U.S. Department of Transportation 

Federal Highway Administration 

Tunnel Load Rating Examples: 

A Supplement to the Reference 

Guide for Load Rating of Tunnel 

Structures 

Publication No. FHWA-HIF-20-058 

Office of Infrastructure, Office of Bridges and Structures

December 2020 

FOREWORD

The National Tunnel Inspection Standards (NTIS, 23 CFR 650.513(g)) require States and 

Federal agencies to “[R]rate each tunnel's safe vehicular load-carrying capacity in accordance 

with the Sections 6 or 8, AASHTO Manual for Bridge Evaluation (incorporated by reference, see 

§ 650.517(b)(1)).” However, the AASHTO Manual for Bridge Evaluation (MBE) lacks specific 

information on the load rating of tunnels. In addition, Section 5.4 of FHWA’s Tunnel 

Operations, Maintenance, Inspection, and Evaluation (TOMIE) Manual, published in 2015 only 

provides limited information. As a result, FHWA published the Reference Guide for Load Rating 

of Tunnel Structures in 2019. 

This report is intended to be used as a supplement to that Reference Guide. It presents two 

practical examples of load rating for actual, in-service tunnel structures. The examples are based 

on the information contained within the Reference Guide and the computations are presented in 

similar detail as the load rating examples included in the Reference Guide. 

Subject matter experts from several State DOTs and FHWA provided technical review of this 

report. Their advice, counsel and contributions during the preparation of this report are greatly 

appreciated. 

Joseph L. Hartmann, PhD, P.E. 

Director, Office of Bridges and Structures 

Office of Infrastructure 

Federal Highway Administration 

Cover photos courtesy: 

Left and bottom right: HDR

Top right: FHWA 
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. They are included for informational purposes only and are not 

intended to reflect a preference, approval, or endorsement of any one product or entity.

The contents of this document do not have the force and effect of law and are not meant to bind 

the public in any way. This document is intended only to provide clarity to the public regarding 

existing requirements under the law or agency policies.

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-HIF-20-058

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

4. Title and Subtitle

Tunnel Load Rating Examples: A Supplement to the

Reference Guide for Load Rating of Tunnel Structures

5. Report Date

December 2020

6. Performing Organization Code:

7. Author(s)

Bernard Frankl, Brian Leshko and Anthony Ream of HDR

Engineering, Inc. and Patrick Wilson of Earth Mechanics,

Inc.

8. Performing Organization Report No.

9. Performing Organization Name and Address

HDR

301 Grant Street, Suite 1700

Pittsburgh, PA 15219

10. Work Unit No.

11. Contract or Grant No.

DTFH6114D00049

12. Sponsoring Agency Name and Address

Office of Bridges and Structures

Federal Highway Administration

200 New Jersey Ave., SE

Washington, DC 20590

13. Type of Report and Period

Supplemental Report

October 2019 to October 2020

14. Sponsoring Agency Code

15. Supplementary Notes

FHWA Task Order COR (Task Order Manager): Lubin Gao, Ph.D., P.E.

FHWA Contracting Officer Representative (COR): Tuonglinh Warren, P.E.

HDR Project Manager: Anthony Ream, P.E.

16. Abstract

This supplement to the FHWA Reference Guide for Load Rating of Tunnel Structures (Guide) presents

two practical examples of real, in-service tunnel structures. The examples are based on the information

contained within the Guide, and computations are presented in similar details as the load rating examples

included in the Guide. The ratings are provided for LRFR only. The examples are rated for the HL-93

and EV vehicles. Additionally, a nominal permit load is applied and rated. For each example, one crosssection with one set of parameters (e.g. dead loads, soil parameters, etc.) is rated. Repetitive hand

calculations are shown once in detail (equations) with the subsequent results summarized in tables.

Elements from the original drawings and inspection reports (where applicable) are used in the examples

where legible and useful. Supplemental sketches are developed as necessary in a similar style as the

Guide. Where actual dimensions or properties are unclear or unavailable, reasonable approximations are

made. The results of the two examples may be used to validate the load ratings in the records of the

tunnels.

17. Key Words

Tunnels, Load Rating, NTIS, TOMIE, SNTI,

LFR, LRFR, ASR

18. Distribution Statement

No restrictions. This document is available to the

public through the National Technical Information

Service, Springfield, VA 22161.

http://www.ntis.gov

19. Security Classif. (of this

report) Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

116

22. Price

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

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH 

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m

2

yd2 square yard 0.836 square meters m

2

ac acres 0.405 hectares ha 

mi2square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m

3 

yd3 cubic yards 0.765 cubic meters m

3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS 

oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) 

o

F Fahrenheit 5 (F-32)/9 Celsius oC 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx 

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH

mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA 

mm2square millimeters 0.0016 square inches in2 

m

2

square meters 10.764 square feet ft2 

m

2

square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac 

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m

3 cubic meters 35.314 cubic feet ft3 

m

3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees)

oC Celsius 1.8C+32 Fahrenheit o

F 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2candela/m20.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for th International e System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
ii

TABLE OF CONTENTS 

LIST OF FIGURES ................................................................................................................................iv

LIST OF TABLES ................................................................................................................................. vi

LIST OF ABBREVIATIONS AND SYMBOLS ................................................................................... viii

CHAPTER 1 - INTRODUCTION ........................................................................................................... 9

1.1 PURPOSE AND SCOPE ............................................................................................................... 9

1.2 APPLICABLE DOCUMENTS ...................................................................................................... 9

1.3 REFERENCE GUIDE FOR LOAD RATING OF TUNNEL STRUCTURES............................... 11

1.4 GENERAL DISCUSSION ........................................................................................................... 11

1.4.1 Example Tunnels .................................................................................................................. 11

1.4.2 Assumptions ......................................................................................................................... 12

1.4.3 Approximations .................................................................................................................... 12

CHAPTER 2 - EXAMPLE 1 – CIRCULAR TUNNEL LINER, RIGID FRAME, & SLAB ................... 13

2.1 STRUCTURE DATA .................................................................................................................. 13

2.1.1 Materials ............................................................................................................................... 13

2.1.2 Dimensions ........................................................................................................................... 14

2.1.3 Example Notes ...................................................................................................................... 15

2.1.4 Rating Approach/Assumptions .............................................................................................. 15

2.2 INTERNAL FRAME ................................................................................................................... 16

2.2.1 Load Calculations ................................................................................................................. 16

2.2.2 Structural Analysis ................................................................................................................ 22

2.2.3 Resistance Calculations ......................................................................................................... 32

2.2.4 LRFR Rating Calculations..................................................................................................... 46

2.3 PRESTRESSED BOTTOM SLAB............................................................................................... 49

2.3.1 Load Calculations ................................................................................................................. 50

2.3.2 Structural Analysis ................................................................................................................ 53

2.3.3 Resistance Calculations ......................................................................................................... 55

2.3.4 LRFR Rating Calculations..................................................................................................... 61

2.4 TUNNEL LINER......................................................................................................................... 62

2.4.1 Materials ............................................................................................................................... 63

2.4.2 FE Model Description ........................................................................................................... 64

2.4.3 FE Analysis Procedure .......................................................................................................... 74

2.4.4 FE Analysis Results .............................................................................................................. 76

2.4.5 Resistance Calculations ......................................................................................................... 89

2.4.6 LRFR Rating Calculations..................................................................................................... 93

CHAPTER 3 - EXAMPLE 2 – BOX TUNNEL ROOF GIRDER ........................................................... 97
iii

3.1 STRUCTURE DATA .................................................................................................................. 97

3.1.1 Materials ............................................................................................................................... 97

3.1.2 Dimensions ........................................................................................................................... 98

3.1.3 Example Notes ...................................................................................................................... 98

3.1.4 Rating Approach/Assumptions .............................................................................................. 99

3.2 STEEL ROOF GIRDER .............................................................................................................. 99

3.2.1 Load Calculations ................................................................................................................. 99

3.2.2 Structural Analysis .............................................................................................................. 102

3.2.3 Resistance Calculations ....................................................................................................... 104

3.2.4 LRFR Rating Calculations................................................................................................... 109

REFERENCES.................................................................................................................................... 111
iv

LIST OF FIGURES

Figure 1. Illustration. Cross-Section Showing Tunnel Geometry. .............................................. 14

Figure 2. Illustration. Permit Truck. .......................................................................................... 16

Figure 3. Illustration. Reinforcing Steel Details of Internal Frame. ............................................ 16

Figure 4. Illustration. Live Load Placement for Internal Frame. ................................................. 22

Figure 5. Illustration. Internal Frame Tunnel Applied Loads...................................................... 23

Figure 6. Illustration. Member Identifiers and Directions........................................................... 24

Figure 7. Illustration. PS Slab Geometry and Prestressing Steel. ................................................ 49

Figure 8. Illustration. Live Load Placement on Slab. ................................................................. 53

Figure 9. Illustration. Slab Direction.......................................................................................... 54

Figure 10. Illustration. Tunnel Cross-Section............................................................................. 62

Figure 11. Illustration. 2D Plane Strain FE Model of Tunnel Cross-Section. Length Units are in 

Feet. .......................................................................................................................................... 65

Figure 12. Illustrations. Node distribution for (Plaxis 2019): a) 15-node triangular continuum 

element and b) 5-node plate element. ........................................................................................ 67

Figure 13. Illustration. Close-Up View of Tunnel in FE Model.................................................. 69

Figure 14. Illustration. Live Load Application for HL-93 Truck at One Location. ..................... 72

Figure 15. Illustration. Generate Stresses Under Original Level Ground Condition. ................... 74

Figure 16. Illustration. Excavate Tunnel, Activate Liner and Interface, Dewater Inside. ............ 75

Figure 17. Illustration. Activate Corbels and Apply Interior Structure Dead Loads. ................... 75

Figure 18. Illustration. Convention Used for Angle from Horizontal to Present the Results. ...... 76

Figure 19. Analysis Results. Permanent Loads Without the Building Surcharge. ....................... 77

Figure 20. Illustration. Building Surcharge Included at the Ground Surface. .............................. 78

Figure 21. Analysis Results. Permanent Loads Including the 7 ksf Building Surcharge. ............ 79

Figure 22. Analysis Results. Permanent Loads + HL-93 Tandem (No Building Surcharge). ...... 80

Figure 23. Analysis Results. Permanent Loads + HL-93 Tandem (No Building Surcharge). ...... 81

Figure 24. Analysis Results. Permanent Loads + HL-93 Truck (No Building Surcharge). .......... 82

Figure 25. Analysis Results. Permanent Loads + HL-93 Truck (No Building Surcharge). .......... 83

Figure 26. Analysis Results. Permanent Loads + EV-2 (No Building Surcharge). ...................... 84

Figure 27. Analysis Results. Permanent Loads + EV-2 (No Building Surcharge). ...................... 85

Figure 28. Analysis Results. Permanent Loads + EV-3 (No Building Surcharge). ...................... 86

Figure 29. Analysis Results. Permanent Loads + EV-3 (No Building Surcharge)....................... 87

Figure 30. Illustration. Cross-Section Showing Tunnel Geometry.............................................. 98
v

Figure 31. Illustration. Live Load Placement for Maximum Moment. ...................................... 103

Figure 32. Illustration. Live Load Placement for Maximum Shear. .......................................... 103

Figure 33. Illustration. Load Placement. .................................................................................. 103

Figure 34. Illustration: Plastic Neutral Axis Offset Dimensions. .............................................. 107

Figure 35. Illustration. Bracket Details. ................................................................................... 109
vi

LIST OF TABLES 

Table 1. Left Wall Axial Forces (kip). ....................................................................................... 25

Table 2. Left Wall Shear (kip). .................................................................................................. 25

Table 3. Left Wall Moment (kip-ft). .......................................................................................... 26

Table 4. Right Wall Axial Forces (kip). ..................................................................................... 26

Table 5. Right Wall Shear (kip). ................................................................................................ 26

Table 6. Right Wall Moment (kip-ft). ........................................................................................ 27

Table 7. Slab Axial Forces (kip). ............................................................................................... 27

Table 8. Slab Shear (kip). .......................................................................................................... 27

Table 9. Slab Moment (kip-ft). .................................................................................................. 28

Table 10. Left Wall Axial Live Load (kip). ............................................................................... 28

Table 11. Left Wall Shear Live Load (kip). ............................................................................... 29

Table 12. Left Wall Moment Live Load (kip-ft). ....................................................................... 29

Table 13. Right Wall Axial Live Load (kip). ............................................................................. 30

Table 14. Right Wall Shear Live Load (kip). ............................................................................. 30

Table 15. Right Wall Moment Live Load (kip-ft). ..................................................................... 31

Table 16. Slab Axial Live Load (kip). ....................................................................................... 31

Table 17. Slab Shear Live Load (kip). ....................................................................................... 32

Table 18. Slab Moment Live Load (kip-ft). ............................................................................... 32

Table 19. Critical Section Data. ................................................................................................. 33

Table 20. Load Factors. ............................................................................................................. 47

Table 21. HL-93 Inventory Rating Factors. ............................................................................... 47

Table 22. HL-93 Operating Rating Factors. ............................................................................... 47

Table 23. EV-2 and EV-3 Rating Factors. ................................................................................. 48

Table 24. Permit Rating Factors. ............................................................................................... 48

Table 25. Slab Shear Forces (kip). ............................................................................................. 54

Table 26. Slab Moment Forces (kip-ft). ..................................................................................... 54

Table 27. Load Factors for Strength Limit States. ...................................................................... 61

Table 28. Slab Resistance Rating Factors for Strength. .............................................................. 61

Table 29. Structure Properties. .................................................................................................. 65

Table 30. Plane Strain Model Inputs. ......................................................................................... 66

Table 31. Soil Parameters. ......................................................................................................... 68
vii

Table 32. Maximum Moment (kip-ft) & Corresponding Shear (kip) & Axial (kip). ................... 88

Table 33. Minimum Moment (kip-ft) & Corresponding Shear (kip) & Axial (kip). .................... 88

Table 34. Maximum Shear (kip) & Corresponding Moment (kip-ft) & Axial (kip). ................... 89

Table 35. Maximum Axial (kip) & Corresponding Moment (kip-ft) & Shear (kip). ................... 89

Table 36. Load Factors. ............................................................................................................. 94

Table 37. HL-93 Inventory Level Rating Factors. ...................................................................... 94

Table 38. HL-93 Operating Level Rating Factors. ..................................................................... 94

Table 39. EV-3 Rating Factors. ................................................................................................. 95

Table 40. Building Performance Ratio. ......................................................................................95

Table 41. Moment Load Placement Dimensions. ..................................................................... 103

Table 42. Shear Load Placement Dimensions. ......................................................................... 104

Table 43. Unfactored Girder Critical Forces. ........................................................................... 104

Table 44. Short Term Composite Properties. ........................................................................... 105

Table 45. Long Term Composite Properties. ........................................................................... 106

Table 46. Load Factors. ........................................................................................................... 110

Table 47. Girder Resistance and Rating Factors....................................................................... 110
viii

LIST OF ABBREVIATIONS AND SYMBOLS

AASHTO American Association of State Highway and Transportation Officials

BDS AASHTO LRFD Bridge Design Specifications, 8th Edition (2017a)

CFR Code of Federal Regulations

FHWA Federal Highway Administration

Guide Reference Guide for Load Rating of Tunnel Structures (FHWA 2019)

LRFDTUN AASHTO LRFD Road Tunnel Design and Construction Guide Specifications, 

First Edition (2017b)

LRFR Load and Resistance Factor Rating

MBE AASHTO Manual for Bridge Evaluation, 3

nd Edition (2018)

NTI National Tunnel Inventory

NTIS National Tunnel Inspection Standards

SNTI Specifications for the National Tunnel Inventory (FHWA 2015b)

SSI Soil-Structure Interaction

TOMIE Tunnel Operations, Maintenance, Inspection, and Evaluation (TOMIE) Manual 

(FHWA 2015a)
9

CHAPTER 1 - INTRODUCTION 

1.1 PURPOSE AND SCOPE 

The purpose of this non-binding Supplement to the voluntary FHWA Reference Guide for Load 

Rating of Tunnel Structures (Guide), created by the Federal Highway Administration (FHWA), 

is to present two practical examples of real, in-service tunnel structures. The examples are based 

on the information contained within the Guide, and computations are presented in similar details 

as the load rating examples included in the Guide. The ratings are provided for LRFR only. The 

examples are rated for the HL-93 and EV vehicles. Additionally, a nominal permit load is 

applied and rated for Example 1. For each example, one cross-section with one set of parameters 

(e.g. dead loads, soil parameters, etc.) is rated. Repetitive hand calculations are shown once in 

detail (equations) with the subsequent results summarized in tables. Elements from the original 

drawings and inspection reports (where applicable) are used in the examples where legible and 

useful. Supplemental sketches are developed as necessary in a similar style as the Guide. Where 

actual dimensions or properties are unclear or unavailable, reasonable approximations are made. 

The results of the two examples may be used to validate the load ratings in the records of the 

tunnels. Unless otherwise specified, all references refer to the voluntary Guide (FHWA 2019), 

the non-binding MBE 3rd Edition (AASHTO 2018) (see Section 1.2 for the components and 

editions incorporated by reference in the Federal regulations), or the TOMIE Manual (FHWA 

2015a) [23 CFR 650.517(c)(1)]. 

The following is an overview of the Supplement’s organizational structure: 

• Chapter 1 comprises introductory information on the two load rating examples, 

applicable standards and specifications, an overview of the guide, and descriptions of 

each example tunnel along with assumptions and approximations. 

• Chapter 2 covers the load rating of Example 1 – Circular Tunnel Liner with Invert 

Structures. 

• Chapter 3 contains the load rating of Example 2 – Box Tunnel Roof Girder. 

1.2 APPLICABLE DOCUMENTS 

National Tunnel Inspection Standards (NTIS) [23 CFR 650.513(g)]. The NTIS requires 

“tunnel owners to establish a program for the inspection of highway tunnels” (Federal Register, 

July 14, 2015, 80 FR 41350) to maintain an inventory, and to provide inspection findings to 

FHWA as well as make corrective actions to critical issues found during inspection. This 

document ultimately sets the rules and regulations regarding tunnel inspection and evaluation. 

Federal regulation at 23 CFR 650.513(g) provides the regulations regarding the load rating of 

tunnels. The NTIS incorporates, by reference, the Specifications for the National Tunnel 

Inventory (SNTI) [23 CFR 650.517((c)(2)] and TOMIE Manual [23 CFR 650.517(c)(1)]. Refer 

to the paragraph on the MBE below for components and editions incorporated by reference into 

the Federal regulations. 
10

Specifications for the National Tunnel Inventory (SNTI) (FHWA 2015b) [23 CFR 650.517 

(c)(2)]. Developed in coordination with the NTIS, the SNTI provides the specifications and 

coding data required to be submitted to the National Tunnel Inventory (similar to the National 

Bridge Inventory (NBI)). Details are provided in each chapter of the SNTI regarding how to log 

information for each item’s code, and how the Engineer decides which codes are applicable for 

the tunnel at hand. Section 2 provides general information about the tunnel, where Section 2.7 

provides the current load rating information. Section 3 discusses the specific tunnel elements, 

with Section 3.2 (structural) and Section 3.3 (civil) being the most relevant. Section 4 provides 

an index of inventory items and elements while Section 5 provides an example tunnel coding that 

helps readers understand the inspection coding process and can be back-referenced to the rest of 

the SNTI for further understanding. There are no references to any load rating process in the 

SNTI.

Tunnel Operations, Maintenance, Inspection and Evaluation (TOMIE) Manual (FHWA 

2015a) [23 CFR 650.517(c)(1)]. The TOMIE Manual expands upon the inspection requirements 

to address and serve as a resource for operations, maintenance, inspection, and evaluation in 

addition to inspection. It provides a wealth of information ranging from initial construction 

techniques to air handlers, to risk assessment, to basic cost estimating; and is considered a 

primer. TOMIE Manual Section 5.4 provides initial discussion of and the requirements for load 

rating tunnels. Section 5.4.2 introduces the concept of LRFR for tunnels, but is limited in scope.

AASHTO Manual for Bridge Evaluation (MBE) (AASHTO 2018). This voluntary edition of 

the MBE provides procedures for determining “the physical condition, maintenance needs, and 

load carrying capacity” for bridges. MBE Chapter 6 through Chapter 8 are of interest because 

they present the concepts of load rating for vehicular live loads. In particular, Chapter 6 

describes non-binding criteria for load rating and posting of bridges and informed several of the 

load rating concepts presented in the Guide. The most current edition at the time of publication 

of the Guide is the 3rd Edition. Subsequent references to the MBE in this document are relevant 

to the voluntary 3rd Edition of the MBE. The portions and editions of the MBE incorporated into 

the Federal regulations by reference in the NTIS, which are binding, are Sections 6 and 8 of the 

2

nd Edition, 2011 [23 CFR 650.517(b)(1)], the 2011 Interim Revisions to Section 6 [23 CFR 

650.517(b)(2)], the 2013 Interim Revisions to Section 6 [23 CFR 650.517(b)(3)], the 2014

Interim Revisions to Section 6 [23 CFR 650.517(b)(4)], and the 2015 Interim Revisions to 

Section 6 [23 CFR 650.517(b)(5)].

AASHTO LRFD Bridge Design Specifications, 8th Edition (BDS) (AASHTO 2017a) [23 

CFR 625.4(d)(1)(v)]. Concepts such as standard loads, load factors, material resistance, and 

resistance factors that are used in the Guide and the MBE are taken directly from the LRFD 

BDS. The document represents an agreement among owners for the proper design of highway 

bridges and is maintained by AASHTO.

AASHTO LRFD Road Tunnel Design and Construction Guide Specifications, 1st Edition 

(LRFDTUN) (AASHTO 2017b). The non-binding LRFDTUN provides specifications for 

design, evaluation, and rehabilitation of highway tunnels and is maintained by AASHTO.

FHWA Memorandum: Load Rating for the FAST Act’s Emergency Vehicles (FHWA 

2016). This voluntary and non-binding memorandum provides explanations of the ratings for the 
11

emergency vehicles enacted by the Fixing America’s Surface Transportation Act (FAST Act) [23 

CFR 650.313(c)].

1.3 REFERENCE GUIDE FOR LOAD RATING OF TUNNEL STRUCTURES

The FHWA Reference Guide for Load Rating of Tunnel Structures (Guide) (FHWA 2019)

is voluntary, non-binding reference guide created by the FHWA to illuminate the technical 

aspects for the load rating of tunnel structures. It provides practical and representative step-bystep examples of load rating that meets the FHWA’s general load rating guidance. It also

provides technical details and breadth appropriate to explain the load rating specifications in the 

non-binding AASHTO MBE (AASHTO 2018) and the FHWA TOMIE Manual (FHWA 2015a)

[23 CFR 650.517(c)(1)]. A set of examples illustrate the specifications, procedures, and methods 

for load rating.

1.4 GENERAL DISCUSSION

The two examples in this Supplement illustrate how to load rate structural components of tunnels 

that support roadway live loads, including invert slabs, walls, and other supporting members as 

well as tunnel liners where applicable, that support live load from nearby highways.

1.4.1 Example Tunnels

Example 1 – The first example is located on the west coast of the United States. The tunnel is a 

single, large diameter circular bored tunnel with a concrete liner. Internally, there are two levels 

of traffic, the lower supported by an invert slab and the upper by a rigid frame. The tunnel liner 

is comprised of precast reinforced concrete segments. The actual tunnel is located a distance 

below the surface that is more than the diameter of the tunnel, which puts it outside the live load 

zone of influence limits for culverts per the MBE (maximum of 8 feet or culvert width). Per 

discussions with the FHWA, it will be assumed the tunnel is closer to the surface for the 

purposes of this example. The invert slab, rigid frame structure, and tunnel liner will be rated for 

this example. The invert slab will be analyzed by hand as a one-way prestressed slab on simple 

supports and rated for shear and moment. The rigid frame will be analyzed with a simple 2D 

analysis and rated for shear, moment, and thrust at controlling locations. The tunnel liner will be 

modeled in Plaxis to determine forces in the liner due to earth loads and surface live loads. Based 

on geotechnical information, a representative soil profile will be selected. The street surface 

above the tunnel will be located a distance less than the tunnel diameter in order to be within the 

zone of influence. Traffic above will be assumed to travel normal to the tunnel. The ring segment 

reinforcement will be checked for bending, shear and thrust. The joints between sections will be 

checked against crushing, shear, and the formation of a mechanism (bending overcoming thrust 

pre-compression and opening joint).

