improved culvert load rating through an evaluation of the

Improved Culvert Load Rating

Through an Evaluation of the Influence of

Cover Soil Depth,

Demand Model Sophistication, and

Live Load Attenuation Method

by

Timothy A. Wood, MSCE

A Dissertation In

Civil Engineering

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved by

William D. Lawson, P.E., Ph.D. Chair of Committee

Priyantha W. Jayawickrama, Ph.D.

Hoyoung Seo, Ph.D., P.E.

James G. Surles, Ph.D.

Mark Sheridan, Ph.D.

Dean of the Graduate School

December 2015

Timothy A. Wood

Copyright 2015

Texas Tech University, Timothy A. Wood, December 2015

i

TABLE OF CONTENTS

Abstract ......................................................................................................................... v

List of Tables .............................................................................................................. vii

List of Figures ............................................................................................................ viii

CHAPTER 1 Introduction ............................................................................................. 1

Policy ........................................................................................................................ 3

Load Rating Concept ................................................................................................ 4

Culvert Load Rating Research at Texas Tech University ......................................... 5

Challenges for CIP RC Box Culvert Load Rating .................................................... 7

Development of Production-Simplified Demand Models ........................................ 9

Factors Influencing Culvert Load Rating ............................................................... 14

Dissertation Outline ................................................................................................ 16

CHAPTER 2 Cover Soil Depth .................................................................................. 17

Chapter Summary ................................................................................................... 17

Introduction and Background ................................................................................. 18

Load Rating Process ........................................................................................... 18

History of Culvert Design and Load Rating Policy ............................................ 20

Use of Standard Designs ..................................................................................... 22

Method .................................................................................................................... 23


ii

Population of Evaluated Culvert Standard Designs ............................................ 23

Load Rating Procedure ....................................................................................... 25

Results ..................................................................................................................... 28

Observations ....................................................................................................... 28

Distribution of load rating vs. cover soil depth relationship in the population ... 36

Implications ......................................................................................................... 39

Conclusions ............................................................................................................. 41

Acknowledgements ................................................................................................. 43

CHAPTER 3 Production-Simplified Demand Model Sophistication ......................... 44

Chapter Summary ................................................................................................... 44

Introduction ............................................................................................................. 45

Literature Review .................................................................................................... 47

Method .................................................................................................................... 51

Field test program ............................................................................................... 51

Loading method .................................................................................................. 55

Comparative analysis .......................................................................................... 56

Findings and Discussion ......................................................................................... 62

Overall performance ........................................................................................... 62

Moment diagrams ............................................................................................... 63


iii

Model performance by cover soil depth ............................................................. 65

Member performance .......................................................................................... 68

Member performance summary .......................................................................... 74

Other Observations ............................................................................................. 76

Conclusions ............................................................................................................. 77

Acknowledgements ................................................................................................. 79

CHAPTER 4 Production-Simplified Live Load Attenuation Method ........................ 80

Chapter Summary ................................................................................................... 80

Introduction and Background ................................................................................. 81

Disconnect Between Observed Structural Performance and Calculated Load

Ratings ...................................................................................................................... 81

Load Rating with Production-Simplified Demand Models ................................ 83

Live Load Attenuation, Past and Present ............................................................ 86

Live Load Attenuation Methods ............................................................................. 89

Current “Top-Slab-Calibrated” Live Load Attenuation Method ........................ 89

New “Depth-Calibrated” Live Load Attenuation Method .................................. 90

Measured Moment Data .......................................................................................... 91

Data Sources ....................................................................................................... 91

Predicted Moment Calculations .......................................................................... 93


iv

Typical Moment Envelopes ................................................................................ 94

Findings and Discussion ......................................................................................... 97

Observations of Moment Bias ............................................................................ 98

Observations of Moment Bias by Section .......................................................... 99

Observations of Moment Bias by Live Load Distribution ................................ 101

Load Rating Case Study .................................................................................... 104

Improved Live Load Distribution ..................................................................... 106

Conclusions ........................................................................................................... 107

Acknowledgements ............................................................................................... 107

CHAPTER 5 Conclusions ........................................................................................ 108

Summary ............................................................................................................... 108

Major Findings ...................................................................................................... 109

Limitations ............................................................................................................ 110

Future Work .......................................................................................................... 111

Work Cited ................................................................................................................ 113

APPENDIX A Distributions of Culvert Designs ...................................................... 121

APPENDIX B Moment Plots Comparing Live Load Attenuation Methods ............ 125


v

ABSTRACT

This dissertation evaluates the influence of three factors – cover soil depth, demand

model sophistication, and live load attenuation method – on the load rating of cast-in-

place (CIP), reinforced-concrete (RC) box culverts. Concrete box culvert load rating

appears simple but is quite complex. The governing federal policy, analysis principles,

challenges, and the disconnect between load rating calculations and field inspection

observations are discussed in detail.

Cover soil depth above the culvert directly influences culvert load rating results in

non-linear ways. A population of Texas Department of Transportation CIP RC culvert

standard designs developed between 1930 and 1980 were load rated using AASHTO

policy guidance and a direct-stiffness demand model for a full range of cover soil depths.

Three typical rating vs. depth relationships are illustrated and described in detail. The

distribution of characteristic rating vs. depth relationships based on culvert geometry,

design cover soil depth, and design era are explored. Cover soil depth is shown to be a

critical parameter that must be explicitly considered for the intelligent load rating and

design of reinforced concrete box culverts.

Demand model sophistication influences the accuracy and precision of culvert load

rating calculations. Two production-simplified culvert load rating demand models were

analyzed using live load test data from three instrumented reinforced concrete box

culverts under four cover soil depths. The demand models were a structural-frame model

and a soil-structure interaction model. As expected, increased sophistication in the soil-

structure interaction model as compared to the structural-frame model resulted in higher


vi

precision and accuracy for predicted moments. Variations in predicted moment accuracy

and precision were not uniform but are a function of the critical section location in the

culvert structure.

The soil-structure interaction model requires an out-of-plane, live load attenuation

method; this method directly affects the accuracy and precision of the culvert load rating

calculation. A new method, called the depth-calibrated method, attenuates out-of-plane

live load to the critical section depths in a culvert. The depth-calibrated method improves

current practice by increasing the accuracy and precision of live load demand predictions,

particularly in culvert walls and bottom slabs. Use of the depth-calibrated method helps

close the disconnect between calculated load rating and observed structural performance

by more accurately predicting both the location of the weakest critical section and the live

load magnitude. The effectiveness of the depth-calibrated method was evaluated by

comparing predicted live load moments to measured live load moments obtained from

published datasets from full-scale culvert load tests. A load rating example shows the

improved alignment between load rating and observed performance.

Understanding the influence of cover soil depth, modeling sophistication, and live

load attenuation allows for more accurate and precise load rating of cast-in-place,

reinforced-concrete, box culverts and better correspondence between load rating

calculations and field inspection observations. This dissertation advances the state of

production-simplified load rating practice and knowledge.


vii

LIST OF TABLES

Table 1. Test culvert parameters ....................................................................................... 53

Table 2. Axle and wheel loads for test dump trucks ......................................................... 55

Table 3. Moment of inertia for specific critical sections .................................................. 58

Table 4. Project data for measured live load moments from field-tested culverts in Texas

(Lawson, et al., 2010; Wood, et al., 2015) and Nebraska (Tadros & Benak, 1989;

Abdel-Karim, et al., 1993) .................................................................................... 93


viii

LIST OF FIGURES

Figure 1. A five span, CIP RC box culvert ......................................................................... 2

Figure 2. Critical section schematic .................................................................................... 5

Figure 3. Production-simplified direct-stiffness model (Lawson, et al., 2009) ................ 10

Figure 4. Production-simplified soil-structure interaction model ..................................... 13

Figure 5. Example standard designs: (a) cross-section view of pre-WWII design with

haunches (Source: TxDOT standard sheet ‘MBC-3-34-F’), and (b) post-WWII

design without haunches (Source: TxDOT standard sheet ‘MC9-1’) .................. 24

Figure 6. Representative load rating vs. cover soil depth plots for increasing, decreasing

and constant relationships (a) Increasing relationship (Source: MBC-2-34-F 1938

2 boxes 1.5mx1.5m (5ftx5ft)); (b) Decreasing relationship (Source: MC9-1 1958

5 boxes 2.7mx2.4m (9ftx8ft)); (c) Constant relationship (Source: MC10-3 1977 3

boxes 3mx2.7m (10ftx9ft)) ................................................................................... 29

Figure 7. (a) Dead load and (b) live load relationship with cover soil depth ................... 32

Figure 8. Trend plot of load rating vs. cover soil depth plot shape by design era ............ 37

Figure 9. Trend plots of load rating vs. cover soil depth relationship by culvert geometry:

(a) aspect ratio, (b) span length, and (c) barrel height .......................................... 38

Figure 10. Trend plot of load rating vs. cover soil depth plot shape by design cover soil

depth ...................................................................................................................... 39

Figure 11. Modeling sophistication illustrations: (a) Level 1, two-dimensional, direct-

stiffness, structural-frame model (Lawson, et al., 2009); (b) Level 3, two-

dimensional, linear-elastic finite-element soil-structure interaction model .......... 51


ix

Figure 12. Test culvert locations and culvert images: (a) Texas county map showing test

culvert locations; (b) Swisher county; (c) Hale county; (d) Lubbock county ....... 52

Figure 13. Typical gage plan: Lubbock County culvert; white circles indicate gage pairs,

black circles indicate single gages, open circles indicate no gages ...................... 54

Figure 14. Live load configurations for the culvert load test: (a) One truck straddling gage

line; (b) Wheel on gage line; (c) Two trucks straddling gage line; (d) Data

acquisition and recording ...................................................................................... 56

Figure 15. Live load moment demand envelopes for each load test: (a) Swisher County

culvert; 0.5m (1.5ft) cover depth; (b) Lubbock County culvert; 0.6m (2ft) cover

depth; (c) Hale County culvert; 1.1m (3.5ft) cover depth; (d) Lubbock County

culvert; 1.2m (4ft) cover depth ............................................................................. 64

Figure 16. Predicted vs. measured moment demand ratios by model and cover soil depth

............................................................................................................................... 66

Figure 17. Predicted vs. measured moment demand ratios by critical section type: (a) Top

slab critical sections; (b) Bottom slab critical sections; (c) Interior wall critical

sections; (d) Exterior wall critical sections ........................................................... 69

Figure 18. Evaluation modeling accuracy for each critical section type in the primary

bending direction .................................................................................................. 75

Figure 19. (a) A five-span reinforced concrete box culvert in Swisher Co., TX; (b) critical

section schematic .................................................................................................. 85

Figure 20. (a) production-simplified, two-dimensional, linear elastic, finite element, soil-

structure interaction model for in-plane live load distribution for a two span


x

reinforced concrete box culvert in Sarpy Co., NE; (b) estimated out-of-plane live

load distribution .................................................................................................... 88

Figure 21. Typical moment envelopes for the 4 span, Hale County, TX culvert. See

Figure 1 for critical section locations .................................................................... 95

Figure 22. Histogram of moment biases from 11 culvert load tests using the (a) top-slab-

calibrated method and (b) depth-calibrated method ............................................. 98

Figure 23. (a) mean and (b) standard deviation of moment bias by critical section ....... 100

Figure 24. Live load attenuation factor, 1/w (ft/ft (m/m)), as a function of depth from

ground surface for a single HS-20 truck for three live load distribution models:

elastic (Poulos & Davis, 1991; Katona, 2015), SSHB (AASHTO, 2002) and

LRFD (AASHTO, 2014) .................................................................................... 102

Figure 25. (a) mean and (b) standard deviation of bias by live load distribution ........... 103

Figure A.1. Distribution plot of load rating vs. cover soil depth plot shape by design era

............................................................................................................................. 122

Figure A.2. Trend plots of load rating vs. cover soil depth relationship by culvert

geometry: (a) aspect ratio, (b) span length, and (c) barrel height ....................... 123

Figure A.3. Distribution plot of load rating vs. cover soil depth plot shape by designcover

soil depth ............................................................................................................. 124

Figure B.1. Moment plot for Swisher Co., TX (Table 4) culvert under 1.5ft of cover soil

............................................................................................................................. 126

Figure B.2. Moment plot for Lubbock Co., TX (Table 4) culvert under 2.0ft of cover soil

............................................................................................................................. 126


xi

Figure B.3. Moment plot for Hale Co., TX (Table 4) culvert under 3.5ft of cover soil . 127


............................................................................................................................. 127

Figure B.5. Moment plot for Sarpy Co., NE (Table 4) culvert under 0ft of cover soil .. 128

Figure B.6. Moment plot for Sarpy Co., NE (Table 4) culvert under 2.0ft of cover soil 128




Figure B.10. Moment plot for Sarpy Co., NE (Table 4) culvert under 10.0ft of cover soil

............................................................................................................................. 130


............................................................................................................................. 131


1

CHAPTER 1

INTRODUCTION

This dissertation evaluates the influence of three factors – cover soil depth, demand

model sophistication, and live load attenuation method – on the load rating of cast-in-

place (CIP), reinforced-concrete (RC) box culverts. Figure 1 shows a typical CIP RC box

culvert of the type explored in this dissertation. Load rating such a structure involves the

analytical determination of the live load capacity of all potential critical sections in the

culvert and reporting the worst-case as the maximum allowable live load for the structure.

Cover soil depth is the distance from the top slab of the culvert to the ground surface and

directly influences the dead and live loads applied to the culvert. Demand models are

analytical methods used to predict the response of the culvert to load. These demand

models vary in degree of sophistication from simple, two-dimensional (2D), production-

simplified, direct-stiffness, structural-frame models to complex, three-dimensional,

research-intensive, non-linear, finite-element, soil-structure interaction models. This

dissertation focuses on two, 2D, linear-elastic, production simplified models: (1) a direct-

stiffness, structural-frame model and (2) a linear-elastic, finite-element soil-structure

interaction model. The live load attenuation method defines how wheel loads on an actual

culvert are evaluated in a 2D demand model, particularly in the out-of-plane direction.

The actual live load must be attenuated in the out-of-plane direction in order to predict

real, three-dimensional, structural response using a 2D model. Historically, the live load

attenuation is top-slab-calibrated as a function of the cover soil depth only. This


2

dissertation introduces a new live load attenuation method specifically to address load

rating challenges of critical sections at various depths. The new, depth-calibrated live

load attenuation method calculates demand magnitude as a function of the critical section

depth from the ground surface. This dissertation discusses the influence of factors in

terms of accuracy and precision. Accuracy is the degree to which the model or method

predicts the true performance and is typically quantified by the mathematical mean.

Precision deals with the scatter in the predictions and is typically quantified by range and

standard deviation. Taken together, increased understanding of cover soil depth,

modeling sophistication, and live load attenuation significantly increases accuracy and

precision in load rating calculations for CIP RC box culverts.

Figure 1. A five span, CIP RC box culvert


3

Policy

Culvert load rating is one component of the National Bridge Inspection Standards

(NBIS) (Bridges, Structures, and Hydraulics, 2009). The NBIS is concerned with “the

proper safety inspection and evaluation of all highway bridges” and establishes a required

system for bridge inspection and evaluation programs (Bridges, Structures, and

Hydraulics, 2009). The NBIS references the AASHTO Manual for Bridge Evaluations

(MBE) as the document of technical authority for all its components. The MBE defines

eight phases for a complete bridge inspection plan: (1) purpose and scope, (2)

documentation, (3) bridge management systems, (4) field inspection types and frequency,

(5) inspection and evaluation methods, (6) load rating, (7) evaluation of structural fatigue

and (8) field load testing (AASHTO, 2013). This dissertation is focused on the load

rating aspect for CIP RC box culverts.

The MBE identifies three methods for load rating: load and resistance factor rating

(LRFR), load factor rating (LFR), and allowable stress rating (ASR). Additional guidance

for LRFR comes from the AASHTO LRFD Bridge Design Specifications (AASHTO,

2014). For ASR and LFR, the MBE references the Standard Specifications for Highway

Bridges (AASHTO, 2002). This dissertation primarily uses the LFR method. Each of the

three load rating methods is concerned with identifying the “live load carrying capacity

of a bridge” (Bridges, Structures, and Hydraulics, 2009).


4

Load Rating Concept

The central idea of the load rating calculation is to compare the live load capacity

(capacity reduced by the dead load demand) to the live load demand. The rating factor

equation for LFR shown in Equation 1 illustrates the relationship.

RF = C − A(DA*L 1 + I

(1)

where: RF = the rating factor C = the structural capacity of the member D = the dead load effect on the member L = the live load effect on the member I = the impact factor, IM A1 = 1.3 = factor for dead loads A2 = 2.17 for Inventory Level = factor for live loads = 1.3 for Operating Level = factor for live loads (AASHTO, 2013)

Equation 1 is applied at each critical section seen in Figure 2. Critical sections are

locations on the structure where load stresses may induce failure, and for a box culvert

include midspans and corners in the top slabs, bottom slabs, and walls. The capacity,

dead load demand and live load demand must be calculated at each critical section for

each type of load (moment, shear, thrust) and each load case (total, reduced) in order to

determine the controlling rating factor for a culvert. A load rating calculation requires

that the rating factor equation be evaluated for every potential critical section location

and load case in a structure, and the lowest rating factor controls the load rating for the

whole structure. As an example, for a typical four-span culvert, the single lowest rating

factor from 468 rating factors determines the load rating for the structure. The tonnage of

the load rating vehicle used to determine the live load demand is multiplied by the lowest


5

rating factor to calculate the load rating. The final load rating is the largest truck tonnage

of a particular pattern that can be carried by the structure. The typical load rating vehicle

is the HS20 truck (AASHTO, 2013). Furthermore, the critical section corresponding to

the controlling rating factor should be where initial damage on the structure would occur

as this is the analytically-identified weakest section. The load rating calculation is the

analytical component of the bridge evaluation process.

Figure 2. Critical section schematic

Culvert Load Rating Research at Texas Tech University

Culvert load rating at Texas Tech University has been funded by the Texas

Department of Transportation (TxDOT) since 2007. The first research project funded by

TxDOT, project 0-5849, explored culvert load rating and the influence of soil-structure

interaction (Lawson, et al., 2010). This project resulted in the repeatable load rating

procedure articulated in the Culvert Rating Guide and included live load field testing of

three in-service culverts. TxDOT project 0-5849 is the backbone of this dissertation. The


6

Culvert Rating Guide establishes the two production-simplified models – the structural-

frame model and soil-structure interaction model – used in this dissertation (Lawson, et

al., 2009). The research report contains details associated with the measured data used to

evaluate the influence of the model sophistication and live load attenuation (Lawson, et

al., 2010).

