Appendix A Investigation of Suitable Soil Constitutive Models for 3-D Finite Element Studies of Live

Limited Use Document

This Final Report is furnished only for review by

members of the NCHRP project panel and is

regarded as fully privileged. Dissemination of

information included herein must be approved by the

NCHRP.

Appendix A Investigation of Suitable Soil Constitutive Models for 3-D Finite Element Studies of Live Load Distribution Through

Fills Onto Culverts

National Cooperative Highway Research Program Project 15-29

CNA Consulting Engineers Simpson, Gumpertz & Heger

April 2009

NCHRP 15-29 Appendix A i

Table of Contents

1. INTRODUCTION 1

2. REVIEW OF AVAILABLE SOIL MODELS 12.1 Linear Elastic 22.2 Elasto-Plastic 32.3 Stress-Dependent Models 6

2.3.1 Duncan-Selig Model 62.3.2 Hardening Soil Model (Plaxis) 10

2.4 Findings from Soil Model Evaluation 22

3. TWO-DIMENSIONAL MODELING OF CULVERTS 223.1 Modeled Structures 223.2 Material Models 23

3.2.1 Linear-Elastic Model 233.2.2 Mohr-Coulomb Model with Perfect Plasticity in Plaxis 233.2.3 Hardening-Soil Model in Plaxis 243.2.4 In-Situ Soil Material 243.2.5 Other Materials 24

3.3 Live Load 243.4 Finite Element Model 263.5 Results of 2D Analysis 35

3.5.1 Concrete Box 353.5.2 Concrete Pipe 443.5.3 Metal Pipe 493.5.4 Thermoplastic Pipe 543.5.5 Concrete Arch 593.5.6 Metal Arch 643.5.7 Summary of Results from 2D Preliminary Analyses 69

3.6 Effect of Interface Strength 713.7 Conclusion 81

4. THREE DIMENSIONAL MODELING OF CULVERTS 824.1 Comparison of Responses to Factored and Unfactored Live Loads 82

4.1.1 Introduction 824.1.2 Method of Approach 824.1.3 Results 83

4.1.3.1 HDPE Pipe in ABAQUS 834.1.3.2 Three-Sided Arch Top Culvert in CANDE 85

4.1.4 Conclusion 864.2 Selected Field Tests for 3D Analysis 86

4.2.1 NCHRP Project 12-45 864.2.2 Minnesota DOT Study 90

4.3 Three-Dimensional Analysis 934.3.1 General Information 934.3.2 Long-Span Concrete Arch Culvert 94

4.3.2.1 Finite Element Model 944.3.2.2 Materials 954.3.2.3 Loading and Boundary Condition 964.3.2.4 Results 97

4.3.3 Long-Span Metal Arch Culvert 101

NCHRP 15-29 Appendix A ii

4.3.3.1 Finite Element Model 1014.3.3.2 Materials 1024.3.3.3 Loading and Boundary Condition 1024.3.3.4 Results 102

4.3.4 60-in. Diameter HDPE Pipe 1084.3.4.1 Finite Element Model 1084.3.4.2 Materials 1094.3.4.3 Loading and Boundary Condition 1094.3.4.4 Results 111

4.3.5 Discussion 1264.4 Comparison between the Mohr-Coulomb and Hardening-Soil Models in Three-

Dimensional Analysis in PLAXIS 1294.4.1 Method of Approach 1304.4.2 Results 130

4.4.2.1 Metal Arch in Test 2 with 3 ft Cover 1304.4.2.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover 130

4.4.3 Conclusion 1354.5 Three-Dimensional Analysis of Field Tests in ABAQUS 135

4.5.1 Introduction 1354.5.2 Method of Approach 1364.5.3 Validation of ABAQUS Model 1394.5.4 Results 140

4.5.4.1 Metal Arch with 3 ft Cover 1404.5.4.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover 142

4.5.5 Conclusion 143

5. DISCUSSION 144

6. CONCLUSIONS AND RECOMMENDATIONS 145

7. REFERENCES 145

List of Tables

Table 1—Elastic soil properties for Backfill (Selig, 1990) ............................................................. 3Table 2—Vertical Stresses and Estimated Horizontal Stresses under Gravity and Corresponding Angle of Friction for SW85 ............................................................................................................ 6Table 3—Parameters for Linear-Elastic and Mohr-Coulomb Models for SW85 ............................ 6Table 4—Soil Properties of Backfill for Duncan-Selig Model (Selig, 1988) ................................... 9Table 5—Input Parameters for Hardening-Soil Model for SW85, SW90, ML85, and CL85 ........ 15Table 6—Structural Types and Cover Depths for 2D Analysis ................................................... 23Table 7—Comparison of Bending Moments and Thrusts in Concrete Box Model ...................... 43Table 8—Comparison of Bending Moments and Thrusts in Concrete Pipe Model ..................... 48Table 9—Comparison of Bending Moments and Thrusts in Metal Pipe Model ........................... 53Table 10—Comparison of Bending Moments and Thrusts in Thermoplastic Pipe Model ........... 58Table 11—Comparison of Bending Moments and Thrusts in Concrete Arch Model ................... 63Table 12—Comparison of Bending Moments and Thrusts in Metal Arch Model ........................ 68Table 13—Ratios of Live Load Moments and Thrusts of Concrete Box ..................................... 69Table 14—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 2 ft ........ 70Table 15—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 6 ft ........ 70Table 16—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 2 ft ...... 70Table 17—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 6 ft ...... 70

NCHRP 15-29 Appendix A iii

Table 18—Comparison of Bending Moments and Thrusts between Concrete Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model) ................................................. 76Table 19—Comparison of Bending Moments and Thrusts Load between Thermoplastic Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model) ............................. 81Table 20—Comparison of Structural Responses between Analyses with Factored and Unfactored Live Loads (HDPE Pipe, A2 Backfill) ........................................................................ 85Table 21—Comparison of Structural Responses between Analyses (Hanson Arch) ................. 86Table 22—Properties of Reinforced Concrete Culvert ................................................................ 88Table 23—Properties of Structural Steel Plate and Culvert ........................................................ 88Table 24—Properties of Type S HDPE Pipe .............................................................................. 91Table 25—Average Trench Measurements for Test Pipes in the MNDOT Study ....................... 92Table 26—Soil Properties Used for the 3D Analyses of Long-Span Arches .............................. 96Table 27—Concrete Properties Used for the 3D Analyses of Long-Span Arches ...................... 96Table 28—Vertical Displacements at Crown of Concrete Arch due to Live Loads ..................... 99Table 29—Chord Extension at Height of 88 in. of Concrete Arch Culvert due to Live Loads ..... 99Table 30—Thrusts at Base of Concrete Arch Culvert due to Live Loads ................................... 99Table 31—Axial and Bending Modulus of Metal Arch in Circumferential and Longitudinal Directions (E=29,000 ksi) .......................................................................................................... 102Table 32—Vertical Displacements at Crown of Metal Arch due to Live Loads ......................... 104Table 33—Chord Extension at Height of 88 in. of Metal Arch Culvert due to Live Loads ......... 104Table 34—Thrusts in Test 1 of Metal Arch Culvert due to Live Loads ...................................... 105Table 35—Thrusts in Test 2 of Metal Arch Culvert due to Live Loads ...................................... 105Table 36—Moments in Test 1 of Metal Arch Culvert due to Live Loads ................................... 106Table 37—Moments in Test 2 of Metal Arch Culvert due to Live Loads ................................... 106Table 38—Axial and Bending Modulus of HDPE Pipe in Circumferential and Longitudinal Directions (E=100,000 psi) ........................................................................................................ 109Table 39—Soil Properties Used for the 3D Analyses of HDPE Pipes ...................................... 110Table 40—Comparison of Vertical Displacements at Crown of HDPE Pipes under Heavy Truck

.................................................................................................................................................. 125Table 41—Comparison of Vertical Displacements at Crown of HDPE Pipes under Light Truck

.................................................................................................................................................. 125Table 42—Comparison of Diametrical Changes at Springline of HDPE Pipes under Heavy Truck

.................................................................................................................................................. 125Table 43—Comparison of Diametrical Changes at Springline of HDPE Pipes under Light Truck

.................................................................................................................................................. 125Table 44—Summary of Displacements under Wheel (Metal Arch, Test 2, 3 ft Cover) ............. 132Table 45—Summary of Thrusts under Wheel (Metal Arch, Test 2, 3 ft Cover) ......................... 132Table 46—Summary of Moments under Wheel (Metal Arch, Test 2, 3 ft Cover) ...................... 133Table 47—Summary of Vertical Displacements under Wheel (HDPE Pipe, A2 Soil, 2.8 ft Cover)

.................................................................................................................................................. 134Table 48—Summary of Horizontal Chord Extensions under Wheel (HDPE Pipe, A2 Soil, 2.8 ft Cover) ....................................................................................................................................... 135Table 49—Summary of Force Results (HDPE Pipe, A2 Soil, 2.8 ft Cover) .............................. 135Table 50—Orthotropic Properties Used in ABAQUS Analyses ................................................ 137Table 51—Orthotropic Stiffness Properties .............................................................................. 137Table 52—Soil Porperties Used for Soft Haunch and Void Areas ............................................ 138Table 53—Summary of Displacements from ABAQUS Analyses with Orthotropic Properties (Metal Arch, 3 ft Cover) ............................................................................................................. 141Table 54—Summary of Displacements from ABAQUS Analyses with Orthotropic Properties (HDPE Pipe, A2 Backfill) ........................................................................................................... 143

List of Figures

NCHRP 15-29 Appendix A iv

Figure 1—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85 in Deviatoric Loading of Triaxial Test .............................................................................................. 16Figure 2—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW90 in Deviatoric Loading of Triaxial Test .............................................................................................. 17Figure 3—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85 in Deviatoric Loading of Triaxial Test .............................................................................................. 18Figure 4—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85 in Deviatoric Loading of Triaxial Test .............................................................................................. 19Figure 5—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85 in Oedometer Loading .................................................................................................................... 20Figure 6—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW90 in Oedometer Loading .................................................................................................................... 20Figure 7—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85 in Oedometer Loading .................................................................................................................... 21Figure 8—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85 in Oedometer Loading .................................................................................................................... 21Figure 9—Live Load per Unit Length of Culvert in 2D Analysis .................................................. 26Figure 10—Conceptual Model for 2D Analysis of Pipes ............................................................. 27Figure 11—Conceptual Model for 2D Analysis of Boxes ............................................................ 28Figure 12—Conceptual Model for 2D Analysis of Arches ........................................................... 28Figure 13—Finite Element Meshes of Concrete Box Model ....................................................... 29Figure 14—Finite Element Meshes of Concrete Pipe Model ...................................................... 30Figure 15—Finite Element Meshes of Metal Pipe Model ............................................................ 31Figure 16—Finite Element Meshes of Plastic Pipe Model .......................................................... 32Figure 17—Finite Element Meshes of Concrete Arch Model ...................................................... 33Figure 18—Finite Element Meshes of Metal Arch Model ............................................................ 34Figure 19—Deformation of Concrete Box due to Live Load ....................................................... 36Figure 20—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box Model (0 ft Cover) .................................................................................................................................. 37Figure 21—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete Box Model (0 ft Cover) ....................................................................................................................... 38Figure 22—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box Model (2 ft Cover) .................................................................................................................................. 39Figure 23—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete Box Model (2 ft Cover) ....................................................................................................................... 40Figure 24—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box Model (6 ft Cover) .................................................................................................................................. 41Figure 25—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete Box Model (6 ft Cover) ....................................................................................................................... 42Figure 26—Deformation of Concrete Pipe due to Live Load ...................................................... 45Figure 27—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (2 ft Cover)

.................................................................................................................................................... 46Figure 28—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (6 ft Cover)

.................................................................................................................................................... 47Figure 29—Deformation of Metal Pipe due to Live Load ............................................................ 50Figure 30—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (2 ft Cover) . 51Figure 31—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (6 ft Cover) . 52Figure 32—Deformation of Thermoplastic Pipe due to Live Load .............................................. 55Figure 33—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model (2 ft Cover) ......................................................................................................................................... 56Figure 34—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model (6 ft Cover) ......................................................................................................................................... 57

NCHRP 15-29 Appendix A v

Figure 35—Deformation of Concrete Arch due to Live Load ...................................................... 60Figure 36—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (2 ft Cover)

.................................................................................................................................................... 61Figure 37—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (6 ft Cover)

.................................................................................................................................................... 62Figure 38—Deformation of Metal Arch due to Live Load ............................................................ 65Figure 39—Bending Moments and Thrusts due to Live Load in Metal Arch Model (2 ft Cover) . 66Figure 40—Bending Moments and Thrusts due to Live Load in Metal Arch Model (6 ft Cover) . 67Figure 41—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover) ......................................................... 72Figure 42—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover) ......................................................... 73Figure 43—Comparison of Bending Moments and Thrusts due to Live Load between Concrete Pipe Models with 50% and 100% Interface Strength (2 ft Cover) ............................................... 74Figure 44—Comparison of Bending Moments and Thrusts due to Live Load between Concrete Pipe Models with 50% and 100% Interface Strength (6 ft Cover) ............................................... 75Figure 45—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover) ......................................................... 77Figure 46—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover) ......................................................... 78Figure 47—Comparison of Bending Moments and Thrusts due to Live Load between Thermoplastic Pipe Models with 50% and 100% Interface Strength (2 ft Cover) ....................... 79Figure 48—Comparison of Bending Moments and Thrusts due to Live Load between Thermoplastic Pipe Models with 50% and 100% Interface Strength (6 ft Cover) ....................... 80Figure 49—Finite Element Model of Three-Sided Arch Top Culvert with 3 ft Cover ................... 83Figure 50—Comparison of Vertical and Horizontal Displacements from Factored Live Load with 1.75 times those from Unfactored Live Load (HDPE Pipe, A2 Backfill) ...................................... 84Figure 51—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times those from Unfactored Live Load (HDPE Pipe, A2 Backfill) ................................................................. 84Figure 52—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times those from Unfactored Live Load (Hanson Arch) ................................................................................. 85Figure 53—Test Setup of NCHRP Project 12-45 ........................................................................ 87Figure 54—Instrumentation for Deformation in Concrete Culvert ............................................... 89Figure 55—Instrumentation for Deformation in Metal Culvert ..................................................... 89Figure 56—Cross Sections of HDPE Pipes: Type D and Type S ............................................... 91Figure 57—Typical Installation of PE Pipe .................................................................................. 91Figure 58—Live Load Vehicle in the MNDOT Study ................................................................... 92Figure 59—Typical Test Pipe Instrumentation in the MNDOT Study .......................................... 93Figure 60—Typical Dimensions of Finite Element Models of Long-Span Arch and HDPE Pipe 94Figure 61—Finite Element Model of Concrete Arch Culvert with a Cover Depth of 3 ft .............. 95Figure 62—Live Load Position in the 3D Analysis of Long-Span Arches ................................... 97Figure 63—Deformed Shapes of Concrete Arch in the Plane of Wheel Loads (Effects of Live Loads only) ................................................................................................................................. 98Figure 64—Displacements Due to Live Loads from the 3D Analyses of Concrete Arch Culvert 98Figure 65—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the 3D Analyses of Concrete Arch Culvert ............................................................................................. 99Figure 66—Plastic Points in Soil Elements in the Plane of Wheel Loads in Concrete Arch Analysis ..................................................................................................................................... 100Figure 67—Finite Element Model of Metal Arch Culvert with a Cover Depth of 3 ft ................. 101Figure 68—Soft Element to Match Longitudinal Stiffness of Metal Arch .................................. 102Figure 69—Deformed Shapes of Metal Arch in the Plane of Wheel Loads (Effects of Live Loads only) .......................................................................................................................................... 103

NCHRP 15-29 Appendix A vi

Figure 70—Displacements due to Live Loads from the 3D Analyses of Metal Arch Culvert .... 104Figure 71—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the 3D Analyses of Metal Arch Culvert ................................................................................................. 104Figure 72—Plastic Points in Soil Elements in the Plane of Wheel Loads in Metal Arch Analysis

.................................................................................................................................................. 107Figure 73—Finite Element Model of HDPE Pipe Culvert for Pipe Run 9 .................................. 108Figure 74—Soft Element to Match Longitudinal Stiffness of HDPE Pipe ................................. 109Figure 75—Positions of Live Load Vehicle Axles in the 3D Analyses of HDPE Pipes .............. 111Figure 76—Deformed Shapes of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads (A-1, 1.4 ft Cover) .......................................................................................................................... 113Figure 77—Vertical Crown Displacements of Pipe Run 1 due to Live Loads ........................... 113Figure 78—Horizontal Displacements of Pipe Run 1 due to Live Loads (A-1, 1.4 ft Cover) ..... 114Figure 79—Thrusts of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads ................. 114Figure 80—Moments of Pipe Run 1 due to Live Loads in Plane of Wheel Loads .................... 114Figure 81—Plastic Points in Soil Elements of Pipe Run 1 in the Plane of Wheel Loads .......... 115Figure 82—Deformed Shapes of Pipe Run 9 due to Live Loads in the Plane of Wheel Loads (A-1, 2.5 ft Cover) .......................................................................................................................... 116Figure 83—Vertical Crown Displacements of Pipe Run 9 due to Live Loads ........................... 116Figure 84—Horizontal Displacements of Pipe Run 9 Due to Live Loads .................................. 117Figure 85—Thrusts of Pipe Run 9 Due to Live Loads in the Plane of Wheel Loads ................ 117Figure 86—Moments of Pipe Run 9 Due to Live Loads in Plane of Wheel Loads .................... 117Figure 87—Plastic Points in Soil Elements of Pipe Run 9 in the Plane of Wheel Loads .......... 118Figure 88—Deformed Shapes of Pipe Run 3 due to Live Loads in the Plane of Wheel Loads (A-2, 1.6 ft Cover) .......................................................................................................................... 119Figure 89—Vertical Crown Displacements of Pipe Run 3 due to Live Loads ........................... 119Figure 90—Horizontal Displacements of Pipe Run 3 due to Live Loads (A-2, 1.6 ft Cover) ..... 120Figure 91—Thrusts of Pipe Run 3 due to Live Loads in the Plane of Wheel Loads (A-2, 1.6 ft Cover) ....................................................................................................................................... 120Figure 92—Moments of Pipe Run 3 due to Live Loads in Plane of Wheel Loads .................... 120Figure 93—Plastic Points in Soil Elements of Pipe Run 3 in the Plane of Wheel Loads .......... 121Figure 94—Deformed Shapes of Pipe Run 7 due to Live Loads in the Plane of Wheel Loads (A-2, 2.8 ft Cover) .......................................................................................................................... 122Figure 95—Vertical Crown Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft Cover)