Example 2 – The second example is located on the east coast of the United States. The tunnel is 

one of several adjacent boxes with steel roof girders connected by shear-only steel bracket

connections to support walls. The tunnel supports a transverse roadway directly above. The 

simply supported composite steel tunnel roof girder will be rated for this example. Forces will be 

determined using hand calculations. The tunnel roof girder will be rated for moment and shear. 

The capacity of the shear connection will be investigated as well.
12

1.4.2 Assumptions 

The following general assumptions were used in the development of the examples and final 

document: 

• The document is entirely separate and acts as a supplement to the Guide with a separate 

report number supplied by the FHWA. 

• The examples are based on the information contained within the Guide. 

• The examples are in a similar format as the ones in the Guide. 

• The ratings are provided for LRFR only. The Guide adequately demonstrates the 

difference between LRFR and LFR with parallel calculations in the examples; the 

purpose of the examples is to apply the methodology to actual structures. 

• The examples are rated for the HL-93 and EV vehicles. Additionally, a nominal permit 

load is applied and rated. 

• For each example, one cross-section with one set of parameters (e.g. dead loads, soil 

parameters, etc.) is rated. 

• Repetitive hand calculations are shown once in detail (equations) with the subsequent 

results summarized in tables. 

• Elements from the original drawings and inspection reports (where applicable) are used 

in the examples where legible and useful. Supplemental sketches are developed as 

necessary in a similar style as the Guide. 

1.4.3 Approximations 

• Where actual dimensions or properties are unclear or unavailable, reasonable 

approximations are made.

• For these examples, the area of steel reinforcement bars for concrete elements are directly 

calculated based on their nominal diameters to the third decimal point. However, and 

alternatively acceptable and common practice, is to use values from design tables 

provided by associations such as the Concrete Reinforcing Steel Institute (CRSI), often 

given to the second decimal place.
13

CHAPTER 2 - EXAMPLE 1 – CIRCULAR TUNNEL LINER, RIGID FRAME, & SLAB 

Example 1 presents a bored tunnel with an internal, reinforced concrete frame and a prestressed 

bottom slab. The internal roadway is a bi-level, reinforced concrete frame. The bottom roadway 

surface consists of a prestressed concrete invert slab. The tunnel is also subjected to surface loads 

influencing the tunnel liner. The tunnel is subjected to vertical dead loads, earth loads, and live 

loads as well as lateral earth, building surcharge and live load surcharge. The tunnel is rated with 

the LRFR method for the design vehicles (HL-93 Inventory and Operating) and emergency 

vehicles (EV-2 and EV-3) at the legal load limit. 

This example will perform the following steps to rate this reinforced concrete box tunnel: 

1. Structure data 

2. Example notes 

3. Rating approach/assumptions 

4. Load application 

5. Structural analysis 

6. Resistance calculations 

7. LRFR rating calculations 

2.1 STRUCTURE DATA 

2.1.1 Materials 

Materials are known, otherwise MBE Articles 6A5.2.1 and 6A5.2.2 may be used. Soil 

parameters were randomly selected for the example. Full soil descriptions and evaluation should 

be performed for actual tunnel ratings so accurate soil parameters can be obtained. This example 

also assumed wet tunnel construction conditions and a 5% soil volume loss. 

 Concrete: f’c = 4.0 ksi (Internal Frame) 

 f’c = 7.0 ksi (Prestressed Bottom Slab) 

 f’c = 7.0 ksi (Tunnel Liner)

 Reinforcing Steel: fy = 60.0 ksi 

Soil: soil = 0.117 kcf

soil = 32°
14

2.1.2 Dimensions 

See Figure 1 for the tunnel cross-section geometry and dimensions. The elements rated in this 

example are outlined in the figure. 

 Source: FWHA 

Figure 1. Illustration. Cross-Section Showing Tunnel Geometry. 

 Top slab thickness: tts = 18 inches 

 Bottom slab thickness: tbs = 14.5 inches 

 Wall thickness: twall = 16 inches 

 Top slab roadway width: wtop = 31.83 feet 

 Bottom slab roadway width: wbot = 32.00 feet 

 Fill depth: hfill = 10.00 feet 
15

2.1.3 Example Notes 

• This cross-section has been selected as representative to demonstrate rating of tunnel 

internal frames, external loading on liners and rating of PS sections. 

• The live load carrying members included the driving surfaces, first level walls, and the 

tunnel liner. The non-live load carrying members need to be included in the structural 

model to account for the load distribution these members may contribute to the overall 

structural behavior. 

• If the roadway has an overlay (in addition to the initial 2-inch design overlay included in 

the prestressed bottom slab ratings), or sidewalk loads, they should be included in the 

dead load analysis. This example assumes that these items are not included in the 

geometry of this reinforced concrete tunnel. 

• Mechanical equipment, signage, and architectural systems should be included in the dead 

load analysis. This example includes fire protection that would be similarly classified. 

• The focus of this example is LRFR. 

• This example focuses on the load rating of an interior section of a reinforced concrete 

tunnel. A section taken near the ends of the tunnel would need to account for increases in 

load effects due to reduced live load distribution near the edge. 

• The rating Engineer should evaluate the existing record drawings to determine if the 

rebar lap lengths and development lengths meet current design code recommendations. 

This example is based on the assumption that rebar lap lengths and development lengths 

meet current design code requirements. In instances where development and lap length 

deficiencies are present, a reduced rebar area proportional to the percent developed 

should be used within the portion of the bar affected by the deficiency. 

• The rating Engineer should review the record drawings and inspection reports carefully to 

properly identify the support condition (pinned, expansion, fixed). Typical conditions, 

and assumed for this example, are a vertically restrained bottom slab and a laterally 

restrained exterior wall for the upper slab. This is achieved by providing a laterally 

restrained roller at the top of the wall. 

• Rating is performed only at maximum shear and moment locations. Typical ratings 

include evaluation at all tenth points and other sectional or material change locations. 

2.1.4 Rating Approach/Assumptions 

• LRFR evaluation is performed for a 1-foot wide strip (perpendicular to direction of traffic 

inside the tunnel and parallel to the direction of traffic above the box tunnel). 

• The pavement on the ground above has approximately the same unit weight as the soil 

and is therefore included with the soil vertical loads (that is, gravel pavement surface). 

• Compacted gravel fill acts along the sides of the tunnel with soil parameters of  = 32° 

and  = 117 pcf. Additionally, the tunnel is partially submerged in ground water.
16

• LRFR live load ratings are evaluated for HL-93 design loading (Truck or Tandem), the 

EV-2 and EV-3 emergency vehicles (FHWA Memo), and a permit truck shown in Figure 

2 (for the internal frame and prestressed bottom slab only). 

9.0'

10 kip18 kip

PERMIT TRUCK (GVW = 220 kip) 18 kip 23 kip

4'

23 kip23 kip

4' 4' 4' 4' 4' 4' 4' 21 kip 21 kip 21 kip 21 kip

21 kip

10.0'

 Source: FHWA 

Figure 2. Illustration. Permit Truck. 

#6 @ 6"

#6 @ 6"

#8 @ 6"

#6 @ 6"

#8 @ 6"

#8 @ 6"

#6 @ 6"

 Source: FHWA 

Figure 3. Illustration. Reinforcing Steel Details of Internal Frame. 

2.2 INTERNAL FRAME 

2.2.1 Load Calculations 

2.2.1.1 Dead Load Component, DC 

Dead load components include the self-weight of the concrete internal frame and barriers on the 

frame. These loads are described in MBE Article 6A.5.12.10.1 and LRFD BDS Article 3.5.1. 

Normal weight of reinforced concrete: 

0.145 kcf 0.005 kcf 0.150 kcf wc= + =

LRFD BDS Table 3.5.1-1 and Article C3.5.1 

The self-weight of the frame is applied to the model by including self-weight on the geometry 

detailed in Figure 1. A 233 plf barrier load is applied on top of the frame at each side of the top 

roadway. 

A 10 psf fire protection load was applied to the interior surface of the upper roadway as indicated 

in the Original Construction Plans (OCPs), this includes the upper walls and the underside of the 
17

upper roadway. In addition to the fire protection load, 10 psf utility loads were applied to the 

upper roadway surface along with the horizontal walkway surface slabs to the left side of the 

internal frame supporting the upper roadway. Additional utility loads of 220 plf and 200 plf were 

applied to the center of the middle and lower of these walking surfaces. Details of these loads 

can be seen in Figure 5. 

2.2.1.2 Wearing Surface, DW 

This internal frame has no wearing surfaces applied and therefore were not included in the load 

calculations. 

2.2.1.3 Air Pressure, AP 

Air pressures were applied to the Egress Areas, roadways and Extraction Duct as shown in 

Figure 1. ±21 psf was applied to the Egress Area, ±10 psf was applied to the roadways and -68 

psf was applied to the Extraction Duct, according to the OCPs. 

2.2.1.4 Temperature, TU 

A temperature range of ±20 degrees was applied to the structure as a whole. This load was 

modelled by applying the Temperature tool within the structural analysis program. 

2.2.1.5 Creep and Shrinkage, CR & SH 

Creep and shrinkage is a displacement based force and needs to be modelled in the program as a 

temperature variation. LRFD BDS Article 5.4.2.3.1 specifies the creep and shrinkage 

displacement is calculated as “c” times “L”. The temperature range can then be backed out of 

the temperature displacement equation. This temperature is then input into the structural analysis 

program to determine the creep and shrinkage forces. Per Article 5.4.2.3.1, in the absence of 

more accurate data, the shrinkage coefficient may be assumed to be 0.0005 after one year of 

drying. The coefficient of thermal expansion, α, is specified in LRFD BDS Article 5.4.2.2 

6

0.0005 6.0

83.4

cL TL

e T

T F

 

−

= = 

= 

 = 

2.2.1.6 Pedestrian Live Load, PL 

A 100 psf pedestrian live load is applied to both the Egress Path Corridor and Utility Corridor 

floor slabs. A 60 psf pedestrian live load is applied to the Ceiling Slab. 

2.2.1.7 Live Load Application, LL 

The internal frame is a reinforced concrete frame and behaves similarly to the concrete deck of a 

slab bridge. The distribution of axle loads based on an equivalent strip width, E, to cast-in-place 

slabs acting perpendicular to the direction of traffic is provided in LRFD BDS Table 4.6.2.1.3-1, 

where S is the span length. Per the LRFD BDS, separate distribution factors are computed for the 
18

positive and negative moments in the roadway slab using a span, S, of 36 feet. The live load 

applied to the walls is dependent on the forces imparted to them due to the fixed end moments 

from the roadway slab. As such, this example determines the live load forces in the walls based 

on the equivalent strip width, E, computed for negative moments in the slab, ESlab-. 

1 1 2 2

26.0 6.6

48.0 3.0

26.0 6.6(36) 263.60 inch 21.97 feet

48.0 3.0(36) 156.00 inch 13.00 feet

13.00 feet

pos

neg

Slab

Slab

Wall Wall Wall Wall Slab

E S

E S

E

E

E E E E E

+

−

+ − + − −

= +

= +

= + = =

= + = =

= = = = =

LRFD BDS Table 4.6.2.1.3-1 

Since the design strip is transverse to the direction of traffic, the wheel loads are included in the 

live load evaluation; however, the lane load is not included per LRFD BDS Article 3.6.1.3.3. The 

maximum number of trucks to be placed on the structure is obtained by taking the roadway width 

and dividing by a 12-foot lane, per LRFD BDS Article 3.6.1.1.1, resulting in two possible trucks 

to be placed parallel. This results in axle loads that are placed transversely such to produce 

maximum forces at the critical locations, identified at mid-span and wall-slab connection. 

Maximum connection forces were obtained by placing the first axle load 2 feet from the edge of 

the driving surface per LRFD BDS Article 3.6.1.3.1. The second truck is placed such that there is 

4 feet between wheel placements. Maximum positive moment is obtained when the centerline of 

the span is midway between the center of gravity of loads and the nearest concentrated load. Live 

load placement is shown in Figure 4. 

Wheel loads for unit, or 1-foot, analysis widths are determined by multiplying the equivalent unit 

width wheel load, P, by the appropriate impact factor (IM) and multiple presence factor (MPF). 

The equivalent wheel load is determined by dividing the full wheel load by the equivalent strip 

width, E, over which the axle load is distributed. However, if the axle spacing is less than the 

equivalent strip width, E, then multiple axles can load the same unit-width analysis strip. In these 

cases, the applied load is not simply the full wheel load divided by E; the overlap of axle loads 

should be considered. For positive bending, the ESlab+ width of 21.97 feet exceeds the axle 

spacing of the HL-93 Truck (14-foot spacing), HL-93 Tandem (4-foot spacing), both EV 

vehicles (15-foot for EV-2 and 15- and 4-foot for EV-3), and the permit truck (4- to 10-foot 

spacings). For negative bending and the walls, the ESlab- width of 13.00 feet exceeds the axle 

spacing of the HL-93 Tandem, EV-2 vehicle (rear axle) and the permit truck. The unit width 

wheel load, P, is computed below using equivalent axle loads where required. For a detailed 

discussion of modified equivalent widths and figures showing the HL-93 and EV vehicle axle 

configurations, refer to Section 9.2.4.1 of the Reference Guide for Load Rating of Tunnel 

Structures (FHWA, 2019). Refer to Figure 2 for the permit truck axle configuration. 

First, compute P for various axle configurations using the roadway slab positive bending 

equivalent slab width, ESlab+, to determine the controlling load for each rating vehicle: 
19

HL-93 Design Truck: 

( ) 93 1

32 kip

2 wheels 16 k/wheel 0.73

21.97 ft HL Truck

Slab

P

E

−

+

     

= = = ( )

93 2

32 kip 32 kip

2 wheels 32 k/wheel 0.89

2 14 2 35.97 ft HL Truck

Slab Slab

P

E E −

+ +

  +

   

= = =

+ + ( )

93 3

32 kip 32 kip 8 kip

2 wheels 36 k/wheel 0.72

2 14 14 2 49.97 ft HL Truck

Slab Slab

P

E E −

+ +

  + +    

= = =

+++

93 ( 93 1 93 2 93 3 )

k/wheel

max , , 0.89

ft

PHL Truck − +

= = P P P HL Truck HL Truck HL Truck − − −

HL-93 Design Tandem: 

( ) 93 1

25 kip

2 wheels 12.5 k/wheel 0.57

21.97 ft HL Tandem

Slab

P

E

−

+

     

= = = ( )

93 2

25 kip 25 kip

2 wheels 25 k/wheel 0.96

2 4 2 25.97 ft HL Tandem

Slab Slab

P

E E −

+ +

  +

   

= = =

+ +

93 ( 93 1 93 2 )

k/wheel

max , 0.96

ft

PHL Tandem − += = P P HL Tandem HL Tandem − −

EV-2:

( ) 2 1

33.5 kip

2 wheels 16.75 k/wheel 0.76

21.97 ft EV

Slab

P

E

−

+

     

= = = ( )

2 2

33.5 kip 24 kip

2 wheels 28.75 k/wheel 0.78

2 15 2 36.97 ft EV

Slab Slab

P

E E −

+ +

  +

   

= = =

+ +

2 ( 2 1 2 2 )

k/wheel max , 0.78

ft

P P P EV += =

EV EV − −
20

EV-3:

( ) 3 1

31 kip

2 wheels 15.5 k/wheel 0.71

21.97 ft EV

Slab

P

E

−

+

     

= = = ( )

3 2

31 kip 31 kip

2 wheels 31 k/wheel 1.19

2 4 2 25.97 ft EV

Slab Slab

P

E E −

+ +

  +

   

= = =

+ + ( )

3 3

31 kip 31 kip 24 kip

2 wheels 43 k/wheel 1.05

2 4 15 2 40.97 ft EV

Slab Slab

P

E E −

+ +

  + +    

= = =

+ + +

3 ( 3 1 3 2 3 3 )

k/wheel max , , 1.19

ft

P P P P EV +

= =

EV EV EV − − −

Permit Truck: 

As stated, all the permit truck axle spacings are less than ESlab+; therefore, all 11 axle load 

possibilities should be considered. The calculations follow the format as above and P

varies from 0.48 to 1.64 over the 11 permutations. The calculation for the controlling 10-

axle configuration is shown below. 

( )

21 21 21 21 21 23 23 23 18 18

2 wheels 105 k/wheel 1.64

2 4 4 4 4 10 4 4 4 4 2 63.97 ft Permit

Slab Slab

P

E E +

+ +

  + + + + + + + + +    

= = =

+ + + + + + + + + +

Next, compute P for various axle configurations using the roadway slab negative bending 

equivalent slab width, ESlab-, which also applies to the walls, to determine the controlling load for 

each rating vehicle: 

HL-93 Design Truck: 

ESlab-;is less than the HL-93 Truck axle spacings; therefore, only one configuration is 

computed. 

( ) 93

32 kip

2 wheels 32 k/wheel 1.23

13.00 ft HL Truck

Slab

P

E

− −

+

     

= = =

HL-93 Design Tandem: 

( ) 93 1

25 kip

2 wheels 25 k/wheel 0.96

17.00 ft HL Tandem

Slab

P

E

−

+

     

= = =
21

( )

93 2

25 kip 25 kip

2 wheels 50 k/wheel 1.47

2 4 2 17.00 ft HL Tandem

Slab Slab

P

E E −

+ +

  +

   

= = =

+ +

93 ( 93 1 93 2 )

k/wheel

max , 1.47

ft

PHL Tandem − − = = P P HL Tandem HL Tandem − −

EV-2:

ESlab-;is less than the EV-2 axle spacing; therefore, only one configuration is computed. 

( ) 2

33.5 kip

2 wheels 33.5 k/wheel 1.29

13.00 ft EV

Slab

P

E

−

+

     

= = =

EV-3:

ESlab-;is less than the front EV-3 axle spacing; therefore, only two configurations are 

computed. 

( ) 3 1

31 kip

2 wheels 31 k/wheel 1.19

13.00 ft EV

Slab

P

E

−

+

     

= = = ( )

3 2

31 kip 31 kip

2 wheels 62 k/wheel 1.82

2 4 2 17.00 ft EV

Slab Slab

P

E E −

+ +

  +

   

= = =

+ +

3 ( 3 1 3 2 )

k/wheel max , 1.82

ft

P P P EV − = =

EV EV − −

Permit Truck: 

Similar to the positive bending calculations, all the permit truck axle spacings are less 

than ESlab-. P varies from 0.81 to 1.91 over the 11 permutations. The calculation for the 

controlling 10-axle configuration is shown below. 

( )

21 21 21 21 21 23 23 23 18 18

2 wheels 105 k/wheel 1.91

2 4 4 4 4 10 4 4 4 4 2 55.00 ft Permit

Slab Slab

P

E E −

+ +

  + + + + + + + + +    

= = =

+ + + + + + + + + +
22

2' 6'

6' 6'

6'

6'

6'

6'

6' 6'

2'

13.50'

9.00'

2'

2'

4'

4'

4'

2' 6'

6' 6'

6'

6'

6'

6'

6' 6'

2'

2'

2'

4'

4'

4'

Right 

Connection

Left 

Connection

Mid-Span

Source: FHWA 

Figure 4. Illustration. Live Load Placement for Internal Frame. 

Single trucks are amplified by a multiple presence factor, MPF, of 1.2 while the two truck effects 

utilize a multiple presence factor of 1.0 per LRFD BDS Table 3.6.1.1.2-1. For this example, 

MBE Article 6A.4.4.3 is used to determine the dynamic impact factor, IM, of 1.33 for all 

vehicles. Additionally, from the equivalent wheel load calculations, it can be seen that the HL-93 

tandem will control over the HL-93 truck, and the EV-3 vehicle controls over the EV-2. 

2.2.2 Structural Analysis 

A frame analysis procedure is applied for the internal frame model with beam-column elements 

based on gross section properties. All the load effects and member resistances are calculated 
23

using this 1-foot wide strip representation. Structural analysis of the internal frame is based on a 

frame model with moment resisting connections between the slab and the wall joints. The frame 

is assumed to be pinned where the concrete frame connects to the tunnel liner, except at the top 

walls where they are assumed to be free to vertically translate with respect to the liner. A P-Delta 

analysis is also included to account for secondary effects. Although not shown in this example, 

consideration for slenderness in the walls may be required in accordance with LRFD BDS 

Article 5.6.4.3 [23 CFR 625.4(d)(1)(v)]. The applied loads are shown in Figure 5, and Table 1

through Table 18 provide the resulting force effects. 

 Source: FHWA 

Figure 5. Illustration. Internal Frame Tunnel Applied Loads. 
24

The LRFD Road Tunnel Design and Construction Guide Specifications, 1st Edition (AASHTO, 

2017b), Table 3.4-1 identifies the applicable minimum and maximum load factors for each load 

specified above. These minimum and maximum load factors are matched with the appropriate 

load direction and summed to identify the critical design forces, including moments, shears, and 

axial forces. The live load factors for the HL-93 are identified in Table 3.4-1, and the live load 

factor for the EV vehicles is 1.3 as identified in the FHWA Emergency Vehicle Memorandum 

(FHWA, 2016) and the live load factor for the permit vehicle was identified as 1.3 from MBE 3rd

Edition (AASHTO, 2018) Table 6A.4.5.4.2a-1, corresponding to unlimited crossings, mixed 

with traffic, two or more lanes, ADTT greater than 5000, and a permit weight factor ratio 

(GVW/AL) greater than 3. 

1.0

1.10

3.0

3.10

1.497 2.0

2.04

2.94

2.10

Left Wall

Right Wall

Slab

Span Point

Span Point

Span Point

 Source: FHWA 

Figure 6. Illustration. Member Identifiers and Directions. 
25

Figure 6 shows the span point identification for the tunnel frame. The whole number represents 

the member identification while the 10ths number indicates the percentage along the span length.

The forces were factored to produce controlling Strength I (or Strength II for permit loads) as 

well as Service I load combinations. These resulting unfactored forces are given in Table 1

through Table 18 and the member identifiers and directions are shown in Figure 6. 

Table 1. Left Wall Axial Forces (kip). 

Sp Pt DC AP+ AP- TU+ TU- CR&SH 

1.0 -12.41 -0.01 0.00 -0.13 0.14 0.56 

1.1 -12.04 -0.01 0.00 -0.13 0.14 0.56 

1.2 -11.68 -0.01 0.00 -0.13 0.14 0.56 

1.3 -11.31 -0.01 0.00 -0.13 0.14 0.56 

1.4 -10.94 -0.01 0.00 -0.13 0.14 0.56 

1.497 -9.94 -0.01 0.00 -0.11 0.11 0.44 

1.6 -9.59 -0.01 0.00 -0.11 0.11 0.44 

1.7 -9.24 -0.01 0.00 -0.11 0.11 0.44 

1.8 -8.89 -0.01 0.00 -0.11 0.11 0.44 

1.9 -8.55 -0.01 0.00 -0.11 0.11 0.44 

1.10 -8.19 -0.01 0.00 -0.11 0.11 0.44 

Table 2. Left Wall Shear (kip). 

Sp Pt DC AP+ AP- TU+ TU- CR&SH 

1.0 0.44 0.04 0.01 0.22 0.22 0.92 

1.1 0.44 0.02 0.01 0.22 0.22 0.92 

1.2 0.44 0.01 0.01 0.22 0.22 0.92 

1.3 0.44 0.01 0.02 0.22 0.22 0.92 

1.4 0.44 0.01 0.04 0.22 0.22 0.92 

1.497 1.99 0.10 0.10 0.11 0.11 0.44 

1.6 1.99 0.06 0.06 0.11 0.11 0.44 

1.7 1.99 0.03 0.03 0.11 0.11 0.44 

1.8 1.99 0.01 0.01 0.11 0.11 0.44 

1.9 1.99 0.05 0.05 0.11 0.11 0.44 

1.10 1.99 0.08 0.08 0.11 0.11 0.44 
26

Table 3. Left Wall Moment (kip-ft).

