Fatigue of Reinforcement Check........................................................................ 238

CHAPTER C10. STRENGTH LIMIT STATE...................................................................... 239

C10.1. Factored Bending Moment Capacity .................................................................... 243

C10.2. Minimum Reinforcement ....................................................................................... 244

CHAPTER C11. SHEAR DESIGN......................................................................................... 247
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C11.1. Contribution of UHPC to Shear Resistance ......................................................... 247

C11.2. Contribution of Steel Reinforcement to Shear Resistance .................................. 250

C.11.2.1. General Approach to Determine θ and fv ........................................................ 250

C.11.2.2. Simplified Approach to Determine θ and fv .................................................... 252

C.11.2.3. Maximum Spacing of Transverse Reinforcement........................................... 252

C11.3. Total Shear Resistance............................................................................................ 252

CHAPTER C12. INTERFACE SHEAR TRANSFER .......................................................... 255

C12.1. Factored Horizontal Shear..................................................................................... 255

C12.2. Required Nominal Resistance ................................................................................ 255

C12.3. Required Interface Shear Reinforcement............................................................. 255

C12.4. Minimum Area of Interface Shear Reinforcement.............................................. 257

C12.5. Maximum Nominal Interface Shear Resistance................................................... 257

CHAPTER C13. MINIMUM LONGITUDINAL REINFORCEMENT 

REQUIREMENT...................................................................................................................... 259

CHAPTER C14. PRETENSIONED ANCHORAGE ZONE................................................ 261

C14.1. Anchorage Zone Reinforcement............................................................................ 261

C14.2. Confinement Reinforcement.................................................................................. 261

CHAPTER C15. DEFLECTION AND CAMBER ................................................................ 263

C15.1. Deflection Due to Prestressing Force at Transfer ................................................ 263

C15.2. Deflection Due to Beam Self-Weight..................................................................... 263

C15.3. Deflection Due to Slab and Haunch Weight......................................................... 264

C15.4. Deflection Due to Barrier and Future Wearing Surface Weights...................... 264

C15.5. Deflection and Camber Summary ......................................................................... 265

C15.6. Deflection Due to Live Load and Impact.............................................................. 265

C15.7. Deflection due to Design Truck Load and Impact............................................... 265

CHAPTER C16. DESIGN SUMMARY ................................................................................. 267

ACKNOWLEDGMENTS ........................................................................................................ 269

REFERENCES.......................................................................................................................... 271
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LIST OF FIGURES

Figure 1.4.2.4.3-1—Idealized compressive stress-strain model for UHPC.................................. 29

Figure 1.4.2.5.4-1—Idealized tensile stress-strain model for UHPC. .......................................... 31

Figure 1.4.2.5.4-2—Idealized tensile stress-strain model for UHPC with ft,loc ≥ 1.20ft,cr. ........... 32

Figure C1.5.4.2-1—Variation of the flexural resistance factor, , with the curvature ductility 

ratio, μ, for UHPC sections reinforced with prestressed steel, nonprestressed steel, or 

both. ...................................................................................................................................... 41

Figure C1.6.3.1-1—Moment-curvature response of UHPC sections reinforced with 

nonprestressed steel failing at localization of cracks (εt = γu εt,loc) and with γu εt,loc > εy. ..... 46

Figure C1.6.3.2.2-1—Stress and strain conditions for UHPC sections in flexure at the onset 

of crack localization, shown for UHPC material exhibiting the tensile stress-strain 

behavior of Figure 1.4.2.5.4-1............................................................................................... 48

Figure C1.6.3.2.2-2—Stress and strain conditions for UHPC sections in flexure at UHPC 

crushing, shown for UHPC material exhibiting the tensile stress-strain behavior of 

Figure 1.4.2.5.4-1.................................................................................................................. 48

Figure C1.6.3.2.3-1—Stress and strain conditions for UHPC sections when the stress in the 

extreme tension steel is equal to fsl, shown for UHPC material exhibiting the tensile 

stress-strain behavior of Figure 1.4.2.5.4-1. ......................................................................... 49

Figure C1.7.2.8-1—Illustration of the terms bv, dv, and de for UHPC sections............................ 60

Figure C1.7.2.8-2—Illustration of the terms bv, dv, and de for circular UHPC sections............... 60

Figure C1.7.2.8-3—Illustration of the terms bv, dv, and de for composite sections with UHPC 

beams and conventional concrete decks showing the maximum allowable dv in 

composite sections. ............................................................................................................... 61

Figure C1.9.4.3.2-1—Idealized relationship between steel stress and distance from the free 

end of the strand.................................................................................................................... 83

Figure C2.6.4.2-1—Normal frequency curve for compressive strength with an a mean f̅ ́cQ

and standard deviation scQ................................................................................................... 109

Figure B1-1. Illustration. Cross-section detail............................................................................ 147

Figure B3.1.1-1. Graphs. Idealized uniaxial stress-strain relationships. .................................... 152

Figure B3.2-1. Graph. Flexural moment-sectional curvature ( – ψ) diagram of the UHPC 

cross section. ....................................................................................................................... 153

Figure B3.2-2. Illustration. Strain and stress profiles at first crack in UHPC. ........................... 154

Figure B3.2-3. Illustration. Strain and stress profiles when the steel reinforcement reaches 

steel service limit. ............................................................................................................... 155

Figure B3.2-4. Illustration. Strain and stress profiles at yielding of steel reinforcement. .......... 155

Figure B3.2-5. Illustration. Strain and stress profiles at UHPC crack localization. ................... 156

Figure B3.2-6. Illustration. Strain and stress profiles at the crushing of UHPC in 

compression. ....................................................................................................................... 156

Figure C1-1. Illustrations. MnDOT “MN54” cross sections. ..................................................... 185
xi

Figure C1-2. Illustration. Bridge cross section (MnDOT 2018)................................................. 186

Figure C3.2.3-1. Illustration. Dimensions of the composite UHPC/conventional concrete 

section. ................................................................................................................................ 190

Figure C5.1-1. Illustration. Strand pattern at midspan................................................................ 204

Figure C5.1-2. Illustration. Strand pattern at beam ends. ........................................................... 204

Figure C10-1. Illustration. Strain compatibility analysis of composite section.......................... 239

Figure C10-2. Illustration. Mode III: UHPC crack localization. ................................................ 242

Figure C10.1-1. Illustration. Strands reach steel service limit.................................................... 243

Figure C11.1-1. Illustration. Debonding pattern of strands. ....................................................... 249

Figure C16-1. Illustration. Conceptual design drawing showing the details of the girder 

reinforcement. ..................................................................................................................... 267
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LIST OF TABLES

Table 2.6.1-1. Material qualification testing methods and objective.......................................... 104

Table 2.6.4.2-1. k-factor for increasing the sample standard deviation based on total number 

of tests considered............................................................................................................... 109

Table 2.7.1-1. Material acceptance testing methods and sampling frequency. .......................... 122

Table A1-1. Typical and minimum mechanical properties for UHPC. ...................................... 131

Table A2.2-1. Values of θ (degrees) for sections without transverse reinforcement (i.e., ρv,α = 

0.0 percent). ........................................................................................................................ 134

Table A2.3-1. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 0.5 percent.................................................................................. 135

Table A2.3-2. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 1.0 percent.................................................................................. 136

Table A2.3-3. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 1.5 percent.................................................................................. 137

Table A2.3-4. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 2.0 percent.................................................................................. 138

Table A2.3-5. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 2.5 percent.................................................................................. 139

Table A2.3-6. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 3.0 percent.................................................................................. 140

Table B3.2-1. Summary of the moment-curvature analysis. ...................................................... 154

Table C3.2.3-1. Properties of the composite UHPC/conventional concrete section. ................. 191

Table C4.1.2-1. Unfactored shear forces and bending moments due to dead loads for a 

typical interior girder. ......................................................................................................... 194

Table C4.2.4.1-1. Unfactored shear forces and bending moments due to live loads for a 

typical interior girder. ......................................................................................................... 199

Table C7.5-1. Stresses at transfer. .............................................................................................. 223

Table C10-1. Summary of the strain compatibility analysis....................................................... 242
1

CHAPTER 1. INTRODUCTION

Ultra-high performance concrete (UHPC) is a class of concrete that has emerged as a compelling 

material for use in the design, construction, and preservation of structures. It is a versatile material 

that can be used in primary structural components, field-cast connections between prefabricated 

components, and repair applications. UHPC provides mechanical and durability properties that are 

enhanced and distinct relative to conventional concrete, thus necessitating the use of design 

methodologies that facilitate the appropriate use of the material. Until now, there has not been any 

formal design guidance in the United States for structural design with UHPC. Recent research and 

development-based advancements related to UHPC, and a growing interest from bridge owners in 

the possibilities presented by this material, have provided an opportunity for the bridge engineering 

community to consider and potentially adopt formal structural design guidance for UHPC 

structural elements.

OBJECTIVE AND USE

This report is intended for bridge owners, designers, and their supporting professionals who are 

interested in the design of UHPC structural components as used, or considered for use, in the 

highway infrastructure. This document aims to support the development of formal UHPC 

structural design guidance by delivering a potential framework for the bridge engineering

community to consider.

REPORT CONTENT

The main body of the report provides an overview of UHPC in the context of structural design. 

Chapter 2 discusses UHPC as a class of materials, along with its availability and use in the United 

States. It also provides some background on directly relevant aspects of the Federal Highway 

Administration’s (FHWA) engagement with the topic of UHPC. The proposed framework for 

UHPC structural design and the associated design examples are also discussed in Chapter 2.

Chapter 3 provides concluding remarks.

3

CHAPTER 2. UHPC STRUCTURAL DESIGN

UHPC is a structural material that exhibits compelling structural behaviors. These behaviors are 

distinct from those exhibited by other more common materials used in the civil infrastructure, such 

as conventional concrete or steel. In comparison to conventional concrete, UHPC offers sustained 

postcracking tensile resistance, along with an increased compressive strength, an increased elastic 

modulus, and a decreased susceptibility to liquid permeation. To effectively engage the enhanced 

behaviors of UHPC, structural design guidance must rationally and conservatively provide a 

framework within which designers can appropriately conceive UHPC structures and proportion 

UHPC elements. With a look toward the future, UHPC can most likely allow for the design of 

novel structures whose composition is efficient, whose functionality is improved, and whose 

lifespan is extended.

UHPC

As with conventional concrete, UHPC is composed of inert and reactive constituents that, when 

combined with water and chemical admixtures, undergoes a hydration reaction to transform from 

a semifluid mixture into a competent structural material. UHPC does not have a standard mixture. 

This class of concrete is defined through prescriptive and performance requirements. UHPC 

commonly contains a high concentration of steel fiber reinforcement, generally near or greater 

than 2 percent per volume. UHPC also commonly contains supplementary cementitious materials 

and graded inert fillers. UHPC rarely contains coarse aggregate.

MATERIAL PROPERTIES

The material properties of UHPC can vary within the ranges of behavior that define UHPC-class 

materials. Choices made in the selection of constituent materials, the mixing process, and the 

curing process will affect the fresh and hardened properties of a particular UHPC. Research 

completed at FHWA’s Turner-Fairbank Highway Research Center (TFHRC) during the past two 

decades has demonstrated some of the material properties that a UHPC might exhibit (Graybeal 

2006a; Haber et al. 2018).

AVAILABILITY

UHPC is available in the U.S. marketplace in a variety of forms and from a variety of sources.

Researchers have developed and published open-source mix designs that allow interested parties 

to develop their own local mixes. Commercial precast concrete fabricators have developed mixes 

that they use within their operations to fabricate finished products composed of UHPC. Finally, 

commercial material suppliers have developed products that they sell as bagged constituents for 

use on construction sites or in fabrication facilities. As long as the delivered solution meets the 

project requirements, the origin of the UHPC mix is generally not relevant.

UHPC USAGE IN BRIDGE CONSTRUCTION

Some of the first uses of UHPC in the civil infrastructure occurred in bridges (Russell and Graybeal 

2013). A pedestrian bridge built in Quebec in 1997 that used UHPC preceded other pedestrian 

bridges in Canada, as well as other countries such as Japan, Germany, and France. A highway 
4

bridge in France built in 2001 using UHPC preceded initial deployments of UHPC in primary 

structural elements in bridges in France, the United States, and other countries during the 2000s. 

The first U.S. roadway bridge was constructed in Iowa using UHPC pretensioned girders and has 

been in service since 2006.

From 2010 through the present, a significant increase has occurred in the number and types of 

applications for which UHPC is engaged. Applications in Switzerland have largely focused on 

using UHPC for repair and rehabilitation of deteriorated structures. Usage in South Korea has 

resulted in construction of some signature bridges. French applications have tended toward 

optimized solutions that demonstrate the benefits of UHPC. Malaysia has been a leader in adopting 

this technology, having more than 180 bridges constructed from UHPC prefabricated components 

(e.g., segmental box girders, pretensioned I-girders) over the past decade. Further international 

perspectives on the use of UHPC in bridge applications can be found in Graybeal et al. (2020).

In the United States, bridge-related applications of UHPC have expanded over time through a 

series of phases. Through 2009, UHPC was largely considered an emerging material that was 

thought to have potential applications in prefabricated components such as pretensioned bridge 

girders. During the 2010–2017 timeframe, UHPC gained traction as a field-cast grout used in 

connections between prefabricated bridge components (Graybeal 2014, 2019). From 2018 through 

the present, UHPC began to be deployed as a material capable of offering an extended lifespan to 

deteriorated structures through repair and preservation applications.

Between 2006 and 2021, UHPC was used in more than 350 bridge construction applications across 

the United States. FHWA has been tracking these deployments, and readers can learn more by 

accessing FHWA’s interactive map (FHWA 2021).

FHWA RESEARCH ON MATERIAL TEST METHODS FOR UHPC

FHWA has a long history of UHPC research, conducted at TFHRC, with an eye toward developing 

innovative solutions that address current challenges in the highway sector. A vertically integrated 

series of research programs have investigated UHPC from the material scale up through the 

structural scale. Individual studies have helped build the knowledge base in topics as varied as 

basic material performance, mechanical property test method development, performance of 

optimized pretensioned bridge girders, field-cast connections for prefabricated bridge systems, and 

rehabilitative overlays for deteriorated bridge decks. 

The tensile response of UHPC is a defining characteristic of this class of concrete, and early 

FHWA research was conducted to investigate methodologies that could be used to quantify the 

behavior (Graybeal 2006a). A bilateral U.S.-French collaborative research project, focused 

specifically on developing a solution to this challenge, began in 2010. An outgrowth of this 

research was a test method akin to that used for tensile assessment of metals, wherein a UHPC 

prism is held in a fixed-end condition and then subjected to a direct tensile load (Graybeal and 

Baby 2013). The test offers significant advantages in that it is minimally susceptible to reporting 

unconservative results due to incorrect boundary conditions during testing or incorrect behavioral 

assumptions during results analysis. This test method has been further exercised and refined in the 

ensuing years, leading to the recent passage of American Association of State Highway and 
5

Transportation Officials (AASHTO) T 397, Standard Method of Test for Uniaxial Tensile 

Response of Ultra-High Performance Concrete (AASHTO 2022b).

The durability properties of UHPC are also distinctly different from those of conventional 

concrete. Concrete durability testing is often time-intensive, and many tests were developed in 

such a way that they cannot differentiate between materials that are more durable than conventional 

concrete. FHWA researchers addressed the need for a simple test method by investigating 

AASHTO TP 119, Standard Method of Test for Electrical Resistivity of a Concrete Cylinder 

Tested in a Uniaxial Resistance Test (AASHTO 2022a). This test method was found to deliver 

results that were indicative of the discontinuous pore structure commonly present in UHPC-class 

materials, a pore structure that resists ingress of liquids and thus enhances durability (Spragg et al.

2022).

FHWA RESEARCH ON STRUCTURAL BEHAVIOR OF UHPC

From the beginning of FHWA’s UHPC research program in 2001, a significant emphasis has been 

placed on assessing the structural behavior of UHPC components and developing compelling 

UHPC-based structural solutions. The first project focused on the structural behavior of AASHTO 

Type II pretensioned girders composed of UHPC and containing no mild steel reinforcement 

(Graybeal 2006b). Researchers investigated the flexural and shear response of this girder shape 

and began to recognize the initial indications of the unique tensile response of UHPC-class 

materials. Further studies in the mid- to late 2000s focused on developing structurally optimized

pretensioned girder cross sections that could better utilize the structural behaviors of UHPC

(Graybeal 2009a, 2009b).

Recently, a significant portion of FHWA’s UHPC research program has focused on the behavior 

of primary structural components. The program’s goal has been to develop foundational 

knowledge that supports an extension of existing U.S.-based bridge design guidance into the realm 

of UHPC. One aspect of the program focused on mechanical behaviors of UHPC and behavioral 

models that could be used in structural design (El-Helou, Haber, and Graybeal 2022). Another 

aspect of the program focused on the flexural behavior of UHPC beams and associated design 

concepts (El-Helou and Graybeal 2022a). A third aspect focused on the shear performance and 

shear behavior modeling of UHPC beams (El-Helou and Graybeal 2022b, 2023). Interface shear 

behavior was investigated in a different part of the program (Muzenski, Haber, and Graybeal 2022, 

2023). Research was also conducted on time-dependent behaviors and prestress loss models 

(Mohebbi and Graybeal 2022; Mohebbi, Graybeal, and Haber 2022).

CONCEPTUAL DRAFT UHPC STRUCTURAL DESIGN GUIDANCE

UHPC provides mechanical and durability properties that are significantly different from those of 

conventional concrete, thus necessitating distinct design methodologies that facilitate the 

appropriate use of the material. These design methodologies must be founded on principles of 

engineering mechanics, while also recognizing where the behaviors of UHPC differ from those of 

more common structural materials. Developing a design framework that engages the behaviors of 

UHPC as viewed through the lens of engineering mechanics should allow for a robust foundation 

on which formal structural design guidance can be developed.
6

As interest in the use of UHPC for structural applications has grown, various jurisdictions around 

the world have developed guidance to assist their communities in appropriately engaging UHPC.

The French have been at the forefront with a standard published as a national addition for UHPC 

design (Association Française de Normalisation 2016) that builds on Eurocode 2: EN 1992–

Design of Concrete Structures (British Standards Institution 2004). Other notable guidance 

includes the recommendations in Switzerland (Swiss Institute of Architects and Engineers 2016)

and the informative annex to Section 8 of the Canadian Highway Bridge Design Code (Canadian 

Standards Association 2019).

Until now, there has not been any formal design guidance in the United States for structural design 

with UHPC. However, given the guidance developed elsewhere and given recent research and 

development-based advancements related to UHPC, the opportunity for developing U.S.-based 

design guidance that is readily adoptable by the bridge design community appears to have arrived.

The AASHTO Committee on Bridges and Structures subcommittee on concrete structural design, 

commonly known as T-10, is aware of the opportunities afforded by UHPC and has expressed an 

interest in UHPC structural design specifications. FHWA staff offered to assist T-10 in taking 

steps toward the anticipated eventual adoption of an AASHTO guide specification of structural 

design with UHPC. FHWA focused research efforts to address identified knowledge gaps and 

develop performance models that facilitate the design of structures. Once sufficient knowledge 

had been developed through internal research and collected from external sources, FHWA staff 

began drafting a proposed framework for structural design.

The proposed framework addresses structural considerations that are integral to the design of 

common structural components, with a focus toward design of reinforced concrete and 

pretensioned concrete components. Although many of the concepts are certainly relevant, the 

framework was not developed with a focus toward post-tensioned components or the design of 

components subjected to seismic demands. The framework parallels the AASHTO LRFD Bridge 

Design Specifications, 9th edition, in particular building on the provisions related to conventional 

concrete (AASHTO 2020). The format of the framework is similar to guide specifications that 

AASHTO has previously published in relation to other innovative technologies.

The framework, developed in the format commonly used for AASHTO guide specifications, is 

presented in Appendix A—Guide Specification for Structural Design with Ultra-High 

Performance Concrete. Section 1 of Appendix A focuses on structural design guidance, while 

Section 2 focuses material conformance guidance. Of note, Section 2 allows potential UHPC-class 

materials to be tested for qualification in advance of use and to be tested for compliance during 

and after use.
7

ANALYSIS AND DESIGN EXAMPLES

To assist readers in understanding the potential application of the proposed structural design 

framework, a pair of design examples has been developed to demonstrate some of the basic 

concepts embedded in the framework. The first example focuses on using the methods in the design 

framework to analyze the behavior of a rectangular beam. This example can be found in 

Appendix B—Analysis of a Rectangular, Mild Steel Reinforced UHPC Beam. The second 

example demonstrates the design of a slab-on-stringer bridge superstructure using pretensioned 

girders. This example can be found in Appendix C—Design Example of a Pretensioned UHPC 

I-Beam Bridge with a Conventional Concrete Deck.

9

CHAPTER 3. CONCLUDING REMARKS

Continuing advancements in concrete technology have led to the widespread availability of a new 

class of concrete, UHPC. This material exhibits properties that, when engaged in structural 

components, can make possible structural solutions that were, until now, not feasible. The creation 

of formal structural design guidance, based on foundational principles of engineering mechanics 

and interpreted through the lens of demonstrated UHPC behaviors, is a critical step on the path 

toward common usage of UHPC in the highway infrastructure. The framework provided here is 

expected to assist the broader bridge engineering community in achieving this goal.

11

APPENDIX A. GUIDE SPECIFICATION FOR STRUCTURAL DESIGN WITH 

ULTRA-HIGH PERFORMANCE CONCRETE

This Appendix is a draft of a potential AASHTO Committee on Bridges and Structures

subcommittee T-10 working agenda item. It shall not be relied upon as design guidance or a design 

standard.

13

SECTION 1. STRUCTURAL DESIGN WITH ULTRA-HIGH PERFORMANCE 

CONCRETE

1.1. SCOPE

1.1.1. General C1.1.1

The provisions in Section 1 of this Appendix 

apply to the design of bridge and ancillary 

structures constructed of ultra-high performance concrete (UHPC). UHPC shall be a 

portland cement composite with a discontinuous pore structure and reinforced with steel 

fiber reinforcement. 

The provisions are based on UHPC materials 

exhibiting a strain-hardening behavior and 

having the following minimum property values 

for use in design determined according to 

Article 1.4 and Section 2 of this Appendix:

• A minimum compressive strength, f ́c, 

of 17.5 ksi,

• A minimum effective cracking strength, 

ft,cr, of 0.75 ksi,

• A minimum crack localization strength, 

ft,loc, greater than or equal to the 

effective cracking strength, ft,cr,

• A minimum crack localization strain, 

εt,loc, of 0.0025, and

• A minimum durability performance as

defined in Article 2.6.5 of this Appendix

or as specified by the owner.

UHPC is a class of concrete that has emerged 

as a compelling material for use in the design, 

construction, and preservation of structures. It 

is a versatile material that can be used in 

primary structural components, field-cast 

connections between prefabricated components, and repair applications. As with 

conventional concrete, UHPC is composed of 

inert and reactive constituents that, when 

combined with water and chemical admixtures, 

undergo a hydration reaction to transform from 

a semifluid mixture into a competent structural 

material.

UHPC is a strain-hardening fiber reinforced 

concrete, meaning that this type of concrete can 

resist tensile loads beyond cracking of the 

cementitious composite (Graybeal 2015b). 

Engagement of this tensile response in 

structural design necessitates a reconsideration 

of some of the fundamental behavioral 

assumptions associated with conventional 

reinforced concrete.

UHPC-class materials have been demonstrated 

to deliver significantly enhanced durability 

compared with conventional concretes (Haber 

et al. 2018). Formulations that meet the 

performance requirements have been demonstrated to have reduced permeability and thus 

are more resistant to liquid permeation and 

associated degradation mechanisms.

There is no standard mixture for UHPC. This 

class of concrete is defined through prescriptive 

and performance requirements stated herein. 

UHPC commonly contains a high concentration 

of steel fiber reinforcement, generally near or 

greater than 2 percent per volume. A mix may 
14

include additional, supplementary non-steel 

fiber reinforcement, but these fibers may not 

supplant the steel fiber reinforcement. UHPC 

commonly contains supplementary cementitious materials and graded inert fillers. UHPC 

rarely contains coarse aggregate.

The recommendations of this Appendix are not 

intended to supplant proper training or the 

exercise of judgment by the design 

professional. They state only the minimum 

requirements necessary to provide public 

safety. The owner or the Designer may require 

the sophistication of the design or the quality of 

materials and construction, or both, to be higher 

than the minimum requirements.

The Designer shall be familiar with the 

provisions of the AASHTO LRFD Bridge 

Design Specifications, 9th edition, hereafter 

referred to as “AASHTO LRFD BDS,” and

with the design of conventional reinforced and 

prestressed concrete structures (AASHTO

2020).

1.1.2. Design Philosophy C1.1.2

The guidance in this Appendix is based on limit 

state design principles where structural 

components shall be proportioned to satisfy the 

requirements of all appropriate limit states. In 

many instances, serviceability limits may 

control the design.

The design professional is referred to AASHTO 

LRFD BDS (AASHTO 2020) for provisions on 

limit state design principles, general design and 

location features, and structural analysis and 

evaluation.

The serviceability and capacity provisions 

herein are based on the observed and 

anticipated performance of example UHPCclass materials when configured for, and 

subjected to, structural loading. In general, 

fundamental structural behaviors (e.g., tension, 

compression, flexure, shear) are treated based 
15

on the intersection of the principles of 

engineering mechanics and the performance of 

UHPC. In situations where the performance of 

UHPC is expected to significantly outperform 

conventional concrete and there is a lack of 

specific UHPC test results, AASHTO LRFD 

BDS (AASHTO 2020) provisions for 

conventional concrete may have been adopted.

The tensile behavior of UHPC at crack

localization is of critical importance. Beyond 

the crack localization strain, UHPC offers

decreasing tensile resistance, causing loads to 

be shed to available alternate load paths. In

general, crack localization within a UHPC 

structural element results in concentrated 

deformations (e.g., wide cracking, flexural 

hinging) that are to be avoided.

Crack localization refers to a point in the tensile 

stress-strain response of UHPC where the 

tensile deformation starts to accumulate into a 

single dominant crack and the tensile resistance 

starts to continuously decline without 

substantial recovery. Crack localization occurs 

when the fiber reinforcement bridging a crack 

debonds and starts to pull out of the 

cementitious matrix. In unreinforced members, 

crack localization coincides with a loss of 

member capacity. In reinforced members, the 

loss of the UHPC tensile resistance and the 

concentrated deformation at a single localized 

crack within the member cause local 

redistribution of the applied stresses, 

potentially straining the tensile reinforcement 

that bridges the crack beyond its capacity to 

resist. This reinforcement straining behavior is 

accentuated by the shorter length over which 

discrete reinforcements can be developed in 

UHPC and should be avoided.

The tensile resistance behavior of UHPC 

depends on the distribution and orientation of 

the fiber reinforcement in the UHPC. These 

provisions rely on the use of appropriate 

construction methods to ensure that the fiber 

reinforcement is evenly dispersed through the 

member and that adverse fiber orientation 

effects have been avoided.

Contract documents should require the use of 

appropriate construction methods. Disturbance 

of fiber distribution, as would occur at a cold 

joint or when fiber flow is restricted from 

reaching a part of the member, will affect the 

structural performance of the member.

1.1.3. Loads and Load Combinations

Refer to AASHTO LRFD BDS (AASHTO 

2020) for provisions on loads, load 

combinations, and load factors.
16

The provisions of AASHTO LRFD BDS

(2020) Table 3.5.1-1 shall be supplemented by 

Article 1.4.2.2 of this Appendix, which defines 

the unit weight of UHPC.

1.1.4. Limitations C1.1.4

The provisions in Section 1 of Appendix A shall 

not apply to: 

• The non-UHPC portion of composite 

structural members, and

This Appendix does not provide guidance on

the design of conventional concrete or 

structural steel portions of a member partially 

composed of UHPC. Refer to Section 5.0 and 

Section 6.0 of AASHTO LRFD BDS for 

provisions applicable to these structural 

materials (AASHTO 2020).

• The design of earthquake resisting

components in Seismic Zones 2, 3, or 4,

as defined within AASHTO LRFD BDS

(AASHTO 2020).

Although many of the concepts contained in 

these provisions may be applicable to the 

design of earthquake resisting components in 

Seismic Zones 2, 3, and 4, these provisions 

were not specifically developed for application 

in these Seismic Zones.

The provisions in Section 1 of Appendix A do

not address the provisions for specific structure 

components and types discussed in AASHTO 

LRFD BDS Article 5.12.

The provisions in Section 1 of Appendix A 

were not developed to address the special 

considerations and detailing inherent in posttensioned structures.

This provision is not intended to prohibit use of 

post-tensioning with UHPC or the structure 

types listed in AASHTO LRFD BDS 

Article 5.12; however, these items are not 

specifically addressed in Section 1 of this 

Appendix, and the guidance provided in 

AASHTO LRFD Article 5.12 may not 

necessarily apply to UHPC.

1.2. DEFINITIONS

Anchor—A steel element either cast into UHPC or post-installed into a hardened UHPC member 

and used to transmit applied loads to the UHPC. Cast-in-place anchors include headed bolts, 

hooked bolts (J-bolt or L-bolt), and headed studs. Post-installed anchors include expansion 

anchors, undercut anchors, and adhesive anchors. Steel elements for adhesive anchors include 

threaded rods, deformed reinforcing bars, or internally threaded steel sleeves with external 

deformations.

Anchorage—In pretensioning, a device used to anchor the tendon until the UHPC has reached a 

predetermined strength, and the prestressing force has been transferred to the UHPC. For 
17

reinforcing bars, a length of reinforcement, or a mechanical anchor or hook, or combination thereof 

at the end of a bar needed to transfer the force carried by the bar into the UHPC.

Anchorage Zone—The portion of the structure in which the prestressing force is transferred from 

the anchorage device onto the local zone of the UHPC and then distributed more widely into the 

general zone of the structure.

At Jacking—At the time of tensioning the prestressing tendons.

At Loading—The maturity of the UHPC when loads are applied. Such loads include prestressing 

forces and permanent loads but generally not live loads.

At Transfer—Immediately after the transfer of prestressing force to the UHPC.

Beam or Bernoulli Region (B-Region)—The regions of UHPC members in which Bernoulli’s 

hypothesis of straight-line strain profiles, linear for bending, and uniform for shear, applies. 

Bonded Tendon—A tendon that is bonded to the UHPC.

Bridge—Any structure with an opening not less than 20.0 ft that forms part of a highway or that 

is located over or under a highway.

Component—Either a discrete element of the bridge or a combination of elements requiring 

individual design consideration; synonymous to member.

Composite Section—UHPC components, UHPC and conventional concrete components, or UHPC 

and steel components, interconnected to respond to force effects as a unit.

Crack Localization—The instance at which the tensile deformation starts to accumulate into a 

single dominant crack and the tensile resistance starts to continuously decline without substantial 

recovery.

Confinement—A condition where the disintegration of the UHPC under compression is prevented 

by the development of lateral and/or circumferential forces such as may be provided by appropriate 

reinforcement, steel or composite tubes, or similar devices.

Creep—The time-dependent deformation of UHPC under permanent load.

Curvature Ductility Ratio—The ratio of sectional curvature at nominal flexural capacity to the 

sectional curvature when the service stress limit in extreme tension steel is reached.

Debonded Strand—A pretensioned, prestressing strand that is bonded for a portion of its length 

and intentionally debonded elsewhere through the use of mechanical or chemical means. Also 

called shielded or blanketed strand.

Deep Component—Components in which the distance from the point of 0.0 shear to the face of 

the support is less than 2d, or components in which a load causing more than one-third of the shear 

at a support is closer than 2d from the face of the support.
18

Deformation—A change in structural geometry due to force effects, including axial displacement, 

shear displacement, and/or rotations.

Design—The process of proportioning and detailing the components and connections of a bridge.

Design Compressive Strength—The nominal compressive strength of UHPC specified for the 

work and assumed for design and analysis of new structures.

Design Crack Localization Strain—The nominal crack localization strain of UHPC specified for 

the work and assumed for design and analysis of new structures.

Design Crack Localization Strength—The nominal crack localization strength of UHPC specified 

for the work and assumed for design and analysis of new structures.

Design Effective Cracking Strength—The nominal effective cracking strength of UHPC specified 

for the work and assumed for design and analysis of new structures.

Design Professional—The architect, engineer, architectural firm, or engineering firm responsible 

for the design of the bridge and issuing contract documents or administering the work under 

contract documents, or both.

Disturbed or Discontinuity Region (D-Region)—The regions of UHPC members encompassing 

abrupt changes in geometry or concentrated forces in which strain profiles more complex than 

straight lines exist.

Ductility—The property of a component or connection that allows inelastic response.

Effective Depth—The depth of a component effective in resisting flexural or shear forces.

Effective Prestress—The stress or force remaining in the prestressing steel after all losses have 

occurred.

Elastic—A structural material behavior in which the ratio of stress to strain is constant; the material 

returns to its original unloaded state on load removal.

Element—A part of a component or member consisting of one material.

Equilibrium—A state where the sum of forces and moments about any point in space is 0.0.

Extreme Event Limit States—Limit states relating to events such as earthquakes, ice load, and 

vehicle and vessel collision, with return periods in excess of the design life of the bridge.

Extreme Tension Steel—The prestressed or nonprestressed reinforcement that is farthest from the 

extreme compression fiber.

Factored Load—The nominal loads multiplied by the appropriate load factors specified for the 

load combination under consideration.

Factored Resistance—The nominal resistance multiplied by a resistance factor.
19

Fiber Reinforcement—Discrete, fine, threadlike steel filament included in the UHPC mix.

Inelastic—Any structural behavior in which the ratio of stress and strain is not constant, and part 

of the deformation remains after load removal.

Limit State—A condition beyond which the bridge or component ceases to satisfy the provisions 

for which it was designed.

Load and Resistance Factor Design (LRFD)—A reliability-based design methodology in which 

force effects caused by factored loads are not permitted to exceed the factored resistance of the 

components.

Maturity of Concrete—The duration of time between time of loading and the time being 

considered for creep effects, or duration of time between the end of concrete curing and the time 

being considered for shrinkage effects.

Member—Same as Component.

Model—A mathematical or physical idealization of a material, structure, or component used for 

analysis.

Monolithic—A portion of an element that was cast through a single, continuous placement of 

UHPC and contains no cold joints.

Net Tensile Strain—The tensile strain at nominal resistance exclusive of strains due to effective 

prestress, creep, shrinkage, and temperature.

Nominal Resistance—The resistance of a component or connection to force effects, as indicated 

by the dimensions specified in the contract documents and by permissible stresses, deformations, 

or specified strength of materials.

Owner—The person or agency with jurisdiction over the bridge.

Precast Members—UHPC elements cast in a location other than their final position.

Prestressed UHPC—UHPC components in which stresses and deformations are introduced by 

application of prestressing forces.

Pretensioning—A method of prestressing in which the strands are tensioned before the UHPC is 

placed.

Reinforced UHPC—Structural UHPC containing prestressing tendons or nonprestressed reinforcement.

Reinforcement—Reinforcing bars, welded wire reinforcement, and/or prestressing steel.

Relaxation—The time-dependent reduction of stress in prestressing tendons.
20

Resistance Factor—A statistically based multiplier applied to nominal resistance, accounting 

primarily for variability of material properties, structural dimensions, and workmanship, as well 

as uncertainty in the prediction of resistance. It is also related to the statistics of the loads through 

the calibration process.

Sectional Curvature—Ratio of the strain in extreme compression fiber to the distance from the 

extreme compression fiber to the neutral axis.

Service Limit States—Limit states relating to stress, deformation, and cracking under regular 

operating conditions.

Slab—A component with a width of at least four times its effective depth.

Spiral—Continuously wound bar or wire in the form of a cylindrical helix.

Strain Hardening—A tensile stress-strain response characterized by increasing tensile resistance 

after initial cracking of the UHPC.

Strain Offset—The strain value added to a linear fit of the elastic stress-strain response to create a 

parallel line that intersects the measured stress-strain response of a test specimen after the elastic 

limit.

Strength Limit States—Limit states relating to strength and stability during the design life.

Stress Range—The algebraic difference between the maximum and minimum stresses due to 

transient loads.

Structural UHPC—All UHPC used for structural purposes.

Strut-and-Tie Method—A procedure used principally in regions of concentrated forces and 

geometric discontinuities to determine UHPC member proportions and reinforcement quantities 

and patterns based on an analytic model consisting of compression struts in the UHPC, tensile ties 

in the reinforcement, and the geometry of nodes at their points of intersection.

Superstructure—Structural parts of the bridge that provide the horizontal span.

Tendon—A high-strength steel element used to prestress the UHPC.

Transfer—The operation of imparting the force in a pretensioning anchoring device to the UHPC.

Transfer Length—The length over which the pretensioning force is transferred to the UHPC by 

bond and friction in a pretensioned member.

Transient Loads—Loads and forces that can vary over a short time interval relative to the lifetime 

of the structure.

Transverse Reinforcement—Reinforcement used to resist shear, torsion, and lateral forces or to 

confine UHPC in a structural member. The terms “stirrups” and “web reinforcement” are usually 
21

applied to transverse reinforcement in flexural members, and the terms “ties,” “cross-ties,” 

“hoops,” and “spirals” are applied to transverse reinforcement in compression members.

UHPC Cover—The specified minimum distance between the surface of the reinforcing bars, 

strands, anchorages, or other embedded items and the surface of the UHPC.

Yield Strength—The specified yield strength of reinforcement.

1.3. NOTATION

Acp = area enclosed by outside perimeter of UHPC cross section (in.2)

Act = area of UHPC on the flexural tension side of the member (in.2)

Acv = area of UHPC considered to be engaged in interface shear transfer (in.2)

Ag = gross area of the cross section of the member (in.2)

Ao = area enclosed by the shear flow path, including any area of holes therein (in.2)

Aps = area of prestressing steel (in.2); area of prestressing steel on the flexural tension side 

of the member (in.2), as shown in AASHTO LRFD BDS Figure 5.7.3.4.2-1 

(AASHTO 2020)

As = area of nonprestressed tension reinforcement (in.2); area of nonprestressed steel on 

the flexural tension side of the member at the section under consideration (in.2), as 

shown in AASHTO LRFD BDS Figure 5.7.3.4.2-1 (AASHTO 2020); total area of 

reinforcement located within the distance h/4 from the end of the beam (in.2)

At = area of one leg of closed transverse torsion reinforcement in solid members, or total 

area of transverse torsion reinforcement in the exterior web and flange of hollow 

members (in.2)

Av = area of transverse reinforcement to resist shear within a distance s (in.2)

Avf = area of interface reinforcement crossing the shear plane within the area Acv (in.2)

be = effective width of the shear flow path taken as the minimum thickness of the exterior 

webs or flanges comprising the closed box section (in.)

bv = effective web width taken as the minimum web width within the depth dv (in.)

bvi = interface width considered to be engaged in shear transfer (in.)

C = total compressive force due to flexure (kip)

C1 = normal clamping force provided by steel reinforcement (kip)

C2 = normal clamping force provided by UHPC (kip)

c = depth of neutral axis (in.); cohesion factor (ksi)

cc = distance from the extreme compression fiber of the member to the neutral axis when 

the UHPC compression strain limit, εcu, at extreme compression fiber is reached (in.)

cL = distance from the extreme compression fiber of the member to the neutral axis when 

the UHPC tensile strain limit, γuεt,loc, at extreme tension fiber is reached (in.)
22

csl = distance from the extreme compression fiber of the member to the neutral axis when 

the steel stress in the extreme tension steel is equal to the steel service stress limit, fsl

(in.)

db = nominal strand diameter (in.); nominal diameter of reinforcing bar (in.); nominal 

diameter of reinforcing bar or wire (in.)

de = effective depth taken as the distance, measured perpendicular to the neutral axis, 

between the extreme compression fiber of the section to the forces in the tensile 

reinforcement (in.)

dl = distance from the extreme compression fiber of the member to the centroid of extreme 

tension steel element (in.)

dp = distance from extreme compression fiber to the centroid of the of the prestressing 

tendons (in.)

ds = distance from extreme compression fiber to the centroid of the nonprestressed tensile 

reinforcement measured along the centerline of the web (in.)

dv = effective shear depth taken as the distance, measured perpendicular to the neutral 

axis, between the resultants of the tensile and compressive forces due to flexure (in.)

Ec = modulus of elasticity of UHPC for use in design (ksi)

Ep = modulus of elasticity of prestressing steel for use in design (ksi)

Es = modulus of elasticity of the nonprestressed steel reinforcement (ksi); modulus of 

elasticity of the steel reinforcement crossing the interface plane (ksi)

f2 = compressive stress in the compression strut (ksi)

fc = compressive stress in the UHPC section at the extreme compression fiber (ksi)

f ́c = compressive strength of UHPC for use in design (ksi)

f ́ci = design UHPC compressive strength at the time of prestressing for pretensioned 

members and at the time of initial loading for nonprestressed members (ksi)

fpc = unfactored compressive stress in UHPC after prestress losses have occurred, either at 

the centroid of the cross section resisting transient loads, or at the junction of the web 

and flange where the centroid lies in the flange (ksi)

fpe = effective stress in prestressing steel after losses (ksi)

fpo = a parameter taken as modulus of elasticity of prestressing steel multiplied by the 

locked-in difference in strain between the prestressing steel and the surrounding 

UHPC (ksi)

fps = average stress in prestressing steel at the time for which the nominal resistance of the 

member is required (ksi)

fpu = specified tensile strength of prestressing steel (ksi)

fpx = design stress in pretensioned strand at nominal flexural strength at the section of the 

member under consideration (ksi)

fpy = specified yield strength of prestressing steel (ksi)
23

fs = stress in the nonprestressed tension reinforcement at nominal flexural resistance (ksi); 

stress in the interface steel reinforcement at the time of UHPC crack localization in 

tension (ksi); stress in longitudinal steel (ksi)

fsl = stress limit in steel at service loads after losses (ksi)

ft,cr = effective cracking strength of UHPC for use in design (ksi)

ft,cri = design UHPC effective cracking strength at the time of prestressing for pretensioned 

members and at the time of initial loading for nonprestressed members (ksi)

ft,loc = crack localization strength of UHPC for use in design (ksi)

fv,α = uniaxial stress in the transverse steel reinforcement at nominal shear resistance (ksi)

fy = specified minimum yield strength of reinforcement (ksi)

H = average annual ambient relative humidity (percent)

h = overall dimension of precast member in the direction in which splitting resistance is 

being evaluated (in.); overall thickness or depth of the member (in.)

K = limiting interface shear resistance (ksi)

K1 = correction factor for modulus of elasticity to be taken as 1.0, unless determined by a 

physical test, and as approved by the owner

K3 = correction factor for creep to be taken as 1.0, unless determined by physical tests, and 

as approved by the owner

K4 = correction factor for shrinkage to be taken as 1.0, unless determined by physical tests,

and as approved by the owner

kf = factor for the effect of UHPC strength

khc = humidity factor for creep

khs = humidity factor for shrinkage

kl = factor for the effect of loading age

ks = factor for the effect of the volume-to-surface ratio of the component

ktd = time development factor

Lvi = interface length considered to be engaged in shear transfer (in.)

ld = development length (in.)

ldh = development length of deformed bars in tension terminating in a standard hook (in.)

lpx = distance from free end of pretensioned strand to the section of member under 

consideration (in.)

lt = transfer length of the prestressing strand (in.)

M = nominal flexural moment (kip-in.)

Mc = nominal crushing moment (kip-in.)

Mcr = nominal cracking moment (kip-in.)
24

ML = nominal crack localization moment (kip-in.)

Mn = nominal flexural resistance (kip-in.)

Mr = factored flexural resistance (kip-in.)

Msl = nominal flexural moment when the steel stress in the extreme tension steel is equal 

to the steel service stress limit, fsl (kip-in.)

Mu = factored moment at the section (kip-in.)

My = nominal yielding moment (kip-in.)

Nu = factored axial force, taken as positive if tensile and negative if compressive (kip)

Pc = permanent net compressive force, normal to the shear plane (kip)

Pn = nominal resistance of a tension member (kip)

Pr = factored axial resistance (kip); factored splitting resistance of pretensioned anchorage 

zones (kip)

Pr,UHPC = splitting resistance of pretensioned anchorage zones provided by the UHPC (kip)

PUHPC = axial compression resistance provided by the UHPC

Ps = axial compression resistance provided by the longitudinal steel reinforcement

pc = length of outside perimeter of the UHPC section (in.)

s = spacing of transverse reinforcement measured in a direction parallel to the 

longitudinal reinforcement (in.)

smax = maximum spacing of transverse reinforcement (in.)

Tcr = torsional cracking moment (kip-in.)

Tn = nominal torsional resistance (kip-in.)

Tr = factored torsional resistance (kip-in.)

Tu = applied factored torsional moment (kip-in.)

t = maturity of UHPC (day)

ti = age of UHPC at time of load application (day)

Veff = effective factored shear resistance (kip)

Vn = nominal shear resistance (kip)

Vni = nominal interface shear resistance (kip)

Vp = component of prestressing force in the direction of the shear force (kip)

Vri = factored interface shear resistance (kip)

Vs = shear resistance provided by transverse reinforcement (kip)

Vu = factored shear force (kip)

VUHPC = nominal shear resistance of the UHPC (kip)
25

vu = shear stress (ksi)

W/CM = water/cementitious materials ratio

α = angle of inclination of transverse reinforcement to longitudinal axis (degrees)

αu = reduction factor to account for the nonlinearity of the UHPC compressive 

stress-strain response

γLL = load factor for live load for Service III load combination

γu = factor to allow for the reduction of UHPC tensile parameter values; it shall not be 

taken greater than 1.0

ε2 = strain in the UHPC diagonal compressive strut (in./in.)

εc = compressive strain in extreme compression fiber of the UHPC section (in./in.)

εcL = strain in the extreme compression fiber of the UHPC section when the UHPC tensile 

strain limit, γuεt,loc, at extreme tension fiber is reached (in./in.)

εcp = elastic compressive strain limit (in./in.)

εcsl = strain in the extreme compression fiber of the UHPC section when the steel stress in 

the extreme tension steel is equal to the steel service stress limit, fsl (in./in.)

εcu = ultimate compressive strain of UHPC for use in design (in./in.)

εs = net longitudinal tensile strain in the section at the centroid of the tension 

reinforcement (in./in.)

εsh = UHPC shrinkage strain at a given time (in./in.)

εsl = net tensile strain in the extreme tension steel when the steel service stress limit, fsl, is 

reached (in./in.)

εt = net tensile strain in extreme tension fiber of the UHPC section (in./in.)

εt,cr = elastic tensile strain limit of UHPC corresponding to a tensile stress of γuft,cr (in./in.)

εt,loc = crack localization strain of UHPC for use in design (in./in.)

εv = strain along the transverse direction of the member (in./in.)

εy = strain in the steel reinforcement corresponding to fy (in./in.)

θ = angle of inclination of diagonal compressive stresses (degrees)

λrl = reinforcement location factor

μ = curvature ductility ratio; friction factor

μl = curvature ductility ratio limit

ξ = multiplier for transfer length of prestressing strand

ρv,α = ratio of area of transverse shear reinforcement to gross UHPC area of a horizontal 

section

 = resistance factor

c = resistance factor for axial resistance
26

f = resistance factor for flexure

v = resistance factor for shear

w = hollow column reduction factor, as defined in AASHTO LRFD BDS Article

5.6.4.7.2c (AASHTO 2020)

Ψ(t, ti) = creep coefficient at time t for loading applied at ti

ψ = sectional curvature at nominal flexural moment, M (1/in.)

ψcr = sectional curvature at nominal cracking moment, Mcr (1/in.)

ψL = sectional curvature at nominal crack localization moment, ML (1/in.)

ψn = sectional curvature at nominal flexural strength, Mn (1/in.)

ψsl = baseline section curvature at Msl (1/in.)

ψy = sectional curvature at nominal yielding moment, My (1/in.)

1.4. MATERIAL PROPERTIES

1.4.1. General

Designs shall be based on the material 

properties cited herein and on the use of 

materials that conform to qualification and 

acceptance testing standards. 

The contract documents shall define the grades 

or properties of all materials to be used.

1.4.2. Ultra-High Performance Concrete

1.4.2.1. General C1.4.2.1

The material properties and idealized 

stress-strain behaviors of UHPC for use in 

design shall be determined according to the 

methods cited in this Article and in Section 2 of 

this Appendix. 

The minimum acceptable material property 

values, including their statistical variability, 

shall be established prior to design and shown 

in the contract documents. 

Typical material property values are given in 

Addendum A1. 

When specifying minimum acceptable material 

properties for UHPC to be used in design, 

designers may refer to material qualification 

results produced in accordance with Section 2 

of this Appendix, supplier certification reports,

and/or the typical values given in 

Addendum A1. Utilizing the minimum 

property values specified in Article 1.1.1 of this 

Appendix, particularly the minimum value of 

the crack localization strain, εt,loc, for products 

with property values that exceed these 

thresholds, may result in overly conservative 

and uneconomical designs.
27

The typical values given in Addendum A1 are 

based on the experimental work of El-Helou,

Haber, and Graybeal (2022) and Haber et al. 

(2018), which included 11 UHPC products 

commercially available in Europe and North 

America. The studies showed that the 

mechanical property values of the considered 

products exceeded the minimum values 

specified in Article 1.1.1.

1.4.2.2. Unit Weight C1.4.2.2

In the absence of more precise information, the 

unit weight of UHPC may be taken equal to 

0.155 kcf.

The unit weight of UHPC includes the weight 

of the fiber reinforcement in the UHPC mix 

design.

For UHPC components reinforced with 

prestressed or nonprestressed steel, the unit 

weight may generally be increased by 0.005 kcf 

to account for the weight of the reinforcement 

in dead load calculations. UHPC components

may contain different proportions of 

reinforcement than commonly occur in 

conventional concrete components, so the 

increased unit weight to account for the dead 

load of the reinforcement may need to be more 

accurately determined.

1.4.2.3. Modulus of Elasticity C1.4.2.3

The modulus of elasticity of UHPC, Ec, is used 

to describe the elastic behavior in tension and 

compression. 

The modulus of elasticity, Ec, may be 

determined by physical tests in accordance with 

ASTM C1856/C1856M (ASTM 2017a). 

ASTM C1856/C1856M provides procedures 

for producing and testing UHPC specimens for 

the purpose of determining the properties of the 

material (ASTM 2017a). The test method to 

determine the modulus of elasticity is based on 

ASTM C469/C469M (ASTM 2022g), with 

exceptions listed in Section 8.3 of ASTM 

C1856/C1856M (ASTM 2017a).

In the absence of measured data, the modulus 

of elasticity, Ec, shall be taken as:

Ec = 2,500 K1 f ́c

0.33 (1.4.2.3-1)

where:

f ́c = compressive strength of UHPC for 

use in design (ksi)

Eq. 1.4.2.3-1 with K1 = 1.0 is similar to 

Eq. C5.4.2.4-1 for normal weight concrete 

specified in AASHTO LRFD BDS (AASHTO 

2020) and is verified to be applicable for 

UHPC-class materials with a typical unit 

weight of approximately 0.155 kcf based on the 

experimental work of El-Helou, Haber, and 

Graybeal (2022).
28

K1 = correction factor for modulus of 

elasticity to be taken as 1.0 unless 

determined by physical test, and as 

approved by the owner

The factor K1 is included to allow the calculated 

modulus to be adjusted for different types of 

UHPC. Unless a value has been determined by 

physical tests, K1 should be taken as 1.0. Use of 

a measured K1 factor permits a more accurate 

prediction of modulus of elasticity and, thus, 

will increase the accuracy of associated design 

calculations. 

1.4.2.4. Compression Behavior 

1.4.2.4.1. Compressive Strength C1.4.2.4.1

The compressive strength of the UHPC, f ́c, for 

use in design shall be based on test cylinders 

produced, tested, and evaluated in accordance 

with ASTM C1856/C1856M (ASTM 2017a). 

For each component, the compressive strength 

for use in design, f ́c, shall be shown in the 

contract documents. 

The testing procedure of the compressive 

strength of UHPC is based on ASTM 

C39/C39M (ASTM 2020d), with exceptions 

listed in Section 8.1 of ASTM C1856/C1856M 

(ASTM 2017a). 

1.4.2.4.2. Ultimate Compressive Strain C1.4.2.4.2 

The ultimate compressive strain for use in 

design, εcu, shall be taken as equal to or less 

than the strain corresponding to the 

compressive strength, f ́c, and may be 

determined by physical tests in accordance with 

ASTM C1856/C1856M (ASTM 2017a). 

The experimental value of the ultimate 

compressive strain, εcu, may be recorded when 

testing for the modulus of elasticity in 

accordance with ASTM C1856/C1856M 

(ASTM 2017a). 

Unless determined by physical tests, the 

ultimate compressive strain of UHPC, εcu, shall 

be taken as the greater of the elastic 

compressive strain limit, εcp, or 0.0035. The 

elastic compressive strain limit shall be taken 

as: 

(1.4.2.4.2-1) 

where: 

αu = reduction factor to account for the 

nonlinearity of the compressive 

stress-strain response; it shall not be 

greater than 0.85

The recommendation for εcu, and αu are based 

on the experimental work of El-Helou, Haber, 

and Graybeal (2022), which included 11 UHPC 

products commercially available in Europe and 

North America. 

The value of εcu = 0.0035, applicable in the 

absence of physical tests, is selected to be lower 

than the average value of εcu for the tested 

products. 

The value of αu coincides with the stress value 

at which the initial elastic compressive 

stress-strain response of the tested products 

deviated not more than 10 percent from 

linearity. 
29 

1.4.2.4.3. Compression Design Model C1.4.2.4.2 

The idealized uniaxial stress-strain model of 

UHPC under compression loading shall be 

defined by modulus of elasticity, Ec, 

compressive strength, f ́c, and ultimate 

compressive strain, εcu, as shown in 

Figure 1.4.2.4.3-1. 

Source: FHWA.

Figure 1.4.2.4.3-1—Idealized compressive 

stress-strain model for UHPC. 

The idealized stress-strain model of UHPC 

under compression loading is based on the work 

of El-Helou, Haber, and Graybeal (2022). The 

model mimics the experimental stress-strain 

response of UHPC in the elastic range and until 

the response deviates more than 10 percent

from linearity at a stress equal to αuf ́c occurring 

at a strain εcp. Beyond this point, the model 

sustains the reduced compressive resistance 

until the strain at the material’s compressive 

strength is reached, εcu. The compression design 

model allows the full compressive strain 

capacity through peak compressive stress 

resistance to be used in design. 

1.4.2.5. Tension Behavior

1.4.2.5.1. Effective Cracking Strength C1.4.2.5.1 

The effective cracking strength for use in 

design, ft,cr, is the stress at the onset of the 

formation of the first crack under uniaxial 

loading. It shall be determined based on direct 

tension testing produced, tested, and evaluated 

in accordance with AASHTO T 397 

(AASHTO 2022b). 

For each component, the minimum effective 

cracking strength for use in design shall be 

shown in the contract documents.

AASHTO T 397 is titled Standard Method of 

Test for Uniaxial Tensile Response of 

Ultra-High Performance Concrete (AASHTO 

2022b). In this method, the effective cracking 

strength, ft,cr, is determined as the stress 

corresponding to the intersection of a 

0.02 percent strain offset to the initial elastic 

portion of the captured stress-strain response 

and the portion of the stress-strain response in 

the inelastic region. As demonstrated by the 

research of El-Helou, Haber, and Graybeal

(2022) and Haber et al. (2018), the effective 

cracking strength value may not exactly 

coincide with the true value of the first cracking 

stress, as it is generally slightly higher than the 

elastic stress limit. The value of the effective 

cracking strength, ft,cr, generally coincides with 

the formation of the first discrete crack over the 
30

full cross section of a prismatic specimen tested 

in accordance with AASHTO T 397.

1.4.2.5.2. Crack Localization Strength C1.4.2.5.2

The crack localization strength for use in 

design, ft,loc, is the first tensile stress value at 

which the tensile stress continuously decreases 

with increasing strain or permanently drops 

below the value of the effective cracking

strength, whichever occurs first. It shall be 

determined based on direct tension testing 

produced, tested, and evaluated in accordance 

with AASHTO T 397 (AASHTO 2022b). 

If the value of the crack localization strength, 

ft,loc, is less than 1.20ft,cr (i.e., ft,loc < 1.20ft,cr), it 

shall be taken as equal to the effective cracking 

strength, ft,cr, defined in Article 1.4.2.5.1 of this 

Appendix (i.e., ft,loc = ft,cr).

For each component, the minimum crack 

localization strength for use in design shall be 

shown in the contract documents.

The localization of cracks corresponds to the 

end of the multicracking phase of the tensile 

behavior. During the multicracking phase, the 

tensile resistance remains approximately equal 

to the effective cracking strength or 

continuously increases with increasing strain. 

The localization of cracks marks the beginning 

of the softening phase, in which the tensile 

resistance decreases with increasing strain. In 

the softening phase, the tensile deformation 

generally accumulates into a single crack 

prompted by the pullout of the crack bridging 

fiber reinforcement.

1.4.2.5.3. Crack Localization Strain C1.4.2.5.3

The crack localization strain for use in design, 

εt,loc, is the strain corresponding to the crack 

localization strength, ft,loc. It shall be 

determined based on direct tension testing 

produced, tested, and evaluated in accordance 

with AASHTO T 397 (AASHTO 2022b). 

For each component, the minimum crack 

localization strain for use in design shall be 

shown in the contract documents.

See commentary on crack localization strength

in Article 1.4.2.5.2 of this Appendix.
31

1.4.2.5.4. Tension Design Models C1.4.2.5.4

The idealized uniaxial stress-strain model of 

UHPC under tension loading shall be defined 

by the modulus of elasticity, Ec; the effective 

cracking strength, ft,cr; the crack localization

strength, ft,loc; and the crack localization strain, 

εt,loc.