In 2012, TxDOT funded two implementation projects with Texas Tech University.

Project 5-5849-01 resulted in the development of a culvert load rating program called

CULVLR. CULVLR allows for the rapid and error-resistant load rating of CIP RC box

culverts using both the structural-frame and soil-structure interaction models (TxDOT,

2013). Along with the development of CULVLR, TxDOT sponsored project 5-5849-03

to perform load rating calculations on a set of TxDOT design standards. The design

standard load ratings resulted in the data set used to explore the influence of cover soil

depth on culvert load rating (Wood, et al., 2013).

Most recently, TxDOT approached Texas Tech University to perform load rating

calculations for 11,000 in-service culverts on Texas roads built prior to 1980. This

research project provided the additional insight, motivation, and funding to develop the

improved depth-calibrated live load attenuation method. This latest study shows that the

improved load rating developed in the dissertation helps close the gap between load

rating results and field inspection observations (TxDOT, 2014).


7

Challenges for CIP RC Box Culvert Load Rating

As simple as the rating factor equation (Equation 1) appears, culvert load rating faces

several challenges. First, CIP RC box culverts appear deceptively simply when they are

actually very complex. Structurally, a box culvert is a three-dimensional, indeterminate,

reinforced concrete structure with many critical sections. Buried structure behavior only

further complicates the prediction of performance. These complications require

specialized knowledge or overly-conservative simplifications of complex soil-structure

interaction. The combination of complex structural response and a massive number of

required calculations make culvert load rating very difficult.

Second, the load rating process for CIP RC box culverts has historically been unclear

and undocumented. A survey of state departments of transportation (DOTs) revealed

widespread confusion about the process for load rating culverts (Lawson, et al., 2010). In

response, the Texas Department of Transportation (TxDOT) generated the Culvert Rating

Guide to articulate a repeatable procedure for culvert load rating (Lawson, et al., 2009).

At the national level, the 2013 interim revisions to the MBE included explicit guidance

for box culvert load rating (AASHTO, 2013). These documents not only define the load

rating process, they also enumerate the factors that impact culvert load rating, including

those discussed in this dissertation.

Third, while improved guidance leads to increased repeatability in the load rating

process, the load rating values from the established load rating processes have failed to

corroborate field inspection observations. Load rating engineers have identified this

phenomenon as a “disconnect” between observed structure performance and calculated


8

load rating. Consider a typical example; inspection of a CIP RC box culvert which shows

mild cracking in the top slab. However, when the load rating is calculated using the

methods and models recommended by the MBE, the calculations suggest load rating is

governed by a bottom slab critical section. Not only does the governing section not match

the location of the observed structural distress, but also the load rating value is low,

indicating a need for replacement. Research at both state and federal levels has attempted

to explain and reduce this disconnect (NCHRP 15-54, 2015; Han, et al., 2013; Orton, et

al., 2013; Lawson, et al., 2010; TxDOT, 2014). This dissertation seeks to shed light on

why the disconnect between observed performance and calculated load ratings exists and

how such variance might be overcome in production load rating.

Finally, load rating is performed on existing culverts. This distinction sets load rating

apart from the design of new culverts. In the case of culvert design, excessive

conservatism in the demand calculations can be “overcome” by increasing the structural

capacity with more steel and concrete. However, for load rating, the capacity is a fixed

quantity; the structure is built, completed, and in the ground. Short of retrofit or other

repair, the load rater has no ability to improve the capacity portion of the load rating

factor equation (Equation 1). If the load rater desires to improve the load rating for a

culvert that performs adequately under field inspection observation, the rater may only

improve the estimate of the dead and live load demands. This dissertation focuses on

these demand calculations.


9

Development of Production-Simplified Demand Models

Additionally, this dissertation is limited to culvert load rating using production-

simplified demand models. Demand models are used to calculate the dead load and live

load demands at critical sections in a culvert structure. Demand models can range in

sophistication from research-intensive models that fully characterize the problem in three

dimensions to production-simplified models that make assumptions such that the

modeling is conservative, repeatable, and expedient. The emphasis in this dissertation is

on production load rating; therefore, this dissertation uses production-simplified models

that intentionally and conservatively simplify the soil-structure system to allow an

engineer to reliably calculate the load rating. The use of production-simplified demand

models carries with it various advantages and disadvantages.

The MBE recommends the use of a production-simplified demand model that treats

the culvert as a two-dimensional (2D) concrete frame with applied dead loads from soil

and self-weight, and applied live load from a truck load attenuated to account for cover

soil to the top slab of the culvert. This direct-stiffness, structural-frame model can be seen

in Figure 3. The benefit of the structural-frame model is that it provides a “quick,

conservative, repeatable load rating” (AASHTO, 2013). The drawback is over-

conservatism in the rating value, “particularly in the bottom slab” (AASHTO, 2013). This

over-conservatism is partially responsible for the disconnect between load rating

calculations and field inspection observations.


10

Figure 3. Production-simplified direct-stiffness model (Lawson, et al., 2009)

This 2D structural-frame model is rightfully the simplest of the production-simplified

models. Culvert load rating is a subset of bridge load rating; therefore, production-

simplified culvert analysis follows bridge analysis. This approach can be described as

loads on a structure. In this case, the structure is the concrete box, and the surrounding

soil is treated as an applied load. Many years of research have attempted to accurately

define loads on buried structures. The goal of this prior research has been to define and

simplify the load conditions such that they can be applied to a structural-frame demand

model (Spangler, et al., 1926; James & Brown, 1987; Tadros, et al., 1989). Much of this

work has focused on installation stresses that are critical to culvert design; however, for


11

load rating, the installation stresses are assumed to have dissipated. At the time of this

dissertation, research on long-term dead load stresses on RC box culverts is not available.

A common approach of the most recent research has been to use research-intensive,

three-dimensional, soil-structure interaction models to characterize the loads on the

structure and then simplify those loads so they may be applied to a production-simplified,

structural-frame model (McGrath, et al., 2005; Petersen, et al., 2010). This approach of

estimating loads and applying them to a culvert is a natural and appropriate way to

approach the analysis of CIP RC box culverts.

The structural-frame model carries with it certain limitations that become particularly

apparent when load rating culverts. When considering dead load, the direct-stiffness

model as applied using AASHTO guidance assumes a generally conservative soil unit

weight (18.9 kN/m3 (120pcf)) for vertical loads. Lateral loads to the exterior walls are

estimated using equivalent fluid weight for the soil based on a range of at-rest lateral

earth pressure coefficients, K0 from 0.25 to 0.5 (φ ≈ 30° - 49°). Some DOTs have found

even this lateral earth pressure to be unreasonably conservative and therefore, assume K0

varies between 0.19 to 0.38 (φ ≈ 38° - 54°) (Lawson, et al., 2010). The use of a two load

cases (total and reduced lateral load cases) to account for the unknown variability in at-

rest lateral earth pressures is helpful; however, such high internal friction angles are

inconsistent with observed construction practice of backfilling culvert installations with

native soil. Nevertheless, taken together, the assumptions for dead load are consistent

with conservative geotechnical design assumptions.


12

Live load demands are very different; here the simplifications required for a

structural-frame model carry many overly-conservative assumptions. First, the live load

is attenuated as a function of depth from the ground surface to the top slab. This is

typically a conservative approximation of the elastic solution. The real challenge is how

live load is supported by the bottom slab. In a structural-frame model, the bottom slab

carries the entire load applied to the top slab, and this leads to the conservatism in the

bottom slab noted by AASHTO (AASHTO, 2013). In real culvert structures, the live load

is further attenuated through the soil and along the flow length and depth of the culvert.

The load rating values produced using the structural-frame model tend to be very

conservative and to poorly predict the regions of a culvert most likely to show damage.

The way to overcome this shortcoming is by increasing the model sophistication.

An enhanced production-simplified model is a 2D, linear-elastic, finite-element, soil-

structure interaction model shown in Figure 4. In this model, the soil-structure system is

used to predict structural response due to self-weight and vehicle loads. This increase in

modeling sophistication increases the analysis effort required, but the goal of a

production-simplified model is to generate “quick, conservative, and repeatable” load

ratings (AASHTO, 2013). The Culvert Rating Guide and the corresponding software,

CULVLR, define a soil-structure interaction model that is production-simplified

(Lawson, et al., 2009; TxDOT, 2013). This model was developed to improve load rating

accuracy and precision. Fundamentally, this model seeks to predict structural response in

the soil-structure system, rather than simply model structural response to loads on a

structure. The simplified soil loads of a structural-frame model greatly reduce the


13

accuracy and precision of the predicted demands. The soil-structure system more

realistically models the actual soil-structure system. Research focused on loads-on-a-

structure has been modified to realistically estimate the structural response when

modeling the whole soil-structure system. Much of this dissertation is focused on

showing the improvement in modeling accuracy and precision that can be achieved by

this soil-structure interaction model.

Figure 4. Production-simplified soil-structure interaction model

For dead load, the soil-structure interaction model uses the unit weight, a soil

stiffness, and Poisson’s ratio to define the soil mass. The self-weight is sufficient to

define the soil-structure response under dead load.

For live load, the in-plane distribution is modeled by applying wheel loads to the

ground surface and letting the finite-element mesh redistribute the load into the structure.


14

The result is a far more accurate in-plane live load distribution to the bottom slab and

walls. The out-of-plane live load distribution is still an approximation derived from

loads-on-a-structure research. The simplest application of the out-of-plane live load

distribution is to attenuate the live load to the top slab, but this approach neglects the

additional, out-of-plane distribution when estimating the bottom slab response. Since the

goal is the accurate prediction of structural response, the more natural method is to

attenuate the live load for the out-of-plane distribution to the depth of the critical section

of interest. This depth-calibrated method in a soil-structure interaction model overcomes

many of the shortcomings of the structural-frame model.

These two production-simplified models – the structural-frame model and the soil-

structure interaction model – are used in this dissertation to explore load rating of CIP RC

box culverts.

Factors Influencing Culvert Load Rating

Load rating is influenced by several factors. Some of these factors are well defined;

though they matter a great deal to the outcome of a load rating calculation, they are

expressed in the load rating problem with little room for interpretation. Other factors

require greater judgment and permit a wider range of variability.

The prime example of a well-defined factor is the culvert design. The design defines

the geometry, quality, and quantity of the steel and concrete used to build the culvert.

Clearly, design is the greatest single factor impacting a load rating; it fully defines culvert

capacity, and capacity represents one-third of the rating factor equation. Another well-


15

defined factor is cover soil depth. For a particular culvert, the cover soil depth is set by

the field conditions, but it has significant impact on culvert load rating.

Other factors that influence load rating require greater engineering judgment to

define. The most significant factor requiring engineering judgment is modeling

sophistication. As discussed previously, the two models used in this dissertation are a

structural-frame model and a soil-structure interaction model, and both of these models

are considered production-simplified.

As part of the specification of these demand models, the live load distribution also

has a significant impact on the load rating. Several different live load distributions have

been developed including the elastic solution (Poulos & Davis, 1991), the SSHB

distribution (AASHTO, 2002), and various iterations on the LRFD solution (AASHTO,

2014; AASHTO, 2012; Han, et al., 2013).

The final factor that requires some engineering judgment is how the lateral soil

pressures are determined. For the structural-frame model, a range of at-rest lateral earth

pressure coefficients could be appropriate. For the soil-structure interaction model, the

main parameter is the soil stiffness. All these parameters require engineering judgment to

correctly define, and the selected values drastically influence the load rating results for a

given culvert (Lawson, et al., 2010). Each of these factors – design, cover soil depth,

modeling sophistication, live load attenuation, and lateral earth pressure – deserve

exploration. This dissertation considers cover soil depth, modeling sophistication and live

load attenuation for the load rating of CIP RC box culverts.


16

Dissertation Outline

Chapter 1 of this dissertation introduces the culvert load rating concept including key

factors impacting load rating. In Chapter 2, the structural-frame model is used to evaluate

the effect of cover soil depth on culvert load rating for a population of culvert standard

designs. Chapter 3 explores the influence of modeling sophistication by comparing the

accuracy and precision of the structural-frame model versus the soil-structure interaction

model using full-scale live load test data. Here the structural-frame model is referred to as

the Level 1 model, and the soil-structure interaction model is referred to as the Level 3

model. In Chapter 4, the soil-structure interaction model is used to explore the influence

of live load attenuation method. That is, a comparison is made between the traditional

top-slab-calibrated method versus the proposed depth-calibrated method. Chapter 5

summarizes the major findings of this research and outlines those factors that remain to

be explored. This dissertation includes two appendices that present data used in the

analyses but which have not been published previously. Appendix A contains plots that

define the population of designs used in Chapter 2. Appendix B contains all moment

diagram plots used in Chapter 4.


17

CHAPTER 2

COVER SOIL DEPTH

Note: Previously published as:

Wood, T. A., Lawson, W. D., & Jayawickrama, P. J. (2015). Influence of Cover Soil Depth on

Reinforced Concrete Box Culvert Load Rating. Transportation Research Record 2511, 61-

71.

Chapter Summary

This chapter describes the influence of cover soil depth on the load rating of multi-

barrel, cast-in-place (CIP), reinforced concrete (RC) box culvert designs and highlights

implications for the load rating and design of culvert structures. The basics of culvert

load rating are discussed followed by the history of culvert design policy and the

challenges created by the use of culvert standard designs. A population of Texas

Department of Transportation (TxDOT) CIP RC culvert standard designs developed

between 1930 and 1980 were load rated using AASHTO policy guidance and a two-

dimensional, direct-stiffness, structural-frame demand model for a full range of cover soil

depths. This analysis resulted in a set of 1081 load rating vs. cover soil depth

relationships. Three typical rating vs. depth relationships are illustrated and described in

detail. The distribution of characteristic rating vs. depth relationships based on culvert

geometry, design cover soil depth, and design era are explored. Cover soil depth is shown

to be a critical parameter which must be explicitly considered for the intelligent load

rating and design of reinforced concrete box culverts.


18

Introduction and Background

State Departments of Transportation (DOTs) are required by federal regulation to

load rate bridge-class, cast-in-place (CIP) reinforced concrete (RC) box culverts as part

of their bridge inspection program (Bridges, Structures, and Hydraulics, 2009). The

AASHTO Manual for Bridge Evaluation (hereafter, MBE) (AASHTO, 2013) is the

document of technical authority providing policy guidance for the load rating process

using both Load and Resistance Factor Rating (LRFR) based on the current AASHTO

LRFD Bridge Design Specifications (AASHTO, 2014) , and Load Factor Rating (LFR)

and Allowable Stress Rating based on the current AASHTO Standard Specifications for

Highway Bridges (AASHTO, 2002). Throughout this chapter the LFR method has been

used consistent with the accepted MBE requirements and current implementation by state

DOTs. Observations and findings from this study may be considered generally applicable

to all three load rating methods. Load ratings are referred to in terms of tractor tonnage

for a design truck loading, often an HS20 truck.

Load Rating Process

The load rating for a culvert structure is calculated by evaluating the rating factor for

each critical section, failure mode, and load case. In box culverts, this requires that a load

rating factor be calculated for all critical sections (corners and midspans for each span

and wall), three failure modes (bending, shear, and thrust), maximum and minimum live

load demands, and, per AASHTO policy, a total and reduced lateral load case. For a two

barrel culvert, the smallest number of spans evaluated in this study, this requires the


19

calculation of 21 x 3 x 2 x 2 = 252 load rating factors. Equation (1) shows the AASHTO

rating factor for the LFR method (AASHTO, 2013).

The lowest rating factor in the comprehensive set of rating factors defines the

controlling load rating for a culvert structure. This controlling rating factor is multiplied

by the tractor tonnage to calculate the load rating. The structure’s load rating essentially

defines the maximum truck load the structure can carry. By using different load factors,

two rating levels have been defined in the MBE LFR approach. The inventory rating (IR)

is the “live load which can safely utilize the bridge [or culvert] for an indefinite period of

time” (AASHTO, 2013). The operating rating (OR) is a larger load intended to identify

“the maximum permissible live load to which the structure [culvert] may be subjected"

(AASHTO, 2013). A culvert or bridge structure with an operating rating below HS20

usually requires load posting or replacement.

Three main factors contribute to the load rating of a RC box culvert: the section

capacity, dead load demand, and live load demand. Each of these factors can be

calculated based on the current policy contained in the Manual for Bridge Evaluation

(AASHTO, 2013) and Standard Specification for Highway Bridges (AASHTO, 2002) for

LFR. The Method section of this chapter provides specifics of these calculations using

LFR.


20

History of Culvert Design and Load Rating Policy

Prior to World War II, culverts were often designed based on H15 and HS15 design

truck loads using allowable stress design (ASD). The modern semi-truck had yet to be

developed, and truck loads in excess of 80,000lbs which have become typical in today’s

rapidly expanding energy sector were unimaginable. Further, typical culvert spans in the

pre-WWII era were relatively small (less than 7ft), and under this situation, minimum

reinforcing steel was more than sufficient to resist design loads. Pre-WWII culvert

designs also featured haunches to reduce the demand moments in the corners. These

factors resulted in robust CIP RC box culvert designs.

After WWII, during construction of the Interstate Highway system, new culvert

standard designs were developed with an emphasis on construction economy. These

designs removed the labor-intensive haunch details and employed thinner slabs. Policy

also changed; the HS20 truck was added to the code in 1944 but was heavily debated for

decades. In 1949, the AASHTO Standard Specifications for Highway Bridges modified

the allowable stress design such that the effective soil unit weight was reduced to 70% of

18.8kN/m3 (120pcf), i.e., 13.2kN/m3 (84pcf). This was intended to produce a 40%

increase in dead load allowable stress over live load as a recognition of the increased

variability in live load compared to dead load (AASHTO, 1949). These factors made

Interstate Highway era culvert designs more economical to construct, but these culvert

structures are more prone to under-perform when evaluated by today’s load rating policy

(Kulicki, 2013).


21

In the early 1970s, AASHTO adopted Load Factor Design that effectively replaced

the 70% reduction in soil dead load. Further, HS20 trucks became the required design

load (AASHTO, 1977). These policy changes required new culvert standard designs that

explicitly met the new requirements. Along with the increases in demand loads, grade 60

reinforcing steel was becoming increasingly common. Some state DOTs took advantage

of this shift by simply reissuing Interstate Highway era designs with grade 60 steel

instead of the previously-required grade 40 steel. This change significantly improved

structural capacity, thereby improving culvert performance under current load rating

requirements.

In 1994, AASHTO adopted its first LRFD Bridge Design Specifications (AASHTO,

1994) which, among other things, changed the live load distribution. In 2007, LRFD

became the exclusive design method for new structures (AASHTO, 2007).