Sp Pt DC AP+ AP- TU+ TU- CR&SH

1.0 0.00 0.00 0.00 0.00 0.00 0.00

1.1 0.80 -0.06 0.02 -0.40 0.40 -1.68

1.2 1.60 -0.08 0.03 -0.81 0.81 -3.36

1.3 2.41 -0.07 0.05 -1.21 1.21 -5.04

1.4 3.21 -0.07 0.07 -1.61 1.61 -6.72

1.497 2.75 -0.09 0.09 -1.71 1.71 -7.13

1.6 -0.71 -0.05 0.05 -1.53 1.53 -6.37

1.7 -4.17 -0.13 0.13 -1.34 1.34 -5.61

1.8 -7.63 -0.14 0.14 -1.16 1.16 -4.85

1.9 -11.09 -0.09 0.09 -0.98 0.98 -4.08

1.10 -14.65 -0.04 0.10 -0.79 0.79 -3.30

Table 4. Right Wall Axial Forces (kip).

Sp Pt DC AP+ AP- TU+ TU- CR&SH

3.0 -9.07 0.00 0.02 -0.07 0.07 0.30

3.1 -8.76 0.00 0.02 -0.07 0.07 0.30

3.2 -8.45 0.00 0.02 -0.07 0.07 0.30

3.3 -8.13 0.00 0.02 -0.07 0.07 0.30

3.4 -7.82 0.00 0.02 -0.07 0.07 0.30

3.5 -7.51 0.00 0.02 -0.07 0.07 0.30

3.6 -7.20 0.00 0.02 -0.07 0.07 0.30

3.7 -6.89 0.00 0.02 -0.07 0.07 0.30

3.8 -6.58 0.00 0.02 -0.07 0.07 0.30

3.9 -6.26 0.00 0.02 -0.07 0.07 0.30

3.10 -5.95 0.00 0.02 -0.07 0.07 0.30

Table 5. Right Wall Shear (kip).

Sp Pt DC AP+ AP- TU+ TU- CR&SH

3.0 1.05 0.00 0.06 0.21 0.21 0.89

3.1 1.05 0.00 0.05 0.21 0.21 0.89

3.2 1.05 0.00 0.03 0.21 0.21 0.89

3.3 1.05 0.00 0.01 0.21 0.21 0.89

3.4 1.05 0.00 0.00 0.21 0.21 0.89

3.5 1.05 0.02 0.00 0.21 0.21 0.89

3.6 1.05 0.03 0.00 0.21 0.21 0.89

3.7 1.05 0.05 0.00 0.21 0.21 0.89

3.8 1.05 0.07 0.00 0.21 0.21 0.89

3.9 1.05 0.08 0.00 0.21 0.21 0.89

3.10 1.05 0.10 0.00 0.21 0.21 0.89
27

Table 6. Right Wall Moment (kip-ft).

Sp Pt DC AP+ AP- TU+ TU- CR&SH

3.0 0.00 0.00 0.00 0.00 0.00 0.00

3.1 1.64 0.00 0.09 -0.33 0.33 -1.38

3.2 3.28 0.00 0.15 -0.66 0.66 -2.77

3.3 4.92 0.00 0.18 -1.00 1.00 -4.15

3.4 6.56 -0.01 0.19 -1.33 1.33 -5.53

3.5 8.20 -0.01 0.18 -1.66 1.66 -6.92

3.6 9.84 -0.01 0.14 -1.99 1.99 -8.30

3.7 11.48 -0.01 0.07 -2.32 2.32 -9.68

3.8 13.12 -0.02 0.01 -2.65 2.65 -11.07

3.9 14.76 -0.14 0.01 -2.99 2.99 -12.45

3.10 16.40 -0.28 0.01 -3.32 3.32 -13.83

Table 7. Slab Axial Forces (kip).

Sp Pt DC AP+ AP- TU+ TU- CR&SH

2.0 1.09 -0.10 0.15 -0.18 0.18 0.74

2.04 -1.00 0.00 0.20 -0.28 0.28 1.17

2.2 -0.97 0.00 0.20 -0.28 0.28 1.17

2.3 -0.95 0.00 0.20 -0.28 0.28 1.17

2.4 -0.93 0.00 0.20 -0.28 0.28 1.17

2.5 -0.91 0.00 0.20 -0.28 0.28 1.17

2.6 -0.90 0.00 0.20 -0.28 0.28 1.17

2.7 -0.88 0.00 0.20 -0.28 0.28 1.17

2.8 -0.86 0.00 0.20 -0.28 0.28 1.17

2.94 -1.05 0.00 0.10 -0.21 0.21 0.89

2.10 -1.05 0.00 0.10 -0.21 0.21 0.89

Table 8. Slab Shear (kip).

Sp Pt DC AP+ AP- TU+ TU- CR&SH

2.0 2.99 0.00 0.00 0.18 0.18 0.74

2.04 4.52 0.00 0.00 0.08 0.08 0.33

2.2 3.08 0.00 0.00 0.08 0.08 0.33

2.3 2.20 0.00 0.00 0.08 0.08 0.33

2.4 1.32 0.00 0.00 0.08 0.08 0.33

2.5 0.44 0.00 0.00 0.08 0.08 0.33

2.6 0.45 0.00 0.00 0.08 0.08 0.33

2.7 1.33 0.00 0.00 0.08 0.08 0.33

2.8 2.21 0.00 0.00 0.08 0.08 0.33

2.94 5.42 0.01 0.00 0.07 0.07 0.30

2.10 5.95 0.00 0.02 0.07 0.07 0.30
28

Table 9. Slab Moment (kip-ft). 

Sp Pt DC AP+ AP- TU+ TU- CR&SH 

2.0 -3.77 -0.09 0.02 -0.01 0.01 0.03 

2.04 -23.88 -0.05 0.05 -0.49 0.49 -2.06 

2.2 -1.78 -0.04 0.04 -0.06 0.06 -0.14 

2.3 7.83 -0.03 0.03 -0.25 0.25 1.03 

2.4 14.24 -0.02 0.02 -0.53 0.53 2.21 

2.5 17.28 -0.02 0.02 -0.81 0.81 3.38 

2.6 17.18 -0.02 0.01 -1.09 1.09 4.56 

2.7 13.92 -0.03 0.00 -1.37 1.37 5.73 

2.8 7.53 -0.04 0.00 -1.66 1.66 6.91 

2.94 -4.08 -0.01 0.27 -3.16 3.16 13.18 

2.10 -16.40 -0.01 0.28 -3.32 3.32 13.83 

Unfactored live load envelopes, including PL, HL-93, EVs, and Permit are presented in Table 10

through Table 18. 

Table 10. Left Wall Axial Live Load (kip). 

Sp Pt PL HL-93 

Negative 

HL-93 

Positive EV Negative EV Positive Permit Negative 

Permit 

Positive 

1.0 -1.46 -6.08 0.00 -7.52 0.00 -7.90 0.00 

1.1 -1.46 -6.08 0.00 -7.52 0.00 -7.90 0.00 

1.2 -1.46 -6.08 0.00 -7.52 0.00 -7.90 0.00 

1.3 -1.46 -6.08 0.00 -7.52 0.00 -7.90 0.00 

1.4 -1.46 -6.08 0.00 -7.52 0.00 -7.90 0.00 

1.497 -0.83 -6.10 0.00 -7.55 0.00 -7.93 0.00 

1.6 -0.83 -6.10 0.00 -7.55 0.00 -7.93 0.00 

1.7 -0.83 -6.10 0.00 -7.55 0.00 -7.93 0.00 

1.8 -0.83 -6.10 0.00 -7.55 0.00 -7.93 0.00 

1.9 -0.83 -6.10 0.00 -7.55 0.00 -7.93 0.00 

1.10 -0.83 -6.10 0.00 -7.55 0.00 -7.93 0.00 
29

Table 11. Left Wall Shear Live Load (kip). 

Sp Pt PL HL-93 Max HL-93 Min EV Max EV Min Permit Max Permit Min

1.0 0.05 0.53 0.14 0.65 0.15 0.69 0.22 

1.1 0.05 0.53 0.14 0.65 0.15 0.69 0.22 

1.2 0.05 0.53 0.14 0.65 0.15 0.69 0.22 

1.3 0.05 0.53 0.14 0.65 0.15 0.69 0.22 

1.4 0.05 0.53 0.14 0.65 0.15 0.69 0.22 

1.497 0.25 3.06 0.81 3.79 0.85 3.98 1.25 

1.6 0.25 3.06 0.81 3.79 0.85 3.98 1.25 

1.7 0.25 3.06 0.81 3.79 0.85 3.98 1.25 

1.8 0.25 3.06 0.81 3.79 0.85 3.98 1.25 

1.9 0.25 3.06 0.81 3.79 0.85 3.98 1.25 

1.10 0.25 3.06 0.81 3.79 0.85 3.98 1.25 

Table 12. Left Wall Moment Live Load (kip-ft). 

Sp Pt PL HL-93 

Negative 

HL-93 

Positive EV Negative EV Positive Permit Negative 

Permit 

Positive 

1.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

1.1 0.07 0.00 0.97 0.00 1.20 0.00 1.26 

1.2 0.14 0.00 1.94 0.00 2.40 0.00 2.52 

1.3 0.20 0.00 2.91 0.00 3.61 0.00 3.78 

1.4 0.27 0.00 3.88 0.00 4.81 0.00 5.04 

1.497 -0.92 0.00 5.09 0.00 6.30 0.00 6.61 

1.6 -0.48 -0.41 0.13 -0.51 0.16 -0.54 0.16 

1.7 -0.04 -5.55 0.00 -6.87 0.00 -7.21 0.00 

1.8 0.40 -10.87 0.00 -13.46 0.00 -14.13 0.00 

1.9 0.84 -16.19 0.00 -20.05 0.00 -21.04 0.00 

1.10 1.29 -21.67 0.00 -26.83 0.00 -28.16 0.00 
30

Table 13. Right Wall Axial Live Load (kip).

Sp Pt PL HL-93 

Negative 

HL-93 

Positive EV Negative EV Positive Permit Negative 

Permit 

Positive 

3.0 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.1 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.2 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.3 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.4 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.5 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.6 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.7 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.8 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.9 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

3.10 0.02 -4.82 0.00 -5.97 0.00 -6.26 0.00 

Table 14. Right Wall Shear Live Load (kip). 

Sp Pt PL HL-93 

Max HL-93 Min EV Max EV Min Permit Max Permit Min 

3.0 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.1 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.2 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.3 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.4 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.5 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.6 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.7 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.8 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.9 0.01 1.22 0.15 1.51 0.16 1.59 0.24 

3.10 0.01 1.22 0.15 1.51 0.16 1.59 0.24 
31

Table 15. Right Wall Moment Live Load (kip-ft). 

Sp Pt PL HL-93 

Negative 

HL-93 

Positive EV Negative EV Positive Permit Negative 

Permit 

Positive 

3.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

3.1 -0.01 0.00 1.90 0.00 2.36 0.00 2.47 

3.2 -0.03 0.00 3.81 0.00 4.71 0.00 4.94 

3.3 -0.04 0.00 5.71 0.00 7.07 0.00 7.42 

3.4 -0.05 0.00 7.61 0.00 9.42 0.00 9.89 

3.5 -0.06 0.00 9.51 0.00 11.78 0.00 12.36 

3.6 -0.08 0.00 11.42 0.00 14.14 0.00 14.83 

3.7 -0.09 0.00 13.32 0.00 16.49 0.00 17.31 

3.8 -0.10 0.00 15.22 0.00 18.85 0.00 19.78 

3.9 -0.11 0.00 17.13 0.00 21.20 0.00 22.25 

3.10 -0.13 0.00 19.03 0.00 23.56 0.00 24.72 

Table 16. Slab Axial Live Load (kip). 

Sp Pt PL HL-93 

Negative 

HL-93 

Positive EV Negative EV Positive Permit Negative 

Permit 

Positive 

2.0 -0.25 0.00 2.18 0.00 2.70 0.00 2.43 

2.04 0.01 -1.11 0.00 -1.37 0.00 -1.44 0.00 

2.2 0.01 -1.11 0.00 -1.37 0.00 -1.44 0.00 

2.3 0.01 -1.11 0.00 -1.37 0.00 -1.44 0.00 

2.4 0.01 -1.11 0.00 -1.37 0.00 -1.44 0.00 

2.5 0.01 -1.06 0.00 -1.32 0.00 -1.38 0.00 

2.6 0.01 -1.03 0.00 -1.27 0.00 -1.33 0.00 

2.7 0.01 -1.01 0.00 -1.25 0.00 -1.32 0.00 

2.8 0.01 -0.99 0.00 -1.22 0.00 -1.28 0.00 

2.94 0.01 -1.22 0.00 -1.51 0.00 -1.59 0.00 

2.10 0.01 -1.22 0.00 -1.51 0.00 -1.59 0.00 
32

Table 17. Slab Shear Live Load (kip).

Sp Pt PL HL-93 

Max HL-93 Min EV Max EV Min Permit Max Permit Min 

2.0 0.81 0.11 0.03 0.14 0.03 0.14 0.05 

2.04 0.02 5.99 0.82 7.42 0.86 7.78 1.27 

2.2 0.02 4.35 0.82 5.38 0.86 5.65 1.27 

2.3 0.02 3.10 0.24 3.83 0.25 4.02 0.37 

2.4 0.02 2.94 0.34 3.64 0.36 3.82 0.53 

2.5 0.02 1.02 0.51 1.26 0.53 1.32 0.79 

2.6 0.02 1.98 0.82 2.45 0.86 2.57 1.27 

2.7 0.02 2.49 0.42 3.09 0.44 3.24 0.65 

2.8 0.02 3.47 0.34 4.30 0.36 4.51 0.53 

2.94 0.02 4.82 0.34 5.97 0.35 6.26 0.52 

2.10 0.02 4.82 0.34 5.97 0.35 6.26 0.52 

Table 18. Slab Moment Live Load (kip-ft). 

Sp Pt PL HL-93 

Negative 

HL-93 

Positive EV Negative EV Positive Permit Negative 

Permit 

Positive 

2.0 -0.78 -3.95 0.00 -3.47 0.00 -5.14 0.00 

2.04 -0.46 -27.29 0.00 -23.95 0.00 -35.46 0.00 

2.2 -0.35 -4.77 2.76 -4.19 5.23 -6.20 4.72 

2.3 -0.29 0.00 8.51 0.00 16.14 0.00 14.55 

2.4 -0.23 0.00 13.18 0.00 24.98 0.00 22.51 

2.5 -0.17 0.00 15.76 0.00 29.87 0.00 26.92 

2.6 -0.10 0.00 15.35 0.00 29.11 0.00 26.23 

2.7 -0.04 0.00 12.06 0.00 22.87 0.00 20.61 

2.8 0.02 -1.08 6.65 -0.95 12.61 -1.41 11.37 

2.94 0.09 -9.89 0.00 -8.68 0.00 -12.85 0.00 

2.10 0.13 -19.02 0.00 -16.69 0.00 -24.72 0.00 

2.2.3 Resistance Calculations 

The depth, analysis section width, and reinforcement for each of the critical sections are 

tabulated below in Table 19. 
33

Table 19. Critical Section Data. 

Section Depth 

(in)

Width 

(in)

Bar 

Number 

Bar Spacing 

(in)

Cover 

(in)

1 - Left Wall Inside 16.00 12.00 6 6.00 2.00 

2 – Left Wall Outside 16.00 12.00 8 6.00 2.00 

3 – Right Wall Outside 16.00 12.00 6 6.00 2.00 

4 – Right Wall Inside 16.00 12.00 6 6.00 2.00 

5 – Slab Mid-Span 18.00 12.00 8 6.00 2.00 

6 – Slab Left Connection 18.00 12.00 8 6.00 2.50 

7 – Slab Right Connection 18.00 12.00 8 6.00 2.50 

Concrete Properties: 

f’c = 4.0 ksi 

∝1 = 0.85 LRFD BDS 5.6.2.2 

β1 = 0.85 if f’c ≤ 4.0 ksi LRFD BDS 5.6.2.2 

λ = 1.0 (normal weight concrete) LRFD BDS 5.4.2.8 

γ3 = 0.67 (AASHTO M31 Grade 60) LRFD BDS 5.6.3.3 

γ1 = 1.6 LRFD BDS 5.6.3.3 

 Modulus of rupture: 

0.24 ' 0.24(1.0) 4.0 0.480 ksi

r c

f f = =  =

LRFD BDS 5.4.2.6 

Compression reinforcement in flexural capacity calculations is conservatively ignored. 

Calculated results are based on a per foot analysis width. 

2.2.3.1 Section 1 – Left Wall Inside Steel 

Rectangular section height: h = 16.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.00 inch 

Area of single rebar: As_bar = 0.442 inch2 

Diameter of rebar: diabar = 0.75 inch 

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of reinforcing bar: 

_ 0.44(12.00) 2

0.884 inch

6.0

s bar

s

A b

A

s

= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

0.75 16.00 2.00 13.63 inches

2 2

bar

s

dia d h clr = − − = − − =
34

Determine distance from the equivalent stress block for tension controlled, nonprestressed tension reinforcement only: 

0.884(60) 1.30 inches

0.85 ' 0.85(4.0)(12.00)

s y

c

A f

a

f b

= = =

LRFD BDS Eq. 5.6.3.1.1-4 and 5.6.2.2 

1

1

1.30 0.85 1.53 inches

0.85

a

 c



=  = = =

LRFD BDS 5.6.2.2 

13.63 1 0.003 1 0.024 0.005

1.53

0.90

s

s c

f

d

c

 



   

= − =     − =     

 =

LRFD BDS 5.5.4.2, 5.6.2.1 

Calculate factored moment resistance: 

2

1.30 1ft 0.9(0.884)(60) 13.63

2 12 inch

51.59 kip-ft

n f s y s

a

  M A f d 

= − =    

     −     

=

LRFD BDS Eq. 5.6.3.2.2-1 

Axial Resistance: 

Walls need to be checked for axial thrust and flexural interaction. Check flexure/axial 

interaction using the largest factored HL-93 Inventory axial load in the wall to check 

applicability: 

'

'

0.1 0.1(0.7)(4.0)(16)(12) 53.76 kip

N 28.28 kip (Strength-I)

0.1 Neglect Axial Thrust

c c g

u

c c g u

f A

f A N





= =

=

 

LRFD BDS 5.6.4.5 

Minimum Steel: 

Determine minimum reinforcement, beginning with the section modulus: 

2 2 12.00(16.00) 3

512 inch

6 6 c

bh S = = =

LRFD BDS 5.6.3.3 

The cracking moment equation simplifies to the following (only monolithic sections and 

no prestress forces), where fr is the modulus of rupture of concrete per LRFD BDS 

Article 5.4.2.6: 
35

3 1

1ft 0.67(1.6)(0.480)(512)

12 inch

21.95 kip-ft

M f S cr r c 

 

= =    

=

LRFD BDS Eq. 5.6.3.3-1 

Since fMn > Mcr, minimum reinforcement at this section is satisfied. 

Shear Resistance: 

Critical section for shear is dv from the face of the support in accordance with LRFD 

BDS 5.7.3.2. 

0.72 0.72(16) 11.52

max 0.9 0.9(13.63) 12.27 12.98 inches

1.30 13.63 12.98 2 2

v s

s

h

d d

a d

   

=

   

   

=    = = = 

   

− − =       

 LRFD BDS 5.7.2.8 

For this example, conservatively combine the largest factored HL-93 Inventory forces 

along the wall (points 1.00 to 1.10 in Figure 6) to compute the minimum shear resistance. 

Concurrent results along the length of the wall could be used to obtain more refined 

results when rating factors are below 1.00: 

60.55 kip-ft

9.00 kip

N 28.28 kip

u,max,LTwall

u,max,LTwall

u,max,LTwall

M

V

=

=

=

The shear resistance of the walls needs to be evaluated using LRFD BDS 5.7.3.3. Shear 

factors  and  need to be calculated since the member thickness is not less than 16 

inches and there is no shear reinforcement. The maximum aggregate size, ag, is set to 

0.75 inches. 

60.55(12) 726.00 kip-in 9.00(12.98) 116.82 kip-inch

M V d u u v  

=  =

 LRFD BDS 5.7.3.4.2 

0.5

60.55(12)

0.5(28.28) 9.00

12.98

0.0031

0.884(29000)

u

u u

v

s

s s

M

N V

d

E A



    + +

 

=

    + +

 

= =

LRFD BDS Eq. 5.7.3.4.2-4 

0 0.006 s

  

29 3500 29 3500(0.0031) 39.85o

  = + = +

s

=

LRFD BDS Eq. 5.7.3.4.2-3 
36 2 12.98 in 16 2(2) 0.75 11.25 inch x v b

s d h cl d =  − − =  − − =

 Where cl is the clear cover and the smaller #6 bar diameter is assumed on each face; 

1.38 12 in 80 inch

0.63

1.38 12 in 11.25 80 in 12.00 inch

0.75 0.63

xe x

g

xe

s s

a

s

=  

+

=   

+

LRFD BDS Eq. 5.7.3.4.2-7 

( ) ( )

   

4.8 51

1 750 39

4.8 51 1.44

1 750(0.0031) 39 12.00

s xe

s





=

+ +

= =

+ +

LRFD BDS Eq. 5.7.3.4.2-2 

' ' 0.0316 0.25

0.90 0.0316(1.44) 4.0 0.25(4.0) 12(12.98)

12.76 kip

Vn v c c v v    f f b d =     

=     

=

LRFD BDS 5.7.3.3 

The following LRFD BDS checks have not been performed for this example but should be 

checked for actual rating calculations: regions requiring transverse reinforcement (5.7.2.3) and 

longitudinal reinforcement checks (5.7.3.5) [23 CFR 625.4(d)(1)(v)]. 

2.2.3.2 Section 2 – Left Wall Outside Steel 

Rectangular section height: h = 16.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.00 inch 

Area of single rebar: As_bar = 0.785 inch2 

Diameter of rebar: diabar = 1.00 inch 

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of reinforcing bar: 

_ 0.785(12.00) 2

1.571 inch

6.0

s bar

s

A b

A

s

= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

1.00 16.00 2.00 13.50 inches

2 2

bar

s

dia d h clr = − − = − − =

Determine distance from the equivalent stress block for tension controlled, nonprestressed tension reinforcement only: 
37 1.571(60) 2.31 inches

0.85 ' 0.85(4.0)(12.00)

s y

c

A f

a

f b

= = =

LRFD BDS Eq. 5.6.3.1.1-4 and 5.6.2.2 

1

1

2.31 0.85 2.72 inches

0.85

a

 c



=  = = =

LRFD BDS 5.6.2.2 

13.50 1 0.003 1 0.012 0.005

2.72

0.90

s

s c

f

d

c

 



   

= − =     − =     

 =

LRFD BDS 5.5.4.2, 5.6.2.1 

Calculate factored moment resistance: 

2

2.31 1ft 0.9(1.571)(60) 13.50

2 12 inch

87.26 kip-ft

n f s y s

a

  M A f d 

= − =    

     −     

=

LRFD BDS Eq. 5.6.3.2.2-1 

Axial Resistance: 

It was shown in Section 1 above that axial thrust can be neglected for the left wall. 

Minimum Steel: 

Determine minimum reinforcement, beginning with the section modulus: 

2 2 12.00(16.00) 3

512 inch

6 6 c

bh S = = =

LRFD BDS 5.6.3.3 

The cracking moment equation simplifies to the following (only monolithic sections and 

no prestress forces), where fr is the modulus of rupture of concrete per LRFD BDS 

Article 5.4.2.6: 

3 1

1ft 0.67(1.6)(0.480)(512)

12 inch

21.95 kip-ft

M f S cr r c 

 

= =    

=

LRFD BDS Eq. 5.6.3.3-1 

Since fMn > Mcr, minimum reinforcement at this section is satisfied. 

Shear Resistance: 

Shear resistance is calculated for the wall with Section 1 above. 