The models include a factor, γu, applied to the 

values of ft,cr, ft,loc, and εt,loc to allow for the 

reduction of tensile response parameter values. 

The value of γu shall not be taken greater than 

1.0.

For UHPC materials with ft,loc < 1.20ft,cr, the 

idealized stress-strain model shall be taken as 

shown in Figure 1.4.2.5.4-1 with ft,loc taken 

equal to ft,cr.

Source: FHWA.

Figure 1.4.2.5.4-1—Idealized tensile 

stress-strain model for UHPC.

For UHPC materials with ft,loc ≥ 1.20ft,cr, the 

idealized stress-strain model shall be taken as 

shown in either Figure 1.4.2.5.4-1 or 

Figure 1.4.2.5.4-2.

The idealized stress-strain model of UHPC 

subjected to tensile loading is based on the work 

of El-Helou, Haber, and Graybeal (2022). 

Two types of tensile stress-strain responses are 

idealized for use in design. 

The first type, shown in the stress-strain model 

of Figure 1.4.2.5.4-1, describes materials 

exhibiting a stress plateau where the 

postcracking strength remains equal to or 

greater than the effective cracking strength, ft,cr, 

until crack localization occurs at εt,loc. This 

behavior corresponds to response Type H-2 as 

classified in AASHTO T 397 and generally 

describes the behavior of UHPC materials 

reinforced with fiber proportions of 2 percent

by volume or less (AASHTO 2022b). 

The second type, shown in the stress-strain 

model of Figure 1.4.2.5.4-2, corresponds to 

materials where the postcracking strength

continuously increases with increasing strain 

until crack localization occurs at a peak stress 

ft,loc equal to or greater than 1.20ft,cr and at a 

strain εt,loc. This behavior corresponds to 

response Type H-1 as classified in AASHTO 

T 397 and generally describes the behavior of 

UHPC materials reinforced with fiber 

proportions greater than 3 percent by volume

(AASHTO 2022b).

The postcrack localization capacity of the 

UHPC (i.e., the softening phase where the strain 

is greater than the crack localization strain) is 

not accounted for in the design models since 

strains in excess of the localization strain are 

not permitted by the provisions in this 

Appendix and the response is a function of 

crack opening, which is not consistent with a 

strain-based design approach.

The factor γu accounts for the cases in which the

values of the tensile properties of the UHPC 

placed in the structural components are
32

Source: FHWA. 

Figure 1.4.2.5.4-2—Idealized tensile 

stress-strain model for UHPC with ft,loc ≥ 

1.20ft,cr.

in which: 

(1.4.2.5.4-1) 

expected to be lower than their respective 

qualified values determined according to 

Section 2 of this Appendix. Casting processes 

should be considered by the Designer to ensure 

that the member is constructed without 

producing undesirable fiber distributions or 

orientations that impair the tensile behavior of 

as-cast UHPC. Different values of γu may be 

assigned to each of the tensile parameters. 

These values may be informed from prototype 

testing of structural components and/or from 

extracted tension specimens tested in 

accordance with AASHTO T 397 (AASHTO 

2022b).

where: 

εt,cr = elastic tensile strain limit 

corresponding to a tensile stress of 

γu ft,cr 

1.4.2.6. Poisson’s Ratio C1.4.2.6

Poisson’s ratio of UHPC may be determined by 

physical tests in accordance with ASTM 

C1856/C1856M (ASTM 2017a). 

The testing procedure for Poisson’s ratio of 

UHPC is in accordance with ASTM 

C469/C469M (ASTM 2022g), with exceptions 

listed in Section 8.3 of ASTM C1856/C1856M 

(ASTM 2017a).

Unless determined by physical tests, Poisson’s 

ratio of UHPC may be assumed as 0.15. 

The proposed value of Poisson’s ratio is based 

on the experimental work of El-Helou, Haber, 

and Graybeal (2022) and Haber et al. (2018). 

1.4.2.7. Coefficient of Thermal Expansion C1.4.2.7

The coefficient of thermal expansion should be 

determined by the laboratory tests on the 

specific product to be used. 

In the absence of more precise data, the thermal 

coefficient of expansion of UHPC-class 

materials may be taken as 7.0 × 10−6/°F. 

The coefficient of thermal expansion should be 

determined from laboratory tests in accordance 

with AASHTO T 336 (AASHTO 2019) for 

hydraulic cement concrete and with the 

following modification: specimens should be 

sealed before testing by applying an epoxy layer 

to their exterior surfaces, with the exception of 

the bearing points of the supports and the 
33

displacement transducer (Graybeal 2006a). 

Sealing the specimens reduces the effects of 

water saturation, given the low permeability of 

UHPC.

The proposed value of the coefficient of thermal 

expansion is based on the experimental work of 

Mohebbi, Graybeal, and Haber (2022).

1.4.2.8. Creep and Shrinkage

1.4.2.8.1. General C1.4.2.8.1

In lieu of alternate models for creep and 

shrinkage strains in UHPC developed based on 

laboratory test data for the specific UHPC-class 

material under consideration, the values of 

creep and shrinkage specified in 

Articles 1.4.2.8.2 and 1.4.2.8.3 of this 

Appendix shall be used. 

The designer shall consider the variability in 

estimates of creep and shrinkage.

Creep and shrinkage of UHPC are variable 

properties that depend on many factors, some of 

which may not be known at the time of design. 

The methods for determining creep and 

shrinkage specified in Articles 1.4.2.8.2 and 

1.4.2.8.3 of this Appendix are based on the 

experimental work of Mohebbi, Graybeal, and 

Haber (2022) and Mohebbi and Graybeal 

(2022). The development of these methods 

parallel the concepts of the creep and shrinkage 

methods specified in AASHTO LRFD BDS

Articles 5.4.2.3.2 and 5.4.2.3.3 (AASHTO 

2020) and recalibrate the different empirical 

factor equations based on experimental data of 

eight UHPC products available in Europe and 

North America.

Predictive relationships for creep and shrinkage 

behavior of UHPC provide an estimate of the 

actual material behavior. An approach that 

considers higher and lower estimates for these 

behaviors should be engaged when designing 

and constructing members sensitive to these 

effects.

1.4.2.8.2. Creep C1.4.2.8.2

Unless verified by physical tests and as 

approved by the owner, the provisions in this 

Article shall apply for compressive strengths at 

the time of prestressing for pretensioned 

members and at the time of initial loading for 

nonprestressed members, f ́ci, equal to or 

The provisions in this Article are based on 

experimental data of Mohebbi, Graybeal, and 

Haber (2022) in which the creep specimens 

were loaded at compressive strengths equal to

or greater than 14.0 ksi. The creep coefficient 

for components loaded at compressive strength 

values less than 14.0 ksi is expected to be 
34

greater than 14.0 ksi. 

The creep coefficient may be taken as: 

Ψ (t, ti) = 1.2ks khc kf ktd kl K3 (1.4.2.8.2-1) 

in which: 

ks = 1.0 (1.4.2.8.2-2) 

khc = 1.12 − 0.0024H (1.4.2.8.2-3) 

(1.4.2.8.2-4) 

(1.4.2.8.2-5) 

where: 

Ψ(t, ti) = creep coefficient at time t for 

loading applied at ti

ks = factor for the effect of the 

volume-to-surface ratio of the 

component 

khc = humidity factor for creep 

kf = factor for the effect of UHPC 

strength 

ktd = time development factor 

kl = factor for the effect of loading 

age determined according to 

Eqs. 1.4.2.8.2-6 and 1.4.2.8.2-7 

K3 = correction factor for creep to be 

taken as 1.0 unless determined 

by physical test, and as approved 

by the owner 

higher than determined by the methods in this 

Article, particularly when subjected to high 

compressive stresses. 

The creep coefficient is applied to the 

compressive strain caused by permanent loads 

to obtain the strain due to creep. 

Creep is influenced by the same factors as 

shrinkage, and by the magnitude and duration 

of stress, the maturity of the UHPC at the time 

of loading, and temperature. 

The factor for the volume-to-surface ratio, ks, is 

taken as 1.0 because UHPC has a dense 

microstructure and a disconnected pore 

structure that significantly reduces the rate of 

moisture exchange with the outside 

environment (Mohebbi and Graybeal 2022).

The factor K3 is included to allow calculated 

creep coefficient values to be adjusted for a 

specific UHPC product. It is assumed to be 1.0, 

except for situations where specific knowledge 

is available. The K3 factor may be determined 

from physical tests on creep performed 

according to ASTM C1856/C1856M and with a 

sustained compressive stress equal to 

65 percent of the compressive strength at the 

time of loading. The testing procedure for the 

compression creep of UHPC is in accordance 

with ASTM C512/C512M (ASTM 2015c), with 

exceptions listed in Section 8.4 of ASTM 

C1856/C1856M (ASTM 2017a). 

In cases where the correction factor for modulus 

of elasticity K1 is different than 1.0 and physical 

tests on creep are not available, K3 may be taken 

as equal to 1/K1. This adjustment accounts for 

the potential impact of the modulus of elasticity 

on the ultimate creep coefficient. 
35

H = average annual ambient relative 

humidity (percent). In the 

absence of better information, H

may be taken from AASHTO 

LRFD BDS Figure 5.4.2.3.3-1

(AASHTO 2020)

f ́ci = design UHPC compressive 

strength at the time of prestressing for pretensioned 

members and at the time of 

initial loading for nonprestressed 

members (ksi)

t = maturity of UHPC (day)

ti = age of UHPC at time of load 

application (day)

The factor for the effect of loading age, kl, shall 

be determined as follows:

If ti < 7.0 days, then kl = 1.0 (1.4.2.8.2-6)

If ti ≥ 7.0 days, then kl = (ti − 6)−0.15 ≥ 0.5

(1.4.2.8.2-7)

1.4.2.8.3. Shrinkage C1.4.2.8.3

The strain due to shrinkage, εsh, at time, t, may 

be taken as:

εsh = 0.6 × 10−3ks khs kf ktd K4 (1.4.2.8.3-1)

in which:

khs = 1.5 − 0.01H (1.4.2.8.3-2)

where:

khs = humidity factor for shrinkage

K4 = correction factor for shrinkage to be 

taken as 1.0 unless determined by 

physical test, and as approved by 

the owner

Shrinkage of UHPC is caused by both drying 

effects and autogenous behaviors associated 

with hydration and other chemical reactions 

occurring during curing. Autogenous shrinkage 

in UHPC is proportionally much larger than that 

occurring in conventional concrete and 

predominantly takes place during the early 

strength development of the material. This

effect is in contract to conventional concrete, 

where most of the shrinkage is related to drying 

effects and occurs later in the curing process.

Additional information on shrinkage in UHPC 

can be found in Haber et al. 2018.
36

The constraining effects of reinforcement and 

composite actions with other elements of the 

structure tend to reduce the dimensional

changes in some components. 

The factor K4 is included to allow calculated 

shrinkage strain values to be adjusted for a 

specific UHPC product. It is assumed to be 1.0, 

except for situations where specific knowledge 

is available. The K4 factor may be determined 

from physical tests on shrinkage performed 

according to ASTM C1856/C1856M. The 

testing procedure for the shrinkage strain of 

UHPC is in accordance with ASTM 

C157/C157M (ASTM 2017e), with exceptions 

listed in Section 8.5 of ASTM C1856/C1856M

(ASTM 2017a).

1.4.3. Reinforcing Steel C1.4.3

The provisions of AASHTO LRFD BDS 

Article 5.4.3 (AASHTO 2020) shall apply.

The total elongation under load for reinforcing 

steel conforming to the requirements of 

AASHTO LRFD BDS Article 5.4.3 (AASHTO 

2020) are defined within the referenced 

documents. The value is contingent on the type 

of steel, the grade of steel, and the size of the 

reinforcing bar.

1.4.4. Prestressing Steel

The provisions of AASHTO LRFD BDS

Article 5.4.4 (AASHTO 2020) shall apply.

When 0.62-in.- or 0.7-in.-diameter strands are 

used, the provisions for 0.6-in.-diameter 

strands in AASHTO LRFD BDS Article 5.4.4 

(AASHTO 2020) shall apply.

The total elongation under load for prestressing 

strands and bars conforming to the material 

standards specified in AASHTO LRFD BDS 

Article 5.4.4.1 (AASHTO 2020) shall not be 

less than 0.035 and 0.04, respectively.
37

1.5. LIMIT STATES AND DESIGN 

METHODOLOGIES

1.5.1. General

The provisions of AASHTO LRFD BDS

Articles 5.5.1.1, 5.5.1.2.1, and 5.5.1.2.2 

(AASHTO 2020) shall apply; reference to 

Articles 5.6 and 5.7 (AASHTO 2020) shall be 

replaced by Articles 1.6 and 1.7 of this 

Appendix, respectively. Design practices for 

D-Regions shall be in accordance with 

Article 1.8.

1.5.2. Service Limit State C1.5.2

Actions to be considered at the service limit 

state shall be cracking, deformations, stresses, 

and strains. The following provisions shall 

apply:

• The cracking strength shall be taken as 

γuft,cr as specified in Articles 1.4.2.5.1

and 1.4.2.5.4 of this Appendix.

• Deformations shall be in accordance 

with Article 1.6.3.5 of this Appendix.

For prestressed components:

• The section properties for service limit 

state may be based on uncracked 

sections when the extreme fiber tensile 

stress due to Service III load 

combination specified in AASHTO 

LRFD BDS Table 3.4.1-1 (AASHTO 

2020) does not exceed γuft,cr.

• The stress limits for prestressing steel 

and stresses at the service limit state 

after losses shall be in accordance with 

Articles 1.9.2.2 and 1.9.2.3 of this 

Appendix, respectively.

For nonprestressed components with or without 

reinforcement:
38

• Service stress limit shall be investigated 

using the Service Limit State Combination I specified in AASHTO LRFD 

BDS Table 3.4.1-1 (AASHTO 2020).

• The section properties for service limit 

state investigations shall be based on 

cracked sections when the extreme 

tensile stress exceeds γuft,cr. 

• The strain in the UHPC at extreme 

tension fiber shall not exceed the lesser 

of 0.25γuεt,loc or 0.001, where εt,loc is the 

crack localization strain specified in 

Article 1.4.2.5.3 of this Appendix.

The upper limit of 0.001 on the service strain 

for nonprestressed UHPC is adopted from the 

Swiss recommendation SIA 2052 UHPFRC: 

Materials, Design, and Application (Swiss 

Institute of Architects and Engineers 2016).

• The compressive stress at extreme 

compression fiber shall not exceed 

0.45f ́c due to permanent loads and 

0.60wf ́c due to permanent and 

transient loads, as well as during 

shipping and handling. The reduction 

factor, w, shall be taken equal to 1.0 

when the web and flange slenderness 

ratio, calculated according to AASHTO 

LRFD BDS Article 5.6.4.7.1 

(AASHTO 2020), is not greater than 

15. When either the web or flange 

slenderness ratio is greater than 15, the 

reduction factor, w, shall be calculated 

according to AASHTO LRFD BDS 

Article 5.6.4.7.2 (AASHTO 2020).

The compressive strength limits for nonprestressed UHPC at the service limit state are 

similar to the limits on prestressed elements as 

specified in AASHTO LRFD BDS Article

5.9.2.3.2a (AASHTO 2020).

• The principal tensile stresses in webs of 

components shall not exceed γuft,cr

when the superstructure element is 

subjected to loadings of Service I load 

combination. The principal tensile 

stresses shall be determined using the 

combination of axial and shear stress 

which produces the greatest principal 

tensile stress, and the shear stresses 

shall be determined using vertical shear 

and concurrent torsion.

The principal tensile stress check addresses

limit web cracking at a service limit state. Shear 

flow may be determined by analytical methods 

and/or finite element analysis. The provisions 

of Article 1.9.2.3.3 of this Appendix may apply.
39

• The stress limit for steel reinforcement 

in nonprestressed components shall be 

taken as 0.80fy, where fy is the steel 

yielding stress.

1.5.3. Fatigue Limit State C1.5.3

The fatigue limit state shall be evaluated as 

follows:

• The tensile stress in UHPC due to the 

Service I load combinations shall not 

exceed 0.95γuft,cr.

Due to lack of sufficient experimental data on 

fatigue of cracked UHPC, members subjected 

to cyclic stresses from live loads must be 

designed to be uncracked. Members that have 

been subjected to tensile stresses exceeding the 

cracking strength (e.g., members subjected to 

loads approaching the Strength I limit state) 

may be susceptible to fatigue damage from 

subsequent cyclic loadings.

The Service I load combination is used for the 

Fatigue Limit State in UHPC in order to reduce 

the likelihood that cracked UHPC will be 

subjected to cyclic loads. This requirement 

often controls the design of nonprestressed 

components; per Article 1.5.2 of this Appendix, 

prestressed components are designed to be 

uncracked at service. 

This tensile stress limit of 0.95γuft,cr applies to 

the tensile stress in members in all directions,

including the longitudinal, transverse, and 

principal directions.

The specified fatigue tensile stress limit is 

adopted from the French Standard for fatigue 

considerations of UHPC components 

(NF P18-710, Association Française de 

Normalisation 2016).

• Discrete steel elements embedded in 

UHPC shall be checked for fatigue in 

accordance with AASHTO LRFD BDS

Eq. 5.5.3.1-1 and Articles 5.5.3.2, 

5.5.3.3, and 5.5.3.4 (AASHTO 2020).

Due to the lack of sufficient experimental data

on fatigue of UHPC, these provisions require all 

discrete steel elements to be checked to ensure 

their stress ranges, in the uncracked section, 

remain less than the fatigue threshold. While 

these provisions require that sections designed 

for fatigue remain uncracked, checking discrete 
40

steel elements is required since the compressive 

and tensile strengths of UHPC are greater than 

that of conventional concrete and certain design 

scenarios may elevate the stress ranges in 

discrete steel elements more than if they were 

embedded in the same design of conventional 

concrete.

• For prestressed components, the 

compressive stress due to the Fatigue I

load combination and one-half the sum 

of the unfactored effective prestress and 

permanent loads shall not exceed 0.40f ́c

after losses.

The compressive stress limit for prestressed

UHPC is adopted from AASHTO LRFD BDS

Article 5.5.3.1 (AASHTO 2020). 

• For nonprestressed components, the 

compressive stress due to the Fatigue I

load combination and one-half of the 

unfactored permanent loads shall not 

exceed 0.40f ́c.

The compressive stress limit for nonprestressed

UHPC is adopted from AASHTO LRFD BDS

Article 5.5.3.1 (AASHTO 2020).

1.5.4. Strength Limit State

1.5.4.1. General

The strength limit state issues to be considered 

shall be those of strength, ductility, and 

stability.

Factored resistance shall be the product of 

nominal resistance as determined in accordance 

with the applicable provisions of Articles 1.6, 

1.7, 1.9, and 1.10 of this Appendix, unless 

another limit state is specifically identified, and 

the resistance factor as specified in Article

1.5.4.2 of this Appendix.

1.5.4.2. Resistance Factors C1.5.4.2

The provisions of this Article are applicable to 

prestressed and nonprestressed UHPC sections. 

Resistance factor  shall be taken as:

The resistance factors for flexural capacity are 

based on sectional ductility considerations. A 

curvature ductility ratio, μ, is defined as the 

ratio of the sectional curvature at the nominal 

moment resistance over a baseline sectional 

curvature as specified in Article 1.6.3.2.3 of this 

Appendix. The background for these provisions 

is given in El-Helou and Graybeal (2022a).
41

• For flexural capacity of sections with 

curvature ductility ratio, μ, greater than 

the curvature ductility ratio limit, μl

 ............................................... 0.90. 

• For flexural capacity of sections with 

curvature ductility ratio, μ, less than 1.0 

................................................ 0.75.

• For flexural capacity of sections with 

no tensile reinforcement ......... 0.75.

• For compression members ..... 0.75.

• For tension members and members 

subjected to combined tension and 

flexure ................................... 0.75.

• For shear and torsion in reinforced and 

unreinforced sections ............ 0.90.

• For bearing on UHPC ............ 0.70.

• For resistance during pile driving 

 ............................................... 1.00. 

For sections in which the curvature ductility 

ratio, μ, is between the curvature ductility ratio 

limit, μl, and 1.0, the value of  associated with 

the curvature ductility ratio may be obtained by 

a linear interpolation from 0.75 to 0.90. 

This variation of  may be computed for 

prestressed and nonprestressed members such 

that: 

(1.5.4.2-1) 

Sections are assumed to be ductile and assigned 

a  factor of 0.90 if their curvature ductility 

ratio, μ, is equal to or greater than the curvature 

ductility ratio limit, μl. 

Sections with μ less than or equal to 1.0 are less 

ductile and are assigned a  factor of 0.75, akin 

to the  factor for compression-controlled 

sections specified in AASHTO LRFD BDS 

Article 5.5.4.2 for conventional concrete 

(AASHTO 2020). 

Sections with μ between 1.0 and μl are 

classified as transitional and assigned a linearly 

varying  factor in the transition zone between 

the two extreme values of μ, as shown in 

Figure C1.5.4.2-1. 

Source: FHWA. 

Figure C1.5.4.2-1—Variation of the flexural 

resistance factor, , with the curvature 

ductility ratio, μ, for UHPC sections 

reinforced with prestressed steel, 

nonprestressed steel, or both. 

The resistance factor  for shear capacity is 

assigned a value of 0.90, in line with AASHTO 

LRFD BDS Article 5.5.4.2 provisions for shear 

in reinforced concrete sections (AASHTO 

2020). 
42

Where combinations of different grades of

reinforcement, or of prestressed and

nonprestressed reinforcement, are used in 

design, the lowest resistance factor calculated 

for each grade or type of reinforcement shall be 

used.

The value of the yield stress of different types 

or grades of reinforcement will result in 

different values of the baseline sectional 

curvature, ψsl, as specified in Article 1.6.3.2.3

of this Appendix, which, consequently, leads to 

different values of μl and . The lowest value of 

 should be used to calculate the flexural 

resistance of the member.

1.5.4.3. Stability

The provisions of AASHTO LRFD BDS 

Article 5.5.4.3 (AASHTO 2020) shall apply.

1.5.5. Extreme Event Limit State

The provisions of AASHTO LRFD BDS 

Article 5.5.5 (AASHTO 2020) shall apply.

1.6. DESIGN FOR FLEXURAL AND

AXIAL FORCE EFFECTS—B REGIONS

1.6.1. Assumptions for Service and Fatigue 

Limit States

The provisions of AASHTO LRFD BDS

Article 5.6.1 shall apply with the following 

amendments:

• Reference to the provisions of 

AASHTO LRFD BDS Article 5.6.6 

(AASHTO 2020) shall not apply.

• Reference to AASHTO LRFD BDS

Article 5.4.2.3.2 (AASHTO 2020) shall 

be replaced by Article 1.4.2.8.2 of this 

Appendix.

Nonprestressed UHPC not subject to fatigue 

loads resists tension when the tensile strain 

does not exceed the lesser of 0.25γuεt,loc or 

0.001. The tensile strains shall be determined 

from a strain compatibility approach, utilizing 

a representative steel stress-strain curve and the 

tensile design models of Article 1.4.2.5.4 of 

this Appendix.
43

Nonprestressed and prestressed UHPC subject 

to fatigue loads resists tension at sections that 

are uncracked when the extreme tension stress 

does not exceed 0.95γuft,cr.

1.6.2. Assumptions for Strength and 

Extreme Event Limit States

C1.6.2

Factored resistance of UHPC components shall 

be based on the conditions of equilibrium and 

strain compatibility as specified in Article 1.6.3

of this Appendix, the resistance factors as 

specified in Article 1.5.4.2 of this Appendix,

and the following assumptions:

• In components with bonded reinforcement or prestressing, or in the bonded 

length of debonded strands, strain is 

directly proportional to the distance 

from the neutral axis, except for deep 

components that shall satisfy the 

requirements for disturbed regions.

• If the UHPC is unconfined, the 

maximum usable strain at extreme 

UHPC compression fiber is not greater 

than the ultimate compression strain, 

εcu, as specified in Article 1.4.2.4.2 of 

this Appendix.

• If the UHPC is confined, a maximum 

usable strain exceeding εcu in the 

confined core may be utilized if verified

by physical test. Calculation of the 

factored resistance shall consider that 

the UHPC cover may be lost at strains 

compatible with those in the confined 

core.

• The UHPC compressive stress-strain 

distribution is assumed to follow the 

uniaxial compression design model 

specified in Article 1.4.2.4.3 of this 

Appendix.

• In components with prestressed and 

nonprestressed reinforcement, the 
44

maximum usable strain at extreme 

UHPC tensile fiber is not greater than 

γuεt,loc.

• In components with no tensile 

reinforcement, the maximum usable 

strain in extreme UHPC tensile fiber is 

not greater than 0.5γuεt,loc.

Components with no tensile reinforcement have 

lesser ability to share tensile loads between the 

UHPC and the reinforcement, thus necessitating extra restrictions on the tensile strain 

capacity.

• The UHPC tensile stress-strain 

distribution is assumed to follow the 

uniaxial tensile design models specified 

in Article 1.4.2.5.4 of this Appendix.

• The strain in the reinforcement is not 

greater than the minimum total 

elongation strain determined according 

to Article 1.4.3 of this Appendix for 

reinforcing steel and Article 1.4.4 of 

this Appendix for prestressing steel. 

• The stress in the reinforcement is based 

on a stress–strain curve representative 

of the steel or on an approved 

mathematical representation, including 

development of reinforcement and 

prestressing elements and transfer of 

pretensioning.

• Except as specified in Article 1.6.3.3 of 

this Appendix, the calculated stress in 

nonprestressed reinforcement shall not 

be taken as greater than the specified 

minimum yield strength.

• The development of reinforcement and 

prestressing elements and transfer of 

pretensioning are considered.

• The curvature ductility ratio limit, μl, 

defined in Article 1.5.4.2 of this 

Appendix, shall be greater than or equal 

to 3.0.

Unless specifically permitted by provisions in 

this Appendix, the strain hardening strength of 

nonprestressed reinforcement may not be used 

in design.
45

• The use of compression reinforcement 

in conjunction with additional tension 

reinforcement is permitted to increase 

the strength of flexural members.

Additional limitations on the maximum usable 

extreme UHPC compressive strain in hollow 

rectangular compression members shall be 

investigated as specified in Article 1.6.4 of this 

Appendix.

1.6.3. Flexural Members C1.6.3

The following methods shall be used for UHPC 

sections with bonded reinforcement.

The proposed framework for flexural members 

is derived from El-Helou and Graybeal (2022a) 

and validated for prestressed and nonprestressed UHPC beams performed by 

El-Helou and Graybeal (2022a), Chen et al. 

(2018), Yoo, Banthia, and Yoon (2017), Qiu et 

al. (2020), Yoo and Yoon (2015), and Graybeal 

(2006b, 2009a).

1.6.3.1. Strain Compatibility Approach C1.6.3.1

The strain compatibility approach shall be used 

to determine the nominal moment resistance, 

Mn, the sectional curvature at nominal moment 

resistance, ψn, and the sectional curvature when 

the stress in the extreme tension steel is equal 

to the steel service stress, ψsl, as specified in 

Article 1.6.3.2 of this Appendix.

The stress and corresponding strain in any 

given layer of the UHPC, prestressing steel, 

and/or nonprestressed reinforcement shall be 

taken from representative stress-strain models 

as specified in Articles 1.4.2.4.3, 1.4.2.5.4, and 

1.6.2 of this Appendix. The conditions of force 

equilibrium in the section shall be satisfied, and 

the nominal flexural strength shall be 

calculated directly from the stresses resulting 

from this analysis.

The idealized moment-curvature (M-ψ) 

response for a nonprestressed flexural member 

utilizing the full tensile strain capacity of UHPC 

is shown in Figure C1.6.3.1-1. The M-ψ 

response can be idealized with the key points 

identified in Figure C1.6.3.1-1. The first point 

in the behavior (Mcr-ψcr) is the nominal moment 

and curvature at the initiation of the first 

flexural crack. At this point, the strain at 

extreme UHPC tensile fiber, εt, is equal to the 

cracking strain εt,cr. The second point (Msl-ψsl) 

is an intermediate point defining the nominal 

moment and curvature when the stress in the 

extreme tensile steel layer is equal to steel 

service stress limit, fsl. This point defines the 

baseline sectional curvature for calculation of 

the section ductility ratio as specified in 

Article 1.6.3.2.3 of this Appendix. The third 

point in the moment-curvature response (My-ψy) 

coincides with the yielding of the extreme 

tension steel layer. The yielding moment occurs 

prior to the localization of cracks, as depicted in 

Figure C1.6.3.1-1 when the UHPC crack 
46

localization strain is greater than the yielding 

strain of flexural steel reinforcement (γuεt,loc > 

εy). The fourth point in the behavior 

corresponds to the nominal moment resistance 

and curvature at localization of cracks (ML-ψL) 

occurring when the strain at extreme UHPC 

tensile fiber, εt, is equal to γuεt,loc, as shown in 

Figure C1.6.3.2.2-1. Note that the case 

demonstrated in the figure shows the UHPC 

tensile strain limit larger than the yielding strain 

of the flexural steel reinforcement (γuεt,loc > εy); 

this may not always be the case.

Source: FHWA.

Figure C1.6.3.1-1—Moment-curvature 

response of UHPC sections reinforced with 

nonprestressed steel failing at localization of 

cracks (εt = γu εt,loc) and with γu εt,loc > εy.

1.6.3.2. Flexural Resistance

1.6.3.2.1. Factored Flexural Resistance

The factored flexural resistance, Mr, shall be 

taken as:

Mr = Mn (1.6.3.2.1-1)

where:

Mn = nominal flexural resistance (kip-in.)

 = resistance factor as specified in 

Article 1.5.4.2 of this Appendix
47

1.6.3.2.2. Nominal Flexural Resistance C1.6.3.2.2

The nominal flexural resistance shall be 

obtained from a strain compatibility analysis 

according to Article 1.6.3.1 of this Appendix

and shall be taken as the moment 

corresponding to the lesser of the sectional 

curvature values calculated when:

• The compressive strain at the extreme 

compression fiber of the UHPC section 

is equal to the compression strain limit, 

εcu,

• The net tensile strain at extreme tension 

fiber of the UHPC section is equal to the 

UHPC tensile strain limit, γuεt,loc, and,

• The strain in the extreme tension steel 

is equal to the minimum total 

elongation strain of reinforcing steel.

The nominal flexural resistance is based on 

strain compatibility and equilibrium 

computations at key points in the behavior 

corresponding to the attainment of the strain 

limits of the UHPC in tension or compression. 

Limit states pertaining to other materials used 

in a composite section, such as the crushing 

limit of a conventional concrete deck acting 

compositely with a UHPC beam, should also be 

considered when relevant. The nominal flexural 

resistance should always be taken at the 

smallest sectional curvature value 

corresponding to all relevant limit states.

For sections fully utilizing the postcracking 

capacity of UHPC, the nominal flexural 

resistance is taken when the strain in the UHPC 

at extreme tension fiber, εt, is equal to the

UHPC tensile strain limit, γuεt,loc, as shown in 

Figure C1.6.3.2.2-1 (i.e., Mn = ML). In this 

failure mode, the UHPC stresses in 

compression remain elastic if the strain at 

extreme compression fiber, εc, is less than εcp, 

as shown in part (a) of Figure C1.6.3.2.2-1, or 

become plastic if εcp < εc < εcu, as shown in part 

(b) of Figure C1.6.3.2.2-1.

The flexural behavior after crack localization 

(εt > γuεt,loc) is not considered for use in design 

because of the formation of a localized crack on 

the tension side of the beam, the loss of UHPC 

fiber bridging capacity, and the hinging of the 

beam about the section with the localized crack. 

The large crack opening at the section with the 

localized crack strains the tensile reinforcement 

over a short distance, increasing the risk of 

reinforcement rupture.

For flexural members with significant axial 

compressive stress or high levels of longitudinal steel reinforcement, the nominal flexural 

resistance may occur at the crushing of UHPC 

(Mn = Mc). In this failure mode, the strain in the 

UHPC at the extreme compression layer, εc, is 

equal to the ultimate compressive strain, εcu, 
48 

while the strain in the extreme tensile fiber, εt, 

is less than the UHPC tensile strain limit, γuεt,loc, 

as shown in Figure C1.6.3.2.2-1. 

Figure C1.6.3.2.2-1 and Figure C1.6.3.2.2-2

depict the tensile stresses of a UHPC material 

by adopting the idealized stress-strain model of 

Figure 1.4.2.5.4-1. When ft,loc > 1.20ft,cr, the 

idealized stress-strain model of 

Figure 1.4.2.5.4-2 may be used instead.

Source: FHWA.

Note: See notation list in Article 1.3 of this Appendix for variable definitions.

Figure C1.6.3.2.2-1—Stress and strain conditions for UHPC sections in flexure at the onset 

of crack localization, shown for UHPC material exhibiting the tensile stress-strain behavior 

of Figure 1.4.2.5.4-1. 

Source: FHWA.

Note: See notation list in Article 1.3 of this Appendix

for variable definitions.

Figure C1.6.3.2.2-2—Stress and strain 

conditions for UHPC sections in flexure at 

UHPC crushing, shown for UHPC material 

exhibiting the tensile stress-strain behavior 

of Figure 1.4.2.5.4-1. 
49 

1.6.3.2.3. Curvature Ductility Ratio C1.6.3.2.3 

The curvature ductility ratio, μ, defined as the 

ratio of the sectional curvature at the nominal 

moment resistance over the baseline sectional 

curvature, shall be calculated at nominal 

flexural strength such as:

 (1.6.3.2.3-1) 

in which: 

 (1.6.3.2.3-2)

where: 

ψn = sectional curvature of the UHPC 

section at nominal flexural strength, 

as specified in Article 1.6.3.2.2 of 

this Appendix (1/in.) 

ψsl = sectional curvature of the UHPC 

section when the steel stress in the 

extreme tension steel is equal to the 

steel service stress limit, fsl (1/in.) 

csl = distance from the extreme 

compression fiber of the UHPC 

section to the neutral axis when the 

steel stress in the extreme tension 

steel is equal to the steel service 

stress limit, fsl (in.) 

εcsl = strain in the extreme compression 

fiber of the UHPC section when the 

steel stress in the extreme tension 

steel is equal to the steel service 

stress limit, fsl (in./in.) 

fsl = stress limit in steel at service loads, 

as defined in Article 1.5.2 of this 

Appendix (ksi)

For sections controlled by the UHPC tensile 

strain limit, γuεt,loc, at extreme tension fiber, the 

sectional curvature at nominal flexural 

strength, ψn, shall be taken as:

(1.6.3.2.3-3) 

The flexural resistance is based on sectional 

curvature ductility considerations to ensure that 

large deformations occur before failure of the 

member. The approach is based on calculating a 

curvature ductility ratio, μ, defined as the ratio 

of the sectional curvature at the nominal 

moment resistance, ψn, over the baseline 

sectional curvature, ψsl. The baseline sectional 

curvature is computed when the stress in the 

extreme tension steel is equal to 80 percent of 

the yielding stress of the reinforcement, as 

shown in Figure C1.6.3.2.3-1. For prestressed 

members, this limit coincides with the stress 

limit for prestressing steel at service limit state,

as specified in AASHTO LRFD BDS Article 

5.9.2.2 (AASHTO 2020). The UHPC 

compression stresses depicted in 

Figure C1.6.3.2.3-1 are assumed to remain 

elastic.

Source: FHWA.

Note: See notation list in Article 1.3 of this Appendix for 

variable definitions.

Figure C1.6.3.2.3-1—Stress and strain 

conditions for UHPC sections when the 

stress in the extreme tension steel is equal to 

fsl, shown for UHPC material exhibiting the 

tensile stress-strain behavior of 

Figure 1.4.2.5.4-1. 

Sections with curvature ductility ratios, μ, 

greater than the ductility ratio limit, μl = 3.0, are 

considered ductile and assumed to provide 

sufficient member deformation before crack 

 

 
50 

For sections controlled by the UHPC 

compression strain limit, εcu, at extreme 

compression fiber, the sectional curvature at 

nominal flexural strength, ψn, shall be taken as:

(1.6.3.2.3-4) 

where: 

εcL = strain in the extreme compression 

fiber of the UHPC section when the 

UHPC tensile strain limit, γuεt,loc, at 

extreme tension fiber is reached

(in./in.) 

cL = distance from the extreme 

compression fiber of the UHPC 

section to the neutral axis when the 

UHPC tensile strain limit, γuεt,loc, at 

extreme tension fiber is reached 

(in.)

cc = distance from the extreme 

compression fiber of the UHPC 

section to the neutral axis when the 

UHPC compression strain limit, εcu, 

at extreme compression fiber is 

reached (in.)

localization. When members have curvature 

ductility less than μl, a reduced resistance 

factor, φ, is imposed in recognition of the 

nonductile behavior, as specified in Article

1.5.4.2 of this Appendix. 

For composite sections made with UHPC beams 

and UHPC or conventional concrete decks: 

• ψsl should be calculated for the UHPC 

beam when the steel service stress limit, 

fsl, is first reached in the extreme tension 

steel of either the UHPC beam or 

composite section, and

• ψn should be computed for the UHPC 

beam at the governing limit state at 

nominal flexural strength pertaining to 

all materials used in the composite 

section.

1.6.3.3. Minimum Reinforcement

The provisions of AASHTO LRFD BDS

Article 5.6.3.3 shall apply with the following 

amendments:

• The modulus of rupture of concrete in 

AASHTO LRFD BDS Eq. 5.6.3.3-1 

(AASHTO 2020) shall be replaced with 

the UHPC effective cracking strength, 

ft,cr, as specified in Article 1.4.2.5.1 of 

this Appendix. 

The factor γu is conservatively not applied to the 

effective cracking stress, ft,cr. 

• Reference to AASHTO LRFD BDS 

Article 5.4.2.6 (AASHTO 2020) shall 

not apply. 

• The provisions of AASHTO LRFD 

BDS Article 5.10.6 (AASHTO 2020) 

need not apply. 
51

1.6.3.4. Moment Redistribution C1.6.3.4

Unless refined analysis is performed and as 

approved by the owner, redistribution of 

moments is not permitted.

Any analysis for moment redistribution should 

consider the UHPC strain capacity.

1.6.3.5. Deformations C1.6.3.5

The provisions of AASHTO LRFD BDS 

Article 5.6.3.5 shall apply with the following 

amendments:

• Reference to AASHTO LRFD BDS 

Articles 5.4.2.2, 5.4.2.3, and 5.4.2.4 

(AASHTO 2020) shall be replaced by 

Articles 1.4.2.7, 1.4.2.8, and 1.4.2.3 of 

this Appendix, respectively.

• The modulus of rupture of concrete in 

AASHTO LRFD BDS Eq. 5.6.3.5.2-2 

(AASHTO 2020) shall be replaced with 

γuft,cr, as specified in Article 1.4.2.5.1 of 

this Appendix.

• Reference to AASHTO LRFD BDS 

Article 5.4.2.6 (AASHTO 2020) shall 

not apply.

Calculated camber values should be treated as 

estimates. The effects of camber variation, 

roadway profile, and roadway cross slope can 

affect key geometric parameters such as haunch 

height, among others. These can affect the 

design of the beams due to the variation of the 

dead loads and can also affect the detailing of 

items such as bearing size, beam seat elevations, 

and the roadway profile. 

The following are the major causes of camber 

variation:

• UHPC modulus of elasticity variation 

based on the mix design, curing, and 

storage.

• Prestress loss variation.

• UHPC strength and stiffness variation at 

the time of prestress application.

These variables, which are part of the normal 

component fabrication process, are not known 

to the designer during design stage when camber 

estimates are calculated. Tolerances on camber 

predictions should be considered. Camber 

predictions should be revisited once initial 

UHPC member fabrication commences.

1.6.4. Compression Members

Except for the limit on design compressive 

strengths, the provisions of AASHTO LRFD 

BDS Articles 5.6.4.1 through 5.6.4.6 and 

Article 5.6.4.7.1 shall apply, in which 

references to AASHTO LRFD BDS 

Articles 5.5.4.2 and 5.10.4 (AASHTO 2020) 

shall be replaced with Articles 1.5.4.2 and 

1.10.4 of this Appendix.
52

1.6.5. Bearing

The provisions of AASHTO LRFD BDS

Article 5.6.5 (AASHTO 2020) shall apply.

1.6.6. Tension Members

1.6.6.1. Resistance to Tension C1.6.6.1

Members in which the factored loads induce 

tensile stresses throughout the cross section 

shall be regarded as tension members.

The factored resistance to uniform tension shall 

be taken as:

Pr = Pn (1.6.6.1-1)

where:

Pn = nominal resistance of a tension 

member determined according to 

Eq. 1.6.6.1-2 (kip)

 = resistance factor specified in 

Article 1.5.4.2 of this Appendix

The nominal resistance of a tension member 

shall be taken as:

Pn = PUHPC + Ps (1.6.6.1-2)

in which:

PUHPC = 0.60γu ft,cr Ag (1.6.6.1-3)

Ps = 0.50Es γu εt,loc As + Aps [fpe + 0.50Es γu εt,loc]

(1.6.6.1-4)

where:

Ag = gross area of the cross section of the 

member (in.2)

Es = modulus of elasticity of the 

nonprestressed steel reinforcement 

(ksi)

As = total area of longitudinal 

nonprestressed reinforcement (in.2)

Aps = area of prestressing steel (in.2)

The design approach for tension members 

applies reduction factors to the UHPC and steel 

mechanical properties because more research is 

needed to investigate the behavior of UHPC in 

tension members.

The nominal resistance of a tension member 

described in Eqs. 1.6.6.1-1 through 1.6.6.1-4 

reduces the effective cracking strength, ft,cr, to 

0.60γuft,cr and ignores the increased 

postcracking resistance for materials exhibiting 

the bilinear behavior shown in 

Figure 1.4.2.5.4-2. The maximum usable tensile 

strain of UHPC is also reduced to 0.50γuεt,loc. 

The stress in nonprestressed reinforcement is 

limited to 0.80fy, and the stress in prestressed 

reinforcement is limited to 0.80fpy.
53

fpe = effective stress in prestressing steel 

after losses (ksi)

In use of Eqs. 1.6.6.1-1 through 1.6.6.1-4, the 

following should be considered:

• The sum of fpe and 0.50Esγuεt,loc shall 

not be taken as greater than 80 percent

of the yield strength of the prestressing 

steel, 0.80fpy.

• The term 0.50Esγuεt,loc shall not be taken 

as greater than 80 percent of the yield 

strength of the nonprestressed longitudinal steel, 0.80fy.

• The nominal resistance of the steel 

contribution, Ps, shall exceed 0.80Pn.

• The provisions of AASHTO LRFD 

BDS Article 5.10.8.4.4 (AASHTO 

2020) shall apply.

1.6.6.2. Resistance to Combined Tension and 

Flexure

Members subjected to eccentric tension 

loading, which induces both tensile and 

compressive stresses in the cross section, shall 

be proportioned in accordance with the 

provisions of Article 1.6.3 of this Appendix

with the following amendments:

• The values of the effective cracking 

strength, ft,cr, and localization strength, 

ft,loc, specified in Articles 1.4.2.5.1 and 

1.4.2.5.2 of this Appendix, shall be 

taken as 0.60γu ft,cr and 0.60γu ft,loc, 

respectively.

• The crack localization strain, εt,loc, 

specified in Article 1.4.2.5.3 of this 

Appendix shall be taken as 0.50εt,loc.
54

• The yield stress of tension reinforcement shall be taken as 0.80fpy for 

prestressed reinforcement and 0.80fy for 

nonprestressed reinforcement.

• The ultimate strength, fpu, of prestressing strands shall be taken as

0.80fpu.

1.7. DESIGN FOR SHEAR AND

TORSION—B-REGIONS

1.7.1. Design Procedures

The provisions of AASHTO LRFD BDS

Articles 5.7.1.1 through 5.7.1.4 (AASHTO 

2020) shall apply with the following 

amendments:

• References to AASHTO LRFD BDS

Articles 5.7.2, 5.7.3, and 5.7.4 

(AASHTO 2020) shall be replaced by 

Articles 1.7.2, 1.7.3, and 1.7.4 of this 

Appendix.

• References to AASHTO LRFD BDS

Articles 5.8.2 and 5.8.4 (AASHTO 

2020) shall be replaced by Article 1.8 of 

this Appendix. 

• The provisions of Articles 5.12.5.3.8 

and 5.12.8.6 (AASHTO 2020) shall not 

apply.

1.7.2. General Requirements

1.7.2.1. General C1.7.2.1

The provisions of AASHTO LRFD BDS

Article 5.7.2.1 (AASHTO 2020) shall apply 

with the following amendments:

• References to AASHTO LRFD BDS

Articles 5.9.2.3.3 (AASHTO 2020) 

shall be replaced by Articles 1.5.2 and 

1.9.2.3.3 of this Appendix.
55

• The upper limit on the compressive 

strength of concrete for use in design 

shall not apply. 

• References to AASHTO LRFD BDS 

Articles 5.5.4.2, 5.7.3, 5.7.3.3, and 

5.7.3.6 (AASHTO 2020) shall be 

replaced by Articles 1.5.4.2, 1.7.3, 

1.7.3.3, and 1.7.3.6 of this Appendix, 

respectively. 

• Torsional moment redistribution is not 

permitted. 

• AASHTO LRFD BDS Eqs. 5.7.2.1-3 

through 5.7.2.1-6 (AASHTO 2020) 

shall be replaced with Eqs. 1.7.2.1-1 

through 1.7.2.1-4 of this Appendix, 

respectively. 

Torsional effects shall be investigated where: 

Tu > 0.10φTcr (1.7.2.1-1) 

The indicated fraction of the factored pure 

torsional cracking moment (i.e., 0.10Tcr) is 

expected to cause a very small reduction in the 

shear capacity or flexural capacity and, hence, 

can be neglected. In the absence of sufficient 

data on torsional resistance of UHPC, the 

indicated fraction of Eq. 1.7.2.1-1 is chosen to 

be 2.5 times smaller than the fraction specified 

in AASHTO LRFD BDS Eq. 5.7.2.1-3 

(AASHTO 2020). This smaller fraction offsets 

the increased tensile resistance of UHPC 

compared to conventional concrete, resulting in 

a comparatively similar requirement for 

torsional transverse reinforcement. This 

reduction is necessary to reserve a portion of the 

tensile resistance to resist shear loads in 

members subjected to combined shear and 

torsion. 

For solid shapes: 

(1.7.2.1-2) 

For hollow shapes: 

Tcr = γu ft,cr K(2Ao be) (1.7.2.1-3) 

Eqs. 1.7.2.1-1 through 1.7.2.1-4 were developed 

by replacing the tensile resistance term of 

conventional concrete in AASHTO LRFD BDS 

Eqs. 5.7.2.1-4 through 5.7.2.1-6 (AASHTO 

2020) with γuft,cr and taking the concrete density 

factor equal to 1.0. 
56

in which: 

(1.7.2.1-4)

The factor, γu, is conservatively not applied to 

the effective cracking strength, ft,cr, in 

Eq. 1.7.2.1-4.

where: 

Tu = applied factored torsional moment 

(kip-in.) 

φ = resistance factor specified in 

Article 1.5.4.2 of this Appendix 

Tcr = torsional cracking moment (kip-in.) 

K = limiting interface shear resistance 

Acp = area enclosed by the outside 

perimeter of the UHPC cross 

section (in.2) 

pc = length of the outside perimeter of 

the UHPC section (in.) 

Ao = area enclosed by the shear flow 

path, including any area of holes 

therein (in.2) 

be = effective width of the shear flow 

path taken as the minimum 

thickness of the exterior webs or 

flanges comprising the closed box 

section (in.); be shall be adjusted to 

account for the presence of ducts, 

voids, or openings 

fpc = unfactored compressive stress in 

UHPC after prestress losses have 

occurred, either at the centroid of 

the cross section resisting transient 

loads, or at the junction of the web 

and flange where the centroid lies in 

the flange (ksi) 

be defined in the preceding list shall not exceed 

Acp/pc, unless a more refined analysis is utilized 

to determine a larger value. 

The effects of any openings or ducts in 

members shall be considered. K shall not be 

taken as greater than 1.0 for any section where 

the stress in the extreme tension fiber, 

calculated on the basis of gross section 

properties, due to factored load and effective 

prestress force after losses, exceeds γuft,cr in 

tension. 

 
57

When calculating K for a section subject to 

factored axial force, Nu, fpc shall be replaced 

with fpc − Nu/Ag. Nu shall be taken as a positive 

value when the axial force is tensile and as a 

negative value when it is compressive.

1.7.2.2. Transfer and Development Length

The provisions of Article 1.9.4.3 of this 

Appendix shall be considered for longitudinal 

reinforcement resisting tension caused by 

shear.

1.7.2.3. Regions Requiring Transverse 

Reinforcement C1.7.2.3

Transverse reinforcement shall be provided 

where:

Vu > φ(VUHPC + Vp) (1.7.2.3-1)

or where consideration of torsion is required by 

Eq. 1.7.2.1-1.

where:

Vu = factored shear force (kip)

 = resistance factor specified in 

Article 1.5.4.2 of this Appendix

VUHPC = nominal shear resistance of the 

UHPC as determined in 

Article 1.7.3.3 of this Appendix

(kip)

Vp = component of prestressing force 

in the direction of the shear 

force; positive if resisting the 

applied shear

Full-scale shear tests on UHPC girders without 

transverse steel reinforcement showed 

significant postcracking ductility (El-Helou and 

Graybeal 2022b). This postcracking performance is attributed to the presence of fibers that 

compensate for the lost resistance of the UHPC 

matrix at cracks and result in the occurrence of 

multiple parallel diagonal cracks before shear 

failure. For this reason, transverse steel 

reinforcement to resist shear is not required 

when the shear demand load does not exceed 

the sum of shear capacity of UHPC and the 

component of prestressing force in the direction 

of the shear force.

1.7.2.4. Types of Transverse Reinforcement

The provisions of AASHTO LRFD BDS

Article 5.7.2.4 (AASHTO 2020) shall apply 

with the following amendments:

• Transverse reinforcement to resist shear 

may consist of anchored individual 

bars.
58

• Transverse reinforcement to resist shear

may not consist of bent longitudinal 

bars in nonprestressed members.

• The referenced provisions of AASHTO 

LRFD BDS Article 5.10.8.2.6d 

(AASHTO 2020) shall be amended by 

the provisions of Article 1.10.8.2.6 of 

this Appendix.

1.7.2.5. Minimum Transverse Reinforcement

Transverse shear reinforcement need not be 

provided where not required, as specified in 

Article 1.7.2.3 of this Appendix.

1.7.2.6. Maximum Spacing of Transverse 

Reinforcement

The spacing of the transverse reinforcement 

shall not exceed the maximum permitted 

spacing, smax, determined as:

smax = 0.25dv cot θ ≤ 24.0 in. (1.7.2.6-1)

where:

dv = effective shear depth, as defined in 

Article 1.7.2.8 of this Appendix

(in.)

θ = angle of inclination of diagonal 

compressive stresses, as determined 

in Article 1.7.3.4 of this Appendix

(degrees)

1.7.2.7. Design and Detailing Requirements

The provisions of AASHTO LRFD BDS

Article 5.7.2.7 (AASHTO 2020) shall apply in 

which the referenced provisions of 

Article 5.10.8.2.6 shall be amended by the 

provisions of Article 1.10.8.2.6 of this 

Appendix.
59

1.7.2.8. Shear Stress on UHPC C1.7.2.8

The shear stress on the concrete shall be 

determined as: 

(1.7.2.8-1) 

where: 

φ = resistance factor for shear as 

specified in Article 1.5.4.2 of this 

Appendix 

bv = effective web width taken as the 

minimum web width, measured 

parallel to the neutral axis, between 

the resultants of the tensile and 

compressive forces due to flexure, 

or for circular sections, the diameter 

of the section, modified for the 

presence of ducts where applicable 

(in.) 

dv = effective shear depth taken as the 

distance, measured perpendicular to 

the neutral axis, between the 

resultants of the tensile and 

compressive forces due to flexure; it 

need not be taken to be less than the 

greater of 0.9de or 0.72h (in.) 

h = overall depth of the component (in.) 

in which: 

(1.7.2.8-2) 

where: 

de = effective depth taken as the distance, 

measured perpendicular to the 

neutral axis, between the extreme 

compression fiber of the section to 

the resultant of the forces in the 

tensile reinforcement (in.) 

fps = average stress in the prestressing 

steel at the time for which nominal 

flexural resistance of the member is 

required (in.) 

Refer to AASHTO LRFD BDS Article C5.7.2.8 

(AASHTO 2020) for commentary with the 

following amendments: 

• AASHTO LRFD BDS Eq. C5.7.2.8-1 

(AASHTO 2020) does not apply; the 

distance between the resultants of the 

tensile and compressive forces due to 

flexure, dv, can be determined utilizing the 

strain compatibility approach described in 

Article 1.6.3.1 of this Appendix. 

Alternatively, it can be taken as the greater 

of 0.9de or 0.72h. For circular sections, dv

can be taken as 0.9de, where de is 

calculated according to AASHTO LRFD 

BDS Eq. C5.7.2.8-2 (AASHTO 2020).

• AASHTO LRFD BDS Figure C5.7.2.8-1 

(AASHTO 2020) does not apply. For 

sections made with UHPC, refer to 

Figure C1.7.2.8-1. 

• AASHTO LRFD BDS Figure C5.7.2.8-2 

(AASHTO 2020) does not apply. For 

circular sections made with UHPC, refer 

to Figure C1.7.2.8-2.
60

fs = average stress in the nonprestressed 

tension reinforcement at the time for 

which nominal flexural resistance of 

the member is required (in.); it shall 

not be taken a value greater than fy

dp = distance from extreme compression 

fiber to the centroid of prestressing 

tendons (in.)

ds = distance from extreme compression 

fiber to the centroid of nonprestressed tensile reinforcement 

measured along the centerline of the 

web (in.)

Source: FHWA.

Figure C1.7.2.8-1—Illustration of the terms 

bv, dv, and de for UHPC sections.

Source: FHWA.

Figure C1.7.2.8-2—Illustration of the terms 

bv, dv, and de for circular UHPC sections.

Note that, due to the additional resistance 

provided by the UHPC on the flexural tensile 

side of the section, the location of the resultant

tensile force may not coincide with the centroid 

of the tensile steel reinforcement, as shown in 

Figure C1.7.2.8-1 and Figure C1.7.2.8-2. This 

UHPC resistance generally raises the centroid 

of the flexural tensile force towards the center 

of the section.

In composite sections made with UHPC beams 

and conventional concrete decks, the effective 

shear depth, dv, shall not exceed the distance, 

measured perpendicular to the neutral axis, 

between the resultant of the forces in the tensile

reinforcement and the extreme UHPC fiber on 

the flexural compression side. In cases where 

tensile reinforcement is not provided on the 

flexural tensile side of the UHPC beam, the 

effective shear depth, dv, shall not exceed the 

depth of the UHPC beam.

In composite UHPC beams with conventional 

concrete decks, the limit on the effective shear 

depth, dv, governs when the distance, measured 

perpendicular to neutral axis, between the 

resultant of the forces in the tensile 

reinforcement in the UHPC beam and the 

extreme UHPC fiber on the flexural 

compression side is less than the greater of 0.9de

and 0.72h, as shown in Figure C1.7.2.8-3.
61

Source: FHWA.

Figure C1.7.2.8-3—Illustration of the terms 

bv, dv, and de for composite sections with 

UHPC beams and conventional concrete 

decks showing the maximum allowable dv in 

composite sections.

In cases where steel reinforcement is not 

provided on the tension side of the UHPC beam 

(e.g., simply supported beams made continuous 

for live loads through a UHPC or conventional 

concrete deck and subjected to bending 

moments causing tensile stresses in the deck), 

the limit on the effective shear depth, dv, 

governs when the height of the UHPC beam is 

less than 0.72h, as shown in Figure C1.7.2.8-3.

1.7.3. Sectional Design Model

1.7.3.1. General

The provisions of AASHTO LRFD BDS

Article 5.7.3.1 (AASHTO 2020) shall apply 

with the following amendments:

• Reference to AASHTO LRFD BDS

Article 5.7.1 (AASHTO 2020) shall be 

replaced by Article 1.7.1 of this 

Appendix.

• The upper limit on the compressive 

strength of concrete for use in design 

shall not apply.

1.7.3.2. Sections Near Supports

The provisions of AASHTO LRFD BDS

Article 5.7.3.2 (AASHTO 2020) shall apply, in 

which references to Articles 5.7.1.2, 5.7.2.8, 

5.8.2, and 5.9.4.4 shall be replaced by 
62

Articles 1.7.1, 1.7.2.8, 1.8, and 1.9.4.4 of this 

Appendix, respectively. 

If the shear stress at the design section 

calculated in accordance with Article 1.7.2.8 of 

this Appendix exceeds 0.18f ́c and the 

beam-type element is not built integrally with 

the support, its end region shall be designed 

according to the provisions of Article 1.8 of this 

Appendix. 

1.7.3.3. Nominal Shear Resistance C1.7.3.3

The nominal shear resistance, Vn, shall be 

determined as the lesser of both of the 

following: 

Vn = VUHPC + Vs + Vp (1.7.3.3-1) 

Vn = 0.25f ́c bv dv + Vp (1.7.3.3-2) 

in which: 

VUHPC = γu ft,loc bv dv cot θ (1.7.3.3-3) 

(1.7.3.3-4) 

where: 

bv = effective web width taken as the 

minimum web width within the 

depth dv, as specified in AASHTO 

LRFD BDS Article 5.7.2.8 

(AASHTO 2020) (in.) 