.................................................................................................................................................. 122Figure 96—Horizontal Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft Cover) ..... 123Figure 97—Thrusts of Pipe Run 7 due to Live Loads in the Plane of Wheel Loads (A-2, 2.8 ft Cover) ....................................................................................................................................... 123Figure 98—Moments of Pipe Run 7 due to Live Loads in Plane of Wheel Loads (A-2, 2.8 ft Cover) ....................................................................................................................................... 123Figure 99—Plastic Points in Soil Elements of Pipe Run 7 in the Plane of Wheel Loads (A-2, 2.8 ft Cover) .................................................................................................................................... 124Figure 100—Ratios of 3D Analysis Results to Field Test Data for Displacements of Concrete Arch ........................................................................................................................................... 128Figure 101—Ratios of 3D Analysis Results to Field Test Data for Displacements of Metal Arch

.................................................................................................................................................. 128Figure 102—Ratios of 3D Analysis Results to Field Test Data for Displacements of HDPE Pipes

.................................................................................................................................................. 129Figure 15—Comparison of Displacements between Cases with Mohr-Coulomb and Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover) ............................................................................. 131Figure 16—Comparison of Thrusts and Moments under Wheel between Cases with Mohr-Coulomb and Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover) ..................................... 132

NCHRP 15-29 Appendix A vii

Figure 17—Comparison of Crown Vertical Displacements between Cases with Mohr-Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................... 133Figure 18—Comparison of Horizontal Diameter Changes between Cases with Mohr-Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................... 134Figure 19—Comparison of Thrusts between Cases with Mohr-Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................................................... 134Figure 20—Comparison of Moments between Cases with Mohr-Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................................................... 134Figure 21—Cross Section of Finite Element Model for HDPE Pipe in ABAQUS ...................... 138Figure 22—ABAQUS Metal Arch Model with 3 ft Cover ........................................................... 139Figure 23—ABAQUS HDPE Pipe Model with 3 ft Cover .......................................................... 139Figure 24—Comparison of Vertical and Horizontal Displacements between PLAXIS 3D and ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover) ................................................................. 140Figure 25—Comparison of Thrusts and Moments under Wheel between PLAXIS 3D and ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover) ................................................................. 140Figure 26—Vertical and Horizontal Displacements from ABAQUS Analyses with Orthotropic Properties (Metal Arch, 3 ft Cover) ........................................................................................... 141Figure 27 –Thrusts and Moments under Wheel from ABAQUS Analyses with Orthotropic Properties (Metal Arch, 3 ft Cover) ........................................................................................... 141Figure 28—Vertical and Horizontal Displacements from ABAQUS Analyses with Orthotropic Properties (HDPE Pipe, A2 Backfill, 2.8 ft Cover) ..................................................................... 142Figure 29—Vertical and Horizontal Displacements from ABAQUS Analyses with Orthotropic Properties (HDPE Pipe, A2 Backfill, 1.6 ft Cover) ..................................................................... 143

NCHRP 15-29 Appendix A 1

1. INTRODUCTION

NCHRP Project 15-29 was funded to investigate the distribution of live loads through fills and

onto culverts. The project is intended to improve AASHTO Specifications for design of buried

structures and to investigate differences between the AASHTO Standard Specifications for

Highway Bridges, 17th Edition (AASHTO, 2002) and the AASHTO LRFD Bridge Design

Specifications, 3th

The scope of project 15-29 is to conduct studies through three-dimensional (3-D) finite element

(FE) modeling of live loads on buried culverts and to develop new AASHTO Specifications

based on the findings. Results and proposed design methods will be evaluated against test

data available in the literature, but no new tests will be conducted as a part of this project.

Edition (AASHTO, 2004).

This report presents an investigation of the available constitutive models for soils that could be

used in the 3-D analyses. Information presented includes initial review of available soil models

and important model features, preliminary two-dimensional FE studies of live load distribution,

and full 3-D FE studies.

2. REVIEW OF AVAILABLE SOIL MODELS

Numerous soil constitutive models have been developed to date and are available for finite

element analysis. Lade (2005) prepared a summary of widely available soil constitutive models.

Each model has different capabilities and requires different experimental data for calibration.

Predicting the response of buried structures to surface live loads in a finite element analysis

requires a soil constitutive model that accurately captures culvert-soil interaction. Research has

been conducted with linear-elastic soil models (for example, Moore and Brachman (1994) and

Fernando and Carter (1998)); with nonlinear models including nonlinear elastic models,

perfectly plastic models, and plastic models with hardening (for example, Pang (1999)). For

typical culvert analysis, which has been historically conducted in 2-D, stress-dependent stiffness

and shear failure have been found to be important characteristics of suitable soil models. The

Duncan-Selig hyperbolic model (Duncan et al., 1980; Selig, 1988) has such features, and has

been implemented in the finite element programs CANDE (Musser, 1989) and SPIDA (Heger et

al., 1985) to analyze soil-structure interaction problems for culverts. Soil properties based on

these models have been used in the development of current AASHTO specifications for

reinforced concrete and thermoplastic pipe. The Duncan-Selig model, consists of the hyperbolic

Young's modulus model developed by Duncan (1980), and the hyperbolic bulk modulus


developed by Selig (1988). As discussed below, the soil properties used with this model were

developed by Selig (1988).

CANDE was developed by the FHWA, and has been widely used to design culverts, but

operates only in two dimensions. For ease of computation and to allow comparison with

CANDE we conducted preliminary analyses in 2-D and then extended these models to 3-D for a

complete investigation of actual live load distribution.

3-D modeling is computationally intensive. Because of this, it is important to select the

computationally simplest soil model that can accurately capture culvert/soil interaction as

resulting from live load. We selected three levels of soil model with varying levels of

sophistication for use on the project:

• linear-elastic (representing the simplest possible model),

• Mohr-Coulomb (linear elastic model with post-failure plasticity), and

• Plaxis 3D hardening-soil (stress dependence plus plasticity, similar to Duncan-Selig).

Features of the linear-elastic model, Mohr-Coulomb model, Hardening-Soil model, and Duncan-

Selig model are briefly discussed below. Compressive stresses are positive throughout this

report.

2.1 Linear Elastic

Modeling soil as linear elastic provides the most basic soil behavior, with no consideration of

non linear stress-strain behavior or plasticity at failure. Linear elastic soil behavior is described

by isotropic linear elasticity. Four elastic constants are used in analysis, but given any two of

the four, the other two can be calculated. The four parameters are: modulus of elasticity, E ,

Poisson’s ratio, ν , bulk modulus, B , and shear modulus, G . In actual soil these elastic

constants vary with soil stress level, and some analysts use elastic properties that vary with

depth. One set of such properties, proposed by Selig (1990) are shown in Table 1. Selig

estimated Young's modulus of elasticity from the hyperbolic model for increasing values of

maximum principal stress, (σ1, typically vertical stress), with the minimum principal stress, (σ3,

typically horizontal stress), equal to one-half to one- times the maximum principal stress. Elastic

constants can be selected by evaluating the soil vertical stress level, usually calculated as the

depth of fill times the soil density, ignoring the presence of a culvert. Procedures to select

elastic constants are described in detail in the following section on the Mohr-Coulomb model.

McGrath (1998) found that the proposed soil properties produced soil stiffnesses higher than


back-calculated from actual projects by Howard (1977), and thus concluded that the properties

are likely achievable but not suitable for routine design where backfill sources may be

undependable and soil gradations variable.

Table 1—Elastic soil properties for Backfill (Selig, 1990) Gravelly Sand (SW)

Maximum 95% Standard Compaction 85% Standard Compaction Principal Stress E B ν E B ν

Level (psi) (psi) (psi) (psi) (psi) 0 to 1 1,600 2,800 0.40 1,300 900 0.26 1 to 5 4,100 3,300 0.29 2,100 1,200 0.21

5 to 10 6,000 3,900 0.24 2,600 1,400 0.19 10 to 20 8,600 5,300 0.23 3,300 1,800 0.19 20 to 40 13,000 8,700 0.25 4,100 2,500 0.23 40 to 60 16,000 13,000 0.29 4,700 3,500 0.28

Sandy Silt (ML) Maximum 95% Standard Compaction 85% Standard Compaction

Principal Stress E B ν E B ν Level (psi) (psi) (psi) (psi) (psi)

0 to 1 1,800 1,900 0.34 600 400 0.25 1 to 5 2,500 2,000 0.29 700 450 0.24

5 to 10 2,900 2,100 0.27 800 500 0.23 10 to 20 3,200 2,500 0.29 850 700 0.30 20 to 40 3,700 3,400 0.32 900 1,200 0.38 40 to 60 4,100 4,500 0.35 1,000 1,800 0.41

Silty Clay (CL) Maximum 95% Standard Compaction 85% Standard Compaction

Principal Stress E B ν E B ν Level (psi) (psi) (psi) (psi) (psi)

0 to 1 400 800 0.42 100 100 0.33 1 to 5 800 900 0.35 250 200 0.29

5 to 10 1,100 1,000 0.32 400 300 0.28 10 to 20 1,300 1,100 0.30 600 400 0.25 20 to 40 1,400 1,600 0.35 700 800 0.35

60 1,500 2,100 0.38 800 1,300 0.40

2.2 Elasto-Plastic

The Mohr-Coulomb failure criterion is used in many geotechnical engineering applications to

describe the shear strength of soil. The principal feature of the Mohr-Coulomb criterion is that

strength is dependent on confining stress. In common applications soil strength is described by

a friction angle and cohesion intercept. In this study, we report on the 3-D Mohr-Coulomb

model as implemented in Plaxis. This model uses an elastic perfectly-plastic constitutive model

(Brinkgreve and Broere, 2004). For stress states within the yield surface, the soil behavior is


elastic and is determined by isotropic linear elasticity, as described in Section 2.1. The Mohr-

Coulomb yield condition consists of six yield functions as shown below:

0cossin22

0cossin22

0cossin22

0cossin22

0cossin22

0cossin22

12123

21213

31312

13132

23231

32321

≤−+

+−

=

≤−+

+−

=

≤−+

+−

=

≤−+

+−

=

≤−+

+−

=

≤−+

+−

=

φφσσσσ

φφσσσσ

φφσσσσ

φφσσσσ

φφσσσσ

φφσσσσ

cf

cf

cf

cf

cf

cf

b

a

b

a

b

a

(1)

For these six yield functions, a non-associated flow rule is used, and six plastic potential

functions are introduced:

ψσσσσ

ψσσσσ

ψσσσσ

ψσσσσ

ψσσσσ

ψσσσσ

sin22

sin22

sin22

sin22

sin22

sin22

12123

21213

31312

13132

23231

32321

++

−=

++

−=

++

−=

++

−=

++

−=

++

−=

b

a

b

a

b

a

g

g

g

g

g

g

(2)

where ψ is a dilatancy angle.

Tensile failure of soil is captured by specifying a tension cut-off. Three yield functions are

defined for the tension cut-off:


000

36

25

14

≤−=≤−=≤−=

t

t

t

fff

σσσσσσ

(3)

where tσ is allowable tensile stress. For these three yield functions for tension cut-off, an

associated flow rule is used.

Basic input parameters required for the Mohr-Coulomb model are modulus of elasticity, E ,

Poisson’s ratio, ν , cohesion, c , angle of friction, φ , dilatancy angle, ψ , and tensile strength,

tσ .

In this study, c for the Mohr-Coulomb model is the same value as in the Duncan-Selig model,

which is given in Table 4. Although cohesion of SW85 is 0 psi for the Duncan-Selig model,

0.001 psi is assigned to cohesion in Plaxis for numerical stability. Elastic constants, E and ν ,

are selected from Table 1 based on the vertical stress at a given depth. Table 2 shows

expected vertical and horizontal stresses in the soil at a given depth under gravity for SW85.

Table 2 also shows corresponding angle of friction at a given depth. In the Duncan-Selig model,

the angle of friction is also a function of confinement stress. To determine the angle of friction at

a certain depth, the horizontal stress was estimated by the empirical formula 13 )sin1( σφσ −=

(Jaky, 1944), using the vertical stress as the first principal stress.

In this study, four sets of elastic constants for SW85 are identified for ranges from 0 ft to 1 ft,

from 1 ft to 6 ft, from 6 ft to 11 ft, and from 11 ft to 18 ft, which correspond to the stress ranges

in Table 1, and are shown in Table 3. Angle of friction in Table 3 is an average value of angle of

friction for each range of depth. Table 3 also gives dilatation angles that were estimated by

subtracting 30 deg from friction angles.


Table 2—Vertical Stresses and Estimated Horizontal Stresses under Gravity and Corresponding Angle of Friction for SW85

Depth 1σ 3σ 13 /σσ φ (ft) (psi) (psi) (psi) (deg) 0.5 0.44 0.14 0.331 42.01 1.5 1.31 0.45 0.344 41.03 2.5 2.19 0.76 0.350 40.57 3.5 3.06 1.08 0.354 40.27 4.5 3.94 1.40 0.357 40.04 5.5 4.81 1.73 0.359 39.86 6.5 5.69 2.05 0.361 39.71 7.5 6.56 2.38 0.363 39.58 8.5 7.44 2.71 0.364 39.47 9.5 8.31 3.04 0.366 39.37 10.5 9.19 3.37 0.367 39.28 11.5 10.06 3.70 0.368 39.20 12.5 10.94 4.04 0.369 39.12 13.5 11.81 4.37 0.370 39.05 14.5 12.69 4.70 0.371 38.99 15.5 13.56 5.04 0.372 38.93 16.5 14.44 5.38 0.372 38.87 17.5 15.31 5.71 0.373 38.82

Table 3—Parameters for Linear-Elastic and Mohr-Coulomb Models for SW85

Depth Modulus of Elasticity

E

Poisson’s Ratio ν

Angle of Friction φ

Dilatation Angle ψ

Cohision c

(ft) (psi) (deg) (deg) (psi) 0 to 1 1,300 0.26 42.0 12.0 0.001

1 to 6 2,100 0.21 40.4 10.4 0.001

6 to 11 2,600 0.19 39.5 9.5 0.001

11 to 18 3,300 0.19 39.0 9.0 0.001

2.3 Stress-Dependent Models

2.3.1 Duncan-Selig Model

The Duncan-Selig model is a composite of the Duncan hyperbolic Young's modulus (Duncan et

al, 1980) and the Selig hyperbolic bulk modulus (Selig, 1988). These models were developed in

2D to specifically address aspects of soil behavior that are important in culvert design. Under


this model Selig developed parameters for the Duncan Young's modulus model (Selig, 1988)

and two sets of parameters for the bulk modulus (Selig, 1988 and Selig, 1990). The 1988

properties for Young's and bulk modulus were used by AASHTO in the development of standard

designs for reinforced concrete pipe and later for the development of the one-dimensional

modulus values adopted by AASHTO for thermoplastic pipe design. The bulk modulus values

proposed by Selig in 1990 are higher and produce an overall soil stiffness about twice as stiff as

the 1988 values (McGrath, 1998).

The Duncan-Selig model provides non-linear behavior and includes the Mohr-Coulomb failure

criterion; however, the formulation is elastic and includes no plasticity. The soil stiffening or

softening is based largely on the confining stress, 3σ , and the ratio of the deviator stress

relative to the ultimate stress.

The Duncan-Selig stress-strain relationship in the triaxial test during deviatoric loading can be

represented by a hyperbolic equation of the form

ui qE

q1

1

1 εε

+=

(4)

where

q : deviator stress (= 31 σσ − , 1σ =maximum principal stress, and 3σ =minimum principal stress)

1ε : maximum principal strain

iE : initial tangent modulus

uq : ultimate deviator stress at large strain

The initial tangent modulus, iE , is assumed to increase with confining pressure as given by

n

aai P

KPE

= 3σ

(5)

where

aP : atmospheric pressure (=14.7 psi, used to non-dimensionalize the parameters K and n )


K : non-dimensional parameter

n : non-dimensional parameter

The hyperbolic soil model is considered to be valid up to soil failure. Thus, the ultimate deviator

stress is defined in terms of the actual failure deviator stress by the failure ratio,

u

ff q

qR =

(6)

The failure envelope is expressed by

φφσφ

sin1sin2cos2 3

−+

=c

q f

(7)

where

fq : deviator stress at failure

c : cohesion

φ : angle of friction

In this model, the angle of friction is a function of the confining stress and expressed as

∆−=

ao P

310log

σφφφ

(8)

where

oφ : value of φ when aP=3σ

φ∆ : reduction in φ for a ten-fold increase in 3σ

By differentiating Eq. 4 with respect to 1ε , the tangent modulus, E , can be expressed as:

n

aa

f

PKP

CqR

qE

+

−−=

∂∂

=

3

2

3

1

sin2cos2)sin1(

1 σ

φσφφ

ε

(9)

The mean stress, mσ , can reasonably be represented by the hyperbolic equation:


uv

vim

Bεε

εσ

/1−=

(10)

where

iB : initial tangent bulk modulus

vε : volumetric strain

uε : ultimate volumetric strain

Therefore, the tangent bulk modulus, B , is determined by

2

1

+=

∂∂

=

ui

mi

vol

m

BB

B

εσ

εσ

(11)

Based on the theory of elasticity, Poisson’s ratio, ν , and shear modulus, G , can be expressed

by using E and B .

The Selig, 1988 parameters for the Duncan-Selig model for backfill are summarized in Table 4.