2.2.3.3 Section 3 – Right Wall Inside Steel 

Rectangular section height: h = 16.00 inch 
38

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.00 inch 

Area of single rebar: As_bar = 0.442 inch2 

Diameter of rebar: diabar = 0.75 inch 

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of reinforcing bar: 

_ 0.44(12.00) 2

0.884 inch

6.0

s bar

s

A b

A

s

= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

0.75 16.00 2.00 13.63 inches

2 2

bar

s

dia d h clr = − − = − − =

Determine distance from the equivalent stress block for tension controlled, nonprestressed tension reinforcement only: 

0.884(60) 1.30 inches

0.85 ' 0.85(4.0)(12.00)

s y

c

A f

a

f b

= = =

LRFD BDS Eq. 5.6.3.1.1-4 and 5.6.2.2 

1

1

1.30 0.85 1.53 inches

0.85

a

 c



=  = = =

LRFD BDS 5.6.2.2 

13.63 1 0.003 1 0.024 0.005

1.53

0.90

s

s c

f

d

c

 



   

= − =     − =     

 =

 LRFD BDS 5.5.4.2, 5.6.2.1 

Calculate factored moment resistance: 

2

1.30 1ft 0.9(0.884)(60) 13.63

2 12 inch

51.59 kip-ft

n f s y s

a

  M A f d 

= − =    

     −     

=

LRFD BDS Eq. 5.6.3.2.2-1 

Axial Resistance: 

Walls need to be checked for axial thrust and flexural interaction. Check flexure/axial 

interaction using the largest factored HL-93 Inventory axial load in the wall to check 

applicability: 
39

'

'

0.1 0.1(0.7)(4.0)(16)(12) 53.76 kip

N 19.54 kip

0.1 Neglect Axial Thrust

c c g

u

c c g u

f A

f A N





= =

=

 

LRFD BDS 5.6.4.5 

Minimum Steel: 

Determine minimum reinforcement, beginning with the section modulus: 

2 2 12.00(16.00) 3

512 inch

6 6 c

bh S = = =

LRFD BDS 5.6.3.3 

The cracking moment equation simplifies to the following (only monolithic sections and 

no prestress forces), where fr is the modulus of rupture of concrete per LRFD BDS 

Article 5.4.2.6: 

3 1

1ft 0.67(1.6)(0.480)(512)

12 inch

21.95 kip-ft

M f S cr r c 

 

= =    

=

LRFD BDS Eq. 5.6.3.3-1 

Since fMn > Mcr, minimum reinforcement at this section is satisfied. 

Shear Resistance: 

Critical section for shear is dv from the face of the support in accordance with LRFD 

BDS 5.7.3.2. 

0.72 0.72(16) 11.52

max 0.9 0.9(13.63) 12.27 12.98 inch

1.30 13.63 12.98 2 2

v

h

d d

a d

   

=

   

   

=    = = = 

   

− − =       

 LRFD BDS 5.7.2.8 

Determining the nominal shear resistance. For this example, conservatively combine the 

largest factored HL-93 Inventory forces along the wall (points 3.00 to 3.10 in Figure 6) to 

compute the minimum shear resistance. Concurrent results along the length of the wall 

could be used to obtain more refined results when rating factors are below 1.00): 

,

,

42.99 kip-ft

4.78 kip

N 19.54 kip

u,max,RTwall

u max,RTwall

u max,RTwall

M

V

=

=

=

The shear resistance of the walls needs to be evaluated using LRFD BDS 5.7.3.3. Shear 

factors  and  need to be calculated since the member thickness is not less than 16 

inches and there is no shear reinforcement. 
40 42.99(12) 515.88 kip-inch

4.78(12.98) 62.04 kip-inch

M V d u u v   =

 =

LRFD BDS5.7.3.4.2 

0.5

515.88

0.5(19.54) 4.78

12.98

0.0021

0.884(29000)

u

u u

v

s

s s

M

N V

d

E A



    + +

 

=

    + +

 

= =

LRFD BDS Eq. 5.7.3.4.2-4 

0 0.006 s

  

29 3500 29 3500(0.0021) 36.35o

  = + = +

s

=

LRFD BDS Eq. 5.7.3.4.2-3 

2 12.98 in 16 2(2) 0.75 11.25 inch x v bs d h cl d =  − − =  − − =

 Where cl is the clear cover; 

1.38 12 in 80 inch

0.63

1.38 12 in 11.25 80 in 12.00 inch

0.75 0.63

xe x

g

xe

s s

a

s

=  

+

=   

+

LRFD BDS Eq. 5.7.3.4.2-7 

( ) ( )

   

4.8 51

1 750 39

4.8 51 1.86

1 750(0.0021) 39 12.00

s xe

s





=

+ +

= =

+ +

LRFD BDS Eq. 5.7.3.4.2-2 

' ' 0.0316 0.25

0.90 0.0316(1.86) 4.0 0.25(4.0) 12(12.98)

16.48 kip

Vn v c c v v    f f b d =     

=     

=

LRFD BDS 5.7.3.3 

The following LRFD BDS checks have not been performed for this example but should be 

checked for actual rating calculations: regions requiring transverse reinforcement (5.7.2.3) and 

longitudinal reinforcement checks (5.7.3.5) [23 CFR 625.4(d)(1)(v)]. 

2.2.3.4 Section 4 – Right Wall Outside Steel 

Rectangular section height: h = 16.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.00 inch 

Area of single rebar: As_bar = 0.442 inch2 

Diameter of rebar: diabar = 0.75 inch 
41

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

The reinforcement and factored moment resistance are the same as calculated for Section 

3 above. 

Axial Resistance: 

It was shown in Section 3 above that axial thrust can be neglected for the left wall. 

Minimum Steel: 

It was shown in Section 3 above that the minimum steel reinforcement is met. 

Shear Resistance: 

Shear resistance is calculated for the wall with Section 3 above.

2.2.3.5 Section 5 – Slab 

Rectangular section height: h = 18.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.00 inch 

Area of single rebar: As_bar = 0.785 inch2 

Diameter of rebar: diabar = 1.00 inch 

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of reinforcing bar: 

_ 0.785(12.00) 2

1.571 inch

6.0

s bar

s

A b

A

s

= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

1.00 18.00 2.00 15.50 inches

2 2

bar

s

dia d h clr = − − = − − =

Determine distance from the equivalent stress block for tension controlled, nonprestressed tension reinforcement only: 

1.571(60) 2.31 inches

0.85 ' 0.85(4.0)(12.00)

s y

c

A f

a

f b

= = =

LRFD BDS Eq. 5.6.3.1.1-4 and 5.6.2.2 

1

1

2.31 0.85 2.72 inches

0.85

a

 c



=  = = =

LRFD BDS 5.6.2.2 
42 15.50 1 0.003 1 0.014 0.005

2.72

0.90

s

s c

f

d

c

 



   

= − =     − =     

 =

LRFD BDS 5.5.4.2, 5.6.2.1 

Calculate factored moment resistance: 

2

2.31 1ft 0.9(1.571)(60) 15.50

2 12 inch

101.40 kip-ft

n f s y s

a

  M A f d = − =    

     −     

=

LRFD BDS Eq. 5.6.3.2.2-1 

Axial Resistance: 

Slab need to be checked for axial thrust and flexural interaction using the largest factored 

HL-93 Inventory axial load in the wall to check applicability: 

'

'

0.1 0.1(0.7)(4.0)(18)(12) 60.48 kip

N 2.76 kip

0.1 Neglect Axial Thrust

c c g

u

c c g u

f A

f A N





= =

=

 

LRFD BDS 5.6.4.5 

Minimum Steel: 

Determine minimum reinforcement, beginning with the section modulus: 

2 2 12.00(18.00) 3

648 inch

6 6 c

bh S = = =

LRFD BDS 5.6.3.3 

The cracking moment equation simplifies to the following (only monolithic sections and 

no prestress forces), where fr is the modulus of rupture of concrete per LRFD BDS 

Article 5.4.2.6: 

3 1

ft 0.67(1.6)(0.480)(648)

12 inch

27.79 kip-ft

M f S cr r c 

 

= =    

=

LRFD BDS Eq. 5.6.3.3-1 

Since fMn > Mcr, minimum reinforcement at this section is satisfied. 

Shear Resistance 

The shear depth, dv, is calculated in accordance with LRFD BDS 5.7.3.2. 

1
43 0.72 0.72(18) 12.96

max 0.9 0.9(15.50) 13.95 14.35 inches

2.31 15.50 14.35 2 2

v

h

d d

a d

   

=

   

   

=    = = = 

   

− − =       

 LRFD BDS 5.7.2.8 

Determining the nominal shear. For this example, conservatively combine the largest 

factored HL-93 Inventory forces along the slab (points 2.00 to 2.10 in Figure 6) to 

compute the minimum shear resistance. Concurrent results along the length of the slab 

could be used to obtain more refined results when rating factors are below 1.00: 

81.27 kip-ft

16.61 kip

N 2.76 kip

u,max,slab

u,max,slab

u,max,slab

M

V

=

=

=

The shear resistance of the slab needs to be evaluated using LRFD BDS 5.7.3.3. Shear 

factors  and  need to be calculated since there is no shear reinforcement. 

81.27(12) 975.24 kip-inch

16.61(14.35) 268.35 kip-inch

M V d u u v   =

 =

LRFD BDS5.7.3.4.2 

0.5

975.24

0.5(2.76) 16.61

14.35

0.0019

1.571(29000)

u

u u

v

s

s s

M

N V

d

E A



    + +

 

=

    + +

 

= =

LRFD BDS Eq. 5.7.3.4.2-4 

0 0.006 s

  

29 3500 29 3500(0.0019) 35.65o

  = + = +

s

=

LRFD BDS Eq. 5.7.3.4.2-3 

( ) ( ) ( ) , , ( ) 0.75 1.00

18 2.50 2.00 12.625 inch

2 2

b top b bot

x v top bot

d d

s d h cl cl

+ +

=  − + − = − + − =

 Where cl is the clear cover; 

1.38 12 in 80 inch

0.63

1.38 12 in 12.625 80 in 12.625 inch

0.75 0.63

xe x

g

xe

s s

a

s

=  

+

=   

+

LRFD BDS Eq. 5.7.3.4.2-7 
44 ( ) ( )

   

4.8 51

1 750 39

4.8 51 1.96

1 750(0.0019) 39 12.625

s xe

s





=

+ +

= =

+ +

LRFD BDS Eq. 5.7.3.4.2-2 

' ' 0.0316 0.25

0.90 0.0316(1.95) 4.0 0.25(4.0) 12(14.35)

19.10 kip

Vn v c c v v    f f b d =     

=     

=

LRFD BDS 5.7.3.3 

The following LRFD BDS checks have not been performed for this example but should be 

checked for actual rating calculations: regions requiring transverse reinforcement (5.7.2.3) and 

longitudinal reinforcement checks (5.7.3.5) [23 CFR 625.4(d)(1)(v)]. 

2.2.3.6 Section 6 –Slab Left Connection 

Rectangular section height: h = 18.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.50 inch 

Area of single rebar: As_bar = 0.785 inch2 

Diameter of rebar: diabar = 1.00 inch 

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of reinforcing bar: 

_ 0.785(12.00) 2

1.571 inch

6.0

s bar

s

A b

A

s

= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

1.00 18.00 2.50 15.00 inches

2 2

bar

s

dia d h clr = − − = − − =

Determine distance from the equivalent stress block for tension controlled, nonprestressed tension reinforcement only: 

1.571(60) 2.31 inches

0.85 ' 0.85(4.0)(12.00)

s y

c

A f

a

f b

= = =

LRFD BDS Eq. 5.6.3.1.1-4 and 5.6.2.2 

1

1

2.31 0.85 2.72 inches

0.85

a

 c



=  = = =

LRFD BDS 5.6.2.2 
45 15.00 1 0.003 1 0.014 0.005

2.72

0.90

s

s c

f

d

c

 



   

= − =     − =     

 =

LRFD BDS 5.5.4.2, 5.6.2.1 

Calculate factored moment resistance: 

2

2.31 1ft 0.9(1.571)(60) 15.00

2 12 inch

97.86 kip-ft

n f s y s

a

  M A f d = − =    

     −     

=

LRFD BDS Eq. 5.6.3.2.2-1 

Axial Resistance: 

It was shown in Section 5 above that axial thrust can be neglected for the left wall. 

Minimum Steel: 

Determine minimum reinforcement, beginning with the section modulus: 

2 2 12.00(18.00) 3

648 inch

6 6 c

bh S = = =

LRFD BDS 5.6.3.3 

The cracking moment equation simplifies to the following (only monolithic sections and 

no prestress forces), where fr is the modulus of rupture of concrete per LRFD BDS 

Article 5.4.2.6: 

3 1

1ft 0.67(1.6)(0.480)(648)

12 inch

27.79 kip-ft

M f S cr r c 

 

= =    

=

LRFD BDS Eq. 5.6.3.3-1 

 Since fMn > Mcr, minimum reinforcement at this section is satisfied. 

Shear Resistance: 

Shear resistance is calculated for the slab with Section 5 above. 

2.2.3.7 Section 7 – Slab Right Connection 

Rectangular section height: h = 18.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.50 inch LRFD BDS 5.10.1 

Area of single rebar: As_bar = 0.785 inch2 

Diameter of rebar: diabar = 1.0 inch 

Spacing of rebar: s = 6.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 
46

The slab right connection details are the same as the left connection details (Section 6); therefore, 

the factored moment and shear resistances are the same. 

2.2.4 LRFR Rating Calculations 

The structural condition of the internal frame tunnel is satisfactory and the system factor falls 

under the category for "All Other Girder Bridges and Slab Bridges." Therefore: 

 Condition factor: c = 1.00 MBE Table 6A.4.2.3-1 

 System factor: s = 1.00 MBE Table 6A.4.2.4-1 

Resistance factors are based on LRFD BDS Section 5 and specified in the calculations of Section 

2.2.3 of this document. 

The equation for calculating the rating factor is based on MBE Equation 6A.4.2.1-1, which has 

been simplified for the load types being applied. 

( )

DC DW EV EH ES

LL

C DC DW EV EH ES RF

LL IM

    



    

=

+

MBE Eq. 6A.4.2.1-1

For the Strength Limit State: 

C R =   c s n

MBE Eq. 6A.4.2.1-2

Table 20 through Table 24 shows the results of the overall capacity based on MBE Equation 

6A.4.2.1-2 as well as the rating load factors based on MBE Equation 6A.4.2.1-1. It is also noted 

that negative moments are designated with a negative sign as well as corresponding negative 

moment capacities. It is imperative to coordinate positive and negative moments and shears 

when calculating the rating factor. For this example, shear forces were taken as positive as a 

simplification because controlling component shear loads acted on the rating sections in the same 

direction. Ratings are performed at the critical section defined in LRFD BDS 12.11.5.2 for 

flexure and 5.7.3.2 for shear. Consequently, the critical moment section (xcr for moment) is at the 

wall to slab interface and the critical shear (xcr for shear) is at a distance dv from the inside face 

of the wall or slab. A summary of these locations are: 

Moment @ 1: Left Wall at centerline of intersection with horizontal Egress 

Corridor Slab 

Moment @ 2: Left Wall at underside of Roadway Slab 

Moment @ 3&4: Right Wall at underside of Roadway Slab 

Moment @ 5: Roadway Slab maximum positive moment based on review of 

tenth point total factored forces 

Moment @ 6: Roadway Slab at inside face of Left Wall 

Moment @ 7: Roadway Slab at inside face of Right Wall 

Shear @ 1&2: Left Wall at dv below Roadway Slab based on review of tenth 

point total factored forces 

Shear @ 3&4: Right Wall at dv below Roadway Slab based on review of tenth 

point total factored forces 
47

Shear @ 5-7: Roadway Slab at dv from inside face of Left Wall based on review 

of tenth point total factored forces 

Maximum and minimum load factors are selectively used to obtain the maximum absolute value 

of the force effects. The selected load factors are shown in Table 20. Note that a minimum load 

factor of 0.00 for CR and SH is defined in case actual forces are less than computed; therefore, 

any beneficial effects of CR and SH will not be counted. Additionally, the design truck controls 

the HL-93 ratings and the EV-3 controls the EV ratings. The moments, shears and respective 

capacities are in kip-ft and kip in Table 21 to Table 24. 

Table 20. Load Factors. 

Max/Min DC AP TU CR&SH PL LL 

Inventory 

LL 

Operating EV Legal Permit 

Max 1.25 1 0.5 1.25 0 1.75 1.35 1.3 1.3 

Min 0.9 1 0.5 0 1.75 1.75 1.35 1.3 1.3 

Table 21. HL-93 Inventory Rating Factors. 

Force Section xcr (in) DC AP+ AP- TU+ TU- CR&SH PL HL-93 C RF

Moment 1 109.50 2.74 -0.09 0.09 -1.71 1.71 -7.13 -0.92 5.07 51.59 5.32 

Moment 2 210.96 -13.21 -0.06 0.09 -0.87 0.87 -3.62 1.10 -19.45 -87.26 1.93 

Moment 3 & 4 178.00 15.61 -0.21 0.01 -3.16 3.16 -13.17 -0.12 18.11 51.59 0.96 

Moment 5 259.20 17.17 -0.02 0.01 -1.09 1.09 4.56 -0.10 15.35 101.4 2.74 

Moment 6 24.00 -21.37 -0.05 0.05 -0.44 0.44 -1.84 -0.44 -24.73 -97.86 1.56 

Moment 7 424.00 -12.61 -0.01 0.28 -3.27 3.27 13.63 0.12 -16.22 -97.86 2.83 

Shear 1 & 2 198.02 1.99 0.05 0.05 0.11 0.11 0.44 0.25 3.06 12.76 1.66 

Shear 3 & 4 16.02 1.05 0.00 0.05 0.21 0.21 0.89 0.01 0.15 16.84 50.62 

Shear 5 - 7 38.35 4.06 0.00 0.00 0.08 0.08 0.33 0.02 5.47 19.1 1.41 

Table 22. HL-93 Operating Rating Factors. 

Force Section xcr (in) DC AP+ AP- TU+ TU- CR&SH PL HL-93 C RF

Moment 1 109.50 2.74 -0.09 0.09 -1.71 1.71 -7.13 -0.92 5.07 51.59 6.90 

Moment 2 210.96 -13.21 -0.06 0.09 -0.87 0.87 -3.62 1.10 -19.45 -87.26 2.50 

Moment 3 & 4 178.00 15.61 -0.21 0.01 -3.16 3.16 -13.17 -0.12 18.11 51.59 1.25 

Moment 5 259.20 17.17 -0.02 0.01 -1.09 1.09 4.56 -0.10 15.35 101.4 3.55 

Moment 6 24.00 -21.37 -0.05 0.05 -0.44 0.44 -1.84 -0.44 -24.73 -97.86 2.01 

Moment 7 424.00 -12.61 -0.01 0.28 -3.27 3.27 13.63 0.12 -16.22 -97.86 3.67 

Shear 1 & 2 198.02 1.99 0.05 0.05 0.11 0.11 0.44 0.25 3.06 12.76 2.15 

Shear 3 & 4 16.02 1.05 0.00 0.05 0.21 0.21 0.89 0.01 0.15 16.84 65.62 

Shear 5 - 7 38.35 4.06 0.00 0.00 0.08 0.08 0.33 0.02 5.47 19.1 1.83 
48

Table 23. EV-2 and EV-3 Rating Factors. 

Force Section xcr (in) DC AP+ AP- TU+ TU- CR&SH PL EV C RF

Moment 1 109.50 2.74 -0.09 0.09 -1.71 1.71 -7.13 -0.92 6.28 51.59 5.79 

Moment 2 210.96 -13.21 -0.06 0.09 -0.87 0.87 -3.62 1.10 -24.08 -87.26 2.10 

Moment 3 & 4 178.00 15.61 -0.21 0.01 -3.16 3.16 -13.17 -0.12 22.42 51.59 1.05 

Moment 5 259.20 17.17 -0.02 0.01 -1.09 1.09 4.56 -0.10 29.11 101.4 1.95 

Moment 6 24.00 -21.37 -0.05 0.05 -0.44 0.44 -1.84 -0.44 -21.70 -97.86 2.38 

Moment 7 424.00 -12.61 -0.01 0.28 -3.27 3.27 13.63 0.12 -14.23 -97.86 4.35 

Shear 1 & 2 198.02 1.99 0.05 0.05 0.11 0.11 0.44 0.25 3.79 12.76 1.83 

Shear 3 & 4 16.02 1.05 0.00 0.05 0.21 0.21 0.89 0.01 0.16 16.84 65.13 

Shear 5 - 7 38.35 4.06 0.00 0.00 0.08 0.08 0.33 0.02 6.77 19.1 1.54 

Table 24. Permit Rating Factors. 

Force Section xcr (in) DC AP+ AP- TU+ TU- CR&SH PL Permit C RF

Moment 1 109.50 2.74 -0.09 0.09 -1.71 1.71 -7.13 -0.92 6.59 51.59 5.51 

Moment 2 210.96 -13.21 -0.06 0.09 -0.87 0.87 -3.62 1.10 -25.27 -87.26 2.00 

Moment 3 & 4 178.00 15.61 -0.21 0.01 -3.16 3.16 -13.17 -0.12 23.53 51.59 1.00 

Moment 5 259.20 17.17 -0.02 0.01 -1.09 1.09 4.56 -0.10 26.23 101.4 2.16 

Moment 6 24.00 -21.37 -0.05 0.05 -0.44 0.44 -1.84 -0.44 -32.13 -97.86 1.62 

Moment 7 424.00 -12.61 -0.01 0.28 -3.27 3.27 13.63 0.12 -21.07 -97.86 2.94 

Shear 1 & 2 198.02 1.99 0.05 0.05 0.11 0.11 0.44 0.25 3.98 12.76 1.75 

Shear 3 & 4 16.02 1.05 0.00 0.05 0.21 0.21 0.89 0.01 0.24 16.84 44.69 

Shear 5 thru 7 38.35 4.06 0.00 0.00 0.08 0.08 0.33 0.02 7.10 19.1 1.47 
49

2.3 PRESTRESSED BOTTOM SLAB 

141⁄2 PS Slab

26 – 0.60"φ strands in 8' wide slab 

AASHTO Grade M203 Grade 270 

Low Relaxation Strands

#4 Stirrups @ 12" 

31⁄2 Clear

(a)

(b)

32' 

31' Span

Prestressed Slab

 Source: FHWA 

Figure 7. Illustration. PS Slab Geometry and Prestressing Steel. 
50

The prestressed bottom slab is a 14 1⁄2-inch slab spanning 32 feet with a 12-inch bearing on each 

end resulting in an effective span of 31 feet. The PS slab is comprised of 7 ksi concrete, (26)-

0.60 inch diameter, AASHTO Grade M203 Low Relaxation strands for each 8 foot slab with #4 

shear bars at 12 inches on center in both transverse and longitudinal directions. The reinforcing 

steel is grade 60 steel. Adjacent precast slabs are connected via shear keys and therefore transfer 

forces between them. 

2.3.1 Load Calculations 

The slab is exposed to self-weight, air pressure, and live load. Other forces such as TU are 

neglected due to the simply supported boundary conditions of the PS slab. Creep and Shrinkage 

as well as prestressing forces are accounted for in the capacity calculations 

2.3.1.1 Dead Load Component, DC 

Dead load components include the self-weight of the concrete prestressed slab. Prestressed 

forces are accounted for in the capacity calculations. 

Use a unit weight, wc, of 0.150 kcf for reinforced concrete. The self-weight of the slab is applied 

to the model by including self-weight on the geometry detailed in Figure 7. 

2.3.1.2 Wearing Surface, DW 

The prestressed slab has a 2-inch asphalt overlay with a unit weight of 140 pcf. This produces a 

load pressure of 23.33 psf. However, since this is a known specified load applied during initial 

construction with field verification as opposed to an allowance for a future overly, the wearing 

surface will be included in the Dead Load Component, DC loads for this example, such that a 

strength load factor of 1.25 is applied per MBE 6A.2.2.3. 

2.3.1.3 Air Pressure, AP 

Air pressure was applied to the roadway of ±10 psf, according to the original construction plans. 