Vs = shear resistance provided by 

transverse reinforcement (kip)

dv = effective shear depth, as specified in 

Article 1.7.2.8 (in.) 

fv,α = uniaxial stress in the transverse steel 

reinforcement at nominal shear 

resistance, as determined in 

Article 1.7.3.4 of this Appendix 

(ksi); it shall not be greater than the 

specified minimum yield strength of 

transverse steel reinforcement 

Av = area of transverse reinforcement to 

resist shear within a distance s (in.2) 

α = angle of inclination of transverse 

The shear design framework in Article 1.7.3.3

of this Appendix was derived from El-Helou 

and Graybeal (2023). It is based on the 

principles of the Modified Compression Field 

Theory (MCFT), originally developed for 

conventional concrete by Vecchio and Collins 

(1986), but integrates modifications to the 

material constitutive models that apply to 

UHPC behavior. The method is validated by 

experiments on prestressed and nonprestressed 

UHPC beams performed by El-Helou and 

Graybeal (2022b) and Baby, Marchand, and 

Toutlemonde (2014). 

The shear failure in UHPC members is 

generally prompted by a localized and 

dominant crack forming from an existing crack 

or a coalescence of closely spaced cracks. The 

localization of the critical crack occurs when 

the bridging fibers start to pull out and the crack 

propagates through the depth of the web, which 

can occur before the yielding of the transverse 

reinforcement or crushing of the compressed 

UHPC in the member. 

The upper limit on Vn given by Eq. 1.7.3.3-2 is 

intended to capture the failure mode in which 

the UHPC in the web of the beam crushes prior 

to, or at the development of, the critical crack. 

It is in line with the limit on Vn specified in 

AASHTO LRFD BDS Article 5.7.3.3 

(AASHTO 2020). For a more detailed analysis, 

the crushing limit in a cracked UHPC member 

subjected to shear forces can be checked by 
63

reinforcement to longitudinal axis 

(degrees) 

s = spacing of transverse reinforcement 

measured in a direction parallel to 

the longitudinal reinforcement (in.) 

ensuring that the stress in the compression strut, 

f2, is less than half of the compression strength, 

f2 ≤ 0.5f ́c. f2 can be computed by multiplying 

the compression principal strain, ε2, calculated 

according to Eq. 1.7.3.4.1-2 by the UHPC 

modulus of elasticity: f2 = 0.5Ecε2. The 

reduction on the modulus of elasticity (i.e., 

0.5Ec) and compressive strength (i.e., 0.5f ́c) 

accounts for the potential weakening of UHPC 

in compression when subjected to a tensile 

strain field in the orthogonal direction. 

Where α = 90 degrees, Eq. 1.7.3.3-4 reduces to: 

(C1.7.3.3-1) 

1.7.3.4. Procedures for Determining Shear 

Resistance Parameters θ and fv,α

C1.7.3.4

Design for shear shall utilize either of the two 

methods identified herein for the determination 

of the angle of inclination of diagonal 

compressive stresses, θ, and the stress in the 

transverse shear reinforcement, fv,α, at nominal 

shear resistance, provided that all requirements 

for usage of the chosen method are satisfied. 

Two complementary approaches are given for 

evaluating the shear resistance of a UHPC 

member, with or without transverse shear 

reinforcement. The first approach, specified in 

Article 1.7.3.4.1 of this Appendix, involves a 

direct evaluation of the resistance parameters θ 

and fv,α at shear failure. The second approach, 

specified in Article 1.7.3.4.2 of this Appendix, 

is a simplified and conservative approach where 

the shear resistance parameters are evaluated 

using tabularized values presented in 

Addendum A2. 

1.7.3.4.1. General Approach C1.7.3.4.1 

The parameters θ and fv,α shall be determined 

by iteratively solving Eqs. 1.7.3.4.1-1 through 

1.7.3.4.1-4: 

(1.7.3.4.1-1) 

The general design approach of Article 

1.7.3.4.1 of this Appendix was derived from the 

comprehensive behavioral model of El-Helou 

and Graybeal (2023) for the response of a 

diagonally cracked UHPC membrane element 

subject to in-plane shear and normal stresses. 

The behavioral model is founded on the 

equilibrium and strain compatibility equations 

while incorporating the material models and 

shear failure modes particular to UHPC. 

 

 
64

(1.7.3.4.1-2) 

εv = γuεt,loc − 0.5εs + ε2 (1.7.3.4.1-3) 

(1.7.3.4.1-4) 

in which: 

(1.7.3.4.1-5) 

In Eqs. 1.7.3.4.1-1 through 1.7.3.4.1-4, εs is the 

net longitudinal tensile strain in the section at 

the centroid of the tension reinforcement, as 

shown in AASHTO LRFD BDS 

Figure 5.7.3.4.2-1 (AASHTO 2020). In lieu of 

more involved procedures, εs may be 

determined by Eq. 1.7.3.4.1-6: 

(1.7.3.4.1-6) 

If the value of εs calculated from 

Eq. 1.7.3.4.1-6 is negative or is positive and 

less than εt,cr, it should be taken as εt,cr, or the 

value should be recalculated according to 

Eq. 1.7.3.4.1-7: 

(1.7.3.4.1-7) 

where: 

ε2 = strain in the UHPC diagonal 

compressive strut (in./in.) 

εv = strain along the transverse direction 

of the member (in./in.) 

The shear stresses of a UHPC member are not 

uniform over the depth of the beam, as shown 

in AASHTO LRFD BDS Figure C5.7.3.4.2-1 

(AASHTO 2020). Similar to the shear 

resistance procedure implemented in AASHTO 

LRFD BDS Article 5.7.3.4.2 (AASHTO 2020), 

the procedure given herein assumes that the 

UHPC shear stresses are uniformly distributed 

over an area bv wide and dv deep, that the 

direction of principal compressive stresses, 

defined by angle θ, remains constant over dv, and 

that the shear strength of the section can be 

determined by considering the biaxial stress 

conditions at just one location in the web, as 

shown in AASHTO LRFD BDS 

Figure C5.7.3.4.2-2 (AASHTO 2020). The axial 

stress at that location is assumed to be equal to 

half the value of εs, calculated using 

Eqs. 1.7.3.4.1-6 and 1.7.3.4.1-7. 

To account for the fact that compressed UHPC 

subjected to tensile strain fields in the 

orthogonal direction is softer than uniaxially 

compressed UHPC, the modulus of elasticity in 

Eqs. 1.7.3.4.1-1 and 1.7.3.4.1-2 is taken as half 

the value of Ec specified in Article 1.4.2.3 of 

this Appendix. 

The shear design procedure described in 

Eqs. 1.7.3.4.1-1 through 1.7.3.4.1-4 calculates 

the resistance parameters θ and fv,α when the 

principal tensile strain in the web of the beam is 

equal to the tensile strain limit, γuεt,loc. The 

formulation accounts for the effect of the 

modulus of elasticity, Ec, the strain in the 

longitudinal steel reinforcement, εs, the crack 

localization strength of UHPC, ft,loc, and the 

ratio of shear reinforcement, ρv,α, on the 

inclination angle of the compressive stresses, θ,

and the stress in the steel bars at shear failure, 

fv,α. 

The solution of Eq. 1.7.3.4.1-1 for θ and fv,α can 

be iteratively solved for a given reinforcement 

ratio, ρv,α, by calculating θ based on an assumed 

value of fv,α. Eqs. 1.7.3.4.1-2 through 

1.7.3.4.1-4 are then evaluated using θ and fv,α. 

 

 
65

|Mu| = absolute value of the factored 

moment at the section, not taken less 

than |Vu − Vp|dv (kip-in.) 

Nu = factored axial force, taken as 

positive if tensile and negative if 

compressive (kip) 

Vu = factored shear force (kip) 

Ep = modulus of elasticity of prestressing 

steel (ksi) 

Aps = area of prestressing steel on the 

flexural tension side of the member 

(in.2), as shown in AASHTO LRFD 

BDS Figure 5.7.3.4.2-1 (AASHTO 

2020) 

fpo = a parameter taken as modulus of 

elasticity of prestressing steel 

multiplied by the locked-in 

difference in strain between the 

prestressing steel and the 

surrounding UHPC (ksi). For the 

usual levels of prestressing, a value 

of 0.7 fpu will be appropriate for 

both pretensioned and posttensioned members 

Act = area of UHPC on the flexural 

tension side of the member (in.2) as 

shown in AASHTO LRFD BDS 

Figure 5.7.3.4.2-1 (AASHTO 2020) 

As = area of nonprestressed steel on the 

flexural tension side of the member 

at the section under consideration 

(in.2), as shown in AASHTO LRFD 

BDS Figure 5.7.3.4.2-1 (AASHTO 

2020) 

The value of fv,α calculated according to 

Eq. 1.7.3.4.1-4 must be checked against the 

assumed value fv,α, and the analysis is repeated 

until the assumed and calculated values of fv,α

converge. If the calculated value of fv,α is greater 

than fy, it must be taken as equal to fy, as 

specified in Eq. 1.7.3.4.1-4, before the 

convergence check between the assumed and 

calculated values of fv,α is performed. A possible 

starting value for fv,α can be set equal to the 

minimum yield strength of the transverse steel, 

fy. The design value of fv,α can be lower than fy, 

especially for UHPC beams with high ratios of 

transverse steel reinforcement, ρv,α, and/or low 

values of crack localization strain capacity, 

εt,loc. 

The factor, γu, is conservatively omitted from 

the tensile stress parameters in Eqs. 1.7.3.4.1-1 

and 1.7.3.4.1-2. Using higher values of the 

crack localization strength, ft,loc, in these 

equations would increase the value of θ, 

resulting in conservative predictions of the 

shear strength. 

For sections without transverse steel 

reinforcement (ρv,α = 0), Eqs. 1.7.3.4.1-1 and 

1.7.3.4.1-2 reduce to: 

(C1.7.3.4.1-1) 

(C1.7.3.4.1-2) 

For sections with transverse steel reinforcement 

perpendicular to the longitudinal axis (α = 

90 degrees), Eqs. 1.7.3.4.1-4 and 1.7.3.4.1-5 

reduce to: 

fv,α = Es εv ≤ fy (C1.7.3.4.1-3) 

(C1.7.3.4.1-4)

Where consideration of torsion is required by 

the provisions of Article 1.7.2.1 of this 

Appendix, the following provisions shall 

apply: 

   
66

• Vu in Eqs. 1.7.3.4.1-6 and 1.7.3.4.1-7 

shall be replaced by the effective 

factored shear resistance, Veff, as 

determined in AASHTO LRFD BDS

Eqs. 5.7.3.4.2-5 and 5.7.3.4.2-6 

(AASHTO 2020).

• Av in Eq. 1.7.3.4.1-5 shall be replaced 

by Av plus the total area of transverse 

torsion reinforcement in the web or 

webs of members.

The value of ρv,α calculated in accordance with 

Eq. 1.7.3.4.1-5 is modified to account for the 

total area of transverse reinforcement provided 

to resist both shear and torsion, because the 

values of θ and fv,α depend on ρv,α.

Within the transfer length, fpo shall be increased 

linearly from zero at the location where the 

bond between the strands and UHPC 

commences to its full value at the end of the 

transfer length.

In the use of Eqs. 1.7.3.4.1-1 through 

1.7.3.4.1-7, the following shall be considered:

• |Mu| should not be taken as less than

|Vu − Vp| dv.

• In calculating As and Aps, the area of 

bars or tendons terminated less than 

their development length from the 

section under consideration should be 

reduced in proportion to their lack of 

full development.

• For sections closer than dv to the face of 

the support, the value of εs calculated at 

dv from the face of the support may be 

used in evaluating θ and fv,α, unless 

there is a concentrated load within dv

from the support, in which case εs

should be calculated at the face of the 

support. The limit on dv in composite 

sections in accordance with 

Article 1.7.3.2 of this Appendix shall 

apply.

The longitudinal strain, εs, is determined 

following the same procedure described in 

AASHTO LRFD BDS Article 5.7.3.4.2 and 

illustrated in AASHTO LRFD BDS

Figure C5.7.3.4.2-3 (AASHTO 2020) but with 

accounting for the UHPC contribution to tensile 

flexural stresses in the longitudinal direction: 

γuft,crAct. Note that Eq. 1.7.3.4.1-6 

conservatively ignores the increase in the 

postcracking strength for materials showing a

tensile stress-strain behavior idealized in

Figure 1.4.2.5.4-2. The value of εs must be 

equal to or less than the UHPC tensile strain 

limit, γuεt,loc, to prevent localization of UHPC 

around the longitudinal tensile reinforcement at 

critical shear sections. The localization of 

UHPC in the extreme tensile fiber of the section 

is considered a flexural failure mode, as 

specified in Article 1.6.3.2.2 of this Appendix.

For pretensioned members, fpo can be taken as 

the stress in the strands when the UHPC is cast 

around them, i.e., approximately equal to the 

jacking stress, as specified in AASHTO LRFD 

BDS Article 5.7.3.4.2 (AASHTO 2020).
67

• In evaluating Eqs. 1.7.3.4.1-6 and 

1.7.3.4.1-7, the limit on dv in composite 

sections made of UHPC beams and 

conventional concrete decks need not 

apply.

• The area of UHPC on the flexural 

tension side of the member, Act, is 

shown in AASHTO LRFD BDS

Figure 5.7.3.4.2-1 (in.2) (AASHTO 

2020). The flexural tension side of the 

member shall be taken as the half-depth 

containing the flexural tension zone.

• If the axial tension is large enough to 

crack the flexural compression face of 

the section, the value calculated from 

Eqs. 1.7.3.4.1-6 and 1.7.3.4.1-7 should 

be doubled.

• Eq. 1.7.3.4.1-6 is not applicable when 

the value of εs is greater than the lesser 

of the yield strain of nonprestressed 

reinforcement or 0.0025. In these 

instances, the value of εs shall be 

determined from a detailed analysis.

• The value of εs must be equal to or less 

than the UHPC tensile strain limit, 

γuεt,loc, for the applicability of 

Eqs. 1.7.3.4.1-1 through 1.7.3.4.1-4.

Eq. 1.7.3.4.1-6 assumes a linear elastic behavior 

of the longitudinal reinforcement and is not 

applicable when the steel strain is greater than 

the yield strain of prestressed or nonprestressed 

reinforcement. For prestressed reinforcement, 

the strain in the strands is assumed to be less 

than the yield strain when εs is less than or equal 

to 0.0025.

1.7.3.4.2. Simplified Approach C1.7.3.4.2

For members made of UHPC with Ec ≥ 

6,500 ksi, ft,loc ≤ 1.80 ksi, and without 

transverse steel reinforcement (i.e., ρv,α = 

0 percent), the values of θ may be determined 

from Table A2.2-1 of Addendum A2 at a 

specific value of γuεt,loc and εs.

For members made of UHPC with Ec ≥ 

6,500 ksi, ft,loc ≤ 1.80 ksi, and reinforced with 

transverse steel with fy ≤ 75.0 ksi, ρv,α ≤ 3.0

percent, and α = 90 degrees, the values of θ and 

fv,α may be determined from the appropriate 

tables of Addendum A2 (i.e., Table A2.3-1

The simplified approach was developed to 

provide an alternative conservative procedure 

for the determination of the resistance 

parameters θ and fv at a known value of εs and 

as a function of the material parameters that 

have the most significant effect on the shear 

capacity.

Considering the shear Eqs. 1.7.3.4.1-1 through 

1.7.3.4.1-4, a parametric study on the effects of 

the UHPC and transverse steel reinforcement 

parameters revealed that the parameter with the

highest overall effect on the shear capacity is 
68

through Table A2.3-6) based on the value of 

ρv,α and at a specific value of γuεt,loc and εs. 

For members with inclined steel reinforcement 

(α ≠ 90 degrees), the general method of 

Article 1.7.3.4.1 of this Appendix should be 

used. 

In lieu of more involved procedures, the value 

of εs may be determined by Eq. 1.7.3.4.1-6 or 

Eq. 1.7.3.4.1-7, as specified in Article 1.7.3.4.1

of this Appendix. 

The value of fv,α used in Eq. 1.7.3.3-4 shall be 

determined as the lesser of the value obtained 

from the tables of Addendum A2 and the 

specified minimum yield strength of the 

transverse steel, fy.

The provisions of Addendum A2 shall apply 

when utilizing the simplified approach for 

determining parameters θ and fv,α. 

the crack localization strain of UHPC, εt,loc, 

followed by the ratio of transverse steel 

reinforcement, ρv,α. An increase in the values of 

the crack localization strength, ft,loc, or a 

decrease in the values of the modulus of 

elasticity, Ec, would increase θ, but this 

variation is much less significant than 

variations in εt,loc, ρv,α, or εs. Therefore, the 

design tables of Addendum A2 were developed 

with fixed values of Ec = 6,500 ksi and ft,loc = 

1.80 ksi and can be conservatively used for any 

UHPC material with higher values of Ec and 

lower values of ft,loc than the ones used to 

develop the design tables. Because increases in 

the values of ρv,α would increase θ, the tables 

pertaining to a specific value of ρv,α can be 

conservatively used for members with values of 

ρv,α lower than the one used to produce the 

design table. 

The design tables of Addendum A2 are 

developed assuming a linearly elastic behavior 

of transverse steel stress, fv,α, up to a minimum 

yield strength, fy, of 75 ksi. Because the value 

of θ decreases with decreased values of fv,α, the 

design tables of Addendum A2 will lead to 

conservative estimates of θ for UHPC members 

reinforced with transverse steel and having a 

minimum yield strength, fy, equal to or less than 

75 ksi. 

1.7.3.5. Longitudinal Reinforcement C1.7.3.5 

Except as specified herein, at each section, the 

tensile capacity of the longitudinal 

reinforcement on the flexural tension side of 

the member shall be proportioned to satisfy: 

(1.7.3.5-1) 

In determining the axial force that the 

longitudinal reinforcement is expected to resist, 

Eqs. 1.7.3.5-1 and 1.7.3.5-2 follow the same 

procedure described in AASHTO LRFD BDS 

Article 5.7.3.5 (AASHTO 2020) but with 

accounting for the UHPC contribution to tensile 

flexural stresses in the longitudinal direction: 

Actγuft,cr. Note that the increase in the 

postcracking strength for materials showing a 

tensile stress-strain behavior idealized in 

Figure 1.4.2.5.4-2 is conservatively ignored.
69

where: 

f, v, c = resistance factors taken from 

Article 1.5.4.2 of this Appendix 

as appropriate for moment, 

shear, and axial resistance, 

respectively

Vs = shear resistance provided by 

transverse reinforcement at the 

section under investigation as 

given by Eq. 1.7.3.3-4, except 

Vs shall not be taken as greater 

than Vu/v in Eqs. 1.7.3.5-1 and 

1.7.3.5-2 (kip)

θ = angle of inclination of diagonal 

compressive stresses used in 

determining the nominal shear 

resistance of the section under 

investigation as determined by 

Article 1.7.3.4 of this Appendix 

(degrees) 

Eq. 1.7.3.5-1 shall be evaluated where simply 

supported girders are made continuous for live 

loads and where longitudinal reinforcement is 

discontinuous. 

At the inside edge of the bearing area of simple 

end supports to the section of critical shear, the 

longitudinal reinforcement on the flexural 

tension side of the member shall satisfy:

(1.7.3.5-2) 

Eqs. 1.7.3.5-1 and 1.7.3.5-2 shall be taken to 

apply to sections not subjected to torsion in 

which the term Esγuεt,loc shall not exceed fy. Any 

lack of full development shall be accounted for. 

In evaluating Eqs. 1.7.3.5-1 and 1.7.3.5-2, the 

limit on dv in composite sections made of 

UHPC beams and conventional concrete decks 

need not apply. 

The 0.6 factor applied to Act is intended to 

conservatively account for the tension 

resistance of the UHPC in determining the 

required longitudinal reinforcement near the 

critical shear section. This approach is similar 

to that taken for tension members in 

Article 1.6.6.1 of this Appendix. 

 
70

For pretensioned sections, including those with 

debonded strands, the tensile force in the 

prestressed reinforcement (i.e., Apsfps) shall 

exceed the tensile forces of the nonprestressed 

reinforcement (i.e., AsEsγuεt,loc ≤ Asfy) at all 

sections. Development of straight and bent-up 

strands as well as overhangs, if present, shall be 

considered for determining the value of Apsfps

and AsEsγuεt,loc ≤ Asfy. 

Except as may be required by Article 1.7.3.6.3

of this Appendix, where the reaction force or 

the load at the maximum moment location 

introduces direct compression into the flexural 

compression face of the member, the area of 

longitudinal reinforcement on the flexural 

tension side of the member need not exceed the 

area required to resist the maximum moment 

acting alone. 

Longitudinal or transverse reinforcing steel, or 

a combination thereof, with specified minimum 

yield strengths up to 100 ksi, may be used in 

elements and connections specified in 

AASHTO LRFD BDS Article 5.4.3.3 

(AASHTO 2020). 

1.7.3.6. Sections Subjected to Combined 

Shear and Torsion 

1.7.3.6.1. Transverse Reinforcement 

The provisions of AASHTO LRFD BDS 

Article 5.7.3.6.1 (AASHTO 2020) shall apply 

in which references to Articles 5.7.2.1, 5.7.3.3, 

and 5.7.3.6.2 shall be replaced by 

Articles 1.7.2.1, 1.7.3.3, and 1.7.3.6.2 of this 

Appendix, respectively. 

1.7.3.6.2. Torsional Resistance C1.7.3.6.2

The nominal torsional resistance shall be taken 

as: 

(1.7.3.6.2-1) 

In the absence of sufficient data on combined 

torsional and shear resistance of UHPC, 

resistance to torsion is assumed to be solely 

provided by additional transverse steel 

reinforcement proportioned to satisfy Tn ≥ Tu. 

The specified torsional resistance equation (i.e., 
71

where:

Ao = area enclosed by the shear flow 

path, including any area of holes 

therein (in.2)

At = area of one leg of the closed 

transverse torsion reinforcement in 

solid members, or total area of the 

transverse torsion reinforcement in 

the exterior web and flange of 

hollow members (in.2)

θ = angle of inclination of diagonal 

compressive stresses, as determined 

in accordance with the provisions of 

Article 1.7.3.4 of this Appendix

(degrees)

fv,α = stress in the transverse steel 

reinforcement at nominal shear 

resistance as determined in 

Article 1.7.3.4 of this Appendix

(ksi)

s = spacing of transverse reinforcement 

measured in a direction parallel to 

the longitudinal reinforcement (in.)

Eq. 1.7.3.6.2-1) is adopted from AASHTO 

LRFD BDS Eq. 5.7.3.6.2-1 (AASHTO 2020)

with the following amendments: 

• The shear strength factor is taken as 

equal to 1.0 because this Appendix does 

not provide guidance on post-tensioned 

UHPC elements. However, the effect of 

voids and/or openings in the section 

must be considered. 

• The yield strength term, fy, is replaced by 

the stress in the transverse reinforcement, fv,α, determined according to the 

provisions of Article 1.7.3.4 of this 

Appendix, because the stress in the steel 

at nominal resistance might be lower 

than the yield stress of reinforcement.

The nominal resistance for shear (Eqs. 1.7.3.3-3 

and 1.7.3.3-4) and torsion (Eq. 1.7.3.6.2-1) 

should be calculated from the same set of values 

for θ and fv,α, determined in accordance with 

Article 1.7.3.4 of this Appendix. In calculating 

ρv,α, Av in Eq. 1.7.3.4.1-5 should be replaced by 

Av plus the total area of additional transverse 

reinforcement provided to resist torsion. This 

modification accounts for the effect of the 

increase in transverse reinforcement on values 

for θ and fv,α.

For composite sections made with UHPC 

beams and conventional concrete decks where 

tensile reinforcement is not provided on the 

tension side of the UHPC beam, εs in Equations 

1.7.3.4.1-1 through 1.7.3.4.1-4 is the net 

longitudinal tensile strain in the extreme 

tension layer of the UHPC beam. The value of 

εs shall account for the effects of bending 

moment and the applied shear and axial forces.

In cases where composite sections made with 

UHPC beams and conventional concrete decks 

have no steel tensile reinforcement on the 

tension side of the UHPC beam (e.g., simply 

supported beams made continuous for live 

loads through a conventional concrete deck and 

subjected to bending moments causing tensile 

stresses in the deck), in lieu of more involved 

procedures, εs may be calculated by subtracting 

the strain in the UHPC beam at the interface 

between the beam and deck at the time of deck 

placement from the value obtained from 

Equation 1.7.3.4.1-6 with Aps = 0 and ft,loc = 0. 

In this computation, the strain in the UHPC 

beam at the interface between the beam and 

deck must be taken as a negative value when in 
72

compression and a positive value when in 

tension.

1.7.3.6.3. Longitudinal Reinforcement

The provisions of Article 1.7.3.5 of this 

Appendix shall be replaced with the provisions 

of AASHTO LRFD BDS Article 5.7.3.6.3 

(AASHTO 2020) with the following 

amendments:

• Reference to AASHTO LRFD BDS

Article 5.7.3.5 (AASHTO 2020) shall 

be replaced by Article 1.7.3.5 of this 

Appendix.

• The expression Apsfps + AsEsγuεt,loc +

Actγuft,loc shall replace the left-hand side 

of AASHTO LRFD BDS Eq.

5.7.3.6.3-1 (i.e., Apsfps + Asfy) 

(AASHTO 2020) in which the term 

Esγuεt,loc shall not exceed fy.

1.7.4. Interface Shear Transfer–Shear 

Friction

1.7.4.1. General C1.7.4.1

The provisions of Article 5.7.4.1 of the 

AASHTO LRFD BDS (AASHTO 2020) shall 

apply.

The commentary of Article 5.7.4.1 of the 

AASHTO LRFD BDS shall apply to UHPC; 

however, an additional source of resistance to 

shear displacement along the interface plane 

shall be provided by UHPC tensile forces 

crossing the plane of the interface (for UHPC

cast monolithically). 

1.7.4.2. Minimum Area of Interface Shear 

Reinforcement

For monolithic UHPC, interface shear 

reinforcement need not be provided if the 

clamping force provided by the fiber 

reinforcement exceeds the following limit:

γu ft,locAcv > 0.05Acv (1.7.4.2-1)

For monolithic UHPC, the fiber reinforcement

should provide a clamping force that exceeds 

the requirement of AASHTO LRFD BDS

Eq. 5.7.4.2-1 (AASHTO 2020).
73

where:

Acv = area of UHPC considered to be 

engaged in interface shear transfer 

(in.2)

ft,loc = the crack localization strength of the 

UHPC in tension specified in 

Article 1.4.2.5.4 of this Appendix

(ksi); it shall not be taken greater

than 1.75 ksi

The value ft.loc is limited to 1.75 ksi due to the 

lack of available data greater than this value in 

the literature.

For all other cases, the provisions of 

Article 5.7.4.2 of the AASHTO LRFD BDS 

(AASHTO 2020) shall apply with the 

following amendments:

• Reference to Eq. 5.7.4.3-3 shall be 

replaced by Eq. 1.7.4.3-4.

• The minimum reinforcement provisions 

of Article 5.7.4.2 of the AASHTO 

LRFD BDS (AASHTO 2020) shall not 

be waived for any non-monolithic 

interfaces.

Article 5.7.4.2 of AASHTO LRFD BDS 

(AASHTO 2020) waives the minimum 

reinforcement requirements for conventional 

concrete girder/slab interfaces when the 

factored interface shear stress is less than 

0.210 ksi and when the minimum transverse 

shear reinforcement required in the web are 

extended across the interface and adequately 

anchored in the slab. Given that UHPC girders 

do not necessarily contain transverse 

reinforcement in the web, the aforementioned 

provision is not applicable.

1.7.4.3. Interface Shear Resistance C1.7.4.3

The factored interface shear resistance, Vri, 

shall be determined in accordance with 

AASHTO LRFD BDS Eq. 5.7.4.3-1

(AASHTO 2020), and the design shall satisfy 

AASHTO LRFD BDS Eq. 5.7.4.3-2 

(AASHTO 2020).

The nominal shear resistance of the interface 

plane for UHPC placed monolithically shall be 

taken as:

Vni = cAcv + μ(C1 + C2 + Pc) (1.7.4.3-1)

in which:

C1 = Avf fs (1.7.4.3-2)

C2 = Acvγu ft,loc (1.7.4.3-3)

The specified nominal shear resistance of the 

interface plane for UHPC is based on the work 

of Muzenski, Haber, and Graybeal (2022, 

2023). It is founded on the same concepts of the 

shear-friction model for conventional concrete 

specified in AASHTO LRFD BDS Eq. 

5.7.4.3-3 (AASHTO 2020) but integrates 

modifications that account for the behaviors 

specific to UHPC. In particular, in addition to 

the traditional clamping force (i.e., Avf fs + Pc), 

the model incorporates an additional clamping 

force (i.e., Acvγu ft,loc) provided by the tensile 

resistance of UHPC.
74

where:

C1 = normal clamping force provided by 

steel reinforcement (kip)

Avf = area of interface reinforcement 

crossing the shear plane within the 

area Acv (in.2)

fs = stress in the interface steel 

reinforcement at the time of UHPC 

crack localization (ksi): fs = Esγuεt,loc

≤ fy

C2 = normal clamping force provided by 

UHPC placed monolithically (kip)

Research conducted by Muzenski, Haber, and 

Graybeal (2022, 2023) on monolithically 

placed UHPC interfaces indicated that the 

interface steel reinforcing bars may not yield 

before or concurrent with shear failure at the 

interface plane; thus, the yield stress of the 

reinforcement is not an appropriate value to use 

for the clamping force provided by the steel 

reinforcement, C1. When the UHPC localizes in 

tension, it is assumed to no longer provide 

clamping force (i.e., C2 = 0). In scenarios where 

the crack localization strain of UHPC is less 

than the yield strain of the interface steel 

reinforcement, a portion of the clamping force 

will be lost before the interface steel yields. 

Therefore, the value fs shall be determined by 

considering the stress in the steel reinforcing 

bars at the time of UHPC crack localization,

assuming compatibility (i.e., fs = Esγuεt,loc ≤ fy). 

This consideration for fs allows for 

contributions to the clamping force from 

reinforcement with yield stresses higher than 

60 ksi.

The nominal shear resistance of the interface 

plane for all other cases shall be taken as:

Vni = cAcv + μ(Avf fy + Pc) (1.7.4.3-4)

The nominal shear resistance, Vni, used in the 

design shall not exceed the following:

Vni ≤ K Acv (1.7.4.3-5)

in which:

Acv = bvi Lvi (1.7.4.3-6)

where:

fy = specified minimum yield strength 

of reinforcement (ksi); it shall not 

be taken greater than 60 ksi for 

cases where UHPC is placed nonmonolithically

bvi = interface width considered to be 

engaged in shear transfer (in.)

Lvi = interface length considered to be 

The limiting interface shear resistance 

established by Eq. 1.7.4.3-5 is necessitated by 

the lack of available experimental data beyond 

the limiting K values provided in Article 1.7.4.4

of this Appendix.
75

engaged in shear transfer (in.)

c = cohesion factor as specified in 

Article 1.7.4.4 of this Appendix

(ksi)

μ = friction factor specified in 

Article 1.7.4.4 of this Appendix

Pc = permanent net compressive force, 

normal to the shear plane; if force is 

tensile, Pc = 0.0 (kip)

K = limiting interface shear resistance 

specified in Article 1.7.4.4 of this 

Appendix (ksi)

1.7.4.4. Cohesion and Friction Factors C1.7.4.4

The following values shall be taken for 

cohesion, c, and friction, μ:

For UHPC placed monolithically:

• c = 1.40 ksi

• μ = 1.00

• K = 4.50 ksi

The values presented for monolithically cast 

UHPC provide a lower bound based on the 

experimental data of Muzenski, Haber, and 

Graybeal (2022, 2023) and Crane (2010). In 

these tests, the interface steel reinforcement 

ratios varied between 0 percent and 

2.76 percent, and the nominal interface 

reinforcement yield stresses varied between 

40 ksi and 120 ksi. 

For UHPC placed against a clean UHPC 

surface, free of laitance, with surface either 

intentionally roughened to an amplitude of 

0.25 in. or cast to have 0.25-in. amplitude 

roughness:

• c = 0.075 ksi

• μ = 1.0

• K = 1.8 ksi

For UHPC placed against a clean UHPC 

surface, free of laitance, but not intentionally 

roughened to an amplitude of 0.25 in. or cast to 

have 0.25-in. amplitude roughness:

• c = 0.075 ksi

• μ = 0.6

• K = 0.8 ksi

For UHPC placed against a clean conventional 

Interfaces consisting of a substrate material and 

a secondarily cast material are defined based on 

the materials and the interface surface type. 

Behavior of these interfaces is informed by the

experimental data presented in Muzenski, 

Haber, and Graybeal (2023). Where limited 

experimental data are available, provisions 

relevant to normal-weight concrete, as provided 

in AASHTO LRFD BDS Article 5.7.4.4

(AASHTO 2020), are adopted. For situations 

where UHPC or conventional concrete is cast 

against a UHPC substrate, the cohesion factor 

is taken as equal to 0.075 ksi. This value is 

consistent with the cohesion factor of concrete 

placed against a clean concrete surface, free of 

laitance, but not intentionally roughened in 

AASHTO LRFD BDS Article 5.7.4.4 

(AASHTO 2020). UHPC tends to replicate the 

formwork surface finish, thus creating a smooth 
76

concrete substrate surface, free of laitance, with 

surface intentionally roughened to an 

amplitude of 0.25 in.:

• c = 0.24 ksi

• μ = 1.0

• K = 1.8 ksi

For UHPC placed against a clean conventional 

concrete substrate surface, free of laitance, but 

not intentionally roughened:

• c = 0.075 ksi

• μ = 0.6

• K = 0.8 ksi

For conventional concrete placed against a 

clean UHPC substrate surface, free of laitance, 

with surface intentionally roughened to an 

amplitude of 0.25 in. or cast to have 0.25-in. 

amplitude roughness:

• c = 0.075 ksi

• μ = 1.0

• K = 1.8 ksi

For conventional concrete placed against a 

clean UHPC substrate surface, free of laitance, 

but not intentionally roughened to an amplitude 

of 0.25 in. or cast to have 0.25-in. amplitude 

roughness:

• c = 0.075 ksi

• μ = 0.6

• K = 0.8 ksi

finish on the microlevel, regardless of the 

macrotexture present in the formwork.

For situations where UHPC is cast against a 

clean concrete surface, free of laitance, with a 

surface intentionally roughened to an amplitude 

of 0.25 in., the cohesion factor is taken as equal 

to 0.24 ksi, and the friction coefficient is taken 

as equal to 1.0. These values are consistent with 

the values for this situation in AASHTO LRFD 

BDS Article 5.7.4.4 (AASHTO 2020). 

For situations where UHPC is cast against a 

clean concrete surface, free of laitance, that is 

not intentionally roughened, the cohesion factor 

is taken as equal to 0.075 ksi, and the friction 

coefficient is taken as equal to 0.6. These values 

are consistent with the values for this situation 

in AASHTO LRFD BDS Article 5.7.4.4 

(AASHTO 2020).

For situations where UHPC is cast against, and 

anchored to, as-rolled structural steel, the 

provisions associated with conventional 

concrete from AASHTO LRFD BDS Article

5.7.4.4 are adopted (AASHTO 2020).

For UHPC anchored to as-rolled structural steel 

by headed studs or by reinforcing bars, where 

all steel in contact with UHPC is clean and free 

of paint:

• c = 0.025 ksi

• μ = 0.7

• K = 0.8 ksi

The relevant parameters for UHPC anchored to 

as-rolled structural steel are adopted from 

AASHTO LRFD BDS Article 5.7.4.4 

(AASHTO 2020) for the same case with 

conventional concrete. 
77

1.7.4.5. Computation of the Factored 

Interface Shear Force for Girder/Slab 

Bridges

The provisions of AASHTO LRFD BDS 

Article 5.7.4.5 (AASHTO 2020) shall apply.

1.7.4.6. Interface Shear in Box Girder 

Bridges

The provisions of AASHTO LRFD BDS 

Article 5.7.4.6 (AASHTO 2020) shall apply.

1.8. DESIGN OF D-REGIONS C1.8

Refined analysis, strut-and-tie, and elastic 

stress analysis methods may be used to 

determine the internal force effects in disturbed 

regions, such as those near supports, and the 

points of application of concentrated loads at 

strength and extreme limit states. The design 

method shall be approved by the owner and

shown in the contract documents.

The analysis method shall consider the internal 

strains in the UHPC and reinforcement at 

strength and extreme limit states, shall use the 

material properties of Article 1.4 of this 

Appendix, and shall consider the potential 

presence of unfavorable fiber distributions in 

the member. The tensile strain in the UHPC at 

any location or orientation shall not be 

permitted to exceed γuεt,loc.

The calculated resistance of a member should 

be justified by performance testing of a 

prototype.

The guidance provided herein relates to 

structural design with UHPC in B-Regions. 

More research is needed to explore the 

applicability of the D-Regions provisions of

AASHTO LRFD BDS Article 5.5.1.2.3 

(AASHTO 2020) to UHPC.
78

1.9. PRESTRESSING

1.9.1. General Design Consideration

1.9.1.1. General C1.9.1.1

The provisions of AASHTO LRFD BDS

Article 5.9.1.1 (AASHTO 2020) shall apply, in 

which references to Articles 5.4, 5.5, 5.6, 5.7, 

and 5.9.2.3 shall be replaced by Articles 1.4, 

1.5, 1.6, 1.7, and 1.9.2.3 of this Appendix, 

respectively.

The load factor for live load for Service III load 

combination, γLL in AASHTO LRFD BDS

Table 3.4.1-4 (AASHTO 2020) shall not be less 

than 1.0.

The applicability of the provisions of AASHTO 

LRFD BDS Article 5.9.3.3 (AASHTO 2020) 

for UHPC materials has not been investigated.

The Service III live load factor, γLL, is keyed to 

the prestress loss method in AASHTO LRFD 

BDS Table 3.4.1-4 (AASHTO 2020). In this 

Appendix, the refined estimates of timedependent losses are in accordance with 

AASHTO LRFD BDS Article 5.9.3.4

(AASHTO 2020), as amended by Article 1.9.3

of this Appendix; thus, the γLL should be taken

as equal to 1.0.

1.9.1.2. Design UHPC Strengths C1.9.1.2

The design strengths, f ́c, ft,cr, ft,loc, f ́ci, and the 

effective cracking strength at the time of 

prestressing for pretensioned members and at 

initial loading for nonprestressed members,

ft,cri, shall be identified in the contract 

documents for each component. Stress limits 

relating to design strengths shall be as specified 

in Article 1.9.2.3 of this Appendix.

UHPC strength at transfer shall be adequate for 

the requirements of the anchorages or for 

transfer through bond as well as for camber or 

deflection requirements.

Unless approved by the owner, the value of f ́ci

shall not be less than 14.0 ksi. 

Unless determined by physical tests and as 

approved by the owner, the value of ft,cri shall 

not be taken as greater than 0.75ft,cr when f ́ci is 

less than or equal to 0.90f ́c.

Uniaxial compression and tension tests 

performed by El-Helou, Haber, and Graybeal

(2022) showed a clear correlation between the 

compressive and tensile strength gains in which 

the effective cracking strength was found to 

increase with increasing compressive strength. 

The upper limit on ft,cri is prescribed to prevent

cracking at transfer when uniaxial tension tests 

to determine ft,cri are not performed. It is based 

on experimental data in which f ́ci was greater 

than 14.0 ksi.

For commentary on a lower limit of f ́ci, refer to 

Article C1.4.2.8.2 of this Appendix.
79

1.9.1.3. Section Properties

The provisions of AASHTO LRFD BDS

Article 5.9.1.3 (AASHTO 2020) shall apply.

1.9.1.4. Crack Control

The provisions of AASHTO LRFD BDS

Article 5.9.1.4 (AASHTO 2020) shall apply, in 

which reference to Articles 5.5 and 5.6 shall be 

replaced by Articles 1.5 and 1.6 of this 

Appendix, respectively.

1.9.1.5. Buckling

The provisions of AASHTO LRFD BDS

Article 5.9.1.5 (AASHTO 2020) shall apply.

1.9.2. Stress Limitations

1.9.2.1. Stresses Due to Imposed 

Deformation

The provisions of AASHTO LRFD BDS

Article 5.9.2.1 (AASHTO 2020) shall apply, in 

which reference to Article 5.4.2.3.2 shall be 

replaced by Article 1.4.2.8.2 of this Appendix.

1.9.2.2. Stress Limitations for Prestressing 

Steel

The provisions of AASHTO LRFD BDS

Article 5.9.2.2 (AASHTO 2020) and 

Article 1.4.4 of this Appendix shall apply.

1.9.2.3. Stress Limits for UHPC

1.9.2.3.1. For Temporary Stresses

1.9.2.3.1.a. Compressive Stresses

The provisions for compressive stress limits of 

AASHTO LRFD BDS Article 5.9.2.3.1a 

(AASHTO 2020) shall apply for UHPC.
80

1.9.2.3.1.b. Tensile Stresses

The tensile stress limit for prestressed UHPC 

components shall be γu ft,cri (ksi) in accordance 

with Articles 1.4.2.5 and 1.9.1.2 of this 

Appendix.

1.9.2.3.2. For Stresses at Service Limit State

1.9.2.3.2.a. Compressive Stresses

Except for the limit on the compressive 

strengths, the provisions for compressive stress 

limits of AASHTO LRFD BDS Article

5.9.2.3.2a (AASHTO 2020) shall apply for 

UHPC.

1.9.2.3.2.b. Tensile Stresses

Other than the provisions for tensile stress 

limits of AASHTO LRFD BDS 

Table 5.9.2.3.2b-1, the provisions of AASHTO 

LRFD BDS Article 5.9.2.3.2b (AASHTO 

2020) shall apply for UHPC.

The tensile stress limit for prestressed UHPC at 

service limit state shall be γu ft,cr (ksi) in 

accordance with Article 1.4.2.5 of this 

Appendix.

1.9.2.3.3. Principal Tensile Stresses in Webs

The provisions of AASHTO LRFD BDS

Article 5.9.2.3.3 (AASHTO 2020) shall apply 

with the following amendments:

• References to Articles 5.7.2 and 5.7.2.1 

(AASHTO 2020) shall be replaced by 

Articles 1.7.2 and 1.7.2.1 of this 

Appendix, respectively.

• The upper limit on principal tensile 

stresses in webs shall be replaced by 

γuft,cr, in accordance with Article 1.4.2.5

of this Appendix.
81

1.9.3. Prestress Losses C1.9.3

The provisions of AASHTO LRFD BDS

Article 5.9.3 (AASHTO 2020) shall apply with 

the following amendments:

• The limit on compressive strength of 

AASHTO LRFD BDS Article 5.9.3.1 

(AASHTO 2020) shall not apply.

• The provisions of AASTHO LRFD

BDS Article 5.9.3.3 (AASHTO 2020) 

shall not apply.

• References to AASHTO LRFD BDS

Eqs. 5.4.2.3.2-1 and 5.4.2.3.3-1 

(AASHTO 2020) shall be replaced by 

Eqs. 1.4.2.8.2-1 and 1.4.2.8.3-1 of this 

Appendix, respectively.

The applicability of the prestress loss method 

for conventional concrete in AASHTO LRFD 

BDS Article 5.9.3 (AASHTO 2020) for UHPC 

is verified by the work of Mohebbi and 

Graybeal (2022).

The applicability of the provisions of AASHTO 

LRFD BDS Article 5.9.3.3 (AASHTO 2020) 

for UHPC materials has not been investigated.

1.9.4. Details for Pretensioning

1.9.4.1. Minimum Spacing of Pretensioning 

Strand

C1.9.4.1

The provisions of AASHTO LRFD BDS

Article 5.9.4.1 (AASHTO 2020) shall apply for 

UHPC components.

When 0.62-in.- or 0.7-in.-diameter strands are 

used, the provisions for 0.6-in. diameter strands 

in AASHTO LRFD BDS Article 5.9.4.1 

(AASHTO 2020) shall apply.

The use of 0.7-in.-diameter strands in UHPC 

structural components has been investigated in 

the research of El-Helou and Graybeal (2022a,

2022b).

In addition to the limits specified in AASHTO 

LRFD BDS Article 5.9.4.1 (AASHTO 2020), 

the minimum spacing between pretensioning 

strands or between groups of bundled strands 

shall not be less than the greater of a clear 

distance taken as 1.50 times the length of the 

longest type of fiber reinforcement in the 

UHPC or 0.75 in.

This provision stipulates a geometry that allows 

for sufficient passing of fibers through 

constricted spaces during casting. Restrictions 

to flow can result in undesirable fiber 

distribution effects within a member.
82

1.9.4.2. Maximum Spacing of Pretensioning 

Strand in Slabs

The provisions of AASHTO LRFD BDS

Article 5.9.4.2 (AASHTO 2020) shall apply.

1.9.4.3. Development of Pretensioning 

Strand

1.9.4.3.1. General C1.9.4.3.1

In determining the resistance of pretensioned 

UHPC components in their end zones, the 

gradual buildup of the strand force in the 

transfer and development lengths shall be taken 

into account.

Between the end of the transfer length and 

development length, the strand stress grows 

from the effective stress in the prestressing steel 

after losses to the stress in the strand at nominal 

resistance of the member.

The stress in the prestressing steel may be 

assumed to vary linearly from 0.0 at the point 

where bonding commences to the effective 

stress after losses, fpe, at the end of the transfer 

length.

Between the end of the transfer length and the 

development length, the strand stress may be 

assumed to increase linearly, reaching the 

stress at nominal resistance, fps, at the 

development length.

The transfer length, lt, shall be taken as:

lt = ξ 24db (1.9.4.3.1-1)

where:

ξ = 0.75 when a shorter transfer length 

value results in more severe stress 

states within the section

ξ = 1.0 when a longer transfer length 

value results in more severe stress 

states within the section

db = nominal strand diameter (in.)

Eq. 1.9.4.3.1-1 is based on the results of 

experimental studies investigating the transfer 

length of pretensioning strands in UHPC (John 

et al. 2011; Bertram and Hegger 2012; 

Mohebbi, El-Helou, and Graybeal 2019).

The transfer length factor, ξ, is typically taken 

as:

• 0.75 when determining the stresses 

and/or strains at service and fatigue limit 

states in accordance with Articles 1.5.2 and 

1.9.2.3 of this Appendix.

• 1.0 when determining the stresses 

and/or strains at strength and extreme event 

limit states in accordance with Articles 1.6

and 1.7 of this Appendix.
83

The development length shall be taken as 

specified in Article 1.9.4.3.2 of this Appendix. 

The effects of debonding shall be considered as 

specified in Article 1.9.4.3.3 of this Appendix.

1.9.4.3.2. Bonded Strands C1.9.4.3.2 

Pretensioning strand shall be bonded beyond 

the section required to develop fps for a 

development length, ld, in inches, where ld

shall satisfy: 

ld ≥ lt + 0.30 (fps − fpe) db (1.9.4.3.2-1) 

where: 

fps = average stress in prestressing steel at 

the time for which the nominal 

resistance of the member is required 

(ksi) 

fpe = effective stress in the prestressing 

steel after losses (ksi) 

The variation of design stress in the 

pretensioned strand from the free end of the 

strand may be calculated as follows: 

• From the point where bonding 

commences to the end of transfer 

length: 

(1.9.4.3.2-2) 

• From the end of the transfer length and 

to the end of the development of the 

strand: 

(1.9.4.3.2-3) 

where: 

fpx = design stress in the pretensioned 

strand at nominal flexural strength at 

a section of the member under 

consideration (ksi) 

Eq. 1.9.4.3.2-1 is based on the results of 

experimental studies that captured information 

on the development length of pretensioning 

strands in UHPC (John et al. 2011; Lubbers 

2003; Graybeal 2006; Graybeal 2015a). The 

development length is the sum of the transfer 

length and a term related to the flexural bond 

length, expressed as a function of the strand 

diameter, similar to that originally proposed by 

Hanson and Kaar (1959). 

The correlation between steel stress and the 

distance over which the strand is bonded to the 

UHPC can be idealized by the relationship 

shown in Figure C1.9.4.3.2-1. This idealized 

variation of strand stress may be used for 

analyzing sections within the transfer and 

development length at the end of pretensioned 

members. 

Source: FHWA. 

Figure C1.9.4.3.2-1—Idealized relationship 

between steel stress and distance from the 

free end of the strand.

 
84

lpx = distance from the free end of the 

pretensioned strand to the section of 

member under consideration (in.)

1.9.4.3.3. Debonded Strands C1.9.4.3.3

The provisions of AASHTO LRFD BDS

Article 5.9.4.3.3 (AASHTO 2020) shall apply 

with the following amendments:

• In restriction C, 60db shall be replaced 

by lt in accordance with

Article 1.9.4.3.1 of this Appendix with 

ξ = 1.00.

• Restriction F shall not apply.

• In Restriction H, the reference to 

guidance in AASHTO LRFD BDS 

Article 5.12.3.3 (AASHTO 2020) shall 

not apply.

The applicability of the provisions of AASHTO 

LRFD BDS Article 5.12.3.3 (AASHTO 2020) 

has not been investigated for UHPC.

1.9.4.4. Pretensioned Anchorage Zones

1.9.4.4.1. Splitting Resistance C1.9.4.4.1

Unless amended herein, the provisions of 

AASHTO LRFD BDS Article 5.9.4.4 

(AASHTO 2020) shall apply for pretensioned 

anchorage zones of UHPC members.

The factored splitting resistance of 

pretensioned anchorage zones provided by the 

reinforcement in the ends of pretensioned 

conventional concrete beams may be replaced 

by:

Pr = fs As + Pr,UHPC (1.9.4.4-1)

in which:

Pr,UHPC = 0.25γu ft,cri bv h (1.9.4.4-2)

where:

fs = stress in steel not to exceed 

20.0 ksi

The tensile resistance of UHPC (i.e., γuft,cri) 

provides a resistance to splitting forces (i.e.,

Pr,UHPC) in anchorage zones of pretensioned 

UHPC beams. In calculating Pr,UHPC according 

to Eq. 1.9.4.4-2, it is assumed that the UHPC 

tensile resistance γuft,cri acts uniformly over an 

area bound by a distance of h/4 from the end of 

the member and a width bv, where h is the 

overall height of the member and bv is the 

effective web width taken as the minimum web 

width within the depth dv. 
85

As = total area of reinforcement 

located within the distance h/4 

from the end of the beam (in.2)

Pr,UHPC = splitting resistance of 

pretensioned anchorage zones 

provided by UHPC (kip)

ft,cri = design UHPC effective cracking 

strength at the time of prestressing determined according 

to Article 1.9.1.2 of this 

Appendix (ksi)

bv = effective web width taken as the 

minimum web width within the 

depth dv (in.)

h = overall dimension of precast 

member in the direction in which 

splitting resistance is being 

evaluated (in.)

1.9.4.4.2. Confinement Reinforcement

The provisions of AASHTO LRFD BDS

Article 5.9.4.4.2 (AASHTO 2020) shall apply.

1.9.4.5. Temporary Strands

The provisions of AASHTO LRFD BDS

Article 5.9.4.5 (AASHTO 2020) shall apply, in 

which reference to Article 5.9.4.3.3 shall be 

replaced by Article 1.9.4.3.3 of this Appendix.

1.10. REINFORCEMENT

Unless verified by physical tests and as 

approved by the owner, the provisions of this 

Article shall apply for reinforcement embedded 

in UHPC.

1.10.1. UHPC Cover C1.10.1

The provisions of AASHTO LRFD BDS

Article 5.10.1 (AASHTO 2020) shall apply in 

which the modification factor for W/CM ratio 

The cover for UHPC is intended to offer 

sufficient durability by limiting crack widths in 

service such that the durability of the structure 

remains minimally affected. The minimum 
86

shall be taken as equal to or greater than 0.6

(AASHTO 2020).

cover is based on the results of an analysis of 

section curvature of UHPC members containing 

main reinforcing steel at the service stress limit 

for common reinforcing steel bars. The analysis 

followed the methodology in Frosch (2001) for 

crack width estimations. To establish limit 

states, a crack width of 0.004 in. was considered 

as having a negligible effect on the transport 

properties of UHPC, while the maximum 

serviceable crack width was considered to be 

0.016 in. These crack widths likely exceed the 

UHPC tensile strain service limits defined in 

Article 1.5.2 of this Appendix. The spacing of 

cracks was considered using the experimentally 

observed relationship between extreme fiber 

tensile strain and crack spacing from Graybeal 

(2006), which is consistent with a weakest link 

model distribution of cracking in a UHPC 

member subjected to flexure, in accordance 

with the Weibull statistical theory for strength.

The design professional should use engineering 

judgment when applying these provisions to 

components that are not expected to crack under 

service loads and to bridge preservation 

applications, wherein the expected service life 

of the repair must be considered alongside that 

of the structure.

The minimum cover shall not be less than the 

greater of 1.5 times the length of the longest 

type of fiber reinforcement included in the 

UHPC or 0.75 in., unless adequate fiber 

distribution can be otherwise demonstrated for 

a specific application.

For each component, the maximum fiber length 

associated with the minimum cover specified in 

design shall be shown in the contract 

documents.

Restrictions to the flow of UHPC during casting

can result in undesirable fiber distribution 

effects. This provision stipulates a geometry 

that allows for sufficient passing of fibers 

through constricted spaces. There may be 

specific applications where this provision could 

be relaxed, but adequate fiber distribution 

would need to be demonstrated.

1.10.2. Hooks and Bends

The provisions of AASHTO LRFD BDS

Article 5.10.2 (AASHTO 2020) shall apply in 

which reference to Article 5.10.8.2.4 shall be 

replaced by Article 1.10.8.2.4 of this Appendix.
87

1.10.3. Spacing of Reinforcement C1.10.3

The provisions of AASHTO LRFD BDS

Article 5.10.3 (AASHTO 2020) shall apply 

with the following amendments:

• The clear distance between parallel bars 

specified in AASHTO LRFD BDS

Articles 5.10.3.1.1 and 5.10.3.1.2 

(AASHTO 2020) shall not be less than 

the greater of 1.5 times the length of the 

longest type of fiber reinforcement 

included in the UHPC or 0.75 in.

• The clear distance between 

reinforcement and an interface shall not 

be less than the greater of 1.5 times the 

length of the longest type of fiber 

reinforcement included in the UHPC or 

0.75 in., unless adequate fiber distribution can be otherwise demonstrated for 

a specific application.

• References to AASHTO LRFD BDS

Articles 5.10.4, 5.10.5, and 5.10.6 

(AASHTO 2020) shall be replaced by 

Articles 1.10.4, 1.10.5, and 1.10.6, 

respectively, of this Appendix.

For each component, the maximum fiber length 

associated with the minimum spacing of 

reinforcement specified in design shall be 

shown in the contract documents.

This provision stipulates a geometry that allows 

for sufficient passing of fibers through 

constricted spaces during casting. Restrictions 

to flow can result in undesirable fiber 

distribution effects within a member.

1.10.4. Transverse Reinforcement for 

Compression Members

C1.10.5

The provisions of AASHTO LRFD BDS

Article 5.10.4 (AASHTO 2020) shall apply 

with the following amendments:

• References to AASHTO LRFD BDS

Articles 5.10.8 (AASHTO 2020) shall 

be replaced by Article 1.10.8 of this 

Appendix.
88

• The clear spacing between the bars of 

spiral reinforcement specified in 

AASHTO LRFD BDS Article 5.10.4.2 

(AASHTO 2020) shall not be less than 

the greater of 1.5 times the length of the 

longest type of fiber reinforcement 

included in the UHPC or 0.75 in. The 

minimum fiber length associated with 

this requirement shall be shown on 

contract documents.

• The maximum limit on the design 

compressive strength specified in 

AASHTO LRFD BDS Article 5.10.4.3 

(AASHTO 2020) shall not apply.

This provision stipulates a geometry that allows 

for sufficient passing of fibers through 

constricted spaces during casting. Restrictions 

to flow can result in undesirable fiber 

distribution effects within a member.

1.10.5. Transverse Reinforcement for 

Flexural Member

The provisions of AASHTO LRFD BDS

Article 5.10.5 (AASHTO 2020) shall apply in 

which reference to Article 5.10.4 shall be 

replaced by Article 1.10.4 of this Appendix.

1.10.6. Shrinkage and Temperature 

Reinforcement

C1.10.6

Reinforcement for shrinkage and temperature 

stresses need not be provided.

Fibers included in UHPC serve the function of 

shrinkage and temperature reinforcement due to 

their ability to sustain tensile loads beyond 

cracking of the cementitious composite.

1.10.7. Reinforcement for Hollow 

Rectangular Compression Members

The provisions of AASHTO LRFD BDS

Article 5.10.7 (AASHTO 2020) shall apply.

1.10.8. Development and Splices of 

Reinforcement

1.10.8.1. General

The provisions of Article 1.10.8.2.1 and 

1.10.8.2.4 of this Appendix, and the provisions 

for lap splices in tension in AASHTO LRFD 

BDS Article 5.10.8.4.3a (AASHTO 2020), as 
89

amended by Article 1.10.8.4 of this Appendix, 

are valid for No. 11 bars or smaller, subject to 

the limitations as specified in each of these 

articles (AASHTO 2020).

The provisions of AASHTO LRFD BDS

Articles 5.10.8.1.1 and 5.10.8.1.2 (AASHTO 

2020) shall apply in which the reference for 

Article 5.10.8.2 shall be replaced by 

Article 1.10.8.2 of this Appendix.

1.10.8.2. Development Length of 

Reinforcement

Development lengths shall be calculated using 

the specified minimum yield strength of the 

reinforcement. Use of nonprestressed 

reinforcement with a specified minimum yield 

strength up to 100 ksi may be permitted for 

elements and connections specified in

AASHTO LRFD BDS Article 5.4.3.3 

(AASHTO 2020) and subject to the provisions 

and limitations specified in each of the 

following articles.