Table 4—Soil Properties of Backfill for Duncan-Selig Model (Selig, 1988)

Soil Type

Standard Compaction Density K n fR ai PB / uε c 0φ φ∆

(%) (pcf) (psi) (deg) (deg)

Gravelly Sand (SW)

95 141 950 0.60 0.70 74.8 0.02 0 48 8 90 134 640 0.43 0.75 40.8 0.05 0 42 4 85 126 450 0.35 0.80 12.7 0.08 0 38 2 80 119 320 0.35 0.83 6.1 0.11 0 36 1 60 91 54 0.85 0.90 1.7 0.23 0 29 0

Sandy Silt

(ML)

95 127 440 0.40 0.95 48.3 0.06 4.0 34 0 90 120 200 0.26 0.89 18.4 0.10 3.5 32 0 85 114 110 0.25 0.85 9.5 0.14 3.0 30 0 80 107 75 0.25 0.80 5.1 0.19 2.5 28 0 60 66 16 0.95 0.55 1.3 0.43 0 23 0

Silty Clay (CL)

95 119 120 0.45 1.00 21.1 0.13 9.0 15 4 90 112 75 0.54 0.94 10.2 0.17 7.0 17 7 85 106 50 0.60 0.90 5.2 0.21 6.0 18 8 80 100 35 0.66 0.87 3.5 0.25 5.0 19 8.5 60 56 16 0.95 0.75 0.7 0.55 0 23 11


2.3.2 Hardening Soil Model (Plaxis)

Two types of hardening can be modeled by the Hardening-Soil model (Brinkgreve and Broere,

2004): shear hardening due to primary deviatoric loading and compression hardening due to

primary compression. A basic feature of the Hardening-Soil model in Plaxis is the stress

dependency of soil stiffness and the hyperbolic relationship between the vertical strain and the

deviatoric stress in primary triaxial loading. This model uses a yield function given below:

pv

ppuru

p

Eq

qqq

Ef

ff

εεγ

γ

−=

−−

=

≤−=

1

50

2

2/1

10

(12)

where

pγ : plastic shear strain as a hardening parameter

50E : confining stress dependent secant modulus at 50% strength for primary loading

urE : confining stress dependent unloading/reloading modulus

p1ε : plastic strain in the 1-principal direction

pvε : plastic volumetric strain

50E and urE are dependent on confinement and evaluated by the following power lows:

m

refref

pcc

EE

++

=φ

σφcotcot 3

5050

(13)

and

m

refrefurur pc

cEE

++

=φ

σφcotcot 3

(14)

where

refp : reference confining pressure

refE50 : reference modulus for primary loading corresponding to the reference confining

pressure refp


refurE : reference modulus for unloading and reloading corresponding to the reference

confining pressure refp

When the stress state is on the yield surface, 0=f , and

uru

pv

pp

Eq

qqq

E2

/11250

1 −−

=−= εεγ (15)

For hard soils, plastic volumetric strain tends to be small; therefore, p1ε can be approximated by

uru

p

Eq

qqq

E−

−≈

/121

501ε (16)

In the triaxial test stress path, urE remains constant since confinement stress is constant, and

elastic strains during the deviatoric loading are evaluated by using urE and urν as follows:

ur

e

Eq

=1ε and ur

uree

Eqνεε −== 32 (17)

When the plastic volumetric strain is small, the axial strain in the deviatoric loading of the triaxial

test can be expressed by a hyperbolic stress-strain curve as follows:

u

pe

qqq

E /121

50111 −≈+= εεε (18)

The relationship between the plastic shear strain rate and the plastic volumetric strain rate is

specified in the linear form:

pm

pv γψε sin= (19)

where mψ is mobilized dilatancy angle. In the Hardening-Soil model, the following expression

is used for mψsin .

cvm

cvmm φφ

φφψ

sinsin1sinsin

sin−

−=

(20)


where cvφ and mφ are the critical state friction angle and the mobilized friction angle. mφsin

and cvφsin are evaluated by the following equations:

φσσσσ

φcot2

sin31

31

cm ++−

=

(21)

and

ψφψφφ

sinsin1sinsinsin

−−

=cv

(22)

When the failure criterion is satisfied ( fqq = ), the yield surface stops increasing in size, and

perfectly plastic yielding occurs.

The Hardening-Soil model used a cap type yield surface to account for plastic volumetric strain

due to primary compression in isotropic compression or oedometer loading. The cap yield

surface is defined by

0~

222

2

≤−+= pc ppqfα

(23)

where α is an auxiliary model parameter that relates to ncK 0 (= 0K -value for normal

consolidation), p is a mean stress, and pp is the isotropic pre-consolidation stress. q~ is a

special stress measure for deviatoric stresses and has the following expression:

φφδ

δσσδσ

sin3sin3

)1(~321

−+

=

−−+=q

(24)

The Hardening-Soil model used the following hardening law relating pp to volumetric cap strain

pcvε (=plastic volumetric strain in isotropic compression):

m

refppc

v pp

m

−

−

=1

1βε

(25)


where β is another model parameter that relates to refoedE (= tangent modulus for primary

oedometer loading at a vertical stress of refp=1σ ). α and β are calculated internally in

Plaxis based on ncK 0 and refoedE , respectively.

The tangent oedometer modulus is also defined by a power low:

m

refrefoedoed pc

cEE

++

=φ

σφcotcot 1

(26)

Tensile failure of soil is captured by specifying a tension cut-off as described for the Mohr-

Coulomb model.

Basic input parameters for the Hardening-Soil model are c , φ , ψ , refE50 , refoedE , m , ref

urE , urν ,

refp , ncK 0 , fR , and tσ . Some of the parameters are assigned to the default values as shown

below:

refurE : refE503

urν : 0.2

refp : 100 stress units

ncK 0 : φsin1−

fR : 0.9

tσ : 0 stress units

To compare the Hardening-Soil model with the Duncan-Selig model, we conducted a set of

analyses in Plaxis 3D to investigate the actual 3D condition. The analyses consisted of

simulation of triaxial and oedometer tests for SW85, SW90, ML85, and CL85 soils. For the

triaxial test, we considered confining pressure of 1 psi, 2 psi, and 5 psi. The analysis model in

Plaxis 3D consists of a cube of soil (1 in. by 1 in. by 1 in.).

To determine input parameters of the Hardening-Soil model, we created general rules. Our goal

was to match the soil behavior between the two models at a confining pressure of 2 psi. The

rules are:

• Use atmospheric pressure, aP , as the reference pressure, refp


• Use a value equal to iE5.0 for refE50 to match the initial tangent modulus in the deviatoric loading of the triaxial test, instead of using the secant modulus of the Duncan-Selig model at a deviator stress of 50% of the failure stress

• Use a value equal to refE50 for refoedE , instead of using the tangent modulus of the

Duncan-Selig model in the oedometer loading at a vertical stress of refp

• Use the same friction angle as the Duncan-Selig model at a confining pressure of 2 psi

• Use n for m

• Use fR of the Duncan-Selig model, instead of using a default value

• Use c of the Duncan-Selig model, but use 0.001 psi when it is 0 psi in the Duncan-Selig model

• Use deg30−φ as ψ

• Use default values for other parameters if possible

Table 5 shows input parameters determined by the general rules described above. For SW90,

Plaxis 3D did not allow us to use refrefoed EE 50= and φsin10 −=ncK . In this case only, we

increased ncK 0 from 0.287 (= φsin1− ) to 0.310 to use refrefoed EE 50= .

Figure 1 through Figure 4 show relationships between deviator stress and vertical strain during

the deviatoric loading of the triaxial test predicted by Plaxis 3D with the Hardening-Soil model

for SW85, SW90, ML85, and CL85. These figures also show the Duncan-Selig hyperbolic

model predictions by Eq. 4 for comparison. Note that failure stress of the Hardening-Soil model

is different from that of the Duncan-Selig for some cases because the angle of friction is not

dependent on confining pressure. For SW85 and SW90, results from the Plaxis Hardening-Soil

model closely match the Duncan-Selig hyperbolic soil model. For ML85, although stiffness

predicted by the Plaxis Hardening-Soil model is slightly lower, results match the Duncan-Selig

model relatively well. However, for CL85, the stiffness predicted by the Plaxis Hardening-Soil

model is significantly lower than that of Duncan-Selig. As explained above, pvε is not small for

the soft soil, which leads to a larger vertical strain for the same stress and causes the stress-

strain relationship to deviate from the hyperbolic equation give in Eq. 18.

Figure 5 through Figure 8 show vertical stress and vertical strain relationships during the

oedometer loading predicted by Plaxis 3D with the Hardening-Soil model for SW85, SW90,

ML85, and CL85. These figures also show the stress-strain relationship of the Duncan-Selig

model as well as actual test data by Lin (1987).


The Duncan-Selig model always predicts a larger strain in the vertical stress range we

examined relative to actual test data, and SW85 is the poorest match among the four soil types

we examined. Lin (1987) also pointed out that the Duncan-Selig model for SW85 predicts the

oedometer stress-strain relationship much different from the actual test data. The Plaxis

Hardening-Soil model predicts much lower strain when compared to the Duncan-Selig model

and the actual test data at the same stress level.

Strains predicted by the Plaxis Hardening-Soil model for ML85 lie between the Duncan-Selig

model and the actual test data. Strains predicted by the Plaxis Hardening-Soil model for CL85

are in good agreement with those of the Duncan-Selig model up to a vertical stress of 5 psi, and

they become significantly larger than those of either the Duncan-Selig model or the actual test

data.

Table 5—Input Parameters for Hardening-Soil Model for SW85, SW90, ML85, and CL85

Input Parameters SW85 SW90 ML85 CL85 c (psi) 0.001 0.001 3 6 φ (deg) 39.7 45.5 30 24.9 ψ (deg) 9.7 15.5 0 0

refE50 (psi) 3,308 4,704 633 161 refoedE (psi) 3,308 4,704 633 161

m 0.35 0.75 0.25 0.6 refurE (psi) 9,924 14,112 1,900 482

urν 0.2 0.2 0.2 0.2

fR 0.8 0.75 0.85 0.9 ncK 0 0.361 0.310 0.500 0.578

tσ (psi) 0 0 0 0 refp (psi) 14.7 14.7 14.7 14.7


Figure 1—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85

in Deviatoric Loading of Triaxial Test

(a) σ3 = 1 psi

(b) σ3 = 2 psi

(c) σ3 = 5 psi

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.000 0.002 0.004 0.006 0.008 0.010

Vertical Strain (in/in)

Dev

iato

r Str

ess

(psi

)

Duncan-Selig

Failure in Duncan-Selig

Plaxis Result

Failure in Plaxis

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.000 0.005 0.010 0.015 0.020


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

2

4

6

8

10

12

14

16

18

20

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis




(a) σ3 = 1 psi

(b) σ3 = 2 psi

(c) σ3 = 5 psi

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.000 0.002 0.004 0.006 0.008 0.010


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

2

4

6

8

10

12

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

5

10

15

20

25

30

0.000 0.005 0.010 0.015 0.020 0.025


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis


Figure 3—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85


(a) σ3 = 1 psi

(b) σ3 = 2 psi

(c) σ3 = 5 psi

0

2

4

6

8

10

12

14

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

2

4

6

8

10

12

14

16

18

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

5

10

15

20

25

0.00 0.05 0.10 0.15 0.20


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis


Figure 4—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85


(a) σ3 = 1 psi

(b) σ3 = 2 psi

(c) σ3 = 5 psi

0

5

10

15

20

25

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

5

10

15

20

25

30

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis

0

5

10

15

20

25

0.00 0.50 1.00 1.50 2.00


Dev

iato

r Str

ess

(psi

)

Duncan-Selig


Plaxis Result

Failure in Plaxis



in Oedometer Loading



0

10

20

30

40

50

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035


Vert

ical

Str

ess

(psi

) Duncan-SeligPlaxis ResultActual Test Data

0

10

20

30

40

50

0.000 0.005 0.010 0.015 0.020 0.025


Vert

ical

Str

ess

(psi

)

Duncan-SeligPlaxis ResultActual Test Data (SW85)Actual Test Data (SW95)


Figure 7—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85


Figure 8—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85


0

10

20

30

40

50

0.00 0.02 0.04 0.06 0.08 0.10


Vert

ical

Str

ess

(psi

)

Duncan-SeligPlaxis ResultActual Test Data

0

10

20

30

40

50

0.00 0.05 0.10 0.15 0.20 0.25


Vert

ical

Str

ess

(psi

)

Duncan-SeligPlaxis ResultActual Test Data


2.4 Findings from Soil Model Evaluation

The soil model comparison shows that the Young's modulus values of the Duncan-Selig and

Hardening Soil models are similar for SW85, SW90, ML85, and CL85 soils, especially SW soils.

The predicted one-dimensional stress-strain curves do not compare as well. We conclude that

the differences are a result of the parameters and not fundamental faults with the model. As

noted above, these differences have been observed previously. In general, having a lower

bound for the stiffness is an appropriate design decision, but may not be the best approach for

establishing design equations. It is possible to modify input parameters of the Hardening-Soil

model to better match the soil behavior in the triaxial and oedometer tests for each soil type and

a given compaction level. However, we used the established design parameters given in Table

5 for the preliminary 2D analysis presented in the next section. Final design parametric study

may require modified parameters if we choose the Hardening-Soil model for the parametric

study.

3. TWO-DIMENSIONAL MODELING OF CULVERTS

Soil models were initially tested in 2D to reduce computational time and allow a broader range

of structure types to be evaluated. We performed a set of preliminary 2D analyses in Plaxis 2D

Version 8 (Brinkgreve, 2002) on six structural types including concrete box, concrete pipe,

concrete arch, metal pipe, metal arch, and thermoplastic pipe for selected cover depths. By

using linear-elastic, Mohr-Coulomb (linear-elastic model with post-failure plasticity), and

Hardening-Soil models (stress-dependent stiffness (=shear hardening) with post-failure

plasticity and compression hardening) available in Plaxis 2D, we examined the effects of

different levels of sophistication of soil models on structural response to surface live loads.

3.1 Modeled Structures

We selected six structures as shown in Table 6. For each structure, we examined cases of

different cover depths as shown in Table 6.

Concrete box: Reinforced concrete box section (12 ft by 6 ft by 12 in.) with a compressive

strength of 5,000 psi specified in ASTM C1433 for HS20 live load conditions.

Concrete pipe: Class II reinforced concrete pipe with an internal diameter of 48 in., a wall

thickness of 5 in., and a compressive strength of 4,000 psi specified in ASTM C76 as Wall B.


Concrete arch: BEBO arch culvert with an inside span of 30 ft, an inside rise of 11 ft 4 in., a wall

thickness of 10 in., and a compressive strength of 4,200 psi, designated as BEBO Type E30/3.

Metal pipe: Corrugated steel pipe with an internal diameter of 48 in. Steel plates have 2-2/3 in.

by 1/2 in. corrugations with an uncoated thickness of 0.0598 in.

Metal arch: Corrugated steel arch culvert with a maximum span of 31 ft 7 in. and a total rise of

12 ft 1 in. (dimensions are to inside crests), designated as Type 108A30 by Contech. Steel

plates have 6 in. by 2 in. corrugations with an uncoated thickness of 0.215 in.

Thermoplastic pipe: Corrugated high-density polyethylene pipe with an internal diameter of 60

in. and Type S corrugation manufactured by Hancor, Inc. of Findlay, Ohio.

Table 6—Structural Types and Cover Depths for 2D Analysis

Structural Type Span Cover Depth Availability of Field Data

Concrete Box 12 ft 0 ft, 2 ft, and 6 ft No

Concrete Pipe 4 ft 2 ft and 6 ft No

Concrete Arch 30 ft 2 ft and 6 ft Yes

Metal Pipe 4 ft 2 ft and 6 ft No

Metal Arch 31 ft 7 in. 2 ft and 6 ft Yes

Thermoplastic Pipe 5 ft 2 ft and 6 ft Yes

3.2 Material Models

For each structural type and for each cover depth, we performed three analyses with the three

soil models: linear-elastic model, Mohr-Coulomb model, and Hardening-Soil model. In this set

of preliminary analyses, we used only SW85 as backfill material. For the linear-elastic

properties of backfill, we varied the soil modulus based on depth of fill.

3.2.1 Linear-Elastic Model

The linear elastic soil model was described in Section 2.1 above. As noted, linear elastic model

is the simplest constitutive soil models. Since elastic constants vary with soil stress level, we

used elastic constants of SW85 recommended by Selig (1990) as shown in Table 1.

3.2.2 Mohr-Coulomb Model with Perfect Plasticity in Plaxis

The Mohr-Coulomb model in Plaxis uses an elastic perfectly-plastic constitutive model as

described in Section 2.2. For stress states within the yield surface, the soil behavior is elastic


and is determined by isotropic linear elasticity. The yield condition is expressed by Mohr-

Coulomb failure condition. Tensile failure of soil is captured by specifying a tension cut-off.

Input parameters of the Mohr-Coulomb model for SW85 are given in Table 3 except for tensile

strength. A tensile strength of 0 psi is used in this study.

3.2.3 Hardening-Soil Model in Plaxis

The Plaxis Hardening-Soil model was described in Section 2.3.2. A basic feature of the

Hardening-Soil model is the stress dependency of soil stiffness and the hyperbolic relationship

between the vertical strain and the deviatoric stress in primary triaxial loading. Two types of

hardening are modeled: shear hardening and compression hardening. The shear yield surface

increases in size until the Mohr-Coulomb failure criterion is satisfied, at which point perfectly

plastic yielding occurs. A cap type yield surface is used to account for plastic volumetric strain

due to primary compression in isotropic compression or oedometer loading. Tensile failure of

soil is captured by specifying a tension cut-off. Basic input parameters of the Hardening-Soil

model for SW85 are given in Table 5.

3.2.4 In-Situ Soil Material

We used a linear-elastic model with E of 3,000 psi, ν of 0.25, and γ of 126 pcf for in-situ

material.

3.2.5 Other Materials

For other materials we used a linear-elastic model with properties as listed:

• steel - E = 29,000,000 psi, ν of 0.3, and γ of 490 pcf,

• concrete - E = 57,000 cf ′ , ν of 0.17, and γ of 150 pcf, where cf ′ is a specified compressive strength in psi, and

• high-density polyethylene - E = 80,000 psi, ν of 0.35, and γ of 59.5 pcf.