This brief account of RC box culvert design and load rating development would be

incomplete without mentioning precast concrete boxes. Precast concrete box culverts

emerged in the 1970s as a viable solution, complete with design guidance (ASTM

Standard C1433-14, 2014; ASTM Standard C789-00, 2000; ASTM Standard C850,

2000) and analysis software (Latona, et al., 1973; FHWA, 2010). Single-cell, precast

concrete box culverts are a major section of concrete box culvert systems today, but as

noted earlier, this chapter focuses on load rating CIP RC multi-barrel box culverts.


22

Use of Standard Designs

In addition to changes in policy, state DOTs, both in the past and now, have relied on

culvert standard designs to build culverts. Typically, standard design sheets present

multiple culvert designs on a single sheet, with the various designs addressing a range of

spans and box geometry but with all designs on a given sheet intended for use under one

defined range of cover soil depth. Such standard designs provide an approved and quick

way to specify a culvert for a construction project.

However, the idea of culvert standard designs has carried with it a unique set of

challenges. Often, the defined range of cover soil depths for a given culvert standard

design is rather wide (i.e., a meter or more). Functionally, a particular culvert design is

presumed to be uniformly appropriate for the entire design soil depth range. However,

this is rarely the case. The relationship between load rating and cover soil depth is highly

nonlinear. Far from being uniformly appropriate over a range of soil depths, as will be

discussed in detail in the next section, a culvert design may swing from unconservative to

overly conservative within just a meter of cover soil depth. This chapter seeks to illustrate

and describe the nonlinear interaction between culvert load rating and the cover soil

depth.


23

Method

This study evaluated a set of standard CIP RC culvert designs. A culvert design will

typically have a load rating vs. cover soil depth relationship, because a design is often

applicable over a range of cover soil depths. In contrast, a culvert structure has an actual

load rating corresponding to its actual cover soil depth. At no point in this study was an

actual culvert load rated, and this is an important distinction. Further, data presented

herein will suggest that certain designs do not perform adequately when evaluated under

contemporary load rating policy. While it is reasonable to describe trends and to observe

that some designs perform better than others, this study should not be construed as the

final determination about the efficacy of a given design. Resolving such issues was

beyond the scope of this study. Other research indicates that more advanced demand

analytical modeling or non-destructive load tests may result in an adequate evaluation of

load rating performance (AASHTO, 2013; Das, 2013; Wood, et al., 2015). The results

presented herein deal only with load rating calculations for CIP RC culvert designs.

Population of Evaluated Culvert Standard Designs

The dataset evaluated in this study consisted of 1081 culvert standard designs

developed by the Texas Department of Transportation (TxDOT) between 1930 and 1980.

These standard designs were generated over three design eras: pre-WWII designs

developed in the 1930s typically with haunches (see Figure 5(a)), Interstate Highway era

designs developed in 1958 typically without haunches (see Figure 5(b)), and modernized

designs (also Figure 5(b)) which are 1977 re-releases of the Interstate Highway era


24

designs with reinforcing steel increased from grade 40 to grade 60. The pre-WWII

designs comprise 30% of the population and the Interstate Highway and modernized

designs each comprise 35% of the population.

(a)

(b)

Figure 5. Example standard designs: (a) cross-section view of pre-WWII design with

haunches (Source: TxDOT standard sheet ‘MBC-3-34-F’), and (b) post-WWII design without haunches (Source: TxDOT standard sheet ‘MC9-1’)

In terms of TxDOT culvert geometry, the designs can be described in terms of span

number (the number of individual boxes comprising the culvert structure), span length

(interior distance from wall to wall for an individual box measured normal to flow

direction), box height (vertical interior distance from top to bottom slab) and aspect ratio


25

(span divided by height). Each era had roughly the same distribution of designs by span

number - ranging from 2 to 6 spans. Aspect ratio was also fairly uniform between design

eras. Pre-WWII designs tend to be smaller, both in terms of span and height, compared to

post-WWII designs. The one exception to this trend is a set of 20 pre-WWII designs with

heights of 3.3m to 3.7m (11ft to 12ft). Otherwise, span length (range - 1.5m to 3m (5ft to

10ft)) and box height (range - 0.6m to 3m (2ft to 10ft)) distributions are consistently

represented in each era. Appendix A provides distributions of these independent variables

within the design standards.

The distribution of culvert designs by design cover soil depth range is also reasonably

uniform between eras. A culvert is considered a “direct traffic” culvert if the cover soil

depth is from 0m to 0.6m (0ft to 2ft). The post-WWII era standards were typically

designed for 0.6m (2ft) soil depth increments, hence designs in the 0 to 0.6m (0ft to 2ft),

0.6m to 1.2m (2ft to 4ft), and 1.2m to 1.8m (4ft to 6ft) categories. The majority of the fill

culvert designs from the pre-WWII era were intended to function over a full range of

cover soil from direct traffic up to the maximum design depth of 1.8m (6ft).

Load Rating Procedure

AASHTO policy provides guidance for load rating. For the LFR method, the capacity

calculations are consistent with civil engineering practice for CIP RC one-way slabs. The

demand guidelines in AASHTO define loads to be applied directly to an analytical

structural culvert model. The loads include dead loads (culvert self-weight plus soil) and

live loads (traffic). Vertical dead loads are determined from soil unit weight of

18.9kN/m3 (120pcf). Lateral dead loads are applied using an equivalent fluid unit weight


26

of 9.4kN/m3 (60pcf). Live loads are HS20 patterned truck loads, distributed, both in-

plane and out-of-plane, through the cover soil to the top slab. AASHTO policy then

requires consideration of two load cases. A total load case applies all the dead load, truck

load, and a 0.6m (2ft) lateral traffic surcharge to the structure. A reduced lateral load case

reduces the total load case by removing the lateral traffic surcharge and half of the lateral

dead load. In this way, AASHTO documentation defines simplified soil-structure

interaction for CIP RC box culverts that can be evaluated using a simple, structural-frame

model. (AASHTO, 2013; AASHTO, 2014)

The AASHTO guidance presumes a reasonable structural-frame model will be used

to calculate demands at critical sections in the culvert. Many states have developed

software programs for the calculation of predicted demands on concrete culverts

including Alabama (Lakmazaheri & Edwards, 1996), Texas (TxDOT, 2003), and

Wyoming (Wyoming DOT, 2008). Each of these programs assumes AASHTO loadings

applied to a direct-stiffness structural-frame model. More complex multi-barrel culvert

analysis tools exist, most notably CANDE (Katona, 2015), but these require specification

of parameters beyond those provided in the AASHTO guidelines. In this study a two-

dimensional direct-stiffness, structural-frame model was used to determine demands from

policy loadings.

The modeling tool used for this study was CULV5, TxDOT's publically available

culvert analysis program (TxDOT, 2003). CULV5 uses a simple 3 line text-based input

(updated from the original punch card system) to quickly calculate demand moments,

shears, and thrusts according to AASHTO policy requirements. Furthermore, the CULV5


27

analysis engine has been incorporated into TxDOT’s load rating specific software

program CULVLR (TxDOT, 2013). CULVLR allows for the digitization of culvert

parameters required for load rating. The program uses CULV5 to rapidly calculate all the

load rating factors and identifies the controlling load rating factor. With valid input

information, the calculation of a load rating, typically tedious and error-prone if done by

hand or spreadsheet, is rapid and accurate. The TxDOT Culvert Rating Guide and

CULVLR documentation provide further details about the application of AASHTO load

rating policy and the load rating method used in this study (TxDOT, 2013; Lawson, et al.,

2009).

The data evaluated in this study were generated by digitizing the population of

TxDOT culvert standard designs described previously and performing load rating

calculations for each design, over a range of soil depths, iterated at 0.15m (0.5ft)

increments. All totaled, the synthesis information described herein is based upon the

calculation of 24,015 unique culvert load ratings and over 15 million load rating factors.


28

Results

The results are presented in three parts. First, observations will be made about the

typical relationships between load rating and cover soil depth. Second, the distribution of

load rating vs. cover soil depth relationship will be explored by design era, culvert

geometry, and design cover soil depth. Third, the implications of these observations for

culvert designers and load raters will be described.

Observations

Each of the 1081 standard culvert designs has a distinct load rating vs. cover soil

depth relationship. Though each culvert standard design is essentially unique, three types

(or forms) of rating vs. depth relationships emerge that are characteristic of the full

population of designs. These are termed “increasing”, “decreasing”, and “constant” as

shown in Figure 6. As the name implies, the “increasing” relationship depicts a culvert

load rating vs. cover soil depth interaction where the load rating increases with increasing

cover soil thickness above the top slab. A “decreasing” relationship is one where the load

rating decreases with increasing cover soil thickness. A “constant” relationship is one

where the culvert load rating remains essentially unchanged, regardless of the amount of

cover soil.


29

(a) (b) (c)Figure 6. Representative load rating vs. cover soil depth plots for increasing, decreasing

and constant relationships (a) Increasing relationship (Source: MBC-2-34-F 1938 2 boxes 1.5mx1.5m (5ftx5ft)); (b) Decreasing relationship (Source: MC9-1 1958 5 boxes 2.7mx2.4m (9ftx8ft)); (c) Constant relationship (Source: MC10-3 1977 3 boxes 3mx2.7m (10ftx9ft))

All 1081 designs analyzed herein can be assigned to one of these three categories.

The criterion used to categorize a design as having an increasing trend required that the

operating rating increase by at least 5 HS truck tons from 0.6m to 1.8m (2ft to 6ft) of soil,

and these types of designs comprise 656/1081, or 61% of the population. The criteria for

the decreasing trend were (a) the rating vs. depth relationship maximized under 0.3m

(1ft) of soil, (b) the rating decreased between 0.6m to 1.8m (2ft and 6ft) of soil, or (c) the

design failed under dead load above 1.8m (6ft) of cover soil. Designs in the decreasing

0

1

2

3

4

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60 70

coversoildep

th,D

(m)

coversoildep

th,D

(ft)

loadrating(HStrucktonnage)

0

1

2

3

4

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60 70

coversoildep

th,D

(m)

coversoildep

th,D

(ft)


0

1

2

3

4

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60 70

coversoildep

th,D

(m)

coversoildep

th,D

(ft)



30

category comprise 314/1081, or 29% of the population. Any design not categorized as

increasing or decreasing was assigned to the “constant” category, and these comprise

111/1081, or 10% of the population.

Figure 6 provides load rating vs. cover soil depth curves for standard designs which

are illustrative of the three categories of rating vs. depth relationship. The charts in Figure

6 share some common characteristics. First, each chart provides load rating relationships

for both the inventory rating (IR) and the operating rating (OR). However, since load

posting and other culvert operational decisions focus more on the OR, this chapter will

implicitly focus on the OR curve. Second, each chart includes a shaded area, and this

shaded area corresponds to the design cover soil depth range for that particular standard

design. For example, the design cover soil depth range for the “constant” curve is 1.2m to

1.8m (4ft to 6ft); whereas, the culvert standard illustrative of the “increasing” category

was designed for a range of soil thickness from 0m to 1.8m (0ft to 6ft). Third, as an aid to

interpreting the magnitude of the OR curve, the HS20 rating is emphasized. Culvert

structures (not designs) with load ratings greater than or equal to HS20 do not require

load posting; however, structures with OR less than HS20 do require load posting.

Finally, it is helpful to think of each rating vs. depth curve in terms of three distinct depth

zones. The upper zone, from 0m to 0.6m (0ft to 2ft) of soil, corresponds to direct traffic

culverts. The deepest zone occurs below the inflection point where the rating starts to

decrease, and the zone between these two is the “low-fill” zone. Again, by way of

example, the upper zone for the “increasing” category is 0m to 0.6m (0ft to 2ft), the low-


31

fill zone is 0.6m to 2.1m (2ft to 7ft), and the deepest zone extends from 2.1m to 4.6m (7ft

to 15ft). General observations can be made about these three characteristic relationships.

The first and most important observation is that each rating vs. depth curve is

nonlinear and, notwithstanding the labels, non-constant for all three categories. This

nonlinearity is expected and can be explained by considering the relationships between

capacity, dead load and live load variables in the rating factor equation (Equation 1). Per

LFR, the capacity for a particular critical section of a particular culvert design is constant;

therefore, the load rating factor can only change through changes in the dead load and

live load. The basic relationship is that an increase in dead load (more cover soil) reduces

the numerator of the rating factor equation, so the rating factor decreases. However,

increased soil attenuates (decreases) live load, and this attenuation simultaneously

increases the rating factor. Figure 7 illustrates this dynamic imbalance in the rating factor

associated with changing distributions of live load and dead load with cover soil depth.


32

(a) (b)

Figure 7. (a) Dead load and (b) live load relationship with cover soil depth

Figure 7(a) illustrates how the dead load increases linearly with depth. If this alone

impacted the culvert load rating, the load rating would decrease linearly. But such is not

the case, and Figure 7(b) shows how additional cover soil depth attenuates the live load.

While the dead load relationship is linear, the live load relationship per AASHTO policy

is highly nonlinear. For 0m to 0.6m (0ft to 2ft) of cover soil, the direct traffic range, the

live load attenuates slowly due to the impact factor and direct traffic requirements. The

discontinuity in the live load attenuation at 0.3m (1.0ft) is due to the step function used to

define the impact factor for LFR (AASHTO, 2002). From 0.6m to 1.2m (2ft to 4ft), the

live load attenuates far more rapidly than the dead load builds. In this range, the load

0 25 50 75 100 125

0

1

2

3

4

0

2

4

6

8

10

12

14

16

0 1 2 3

soilpressure,σ(kPa)

coversoildep

th,D

(m)

coversoildep

th,D

(ft)

soilpressure,σ(ksf)

verticalsoilstresstotalloadcaselateralstressreducedlateralloadcaselateralstress

0

1

2

3

4

0

2

4

6

8

10

12

14

16

0% 5% 10% 15% 20%

coversoildep

th,D

(m)

coversoildep

th,D

(ft)

%ofgrossHStruckweightcarriedbya0.3m(1ft)crosssectionofculvert

liveloadattenuation


33

rating is expected to rise. At 1.2m (4ft) of cover soil, live load attenuation begins to taper

off, dead load continues to increase linearly, and for 1.2m (4ft) or more of cover soil, the

load rating tends to level off until the dead load increase overshadows the live load

attenuation. It is highly unusual for a culvert design to yield anything approaching a

constant load rating vs. depth relationship. Nonlinearity in the load rating should be

expected.

Another commonality between the rating vs. depth relationships for the evaluated

culvert designs is their behavior under direct traffic. Recall that “direct traffic culverts”

are defined as having cover soil depths from 0m to 0.6m (0ft to 2ft). In this range, the live

load is completely un-attenuated by cover soil, yet dead load is increasing. The impact

factor does decrease with increasing cover soil thickness, but the net result is that through

the upper soil depth range, the load rating tends to be relatively constant. This

phenomenon holds for all three categories of rating vs. depth.

A final universal trend, already mentioned, is that for each rating vs. depth

relationship, there is an inflection point, typically at a slightly greater cover soil depth

than the maximum load rating, after which the load rating precipitously decreases to less

than zero. Through this range the factored dead load demands increase faster than the

factored live load demands attenuate until the dead load demands overwhelm the capacity

at the controlling critical section. This behavior occurs for every design, regardless of the

trend in the rating vs. depth relationship prior to the inflection point. Having made these

general observations, unique aspects of each type of relationship can be discussed.


34

Increasing Load Rating vs. Cover Soil Depth Relationship

Figure 6(a) shows a representative example of a culvert design with an increasing

relationship between load rating and cover soil depth. The increasing relationship

between load rating and cover soil depth is the intuitively expected relationship for

culvert structures: lowest load rating around 0.3m (1ft), constant load rating from 0m to

0.6m (0ft to 2ft), increasing from 0.6m to 2.4m (2ft to 8ft) of fill, and precipitous

decrease in load rating as dead load overwhelms the capacity. The load rating factor

equation, capacity minus dead load divided by live load, suggests that this relationship

would be typical for cases where the capacity is adequate to resist dead load. A design

with an increasing rating vs. depth relationship is particularly helpful for culverts

intended for use under more than 0.6m (2ft) of soil.

Figure 6(a) illustrates another key load rating issue. Here, the culvert performs well

over a range from 0.9m to 4.0m (3ft to 13ft) of cover soil. However, the design was

intended for use as a direct traffic culvert. Note that even though the culvert design

exhibits suitable performance over a wide range of cover soil depths, by virtue of its

stated design soil depth range from 0m to 1.8m (0ft to 6ft), a structure based on this

design may not load rate adequately if the cover soil depth is less than 0.9m (3ft). The

nonlinear rating vs. soil depth relationship highlights how a particular culvert design may

work well for some cover soil depths, but not for others. In particular, the direct traffic

condition tends to represent the most severe loading.


35

Decreasing Load Rating vs. Cover Soil Depth Relationship

Figure 6(b) shows a typical decreasing relationship between load rating and cover

soil depth. Again, in the direct traffic range from 0m to 0.6m (0ft to 2ft) of cover soil, the

load rating is relatively constant. Culvert designs characterized by this decreasing rating

vs. depth relationship have such a narrow gap between capacity and dead load that, as

dead load increases, the load rating plummets.

Designs with a decreasing relationship between load rating and cover soil depth may

indicate an economical culvert design where the design is intended for use as a direct

traffic culvert. In this case, there is no need for additional capacity at greater cover soil

depths. However, a load rating is defined by the worst-case critical section; other sections

may not be so economically designed. Unfortunately, designs characterized by decreasing

relationships often do not load rate acceptably for any cover soil depth, and such is the

case for the example depicted in Figure 6(b). In general, a decreasing relationship

between rating and depth is undesirable.

Constant Load Rating vs. Cover Soil Depth Relationship

In between culvert designs with increasing relationships and designs with decreasing

relationships are those with a constant rating vs. depth relationship. Figure 6(c) shows a

representative example. For this case, the capacity is such that the increases in dead load

and decreases in live load balance each other, providing a relatively constant load rating.

Such designs, while rare, might be very economical across a wide range of soil

depths, assuming the culvert load rates adequately (above HS20) for the intended design

cover soil depth. The example in Figure 6(c) is a case in point. Unfortunately, most of the


36

curves in this category do not load rate adequately. Even if the design has an adequate,

constant relationship between rating and depth, it might only have one economically

designed critical section. Other sections might be over conservative, and not reflected in

the load rating.