1.10.8.2.1. Deformed Bars and Deformed 

Wire in Tension

C1.10.8.2.1

For No. 8 bars and smaller embedded in UHPC, 

having a minimum f ́ci of 14.0 ksi and a 

minimum cover of two times the diameter of

the bar, i.e., 2db, the development length, ld, 

may be taken as:

• 10db for bars with yield strength less 

than or equal to 75 ksi.

• 12db for bars with yield strength greater 

than 75 ksi and less than or equal to 

100 ksi.

For all other cases, the provisions of AASHTO 

LRFD BDS Article 5.10.8.2.1 (AASHTO 

2020) shall apply with the following 

amendments:

FHWA published guidance on reinforcing bar 

development length within a document on the 

design of field-cast UHPC connections (Design 

and Construction of Field-Cast UHPC 

Connections, FHWA-HRT-19-011) (Graybeal 

2019). This guidance is based on testing of 

UHPC-class materials that meet the baseline 

requirements for UHPC stated herein. The 

guidance addresses a subset of deformed bar 

sizes and covers.
90

• The maximum limit on the design 

compressive strength shall not apply.

• References to AASHTO LRFD BDS

Articles 5.7.2.5 and 5.10.8.2.6 

(AASHTO 2020) shall be replaced by 

Articles 1.7.2.5 and 1.10.8.2.6 of this 

Appendix, respectively.

• The referenced provisions of AASHTO 

LRFD BDS Article 5.10.4.3 (AASHTO 

2020) shall be amended by the 

provisions of Article 1.10.4 of this 

Appendix.

• The concrete density modification 

factor shall be taken equal to 1.0.

• The UHPC compressive strength, f ́c, in 

AASHTO LRFD BDS Eq.

5.10.8.2.1a-2 (AASHTO 2020) shall 

not be taken as a value greater than 15.0 

ksi.

• The reinforcement location factor, λrl, 

in AASHTO LRFD BDS Eq.

5.10.8.2.1a-2 (AASHTO 2020) may be 

taken equal to 1.0.

1.10.8.2.2. Deformed Bars in Compression

The provisions of AASHTO LRFD BDS

Article 5.10.8.2.2 (AASHTO 2020), except for 

Eq. 5.10.8.2.2a-2, shall apply.

1.10.8.2.3. Bundled Bars

The provisions of AASHTO LRFD BDS

Article 5.10.8.2.3 shall apply, in which the 

referenced provisions of Articles 5.10.8.2.1b 

and 5.10.8.2.1c shall be amended by the 

provisions of Article 1.10.8.2.1 of this 

Appendix.

1.10.8.2.4. Standard Hooks in Tension C1.10.8.2.4

For standard hooks, made of No. 8 deformed 

bars and smaller and embedded in UHPC 

FHWA published guidance on reinforcing bar 

development length within a document on the 
91

having a minimum f ́ci of 14.0 ksi and a 

minimum cover of 2db, the modified 

development length, ldh, may be taken as the 

greater of the following:

• 8db.

• 6.0 in.

design of field-cast UHPC connections (Design 

and Construction of Field-Cast UHPC 

Connections, FHWA-HRT-19-011) (Graybeal 

2019). This guidance is based on testing of 

UHPC-class materials that meet the baseline 

requirements for UHPC stated herein. The 

guidance addresses a subset of deformed bar 

sizes and covers. The provisions in Article

1.10.8.2.1 of this Appendix are combined here 

with the detailing requirements for standard 

hooks in tension.

For all other cases, the provisions of AASHTO 

LRFD BDS Article 5.10.8.2.4 shall apply with 

the following amendments:

• The maximum limit on the design 

compressive strength shall not apply.

• The referenced provisions of AASHTO 

LRFD BDS Article 5.10.2.1 (AASHTO 

2020) shall be amended by the 

provisions of Article 1.10.2 of this 

Appendix.

• The referenced provisions of AASHTO 

LRFD BDS Article 5.10.4.3 (AASHTO 

2020) shall be amended by the 

provisions of Article 1.10.4 of this 

Appendix.

• The concrete density modification 

factor shall be taken as equal to 1.0.

• The UHPC compressive strength, f ́c, in 

AASHTO LRFD BDS Eq.

5.10.8.2.4a-2 (AASHTO 2020) shall 

not be taken as a value greater than 15.0 

ksi.

1.10.8.2.5. Welded Wire Reinforcement

The provisions of AASHTO LRFD BDS

Article 5.10.8.2.5 (AASHTO 2020) shall apply 

with the following amendments:
92

• Reference to AASHTO LRFD BDS

Article 5.10.8.2.6 (AASHTO 2020) 

shall be replaced by Article 1.10.8.2.6

of this Appendix.

• The referenced provisions of AASHTO 

LRFD BDS Articles 5.10.8.2.1a and 

5.10.8.2.1c (AASHTO 2020) shall be 

amended by the provisions of 

Article 1.10.8.2.1 of this Appendix.

• The maximum limit on the design 

compressive strength shall not apply.

• The UHPC compressive strength, f ́c, in 

AASHTO LRFD BDS Eqs.

5.10.8.2.5-2, 5.10.8.2.5-3, and 

5.10.8.2.5-4 (AASHTO 2020) shall not 

be taken as a value greater than 15.0 ksi.

• The concrete density modification 

factor shall be taken as equal to 1.0.

1.10.8.2.6. Shear Reinforcement

The provisions of AASHTO LRFD BDS

Article 5.10.8.2.6 (AASHTO 2020) shall apply 

with the following amendments:

• Shear reinforcement consisting of 

single bars shall be placed equidistantly 

to the surfaces of members.

• Bent longitudinal bars shall not act as 

transverse reinforcement.

• The maximum limit on the design 

compressive strength shall not apply.

• The UHPC compressive strength, f ́c, in 

AASHTO LRFD BDS Eq.

5.10.8.2.6b-1 (AASHTO 2020) shall 

not be taken as a value greater than 

15.0 ksi.

• The concrete density modification 

factor shall be taken as equal to 1.0.
93

1.10.8.3. Development by Mechanical 

Anchorages

The provisions of AASHTO LRFD BDS

Article 5.10.8.3 (AASHTO 2020) shall apply.

1.10.8.4. Splices of Bar Reinforcement C1.10.8.4

The provisions of AASHTO LRFD BDS

Article 5.10.8.4 (AASHTO 2020) shall apply 

with the following amendments:

• Reference to AASHTO LRFD BDS

Articles 5.7.2.5 and 5.10.8.2.3 

(AASHTO 2020) shall be replaced by 

Articles 1.7.2.5 and 1.10.8.2.3 of this 

Appendix, respectively.

• The referenced provisions of 

Articles 5.10.4.3 and 5.10.8.2.1a shall 

be amended by the provisions of 

Articles 1.10.4 and 1.10.8.2.1 of this 

Appendix, respectively.

• The maximum limit on the design 

compressive strength shall not apply.

• No. 8 bars and smaller embedded in 

UHPC, having a minimum f ́ci of

14.0 ksi and a minimum cover of 2db,

and spliced by noncontact lap splices in 

flexural members may be spaced farther 

apart transversely at three-fourths the 

required lap splice but not greater than 

6.0 in.

• Lap splices in tension of No. 8 bars and 

smaller embedded in UHPC, having a 

minimum f ́ci of 14.0 ksi and at a 

minimum cover of 2db, may be 

classified as Class A splice for the 

purposes of AASHTO LRFD BDS

Article 5.10.8.4.3a (AASHTO 2020).

FHWA published guidance on splices of bar 

reinforcement within a document on the design

of field-cast UHPC connections (Design and 

Construction of Field-Cast UHPC Connections, FHWA-HRT-19-011) (Graybeal 2019). 

This guidance is based on testing of UHPCclass materials that meet the baseline 

requirements for UHPC stated herein. The 

guidance addresses a subset of deformed bar 

sizes and covers. 
94

1.10.8.5. Splices of Welded Wire 

Reinforcement

The provisions of AASHTO LRFD BDS

Article 5.10.8.5 (AASHTO 2020) shall apply in 

which the referenced provisions of Article

5.10.8.4.3a and Eq. 5.10.8.2.5-1 shall be 

amended by the provisions of Articles 1.10.8.4

and 1.10.8.2.5 of this Appendix, respectively.
95

SECTION 2. MATERIAL CONFORMANCE GUIDANCE FOR ULTRA-HIGH 

PERFORMANCE CONCRETE

2.1. SCOPE

2.1.1. General C2.1.1

The provisions in Section 2 of this Appendix 

relate to material constituents, material 

qualification, and material acceptance for 

UHPC.

The guidance in Section 2 of this Appendix

applies to UHPC materials intended for use in 

structural design following Section 1 of this 

Appendix. 

The material behavior of UHPC is directly 

related to the mixture ingredients, mixture 

proportions, mixing procedures, curing 

regimes, and casting methods. Section 2 of this 

Appendix delineates the requirements for 

material constituents of UHPC and provides 

guidance on the qualification and acceptance 

testing for a UHPC mixture.

The provisions of Section 2 of this Appendix

provide owners, designers, constructors, and 

material suppliers with a process through which 

to determine the qualified design values of 

individual material properties for a particular 

UHPC mixture. These qualified values are 

expected to be instrumental in an owner’s 

determination of the specified values that are 

defined within project procurement documents. 

At the project execution phase, these specified 

values are then used to determine the required 

values that are engaged during acceptance 

testing to ensure material compliance.

2.1.2. Limitations C2.1.1

The provisions of Section 2 of Appendix A do 

not address the fabrication of components. 

Materials manufactured, qualified, and 

accepted according to Section 2 of this 

Appendix may express different properties 

resulting from the means and method of 

fabrication.

Fabrication processes may affect the fresh and 

hardened properties of the UHPC mixture. 

Some mixing and placing methodologies may

result in undesirable entrapment of air within 

the placed UHPC, as well as undesirable 

distribution and orientation of fiber reinforcement.
96

2.2. DEFINITIONS

Acceptance Testing—Testing performed during the construction of structural components to 

determine compliance of the material properties of the supplied UHPC with the requirements of 

the project specifications or contract documents.

Design Value—Value of a material property used in structural design.

Qualification Testing—Testing performed before construction of structural components to 

determine the expected material properties of a UHPC mixture.

Qualified Design Value—Maximum value of a material property that can be used in structural 

design.

Required Value—Minimum average value of a material property required to be obtained from 

acceptance testing.

Specified Value—Average value of a material property stipulated in the project specifications or 

contract documents, which may be greater than the value used in design.

Test Result—The material property value obtained from a standard test conducted on an individual 

specimen for qualification or acceptance of a UHPC mixture.

2.3. NOTATION

Au,i = cross-sectional area of test specimen i, measured with an accuracy of less than 1 percent 

(m2)

Ec = modulus of elasticity for use in design (ksi)

Ec,i = modulus of elasticity test result of specimen i (ksi)

E̅cQ = average value of the modulus of elasticity test results for a mixture obtained from 

qualification testing (ksi)

f ́c = compressive strength of UHPC for use in design (ksi)

f ́c,i = compression-strength test result of specimen i (ksi)

f ́cQ = qualified design value of the compressive strength (ksi)

f̅ ́cQ = average value of the compressive strength test results for a UHPC mixture obtained 

from qualification testing (ksi)

f ́cR = required average value of the compressive strength obtained from acceptance testing 

(ksi)

f t,cr = effective cracking strength of UHPC for use in design (ksi)

f t,crQ = qualified design value of the effective cracking strength (ksi)

f̅t,crQ = average value of the effective cracking strength test results for a mixture obtained from 

qualification testing (ksi)
97

f t,crR = required average value of the effective cracking strength obtained from acceptance 

testing (ksi)

f t,cr,i = effective cracking strength value obtained from tension-test specimen i (ksi)

f t,loc = crack localization strength of UHPC for use in design (ksi)

f t,loc,i = crack localization strength value obtained from tension-test specimen i (ksi)

f t,locQ = qualified design value of the crack localization strength (ksi)

f̅t,locQ = average value of the crack localization strength test results for a UHPC mixture 

obtained from qualification testing (ksi)

f t,locR = required average value of the crack localization strength obtained from acceptance 

testing (ksi)

kcQ = modification factor for the total number of compression strength test results, ncQ, 

considered in calculating the sample standard deviation determined from 

Table 2.6.4.2-1 of this Appendix

kfQ = modification factor for the total number of flexural test results, nfQ, considered in 

calculating the sample standard deviation determined from Table 2.6.4.2-1 of this 

Appendix

ki = individual specimen geometry factor of test specimen i (m)

ktQ = modification factor for the total number of tension test results, ntQ, considered in 

calculating the sample standard deviation determined from Table 2.6.4.2-1 of this 

Appendix

kρQ = modification factor for the total number of uniaxial resistivity test results, nρQ, 

considered in calculating the sample standard deviation determined from 

Table 2.6.4.2-1 of this Appendix

Lu,i = length of test specimen i, measured with an accuracy less than 1 percent (m)

ncQ = total number of compression strength test results obtained from qualification testing

nEQ = total number of modulus of elasticity test results obtained from qualification testing

nfQ = total number of flexural test results obtained from qualification testing

ntQ = total number of tensile test results exhibiting tension responses of Types H-1 or H-2 

(AASHTO T 397) obtained from qualification testing

nρQ = total number of uniaxial resistivity test results obtained from qualification testing

PpQ = qualified value of the load Pp for a UHPC mixture obtained from qualification testing 

(ksi)

Pp,i = peak load obtained from the flexural prism test results of specimen i (kip)

P̅pQ = average value of the peak load obtained from qualification testing (ksi)

Pδ = load measured during flexural prism test at δ (kip)

Pδ,i = individual load measured at each increment value of net midspan displacement, δ, 

obtained from the flexural prism test results of specimen i (kip)
98

PδQ = qualified value of the load Pδ for the mixture obtained from qualification testing (kip)

P̅δQ = average value of Pδ for a UHPC mixture obtained from qualification testing (kip)

Ri = electrical resistance obtained from uniaxial resistivity-test specimen i (Ω)

scQ = sample standard deviation of the compressive strength test results for a UHPC mixture 

obtained from qualification testing (ksi)

sEQ = sample standard deviation of the modulus of elasticity test results for a UHPC mixture 

obtained from qualification testing (ksi)

spQ = sample standard deviation of P̅pQ for a UHPC mixture obtained from qualification 

testing (ksi)

st,crQ = sample standard deviation of the effective crack localization strength test results for a 

UHPC mixture obtained from qualification testing (ksi)

st,locQ = sample standard deviation of the crack localization strength test results for a UHPC 

mixture obtained from qualification testing (ksi)

sδQ = sample standard deviation of P̅δQ for a UHPC mixture obtained from qualification 

testing (ksi)

sεt,locQ = sample standard deviation of the crack localization strain test results for a UHPC 

mixture obtained from qualification testing (in./in.)

δp,i = net midspan deflection recorded at peak load, Pp,i, from the flexural prism test result of 

specimen i (in.)

δ̅pQ = average value of the net midspan deflection at peak load, δp,i, obtained from 

qualification testing (ksi)

sρQ = sample standard deviation of the uniaxial resistivity test results for a UHPC mixture 

obtained from qualification testing (Ω)

εt,loc = crack localization strain of UHPC for use in design (in./in.)

εt,loc,i = crack localization strain value obtained from tension-test specimen i (in./in.)

εt,locQ = qualified design value of the crack localization strain (in./in.)

ε̅t,locQ = average value of the crack localization strain test results for a UHPC mixture obtained 

from qualification testing (in./in.)

εt,locR = required average value of the crack localization strain from acceptance testing (in./in.)

ρi = uniaxial electrical resistivity test result obtained from uniaxial resistivity test specimen 

i (Ω∙m)

ρQ = qualified value of uniaxial resistivity for a UHPC mixture obtained from qualification 

testing (Ω∙m)

ρ̅Q = average value of uniaxial resistivity for a UHPC mixture obtained from qualification 

testing (Ω∙m)
99

2.4. REFERENCED STANDARDS, TEST METHODS, AND REPORTS

AASHTO M 201-21, Standard Specification for Mixing Rooms, Moist Cabinets, Moist Rooms, 

and Water Storage Tanks Used in the Testing of Hydraulic Cements and Concretes

(AASHTO 2021).

AASHTO T 397-22, Standard Method of Test for Uniaxial Tensile Response of Ultra-High

Performance Concrete (AASHTO 2022b).

AASHTO TP 119-22, Standard Method of Test for Electrical Resistivity of a Concrete Cylinder 

Tested in a Uniaxial Resistance Test (AASHTO 2022a).

ACI 214R-11, Guide to Evaluation of Strength Test Results of Concrete (ACI Committee 214 

2011).

ACI 301-20, Specifications for Concrete Construction (ACI Committee 301 2020).

API Specification 10A, Cements and Materials for Well Cementing (American Petroleum 

Institute 2022).

ASTM A820/A820M-22, Standard Specification for Steel Fibers for Fiber-Reinforced Concrete

(ASTM 2022e).

ASTM C31/C31M-12, Standard Practice for Making and Curing Concrete Test Specimens in the 

Field (ASTM 2012).

ASTM C33/C33M-18, Standard Specification for Concrete Aggregates (ASTM 2018b).

ASTM C39/C39M-20, Standard Test Method for Compressive Strength of Cylindrical Concrete 

Specimens (ASTM 2020d).

ASTM C138/C138M-17a, Standard Test Method for Density (Unit Weight), Yield, and Air Content 

(Gravimetric) of Concrete (ASTM 2017d).

ASTM C144-18, Standard Specification for Aggregate for Masonry Mortar (ASTM 2018a).

ASTM C150/C150M-22, Standard Specification for Portland Cement (ASTM 2022c).

ASTM C157/C157M-17, Standard Test Method for Length Change of Hardened HydraulicCement Mortar and Concrete (ASTM 2017e).

ASTM C192/C192M-19, Standard Practice for Making and Curing Concrete Test Specimens in 

the Laboratory (ASTM 2019a).

ASTM C494/C494M-19, Standard Specification for Chemical Admixtures for Concrete (ASTM 

2019b).

ASTM C469/C469M-22, Standard Test Method for Static Modulus of Elasticity and Poisson’s 

Ratio of Concrete in Compression (ASTM 2022g).
100

ASTM C595/C595M-21, Standard Specification for Blended Hydraulic Cements (ASTM 2021).

ASTM C618-22, Standard Specification for Coal Ash and Raw or Calcined Pozzolan for Use in 

Concrete (ASTM 2022a).

ASTM C989/C989M-22, Standard Specification for Slag Cement for Use in Concrete and Mortars

(ASTM 2022d).

ASTM C1064/C1064M-17, Standard Practice for Temperature of Freshly Mixed HydraulicCement Concrete (ASTM 2017b).

ASTM C1240-20, Standard Specification for Silica Fume Used in Cementitious Mixtures

(ASTM 2020c).

ASTM C1437-20, Standard Test Method for Flow of Hydraulic Cement Mortar (ASTM 2020e).

ASTM C1556-22, Standard Test Method for Determining the Apparent Chloride Diffusion 

Coefficient of Cementitious Mixtures by Bulk Diffusion (ASTM 2022f).

ASTM C1602/C1602M-22, Standard Specification for Mixing Water Used in the Production of 

Hydraulic Cement Concrete (ASTM 2022b).

ASTM C1609/C1609M-19a, Standard Test Method for Flexural Performance of Fiber-Reinforced 

Concrete (Using Beam With Third-Point Loading) (ASTM 2019c).

ASTM C1666/C1666M-08, Standard Specification for Alkali Resistant (AR) Glass Fiber for 

GFRC and Fiber-Reinforced Concrete and Cement (ASTM 2008).

ASTM C1712-20, Standard Test Method for Rapid Assessment of Static Segregation Resistance 

of Self-Consolidating Concrete Using Penetration Test (ASTM 2020g).

ASTM C1758/C1758M-15, Standard Practice for Fabricating Test Specimens with SelfConsolidating Concrete (ASTM 2015b).

ASTM C1797-17, Standard Specification for Ground Calcium Carbonate and Aggregate Mineral 

Fillers for use in Hydraulic Cement Concrete (ASTM 2017c).

ASTM C1812/C1812M-15, Standard Practice for Design of Journal Bearing Supports to be Used 

in Fiber Reinforced Concrete Beam Tests (ASTM 2015a).

ASTM C1856/C1856-17, Standard Practice for Fabricating and Testing Specimens of Ultra-High 

Performance Concrete (ASTM 2017a).

ASTM D7357-07, Standard Specification for Cellulose Fibers for Fiber-Reinforced Concrete

(ASTM 2007).
101

ASTM D7508/D7508M-20, Standard Specification for Polyolefin Chopped Strands for Use in 

Concrete (ASTM 2020b).

PCI TR-9-22, Guidelines for the Use of Ultra High Performance Concrete (UHPC) in Precast and 

Prestressed Concrete (PCI Concrete Materials Technology Committee 2022). 

2.5. MATERIAL CONSTITUENTS

Each constituent material used in the 

manufacture of the UHPC shall be of a 

consistent type or mill source.

The constituent materials included in a UHPC 

mixture shall be reported to the owner.

2.5.1. Granular Constituents

2.5.1.1. Hydraulic Cement

The UHPC shall contain hydraulic cement. The 

hydraulic cement shall be compliant with the 

requirements of ASTM C150 (ASTM 2022c), 

ASTM C595 (ASTM 2021), or API 

Specification 10A (American Petroleum 

Institute 2022).

2.5.1.2. Fine Aggregates

The fine aggregates shall be compliant with 

ASTM C33 (ASTM 2018b) or ASTM C144 

(ASTM 2018a), except that the grading 

requirements need not apply. 

Constituent materials that are potentially 

susceptible to alkali-aggregate reactions shall 

not be used.

2.5.1.3. Supplementary Cementitious 

Materials, Mineral Fillers, and Other 

Granular Constituents

2.5.1.3.1. Silica Fume

If included as a constituent material, silica fume 

shall be compliant with ASTM C1240 (ASTM 

2020c).
102

2.5.1.3.2. Fly Ash

If included as a constituent material, fly ash 

shall be compliant with ASTM C618 (ASTM

2022a).

2.5.1.3.3. Slag Cement

If included as a constituent material, slag 

cement shall be compliant with ASTM C989 

(ASTM 2022d).

2.5.1.3.4. Ground Calcium Carbonate

If included as a constituent material, ground 

calcium carbonate shall be compliant with 

ASTM C1797 (ASTM 2017c).

2.5.1.3.5. Other Natural Pozzolan

If included as a constituent material, other 

natural pozzolans shall be compliant with 

ASTM C618 (ASTM 2022a).

2.5.1.3.6. Other Granular Constituents

The suitability of all other granular constituents 

included in the UHPC mixture shall be 

demonstrated and approved by the owner.

2.5.2. Liquid and Frozen Water

Liquid and frozen water shall be compliant with 

ASTM C1602 (ASTM 2022b).

2.5.3. Chemical Admixtures

Chemical admixtures shall be compliant with 

ASTM C494 (ASTM 2019b). 

Chemical admixtures shall not contain calcium 

chloride. Chemical admixtures shall not contain 

more than 0.15 percent chloride ions by weight 

of admixture.

For UHPC mixtures containing more than one 

chemical admixture, the chemical admixtures 
103

shall be certified by the manufacturer to be 

compatible.

2.5.4. Fiber Reinforcement

2.5.4.1. Steel Fiber Reinforcement C2.5.4.1

The UHPC mixture shall contain steel fiber 

reinforcement as primary fiber reinforcement.

Steel fiber reinforcement shall be compliant 

with ASTM A820 (ASTM 2022e).

The following properties of each steel fiber 

reinforcement type shall be reported to the 

owner:

• Chemical composition

• Tensile yield and ultimate strength

• Length

• Cross-sectional geometry

• Deformations and/or end anchorages

• Coating (if applicable)

The volumetric proportion of steel fiber 

reinforcement in the UHPC mixture design 

shall be reported to the owner.

Various compositions of steel, including grades 

of stainless steel, are allowable.

2.5.4.2. Non-steel Fiber Reinforcement C2.5.4.2

Non-steel fiber reinforcement may be included 

in a UHPC mixture but shall not be the primary 

fiber reinforcement.

For each type of non-steel fiber reinforcement 

included in the mixture, the following 

properties shall be reported to the owner:

• Chemical composition

• Tensile strength

• Length

• Cross-sectional geometry

• Deformations and/or end anchorages

• Coating (if applicable)

Non-steel fiber reinforcements may be included 

to enhance certain performance characteristics

of a UHPC mixture, such as to increase fire 

resistance. When non-steel fiber reinforcements 

are included in a UHPC mixture, unfavorable 

effects should be considered. Among other 

considerations, non-steel fiber reinforcements 

may segregate relative to the steel fibers due to 

their different density, may exhibit creep 

behaviors under sustained load, may degrade 

more rapidly under certain environmental 

conditions, and may result in wider cracks that 

increase permeability.
104

The volumetric proportion of each type of other 

fiber reinforcement in the UHPC mixture shall 

be reported to the owner.

If included in the UHPC mixture, alkali 

resistant glass fibers shall be compliant with 

ASTM C1666 (ASTM 2008).

If included in the UHPC mixture, polyolefin 

fibers shall be compliant with ASTM D7508 

(ASTM 2020b).

If included in the UHPC mixture, cellulose 

fibers shall be compliant with ASTM D7357 

(ASTM 2007).

2.6. MATERIAL QUALIFICATION

2.6.1. General C2.6.1

Material qualification testing shall be 

conducted in accordance with Table 2.6.1-1 and 

prior to the selection of a UHPC mixture for use 

in the fabrication of structural components. 

Qualification testing is performed to determine 

the suitability of a particular UHPC mixture for 

use in structures designed according to the 

provisions of Section 1 of this Appendix. 

Material qualification results pertain to the 

specific UHPC mixture that was mixed, cured, 

and tested. Modifications to the constituent 

materials, mixture design, or curing process 

necessitate requalification of the UHPC 

product.

Material qualification should be based on 

field-test specimens that represent materials, 

mixture proportions, batching procedures, and 

climatic conditions similar to those expected 

during the fabrication of UHPC elements.

Table 2.6.1-1. Material qualification testing methods and objective.

Property Article Objective Requirement

Flow 2.6.3.2 Qualification Yes

Fiber segregation 2.6.3.3 Qualification Yes

Unit weight 2.6.3.4 Information Yes

Compressive strength 2.6.4.2 Qualification Yes

Tensile response 2.6.4.3 Qualification Yes

Durability 2.6.5 Qualification Yes

Modulus of elasticity 2.6.4.4 Information Optional

Flexural response 2.6.6 Correlation to acceptance testing Optional
105

2.6.2. Qualification Frequency

A complete set of material qualification testing 

shall be completed at an interval not to exceed 

3 years.

2.6.3. Plastic Properties

2.6.3.1. General

The plastic property values of a particular 

UHPC mixture shall be established according 

to the methods cited herein and shown in the 

contract documents.

2.6.3.2. Flow C2.6.3.2

The flow shall be measured and reported for 

each batch from which specimens for 

qualification testing are cast. Flow shall be 

measured in accordance with ASTM 

C1856/1856M (ASTM 2017a), which provides 

direction on the use of ASTM C1437 (ASTM 

2020e) when testing UHPC.

Flow shall be measured not more than 

10 minutes before the time of specimen casting.

Flow spread values typically range from 8 to 

10 in. Lesser values may indicate a stiffer

mixture that is difficult to place. Greater values 

may indicate a fluid mixture that could be more 

susceptible to fiber segregation.

2.6.3.3. Fiber Segregation C2.6.3.3

Susceptibility to fiber segregation shall be 

measured and reported for each batch from 

which specimens for tensile response 

qualification testing are cast.

The test shall be completed according to ASTM 

C1712 (ASTM 2020g) with the following 

modifications:

• Use of inverted slump mold is not 

required.

• Minimum mold dimensions are a 

cylinder with a 4-in. diameter and an 

8-in. height.

A 4-in. diameter by 8-in. tall cylinder may be 

used as the test specimen mold.
106

Static fiber segregation resistance shall be 

classified according to the criteria defined in 

Table X1.1 of ASTM C1712. To qualify as 

having acceptable fiber segregation resistance, 

the tested material must perform in the 

“resistant” category.

Both the penetration depth and the static 

segregation resistance classification shall be 

reported. 

The elapsed time between the conclusion of 

mixing and the final reading, defined in ASTM 

C1712 Section 8.8.2, shall not exceed 10 

minutes. The elapsed time shall be reported.

2.6.3.4. Unit Weight

Unit weight shall be measured and recorded 

according to ASTM C138-17a (ASTM 2017d) 

with the following modifications:

• The sample container shall be filled in a 

single, continuous pour as described in 

ASTM C1758 Section 7.4 

(ASTM 2015b).

• The sample shall be consolidated by 

tapping the outside of the sample 

container 30 times with a rubber mallet.

The sample shall not be rodded or 

internally vibrated.

2.6.4. Hardened Properties

2.6.4.1. General C2.6.4.1

The hardened property values of a particular 

UHPC mixture and their statistical variability 

shall be established according to the methods 

cited herein and shown in the contract 

documents.

Given that the material behaviors of different 

UHPC mixtures are not unique, qualification 

testing establishes the maximum property 

values that can be used in design with a 

particular UHPC mixture, henceforth referred 

to as “qualified” values.

The qualified property values are obtained as a 

function of variability of the test results and the 
107

proportion of test results allowed to fall below 

the qualified property values, such as:

• No more than the 1 percent of the average 

of three consecutive property test results 

fall below the qualified value, and

• No more than 1 percent of individual 

property test results fall below the 

qualified value by more than 10 percent.

The statistical formulations in this Article 

assume a normal distribution of the test results 

of each hardened property and are based on the 

concepts engaged in the American Concrete 

Institute Specification for Concrete

Construction (ACI Committee 301 2020), 

Guide to Evaluation of Strength Test Results of 

Concrete (ACI Committee 214 2011), and 

Building Code Requirements for Structural 

Concrete (ACI Committee 318 2019).

The determination of the hardened property 

values from statistical procedures other than 

those cited in this Article shall be approved by 

the owner.

When the test results of a particular property 

show evidence of skewness or kurtosis, the data 

may not be normally distributed (ACI 

Committee 214 2011). The distribution is said 

to be skewed when the data are not symmetrical 

about the mean but concentrated to the right or 

left. Kurtosis is said to exist when the 

distribution curve is either too peaked or too 

flat. Simplified equations to calculate the 

relative skewness and kurtosis for a particular 

data set can be found in Cook (1989).

Test specimens used for qualification testing 

shall be fabricated according to ASTM C31 

(ASTM 2012) or ASTM C192 (ASTM 2019a), 

as modified by ASTM C1856 (ASTM 2017a), 

AASHTO T 397 (AASHTO 2022b), and 

accompanying articles in this Appendix.

The casting method of the test specimens shall 

follow the procedures described in the standard 

of each property test.

The curing of all specimens used for 

qualification shall follow a defined regime and 

shall be reported to the owner. The curing 

regime shall not be more favorable than that 

anticipated for structural elements.

The curing regime during qualification should 

represent the curing regime expected during 

fabrication. Additional qualification testing 

should be required when field climatic 

conditions are appreciably different than the 

climatic conditions during qualification.
108

The testing of the hardened properties shall be 

conducted at 28 days or as approved by the 

owner.

The qualification testing should be performed 

when the material has reached a mature age 

where minimum changes of properties are 

expected to change with time, commonly 

assumed at 28 days. Other maturity ages may be 

assumed for components expected to receive 

loads at times appreciably different than 28 

days after placement. 

2.6.4.2. Compressive Strength C2.6.4.2 

The qualified design value of the compressive 

strength of a UHPC mixture, f̅ ́cQ, shall be 

determined from the results of a minimum of 15 

cylinders produced, tested, and evaluated in 

accordance with ASTM C1856/C1856M 

(ASTM 2017a). 

A minimum of 30 compression test results are 

recommended to capture the variation (i.e., 

mean and sample standard deviation) in the 

compressive strength results of a particular 

UHPC mixture. 

A test result shall not be discarded, except that 

if any cylinder shows evidence of improper 

sampling, molding, or testing, said cylinder 

shall be discarded, and the strength shall be 

determined from the remaining cylinders.

Qualification testing is performed to capture the 

variation between the test results due to mixing, 

sampling, fabrication, curing, or testing. The 

test result of any specimen should not be 

discarded unless improper sampling, molding, 

or testing is evident.

Specimens shall be sampled from at least three 

separate batches. A minimum of two but not 

more than half of the tested cylinders shall be 

obtained from a single batch.

Qualification testing is intended to capture the 

variation within a batch as well as between 

different batches of a UHPC mixture.

The average and sample standard deviation of 

the compression strength, f̅ ́cQ and scQ, 

respectively, shall be calculated based on the 

total number of tested specimens as follows:

(2.6.4.2-1) 

(2.6.4.2-2)

where: 

ncQ = total number of compression 

strength test results obtained from 

qualification testing. 

Assuming that the strength results follow a 

normal distribution and that the number of tests 

performed during qualification is large enough 

to provide sufficient data for statistical analysis, 

the probability density function for the 

compressive strength, shown in Figure 

C2.6.4.2-1, is mathematically defined by two 

statistical parameters: the population mean and 

the standard deviation, taken as f̅ ́cQ and scQ, 

respectively.

 

 

 
109

f ́c,i = compression-strength test result of 

specimen i (ksi).

Source: FHWA.

Figure C2.6.4.2-1—Normal frequency curve 

for compressive strength with an a mean 

f̅ ́cQ and standard deviation scQ.

The qualified design value of the compression 

strength, f ́cQ, shall be determined as the lesser 

of the following: 

f ́cQ = f̅ ́cQ − 1.34 kcQ scQ (2.6.4.2-3)

f ́cQ = 1.11 f̅ ́cQ − 2.59 kcQ scQ (2.6.4.2-4)

where:

kcQ = modification factor for the total 

number of compression strength test 

results, ncQ, considered in calculating 

the sample standard deviation 

determined from Table 2.6.4.2-1 of 

this Appendix

Table 2.6.4.2-1. k-factor for increasing the 

sample standard deviation based on total 

number of tests considered.

Total Number of 

Tests Considered

k-Factor for 

Increasing the Sample 

Standard Deviation

15 1.16

20 1.08

25 1.03

30 or more 1.00

Note: Linear interpolation for the intermediate number of 

tests is acceptable.

The qualified design value of the compressive 

strength, f ́cQ, shown in Figure C2.6.4.2-1, is 

computed following Eqs. 2.6.4.2-3 and 2.6.4.2-

4 to satisfy the two criteria defined in Article

C2.6.4.1 of this Appendix.

Eqs. 2.6.4.2-3 and 2.6.4.2-4 are derived from 

the equations of Table 4.2.3.3(a)1 of 

ACI SPEC-301 (ACI Committee 301 2020), 

originally developed for conventional concrete 

with compressive strength greater than 5,000 

psi, to obtain the qualified design value of 

UHPC mixture, f ́cQ, as a function of the average 

value, f̅ ́cQ, and the sample standard deviation, 

scQ. In Specification for Concrete Construction

(ACI Committee 301 2020), f̅ ́cQ and f ́cQ are 

akin to the required average and the design

value of the compressive strength, respectively, 

when data are available to establish the sample 

standard deviation.

Confidence in the statistical estimations is a 

function of the number of test results used to 

establish the statistical parameters. When fewer 

than 30 test results are considered, the standard 

deviation, scQ, should be multiplied by a 

modification factor, kcQ, obtained from 

Table 2.6.4.2-1, to account for the uncertainty 

in the calculated standard deviation 

Specification for Concrete Construction (ACI 

Committee 301 2020).
110

2.6.4.3. Tensile Response C2.6.4.3

The qualified design values of the effective 

cracking strength, ft,crQ, the design crack 

localization stress, ft,locQ, and the design crack 

localization strain, εt,locQ, of a UHPC mixture 

shall be determined from a minimum of 15 

tension response test results classified as Type 

H-1 or H-2, as defined in AASHTO T 397 

(AASHTO 2022b). 

The tensile qualification described in this 

Article pertains to UHPC mixtures exhibiting a 

strain-hardening stress-strain tensile behavior 

of Types H-1, H-2, H-3, and H-4, as defined in 

AASHTO T 397 (AASHTO 2022b). However, 

only test results where the localization crack 

occurs within the gauge length, i.e., tensile 

behavior of Types H-1 and H-2, are permitted 

to be used to determine the tensile properties.

A minimum of 30 tension test results, with 

tensile behavior of Type H-1 or H-2, are 

recommended to establish the variation (i.e., 

mean and sample standard deviation) in the 

tensile properties of a particular UHPC mixture.

Specimens shall be produced, tested, and 

evaluated in accordance with AASHTO T 397 

(AASHTO 2022b) with the following 

restrictions:

• Specimens shall have a square cross

section with 2-in. minimum dimension.

• Deformation shall be recorded over a 

minimum gauge length of 4 in.

To enable testing of both cast and extracted test 

specimens, AASHTO T 397 (AASHTO 2022b) 

does not stipulate strict restrictions on specimen 

geometry. To facilitate consistency in 

qualification testing, restrictions on test 

specimen geometry are provided in this Article.

A test specimen geometry with a 2-in.-square 

cross section and a 17-in. length has been 

commonly used in the execution of AASHTO

T 397. This geometry allows for fiber 

reinforcement that is up to 0.8-in. long. It also 

provides for a 4-in. gauge length that is more 

than 2.5-in. away from the gripped portion of 

the test specimen.

A test result shall not be discarded, except that 

if any sample shows evidence of improper 

sampling, molding, or testing, said specimen 

shall be discarded and the tensile parameters 

shall be determined from the remaining 

specimens. A properly sampled, molded, and 

tested specimen exhibiting a tensile behavior of 

types other than H-1 or H-2, as defined in 

AASHTO T 397 (AASHTO 2022b), shall not 

be considered discarded, but its test results shall 

not be used to determine the tensile properties.
111

Specimens shall be sampled from at least three 

separate batches. Each test result shall be 

classified based on the tensile behaviors 

described in Section 9.0 of AASHTO T 397 

(AASHTO 2022b) and reported to the owner. A 

minimum of two but not more than half of the 

tested specimens used for qualification, i.e., 

exhibiting tensile behavior of Type H-1 or H-2,

as defined in AASHTO T 397 (AASHTO 

2022b), shall be obtained from a single batch.

Given that the localized crack of a tested 

specimen might occur outside the gauge length, 

i.e., tensile behavior of Type H-3 or H-4, as 

defined in AASHTO T 397 (AASHTO 2022b), 

a minimum of 10 to 14 specimens are 

recommended to be sampled from each batch. 

The testing of the sampled specimens may be 

halted when the minimum or required number 

of test results with tensile behaviors of Type H1 or H-2 is achieved. 

If more than one out of five tested specimens 

from a single batch result in tension responses 

classified as Type S, as defined in AASHTO

T 397 (AASHTO 2022b), all specimens 

sampled from the same batch shall not be used 

for qualification.

UHPC mixtures exhibiting a tensile response of 

Type S, as defined in AASHTO T 397 

(AASHTO 2022b), are strain-softening 

materials and are not for use in structures 

designed according to the provisions of Section 

1 of this Appendix. 

If more than one out of five tested specimens 

from a single batch result in tension responses 

of Type S, it can indicate a problem with the 

batching procedure or with the production of 

the samples. Steps must be taken to correct the 

circumstances that resulted in such incompliant 

behavior.

A UHPC mixture shall be disqualified if more 

than one out of five tested specimens from two 

or more batches result in a tension response of 

Type S or if any tested specimen from any batch

results in a tension response of Type N, as 

defined in AASHTO T 397 (AASHTO 2022b).

When stress-strain responses of Type S are 

manifested in the test results of specimens 

sampled from more than one batch, the tension 

response of the mixture should be classified as 

strain softening, and the mixture is not for use 

in structures designed according to the 

provisions of Section 1 of this Appendix. 

Mixture design improvement shall be 

undertaken before qualification testing is 

repeated.

Tensile stress-strain results of Type N indicate 

regions of the specimens without fibers and 

with no resistance to postcracking tensile loads. 

Mixtures exhibiting behaviors of Type N are

not for use in structures designed according to 

the provisions of Section 1 of this Appendix. 

Mixture design improvement shall be 

undertaken before qualification testing is 

repeated.
112

The average and sample standard deviation of 

the effective cracking strength, f̅t,crQ and st,crQ, 

respectively, the crack localization strength, 

f̅t,locQ and st,locQ, respectively, and the crack 

localization strain, ε̅t,locQ and sεt,locQ, 

respectively, shall be calculated based on the 

total number of tested specimens exhibiting 

Type H-1 or H-2 tension responses (AASHTO 

T 397 in AASHTO 2022b) as follows: 

(2.6.4.3-1)

(2.6.4.3-2)

 (2.6.4.3-3)

(2.6.4.3-4) 

(2.6.4.3-5) 

(2.6.4.3-6) 

where: 

ntQ = total number of tensile test results 

exhibiting tension responses of 

Types H-1 or H-2 (AASHTO T 397 

in AASHTO 2022b) obtained from 

qualification testing 

ft,cr,i = effective cracking strength value 

obtained from tension test specimen 

i (ksi)

Similar to the statistical treatments of the compressive strength test results described in 

Article C2.6.4.2 of this Appendix, the results of 

each of the tensile parameters, namely, the 

effective cracking strength, the crack 

localization strength, and the crack localization 

strain, are assumed to follow a normal 

distribution. The probability density function 

for each of the three aforementioned property 

test results is established from the average and 

standard deviation calculated from a sample 

assumed to be large enough to provide 

sufficient data for statistical analysis.

 

 

 

 

 

 

 

 

 
113

ft,loc,i = crack localization strength value 

obtained from tension test specimen 

i (ksi)

εt,loc,i = crack localization strain value 

obtained from tension test specimen 

i (in./in.)

The qualified design value of the effective 

cracking strength, ft,crQ, of the UHPC mixture 

shall be determined as the lesser of the 

following:

ft,crQ = f̅t,crQ − 1.34 ktQ st,crQ (2.6.4.3-7)

ft,crQ = 1.11 f̅t,crQ − 2.59 ktQ st,crQ (2.6.4.3-8)

where:

ktQ = modification factor for the total 

number of tension test results, ntQ, 

considered in calculating the sample 

standard deviation determined from 

Table 2.6.4.2-1 of this Appendix

Following the same statistical considerations 

used to compute the qualified design value of 

the compressive strength, f ́cQ, described in 

Article C2.6.4.2 of this Appendix, the qualified 

values of each of the tensile properties are 

computed such that the two criteria defined in 

Article C2.6.4.1 of this Appendix are satisfied.

The qualified design value of the crack 

localization strength, ft,locQ, of the UHPC 

mixture shall be determined as the lesser of the 

following: 

ft,locQ = f̅t,locQ − 1.34 ktQ st,locQ (2.6.4.3-9)

ft,locQ = 1.11 f̅t,locQ − 2.59 ktQ st,locQ

(2.6.4.3-10)

The qualified design value of the crack 

localization strain, εt,locQ, of the UHPC mixture 

shall be determined as the lesser of the values 

obtained from Eqs. 2.6.4.3-11 and 2.6.4.3-12

but need not be taken less than 0.80ε̅t,locQ:

εt,locQ = ε̅t,locQ − 1.34 ktQ sεt,locQ (2.6.4.3-11)

εt,locQ = 1.11ε̅t,locQ − 2.59 ktQ sεt,locQ

(2.6.4.3-12)

Given the variability that may be observed in 

crack localization strain test results, a maximum 

reduction of 20 percent from the average 

qualification result is stated in this Article.
114

2.6.4.4. Modulus of Elasticity C2.6.4.4 

The qualified design value of the modulus of 

elasticity of a UHPC mixture, EcQ, shall be 

determined from the results of a minimum of 15 

cylinders produced, tested, and evaluated in 

accordance with ASTM C469/C469M (ASTM 

2022g) as modified by ASTM C1856 (ASTM 

2017a).

As indicated in Article 2.6.1 of this Appendix, 

qualification testing of the modulus of elasticity 

is optional. However, when the modulus of 

elasticity is assessed during the qualification 

phase, a minimum of 15 test results are to be 

included to establish the statistical parameters 

of this property. Similar to the suggested 

number of compression strength and tensile 

response tests, a minimum of 30 modulus of 

elasticity test results are recommended to 

establish confidence in the statistical 

parameters.

A test result shall not be discarded, except that 

if any cylinder shows evidence of improper 

sampling, molding, or testing, said cylinder 

shall be discarded, and the modulus of elasticity 

shall be determined from the remaining 

cylinders. 

Specimens shall be sampled from at least three 

separate batches. A minimum of two but not 

more than half of the tested cylinders shall be 

obtained from a single batch. 

The average and sample standard deviation of 

the modulus of elasticity, E̅cQ and sEQ, 

respectively, shall be calculated based on the 

total number of tested specimens as follows:

(2.6.4.4-1)

(2.6.4.4-2)

where: 

nEQ = total number of modulus of 

elasticity test results obtained from 

qualification testing 

Ec,i = modulus of elasticity test result of 

specimen i (ksi)

When determined from physical tests, the 

design value of the modulus of elasticity, Ec, 

should be selected considering its statistical 

variability and should lead to conservative 

designs of the UHPC component or structure. 

When a lower bound of the modulus of 

elasticity value is desired for design, the 

qualified value can be obtained using the same 

statistical considerations adopted for the 

qualification of the compression strength 

property described in Article 2.6.4.2 of this 

Appendix. 

 
115

2.6.5. Durability

Unless otherwise approved by the owner, the 

durability of UHPC shall be assessed based on 

electrical resistivity, as described in Article 

2.6.5.1 of this Appendix.

2.6.5.1. Electrical Resistivity C2.6.5.1

The electrical resistivity of a UHPC mixture 

shall be determined from the results of a 

minimum of six cylinders produced, tested, and 

evaluated in accordance with AASHTO

TP 119-22 (AASHTO 2022a) and as amended 

herein.

The testing shall be conducted at 28 days after 

mixing or as approved by the owner.

AASHTO TP 119-22 is entitled, Standard 

Method of Test for Electrical Resistivity of a 

Concrete Cylinder Tested in a Uniaxial 

Resistance Test (AASHTO 2022a). Electrical 

resistivity provides information on how easily 

electrical charges move through a material 

when an electric field is applied. This property 

is commonly used as an indicator of concrete 

durability (Snyder 2001; Rajabipour, Sant, and 

Weiss 2007; McCarter et al. 2015; and Spragg 

et al. 2016). The amendments to AASHTO 

TP 119-22 listed in this Article are based on the 

work of Spragg et al. (2022).

Specimens shall be produced, tested, and 

evaluated in accordance with AASHTO

TP 119-22 (AASHTO 2022a) with the 

following amendments and/or restrictions:

The standard test specimen is a 4-in.-diameter 

by 8-in.-high cylinder. Three-inch-diameter by 

6-in.-high cylinders can also be used for 

evaluating resistivity. 

• The UHPC mixture being evaluated

shall be proportioned and mixed 

according to the predefined procedure 

provided by the product developer or 

supplier. Fiber reinforcement shall be 

excluded from the mixture.

The cementitious composite material, inclusive 

of any binders, admixtures, and inert granular 

constituents and aggregates, shall be mixed 

without modification of the product batch 

proportions or the mixing process. Fiber

reinforcement is excluded from the mixture 

because it can act as a secondary conductive 

phase within the UHPC matrix, resulting in 

increased conductivity.

• The default conditioning option 

described in Section 10.2 of AASHTO 

TP 119-22 shall be used.

The use of a different conditioning solution will 

affect the results of the test.

The use of the required conditioning solution 

may prove challenging for some laboratories. 

Owner approval is required for the use of an 

alternate conditioning solution. Spragg et al. 

(2022) have investigated the use of a saturated 

limewater solution in accordance with Section 
116

7 of AASHTO M 201-21 (AASHTO 2021). 

This alternate conditioning solution will 

necessitate a revision to the UHPC qualification 

threshold; Spragg et al. (2022) suggest a 

uniaxial resistivity, ρQ, threshold value of 

1,400 Ω∙m.

• The specimens shall be demolded 

within 48 hours from the time of mixing 

and immediately placed into the 

conditioning solution as specified in 

AASHTO TP 119-22 (AASHTO 

2022a). The specimens shall be 

continuously submerged in the conditioning solution until immediately 

before resistivity testing. 

• The test equipment shall be capable of 

reading electrical resistance values 

across a range from 10 to 10,000 Ω.

Equipment capable of measuring a large 

resistance range is necessary, given the large 

range of values that are observed in 

conventional concrete and UHPC-class 

materials.

A test result shall not be discarded, except that 

if any cylinder shows evidence of improper 

sampling, molding, or testing, said cylinder 

shall be discarded, and the electrical resistivity 

shall be determined from the remaining 

cylinders. 

Specimens shall be sampled from at least three 

separate batches. A minimum of two but not 

more than half of the tested cylinders shall be 

obtained from a single batch. 

The uniaxial resistivity obtained from each 

tested specimen, ρi, shall be calculated as: 

ρi = Riki (2.6.5.1-1) 

in which: 

(2.6.5.1-2) 

where: 

Ri = electrical resistance obtained from 

test specimen i (Ω)

Some resistivity devices report resistance in 

ohms (Ω), while others report uniaxial 

resistivity in ohm-meters (Ω∙m). Devices that 

report uniaxial resistivity have integrated a 

specimen geometry factor into their 

measurement process. Confirmation of the 

appropriateness of an integrated specimen 

geometry factor is required.
117

ki = specimen geometry factor of test 

specimen i (m) 

Au,i = cross-sectional area test specimen i, 

measured with an accuracy of less 

than 1 percent (m2) 

Lu,i = length of test specimen i, measured 

with an accuracy of less than 1 

percent (m)

The average and sample standard deviation of 

the uniaxial resistivity, ρ̅Q and sρQ, respectively, 

shall be calculated and reported to the owner 

based on the total number of tested specimens 

as follows: 

(2.6.5.1-3)

(2.6.5.1-4)

where: 

nρQ = total number of uniaxial resistivity 

test results obtained from 

qualification testing

Similar to the statistical treatments of the 

compressive strength test results described in 

Article C2.6.4.2 of this Appendix, the results of 

the electrical resistivity tests are assumed to 

follow a normal distribution, in which their

probability density function is established from 

the average and standard deviation calculated 

from a sample assumed to be large enough to 

provide sufficient data for statistical analysis.

The qualified value of the uniaxial resistivity, 

ρQ, of the UHPC mixture shall be determined as 

the lesser of the following: 

ρQ = ρ̅Q − 1.34 kρQ sρQ (2.6.5.1-3) 

ρQ = 1.11 ρ̅Q − 2.59 kρQ sρQ (2.6.5.1-4) 

where: 

kρQ = modification factor for the total 

number of uniaxial resistivity test 

results, nρQ, considered in 

calculating the sample standard 

deviation determined from 

Table 2.6.4.2-1 of this Appendix

The qualified values of each of the tensile 

properties are computed such that the two 

criteria defined in Article C2.6.4.1 of this 

Appendix are satisfied.

To be classified as a UHPC-class material, the 

qualified value of the uniaxial resistivity, ρQ, 

Refer to Spragg et al. (2022) for information on 

determination of the threshold value. 

 

 
118

shall meet or exceed a threshold value of 1,500 

Ω∙m.

2.6.5.2. Alternative Testing C2.6.5.2

UHPC mixtures may contain conductive 

inclusions apart from steel fiber reinforcement. 

The use of other durability test methods may be 

acceptable for defining the durability of a 

material containing conductive inclusions.

The use of alternative testing methods shall be 

approved by the owner.

Conductive inclusions will affect the results of 

the AASHTO TP 119-22 test method 

(AASHTO 2022a). If these inclusions cannot be 

excluded from the mix design of the product 

being tested, the test result may indicate an

electrical conductivity greater than what can be 

attributed to the cementitious matrix. Water 

absorption performed in accordance with 

ASTM C1585 (ASTM 2020f), chloride 

migration performed in accordance with NT 

BUILD 492 (Nordtest 1999), and chloride 

ponding performed in accordance with ASTM 

C1556 (ASTM 2022f) are alternative test 

methods to consider when evaluating durability 

performance. Threshold limits for UHPC 

durability classification based on alternative 

durability tests have not yet been established.

2.6.6. Supplemental Flexural Testing 

Supporting Alternative Tensile Response 

Acceptance Methods

C2.6.6

Article 2.7.3.3.2 of this Appendix allows the 

use of a modified version of ASTM C1609 

Revision A (ASTM 2019c) for tensile property 

material acceptance through flexural response 

testing. The testing described in this Article 

shall be completed prior to the engagement of

Article 2.7.3.3.2 of this Appendix.

The qualification testing requirements of this 

Article are optional, unless acceptance testing 

based on the provisions of Article 2.7.3.3.2 of 

this Appendix is desired. The test results 

obtained from flexural response testing are not 

used in structural design.

The procedures described in this Article 

establish the qualified load-deflection response 

of a beam made of a particular UHPC mixture 

and tested according to the method described in 

this Article. The qualified load deflection is 

established such that the two criteria defined in 

Article C2.6.4.1 of this Appendix are satisfied

and is used as the basis for acceptance testing 

when Article 2.7.3.3.2 is engaged.

The applicability of the procedures of this 

Article is limited to flexural test results showing 

small statistical variability. A large scatter in 
119

the individual load-deflection responses will

result in objectionably low qualified loaddeflection responses that should not be used for 

acceptance.

The qualified flexural response of a particular 

UHPC mixture shall be defined by the qualified 

value of the peak load, PpQ, and the qualified 

values of the loads, PδQ, from the 

load-deflection responses of at least 15 tested 

specimens. The qualified values of the loads, 

PδQ, shall be obtained at 0.005-in. increments of 

net midspan deflection up to the average value 

of net midspan deflection at peak load, δ̅pQ,

rounded up to the next 0.005 in.

The qualified values of the loads, PδQ, obtained 

from flexural testing are intended to fall within 

the pre-peak portion of the load-deflection 

response to avoid the variability expected 

during the post-peak response after the 

development of the localized crack. The last 

load value of PδQ must be taken at the next 

incremental value of net midspan deflection

that is greater than the average net midspan 

deflection at peak load, δ̅pQ.

A minimum of 30 flexural response test results 

are recommended to establish the variation (i.e., 

mean and sample standard deviation) in the test 

results of a particular UHPC mixture.

Specimens shall be produced, tested, and 

evaluated in accordance with ASTM C1609 

Revision A (ASTM 2019c) with the following 

modifications:

• Specimens shall be cast following the 

procedure described in AASHTO T 397 

Section 7.3 (AASHTO 2022b).

• Specimens shall have a 4-in. by 4-in. 

cross section and shall be 14-in. long.

• The test machine shall be operated such 

that the net midspan deflection 

increases at a constant rate between 

0.001 and 0.003 in./min until the peak 

load has been reached and the load 

subsequently decreased to less than 

50 percent of the maximum load 

attained.

As described in ASTM C1609 Revision A 

(ASTM 2019c), test specimens shall be 

supported on rollers that are free to rotate on 

their axes in accordance with ASTM C1812 

(ASTM 2015a).

The UHPC prismatic test specimen fabrication 

procedures in AASHTO T 397 (AASHTO

2022b) describes a consistent casting method of 

the test specimens.

The results obtained from an ASTM C1609

Revision A (ASTM 2019c) test are dependent 

on the test specimen geometry. A prescribed 

geometry is necessary to ensure that test results 

represent data points in the same population.
120

Specimens shall be sampled from at least three 

separate batches. A minimum of two but not 

more than half of the tested specimens shall be 

obtained from a single batch. 

The load-deflection response of each tested 

specimen shall be measured and recorded. The 

load, Pδ,i, of each tested specimen i shall be 

recorded at 0.005-in. increments of net midspan 

deflection, until the peak load has been reached 

and the load has subsequently decreased to less 

than 50 percent of the maximum load attained.

The peak load resisted by each specimen, Pp,i, 

and its corresponding net midspan deflection, 

δp,i, shall be measured and recorded. This load 

may occur at a net midspan deflection other 

than the prescribed increments.

The load, Pδ,i, should be recorded at a large 

number of 0.005-in. increments so that the total 

net midspan deflection of each tested specimen 

surpasses the expected average net midspan 

deflection at peak load, δ̅pQ.

The average and standard deviation of the load 

recorded at each net midspan deflection 

increment, P̅δQ and sδQ, respectively, shall be 

calculated based on the total number of tested 

specimens from all batches as follows:

(2.6.6-1)

(2.6.6-2)

where: 

nfQ = total number of flexural test results 

obtained from qualification testing 

The average and sample standard deviation of 

the peak load, P̅pQ and spQ, respectively, shall 

be calculated based on the total number of 

tested specimens as follows: 

(2.6.6-3) 

Similar to the statistical treatments of the 

compressive strength test results described in 

Article 2.6.4.2 of this Appendix and the tensile 

response test results described in Article 2.6.4.3

of this Appendix, the loads captured at the 

prescribed midspan deflection increments, Pδ,i, 

as well as the peak loads, Pp,i, are assumed to 

follow a normal distribution. The probability 

density functions of Pδ,i and Pp,i are established 

from their average and sample standard 

deviation, P̅δQ and sδQ, and P̅pQ and spQ, 

respectively, calculated from a sample assumed 

to be large enough to provide sufficient data for 

statistical analysis.