3.3 Live Load

One of the key shortcomings of performing a 2D analysis to examine culvert structural response

is that the distribution of live load along the length of the culvert cannot be modeled. 3D

behavior must be addressed by modifying the load applied to the surface of the 2D finite

element mesh. While developing equations for this purpose is one of the goals of this project,

we have taken equations from codes and the literature that are suitable for the immediate


purpose of evaluating soil models. We used Eq. 27 for box culverts, Eq. 28 for pipes, and Eq.

29 for arches to calculate the live load per unit length of culvert.

≥++

+

<+

+

=ftH

HSPIMm

ftHS

PIMm

Wmpf

mpf

LL

0.2for 15.172.04.20

)100/1(

0.2for 72.048

)100/1(

(27)

++

+=

HLR

HwPIMm

Wt

t

t

mpfLL 15.1

7.015.1

)100/1(

(28)

++

+=

HLR

HwPIMm

Wt

t

t

mpfLL 15.1

7.0)15.1(3)100/1(

(29)

where

LLW : live load per unit length of culvert, lb/in.

mpfm : multiple presence factor (=1.2, AASHTO LRFD 3.6.1.1.2)

IM : dynamic load allowance (=33(1.0-0.125H/12)≥0%, AASHTO LRFD 3.6.2.2), %

P : wheel load magnitude (=16,000 lb, AASHTO LRFD 3.6.1.2.2), lb

S : clear span, ft

H : depth of cover from road surface to top of culvert, in.

tw : width of tire footprint at surface (=20 in., AASHTO LRFD 3.6.1.2.5), in.

tL : length of tire footprint at surface (=10 in., AASHTO LRFD 3.6.1.2.5), in.

tR : mean culvert radius (top radius for arches), in.

According to NCHRP 473 (McGrath, 2002), the live load calculated by Eq. 28 results in

reasonable thrusts but greater moments and deflections when compared to those from 3D

analysis, and the live load calculated by Eq. 29 results in reasonable moments but inaccurate

deflections and thrusts. For the purpose of comparison of different soil models, we used Eq. 28

for pipes and Eq. 29 for arches. Figure 9 shows live load per unit length of a culvert to be used

for 2D analyses of structures listed in Table 6.


Figure 9—Live Load per Unit Length of Culvert in 2D Analysis

3.4 Finite Element Model

The finite element model includes the buried structure, in-situ soil, and backfill. Conceptual

models are shown in Figure 10 through Figure 12. Bedding thickness, bH , in this figure is

specified in Section 27.5 of AASHTO LRFD Bridge Construction Specifications (AASHTO,

2004). Thirty-nine models were created for six types of structures, 2 ft and 6 ft of cover depth

(in addition, 0 ft for concrete box culvert), and three soil constitutive models (linear-elastic,

Mohr-Coulomb, and Hardening-Soil models). The first 4 in. of backfill from the surface is always

modeled by linear-elastic soil model to prevent the soil from failing under the applied live loads.

The bottom of the model is restrained in the vertical and horizontal directions and the sides of

the model are restrained in the horizontal direction.

For the finite element models with either the Mohr-Coulomb soil model or the Hardening-Soil

model, soil was placed incrementally. In-situ soil was placed at once in the first stage. Backfill

soil was placed with about 1 ft increments.

For the finite element models with the linear-elastic soil model, full bonding was assumed at the

interface between the soil and the structure. For the models with either the Mohr-Coulomb soil

model or the Hardening-Soil model, the interface strength (friction and adhesion) was

considered. In Plaxis 2D, the interface strength is specified by a fraction of the soil strength as

follows:

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8

Cover depth (ft)

Live

load

per

uni

t len

gth

of c

ulve

rt

(lb/in

)

Concrete BoxConcrete PipeConcrete ArchMetal PipeMetal ArchThermoplastic Pipe


)tan(tan int φσφστ nerinii cRc +=+= (30)

where

iτ : interface strength

ic : cohesion of the interface

iφ : friction angle of the interface

nσ : normal stress

erRint : strength reduction factor for the interface

c : cohesion of the soil

φ : friction angle of the soil

We used 0.5 for erRint in this study.

Figure 13 through Figure 18 show finite element meshes for the six different structures.

Figure 10—Conceptual Model for 2D Analysis of Pipes

Wst

Hst

1.5Hst

6Wst

10 in.

Hb

Wst

Hst

1.5Hst

6Wst

10 in.

Hb

In-situ soil

Backfill (SW85)


Figure 11—Conceptual Model for 2D Analysis of Boxes

Figure 12—Conceptual Model for 2D Analysis of Arches

Wst

Hst

1.5Hst

6Wst

10 in.

Hb

Wst

Hst

1.5Hst

6Wst

10 in.

Hb

In-situ soil

Backfill (SW85)

46 ft

192 ft

28 ft

2 in

.

10 ft 2 in.

18 ft

7724

10 in.

46 ft

192 ft

28 ft

2 in

.

10 ft 2 in.

18 ft

7724

46 ft

192 ft

28 ft

2 in

.

10 ft 2 in.

18 ft

7724

10 in.10 in.

In-situ soil

SW85


Figure 13—Finite Element Meshes of Concrete Box Model

(a) 0 ft cover depth

(b) 2 ft cover depth

(c) 6 ft cover depth

72 ft

9 ft

13 ft 7 ft


Figure 14—Finite Element Meshes of Concrete Pipe Model



24 ft

6 ft

26.5 in.


Figure 15—Finite Element Meshes of Metal Pipe Model



24 ft

6 ft

24.3 in.


Figure 16—Finite Element Meshes of Plastic Pipe Model



30 ft

7.5 ft

31.4 in.


Figure 17—Finite Element Meshes of Concrete Arch Model


(b) 2 ft cover depth (close-up)

192 ft

18 ft

10 ft 2 in. 11 ft 9 in.

30 ft 10 in.

46 ft

83 ft 8 in.


(d) 6 ft cover depth (close-up)


Figure 18—Finite Element Meshes of Metal Arch Model


(b) 2 ft cover depth (close-up)

192 ft

18 ft

10 ft 2 in. 12 ft 2.5 in.

31 ft 4 in.

46 ft

83 ft 8 in.


(d) 6 ft cover depth (close-up)


3.5 Results of 2D Analysis

3.5.1 Concrete Box

Figure 19 shows deformations of the concrete box due to live load. Figure 20 through Figure 25

show bending moments and thrusts in the concrete box due to the live load for different the soil

models and cover depths considered. Table 7 summarizes the results. The linear-elastic soil

model resulted in greater positive bending moments at the center of the top slab and greater

negative bending moments at the tip of the upper haunch in the wall than the other two soil

models except for a cover depth of 0 ft. The linear-elastic soil model resulted in thrusts in the

top slab and the walls that are between Mohr-Coulomb and Hardening-Soil models in size for a

cover depth of 0 ft, and resulted in greater thrusts in the top slab and the walls for cover depths

of 2 ft and 6ft. The Mohr-Coulomb model resulted in greater positive and negative moments

than the Hardening-Soil model, except for the positive moment in the top slab for a cover depth

of 2 ft.


Figure 19—Deformation of Concrete Box due to Live Load

Undeformed

Linear-Elastic Model

Mohr-Coulomb Model

Hardening-Soil Model 300X Deformation

Undeformed


Mohr-Coulomb Model


Undeformed


Mohr-Coulomb Model


(a) 0 ft cover

(b) 2 ft cover

(c) 6 ft cover


Figure 20—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box

Model (0 ft Cover)

(a) Bending Moment

(b) Thrust

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

12000

-80 -60 -40 -20 0 20 40 60 80X-Coordinates (in.)

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model

Hardening-Soil Model

-106

-104

-102

-100

-98

-96

-94

-92

-90

-88

-80 -60 -40 -20 0 20 40 60 80X-Coordinates (in.)

Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)


Mohr-Coulomb Model



Figure 21—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete

Box Model (0 ft Cover)

(a) Bending Moment

(b) Thrust

-9000

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

-50 -40 -30 -20 -10 0 10 20 30 40 50Y-Coordinates (in.)

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-230

-220

-210

-200

-190

-180

-170

-160

-150

-50 -40 -30 -20 -10 0 10 20 30 40 50Y-Coordinates (in.)

Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)


Mohr-Coulomb Model




Model (2 ft Cover)

(a) Bending Moment

(b) Thrust

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-80 -60 -40 -20 0 20 40 60 80X-Coordinates (in.)

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-100

-80

-60

-40

-20

0

20

40

60

80

-80 -60 -40 -20 0 20 40 60 80X-Coordinates (in.)

Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)


Mohr-Coulomb Model





(a) Bending Moment

(b) Thrust

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

-50 -40 -30 -20 -10 0 10 20 30 40 50Y-Coordinates (in.)

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-220

-210

-200

-190

-180

-170

-160

-150

-140

-130

-120

-50 -40 -30 -20 -10 0 10 20 30 40 50Y-Coordinates (in.)

Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)


Mohr-Coulomb Model




Model (6 ft Cover)

(a) Bending Moment

(b) Thrust

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

-80 -60 -40 -20 0 20 40 60 80X-Coordinates (in.)

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-30

-20

-10

0

10

20

30

40

50

60

-80 -60 -40 -20 0 20 40 60 80X-Coordinates (in.)

Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)


Mohr-Coulomb Model





(a) Bending Moment

(b) Thrust

-2500

-2000

-1500

-1000

-500

0

-50 -40 -30 -20 -10 0 10 20 30 40 50Y-Coordinates (in.)

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-90

-85

-80

-75

-70

-65

-60

-55

-50

-45

-40

-50 -40 -30 -20 -10 0 10 20 30 40 50Y-Coordinates (in.)

Thru

st d

ue to

Liv

e Lo

ad (l

b/in

) Linear-Elastic Model

Mohr-Coulomb Model



Table 7—Comparison of Bending Moments and Thrusts in Concrete Box Model

(a) Bending Moments and Thrusts due to Earth Load

(b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

(c) Bending Moments and Thrusts due to Surface Live Load

Cover Soil Moment (lb*in/in) Thrust (lb/in)Depth Model Top Center Tip of Haunch Top Center Tip of Haunch

(ft) Top Wall Wall0 Linear-Elastic 2,208 333 -2,408 84 -134

Mohr-Coulomb 1,908 32 -1,956 33 -120Hardening-Soil 1,881 5 -1,908 29 -120

2 Linear-Elastic 5,171 -532 -5,520 75 -341Mohr-Coulomb 4,940 -812 -5,215 12 -308Hardening-Soil 4,844 -909 -5,047 -1 -309

6 Linear-Elastic 9,774 -1,829 -10,494 49 -685Mohr-Coulomb 9,748 -2,131 -10,474 -29 -618Hardening-Soil 9,567 -2,310 -10,134 -52 -621


(ft) Top Wall Wall0 Linear-Elastic 11,493 -3,344 -8,583 -12 -353

Mohr-Coulomb 11,233 -3,605 -8,175 -57 -338Hardening-Soil 11,134 -3,703 -8,001 -75 -338

2 Linear-Elastic 12,106 -3,661 -11,118 63 -542Mohr-Coulomb 11,536 -3,733 -10,635 78 -502Hardening-Soil 11,568 -3,947 -10,456 58 -499

6 Linear-Elastic 11,734 -2,674 -12,425 55 -767Mohr-Coulomb 11,102 -2,602 -12,036 10 -690Hardening-Soil 10,815 -2,711 -11,516 2 -687


(ft) Top Wall Wall0 Linear-Elastic 9,285 -3,677 -6,176 -96 -219

Mohr-Coulomb 9,325 -3,637 -6,220 -90 -218Hardening-Soil 9,253 -3,708 -6,092 -103 -219

2 Linear-Elastic 6,935 -3,130 -5,598 -13 -201Mohr-Coulomb 6,595 -2,921 -5,420 66 -195Hardening-Soil 6,724 -3,038 -5,409 59 -191

6 Linear-Elastic 1,959 -844 -1,931 5 -82Mohr-Coulomb 1,354 -471 -1,562 39 -72Hardening-Soil 1,248 -400 -1,382 54 -66


3.5.2 Concrete Pipe

Figure 26 shows deformations of the concrete pipe due to the live load. Figure 27 and Figure

28 show bending moments and thrusts in the concrete pipe due to the live load for different soil

models and cover depths, and Table 8 summarizes the results. The linear-elastic model

resulted in greater positive moments at the crown and invert and greater negative moments at

springlines than did the other two soil models for both cover depths of 2 ft and 6 ft. The linear-

elastic model resulted in greater thrusts at springlines for a cover depth of 2 ft and everywhere

for a cover depth of 6 ft. Between the Mohr-Coulomb and Hardening-Soil models, the Mohr-

Coulomb model resulted in greater peak positive and negative moments for both cover depths

of 2 ft and 6 ft. The Mohr-Coulomb model resulted in less thrusts except at the crown for a

cover depth of 6 ft.


Figure 26—Deformation of Concrete Pipe due to Live Load

(a) 2 ft cover

(b) 6 ft cover

Undeformed


Mohr-Coulomb Model


150X Deformation

Undeformed


Mohr-Coulomb Model


150X Deformation


Figure 27—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (2 ft

Cover)

(a) Bending Moment

(b) Thrust

-800

-600

-400

-200

0

200

400

600

800

1000

1200

-180 -135 -90 -45 0 45 90 135 180Degrees from Crown

Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-120

-100

-80

-60

-40

-20

0

20


Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)

Linear-Elastic Model Mohr-Coulomb ModelHardening-Soil Model


Figure 28—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (6 ft

Cover)

(a) Bending Moment

(b) Thrust

-80

-60

-40

-20

0

20

40

60

80

100


Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)

Linear-Elastic ModelMohr-Coulomb ModelHardening-Soil Model

-12

-10

-8

-6

-4

-2

0

2

4


Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)



Table 8—Comparison of Bending Moments and Thrusts in Concrete Pipe Model




Cover Soil Moment (lb*in/in) Thrust (lb/in)Depth (ft) Model Peak Pos Peak Neg Crown Springline

2 Linear-Elastic 655 -615 -5 -115Mohr-Coulomb 569 -503 -22 -90Hardening-Soil 495 -427 -27 -91

6 Linear-Elastic 1,282 -1,235 -49 -239Mohr-Coulomb 1,158 -1,096 -59 -194Hardening-Soil 1,020 -942 -72 -196


2 Linear-Elastic 1,439 -1,266 -5 -208Mohr-Coulomb 1,247 -1,113 -21 -160Hardening-Soil 1,191 -1,029 -28 -164

6 Linear-Elastic 1,344 -1,301 -50 -248Mohr-Coulomb 1,196 -1,141 -57 -197Hardening-Soil 1,053 -976 -71 -200


2 Linear-Elastic 933 -693 0.68 -92.61Mohr-Coulomb 856 -651 0.95 -69.52Hardening-Soil 852 -640 -1.30 -72.98

6 Linear-Elastic 77 -68 -0.30 -9.51Mohr-Coulomb 39 -45 1.52 -2.78Hardening-Soil 34 -35 1.57 -3.47


3.5.3 Metal Pipe

Figure 29 shows deformations of the metal pipe due to the live load. Figure 30 and Figure 31

show bending moments and thrusts in the metal pipe due to the live load for different soil


resulted in less peak positive and negative moments than did the other two soil models for both

cover depths of 2 ft and 6 ft. For a cover depth of 2 ft, the linear-elastic model resulted in less

thrusts at the crown and invert than did the other two models and greater thrusts at springlines

than did the Mohr-Coulomb model. For a cover depth of 6 ft, the linear-elastic model resulted in

greater thrusts at springlines than did the other two soil models and less thrusts at the crown

and invert than did the Mohr-Coulomb model. Between the Mohr-Coulomb and Hardening-Soil

models, the Mohr-Coulomb model resulted in greater peak positive and negative moments for

both cover depths of 2 ft and 6 ft. The Mohr-Coulomb model resulted in less thrusts everywhere

for a cover depth of 2 ft, whereas it resulted in greater thrusts everywhere for a cover depth of

6 ft.


Figure 29—Deformation of Metal Pipe due to Live Load

(a) 2 ft cover

(b) 6 ft cover

Undeformed


Mohr-Coulomb Model


50X Deformation

Undeformed


Mohr-Coulomb Model


50X Deformation


Figure 30—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (2 ft

Cover)

(a) Bending Moment

(b) Thrust

-150

-100

-50

0

50

100

150

200


Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-120

-100

-80

-60

-40

-20

0


Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)



Figure 31—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (6 ft

Cover)

(a) Bending Moment

(b) Thrust

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0


Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0


Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)



Table 9—Comparison of Bending Moments and Thrusts in Metal Pipe Model











2 Linear-Elastic 61 -27 -22.80 -82.59Mohr-Coulomb 166 -99 -59.55 -57.03Hardening-Soil 151 -91 -72.45 -68.27



3.5.4 Thermoplastic Pipe

Figure 32 shows deformations of the thermoplastic pipe due to the live load. Figure 33 and

Figure 34 show bending moments and thrusts in the thermoplastic pipe due to the live load for

different soil models and cover depths, and Table 10 summarizes the results. The linear-elastic

model resulted in less peak positive and negative moments than did the other two soil models

for both cover depths of 2 ft and 6 ft. For a cover depth of 2 ft, the linear-elastic model resulted

in less thrusts except a small portion near the springlines. For a cover depth of 6 ft, the linear-

elastic model resulted in greater thrusts at springlines and less thrusts at the crown and invert

than did the other two soil models. Between the Mohr-Coulomb and Hardening-Soil models, the

Mohr-Coulomb model resulted in greater peak positive and negative moments except for the

peak positive moment at a cover depth of 6 ft. The Mohr-Coulomb model resulted in greater

thrusts except at the crown for a cover depth of 2 ft.