Distribution of load rating vs. cover soil depth relationship in the

population

Additional analyses were performed to identify the distribution of the three

characteristic rating vs. depth relationships by design era, culvert geometry, and design

cover soil depth. Figure 8 illustrates the distribution between rating vs. depth

relationships and the design era. The modernized designs are the most likely to have an

increasing relationship between rating and depth. This better performance is expected due

to higher capacities associated with the wholesale shift from grade 40 reinforcing steel to

grade 60 and the use of design truck loads which are very similar to the current policy

loads. The pre-WWII era culvert designs are twice as likely to have an increasing

relationship rather than a decreasing relationship between rating and depth. Again, this is

not inconsistent with expected performance due to excess conservatism in the analysis

methods and the design philosophy of that era. The Interstate Highway era designs are

the most likely to have a decreasing rating vs. depth relationship.


37

Figure 8. Trend plot of load rating vs. cover soil depth plot shape by design era

Figure 9 illustrates the distribution of typical rating vs. depth relationships by culvert

geometry. The likelihood of a design having an increasing rating vs. depth relationship

increases with increasing aspect ratio, as shown in Figure 9(a). Square box culvert

designs have roughly similar numbers of decreasing and increasing rating vs. depth

relationships. However, rectangular culvert designs (long span, low height) are most

likely to have an increasing relationship between rating and depth.

208

143

305

12

56

43101181

32

0%10%20%30%40%50%60%70%80%90%

100%

pre-WWII InterstateHighway

modernized%ofculvertdesignsby

desig

nera

designera

increasing constant decreasing


38

(a) (b)

(c)

Figure 9. Trend plots of load rating vs. cover soil depth relationship by culvert geometry: (a) aspect ratio, (b) span length, and (c) barrel height

Figure 9(b) shows the strong correlation between the rating vs. depth relationship and

the span length. The smallest culvert designs (shorter span, lowest height) almost always

have increasing relationships. As the span increases, the likelihood of an increasing

relationship reduces. Figure 9(c) shows a similar trend based on box height. Culvert

124171 120

177

6430

3926

15

1

11885 47 59

5

0%10%20%30%40%50%60%70%80%90%

100%

1 1.25 1.5 2 2.5

%ofculvertdesignsbyaspe

ctra

tio

aspectratio,S/H

157142

13996

6557

317

9

21

38

23

3

5262 71

126

1.5 1.8 2.1 2.4 2.7 3.0

0%10%20%30%40%50%60%70%80%90%

100%

5 6 7 8 9 10

span,S(m)

%ofculvertdesignsbyspan

span,S(ft)

40102

128 127111

75 62

11

68 15

17

17 17

29

2

525 26

4659 49

44

428

10

0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6

0%10%20%30%40%50%60%70%80%90%

100%

2 3 4 5 6 7 8 9 10 11 12

height,H(m)

%ofculvertdesignsbyhe

ight

height,H(ft)



39

designs with taller walls often have decreasing rating vs. depth relationships while culvert

designs with shorter walls often have increasing relationships between rating and depth.

Figure 10 illustrates the relationship between load rating vs. depth trend and the

design cover soil depth. That is, this chart compares how culverts perform vs. how they

were designed. As the maximum design cover soil depth increases, the likelihood of a

culvert design having an increasing rating vs. depth relationship improves. This is

consistent with the notion that a culvert designer would, at a minimum, check for

adequate capacity at the maximum and minimum design cover soil depths.

Figure 10. Trend plot of load rating vs. cover soil depth plot shape by design cover soil

depth

Implications

The nonlinearity of the load rating vs. cover soil depth relationship has profound

implications for engineers working with culvert standard designs. For load rating

engineers, this nonlinearity must be considered from the beginning. It is not sufficient to

114101 69

240132

28

3130

20

2

195

6831

20

DT(0-0.6m) 0.6-1.2m 0-1.2m 1.2-1.8m 0-1.8m

0%10%20%30%40%50%60%70%80%90%

100%

DT(0-2ft) 2-4ft 0-4ft 4-6ft 0-6ft

designcoversoildepthrange

%ofculvertdesignsby

desig

ncoversoildep

thra

nge




40

simply load rate a design for the “worst case” and assume that every application of the

design is acceptable. There is no way to identify the worst case cover soil depth for a

design without investigating the full range of depths. Rather, if load rating calculations

for a standard design are required, a load rating vs. cover soil depth curve is needed. In

practice, a load rating should be performed for an actual culvert considering the actual

cover soil depth.

Even if the CIP RC box culvert designs can be shown to load rate adequately

throughout the design range, these designs can still create challenges for engineers

involved in roadway rehabilitation or road widening projects. If the cover soil depth is

changed, a culvert which has stood the test of time under one cover soil depth may no

longer perform adequately under a new (thicker or thinner) cover soil depth. Engineers in

this position must consider the final cover soil depth explicitly.

Though culvert design was outside the scope of this study, these findings have

implications for the design process as well. For culvert designers, knowledge of the

relationship between rating and depth will help them more intelligently design

economical and serviceable culverts. Economical standards could be achieved through

tailoring designs to cover specific soil depth ranges. Improved serviceability can be

achieved by explicitly identifying the complete soil range for which the design is

appropriate.

This chapter does not explicitly address the implications of the LRFR method for load

rating CIP RC box culverts. Rather the LFR method has been used. The influence of

cover soil depth over box culverts is a function of soil-structure mechanics and therefore


41

the observations made in this study concerning LFR analysis of CIP RC box culverts will

have similar implications for LRFR and design for box culvert systems.

Conclusions

The following conclusions have been demonstrated through the evaluation of 1081

standard CIP RC box culvert designs developed in Texas between 1930 and 1980.

1. The relationship between culvert load rating and cover soil depth is highly

nonlinear and non-constant. This derives from the simultaneous, yet unbalanced,

linear increase in dead load and nonlinear decrease in live load with increasing

cover soil depth.

2. Culvert designs can be characterized by three typical rating vs. depth relationships:

a. The increasing relationship between rating and depth is the preferable and

expected relationship, particularly for culvert designs intended for use with

more than 0.6m (2ft) of cover soil. The increasing relationship comprises 61%

of the evaluated standard design population.

b. The decreasing relationship between rating and depth may be acceptable for

direct traffic (0m to 0.6m (0ft to 2ft) of cover soil) culverts. However, for the

observed population, most of the time, decreasing culvert designs did not rate

acceptably at any cover soil depth. The decreasing relationship comprises 29%

of the evaluated standard design population.


42

c. The constant relationship between rating and depth represents a dynamic

balance in which the increase in dead load demand and decrease in live load

demand create a near-constant load rating. The constant relationship comprises

10% of the evaluated standard design population. This rare situation is not

readily predicted by other culvert parameters. Culvert designs in this class

rarely rate acceptably at any depth.

3. Culvert design era is associated with the characteristic rating vs. depth

relationships. Modernized designs are the most likely to have an increasing

relationship and the least likely to have a decreasing relationship. The Interstate

Highway era designs are the most likely to have a decreasing relationship and the

least likely to have an increasing relationship. The pre-WWII era designs are

slightly more likely to have an increasing relationship and slightly less likely to

have a decreasing relationship.

4. Trends exist between rating vs. depth relationship and culvert aspect ratio, span

length, box height, and design cover soil depth. The likelihood of a standard design

having an increasing relationship between rating and depth improves with

increasing aspect ratio, decreasing span length, decreasing box height, and

increasing maximum design cover soil depth.

The nonlinear nature of the relationship between load rating and cover soil depth is

not always appreciated by the culvert load rating and design community. This chapter


43

argues that cover soil depth must be carefully considered to effectively and intelligently

address current culvert load rating and design needs.

Acknowledgements

The Texas Department of Transportation sponsored the research work described in

this chapter. The author thanks Bernie Carrasco, P.E., Keith Ramsey, P.E., and Gregg

Freeby, P.E. of TxDOT for their technical guidance during this research.


44

CHAPTER 3

PRODUCTION-SIMPLIFIED DEMAND MODEL

SOPHISTICATION

Note: Previously published as:

Wood, T. A., Lawson, W. D., Jayawickrama, P. W., & Newhouse, C. D. (2015). Evaluation of Production

Models for Load Rating Reinforced Concrete Box Culverts. J. Bridge Engineering , 20 (1),

04014057.

Chapter Summary

Analyses of two production-simplified culvert load rating demand models were

performed using live load test data from three instrumented reinforced concrete box

culverts under four cover soil depths. The demand models were a two-dimensional,

direct-stiffness, structural-frame model represented by the program, CULV5, and a two-

dimensional soil-structure interaction model represented by the program, RISA. As

expected, increased sophistication in the soil-structure model as compared to the

structural-frame model resulted in higher precision and accuracy for predicted moments.

The impact of modeling accuracy for sections in a culvert where the demand moments

approach zero was deemed practically insignificant; when evaluating model accuracy, it

is of first importance that the models predict meaningful load magnitudes. Variations in

predicted moment accuracy and precision were not uniform but are a function of the

critical section location in the culvert structure. Improvements in modeling prediction

associated with increased modeling sophistication were only seen when the structural-

frame model was very imprecise.


45

Introduction

This chapter describes an evaluation of the precision and accuracy of two production-

simplified models for load rating reinforced concrete box culverts. State Departments of

Transportation (DOTs) are required by federal law (Bridges, Structures, and Hydraulics,

2009) to load rate all bridge-class structures, including bridge-class culverts, in their

systems using the American Association of State Highway and Transportation Officials

the Manual for Bridge Evaluation (MBE) (AASHTO, 2013). Load rating is an analytical

component of the structural condition evaluation process and consists of determining the

safe load carrying capacity of the culvert, determining whether specific legal or

overweight vehicles can safely cross the culvert, and determining if the culvert needs to

be restricted and, if so, what level of posting is required. The load rating process first

requires an evaluation of the structural capacity at sections that are deemed critical,

followed by estimates of the dead and live load demands under specified loading

conditions at those same sections.

Determination of structural capacities for sections within a reinforced concrete box

culvert is relatively straightforward and requires knowledge of the section and material

properties. Calculation of the demands, on the other hand, is more challenging as many

uncertainties must be considered. The degree to which the soil surrounding the culvert

loads and/or supports the culvert, the three-dimensional nature of the culvert, and the

distribution of traffic loads through the soil into the structure are just a few of the factors

which influence demand predictions.


46

In routine culvert load rating applications, engineers use simple direct-stiffness

structural-frame models to evaluate the demand loads according to guidance published in

AASHTO’s Standard Specification for Highway Bridges (AASHTO, 2002) and LRFD

Bridge Design Specification (AASHTO, 2014). Such basic models are expedient and

conservative; however, over-conservatism (lack of accuracy) can lead to load rating

results that do not correspond with observed culvert performance. When culverts are

determined to be structurally deficient based on the load rating, the DOTs must load-post

or replace the structure.

This chapter is based on the idea that critical review of the culvert load rating process

– in particular, the precision and accuracy inherent in production-ready culvert models –

may offer cost-effective analytical alternatives to load-posting or replacement. The

literature review describes the establishment of a repeatable culvert load rating procedure

as derived from various policy sources. Two production-simplified box culvert load

rating models of differing sophistication are evaluated by comparing predicted moment

demands obtained from the models to measured moment demands obtained from full-

scale live load testing performed on three reinforced concrete box culverts under four

cover soil depths. Research findings will demonstrate that a soil-structure model more

precisely and accurately predicts measured structural response than a structural-frame

model. The increase in precision and accuracy is not uniform throughout the structure,

however. The implication is that when the initial load rating for a culvert is lower than

desired, a higher-order but still production-simplified model may be used to establish the


47

load rating. From the findings, conclusions about the value of increasing modeling

sophistication for culvert load rating are made.

Literature Review

Culvert load rating is a subset of bridge load rating. AASHTO defines load rating as

the maximum truck tractor tonnage, typically expressed in terms of HS load designation

for older structures, permitted across a bridge [or culvert]. The load rating is expressed in

terms of two separate ratings – an Inventory Rating and an Operating Rating. The

Inventory Rating (IR) is the “live load which can safely utilize the bridge [or culvert] for

an indefinite period of time” (AASHTO, 2013). The Operating Rating (OR) is “the

maximum permissible live load to which the structure [culvert] may be subjected”

(AASHTO, 2013). The AASHTO MBE (AASHTO, 2013) provides a complex load and

resistance factor rating equation, but for the purposes of this chapter the general load

rating factor equation (Equation 1) based on the load factor rating method will suffice.

Equation 1 can be thought of as a weighted live load factor of safety calculated from

estimates of section capacity and corresponding dead and live load demands. A rating

factor must be calculated for each critical section and load combination in the structure.

In typical bridges, calculations are performed at midspans, supports, and other unique

cross-sections. The same applies to box culverts, and although they may seem simpler

than bridges, box culverts are complex indeterminate structures with many critical

sections. Each slab and wall element contains three critical sections (two corners and

midspan). Furthermore, each critical section must be checked for bending, shear and axial

thrust and each of these conditions must be checked at the maximum and minimum live


48

load for a total lateral load case and a reduced lateral load case. This means that for a

culvert with N number of barrels, the number of load rating factors which must be

calculated is (9*N + 3)*3*2*2. For a single barrel culvert, 144 load rating factors must be

evaluated, and for a four barrel culvert, the number of load rating factors increases to 468.

Having calculated all these load rating factors for the critical sections, the lowest factor is

the one which controls the load rating for a culvert.

Beyond the sheer number of load rating factor evaluations, calculation of the

individual components in the load rating equation can be computationally difficult. The

capacity of each section in a reinforced concrete box culvert can be conveniently

established by interpreting the culvert members as doubly-reinforced, one-way slabs.

Demand modeling is more nuanced. The AASHTO bridge design standards (AASHTO,

2002; AASHTO, 2014) and the MBE (AASHTO, 2013) contain guidance on soil weights,

equivalent earth fluid weights and live load distributions. “In lieu of a more precise

analysis,” the default parameters are applied directly to a structural-frame model

(AASHTO, 2014). In fact, the 2013 Interim Revision to the MBE explicitly defines a

two-dimensional, direct-stiffness, structural-frame model as the preferred, conservative

load rating model (AASHTO, 2013).

Research has been done on many aspects of culvert inspection and management of

which load rating is one small part (Cahoon, et al., 2002; Salem, et al., 2012; Wissink, et

al., 2005). Still more work has been done related to culvert demands including the topics

of soil-structure interaction (Duane, et al., 1986; Gardner & Jeyapalan, 1982; Katona &

Vittes, 1982; Roschke & Davis, 1986), pressure distributions (Bennett, 2005; Kim &


49

Yoo, 2005; Tadros, et al., 1989; Yang, 1999) and live load distributions (Abdel-Karim, et

al., 1990; Kitane & McGrath, 2006; Tadros & Benak, 1989). These studies focus on

demand predictions as part of design and analysis, but have not considered the impact of

demands on load rating explicitly. This chapter seeks to build upon these contributions by

considering the conceptual intersection of culvert demand modeling and production

culvert load rating.

The state DOTS have interpreted, adjusted and applied the default AASHTO

parameters in different ways. A nationwide survey of state DOTs indicates that a great

deal of confusion surrounds load rating culverts (Lawson, et al., 2010). Many states

claimed to use AASHTO, but when asked if their analysis includes soil-structure

interaction, these same states offered a mixed response. Some states modify the

AASHTO parameters in an attempt to make the analysis correspond better with their

visual inspections. The majority of responding DOTs in the study appear to only replace

culverts based on hydraulic functionality, not structural load rating. Several states quoted

the MCEB Section 7.4.1 saying, “A concrete bridge need not be posted for restricted

loading when it has been carrying normal traffic for an appreciable amount of time and

shows no distress” (AASHTO, 2003). The repeatability of culvert load rating analyses is

a concern because there are simply too many poorly-defined or questioned parameters to

be confident that two engineers will calculate the same load rating for a single culvert

structure.

The Texas Department of Transportation (TxDOT) funded research for the

development and validation of a repeatable culvert load rating procedure that fully


50

synthesized and applied the available guidance from the various federal policy sources.

The Culvert Rating Guide (Lawson, et al., 2009) was the result. The Culvert Rating

Guide provides details concerning applicable AASHTO sources of policy guidance and

presents a synthesized culvert load rating flow chart including detailed guidance for

calculating section capacities, dead load demands and live load demands. The Culvert

Rating Guide specifies four tiers of analytical rigor associated with demand modeling:

Level 1 production-simplified, two-dimensional, structural-frame model; Level 2

production-simplified, two-dimensional, direct-stiffness, structural-frame model with soil

spring supports; Level 3 production-simplified, two-dimensional, linear-elastic finite-

element, soil-structure interaction model; and Level 4 research-intensive soil-structure

interaction models. The research report in support of the Culvert Rating Guide provides

in-depth details related to model selection and procedure validation (Lawson, et al.,

2010). This chapter follows the Culvert Rating Guide as an expression of the AASHTO

load rating policy relative to reinforced concrete box culverts. Only the Level 1, two-

dimensional, direct-stiffness, structural-frame model and the Level 3, two-dimensional,

linear-elastic finite-element soil-structure interaction model are evaluated in this chapter

(Figure 11). Full-scale live load tests were undertaken to evaluate the precision and

accuracy of these two demand models as one aspect of the load rating method validation

process.


51

(a) (b)

Figure 11. Modeling sophistication illustrations: (a) Level 1, two-dimensional, direct-stiffness, structural-frame model (Lawson, et al., 2009); (b) Level 3, two-dimensional, linear-elastic finite-element soil-structure interaction model

Method

Beginning with two, well-defined analytical models, a research method was

developed to evaluate the demand models. The research method included identifying and

instrumenting test culverts, measuring strains while loading the culverts with dump

trucks, and then converting the strains into moments comparable to moment predictions

from the demand models.

Field test program

Test culverts

Three in-service culverts were selected for live load testing. The sample of culverts

represents a range of design eras, barrel sizes and cover soil depths. Each culvert was dry

and the surrounding soil well drained. Figure 12 shows an image and the location of each

culvert. Table 1 summarizes the culvert parameters and soil investigation findings. The


52

research report presents additional detail concerning culvert designs, properties, location

and selection (Lawson, et al., 2010).