 

 
121

(2.6.6-4)

The average of the net midspan deflection at 

peak load, δ̅pQ, shall be calculated based on the 

total number of tested specimens as follows:

(2.6.6-5)

The qualified value of the loads, PδQ, at each 

increment of net midspan deflection shall be 

determined as the lesser of the following: 

PδQ = P̅δQ − 1.34 kfQ sδQ (2.6.6-6) 

PδQ = 1.11P̅δQ − 2.59 kfQ sδQ (2.6.6-7) 

where: 

kfQ = modification factor for the total 

number of flexural test results, nfQ, 

considered in calculating the 

sample standard deviation 

determined from Table 2.6.4.2-1 of 

this Appendix 

Following the same statistical considerations 

used to compute the design compressive 

strength, f ́cQ, described in Article C2.6.4.2 of 

this Appendix, and the design tensile 

parameters, ft,crQ, ft,locQ, and εt,locQ, the qualified 

values of each of the captured loads are 

computed such that the two criteria defined in 

Article C2.6.4.1 of this Appendix are satisfied.

The qualified value of the peak load, PpQ, shall 

be determined as the lesser of the following: 

PpQ = P̅pQ − 1.34 kfQ spQ (2.6.6-8) 

PpQ = 1.11P̅pQ − 2.59 kfQ spQ (2.6.6-9) 

2.7. MATERIAL ACCEPTANCE 

2.7.1. General 

Material acceptance testing shall be conducted 

on the UHPC material being used to construct 

structural components. 

Test specimens shall be fabricated according to 

ASTM C31 (ASTM 2012) or ASTM C192 

(ASTM 2019a), as modified by ASTM C1856 

(ASTM 2017a), AASHTO T 397 (AASHTO 

 

 
122

2022b), and accompanying articles in this 

Appendix.

Test specimen curing shall match the curing 

process implemented during material 

qualification and specified for structural 

product manufacture.

Hardened property testing shall be conducted at 

the same age that the associated material 

qualification testing was completed.

Testing method and sampling frequency shall 

be in accordance with Table 2.7.1-1.

Table 2.7.1-1. Material acceptance testing methods and sampling frequency.

Property Article Minimum Frequency

Flow 2.7.2.1 Every batch.

Fiber segregation 2.7.2.2

First batch each day.

Whenever compression or tension test 

specimens are cast.

Temperature 2.7.2.3 Every batch.

Compressive strength 2.7.3.2

One set per element cast.

If element volume exceeds 25 yd3, once per 

25 yd3cast.

Tensile response: Option 1 2.7.3.3.1

One set per element cast.

If element volume exceeds 25 yd3, once per 

25 yd3cast.

Tensile response: Option 2 2.7.3.3.2

One set per element cast.

If element volume exceeds 25 yd3, once per 

25 yd3cast.

2.7.2. Plastic Properties

2.7.2.1. Flow

The flow shall be measured and reported for 

each batch. Flow shall be measured in 

accordance with ASTM C1856/1856M (ASTM 

2017a) which provides direction on the use of 

ASTM C1437 (ASTM 2020e) when testing 

UHPC.

Flow shall be measured after the addition and 

mixing of all constituent materials. If water or 

admixture dosages are adjusted to achieve a 
123

target flow property, flow shall be measured 

after the adjustments are implemented. 

Flow shall be not more than 10 minutes from

the start of component casting. For batches 

whose casting lasts for more than 30 minutes, 

flow shall be measured and reported at 

30-minute intervals.

2.7.2.2. Fiber Segregation

Susceptibility to fiber segregation shall be 

measured and reported at the frequency defined 

in Table 2.7.1-1.

The test shall be completed according to ASTM 

C1712 (ASTM 2020g) with the modifications 

and procedures described in Article 2.6.3.3.

2.7.2.3. Temperature

The temperature of each batch shall be 

measured and reported in accordance with 

ASTM C1064 (ASTM 2017b). The 

temperature shall be measured in the mixer 

after the conclusion of mixing, in a convenance 

device transporting the UHPC to the placement 

forms, or in a sample container.

The temperature shall be measured no more 

than 5 minutes prior to the start of component 

casting.

Additional measurement times and locations 

may be defined by the owner.

2.7.3. Hardened Properties

2.7.3.1. General C2.7.3.1

The material acceptance of the hardened 

properties used for design—namely, design 

compressive strength, design effective cracking 

strength, design crack localization stress, and 

design crack localization strain—shall be 

established based on the required values 

The material acceptance procedures described 

herein establish required values of each 

property based on information particular to 

each UHPC mixture assessed in the 

qualification phase.

To optimize use of UHPC in structural 

components and to achieve efficient and 
124

obtained following the provisions of this 

Article.

economical designs, the value of each hardened 

property used in design should be equal to the 

qualified value obtained from the qualification 

phase of a particular UHPC mixture according 

to the provisions of Article 2.6.4 of this 

Appendix. If the values of the hardened 

properties used in design, particularly the 

values of the tensile properties, are significantly 

lower than the qualified values of the UHPC 

mixture selected for the project, significant 

reduction in the UHPC component size and 

steel reinforcement may be achieved through 

design refinements based on the qualified 

properties.

Each hardened property shall be acceptable if 

both of the following criteria are met:

• Every average of three consecutive test 

results equals or exceeds their 

respective required value.

• No single property test result falls 

below their respective required value by 

more than 10 percent.

The acceptance criteria for each hardened 

property specified in this Article are based on 

the acceptance criteria for the compressive 

strength of conventional concrete stated in 

Article 26.12.3.1 of Building Code 

Requirements for Structural Concrete

(ACI Committee 318 2019). 

Given that the qualified value of each hardened 

property is computed based on the provisions of 

Article 2.6.4 of this Appendix, the acceptance 

criteria are expected to be satisfied with a 

probability of failure of 1 percent. Allowance 

should be made for statistically expected 

variations in deciding whether the hardened 

property of the UHPC mixture being produced 

is adequate, particularly when the qualified 

value of the hardened property is equal to the 

design value.

2.7.3.2. Compressive Strength C2.7.3.2

The compressive strength for acceptance of a 

UHPC mixture shall consist of the compressive 

strength test results of at least three cylinders 

fabricated from material taken from a single 

batch of UHPC. A minimum of four cylinders 

shall be fabricated for each compressive 

strength acceptance requirement. The cylinders 

shall be produced, tested, and evaluated in 
125

accordance with ASTM C1856/C1856M 

(ASTM 2017a). 

A test result shall not be discarded, except that 

if any cylinder shows evidence of improper 

sampling, molding, or testing, said cylinder 

shall be discarded and the strength shall be 

determined from the remaining cylinders.

The required value of the compressive strength 

shall be taken as the greater of the following:

f ́cR = f ́c + 1.34 kcQ scQ ≤ f ́cQ (2.7.3.2-1)

f ́cR = 0.90 f ́c + 2.33 kcQ scQ ≤ f ́cQ (2.7.3.2-2)

where:

f ́cR = required average value of the 

compressive strength obtained 

from acceptance testing (ksi)

f ́c = compressive strength of UHPC for 

use in design (ksi)

kcQ = modification factor for the total 

number of compression strength 

test results, ncQ, considered in 

calculating scQ according to 

Article 2.6.4.2 of this Appendix

scQ = sample standard deviation of the 

compressive strength for the 

mixture obtained from the 

qualification testing according to 

Article 2.6.4.2 of this Appendix

(ksi)

f ́cQ = qualified design value of the 

compressive strength obtained 

from qualification testing 

according to Article 2.6.4.2 of this 

Appendix (ksi)

When the value of the compressive strength 

used in design, f ́c, is lower than the qualified 

value of a particular UHPC mixture, Eqs.

2.7.3.2-1 and 2.7.3.2-2 establish a required 

value greater than the design value based on the 

statistical variability of the UHPC mixture. This 

consideration is intended to restrict the average 

compression strength values obtained from 

acceptance testing from being significantly 

lower than their qualified values obtained 

during qualification of the same UHPC mixture.

The specified value of the compressive strength 

required for acceptance is recommended to be 

set equal to its qualified value, f ́cQ, when both 

the design value, f ́c, and the required value f ́cR

are less than the qualified value, f ́cQ. This 

requirement ensures that the UHPC mixture 

being produced on the project is identical to the 

UHPC mixture qualified based on the 

provisions of Article 2.6.4.2 of this Appendix. 

When the acceptance value of the compressive 

strength is specified to be equal to the required 

value, f ́cQ, the acceptance criteria are expected 

to be satisfied with a probability of failure of 1 

percent.
126

2.7.3.3. Tensile Response

Acceptance testing for the tension response 

shall be performed according to either of the 

two procedures described in this Article.

2.7.3.3.1. Option 1: Evaluation by Tension 

Testing

C2.7.3.3.1

The tensile properties for acceptance of a 

UHPC mixture shall consist of the effective 

cracking strength, crack localization stress, and

the crack localization strain obtained from the 

tension response test results of at least three 

specimens fabricated from material taken from 

a single batch of UHPC. The tension response

test result of each specimen used to determine 

the tensile properties must be classified as Type 

H-1 or H-2 as defined in AASHTO T 397 

(AASHTO 2022b). A minimum of six 

specimens shall be sampled for each tension 

response acceptance requirement. The 

specimens shall be produced, tested, and 

evaluated in accordance with AASHTO T 397 

(AASHTO 2022b) with the modifications 

described in Article 2.6.4.3 of this Appendix. 

The acceptance testing for the tensile response 

verifies that the produced UHPC mixture 

exhibits a strain-hardening, stress-strain tensile 

behavior of Types H-1, H-2, H-3, and H-4 as 

defined in AASHTO T 397 (AASHTO 2022b). 

However, only test results where the 

localization crack occurs within the gauge 

length, i.e., tensile behavior of Types H-1 and 

H-2, are permitted to be used to determine the 

tensile properties.

A test result shall not be discarded, except that 

if any specimen shows evidence of improper 

sampling, molding, or testing, said specimen 

shall be discarded, and the tensile parameters

shall be determined from the remaining 

specimens. A properly sampled, molded, and 

tested specimen exhibiting a tensile behavior of 

types other than H-1 or H-2, as defined in 

AASHTO T 397 (AASHTO 2022b), shall not 

be considered discarded, but its test results shall 

not be used for acceptance.

Each test result shall be classified based on the 

tensile behaviors described in Section 9.0 of 

AASHTO T 397 (AASHTO 2022b) and 

reported to the owner.

The minimum number of specimens considered 

for acceptance may be reduced, if, after testing 

all the sampled specimens, the tension response 

The minimum number of the specimens 

considered for acceptance can be reduced only 

after the testing of at least the six specimens 
127

of less than three specimens resulted in Type H1 or H-2 response as defined in AASHTO

T 397 (AASHTO 2022b). 

required to be sampled for each acceptance 

requirement.

If more than one out of every six tested 

specimens from a single batch result in a 

tension response of Type S, or if any tested 

specimen resulted in a tension response of Type 

N as defined in AASHTO T 397 (AASHTO

2022b), the mixture shall be considered as not 

meeting the acceptance criteria.

When tensile behaviors of Type S are 

manifested in the test results of sampled 

specimens, the tension response of the mixture 

is classified as strain softening, and the mixture 

shall not be used in design according to this 

Appendix.

Tensile behaviors of Type N indicate regions of 

the specimens without fibers and with no 

resistance to postcracking tensile loads. 

Mixtures exhibiting behaviors of Type N are 

not for use in structures designed according to 

the provisions of Section 1 of this Appendix.

When tensile behaviors of Type S are 

manifested at a frequency greater than one out 

of every six tested specimens from a single 

batch or when tensile behaviors of Type N are 

manifested, UHPC production must be halted 

until corrective action is taken to improve the 

fabrication procedures.

The required value for the effective cracking 

strength, ft,crR, shall be taken as the greater of 

the following:

ft,crR = ft,cr + 1.34 ktQ st,crQ ≤ ft,crQ

(2.7.3.3.1-1)

ft,crR = 0.90 ft,cr + 2.33 ktQ st,crQ ≤ ft,crQ

(2.7.3.3.1-2)

where:

ft,cr = effective cracking strength for use 

in design (ksi)

ktQ = modification factor for the total 

number of tension test results, ntQ,

considered in calculating the sample 

standard deviation according to 

Article 2.6.4.3 of this Appendix

When the values of the tensile properties used 

in design are lower than their respective 

qualified values of a particular UHPC mixture, 

Eqs. 2.7.3.3.1-1 through 2.7.3.3.1-6 establish a 

required value greater than the design value for 

each tensile property based on the statistical 

variability of the selected UHPC mixture. This 

consideration is intended to restrict the average 

tensile property values obtained during 

acceptance testing from being significantly 

lower that their corresponding qualified 

property values obtained during qualification of 

the same UHPC mixture.

The specified value for acceptance of each 

tensile property is recommended to be set equal 

to its qualified value when both the design and 

required values are less than the qualified value. 

This requirement ensures that the UHPC 

mixture being produced on the project is 

identical to the UHPC mixture qualified based 

on the provisions of Article 2.6.4.3 of this 
128

st,crQ = sample standard deviation of the 

effective crack localization strength 

test results for a UHPC mixture 

obtained from qualification testing 

according to Article 2.6.4.3 of this 

Appendix (ksi)

ft,crQ = qualified design value of the 

effective cracking strength from 

qualification testing according to 

Article 2.6.4.3 of this Appendix

(ksi).

Appendix. When the acceptance value of each

tensile property is specified to be equal to the 

qualified value, the acceptance criteria are 

expected to be satisfied with a probability of 

failure of 1 percent.

The required value for the crack localization 

strength, ft,locR, shall be taken as the greater of 

the following:

ft,locR = ft,loc + 1.34 ktQ st,locQ ≤ ft,locQ

(2.7.3.3.1-3)

ft,locR = 0.90 ft,loc + 2.33 ktQ st,locQ ≤ ft,locQ

(2.7.3.3.1-4)

where:

ft,loc = crack localization strength for use in 

design (ksi)

st,locQ =sample standard deviation of the 

crack localization strength test 

results for a UHPC mixture 

obtained from qualification testing 

according to Article 2.6.4.3 of this 

Appendix (ksi)

ft,locQ = qualified design value of the crack 

localization strength from 

qualification testing according to 

Article 2.6.4.3 of this Appendix

(ksi).

The required value for the crack localization 

strain, εt,locR, shall be taken as the greater of the 

values obtained from Eqs. 2.7.3.3.1-5 and 

2.7.3.3.1-6 but not greater than the lesser of 

1.25εt,loc and εt,locQ:

As described in Article 2.6.4.3 of this 

Appendix, the qualified values of crack 

localization strain test results are capped to a 

maximum reduction of 20 percent from the 

average qualification result. In this case, the 

qualified value of the crack localization strain 
129

εt,locR = εt,loc + 1.34 ktQ sεt,locQ (2.7.3.3.1-5)

εt,locR = 0.90 εt,loc + 2.33 ktQ sεt,locQ

(2.7.3.3.1-6)

where:

εt,loc = crack localization strain for use in 

design (in./in.)

sεt,locQ = sample standard deviation of the 

crack localization strain test 

results for a UHPC mixture 

obtained from qualification 

testing according to Article

2.6.4.3 of this Appendix (in./in.)

εt,locQ = qualified design value of the crack 

localization strain from 

qualification testing according to 

Article 2.6.4.3 of this Appendix

(in./in.).

may be expected to be satisfied with a 

probability of failure of more than 1 percent. 

Therefore, allowance may be made for 

variations in deciding whether the average 

value of the crack localization being produced

is adequate, particularly when the required 

design value is equal to the qualified value and 

when the qualified value was determined by the 

20 percent reduction cap from the average 

qualification result.

2.7.3.3.2. Option 2: Evaluation by Flexural 

Testing 

C2.7.3.3.2

The acceptance of the tensile response of a 

UHPC mixture by flexural testing shall consist 

of the flexural responses defined by the peak 

loads and the loads corresponding to the same 

net midspan deflection increments determined

during qualification testing of the mixture 

according to Article 2.6.6 of this Appendix. The 

flexural responses shall be obtained from the 

load-deflection test results of at least three 

specimens fabricated from a single batch of 

UHPC. A minimum of four specimens shall be 

sampled for each tension response acceptance 

criteria. The test specimens shall be produced, 

tested, and evaluated in accordance with ASTM 

C1609 Revision A (ASTM 2019c) with the 

modifications described in Article 2.6.6 of this 

Appendix.

Given that the load-deflection results of 

specimens tested in flexure are not a direct 

measurement of the uniaxial tensile properties 

used in structural design, the acceptance criteria 

based on flexural testing must be evaluated with 

respect to the qualified values of the peak load, 

PpQ, and the loads, PδQ, at the same net midspan 

deflection increments determined during qualification testing of the mixture.

The applicability of the procedures of this 

Article are limited to flexural test results 

showing small statistical variability during 

qualification testing. A large scatter in the

individual load-deflection responses during 

qualification will result in objectionably low 

qualified load-deflection responses that should 

not be used for acceptance.
130

A test result shall not be discarded, except that, 

if any specimen shows evidence of improper 

sampling, molding, or testing, said specimen 

shall be discarded, and the tensile test shall 

consist of the remaining specimens.

The tensile response of a UHPC mixture shall 

be acceptable if the two criteria specified in 

Article 2.7.3.1 of this Appendix are met for 

each of the loads recorded at the prescribed net 

midspan deflection increments and at peak

load.
131

ADDENDUM A1. TYPICAL MATERIAL PROPERTIES OF ULTRA-HIGH 

PERFORMANCE CONCRETE

Typical UHPC mechanical properties and associated test methods are presented in Table A1-1.

Additionally, minimum properties as defined in Article 1.1.1 of this Appendix are also provided.

Table A1-1. Typical and minimum mechanical properties for UHPC.

Property Test Method Typical Values

Minimum 

Value

Modulus of elasticity, Ec (ksi) ASTM C1856/C1856M

(ASTM 2017a)

6,500–9,400 N/A

Compressive strength, f ́c (ksi) ASTM C1856/C1856M

(ASTM 2017a)

20.0–36.0 17.5

Ultimate compressive strain, εcu

ASTM C1856/C1856M

(ASTM 2017a)

0.003–0.005 N/A

Poisson’s ratio ASTM C1856

(ASTM 2017a) 0.1–0.2 N/A

Effective cracking strength, ft,cr

(ksi)

AASHTO T 397

(AASHTO 2022b)

0.90–1.80 0.75

Crack localization strength, ft,loc

(ksi)

AASHTO T 397

(AASHTO 2022b)

0.90–1.80 ≥ ft,cr

Crack localization strain, εt,loc

AASHTO T 397

(AASHTO 2022b)

0.003–0.008 0.0025

N/A = not applicable.

133

ADDENDUM A2. SHEAR DESIGN TABLES FOR θ AND fv,α

A2.1. GENERAL

In lieu of the general method of 

Article 1.7.3.4.1 of this Appendix, the shear 

design parameters θ and fv,α may be obtained 

from the tables provided in this Article as 

specified in Article 1.7.3.4.2 of this Appendix. 

The provisions of this Addendum are applicable 

for members made of UHPC with or without 

transverse steel reinforcement having Ec ≥ 

6,500 ksi and ft,loc ≤ 1.80 ksi.

A2.2. MEMBERS WITHOUT

TRANSVERSE STEEL

REINFORCEMENT

For members without transverse steel 

reinforcement (i.e., ρv,α = 0.0 percent), the 

values of θ may be determined from 

Table A2.2-1 at a specific value of γuεt,loc and 

εs.

Interpolations to calculate θ between the values 

of εs and γuεt,loc in Table A2.2-1 shall not be 

permitted.
134

Table A2.2-1. Values of θ (degrees) for sections without transverse reinforcement (i.e., 

ρv,α = 0.0 percent).

εs ×

1,000

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 30.8 30.0 29.3 28.7 28.2 27.7 27.2 26.8 26.5 26.1 25.8 25.5

≤ −0.5 32.5 31.5 30.7 30.0 29.3 28.8 28.3 27.8 27.4 27.0 26.6 26.3

≤0.0 34.5 33.2 32.2 31.4 30.6 30.0 29.4 28.9 28.4 27.9 27.5 27.2

≤0.5 36.7 35.2 33.9 32.9 32.0 31.3 30.6 30.0 29.4 28.9 28.5 28.1

≤1.0 39.2 37.3 35.8 34.6 33.6 32.7 31.9 31.2 30.5 30.0 29.5 29.0

≤1.5 42.1 39.7 37.9 36.4 35.2 34.1 33.2 32.4 31.7 31.1 30.5 30.0

≤2.0 45.4 42.4 40.1 38.4 36.9 35.7 34.7 33.8 33.0 32.3 31.6 31.0

≤2.5 49.1 45.3 42.6 40.5 38.8 37.4 36.2 35.2 34.3 33.5 32.8 32.1

≤3.0 – 48.6 45.3 42.8 40.8 39.2 37.9 36.7 35.7 34.8 34.0 33.2

≤3.5 – – 48.2 45.3 43.0 41.1 39.6 38.2 37.1 36.1 35.2 34.4

≤4.0 – – – 47.9 45.3 43.1 41.4 39.9 38.6 37.5 36.5 35.6

≤4.5 – – – – 47.7 45.2 43.2 41.6 40.2 38.9 37.8 36.9

≤5.0 – – – – – 47.5 45.2 43.4 41.8 40.4 39.2 38.2

≤5.5 – – – – – – 47.5 45.2 43.5 42.0 40.6 39.5

≤6.0 – – – – – – – 47.1 45.2 43.5 42.1 40.9

≤6.5 – – – – – – – – 47.0 45.2 43.6 42.3

≤7.0 – – – – – – – – – 46.9 45.2 43.7

≤7.5 – – – – – – – – – – 46.8 45.2

≤8.0 – – – – – – – – – – – 46.7

Note: Values for θ where εs is greater than γuεt,loc are not relevant. These combinations are indicated by cells containing 

the “–” symbol.

A2.3. MEMBERS WITH TRANSVERSE 

STEEL REINFORCEMENT

CA2.3

For members with transverse steel 

reinforcement having fy ≤ 75.0 ksi, ρv,α

≤ 3.0 percent, and α = 90 degrees, the values of 

the upper limit of fv,α and θ may be determined 

from Table A2.3-1 through Table A2.3-6 based 

on the value of ρv,α and at a specific value of 

γuεt,loc and εs.

Interpolations to calculate θ and fv,α between the 

values of ρv,α, εs, and γuεt,loc in Table A2.3-1

through Table A2.3-6 shall not be permitted.

As specified in Article 1.7.3.4.2 of this 

Appendix, the value of fv,α used in Eq. 1.7.3.3-4 

shall be determined as the lesser of the value 

obtained from the tables of this Addendum and 

the specified minimum yield strength of the 

transverse steel, fy.
135

Table A2.3-1. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 0.5 percent.

εs × 

1,000

Parameter

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 θ (deg) 31.8 31.2 30.8 30.4 29.9 29.3 28.8 28.4 28.0 27.6 27.3 27.0

fv,α (ksi) ≤39.0 ≤49.6 ≤60.4 ≤71.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤ −0.5 θ (deg) 33.5 32.7 32.1 31.6 31.0 30.4 29.9 29.4 28.9 28.5 28.1 27.7

fv,α (ksi) ≤37.6 ≤48.1 ≤58.7 ≤69.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.0 θ (deg) 35.4 34.4 33.6 32.9 32.3 31.6 30.9 30.4 29.9 29.4 29.0 28.6

fv,α (ksi) ≤35.9 ≤46.2 ≤56.8 ≤67.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.5 θ (deg) 37.5 36.2 35.2 34.3 33.6 32.8 32.1 31.5 30.9 30.4 29.9 29.4

fv,α (ksi) ≤33.9 ≤44.2 ≤54.6 ≤65.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.0 θ (deg) 40.0 38.3 37.0 35.9 35.0 34.1 33.3 32.6 32.0 31.4 30.8 30.3

fv,α (ksi) ≤31.7 ≤41.8 ≤52.2 ≤62.8 ≤73.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.5 θ (deg) 42.8 40.6 38.9 37.6 36.6 35.6 34.6 33.8 33.1 32.4 31.8 31.3

fv,α (ksi) ≤29.1 ≤39.2 ≤49.5 ≤60.0 ≤70.7 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.0 θ (deg) 45.9 43.1 41.1 39.5 38.2 37.1 36.0 35.1 34.3 33.5 32.9 32.3

fv,α (ksi) ≤26.2 ≤36.2 ≤46.5 ≤57.0 ≤67.7 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.5 θ (deg) 49.5 45.9 43.4 41.5 39.9 38.7 37.5 36.5 35.5 34.7 34.0 33.3

fv,α (ksi) ≤22.8 ≤32.8 ≤43.2 ≤53.7 ≤64.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.0 θ (deg) – 49.1 45.9 43.6 41.8 40.4 39.0 37.9 36.8 35.9 35.1 34.4

fv,α (ksi) – ≤29.1 ≤39.6 ≤50.1 ≤60.8 ≤71.7 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.5 θ (deg) – – 48.7 46.0 43.8 42.1 40.7 39.4 38.2 37.2 36.3 35.5

fv,α (ksi) – – ≤35.6 ≤46.2 ≤57.0 ≤67.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.0 θ (deg) – – – 48.5 46.0 44.0 42.4 40.9 39.6 38.5 37.5 36.7

fv,α (ksi) – – – ≤42.1 ≤52.9 ≤63.9 ≤74.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.5 θ (deg) – – – – 48.3 46.0 44.1 42.5 41.1 39.9 38.8 37.9

fv,α (ksi) – – – – ≤48.6 ≤59.6 ≤70.7 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.0 θ (deg) – – – – – 48.1 46.0 44.2 42.7 41.3 40.1 39.1

fv,α (ksi) – – – – – ≤55.1 ≤66.3 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.5 θ (deg) – – – – – – 47.9 46.0 44.3 42.8 41.5 40.4

fv,α (ksi) – – – – – – ≤61.7 ≤73.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤6.0 θ (deg) – – – – – – – 47.8 46.0 44.3 42.9 41.7

fv,α (ksi) – – – – – – – ≤68.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤6.5 θ (deg) – – – – – – – – 47.7 45.9 44.4 43.0

fv,α (ksi) – – – – – – – – ≤74.8 ≤75.0 ≤75.0 ≤75.0

≤7.0 θ (deg) – – – – – – – – – 47.5 45.9 44.4

fv,α (ksi) – – – – – – – – – ≤75.0 ≤75.0 ≤75.0

≤7.5 θ (deg) – – – – – – – – – – 47.4 45.8

fv,α (ksi) – – – – – – – – – – ≤75.0 ≤75.0

≤8.0 θ (deg) – – – – – – – – – – – 46.6

fv,α (ksi) – – – – – – – – – – – ≤75.0

Note: Values for θ and fv,α where εs is greater than γu εt,loc are not relevant. These combinations are indicated by cells 

containing the “–” symbol.
136

Table A2.3-2. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 1.0 percent.

εs × 

1,000

Parameter

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 θ (deg) 32.7 32.3 31.9 31.6 31.3 30.8 30.2 29.8 29.3 28.9 28.6 28.2

fv,α (ksi) ≤36.7 ≤46.6 ≤56.5 ≤66.5 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤ −0.5 θ (deg) 34.3 33.7 33.2 32.8 32.4 31.8 31.2 30.7 30.2 29.8 29.4 29.0

fv,α (ksi) ≤35.3 ≤45.1 ≤54.9 ≤64.8 ≤74.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.0 θ (deg) 36.2 35.3 34.6 34.1 33.6 32.9 32.3 31.7 31.2 30.7 30.2 29.8

fv,α (ksi) ≤33.8 ≤43.4 ≤53.1 ≤63.0 ≤72.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.5 θ (deg) 38.2 37.1 36.2 35.4 34.8 34.2 33.4 32.7 32.1 31.6 31.1 30.6

fv,α (ksi) ≤32.0 ≤41.4 ≤51.1 ≤60.9 ≤70.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.0 θ (deg) 40.6 39.1 37.9 36.9 36.2 35.4 34.6 33.9 33.2 32.6 32.0 31.5

fv,α (ksi) ≤29.9 ≤39.3 ≤48.9 ≤58.6 ≤68.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.5 θ (deg) 43.3 41.3 39.7 38.6 37.6 36.8 35.9 35.0 34.3 33.6 33.0 32.4

fv,α (ksi) ≤27.4 ≤36.8 ≤46.4 ≤56.1 ≤65.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.0 θ (deg) 46.4 43.7 41.8 40.3 39.2 38.2 37.2 36.3 35.4 34.7 34.0 33.4

fv,α (ksi) ≤24.7 ≤34.0 ≤43.6 ≤53.3 ≤63.2 ≤73.1 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.5 θ (deg) 49.9 46.4 44.0 42.2 40.8 39.7 38.6 37.6 36.6 35.8 35.0 34.4

fv,α (ksi) ≤21.5 ≤30.9 ≤40.6 ≤50.3 ≤60.2 ≤70.1 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.0 θ (deg) – 49.5 46.5 44.3 42.6 41.3 40.1 38.9 37.9 37.0 36.1 35.4

fv,α (ksi) – ≤27.5 ≤37.2 ≤47.0 ≤56.9 ≤66.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.5 θ (deg) – – 49.2 46.5 44.5 42.9 41.6 40.4 39.2 38.2 37.3 36.5

fv,α (ksi) – – ≤33.5 ≤43.5 ≤53.4 ≤63.5 ≤73.5 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.0 θ (deg) – – – 48.9 46.6 44.7 43.2 41.8 40.6 39.5 38.5 37.6

fv,α (ksi) – – – ≤39.6 ≤49.7 ≤59.8 ≤70.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.5 θ (deg) – – – – 48.7 46.6 44.9 43.4 42.0 40.8 39.7 38.7

fv,α (ksi) – – – – ≤45.7 ≤55.9 ≤66.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.0 θ (deg) – – – – – 48.6 46.6 45.0 43.5 42.2 41.0 39.9

fv,α (ksi) – – – – – ≤51.8 ≤62.1 ≤72.5 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.5 θ (deg) – – – – – – 48.5 46.6 45.0 43.6 42.3 41.1

fv,α (ksi) – – – – – – ≤57.9 ≤68.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤6.0 θ (deg) – – – – – – – 48.3 46.6 45.0 43.6 42.4

fv,α (ksi) – – – – – – – ≤64.1 ≤74.6 ≤75.0 ≤75.0 ≤75.0

≤6.5 θ (deg) – – – – – – – – 48.3 46.6 45.0 43.7

fv,α (ksi) – – – – – – – – ≤70.2 ≤75.0 ≤75.0 ≤75.0

≤7.0 θ (deg) – – – – – – – – – 48.1 46.5 45.0

fv,α (ksi) – – – – – – – – – ≤75.0 ≤75.0 ≤75.0

≤7.5 θ (deg) – – – – – – – – – – 48.0 46.4

fv,α (ksi) – – – – – – – – – – ≤75.0 ≤75.0

≤8.0 θ (deg) – – – – – – – – – – – 47.8

fv,α (ksi) – – – – – – – – – – – ≤75.0

Note: Values for θ and fv,α where εs is greater than γu εt,loc are not relevant. These combinations are indicated by cells 

containing the “–” symbol.
137

Table A2.3-3. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 1.5 percent.

εs× 

1,000

Parameter

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 θ (deg) 33.4 33.1 32.8 32.6 32.5 32.0 31.5 31.0 30.5 30.1 29.7 29.4

fv,α (ksi) ≤34.7 ≤43.9 ≤53.2 ≤62.5 ≤71.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤ −0.5 θ (deg) 35.0 34.5 34.1 33.8 33.5 33.0 32.4 31.9 31.4 30.9 30.5 30.1

fv,α (ksi) ≤33.4 ≤42.5 ≤51.7 ≤60.9 ≤70.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.0 θ (deg) 36.8 36.1 35.5 35.0 34.6 34.2 33.5 32.9 32.3 31.8 31.3 30.9

fv,α (ksi) ≤31.9 ≤40.9 ≤50.0 ≤59.2 ≤68.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.5 θ (deg) 38.8 37.8 37.0 36.3 35.8 35.3 34.6 33.9 33.3 32.7 32.2 31.7

fv,α (ksi) ≤30.2 ≤39.1 ≤48.1 ≤57.2 ≤66.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.0 θ (deg) 41.1 39.7 38.6 37.8 37.1 36.5 35.7 35.0 34.3 33.6 33.1 32.5

fv,α (ksi) ≤28.2 ≤37.0 ≤46.0 ≤55.1 ≤64.2 ≤73.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.5 θ (deg) 43.8 41.8 40.4 39.3 38.5 37.8 36.9 36.1 35.3 34.6 34.0 33.4

fv,α (ksi) ≤26.0 ≤34.7 ≤43.7 ≤52.7 ≤61.9 ≤71.1 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.0 θ (deg) 46.7 44.2 42.4 41.0 39.9 39.1 38.2 37.3 36.4 35.7 35.0 34.4

fv,α (ksi) ≤23.4 ≤32.2 ≤41.1 ≤50.2 ≤59.3 ≤68.5 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.5 θ (deg) 50.2 46.8 44.5 42.9 41.5 40.5 39.6 38.5 37.6 36.8 36.0 35.3

fv,α (ksi) ≤20.4 ≤29.3 ≤38.3 ≤47.4 ≤56.6 ≤65.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.0 θ (deg) – 49.8 46.9 44.8 43.2 42.0 40.9 39.9 38.8 37.9 37.1 36.3

fv,α (ksi) – ≤26.1 ≤35.1 ≤44.3 ≤53.6 ≤62.8 ≤72.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.5 θ (deg) – – 49.5 47.0 45.1 43.6 42.4 41.2 40.1 39.1 38.2 37.4

fv,α (ksi) – – ≤31.7 ≤41.0 ≤50.3 ≤59.7 ≤69.1 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.0 θ (deg) – – – 49.3 47.0 45.3 43.9 42.7 41.4 40.3 39.3 38.4

fv,α (ksi) – – – ≤37.4 ≤46.9 ≤56.3 ≤65.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.5 θ (deg) – – – – 49.2 47.1 45.4 44.1 42.8 41.6 40.5 39.5

fv,α (ksi) – – – – ≤43.2 ≤52.7 ≤62.3 ≤71.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.0 θ (deg) – – – – – 49.0 47.1 45.6 44.2 42.9 41.7 40.7

fv,α (ksi) – – – – – ≤48.9 ≤58.6 ≤68.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.5 θ (deg) – – – – – – 48.9 47.2 45.7 44.3 43.0 41.9

fv,α (ksi) – – – – – – ≤54.7 ≤64.4 ≤74.1 ≤75.0 ≤75.0 ≤75.0

≤6.0 θ (deg) – – – – – – – 48.8 47.2 45.7 44.3 43.1

fv,α (ksi) – – – – – – – ≤60.4 ≤70.2 ≤75.0 ≤75.0 ≤75.0

≤6.5 θ (deg) – – – – – – – – 48.7 47.2 45.7 44.3

fv,α (ksi) – – – – – – – – ≤66.2 ≤75.0 ≤75.0 ≤75.0

≤7.0 θ (deg) – – – – – – – – – 48.6 47.1 45.6

fv,α (ksi) – – – – – – – – – ≤72.0 ≤75.0 ≤75.0

≤7.5 θ (deg) – – – – – – – – – – 48.5 47.0

fv,α (ksi) – – – – – – – – – – ≤75.0 ≤75.0

≤8.0 θ (deg) – – – – – – – – – – – 48.4

fv,α (ksi) – – – – – – – – – – – ≤75.0

Note: Values for θ and fv,α where εs is greater than γu εt,loc are not relevant. These combinations are indicated by cells 

containing the “–” symbol.
138

Table A2.3-4. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 2.0 percent.

εs × 

1,000

Parameter

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 θ (deg) 34.0 33.8 33.6 33.5 33.3 33.1 32.6 32.1 31.6 31.2 30.7 30.4

fv,α (ksi) ≤32.9 ≤41.6 ≤50.3 ≤59.0 ≤67.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤ −0.5 θ (deg) 35.6 35.2 34.8 34.6 34.3 34.2 33.5 33.0 32.4 32.0 31.5 31.1

fv,α (ksi) ≤31.7 ≤40.2 ≤48.9 ≤57.5 ≤66.2 ≤74.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.0 θ (deg) 37.4 36.7 36.2 35.8 35.4 35.1 34.5 33.9 33.3 32.8 32.3 31.9

fv,α (ksi) ≤30.3 ≤38.7 ≤47.3 ≤55.8 ≤64.5 ≤73.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.5 θ (deg) 39.3 38.4 37.6 37.1 36.6 36.2 35.6 34.9 34.3 33.7 33.1 32.7

fv,α (ksi) ≤28.6 ≤37.0 ≤45.5 ≤54.0 ≤62.6 ≤71.3 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.0 θ (deg) 41.6 40.2 39.2 38.4 37.8 37.3 36.7 35.9 35.2 34.6 34.0 33.5

fv,α (ksi) ≤26.8 ≤35.1 ≤43.5 ≤52.0 ≤60.6 ≤69.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.5 θ (deg) 44.2 42.3 41.0 40.0 39.2 38.5 37.9 37.1 36.3 35.6 34.9 34.3

fv,α (ksi) ≤24.7 ≤32.9 ≤41.3 ≤49.8 ≤58.4 ≤67.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.0 θ (deg) 47.1 44.6 42.9 41.6 40.6 39.8 39.1 38.2 37.4 36.6 35.9 35.2

fv,α (ksi) ≤22.2 ≤30.5 ≤38.9 ≤47.4 ≤56.0 ≤64.6 ≤73.3 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.5 θ (deg) 50.4 47.2 45.0 43.4 42.1 41.1 40.3 39.4 38.5 37.6 36.9 36.2

fv,α (ksi) ≤19.4 ≤27.8 ≤36.2 ≤44.8 ≤53.4 ≤62.1 ≤70.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.0 θ (deg) – 50.1 47.3 45.3 43.8 42.6 41.6 40.7 39.7 38.7 37.9 37.1

fv,α (ksi) – ≤24.8 ≤33.3 ≤42.0 ≤50.6 ≤59.3 ≤68.1 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.5 θ (deg) – – 49.9 47.4 45.6 44.1 43.0 42.0 40.9 39.9 39.0 38.2

fv,α (ksi) – – ≤30.1 ≤38.9 ≤47.6 ≤56.4 ≤65.2 ≤74.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.0 θ (deg) – – – 49.7 47.5 45.8 44.4 43.3 42.2 41.1 40.1 39.2

fv,α (ksi) – – – ≤35.5 ≤44.4 ≤53.2 ≤62.1 ≤71.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.5 θ (deg) – – – – 49.5 47.5 45.9 44.7 43.5 42.3 41.2 40.3

fv,α (ksi) – – – – ≤40.9 ≤49.9 ≤58.9 ≤67.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤5.0 θ (deg) – – – – – 49.4 47.6 46.1 44.9 43.6 42.4 41.4

fv,α (ksi) – – – – – ≤46.4 ≤55.4 ≤64.5 ≤73.5 ≤75.0 ≤75.0 ≤75.0

≤5.5 θ (deg) – – – – – – 49.3 47.6 46.2 44.9 43.7 42.5

fv,α (ksi) – – – – – – ≤51.8 ≤60.9 ≤70.0 ≤75.0 ≤75.0 ≤75.0

≤6.0 θ (deg) – – – – – – – 49.2 47.6 46.3 44.9 43.7

fv,α (ksi) – – – – – – – ≤57.2 ≤66.4 ≤75.0 ≤75.0 ≤75.0

≤6.5 θ (deg) – – – – – – – – 49.1 47.7 46.3 44.9

fv,α (ksi) – – – – – – – – ≤62.7 ≤71.9 ≤75.0 ≤75.0

≤7.0 θ (deg) – – – – – – – – – 49.1 47.6 46.2

fv,α (ksi) – – – – – – – – – ≤68.1 ≤75.0 ≤75.0

≤7.5 θ (deg) – – – – – – – – – – 49.0 47.5

fv,α (ksi) – – – – – – – – – – ≤73.6 ≤75.0

≤8.0 θ (deg) – – – – – – – – – – – 48.8

fv,α (ksi) – – – – – – – – – – – ≤75.0

Note: Values for θ and fv,α where εs is greater than γu εt,loc are not relevant. These combinations are indicated by cells 

containing the “–” symbol.
139

Table A2.3-5. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 2.5 percent.

εs × 

1,000

Parameter

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 θ (deg) 34.5 34.4 34.2 34.1 34.1 34.0 33.6 33.0 32.6 32.1 31.7 31.3

fv,α (ksi) ≤31.3 ≤39.5 ≤47.7 ≤56.0 ≤64.2 ≤72.5 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤ −0.5 θ (deg) 36.1 35.7 35.5 35.2 35.1 34.9 34.5 33.9 33.4 32.9 32.4 32.0

fv,α (ksi) ≤30.1 ≤38.2 ≤46.4 ≤54.5 ≤62.7 ≤70.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.0 θ (deg) 37.8 37.2 36.8 36.4 36.1 35.9 35.5 34.9 34.3 33.7 33.2 32.8

fv,α (ksi) ≤28.8 ≤36.8 ≤44.8 ≤52.9 ≤61.1 ≤69.3 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.5 θ (deg) 39.8 38.9 38.2 37.7 37.2 36.9 36.5 35.8 35.2 34.6 34.0 33.5

fv,α (ksi) ≤27.3 ≤35.1 ≤43.2 ≤51.2 ≤59.3 ≤67.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.0 θ (deg) 42.0 40.7 39.8 39.0 38.4 38.0 37.6 36.9 36.1 35.5 34.9 34.3

fv,α (ksi) ≤25.5 ≤33.3 ≤41.3 ≤49.3 ≤57.4 ≤65.5 ≤73.7 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.5 θ (deg) 44.5 42.7 41.5 40.5 39.7 39.1 38.6 37.9 37.1 36.4 35.8 35.2

fv,α (ksi) ≤23.5 ≤31.3 ≤39.2 ≤47.2 ≤55.3 ≤63.4 ≤71.6 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.0 θ (deg) 47.4 45.0 43.3 42.1 41.1 40.4 39.7 39.1 38.2 37.4 36.7 36.1

fv,α (ksi) ≤21.2 ≤29.0 ≤37.0 ≤45.0 ≤53.1 ≤61.2 ≤69.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.5 θ (deg) 50.7 47.5 45.4 43.8 42.6 41.7 40.9 40.3 39.3 38.4 37.7 37.0

fv,α (ksi) ≤18.5 ≤26.4 ≤34.4 ≤42.5 ≤50.7 ≤58.8 ≤67.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.0 θ (deg) – 50.4 47.6 45.7 44.2 43.1 42.1 41.4 40.4 39.5 38.7 37.9

fv,α (ksi) – ≤23.6 ≤31.7 ≤39.8 ≤48.0 ≤56.2 ≤64.5 ≤72.7 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.5 θ (deg) – – 50.1 47.7 46.0 44.6 43.5 42.6 41.6 40.6 39.7 38.9

fv,α (ksi) – – ≤28.7 ≤36.9 ≤45.2 ≤53.5 ≤61.8 ≤70.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.0 θ (deg) – – – 50.0 47.8 46.2 44.9 43.8 42.9 41.8 40.8 39.9

fv,α (ksi) – – – ≤33.8 ≤42.2 ≤50.5 ≤58.9 ≤67.2 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤4.5 θ (deg) – – – – 49.8 47.9 46.4 45.1 44.1 43.0 41.9 40.9

fv,α (ksi) – – – – ≤38.9 ≤47.4 ≤55.9 ≤64.3 ≤72.7 ≤75.0 ≤75.0 ≤75.0

≤5.0 θ (deg) – – – – – 49.7 47.9 46.5 45.3 44.2 43.1 42.0

fv,α (ksi) – – – – – ≤44.1 ≤52.6 ≤61.2 ≤69.6 ≤75.0 ≤75.0 ≤75.0

≤5.5 θ (deg) – – – – – – 49.6 48.0 46.7 45.5 44.3 43.2

fv,α (ksi) – – – – – – ≤49.2 ≤57.8 ≤66.4 ≤75.0 ≤75.0 ≤75.0

≤6.0 θ (deg) – – – – – – – 49.5 48.0 46.8 45.5 44.3

fv,α (ksi) – – – – – – – ≤54.4 ≤63.1 ≤71.7 ≤75.0 ≤75.0

≤6.5 θ (deg) – – – – – – – – 49.5 48.1 46.8 45.5

fv,α (ksi) – – – – – – – – ≤59.5 ≤68.3 ≤75.0 ≤75.0

≤7.0 θ (deg) – – – – – – – – – 49.4 48.1 46.7

fv,α (ksi) – – – – – – – – – ≤64.7 ≤73.5 ≤75.0

≤7.5 θ (deg) – – – – – – – – – – 49.4 48.0

fv,α (ksi) – – – – – – – – – – ≤69.9 ≤75.0

≤8.0 θ (deg) – – – – – – – – – – – 49.3

fv,α (ksi) – – – – – – – – – – – ≤75.0

Note: Values for θ and fv,α where εs is greater than γu εt,loc are not relevant. These combinations are indicated by cells 

containing the “–” symbol.
140

Table A2.3-6. Values of θ (degrees) and upper limit of fv,α (ksi) for sections with transverse 

reinforcement with ρv,α ≤ 3.0 percent.

εs ×

1,000

Parameter

γu εt,loc × 1,000

≥2.5 ≥3.0 ≥3.5 ≥4.0 ≥4.5 ≥5.0 ≥5.5 ≥6.0 ≥6.5 ≥7.0 ≥7.5 ≥8.0

≤ −1.0 θ (deg) 35.0 34.9 34.8 34.7 34.7 34.6 34.5 33.9 33.4 33.0 32.5 32.1

fv,α (ksi) ≤29.9 ≤37.6 ≤45.4 ≤53.2 ≤61.0 ≤68.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤ −0.5 θ (deg) 36.5 36.2 36.0 35.8 35.7 35.5 35.4 34.8 34.3 33.7 33.3 32.8

fv,α (ksi) ≤28.8 ≤36.4 ≤44.1 ≤51.9 ≤59.6 ≤67.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.0 θ (deg) 38.2 37.7 37.3 37.0 36.7 36.5 36.3 35.7 35.1 34.6 34.0 33.6

fv,α (ksi) ≤27.5 ≤35.0 ≤42.7 ≤50.3 ≤58.0 ≤65.8 ≤73.5 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤0.5 θ (deg) 40.2 39.3 38.7 38.2 37.8 37.5 37.2 36.7 36.0 35.4 34.8 34.3

fv,α (ksi) ≤26.0 ≤33.5 ≤41.1 ≤48.7 ≤56.4 ≤64.1 ≤71.8 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.0 θ (deg) 42.3 41.1 40.2 39.5 39.0 38.5 38.2 37.7 37.0 36.3 35.7 35.1

fv,α (ksi) ≤24.3 ≤31.8 ≤39.3 ≤46.9 ≤54.5 ≤62.2 ≤69.9 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤1.5 θ (deg) 44.8 43.1 41.9 41.0 40.2 39.7 39.2 38.7 37.9 37.2 36.6 35.9

fv,α (ksi) ≤22.4 ≤29.8 ≤37.3 ≤45.0 ≤52.6 ≤60.3 ≤68.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.0 θ (deg) 47.6 45.3 43.7 42.5 41.6 40.9 40.3 39.8 39.0 38.2 37.5 36.8

fv,α (ksi) ≤20.2 ≤27.7 ≤35.2 ≤42.8 ≤50.5 ≤58.2 ≤65.9 ≤73.6 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤2.5 θ (deg) 50.9 47.8 45.7 44.2 43.1 42.1 41.4 40.8 40.0 39.2 38.4 37.7

fv,α (ksi) ≤17.7 ≤25.2 ≤32.8 ≤40.5 ≤48.2 ≤55.9 ≤63.7 ≤71.4 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.0 θ (deg) – 50.6 47.9 46.1 44.6 43.5 42.6 41.9 41.2 40.2 39.4 38.6

fv,α (ksi) – ≤22.5 ≤30.2 ≤38.0 ≤45.7 ≤53.5 ≤61.3 ≤69.1 ≤75.0 ≤75.0 ≤75.0 ≤75.0

≤3.5 θ (deg) – – 50.4 48.1 46.3 45.0 43.9 43.0 42.3 41.3 40.4 39.6

fv,α (ksi) – – ≤27.4 ≤35.2 ≤43.1 ≤50.9 ≤58.7 ≤66.6 ≤74.4 ≤75.0 ≤75.0 ≤75.0

≤4.0 θ (deg) – – – 50.2 48.1 46.5 45.3 44.2 43.4 42.4 41.5 40.6

fv,α (ksi) – – – ≤32.3 ≤40.2 ≤48.1 ≤56.0 ≤63.9 ≤71.8 ≤75.0 ≤75.0 ≤75.0

≤4.5 θ (deg) – – – – 50.1 48.2 46.7 45.5 44.5 43.6 42.5 41.6

fv,α (ksi) – – – – ≤37.1 ≤45.2 ≤53.2 ≤61.1 ≤69.1 ≤75.0 ≤75.0 ≤75.0

≤5.0 θ (deg) – – – – – 50.0 48.3 46.9 45.8 44.8 43.7 42.6

fv,α (ksi) – – – – – ≤42.0 ≤50.1 ≤58.2 ≤66.2 ≤74.2 ≤75.0 ≤75.0

≤5.5 θ (deg) – – – – – – 49.9 48.3 47.0 46.0 44.9 43.7

fv,α (ksi) – – – – – – ≤46.9 ≤55.1 ≤63.2 ≤71.3 ≤75.0 ≤75.0

≤6.0 θ (deg) – – – – – – – 49.8 48.4 47.1 46.1 44.9

fv,α (ksi) – – – – – – – ≤51.8 ≤60.0 ≤68.2 ≤75.0 ≤75.0

≤6.5 θ (deg) – – – – – – – – 49.8 48.4 47.3 46.0

fv,α (ksi) – – – – – – – – ≤56.7 ≤65.0 ≤73.2 ≤75.0

≤7.0 θ (deg) – – – – – – – – – 49.7 48.4 47.2

fv,α (ksi) – – – – – – – – – ≤61.6 ≤69.9 ≤75.0

≤7.5 θ (deg) – – – – – – – – – – 49.7 48.5

fv,α (ksi) – – – – – – – – – – ≤66.5 ≤74.9

≤8.0 θ (deg) – – – – – – – – – – – 49.6

fv,α (ksi) – – – – – – – – – – – ≤71.4

Note: values for θ and fv,α where εs is greater than γu εt,loc are not relevant. These combinations are indicated by cells 

containing the “–” symbol.
141

APPENDIX B. ANALYSIS OF A RECTANGULAR, MILD STEEL REINFORCED 

UHPC BEAM

Appendix B is based on the content presented in Appendix A of this report.

143

LIST OF NOTATIONS

Act = area of UHPC on the flexural tension side of the member (in.2)

Ag = gross area of section (in.2)

Aps = area of prestressing steel on the flexural tension side of the member (in.2), as shown in 

AASHTO LRFD BDS Figure 5.7.3.4.2-1 (AASHTO 2020)

As = area of nonprestressed steel on the flexural tension side of the member at the section 

under consideration (in.2), as shown in AASHTO LRFD BDS Figure 5.7.3.4.2-1

(AASHTO 2020) (in.2)

As,bar = area of one nonprestressed steel bar (in.2)

Av = area of transverse reinforcement to resist shear within a distance s (in.2)

bv = effective web width taken as the minimum web width within the depth dv (in.)

bw = web width (in.)

c = distance from the extreme compression fiber to the neutral axis (in.)

cL = distance from the extreme compression fiber of the member to the neutral axis when 

the UHPC tensile strain limit, γuεt,loc, at extreme tension fiber is reached (in.)

db = nominal diameter of reinforcing bar (in.)

de = effective depth taken as taken as the distance, measured perpendicular to the neutral 

axis, between the extreme compression fiber to the centroid of the flexural tensile 

reinforcement (in.)

dv = effective shear depth taken as the distance, measured perpendicular to the neutral axis, 

between the resultants of the tensile and compressive forces due to flexure (in.)

Ec = modulus of elasticity of UHPC for use in design (ksi)

Es = modulus of elasticity of the mild steel reinforcement (ksi)

f ́c = compressive strength of UHPC for use in design (ksi)

fpo = a parameter taken as modulus of elasticity of prestressing steel multiplied by the 

locked-in difference in strain between the prestressing steel and the surrounding UHPC 

(ksi)

fpu = specified tensile strength of prestressing steel (ksi)

fsl = stress limit in steel at service loads after losses (ksi)

ft,cr = effective cracking strength of UHPC for use in design (ksi)

ft,loc = crack localization strength of UHPC for use in design (ksi)

fv = stress in the transverse steel reinforcement at nominal shear resistance (ksi)

fv,Check = value of fv at the end of an iteration step during analysis of nominal shear resistance 

(ksi)

fy = specified minimum yield strength of reinforcement (ksi)
144

h = overall depth of the member (in.)

L = span length measured between center of supports (in.)

M = nominal flexural moment (kip-ft)

Mc = nominal crushing moment (kip-ft)

Mcr = nominal cracking moment (kip-ft)

ML = nominal crack localization moment (kip-ft)

Mn = nominal flexural resistance (kip-ft)

Mr = factored flexural resistance (kip-ft)

Msl = nominal flexural moment when the steel stress in the extreme tension steel is equal to 

the steel service stress limit, fsl (kip-ft)

Msu = nominal flexural moment at the rupture of tension steel (kip-ft)

Msy = nominal flexural moment when the steel stress in the extreme tension steel is equal to 

the steel yield strength, fsy (kip-ft)

Mu = factored moment at the section (kip-ft)

Nu = factored axial force, taken as positive if tensile and negative if compressive (kip)

s = spacing of transverse reinforcement measured in a direction parallel to the longitudinal 

reinforcement (in.)

Vn = nominal shear resistance (kip)

Vn,max = upper limit on factored shear force at critical shear section (kip)

Vp = component of prestressing force in the direction of the shear force (kip)

Vs = shear resistance provided by transverse reinforcement (kip)

Vu = factored shear force (kip)

VUHPC = nominal shear resistance of the UHPC (kip)

Vu1 = factored shear force at a distance dv from face of support (kip)

Vu,max = maximum factored shear force applied on the beam (kip)

wu,max = maximum factored applied load that the beam can carry (kip/ft)

ybs = distance between centroid of steel reinforcement and extreme tension fiber of the beam 

(in.)

α = angle of inclination of transverse reinforcement to longitudinal axis (degrees)

αu = reduction factor to account for the nonlinearity of the UHPC compressive stress-strain 

response

γu = factor to allow for the reduction of UHPC tensile parameter values; it shall not be taken 

greater than 1.0

ε2 = strain in the UHPC diagonal compressive strut (in./in.)
145

εc = compressive strain in extreme compression fiber of the UHPC section (in./in.)

εcp = elastic compressive strain limit of UHPC (in./in.)

εcu = ultimate compressive strain of UHPC for use in design (in./in.)

εs = net longitudinal tensile strain in the section at the centroid of the tension reinforcement 

(in./in.)

εsu = ultimate tensile strain capacity of steel reinforcement

εsl = net tensile strain in the extreme tension steel when the steel service stress limit, fsl, is 

reached (in./in.)

εt = net tensile strain in extreme tension fiber of the UHPC section (in./in.)

εt,cr = elastic tensile strain limit of UHPC corresponding to a tensile stress of γuft,cr (in./in.)

εt,loc = crack localization strain of UHPC for use in design (in./in.)

εv = strain in the transverse steel reinforcement at nominal shear resistance (in./in.)

εy = strain in the steel reinforcement corresponding to fy (in./in.)

θ = angle of inclination of diagonal compressive stresses (degrees)

μ = curvature ductility ratio

μl = curvature ductility ratio limit

μsl = curvature ductility ratio at steel service limit

ρv = ratio of area of transverse shear reinforcement to area of gross UHPC area of a 

horizontal section

 = resistance factor

f = resistance factor for flexure

ψ = sectional curvature at nominal flexural moment, M (1/in.)

ψc = sectional curvature at nominal crushing moment, Mc (1/in.)

ψcr = sectional curvature at nominal cracking moment, Mcr (1/in.)

ψL = sectional curvature at nominal crack localization moment, ML (1/in.)

ψn = sectional curvature at nominal flexural strength (1/in.)

ψsl = baseline sectional curvature at Msl (1/in.)

ψsy = sectional curvature at nominal steel yielding moment, Msy (1/in.)

ωc = unit weight of UHPC

147

CHAPTER B1. INTRODUCTION

The example in this Appendix illustrates the flexural and shear analysis of a beam made of 

ultra-high performance concrete (UHPC). The beam is rectangular in shape with a width of 12 in. 

and a height of 24 in.; it is simply supported over a span of 30 ft (center-to-center of supports) and 

subjected to a uniformly distributed total factored load of 4.55 kip/ft (including self-weight). The 

width of each support is 1 ft. The beam is reinforced with one layer of three No. 11 Grade 60 steel 

reinforcing bars (ASTM A615/A615M (ASTM 2020a)) in the longitudinal direction and No. 4 

Grade 60 steel two-legged stirrups at 6-in. spacing, as shown in Figure B1-1. This analysis 

example explains in detail the flexural behavior of the reinforced UHPC beam at cracking, yield, 

crack localization, and compression failure, and presents the shear capacity calculations at one 

location within the beam’s span. The analysis is accomplished in accordance with the AASHTO 

LRFD Bridge Design Specifications, 9th edition (AASHTO 2020), hereafter referred to “AASHTO 

LRFD BDS,” and Appendix A of this document, entitled, “Guide Specification for Structural 

Design with Ultra-High Performance Concrete.” Appendix A, hereafter referred to as the “UHPC 

Guide,” has been proposed to AASHTO T-10 for their consideration by the Federal Highway 

Administration’s Turner-Fairbank Highway Research Center.

Source: FHWA.

Figure B1-1. Illustration. Cross-section detail.

149 

CHAPTER B2. MATERIALS

The material properties for the UHPC in the beam are as follows: 

• Compressive strength for use in design, f ́c = 22.0 ksi. 