Figure 32—Deformation of Thermoplastic Pipe due to Live Load

(a) 2 ft cover

(b) 6 ft cover

Undeformed


Mohr-Coulomb Model


30X Deformation

Undeformed


Mohr-Coulomb Model


30X Deformation


Figure 33—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model

(2 ft Cover)

(a) Bending Moment

(b) Thrust

-300

-200

-100

0

100

200

300

400


Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-160

-140

-120

-100

-80

-60

-40

-20

0

20


Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)



Figure 34—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model

(6 ft Cover)

(a) Bending Moment

(b) Thrust

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0


Ben

ding

Mom

ent d

ue to

Liv

e Lo

ad

(lb*in

/in)


Mohr-Coulomb Model


-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0


Thru

st d

ue to

Liv

e Lo

ad (l

b/in

)

Linear-Elastic ModelMohr-Coulomb ModelHardening-Soil Model


Table 10—Comparison of Bending Moments and Thrusts in Thermoplastic Pipe Model











2 Linear-Elastic 95 -43 7.80 -73.48Mohr-Coulomb 323 -210 -91.42 -71.47Hardening-Soil 269 -174 -95.22 -71.16

6 Linear-Elastic 4 -2 0.60 -6.44Mohr-Coulomb 5 -6 -3.42 -3.92Hardening-Soil 7 -5 -2.67 -3.20


3.5.5 Concrete Arch

Figure 35 shows deformations of the concrete arch due to the live load. Figure 36 and Figure

37 show bending moments and thrusts in the concrete arch due to the live load for different soil


resulted in less peak positive and negative moments for a cover depth of 2 ft than did the other

two soil models. For a cover depth of 6 ft, it resulted in a greater peak moment than did the

other two soil models and an intermediate peak negative moment among the three soil models.

The linear-elastic model always resulted in greater thrusts for both cover depths of 2 ft and 6 ft.

Between the Mohr-Coulomb and Hardening-Soil models, the Mohr-Coulomb model resulted in

greater peak positive and negative moments for both cover depths of 2 ft and 6 ft, and it

resulted in less thrusts everywhere for both cover depths of 2 ft and 6 ft.


Figure 35—Deformation of Concrete Arch due to Live Load

(a) 2 ft cover

(b) 6 ft cover

Undeformed


Mohr-Coulomb Model


Undeformed


Mohr-Coulomb Model



Figure 36—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (2 ft

Cover)

(a) Bending Moment

(b) Thrust

-15000

-10000

-5000

0

5000

10000

15000

20000

25000

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Figure 37—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (6 ft

Cover)

(a) Bending Moment

(b) Thrust

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Table 11—Comparison of Bending Moments and Thrusts in Concrete Arch Model





2 Linear-Elastic 6,630 -5,541 -546 -1,363Mohr-Coulomb 5,719 -4,125 -629 -1,088Hardening-Soil 5,388 -3,376 -644 -1,105

6 Linear-Elastic 14,343 -11,541 -973 -2,129Mohr-Coulomb 13,802 -10,131 -1,104 -1,798Hardening-Soil 13,744 -9,395 -1,118 -1,796


2 Linear-Elastic 26,532 -12,969 -788 -1,786Mohr-Coulomb 27,197 -11,849 -811 -1,457Hardening-Soil 26,081 -10,084 -843 -1,498

6 Linear-Elastic 16,522 -12,532 -1,001 -2,197Mohr-Coulomb 15,765 -11,139 -1,118 -1,857Hardening-Soil 15,572 -10,267 -1,132 -1,858


2 Linear-Elastic 19,911 -8,961 -241.76 -423.16Mohr-Coulomb 21,491 -9,891 -181.49 -368.86Hardening-Soil 20,707 -8,989 -199.11 -392.64



3.5.6 Metal Arch

Figure 38 shows deformations of the metal arch due to the live load. Figure 39 and Figure 40

show bending moments and thrusts in the metal arch due to the live load for different soil


resulted in less peak positive and negative moments for both cover depth of 2 ft and 6 ft than

did the other two soil models. The linear-elastic model resulted in less thrusts near the crown

and greater thrusts at other locations for both cover depths of 2 ft and 6 ft. Between the Mohr-

Coulomb and Hardening-Soil models, the Mohr-Coulomb model resulted in greater peak

positive and negative moments except for the peak negative moment for cover depths of 6 ft.

For a cover depth of 2 ft, the Mohr-Coulomb model resulted in greater thrusts except near the

crown, whereas for a cover depth of 6 ft, it resulted in less thrust except near the crown.


Figure 38—Deformation of Metal Arch due to Live Load

(a) 2 ft cover

(b) 6 ft cover

Undeformed


Mohr-Coulomb Model


Undeformed


Mohr-Coulomb Model



Figure 39—Bending Moments and Thrusts due to Live Load in Metal Arch Model (2 ft

Cover)

(a) Bending Moment

(b) Thrust

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Figure 40—Bending Moments and Thrusts due to Live Load in Metal Arch Model (6 ft

Cover)

(a) Bending Moment

(b) Thrust

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Table 12—Comparison of Bending Moments and Thrusts in Metal Arch Model





2 Linear-Elastic 496 -1,040 -445 -1,043Mohr-Coulomb 1,508 -1,727 -584 -771Hardening-Soil 1,996 -2,034 -602 -802

6 Linear-Elastic 489 -1,527 -901 -1,804Mohr-Coulomb 1,541 -1,461 -1,199 -1,463Hardening-Soil 1,452 -1,563 -1,210 -1,508


2 Linear-Elastic 5,866 -1,528 -820 -1,417Mohr-Coulomb 9,062 -4,294 -1,406 -1,118Hardening-Soil 8,791 -4,358 -1,453 -1,132

6 Linear-Elastic 605 -1,566 -924 -1,872Mohr-Coulomb 1,432 -1,359 -1,266 -1,518Hardening-Soil 1,376 -1,735 -1,275 -1,566





3.5.7 Summary of Results from 2D Preliminary Analyses

Table 13 compares bending moments and thrusts produced by different soil models for concrete

box culvert. Table 14 and Table 15 compare bending moments and thrusts produced by

different soil models for pipe culverts. Table 16 and Table 17 compare bending moments and

thrusts produced by different soil models for long-span arches.

The linear-elastic soil model produced similar bending moments to those produced by the other

two soil models in concrete culverts for a shallow cover depth. However, thrusts in concrete

culverts produced with the linear-elastic model were slightly different from those with the other

soil models because the interface strength was not considered in the linear-elastic model.

Bending moments and thrusts in flexible culverts produced with the linear-elastic soil model

were not close to those with the other two soil models.

Comparing the Mohr-Coulomb and Hardening-Soil models, they produced very similar structural

responses. In many cases, but not always, the Mohr-Coulomb model produced greater peak

moments and less thrust, which resulted from stress-dependent stiffness in the Hardening-Soil

model.

Table 13—Ratios of Live Load Moments and Thrusts of Concrete Box Cover Soil Ratio to Linear-Elastic Soil Model Ratio to Earth Load CaseDepth Model Moment Thrust Moment Thrust

(ft) Top Center Tip of Haunch Top Center Tip of Haunch Top Center Tip of Haunch Top Center Tip of HaunchTop Wall Wall Top Wall Wall

0 Linear-Elastic 1.000 1.000 1.000 1.000 1.000 4.205 -11.037 2.565 -1.139 1.636Mohr-Coulomb 1.004 0.989 1.007 0.943 0.996 4.887 -114.162 3.180 -2.730 1.824Hardening-Soil 0.997 1.008 0.987 1.079 0.999 4.920 -743.613 3.193 -3.587 1.827

2 Linear-Elastic 1.000 1.000 1.000 1.000 1.000 1.341 5.888 1.014 -0.166 0.589Mohr-Coulomb 0.951 0.933 0.968 -5.311 0.970 1.335 3.595 1.039 5.775 0.633Hardening-Soil 0.970 0.971 0.966 -4.699 0.949 1.388 3.340 1.072 -82.777 0.617

6 Linear-Elastic 1.000 1.000 1.000 1.000 1.000 0.200 0.462 0.184 0.106 0.120Mohr-Coulomb 0.691 0.558 0.809 7.454 0.876 0.139 0.221 0.149 -1.360 0.117Hardening-Soil 0.637 0.474 0.716 10.445 0.797 0.130 0.173 0.136 -1.042 0.105


Table 14—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 2 ft Structural Soil Ratio to Linear-Elastic Soil Model Ratio to Earth Load CaseMaterial Model Moment Thrust Moment Thrust

Peak Pos Peak Neg Crown Springline Peak Pos Peak Neg Crown SpringlineConcrete Linear-Elastic 1.000 1.000 1.000 1.000 1.423 1.128 -0.128 0.805

Mohr-Coulomb 0.917 0.939 1.398 0.751 1.504 1.293 -0.043 0.773Hardening-Soil 0.913 0.923 -1.905 0.788 1.720 1.498 0.049 0.803

Metal Linear-Elastic 1.000 1.000 1.000 1.000 3.763 1.836 0.887 1.060Mohr-Coulomb 2.717 3.692 2.612 0.691 13.481 15.526 1.614 1.009Hardening-Soil 2.477 3.396 3.178 0.827 14.058 10.425 1.906 1.195

Thermoplastic Linear-Elastic 1.000 1.000 1.000 1.000 3.377 1.512 -0.414 0.965Mohr-Coulomb 3.382 4.845 -11.728 0.973 18.700 10.748 2.417 1.220Hardening-Soil 2.814 4.002 -12.214 0.968 18.333 9.398 2.379 1.185

Table 15—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 6 ft Structural Soil Ratio to Linear-Elastic Soil Model Ratio to Earth Load CaseMaterial Model Moment Thrust Moment Thrust

Peak Pos Peak Neg Crown Springline Peak Pos Peak Neg Crown SpringlineConcrete Linear-Elastic 1.000 1.000 1.000 1.000 0.060 0.055 0.006 0.040

Mohr-Coulomb 0.500 0.667 -5.094 0.293 0.033 0.041 -0.026 0.014Hardening-Soil 0.435 0.514 -5.264 0.365 0.033 0.037 -0.022 0.018


Thermoplastic Linear-Elastic 1.000 1.000 1.000 1.000 0.080 0.051 -0.016 0.043Mohr-Coulomb 1.408 2.471 -5.711 0.610 0.181 0.195 0.045 0.034Hardening-Soil 1.885 2.273 -4.454 0.497 0.299 0.147 0.032 0.026

Table 16—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 2 ft Structural Soil Ratio to Linear-Elastic Soil Model Ratio to Earth Load CaseMaterial Model Moment Thrust Moment Thrust


Mohr-Coulomb 1.079 1.104 0.751 0.872 3.758 2.398 0.288 0.339Hardening-Soil 1.040 1.003 0.824 0.928 3.843 2.663 0.309 0.355


Table 17—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 6 ft Structural Soil Ratio to Linear-Elastic Soil Model Ratio to Earth Load CaseMaterial Model Moment Thrust Moment Thrust


Mohr-Coulomb 0.901 1.035 0.498 0.885 0.142 0.120 0.013 0.033Hardening-Soil 0.839 0.892 0.498 0.919 0.133 0.111 0.013 0.035



3.6 Effect of Interface Strength

In the 2D preliminary analyses described above, the interface strength was set 50% of the soil

shear strength. To examine the effect of interface strength on structural response, we analyzed

the concrete and thermoplastic pipe with backfill modeled by the Mohr-Coulomb constitutive

model with the interface strength equal to 100% of the soil shear strength.

Figure 41 and Figure 42 show plastic points in soil elements modeled by Mohr-Coulomb

constitutive model in the concrete pipe models with interface strengths of 50% and 100% of the

soil shear strength. Figure 43 and Figure 44 compares thrusts and bending moments in the

concrete pipe model with an interface strength of 100% of the soil shear strength with those

from the previous analyses. Table 18 compares thrusts and bending moments in the concrete

pipe models with backfill modeled by Mohr-Coulomb model between the cases with interface

strengths of 50% and 100% of the soil shear strength. By changing the interface strength from

100% of the soil shear strength to 50%, peak live load moments decreased by a few percent for

the 2 ft cover case and by 15% for the 6 ft cover case. Live load thrusts were affected more by

this change. Peak thrusts decreased by 10% for the 2 ft cover case and by 21% for the 6 ft

cover case.

Results for the thermoplastic pipe are shown in Figure 45 to Figure 48 and Table 19. By

changing the interface strength from 100% of the soil shear strength to 50%, peak live load

moments in the thermoplastic pipe increased by 18% for the 2 ft cover case and decreased by

14% for the 6 ft cover case. Peak thrusts decreased by 10% for the 2 ft cover case and by 24%

for the 6 ft cover case.

In summary, structural responses to live loads did not change significantly when the interface

strength was changed from 50% of the soil shear strength to 100% although the cases with the

100% strength showed slightly larger peak responses than those with the 50% strength except

for moments of the thermoplastic pipe with 2 ft cover. A change in the Interface strength

affected thrusts more than moments. Structural responses of the thermoplastic pipe were

affected more by a change of interface strength than those of the concrete pipe. Structural

responses of the 6 ft cover cases were affected more by a change of interface strength than

those of the 2 ft cover cases; however, it should be noted that responses of the 6 ft cover cases

were much smaller than those for the 2 ft cover cases.


Figure 41—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100%

Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover)

(a) 50% strength

(b) 100% strength

Tension cut-off point

Mohr-Coulomb point

Without live load

With live load

Without live load

With live load


Figure 42—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100%

Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover)

(a) 50% strength

(b) 100% strength

Without live load

With live load


Mohr-Coulomb point

Without live load

With live load


Figure 43—Comparison of Bending Moments and Thrusts due to Live Load between

Concrete Pipe Models with 50% and 100% Interface Strength (2 ft Cover)

(a) Bending Moment

(b) Thrust

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Linear-Elastic Model (Full Bonding)Mohr-Coulomb Model (50% Strength)Hardening-Soil Model (50% Strength)Mohr-Coulomb Model (100% Strength)

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Linear-Elastic Model(Full Bonding)Mohr-Coulomb Model(50% Strength)Hardening-Soil Model(50% Strength)Mohr-Coulomb Model(100% Strength)



Concrete Pipe Models with 50% and 100% Interface Strength (6 ft Cover)

(a) Bending Moment

(b) Thrust

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Table 18—Comparison of Bending Moments and Thrusts between Concrete Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model)

(a) Moments and thrusts

(b) Ratios of moments and thrusts of the model with 50% strength to those of the model with 100% strength

Cover Loads Interface Moment (lb*in/in) Thrust (lb/in)Depth (ft) Strength Peak Pos Peak Neg Crown Springline Peak Neg

2 Dead 50% 568.8 -503.2 -22.1 -90.0 -94.1100% 574.7 -507.9 -17.1 -97.5 -103.5

Dead plus 50% 1246.8 -1113.0 -21.1 -159.5 -159.5Live 100% 1269.9 -1125.8 -9.6 -170.2 -170.3Live 50% 855.6 -650.8 1.0 -69.5 -72.2

100% 865.4 -664.0 7.5 -72.7 -80.26 Dead 50% 1158.2 -1096.3 -59.0 -194.4 -196.6

100% 1170.3 -1098.0 -51.7 -214.8 -218.6Dead plus 50% 1196.3 -1141.0 -57.5 -197.2 -199.4

Live 100% 1215.8 -1146.7 -49.2 -218.5 -222.5Live 50% 38.7 -45.5 1.5 -2.8 -3.1

100% 45.5 -49.4 2.5 -3.7 -3.9

Cover Moment ThrustDepth (ft) Peak Pos Peak Neg Crown Springline Peak Neg

2 0.99 0.98 0.13 0.96 0.906 0.85 0.92 0.60 0.75 0.79


Figure 45—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and

100% Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover)

(a) 50% strength

(b) 100% strength


Mohr-Coulomb point

Without live load

With live load

Without live load

With live load


Figure 46—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and

100% Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover)

(a) 50% strength

(b) 100% strength

Without live load

With live load


Mohr-Coulomb point

Without live load

With live load



Thermoplastic Pipe Models with 50% and 100% Interface Strength (2 ft Cover)

(a) Bending Moment

(b) Thrust

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Thermoplastic Pipe Models with 50% and 100% Interface Strength (6 ft Cover)

(a) Bending Moment

(b) Thrust

-8.0

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Table 19—Comparison of Bending Moments and Thrusts Load between Thermoplastic Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model)

3.7 Conclusion

The Mohr-Coulomb and Hardening-Soil models produced similar structural responses of

culverts to surface live loads in the preliminary 2D analysis. Since the Hardening-Soil model is

more sophisticated model and it has more parameters to be determined, the Mohr-Coulomb

model is the best candidate soil model for the parametric study of NCHRP 15-29.

The interface strength did not significantly affect structural response of culverts to surface live

loads in the 2D analysis with the Mohr-Coulomb soil model, especially for the cases with a

shallow cover. This suggests that soil failure is most important effect in capturing live load

effects than slippage at the interface.

(a) Moments and thrusts

(b) Ratios of moments and thrusts of the model with 50% strength to those of the model with 100% strength

Cover Loads Interface Moment (lb*in/in) Thrust (lb/in)Depth (ft) Strength Peak Pos Peak Neg Crown Springline Peak Neg

2 Dead 50% 17.3 -19.6 -37.8 -58.6 -59.2100% 15.7 -24.0 -29.2 -68.5 -68.8

Dead plus 50% 313.4 -208.9 -129.2 -130.1 -181.9Live 100% 275.4 -191.2 -112.3 -124.7 -198.6Live 50% 322.8 -210.5 -91.4 -71.5 -135.1

100% 273.2 -189.6 -83.0 -56.2 -150.76 Dead 50% 28.0 -29.2 -75.5 -116.9 -118.3

100% 24.7 -40.8 -59.8 -140.3 -141.2Dead plus 50% 29.2 -29.9 -78.9 -120.8 -122.8

Live 100% 28.2 -42.4 -63.0 -145.7 -147.0Live 50% 5.1 -5.7 -3.4 -3.9 -5.1

100% 5.9 -6.5 -3.2 -5.4 -6.7

Cover Moment ThrustDepth (ft) Peak Pos Peak Neg Crown Springline Peak Neg

2 1.18 1.11 1.10 1.27 0.906 0.86 0.88 1.08 0.73 0.76


4. THREE DIMENSIONAL MODELING OF CULVERTS

4.1 Comparison of Responses to Factored and Unfactored Live Loads

4.1.1 Introduction

In the panel comments on early reports, a few panel members showed their interest in a

comparison of structural responses to unfactored and factored live loads. If the structure and

surrounding soil have linear-elastic material properties, structural responses to the factored live

loads will differ from those to the unfactored live loads by a load factor. However, backfill

surrounding the structure is nonlinear, and the ratio of structural response to the factored load to

the response to the unfactored live load will not be exactly equal to the load factor. To examine

the effect of soil nonlinearity, SGH performed soil-structure interaction analyses of culverts

subjected to factored and unfactored live loads, and compared responses between the factored

and unfactored load cases.