2.3.1.4 Temperature, TU 

No temperature loads were applied since this prestressed slab behaves as a simply supported 

slab. 

2.3.1.5 Creep and Shrinkage, CR & SH 

Creep and shrinkage is accounted for in the prestress losses within the capacity calculations. 

2.3.1.6 Live Load Application, LL 

The live load travels perpendicular to the span of the prestressed slab. The distribution of axle 

loads based on an equivalent strip width, E, to cast-in-place slabs acting perpendicular to the 

direction of traffic is provided in LRFD BDS Table 4.6.2.1.3-1, where S is the span length of 31 

feet. 
51 26.0 6.6

26.0 6.6(31) 230.60 inches 19.22 feet

pos

Slab

E S

E +

= +

= + = =

Since the design strip is transverse to the direction of traffic, the wheel loads are included in the 

live load evaluation; however, the lane load is not included per LRFD BDS Article 3.6.1.3.3. The 

maximum number of trucks to be placed on the structure is obtained by taking the roadway width 

and dividing by a 12-foot lane per LRFD BDS Article 3.6.1.1.1 resulting in two possible trucks 

to be placed parallel. This results in axle loads that are placed transversely such to produce 

maximum forces at the critical locations, identified at mid-span and end of span. Maximum end 

forces (shear) were obtained by placing the first axle load 2 feet from the barrier per LRFD BDS 

Article 3.6.1.3.1. The second truck is placed such that there is 4 feet between wheel placements. 

Maximum positive moment is obtained when the centerline of the span is midway between the 

center of gravity of loads and the nearest concentrated load. Live load placement is shown in 

Figure 8. 

Wheel loads for unit, or 1-foot, analysis widths are determined by multiplying the equivalent unit 

width wheel load, P, by the appropriate impact factor (IM) and multiple presence factor (MPF). 

The equivalent wheel load is determined by dividing the full wheel load by the equivalent strip 

width, E, over which the axle load is distributed. Calculate the wheel loads in the same manner 

as the internal frame. Refer to Section 2.2.1.7 for a detailed discussion. 

HL-93 Design Truck: 

( ) 93 1

32 kip

2 wheels 16 k/wheel 0.83

19.22 ft HL Truck

Slab

P

E

−

+

     

= = = ( )

93 2

32 kip 32 kip

2 wheels 32 k/wheel 0.96

2 14 2 33.22 ft HL Truck

Slab Slab

P

E E −

+ +

  +

   

= = =

+ + ( )

93 3

32 kip 32 kip 8 kip

2 wheels 36 k/wheel 0.76

2 14 14 2 47.22 ft HL Truck

Slab Slab

P

E E −

+ +

  + +    

= = =

+++

93 ( 93 1 93 2 93 3 )

k/wheel

max , , 0.96

ft

PHL Truck − += = P P P HL Truck HL Truck HL Truck − − −

HL-93 Design Tandem: 

( ) 93 1

25 kip

2 wheels 12.5 k/wheel 0.65

19.22 ft HL Tandem

Slab

P

E

−

+

     

= = =
52

( )

93 2

25 kip 25 kip

2 wheels 25 k/wheel 1.08

2 4 2 23.22 ft HL Tandem

Slab Slab

P

E E −

+ +

  +

   

= = =

+ +

93 ( 93 1 93 2 )

k/wheel

max , 1.08

ft

PHL Tandem − +

= = P P HL Tandem HL Tandem − −

EV-2:

( ) 2 1

33.5 kip

2 wheels 16.75 k/wheel 0.87

19.22 ft EV

Slab

P

E

−

+

     

= = = ( )

2 2

33.5 kip 24 kip

2 wheels 28.75 k/wheel 0.84

2 15 2 34.22 ft EV

Slab Slab

P

E E −

+ +

  +

   

= = =

+ +

2 ( 2 1 2 2 )

k/wheel max , 0.87

ft

P P P EV += =

EV EV − −

EV-3:

( ) 3 1

31 kip

2 wheels 15.5 k/wheel 0.81

19.22 ft EV

Slab

P

E

−

+

     

= = = ( )

3 2

31 kip 31 kip

2 wheels 31 k/wheel 1.34

2 4 2 23.22 ft EV

Slab Slab

P

E E −

+ +

  +

   

= = =

+ + ( )

3 3

31 kip 31 kip 24 kip

2 wheels 43 k/wheel 1.13

2 4 15 2 38.22 ft EV

Slab Slab

P

E E −

+ +

  + +    

= = =

+ + +

3 ( 3 1 3 2 3 3 )

k/wheel max , , 1.34

ft

P P P P EV += =

EV EV EV − − −

Permit Truck: 

All the permit truck axle spacings are less than ESlab+; therefore, all 11 axle load 

possibilities should be considered. The calculations follow the format as above and P

varies from 0.55 to 1.72 over the 11 permutations. The calculation for the controlling 10-

axle configuration is shown below. 
53

( )

21 21 21 21 21 23 23 23 18 18

2 wheels 105 k/wheel 1.72

2 4 4 4 4 10 4 4 4 4 2 61.22 ft Permit

Slab Slab

P

E E +

+ +

  + + + + + + + + +    

= = =

+ + + + + + + + + +

Mid-Span

2' 6'

6' 6'

6'

6'

6'

2'

10.50'

6.00'

4'

4'

End-Span

Mid-Span

2' 6'

6' 6'

6'

6'

6'

2' 4'

4'

 Source: FHWA 

Figure 8. Illustration. Live Load Placement on Slab. 

Single trucks are amplified by a multiple presence factor, MPF, of 1.2 while the two truck effects 

utilize a multiple presence factor of 1.0 per LRFD BDS Table 3.6.1.1.2-1. For this example, 

MBE Article 6A.4.4.3 is used to determine the dynamic impact factor, IM, of 1.33 for all 

vehicles.. Additionally, from the equivalent wheel load calculations, it can be seen that the HL93 tandem will control over the HL-93 truck and the EV-3 vehicle controls over the EV-2 for 

this example. 

2.3.2 Structural Analysis 

A beam analysis is applied for the bottom slab. All the load effects and member resistances are 

calculated using this 1-foot wide strip representation. The slab is assumed to be simply supported 

and therefore can be evaluated with simple hand calculations or by a structural analysis program. 

For the purpose of this example, a structural analysis program was used for simplicity. 

The LRFD Road Tunnel Design and Construction Guide Specifications, 1st Ed. (AASHTO, 

2017b), Table 3.4-1 identifies the applicable minimum and maximum load factors for each load 

specified above. Since the slab is simply supported, only maximum load factors were necessary 

to produce critical design forces. The live load factors for the HL-93 is identified in Table 3.4-1, 

and the live load factor for the EV vehicles is 1.3 as identified in the FHWA Emergency Vehicle 

Memorandum and the live load factor for the permit vehicle is identified as 1.3 from MBE 3rd

Edition Table 6A.4.5.4.2a-1, corresponding to unlimited crossings, mixed with traffic, two or 

more lanes, ADTT greater than 5000, and a permit weight factor ratio (GVW/AL) greater than 3. 

The forces were factored to produce controlling Strength I (or Strength II for permit loads) as 

well as Service I and Service III load combinations. These resulting unfactored forces are given 

in Table 25 and Table 26. Note shears are shown as absolute values. 
54

Table 25. Slab Shear Forces (kip). 

Sp Pt DC AP HL-93 EV Permit 

1.0 3.45 0.16 3.89 5.22 6.20 

1.1 2.76 0.13 3.15 4.22 5.02 

1.2 2.07 0.10 2.46 3.29 3.91 

1.3 1.38 0.06 1.95 2.61 3.10 

1.4 0.69 0.03 0.28 0.37 0.44 

1.5 0.00 0.00 0.28 0.37 0.44 

1.6 -0.69 -0.03 -0.28 -0.37 -0.44 

1.7 -1.38 -0.06 -1.95 -2.61 -3.10 

1.8 -2.07 -0.10 -2.46 -3.29 -3.91 

1.9 -2.76 -0.13 -3.15 -4.22 -5.02 

1.10 -3.45 -0.16 -3.89 -5.22 -6.20 

Table 26. Slab Moment Forces (kip-ft). 

Sp Pt DC AP HL-93 EV Permit 

1.0 0.00 0.00 0.00 0.00 0.00 

1.1 9.62 0.45 10.49 14.06 16.70 

1.2 17.10 0.80 19.25 25.81 30.65 

1.3 22.44 1.05 24.56 32.93 39.12 

1.4 25.64 1.20 29.30 39.29 46.67 

1.5 26.71 1.25 30.16 40.44 48.04 

1.6 25.64 1.20 27.29 36.59 43.46 

1.7 22.44 1.05 23.70 31.78 37.75 

1.8 17.10 0.80 16.09 21.57 25.62 

1.9 9.62 0.45 8.04 10.79 12.81 

1.10 0.00 0.00 0.00 0.00 0.00 

1.0

1.10

Span Point

Prestressed Slab

 Source: FHWA 

Figure 9. Illustration. Slab Direction. 

Figure 9 shows the span point identification for the prestressed slab. The whole number 

represents the member identification while the 10ths number indicates the percentage along the 

span length. 
55

2.3.3 Resistance Calculations 

2.3.3.1 Concrete Properties 

f’c = 7.0 ksi 

f’ci = 5.6 ksi LRFD BDS 5.4.2.3.2 

∝1 = 0.85 LRFD BDS 5.6.2.2 

β1 = 0.70 if f’c = 7.0 ksi LRFD BDS 5.6.2.2 

λ = 1.0 (normal weight concrete) LRFD BDS 5.4.2.8 

γ2 = 1.1 (bonded strands) LRFD BDS 5.6.3.3 

γ1 = 1.6 LRFD BDS 5.6.3.3 

γ3 = 1.0 (prestressed girders) LRFD BDS 5.6.3.3 

 Modulus of rupture: 

0.24 ' 0.24(1.0) 7.0 0.635 ksi

r c

f f = =  =

LRFD BDS 5.4.2.6 

Compression reinforcement in flexural capacity calculations is conservatively ignored. 

Calculated results are based on a per foot analysis width. 

2.3.3.2 General Properties 

Rectangular section height: h = 14.50 inch 

Rectangular section width: b = 12.00 inch 

Slab width: bslab = 8.00 feet 

Distance to PS strand from tension face: clr = 3.50 inch 

Area of PS strand: As_PS = 0.217 inch2 

Diameter of PS strand: diaPS = 0.60 inch 

Area of single rebar: As_bar = 0.196 inch2 

Diameter of rebar: diabar = 0.500 inch 

Spacing of rebar shear steel: s = 12.0 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Shear resistance factor: v = 0.9 LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of prestressing: 

_ 0.217(26) 2

0.705 in /ft

8

s PS

s

slab

A n

A

b

= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

3.50 14.50 3.5 11.00 inches sd h = − = − =
56

Eccentricity of prestressing steel: 

14.50 11.00 3.75 inches

2 2 s

h

e d = − = − =

Modulus of Elasticity: 

0.140 0.001 ' 0.140 0.001(7.0) 0.147 kcf

c c

 = + = + = f

LRFD BDS Table 3.5.1-1 

0.140 0.001 ' 0.140 0.001(5.6) 0.1456 kcf ci ci  = + = + = f

LRFD BDS Table 3.5.1-1 

2 '0.33 2 0.33

1

120,000 ( ) 120,000(1.0)(0.147) (7.0)

4928 ksi

E K f

c c c

= = 

=

LRFD BDS Eq. 5.4.2.4-1 

2 ' 0.33 2 0.33

1

120,000 ( ) 120,000(1.0)(0.1456) (5.6)

4492 ksi

E K f ci ci ci= = 

=

LRFD BDS Eq. 5.4.2.4-1 

28,500 ksi E

p

=

LRFD BDS 5.4.4.2 

Section Properties: 

2

3 3

4

2 2

3

12(14.50) 174 inch

14.50 7.25 inches

2 2

12(14.50) 3049 inch

12 12

12(14.50) 420.5 inch

6 6

c

x

x

A bh

h

y

bh I

bh S

= = =

= = =

= = =

= = =

2.3.3.3 Prestressed Losses 

Jacking Stress: LRFD BDS Table 5.9.2.2-1 

0.75 0.75(270) 202.5 ksi j puf f = = =

Elastic Shortening Losses: LRFD BDS Eq. C5.9.3.2.3a-1

( )

( )

2

2

2

2

/

0.705(202.5)(3049 3.75 174) 3.75(23.95 12)(174)

6.82 ksi

0.705(3049 3.75 174) 174(3049)(4492) / (28,500)

ps j x c SW c

pES

ps x c c x ci p

A f I e A eM A

f

A I e A A I E E

  + −    =   + +  

  +  −   

= =   +  +  
57 202.5 6.82 195.68 ksi

P 195.68(0.705) 138 kip

pi pj pES

i pi ps

f f

f A

= −  = − =

= = =

Elastic gains are neglected for this example for conservatism. Long term losses are calculated 

utilizing the Approximate Method as defined in LRFD BDS Article 5.9.3.3. These losses assume 

the specimens were cured with 70% humidity. 

'

2.4 ksi

1.7 0.01 1.7 0.01(70) 1.0

5 5 0.76

1 1 5.6

10.0

12

10.0(202.5)(0.705)(1.0)(0.76) 12(1.0)(0.76) 2.4 17.76 ksi

174

pR

h

st

ci

pj ps h st

p T h st pR

c

f

RH

f

f A

f f

A





 

  

 =

= − = − =

= = =

+ +

 = + + 

= + + =

195.68 17.76 177.92 ksi

P 177.92(0.705) 125.4 kip

pe pi p T

e pi ps

f f f

f A

= −  = − = 

= = =

2.3.3.4 Concrete Stresses 

Allowable concrete stresses after losses are per LRFD BDS Table 5.9.2.3.2a-1 and Table 

5.9.2.3.2b-1. Note compression is represented by negative (-) and tension is represented by 

positive (+). 

'

e

'

e

'

0.45 0.45(7.0) 3.15 ksi (P +DL)

0.60 0.60(7.0) 4.20 ksi (P +DL+LL)

0.19 0.6 0.19 7.0 0.6 0.503 ksi

c c

c c

t c

f f

f f

f f

 − = − = −

 − = − = −

  =  =

HL-93 stresses: 

26.7 kip-ft

1.3 kip-ft

30.2 kip-ft

NC

C

LL

M

M

M

=

=

=

 
58

 Top of section stresses: 

, 93

125.4 125.4(3.75) 26.7(12) 1.3(12) 0.401 ksi

174 420.5 420.5

(30.2)(12) 0.862 ksi

420.5

e e NC C

PE DL

c x x

LL

LL HL

x

P P e M M f

A S S

M

f

S

+

− + − +

= + − = + − = −

= − = − = −

 Bottom of section stresses: 

, 93

125.4 125.4(3.75) 26.7(12) 1.3(12) 1.040 ksi

174 420.5 420.5

(30.2)(12) 0.861 ksi

420.5

e e NC C

PE DL

c x x

LL

LL HL

x

P P e M M f

A S S

M

f

S

+

− + − +

= − + = − + = −

= = =

EV-2 and EV-3 stresses: 

40.4 kip-ft MLL=

 Top of section stresses: 

,

(40.4)(12) 1.153 ksi

420.5

LL

LL EV

x

M

f

S

= − = − = −

 Bottom of section stresses: 

,

(40.4)(12) 1.153 ksi

420.5

LL

LL EV

x

M

f

S

= = =

Permit truck stresses: 

48.0 kip-ft MLL=

 Top of section stresses: 

,

(48.0)(12) 1.370ksi

420.5

LL

LL Permit

x

M

f

S

= − = − = −

 Bottom of section stresses: 

,

(48.0)(12) 1.370ksi

420.5

LL

LL Permit

x

M

f

S

= = =
59

2.3.3.5 Ultimate Flexural Resistance 

The ultimate moment strength for one foot strip of the prestressed slab is calculated below 

conservatively ignoring the additional capacity due to the minimal area of non-prestressed 

tension (or compression) reinforcement. The following calculations are outlined in LRFD BDS 

Eq. 5.6.3.1.1-1 and Eq. 5.6.3.1.1-4. The flexural resistance factor, which is based on a tensioncontrolled section, is calculated in accordance with LRFD BDS 5.6.2.1 & 5.5.4.2. 

k = 0.28

LRFD BDS Table C5.6.3.1.1-1 

'

1

0.705(270) 3.47 in

0.85 / 0.85(7.0)(0.7)12 0.28(0.705)270 /11.00

ps pu

c ps pu p

A f

c

f b kA f d 

= = =

+ +

11.00 1 0.003 1 0.0065 0.005 1.0

3.47

s

s c f

d

c

  

   

= − =   − =   =       1

a c = = =  0.7(3.47) 2.43in0.28 3.47 1 270 1 246.15 ksi

11.00 ps pu

s

kc f f

d

    

= − = −     =

   

2.43 1ft M 1.0(0.705)(246.15) 11.00 142 kip-ft

2 2 12 in n f ps ps s

a

  A f d    =   − =  −   =

    

The minimum steel per LRFD BDS 5.6.3.3 should be checked to ensure minimum strength is 

achieved. Since the composite and non-composite section is the same, the second term within the 

brackets becomes zero and can therefore be neglected. 

( )

( )

3 1 2

125.4 125.4(3.75) 1.84 ksi

174 420.5

1

1ft 1.0 1.6 0.635 1.1 1.84 420.5 107 kip-ft

12 in

OK

e e

cps

c x

c

cr r cps x dnc

nc

n cr

P P e f

A S

S

M f f S M

S

M M

  



− −

= − = − = −

   

= + − −        

 

=  +     =   

 

2.3.3.6 Shear Resistance 

Critical section for shear is dv from the face of the support in accordance with LRFD BDS 

5.7.3.2. 
60 0.72 0.72(14.50) 10.44

max 0.9 0.9(11.00) 9.90 10.44 inches

2.43 11.00 9.79 2 2

v

h

d d

a d

   

=

   

   

=    = = = 

   

− − =       

 LRFD BDS 5.7.2.8 

Applied shears and moments at the section of interest located a distance dv from the face of the 

support are used to determine the nominal shear resistance. It was revealed that the shear 

resistance is significantly larger than any of the live load shears; therefore, for simplicity and 

conservatism, the maximum shear resistance will be based on the maximum shear force due to 

the critical live load of HL-93 Inventory. The maximum shear force and moment is: 

max

max

87.4 kip-ft

11.3 kip

u

u

M

V

−

−

=

=

The shear resistance of the slab needs to be evaluated using LRFD BDS 5.7.3.3. Shear factors 

and  need to be calculated per LRFD BDS Article 5.7.3.4.2. #4 shear reinforcement is provided 

on 12-inch centers along the span and 16.5-inch centers in the direction of traffic and therefore 

can be included in the shear resistance. 

87.4(12) 1048.8 kip-inch 11.3(10.44) 118.0 kip-inch M V d u u v   =  =

1048.8

0.5 0.5(0) 11.3 0.705(189)

10.44

0.0011 0.000

0.705(28500)

u

u u ps po

v

s

ps ps

M

N V A f

d

E A



   

 + + −   + + − 

   

= = = −  0 0.006 s

  

29 3500 29 3500(0.0000) 29.00o

  = + = +

s

=

 

( )  

4.8 4.8 4.80

1 750 1 750(0.0000)

s





= = =

+ +

'

0.0316 0.0316(4.80) 7.012(10.44) 50.28 kip Vc c v v

=  f b d = =

( )

12 0.196 (60)(10.44)cot(29 ) cot 29 16.5 10.96 kip

12

v y v

s

A f d

V

s

    



 

= = =

  V V V n v c s= + = + =    0.90 50.28 10.96 55.12 kip 

The following LRFD BDS checks have not been performed for this example but should be 

checked for actual rating calculations: regions requiring transverse reinforcement (5.7.2.3); 

detailing of transverse reinforcement (5.7.2.4); minimum shear reinforcement (5.7.2.5); and 

longitudinal reinforcement checks (5.7.3.5). 
61

2.3.4 LRFR Rating Calculations 

The structural condition of the prestressed slab is satisfactory and the system factor falls under 

the category for "All Other Girder Bridges and Slab Bridges." Therefore: 

 Condition factor: c = 1.00 MBE Table 6A.4.2.3-1 

 System factor: s = 1.00 MBE Table 6A.4.2.4-1 

Resistance factors are based on LRFD BDS Section 5 and specified in the calculations of Section 

2.3.3 of this document. 

The equation for calculating the rating factor is based on MBE Equation 6A.4.2.1-1, which has 

been simplified for the load types being applied. 

( )

DC DW EV EH ES

LL

C DC DW EV EH ES RF

LL IM

    



    

=

+

MBE Eq. 6A.4.2.1-1

For the Strength Limit State: 

C R =   c s n

MBE Eq. 6A.4.2.1-2

Table 28 shows the results of the overall capacity based on MBE Equation 6A.4.2.1-2 as well as 

the load rating factors based on MBE Equation 6A.4.2.1-1. Maximum and minimum load factors 

are selectively used to obtain the maximum absolute value of the force effects. The load factors 

for strength limit states are as shown in Table 27. In Table 28, the load effects are unfactored 

having a unit of kip-ft or kip for moment and shear, respectively. 

Table 27. Load Factors for Strength Limit States. 

Max/Min DC AP

HL-93 

Inventory 

Level 

HL-93 

Operating 

Level 

EV 

Legal Permit 

Max 1.25 1 1.75 1.35 1.3 1.3 

Min 0.9 N/A N/A N/A N/A N/A 

Table 28. Slab Resistance Rating Factors for Strength. 

Force Rating DC AP LL C RF

Moment HL-93 Inventory 26.71 1.25 30.16 142 2.03 

Moment HL-93 Operating 26.71 1.25 30.16 142 2.64 

Moment EV Legal 26.71 1.25 40.44 142 2.04 

Moment Permit 26.71 1.25 48.04 142 1.72 

Shear HL-93 Inventory 3.45 0.16 3.89 55.12 7.44 

Shear HL-93 Operating 3.45 0.16 3.89 55.12 9.64 

Shear EV Legal 3.45 0.16 5.22 55.12 7.47 

Shear Permit 3.45 0.16 6.20 55.12 6.29 
62

For prestressed concrete, rating for stresses at Service limit states is optional per MBE Table 

6A.4.2.2-1 at the legal load level (EV vehicle at Service-III limit state) and permit load level 

(permit truck at Service-I limit state) but may be included upon the owners request. However, 

Service-III limit state ratings for the design load (HL-93) at the Inventory level are applicable for 

prestressed concrete. Per MBE Table 6A.4.2.2-1, the dead load factor (DC and AP) is 1.00 and 

the live load factor is 0.80. Per LRFD BDS 5.9.2.3.2, tensile stresses in prestressed slabs apply to 

the Service-III limit state (LRFD BDS 5.9.2.3.2b). Using the factored stresses and stress limits 

calculated in Section 2.3.3.4, compute the HL-93 Inventory level rating factor for tensile stress 

on the bottom face of the slab: 

( )

( ) ( )

1.0 0.503 1.0( 1.040) 2.24

0.80 0.80 0.861

t PE DL

bot

LL

f f

RF

f

− + − −

= = =

2.4 TUNNEL LINER 

This example considers a circular bored tunnel with a precast segmental reinforced concrete 

lining. An earth pressure balance tunnel boring machine (EPB TBM) was used to excavate the 

tunnel and install the lining. The internal diameter is 52 feet and the liner segment thickness is 2 

feet. A typical cross-section of the tunnel is presented in Figure 10. 

For the purpose of performing a load rating of the liner, the crown of the tunnel is assumed to be 

only 20 feet below the at-grade roadway. In reality, such a large diameter tunnel in shallow soft 

ground is unlikely. 

Source: FHWA 

Figure 10. Illustration. Tunnel Cross-Section. 
63

Underground conditions were modeled after the north portal area of the tunnel, where the tunnel 

is relatively shallow and is not fully submerged. A submerged tunnel at such shallow depths 

would experience instability due to buoyancy. As illustrated in Figure 11, soils were simplified 

into two representative layers including Recent Granular Deposits (RGD) above the tunnel and 

glacial Till-Like Deposits (TLD) surrounding and beneath the tunnel. The yellow color 

represents the RGD (upper 15 feet) and the bluish color represents the TLD (below 15 feet).