• Ultimate compressive strain for use in design, εcu = 0.0035. 

• UHPC unit weight, ωc = 0.155 kip/ft3. 

• Modulus of elasticity for use in design, Ec = 2,500(f ́c)

0.33 = 6,933 ksi. 

[UHPC Guide Article 1.4.2.3] 

• Cracking strength for use in design, ft,cr = 1.00 ksi. 

• Crack localization stress for use in design, ft,loc = 1.00 ksi. 

• Ultimate tension strain for use in design, εt,loc = 0.003. 

• Reduction factor on tensile parameters, γu = 1.00. 

The material properties for the steel reinforcement are as follows: 

• Modulus of elasticity, Es = 29,000 ksi. 

• Yield strength, fy = 60 ksi. 

• Yield strain for use in design, εy = 0.00207. 

• Service stress limit, fsl = 0.80fy = 48 ksi. [UHPC Guide Article 1.5.2] 

• Ultimate strain for use in design, εsu = 0.09. [ASTM A615/A615M (ASTM 2020a)] 

• No. 11 bar diameter, db = 1.41 in. 

• No. 11 bar area, As,bar = 1.56 in.2. 

• No. 4 bar area, As,bar = 0.20 in.2. 

151

CHAPTER B3. FLEXURAL ANALYSIS

According to UHPC Guide Article 1.6.3.1, the flexural behavior of UHPC beams shall be 

determined from a strain compatibility approach by employing representative stress-strain models 

to determine the stresses based on strains in any cross-sectional layer of UHPC or steel 

reinforcement.

The nominal flexural moment is taken as the moment corresponding to the lesser of (1) the 

sectional curvature values calculated when the compressive strain at the extreme compression fiber 

of the UHPC section is equal to the compression strain limit, i.e., εcu = 0.0035; (2) the net tensile 

strain at extreme tension fiber of the UHPC section is equal to the UHPC tensile strain limit, i.e.,

γuεt,loc = 0.003; and (3) the strain in the extreme tension steel is equal to the minimum total 

elongation strain of the reinforcing steel, i.e., εsu = 0.09 [UHPC Guide Article 1.6.3.2.2].

The resistance factor for flexural capacity is based on a sectional curvature ductility ratio, where a 

minimum ratio, μsl = ψn/ψsl, of 3.0 is specified in UHPC Guide Article 1.6.2. The baseline sectional 

ductility is calculated at equilibrium when the stress in the extreme layer of steel is equal to the 

steel service limit stress, fsl = 48 ksi, which corresponds to a strain, εsl, of εsl = fsl/Es = 0.00166. 

The sectional curvature at steel service stress limit, ψsl, shall also be calculated from a strain 

compatibility analysis [UHPC Guide Article 1.6.3.1].

In many cases, the nominal capacity of a UHPC beam occurs when the strain in the extreme tension 

layer of UHPC reaches the UHPC tensile strain limit, i.e., εt = γuεt,loc, while the strain in the extreme 

compression layer, εc, is less than the compression strain limit, εcu = 0.0035. Therefore, a good 

practice is to first determine the moment at crack localization from a strain compatibility analysis 

and assess the state of strain in the section compared with the strain limits of each material. If none 

of the strains in UHPC and steel layers is greater than its respective strain limit, then the nominal 

moment, Mn, is the crack localization moment, ML. The next step would be to determine the 

sectional curvature, ψsl, when the extreme steel layer strain reaches εsl from a second run of the 

strain compatibility analysis. ψsl will be used to determine the flexural resistance factor, as 

described in UHPC Guide Article 1.5.4.2.

Note that the case where the compression strain limit is reached before the tensile strain limit of 

UHPC, γuεt,loc, may generally occur in compression members or heavily reinforced beams with a 

shallow compression zone and/or made from a UHPC material with a high localization strain 

capacity. The third strain limit described in UHPC Guide Article 1.6.3.2.2 pertains to the rupture 

of tensile reinforcement and may generally occur in heavily prestressed members.

In the analysis of the rectangular beam described in Chapter B1 of this Appendix, a full strain 

compatibility analysis is performed in accordance with UHPC Guide Article 1.6.3.1. The objective 

is to illustrate the full moment-curvature diagram of the beam and better demonstrate the flexural 

behavior under various loading conditions. 
152

B3.1. STRAIN COMPATIBILITY ANALYSIS

The strain compatibility analysis of the beam shown in Figure B1-1 was initiated by discretizing 

the beam’s cross section into 0.1-in.-thick horizontal layers through the height and assuming 

constant strains within each layer. In calculating the moment corresponding to each possible strain 

profile, various neutral axis depths were assumed for a chosen extreme compression strain value,

εc, until equilibrium of forces was established, at which point the flexural bending moment at this 

value was calculated. The process was repeated for 110 values of εc ranging from 0 to εcu = 0.0035.

B3.1.1. Material Models for UHPC and Reinforcing Steel

In calculating the moment and forces at equilibrium, the compression and tension stress-strain 

models specified in UHPC Guide Articles 1.4.2.4.3 and 1.4.2.5.4 for UHPC were employed, 

respectively, as shown in Figure B3.1.1-1-A and Figure B3.1.1-1-B.

Source: FHWA.

A. UHPC in compression.

Source: FHWA.

B. UHPC in tension.

Source: FHWA.

C. Conventional steel bars in 

tension.

Figure B3.1.1-1. Graphs. Idealized uniaxial stress-strain relationships.

The idealized uniaxial stress-strain model of UHPC under compression is defined by the modulus 

of elasticity of UHPC, Ec = 6,933 ksi, a reduced value of the compressive strength, αu f ́c = 18.7 

ksi, where αu = 0.85 [UHPC Guide Article 1.4.2.4.2], the elastic compressive strain limit, εcp = αu

f ́c/Ec = 0.00270, and the ultimate compressive strain of UHPC, εcu = 0.0035 [UHPC Guide Article 

1.4.2.4.2]. Note that αu is a reduction factor to account for the nonlinearity of the UHPC 

compressive stress-strain response.

The idealized elastic, perfectly plastic uniaxial stress-strain model of UHPC defined in UHPC 

Guide Figure 1.4.2.5.4-1 is employed in this analysis since the UHPC material used to make the 

beam has an effective cracking strength, ft,cr, equal to or less than 1.2ft,loc (i.e., ft,cr = ft,loc = 1.00 ksi) 

[UHPC Guide Article 1.4.2.5.4]. The tensile stress-strain model of Figure B3.1.1-1-B is defined 

by the modulus of elasticity of UHPC, Ec = 6,933 ksi, the effective cracking and crack localization 

strength limits, γu ft,cr = γu ft,loc = 1.00 ksi, the cracking strain, εt,cr = γu ft,cr/Ec = 0.000144, and the 

and the tensile strain limit, γu ft,loc = 0.003. Note that γu = 1.00.
153

For the tensile steel reinforcement, an elastic, perfectly plastic stress-strain model was 

implemented as shown in Figure B3.1.1-1-C, where Es = 29,000 ksi, fsy = 60 ksi, εsy = 0.00207, 

and εsu = 0.09.

B3.2. MOMENT CURVATURE DIAGRAM

The flexural moment-sectional curvature (M − ψ) diagram obtained from strain compatibility is 

shown in Figure B3.2-1 and includes the intermediate and final behaviors of the beam, such as 

first cracking, service steel stress limit, yielding of reinforcement, UHPC crack localization, and 

UHPC crushing. Note that the M − ψ diagram of Figure B3.2-1 includes the moments and 

curvatures corresponding to strain profiles in which the crack localization strain of UHPC is 

exceeded. This information is provided to illustrate sectional behaviors after crack localization.

Source: FHWA.

Figure B3.2-1. Graph. Flexural moment-sectional curvature ( – ψ) diagram of the UHPC 

cross section.

The first key point (Mcr, ψcr) in the M − ψ diagram of the UHPC section represents the initiation 

of cracking, which occurs when the strain in the extreme tension fiber of the section reaches the 

UHPC effective cracking strain limit, εt,cr = γu ft,cr/Ec = 0.000144 (tension). At this point, the 

compressive and tensile stresses of the UHPC and steel remained within the elastic region of their 

respective constitutive models. The strain and stress profiles obtained at the end of this analysis 

are shown in Figure B3.2-2. The details of the results are presented in Table B3.2-1.
154

Source: FHWA.

Figure B3.2-2. Illustration. Strain and stress profiles at first crack in UHPC.

Table B3.2-1. Summary of the moment-curvature analysis.

Parameter

(Mcr, ψcr)

First Crack

(Msl/ψsl)

Steel Service

(Msy, ψsy)

Steel Yield

(ML, ψL)

Localization

(Mc, ψc)

Crushing

M (kip-ft) 109.8 548.8 642.8 655.3 496.5

c (in.) 12.48 8.69 8.43 7.78 2.20

εc 0.000156 0.00110 0.00130 0.00144 εcu = 0.0035

εt εt,cr = 0.000144 0.00193 0.00241 γuεt,loc = 0.00300 0.0348

εs 0.000117 εsl = 0.00166 εsy = 0.00207 0.00259 0.0312

ψ (1/in.) × 

10−3

0.0125 0.127 0.154 0.185 1.591

Note: See List of Notations for variable definitions.

The second point (Msl/ψsl) shown in the M – ψ diagram of Figure B3.2-1 corresponds to the 

baseline sectional curvature for ductility calculations in which the extreme tension layer of the 

steel reinforcement reached the steel service stress and strain limits, εsl and fsl, of 0.00166 and 

48 ksi, respectively. The strain and stress profiles obtained at the end of this analysis are shown in 

Figure B3.2-3. At this point, the compressive stresses and strains remained within the elastic 

portions of their constitutive law, while the tensile strains reached a strain of 0.00193 within the 

plastic zone. The details of the results are presented in Table B3.2-1 in which the sectional 

curvature was calculated as follows: ψsl = 0.00110/8.69 = 0.127 × 10−3[UHPC Guide Eq. 

1.6.3.2.3-2].
155

Source: FHWA.

Figure B3.2-3. Illustration. Strain and stress profiles when the steel reinforcement reaches 

steel service limit.

The third point (Msy, ψsy) corresponds to the yielding of reinforcement when the strain in the 

extreme tension steel layer reaches the steel yield strain, εy = 0.00207.

Source: FHWA.

Figure B3.2-4. Illustration. Strain and stress profiles at yielding of steel reinforcement.

The fourth point (ML, ψL) shown in the M – ψ diagram of Figure B3.2-1 corresponds to the UHPC 

crack localization. At this point, the tensile contribution of the UHPC is exhausted and the strain 

in the extreme tension fiber of the section is equal to the UHPC tensile strain limit, γuεt,loc = 0.003. 

The strain and stress profiles obtained at the end of this analysis are shown in Figure B3.2-5, and 

the details of the results are presented in Table B3.2-1. The moment corresponding to this point 

shall be taken as the nominal moment capacity of the section, i.e., Mn = ML = 655.3 kip-ft, because 

the tensile strain limit of UHPC was reached before the strains in the steel or the compressed 

UHPC reached their ultimate limits (i.e., εs = 0.00259 < εsu = 0.09 and εc = 0.00144 < εcu = 0.0035

[UHPC Guide Article 1.6.3.2.2]. The nominal curvature ductility can be calculated as follows:

ψn = ψL = 0.00144/7.78 = 0.185×10−3[UHPC Guide Eq. 1.6.3.2.3-3]
156

Source: FHWA.

Figure B3.2-5. Illustration. Strain and stress profiles at UHPC crack localization.

The fifth and final point (Mc, ψc) shown in the M – ψ diagram of Figure B3.2-1 corresponds to the 

crushing of the UHPC and is calculated herein for illustration purposes. The crushing of UHPC in 

compression occurs when the strain in the extreme compression fiber of the section reaches the 

ultimate compression strain limit of UHPC, εcu = 0.0035. The portion of the UHPC within the 

tension zone with a strain lower than the UHPC localization strain is assumed to resist tension, as 

shown in Figure B3.2-6. Note that for all points in the M – ψ diagram of Figure B3.2-1 calculated 

after the localization of cracks in UHPC (i.e., all points with sectional curvature greater than the 

sectional curvature at UHPC crack localization, ψ > ψL), the sectional behavior violates UHPC 

Guide Article 1.6.3.2.2, because the strain in the UHPC at the extreme tension layer at each of 

these points is greater than the UHPC crack localization capacity. The flexural behavior after 

UHPC crack localization is not recommended to be considered in the design, because the loss of 

UHPC fiber bridging capacity may result in the hinging of the beam at the localized crack section 

and the straining of the tensile reinforcement over a short distance, increasing the risk of 

reinforcement rupture [UHPC Guide Article C1.6.3.2.2]. Note that the hinging and local straining 

of the tensile reinforcement might occur in cases where the calculated postcrack localization 

moment (or moment at crushing or rupture of tension steel) is greater than the localization moment 

capacity (i.e., Mc > ML or Msu > ML). Regardless of the moment capacity calculated after UHPC 

crack localization, the provisions of Article 1.6.3.2.2 limit the flexural capacity to the moment 

calculated at the UHPC crack localization. 

Source: FHWA.

Figure B3.2-6. Illustration. Strain and stress profiles at the crushing of UHPC in 

compression.
157

B3.3. FACTORED FLEXURAL RESISTANCE 

The factored flexural resistance shall be taken as: 

Mr = φMn [UHPC Guide Eq. 1.6.3.2.1-1] 

where: 

Mn = nominal flexural resistance (kip-ft.) 

φ = resistance factor specified in UHPC Guide Article 1.5.4.2 

The resistance factor for flexural capacity is based on a ductility performance threshold expressed 

in terms of a curvature ductility ratio. [UHPC Guide Article 1.5.4.2] 

The curvature ductility ratio is defined as: 

[UHPC Guide Eq. 1.6.3.2.3-1] 

where: 

ψn = sectional curvature at nominal flexural strength 

ψsl = sectional curvature when the steel stress in the extreme tension steel is equal to the steel 

service stress limit, fsl 

in which fsl = stress limit in reinforcing steel = 0.80fsy = 48 ksi. 

The resistance factor can be calculated in accordance with UHPC Guide Articles 1.5.4.2 and 

1.6.3.2.3: 

Ultimate moment: Mn = ML = 659.0 kip-ft

Curvature ductility ratio limit: μl = 3.0 [UHPC Guide Article 1.6.2] 

Curvature ductility ratio: 

Resistance factor: 

Therefore: 

φf Mn = φf ML = (0.785)(655.3) = 514.4 kip-ft
158

B3.4. FLEXURE DEMAND VS. CAPACITY CHECK

The simply supported beam in this example is subjected to a uniformly distributed total factored 

load of 4.55 kip/ft (including self-weight) over a span of 30 ft. Therefore, the maximum total

factored moment the beam is subjected is calculated at midspan as follows:

Mu,max = wu L

2

/8 = 4.55 (30)2/8 = 511.9 kip-ft

Since φMn = 514.4 kip-ft is less than the maximum applied moment Mu,max = 511.9 kip-ft, the beam

passes the ultimate limit state check in flexure.
159

CHAPTER B4. SHEAR ANALYSIS 

The area and spacing of shear reinforcement must be determined at regular intervals along the 

length of the beam. Transverse reinforcement shall be provided where: 

Vu > φ(VUHPC + Vp) [UHPC Guide Eq. 1.7.2.3-1] 

where: 

Vu = factored shear force 

φ = resistance factor = 0.9 [UHPC Guide Article 1.5.4.2] 

VUHPC = nominal shear resistance of the UHPC 

Vp = component of prestressing force in the direction of shear force; positive if resisting 

the applied shear 

Vp = 0 kips for this nonprestressed example 

or where consideration of torsion is required. 

In this example, torsion is not being considered, and the ultimate limit state check in shear is only 

performed at the critical section for shear located at a distance equal to the effective shear depth, 

dv, from the internal face of support [UHPC Guide Article 1.7.3.2]. Note that the factored shear 

capacity must be verified not to exceed the factored demand force at each section within the length 

of the beam. 

B4.1. CONTRIBUTION OF UHPC TO SHEAR RESISTANCE 

The effective shear depth, dv, is taken as the distance, measured perpendicular to the neutral axis, 

between the resultants of the tensile and compressive forces due to flexure. It need not be taken to 

be less than the greater of 0.9de or 0.72h [AASHTO LRFD BDS Article 5.7.2.8]. In this design 

example, it is taken at the specified maximum limit shown as follows: 

de = h – ybs = 24 − 2.205 = 21.795 in.

dv = max(0.9de, 0.72h) = 19.62 in.=1.64 ft 

For a simply supported span of 30 ft subjected to 4.55 kip/ft uniformly distrusted factored load: 

Factored shear at the end of the beam: 

Vu,max = wu L/2 = 4.55(30)/2 = 68.25 kip 

Factored shear demand at the critical shear section (distance dv from internal face of support 

with a support width, Wsup = 1 ft): 

 
160

Factored moment at critical shear section: 

Mu = Vu1 (dv + (wsup/2)) = 125.0 kip-ft

The net longitudinal tensile strain at the centroid of tensile reinforcement at the critical shear 

section is determined by: 

[UHPC Guide Eq. 1.7.3.4.1-6]

where: 

Nu = factored axial force = 0 kip 

fpo = parameter equal to modulus of elasticity multiplied by the locked-in difference in 

strain and surrounding concrete 

= 0.7fpu = 0 ksi for a nonprestressed beam 

Act = area of UHPC on the flexural tension side of the member 

= (Ag/2) – As = (12(24)/2) – 3(1.56) = 139.32 in.2

Aps = area of prestressing steel on the flexural tension side of the member 

= 0 in.2

As = area of nonprestressed steel on the flexural tension side of the member 

= 33(1.56) = 4.68 in.2

Therefore: 

Since the value of εs calculated from UHPC Guide Eq. 1.7.3.4.1-6, is less than εt,cr = 0.000144, 

Eq. 1.7.3.4.1-7 of the UHPC Guide shall be used to determine εs: 

[UHPC Guide Eq. 1.7.3.4.1-7] 

 

 
161

B4.2. CALCULATION OF INCLINATION ANGLE AND STRESS IN TRANSVERSE 

STEEL 

B4.2.1. General Approach 

The general approach for determining shear resistance parameters, θ and fv, involves solving 

UHPC Guide Eqs. 1.7.3.4.1-1 through 1.7.3.4.1-4 iteratively. For sections with transverse steel 

reinforcement perpendicular to the longitudinal axis (α = 90 degrees), Eqs. 1.7.3.4.1-1 and 

1.7.3.4.1-2 reduce to: 

[UHPC Guide Eq. C1.7.3.4.1-3] 

[UHPC Guide Eq. C1.7.3.4.1-4]

Strain in vertical steel: εv = γuεt,loc − 0.5εs + ε2 [UHPC Guide Eq. 1.7.3.4.1-3] 

Stress in vertical steel: fv = Es εv ≤ fy [UHPC Guide Eq. 1.7.3.4.1-4] 

in which: [UHPC Guide Eq. 1.7.3.4.1-5] 

For this design example, the beam is reinforced with two legs of No. 4 stirrups at a spacing of 

6 in.: 

Spacing of transverse reinforcement s = 6 in. 

Area of transverse reinforcement Av = 2(0.20) = 0.40 in.2

Yield stress of transverse steel fy = 60 ksi 

Modulus of elasticity of transverse steel Es = 29,000 ksi 

Shear reinforcement ratio ρv = 0.40/(12(6)) = 0.0056 = 0.56 percent 

The solution to Eq. C1.7.3.4.1-3 shown in the preceding equations can be solved by setting 

εt,loc = 0.0030 and assuming a value of fv. Eqs. C1.7.3.4.1-4, 1.7.3.4.1-3, and 1.7.3.4.1-4 are then 

evaluated using the assumed value of fv and the calculated value of θ. This process is repeated until 

the assumed and calculated value of fv converges. 

Trial 1: 

Assume fv = fy = 60 ksi. 

By solving Eq. C1.7.3.4.1-3 with fv = 60 ksi, the inclination angle θ is determined to be 

31.98 degrees. 

Using θ = 31.98 degrees and fv = fy = 60 ksi: 

  
162

εv = γuεt,loc − 0.5εs + ε2 = 0.00186 

Check fv: fv,Check = min(Es εv, fy) = 53.8 ksi. Repeat with fv = 53.8 ksi. 

Trial 2: 

Assume fv = 53.8 ksi. 

By solving Eq. C1.7.3.4.1-3 with fv = 53.8 ksi, the inclination angle θ is determined to be 

31.75 degrees. 

Using θ = 31.75 degrees and fv = 53.8 ksi: 

εv = γuεt,loc − 0.5εs + ε2 = 0.00187

Check fv: fv,Check = min(Es εv, fy) = 54.3 ksi. Repeat with fv = 53.8 ksi. 

Trial 3: 

Assume fv = 54.3 ksi. 

By solving Eq. C1.7.3.4.1-3 with fv = 54.3 ksi, the inclination angle θ is determined to be 

31.76 degrees. 

Using θ = 31.76° and fv = 54.3 ksi: 

        
163

εv = γuεt,loc − 0.5εs + ε2 = 0.00187

Check fv: fv,Check = min(Es εv, fy) = 54.3 ksi. 

OK.

The angle of inclination of diagonal compressive stresses and the stress in the steel at nominal 

flexural resistance are: 

fv = 54.3 ksi and θ = 31.8 degrees 

B4.2.2. Simplified Approach 

The simplified approach detailed in UHPC Guide Article 1.7.3.4.2 was developed to provide a 

conservative alternative procedure for determining shear resistance parameters, θ and fv, without 

the need for iteratively solving Eqs. 1.7.3.4.1-1 through 1.7.3.4.1-4 [UHPC Guide Article 

C1.7.3.4.2]. The shear capacity determined from the simplified approach yields to a lower estimate 

of the shear capacity of the beam compared to the general approach. 

To use the simplified approach, the following should apply: 

• A UHPC modulus of elasticity: Ec ≥ 6,500 ksi: Ec = 6,933 ksi > 6,500 ksi. OK. 

• A UHPC localization stress: ft,loc ≤ 1.8 ksi: ft,loc = 1.00 ksi < 1.8 ksi. OK.

• A transverse steel yield stress: fy ≤ 75.0 ksi: fy = 60.0 ksi < 75.0 ksi. OK.

• A transverse steel reinforcement ratio: ρv ≤ 3.0 percent: ρv = 0.56 percent 

< 3.0 percent. OK.

[UHPC Guide Article 1.7.3.4.2] 

Therefore, the tables of Addendum A2 of the UHPC Guide may be used to estimate θ and fv.

Note that the beam is reinforced with vertical transverse bars, and thus, α=90° and no modification 

factor for ρv need be applied [UHPC Guide Article 1.7.3.4.2]. For ρv = 0.56 percent ≤ 1.0 percent, 

Table A2.3-2 of Addendum A2 of the UHPC Guide can be used to determine θ and fv.

For εs = 0.000123 ≤ 0.5 × 10−3and γuεt,loc = 0.003 ≥ 0.003, and using UHPC Guide Table A2.3-2, 

θ = 37.1° and fv ≤ 41.4 ksi (maximum stress lower than yield). Therefore, the shear capacity can 

be calculated with θ = 37.1° and fv = 41 ksi. Note that if the value of fv in the table is greater than 

fy, it must be taken equal to fy [UHPC Guide Article 1.7.3.4.2].

 
164

B4.3. TOTAL SHEAR RESISTANCE

UHPC shear resistance based on the shear resistance parameters calculated from the general 

method: 

VUHPC = γu ft,loc dv bv cot θ = 1.00(1.00)(19.62)(12)cot(31.8) = 379.6 kips 

[UHPC Guide Eq. 1.7.3.3-3] 

Transverse reinforcement shear contribution based on the shear resistance parameters calculated 

from the general method: 

[UHPC Guide Eq. 1.7.3.3-4] 

Total shear resistance: Vn = VUHPC + Vs = 494.1 kips [UHPC Guide Eq. 1.7.3.3-1] 

Factored shear resistance: φVn = (0.9)494.1 = 444.7 kips 

Maximum shear resistance: 

Vn,max = 0.25f ́c dv bv = 0.25(22)(19.62)(12) = 1,294.9 kips > Vn = 494.1 kip OK.

[UHPC Guide Eq. 1.7.3.3-2] 

Using the shear resistance parameters obtained from the simplified approach, the total factored 

shear resistance can be calculated as: 

B4.4. SHEAR DEMAND VS. CAPACITY CHECK 

The shear demand at the critical shear location (i.e., distance dv = 1.64 ft from internal face of 

support) is Vu = 58.5 kips. Given that φVn = 494.1 kips (or 344.6 kips if using the simplified 

approach) is greater than Vu = 58.5 kips, the beam can sustain the applied shear load at this location. 

The maximum shear spacing at the critical shear location, smax, can be calculated using UHPC 

Guide Eq. 1.7.2.6-1 as follows: 

Using θ = 31.8 degrees obtained from the general approach 

smax = 0.25dv cot θ = 0.25(19.62)cot(31.8) = 7.92 in. ≤ 24.0 in.
165

Using θ = 37.1 degrees obtained from the simplified approach 

smax = 0.25dv cot θ = 0.25(19.62)cot(37.1) = 6.48 in. ≤ 24.0 in. 

Given that the provided spacing of transverse reinforcement, s = 6 in., is less than smax = 7.92 in. 

(or 6.48 in. if using the simplified approach), the reinforcement spacing is compliant with the 

provisions of UHPC Guide Article 1.7.2.6.

Note that the shear demand, capacity calculations, and maximum spacing of transverse 

reinforcement checks presented herein are shown at one location for illustration purposes. To 

complete the ultimate limit state check in shear, the factored shear capacity and the maximum 

spacing of transverse reinforcement must be determined at regular intervals along the length of the 

beam and compared to the factored shear demand and the provided spacing at the same location, 

respectively. The factored shear capacity must be greater than the factored demand at each section 

along the length of the beam. The spacing of transverse reinforcement must also be lower than the 

provided spacing at each section.

167

APPENDIX C. DESIGN EXAMPLE OF A PRETENSIONED UHPC I-BEAM BRIDGE 

WITH A CONVENTIONAL CONCRETE DECK

Appendix C is based on the content presented in Appendix A of this report.

169

LIST OF NOTATIONS

A = area of a portion of the cross section (in.2)

Ac = total area of the composite section (in.2)

Act = area of UHPC on the flexural tension side of the member (in.2)

Acv = area of UHPC considered to be engaged in interface shear transfer (in.2)

Ad = area of conventional concrete deck (in.2)

Ag = gross area of cross section of the UHPC girder (in.2)

Ap = area of one prestressing strand (in.2)

Aps = total area of prestressing strands (in.2)

Aps,crs = total area of prestressing strands on the flexural tension side at critical shear location 

(in.2)

ApsE = total area of prestressing strands at girder ends (in.2)

As = area of nonprestressed reinforcement (in.2)

As,crs = area of nonprestressed steel on the flexural tension side of the member at critical 

shear location (in.2)

As,r = total area of pretensioned anchorage zone reinforcement located within the distance 

h/4 from the end of the beam (in.2)

Atc = area of composite transformed section at midspan at final time (in.2)

Atf = area of noncomposite transformed section at midspan at final time (in.2)

Ati = area of transformed section at midspan at time of prestressing (in.2)

AtiE = area of transformed section at girder ends at time of prestressing (in.2)

Av = area of transverse reinforcement to resist shear within a distance s (in.2)

Avf = area of interface reinforcement crossing the shear plane within the area Acv (in.2)

Avf,min = minimum required area of interface reinforcement crossing the shear plane within 

the area Acv (in.2)

a = distance from the girder end to the harp point of strands (in.)

be = effective flange width in composite girder system (in.)

bh = width of haunch (in.)

bv = effective web width taken as the minimum web width within the depth dv (in.)

bvi = interface width considered to be engaged in shear transfer (in.)

bw = web width (in.)

c = depth of the compression zone measured from the extreme compression fiber of the 

composite section at the completion of strain compatibility analysis (in.)
170

ci = cohesion factor for interface shear resistance calculation (ksi)

DC = dead load of structural components and nonstructural attachments

DFD = distribution factor for deflection

DFM = distribution factor for bending moment for interior girder

DFV = distribution factor for shear force for interior girder

DLws = dead load of future wearing surface (kip/in.2)

Dp = diameter of prestressing strand (in.)

DW = dead load of wearing surfaces and utilities

dcrack = horizontal distance from the girder end to the location where the critical shear crack 

plane crosses the centroid of the bottom strands (in.)

de = effective depth taken as the distance, measured perpendicular to the neutral axis, 

between the extreme compression fiber of the UHPC section to the resultant of the 

forces in the tensile reinforcement (in.)

dend = distance between the centroid of the strands in a row and the bottom of the section 

at girder ends (in.)

dmid = distance between the centroid of the strands in a row and the bottom of the section 

at midspan (in.)

dv = effective shear depth taken as the distance, measured perpendicular to the neutral 

axis, between the resultants of the tensile and compressive forces due to flexure (in.)

dvi = distance between the centroid of the tension steel and the midthickness of the deck 

(in.)

dv,limit = maximum value of dv in composite sections made of a UHPC girder and a 

conventional concrete deck (in.)

Ec = modulus of elasticity of UHPC for use in design (ksi)

EcD = modulus of elasticity of conventional concrete deck for use in design (ksi)

Eci = modulus of elasticity of UHPC at the time of prestressing for use in design (ksi)

Ep = modulus of elasticity of prestressing steel for use in design (ksi)

Es = modulus of elasticity of the mild steel reinforcement (ksi)

e′ = difference between eccentricity of the prestressing strand at midspan and at end of 

the girder

ed = eccentricity to centroid of the conventional concrete deck with respect to the gross 

composite section (in.)

eg = distance between the centers of gravity of the UHPC girder and the conventional 

concrete deck (in.)

epc = eccentricity of strands with respect to the centroid of composite section (in.)
171

epg = eccentricity of strands in noncomposite section at midspan (in.)

epgE = eccentricity of strands in noncomposite section at ends (in.)

etc = eccentricity of strands in composite transformed section at midspan at final time (in.)

etf = eccentricity of strands in noncomposite transformed section at midspan at final time 

(in.)

eti = eccentricity of strands in noncomposite transformed section at midspan at time of 

prestressing (in.)

etiE = eccentricity of strands in noncomposite transformed section at girder ends at time of 

prestressing (in.)

fbi = stress in the extreme bottom fiber of UHPC girder at prestress transfer at midspan 

(ksi)

fbiE = stress in the extreme bottom fiber of UHPC girder at prestress transfer at transfer 

length section (ksi)

f ′c = compressive strength of UHPC for use in design (ksi)

f ′cD = compressive strength of conventional concrete for use in design (ksi)

f ′cDi = compressive strength of conventional concrete deck at initial loading (ksi)

f ′cd = UHPC girder compressive strength at time of deck placement (ksi)

fcgp = sum of stresses at the center of gravity of prestressing strands due to prestressing 

force at transfer and the self-weight of the member at maximum moment sections 

(ksi)

fci = stress in conventional concrete layer as obtained by stress-strain constitutive model 

(ksi)

f ′ci = compressive strength of UHPC at time of prestressing for use in design (ksi)

fcpe = compressive stress in UHPC due to effective prestress forces only at extreme fiber 

section, where tensile stress is caused by externally applied loads

fin,1 = initial stress in conventional concrete at top of the deck at the start of strain 

compatibility analysis (ksi)

fin,2 = initial stress in conventional concrete at bottom of the haunch at the start of strain 

compatibility analysis (ksi)

fin,3 = initial stress in UHPC at top of the girder at the start of strain compatibility analysis 

(ksi)

fin,4 = initial stress in UHPC at bottom of the girder at the start of strain compatibility 

analysis (ksi)

fmin = minimum principal stress in the web with the tension taken as a negative value (ksi)

fpcx = horizontal stress in the web (ksi)

fpcy = vertical stress in the web (ksi)
172

fpe = stress in prestressing steel after all losses and permanent gains (ksi)

fpee = effective prestress in strands after allowance of all losses (ksi)

fpe,ng = stress in prestressing steel after all losses (no gains) (ksi)

fpeS = stress in prestressing steel at service after all losses and gains (ksi)

fpi = stress limit in prestressing steel at time of prestressing (ksi)

fpo = a parameter taken as modulus of elasticity of prestressing steel multiplied by the 

locked-in difference in strain between the prestressing steel and the surrounding 

UHPC (ksi)

fps = average stress in prestressing steel at the time for which the nominal resistance of 

the member is required (ksi)

fpsT,j = total stress in prestressing steel layer j at the completion of strain compatibility 

analysis (ksi)

fpt = stress in prestressing strands immediately after transfer (ksi)

fpu = specified tensile strength of prestressing steel (ksi)

fpy = specified yield strength of prestressing steel (ksi)

fsl = stress limit in prestressing steel at service loads after losses (ksi)

fs,r = stress in the interface steel reinforcement not to exceed 20.0 ksi (ksi)

fT,i = total stress in UHPC or conventional concrete layer i at the completion of strain 

compatibility analysis (ksi)

fti = stress in the extreme top fiber of the UHPC girder at prestress transfer at midspan 

(ksi)

ft,cr = effective cracking strength of UHPC for use in design (ksi)

ft,cri = effective cracking strength of UHPC at time of prestressing for use in design (ksi)

ftiE = stress in the extreme top fiber of the UHPC girder at prestress transfer at transfer 

length section (ksi)

ft,loc = crack localization strength of UHPC for use in design (ksi)

fv = stress in the transverse steel reinforcement at nominal shear resistance (ksi)

fv,Check = value of fv at the end of an iteration step during analysis of nominal shear resistance 

(ksi)

fv,max = maximum value of fv determined according to UHPC Guide Article 1.7.3.4.2 (ksi)

fy = specified minimum yield strength of reinforcement (ksi)

H = average annual ambient relative humidity (percent)

h = overall depth of the UHPC beam (in.)

hc = overall depth of the composite section (in.)

I = moment of inertia of a portion of the cross section (in.4)
173

Ic = moment of inertia of the composite section (in.4)

Ig = moment of inertia about the centroid of the noncomposite UHPC girder (in.4)

IM = dynamic allowance applied to the design truck load

Itc = composite transformed section moment of inertia at midspan at final time (in.4)

Ite = transformed flange moment of inertia (in.4)

Itf = noncomposite transformed section moment of inertia at midspan at final time (in.4)

Ith = transformed haunch moment of inertia (in.4)

Iti = noncomposite transformed section moment of inertia at midspan at time of 

prestressing (in.4)

ItiE = noncomposite transformed section moment of inertia at girder ends at time of 

prestressing (in.4)

K = limiting interface shear resistance (ksi)

Kdf = transformed section coefficient that accounts for time-dependent interaction 

between UHPC and bonded steel in the section considered for the time period 

between deck placement and final time

Kg = longitudinal stiffness parameter (in.4)

Kid = transformed section coefficient that accounts for time-dependent interaction 

between UHPC and bonded steel in the section considered for the time period 

between prestress transfer and deck placement

KL = factor accounting for the type of steel taken as 30 for low-relaxation strands and 7 

for other prestressing steel, unless more accurate manufacturer’s data are available

K1 = correction factor for modulus of elasticity of UHPC

K1D = correction factor for modulus of elasticity of conventional concrete

K3 = correction factor for creep of UHPC

K4 = correction factor for shrinkage of UHPC

k = factor for conventional concrete stress-strain constitutive model

kf = factor for the effect of UHPC strength

kf,D = factor for the effect of conventional concrete strength

khc = humidity factor for creep of UHPC

khc,D = humidity factor for creep of conventional concrete deck

khs = humidity factor for shrinkage of UHPC

khs,D = humidity factor for shrinkage of conventional concrete deck

kld = factor for the effect of loading age at time of deck placement

kli = factor for the effect of loading age at time of prestress transfer

ks = factor for the effect of the volume-to-surface ratio of the UHPC girder
174

ks,D = factor for the effect of the volume-to-surface ratio of the conventional concrete deck

ktdd = time development factor between prestress transfer and deck placement

ktdf = time development factor between prestress transfer and final time

ktd,D = time development factor for conventional concrete deck between deck placement 

and final time

L = design span of UHPC girder (ft)

LL = vehicular live load

LT = overall length of UHPC girder (ft)

Lvi = interface length considered to be engaged in shear transfer (in.)

ld = development length of the prestressing strand (in.)

lt = transfer length of the prestressing strand (in.)

Mb = bending moment due to barrier weight (kip-ft)

Mb,crs = bending moment at critical shear location due to barrier weight (kip-ft)

Mb,m = bending moment at midspan due to barrier weight (kip-ft)

Mcr = nominal cracking moment (kip-in.)

MD = bending moment due to deck and haunch weight (kip-ft)

MD,crs = bending moment at critical shear location due to deck and haunch weight (kip-ft)

MD,m = bending moment at midspan due to deck and haunch weight (kip-ft)

Mdnc = total unfactored dead load moment acting on the monolithic or noncomposite section 

(kip-in.)

MFI = moment due to Fatigue I load combination (kip-ft)

Mf = bending moment due to fatigue truck (kip-ft)

Mf,m = bending moment at midspan due to fatigue truck (kip-ft)

Mg = bending moment due to UHPC girder self-weight based on girder span length, L

(kip-ft)

Mg,crs = bending moment at critical shear location due to UHPC girder self-weight based on 

girder span length, L (kip-ft)

Mg,m = bending moment at midspan due to UHPC girder self-weight based on girder span 

length, L (kip-ft)

MgT,m = bending moment at midspan due to UHPC girder self-weight based on overall length 

of girder, LT (kip-ft)

MgT,tr = bending moment at transfer length section due to UHPC girder self-weight based on 

overall length of girder, LT (kip-ft)
175

ML = bending moment when the strain in the extreme tension layer of UHPC is equal to 

the UHPC tensile strain limit, γuεt,loc (kip-ft)

MLL = bending moment due to lane load (kip-ft)

MLL,crs = bending moment at critical shear location due to lane load (kip-ft)

MLL,m = bending moment at midspan due to lane load (kip-ft)

MLT = bending moment due to truck load (kip-ft)

MLT,crs = bending moment at critical shear location due to truck load (kip-ft)

MLT,m = bending moment at midspan due to truck load (kip-ft)

Mn = nominal flexural resistance (kip-in.)

Mr = factored flexural resistance (kip-in.)

MSIDL = bending moment at midspan due to superimposed dead load (kip-ft)

MSL = bending moment at midspan due to service load (kip-ft)

Msl = nominal flexural moment when the steel stress in the extreme tension layer is equal 

to the steel service stress limit, fsl (kip-in.)

Mu = factored bending moment at a location along the span of the beam (kip-in.)

Mu,crs = factored moment at the critical shear location due to dead load and live load (kip-in.)

Mu,m = factored moment at midspan due to dead load and live load (kip-in.)

Mws = bending moment due to wearing surface weight (kip-ft)

Mws,crs = bending moment at critical shear location due to wearing surface weight (kip-ft)

Mws,m = bending moment at midspan due to wearing surface weight (kip-ft)

Mx = bending moment of a simply supported beam at distance x along the span (kip-in.)

Nb = number of girders in the bridge

Nend = quantity of strands per row at girder ends

Nmid = quantity of strands per row at girder ends

Nu,crs = factored axial force at the critical shear location (kip)

n = modular ratio between UHPC and prestressing strands

nbD = modular ratio between UHPC and conventional concrete 

nc = factor for conventional concrete stress-strain constitutive model

nDb = modular ratio between conventional concrete and UHPC 

ntf = modular ratio modular ratio between UHPC and conventional concrete at final time

nti = modular ratio modular ratio between UHPC and conventional concrete at time of 

prestressing

Pc = permanent net compressive force, normal to the shear plane (kip)
176

Ppe = force in prestressing steel after all losses and permanent gains (ksi)

Ppee = force in prestressing steel due to effective prestress only (kips)

Ppi = total prestressing force before transfer at midspan (kip)

PpiE = total prestressing force before transfer at girder ends (kip)

Ppt = total prestressing force after transfer at midspan (kips)

Pr = factored splitting resistance of pretensioned anchorage zones (kip)

Pr,UHPC = splitting resistance of pretensioned anchorage zones provided by the UHPC (kip)

Q = total factored load for a particular load combination

Qg = first moment of area about the centroid of the gross concrete area above or below 

the height of the web where the principal tension is being calculated (in.3)

Qgc = first moment of area about the centroid of the composite nontransformed section 

above or below the height of the web where the principal tension is being calculated 

(in.3)

Qi = force effects from specified loads for case i

S = spacing of girders in a bridge (ft)

Sb = section modulus for the extreme bottom fiber of the noncomposite UHPC section 

(in.

3

)

Sbc = section modulus for the extreme bottom fiber of the composite section (in.3)

Sbtc = section modulus for the extreme bottom fiber of the composite transformed at 

midspan at final time (in.3)

Sbtf = section modulus for the extreme bottom fiber of the noncomposite transformed 

section at midspan at final time (in.3)

Sbti = section modulus for the extreme bottom fiber of the noncomposite transformed 

section at midspan at time of prestressing (in.3)

SbtiE = section modulus for the extreme bottom fiber of the noncomposite transformed 

section at girder ends at time of prestressing (in.3)

Sdtc = section modulus for extreme top fiber of the composite transformed section at 

midspan at final time (in.3)

St = section modulus for the extreme top fiber of the noncomposite UHPC section (in.3)

Stc = section modulus for the extreme top fiber of composite section (in.3)

Stg = section modulus for the extreme top fiber of the noncomposite UHPC girder (in.3)

Sttc = section modulus for extreme top fiber of the UHPC section in the composite 

transformed section at midspan at final time (in.3)

Sttf = section modulus for extreme top fiber of the noncomposite section at midspan at 

final time (in.3)
177

Stti = section modulus for extreme top fiber of the noncomposite section at midspan at 

time of prestressing (in.3)

SttiE = section modulus for extreme top fiber of the noncomposite section at girder ends at 

time of prestressing (in.3)

s = spacing of transverse reinforcement measured in a direction parallel to the 

longitudinal reinforcement (in.)

smax = maximum spacing of transverse reinforcement (in.)

Ts = total number of prestressing strands at midspan

TsE = total number of prestressing strands at girder ends

t = maturity of conventional concrete or UHPC, taken as the time being considered for 

creep or shrinkage effects (days)

tD = thickness of deck at time of placement (days)

td = UHPC age at time of deck placement (days)

tf = UHPC age at final time (days)

th = structural thickness of haunch (in.)

ti = age of UHPC at time of prestress transfer (days)

tiD = age of conventional concrete deck at time of loading (days)

ts = structural thickness of cast-in-place conventional concrete deck (in.)

V = shear force from Service III load combination (kips)

Vb = shear force due to barrier weight (kip)

Vb,crs = shear force at critical shear location due to barrier weight (kip)

VD = shear force due to deck and haunch weight (kip)

VD,crs = shear force at critical shear location due to deck and haunch weight (kip)

Vg = shear force due to girder weight based on girder span length, L (kip)

Vg1 = shear force applied to the UHPC gross section at the critical shear location at service 

(kip)

Vg2 = shear force applied to the composite nontransformed gross section at the critical 

shear location at service (kip)

Vg,crs = shear force at critical shear location due to girder weight based on girder span length, 

L (kip)

Vhi = horizontal factored shear force per unit length of girder (kip/in.)

VLL = shear force due to lane load (kip)

VLL,crs = shear force at critical shear location due to lane load (kip)

VLT = shear force due to truck load (kip)

VLT,crs = shear force at critical shear location due to truck load (kip)
178

Vn = nominal shear resistance (kip)

Vni = nominal interface shear resistance (kip)

Vp = component of prestressing force in the direction of the shear force (kip)

V/S = volume to surface ratio of the conventional concrete deck (in.)

Vs = shear resistance provided by transverse reinforcement (kip)

VUHPC = nominal shear resistance of the UHPC (kip)

Vu = factored shear force (kip)

Vu,crs = factored shear force at the critical shear section (kip)

Vui = horizontal factored shear force per unit length of the girder (kip)

Vu,max = maximum nominal shear resistance (kip)

Vws = shear due to wearing surface weight (kip)

Vws,crs = shear due to wearing surface weight (kip)

Vx = shear force of a simply supported beam at distance x along the span (kip)

WR = clear roadway width between curbs of a bridge (ft)

w = uniform distributed load (kip/ft)

wb = linear weight of the barrier (kip/ft)

wD = linear weight of the conventional concrete deck (kip/ft)

wdl = design lane load for live load deflection calculations per girder (kip/ft)

wg = linear weight of the noncomposite UHPC girder (kip/ft)

wh = linear weight of the conventional concrete haunch (kip/ft)

wp = nominal weight of 0.7-in.-diameter prestressing strand per unit length (kip/ft)

ws = total linear weight of conventional concrete deck and haunch (kip/ft)

wws = linear weight of future wearing surface (kip/ft)

x = distance along span of a simply supported beam where sectional analysis is 

completed (ft)

yb = distance from the centroid to the extreme bottom fiber of the UHPC girder (in.)

yb* = distance from the centroid of a portion of the cross section to the extreme bottom 

fiber of the composite section (in.)

ybc = distance from the centroid of the composite section to the extreme bottom fiber of 

the UHPC girder (in.)

ybcb = distance between the extreme tension fiber and the centroid of the strands on the 

flexural tension side at beam ends and within the debonded length (in.)
179

ybs = distance between the centroid of the strands and the extreme bottom fiber of the 

UHPC girder at midspan (in.)

ybsE = distance between the centroid of the strands and the extreme bottom fiber of the 

UHPC girder at ends (in.)

ybtc = distance from centroid of composite transformed section to extreme bottom fiber of 

composite section at midspan at final time (in.)

ybtf = distance from centroid of noncomposite transformed section to extreme bottom fiber 

of UHPC section at midspan at final time (in.)

ybti = distance from centroid of noncomposite transformed section to extreme bottom fiber 

of UHPC section at midspan at time of prestressing (in.)

ybtiE = distance from centroid of noncomposite transformed section to extreme bottom fiber 

of UHPC section at girder ends at time of prestressing (in.)

yt = distance from the centroid to the extreme top fiber of the UHPC girder (in.)

ytc = distance from the centroid of the composite section to the extreme top fiber of the 

structural deck (in.)

ytg = distance from the centroid of the composite section to the extreme top fiber of the 

UHPC section (in.)

yttc = distance from centroid of composite transformed section to extreme top fiber of 

composite section at midspan at final time (in.)

yttf = distance from centroid of noncomposite transformed section to extreme top fiber of 

UHPC section at midpan at final time (in.)

ytti = distance from centroid of noncomposite transformed section to extreme top fiber of 

UHPC section at midpan at time of prestressing (in.)

yttiE = distance from centroid of noncomposite transformed section to extreme top fiber of 

UHPC section at girder ends at time of prestressing (in.)

α = angle of inclination of transverse reinforcement to the longitudinal axis (degrees)

γ = load factor specified in AASHTO LRFD BDS Table 3.4.1-1 for Fatigue I load 

combination

γi = load factor for case i

γLL = load factor for live load for Service III load combination

γu = factor to allow for the reduction of UHPC tensile parameter values

γ1 = flexural cracking variability factor

γ2 = prestress variability factor

γ3 = steel yield to ultimate strength factor

∆b+ws = deflection due to barrier and future wearing surface weights (in.)

∆c = depth of the compression zone measured from the extreme compression fiber of the 

composite section during strain compatibility analysis (in.)
180 

∆D = deflection due to deck and haunch weights (in.) 

(∆F)TH = constant-amplitude fatigue threshold, as specified in AASHTO LRFD BDS Article

5.5.3.3 for prestressing steel (ksi) 

∆f = force effect, live load stress range at the bottom layer of strands due to the passage 

of the fatigue load as specified AASHTO LRFD BDS Article 3.6.1.4 (ksi) 

∆fbD = change in stress at bottom of UHPC girder due to conventional concrete deck weight 

(ksi)

∆fbLDF = change in stress at bottom of UHPC girder between deck placement and final time 

(ksi)

∆fbLTD = change in stress at bottom of UHPC girder between transfer and deck placement 

(ksi)

∆fbSIDL = change in stress at bottom of UHPC girder due to superimposed dead load (ksi) 

∆fbSL = change in stress at bottom of UHPC girder due to service load (ksi) 

∆fbSS = change in stress at bottom of the UHPC girder due to shrinkage of conventional 

concrete deck at final time (ksi)

∆fcD = change in stress in UHPC at the level of the centroid of prestressing steel due to 

conventional concrete weight (ksi) 

∆fcd = change in UHPC stress at the centroid of the prestressing strands due to long-term 

losses between transfer and deck placement, combined with deck weight and 

superimposed loads (ksi) 

∆fcdf = change in UHPC stress at the centroid of the prestressing strands due to shrinkage 

of the conventional concrete deck (ksi) 

∆fcSIDL = change in UHPC stress at level of the centroid of prestressing steel due to 

superimposed dead load (ksi) 

∆fcSL = change in stress in UHPC at the level of the centroid of prestressing steel due to the 

service load (ksi)

∆fdbLDF = change in stress at bottom of the conventional concrete deck between deck 

placement and final time (ksi)

∆fdbSIDL = change in stress at bottom of the concrete deck due to superimposed dead load (ksi) 

∆fdbSL = change in stress at bottom of conventional concrete deck due to the service load (ksi)

∆fdtLDF = change in stress at top of the concrete deck between deck placement and final time 

(ksi)

∆fdtSL = change in stress at top of conventional concrete deck due to the service load (ksi) 

∆fdtSIDL = change in stress at top of the concrete deck due to superimposed dead load (ksi) 

∆fdtSS = change in stress at top of conventional concrete deck due to shrinkage at final time 

(ksi)
181

∆fpCD = prestress loss due to creep of girder UHPC between the time of deck placement and 

final time (ksi)

∆fpCR = prestress loss due to creep of UHPC girder between the time of prestress transfer 

and deck placement (ksi)

∆fpD = prestress gain in prestressing steel due to conventional concrete deck weight (ksi)

∆fpES = sum of all prestress losses or gains due to elastic shortening or extension at the time 

of prestressing (ksi)

∆fpi = total prestress loss after transfer (ksi)

∆fpLDF = change in steel stress due to long-term effects between deck placement and final 

time, exclusive of deck shrinkage (ksi)

∆fpLT = long-term prestress losses due to shrinkage and creep of UHPC and conventional 

concrete, and relaxation of steel after transfer (ksi)

∆fpLTD = change in steel stress due to long-term effects between transfer and deck placement 

(ksi)

∆fpR1 = prestress loss due to relaxation of prestressing strands between prestress transfer and 

deck placement (ksi)

∆fpR2 = prestress loss due to relaxation of prestressing strands between deck placement and 

final time (ksi)

∆fpSD = prestress loss due to shrinkage of UHPC between time of deck placement and final 

time (ksi)

∆fpSIDL = prestress gain in steel due to superimposed dead load (ksi)

∆fpSL = prestress gain in steel due to the service load (ksi)

∆fpSR = prestress loss due to shrinkage of UHPC girder between the time of prestress transfer 

and deck placement

∆fpSS = prestress gain due to shrinkage of the conventional concrete deck between initial 

loading and final time (ksi)

∆fpT = total prestress loss in prestressing steel (ksi)

∆ftD = change in stress at top of UHPC girder due to conventional concrete deck weight 

(ksi)

∆ftLDF = change in stress at top of UHPC girder between deck placement and final time (ksi)

∆ftLTD = change in stress at top of UHPC girder between transfer and deck placement (ksi)

∆ftSIDL = change in stress at top of UHPC girder due to superimposed dead load (ksi)

∆ftSL = change in stress at top of UHPC girder due to service load (ksi)

∆ftSS = change in stress at top of the UHPC girder due to shrinkage of conventional concrete 

deck at final time (ksi)

∆g = deflection due to the beam self-weight (in.)
182

∆LL = deflection due to live load and impact (in.)

∆LT = deflection due to truck load (in.)

∆Pds = restraining force due to shrinkage of conventional concrete deck (kips)

∆PpLDF = change in steel force between deck placement and final time (ksi)

∆PpLTD = change in steel force between transfer and deck placement (kip)

∆p = camber due to prestressing force at transfer (in.)

∆εcs = change in strain in UHPC at centroid of prestressing steel during strain compatibility 

analysis (in./in.)

∆εcs,j = change in strain in prestressing steel layer j during strain compatibility analysis 

(in./in.)

∆εi = change in strain in UHPC or conventional concrete layer i during strain compatibility 

analysis (in./in.)

∆ε1 = change in strain in conventional concrete at top of the deck during strain 

compatibility analysis (in./in.)

∆ε2 = change in strain in conventional concrete at bottom of the haunch during strain 

compatibility analysis (in./in.)

∆ε3 = change in strain in UHPC at top of the girder during strain compatibility analysis 

(in./in.)

∆ε4 = change in strain in UHPC at bottom of the girder during strain compatibility analysis 

(in./in.)

εbdf = UHPC shrinkage strain for the period between deck placement and final time (in./in.)

εbid = UHPC shrinkage strain for the period between prestress transfer and deck placement 

(in./in.)

εbif = UHPC shrinkage strain for the period between prestress transfer and final time 

(in./in.)

ε′c = strain at peak strength for use conventional concrete stress-strain constitutive model

(in./in.)

εcsin,j = initial strain in UHPC at prestressing steel layer j at the start of strain compatibility 

analysis (in./in.)

εcsT,b = total strain in UHPC at extreme bottom prestressing steel layer at the completion of 

strain compatibility analysis (in./in.)

εcsT,i = total strain in UHPC at prestressing steel layer j at the completion of strain 

compatibility analysis (in./in.)

εcu = ultimate compressive strain of UHPC for use in design (in./in.)

εddf = shrinkage strain of conventional concrete deck for the time period between deck 

placement and final time (in./in.)
183

εin,i = initial strain in UHPC or conventional concrete layer i at the start of strain 

compatibility analysis (in./in.)

εin,1 = initial strain in conventional concrete at top of the deck at the start of strain 

compatibility analysis (in./in.)

εin,2 = initial strain in conventional concrete at bottom of the haunch at the start of strain 

compatibility analysis (in./in.)

εin,3 = initial strain in UHPC at top of girder at the start of strain compatibility analysis 

(in./in.)

εin,4 = initial strain in UHPC at bottom of girder at the start of strain compatibility analysis 

(in./in.)

εpin,j = initial strain in prestressing steel layer j at the start of strain compatibility analysis 

(in./in.)

εpsT,j = total strain in prestressing steel layer j at the completion of strain compatibility 

analysis (in./in.)

εs = net longitudinal tensile strain in the section at the centroid of the tension 

reinforcement (in./in.)

εsh = UHPC shrinkage strain at a given time (in./in.)

εsh,D = conventional concrete deck shrinkage strain at final time (in./in.)

εsl = tensile strain in the extreme tension steel when the steel service stress limit, fsl, is 

reached (in./in.)

εT,i = total strain in UHPC or conventional concrete layer i at the completion of strain 

compatibility analysis (in./in.)

εT,1 = total strain in conventional concrete at top of the deck at the completion of strain 

compatibility analysis (in./in.)

εT,2 = total strain in conventional concrete at bottom of haunch at the completion of strain 

compatibility analysis (in./in.)

εT,3 = total strain in UHPC at top of girder at the completion of strain compatibility analysis 

(in./in.)

εT,4 = total strain in UHPC at bottom of girder at the completion of strain compatibility 

analysis (in./in.)

εt,cr = elastic tensile strain limit of UHPC corresponding to a tensile stress of γuft,cr (in./in.)

εt,loc = crack localization strain of UHPC for use in design (in./in.)

εv = strain in the transverse steel reinforcement at nominal shear resistance (in./in.)

ηi = load modification factor for case i relating to ductility, redundancy, and operational 

importance 

θ = angle of inclination of diagonal compressive stresses (degrees)

μ = curvature ductility ratio
184

μi = friction factor for interface shear resistance calculation

μl = curvature ductility ratio limit

ξ = multiplier for transfer length of prestressing strand

ρv = ratio of area of transverse shear reinforcement to area of gross UHPC area of a 

horizontal section

τ = shear stress in web (ksi)

c = resistance factor for axial resistance

f = resistance factor for flexure

v = resistance factor for shear

Ψ(td,ti) = creep coefficient of UHPC at time of deck placement, td, due to loading at prestress 

transfer, ti

Ψ(tf,td) = creep coefficient of UHPC at final time, tf, due to loading at deck placement, td

Ψ(tf,ti) = creep coefficient of UHPC at final time, tf, due to loading at prestress transfer, ti

ΨD(tf,td) = creep coefficient of conventional concrete deck at the final time, tf, due to loading 

shortly after deck placement, td

ψ = sectional curvature at nominal flexural moment (1/in.)

ψL = sectional curvature when the strain at extreme tension layer of UHPC is equal to the 

UHPC tensile strain limit, γuεt,loc (1/in.)

ψn = sectional curvature at nominal flexural strength (1/in.)

ψsl = sectional curvature when the steel stress in the extreme tension steel is equal to the 

steel service stress limit, fsl (1/in.)

ωc = unit weight of UHPC in girder (kip/ft3)

ωcD = unit weight of conventional concrete in deck (kip/ft3)
185

CHAPTER C1. INTRODUCTION

The example in this Appendix illustrates the design of a pretensioned girder for a 150-ft-long, 

single-span bridge with no skew (overall beam length is 151 ft). The bulb tee girder of this example 

is a modified Minnesota Department of Transportation (MnDOT) “MN54” beam made of 

ultra-high performance concrete (UHPC) in which the web and flange widths are decreased by 

sliding the forms inward and utilizing a 27-in. soffit width and 5-in.-wide blocks in the top flange,

as shown in Figure C1-1. This design example explains in detail the design of a typical interior 

beam at the critical sections in positive flexure, shear, and deflection due to prestress, dead loads, 

and live load. The bridge dimensions are adapted from the design example for a MnDOT “MN63” 

beam from the MnDOT (2018) LRFD Bridge Design Manual. 