4.1.2 Method of Approach

Three-dimensional soil-structure interaction analysis of HDPE pipe subjected to the surface live

load was performed using ABAQUS. HDPE pipes tested in the MNDOT study were selected

(Pipe Run 7 with A-2 backfill and 2.8 ft cover and Pipe Run 3 with A-2 backfill and 1.6 ft). For

each case, one analysis was performed with unfactored live load; another was performed with

factored live load. Design tandem (a pair of 25 kip axles spaced 4 ft apart with a transverse

spacing of 6 ft) specified in AASHTO LRFD specifications was used in the analysis with a tire

contact area of 20 in. x 10 in., a multiple presence factor of 1.2, and dynamic load allowance

calculated as 33% x (1.0—0.125 x cover height). A load factor of 1.75 was used for live load,

corresponding to Strength Limit I in AASHTO LRFD specifications. Only live load was factored

in the analysis. FEA models and soil properties used were described in Section 4.5.

We also performed 2D analyses of an three-sided arch top culvert (24 ft x 6 ft) with 3 ft cover.

SW95 properties were used for backfill soil, which was modeled by Duncan-Selig soil model.

The finite element model is shown in Figure 49. HS20 truck was assumed for live load truck,

and live load distribution along the length of the arch was calculated per AASHTO LRFD

specifications. A multiple presence factor of 1.2 and dynamic load allowance calculated as

33% x (1.0—0.125 x cover height) were also included. Equivalent 2D service live load was

calculated as 377 lb for a 1-in. thick slice as shown in Figure 49. With this model, we performed

three analyses for the following cases: Case 1 with unfactored earth load and unfactored live

load, Case 2 with unfactored earth load and factored live load, and Case 3 with factored earth


load and unfactored live load. Load factors of 1.35 and 1.75 were used for earth load and live

load, respectively.

Figure 49—Finite Element Model of Three-Sided Arch Top Culvert with 3 ft Cover

4.1.3 Results

4.1.3.1 HDPE Pipe in ABAQUS

Figure 50 and Figure 51 compare displacement and force results between the cases with

unfactored and factored live loads. To make the comparison easier, displacements and forces

in Figure 50 and Figure 51 for the unfactored live load are 1.75 times those from the analysis

with the unfactored live load. Table 20 summarizes peak responses and shows ratios of

responses to the factored live load to responses to the unfactored live load. It is apparent in

Table 20 that ratios between the two cases are very close to 1.75. The maximum deviation

from 1.75 among the responses compared in Table 20 is 1.805, which is only 3 percent greater

than 1.75.

Soil zones: : In-situ soil (linear elastic) : Concrete footing (linear elastic) : Back fill (SW95 Duncan-Selig)

P=377 lb

28 ft

80 ft

Span = 24 ft Rise = 6 ft


Figure 50—Comparison of Vertical and Horizontal Displacements from Factored Live

Load with 1.75 times those from Unfactored Live Load (HDPE Pipe, A2 Backfill)

Figure 51—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times

those from Unfactored Live Load (HDPE Pipe, A2 Backfill)

(a) Thrust (b) Moment

0

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-180-135-90-4504590135180Degrees from Crown

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-30

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./in.

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2.8 ft Cover Unfactored LL X 1.75 2.8 ft Cover Factored LL1.6 ft Cover Unfactored LL X 1.75 1.6 ft Cover Factored LL

(a) Vertical crown displacement (b) Horizontal chord extension

0.00

0.05

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0 2 4 6 8 10 12

Distance from Symmetry Line of Tandem Axles (ft)

Vert

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.)2.8 ft Cover Unfactored LL X 1.752.8 ft Cover Factored LL1.6 ft Cover Unfactored LL X 1.751.6 ft Cover Factored LL

0.00

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2.8 ft Cover Unfactored LL X 1.752.8 ft Cover Factored LL1.6 ft Cover Unfactored LL X 1.751.6 ft Cover Factored LL


Table 20—Comparison of Structural Responses between Analyses with Factored and Unfactored Live Loads (HDPE Pipe, A2 Backfill)

Case Displacement under wheel (in.) Thrust (lb/in.) Moment (lb-in./in.)Cover Live Load Vertical Horizontal Crown Springline Peak Peak Pos. Peak Neg.

Unfactored 0.096 0.025 27.3 30.6 35.5 14.0 -10.92.8 ft Factored 0.169 0.044 49.3 53.6 63.3 24.2 -19.1

Ratio: Factored / Unfactored

1.757 1.744 1.805 1.752 1.781 1.730 1.750

Unfactored 0.126 0.033 43.2 43.8 51.7 13.9 -14.81.6 ft Factored 0.220 0.057 76.2 75.8 91.4 24.6 -25.8

Ratio: Factored / Unfactored

1.749 1.750 1.764 1.731 1.769 1.769 1.740

4.1.3.2 Three-Sided Arch Top Culvert in CANDE

Figure 52 compares force results from the three cases examined for the Hanson arch. To make

the comparison easier, forces in Figure 52 for the unfactored live load are 1.75 times those from

the analysis with the unfactored live load. Table 21 summarizes peak responses and shows

ratios of responses to the factored live load to responses to the unfactored live load. As is the

case with HDPE pipes analyzed in ABAQUS, ratios between factored and unfactored live load

cases are very close to 1.75. The maximum deviation from 1.75 among the responses

compared in Table 21 is 1.792, which is only 2 percent greater than 1.75.

Figure 52—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times

those from Unfactored Live Load (Hanson Arch)


0

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b/in

.)

Factored DL & LL

Factored LL & Unfactored DL

Unfactored DL & LL X 1.75-15000

-10000

-5000

0

5000

10000

15000

0 10 20 30 40 50 60 70

Element Number

Mom

ent (

in.-l

b/in

.)

Factored DL & LL

Factored LL & Unfactored DL

Unfactored DL & LL X 1.75


Table 21—Comparison of Structural Responses between Analyses (Hanson Arch)

Case Vertical Crown Peak Thrust Moment (lb-in./in.)Displacement (in.) (lb/in.) Peak Pos. Peak Neg.

Case 1: Unfactored DL & LL 0.28 304 -6,180 6,298 Case 2: Unfactored DL & Factored LL 0.48 527 -10,640 11,175 Case 3: Factored DL & LL 0.49 527 -10,750 11,287 Ratio: Case 2 / Case 1 1.741 1.735 1.722 1.774 Ratio: Case 3 / Case 1 1.755 1.735 1.739 1.792

4.1.4 Conclusion

Based on the limited cases we examined in this study, structural responses to the factored live

load can be estimated by scaling those to the unfactored live load by a load factor.

4.2 Selected Field Tests for 3D Analysis

As one final validation of the suitability of the Mohr-Coulomb model for evaluating structural

response of culverts subjected to surface live loads, we conducted a set of 3D soil-structure

interaction analyses using Plaxis 3D Tunnel Version 2 (Brinkgreve, 2004) to simulate selected

field tests and compared predicted responses with the existing field test data.

We selected two studies: NCHRP Project 12-45 (Webb, 1999; Taleb, 2000; McGrath et al.,

2002) and Minnesota DOT study (McGrath et al., 2002; McGrath and Beaver, 2005). These

studies include a reinforced concrete arch culvert, a corrugated structural plate metal culvert,

and a corrugated polyethylene pipe. Details of each culvert are provided below.

4.2.1 NCHRP Project 12-45

Long-span reinforced concrete and metal arch culverts were tested at the University of

Massachusetts, Amherst to investigate the structural behavior when subjected to live loads with

shallow fills (1 ft to 3 ft cover). A 30-ft span x 11-ft 4-in. rise x 42-ft long reinforced concrete

arch culvert and a 31-ft 2-in. span at the footing x 12-ft 1-in. rise x 40-ft long structural plate

metal arch culvert were installed end to end in a pre-excavated wide trench as shown in Figure

53. The trench was backfilled with existing site material, a well-graded sand with gravel.

The concrete arch culvert was the BEBO type arch, designated BEBO Type E30/3. Properties

of the concrete arch culvert are summarized in Table 22. The metal culvert was a Contech

Construction Products Type 108A30 nongalvanized corrugated steel arch culvert. Properties of


the metal arch culvert are summarized in Table 23. Each culvert was supported on 4.9 ft wide x

2 ft deep continuous reinforced concrete spread footings.

Live load testing was conducted with a tandem-axle truck with 70,000 lb on the tandem axles at

depths of fill of 3 ft, 2 ft, and 1 ft. Center-to-center spacing of the tandem axles was 4 ft 7 in.,

and center-to-center spacing between the wheels of the tandem axles was 6 ft 5 in. Live load

testing was conducted twice: once with backfill compacted to 92 percent of maximum density

(Test 1) and once with backfill compacted to 87 percent of maximum density (Test 2).

Figure 54 and Figure 55 summarize locations of displacement measurements in the concrete

and metal arch culverts, respectively.

Figure 53—Test Setup of NCHRP Project 12-45

(a) Cross-sectional view

In-Situ Soil

(b) Elevation view


Table 22—Properties of Reinforced Concrete Culvert

Culvert Properties

Span 30 ft (inside) 31 ft 8 in. (outside)

Rise 11 ft 4 in. (inside) 12 ft 2 in. (outside)

Wall Thickness 10 in. Cross Sectional Area per Unit Length 10 in.2

Moment of Inertia per Unit Length

/in.

83.3 in.4

Specified Compressive Strength

/in.

4,200 psi Poisson’s Ratio 0.17 Density 150 pcf

Circumferential Reinforcement Details Area of Inside Steel 0.0451 in.2

Area of Outside Steel /in.

0.0451 in.2

Specified Yield Strength /in.

70 ksi Inside Cover 1.5 in. Outside Cover 2.0 in.

Table 23—Properties of Structural Steel Plate and Culvert

Culvert Properties Bottom Span 31 ft 2 in. Maximum Span 31 ft 7 in. Total Rise 12 ft 1 in. Top Radius 20 ft 7 in. Side Radius 7 ft 3 in. Angle below Horizontal 14˚ 3’

Plate Properties Corrugation Pitch and Depth 6 in. x 2 in. Uncoated Plate Thickness 0.215 in. Nominal Uncoated Section Depth 2.215 in. Cross Sectional Area per Unit Length 0.267 in.2

Moment of Inertia per Unit Length /in.

0.127 in.4

Section Modulus /in.

0.115 in.3

Modulus of Elasticity /in.

29,000 ksi Poisson’s Ratio 0.3 Yield Strength 40.9 ksi Ultimate Strength 55.0 ksi Density 490 pcf


Figure 54—Instrumentation for Deformation in Concrete Culvert

Figure 55—Instrumentation for Deformation in Metal Culvert


4.2.2 Minnesota DOT Study

Corrugated high density polyethylene (HDPE) pipes with a diameter of 60 in. were tested at the

MnRoad Research Center to investigate performance of PE pipes under live loads with shallow

fills (1 ft to 3 ft). The PE pipes were either Type S or Type D corrugation as shown in Figure 56.

They both met the requirements of AASHTO M294. The Type S pipes were manufactured by

Hancor, Inc. of Findlay, Ohio, and The Type D pipes were manufactured by Advanced Drainage

Systems, Inc. of Hilliard, Ohio. As discussed in Section 4.3.4, only pipes with the Type S

corrugation were modeled in the 3D analysis. Properties of the Type S pipe are summarized in

Table 24.

Figure 57 shows typical installation of a test pipe. The test pipes were installed in a rectangular

trench. They were then backfilled with soils meeting the requirements of either Group

Classification A-1 or A-2 per AASHTO M145. The target compaction of backfill was 90 percent

of the maximum standard Proctor density. The road surface consisted of 8 in. of gravel base

and 4 in. of asphalt pavement. The average trench width, cover depth, backfill material, and

percent compaction of backfill for each test pipe are shown in Table 25.

Two test vehicles were used: a truck with a maximum axle load of 24,000 lb (heavy truck) and a

truck with a maximum axle load of 18,000 lb (light truck). Axle loads of the test vehicles are

shown in Figure 58.

Figure 59 shows typical test pipe instrumentation for extensively instrumented sections. The

accuracy of LVDTs was 0.005 in. Some pipe sections were instrumented to collect static data,

and others were instrumented to collect dynamic data. Static tests were conducted by placing

the truck wheels over the instrumented pipe cross sections. Dynamic tests were conducted by

recording data at a sampling rate of 200 Hz while the test vehicle passed over the pipes. Only

static tests were modeled in the 3D analysis in the current study.


Type S

Type D

Figure 56—Cross Sections of HDPE Pipes: Type D and Type S

Table 24—Properties of Type S HDPE Pipe

Inside Diameter 60 in. Outside Diameter 67.3 in. Cross Sectional Area per Unit Length 0.538 in.2/in. Moment of Inertia per Unit Length 0.798 in.4/in. Distance from Inside Diameter to Neutral Axis 1.37 in.

Modulus of Elasticity 100 ksi Poisson’s Ratio 0.35 Density 0.0344 pci

Figure 57—Typical Installation of PE Pipe

Pipe

~ 65 ft

Depth of Fill Open End (typ.)

CL of Truck

60 in. nominal


Table 25—Average Trench Measurements for Test Pipes in the MNDOT Study

Pipe Run Pipe Type Backfill

Material Average

Cover Depth (ft)

Average Trench

Width (ft)

% of Maximum Standard Proctor Dry

Density (Average) 1 Type S A-1 1.4 8.0 97 2 Type D 1.4 9.2 93 3 Type S A-2 1.6 8.8 91 4 Type D 1.7 8.8 88 7 Type S 2.8 9.5 82 8 Type D 2.8 8.5 85 9 Type S A-1 2.5 9.2 85 10 Type D 2.4 9.0 87

Figure 58—Live Load Vehicle in the MNDOT Study

Reference Axle

Axle No:Light Truck (80,000 lb):

Heavy Truck (102,000 lb):

4’-0”34’-7 5/8”4’-4”19’-0 3/4”

Reference Axle

Axle No:Light Truck (80,000 lb):

Heavy Truck (102,000 lb):

4’-0”34’-7 5/8”4’-4”19’-0 3/4”


Figure 59—Typical Test Pipe Instrumentation in the MNDOT Study

4.3 Three-Dimensional Analysis

4.3.1 General Information

The 3D finite element analysis was performed using a commercial soil-structure interaction finite

element software, Plaxis 3D Tunnel Version 2 (Plaxis 3D). Plaxis 3D uses 15-node wedge

elements for soils and 8-node plate elements for structures.

Figure 60 shows typical dimensions of finite element models of long-span arches and HDPE

pipes. By using symmetry conditions, only one side of axles was modeled. The long-span

concrete and metal arch models had a length of 20 ft in the longitudinal direction, and the HDPE

pipe model had a length of 12 ft.

ADS4 gages

OCVOCP

ICVICP

ADS4 gages

OCVOCP

ICVICP

LVDTs

30°

Soil Pressure Cell

Strain Gages

Thermocouples

Settlement Gage

Pavement Surface

Hancor5 gages

OTEOTC

OW

IF2IF1

Hancor5 gages

OTEOTC

OW

IF2IF1

OCP=outside center pipeOCV=outside center valleyICP=inside center pipeICV=inside center valley

OTC=outside top centerOTE=outside top edgeOW=outside webIF1=inside foot 1IF2=inside foot 2

Type D4 gages

Type S5 gages


Linear-elastic properties were assigned to structures and in-situ soils. The Mohr-Coulomb soil

model was used to describe constitutive models of backfill soils. Parameters for Mohr-Coulomb

model were determined based on Duncan-Selig parameters (Duncan et al., 1980; Selig, 1988)

and elastic parameter recommended by Selig (1990). Procedures to determine Mohr-Coulomb

soil parameters were described in Section 2.2. A tension cut-off stress of 0 psi was used for all

backfill soil types. Actual soil parameters used in the analysis are described in the sections

below.

Figure 60—Typical Dimensions of Finite Element Models of Long-Span Arch and HDPE

Pipe

4.3.2 Long-Span Concrete Arch Culvert

4.3.2.1 Finite Element Model

Figure 61 shows a finite element model of the concrete arch culvert with a cover depth of 3 ft.

The coordinate system of the model was oriented so that the x-axis aligns with the transverse

direction of the culvert (parallel to span), the y-axis aligns with the vertical direction, and the z-

46 ft

192 ft

28 ft

2 in

. 10 ft 2 in.

18 ft

7724

In-Situ Soil

83 ft 8 in.

Embankment

Backfill

Cover Depth(varies)

46 ft

192 ft

28 ft

2 in

. 10 ft 2 in.

18 ft

7724

In-Situ Soil

83 ft 8 in.

Embankment

Backfill

Cover Depth(varies)

60 in.Nominal

AASHTO Backfill(A-1 or A-2)

30 ft

In-Situ Soil

6 in. Bedding

Cover Depth(varies)

12 in.

4 in. Pavement 8 in. Gravel

8 ft 8 in. 84 in.

60 in.Nominal

AASHTO Backfill(A-1 or A-2)

30 ft

In-Situ Soil

6 in. Bedding

Cover Depth(varies)

12 in.


8 ft 8 in. 84 in.

(a) Long-span arch

(b) HDPE pipe


axis aligns with the longitudinal direction. The model had a length of 20 ft in the longitudinal

direction of the culvert. The model with a cover depth of 3 ft had 7,623 elements and 23,086

nodes.

Four models were created in total: (1) Test 1 with a cover depth of 3 ft, (2) Test 1 with a cover

depth of 1 ft, (3) Test 2 with a cover depth of 3 ft, and (4) Test 2 with a cover depth of 1 ft.