(b)

(c)

(a) (d)

Figure 12. Test culvert locations and culvert images: (a) Texas county map showing test culvert locations; (b) Swisher county; (c) Hale county; (d) Lubbock county


53

Table 1. Test culvert parameters

Location Swisher County, TX

Lubbock County, TX

Hale County, TX

Facility carried FM-1318 US-84 SH-194 Annual average daily traffic, AADT,

(vehicles/day) 240 7400 2900

Culvert Properties

Cover depth of soil 0.5m (1.5ft)

0.6m-1.2m (2ft-4ft)

1.1m (3.5ft)

No. of barrels 5 4 4 Barrel span 1.8m (6ft) 3m (10ft) 3m (10ft) Height 1.8m (6ft) 2.4m (8ft) 1.8m (6ft) Constructed year 1951 1963 1991 AASHTO default soil unit weight, γ

18.9kN/m3 (120pcf)

18.9kN/m3 (120pcf)

18.9kN/m3 (120pcf)

Specified reinforcing steel strength, Fy 227MPa (33ksi)

276MPa (40ksi)

414MPa (60ksi)

Measured concrete compressive strength, f’c

67MPa (9750psi)

41MPa (6000psi)

55MPa (8000psi)

Cover Soil Properties USCS soil classification sandy clay clayey sand fat clay USCS group symbol CL SC CH

Level 1 Two-Dimensional Structural-Frame Soil Properties

Total equivalent soil fluid unit weight

9.4kN/m3 (60pcf)

9.4kN/m3 (60pcf)

9.4kN/m3 (60pcf)

Reduced equivalent soil fluid unit weight

4.7kN/m3 (30pcf)

4.7kN/m3 (30pcf)

4.7kN/m3 (30pcf)

Level 3 Two-Dimensional, Linear-Elastic, Finite-Element, Soil-Structure Interaction Soil Properties

Soil modulus of elasticity, E 62MPa (9.0ksi)

83MPa (12.0ksi)

55MPa (8.0ksi)

Assumed soil Poisson’s ratio, ν 0.3 0.3 0.3


54

Instrumentation design

The instrumentation plan centered on the installation of 10.2cm (4in.) electrical resistance

strain gages at critical sections for each selected culvert. Gages were placed along a

single gage line running perpendicular to the culvert flow direction. Taking advantage of

the symmetrical nature of culverts, only one half of each culvert was gaged. Figure 13

shows a typical gage plan. Where possible, strain gages were placed on opposing faces of

the concrete slabs as shown in white on the figure. In other locations where this was not

possible, the black circles indicate critical sections with a single strain gage. Figure 13

also identifies the critical section labeling scheme used in the presentation of results.

Displacement gages were also used along the gage line to measure top and bottom

midspan deflections within the gaged culvert barrels.

Figure 13. Typical gage plan: Lubbock County culvert; white circles indicate gage pairs, black circles indicate single gages, open circles indicate no gages


55

Loading method

The load test design was limited to live load testing of in-service culverts in an effort

to minimize costs. The live load consisted of loaded 7.6m3 (10yd3) dump trucks weighing

approximately 222kN (50kips). Each truck had a front single axle and tandem rear axles.

The measured axle and wheel loads for the test trucks are shown in Table 2.

Table 2. Axle and wheel loads for test dump trucks Culvert Swisher County, TX Lubbock County, TX Hale County, TX

Front Axle (single) 55kN (12.3kips) 62kN (14.0kips) 51kN (11.5kips)

Rear Axles (tandem) 172kN (38.7kips) 178kN (40.0kips) 158kN (35.5kips)

Front Wheel 28kN (6.2kips) 31kN (7.0kips) 26kN (5.8kips)

Rear Wheels 43kN (9.7kips) 44kN (10.0kips) 40kN (8.9kips)

The loading design used three truck configurations, with the trucks traveling back and

forth across the culvert to create moment envelopes (Figure 14). Static measurements

were taken with the truck(s) stopped at 0.6m (2ft) intervals along the established gage

line. The goal was to measure the worst case maximum and minimum moment demands

for each critical section.


56

(a) (b)

(c) (d)

Figure 14. Live load configurations for the culvert load test: (a) One truck straddling gage line; (b) Wheel on gage line; (c) Two trucks straddling gage line; (d) Data acquisition and recording

Comparative analysis

Converting strains to moment

The load tests facilitated direct measurement of strains at the culvert critical sections;

whereas, the load rating equation requires demand loads at each critical section. Moment

demands are the primary loading for most reinforced concrete box culverts. Therefore,

the measured strains were transformed into moment demands. Converting the strain


57

profile into moment requires approximations of the elastic modulus and moment of

inertia for each section.

The concrete elastic modulus was calculated using the widely accepted estimate

based on the measured compressive strength from ACI 318 Section R8.5.1. The actual

concrete elastic modulus can vary from 80% to 120% of the estimated value (ACI

Committee 318, 2011). Throughout the analysis, this variance, ±20%, is presented as the

error for the measured moments.

The effective moment of inertia is a function of the cross-section thickness, amount of

reinforcing, the ratio of steel to concrete stiffness, the degree of cracking and the state of

stress in the section. Most of these parameters are unknown for an in-service culvert, and

incorporating them into an effective moment of inertia is complex. ACI 318 provides

simplified guidance based on the gross moment of inertia of the concrete section,

neglecting steel. Furthermore, ACI 318 Section R10.10.4.1 states that a stiffness

reduction factor of 0.875 was used to determine these values. From this guidance, Table 3

shows the EI stiffness parameters used to convert strain to moment.


58

Table 3. Moment of inertia for specific critical sections Section EI Crack Condition

Exterior Walls 0.35𝐼4𝐸60.875 Cracked Wall

Interior Walls 0.70𝐼4𝐸60.875 Uncracked Wall

Slabs 0.25𝐼4𝐸60.875 Cracked Flat Slab

Note: Ig = the gross moment of inertia for the concrete

section, neglecting reinforcing steel Ec = concrete elastic modulus

Strain gages were placed on the inside surface of the culvert walls and slabs. In some

locations, particularly the interior culvert walls, gages were placed on both sides. For

each strain measurement with only one gage at the critical section, the curvature was

calculated assuming no axial load. For each gage couple (inside and outside gage at the

same location) the curvature was calculated based on the difference between the

measured strain on the inside and outside face of the slab.

From these inputs, the moment envelope was calculated for each gaged critical

section. The maximum and minimum moment have an estimated ±20% error based

primarily on variations in the concrete modulus of elasticity.


59

Calculation of Level 1 predicted moments

AASHTO’s guidance provides loads which can be directly applied to a structural

model of a concrete culvert. In this way, AASHTO encourages the use of a Level 1 two-

dimensional, direct-stiffness, structural-frame model. State DOTs and researchers use

several programs and methods when performing a Level 1 analysis. Several DOTs have

developed their own design programs including Virginia (Latona, et al., 1973), Alabama

(Lakmazaheri & Edwards, 1996), Wyoming (Wyoming DOT, 2008) and Texas (TxDOT,

2003). In this chapter, TxDOT’s custom concrete box culvert analysis program, CULV5,

has been applied as a representative Level 1 structural-frame model (TxDOT, 2003).

The Level 1 structural-frame analysis requires AASHTO-provided soil unit weight

and lateral earth pressure input parameters as per Table 1. Furthermore, AASHTO

specifies two load cases: a total load case and a reduced lateral load case. The total load

case live load applies a 0.6m (2ft) equivalent surcharge on the exterior walls; the reduced

lateral case live load applies no lateral loads to the culvert.

The truck loads applied to the culvert are also an important Level 1 component.

CULV5 is relatively limited in this regard with only a select number of design loads. The

truck designation which most closely approximates the measured dump truck loads is the

HS25 truck. The CULV5 program automatically converts the wheel loads into distributed

loads. These distributed loads take into account load dissipation in the in-plane and out-

of-plane directions in the soil between the pavement surface and the top slab of the

culvert. They do not account for additional distribution of the wheel loads through the

soil into the walls and bottom slab of the culvert.


60

Calculation of Level 3 predicted moments

The Culvert Rating Guide identifies a Level 3 two-dimensional, linear-elastic finite-

element, soil-structure interaction model of greater analytical rigor compared to the

Level 1 structural-frame model. Several types of programs representative of a Level 3

model have been used in research. The most basic and production-simplified types of

programs are represented by two-dimensional, structural-frame programs which are

capable of in-plane, finite-element, plate modeling. RISA is of this type (RISA

Technologies, LLC, 2012).

Other general-use, two-dimensional, structural finite-element programs can also be

used for a Level 3 analysis. These use true finite-element methods (not frame methods) to

model both the soil and the structure. Model generation for such programs tends to be

more text-oriented and less graphically-oriented, therefore, they tend to be more difficult

to use. These are the programs most typically associated with the Level 3 soil-structure

modeling level.

Yet another class of programs capable of Level 3 soil-structure modeling are the soil-

structure specific programs such as the two-dimensional versions of Plaxis (Kitane &

McGrath, 2006), ABAQUS (Kim & Yoo, 2005; Kitane & McGrath, 2006) and CANDE

(Katona & McGrath, 2007; Katona & Vittes, 1982; Tadros & Benak, 1989; Tadros, et al.,

1989; Abdel-Karim, et al., 1990). Though these programs are typically used for non-

linear, soil-structural models, they can be used with simple, linear-elastic soil models.

When used in this way, they are a Level 3, soil-structure model.


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The most popular of these models for culvert analysis is the specially designed

CANDE software. CANDE was created in 1975 and has been updated continuously since

that time. CANDE’s strengths are primarily in construction and dead load calculations

using non-linear soil models. However, for load rating purposes, CANDE is not user-

friendly in that moving live loads must be added as individual load cases (Katona, 2015).

When CANDE is used in conjunction with a linear-elastic soil model, the differences

between RISA with linear-elastic finite-elements and CANDE become negligible.

Therefore, though CANDE has an established history of reliability and validity, for

production load rating, substantially similar results can be achieved using simpler

programs with graphical interfaces.

The bottom line is that though many program options are available for Level 3, soil-

structure modeling, all these programs will produce approximately the same load rating

when used within the context of a two-dimensional, linear-elastic, soil-structural model.

Because the first type is the easiest to use, RISA was selected as the representative

program for a Level 3 model.

The increase in modeling sophistication of a Level 3 analysis requires additional soil

parameters to model the soil-structure interaction. Rather than using estimated lateral

earth pressures, the Level 3 model uses a soil elastic modulus and Poisson’s ratio, both of

which are shown in Table 1.

The equations for live load distribution have been derived from AASHTO’s wheel

load distributions onto a slab. For the Level 3 model, RISA’s algorithm for handling of

moving loads was implemented to apply moving point loads (RISA Technologies, LLC,


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2012). These point loads are unit lengths of equivalent line loads for the out-of-plane

distribution in the soil between the pavement surface and the top slab of the culvert. The

distribution in the in-plane direction is handled by the soil finite-element mesh. The

implication is that in-plane distribution and soil-arching of the live load are applied to the

walls and bottom slab rather than a coarse live load surcharge as used in a Level 1

structural-frame analysis. Out-of-plane distribution of the wheel loads is not accounted

for below the top slab. The measured wheel loads and spacing (Table 2) were used, as

appropriate, for each culvert.

Findings and Discussion

Overall performance

Data collected from the load tests function as reference points for the evaluation of

the models. The first consideration is the precision of the model. For each culvert section

and bending direction, the ratio of predicted moment demand vs. the measured moment

demand becomes a normalized comparator. When a subset of similar ratios is considered,

the model precision can be evaluated. A simple indicator of precision is range; as the

difference between the maximum and minimum predicted vs. measured moment becomes

smaller, the model for that subset of data increases in precision. Another indicator of

precision is scatter or distribution; if the ratios are clustered together or show very tight

inner quartiles, the precision is high. Conversely, if the distribution is uniform and the

quartiles are widely spaced, the precision is low. If the precision is reasonable, then

accuracy can be considered.


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For the purpose of this chapter, accuracy is defined as proximity of the predicted to

the actual (measured) value. As the ratio of predicted moment demand vs. measured

moment demand approaches 1.0, the prediction increases in accuracy. A single data point

can be more or less accurate. By considering the median of a subset of data, the accuracy

for a certain subset of data can be evaluated. Furthermore, if the ratio of predicted vs.

measured moment is greater than 1.0 the model is conservative.

An expected finding is that both models will be conservative. Furthermore, the

increase in sophistication between the Level 1 structural-frame model and the Level 3

soil-structure model should result in an increase in modeling accuracy. The increase in

precision and accuracy is expected for every culvert section, but the degree of precision

and accuracy may not be uniform. Additionally, the measured moments will indicate that

certain sections and bending directions have bending moments which approach zero and

therefore are less significant than larger moments for other sections and bending

directions.

Moment diagrams

Relative to overall performance, Figure 15 shows the measured moment envelope as

per the load tests, and the predicted moment envelopes for both the Level 1 structural-

frame model and the Level 3 soil-structure model for each culvert. The critical section

labels correspond to critical section locations shown in Figure 13.


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(a) (b)

(c) (d)

Figure 15. Live load moment demand envelopes for each load test: (a) Swisher County culvert; 0.5m (1.5ft) cover depth; (b) Lubbock County culvert; 0.6m (2ft) cover depth; (c) Hale County culvert; 1.1m (3.5ft) cover depth; (d) Lubbock County culvert; 1.2m (4ft) cover depth

By inspection, it does appear that both models are conservative. For most sections,

the predicted moments in each bending direction are of greater magnitude than the

measured moments. The Level 1 structural-frame model appears to be more conservative


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and less accurate than the Level 3 soil-structure model as shown by the Level 3 moment

envelope located between the Level 1 structural-frame moment envelope and the

measured moment envelope. This supports the assumption that increases in modeling

sophistication increase the modeling accuracy.

Though all sections and bending directions must be considered in a complete load

rating, Figure 15 illustrates that not all sections and bending directions experience

significant live load bending moment. For example, negative bending moment the top

and bottom midspans are always very nearly zero. These low magnitude live load

demand cases will only govern the load rating when the dead load exceeds the capacity;

otherwise, the load rating factor will be very large (consider the effect of very small live

load on Eq. 1). Therefore, these low magnitude moments should not drastically skew an

understanding of the precision or accuracy of the model. It does not really matter if the

models can accurately predict zero load; instead, the models should accurately predict

meaningful load magnitudes.

Model performance by cover soil depth

Figure 16 presents the ratios of predicted live load moment to measured live load

moment for each critical section by model and cover soil depth. Both positive and

negative bending moments are shown for each critical section (symbol Í). Quartiles for

each set of data are also identified to illustrate distribution and range. The log scale on the

x-axis indicates the ratio of predicted vs. measured moment. This means that a quartile

range to the right side of the plot is actually much larger than a similarly sized quartile


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range toward the left side of the plot. Additionally, the gray circles indicate the live load

moment which is part of the controlling load rating factor for each culvert’s load rating.

Figure 16. Predicted vs. measured moment demand ratios by model and cover soil depth

Several observations can be made. First, for each culvert, the scatter decreases

substantially, by as much as one half, by considering the Level 3 soil-structure model

instead of the Level 1 structural-frame model. This decrease in range shows that the more

sophisticated Level 3 soil-structure model will predict culvert response with greater

precision than the Level 1 structural-frame model.


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A second observation is that the median predicted vs. measured demand ratio for each

culvert decreases by approximately 55% from the Level 1 structural-frame model to the

Level 3 soil-structure model. This illustrates the increased accuracy expected from an

increase in modeling sophistication.

Third, the data suggest that with increasing cover soil depth, the range and scatter

decrease in both the Level 1 and Level 3 models. In particular, the scatter in the 1.2m

(4ft) test is less than the 0.6m (2ft) test; both were performed on the same culvert

(Lubbock County). Comparisons with the 0.5m (1.5ft) and 0.6m (3.5ft) culverts are less

conclusive due to interactions between depth of fill and culvert design. The implication is

that live load demand predictions are more erratic for shallow-fill culverts.

Most importantly, the vast majority (greater than 96% for Level 1 and 91% for Level

3) of the moment ratios are conservative (greater than 1.0). This is expected for any

predictive engineering model. For load rating, however, the final answer is not directly

dependent on the performance of every location in the structure. Rather it the critical

section with the lowest load rating factor which governs the final culvert load rating.

Further, load rating is not only a function of live load, but also of dead load and capacity.

Those locations where live load moment is under-predicted do not necessarily control the

load rating for the culvert. Rather for these load tests, the controlling critical sections

(e.g., the gray circles in Figure 16) all depict predicted vs. measured moment ratios

greater than 1.0. In the case of load rating, the occasional under-prediction of the live

load in one critical section may not result in an under-predicted load rating. Here, the


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conservatism included in the dead load and culvert section capacity values is sufficient to

indicate a load rating based on conservative live load demand predictions.

Member performance

When discussing the precision and accuracy of production-simplified culvert demand

models, it is helpful to compare predicted vs. measured moment demands at similar

critical sections for all four load tests. Though the models do not show uniform accuracy

(as would be illustrated by high precision or no scatter) throughout a single culvert, they

may show a similar level of precision and accuracy for similar sections on all the test

culverts. Such comparisons are made by exploring the ratio of predicted vs. measured

moment for subsets of data, as per Figure 17. Figure 17 must be interpreted in light of

Figure 15, recalling that the precision and accuracy with which a model predicts low

moment is not of practical significance.


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(a) (b)

(c) (d)

Figure 17. Predicted vs. measured moment demand ratios by critical section type: (a) Top slab critical sections; (b) Bottom slab critical sections; (c) Interior wall critical sections; (d) Exterior wall critical sections

Top slab

The top slab of a reinforced box culvert structure typically runs continuously over all

the walls. As a continuous member, positive bending (tension on the inside face of the

culvert) at the midspan and negative bending (tension on the outside face of the culvert)


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at the corners are of significant magnitude. However, negative moments at the midspan

and positive moments at the corners are relatively low and approach zero.

Figure 17(a) shows the ratio of predicted vs. measured moments for each model

(Level 1 structural-frame and Level 3 soil-structure), critical section type (corner or

midspan) and bending direction (positive or negative) for every top slab tested in this

study. For the large magnitude moments (positive bending at the midspans and negative

bending at the corners), the scatter is relatively low. The Level 1 structural-frame model

moment ratios vary from 1.7 to 19.6 and have very tight distributions particularly for the

top slab. The Level 3 soil-structure model moment ratios vary from 0.8 to 15.9 but the

distribution is roughly equivalent for both the midspans and corners. Though the

precision is not much increased, the accuracy (based on the median values) nearly

doubles in the Level 3 soil-structure model over the Level 1 structural-frame model.

For the low magnitude bending moments in the top slab, the precision is quite poor

for the Level 1 structural-frame model (0.9 to 69.9). Given such low precision, it is

unsurprising that the accuracy is also low. The Level 3 soil-structure model has roughly

the same range and accuracy in the low bending direction as the high bending direction.

For the low magnitude subset of data, the increase in modeling sophistication results in a

clear increase in modeling precision and accuracy. However, the ability of a model to

predict low bending moment magnitudes is not of practical interest when evaluating

production load rating demand models.