The superstructure of the bridge consists of six beams spaced at 9-ft centers. A typical transverse 

superstructure section is provided in Figure C1-2. The roadway section is composed of two 12-ft 

traffic lanes and two 12-ft shoulders. Beams are designed to act compositely with the 9-in.-thick 

cast-in-place conventional concrete deck to resist all superimposed dead loads, live loads, and 

impact. A 1⁄2-in.-thick wearing surface is assumed. A deck thickness of 8.5 in. is used for composite 

section properties. The haunch is assumed to have an average thickness of 21⁄2 in. for dead load 

computations and 11⁄2 in. for section property computations. The design live load is HL-93. Elastic 

stresses from external loads are calculated using transformed sections.

The design is accomplished in accordance with the AASHTO LRFD Bridge Design Specification, 

9th edition, hereafter referred to as “AASHTO LRFD BDS,” and Appendix A of this document, 

entitled, “Guide Specifications for Structural Design with Ultra-High Performance Concrete,”

which has been proposed to AASHTO T-10 for their consideration by the Federal Highway 

Administration’s Turner-Fairbank Highway Research Center, hereafter referred to as the “UHPC 

Guide” (AASHTO 2020). 

Source: FHWA.

A. Conventional concrete.

Source: FHWA.

B. With 27-in. soffit.

Source: FHWA.

C. Modified for UHPC.

Figure C1-1. Illustrations. MnDOT “MN54” cross sections.
186

Original illustration: © 2018 Minnesota Department of Transportation. Modified by FHWA to show 

new annotations and dimensions.

Figure C1-2. Illustration. Bridge cross section (MnDOT 2018).

C1.1. TERMINOLOGY

The following terminology is used to describe cross sections in this design example: 

• Noncomposite Section—The UHPC beam cross section. 

• Noncomposite, Nontransformed Section—The UHPC beam cross section without the 

strands transformed (also called the gross section). 

• Noncomposite, Transformed Section—The UHPC beam cross section with the strands 

transformed to provide cross-sectional properties equivalent to the UHPC used in the 

girder. 

• Composite Section—The UHPC beam plus the conventional concrete deck and haunch 

transformed to provide cross-sectional properties equivalent to the UHPC used in the 

girder. 

• Composite, Nontransformed Section—The UHPC beam plus the conventional concrete 

deck and haunch transformed to provide cross-sectional properties equivalent to the UHPC 

used in the girder but without the strands transformed. 

• Composite, Transformed Section—The UHPC beam plus the conventional concrete deck 

and haunch and the strands transformed to provide cross-sectional properties equivalent to 

the UHPC used in the girder. 

The term “composite” implicitly includes the transformation of the conventional concrete deck 

and haunch. The term “transformed” generally refers to transformation of the strands.
187

CHAPTER C2. MATERIALS

The material properties and key geometric details for the bridge being designed are listed below.

Cast-in-place conventional concrete deck:

• Actual thickness, tD = 9.0 in.

• Structural thickness, ts = 8.5 in.

(Note: a 1⁄2-in. wearing surface is considered to be part of the 9-in.-thick deck.)

• Specified concrete compressive strength for use in design: f ́cD = 4.0 ksi.

• Concrete unit weight, ωcD = 0.145 kip/ft3.

• Aggregate correction factor, K1D = 1.0.

• Modulus of elasticity for use in design,

EcD = 120,000K1D(ωcD)

2.0(f ́cD)0.33 = 3,987 ksi. [AASHTO LRFD BDS Eq. 5.4.2.4-1]

• Average thickness of haunch = 2.5 in.

• Structural thickness of haunch, th = 1.5 in.

• Width of haunch, bh = 21 in.

Precast UHPC girder—Modified MN54 bulb tee, as shown in Figure C1-1-C:

• UHPC compressive strength at transfer, f ́ci = 14.0 ksi.

• Compressive strength for use in design, f ́c = 22.0 ksi.

• Ultimate compressive strain for use in design, εcu = 0.0035.

• UHPC unit weight, ωc = 0.160 kip/ft3.

(Note: Includes 0.005 kip/ft3to account for the steel strands.)

• Overall girder length, LT = 151 ft.

• Girder design span, L = 150 ft.

• Correction factor for modulus of elasticity, K1 = 1.0.

• Modulus of elasticity at transfer, Eci = 2,500K1 f ́ci

0.33 = 5,973 ksi.

[UHPC Guide Article 1.4.2.3]
188

• Modulus of elasticity for use in design, Ec = 2,500K1 f ́c

0.33 = 6,933 ksi.

• Cracking strength for use in design, ft,cr = 1.00 ksi.

• Cracking strength at transfer, ft,cri = 0.75ft,cr = 0.75 ksi. [UHPC Guide Article 1.9.1.2]

• Ultimate tension strength for use in design, ft,loc = 1.00 ksi.

• Ultimate tension strain for use in design, εt,loc = 0.0040.

• Reduction factor on tensile parameters, γu = 1.00.

• Correction factor for creep, K3 = 0.62.

• Correction factor for shrinkage, K4 = 0.41.

Prestressing strands: 0.7-in.-diameter, seven-wire, low-relaxation:

• Diameter of strand, Dp = 0.7 in.

• Area of one strand, Ap = 0.294 in.2.

• Specified tensile strength, fpu = 270 ksi.

• Yield strength, fpy = 0.9·fpu = 243.0 ksi.

• Stress limit at transfer, fpi = 0.75·fpu = 202.5 ksi.

• Stress limit at service after losses, fsl = 0.80·fpy = 194.4 ksi.

• Ultimate tension strain for use in design, εps = 0.035.

• Modulus of elasticity, Ep = 28,500 ksi.

• Nominal weight, ωp = 0.001 kip/ft.

Future wearing surface: 2 in. of additional concrete, unit weight = 0.145 kcf.

New Jersey-type barrier: unit weight = 0.300 kips/ft/side.
189

CHAPTER C3. CROSS-SECTIONAL PROPERTIES FOR A TYPICAL INTERIOR 

GIRDER 

C3.1. NONCOMPOSITE, NONTRANSFORMED GIRDER SECTION 

The noncomposite, nontransformed cross-sectional properties of the UHPC girder section are as 

follows: 

• Gross cross-sectional area of UHPC girder, Ag = 544.35 in.2. 

• Overall depth of girder, h = 54 in. 

• Moment of inertia about the centroid of the noncomposite girder, Ig = 209,570.6 in.4. 

• Distance from the centroid to extreme bottom fiber of the noncomposite beam, 

yb = 21.89 in. 

• Distance from the centroid to extreme top fiber of the noncomposite beam, yt = 32.11 in. 

• Section modulus for extreme bottom fiber of the noncomposite beam, .

• Section modulus for extreme top fiber of the noncomposite beam, .

• Beam weight per unit length, wg = (544.35 in.2)(1 ft2/144 in.2)(0.160 kip/ft3) = 

0.6048 kip/ft. 

C3.2. COMPOSITE GIRDER SECTION 

C3.2.1. Effective Flange Width 

Article 4.6.2.6 of AASHTO LRFD BDS states the deck effective flange width in composite girder 

systems may be taken as one-half the distance to the adjacent girder on each side of the component 

(AASHTO 2020). 

Therefore, the effective flange width, be = 2(9/2) = 9.0 ft = 108 in. 

C3.2.2. Modular Ratio Between Deck and Girder UHPC 

Modular ratio between concrete deck and UHPC girder, .

C3.2.3. Section Properties 

To obtain cross-sectional properties equivalent to the girder UHPC, the effective flange width must 

be transformed by the modular ratio. Only the structural thickness of the deck, 8.5 in., is 

considered. 
190

Transformed flange width = nDb·be = (0.575)(108) = 62.10 in.

Transformed flange area = nDb·be·ts = (0.575)(108)(8.5) = 527.84 in.2.

Transformed flange moment of inertia = (62.1)(8.53)/12 = 3,178.0 in.4.

The width of the haunch must also be transformed by the modular ratio.

Transformed width of haunch = (0.575)(21.00) = 12.07 in.

Transformed area of haunch = (0.575)(21.00)(1.5) = 18.11 in.2.

Transformed moment of inertia of haunch = (12.07)(1.53)/12 = 3.40 in.4.

Note that the haunch should only be considered to contribute to section properties if it is required 

to be provided in the complete structure. For this example, haunch contributions to section 

properties will be used.

Figure C3.2.3-1 illustrates the dimensions of the composite UHPC/conventional concrete section. 

Table C3.2.3-1 provides calculated properties.

Source: FHWA.

Figure C3.2.3-1. Illustration. Dimensions of the composite UHPC/conventional concrete 

section.
191

Table C3.2.3-1. Properties of the composite UHPC/conventional concrete section. 

Area, A

(in.2) 

yb* 

(in.) 

Ayb* 

(in.3)

A(ybc − yb*)2

(in.4) 

I

(in.4)

I + A(ybc − yb*)2

(in.4) 

Girder 544.35 21.89 11,915.8 193,927.3 209,570.6 403,497.9 

Haunch 18.11 54.75 991.6 3,542.5 3.4 3,545.9 

Deck 527.84 59.75 31,538.3 190,254.4 3,178.0 193,432.4 

Summation 1,090.30 — 44,445.7 — — 600,476.2 

—Not calculated. 

Note: See List of Notations for variable definitions. 

Total area of the composite section, Ac = 1,090.30 in.2

. 

Overall depth of the composite section, hc = 64.00 in. 

Moment of inertia of the composite section, Ic = 600,476.2 in.4. 

Distance from the centroid of the composite section to the extreme bottom fiber of the UHPC 

girder, 

ybc = 44,445.7 in.3/1,090.30 in.2 = 40.76 in. 

Distance from the centroid of the composite section to the extreme top fiber of the UHPC girder, 

ytg = h − ybc = 13.24 in. 

Distance from the centroid of the composite section to the extreme top fiber of the structural deck, 

ytc = hc − ybc = 23.24 in. 

Composite section modulus for the extreme bottom fiber of the UHPC girder, 

Composite section modulus for the extreme top fiber of the UHPC girder, 

Composite section modulus for extreme top fiber of the structural deck, 

 

193

CHAPTER C4. SHEAR FORCES AND BENDING MOMENTS

C4.1. SHEAR FORCES AND BENDING MOMENTS DUE TO DEAD LOADS

C4.1.1. Dead Loads

The dead loads acting on the structure are as follows. Note that the haunch thickness is taken as

2.5 in. for load calculations.

Dead loads acting on the noncomposite structure:

• Girder self-weight, wg = 0.6048 kip/ft.

• 9-in. deck weight, wD = (9 in./(12 in./ft))(9 ft.)(0.145 kip/ft3) = 0.9788 kip/ft.

• Haunch weight, wh = (2.5 in./(12 in./ft))(21 in./(12 in./ft))(0.145 kip/ft3) = 0.0529 kip/ft.

• Total weight of deck and haunch, ws = wD + wh = 1.0316 kip/ft.

Dead loads acting on the composite structure:

AASHTO LRFD BDS Article 4.6.2.2.1 (AASHTO 2020) specifies that permanent loads may 

be distributed uniformly among the girders if the following criteria are met:

• Width of deck is constant.

OK.

• Number of girders is not less than four.

(Nb = 6).

OK.

• Beams are parallel and have approximately the same stiffness.

OK.

• The roadway part of the overhang is less than or equal to 3.0 ft.

(3.0 − 1.5 − 0.5(3.5/12) =1.854 ft).

OK.

• Curvature is less than specified in the AASHTO LRFD BDS (AASHTO 2020).

(curvature = 0.0 degrees).

OK.

• Cross section of the bridge is consistent with one of the cross-sections given in 

Table 4.6.2.2.1-1 of AASHTO LRFD BDS (AASHTO 2020). For precast concrete 

I-beams or bulb-tee beams with cast-in-place concrete deck, the bridge is type (k).

OK.
194

This example meets the criteria, and, therefore, the permanent loading is distributed uniformly.

• Barrier weight,

wb = (2)(3 kip/ft)/6 = 0.1000 kip/ft/beam

• Dead load of future wearing surface,

DLws = (2 in.)(0.145 kip/ft3) = 0.1678 kip/in.2

• Future wearing surface weight,

wws = (DLws)(48 ft)/6 = 0.1933 kip/ft/beam

C4.1.2. Unfactored Shear Forces and Bending Moments

For the simply supported beam of length, L, under a uniform distributed load, w, shear force, Vx,

and bending moment, Mx, at any distance, x, from the support are given as:

Vx = w (0.5L − x)

Mx = 0.5wx (L − x)

Shear forces and bending moments for a typical interior beam under self-weight, weight of slab 

and haunch, and weight of barriers and wearing surface are calculated and shown in 

Table C4.1.2-1. A span length of 150 ft is used for these calculations, but the total span length of 

151 ft will be used for stress and deflection calculations at the time of prestress transfer.

Table C4.1.2-1. Unfactored shear forces and bending moments due to dead loads for a 

typical interior girder.

Distance

x (ft)

Section

x /L

Girder Weight

Deck + Haunch 

Weight Barrier Weight

Wearing Surface 

Weight

Shear

Vg

(kip)

Moment

Mg

(kip-ft)

Shear

VD

(kip)

Moment

MD

(kip-ft)

Shear

Vb

(kip)

Moment

Mb

(kip-ft)

Shear

Vws

(kip)

Moment

Mws

(kip-ft)

4.60* 0.031 42.6 202.3 72.6 345 7.0 33.4 13.6 64.7

0 0.0 45.4 0.0 77.4 0.0 7.5 0.0 14.5 0.0

15 0.1 36.3 612.4 61.9 1,044.5 6.0 101.3 11.6 195.8

30 0.2 27.2 1088.7 46.4 1,856.9 4.5 180.0 8.7 348.0

45 0.3 18.1 1428.9 30.9 2,437.2 3.0 236.3 5.8 456.8

60 0.4 9.1 1633.1 15.5 2,785.4 1.5 270.0 2.9 522.0

75 0.5 0.0 1701.1 0.0 2,901.4 0.0 281.3 0.0 543.8

*Critical section for shear according to Section C11.1 of this Appendix.

Note: See List of Notations for variable definitions.
195

C4.2. Shear Forces and Bending Moments Due to Live Loads

C4.2.1. Live Loads 

The design live load is HL-93, which consists of:

1. Design truck or design tandem with dynamic allowance: The design truck consists of 8.0-, 

32.0-, and 32.0-kip axles, with the first pair spaced at 14 ft and the second pair spaced at 

14–30 ft. The design tandem consists of a pair of 25.0-kip axles spaced at 4 ft.

2. Design lane load of 0.64 kips/ft without dynamic allowance.

[AASHTO LRFD BDS Article 3.6.1.2.1-4]

C4.2.2. Live Load Distribution Factors for a Typical Interior Beam

The live load bending moments and shear forces are determined by using the simplified 

distribution factor formulas specified in AASHTO LRFD BDS (AASHTO 2020) Article 4.6.2.2. 

To use the simplified live load distribution factor formulas, the following conditions must be met:

• Width of deck is constant.

OK.

• Number of beams is not less than four (Nb = 6).

OK.

• Beams are parallel and have approximately the same stiffness.

OK.

• The roadway part of the overhang is less than or equal to 3.0 ft.

(3.0 – 1.5 – 0.5(9 ⁄ 12) = 1.854 ft).

OK.

• Curvature is less than specified in the AASHTO LRFD BDS (AASHTO 2020).

(curvature = 0.0°).

OK.

• Cross section of the bridge is consistent with one of the cross-sections given in 

Table 4.6.2.2.1-1 of AASHTO LRFD BDS (AASHTO 2020). For precast concrete I-beams 

or bulb-tee beams with cast-in-place concrete deck, the bridge is type (k).

OK.

The number of design lanes = the integer part of the ratio of WR/12, where WR is the clear roadway 

width, in feet, between the curbs. From Figure C1-2, WR = 48 ft. Number of design lanes = integer 

part of (48 ⁄ 12) = 4 lanes.
196

C4.2.2.1. Distribution Factor for Bending Moment 

For all limit states except fatigue limit state, the distribution factor for bending moment is 

calculated as follows. 

• For two or more lanes loaded: 

provided: 

3.5 ≤ S ≤ 16.0; S = 9.0 ft OK.

4.5 ≤ ts ≤ 12.0; ts = 8.5 in. or 9.0 in. with wearing surface OK.

20 ≤ L ≤ 240; L = 150 ft OK. 

Nb ≥ 4; Nb = 6 OK.

10,000 ≤ Kg ≤ 7,000,000 OK. 

Where: 

DFM = distribution factor for bending moment for interior girder 

S = girder spacing, ft 

L = design span, ft 

ts = structural depth of concrete deck, in. 

Kg = longitudinal stiffness parameter, in.4 = nbD (Ig + Ag eg

2

) 

[AASHTO LRFD BDS Eq. 4.6.2.2.1-1] 

in which: 

nbD = modular ratio between UHPC girder and conventional concrete deck 

= Ec / EcD = 1.739

Ag = gross area of UHPC girder = 544.35 in.2

Ig = moment of inertia of the UHPC girder = 209,570.6 in.4

eg = distance between the centers of gravity of the UHPC girder and 

conventional concrete deck 

= 9 in./2 + 2.5 in. + 32.11 in. = 39.11 in. 

(Note: eg is conservatively calculated using the total thickness of deck including the 

wearing surface and the average thickness of the haunch.) 

Therefore: 

Kg = nbD (Ig + Ag eg

2

) = 1.739 (209,570.6 + 544.35(39.112)) = 1,812,572 in.4

(Note: DFM is conservatively calculated using the structural thickness of deck.) 

 

 
197

• For one design lane loaded: 

(Note: DFM is conservatively calculated using the structural thickness of deck.)

The situation where two or more lanes are loaded controls; therefore, DFM = 0.654 lanes/beam. 

For the fatigue limit state, AASHTO LRFD BDS Article C3.4.1 (AASHTO 2020) states that a 

single design truck should be used and that the live load distribution factors take into consideration 

the multiple presence factor, m. The multiple presence factor for one design lane loaded is 1.2. 

Therefore, the distribution factor for one design lane loaded with the multiple presence factor 

removed should be used. The distribution factor for fatigue limit state is DFM = 0.439/1.2 = 

0.365 lanes/beam. 

C4.2.2.2. Distribution Factor for Shear Force 

The distribution factor for shear is calculated as follows. 

• For two or more lanes loaded: 

provided: 

3.5 ≤ S ≤ 16.0; S = 9.0 ft OK.

4.5 ≤ ts ≤ 12.0; ts = 8.5 in. or 9.0 in. with wearing surface OK.

20 ≤ L ≤ 240; L = 150 ft OK. 

Nb ≥ 4; Nb = 6 OK.

Where: 

DFV = distribution factor for shear force for interior girder 

S = girder spacing, ft 

Therefore: 

• For one design lane loaded: 

The situation where two or more lanes are loaded controls, with DFM = 0.884 lanes/beam. 

 
198

C4.2.3. Dynamic Allowance

The dynamic allowance parameters are as follows, where IM is the dynamic allowance, applied to 

the design truck load only.

IM = 15 percent for fatigue limit state.

IM = 33 percent for all other limit states.

C4.2.4. Unfactored Shear Forces and Bending Moments

C4.2.4.1. Due to Truck Load: VLT and MLT

For all limit states except for the fatigue limit state, shear force and bending moment envelopes 

per lane are calculated at tenth points of the span using commercially available computer software 

that analyzes moving loads. The values of the truck load shear forces, VLT, and the bending 

moments, MLT, at selected locations along the girder length are given in Table C4.2.4.1-1 and 

calculated as follows:

VLT = (shear force per lane)(DFV)(1 + IM).

= (shear force per lane)(0.884)(1 + 0.33).

= (shear force per lane)(1.176).

MLT = (bending moment per lane)(DFM)(1 + IM).

= (bending moment per lane)(0.654)(1 + 0.33).

= (bending moment per lane)(0.869).

For the fatigue limit state, AASHTO LRFD BDS Article 3.6.1.4.1 states that fatigue load is a single 

truck that has the same axle weight used in all other limit states but with a constant spacing of 30 ft 

between the 32-kip axles. The bending moment envelope per lane is calculated using computer 

software. The values of the bending moment of the fatigue truck, Mf, at selected locations along 

the girder’s length are given in Table C4.2.4.1-1 and calculated as follows:

Mf = (bending moment per lane)(DFM)(1 + IM).

= (bending moment per lane)(0.365)(1 + 0.15).

= (bending moment per lane)(0.420).
199

Table C4.2.4.1-1. Unfactored shear forces and bending moments due to live loads for a 

typical interior girder. 

Distance, 

x, ft 

Section 

x /L 

Truckload with Impact Lane Load 

Fatigue I Truck 

with Impact 

Shear 

VLT

(kips) 

Moment 

MLT

(kip-ft) 

Shear 

VLL

(kips) 

Moment 

MLL

(kip-ft) 

Moment 

Mf 

(kip-ft) 

4.60* 0.031 76.6 260.7 39.9 140.1 117.7 

0 0.0 78.8 0.0 42.4 0.0 0.0 

15 0.1 70.3 787.6 34.4 424.1 353.4 

30 0.2 61.9 1387.1 27.2 753.9 616.0 

45 0.3 53.4 1798.6 20.8 989.5 787.8 

60 0.4 45.0 2022.1 15.3 1130.9 868.9 

75 0.5 36.5 2106.4 10.6 1178.0 909.6 

*Critical section for shear according to Section C11.1 of this Appendix. 

Note: See List of Notations for variable definitions. 

C4.2.4.2. Due to Design Lane Load: VLL and MLL

To obtain the maximum shear force at a section located at a distance (x) from the left support under 

a uniformly distributed load of 0.64 kips/ft, the member should be loaded to the right of the section 

under consideration. Therefore, the maximum shear force per lane at location x along the girder’s 

length is: 

where: 

VLL = (shear force per lane)(DFV) 

 = (shear force per lane)(0.884) 

For all limit states except for the fatigue limit state: 

MLL = (bending moment per lane)(DFM) 

 = (bending moment per lane)(0.654) 

The values of VLL and MLL are given along selected location of the girder’s length in 

Table C4.2.4.1-1. 

C4.3. LOAD COMBINATIONS 

Total factored load is taken as: 

Q = Σηi γi Qi [AASHTO LRFD BDS Eq. 3.4.1-1] 
200

where:

ηi = a load modifier relating to ductility, redundancy, and operational importance (here ηi is 

taken as 1.0 for typical bridges) [AASHTO LRFD BDS Article 1.3.2]

γi = load factor [AASHTO LRFD BDS Table 3.4.1-1]

Qi = force effects from specified loads

The following limit states given in AASHTO LRFD BDS Article 3.4.1 are applicable:

• Service I: Check compressive stresses in prestressed UHPC components and tensile stress 

limits for prestressed UHPC components subject to fatigue loads:

Q = 1.00(DC + DW) + 1.00(LL + IM) [AASHTO LRFD BDS Table 3.4.1-1]

where:

DC = dead load of structural components and nonstructural attachments

DW = dead load of wearing surfaces and utilities

LL = vehicular live load

IM = dynamic allowance

This load combination is the general combination for service limit state stress checks and 

applies to all conditions other than Service III.

• Service III: Check tensile stresses in prestressed UHPC components:

Q = 1.00(DC + DW) + γLL (LL + IM) [AASHTO LRFD BDS Table 3.4.1-1]

where:

γLL = load factor for live load for Service III load combination

In prestressed conventional concrete, this load combination is a special combination for 

service limit state stress checks that applies only in tension to control cracks. However, γLL

for UHPC is always equal to 1.0 because only the refined estimates of time-dependent 

losses are allowed.

[AASHTO LRFD BDS Table 3.4.1-4; UHPC Guide Article 1.9.1.1]

Note that, in this design example, the bending stresses due to Service I and Service III load 

combinations are assumed to be similar.

• Strength I: Check ultimate strength:

Maximum Q = 1.25(DC) + 1.50(DW) + 1.75(LL + IM)

[AASHTO LRFD BDS Table 3.4.1-1 and Table 3.4.1-2]

Minimum Q = 0.90(DC) + 0.65(DW) + 1.75(LL + IM)
201

This combination is the general load combination for strength limit state design.

Note: For simple span bridges, the maximum load factors produce maximum effects. 

However, use minimum load factors for dead load (DC) and wearing surface (DW) when 

dead load and wearing surface stresses are opposite to those of live load. 

• Fatigue I: Check the stress range in strands and UHPC compression stresses:

Q = 1.75(LL + IM) [AASHTO LRFD BDS Table 3.4.1-1]

This load combination is a special load combination to check tensile stress range in the 

strands due to live load and dynamic allowance. 

203

CHAPTER C5. REQUIRED PRESTRESSING 

UHPC Guide Article 1.9.1 states prestressing strands shall satisfy requirements at service, fatigue, 

strength, and extreme event limit states, as specified in Articles 1.5 and 1.9.2. For this design 

example, a strand profile has been predetermined. 

C5.1. STRAND PATTERN 

Fifty 0.7-in.-diameter, Grade 270 low-relaxation strands are selected to reinforce the girder. 

Forty-eight strands are placed in the bottom flange at a horizontal and vertical spacing of 2 in. 

center-to-center, as shown in Figure C5.1-1. Two strands are placed in the top flange at a distance 

of 2 in. between the top of the girder and the center of the strands; the top strands were placed at a 

horizontal spacing of 4 in. center to center. At each end of the girder, 14 strands are debonded near 

girder ends to satisfy stress limit requirement at release at the transfer length section, as described 

in Section C7.2 of this Appendix, and to adequately reinforce the girder at the pretensioned 

anchorage zones, as described in Section C14.1 of this Appendix. The strand pattern at the girder 

ends is selected following the provisions of UHPC Guide Article 1.9.4.3.3 and is shown in 

Figure C5.1-2. 

Total area of prestressing strands at midspan, Aps = TsAp = 14.700 in.2. 

Total area of prestressing strands at girder ends, ApsE = TsEAp = 10.584 in.2

. 

Distance between the centroid of the strands and the extreme bottom fiber of the UHPC girder at 

midspan, 

Distance between the centroid of the strands and the extreme bottom fiber of the UHPC girder at 

ends, 

where: 

Nmid = quantity of strands per row at midspan = [2, 9, 13, 13, 13] 

dmid = distance between the centroid of the strands in a row and the bottom of the section at 

midspan = [52, 8, 6, 4, 2] 

Ts = total number of prestressing strands at midspan = 50 

TsE = total number of prestressing strands at girder ends = 36

Nend = quantity of strands per row at girder ends = [2, 7, 9, 9, 9] 

dend = distance between the centroid of the strands in a row and the bottom of the section at 

girder ends = dmid

Strand eccentricity in noncomposite section at midspan, epg = yb − ybs = 21.89 − 6.64 = 15.25 in. 

Strand eccentricity in noncomposite section at ends, epgE = yb − ybsE = 21.89 − 7.44 = 14.45 in. 

Total prestressing force before transfer at midspan, Ppi = Ts Ap fpi = 2,976.75 kip. 

  
204

Total prestressing force before transfer at girder ends, PpiE = TsE Ap fpi = 2,143.26 kip.

Source: FHWA.

Figure C5.1-1. Illustration. Strand pattern at midspan.

Source: FHWA.

Figure C5.1-2. Illustration. Strand pattern at beam ends.

C5.2. STEEL TRANSFORMED SECTION PROPERTIES

Transformed sections are obtained from the gross sections by adding the transformed area of steel. 

Transformed sectional analysis yields more accurate results of prestressing forces than does the 

gross section analysis. To obtain the transformed area of the reinforcing steel, the steel area in each 

row of prestressing strands shown in Figure C5.1-1 is multiplied by (n − 1), where n is the modular 

ratio between UHPC and prestressing strands. The modulus of elasticity of UHPC is different at 

Number 

of strands 

2 

9 

13 

13 

13 

Number 

of strands 

2 

7 

9 

9 

9 

Distance from 

bottom (in.) 

52 

8 

6 

4 

2 

Distance from 

bottom (in.) 

52 

8 

6 

4 

2 
205

transfer and final time; therefore, the transformed section properties are calculated separately for 

the two periods. 

C5.2.1. Noncomposite Transformed Section at Transfer at Midspan 

The noncomposite transformed section properties at transfer at midspan are as follows: 

n − 1 at transfer, denoted by nti,

Area of transformed section, Ati = Ag + ∑ntiNmidAp = 599.8 in.2

Distance from centroid to bottom 

of section, 

Moment of inertia, Iti = Ig + Ag (ybti − yb)

2 +∑ntiNmid Ap (ybti − dmid)2

Iti = 226,271.6 in.4

Distance from centroid to top of 

section, 

ytti = h − ybti = 33.52 in. 

Eccentricity of strands, eti = ybti − ybs = 13.84 in. 

Section modulus for the bottom 

of section, 

Section modulus for the top of 

section, 

C5.2.2. Noncomposite Transformed Section at Girder Ends at Transfer (14 Strands 

Debonded) 

The noncomposite transformed section properties at girder ends at transfer are as follows: 

n − 1 at transfer, denoted by nti, nti = 3.772 

Area of transformed section, AtiE = Ag + ∑ nti Nend Ap = 584.3 in.2

 

Distance from centroid to bottom 

of section, 

Moment of inertia, ItiE = Ig + Ag (ybtiE − yb)

2 + ∑ntiNendAp(ybtiE − dend)2

 

 

 
206

ItiE = 222,172.1 in.4 

Distance from centroid to top of 

section, 

yttiE = h − ybtiE = 33.10 in. 

Eccentricity of strands, etiE = ybtiE − ybsE = 13.46 in. 

Section modulus for bottom of 

section, 

Section modulus for top of 

section, 

C5.2.3. Noncomposite Transformed Section at Midspan at Final Time 

The noncomposite transformed section properties at midspan at final time are as follows: 

n − 1 at transfer, denoted by ntf,

Area of transformed section, Atf = Ag + ∑ntfNmid Ap = 590.1 in.2

Distance from centroid to bottom 

of section, 

Moment of inertia, Itf = Ig + Ag (ybtf − yb)

2 + ∑ntfNmidAp(ybtf − dmid)2

Itf = 223,502.9 in.4

Distance from centroid to top of 

section, 

yttf = h − ybtf = 33.29 in. 

Eccentricity of strands, etf = ybtf − ybs = 14.07 in. 

Section modulus for bottom of 

section, 

Section modulus for top of 

section, 

 
207

C5.2.4. Composite Transformed Section at Midspan at Final Time 

The composite transformed section properties at midspan at final time are as follows: 

n − 1 at transfer, denoted by ntf,

Area of transformed section, Atc = Ac + ∑ntf Nmid Ap = 1,136.0 in.2

Distance from centroid to bottom 

of section, 

Moment of inertia, Itc = Ite + Ith + Ig + Atc(ybtc − ybc)

2 + ∑ntf Nmid Ap(ybtc − dmid)2

Itc = 655,702.7 in.4

 

Distance from centroid to top of 

composite section, yttc = hc − ybtc = 24.61 in. 

Eccentricity of strands, etc = ybtc − ybs = 32.75 in. 

Section modulus for the bottom 

of composite section, 

Section modulus for the top of 

UHPC section,

Section modulus for the top of 

composite section,

 

 

209

CHAPTER C6. PRESTRESS LOSSES 

The total prestress loss is calculated as follows. 

∆fpT = ∆fpES + ∆fpLT [AASHTO LRFD BDS Eq. 5.9.5.1-1] 

where: 

∆fpT = total prestress loss in prestressing steel (ksi) 

∆fpES = sum of all prestress losses or gains due to elastic shortening or extension at time of 

prestressing (ksi) 

∆fpLT = long-term prestress losses due to shrinkage and creep of UHPC and conventional 

concrete, and relaxation of steel after transfer (ksi) 

C6.1. ELASTIC SHORTENING 

The prestress loss due to elastic shortening is calculated as follows: 

where: 

Ep = modulus of elasticity of prestressing strands = 28,500 ksi 

Eci = modulus of elasticity of beam UHPC at transfer = 5,973 ksi 

fcgp = sum of stresses at the center of gravity of prestressing strands due to prestressing 

force at transfer and the self-weight of the member at maximum moment sections 

(ksi) 

When gross section properties are used, the elastic loss, ∆fpES, is usually assumed to be 10 percent 

of the initial prestress to calculate fcgp to calculate a more refined ∆fpES. This iteration continues 

until the values of ∆fpES and fcgp converge. 

By using transformed properties to calculate stress, effects of losses and gains due to elastic 

deformations are accounted for. Therefore, for this example, ∆fpES will not be included in 

calculating fcgp. 

Force per strand before transfer = area of strand × prestress before transfer = (0.294)(202.5) = 

59.54 kips.

where:

eti = eccentricity of strands at midspan with respect to the transformed section at transfer 

 = 13.84 in. 

Ppi = total prestressing force before transfer 

 = 2,976.8 kips 

 
210

MgT,m = unfactored bending at midspan moment due to UHPC girder self-weight, which is 

calculated based on overall girder length 151 ft 

 = 1,723.9 kip-ft

Therefore, prestress loss due to elastic shortening: 

AASHTO LRFD BDS Article C5.9.5.3 (AASHTO 2020) indicates that the loss due to elastic 

shortening at transfer should be added to the time-dependent losses to determine total losses. 

However, this loss at transfer is directly accounted for when transformed properties are used in 

stress analysis. 

C6.2. TIME-DEPENDENT LOSSES BETWEEN TRANSFER AND DECK PLACEMENT

The time-dependent prestress loss between prestress transfer and deck placement is calculated as 

follows: 

The following construction schedule is assumed for calculating time-dependent losses: 

• UHPC age at time of prestress transfer: ti = 1 day. 

• UHPC age at time of deck placement: td = 90 days. 

• UHPC age at final time: tf = 20,000 days. 

The total time-dependent losses between transfer and deck placement are the sum of prestress 

losses incurred during all stages listed above. 

C6.2.1. Creep of UHPC 

The prestress loss due to creep of UHPC girder between the time of prestress transfer, ti, and deck 

placement, td, is: 

where: 

Ψ(td,ti) = creep coefficient of UHPC at time of deck placement, td, due to loading at prestress 

transfer, ti

Kid = transformed section coefficient that accounts for time-dependent interaction 

between UHPC and bonded steel in the section considered for the time period 

between prestress transfer and deck placement 

  

 

 
211

The creep coefficient Ψ(td, ti) is taken as: 

Ψ(td,ti) = 1.2ks khc kf ktdd kli K3 [AASHTO LRFD BDS Eq. 1.4.2.8.2-1] 

where: 

ks = factor for the effect of volume-to-surface ratio of the UHPC girder = 1.0 

khc = humidity factor for creep of UHPC = 1.12 − 0.0024H = 0.945 

kf = factor for the effect of UHPC strength 

ktdd = time development factor between prestress transfer and deck placement 

kli = factor for the effect of loading age at prestress transfer = 1.0, because ti < 7 

K3 = correction factor for creep of UHPC = 0.62 

in which: 

H = average annual ambient relative humidity = 73 percent 

f ́ci = compressive strength of UHPC at the time of prestressing for use in design = 14 ksi 

t = maturity of UHPC in days, taken as the time being considered for analysis of creep or 

shrinkage effects = td – ti = 90 – 1 = 89 days 

Therefore: 

Ψ(td,ti) = 1.2(1.0)(0.945)(1.0)(1.238)(1.0)(0.62) = 0.870

The transformed section coefficient is taken as: 

where: 

epg = eccentricity of strands in noncomposite section at midspan = 15.25 in. 

Ψ(tf,ti) = girder creep coefficient at final time, tf, due to loading at prestress transfer, ti

= 1.2ks khc kf ktdf kli K3 = 1.071

in which: 

ktdf = time development factor between prestress transfer and final time 

 

 

 

 c
212

t = maturity of UHPC in days, taken as the time being considered for analysis of creep or 

shrinkage effects = tf – ti = 20,000 – 1 = 19,999 days. 

Therefore: 

and

C6.2.2. Shrinkage of UHPC 

The prestress loss due to shrinkage of UHPC girder between the time of prestress transfer, ti, and 

the time of deck placement, td, is calculated as: 

∆fpSR = εsh Ep Kid [AASHTO LRFD BDS Eq. 5.9.5.4.2a-1] 

where the strain due to shrinkage, εsh, is taken as: 

εsh = εbid = 0.6 × 10−3ks khs kf ktdd K4 [AASHTO LRFD BDS Eq. 1.4.2.8.3-1] 

in which: 

εbid = UHPC shrinkage strain for the period between prestress transfer and deck placement 

(in./in.) 

khs = humidity factor for shrinkage of UHPC = 1.5 – 0.01H = 0.770 

K4 = correction factor for shrinkage = 0.41 

ktdd = 1.238 

Therefore: 

εbid = (0.6 × 10−3

)(1.0)(0.770)(1.0)(1.238)(0.41) = 0.000234 

Prestress losses due to shrinkage between transfer and deck placement is: 

∆fpSR = (0.000234)(28,500)(0.734) = 4.907 ksi 

C6.2.3. Relaxation of Prestressing Strands 

Prestress loss due to the relaxation of prestressing strands between prestress transfer and deck 

placement is calculated by: 

[AASHTO LRFD BDS Eq. 5.9.5.4.2c-1] 

 

 

 

 
213

where: 

fpt = stress in prestressing strands immediately after transfer, taken as not less than 0.55fy

(ksi) 

KL = factor accounting for the type of steel taken as 30 for low-relaxation strands and 7 

for other prestressing steel, unless more accurate manufacturer’s data are available 

Therefore: 

C6.3. TIME-DEPENDENT LOSSES BETWEEN DECK PLACEMENT AND FINAL 

TIME 

The total time-dependent losses between deck placement and final time are the sum of prestress 

losses due to shrinkage and creep of the UHPC beam, relaxation of prestressing strands, and 

shrinkage of the deck concrete. The calculation of each of these prestress losses are detailed here 

in sections C6.3.1 to C6.3.4.

C6.3.1. Creep of UHPC 

The prestress loss due to creep of the girder UHPC between the time of deck placement, td, and 

final time, tf, is determined as: 

[AASHTO LRFD BDS Eq. 5.9.3.4.3b-1] 

where: 

Kdf = transformed section coefficient that accounts for time-dependent interaction 

between UHPC and bonded steel in the section considered for the time period 

between deck placement and final time 

Ψ(tf,td) = girder creep coefficient at final time, tf, due to loading at time of deck placement, 

td 

∆fcd = change in UHPC stress at the centroid of the prestressing strands due to long-term 

losses between transfer and deck placement, combined with deck weight and 

superimposed loads (ksi) 

The UHPC girder creep coefficient Ψ(tf,td) is calculated as follows: 

Ψ(tf,td) = 1.2ks khc kf ktdfd kld K3 [AASHTO LRFD BDS Eq. 5.4.2.3.2-1] 

where: 

kld = factor for the effect of loading age at time of deck placement, td ≥ 7 days 

 

 
214

= (td − 6)−0.15 = 0.5145 ≥ 0.5 [UHPC Guide Eq. 1.4.2.8.2-7] 

ktdfd = time development factor between deck placement and final time 

in which: 

t = maturity of UHPC in days, taken as the time being considered for analysis of creep or 

shrinkage effects = tf – td = 20,000 – 90 = 19,910 days 

f ́cd = UHPC girder strength at time of deck loading placement = 18 ksi 

Therefore: 

Ψ(tf, td) = 1.2(1.0)(0.945)(1.0)(1.523)(0.5145)(0.62) = 0.551. 

The change in UHPC stress at the centroid of the prestressing strands, ∆fcd, can be calculated as 

follows: 

where: 

MD,m = unfactored bending moment at midspan due to deck and haunch weight = 

2,901.4 kip-ft

Mb,m = unfactored bending moment at midspan due to barrier weight = 281.3 kip-ft

Mws,m = unfactored bending moment at midspan due to wearing surface weight = 543.8 kip-ft

Therefore: 

The transformed section coefficient of the UHPC girder, Kdf, can be calculated according to the 

following equation: 

[AASHTO LRFD BDS Eq. 5.9.3.4.3a-2] 

 

 

 

2

 
215

where: 

epc = eccentricity of strands with respect to the centroid of composite section (in.) 

 = ybc − ybs = 34.13 in. 

Therefore: 

The prestress loss due to creep, ΔfpCD, can be calculated as: 

Note that the negative sign indicates a prestress gain. 

C6.3.2. Shrinkage of UHPC 

The prestress loss due to shrinkage of UHPC between time of deck placement, td, and final time, 

tf, can be computed based on the following equation: 

ΔfpSD = εbdf Ep Kdf [AASHTO LRFD BDS Eq. 5.9.3.4.3a-1] 

where: 

εbdf = εbif − εbid 

in which: 

εbdf = UHPC shrinkage strain for the period between deck placement and final time 

εbif = UHPC shrinkage strain for the period between prestress transfer and final time 

(in./in.) 

= 0.6 × 10−3

ks khs kf ktdf K4

Therefore: 

εbif = 0.6 × 10−3(1.0)(0.770)(1.0)(1.523)(0.41) = 0.000288 

εbdf = εbif − εbid = 0.000054 

ΔfpSD = (0.000054)(28,500)(0.740) = 1.140 ksi 

 

 

 

 
216

C6.3.3. Relaxation of Prestressing Strands 

The prestress loss due to relaxation of the prestressing strands in the composite section between 

time of deck placement and final time is taken equal to the loss due to the relaxation between time 

of prestress transfer and deck placement. Therefore: 

∆fpR2 = ∆fpR1 = 0.929 ksi [AASHTO LRFD BDS Eq. 5.9.3.4.3c-1] 

C6.3.4. Shrinkage of Deck Concrete 

The prestress gain due to shrinkage of deck concrete is: 

[AASHTO LRFD BDS Eq. 5.9.3.4.3d-1] 

where ∆fcdf is the change in UHPC stress at the centroid of prestressing strands due to shrinkage 

of conventional concrete deck (ksi), calculated according to the following equation: 

[AASHTO LRFD BDS Eq. 5.9.3.4.3d-2] 

in which: 

εddf,D = shrinkage strain of conventional concrete deck for the period between deck 

placement and final time (in./in.) 

Ad = area of conventional concrete deck = 9 in. × 108 in. = 972 in.2

ΨD(tf,td) = creep coefficient of conventional concrete at the final time, tf, due to loading 

shortly after deck placement, td

ed = eccentricity to the centroid of the conventional concrete deck with respect to the 

gross composite section (in.)

= ytc – (tD/2)= 23.24 – (9.0/2) = 18.74 in. 

Assuming that the strength of the concrete at initial loading, f ́cDi, is 0.8(4.0) = 3.2 ksi, the strain 

due to shrinkage for the conventional concrete deck at final time, εsh,D, can be calculated as follows: 

εsh,D = ks,D khs,D kf,D ktd,D 0.48 × 10−3[AASHTO LRFD BDS Eq. 5.4.2.3.3-1] 

where: 

ks,D = factor for the effect of the volume to surface ratio of the deck, calculated according 

to AASHTO LRFD BDS Eq. 5.4.2.3.2-2

= 1.45 – 0.13(V/S) = 1.45 − 0.13(3.940) = 0.938 ≥ 1.0, thus ks,D = 1.0 

khs,D = humidity factor for shrinkage of conventional concrete deck, calculated according 

to AASHTO LRFD BDS Eq. 5.4.2.3.3-2 

= 2 − 0.014 H = 2 – 0.014 × 73 = 0.978 

kf,D = factor for the effect of conventional concrete strength, calculated according to 

AASHTO LRFD BDS Eq. 5.4.2.3.2-4

 

 
217

ktd,D = time development factor for the conventional concrete deck between deck 

placement and final time, calculated according to AASHTO LRFD BDS 

Eq. 5.4.2.3.2-5 

in which: 

V/S = volume to surface ratio of the conventional concrete deck (in.) 

t = maturity of conventional concrete in days, taken as the time being considered for 

analysis of creep or shrinkage effects = tf – td = 20,000 – 90 = 19,910 days 

Therefore: 

εsh,D = (1.0)(0.978)(1.190)(0.998)(0.48 × 10−3

) = 0.000558

The creep coefficient of the conventional concrete deck at final time due to loading shortly after 

deck placement, ΨD(tf, td), can be calculated according to the following equation: 

ΨD(tf, td) = 1.9ks,D khc,D kf ktd,D tiD

−0.118 [AASHTO LRFD BDS Eq. 5.4.2.3.2-1]

where: 

khc,D = humidity factor for creep of conventional concrete deck, calculated according to 

AASHTO LRFD BDS Eq. 5.4.2.3.2-3 

= 2 − 0.008 H = 2 – 0.008(73) = 0.976 

tiD = age of conventional concrete deck at time of loading, taken as 1 day after casting 

Therefore: 

ΨD(tf, td) = 1.9(1.0)(0.976)(1.190)(0.998)(1−0.118) = 2.203. 

The change in UHPC stress at the centroid of prestressing strands, ∆fcdf, can now be calculated as: 

Note that the negative sign indicates a prestress gain. 

Finally, the prestress gain due to shrinkage of the deck can be computed as follows: 

 

 

 

 
218

The effect of deck shrinkage on the UHPC stresses should be analyzed by considering it as an 

external force applied to the composite nontransformed section. One hundred percent of the effect 

due to deck shrinkage on the prestress losses and UHPC stresses is considered according to 

AASHTO LRFD BDS Eq. 5.9.3.4.3d-2.

C6.4. TOTAL TIME-DEPENDENT LOSSES

The total time-dependent loss is:

∆fpLT = (∆fpSR + ∆fpCR + ∆fpR1) + (∆fpSD + ∆fpCD + ∆fpR2 + ∆fpSS)

[AASHTO LRFD BDS Eq. 5.9.3.4.1-1]

= (4.907 + 18.958 + 0.929) + (1.140 − 1.898 + 0.929 − 0.529)

= 24.44 ksi

C6.5. TOTAL LOSSES AT PRESTRESS TRANSFER

AASHTO LRFD BDS Articles C5.9.5.2.3a and C5.9.5.3 state that the losses or gains due to elastic 

deformation must be taken equal to zero if transformed section properties are used in stress 

analysis. However, the losses or gains due to elastic deformation must be included in determining 

the total prestress losses and effective stress in the prestressing strands.

The stresses and forces in the prestressing strand immediately after transfer are as follows:

Total prestress loss: ∆fpi = ∆fpES = 29.67 ksi.

Stress in strands: fpt = fpi − ∆fpi = 202.5 – 29.67 = 172.83 ksi.

Force in one strand = (fpt)(As) = 172.83(0.294) = 50.81 kips.

Total prestressing force after transfer, Ppt = 50.81(50) = 2,540.6 kips.

Initial loss, in percent = (∆fpi)/fpi = 29.67/202.5 = 14.65 percent.

For determining concrete stresses using transformed section properties, strand force is the force

before transfer:

Force per strand = (202.5)(0.294) = 59.54 kips.

Total prestressing force before transfer, Ppi = 2,976.75 kips.

C6.6. TOTAL LOSSES AT SERVICE LOADS

The total prestress loss due to elastic shortening and long-term losses is:

∆fpT = ∆fpES + ∆fpLT = 29.67 + 24.44 = 54.11 ksi

The elastic gain due to deck weight, superimposed dead load, and live load is:
219

in which: 

MLT,m = bending moment at midspan due to truck load = 2,106.4 kip-ft

MLL,m = bending moment at midspan due to lane load = 1,178.0 kip-ft

The stresses and forces in the prestressing strand after all losses and permanent gains (gains due 

to MD, Mb, and Mws) are as follows: 

Effective prestress: fpe = fpi − ∆fpT + 11.04 = 202.5 − 54.11 + 11.04 = 159.43 ksi.

Force per strand = (fpe)(As) = (159.43)(0.294) = 46.87 kips. 

Total prestressing force, Ppe = 46.87(50) = 2,343.6 kips. 

Final loss, in percent = (total losses and permanent gains)/(fpi) = (202.5 – 159.43)/(202.5) = 

21.27 percent. 

Final loss without consideration of prestressing gain at deck placement = ∆fpT/fpi = 54.11/202.5 

= 26.72 percent.

 

  

221

CHAPTER C7. UHPC STRESSES AT PRESTRESS TRANSFER 

C7.1. STRESS LIMITS FOR UHPC AT PRESTRESS TRANSFER, BEFORE LOSSES 

The stress limits for UHPC at transfer and before losses are defined as follows: 

Compression: 0.65 f ́ci = 0.65(14) = +9.100 ksi [UHPC Guide Article 1.9.2.3.1a]

Tension: − γu ft,cri = −1.00(0.750) = −0.750 ksi [UHPC Guide Article 1.9.2.3.1b] 

C7.2. STRESSES AT TRANSFER LENGTH SECTION 

Stresses at the transfer length section will only be checked at transfer in this example because this 

stage almost always controls. 

Transfer length, lt = ξ24Dp = 0.75(24)(0.7) = 12.6 in. = 1.05 ft 

where ξ is the factor for transfer length, taken as 0.75 because shorter transfer lengths result in 

more severe stress states within the section. [UHPC Guide Eq. 1.9.4.3.1-1] 

The moment due to self-weight of the beam at the transfer length section is: 

MgT,tr = 0.5wg lt (LT − lt) = (0.5)(0.6048)(1.05)(151−1.05) = 47.61 kip-ft

where: 

wg = ωc Ag = 0.6048 kip/ft. 

The stress in the extreme top fiber of the UHPC girder, assuming all strands are bonded at the 

ends: 

Tensile stress limit for UHPC is −0.750 ksi. 

Not acceptable. 

The stress in the extreme bottom fiber of the UHPC girder, assuming all strands are bonded at the 

ends: 

Compressive stress limit for UHPC is +9.100 ksi. 

OK.

Since the stresses at the top of the beam exceed the tensile stress limit of −0.750 ksi, a number of 

strands must be debonded to reduce the top tensile stresses. In this example, six strands must be 

    
222

debonded in the girder bottom flange near the ends to meet the tensile stress limit at release at the 

transfer length section. However, 14 strands are debonded to lower the bursting stresses and 

adequately reinforce the girder at the pretensioned anchorage zones, as described in Section C14.1

of Appendix C. 

C7.3. RECOMPUTED STRESSES AT TRANSFER LENGTH SECTION WITH 14

STRANDS DEBONDED 

The stress at extreme top fiber of the UHPC girder, after debonding 14 strands at ends: 

Tensile stress limit for UHPC is −0.750 ksi 

OK.

Compute stress in the bottom of the beam: 

Compressive stress limit for UHPC is +9.100 ksi 

OK.

C7.4. STRESSES AT MIDSPAN 

The stress in the extreme top fiber of the UHPC girder is calculated as follows: 

where MgT,m is the moment due to self-weight of the beam at midspan, calculated as: 

MgT,m = 0.5wg x(LT − x) = (0.5)(0.6048)(75.5)(151 − 75.5) = 1,724 kip-ft.

Therefore: 

Tensile stress limit for UHPC is −0.750 ksi. 

OK.

The compressive stress in the bottom of the beam is calculated as follows: 

       
223

Compressive stress limit for UHPC is +9.100 ksi. 

OK.

C7.5. SUMMARY OF STRESSES AT TRANSFER 

Table C7.5-1 shows a summary of the stresses at transfer. 

Table C7.5-1. Stresses at transfer. 

Location Top Fiber Stresses (ksi) Bottom Fiber Stresses (ksi) 

At transfer length −0.544 6.328 

At midspan 1.924 6.820 

C7.6. PRINCIPAL TENSILE STRESS IN THE WEB OF THE GIRDER 

The provisions of UHPC Guide Article 1.9.2.3.3 and AASHTO LRFD BDS Article 5.9.2.3.3 state 

that the maximum principal stress at any location along the height of the web shall not exceed 

γu ft,cr when the superstructure element is subject to the loadings of Service III limit state of 

AASHTO LRFD BDS Article 3.4.1, both before and after all losses. 

The principal tensile stress limit before losses is γu ft,cri = 1.00(−0.750) = −0.750 ksi. 

The principal stress determination may be based on the analysis using Mohr’s circle: 

[AASHTO LRFD BDS Article 5.9.2.3.3-4] 

where: 

fmin = minimum principal stress in the web with the tension taken as a negative value (ksi) 

fpcx = horizontal stress in the web (ksi) 

fpcy = vertical stress in the web = 0 ksi 

τ = shear stress in web (ksi) 

For open sections that may be considered thin walled, such as typical I-girders and bulb-tee girders: 

[AASHTO LRFD BDS Article 5.9.2.3.3-1] 

where: 

V = shear force for Service III load combination (kip) 

Qg = first moment about the neutral axis of the gross concrete area above or below the 

height of the web where the principal tension is being checked (in.3) 

bw = web width at the height of the web where principal tension is being checked (in.) 

AASHTO LRFD BDS Article 5.9.2.3.3 recommends calculating the principal stress at multiple 

locations in the web. 

  
224

In this design example, the principal tensile stress check will only be computed at the transfer 

length section and at two locations in the web:(1) at the centroid of the UHPC gross section 

(location of maximum shear stress); and (2) at the top of the web are computed (location of 

maximum longitudinal tensile stress in the web). Note that principal tensile stress calculations are 

presented herein for illustration purposes. To complete the principal tensile stress check at transfer 

of prestress, the principal stresses in the web must be computed at multiple locations within the 

web and at regular intervals along the length of the beam. 

C7.6.1. Principal Stress at the Centroid of the UHPC Gross Section 

The first moment of the area is calculated using the UHPC area below or above the centroid of the 

gross UHPC area (yb = 21.89 in.): Qg = 4,892.91 in.3. 

The applied shear force at the transfer length section at release is conservatively computed at beam 

ends such as V = 0.5wg LT = (0.5)(0.6048)(151) = 45.66 kips. 

The shear stress in the web at the centroid of the UHPC gross section, yb = 21.89 in., can be 

calculated as follows: 

The horizontal stress in the web at the neutral axis of the gross UHPC section, yb = 21.89 in., can 

be determined from elastic analysis as follows: 

The maximum principal tension stress is: 

Tensile stress limit for UHPC is −0.750 ksi. 

OK.

C7.6.2. Principal Stress at Top of the Web 

The top of the web is located at a distance of 41.44 in. from the bottom of the girder. The first 

moment of area at this location is calculated using the UHPC area of the section above or below 

the top of the web: Qg = 4,223.99 in.3. 

 

 

 
225

The maximum principal tension stress is: 

Tensile stress limit for UHPC is −0.750 ksi. 

OK.

 

227

CHAPTER C8. CONCRETE STRESSES AT SERVICE LOADS

C8.1. STRESS LIMITS FOR UHPC GIRDER AND CONVENTIONAL CONCRETE 

DECK AT FINAL TIME, AFTER LOSSES

The stress limits for UHPC girder and conventional concrete deck at final time after losses are 

calculated as follows:

• Compression:

Due to the sum of effective prestress and permanent loads, for load combination Service I

the following values area selected.

For precast beams: 0.45 f ́c = 0.45(22) = +9.900 ksi.

For deck: 0.45 f ́cD = 0.45(4) = +1.800 ksi.

Due to permanent and transient loads, for load combination Service I the following values 

are selected.

For precast beams: 0.60 f ́c = 0.60(22) = +13.20 ksi.

For deck: 0.60 f ́cD = 0.60(4) = +2.400 ksi.

• Tension:

For components with bonded prestressing tendons the following values are selected.

For load combination Service III: − γu ft,cr.

For precast beam: −1.00(1.00) = −1.00 ksi.

For deck concrete, a stress limit of 0.000 ksi is selected.

Note that for members subject to fatigue loads, the UHPC tensile stress due to Service I 

load combination must not exceed 0.95γu ft,cr = 0.95(1.00)(1.00) = 0.95 ksi, as described in 

Section C9.2.2 of this Appendix and UHPC Guide Article 1.5.3. In this design example, 

the tensile stresses due to Service I and Service III are identical; therefore, the tension stress 

due to Service III must not exceed 0.95 ksi for the member to pass the fatigue limit state 

requirements of UHPC Guide Article 1.5.3.

C8.2. STRESSES AT MIDSPAN

UHPC stresses at service loads are found by superposition of the following components:

• Instantaneous elastic stresses at transfer, using the precast transformed section properties.

• Long-term effects between transfer and deck placement, applied to the precast gross section 

properties.
228

• Instantaneous elastic stresses at deck placement using precast transformed section 

properties.

• Instantaneous elastic stresses due to superimposed dead loads, using composite 

transformed section properties.

• Long-term effects between deck placement and final time, except for effects of deck 

shrinkage, applied to the composite section properties.

• Long-term effects of deck shrinkage.

• Instantaneous elastic stresses due to live loads using composite transformed section 

properties.

The calculations for each of these items are detailed as follows.

A. At transfer, as previously calculated in Section C7.4 of this Appendix:

Stress at top of UHPC girder = fti = +1.924 ksi.

Stress at bottom of UHPC girder = fbi = +6.820 ksi.

Stress in UHPC at the level of the steel centroid, fcgp = 6.218 ksi.

Instantaneous prestress losses at transfer, ∆fpi = 29.67 ksi.

Stress in steel at transfer, fpt = fpi − ∆fpi = 173.83 ksi.

B. Between transfer and deck placement:

Long-term loss of steel stress, ∆fpLTD = ∆fpSR + ∆fpCR + ∆fpR1 = 24.80 ksi.

Long-term loss of steel force, ∆PpLTD = ∆fpLTD (Aps) = 364.5 kips.

This force is applied as a negative prestress to the precast UHPC section.

Change in stress at top of UHPC girder, ∆ftLTD = (−∆PpLTD)/Ag + ∆PpLTD (epg)/St = +0.182 ksi.