Figure 61—Finite Element Model of Concrete Arch Culvert with a Cover Depth of 3 ft

4.3.2.2 Materials

Table 26 shows soil properties used in the 3D analyses of the concrete culvert. Backfill and

embankment of Test 1 were assigned properties of SW95, and those of Test 2 were assigned to

properties of SW85. Densities of the backfill soils were from those used in the computational

models of the NCHRP project 12-45 (McGrath et al., 2002). Table 27 gives properties of

concrete used in the 3D analyses.

192 ft20 ft

28 ft 2 in.

Plane of symmetry(a) 3D model

(b) Cross section

x

y

x

y

z


Table 26—Soil Properties Used for the 3D Analyses of Long-Span Arches

Soil Type Density (pcf)

Depth (ft)

Modulus of

Elasticity (psi)

Poisson’s Ratio

Angle of Friction

Angle of Dilatancy

Cohesion (psi)

Backfill 121 0 to 1 1,600 0.40 57.8 27.8 0.001 SW95 1 to 5 4,100 0.29 54.3 24.3

(Mohr-Coulomb) 5 to 10 6,000 0.24 53.2 23.2 10 to 18 8,600 0.23 52.2 22.2

Backfill 111 0 to 1 1,300 0.26 42.0 12.0 0.001 SW85 1 to 6 2,100 0.21 40.4 10.4


In-Situ (Linear-Elastic) 127 any 6,000 0.25

Table 27—Concrete Properties Used for the 3D Analyses of Long-Span Arches

Density (pcf)

Modulus of Elasticity (ksi) Poisson’s Ratio

Concrete Arch 150 3,694 0.17 Footing 150 3,916 0.17

4.3.2.3 Loading and Boundary Condition

Side planes of the model (y-z planes) were fixed in the x-direction. The bottom plane of the

model (x-z plane) was fixed in the y-direction. The front and rear planes of the model (x-z

planes) were fixed in the z-direction. Plate elements that extended to the front and rear planes

were fixed about rotations around the x- and y-axes.

Loading steps in the analysis were: (1) in-situ soil under gravity, (2) in-situ soil and the culvert

under gravity, (3) in-situ soil, the culvert, and backfill under gravity, and (4) gravity and live

loads. Effects of construction sequence during backfilling were not considered. We did not

assign horizontal stresses in the backfill as initial conditions, but let the horizontal stresses be

those due to gravity effects; therefore, the horizontal stresses in the analysis may be different

from those in the field tests after compaction.

In the analysis, the tandem axles were placed symmetrically over the crown of the arch as

shown in Figure 62. The footprint of two wheels was assumed to be 12 in. x 24 in.


Figure 62—Live Load Position in the 3D Analysis of Long-Span Arches

4.3.2.4 Results

Figure 63 shows deformed shapes of the concrete culvert due to live loads for the four different

cases. Figure 64 shows vertical and horizontal displacement results due to live loads along the

length of the concrete culvert for the four cases. Figure 65 shows thrusts and moments in the

concrete culvert due to live loads for the four cases. Table 28, Table 29, and Table 30 compare

vertical displacements at crown, chord extensions, thrusts at base between the field tests and

the 3D analyses. In the presentation of results, the downward displacement of the crown is

taken as positive, and the vertical displacement of the crown is the relative displacement

between the crown and footings. Compressive force is taken as positive for thrusts, and

moment that produces tension on the inside of the culvert wall is taken as positive.

Vertical crown displacements of the concrete arch estimated by the 3D analysis were much

larger than those measured in the field tests. Horizontal chord extensions of the concrete arch

estimated by the 3D analysis were also larger than those measured in the field test except for

the case of Test 1 with a cover depth of 1 ft. Thrusts at the base of the concrete arch from the

3D analyses were significantly smaller than those measured in the field test, especially for

Test 2.

Figure 66 shows plastic points in the soil elements after the surface live loads were applied.

Plastic points are the integration points in a plastic state. Two types of plastic points are

6 ft

14 ft (assumed) 4 ft 7 in.

12 in.

24 in.

20 ft

Longitudinal direction

Axle Load = 14.4 kip 35.1 kip 33.7 kip

North South

Plane of symmetry

Crown of culvert


defined: tension cut-off point and Mohr-Coulomb point. Tension cut-off point indicates that the

tension cut-off criterion was applied to the integration point. Mohr-Coulomb point indicates that

the integration point lies on the Mohr-Coulomb failure surface. Tension failure occurred near

the wheels at the surface. Since Test 2 was conducted with less compacted backfill (SW85)

than Test 1, more Mohr-Coulomb plastic points were found in Test 2 cases than in Test 1 cases.

UndeformedTest 1 (Cover = 3 ft)Test 1 (Cover = 1 ft)Test 2 (Cover = 3 ft)Test 2 (Cover = 1 ft) 400X Deformation

Figure 63—Deformed Shapes of Concrete Arch in the Plane of Wheel Loads

(Effects of Live Loads only)

Figure 64—Displacements Due to Live Loads from the 3D Analyses of Concrete Arch

Culvert

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20


Vert

ical

Dis

plac

emen

t at C

row

n (in

)

Test 1 (Cover = 3 ft)



Test 2 (Cover = 1 ft)Wheel Load Location0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 5 10 15 20


Hor

iz. E

xten

sion

of C

hord

at H

eigh

t of 8

8 in

. (in

)




Test 2 (Cover = 1 ft)Wheel Load Location

(a) Vertical displacement at crown (b) Chord extension at a height of 88 in.


Figure 65—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the

3D Analyses of Concrete Arch Culvert

Table 28—Vertical Displacements at Crown of Concrete Arch due to Live Loads

Cover Test 1 Test 2

(ft) Field Test (in.)

Plaxis 3D (in.)

Ratio: Plaxis 3D / Field Test

Field Test (in.)

Plaxis 3D (in.)


3 0.008 0.046 5.83 0.024 0.052 2.19 1 0.028 0.050 1.83 0.012 0.056 4.71

Table 29—Chord Extension at Height of 88 in. of Concrete Arch Culvert due to Live Loads

Cover Test 1 Test 2


Plaxis 3D (in.)


Field Test (in.)

Plaxis 3D (in.)


3 0.016 0.031 1.98 0.008 0.037 4.66 1 0.035 0.034 0.96 0.016 0.039 2.46

Table 30—Thrusts at Base of Concrete Arch Culvert due to Live Loads

Cover Test 1 Test 2

(ft) Field Test (kip/ft)

Plaxis 3D (kip/ft)


Field Test (kip/ft)

Plaxis 3D (kip/ft)


3 1.37 - 2.40 0.923 0.38 - 0.67 2.672 0.843 0.32 1 1.37 - 2.40 0.938 0.39 - 0.68 6.715 0.845 0.13

Test 1 (Cover = 3 ft)Test 1 (Cover = 1 ft)Test 2 (Cover = 3 ft)Test 2 (Cover = 1 ft)

-4

8

kip/ftkip/ft


-40

80

kip*in/ftkip-in/ft



Figure 66—Plastic Points in Soil Elements in the Plane of Wheel Loads in Concrete Arch

Analysis

(a) Test 1 and 3 ft cover

(b) Test 1 and 1 ft cover

(c) Test 2 and 3 ft cover

(d) Test 2 and 1 ft cover


Mohr-Coulomb point


4.3.3 Long-Span Metal Arch Culvert


Figure 67 shows a finite element model of the metal arch culvert with a cover depth of 3 ft. The

coordinate system of the model was oriented as reported above for the concrete culvert. The

model had a length of 20 ft in the longitudinal direction of the culvert. The model with a cover

depth of 3 ft had 17,512 elements and 49,928 nodes.

The metal arch made of corrugated structural metal plates has different axial and bending

stiffnesses in the circumferential and longitudinal directions. EA and EI are summarized in

Table 31 for the circumferential and longitudinal directions. However, since orthotropic material

properties could not be specified for plate elements in Plaxis 3D, short elements with low

section properties were inserted between elements with section properties close to

circumferential properties to match stiffnesses as shown in Figure 68. A1, I1, A2, and I2 for

L2/L1 Figure 68=1/11 are given in .

Four models were created in total: (1) Test 1 with a cover depth of 3 ft, (2) Test 1 with a cover

depth of 1 ft, (3) Test 2 with a cover depth of 3 ft, and (4) Test 2 with a cover depth of 1 ft.

Figure 67—Finite Element Model of Metal Arch Culvert with a Cover Depth of 3 ft

(a) 3D model

(b) Cross section

192 ft20 ft

28 ft 2 in.

Plane of symmetry

x

y

x

y

z


Table 31—Axial and Bending Modulus of Metal Arch in Circumferential and Longitudinal Directions (E=29,000 ksi)

Direction EA (lb/in.) EI (lb-in.2

Circumferential /in.)

7,731,400 3,680,100 Longitudinal 47,908 201,057

Figure 68—Soft Element to Match Longitudinal Stiffness of Metal Arch

4.3.3.2 Materials

Table 26 shows soil properties used in the 3D analyses of the metal culvert. Backfill and

embankment of Test 1 were assigned to properties of SW95, and those of Test 2 were assigned

to properties of SW85. For the metal arch, modulus of elasticity of 29,000 ksi and Poisson’s

ratio of 0.3 were used.


Loading and boundary conditions for the 3D analyses of the metal culvert were the same as

those used for the concrete culvert, which were described in Section 4.3.2.3.

4.3.3.4 Results

For the 4 cases analyzed: Figure 69 shows deformed shapes of the metal culvert due to live

loads, Figure 70 shows vertical and horizontal displacement results due to live loads along the

length of the concrete culvert, and Figure 71 shows thrusts and moments in the concrete culvert

due to live loads. Table 32 and Table 33 compare vertical displacements at crown and chord

extensions between the field tests and the 3D analyses. Table 34 through Table 37 compare

thrusts and moments at various locations of the metal culvert between the field tests and the 3D

L2/L1=1/11

0.2908

A1

(in2/in)

0.1384

I1(in4/in)

1.112x10-41.384x10-4

I2(in4/in)

A2

(in2/in)

0.2908

A1

(in2/in)

0.1384

I1(in4/in)

1.112x10-41.384x10-4

I2(in4/in)

A2

(in2/in)


analyses. Designations of measurement locations are shown in Figure 55. In the presentation

of results, the downward displacement of the crown is taken as positive, and the vertical

displacement of the crown is the relative displacement between the crown and footings.

Compressive force is taken as positive for thrust, and a moment that produces tension on the

inside of the culvert wall is taken as positive.

Vertical crown displacements of the metal arch estimated by the 3D analysis were larger than

those measured in the field test except for Test 1 with a cover depth of 1 ft. The 3D analysis

estimates of vertical displacements of the metal culvert were much closer to the field test data

than those of the concrete culvert. However, the 3D analysis significantly overestimated

horizontal chord extensions of the metal arch.

Thrusts at the springlines of the metal arch from the 3D analyses were significantly larger than

those of the field test. Thrusts at the shoulders of the metal arch from the 3D analyses were

significantly smaller than those measured in the field test. Moments of the metal arch from the

3D analyses were in good agreement with those measured in the field test except for the crown.

Figure 72 shows plastic points in the soil elements after the surface live loads were applied.

Tension failure occurred near the wheels at the surface. More Mohr-Coulomb plastic points

were found in Test 2 cases than in Test 1 cases. Mohr-Coulomb plastic points spread in a

wider area in the metal arch analysis when compared to the concrete arch analysis.

UndeformedTest 1 (Cover = 3 ft)Test 1 (Cover = 1 ft)Test 2 (Cover = 3 ft)Test 2 (Cover = 1 ft) 30X Deformation

Figure 69—Deformed Shapes of Metal Arch in the Plane of Wheel Loads

(Effects of Live Loads only)


Figure 70—Displacements due to Live Loads from the 3D Analyses of Metal Arch Culvert

Figure 71—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the

3D Analyses of Metal Arch Culvert

Table 32—Vertical Displacements at Crown of Metal Arch due to Live Loads

Cover Test 1 Test 2


Plaxis 3D (in.)


Field Test (in.)

Plaxis 3D (in.)


3 0.413 0.509 1.23 0.453 0.756 1.67 1 1.130 1.104 0.98 1.055 1.540 1.46

Table 33—Chord Extension at Height of 88 in. of Metal Arch Culvert due to Live Loads

Cover Test 1 Test 2


Plaxis 3D (in.)


Field Test (in.)

Plaxis 3D (in.)


3 0.035 0.225 6.36 0.118 0.383 3.24 1 0.091 0.375 4.14 0.094 0.645 6.83


-5

10

kip/ft


-30

60

kip*in/ftkip/ft kip-in/ft


(a) Vertical displacement at crown (b) Chord extension at a height of 88 in.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 20


Vert

ical

Dis

plac

emen

t at C

row

n (in

) Test 1 (Cover = 3 ft)




Wheel Load Location0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20


Hor

iz. E

xten

sion

of C

hord

at H

eigh

t of 8

8 in

. (in

)





Wheel Load Location


Table 34—Thrusts in Test 1 of Metal Arch Culvert due to Live Loads

Cover Location Field Test (kip/ft)

Plaxis 3D (kip/ft)


(ft) P1 P2 P1 P2 3 NS 0.71 0.35 2.04 2.87 5.86 NC 1.22 0.82 2.48 2.04 3.03 NH 8.61 7.99 2.55 0.30 0.32 CR -1.23 1.90 -1.55 SH 9.07 9.51 2.60 0.29 0.27 SC 3.69 3.83 2.41 0.65 0.63 SS 1.60 -1.68 1.80 1.13 -1.07 1 NS 0.42 0.57 2.82 6.69 4.91 NC -2.58 -3.02 3.74 -1.45 -1.24 NH 32.64 24.69 5.22 0.16 0.21 CR 10.33 8.48 0.82 SH 26.48 17.72 5.19 0.20 0.29 SC -0.08 3.66 3.63 -45.06 0.99 SS 0.28 -0.55 2.49 8.87 -4.51

Table 35—Thrusts in Test 2 of Metal Arch Culvert due to Live Loads

Cover Location Field Test (kip/ft)

Plaxis 3D (kip/ft)


(ft) P1 P2 P1 P2 3 NS 0.00 0.69 2.05 2.98 NC 1.47 1.37 2.23 1.52 1.63 NH 10.62 7.52 2.06 0.19 0.27 CR 3.08 1.33 0.43 SH 5.49 6.19 2.16 0.39 0.35 SC 0.48 4.43 2.23 4.65 0.50 SS -1.48 1.87 -1.26 1 NS 0.00 0.18 2.68 14.81 NC -3.98 -3.79 3.15 -0.79 -0.83 NH 27.32 19.99 4.57 0.17 0.23 CR 8.97 8.01 0.89 SH 24.75 16.62 4.56 0.18 0.27 SC -1.80 3.66 3.11 -1.73 0.85 SS -2.30 2.36 -1.03


Table 36—Moments in Test 1 of Metal Arch Culvert due to Live Loads

Cover Location Field Test (kip-in./ft)

Plaxis 3D (kip-in./ft)


(ft) P1 P2 P1 P2 3 NS -0.76 -0.52 -1.25 1.64 2.40 NC -0.21 -0.76 -1.83 8.74 2.40 NH -5.91 -5.80 -4.36 0.74 0.75 CR 4.88 9.68 1.98 SH -5.90 -4.85 -4.91 0.83 1.01 SC -1.39 -2.26 -1.62 1.17 0.72 SS -1.78 0.10 -0.97 0.54 -9.67 1 NS -0.12 0.00 -0.81 6.53 NC 2.84 2.24 -1.01 -0.36 -0.45 NH -19.77 -20.25 -14.39 0.73 0.71 CR 3.05 17.18 5.63 SH -16.65 -13.16 -14.51 0.87 1.10 SC -0.54 -0.63 -0.96 1.80 1.53 SS -0.01 0.15 -0.45 49.42 -3.09

Table 37—Moments in Test 2 of Metal Arch Culvert due to Live Loads

Cover Location Field Test (kip-in./ft)

Plaxis 3D (kip-in./ft)


(ft) P1 P2 P1 P2 3 NS 0.00 -0.03 -3.22 96.92 NC -0.24 -0.51 -2.58 10.66 5.03 NH -5.52 -4.96 -5.65 1.02 1.14 CR 2.04 13.39 6.58 SH -3.64 -2.91 -6.18 1.70 2.12 SC -0.53 -3.40 -2.46 4.63 0.72 SS 0.52 -2.46 -4.72 1 NS 0.00 0.26 -1.55 -5.96 NC 3.39 2.78 -2.68 -0.79 -0.96 NH -16.79 -17.41 -18.80 1.12 1.08 CR 4.28 22.39 5.24 SH -15.15 -10.34 -19.24 1.27 1.86 SC -0.95 -4.26 -2.71 2.86 0.64 SS 1.29 -0.77 -0.60


Figure 72—Plastic Points in Soil Elements in the Plane of Wheel Loads in Metal Arch

Analysis

(a) Test 1 and 3 ft cover

(b) Test 1 and 1 ft cover

(c) Test 2 and 3 ft cover

(d) Test 2 and 1 ft cover


Mohr-Coulomb point


4.3.4 60-in. Diameter HDPE Pipe


Four pipe runs with Type S pipes were modeled: (1) Pipe Run 1 (A-1 backfill and 1.4 ft cover),

(2) Pipe Run 9 (A-1 backfill and 2.5 ft cover), (3) Pipe Run 3 (A-2 backfill and 1.6 ft cover), and

(4) Pipe Run 7 (A-2 backfill and 2.8 ft cover). An average trench width varied from pipe to pipe

as shown in Table 25. A trench width of 8 ft 8 in. was used for all four models as shown in

Figure 60.

Figure 73 shows a finite element model of the HDPE pipe culvert for Pipe Run 9 (A-1 backfill

and 2.5 ft cover). Soils and pavement were modeled by wedge elements, and the concrete

culvert was modeled by plate elements. The coordinate system of the model was oriented so

that the x-axis aligns with the transverse direction of the culvert, the y-axis aligns with the

vertical direction, and the z-axis aligns with the longitudinal direction. The model had a length of

12 ft in the longitudinal direction of the culvert. The model for Pipe Run 9 had 19,888 elements

and 55,993 nodes.