In this analysis, the only structural component subject to load rating analysis is the tunnel liner. 

Structure demands used in the example calculations are assumed to come from: self-weight of 

the structure or dead load component (DC); vertical earth loads (EV); horizontal earth loads 

(EH); at-grade building surcharge (BS); and live loads (LL). 

Two-Dimensional (2D) Finite Element (FE) analysis is used to determine structural demands in 

the form of moment, shear, and axial forces along the tunnel liner. The commercially-available 

FE software Plaxis 2D (2019) is used for this example. A 2D model is generated which includes 

the structural elements surrounded by a continuum representing the soil, allowing the model to 

account for effects of soil-structure-interaction (SSI). 

2.4.1 Materials 

Concrete and section properties are listed below. In order to include the liner as a linear elastic 

plate element in the FE model, an effective moment of inertia was assumed (FHWA, 2009). The 

effective moment of inertia is significantly lower than the gross value of a 2-foot thick segment 

because it accounts for the joints in between the precast liner segments. It should be recognized 

that this effective moment of inertia is only an approximation; the actual structural behavior of a 

precast segmental liner with bolted connections would be more complex and is beyond the scope 

of this geotechnical analysis example. 

2.4.1.1 Concrete Properties 

Specified tunnel liner concrete properties: 

Concrete strength: f’c = 7.0 ksi (1,008 ksf) 

Unit weight of concrete: wc = 0.155 kcf 

Design modulus of elasticity: Ecd = 5,328 ksi (767,232 ksf) 

2.4.1.2 Section Properties 

Liner thickness: t = 24 inches (2 feet)

Section area: A = 2 ft2/ft (per foot of tunnel length) 

Gross moment of inertia: Ig = 0.667 ft4/ft (per foot of tunnel length) 

Number of joints: n = 10 (assumed for this example) 

Joint moment of inertia: Ij = 0 (no moment resistance is provided by the joints) 

Effective moment of inertia: Ieff = 0.107 ft4/ft (per foot of tunnel length) 

( )2

4 / eff j g I I I n = +

 FHWA-NHI-10-034 Equation 10-11 
64

2.4.1.3 Example Notes 

• This cross-section is fictitious and is only for the purpose of demonstrating the principles 

of using FE models to perform analysis to aid in rating tunnels. 

• Live loading on the ground surface over the tunnel may increase force effects and is 

therefore included in the load rating analysis. 

• Dead loads from the roadway and other structures interior to the liner are included in the 

analysis. 

• In general, if the surface roadway passing over the tunnel has an overlay, sidewalk, or 

other loads, they should be included in the dead load analysis for tunnel load rating. This 

example assumes that these items are not included in the geometry of the tunnel. 

• While not assumed here, any other mechanical (including dynamic equipment loads) and 

architectural systems supported by the tunnel should be included in the dead load. 

• The liner is modeled in this example with the approximated effective moment of inertia 

described above. 

2.4.1.4 Rating Approach/Assumptions 

• The load rating evaluation is performed for a 1-foot wide strip of tunnel and surrounding 

soil (parallel to the direction of traffic above the tunnel). 

• Pavement has approximately the same unit weight as the soil and is therefore included 

with the soil vertical loads. 

• Groundwater is assumed to be 10 feet above the tunnel invert, and is included in the FE 

model accordingly. The location of groundwater is shown in Figure 11. 

• In this example, the HL-93 Truck, Tandem, EV-2, and EV-3 live loads are considered. 

All relevant live loads may be considered in an actual load rating analysis. 

• There are no signs of distress or deterioration as the tunnel construction was recently 

completed; therefore, the tunnel is considered to be in satisfactory physical condition. 

• Structural demands are calculated by performing 2D FE analysis including the soil and 

structure. 

2.4.2 FE Model Description 

A 2D plane strain model was developed to assess the load demands on the tunnel due to dead 

loads, earth pressures, and live loads. The model is illustrated in Figure 11, representing a 1-foot 

wide strip (into the page) of the tunnel and surrounding soil. Structural elements are used to 

represent the tunnel and continuum elements are used to represent the soil. Interface elements are 

included around the outside of the tunnel to simulate the interaction between the soil and 

structure. Model length units are feet, and force units are kips. 
65 

Source: FHWA

Figure 11. Illustration. 2D Plane Strain FE Model of Tunnel Cross-Section. Length Units 

are in Feet. 

2.4.2.1 Structural Elements

The tunnel liner is modeled with linear-elastic curved plate (i.e., shell) elements (Plaxis, 2019), 

which are used to model slender structures in the ground with a significant flexural rigidity (EI) 

and normal stiffness (EA). The assumption of linear-elastic structural behavior is expected to be 

sufficient for load rating purposes, where loading is restricted to typical service-type conditions. 

However, after the FE analysis is completed, the resulting structural demands should be checked 

against capacities to verify this assumption.

Properties are assigned to the liner based on the structure data listed in Section 2.4.1. The 

assigned properties are listed in Table 29 and Table 30. In the 1-foot wide plane strain model, 

structure properties are defined per foot of model width. Plate elements have zero thickness, 

while the actual tunnel liner is 2 feet thick. The plate elements are modeled along the centerline 

of the actual tunnel liner elements, with a diameter of 54 feet. The liner weight is based on the 

actual liner thickness of 2 feet times the unit weight of 0.155 kcf for the high strength reinforced 

concrete. The tunnel is centered horizontally at an X coordinate of zero feet. The tunnel crown is 

at a Y coordinate of -20 feet (20 feet below grade), and the invert is as -74 feet (74 feet below 

grade).

Table 29. Structure Properties. 

E 

(ksf)

A 

(ft2/ft)

Ieff

(ft4/ft)

767,232 2 0.107
66

Table 30. Plane Strain Model Inputs. 

EA 

(kip/ft) 

EIeff

(kip-ft2/ft) 

Weight, w 

(kip/ft/ft) 

1,534,464 82,094 0.31 

Notes for Table 29 and Table 30: 

1. E = Young’s modulus.

2. A = Area. 

3. Ieff = Effective moment of inertia.

4. Precast Liner (Plate Element)

2.4.2.2 Soil Domain 

The model soil domain surrounding the tunnel (Figure 11) has overall dimensions of 400 feet in 

width (X coordinates of -200 feet to +200 feet), and 200 feet in depth (Y coordinates of +0 to -

200 feet). In soil-structure interaction (SSI) modeling, overall dimensions should be sufficiently 

large such that they do not significantly influence the model response and the results of interest. 

The soil dimensions were determined for this example by first starting with an initial assumption 

with relatively small dimensions and running the model to determine the structure forces 

(moment, shear, and axial). Then, the dimensions were increased incrementally and the model 

was re-run until the forces became insensitive to further increases. 

2.4.2.3 FE Mesh 

Fifteen-node triangular continuum elements were used to discretize the soil domain and the plate 

elements have five nodes per adjacent triangular continuum element as shown in Figure 12. 

Plaxis (2019) generates the mesh automatically based on the settings and adjustments selected by 

the user. Similar to selecting the domain size, the fineness of the mesh should be selected such 

that it does not significantly influence the model response and the results of interest. An 

appropriate mesh was selected for this example by comparing the results for different mesh sizes. 
67 

(A)15-node triangle (B) 5-node plate 

Source: FHWA 

Figure 12. Illustrations. Node distribution for (Plaxis 2019): a) 15-node triangular 

continuum element and b) 5-node plate element. 

2.4.2.4 Soil Properties 

The soil domain in the FE model is described by constitutive models (stress-strain relationships) 

and soil properties which are assigned by the user. Actual stress-strain behavior of soil tends to 

involve significant variability and uncertainty. In order to address this variability in SSI analyses 

and ensure a reliable load rating, a range of soil properties may need to be considered. 

Accordingly, sets of Upper and Lower Estimate (UE and LE) soil parameters (listed in Table 31) 

were used in this example for the Recent Granular Deposits (upper 15 feet) and Till-Like 

Deposits (below 15 feet). 

The Mohr-Coulomb (Plaxis, 2019) linear elastic-perfectly plastic model is used to approximate 

nonlinear soil behavior in this example. Five parameters are typically considered to define the 

Mohr-Coulomb model for each soil type, including: unit weight, friction angle (), cohesion 

intercept (c), Young’s modulus (E), and Poisson’s ratio (). An advanced feature was also used 

to allow for increasing soil stiffness with depth based on a sixth input parameter (Eincr). As 

indicated in Table 31, the value for Young’s modulus was defined at the top of the layer (Etop) 

and limited to a maximum value (Emax).

The at-rest earth pressure coefficient (K0) is used to define the initial horizontal stress conditions 

in the model. In FE analysis of underground structures, K0 can have a significant impact, because 

it significantly influences horizontal earth pressures. Construction sequencing may also affect 

horizontal earth pressures in FE models, which do not always exactly simulate reality. In order to 

account for variability, uncertainty, and potential effects of construction, a relevant range of K0

should be considered in the load rating analysis. 
68 

The Recent Granular Deposits are modeled after loose to dense sandy soils encountered near the 

ground surface. These recent deposits are normally consolidated, with K0 assigned as 1-sin() 

based on the Plaxis (2019) default recommendation. It is noted that the Recent Granular Deposits 

are only located above the tunnel crown (Figure 11), so the K0 value in this layer does not have 

as significant an impact as the Till-Like Deposits surrounding the tunnel. 

Glacial till tends to be over-consolidated, strong, and stiff. In this example, the Till-Like 

Deposits were modeled as primarily coarse grained, with relatively high  values and stiffness 

parameters. Lower and Upper Estimate K0 values of 0.5 and 1.0, respectively, were considered in 

order to cover a typical range of over-consolidated conditions in such materials. 

Table 31. Soil Parameters. 

Soil Type 

Unit 

Wt. 

(pcf)

 (deg) c (ksf) Etop

(ksf) 1 

Einc

(ksf/ft) 2

Emax

(ksf) 3 

Poisson

’s Ratio K0

4 

Recent Granular 

Deposits 

Lower Estimate 

105 30 0.1 500 25 875 0.3 0.50 

Recent Granular 

Deposits 

Upper Estimate 

125 36 0.1 1,000 50 1,750 0.3 0.41 

Till-Like Deposits 

Lower Estimate 125 36 0.1 1,000 300 8,500 0.35 0.50 

Till-Like Deposits 

Upper Estimate 145 42 0.1 2,000 600 17,000 0.35 1.00 

Notes: 

1. Etop = Young’s modulus at top of layer.

2. Einc = Rate of change of stiffness with depth below top of layer. 

3. Emax = Maximum value for E (no further increase with depth beyond this value). 

4. K0 = At-rest earth pressure coefficient.

2.4.2.5 Soil-Structure Interface 

A soil-structure interface is included around the outside perimeter of the tunnel with strength 

equal to two-thirds (0.67) that of the adjacent soil. The interface limits the amount of stress that 

can be transferred between the structure and the soil. Small circles with a minus sign inside them 

graphically indicate the interface around the perimeter of the tunnel in Figure 11 and in later 

figures. 

Permanent loads including the dead weight of the tunnel (DC), vertical earth pressures (EV), and 

horizontal earth pressures (EH) are generated automatically in the model based on the assigned 

structure and soil properties. Dead load reactions from the internal tunnel structures such as the 

roadway boxes (Figure 10) are applied as downward point loads acting on the corbels, as 

illustrated in Figure 13. Loads of 15.6 kip/ft (kips per foot model width) and 12.3 kip/ft are 

applied to the left and right corbels, respectively. The corbels are included as linear elastic 

continuum elements with concrete stiffness in order to distribute the point loads over their 

contact area with the tunnel liner. 
69 

Source: FHWA

Figure 13. Illustration. Close-Up View of Tunnel in FE Model. 

Live loads are applied as downward distributed loads on the surface of the model (Figure 13). 

Surface traffic is assumed to be moving in the direction from right to left across the tunnel. 

Based on AASHTO BDS Article 3.6.1.2.5, a tire contact length (lt) of 10 inches is assumed along 

the direction of traffic. In order to consider various vehicle positions above the tunnel and find 

the most critical location, a distributed load with a length of approximately 10 inches was placed 

(but not always activated) at every foot on-center from X = -30 feet to X = +30 feet (lighter

arrows in Figure 13). Vehicle loads are simulated by activating the distributed loads at wheel 

locations above the tunnel for various vehicle positions. For instance, the two activated loads 

(dark arrows) in Figure 13 represent HL-93 Tandem loading, which consists of a pair of wheel 

loads spaced at 4 feet.

The four live loads included in this example for demonstration purposes are the HL-93 Truck 

and HL-93 Tandem loads, EV-2, and EV-3. The HL-93 Truck consists of an 8-kip load, and two 

32-kip loads, each spaced at 14 feet. The HL-93 Tandem consists of two 25-kip loads spaced at 4

feet, as mentioned above. EV-2 consists of a 24-kip load and a 33.5-kip load spaced at 15 feet.

EV-3 includes a 24-kip wheel load followed by a 15-ft spacing and a pair of 31-kip loads spaced

at 4 feet. Lane loads for the HL-93 Tandem and Truck loads were not applied, refer to Section

2.4.2.5.1 for a discussion.

2.4.2.5.1 Distribution of Wheel Loads through Fill 

Axle loads will distribute outward laterally as they pass downward through the fill until they 

reach the tunnel. This leads to lower magnitude pressures (spread out over a larger area) on the 

tunnel than at the ground surface. Distribution transverse to the tunnel’s longitudinal axis (to the 

left and right in Figure 13) will occur automatically in the 2D model. However, the 2D plane 
70 

strain model cannot automatically account for the distribution along the tunnel’s longitudinal 

direction (in and out of the page in Figure 13).

The guidelines in LRFD BDS Article 3.6.1.2.6 were used to account for the distribution along 

the tunnel’s longitudinal direction as follows: 

• For traffic running transverse to the tunnel’s longitudinal axis (crossing over the tunnel), 

and wheel load distribution along the tunnel’s longitudinal direction, 

0.06 20 0.06(648) 6

12 12 12 12 0.95 feet

1.15

t i

w

int-t

w D

s

H

LLDF

− − − −

= = =

LRFD BDS Eq. 3.6.1.2.6b-1 

Where: 

sw = 6 feet 

wt = 20 inches 

Di = 54 feet (648 inches) 

LLDF = 1.15 LRFD BDS Table 3.6.1.2.6a-1 

• Conservatively assume H = 20 feet above the tunnel (H = 20 feet at the crown and H > 20 

feet away from the crown). 

• Since H > Hint-t: 

( ) 0.06

12 12

20 648 6 1.15(20) 0.06 33.9 feet

12 12

t i

w w

w D w s LLDF H = + + +

= + + + =

LRFD BDS Eq. 3.6.1.2.6b-3 

• The length of the distributed axle loads in the model is 0.84 feet (approx. 10 inches / 12 

inches per foot). 

• The distributed loads to be applied in the 2D FE model are: 

o Each axle load for HL-93 Tandem: 25 kips / 33.9 feet / 0.84 feet = 0.88 ksf 

o Front axle load for HL-93 Truck: 8 kips / 33.9 feet / 0.84 feet = 0.28 ksf 

o Middle and rear axle loads for HL-93 Truck: 32 kips / 33.9 feet / 0.84 feet = 1.12 

ksf 

o Front axle load for EV-2 and EV-3: 24 kips / 33.9 feet / 0.84 feet = 0.84 ksf 

o Rear axle load for EV-2: 33.5 kips / 33.9 feet / 0.84 feet = 1.18 ksf 

o Middle and rear axle loads for EV-3: 31 kips / 33.9 feet / 0.84 feet = 1.09 ksf 

The above magnitudes are assigned to the appropriate discrete distributed load in the model to 

simulate each wheel. For instance, to simulate HL-93 Tandem loading at the arch mid-span as 

illustrated in Figure 13, the distributed loads centered at X-coordinates -2 feet and +2 feet were 

activated and assigned a magnitude of 0.88 ksf. All other distributed loads were left unactivated. 
71 

As mentioned previously, lane loading was not applied with the HL-93 vehicles. LRFD BDS 

Article 3.6.1.2.6a discusses the use of axle loads only for buried structures. However, for larger 

tunnel structures, lane loading should be considered unless it is negligible. In the case of this 

example, comparing the pressure due to the lane loading, pland, at the top of the tunnel (under 20 

feet of fill) with the soil pressure at that depth, psoil, shows it is negligible (less than 1% of soil 

load): 

( )

0.640 klf 0.640 0.02 ksf

w + 10 1.15(20) lane

land

p

LLDF H

= = =

+

(0.125 kcf )(20) 2.50 ksf 0.02 ksf soil soil lane p H = =  = ? p =

Per LRFD BDS Article 3.6.2.2, the dynamic load allowance at depth of 20 feet is 0% (no 

increase in wheel loads). Additionally, per LRFD BDS Article 3.6.1.2.6a, the single lane 

multiple presence factor of 1.20 should be applied to traffic parallel to the span (perpendicular to 

longitudinal direction of tunnel). The 1.20 factor is applied to the FE results prior to computing 

load factors. 

2.4.2.5.2 Determination of Critical Location for Live Loads 

Live load placements need to be considered in order to capture the maximum and minimum 

force envelopes. Influence lines can sometimes be used to determine placement of live loads. 

However for more complex SSI analyses, alternative methods may be more appropriate. 

A relatively simple, but time consuming method is to consider all relevant vehicle locations one 

at a time in separate FE analyses. Figure 14 presents an example for the HL-93 Truck at one 

specific location. The three activated loads (dark arrows) can be moved incrementally to any 

other position. Structural demands can be compared for each load position in order to identify the 

maximum demands. 
72 

Source: FHWA

Figure 14. Illustration. Live Load Application for HL-93 Truck at One Location. 

Alternatively, a moving load analysis can be performed to identify the critical load location. 

Moving load analysis is more complex, but can save time if performed by an experienced 

modeling professional. A moving load analysis is demonstrated in Example 4 of the Reference 

Guide for Load Rating of Tunnel Structures (FHWA, 2019) using the Dynamics feature of Plaxis 

(2019). Moving load analyses were performed to determine the location of each live load that 

results in the maximum moment in the tunnel liner (at the crown) for this example. 

2.4.2.6 Load Factors 

Structural dead loads and earth pressures are generated automatically in the FE model. As a 

result, it is not possible to apply a load factor directly to the EV or EH loads. It is possible to 

apply load factors to live loads in the FE model. However, for consistency with the handling of 

permanent loads, load factors were not applied to live loads in the FE analysis. Therefore, the FE 

model results are for unfactored permanent and live loads. Load factors should be applied to the 

resulting demands later in the load rating procedure. 
73 

2.4.2.7 Volume Loss 

During earth pressure balanced tunnel boring machine (EPB TBM) tunnel construction, there is 

typically a small volume of soil that is over-excavated in excess of the theoretical volume of the 

excavation, leading to ground or volume loss. This volume loss can lead to ground settlements at 

the surface and can also impact the tunnel liner forces. Volume loss is typically lumped into 

three groups (FHWA, 2009): face loss, shield loss, and tail loss. Face loss consists of ground 

movements into the shield face. Shield loss occurs between the cutting edge and the shield tail, 

due in part to the small amount of overcut that occurs for maneuverability. Tail loss occurs 

behind the tail as the ground support mechanism moves to the liner itself and the grout that fills 

the annular space between the cut and the liner. In the tunneling industry, the volume loss (i.e., 

amount of over-excavation) is typically expressed as a percentage of the tunnel size. 

EPB TBMs are able to minimize the magnitude of volume loss by applying pressure to the tunnel 

face, minimizing the overcut, and grouting the tail void (FHWA, 2009). However, the volume 

loss is not likely to be eliminated entirely. Ultimately, the amount of volume loss is dependent on 

the quality of workmanship during construction. As a result, it is good practice to consider a 

range of volume loss in the load rating analysis in order to address this uncertainty. However, if 

ground loss levels were measured and recorded during the construction of a tunnel, a more 

precise value can be used for the load rating. 

Modern EPB TBM techniques have significantly reduced the ground loss on recent projects. For 

example, the LA Metro Regional Connector, Sao Paulo Metro Line 4, Heathrow Airside Road 

Tunnel, and Madrid South Bypass M-30 Tunnels have reported ground loss in the range of 0.1% 

to 0.4% for tunnel diameters of approximately 20 to 50 feet (Metro, 2019). 

Volume loss is applied in this example using the line contraction feature in Plaxis (2019). The 

circumference of the liner plate element is reduced to achieve the specified volume loss. As such, 

it is a simplification of the actual volume loss mechanism described above. In this example, two 

volume loss scenarios are considered as lower and upper bounds in order to address uncertainties 

and numerical simplifications: zero volume loss; and 0.5% volume loss. 

2.4.2.8 Sensitivity Scenarios 

FE analyses were performed for the following scenarios to address uncertainties discussed in the 

previous sections: 

• LE soil properties with no volume loss.

• LE soil properties with 0.5% volume loss. 

• UE soil properties with no volume loss. 

• UE soil properties with 0.5% volume loss. 

The load rating analysis for this example was based on the highest structure demands from these 

four scenarios. 
74 

2.4.3 FE Analysis Procedure

The FE analysis was performed in phases in order to consider a realistic construction sequence 

and to separately evaluate demands under permanent loads only as well as permanent plus live 

loads. An initial phase was performed with level ground conditions (without the tunnel present) 

to generate stresses in the ground prior to the start of construction, as illustrated in Figure 15. 

Vertical stresses were generated based on the assigned soil unit weights, and horizontal stresses 

were generated based on a K0 procedure (horizontal stress = vertical stress times K0). The 

following staged construction phases were performed next: 

Tunnel Construction (Figure 16): Excavate tunnel, activate liner and interface, and dewater 

inside tunnel by setting soil clusters inside tunnel to dry. 

Volume Loss (not shown): Activate the line contraction to apply the volume loss (for scenarios 

that include non-zero volume loss). 

Interior Dead Loads (Figure 17): Activate corbels and point loads representing the dead load 

from the structures inside the tunnel lining. All unfactored permanent loads (DC, EV, and EH) 

are accounted for at the end of this phase, with the exception of the building surcharge which is 

discussed in Section 2.4.4.1.2. 

Live Loads (Figure 13 and Figure 14): Activate vehicle live load. All unfactored permanent 

loads and live loads (DC, EV, EH, and LL) are accounted for at the end of this phase, with the 

exception of the building surcharge which is discussed in Section 2.4.4.1.2. 

Source: FHWA

Figure 15. Illustration. Generate Stresses Under Original Level Ground Condition. 
75 

Source: FHWA

Figure 16. Illustration. Excavate Tunnel, Activate Liner and Interface, Dewater Inside. 

Source: FHWA

Figure 17. Illustration. Activate Corbels and Apply Interior Structure Dead Loads. 

All of the analysis phases listed above were performed for each of the four sensitivity scenarios 

described in Section 2.4.2.8. 
76 

2.4.4 FE Analysis Results

Results are presented in the following sections for use in the load rating analysis in the form of 

moment, shear, and axial force plots along the liner. The results are plotted as a function of the 

angle from the horizontal, with zero degrees being at the right-most point on the liner (3 o’clock 

position), 90 degrees at the crown (12 o’clock position), and 270 degrees at the invert (6 o’clock 

position), as illustrated in Figure 18. Axial compression loads are negative. 

Source: FHWA

Figure 18. Illustration. Convention Used for Angle from Horizontal to Present the Results. 

2.4.4.1 Structure Demands due to Permanent Loads 

2.4.4.1.1 Without Building Surcharge 

Results of the analysis including all of the permanent loads except for the building surcharge are 

presented in Figure 19 for each of the four sensitivity scenarios described in Section 2.4.2.8. 
77 

 Source: FHWA 

Figure 19. Analysis Results. Permanent Loads Without the Building Surcharge. 
78 

2.4.4.1.2 With Building Surcharge 

The presence of large buildings at the surface above the tunnel was considered by applying a 

vertical surcharge of 7 ksf across a width of 84 feet centered about the tunnel centerline, as 

illustrated in Figure 20. The 7 ksf is a defined building load prescribed within the project 

parameters. Based on the assumption that the buildings were in place before the tunnel, the 

building surcharge was activated prior to tunnel excavation and construction. Results of the 

analysis including all of the permanent loads plus the building surcharge are presented in Figure 

21 for each of the four sensitivity scenarios described in Section 2.4.2.8. 