Change in stress at bottom of UHPC girder,

∆fbLTD = (−∆PpLTD)/Ag − ∆PpLTD (epg)/Sb = −1.250 ksi.

C. At deck placement:

MD,m = 2,901.4 kip-ft.

The moment due to deck weight is applied to the noncomposite transformed section at the final 

time.

Change in stress at top of UHPC girder, ∆ftD = MD,m/Sttf = 2,901.4(12)/6,713 = +5.186 ksi.
229

Change in stress at bottom of UHPC girder, 

∆fbD = −MD,m/Sbtf = −2,901.4(12)/10,793 = −3.226 ksi. 

Change in stress in UHPC at the level of steel, ∆fcD = (−MD,m etf)/Itf = −2.192 ksi. 

Prestress gain at centroid of prestressing steel, ∆fpD = ∆fcD Ep/Ec = −9.01 ksi. 

D. Due to superimposed dead load: 

MSIDL = Mb,m + Mws,m = 825.0 kip-ft.

Moment due to superimposed dead load is applied to the composite transformed section at final 

time.

Change in stress at top of UHPC girder, ∆ftSIDL = MSIDL/Sttc = +0.221 ksi. 

Change in stress at bottom of UHPC girder, ∆fbSIDL = −MSIDL/Sbtc = −0.595 ksi. 

Change in stress in UHPC at the level of steel, ∆fcSIDL = −MSIDL etc/Itc = −0.494 ksi. 

Prestress gain in steel at centroid, ∆fpSIDL = ∆fcSIDL Ep/Ec = −2.033 ksi. 

Change in stress at top of the concrete deck, ∆fdtSIDL = MSIDL/Sdtc = 0.214 ksi.

Change in stress at bottom of the concrete deck, 

∆fdbSIDL = (EcD/Ec)(MSIDL (yttc − ts − th)/Itc) = 0.127 ksi. 

E. Long-term effects between deck placement and final time, except for deck shrinkage: 

Change in steel stress due to shrinkage and creep of girder and relaxation strand, 

∆fpLDF = ∆fpSD + ∆fpCD + ∆fpR2 = 0.172 ksi. 

Long term loss of prestress force, ∆PpLDF = ∆fpLDF Aps = 2.524 kip.

This force is applied as a positive prestress to the composite section. 

Change in stress at top of UHPC girder, 

∆ftLDF = −∆PpLDF/Ac −(−∆PpLDF) (ybc − ybs)/Stc = −0.00040 ksi. 

Change in stress at bottom of UHPC girder, 

∆fbLDF = −∆PpLDF/Ac + (−∆PpLDF)(ybc − ybs)/Sbc = −0.008 ksi. 

Change in stress at top of the concrete deck, 

 

 
230

Change in stress at bottom of the concrete deck, 

F. Long-term effect of deck shrinkage: 

The change in steel stress due to deck shrinkage is a stress gain. The deck shrinkage creates a 

sagging of the girder. Effects on the girder are represented by a compressive force acting at the 

center of the deck. AASHTO LRFD BDS Eq. 5.9.5.4.3d-2 can be used to calculate the stress 

at the top and bottom fibers by replacing eccentricity of prestressing steel with those of the top 

and bottom fibers of the girder. 

Restraining force due to shrinkage of conventional concrete is calculated as: 

This force is applied at the center of the deck with an eccentricity from the center of the deck 

to the composite center, ed = ytc – tD/2 = 18.74 in. 

The corresponding change in top and bottom fiber stresses in the UHPC girder, ∆ftSS and ∆fbSS, 

respectively, are calculated below using composite nontransformed section and assuming the 

force is 100 percent effective: 

Change in stress at level of steel centroid, ∆fcdf = –0.125 ksi. 

Stress gain in steel at centroid from Section C6.3.4 of this Appendix, ∆fpSS = −0.529 ksi. 

The change in stresses in the deck due to deck shrinkage are considered in this design example. 

G. Due to live load (service load): 

Bending moment due to service load is applied to the composite transformed section at final 

time, 

MSL = MLT,m + MLL,m = 3,284.4 kip-ft

Change in stress at top of UHPC girder ΔftSL = MSL/Sttc = 0.878 ksi. 

Change in stress at bottom of UHPC girder ΔfbSL = −MSL/Sbtc = −2.368 ksi. 

 

 

 

  

 

  
231

Change in stress in UHPC at the level of steel, ∆fcSL = (−MSL etc)/Itc = −1.969 ksi. 

Stress gain in steel at centroid, ∆fpSL = (∆fcSL Ep)/Ec = −8.092 ksi. 

Change in stress at top of the concrete deck, ∆fdtSL = MSL/Sdtc = 0.851 ksi. 

Change in stress at bottom of concrete deck, 

C8.2.1. Net UHPC Girder Stresses 

Based on these calculations, the net UHPC girder stresses at midspan are as follows. 

At deck 

placement: 

Top fiber stress = fti + ∆ftLTD + ∆ftD = +7.292 ksi 

Bottom fiber stress = fbi + ∆fbLTD + ∆fbD = +2.344 ksi 

No stress limits need to be checked for this loading stage. 

At service 

(no live load):

Top fiber stress = fti + ∆ftLTD + ∆ftD + ∆ftSIDL + ∆ftLDF + ∆ftSS 

 = +8.643 ksi < +9.900 ksi OK.

Bottom fiber stress = fbi + ∆fbLTD + ∆fbD + ∆fbSIDL + ∆fbLDF + ∆fbSS 

 = +1.439 ksi > −1.00 ksi OK.

At service 

(including live 

load):

Top fiber stress = fti + ∆ftLTD + ∆ftD + ∆ftSIDL + ∆ftLDF + ∆ftSS + ftSL 

 = +9.521 ksi < +13.20 ksi OK.

Bottom fiber stress = fbi + ∆fbLTD + ∆fbD + ∆fbSIDL + ∆fbLDF + ∆fbSS + fbSL

 = −0.929 ksi > −1.00 ksi OK.

Tensile stress limit due to Service I for members subjected to fatigue loads is 

−0.95 ksi (see Section C9.2.2 of this Appendix for details). OK.

C8.2.2. Net Conventional Concrete Deck Stresses 

Based on these calculations, the net conventional concrete deck stresses at midspan are as follows. 

At service 

(no live load):

Stress at top of deck, ∆fdtSIDL + ∆fdtLDF = +0.215 ksi < +1.800 ksi OK.

Stress at bottom of deck/haunch, 

∆fdbSIDL + ∆fdbLDF = +0.127 ksi > 0.000 ksi; < +1.800 ksi OK.

 

 
232

At service 

(including live 

load):

Stress at top of deck,

∆fdtSIDL + ∆fdtLDF + fdtSL = +1.066 ksi < +2.400 ksi OK.

Stress at bottom of deck/haunch,

∆fdbSIDL + ∆fdbLDF + fdbSL = +0.632 ksi > 0.000 ksi OK.

C8.3. PRINCIPAL TENSILE STRESS IN WEBS

The provisions of UHPC Guide Article 1.9.2.3.3 and AASHTO LRFD BDS Article 5.9.2.3.3 state 

that the maximum principal stress at any location along the height of the web shall not exceed

γu ft,cr when the superstructure element is subject to the loadings of Service III limit state of 

AASHTO LRFD BDS Article 3.4.1, both before and after all losses.

The principal tensile stress limit at service limit state after losses is:

γu ft,cr = 1.00(−1.00) = −1.00 ksi

For precast sections made composite with a cast-in-place deck, AASHTO LRFD BDS

Article C5.9.2.3.3 recommends calculating the principal tensile stress at both the noncomposite 

and the composite neutral axes using the shear stress and axial stress at each location.

In this design example, the principal tensile stress check will only be computed at the critical shear 

location (location of highest shear force per UHPC Guide Article 1.7.3.2; See Section C11.1 of 

this Appendix for critical shear force location calculations) and at two locations in the web: (1) at 

the centroid of the noncomposite gross section; and (2) at the centroid of the noncomposite,

nontransformed cross section. Note that principal tensile stress calculations are presented herein 

for illustration purposes. To complete the principal tensile stress check at transfer of prestress, the 

principal stresses in the web must be computed at multiple locations within the web and at regular

intervals along the length of the beam.

C8.3.1. Principal Stress at the Centroid of the Noncomposite Gross Section

The first moment of area of the composite nontransformed section at the centroid of the 

noncomposite UHPC gross section (yb = 21.89 in.) is Qgc = 11,021.13 in.3.

The shear force applied to the UHPC gross section at the critical shear location at service (see 

Table C4.1.2-1) is:

Vg1 = Vg,crs + VD,crs = 42.6 + 72.6 = 115.2 kip

The shear force applied to the composite nontransformed gross section at the critical shear location

at service (see Table C4.1.2-1 and Table C4.2.4.1-1) is:

Vg2 = Vb,crs + Vws,crs + VLT,crs + VLL,crs = 7.0 + 13.6 + 76.6 + 39.9 = 137.1 kip

The shear stress at the centroid of the noncomposite UHPC gross section, yb = 21.89 in., is as 

follows: 
233

The horizontal stress in the web at yb = 21.89 in. at the critical shear location at service can be 

determined from an elastic analysis following the procedures of Section C8.2.1 of this Appendix 

but using the applied moments at the critical shear location given in Table C4.1.2-1 and 

Table C4.2.4.1-1. 

Assuming that (1) all 14 strands are debonded at the critical shear section (see Section C11.1 of 

this Appendix for details), (2) the change in stresses due to all prestress losses and gain due to 

conventional concrete deck shrinkage at the critical shear location is the same as those calculated 

at midspan in Section C8.2.1, and (3) that using the gross section properties of the girder and 

composite sections to determine the stresses due to the applied loads is sufficiently adequate 

(instead of transformed section properties at ends at final time), the top and bottom fiber stresses 

at the critical shear location can be calculated as follows: 

Top fiber stress: 

Bottom fiber stress: 

 

 

 

     

 

 

 

  

  
234

Horizontal stress at yb = 21.89 in.: 

The principal tension stress is: 

Tensile stress limit for UHPC is −1.00 ksi. 

Tensile stress limit due to Service I for members subjected to fatigue loads is −0.95 ksi (See 

Section C9.2.2 of this Appendix for details). 

OK.

Note that for sections controlled by the principal tension stress limit in webs at Service III load 

combination or at Service I for members subjected to fatigue loads, the transformed section 

properties at ends at final time are recommended be used to calculate the horizontal stress, fpcx, 

when the overall section is subjected to compression stresses. Such calculation will result in lower 

values of compression stresses in the section leading to lower values of fpcx, and thus a more 

conservative prediction of principal tension stresses in webs. 

C8.3.2. Principal Stress at the Centroid of the Composite Nontransformed Section 

The principal tensile stress at the location of the centroid of the composite nontransformed section 

(ybc = 40.76 in.) may control since it is closer to the top fiber of the UHPC section where the 

horizontal compression stress is lower than the stress at the bottom fiber. 

At ybc = 40.76 in.: Qgc = 11,644.55 in.3

, and Qg = 4,269.71 in.3

 

 

 
235

Tensile stress limit for UHPC is −1.00 ksi.

Tensile stress limit due to Service I for members subjected to fatigue loads is −0.95 ksi (see 

Section C9.2.2 of this Appendix for details).

OK.

Note that if the tension zone at service due to Service III extends to the web, checking the principal 

stresses at that location is recommended as well (i.e., location of maximum longitudinal stress in 

the web). In this example, the section at critical shear location is under compressive stresses over 

the whole depth.

237

CHAPTER C9. STEEL STRESS AT SERVICE LOADS

C9.1. SERVICE LIMIT STATE

The prestress limit at service is as follows:

For bonded prestressing tendons, for load combination Service I:

0.80fpy = 0.80(243 ksi) = 194.4 ksi

Stress in prestressing strands after all losses and gains at Service I limit state at level of steel 

centroid is as follows:

fpeS = fpi – ΔfpES – ΔfpLTD – ΔfpD – ΔfpSIDL – ΔfpLDF – ΔfpSS – ΔfpSL

= 202.5 – 29.67 – 24.80 – (−9.01) – (−2.03) – 0.172 – (−0.529) – (−8.09)

= 167.51 ksi

fpeS = 167.51 ksi ≤ 194.4 ksi OK.

Note that in this example Service I and Service III load combinations are identical.

C9.2. FATIGUE LIMIT STATE

C9.2.1. UHPC Compression Stress Check

UHPC Guide Article 1.5.3 states that the compressive stress due to Fatigue I load combination and 

one-half the sum of effective prestress and permanent loads shall not exceed 0.40f ́c after losses. 

0.4f ́c = 0.4(22) = 8.800 ksi

Compression stress due to Fatigue I load combination is calculated at midspan as follows:

Moment due to Fatigue I load combination is applied to the composite transformed section at 

the final time.

MFI = 1.75Mf,m = 1.75(909.6) = 1,591.7 kip-ft

Change in stress in at top of UHPC girder = MFI/Sttc = +0.426 ksi

The compression stress due to one-half of effective prestress and permanent loads is taken as half 

of the net stresses at service with no live load summarized in Section C8.2.1 of this Appendix, 

such as:

Top fiber stress = 0.5 (fti + ∆ftLTD + ∆ftD + ∆ftSIDL + ∆ftLDF + ∆ftSS) = 0.5(+8.643) = +4.322

Total stress for fatigue limit state requirement for compression stress is calculated as follows:
238

Top fiber stress = 0.426 + 4.322 = +4.747 < 8.800 ksi 

OK.

C9.2.2. UHPC Tension Stress Check 

For tension stresses, UHPC Guide Article 1.5.3 states that regions of members subjected to fatigue 

loads shall be designed such that the UHPC tensile stress due to Service I load combination does 

not exceed 0.95γu ft,cr = 0.95(1.00)(1.00)= 0.95 ksi. 

In this design example, the tensile stresses due to Service I and Service III are identical, and the 

maximum tensile stresses at midspan due to Service III calculated in Section C8.2.1 of this 

Appendix, i.e. 0.929 ksi, and Section C8.3 of this Appendix, i.e., 0.696 ksi, are lower than the 

fatigue limit of 0.95 ksi. Therefore, the stress requirement for fatigue is satisfied. 

C9.2.3. Fatigue of Reinforcement Check 

Given that UHPC tensile stress is greater than conventional concrete, UHPC Guide Article 1.5.3

states that discrete steel elements embedded in UHPC should be checked for fatigue in accordance 

with AASHTO LRFD BDS Eq. 5.5.3.1-1 and Articles 5.5.3.2, 5.5.3.3, and 5.5.3.4. This provision 

ensures that the fatigue limit on the steel reinforcement in UHPC is met even when the UHPC 

girder is uncracked. 

For fatigue considerations, the prestressing steel in UHPC members shall satisfy: 

γ(∆f ) ≤ (∆F)TH [UHPC Guide Article 1.5.3; AASHTO LRFD BDS Article 5.5.3.1] 

where: 

γ = load factor specified in AASHTO LRFD BDS Table 3.4.1-1 for Fatigue I load 

combination = 1.75

∆f = force effect, live load stress range at the bottom layer of strands due to the passage 

of the fatigue load as specified in AASHTO LRFD BDS Article 3.6.1.4 (ksi) 

(∆F)TH = constant-amplitude fatigue threshold, as specified in AASHTO LRFD BDS 

Articles 5.5.3.3 for prestressing steel 

= 18.0 ksi because the radius of the curvature of the girder exceeds 30.0 ft 

γ(∆f ) = 1.75(2.56) = 4.48 ksi < (∆F)TH = 18.0 ksi 

OK.

 
239

CHAPTER C10. STRENGTH LIMIT STATE

The factored bending moment at midspan due to dead load and live load is computed as follows:

Mu,m = 1.25(Mg,m + MD,m + Mb,m) + 1.5(Mws,m) + 1.75(MLL.m + MLT,m) = 12,668 kip-ft

In this design example, strain compatibility between the conventional concrete deck, UHPC girder, 

and prestressing steel is assumed; the strain compatibility analysis, specified in UHPC Guide 

Article 1.6.3.1, is utilized to determine the ultimate flexural capacity. Figure C10-1 shows the steps 

followed in the strain compatibility analysis. 

Source: FHWA.

Note: See List of Notations for variable definitions.

Figure C10-1. Illustration. Strain compatibility analysis of composite section.

Note that in strain compatibility of conventional concrete girders, the locked-in stress between the 

top of the UHPC girder and the concrete deck is sometimes ignored. The final strain diagram in 

the composite section is then assumed to be linear (i.e., no discontinuity at the top of the girder), 

and the precompression strain is added to the change in the steel strain in the final calculations of 

stresses. This assumption may be acceptable in some cases where the neutral axis is in the deck. 

However, in UHPC girders, this assumption will not be made because the neutral axis may not be 

in the deck at ultimate flexural strength. 

Given the stresses in the UHPC girder at the deck placement may be significant, the initial strains 

due to girder self-weight and all losses and gains are calculated in this example. The strain 

compatibility analysis is performed by assuming a strain profile due to applied load and adding it 

to the initial strain profile to obtain the total strains as shown in Figure C10-1. 

The initial stress profile in the UHPC can be found by calculating the stresses at the top and bottom 

of the conventional concrete deck (denoted by the subscripts 1 and 2, respectively) and at top and 

bottom of the UHPC girder (denoted by the subscripts 3 and 4, respectively) (Figure C10-1). The 

initial strain in the UHPC at each of these four locations and the initial tensile strain at the centroid 

of the strands due to prestressing can be found by the superposition of the following components:
240

• Instantaneous elastic stresses at transfer, using the precast transformed section properties.

• Long-term effects between transfer and deck placement, applied to the precast gross section 

properties.

• Instantaneous elastic stresses at deck placement due to deck self-weight, using precast 

transformed section properties.

• Long-term effects between deck placement and final time, except for effects of deck 

shrinkage, applied to the composite section properties.

• Long-term effects of deck shrinkage.

Note that the calculation of each of the preceding listed stresses can be found in Section C8.2.

Stresses and strains in the composite section for strain compatibility calculations are calculated as

follows:

At the top of deck, fin,1 = ΔfdtLDF = 0.001 ksi

εin,1 = fin,1 /EcD = 0.0000001

At the bottom of the 

deck/haunch,

fin,2 = ΔfdbLDF = −0.00024 ksi

εin,2 = fin,2 /EcD = 0.0000000

At the top of the 

girder,

fin,3 = ft + ΔftLTD + ΔftD + ΔftLDF + ΔftSS = 8.423 ksi

εin,3 = fin,3 /Ec = 0.0012148

At the bottom of the 

girder,

fin,4 = fb + ΔfbLTD + ΔfbD + ΔfbLDF + ΔfbSS = 2.034 ksi

εin,4 = fin,4 /Ec = 0.0002933

In each layer of 

prestressing strands

(taken equal to initial 

strain at centroid),

fpin,j = fpi − ΔfpES − ΔfpLTD − ΔfpD − ΔfpLDF − ΔfpSS = 157.4 ksi

εpin,j = fpin /Ep = 0.0055230

In calculating the bending moment at a desired strain profile, the section is discretized into 

0.5-in.-thick horizontal layers (denoted by i) through the overall height of the composite section; 

the stress and strain in each layer are assumed to be constant. The strains in each layer are inferred 

from the strain profiles of Figure C10-1. The total strain in each layer of steel, UHPC, and 

conventional concrete is calculated as follows:

εT,i = εin,i + ∆εi at each layer i of the UHPC or conventional concrete

εpsT,j = εpin,j + ∆εcs,j at each layer j of prestressing steel
241

The stresses in each of the conventional concrete, UHPC, and steel layers are calculated by 

employing the mechanical models of conventional concrete, UHPC, and steel and utilizing the 

total strains of Figure C10-1: 

• For UHPC, the mechanical models described in UHPC Guide Articles 1.4.2.4.3 and 

1.4.2.5.4 are employed. 

• For conventional concrete, the model suggested by Popovics (1973) and later modified by 

Thorenfeldt, Tomaszewicz, and Jensen (1987), with the curve fitting parameters proposed 

by Collins and Mitchell (1991), is utilized to determine the stress in each of the layers in 

the deck, fci: 

where: 

• For prestressing steel, the power formula recommended by the PCI Bridge Design Manual

is utilized to determine the total stress in each layer of strands (PCI 2014): 

The stresses in each of the conventional concrete, UHPC, and steel are then converted into 

compression and tension forces, which are summed to check force equilibrium in the section. The 

process is repeated, assuming different change in strain profiles, until equilibrium is established, 

after which the flexural moment is computed. 

In this design example, four failure modes exist for the prestressed UHPC girder with a 

conventional concrete deck: 

• Mode I: Deck concrete crushing, εT,1 = 0.003. 

• Mode II: UHPC crushing, εT,3 = εcu = 0.0035. 

• Mode III: UHPC tensile strain limit, εT,4 = γuεt,loc = −0.0040. 

• Mode IV: Steel strand rupture at bottom steel layer, εpsT = −0.035.

 

 

 

 
242

In prestressed UHPC girders, the third mode of failure (i.e., UHPC crack localization) typically 

controls; therefore, the strain compatibility analysis is first performed to find the stress and strain 

profiles at this failure mode. Figure C10-2 shows the results of this analysis.

Source: FHWA.

Figure C10-2. Illustration. Mode III: UHPC crack localization.

As shown in Figure C10-2, when the UHPC strain in the extreme tensile layer reaches the UHPC 

tensile strain limit (εT,4 = γuεt,loc = −0.0040), the values of the tensile strains in the extreme 

compression layer of the conventional concrete deck (εT,1 = +0.00118) and the UHPC girder 

(εT,3 = +0.00153) are lower than their respective ultimate limit compression strain of 0.003 and 

0.0035, respectively. Therefore, the nominal bending capacity of the section is equal to the moment 

at UHPC crack localization (Mn = ML). Note that any further increases in the top strain will cause 

the strains in the tension to increase beyond the localization strain capacity of UHPC, which is not 

allowed per UHPC Guide Article 1.6.3.2.2. The postlocalization behavior is not recommended to 

be considered in the design because the loss of UHPC fiber bridging capacity may cause hinging 

of the beam at the localized crack section and the straining of the tensile reinforcement over a short 

distance, thus increasing the risk of reinforcement rupture. The details of the strain compatibility 

analysis results and the bending moment at localization are presented in Table C10-1.

Table C10-1. Summary of the strain compatibility analysis.

Parameter

(Msl, ψsl)

Steel Service

(ML, ψL) or (Mn, ψn)

Crack Localization

M (kip-ft) 12,502 16,131

c (in.) 39.67 24.98

εT,1 0.00058 0.00118

εT,2 0.00026 0.00032

εT,3 0.00147 0.00153

εT,4 −0.00121 εt,loc = −0.0040

εpsT,b (bottom steel layer) εsl = −0.00696 −0.00964

ψ (in.−1) × 10−40.4965 1.0254

Note: Neutral axis, c, is measured from the top of the composite section.

Note: See List of Notations for variable definitions.
243

C10.1. FACTORED BENDING MOMENT CAPACITY 

The resistance factor for flexural capacity is based on a ductility performance threshold expressed 

in terms of a curvature ductility ratio [UHPC Guide Article 1.5.4.2]. 

The curvature ductility ratio is defined as: 

[UHPC Guide Eq. 1.6.3.2.3-1] 

where: 

ψn = sectional curvature at nominal flexural strength 

ψsl = sectional curvature when the steel stress in the extreme tension steel is equal to the steel 

service stress limit, fsl

in which fsl is the stress limit in steel at service loads after losses = 0.80 fpy = 194.4 ksi.

[AASHTO LRFD BDS Table 5.9.2.2-1] 

To determine the sectional curvature when the steel stress in the extreme tension steel is equal to 

the steel service stress limit, a strain compatibility analysis was performed to find the strain profile 

at equilibrium of forces when the total strain in the extreme prestressing steel layer is equal to 

εsl = −0.00696 (determined from the strand power formula equation presented above). The strain 

and stress profiles obtained at the end of this analysis are shown in Figure C10.1-1. The details of 

the results are presented in Table C10-1.

Source: FHWA. 

Figure C10.1-1. Illustration. Strands reach steel service limit. 

The resistance factor can be calculated in accordance with UHPC Guide Articles 1.5.4.2 and 

1.6.3.2.3: 

Ultimate 

moment: Mn = ML = 16,131 kip-ft
244

Curvature at 

localization: 

Curvature at steel 

service: 

Curvature 

ductility ratio 

limit:

μl = 3.0 [UHPC Guide Article 1.6.2]

Curvature 

ductility ratio: 

Resistance 

factor: 

Therefore: 

φMn = φf ML = (0.830)(16,131) = 13,387 kip-ft > 12,668 kip-ft OK. 

C10.2. MINIMUM REINFORCEMENT 

Article 1.6.3.3 of the proposed UHPC Guide and Article 5.6.3.3 of AASHTO LRFD BDS states 

that the amount of prestressed and nonprestressed tensile reinforcement shall be adequate to 

develop a factored flexural resistance, Mr, greater than or equal to the lesser of the following: 

• 1.33 times the factored moment required by the applicable strength load combination, Mu. 

•

where: 

Mcr = cracking moment (kip-ft) 

ft,cr = UHPC effective cracking strength = 1.00 ksi 

fcpe = compressive stress in UHPC due to effective prestress forces only at extreme fiber 

section, where tensile stress is caused by externally applied loads 

Mdnc = total unfactored dead load moment acting on the monolithic or noncomposite section 

 = Mg,m + MD,m = (1,701.1 + 2,901.4)(12) = 55,230 kip-in. 

γ1 = flexural cracking variability factor 

 = 1.6 for concrete structures other than precast segmental structures 

γ2 = prestress variability factor 

 = 1.1 for bonded tendons 

γ3 = ratio of specified minimum yield strength to ultimate tensile strength of the 

nonprestressed reinforcement 

 

   

 
245

= 1.0 for prestressing steel 

in which: 

Ppee = force in prestressing strands due to effective prestress only 

= fpee ApT = (178.05)(14.70) = 2,617.3 kips 

fpee = effective prestress in strands after allowance of all losses 

= fpi − ∆fpLTD − ∆fpLDF − ∆fpSS = 178.05 ksi 

Therefore: 

= 140,368 kip-in. = 11,698.4 kip-ft

and 1.33Mu = (1.33)(12,668) = 16,848.5 kip-ft

Since 1.33Mu = 16,848.5 kip-ft > Mcr = 11,697.3 kip-ft, the factored flexural resistance, Mr, must 

be greater than Mcr = 11,698.4 kip-ft.

In this design, Mr = 13,387 kip-ft is greater than Mcr = 11,698.4 kip-ft; therefore, the section 

satisfied the minimum reinforcement requirements. 

 

247

CHAPTER C11. SHEAR DESIGN

The area and spacing of shear reinforcement must be determined at regular intervals along the 

length of the beam. Transverse reinforcement shall be provided where:

Vu > φv (VUHPC + Vp) [UHPC Guide Eq. 1.7.2.3-1]

where:

Vu = factored shear force (kip)

φv = resistance factor for shear = 0.9 [UHPC Guide Article 1.5.4.2]

VUHPC = nominal shear resistance of the UHPC (kip)

Vp = component of prestressing force in the direction of shear force; positive if resisting 

the applied shear = 0 kips

or where consideration of torsion is required. In this design example, torsion is not considered and 

the shear design is only performed at the critical shear location.

C11.1. CONTRIBUTION OF UHPC TO SHEAR RESISTANCE

Effective shear depth, dv, is taken as the distance, measured perpendicular to the neutral axis, 

between the resultants of the tensile and compressive forces due to flexure. It need not be taken 

less than the greater of 0.9de or 0.72hc, in which de is the distance between the extreme compression 

fiber of the UHPC section to the resultant of the forces in the tensile reinforcement.

[AASHTO LRFD BDS Article 5.7.2.8]

However, according to UHPC Guide Article 1.7.2.8, in composite sections made with UHPC 

beams and conventional concrete decks, the effective shear depth, dv, shall not exceed the distance, 

dv,limit, measured perpendicular to the neutral axis, between the resultant of the forces in the tensile

reinforcement and the extreme UHPC fiber on the flexural compression side.

The distance from the extreme bottom fiber of the UHPC girder to resultant of the forces in the 

tensile reinforcement is (9 × 2 + 9 × 4 + 9 × 6 + 7 × 8)/(9 + 9 + 9 + 7) = 4.82 in. Note that this 

calculation only considers the strands in the flexural tensile zone and assumes all 14 strands are 

debonded at the critical shear location (which is a conservative estimation based on the debonding 

pattern of Figure C5.1-2.

Therefore, the effective shear depth, dv, is calculated as follows:

de = hc − ybsE = 56.56 in.

max (0.9de, 0.72hc) = 50.90 in. = 4.24 ft

dv,limit = h − 4.82 = 49.18 in. = 4.10 ft [UHPC Guide Article 1.7.2.8]

The value of dv,limit = 4.10 ft is less than the value of the greater of 0.9de and 0.72hc and thus, the 

effective shear depth need not be less than 4.10 ft. Therefore, for shear calculations, dv = 4.10 ft.
248

According to UHPC Guide Article 1.7.3.2, the location of the critical shear section is taken at 

dv = 4.10 ft from the face of the support. Assuming a support width of 1 ft, the location of the 

critical section for shear is at 4.10 ft + 0.5 ft = 4.60 ft from the centerline of the beam’s supports. 

The factored shear force due to dead load and live load at critical section is: 

Vu,crs = 1.25(Vg,crs + VD,crs + Vb,crs) + 1.5Vws,crs + 1.75(VLL,crs + VLT,crs) 

= 1.25(42.6 + 72.6 + 7.0) + 1.5(13.6) + 1.75(39.9 + 76.6)

= 377.0 kips 

Factored moment due to dead load and live load at critical shear section is: 

Mu,crs = 1.25(Mg,crs + MD,crs + Mb,crs) + 1.5Mws,crs + 1.75(MLL,crs + MLT,crs) 

= 1.25(202.3 + 345.0 + 33.4) + 1.5(64.7) + 1.75(140.1 + 260.7) 

= 1,524.3 kip-ft

The net longitudinal tensile strain at the centroid of tensile reinforcement at the critical shear 

section is determined by: 

[UHPC Guide Eq. 1.7.3.4.1-6]

where: 

Mu,crs = factored bending moment at section at the critical shear location, not taken less than 

|Vu – Vp| dv = 377.0 × 4.10 = 1,545.8 kip-ft; thus, Mu,crs = 1,545.8 kip-ft

Nu,crs = factored axial force at the critical shear location = 0 kip 

fpo = parameter equal to modulus of elasticity multiplied by the locked-in difference in 

strain and surrounding UHPC which may be taken equal to 0.7fpu = 189 ksi for 

prestressed strands 

Act = area of UHPC on the flexural tension side of the member 

 = 360.05 – 48(0.294) = 345.94 in.2

Aps,crs = area of prestressing steel on the flexural tension side at the critical shear location 

As,crs = area of nonprestressed steel on the flexural tension side of the member at the critical 

shear location = 0 in.2

According to UHPC Guide Article 1.7.3.4.1, if the location of the critical shear section is within 

the transfer length section, fpo in Eqs. 1.7.3.4.1-6 and 1.7.3.4.1-7 shall be increased linearly from 

zero at the location where the bond between the strands and UHPC commences to its full value at 

the end of the transfer length. Moreover, if the critical shear location falls within the development 

length of the prestressing strands, the area of the strands, Aps,crs, shall be reduced in proportion of 

the strands’ lack of development. This condition will be checked by calculating the transfer and 

development length of the strands. 

Given that 14 strands are debonded at each end of the girder, a debonding pattern must be 

determined based on the restrictions of UHPC Guide Article 1.9.4.3.3 before the values of fpo and 

Aps,crs can be computed. The debonding pattern shown in Figure C11.1-1 is chosen for this example 

in which the debonding of six, four, and four strands is terminated at a longitudinal spacing equal 

to the transfer length for shear considerations, lt.

 
249

Source: FHWA. 

Figure C11.1-1. Illustration. Debonding pattern of strands. 

In calculating the transfer length for shear design considerations, longer transfer and development 

length values will result in more critical stress states generated by the applied shear demand. This 

result is because longer development length values will increase the likelihood of the critical shear 

location to fall at a location where the strands are not fully developed. Therefore, the factor for 

transfer length, ξ, should be taken as 1.0. 

Transfer length, lt = ξ24Dp = 1.00(24)(0.7) = 16.8 in. or 1.4 ft 

[UHPC Guide Eq. 1.9.4.3.1-1] 

As shown in Figure C11.1-1, the location of the critical shear section is within the transfer length 

of the last 4 bonded strands; therefore: 

The development length of the strands, ld, can be calculated according to UHPC Guide 

Article 1.9.4.3.2 as follows: 

ld ≥ lt + 0.30(fps − fpe,ng) Dp [UHPC Guide Eq. 1.9.4.3.2-1] 

where: 

fps = average stress in prestressing steel at the time for which the nominal resistance of the 

member is required, taken equal to the ultimate capacity of the strands = 270 ksi 

fpe,ng = effective stress in the prestressing steel after losses (no gains) 

= fpi − ∆fpT = 202.5 − 54.11 = 148.39 ksi 

Therefore: 

ld ≥ lt + 0.30(fps − fpe,ng) Dp = 16.8 + 0.30(270 – 148.39)(0.7) = 42.34 in. or 3.53 ft

As shown in Figure C11.1-1, the location of the critical shear section is within various locations 

of the development length of the debonded strands; therefore: 
250

The value of the εs can be now calculated according to UHPC Guide Eq. 1.7.3.4.1-6 as: 

Note that in calculating εs according to Eq. 1.7.3.4.1-6, the limit on dv in composite sections need 

not apply [UHPC Guide Article 1.7.3.4.1], and, thus, the value of dv in the calculation above was 

taken equal to the greater of 0.9de or 0.72hc, i.e., 4.24 ft. 

Since the value of εs calculated from UHPC Guide Eq. 1.7.3.4.1-6 is negative, it should be either 

taken as εt,cr, or recalculated using Eq. 1.7.3.4.1-7: 

[UHPC Guide Eq. 1.7.3.4.1-7] 

C11.2. CONTRIBUTION OF STEEL REINFORCEMENT TO SHEAR RESISTANCE 

C.11.2.1. General Approach to Determine θ and fv

The general approach for determining shear resistance parameters θ and fv is to iteratively solve 

UHPC Guide Eqs. 1.7.3.4.1-1 through 1.7.3.4.1-4. For sections with transverse steel reinforcement 

perpendicular to the longitudinal axis (α = 90 degrees), equations reduce to: 

[UHPC Guide Eq. C1.7.3.4.1-3] 

[UHPC Guide Eq. C1.7.3.4.1-2] 

Strain in vertical steel, εv = γu εt,loc − 0.5εs + ε2 [UHPC Guide Eq. 1.7.3.4.1-3] 

Stress in vertical steel, fv = Es εv ≤ fy [UHPC Guide Eq. 1.7.3.4.1-4] 

in which: 

[UHPC Guide Eq. 1.7.3.4.1-5] 

For this design example, try No. 4 Grade 60 reinforcing bars at a spacing of 6 in.: 

Spacing of transverse reinforcement s = 6 in. 

  

 

 

 
251

Area of transverse reinforcement Av = 0.20 in.2

Yield stress of transverse steel fy = 60 ksi 

Modulus of elasticity of transverse steel Es = 29,000 ksi 

Shear reinforcement ratio 

Set the steel stress at shear failure to steel yield stress, fv = fy = 60 ksi. 

Assume a principal angle direction and begin the iterative process to obtain a localization strain 

equal to the localization strain of UHPC. 

Using θ = 29.13 degrees: 

  OK

ε2 = 

 = −0.00162

εv = εt,loc − 0.5εs + ε2 = 0.00267

Check fv, fv,Check = min(Es εv, fy) = 60.0 ksi OK. 

Note: If fv,Check was not equal to the assumed value of fv, a new value of fv should be assumed and 

the iteration process repeated until the assumed values of θ and fv converge. Convergence occurs 

when both the pair of assumed values for θ and fv results in a value of γuεt,loc equal to 0.004 [UHPC 

Guide Eq. C1.7.3.4.1-3] and the assumed value of fv is equal to the calculated value [UHPC Guide 

Eq. 1.7.3.4.1-4].

Transverse reinforcement shear contribution, 

[UHPC Guide Eq. 1.7.3.3-4] 

  

  
252

C.11.2.2. Simplified Approach to Determine θ and fv

If a simplified approach to determine the shear resistance parameters θ and fv,α described in UHPC 

Guide Article 1.7.3.4.2 is used, the contribution of the steel reinforcement to the shear resistance 

can be calculated as follows: 

• The reinforcement ratio ρv = 0.0095 < 0.01. Therefore, Table A2.3-2 of the UHPC Guide 

can be used to estimate the value of the inclination angle θ and the maximum allowable 

stress in steel, fv,max. For γuεt,loc = 0.004 ≥ 0.004 and εs = −0.00058 ≤ −0.0005, θ = 32.8 and 

fv,max = 64.8 ksi. Since fv = 60.0 ksi < fv,max, use fv = 60.00 ksi. 

• Transverse reinforcement shear contribution, 

[UHPC Guide Eq. 1.7.3.3-4] 

C.11.2.3. Maximum Spacing of Transverse Reinforcement 

The maximum spacing of transverse reinforcement cannot exceed smax, determined as: 

smax = 0.25dv,limit cot θ ≤ 24.0 in. [UHPC Guide Eq. 1.7.2.5-1] 

Using θ = 29.13 degrees obtained from the general approach: 

smax = 0.25(49.18) cot(29.13) = 22.07 in. ≤ 24.0 in. 

Using θ = 32.8 degrees obtained from the general approach: 

smax = 0.25(49.18) cot(32.8) = 19.08 in. ≤ 24.0 in. 

Given that the provided spacing of transverse reinforcement, s = 6 in., is less than smax = 22.07 in. 

using general approach or smax = 19.08 in. from the simplified approach, the reinforcement spacing 

is compliant with the provisions of UHPC Guide Article 1.7.2.6. 

C11.3. TOTAL SHEAR RESISTANCE

The total shear resistance of the beam is calculated from the general approach as follows: 

UHPC shear resistance, VUHPC = γu ft,loc dv,limit bv cot θ = 308.9 kip [UHPC Guide Eq. 1.7.3.3-3] 

Total shear resistance, Vn = VUHPC + Vs = 485.3 kip [UHPC Guide Eq. 1.7.3.3-1] 

Factored shear resistance, φvVn = 436.8 kip > Vu = 377.0 kip OK. 

Maximum shear resistance: Vu,max = 0.25 f ́c dv bv = 946.6 kip > Vn = 485.3 kip OK.

[UHPC Guide Eq. 1.7.3.3-2] 
253

The total shear resistance of the beam is calculated from the simplified approach as follows:

• UHPC shear resistance, VUHPC = γu ft,loc dv bv cot θ = 267.1 kip

[UHPC Guide Eq. 1.7.3.3-3]

• Total shear resistance, Vn = VUHPC + Vs = 419.7 kip [UHPC Guide Eq. 1.7.3.3-1]

• Factored shear resistance, φvVn = 377.7 kip > Vu = 377.0 kip

OK.

255

CHAPTER C12. INTERFACE SHEAR TRANSFER 

C12.1. FACTORED HORIZONTAL SHEAR 

At the strength limit state, horizontal shear at a section per unit can be taken as: 

[AASHTO LRFD BDS Eq. C5.7.4.5-7] 

where: 

Vui = horizontal factored shear force per unit length of the girder (kip/in.) 

Vu = factored shear force due to total load based on the applicable strength and extreme event 

load combinations, kip 

dvi = distance between the centroid of the tension steel and the mid-thickness of the deck. 

= de – ts/2 = 56.56 − 8.5/2 = 52.31 in. 

The location of the critical section is assumed to be the same location as the critical section for 

vertical shear, 3.88 ft from face of support. 

The factored shear force due to Strength I load combination at the critical shear section is:

Vu,crs = 377.0 kip 

Applied factored horizontal shear is: 

Vui = 377.0 kip/52.31 in. = 7.21 kip/in. 

C12.2. REQUIRED NOMINAL RESISTANCE 

Required nominal shear resistance Vni = Vui/φv = 7.21/0.9 = 8.01 kip/in. 

[AASHTO LRFD BDS Eq. 5.7.4.3-1] 

C12.3. REQUIRED INTERFACE SHEAR REINFORCEMENT 

The nominal shear resistance of the surface is: 

Vni = ciAcv + μi(Avf fy + Pc) [UHPC Guide Eq. 1.7.4.3-4] 

where: 

ci = cohesion factor (ksi) 

μi = friction factor 

Acv = area of UHPC considered to be engaged in interface shear transfer (in.2)

Avf = area of interface reinforcement crossing the shear plane within the area Acv (in.2) 

fy = specified minimum yield strength of reinforcement; it shall not be taken greater than 

60 ksi (ksi) 

Pc = permanent net compressive force normal to the horizontal shear plane; if force is 

tensile, Pc = 0.0 kip 
256

The nominal shear resistance shall not exceed Vni ≤ K Acv [UHPC Guide Eq. 1.7.4.3-5]

where: Acv = bvi Lvi [UHPC Guide Eq. 1.7.4.3-6]

in which:

Acv = area of UHPC considered to be engaged in interface shear transfer

bvi = interface width considered to be engaged in shear transfer

K = limiting interface shear resistance

Lvi = interface length considered to be engaged in shear transfer

For this example:

bvi = 21 in.

Lvi = 1 in.

Acv = 21 in.2

Pc = 0 kip

For conventional concrete placed against a clean UHPC substrate surface, free of laitance, with 

surface intentionally roughened to an amplitude of 0.25 in. or cast to have 0.25-in. amplitude 

roughness: [UHPC Guide Article 1.7.4.4]

ci = 0.075 ksi

μi = 1.00

K = 1.8 ksi

Solving UHPC Guide Eq. 1.7.4.3-4 for Avf :

8.01 kip/in. = (0.075)(21) + 1.00(Avf fy + 0) → Avf fy = 6.435 kip/in.

Avf (required) = (6.435 kip/in.)/60 ksi = 0.107 in.2/in.

Provide two legs of No. 5 Grade 60 stirrup at 6-in. spacing and extend the No. 4 shear 

reinforcement bars across the interface zone.

Spacing of transverse reinforcement, s = 6 in.

Area of interface shear reinforcement = 2(0.31 in.2) + 1(0.20 in.2) = 0.82 in.2

Therefore, Avf (provided) = 0.82/6 = 0.137 in.2/in. ≥ 0.107 in.2/in. OK.

Therefore, the nominal shear resistance of the interface plane is:

Vni = (0.075)(21) + 1.00((0.137)(60) + 0) = 9.775 kip/in. and φvVni = 8.79 kip/in.

Vui = 7.21 kip/in. < φvVni = 8.789 kip/in. OK.
257

C12.4. MINIMUM AREA OF INTERFACE SHEAR REINFORCEMENT

For a cast-in-place concrete slab on clean concrete girder surfaces free of laitance, Article 1.7.4.2

of UHPC Guide and Article 5.7.4.2 of AASHTO LRFD BDS (AASHTO 2020) state that the 

minimum interface shear reinforcement, Avf, need not exceed the lesser of the amount determined 

using Eq. 5.7.4.2-1 and the amount needed to resist 1.33Vui/φv as determined using Eq. 1.7.4.3-4.

The minimum shear resistance calculated according to Eq. 5.7.4.2-1 of AASHTO LRFD BDS 

(AASHTO 2020) is:

Avf ,min = 0.05 Acv/fy = 0.05(21)/60 = 0.0175 in.2/in.

The amount of reinforcement needed to resist 1.33Vui/φv = 1.33 (7.21)/0.9 = 10.65 kip/in. is

calculated as follows:

10.65 kip/in. = (0.075)(21) + 1.00(Avf fy + 0) → Avf fy = 9.080 kip/in.

Avf = 9.080 kip/in. ⁄ 60 ksi = 0.151 in.2/in.

Therefore, the minimum area of interface shear reinforcement shall be Avf,min = 0.0175 in.2/in. The 

provided area of interface shear reinforcement, Avf = 0.151 in.2/in., exceeds the minimum and 

therefore the provisions of Article 1.7.4.2 of UHPC Guide are satisfied.

C12.5. MAXIMUM NOMINAL INTERFACE SHEAR RESISTANCE

The maximum nominal shear resistance is checked as follows:

Vni ≤ KAcv

Vni = 8.789 kip/in. ≤ K Acv = (1.8)(21) = 37.80 kip/in. OK.

259

CHAPTER C13. MINIMUM LONGITUDINAL REINFORCEMENT REQUIREMENT

According to UHPC Guide Article 1.7.3.5, at each section, the tensile capacity of the longitudinal 

reinforcement on the flexural tension side shall be proportioned to satisfy: 

[UHPC Guide Eq. 1.7.3.5-1]

where: 

φf, φc, φv = resistance factors taken from UHPC Guide Article 1.5.4.2 as appropriate for 

moment, axial resistance, and shear, respectively 

Vs = shear resistance provided by transverse reinforcement at the section under 

investigation, as given by UHPC Guide Eq. 1.7.3.3-4, except Vs shall not be taken 

as greater than Vu/φv

θ = angle of inclination of diagonal compressive stresses used in determining the 

nominal shear resistance of the section under investigation 

Mu = factored moment at the section corresponding to the factored shear force 

Vp = component in the direction of the applied shear of the effective prestressing force 

As = area of nonprestressed tensile reinforcement 

Note that the term Es εt,loc shall not exceed the specified yield strength of reinforcing bars, fy. 

In this design example, the required longitudinal tensile reinforcement check is performed only at 

the inside edge of the bearing area of the supports. To complete the design, the longitudinal 

reinforcement requirement must be checked at regular intervals within the span of the beam. 

According to UHPC Guide Article 1.7.3.5, at the inside edge of the bearing area of simple end 

supports to the section of critical shear, the longitudinal reinforcement on the flexural tension side 

of the member shall satisfy: 

[UHPC Guide Eq. 1.7.3.5-2]

Note that the term Es εt,loc shall not exceed the specified yield strength of reinforcing bars, fy. 

According to AASHTO LRFD BDS Article C5.7.3.5, the values of Vu, Vs, Vp, and θ calculated for 

the section located at dv from the face of support may be used for determining the tensile force that 

the reinforcement is expected to resist at the inside edge of the bearing area. Therefore, Vu = Vu,crs

= 377.025 kip, Vp = 0 kip, Vs = 176.57 kip (using the general approach of Article C.11.2.1), and θ

= 29.13 degrees. 

When the required longitudinal reinforcement at the face of the bearing is checked, longer transfer 

lengths will necessitate the use of more reinforcement; therefore, the factor for transfer length, ξ, 

is taken equal to 1.0. Therefore, lt = 16.8 in. [UHPC Guide Article 1.9.4.3.1] 

  

 

  
260

Assuming a distance of 6 in. between the end of the girder and the center of the support (centerline 

of bearing), the horizontal distance from the end of the girder to the location where the critical 

shear crack plane crosses the centroid of the bottom strands (within the flexural tension zone) can 

be calculated as: 

dcrack = 6 in. + ybcb cot θ = 6 + 4.82 cot(29.13) = 14.65 in. 

where, ybcb = 4.82 in. is the distance between the extreme tension fiber and the centroid of the 

strands on the flexural tension side at beam ends and within the debonded length (i.e., 14 debonded 

strands). 

Therefore, the average stress in the prestressing strand at the time that nominal resistance is 

required can be taken as: 

The right-hand side of UHPC Guide Eq. 1.7.3.5-2 can be computed as follows: 

Aps fps + As Es γu εt,loc + 0.60Act γu ft,cr = (34)(0.294)(129.4) + 0 + 0.60(345.94)(1.00)(1.00) 

= 1,501.58 kip 

The left-hand side of UHPC Guide Eq. 1.7.3.5-2 can be computed as follows: 

Given that, capacity of the tensile reinforcement and UHPC, i.e., right-hand side of UHPC Guide 

Eq. 1.7.3.5-2, is greater than the demand, i.e., the left hand side of UHPC Guide Eq. 1.7.3.5-2, 

additional flexural tensile steel reinforcements are not needed. 

 

 
261

CHAPTER C14. PRETENSIONED ANCHORAGE ZONE 

C14.1. ANCHORAGE ZONE REINFORCEMENT 

The design of the anchorage zone reinforcement is computed using the force in the strands just 

before transfer: 

PpiE = 2,143.26 kip 

The factored splitting resistance, Pr, should not be less than 4.0 percent of Ppi. 

Pr = fs As,r + Pr,UHPC [UHPC Guide Eq. 1.9.4.4-1] 

in which: 

Pr,UHPC = 0.25γu ft,cri bv h [UHPC Guide Eq. 1.9.4.4-2] 

where: 

fs,r = stress in steel not to exceed 20.0 ksi 

As,r = total area of reinforcement within the distance h/4 from the end of the beam 

h = overall height of the member = 54 in. 

bv = effective web width taken as the minimum web width within the depth = 3.5 in. 

Therefore: 

Pr = 0.04PpiE = 85.73 kip 

Pr,UHPC = 0.25(1.00)(0.75)(3.5)(54) = 35.44 kip 

At least 2.52 in.2

 of vertical transverse reinforcement should be provided within a distance of 

h/4 = 13.5 in. from the end of the beam. 

Use five bundles of two No. 5 bars at 3 in. spacing center to center. 

Provided As = 5(2)(0.31) = 3.10 in.2 > 2.52 in.2 OK.

C14.2. CONFINEMENT REINFORCEMENT 

For the distance of 1.5h = 81 in. from the end of the beam, reinforcement shall be placed to confine 

the prestressing steel in the bottom flange. The reinforcement shall not be less than No. 3 deformed 

bars with spacing not exceeding 6.0 in. and shaped to enclose the strands. 

[UHPC Guide Article 1.9.4.4.2; AASHTO LRFD BDS Article 5.9.4.4.2] 

 

263

CHAPTER C15. DEFLECTION AND CAMBER 

Deflections are calculated using the modulus of elasticity of UHPC and the gross section properties 

of the noncomposite girder section. [UHPC Guide Article 1.6.3.5] 

C15.1. DEFLECTION DUE TO PRESTRESSING FORCE AT TRANSFER 

Force per strands after transfer, Pp = fpt Ap = 50.81 kip.

in which: 

e′ = difference between eccentricity of the prestressing strand at midspan and at end of the 

beam = epg – epgE = 15.25 – 14.45 = 0.80 in.

where: 

Δp = camber due to prestressing force at transfer 

Ppt = total prestressing force after transfer at midspan = 50 (50.8) = 2,540.6 kip 

Eci = modulus of elasticity of UHPC at transfer = 5,973 ksi 

Ig = moment of inertia of the noncomposite precast beam = 209,570.6 in.2 

epg = eccentricity of strands in noncomposite section at midspan = 15.25 in.2 

a = distance from the girder end to the harp point of strands = 0 ft 

LT = overall beam length = 151 ft 

Therefore: 

C15.2. DEFLECTION DUE TO BEAM SELF-WEIGHT 

The deflection due to the beam self-weight is calculated as follows: 

where: 

Δg = deflection due to beam self-weight 

wg = girder linear weight = 0.6058 kip/ft 

Deflection due to beam self-weight after transfer is: 

 

 
264

Deflection due to beam self-weight used to compute deflection at erection is: 

C15.3. DEFLECTION DUE TO SLAB AND HAUNCH WEIGHT

The deflection due to the slab and haunch weight is calculated as follows: 

where: 

ΔD = deflection due to deck and haunch weights (in.) 

wD = deck linear weight = 0.9788 kip/ft 

wh = haunch linear weight = 0.0529 kip/ft 

L = design span = 150 ft 

Ec = modulus of elasticity of precast beam at service loads = 6,933 ksi 

Therefore: 

C15.4. DEFLECTION DUE TO BARRIER AND FUTURE WEARING SURFACE 

WEIGHTS 

The deflection due to barrier and future wearing surface weight is calculated as follows: 

where: 

Δb+ws = deflection due to barrier and future wearing surface weights (in.) 

wb = barrier weight = 0.100 kip/ft

wws = wearing surface weight = 0.1933 kip/ft

Ic = moment of inertia of the composite section = 600,476.2 in.2

Therefore: 

 

 

 

 

 
265

C15.5. DEFLECTION AND CAMBER SUMMARY 

After transfer, Δp − Δg = 7.20 in.↑. 

The Precast/Prestressed Concrete Institute has proposed multipliers that are used to predict the 

long-term deflection of a pretensioned beam (PCI 2014). These multipliers are used herein to 

estimate the total long-term deflection of the girder. 

Estimate of total long-term deflection calculated using PCI multipliers = 1.8(12.70) − 1.85(5.65) = 

12.68 in.↑. 

C15.6. DEFLECTION DUE TO LIVE LOAD AND IMPACT 

Live load deflection limit = span/800 = 2.25 in. [AASHTO LRFD BDS Article 2.5.2.6.2] 

If the owner invokes the optional live load deflection criteria, the deflection is the greater of the 

value resulting from the design truck plus impact ΔLT, or that resulting from 25 percent of the 

design truck plus impact ΔLT, taken together with the design lane load, ΔLL. 

AASHTO LRFD BDS states that all beams should be assumed to deflect equally under the applied 

live load and impact. Therefore, the distribution factor for deflection is calculated as follows: 

DFD = (number of lanes)/(number of beams) = 4/6 = 0.667

[AASHTO LRFD BDS Article C2.5.2.6.2] 

Deflection due to lane load is as follows: 

Design lane load, wdl = 0.64(DFD) = 0.4267 kip/ft/girder 

Therefore: 

C15.7. DEFLECTION DUE TO DESIGN TRUCK LOAD AND IMPACT 

For maximum moment and deflection at the midspan due to the truck load, assume the centerline 

of the beam coincides with the middle point of the distance between the inner 32-kip axle and the 

resultant of the truck load. 

Using elastic moment area or influence lines, deflection at midspan is: 

∆LT = (1.044)(1 + IM)(DFD) = (1.044)(1.33)(0.667) = 0.93 in.↓

Live load deflection is the larger of: 

∆LT = 0.93 in.↓

0.25∆LT + ∆LL = 0.25(0.93) + 1.17 = 1.40 in.↓ (Controls) 

 

 

 
266

Therefore:

Live load deflection = 1.40 in. < 2.25 in. allowable deflection. OK.
267

CHAPTER C16. DESIGN SUMMARY

This example illustrated the design of a pretensioned I-beam for a 150-ft-long, single-span I-beam 

bridge with no skew (overall beam length is 151 ft). The I-beam cross section is modified from the 

MnDOT “MN54” beam and is made of UHPC. The superstructure of the bridge consists of six 

beams spaced at 9-ft centers. The roadway section is composed of two 12-ft traffic lanes and two 

12-ft shoulders. Beams are designed to act compositely with the 9-in.-thick cast-in-place 

conventional concrete deck to resist all superimposed dead loads, live loads, and impact. A 

1⁄2-in.-thick wearing surface is assumed. A deck thickness of 8.5 in. is used for composite section 

properties. The haunch is assumed to have an average thickness of 21⁄2 in. for dead load 

computations and 11⁄2 in. for section property computations. Design live load is HL-93. The 

conceptual reinforcement design drawing for the UHPC beam is shown in Figure C16-1. Note that 

the reinforcement of the deck is not shown.

Source: FHWA.

Figure C16-1. Illustration. Conceptual design drawing showing the details of the girder 

reinforcement.

UHPC Guide Articles 1.10.1 and 1.10.3 state that the maximum fiber length associated with the 

minimum cover or reinforcement spacing requirements shall be shown in contract documents. The 

minimum cover and the clear distance between reinforcement bars shall not be less than the greater 

of 1.5 times the length of the longest fiber reinforcement included in the UHPC or 0.75 in. In this 

design example, the cover requirement to the transverse bars or the clear spacing between the 

vertical reinforcement at the interface between the girder and the deck at ends are critical to the

fiber length requirement of the UHPC beam. Selecting a minimum cover of 0.75 in. would result 

in a maximum allowable fiber length of 0.5 in. in the UHPC mixture used to cast the girder.

269

ACKNOWLEDGMENTS

Many people assisted with various aspects of the content delivered in this report. This assistance 

is graciously appreciated.

Special thanks are given to these individuals who assisted with the conduct of research studies and 

the development of content in Appendix A.

• Zachary Haber contributed to the development of the compression behavior model. He and 

Scott Muzenski contributed to the development of the interface shear guidance.

• Alireza Mohebbi contributed to the development of the creep, shrinkage, and prestress loss 

models.

• Robert Spragg, Naveen Saladi, Luca Montanari, and Igor De la Varga contributed to the 

development of the resistivity testing framework.

• Timothy Barrett contributed to the cover requirements model.

• Alireza Valikhani contributed to the prestressing strand transfer and development length 

models.

Review and insight were obtained from a technical review panel of external experts during the 

development of Appendix A. Their efforts are greatly appreciated. The technical review panel 

included Theresa Ahlborn, Oguzhan Bayrak, J. P. Binard, Richard Brice, Reid Castrodale, Katrin 

Habel, Fouad Jaber, Bijan Khaleghi, Michael McDonagh, Gregory Nault, James Nelson, Tanarat 

Potisuk, William Potter, and Eric Steinberg.

Special thanks are also offered to Elizabeth Deller who, through her internship and subsequent 

work with the team, assisted with the development of Appendix B and Appendix C. 

Finally, thanks are offered to the former and current chairs of AASHTO T-10, Kevin Western and 

William Potter, respectively, and to the members and friends of AASHTO T-10 for their reviews, 

comments, and advice during the development and refinement of Appendix A, Appendix B, and 

Appendix C.

271

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HRDI-40/10-23(WEB)E Recycled

Recyclable

Recommended citation: Federal Highway Administration,

Structural Design with Ultra-High Performance Concrete 

(Washington, DC: 2023) https://doi.org/10.21949/1521382