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Bottom slab

The bottom slab performs much like the top slab. The slab is continuous, and the

larger magnitude moments occur in the positive bending direction for the midspans and

negative bending direction in the corners. However, in contrast to the top slab, all

measured moments in the bottom slab are very low magnitude (see Figure 15). This is

consistent with the three-dimensional nature of the actual culvert. Wheel loads from

traffic are distributed through the soil mass into the top slab of the culvert. The load is

further distributed along the length of the culvert (out-of-plane relative to the model)

before reaction pressures develop in the bottom slab. For both Level 1 and Level 3, the

two-dimensional nature of the models does not additionally distribute the load in the out-

of-plane direction beyond the top slab.

Figure 17(b) shows the ratio of predicted vs. measured moment for all bottom slab

sections. For the Level 1 structural-frame model, there is wide range for every critical

section and every bending direction. The overall range extends from 0.9 to 210.6 times

over-predicted. Even for the most precise critical sections, negative bending in the

corners, the Level 1 structural-frame model moment ratios vary from 5.9 to 42.8. The

Level 3 soil-structure model is more precise in that the ratios range from 0.1 to 44.5, and

therefore are also slightly more accurate. The negative bending moments in the midspans

are all less than 1.0 (un-conservative) in the Level 3 soil-structure model; however, the

actual moment in this direction is very nearly zero. Though bottom slab moments are not

well predicted by either model, it is in this part of the culvert that the improvement in


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precision and accuracy by using the Level 3 soil-structure model instead of the Level 1

structural-frame model is most pronounced.

The bottom slab comparisons of predicted vs. measured moments illustrates that the

Level 1, structural-frame model over-predicts the moment by a greater degree and with

greater scatter than the Level 3, soil-structure model. More noticeably, both models are

far less accurate or precise in the bottom slab than they are in the top slab. However, the

lower magnitude moments in the bottom slab mean that the bottom slab is less likely to

control a load rating.

Interior walls

The interior walls are unique among culvert elements tested in this study because the

interior walls experience no directly-applied load. Rather, rotation in the top and bottom

slabs induces bending in the wall ends. The bending moment in interior wall sections is

therefore much smaller than bending moment in the top and bottom slabs. Because there

is no directly applied load, the bending direction is not critical for the interior walls.

Figure 17(c) shows the ratio of predicted to measured live load demand for the top,

middle and bottom interior wall sections for both the Level 1 and Level 3 analysis. The

top corners of the interior walls are reasonably well modeled by both Level 1 and Level 3

models. For both cases, the ratio between predicted and measured moments varies from

slightly less than 1.0 (under-predicted) to around 4.0 (over-predicted). The median

predicted moment for both models is around 1.8 times the measured moment. Therefore,

the increase in modeling sophistication does not noticeably increase either the accuracy

or precision of the moment prediction for this part of the culvert.


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The midspans and bottom corners in the interior walls are more typical with the

Level 1 structural-frame model having less precision or accuracy than the Level 3 soil-

structure model. The actual moment in these sections is relatively small, therefore any

lack of accuracy in these predictions is unlikely to greatly influence the load rating for the

whole structure.

Exterior walls

The main difference between moments in the exterior walls vs. moments in the

interior walls is the importance of bending direction. For interior walls, the bending is of

the same magnitude in both the positive and negative bending directions due to the lack

of direct load applied to the wall. The exterior wall on the other hand has external soil

load, therefore, the moment is not symmetric. Figure 17(d) illustrates this point by

showing the ratios of predicted to measured live load moment.

For the top corner in the negative bending direction (the direction that matters), both

models are relatively precise and accurate. Where the Level 1 structural-frame model is

precise and accurate, the Level 3 soil-structure model cannot offer much improvement. In

the positive bending direction, the moment predictions for the top corner are less precise,

but the net effect on the final load rating is insignificant since the positive bending

magnitude is so low. For the bottom corner, the Level 3 soil-structure model shows a

moderate degree of precision and accuracy in both bending directions and is slightly

more precise than the Level 1 structural-frame model.

The predicted vs. measured moment ratios in the exterior wall midspans show a great

degree of scatter in both models. Due to dead load, the positive bending case generally


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sees more cumulative (dead and live) moment. In the positive bending direction, the

measured live load moments in the exterior wall midpans are relatively low, therefore,

the predicted moment precision is very poor. For the Level 1 soil-structure model, the

ratios range from 9.0 to 127.8. The Level 3 soil-structure model predicts much more

precisely and accurately, though the ratio range between 1.6 and 46.7 is not excellent.

The attenuation of wheel loads with depth may be affecting the accuracy of the

predictions in much the same way it impacts the relative accuracy between the top and

bottom slabs. As a microcosm of the whole culvert, the exterior walls see great variations

in modeling precision and accuracy between critical sections and bending directions and

mixed degrees of improvement from the Level 3 soil-structure model as opposed to the

Level 1 structural-frame model.

Member performance summary

Figure 18 summarizes the findings from the analysis of modeling accuracy and

precision in each part of the culvert. Data are shown only for the significant bending

direction for each critical section. The y-axis shows the ratio of predicted moment vs.

measured moment in the normal scale. Along this axis, the quartiles for the ratios

associated with the primary bending direction are shown. The x-axis identifies the critical

section classes.


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Figure 18. Evaluation modeling accuracy for each critical section type in the primary bending direction

Figure 18 supports some useful observations. The Level 1 model offers a large level

of conservatism throughout (predicted/measured moment rations greater than 1.0), but the

bottom slab midspan and the exterior wall midspan show excessively large scatter. The

scatter is also proportionally larger for these same sections in the Level 3 analysis even

though the ratios are much improved over the Level 1 analysis. In contrast, the top

exterior wall corner is well predicted by both a Level 1 and Level 3 model. If the top

exterior wall corner controls the load rating, then an increase in modeling sophistication


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is unlikely to achieve an increase in precision or accuracy. The other critical sections

show degrees of improvement with the shift from Level 1 to Level 3.

The clearest observation from Figure 18 is that the precision and accuracy for an

individual analysis method is not uniform for all sections in their most significant

bending directions. Interestingly, only the degree of accuracy and precision is dependent

on model sophistication. Trends for the variation with critical section appear the same for

each model: low accuracy and precision in the bottom slab midspans and exterior wall

midspans, high accuracy and precision in the top slabs and top wall corners.

Other Observations

Throughout this chapter, the findings have been expressed in terms of live load

demand model precision (degree of scatter) and accuracy (proximity of predicted to

measured demands). However, it should be emphasized that live load demands tell only

part of the load rating story. The load rating for a particular culvert structure is intended

to provide a quantitative measure of the condition of the structure. DOTs seldom witness

structural failures of older, in-service reinforced concrete box culverts, yet when these

culverts are load-rated using a Level 1 type demand model, it is not uncommon for the

load rating results to show a need for load-posting or culvert replacement.

The apparent disconnect between observed reinforced concrete box culvert

performance (from a visual inspection) and reinforced concrete box culvert load rating

(from analysis) may be due to over-conservatism stemming from a lack of modeling

precision and accuracy. Alternatively, such a disconnect may be due to limitations of


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visual observations to detect structural damage. For example, cracks forming on the soil

side of the culvert slabs might not be visible during an inspection.

What can meaningfully be extracted from this research is a culvert load rating method

that is repeatable and offers increased precision and accuracy by taking advantage of

increasing sophistication in the production demand models selected for load rating. More

work remains to fully establish correspondence between the actual structural performance

and all aspects of the load rating process. Variations in culvert input parameters, material

strengths, structural damage, backfill conditions, loading history, capacity prediction

models and soil-structure interaction models contribute to the validity of the demand

model and load rating.

Conclusions

In this evaluation of production-simplified culvert load rating models, predicted live

load demand moments have been compared to measured live load demand moments in

order to comment on the precision and accuracy of a Level 1, two-dimensional, direct-

stiffness, structural-frame model and a Level 3, two-dimensional, linear-elastic finite-

element, soil-structure interaction model. The following conclusions are supported with

data from live load testing of three in-service box culverts under four cover soil depths.

1. The Level 1 structural-frame model and the Level 3 soil-structure model provide

conservative predictions for live load demands used in load rating calculations.

Reasonable confidence can be placed in the conservatism of the demand

predictions, particularly when considering the complex and interdependent nature

of load rating as a function of live load, dead load and capacity.


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2. The increase in modeling sophistication from Level 1 to Level 3 improves both the

precision and accuracy of the demand predictions, i.e. the Level 3 soil-structure

model predicts demands better than the Level 1 structural-frame model.

3. Careful analysis of predicted and measured moment demands by culvert section

and bending direction reveals that all culvert sections do not hold the same practical

significance for load rating. When the moment demand for a section and bending

direction approaches zero, the corresponding load rating factor will not control the

load rating; therefore, the relative accuracy of this predicted demand does not

impact the evaluation of the model.

4. The precision and accuracy of both the Level 1 structural-frame model and the

Level 3 soil-structure model vary relative to culvert section (top slab midspans,

bottom slab corners, etc.). Predicted moment demands for top slabs and corners are

far more precise and accurate than predicted moment demands for bottom slabs and

corners.

5. For the top exterior wall corners and top midspans of culverts, the precision and

accuracy of Level 1 structural-frame demand predictions are high and offer little

margin for improvement through reanalysis using the Level 3 soil-structure model.

Conversely, sections where the Level 1 structural-frame modeling precision is low,

significant potential for improvement through a Level 3 soil-structure reanalysis

exists.

Two-dimensional, structural-frame models can be used to quickly predict safe culvert

load ratings. However, when the simplest model identifies a load rating value which calls


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for load-posting or replacement of the culvert, the extra effort needed to identify

parameters and create a two-dimensional, linear-elastic, finite-element, soil-structure

model will likely increase the precision and accuracy of the demand predictions while

still maintaining an appropriate degree of conservatism for the load rating. In states with

thousands of in-service reinforced concrete box culverts, older, adequately-performing

culverts (based on inspection) may be shown sufficient (based on load rating) through

increased, production-simplified, analytical effort at a much lower cost than load-posting

or replacement.

Acknowledgements

The authors thank the Texas Department of Transportation for funding this work as

part of the research project 0-5849 Evaluating Existing Culverts for Load Capacity

Allowing for Soil Structure Interaction.


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CHAPTER 4

PRODUCTION-SIMPLIFIED LIVE LOAD ATTENUATION

METHOD

Note: Submitted to Standing Committee on Culverts and Hydraulic Structures of the Transportation

Research Board for presentation and publication as:

Wood, T. A., Lawson, W. D., Surles, J. G., Jayawickrama, P. J., & Seo, H. (2015). Improved Load

Rating of Reinforced Concrete Box Culverts through Depth-Calibrated Live Load

Attenuation.

Chapter Summary

This chapter describes an out-of-plane live load attenuation method for the load rating

of reinforced concrete box culverts using a production-simplified, two-dimensional,

linear-elastic, finite-element, soil-structure interaction model. The new method, called the

depth-calibrated method, attenuates out-of-plane live load to the critical section depths in

a culvert. The method improves current practice by increasing the accuracy and precision

of live load demand predictions, particularly in culvert walls and bottom slabs. Use of the

depth-calibrated method helps close the disconnect between calculated load rating and

observed structural performance by more accurately predicting both the location of the

weakest critical section and the live load magnitude. The depth-calibrated method also

moves the model toward more uniform accuracy and precision across all critical sections.

This chapter illustrates the effectiveness of the depth-calibrated method by comparing

predicted live load moments to measured live load moments obtained from published


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datasets from full-scale culvert load tests. A load rating case study shows the improved

alignment between load rating and observed performance.

Introduction and Background

This chapter describes a depth-calibrated live load attenuation method that improves

the accuracy and precision of demand predictions for culvert load rating by attenuating

out-of-plane live load to each critical section depth in a culvert. The depth-calibrated

method is specific to cast-in-place (CIP), reinforced concrete (RC) box culverts, and

functions within the following context. First, culvert load rating requires a structural

analysis model, and a valid model should predict both the form and the magnitude of

structural response. Alignment between predicted and measured performance is a

fundamental requirement. Second, engineers typically use production-simplified demand

models for routine culvert load rating as opposed to research-intensive models in an

attempt to balance work effort with analysis sophistication. Third, the state of practice for

culvert load rating with production-simplified models focuses on live load induced

pressures on the top slab. However, load rating of RC box culverts requires evaluation of

all sections of the culvert structure, not just the top slab. Load rating benefits from a focus

on live load induced structural response predicted throughout a culvert.

Disconnect Between Observed Structural Performance and

Calculated Load Ratings

Federal law requires state DOTs to conform to the National Bridge Inspection

Standards (NBIS) for “the proper safety inspection and evaluation of all highway


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bridges” (Bridges, Structures, and Hydraulics, 2009). By NBIS definition, “bridge”

includes RC box culverts with a total span of 6m (20ft) or greater, of which thousands are

in service in the United States. Further, the NBIS incorporates the AASHTO Manual for

Bridge Evaluation (MBE) by reference (AASHTO, 2013). The MBE outlines a system of

documentation, field inspection, load rating, and field-testing that together satisfy the

requirements of the NBIS.

Per the NBIS, typically, routine bridge inspections are performed every 24 months

(Bridges, Structures, and Hydraulics, 2009). A qualified engineer visits and carefully

examines the culvert structure, notes any damage, and assigns condition ratings to the

culvert and its elements (AASHTO, 2013). Culvert elements include top slabs, bottom

slabs and walls. Typically, field inspections show that in-service RC box culverts perform

very well. For example, the Texas Department of Transportation (TxDOT) maintains an

inventory of 11,000 pre-1980 bridge-class culverts, and these structures show an average

overall condition rating of 7 out of 9, which recognizes “light” to “insignificant” damage

“not requiring corrective action” (FHWA, 1995). Structural condition ratings greater than

4 to 5 are typically adequate so as not to require load posting, replacement, or retrofit.

Good structural performance per routine bridge inspections is expected and is

consistent with NBIS goals and objectives. However, highway officials at both the state

and federal levels have noted a disconnect between field inspection results and calculated

load rating values (NCHRP, 2013). The typical case is that an older, in-service RC box

culvert shows little structural damage, but the calculated load rating for the structure

indicates that the culvert does not have adequate capacity for an HS20 truck and would


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require load posting or possibly replacement. The significant problem is that “overly

conservative rating procedures result in expensive replacements or upgrades, while

unconservative rating procedures could result in future highway load limitations,

premature deterioration, and even sudden failures” (NCHRP, 2013). Sponsored research

projects (NCHRP 15-54, 2015; Lawson, et al., 2010; Han, et al., 2013; Orton, et al.,

2013) have sought to overcome this disconnect within the framework of existing policy.

The live load attenuation method introduced in this chapter helps close the disconnect by

improving the accuracy and precision of the demand model.

Load Rating with Production-Simplified Demand Models

Culvert load rating is a component of the NBIS and involves numerical calculations

to determine the safe load carrying capacity of the culvert structure, whether specific

legal or overweight vehicles can safely cross the culvert, and the level of posting required

if needed. The MBE provides three methods for load rating: load and resistance factor

rating (LRFR), load factor rating (LFR), and allowable stress rating (ASR). For this

chapter, the LFR rating factor equation will suffice to summarize the principles of load

rating. Equation 1 shows the rating factor equation.

The rating factor is essentially a live load factor of safety for a particular section in a

structure. If the rating factor is greater than 1.0, the section can carry the applied live load

at that section; if less than 1.0, the section does not have adequate capacity. The LFR

rating factor equation accommodates two rating levels, the inventory rating level (IR) and

operating rating level (OR). The IR represents the design capacity of the structure given

its current condition. The OR represents the maximum load carrying capacity of the


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structure. Repeated loading at the OR level is expected to cause damage to the structure.

The rating for the section is calculated by multiplying the rating factor by the nominal

tonnage of the load rating truck. The typical live load for the LFR method is the HS20

truck, nominally, a 20 ton load rating truck. The lowest rating from all sections governs

the load rating for the structure. (AASHTO, 2013)

The MBE explicitly identifies those critical sections on an RC box culvert where

rating factors must be calculated. Figure 19(a) shows an in-service culvert. Figure 19(b)

illustrates critical sections where demand moments, shears, or thrusts would maximize on

a unique cross section of a culvert element; i.e. corners and midspans. Unique rating

factors must be calculated for all critical sections, demand types, bending directions, and

load cases. For example, a typical 4-span RC box culvert can have as many as 468 rating

factor calculations; the lowest one rating factor will govern the load rating. In this way,

the load rating not only indicates the safe carrying live load capacity of the culvert, but

also the location of the weakest section. Ideally, calculated load rating results will

corroborate field performance by correctly predicting the weakest section where damage

would first occur.


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(a)

(b)

Figure 19. (a) A five-span reinforced concrete box culvert in Swisher Co., TX; (b)

critical section schematic

Additionally, the MBE recommends the use of production-simplified demand models

to achieve “consistent and repeatable” load ratings (AASHTO, 2013). The alternative to

production-simplified models is research-intensive models that typically use finite

element analysis with non-linear constitutive models for soil and concrete. Research-

intensive models promise more accurate and precise demand predictions, but the

additional complexity makes them onerous for routine load rating.


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Live Load Attenuation, Past and Present

The top slabs of RC box culverts have historically been of primary concern for

culvert load rating. From the early days of culvert design and analysis, engineers

recognized that the culvert top slab – especially for structures with shallow cover soil –

directly receives the most intense loading (Spangler, et al., 1926). Therefore, good

performance in the top slab became an obvious structural requirement. The general

approach to culvert design and analysis has been to apply loads from soil and vehicles to

a structure. This has naturally led to a research emphasis on live load induced pressures

on the culvert structure, especially the top slab (James & Brown, 1987; Tadros & Benak,

1989). Generally, top slab pressures are estimated by uniformly distributing wheel loads

over a rectangular area calculated using a variation of the 60° or 2-to-1 rule for discrete

surface loading on soil. Each side of the distributed rectangle is referred to as a live load

patch width (AASHTO, 2014).

AASHTO policy dictates how a production-simplified model should handle live load

attenuation. The LRFD Bridge Design Specifications (hereafter, LRFD Specifications)

states that “live load shall be distributed to the top slabs” of culverts (AASHTO, 2014).

Further, the LRFD Specifications define live load patch widths both parallel to the culvert

span (in-plane attenuation) and transverse to the culvert span (out-of-plane attenuation).

Uniform pressure on the bottom slab balances vertical loads applied to the top slab

(AASHTO, 2013). The live load patch widths, both parallel and transverse to the culvert

span, are a function of the distance from the ground surface to the top slab, called cover

soil depth. The live load distribution attenuates the live load induced pressure to create a


87

calibrated culvert response in the top slab. The MBE recommends that this live load

pressure be applied directly to a production-simplified demand model, a unit-width, two-

dimensional, linear-elastic, direct-stiffness, structural-frame demand model.