It should be noted that for tunnels at this depth below soil, building loads as shown in Figure 20

and live loads over the top are not coincident. As a result, the ratings are not calculated including 

the building surcharge. At increased depths where adjacent building surcharges would overlap 

with overhead vehicular traffic, the live load effects would be nonexistent. 

 Source: FHWA 

Figure 20. Illustration. Building Surcharge Included at the Ground Surface. 
79 

Source: FHWA

Figure 21. Analysis Results. Permanent Loads Including the 7 ksf Building Surcharge. 
80 

2.4.4.2 Structure Demands due to Permanent and Live Loads 

Live load analysis results are presented in this section. The results indicate that the impacts of 

surface live loads on the tunnel liner are relatively small, and the difference between each vehicle 

is hardly noticeable. As explained in Section 2.4.2.5.1, the wheel loads are distributed 

horizontally through the overlying soil before reaching the tunnel, which significantly lessens 

their impact on the liner as compared with the weight of the soil itself.

2.4.4.2.1 HL-93 Tandem 

The deformed mesh and liner moments for the HL-93 Tandem live load plus permanent loads 

(excluding the building surcharge) are illustrated in Figure 22 for the LE with 0.5% volume loss 

scenario. Moment, shear, and axial force diagrams are presented in Figure 23 for each of the four 

sensitivity scenarios described in Section 2.4.2.8. 

a) Exaggerated (50x) Deformed Mesh (LE, 0.5% Volume Loss) 

b) Liner Moments (LE, 0.5% Volume Loss) 

Source: FHWA

Figure 22. Analysis Results. Permanent Loads + HL-93 Tandem (No Building Surcharge).
81 

 Source: FHWA 

Figure 23. Analysis Results. Permanent Loads + HL-93 Tandem (No Building Surcharge).
82 

2.4.4.2.2 HL-93 Truck 

The deformed mesh and liner moments for the HL-93 Truck live load plus permanent loads 

(excluding the building surcharge) are illustrated in Figure 24 for the LE with 0.5% volume loss 

scenario. Moment, shear, and axial force diagrams are presented in Figure 25 for each of the four 

sensitivity scenarios described in Section 2.4.2.8. 

a) Exaggerated (50x) Deformed Mesh (LE, 0.5% Volume Loss) 

b) Liner Moments (LE, 0.5% Volume Loss) 

Source: FHWA

Figure 24. Analysis Results. Permanent Loads + HL-93 Truck (No Building Surcharge).
83 

 Source: FHWA 

Figure 25. Analysis Results. Permanent Loads + HL-93 Truck (No Building Surcharge).
84 

2.4.4.2.3 EV-2 

The deformed mesh and liner moments for the EV-2 live load plus permanent loads (excluding 

the building surcharge) are illustrated in Figure 26 for the LE with 0.5% volume loss scenario. 

Moment, shear, and axial force diagrams are presented in Figure 27 for each of the four 

sensitivity scenarios described in Section 2.4.2.8. 

a) Exaggerated (50x) Deformed Mesh (LE, 0.5% Volume Loss) 

b) Liner Moments (LE, 0.5% Volume Loss) 

Source: FHWA

Figure 26. Analysis Results. Permanent Loads + EV-2 (No Building Surcharge).
85 

 Source: FHWA 

Figure 27. Analysis Results. Permanent Loads + EV-2 (No Building Surcharge).
86 

2.4.4.2.4 EV-3 

The deformed mesh and liner moments for the EV-3 live load plus permanent loads (excluding 

the building surcharge) are illustrated in Figure 28 for the LE with 0.5% volume loss scenario. 

Moment, shear, and axial force diagrams are presented in Figure 29 for each of the four 

sensitivity scenarios described in Section 2.4.2.8.

a) Exaggerated (50x) Deformed Mesh (LE, 0.5% Volume Loss) 

b) Liner Moments (LE, 0.5% Volume Loss) 

Source: FHWA

Figure 28. Analysis Results. Permanent Loads + EV-3 (No Building Surcharge).
87 

 Source: FHWA

Figure 29. Analysis Results. Permanent Loads + EV-3 (No Building Surcharge).
88

2.4.4.3 Critical Forces

The critical load forces were extracted from the shear, moment and axial force diagrams. 

Factored loads were reviewed around the perimeter to find the maximum load locations. The 

load combination and location for the maximum and minimum moments, shears and axial forces 

are identified and the subsequent forces are extracted and reported in Table 32 through Table 35. 

These loads are unfactored. Live load results were obtained by subtracting the demands due to 

permanent loads only in Section 2.4.4.1 from the corresponding permanent plus live load 

demands in Section 2.4.4.2. Live load values in the following tables include the 1.20 single lane 

multiple presence factor.

The maximum moment shown in Table 32 was obtained when subjected to the LE-Contraction

soil condition at 90° with respect to Figure 18, which occurs at the top of the tunnel. Positive 

moment causes tension on the inside face of the tunnel liner (in the absence of axial 

compression). It can be seen that the shear is effectively zero at this location and the live load 

components of the moment and axial thrust is small in relation to dead load.

Table 32. Maximum Moment (kip-ft) & Corresponding Shear (kip) & Axial (kip).

Force Perm Perm

+Bldg Truck Tandem EV-2 EV-3

Moment 20.95 74.80 0.82 0.89 0.61 0.96

Shear 0.02 0.10 0.00 0.00 0.00 0.01

Axial -27.91 -116.23 -0.71 -0.50 -0.48 -0.71

The minimum moment as shown in Table 33 was obtained when subjected to the UEContraction soil condition at 354° with respect to Figure 18, which occurs at the right side of the 

tunnel and is further from the surface than the positive moments above. It can be seen that shear 

live load forces at this location are essentially zero. Although this location is the maximum 

factored negative (or minimum) moment location, only negligible live loads moments, which are 

positive, exist there. To compare against the maximum positive moments above, maximum 

negative live load moments from around the perimeter are shown in Table 33 to envelope the 

results. The maximum negative live load moments occur in the LE-Contraction soil condition at 

52°. It will be shown later the reinforcement is the same on both faces and the concrete cover on 

each face is similar (2 inches on outside versus 1.5 inches on inside) so the larger positive 

moments above will control the rating. As such, no negative moment rating factors will be 

calculated.

Table 33. Minimum Moment (kip-ft) & Corresponding Shear (kip) & Axial (kip).

Force Perm Perm

+Bldg Truck Tandem EV-2 EV-3

Moment* -15.93 -12.47 -0.27 -0.27 -0.21 -0.32

Shear 0.07 -2.73 0.02 0.01 0.01 0.02

Axial -31.94 -72.06 -0.82 -0.64 -0.57 -0.77

*Note: Live load moments are minimums from entire perimeter.
89 

The location of maximum factored shear, whether it be positive or negative, as shown in 

Table 34 was obtained when subjected to the LE-No Contraction soil condition at 245° with 

respect to Figure 18, which is located near the bottom of the tunnel liner at the left corbel. It can 

be seen that live load forces (moment, shear or axial) at this location are essentially zero. This 

location, at the bottom of the tunnel is not influenced by live load above. Other locations do have 

live load shears but they are not the maximum factored shear location. It will be shown later, that 

the shear capacity is much greater than the factored maximum shear (1.35 times dead load) so 

ratings will not be calculated. The shear resistance will be conservatively based on maximum 

positive or negative moments and axial thrust to ensure capacity envelopes all possible 

combinations. For reference, the maximum HL-93 Truck shear is ±0.12 kips for the LEContraction soil condition on either side of the tunnel crown (90°).

Table 34. Maximum Shear (kip) & Corresponding Moment (kip-ft) & Axial (kip). 

Force Perm Perm

+Bldg Truck Tandem EV-2 EV-3 

Moment -5.50 1.23 0.06 0.05 0.04 0.06 

Shear 4.00 4.83 0.03 0.03 0.02 0.03 

Axial -93.99 -143.16 -0.06 -0.01 -0.04 -0.07 

The minimum axial thrust (negative axial load) as shown in Table 35 was obtained when 

subjected to the UE-No Contraction soil condition at 270° with respect to Figure 18, which is 

located at the bottom of the tunnel. As expected, there are no factored axial tension forces in the 

liner. For similar reasons to the maximum shear location above, it can be seen that live load 

forces (shear or axial) at this location are essentially zero. This location, at the bottom of the 

tunnel is not influenced by live load above. Similar to the minimum (negative) moment location, 

the maximum factored thrust location is not subjected to compression live load. The live load 

values show in Table 35 are the maximum values around the perimeter to envelope the results. 

The maximum live load axial thrust is for the LE-Contraction soil condition on the left and right 

sides of the tunnel. It will be shown later, that the axial capacity is much greater than the factored 

maximum axial thrust (1.35 times dead load) so ratings will not be calculated. 

Table 35. Maximum Axial (kip) & Corresponding Moment (kip-ft) & Shear (kip). 

Force Perm Perm

+Bldg Truck Tandem EV-2 EV-3 

Moment -0.43 1.64 0.02 0.01 0.01 0.02 

Shear -0.07 -0.07 0.00 0.00 0.00 0.00 

Axial* -175.57 -195.19 -1.32 -1.06 -0.92 1.35

*Note: Live load thrusts are minimums from entire perimeter.

2.4.5 Resistance Calculations

The tunnel liner was constructed from 24-inch thick precast concrete sections that are 77.86 

inches wide at the outside. These sections have 2 inches of clear to the ASTM A496 deformed 

steel wire size D-31 transverse bars on the outside face and 1.5 inches on the inside face. The 
90 

size D-30 longitudinal bars, which resist the bending moments, are inside the transverse 

reinforcement. Structural resistances are calculated assuming the 16 longitudinal bars in each 

section equally resist the force demands in that precast section. This results in 2.47 bars per 

linear foot of the tunnel liner. The deformed steel wire size D-30 longitudinal bars have a 

diameter of 0.618 inch while the size D-31 transverse bars have a diameter of 0.628 inch. These 

sections are constructed utilizing 7.0 ksi concrete and 75 ksi reinforcing steel. 

Concrete Properties: 

f’c = 7.0 ksi 

∝1 = 0.85 LRFD BDS 5.6.2.2 

β1 = 0.70 for f’c = 7.0 ksi LRFD BDS 5.6.2.2 

λ = 1.0 (normal weight concrete) LRFD BDS 5.4.2.8 

γ3 = 0.75 (AASHTO M31 Grade 75) LRFD BDS 5.6.3.3 

γ1 = 1.6 LRFD BDS 5.6.3.3 

 Modulus of rupture: 

0.24 ' 0.24(1.0) 7.0 0.635 ksi

r c

f f = =  =

 

Compression reinforcement in flexural capacity calculations is conservatively ignored. The same 

reinforcement is used for both faces. Compute minimum moment resistance based on direction 

with maximum reinforcement cover (outside face). Calculated results are based on a per foot 

analysis width. 

2.4.5.1 Moment Resistance 

Rectangular section height: h = 24.00 inch 

Rectangular section width: b = 12.00 inch 

Clear distance to rebar from tension face: clr = 2.00 inch 

Area of single rebar: As_bar = 0.300 inch2 

Diameter of rebar: diabar = 0.618 inch 

Spacing of rebar: s = 4.87 inch 

Flexural resistance factor: f = TBD LRFD BDS 5.5.4.2 

Moment Resistance: 

Determine equivalent area of reinforcing bar: 

2

_

. 0.300(12"/ 4.87" bar spacing) 0.740 inch A A No Bar s s bar= = =

Determine distance of the extreme compression fiber to the centroid of the reinforcement: 

0.618 24.00 0.628 2.00 21.06 inches

2 2

bar

s trans

dia d h d clr = − − − = − − − =

Determine distance from the equivalent stress block for tension controlled, nonprestressed tension reinforcement only: 
91 

0.740(75) 0.78 inches

0.85 ' 0.85(7.0)(12.00)

s y

c

A f

a

f b

= = =

LRFD BDS Eq. 5.6.3.1.1-4 and 5.6.2.2 

1

1

0.78 0.7 1.11inches

0.7

a

 c



=  = = =

LRFD BDS 5.6.2.2 

21.06 1 0.003 1 0.054 0.005

1.11

0.90

s

s c

f

d

c

 



   

= − =     − =     

 =

LRFD BDS 5.5.4.2, 5.6.2.1 

Accounting for the level of applied axial compression at the location of maximum 

factored moment in the equation above would not shift the strain out of the tensioncontrolled zone. 

Calculate nominal moment resistance: 

2

0.78 1ft 0.9(0.740)(75) 21.06

2 12 inch

86.02 kip-ft

n f s y s

a

  M A f d 

= − =    

     −     

=

LRFD BDS Eq. 5.6.3.2.2-1 

The calculation of moment resistance conservatively ignores the presence of axial thrust, which 

generally increase the moment resistance up to the levels of axial compression present in the 

analysis. This can be seen in Figure G6A-1 in Appendix G6A of the MBE (AASHTO, 2018). If 

the resulting moment ratings were inadequate, the refined moment-axial interaction procedure 

could be employed. However, as can be seen in the figure, very high loads of axial compression 

may reduce the moment resistance. 

Axial Resistance: 

Tunnel liner need to be checked for axial thrust and flexural interaction. Check 

flexure/axial interaction: 

'

0.1 0.1(0.7)(7.0)(24)(12) 141.12 kip c c g  f A = =

LRFD BDS 5.6.4.5 

Since the axial forces associated with the maximum factored moments (permanent dead 

load plus live load) do not exceed 0.1cfc’Ag, the axial thrust can be neglected. 

Furthermore, at the locations where the dead load effects are approaching this limit, the 

live load moment effect is approaching zero; as a result, the load rating can be neglected 

when axial forces are in excess of this limit. 

Minimum Steel: 

Determine minimum reinforcement, beginning with the section modulus: 
92 2 2 12.00(24.00) 3

1152 inch

6 6 c

bh S = = =

LRFD BDS 5.6.3.3 

The cracking moment equation simplifies to the following (only monolithic sections and 

no prestress forces): 

3 1

1ft 0.75(1.6)(0.635)(1152)

12 inch

73.15 kip-ft

M f S cr r c 

 

= =    

=

LRFD BDS Eq. 5.6.3.3-1 

Since fMn > Mcr, minimum reinforcement at this section is satisfied 

2.4.5.2 Shear Resistance 

The shear resistance of the liner is also evaluated. The liner does have shear bars dispersed 

through the precast sections. However, for the purpose of simplification, only the concrete shear 

capacity will be utilized for the shear resistance. If the resistance is not sufficient with just the 

concrete shear resistance, the shear bars can be included in the resistance calculations. 

The shear depth, dv, in accordance with LRFD BDS 5.7.3.2. 

0.72 0.72(24) 17.28

max 0.9 0.9(21.06) 18.95 20.67 inches

0.78 21.06 20.67 2 2

v

h

d d

a d

   

=

   

   

=    = = = 

   

− − =       

 LRFD BDS 5.7.2.8 

Determining the nominal shear resistance involves the use of the applied shears and moments at 

the section of interest. To envelope the possible combinations of forces, use the non-concurrent 

maximum factored HL-93 Inventory loadings.: 

29.85 kip-ft

5.38 kip

N 236.96 kip

u

u

u

M

V

=

=

=

The shear resistance of the liner needs to be evaluated using LRFD BDS 5.7.3.3. Shear factors 

and  need to be calculated since the member thickness is not less than 16 inches and the shear 

steel is not being considered in the resistance calculations. 

29.85(12) 358.2kip-in 5.38(20.67) 111.2 kip-inch

M V d u u v  

=  =

 LRFD BDS5.7.3.4.2 
93 

0.5

358.2

0.5(239.96) 5.38

20.67

0.0066

0.740(29000)

u

u u

v

s

s s

M

N V

d

E A



    + +

 

=

    + +

 

= =

LRFD BDS Eq. 5.7.3.4.2-4 

0 0.006 s

  

29 3500 29 3500(0.006) 50.00o

  = + = +

s

=

s d h cl d d x v=  − − − =  − + − − = 2 20.67 in 24 2.00 1.50 2 0.628 0.618 18.63 inch trans b ( ) ( )

 Where cl is the clear cover; 

1.38 12 in 80 inch

0.63

1.38 12 in 18.63 80 in 18.63 inch

0.75 0.63

xe x

g

xe

s s

a

s

=  

+

=   

+

LRFD BDS Eq. 5.7.3.4.2-7 

( ) ( )

   

4.8 51

1 750 39

4.8 51 0.88

1 750(0.006) 39 18.63

s xe

s





=

+ +

= =

+ +

LRFD BDS Eq. 5.7.3.4.2-2 

' ' 0.0316 0.25

0.85 0.0316(0.88)(1.0) 7.0 0.25(7.0) 12(20.67)

15.51 kip

Vn v c c v v    f f b d =     

=     

=

LRFD BDS 5.7.3.3 

It can be seen that even with very conservative calculation of shear resistance using nonconcurrent loads and ignoring shear reinforcement, the resistance is significantly higher than the 

maximum factored HL-93 Inventory shear force. The following LRFD BDS checks have not 

been performed for this example but should be checked for actual rating calculations: regions 

requiring transverse reinforcement (5.7.2.3) and longitudinal reinforcement checks (5.7.3.5) [23 

CFR 625.4(d)(1)(v)]. 

2.4.6 LRFR Rating Calculations 

The structural condition of the tunnel liner is satisfactory and the system factor falls under the 

category for "All Other Girder Bridges and Slab Bridges." Therefore: 

 Condition factor: c = 1.00 MBE Table 6A.4.2.3-1 
94 

 System factor: s = 1.00 MBE Table 6A.4.2.4-1 

Resistance factors are based on LRFD BDS Section 5 and specified in the calculations of Section 

2.4.5 of this document. 

The equation for calculating the rating factor is based on MBE Equation 6A.5.12.4-1, which has 

been simplified for the load types being applied. 

( )

DC DW EV EH ES

LL LS

C DC DW EV EH ES RF

LL IM LS

    

 

    

=

+ 

MBE Eq. 6A.5.12.4-1

For the Strength Limit State: 

C R =   c s n

MBE Eq. 6A.5.12.4-2

Table 37 though Table 39 show the results of the overall capacity based on MBE Equation 

6A.5.12.4-2 as well as the rating factors based on MBE Equation 6A.5.12.4-1. Ratings are 

performed at the critical section. As discussed in Section 2.4.4.3, only ratings for maximum 

positive moment will be calculated. It was shown that the total factored loads and low levels of 

live load forces indicate the negative moment, shear and axial loads do not need to be 

investigated. 

Maximum and minimum load factors are selectively used to obtain the maximum absolute value 

of the force effects. Dead loads and earth loads are not able to be de-coupled due to the nature of 

the FEM. Therefore, a conservative load factor of 1.35 is used for all permanent loads. This load 

factor is the larger of the DC, EV and EH load factors. The live load factor for the EV vehicles is 

2.0 as identified in the FHWA Emergency Vehicle Memorandum (FHWA, 2016) for buried 

structures. The load factors used are shown in Table 36. The moments, shears and their 

corresponding capacities in Table 37 to Table 39 are in kip-ft and kip, respectively. The load 

effects are unfactored, but the capacities are factored resistances. Note that Table 39 is for the 

EV-3 vehicle because it controls over EV-2. 

Table 36. Load Factors. 

Permanent 

Loads 

HL-93 LL 

Inventory 

HL-93 LL 

Operating EV Legal 

1.35 1.75 1.35 2.0 

Table 37. HL-93 Inventory Level Rating Factors. 

Section Perm. Live Load Capacity R.F. 

Mpos 20.95 0.89 86.02 37.1 

Table 38. HL-93 Operating Level Rating Factors. 

Section Perm. Live Load Capacity R.F. 

Mpos 20.95 0.89 86.02 48.1
95 

Table 39. EV-3 Rating Factors. 

Section Perm. Live Load Capacity R.F.

Mpos 20.95 0.96 86.02 30.07

The rating factors are all well above 1 indicating that the tunnel liner is sufficient for all the 

evaluated live loads. In addition, it is observed that the live load effects induce a minor effect on 

the tunnel liner force demands in relation to permanent loads. 

The flexural and shear resistance of the joints between panels around the circumference of the 

liner are not investigated in this example but should be reviewed for an actual rating. This 

includes resistance to crushing and opening of the joint under axial thrust and bending. In 

addition to the thrust due to external loads, the panels are attached to each other with torqued 

bolts creating additional compression across the joint providing additional resistance to joint 

opening. The shear resistance of the interface also needs to be investigated. Shear is resisted 

through friction and steel guide rods and the previously mentioned bolts. 

A structural adequacy evaluation under a 7 ksf building load is investigated at other locations in 

addition to the rating of the liner under live load effects. This loading does not include a live load 

effect and therefore cannot produce a rating factor; however, a performance ratio can be 

evaluated by dividing the resistance by the demand. This ratio will therefore correlate in the 

same manner as the rating factor in which a ratio greater than 1 indicates adequate strength while 

a ratio less than 1 indicates insufficient strength. The moments, shears and their corresponding 

capacities in Table 40 are in kip-ft and kip, respectively. The permanent loads including the 

building load are factored per Table 36 and the capacities are factored resistances.

Table 40. Building Performance Ratio. 

Force Perm. Capacity P.R.

Mpos 72.82 86.02 1.18

Mneg 45.64 86.02 1.88

Shear 8.05 15.51 1.93

The performance ratio is greater than 1, indicating this tunnel liner has sufficient resistance to 

support the 7 ksf building load when not subjected to concurrent live load at this depth below the 

surface. 
96 
97 

CHAPTER 3 - EXAMPLE 2 – BOX TUNNEL ROOF GIRDER 

Example 2 presents a double cell tunnel consisting of simply supported, composite, steel roof 

girders supported by walls and a slab-on-grade bottom slab. The roof girder supports surface 

roadway loads. The lower roadway slab is on-grade and therefore does not need to be rated. The 

walls are soldier pile tremie concrete walls, which resist the lateral earth loads and the vertical 

loads from the roof girders. The tunnel roof girder is subjected to vertical dead loads, earth loads 

and live loads. It is assumed that little or no lateral loads are imparted to the roof girders due to 

the simple shear connections of the girders to the soldier piles. The box tunnel roof girder is rated 

with the LRFR method for the design vehicles (HL-93 Inventory and Operating Level) and 

emergency vehicles (EV-2 and EV-3) at the legal load level. 

This example will perform the following steps to rate this composite steel roof girder: 

1. Structure data 

2. Example notes 

3. Rating approach/assumptions 

4. Load application 

5. Structural analysis 

6. Resistance calculations 

7. LRFR rating calculations 

3.1 STRUCTURE DATA 

3.1.1 Materials 

Materials are known, otherwise use MBE Articles 6A5.2.1 and 6A5.2.2. Soil parameters were 

randomly selected for the example. Full soil descriptions and evaluation should be performed for 

actual tunnel ratings so accurate soil parameters can be obtained. This example also assumed dry, 

or drained, soil above the roof girder. 

 Concrete: f’c = 4.0 ksi 

 Reinforcing Steel: fy = 60.0 ksi 

 Structural Steel: Fy = 50.0 ksi 

 Soil: soil = 0.125 kcf 

soil = 30°
98 

3.1.2 Dimensions 

60.77' 

Simply Supported Steel Composite Girder

14'-30' 32 kip

32 kip

8 kip

14'

Ground Surface

4.0' Fill Depth

12" Slab with Stay-In-Place Forms

W36x280 Steel Girder @ 6.0' Spacing

Steel Bracket (Typ.)

 Source: FHWA 

Figure 30. Illustration. Cross-Section Showing Tunnel Geometry. 

3.1.3 Example Notes 

• This cross-section has been selected from this project as representative to demonstrate 

rating of the tunnel roof girder under external loading. 

• The live load carrying member considered in this example is the simply supported 

composite steel I- girder. 

• If the roadway above the tunnel has overlay or sidewalk loads, they should be included in 

the dead load analysis. This example assumes that these items are not included in the 

geometry of this roof girder. 

• The focus of this example is LRFR. 

• This example focuses on the load rating of a critical and typical roof girder. A section 

taken near the ends of the tunnel would need to account for increases in loading due to 

edge of slab effects. Additionally, critical connection details may need to be rated in a 

full rating project. 

• The rating Engineer should review the record drawings and inspection reports carefully to 

properly identify the support condition (pinned, expansion, fixed). Typical conditions, 

assumed for this example are simply supported conditions for the roof girder. 

• Rating is performed only at maximum shear and moment locations. The capacity of the 

bracket is also evaluated. Typical ratings include evaluation at all tenth points and full 

connection evaluation. 
99 

3.1.4 Rating Approach/Assumptions 

• LRFR evaluation is performed for a single girder (perpendicular to direction of traffic 

inside the tunnel and parallel to the direction of traffic above the box tunnel). 

• The steel girder is composite with the concrete roof slab. 

• The pavement on the ground above has approximately the same unit weight as the soil 

and is therefore included with the soil vertical loads (that is, gravel pavement surface). 

• Compacted gravel fill acts along the sides of the tunnel with soil parameter  = 125 pcf. 

Additionally, this box tunnel is in a dry soil condition above the roof girder. 

• LRFR live load ratings are evaluated for HL-93 design loading (Truck or Tandem) and 

the EV-2 and EV-3 emergency vehicles per the associated FHWA Memo (FHWA, 2016). 

3.2 STEEL ROOF GIRDER 

3.2.1 Load Calculations 

The roof girder is subject to self-weight, air pressure, soil/pavement, and live load. There is no 

wearing surface applied and is therefore neglected in this analysis. Other forces such as TU are 

neglected due to the simply supported boundary conditions of the composite girder. 

3.2.1.1 Dead Load Component, DC 

Dead load components include the self-weight of the reinforced concrete slab, steel girder and 

stay-in-place (SIP) forms. 

 Normal weight of reinforced concrete: 

0.145 kcf 0.005 kcf 0.150 kcf

w 0.150(1.0)(6.0) 0.900 klf

c

c c s eff t b





= + =

= = =

 Steel girder weight: 

36 280 0.280 klf wW x

=

Stay-in-place forms: 

0.015 ksf(6.0 ft Spa.) 0.090 klf wSIP= =

3.2.1.2 Wearing Surface, DW 

This tunnel assumed no wearing surfaces were applied and therefore were not included in the 

load calculations. 

LRFD BDS Table 3.5.1-1 

and Article C3.5.1 
100 

3.2.1.3 Vertical Earth Pressure, EV 

The construction method is assumed to be an embankment installation. Therefore, the vertical 

earth pressure can utilize the “embankment installation” in LRFD BDS equations 12.11.2.2.1-1 

and 12.11.2.2.1-2. 

4.0 1 0.20 1 0.2 1.013

60.77 e

c

H

F

B

= + = + =

LRFD BDS Eq. 12.11.2.2.1-1 

 Where H is the fill depth of 4 feet and Bc is the supported width of 60.77 feet. 

1.013(0.125)( )(4.0) 0.507 klf W F B H E e s c B

c

B

c

= =  =

LRFD BDS Eq. 12.11.2.2.1-1 

 Convert WE to a distributed earth load, wev, along the length of girder: 

0.507 (6.0) 3.042 klf E girder c

ev

c c

W S B

w

B B

= = =

3.2.1.4 Live Load Application, LL 

The live load travels parallel to the span of the steel girder over a 4 feet fill and therefore the 

wheel load distributes through the earth fill. The transverse live load distribution should be 

calculated in accordance to LRFD BDS Article 3.6.1.2.6b for traffic parallel to span. The LLDF 

is specified as 1.15 from LRFD BDS Table 3.6.1.2.6a-1. 

int

4.0 ft

0.06 20 0.06(60.77 12) 6

12 12 12 12

1.15

0.598 ft

j t

w

H

w D

s

H

LLDF

=



− − − −

= =

=

LRFD BDS Eq. 3.6.1.2.6b-1 

The effective width the load is distributed over can be calculated by LRFD BDS Eq. 3.6.1.2.6b-3 

since H is greater than Hint. 

20 60.77 12 ( ) 0.06 6.0 1.15(4.0) 0.06 15.913 ft

12 12 12 12

t i

w w

w D w s LLDF H   

= + + + = + + +   =

 

The longitudinal live load distribution should be calculated in accordance to LRFD BDS Article 

3.6.1.2.6b for traffic parallel to the span. Compute the distribution widths for the various axle 

spacings of the HL-93 and EV vehicles (4’ for HL-93 Tandem and EV-3 vehicle, 14 feet for HL93 Truck, and 15’ for EV-2 and EV-3 vehicles). Note the HL-93 Truck produces the critical 

force effects when the rear axle is at 14-foot spacing. Additionally, where ratings are adequate, 

the longitudinal distribution of wheel loads may conservatively and conveniently be ignored by 

applying the axle point loads directly to the beam. 
101 

int (4')

10 4

12 12 2.754 ft

1.15

10 ( ) 4 1.15(4) 9.43 ft

12 12

t

a

p

t

w a

l

s

H

LLDF

l

l s LLDF H

−

− −

= = =

= + + = + + =

LRFD BDS Eq. 3.6.1.2.6b-4 & 5 

int (14')

10 14

12 12 11.449 ft

1.15

10 ( ) 1.15(4) 5.43 ft

12 12

t

a

p

t

w

l

s

H

LLDF

l

l LLDF H

−

− −

= = =

= + = + =

LRFD BDS Eq. 3.6.1.2.6b-4 & 5 

int (15')

10 15

12 12 12.319 ft

1.15

10 ( ) 1.15(4) 5.43 ft

12 12

t

a

p

t

w

l

s

H

LLDF

l

l LLDF H

−

− −

= = =

= + = + =

LRFD BDS Eq. 3.6.1.2.6b-4 & 5 

Each axle load is distributed over the transverse and longitudinal areas and normalized over the 

girder spacing. For calculation simplicity, a unit load pressure is calculated then multiplied by 

the axle loads of the various vehicles. 

( )

1kip

1kip 2

2

1axle 0.063 axle

15.913 ft

0.063 axle 0.377 axle

w 6 ft girder spacing

ft ft LL

LL

w w

w w

p

l l

l l −

−

 

= =    

   

= =        

This distributed load (axle) per length along the girder can then be multiplied by the various axle 

loads. However, the live load also needs to be multiplied by the dynamic allowance for buried 

structures specified in LRFD BDS Article 3.6.2.2 and a single multiple presence factor of 1.2 as 

specified in LRFD BDS Article 3.6.1.1.2. For this example, a multiple presence factor of 1.0 was 

used for the EV legal loads based on MBE Article 6A5.12.10.3 for buried structures with traffic 

traveling parallel to the span. Where the axle loads are 14 feet or 15 feet, the axle loads shall be 

distributed over 5.43 feet. When the axle spacing is 4 feet, the axle loads shall be combined and 

distributed over 9.43 feet. 

I.M. 33(1.0 0.125 ) 0% 33(1 0.125 4) 16.5% = −  = −  = DE

LRFD BDS Eq. 3.6.2.2-1 
102 

( )

( )

93

93

93

0.377 axle (8 kip/axle) (1.165)(1.2) 0.776 klf per girder

5.43 ft

0.377 axle 32 kip/axle (1.165)(1.2) 3.106 klf per girder

5.43 ft

0.377 axle (2) 25 kip/axle

9.43 ft

HL Truck

HL Truck

HL Tandem

P

P

P

−

−

−

 

= =    

 

= =    



=



(1.165)(1.2) 2.795 klf per girder 

  =



( )

( )

2

2

0.377 axle 24 kip/axle (1.165)(1.0) 1.941 klf per girder

5.43 ft

0.377 axle 33.5 kip/axle (1.165)(1.0) 2.710 klf per girder

5.43 ft

EV

EV

P

P

 

= =    

 

= =    

( )

( )

3

3

0.377 axle 24 kip/axle (1.165)(1.0) 1.941 klf per girder

5.43 ft

0.377 axle (2) 31 kip/axle (1.165)(1.0) 2.888 klf per girder

9.43 ft

EV

EV

P

P

 

= =    

 

= =    

For this example, only axle loads are applied based on MBE Article 6A5.12.10.3 for reinforced 

concrete box culverts,. However, this is a steel roof girder with a span in excess of 60 feet; 

therefore, it is prudent to include the lane load for HL-93 ratings. As a result, the distribution of 

the lane load also needs to be determined, assuming the load distributes at a 1.15 factor as 

specified in LRFD BDS Table 3.6.1.2.6a-1. Therefore, the bottom width of the load application 

at the structure is 1.15 times wider than the width at the ground surface. 

93

( ) 10 2(1.15)(4) 19.2 feet

6 feet . . (0.64 klf ) (1.2) 0.240 klf per girder

19.2 feet

lane buried lane

girder

LL HL Lane

lane buried

W W LLDF H

s

w w M P

W

−

−

−

= + = + =

   

=   =   =

   

3.2.2 Structural Analysis 

A beam analysis is utilized for the structural analysis. All the load effects and member 

resistances are calculated using the tributary, one-dimensional analysis. The composite girder is 

assumed to be simply supported and therefore can be evaluated with simple hand calculations or 

by a structural analysis program. 

The maximum moment can be obtained by positioning the axles such that the centerline of the 

span is midway between the center of gravity of the load and the nearest wheel load, as depicted 

in Figure 31. The maximum shear is obtained when the axle loads are placed adjacent to the end 

of the span, as depicted in Figure 32. 
103 

C.G.load

C.L.span

P1 P2 P3

 Source: FHWA 

Figure 31. Illustration. Live Load Placement for Maximum Moment. 

P1 P2 P3

 Source: FHWA 

Figure 32. Illustration. Live Load Placement for Maximum Shear. 

Using this load placement theory, the distributed dead and live loads are placed on the beam as 

depicted in Figure 33. 

w1 w2

60.77'

wDC, wEV, wLane

wAxle wAxle

x1 x2 x3

wAxle

w4 x4

 Source: FHWA 

Figure 33. Illustration. Load Placement. 

Where the variables xi and wi are defined in Table 41. Results show the Tandem controls the HL93 loading due to the convergence of the axle load effects over such a concentrated portion of the 

girder. 

Table 41. Moment Load Placement Dimensions. 

Load x1 (ft) w1 (ft) x2 (ft) w2 (ft) x3 (ft) w3 (ft) x4 (ft) 

HL-93 Truck 9.00 5.43 8.57 5.43 8.57 5.43 18.34 

HL-93 Tandem 25.67 9.43 25.67 N/A N/A N/A N/A 

EV-2 18.93 5.43 9.57 5.43 21.41 N/A N/A 

EV-3 15.41 5.43 9.57 9.43 20.93 N/A N/A 
104 

Table 42. Shear Load Placement Dimensions. 

Load x1 (ft) w1 (ft) x2 (ft) w2 (ft) x3 (ft) w3 (ft) x4 (ft) 

HL-93 Truck 27.34 5.43 8.57 5.43 8.57 5.43 0.00 

HL-93 Tandem 51.34 9.43 0.00 N/A N/A N/A N/A 

EV-2 40.34 5.43 9.57 5.43 0.00 N/A N/A 

EV-3 36.34 5.43 9.57 9.43 0.00 N/A N/A 

LRFD Road Tunnel Design and Construction Guide Specifications, 1st Ed. Table 3.4-1 identifies 

the applicable minimum and maximum load factors for each load specified above. Since the 

girder is simply supported, only maximum load factors were necessary to produce critical forces. 

The live load factors for the HL-93 are identified in Table 3.4-1; however, the live load factor for 

the EV vehicles are 2.0 as identified in the FHWA Emergency Vehicle Memorandum. The forces 

are given in Table 43. 

Table 43. Unfactored Girder Critical Forces. 

Load Shear (kip) Moment (k-ft) 

DC 39 586 

Earth Load, EV 92 1404 

HL-93 Lane 7 111 

HL-93 Truck 30 424 

HL-93 Tandem 24 369 

EV-2 22 300 

EV-3 32 457 

3.2.3 Resistance Calculations 

The depth, analysis section width, and reinforcement for each of the critical sections are 

tabulated below in Table 44 and Table 45. 

3.2.3.1 Material Properties 

f’c = 4.0 ksi 

Fy = 50.0 ksi 

∝1 = 0.85 LRFD BDS 5.6.2.2 

β1 = 0.85 if f’c ≤ 4.0 ksi LRFD BDS 5.6.2.2 

 

Slab reinforcement, which is in compression, is conservatively ignored in the flexural capacity 

calculations. 

3.2.3.2 General Properties 

Beam section: W36x280

Beam spacing: s=beff = 72.00 inch 
105 

Slab thickness: tslab = 12.00 inch 

Flexural resistance factor: f = 1.0 LRFD BDS 6.5.4.2 

Shear resistance factor: v = 1.0 LRFD BDS 6.5.4.2 

Modulus of Elasticity: LRFD BDS 5.4.2.4 & 6.4.1 

2 '0.33 2 0.33 120,000( ) 120,000(0.145) (4.0) 3987 ksi

29,000 ksi

=8.0 (assumed)

c c c

p

E f

E

n

=  = =

=

3.2.3.3 Section Properties 

Non-Composite Section: 

2

36 280

4

3

82.4 inch

18.26 inches

18900 inch

1030 inch

W x

b t

x

x top x bot

A

y y

I

S S − −

=

= =

=

= =

Short Term Properties (assume n=8.0): 

Table 44. Short Term Composite Properties. 

Member A (inch2) Ybot

(inches) 

AYbot

(inch3) Ῡ (inches) AῩ

2

 (inch4) Imember

(inch4) 

Slab 108 42.52 4592 -10.44 11782 1296 

Top Flange 26.1 35.74 931 -3.66 349 5 

Web 29.5 18.26 539 13.82 5638 2743 

Bottom Flange 26.1 0.79 20 31.29 25509 5 

 190 6083 43278 4050 

2

2 4

3

3

190 inch

6083 =32.08 inches

190

y 36.52 12 32.08 16.64 inches

43278 4050=47328 inch

47328 1476 inch

32.08

47328 2844 inch

16.64

ST

bot

bot

top grd slab bot

x member

x

x bot

bot

x

x top

top

A

Ay

y

A

d t y

I Ay I

I

S

y

I

S

y

−

−

=



= =



= + − = + − =

=  +  = +

= = =

= = =
106 

Long Term Properties (assume 3n=24.0): 

Table 45. Long Term Composite Properties. 

Member A (inch2) Ybot

(inches) 

AYbot

(inch3) Ῡ (inches) AῩ

2

 (inch4) Imember

(inch4) 

Slab 36 42.52 1531 -16.84 10205 432 

Top Flange 26.1 35.74 931 -10.05 2632 5 

Web 29.5 18.26 539 7.42 1628 2743 

Bottom Flange 26.1 0.79 20 24.90 16152 5 

 118 3022 30617 3186 

2

2 4

3

3

118 inch

3022 =25.68 inches

118

y 36.52 12 25.68 22.84 inch

30617 3186=33803 inch

33803 1316 inch

25.68

33803 1480 inch

22.84

LT

bot

bot

top grd slab bot

x member

x

x bot

bot

x

x top

top

A

Ay

y

A

d t y

I Ay I

I

S

y

I

S

y

−

−

=



= =



= + − = + − =

=  +  = +

= = =

= = =

3.2.3.4 Plastic Moment 

The plastic moment of the composite section is calculated in accordance to LRFD BDS Article 

D6.1. To accomplish this, the rolled W36x280 is discretized into top flange, web and bottom 

flange areas. 

16.595 inches

1.57 inches

=33.38 inches

0.885 inches

ft fb

ft fb

w

b b

t t

D

t

= =

= =

=

The section plastic forces are then calculated. 

'

0.85 0.85(4.0)(72)(12) 2938 kip

50(16.595)(1.57) 1303 kip

50(33.38)(0.885) 1477 kip

s c eff s

c t y f f

w y w

P f b t

P P F b t

P F Dt

= = =

= = = =

= = =

The relative magnitudes of these plastic forces are compared as outlined in LRFD BDS Table 

D6.1-1. This reveals this example is Case II. The corresponding plastic section properties are 

then calculated. 
107 

( )  

( )

2

2

2 2

1303 1477 1303 4082 2938 kip

1.57 1477 1303 2938

= 1 1 0.69 inches

2 2 1303

=

2

1303 0.69 1.57 0.69 2938

2 1.57

t w c s

c w t s

c

c

p c s s w w t t

c

P P P P

t P P P Y

P

P

M Y t Y P d P d Pd

t

+ +   + + = 

   + −    + − 

  + =   + =           

     + − + + +    

 

=    + − +

      

1ft 6.69 1477 17.57 1303 35.05

12 inch

7648 kip-ft

0.69 12 12.69 inches D Y t

p s

  

   +  +      

=

= + = + =

 Source: FHWA 

Figure 34. Illustration: Plastic Neutral Axis Offset Dimensions. 

3.2.3.5 Moment Resistance 

The section has been determined to be compact for composite positive flexure per LRFD BDS 

Article 6.10.6.2.2. The plastic depth is compared against the total depth to ensure the concrete 

deck will not crush prior to steel plastic capacity is reached per LRFD BDS Article 6.10.7.12. 

( )

36.52 12 48.52 inches

0.1 12.69 0.1 48.52 4.85 inches

t grd s

p t

D h t

D D

= + = + =

   =

Since the plastic compressive depth is greater than 10% of the total depth and this is a simply 

supported section, the plastic moment capacity needs to be determined per LRFD BDS Eq. 

6.10.7.1.2-2. 

( ) 12.69 1.07 0.7 1.0 7648 1.07 0.7 6783 kip-ft

48.52

p

f n f p

t

D

M M

D

     = − =     − =

   
108 

3.2.3.6 Shear Resistance 

The shear resistance also needs to be determined. The maximum shear force of this simply 

supported girder is within the end panel of the girder. Therefore, tension field action cannot be 

included in the shear resistance of the girder. 

The slenderness of the web first needs to be determined to calculate the shear buckling 

coefficient per LRFD BDS Article 6.10.9.3.2. The panel length, do, is 15.19 feet since the cross 

frames with transverse stiffeners are located at quarter points of the span. 

33.38 37.72

0.885

5

29000 5 1.12 1.12 60.32

50

w

y

D

t

k

Ek

F

= =

=



= =

Since the web slenderness ratio, D/tw, is less than the web compact limit, C=1.0. Therefore, the 

shear resistance is the plastic shear limit per LRFD BDS Article 6.10.9.3.3. 

0.58 1.0(1.0)(0.58)(50)(33.38)(0.885) 857 kip   v n v V C F Dt =

y w

= =

3.2.3.7 Bracket Resistance 

The bracket connection needs to be rated along with the girder capacities. The bracket buckling 

resistance is often prudent to rate along with the connection resistance. However, for simplicity 

of this example, the weld connection will only be shown as that is the controlling limit state on 

this particular bracket plate. 

The bracket is composed of 3~8-inch x 21-inch vertical plates with vertical 1⁄2-inch fillet welds 

on both sides of each plate. The weld is composed of E70 electrodes. 

2

6(21inches) 126 inches

0.707( ) 0.707(0.5") 0.354 inches

0.6 0.6(0.80)(70)(126)(0.354)

1497 kip

w

w weld

n e exx w w

l

a w

  R F l a

= =

= = =

= =

=

BDS Article 6.13.3.2.4 & 6.5.4.2 

It can be seen that the capacity of the bracket exceeds the shear capacity of the girder; therefore, 

the girder will control the ratings. 
109 

 Source: FHWA 

Figure 35. Illustration. Bracket Details. 

3.2.4 LRFR Rating Calculations 

The structural condition of the steel girder is satisfactory and the system factor falls under the 

category for "All Other Girder Bridges and Slab Bridges." Therefore: 

 Condition factor: c = 1.00 MBE Table 6A.4.2.3-1 

 System factor: s = 1.00 MBE Table 6A.4.2.4-1 

Resistance factors are based on LRFD BDS Article 6.5.4.2: 

 Flexure resistance factor: f = 1.00 LRFD BDS 6.5.4.2 

 Shear resistance factor: v = 1.00 LRFD BDS 6.5.4.2 

The equation for calculating the rating factor is based on MBE Equation 6A.4.2.1-1, which has 

been simplified for the load types being applied. 

( )

DC EV

LL

C DC EV RF

LL IM

 



 

=

+

MBE Eq. 6A.4.2.1-1

For the Strength Limit State: 

C R =   c s n

MBE Eq. 6A.4.2.1-2

Table 47 shows the results of the overall capacity based on MBE Equation 6A.4.2.1-2 as well as 

the rating factors based on MBE Equation 6A.4.2.1-1. Maximum and minimum load factors are 

selectively used to obtain the maximum absolute value of the force effects. The load factors used 

are shown in Table 46. Moments, shears and their corresponding capacities in Table 47 are in 

kip-ft and kip, respectively. 
110 

Table 46. Load Factors. 

DC

Earth 

Load, 

EV

HL-93 LL 

Inventory 

HL-93 LL 

Operating 

Emergency 

Vehicle, 

EV LL

Legal 

1.25 1.35 1.75 1.35 2.00 

Table 47. Girder Resistance and Rating Factors. 

Force Rating DC Earth 

Load, EV LL C RF

Moment HL-93 Inventory 586 1404 535 6783 4.44

Moment HL-93 Operating 586 1404 535 6783 5.75

Moment EV-2 586 1404 300 6783 6.92

Moment EV-3 586 1404 457 6783 4.55

Shear HL-93 Inventory 39 92 37 857 10.57

Shear HL-93 Operating 39 92 37 857 13.70

Shear EV-2 39 92 22 857 15.56

Shear EV-3 39 92 32 857 10.70

Bracket HL-93 Inventory 39 92 37 1497 20.46

Bracket HL-93 Operating 39 92 37 1497 26.52

Bracket EV-2 39 92 22 1497 30.10

Bracket EV-3 39 92 32 1497 20.70
111

REFERENCES

AASHTO. (2017a). AASHTO LRFD Bridge Design Specifications, 8

th Edition, LRFD-8.

American Association of State Highway and Transportation Officials, Washington, DC.

AASHTO. (2017b). LRFD Road Tunnel Design and Construction Guide Specifications, 1st

Edition, LRFDTUN-1. American Association of State Highway and Transportation Officials,

Washington, DC.

AASHTO. (2018). Manual for Bridge Evaluation, 3

rd Edition. American Association of State 

Highway and Transportation Officials, Washington, DC.

FHWA. (2004). FHWA Road Tunnel Design Guidelines, FHWA-IF-05-023. Federal Highway 

Administration, Washington, DC.

FHWA. (2009). Technical Manual for Design and Construction of Road Tunnel—Civil 

Elements, FHWA-NHI-10-034. Federal Highway Administration, Washington, DC.

FHWA. (2013). Information: Load Rating of Specialized Hauling Vehicles. Federal Highway 

Administration, Washington, DC.

FHWA. (2015a). Specifications for the National Tunnel Inventory, FHWA-HIF-15-006. Federal 

Highway Administration, Washington, DC.

FHWA. (2015b). Tunnel Operations, Maintenance, Inspection, and Evaluation (TOMIE) 

Manual, FHWA-HIF-15-005. Federal Highway Administration, Washington, DC.

FHWA. (2016). Memorandum: Load Rating for the FAST Act’s Emergency Vehicles. Federal 

Highway Administration, Washington, DC.

FHWA. (2019). Reference Guide for Load Rating of Tunnel Structures, FHWA-HIF-19-010. 

Federal Highway Administration, Washington, DC.

Los Angeles County Metropolitan Transit Authority (Metro). (2019). Westside Purple Line 

Extension, Section 3, Structure Impact Assessment Report for Tunnels and Cross Passages. 

Prepared by ALDEA, November 26.

Plaxis. (2019). Plaxis 2D Reference Manual. Plaxis BV. The Netherlands.