However, rather than a structural-frame model, the enhanced production-simplified

model used to predict demands in this chapter is a two-dimensional, linear-elastic, finite-

element, soil-structure interaction model. This soil-structure interaction (SSI) model is

repeatable and production-simplified due to clear documentation in Texas Department of

Transportation (TxDOT) Culvert Rating Guide (Lawson, et al., 2009). Furthermore,

published research has illustrated a substantial increase in accuracy and precision

achieved by this SSI model over a structural-frame model (Wood, et al., 2015).

The SSI model treats the soil-culvert system as a whole. As with the structural-frame

model, the in-plane live load distribution runs parallel to culvert span. However, rather

than deriving the in-plane distribution that the direct-stiffness model applies to the top

slab, the SSI model directly handles the in-plane loads applied at the ground surface,

through the cover soil, to the top slab, and on around the entire culvert structure.

Figure 20(a) illustrates this soil-structure in-plane live load distribution. The model itself

responsively predicts demands without the need to calculate intermediate pressures

induced by live load. This allows for a subtle but critical shift in approaching the live

load distribution to a culvert: load rating benefits from calibrated structural response, not

calibrated live load pressures.


88

(a)

(b)

Notes: wt = tire patch width

ww = live load patch width at top slab depth per policy (AASHTO, 2014) wtop = live load patch width at top slab critical section depth wwall top = live load patch width at wall top corner critical section depth wwall mid = live load patch width at wall midspan critical section depth wwall bot = live load patch width at wall bottom corner critical section depth wbot = live load patch width at bottom slab critical section depth

Figure 20. (a) production-simplified, two-dimensional, linear elastic, finite element, soil-structure interaction model for in-plane live load distribution for a two span reinforced concrete box culvert in Sarpy Co., NE; (b) estimated out-of-plane live load distribution

The SSI model does require out-of-plane live load distribution such as that shown in

Figure 20(b). The out-of-plane load distribution for the SSI model is the same out-of-

culvert model

wheel loads

production-simplified, 2D, LE, FE, soil model

ground surface

culvert top slab

transverse live load distribution

w

w

w

w

w

w

w

top

wall top

wall mid

wall bot

bot

w t cover soil depth

box culvert height

culvert bottom slab

wheel load

In-Plane Distribution

Out-Of-Plane Distribution


89

plane distribution prescribed by the LRFD Specifications. Several different out-of-plane

distributions have been developed to estimate live load attenuation to the top slab

including the elastic distribution (Poulos & Davis, 1991; Katona, 2015), the AASHTO

Standard Specification for Highway Bridges (SSHB) distribution (AASHTO, 2002), and a

family of LRFD Specifications distributions (AASHTO, 2012; AASHTO, 2014; Han, et

al., 2013). A 2D model fundamentally assumes that the conditions modeled in-plane

extend infinitely in the out-of-plane direction. Each live load distribution was developed

to estimate the infinite strip or line load that produces the same live load pressure at the

top slab as the discrete wheel loading does in an actual culvert.

Live Load Attenuation Methods

Current “Top-Slab-Calibrated” Live Load Attenuation Method

The current live load attenuation method is intended to estimate live load induced soil

pressures on the top slab of the culvert. For the out-of-plane loads (transverse to the

culvert span), the LRFD Specifications attenuate live load by dividing wheel loads by a

live load patch width at the depth of the top slab. The live load patch width, ww in Figure

20(b), is a function of cover soil depth. This effectively calibrates the live load

attenuation such that the SSI model will accurately predict the live load induced pressure

on the top slab. Therefore, this chapter refers to this live load attenuation method as the

“top-slab-calibrated method.”

However, the top-slab-calibrated method “tends to produce conservative force

demands, particularly in the bottom slab” (AASHTO, 2013). Any 2D model assumes that


90

the same unit width on the bottom slab is all that will resist the loading to the top slab.

But in a 3D loading, the live load continues to be distributed through the top slab, the

walls, and the bottom slab. Once the load has reached the bottom slab, the live load has

attenuated over a much longer out-of-plane length. Because the top-slab-calibrated

method does not consider this behavior, predictions tends to be more accurate and precise

in the top slab than in the bottom slab. Researchers have considered the bottom slab, but

the conservatism in the bottom slab is desirable for design (McGrath, et al., 2005). This

adversely affects the load rating analysis consideration of every critical section. The top-

slab-calibrated method is adequate for the top slab, but a new method should reduce

excess conservatism in the bottom slab and walls.

New “Depth-Calibrated” Live Load Attenuation Method

The new live load attenuation method is intended to estimate the live load induced

structural response at critical section locations. The live load attenuation method

introduced by this chapter considers additional out-of-plane attenuation with depth. As

has been stated, the out-of-plane distribution should provide an equivalent infinite strip or

line load that induces the same response in a 2D model as the wheel load would on an

actual culvert. The out-of-plane distribution is a function of the critical section depth, the

distance from the ground surface to the center of the critical section. Therefore, each

critical section has its own live load patch width. This chapter refers to the new live load

attenuation method as the depth-calibrated method.

Figure 20(b) shows five live load patch widths at unique critical section depths for a

RC box culvert. Rather than one live load patch width, ww, based on cover soil depth for


91

the whole culvert as in the top-slab-calibrated method, each critical section has its own

live load patch width based on critical section depth: wtop for top slab critical sections,

wwall top, wwall mid, and wwall bot for wall critical sections, and wbot for bottom slab critical

sections. To estimate the appropriate live load patch widths for each critical section

depth, a live load distribution is needed that attenuates the live load through the soil

above the culvert, along the top slab, through the wall-soil system, and along the bottom

slab. This chapter has not attempted to develop this live load distribution. Rather, the live

load distribution used in this chapter assumes, as a first-order approximation, that the

culvert-soil system is at least as efficient as the soil above the culvert at distributing the

live load out-of-plane. Therefore, the LRFD Specifications distribution has been modified

to consider critical section depth rather than cover soil depth. In the top slab, the

difference between the top-slab-calibrated method and the depth-calibrated method is

minimal. But, as critical section depth increases, the depth-calibrated method continues to

attenuate live load in the out-of-plane direction.

Measured Moment Data

Data Sources

This chapter evaluates predicted live load moment response using measured live load

moments from two published studies of full-scale culvert load tests. Researchers at Texas

Tech University instrumented three culverts and load tested them under a total of four

cover soil depths (Lawson, et al., 2010; Wood, et al., 2015). Researchers at University of

Nebraska – Lincoln instrumented a single culvert and load tested this culvert under seven


92

cover soil depths (Tadros & Benak, 1989; Abdel-Karim, et al., 1993). Both studies

instrumented each culvert with strain gages at critical sections. Static truck loads were

placed over the top of the culvert to induce worst-case structural responses, and both

studies reported truck axle loads and wheel spacings. Live load induced moments were

back-calculated from measured strains using ACI recommended effective moment of

inertia calculations for cracked members. Each load test produced a measured moment

envelope by critical section. The extreme values from the measured live load envelopes

are referred to as measured moments. In total, the 11 load tests on four physical culverts

provided 241 measured moments. Of the 241 measured moments, 169 measured

moments were meaningful, that is, significantly different from zero.

Both research studies included geotechnical site investigations. The site investigations

identified the USCS soil classification and group symbol for the surrounding soil (ASTM

Standard D2487-11, 2011). The Texas Tech University study obtained soil stiffness

values using a falling-weight deflectometer (ASTM Standard D4694-09, 2009). Soil

stiffness for the University of Nebraska – Lincoln culvert was assigned using published

soil type / stiffness correlations (Lawson, et al., 2009). Table 4 summarizes the project

data from these published studies including test locations, culvert dimensions, cover soil

depths, soil types, soil stiffness values, truckloads, and the number of measured moments

from each test.


93

Table 4. Project data for measured live load moments from field-tested culverts in Texas (Lawson, et al., 2010; Wood, et al., 2015) and Nebraska (Tadros & Benak, 1989; Abdel-Karim, et al., 1993)

location num

ber

of sp

ans

box

span

, m (f

t)

culv

ert h

eigh

t, m

(ft)

cons

truc

ted

year

cove

r so

il de

pth,

m (f

t)

USC

S gr

oup

sym

bol a

soil

mod

ulus

of

elas

ticity

, E, M

Pa (k

si)

live load mea

sure

d m

omen

ts

Swisher Co., TX 5 1.8 (6)

1.8 (6) 1951 0.5 (1.5) CL 62 b

(9) 2, 227kN (51 =kip),

three axle trucks 26

Lubbock Co., TX 4 3 (10)

2.4 (8) 1963 0.6 (2.0) SC 83 b

(12) 2, 240kN (54 =kip),

three axle trucks 28

“ “ “ “ “ 1.2 (4.0) “ “ “ 28

Hale Co., TX 4 3 (10)

1.8 (6) 1991 1.1

(3.5) CH 55 b

(8) 2, 209kN (47kip), three axle trucks 28

Sarpy Co., NE 2 3.7 (12)

3.7 (12) 1987 0.0 (0.0) CL 138 c

(20) 1, 120kN (27kip),

two axle truck 9

“ “ “ “ “ 0.6 (2.0) “ “ “ 9 “ “ “ “ “ 1.1 (3.5) “ “ “ 8 “ “ “ “ “ 1.8 (6.8) “ “ “ 8 “ “ “ “ “ 2.4 (8.0) “ “ “ 8 “ “ “ “ “ 3.0 (10.0) “ “ “ 8 “ “ “ “ “ 3.7 (12.0) “ “ “ 9

Notes: a (ASTM Standard D2487-11, 2011)

b composite soil stiffness from falling weight deflectometer test (ASTM Standard D4694-09, 2009) c assigned soil stiffness based on published correlation with soil classification (Lawson, et al., 2009)

Predicted Moment Calculations

The SSI model was used to predict live load moment envelopes for each load test

identified in Table 4. All culverts were modeled for demand calculations using concrete

stiffness calculated from measured strengths, f’c, and gross section properties. The

models used soil stiffness values shown in Table 4 for the soil mass surrounding each

culvert. The moving live load feature in RISA-3D applied actual truck wheel loads (RISA

Technologies, LLC, 2012). The LRFD Specification out-of-plane distribution accounted


94

for overlapping wheel loads at greater depth based on the actual number of trucks applied

to the culvert and the test truck wheel and axle spacings. Additional detail on load rating-

centric modeling using SSI models can be found in TxDOT policy (Lawson, et al., 2009).

Typical Moment Envelopes

The measured moment envelope, the top-slab-calibrated predicted moment envelope,

and the depth-calibrated predicted moment envelope show similar trends for all 11 load

tests. For the purpose of illustration, the moment envelopes for one load test are provided

in this chapter. The Hale Co., TX culvert moment envelope was selected because the

culvert was tested under moderate fill, suffered no gage failures, and had a full set of

measured midspan moments including the interior walls. The Hale Co., TX culvert is

representative of the moment envelopes for all 11 culvert load tests. Similar figures for

the other 10 culvert / cover soil depth test combinations may be found in Appendix B.

Figure 21 shows moment envelopes for the Hale Co, TX culvert, arranged by culvert

element. For this chapter, positive bending induces tensile stress on the inside of the

culvert element. Negative bending induces tensile stress on the outside, typically the soil

side, of the culvert element.

Several observations can be made about both the top-slab-calibrated and depth-

calibrated methods. First, predicted moment envelopes for both methods follow the trend

of the measured moments; that is, the predicted moment magnitude is large where the

measured moment is large and small where the measured moment is small. Second, the

predicted moments appear to be generally conservative, i.e. the predicted moment

envelopes fall mostly outside the measured envelope. In some few cases, such as negative


95

bending at the exterior wall top corner, the model under-predicts the moment. This may

be due to variability in the test data or error in the predictive model. However, this

behavior is rare, and the single, weakest critical section in a culvert governs the load

rating. Therefore, the occasional under-prediction typically will not influence the load

rating.

Figure 21. Typical moment envelopes for the 4 span, Hale County, TX culvert. See

Figure 19(b) for critical section locations

Of greater interest is the comparison of the two predictive methods by culvert

element. In the top slab (leftmost moment envelope in Figure 21), the top-slab-calibrated

and the depth-calibrated methods generate very similar live load envelopes. This is

consistent with the expected behavior and can be illustrated by considering the live load

patch widths in Figure 20(b). The depth-calibrated live load patch width, wtop, is only

marginally longer (slightly greater attenuation) than the top-slab-calibrated live load

topslab ext.wall int.wall center_wall bottomslab

-13.5

-9

-4.5

0

4.5

9

13.5

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

culvertelement

mom

ent(kN

-m/m

)

mom

ent(k-ft/

ft)

criticalsection

measuredtop-slab-calibrateddepth-calibrated


96

patch width, ww. Because the live load path widths are so close, the predicted moments in

the top slab are very similar for both the top-slab-calibrated and depth-calibrated

methods.

In contrast, the bottom slab (rightmost moment envelope in Figure 21) most clearly

illustrates the improvement provided by the depth-calibrated method. At the bottom slab

interior corners, the negative bending moments predicted by top-slab-calibrated method

are of the greatest magnitude for the whole structure. This does not match expected or

measured behavior. The depth-calibrated method reduces the predicted moment by half in

the bottom slab interior corners. This matches expected behavior; the depth-calibrated

live load patch width, wbot, is much larger (significantly greater attenuation) than the top-

slab-calibrated live load patch width, ww. The greatest improvement in predictive

accuracy provided by the depth-calibrated method is in the bottom slab.

Each of the walls (the middle three moment envelopes in Figure 21) serves as a case

study of the difference between the top-slab-calibrated and depth-calibrated methods. At

the wall top corners, the difference between the two methods is slight, much like the top

slab. The wall midspans show a divergence between the predictive methods; the depth-

calibrated method predicts live load moment about half way between the top-slab-

calibrated moments and the measured moments. The wall bottom corners show

performance similar to the bottom slab with marked improvement in predicted moments

provided by the depth-calibrated method.

Figure 21 also clearly illustrates the difference between meaningful moments and

practically insignificant moments. As an example, for negative bending in the bottom


97

slab midspans, the measured moments, top-slab-calibrated moments and depth-calibrated

moments are all approximately zero. Given the low magnitude, this combination of

critical section and bending direction is unlikely to control the culvert load rating. It

simply does not matter how well the live load method can predict zero. When evaluating

the methods from all load tests, this chapter ignores these practically insignificant

moments. Rather, all further analysis and discussion focuses on meaningful moments:

positive moment in the top and bottom slab midspans, negative moment at top and

bottom slab corners, negative moment at exterior wall corners, and all moments for

exterior wall midspans and interior walls.

Findings and Discussion

The moment envelopes provide valuable insight into the differences between the top-

slab-calibrated and depth-calibrated methods. The influence of method choice can be

further illustrated by evaluating the accuracy and precision of moment results from all 11

load tests. The ratio of the predicted vs. measured live load moment, referred to as the

moment bias, will be evaluated. When moment bias is greater than 1.0, the model over-

predicts the live load moment (conservative); when less than 1.0, the model under-

predicts (unconservative).

Analysis of the moment bias mean and standard deviation quantifies the concepts of

accuracy and precision. Qualitatively, accuracy is the model’s ability to predict the true

moment in the culvert, and precision is the scatter in those predictions. Quantitatively, an

accurate and precise method produces a moment bias mean close to 1.0, a small standard

deviation, and uniform mean and standard deviation at all critical sections. These


98

definitions provide an interpretive framework for all 169 moment biases from 11 load

tests.

Observations of Moment Bias

Figure 22 shows the moment bias histogram on the log-scale for both the top-slab-

calibrated and depth-calibrated methods. In both plots the mean, 𝑥, and standard

deviation, s, are shown. Both methods show a precipitous drop in the number of moment

biases less than 1.0. The few moment bias values that are less than 1.0 may be due to

variations in the measured values and are unlikely to control a culvert load rating.

(a)

(b)

Figure 22. Histogram of moment biases from 11 culvert load tests using the (a) top-slab-

calibrated method and (b) depth-calibrated method

0

10

20

30

40

50

60

0.10 1.00 10.00 100.00

numbe

rofm

omen

tbiases

momentbias(predicted/measured)

s=8.9

0

10

20

30

40

50

60

0.10 1.00 10.00 100.00

numbe

rofm

omen

tbiases

momentbias(predicted/measured)

s =4.6

Top-Slab-CalibratedMethod

Depth-CalibratedMethod


99

Figure 22(a) shows the moment bias histogram for the top-slab-calibrated method.

The moment bias mean is 6.1, meaning that on average, the method predicts 6.1 times the

live load moment actually in the culvert. The moment bias standard deviation (s=8.9) is

quite large as well. Some of the predicted moments are greater than 50 times the

measured values.

Figure 22(b) shows the moment bias histogram for the depth-calibrated method. The

mean and standard deviation have both improved dramatically (𝑥=3.8, s=4.6). The mean

reduced by almost half and moved closer to 1.0. The standard deviation also reduced by

nearly half. Furthermore, the mode (the peak point on the histogram) is around one for

the depth-calibrated method rather than two for the top-slab-calibrated method. The

improvement in mean and standard deviation indicates that the depth-calibrated method

improves the accuracy and precision of the SSI model.

Observations of Moment Bias by Section

This section considers whether the top-slab-calibrated method or the depth-calibrated

method have uniform moment bias mean and standard deviation between critical

sections. Figure 23 plots the mean and standard deviation of the moment bias by critical

section. The first columns show the mean and standard deviation for all data previously

presented in Figure 22.

The next two columns sets, the top slab and top wall corners, are consistent with

observations from the moment envelope (Figure 21), and show similar moment bias

mean and standard deviation for each method. As expected for the top critical sections,

the difference in live load patch widths between the methods does not significantly


100

change in predicted moment. In addition, the mean and standard deviation for top critical

sections are lower than the overall mean and standard deviation for the culvert structure.

(a)

(b)

Figure 23. (a) mean and (b) standard deviation of moment bias by critical section

The bottom slab (far right in Figure 23) has a larger mean and standard deviation than

the top slab counterparts. For both methods, the moment bias mean and standard

deviation are clearly not uniform for the overall culvert structure. However, for the

bottom slab, the depth-calibrated method reduces the moment bias mean and standard

deviation to almost half the top-slab-calibrated values. The improvement in accuracy and

precision in the bottom slab is substantial.

The wall midspan and bottom corner bias data are similar to the bottom slab. Though

the top-slab-calibrated method mean is lower for the wall sections than for the bottom

slab, the depth-calibrated method improves the moment bias mean, again, reducing it by

6.1

3.31.6

3.9

7.1

12.6

3.7 3.21.5

2.7 3.4

6.5

02468101214

all topslab walltopcorner wallmidspan wallbottomcorner

bottomslab

meanmom

entb

ias

(predicted

/measured)

8.9

2.50.8

8.6

12.6

9.8

4.6

2.40.8

6.44.8 5.1

0.02.04.06.08.010.012.014.0

all topslab walltopcorner wallmidspan wallbottomcorner

bottomslab

st.dev.o

fmom

entb

ias

(predicted

/measured)

topslabcalibrated depthcalibrated


101

half. The moment bias standard deviation decreased by more than half by using the

depth-calibrated method. The wall midspan has the greatest standard deviation in both

methods; this may be due to a shortcoming in the production-simplified model since this

variation appears in both methods.

Nevertheless, the trend is clear; the depth-calibrated model significantly improves

moment bias mean and standard deviation, particularly in the bottom slab and walls.

Therefore, the depth-calibrated method is the more accurate and precise of the two live

load attenuation methods. Neither method achieves uniform mean or standard deviation.

However, the depth-calibrated method more closely approaches uniform mean and

standard deviation than the top-slab-calibrated method.

Observations of Moment Bias by Live Load Distribution

Throughout this chapter, the top-slab-calibrated and depth-calibrated methods have

been evaluated using the current policy live load distribution, the AASHTO LRFD

distribution. But, the same live load attenuation methods could be used with other

published live load distributions. Figure 24 shows three live load distributions for a single

HS20 truck. The elastic distribution was derived for a single discrete rectangular surface

loading (one wheel) on a homogeneous mass (Poulos & Davis, 1991) and adapted for

out-of-plane distribution (Katona, 2015). Both the SSHB and the LRFD distributions

assume that slab bending behavior controls when the cover soil depth is less than 0.6m

(2.0ft) (McGrath, et al., 2005). Below 0.6 m (2 ft) of cover soil, the SSHB distribution

prescribes an out-of-plane, live load patch width of 1.75 times depth (AASHTO, 2002).

The LRFD distribution prescribes an out-of-plane, live load patch width consisting of the


102

initial tire width (0.5m (20in.) for an HS20 truck), a soil distribution width of 1.15 times

depth and a width to account for span length. The LRFD method is consistent with the

60° or 2:1 rule found in soil mechanics textbooks (AASHTO, 2014). At greater depth,

both the SSHB and the LRFD distributions account for overlapping wheel load effects by

distributing the sum of the wheel loads over a cumulative live load patch width.

Figure 24. Live load attenuation factor, 1/w (ft/ft (m/m)), as a function of depth from

ground surface for a single HS-20 truck for three live load distribution models: elastic (Poulos & Davis, 1991; Katona, 2015), SSHB (AASHTO, 2002) and LRFD (AASHTO, 2014)

Figure 25 shows the overall moment bias mean and standard deviation for the live

load distributions shown in Figure 24. For each case, the depth-calibrated method

decreases both the moment bias mean and standard deviation. Regardless of live load

distribution, the depth-calibrated method improves accuracy and precision.

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0.00 0.05 0.10 0.15 0.20 0.25

depth,H(ft)

attenuationfactor,1/w(ft/ft(m/m))

Elastic

SSHB

LRFD(7th)


103

(a)

(b)

Figure 25. (a) mean and (b) standard deviation of bias by live load distribution

The elastic distribution improves the most. This improvement is due to differences in

assumptions between the elastic distributions and the policy distributions. First, the

elastic method models only soil attenuation. Therefore, below 0.6m (2ft) of cover soil,

the live load is much larger than the policy distributions that consider slab distributions at

this depth (see Figure 24). For the top-slab-calibrated method, this method grossly over

predicts live load moment particularly in the bottom slab. Since nearly 42% of the

meaningful moment biases come from such low fill depths, the moment bias mean and

standard deviation are very high for the top-slab-calibrated method. However, when the

elastic distribution is used with the depth-calibrated method, the bottom slab and walls

13.3

6.5 6.1

2.9 3.4 3.7

0

2

4

6

8

10

12

14

elastic SSHB LRFD

meanmom

entb

ias

(predicted

/measured)

34.3

10.3 8.94.7 4.1 4.6

0510152025303540

elastic SSHB LRFD

st.dev.o

fmom

entb

ias

(predicted

/measured)

topslabcalibrated depthcalibrated


104

see less live load (greater attenuation) than the policy distributions (Figure 24).

Additionally, the elastic method only considers a single wheel load, whereas, the policy

distributions consider overlapping influence from multiple wheel loads at greater depth;

therefore, the elastic distribution is less conservative at greater depth. These factors

explain why the elastic distribution improves so much when using the depth-calibrated

method.

The SSHB and LRFD distributions experience roughly the same improvement. These

two methods use different distributions, but use the same basic assumptions about slab

distribution for shallow cover soil depths, linear increases in live load patch width with

depth, and overlapping influence from multiple wheel loads at depth. Therefore, the

trends observed in the LRFD distribution hold true for the SSHB distribution; both mean

and standard deviation of the moment bias are reduced by roughly half by using the

depth-calibrated method instead of the top-slab-calibrated method. Though the methods

are different, the live load fit appears to be essentially the same for the SSHB and LRFD

live load distributions.

Figure 25 shows that regardless of the live load distribution used to estimate the live

load attenuation, the depth-calibrated method will improve the modeling accuracy and

precision. The depth-calibrated live load attenuation method can improve the modeling

accuracy and precision of any reasonable live load distribution.

Load Rating Case Study

The depth-calibrated live load attenuation method improves the accuracy and

precision of the moment predictions obtained from a production-simplified demand


105

model. However, live load moment is only one component of the rating factor equation

(Equation 1), the others being capacity and dead load demands. Therefore, the influence

of the live load attenuation method on overall culvert load rating is indirect and varies by

structure. An Case Study will illustrate the improvement in overall load rating.

The test culvert in Lubbock Co., TX (Table 4) has approximately 0.6m (2.0ft) of

cover soil under the traffic lanes. Field inspections show this structure has performed

reasonably well with minor cracking and leaching appearing in the top slab and an overall

condition rating of 6. An SSI model using the top-slab-calibrated method calculates an IR

of HS7 and OR of HS12. The design load was an HS20 truck. The analysis identified the

exterior bottom slab corner as governing the load rating. Yet, field inspection did not

show distress in the bottom slab corner, and the magnitude of the load rating values do

not correspond well with the observed performance under normal traffic loads. This

illustrates the DOT-observed disconnect between load rating and field inspection

performance.

Using the depth-calibrated method, the bottom slab critical sections experience a

decrease in live load demand moments. Therefore, the rating factors for the bottom slab

increase. In so doing, the location of the governing rating factor shifts from the bottom

slab corner to the top slab midspan where the field inspections reported cracking.

Furthermore, the magnitude of the load rating increased (IR=HS11, OR=HS19). In this

example, the depth-calibrated method yielded load rating results that more reasonably

correspond with field inspection performance.


106

As noted, live load comprises one part of the load rating equation, so the depth-

calibrated method will not provide significant load rating improvement in every case.

Sometimes capacity will drive the load rating process. If the top slab governs the load

rating using the top-slab-calibrated method, the top slab will still govern if the depth-

calibrated method is used. In other cases, the predicted dead load may exceed the critical

section capacity, and this behavior will govern the load rating. But, for cases where the

top-slab-calibrated method shows that a bottom slab, wall midspan, or wall bottom corner

critical section governs the load rating, the depth-calibrated method will improve the load

rating outcome, in some cases dramatically.

Improved Live Load Distribution

Most published research on live loads to RC box culverts has focused on live load

induced pressures on the culvert, particularly loads on the top slab. However, if the

research question were reframed to look at live load induced structural response, a better

out-of-plane live load distribution could be developed. This better distribution should

consider the many sources of live load attenuation including pavement, cover soil, slab

behavior, and soil-culvert interaction. The goal would be an even more accurate and

precise out-of-plane live load distribution that achieves uniform moment bias mean and

standard deviation by critical section.

Additionally, current LRFR calibration methods implicitly assume independent

measurements with uniform bias mean and variance (related to standard deviation).

However, measurements at each critical section on a single culvert are not statistically

independent. Further, the biases at each critical section for a given culvert do not have


107

uniform bias mean or variance; rather a relationship exists between bias distribution and

critical section location. Achieving uniform bias through a better out-of-plane live load

distribution would decrease variance and improve calibrated load factors. These issues

should be addressed by a robust LRFR load calibration effort for RC box culverts.

Conclusions

The depth-calibrated live load attenuation method represents a step forward in closing

the disconnect between calculated load rating values and field inspection performance for

RC box culverts. This method reduces over-prediction of live load demand in the bottom

slab and walls of a culvert. Further, existing live load distributions already prescribed by

state and federal policy are compatible with the depth-calibrated method. A better out-of-

plane live load distribution could be determined by using the depth-calibrated method to

estimate live load induced structural response in each portion of the culvert system. The

depth-calibrated live load attenuation method can provide more accurate and precise load

ratings for RC box culverts.

Acknowledgements

The Texas Department of Transportation sponsored the research work described in

this chapter.


108

CHAPTER 5

CONCLUSIONS

Summary

This dissertation investigated the influence of three factors on load rating of

reinforced concrete box culverts. The factors explored were (1) cover soil depth in a

structural-frame model, (2) production-simplified modeling sophistication, and (3) live

load attenuation method. This dissertation builds directly upon research performed at

Texas Tech University for TxDOT. Additional information on load rating CIP RC box

culverts can be found in the published research reports (Lawson, et al., 2010; Lawson, et

al., 2009; TxDOT, 2013; TxDOT, 2014).

For each explored factor, the influence was significant. Cover soil depth strongly

influences the load rating results in three characteristic and non-linear ways. Load rating

of a particular culvert must consider the cover soil depths that actually exist on the

culvert. Using field live load tests, a production-simplified soil-structure interaction

model was shown to be an order of magnitude more accurate and precise than the

AASHTO-recommended production-simplified structural-frame model. By using a

depth-calibrated, out-of-plane, live load attenuation model, the live load prediction

accuracy and precision was nearly doubled compared to the traditional top-slab-calibrated

live load attenuation method. Taken together, the improvements in load rating

calculations described in this dissertation help to reduce the disconnect between load

rating calculations and observed field inspection performance of the culvert structure.


109

Major Findings

Key findings are defined by factor.

• Cover Soil Depth

o Interaction of live load and dead load with depth causes a non-linear interaction

between cover soil depth and load rating for a structure.

o There is no a priori worst case cover soil depth.

• Demand Model Sophistication

o The structural-frame model is conservative, production-simplified, and reasonable

for the top slab.

o The structural-frame model significantly over-predicts live loads in the bottom slab

and walls.

o The soil-structure interaction model improves in-plane live load attenuation.

o The soil-structure interaction model significantly improves live load accuracy and

precision particularly in the bottom slab and walls.

• Live Load Attenuation Methods

o Depth-calibrated live load attenuation improves out-of-plane live load attenuation.

o Depth-calibrated live load attenuation improves the accuracy and precision of the

soil-structure model in the bottom slab and walls.


110

o Depth-calibrated live load attenuation improves correspondence between load

rating values and field inspection observations.

Limitations

This work is specifically limited to load rating cast-in-place, reinforced-concrete box

culverts. First, these findings are specific to load rating. No attempt was made to predict

the impact of applying these findings to culvert design. Current methods for design are

appropriately conservative and can be used without reservations. Important corollaries

exist between load rating and design, but findings from this dissertation must be applied

to design with extreme care as design is outside the scope of this dissertation.

Additionally, the research is focused on CIP RC box culverts. The demand calculation

methods used in this work may be appropriate for certain precast box culverts, but the

capacity calculations for such structures are slightly different. Other culvert types

including rigid and flexible round pipes, arches, and three-sided boxes behave in

fundamentally different ways from CIP RC box culverts. The dissertation makes no

attempt to predict the performance of these structures.

Finally, the dissertation evaluates the structural capacity of culverts in drained soils.

No work has been presented addressing hydraulic performance or the significant

influence of water in the culvert and surrounding soils.


111

Future Work

Several factors with significant impact on load rating of box culverts were not

explored in this dissertation including cover soil depth in a soil-structure interaction

model, lateral loading parameters, long-term dead load behavior, and an improved depth-

calibrated live load distribution.

The influence of cover soil depth has not been explored using the soil-structure

interaction model. A non-linear relationship between cover soil depth and load rating

when using the soil-structure interaction model is expected. However, the significant

differences between the structural-frame and soil-structure interaction models mean that

the relationship is unlikely to be identical to the relationships identified for the structural-

frame model. An automated soil-structure interaction model should be developed and

implemented to evaluate the full range of cover soil depths on culvert designs. This

would improve understanding of nuances associated with the soil-structure interaction

model.

The influence of those parameters that define the lateral loading to the culvert –

namely, at-rest lateral earth pressure coefficient for the structural-frame model and soil

modulus for the soil-structure interaction model – have not been fully explored. The

structural-frame model uses a range of at-rest lateral earth pressures that are likely

conservative. No such analysis has been performed using the soil-structure interaction

model. A small scale parametric study from the original research indicates that load

rating is strongly dependent on the soil stiffness (Lawson, et al., 2010). Further, the

nature of the dead loads and live loads suggest that different soil stiffnesses may be


112

appropriate for each loading. Dead load may be modeled better using static soil modulus

values while the live load may be better modeled using dynamic or resilient elastic

moduli. Further literature review, field and laboratory testing, and analytical analysis are

required to fully explore the issue of soil stiffness and its impact on lateral loading.

Additionally, long-term dead load behavior for load rating purposes is poorly

understood. Much research has been performed to examine installation-induced soil

stresses. No long-term tests have been performed to identify long-term dead load

behavior. This would require carefully designed, long-term monitoring of full-scale test

culverts or careful scale-model testing.

Finally, an improved depth-calibrated live load distribution is needed that accounts

for out-of-plane distribution from pavement, soil, top slab, culvert-soil interaction, and

bottom slab. The depth-calibrated live load attenuation method introduced here is shown

to improve live load precision and accuracy, particularly in the bottom slab and walls.

The underlying assumption is that the culvert-soil system is at least as effective as the soil

itself in redistributing the live load in the out-of-plane direction. However, additional

analytical work is required to develop a more rational distribution that considers the

greater out-of-plane attenuation likely generated by the culvert top slab, walls, and

bottom slab.


113

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121

APPENDIX A

DISTRIBUTIONS OF CULVERT DESIGNS


122

Figure A.1. Distribution plot of load rating vs. cover soil depth plot shape by design

era

208143

305

12

56

43

101181

32

050

100150200250300350400

pre-WWII InterstateHighway

modernizedculvertd

esignsby

desig

nera

designera



123

(a) (b)

(c)

Figure A.2. Trend plots of load rating vs. cover soil depth relationship by culvert geometry: (a) aspect ratio, (b) span length, and (c) barrel height

124171

120177

64

30

39

26

15

1

118

85

47

59

5

0

50

100

150

200

250

300

350

1 1.25 1.5 2 2.5

culvertd

esignsbyaspe

ctra

tio

aspectratio,S/H

157 142 139

9665 57

3 17 9

21

3823

352

6271 126

1.5 1.8 2.1 2.4 2.7 3.0

0

50

100

150

200

250

5 6 7 8 9 10

span,S(m)

%ofculvertdesignsbyspan

span,S(ft)

40

102128 127 111

75 62

11

6

8 1517

1717

29

2

5

25 26 46

5949

44

428 10

0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6

020406080

100120140160180200

2 3 4 5 6 7 8 9 10 11 12

height,H(m)

%ofculvertdesignsbyhe

ight

height,H(ft)



124

Figure A.3. Distribution plot of load rating vs. cover soil depth plot shape by design

cover soil depth

114 101 69

240

132

28 3130

20

2

195

68

31

20

DT(0-0.6m) 0.6-1.2m 0-1.2m 1.2-1.8m 0-1.8m

050

100150200250300350400

DT(0-2ft) 2-4ft 0-4ft 4-6ft 0-6ft


%ofculvertdesignsby

desig

ncoversoildep

thra

nge




125

APPENDIX B

MOMENT PLOTS COMPARING LIVE LOAD ATTENUATION

METHODS


126

Figure B.1. Moment plot for Swisher Co., TX (Table 4) culvert under 1.5ft of cover soil


-18.0

-13.5

-9.0

-4.5

0.0

4.5

9.0

13.5

18.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

B2O

B2M B2I

B4O

B4M B4I

B6O

B6M

T2O

T2M T2I

T4M T4I

T6O

T6M

W1B

W1M W1T

W3B

W3T

W5B

W5T

mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)

criticalsectionlabel

measuredtop-slabcalibrateddepth-calibrated

-18

-13.5

-9

-4.5

0

4.5

9

13.5

18

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

B2O

B2M B2I

B4O

B4M B4I

T2O

T2M T2I

T4O

T4M T4I

W1B

W1M W1T

W3B

W3M W3T

W5B

W5M W5T

mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)




127

Figure B.3. Moment plot for Hale Co., TX (Table 4) culvert under 3.5ft of cover soil


-13.5

-9

-4.5

0

4.5

9

13.5

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

B2O

B2M B2I

B4O

B4M B4I

T2O

T2M T2I

T4O

T4M T4I

W1B

W1M W1T

W3B

W3M W3T

W5B

W5M W5T

mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)



-13.5

-9

-4.5

0

4.5

9

13.5

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

B2O

B2M B2I

B4O

B4M B4I

T2O

T2M T2I

T4O

T4M T4I

W1B

W1M W1T

W3B

W3M W3T

W5B

W5M W5T

mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)




128

Figure B.5. Moment plot for Sarpy Co., NE (Table 4) culvert under 0ft of cover soil


-13.5

-9

-4.5

0

4.5

9

13.5

18

22.5

27

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

B2O B2M B2I T2O T2M T2I W1B W1M W1T

mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)



-13.5

-9

-4.5

0

4.5

9

13.5

18

22.5

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0


mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)




129



-9

-4.5

0

4.5

9

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)



-6.75

-5.5

-4.25

-3

-1.75

-0.5

0.75

2

3.25

4.5

5.75

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5


mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)




130



-4.5

-3

-1.5

0

1.5

3

4.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0


mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)



-4.5

0

4.5

9

13.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0


mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)




131


-4.5

-2.25

0

2.25

4.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0


mom

ent(kN

-m/m

)

mom

ent(k-ft/ft)



improved culvert load rating through an evaluation of the

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