HDPE pipes of Type S corrugation have different axial and bending stiffnesses in the

circumferential and longitudinal directions. EA and EI are summarized in Table 38 for the

circumferential and longitudinal directions. As discussed in Section 4.3.3.1 for the metal arch

culvert, short elements with low section properties were inserted between elements with section

properties close to circumferential properties to match stiffnesses as shown in Figure 74. A1, I1,

A2, and I2 for L2/L1 Figure 74=1/20 are given in .

Figure 73—Finite Element Model of HDPE Pipe Culvert for Pipe Run 9

30 ft 12 ft

15 ft 7 in.

Plane of symmetry 30 ft 12 ft

15 ft 7 in.

Plane of symmetryx

y

z


Table 38—Axial and Bending Modulus of HDPE Pipe in Circumferential and Longitudinal Directions (E=100,000 psi)

Direction EA (lb/in.) EI (lb-in.2


53,600 79,800 Longitudinal 32,700 1,020

Figure 74—Soft Element to Match Longitudinal Stiffness of HDPE Pipe

4.3.4.2 Materials

Table 39 shows properties of soils and pavement used in the 3D analyses of the HDPE pipe

culverts. The A-1 backfill was assigned properties of SW95, and the A-2 backfill was assigned

properties of ML95. A target compaction of the backfill was 90 percent of maximum standard

Proctor density. As shown in Table 25, backfill densities ranged from 82 percent to 97 percent

of the maximum standard Proctor density. Since the backfill was compacted more when the

pavement was placed, 95 percent compaction was selected for the 3D analysis. Properties of

pavement were those used in the MNDOT study (McGrath, 2005). Gravel was assigned

properties of SW95.

For the HDPE pipes, modulus of elasticity of 100,000 psi and Poisson’s ratio of 0.35 were used.


Boundary conditions of the model were the same as those described in Section 4.3.2.3.

Loading steps in the analysis were: (1) in-situ soil under gravity, (2) in-situ soil, bedding, and the

culvert under gravity, (3) in-situ soil, bedding, the culvert, and backfill under gravity, (4) in-situ

soil, bedding, the culvert, backfill, gravel, and pavement under gravity, and (5) all gravity and

A1, I1

L1 L2

MM A2, I2

A1, I1

L1 L2

MM A2, I2

L2/L1=1/20

0.562

A1

(in2/in)

0.840

I1(in4/in)

1.005x10-30.0365

I2(in4/in)

A2

(in2/in)

0.562

A1

(in2/in)

0.840

I1(in4/in)

1.005x10-30.0365

I2(in4/in)

A2

(in2/in)


live loads. Effects of construction sequence during backfilling were not considered. We did not

assign horizontal stresses in the backfill as initial conditions, but let the horizontal stresses be

those due to gravity effects; therefore, the horizontal stresses in the analysis may be different

from those in the field tests after compaction.

In the 3D analysis, two positions of a live load vehicle were examined as shown in Figure 75: (1)

the reference axle of the tandem axles was placed over the crown, and (2) the tandem axles

were placed symmetrically over the crown. These two positions were designated as Position 3

and Position 4 in the field test. The footprint of two wheels was assumed to be 10 in. x 20 in.

For each of the four pipe runs, four cases of live loads were analyzed: (1) heavy truck at

Position 3, (2) heavy truck at Position 4, (3) light truck at Position 3, and (4) light truck at

Position 4. Differences of axle loads between the heavy and light trucks are shown in Figure

58.

Table 39—Soil Properties Used for the 3D Analyses of HDPE Pipes

Soil Type Density (pcf)

Depth (ft)

Modulus of

Elasticity (psi)

Poisson’s Ratio

Angle of Friction

Angle of Dilatancy

Cohesion (psi)

A-1 Backfill 141 0 to 1 1,600 0.40 57.8 27.8 0.001 SW95 1 to 5 4,100 0.29 54.3 24.3


A-2 Backfill 127 0 to 1 1,800 0.34 34.0 4.0 4.0 ML95 1 to 6 2,500 0.29

(Mohr-Coulomb) 6 to 11 2,900 0.27 11 to 18 3,200 0.29

In-Situ (Linear-Elastic) 145 any 15,000 0.30

Pavement (Linear-Elastic) 150 0 to

0.33 400,000 0.35

Gravel SW95

(Mohr-Coulomb) 141 0.33 to

1.0 1,600 0.40 57.8 27.8 0.001


Figure 75—Positions of Live Load Vehicle Axles in the 3D Analyses of HDPE Pipes

4.3.4.4 Results

Figure 76 shows deformed shapes of Pipe Run 1 due to live loads for the four different cases.

Figure 77 shows vertical crown displacements due to live loads along the length of Pipe Run 1

for the four cases. Figure 78 shows horizontal displacements of culvert (diametrical change at

the springline) due to live loads along the length of Pipe Run 1 for the four cases. Figure 79

shows thrusts of Pipe Run 1 in the plane of wheel loads due to live loads for the four cases.

Figure 80 shows moments of Pipe Run 1 in the plane of wheel loads due to live loads for the

four cases. Figure 81 shows plastic points in the soil elements in the analysis of Pipe Run 1. In

these figures for analysis results, field test data are also shown whenever available. The field

test data are designated by the month when the tests were conducted, such as Oct 00, May 01,

and Aug 02. These three tests were conducted right after the installation and seven months

and 22 months after the installation. Figure 82 through Figure 87 show results of Pipe Run 9.

Figure 88 though Figure 93 show results of Pipe Run 3. Figure 94 through Figure 99 show

results of Pipe Run 7. Table 40 and Table 41 compare vertical displacements at the crown

between the field tests and the 3D analyses for the heavy and light trucks. Table 42 and Table

43 compare horizontal displacements (diametrical changes at the springline) between the field

tests and the 3D analyses for the heavy and light trucks. In the presentation of results, the

downward displacement of the crown is taken as positive, and the vertical displacement of the

crown is the relative displacement between the crown and the invert. A diametrical change is

positive when there is an extension. Compressive force is taken as positive for thrusts, and

moment that produces tension on the inside of the culvert wall is taken as positive.

0’-0”0’-0”0’-0”0’-0” Symmetric = 2’-2” each sideSymmetric = 2’-2” each sideSymmetric = 2’-2” each sideSymmetric = 2’-2” each side

(a) Position 3 (b) Position 4

Note: Axles colored in pink were modeled in the 3D analysis.


Responses of the pipes with a nominal cover of 1 ft were estimated by the 3D analyses better

than those of the pipes with a nominal cover of 2 ft. The 3D analysis tends to overestimate as

the cover height increases. The 3D analysis estimated the vertical displacements better than

the horizontal displacements.

In general, moments were estimated by the 3D analyses better than thrusts.

For the A-1 backfill cases (Pipe Runs 1 and 9), shear failure indicated by Mohr-Coulomb plastic

points occurred at the invert and at the soil-structure interface below the springlines. Shear

failure was also observed at boundaries between in-situ soil and backfill up to a depth of about

3 ft. However, for the A-2 backfill cases (Pipe Runs 3 and 7), not many Mohr-Coulomb points

were found. Instead, tension failure occurred at the invert and at the boundaries of in-situ soil

and backfill up to a depth of 3 ft. The difference in soil failure profile between A-1 and A-2

backfills stems from cohesion assigned to each soil models. A cohesion value of SW95 for A-1

back fill was 0.001 psi, and that of ML95 for A-2 backfill was 4.0 psi. Owing to the cohesion of

ML95, tension failure occurred prior to shear failure at the invert and at the boundaries of in-situ

soil and backfill up to a depth of 3 ft for the A-2 backfill cases.


UndeformedHeavy P3Heavy P4Light P3Light P4

100X deformation

Figure 76—Deformed Shapes of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads (A-1, 1.4 ft Cover)

Figure 77—Vertical Crown Displacements of Pipe Run 1 due to Live Loads

(A-1, 1.4 ft Cover)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)

Heavy P3 (Plaxis)Heavy P4 (Plaxis)Heavy P3 (Oct-00)Heavy P3 (May-01)Heavy P3 (Aug-02)Heavy P4 (Oct-00)Heavy P4 (May-01)Heavy P4 (Aug-02)

(a) Heavy truck (b) Light truck

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)

Light P3 (Plaxis)Light P4 (Plaxis)Light P3 (Oct-00)Light P3 (May-01)Light P3 (Aug-02)Light P4 (Oct-00)Light P4 (May-01)Light P4 (Aug-02)


Figure 78—Horizontal Displacements of Pipe Run 1 due to Live Loads (A-1, 1.4 ft Cover)

Figure 79—Thrusts of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads

(A-1, 1.4 ft Cover)

Figure 80—Moments of Pipe Run 1 due to Live Loads in Plane of Wheel Loads

(A-1, 1.4 ft Cover)


-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.) Heavy P3 (Plaxis)

Heavy P4 (Plaxis)Heavy P3 (Oct-00)Heavy P3 (May-01)Heavy P3 (Aug-02)Heavy P4 (Oct-00)Heavy P4 (May-01)Heavy P4 (Aug-02)

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.) Light P3 (Plaxis)

Light P4 (Plaxis)Light P3 (Oct-00)Light P3 (May-01)Light P3 (Aug-02)Light P4 (Oct-00)Light P4 (May-01)Light P4 (Aug-02)


0

10

20

30

40

50

60

70


Thru

st (l

b/in

.)


0

10

20

30

40

50

60

70


Thru

st (l

b/in

.)Light P3 (Plaxis)

Light P4 (Plaxis)


-30

-20

-10

0

10

20

30

40


Ben

ding

Mom

ent (

lb-in

./in.

)


-30

-20

-10

0

10

20

30

40


Ben

ding

Mom

ent (

lb-in

./in.

)

Light P3 (Plaxis)

Light P4 (Plaxis)


Figure 81—Plastic Points in Soil Elements of Pipe Run 1 in the Plane of Wheel Loads

(A-1, 1.4 ft Cover)

(a) Heavy Truck Position 3


Mohr-Coulomb point

(b) Heavy Truck Position 4

(c) Light Truck Position 3 (c) Light Truck Position 4



100X deformation



(A-1, 2.5 ft Cover)


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)



Figure 84—Horizontal Displacements of Pipe Run 9 Due to Live Loads

(A-1, 2.5 ft Cover)

Figure 85—Thrusts of Pipe Run 9 Due to Live Loads in the Plane of Wheel Loads

(A-1, 2.5 ft Cover)

Figure 86—Moments of Pipe Run 9 Due to Live Loads in Plane of Wheel Loads

(A-1, 2.5 ft Cover)


-15

-10

-5

0

5

10

15


Ben

ding

Mom

ent (

lb-in

./in.

)

Heavy P3 (Plaxis)

Heavy P4 (Plaxis)

-15

-10

-5

0

5

10

15


Ben

ding

Mom

ent (

lb-in

./in.

)



-5

0

5

10

15

20

25

30

35


Thru

st (l

b/in

.)

Heavy P3 (Plaxis)

Heavy P4 (Plaxis)

-5

0

5

10

15

20

25

30

35


Thru

st (l

b/in

.)



0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)


0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)




(A-1, 2.5 ft Cover)



Mohr-Coulomb point





100X deformation



(A-2, 1.6 ft Cover)


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)




Figure 91—Thrusts of Pipe Run 3 due to Live Loads in the Plane of Wheel Loads (A-2, 1.6

ft Cover)

Figure 92—Moments of Pipe Run 3 due to Live Loads in Plane of Wheel Loads

(A-2, 1.6 ft Cover)


-20

-15

-10

-5

0

5

10

15

20

25

30


Ben

ding

Mom

ent (

lb-in

./in.

)

Heavy P3 (Plaxis)

Heavy P4 (Plaxis)

-20

-15

-10

-5

0

5

10

15

20

25

30


Ben

ding

Mom

ent (

lb-in

./in.

)

Light P3 (Plaxis)

Light P4 (Plaxis)


0

5

10

15

20

25

30

35

40

45

50


Thru

st (l

b/in

.)

Heavy P3 (Plaxis)

Heavy P4 (Plaxis)

0

5

10

15

20

25

30

35

40

45

50


Thru

st (l

b/in

.)Light P3 (Plaxis)

Light P4 (Plaxis)


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)




(A-2, 1.6 ft Cover)



Mohr-Coulomb point





100X deformation


Figure 95—Vertical Crown Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft

Cover)


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)




Figure 97—Thrusts of Pipe Run 7 due to Live Loads in the Plane of Wheel Loads (A-2, 2.8

ft Cover)

Figure 98—Moments of Pipe Run 7 due to Live Loads in Plane of Wheel Loads (A-2, 2.8 ft

Cover)


-15

-10

-5

0

5

10

15

20


Ben

ding

Mom

ent (

lb-in

./in.

)

Heavy P3 (Plaxis)

Heavy P4 (Plaxis)

-15

-10

-5

0

5

10

15

20


Ben

ding

Mom

ent (

lb-in

./in.

)

Light P3 (Plaxis)

Light P4 (Plaxis)


0

5

10

15

20

25

30

35


Thru

st (l

b/in

.)

Heavy P3 (Plaxis)

Heavy P4 (Plaxis)

0

5

10

15

20

25

30

35


Thru

st (l

b/in

.)

Light P3 (Plaxis)

Light P4 (Plaxis)


0.000

0.005

0.010

0.015

0.020

0.025

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)


0.000

0.005

0.010

0.015

0.020

0.025

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)



Figure 99—Plastic Points in Soil Elements of Pipe Run 7 in the Plane of Wheel Loads (A-

2, 2.8 ft Cover)



Mohr-Coulomb point




Table 40—Comparison of Vertical Displacements at Crown of HDPE Pipes under Heavy Truck

Pipe Backfill Average Truck Plaxis 3D Field Test (in.) Ratio: Plaxis 3D / Field TestRun Cover (ft) Position (in.) Oct-00 May-01 Aug-02 Oct-00 May-01 Aug-02

1 A-1 1.4 3 0.092 0.106 0.117 0.119 0.87 0.79 0.784 0.085 0.074 0.095 0.088 1.15 0.90 0.97

3 A-2 1.6 3 0.101 0.174 0.122 0.091 0.58 0.83 1.114 0.100 0.137 0.097 0.072 0.73 1.03 1.39

7 A-2 2.8 3 0.069 0.061 0.036 0.039 1.13 1.91 1.764 0.076 0.065 0.040 0.039 1.18 1.91 1.96

9 A-1 2.5 3 0.060 0.075 0.051 0.031 0.81 1.19 1.954 0.065 0.071 0.050 0.037 0.91 1.30 1.75

Table 41—Comparison of Vertical Displacements at Crown of HDPE Pipes under Light Truck


1 A-1 1.4 3 0.071 0.093 0.103 0.088 0.76 0.69 0.814 0.065 0.101 0.122 0.063 0.64 0.53 1.02

3 A-2 1.6 3 0.078 0.075 0.050 0.056 1.04 1.55 1.394 0.076 0.059 0.045 0.045 1.28 1.68 1.68

7 A-2 2.8 3 0.053 0.049 0.018 0.022 1.07 2.92 2.394 0.058 0.049 0.026 0.026 1.18 2.23 2.23

9 A-1 2.5 3 0.046 0.030 0.024 0.024 1.53 1.92 1.924 0.049 0.031 0.023 0.023 1.58 2.13 2.13

Table 42—Comparison of Diametrical Changes at Springline of HDPE Pipes under Heavy Truck


1 A-1 1.4 3 0.014 -0.009 0.015 0.013 -1.58 0.95 1.104 0.017 -0.006 0.017 0.014 -2.91 1.03 1.25

3 A-2 1.6 3 0.021 0.014 0.006 0.012 1.48 3.44 1.724 0.026 0.027 0.016 0.014 0.95 1.60 1.83

7 A-2 2.8 3 0.016 0.009 0.004 0.005 1.74 3.91 3.134 0.019 0.016 0.008 0.006 1.21 2.43 3.24

9 A-1 2.5 3 0.011 0.005 0.005 0.002 2.13 2.13 5.334 0.013 0.007 0.007 0.003 1.89 1.89 4.40

Table 43—Comparison of Diametrical Changes at Springline of HDPE Pipes under Light Truck


1 A-1 1.4 3 0.011 0.012 0.013 0.002 0.91 0.84 5.494 0.013 0.016 0.017 0.005 0.83 0.78 2.66

3 A-2 1.6 3 0.016 0.010 0.009 0.005 1.59 1.77 3.184 0.019 0.011 0.011 0.005 1.77 1.77 3.89

7 A-2 2.8 3 0.012 0.005 0.003 0.003 2.40 4.00 4.004 0.015 0.007 0.006 0.004 2.11 2.46 3.69

9 A-1 2.5 3 0.008 0.001 0.000 0.002 8.16 --- 4.084 0.010 0.001 0.001 0.002 10.03 10.03 5.02


4.3.5 Discussion

Figure 100 shows ratios of the 3D analysis results to the field test data of displacements of the

concrete arch culvert due to live load. These ratios are tabulated in Table 28 and Table 29. For

the concrete arch culvert, a ratio of the vertical displacement at the crown of the 3D analysis to

that of the field test ranged from 1.83 to 5.83. A ratio of the chord extension of the 3D analysis

to that of the field test ranged from 0.96 to 4.66. The 3D analysis overestimated displacements

in most cases. Between the 3 ft and 1 ft cover cases, the 3D analysis estimated displacements

of the 1 ft cover case better. A ratio of thrust at base of the 3D analysis to that of the field test

ranged from 0.13 to 0.68. The 3D analysis underestimated the thrusts at base.


metal arch culvert due to live load. These ratios are tabulated in Table 32 and Table 33. For

the metal arch culvert, a ratio of the vertical displacement at the crown of the 3D analysis to that

of the field test ranged from 0.98 to 1.67. A ratio of the chord extension of the 3D analysis to

that of the field test ranged from 3.24 to 6.83. The 3D analysis estimated the vertical

displacements of the metal arch much better than those of the concrete arch although there is

still a tendency of overestimation by the 3D analysis. The 3D analysis significantly

overestimated the chord extension of the metal arch. The 3D analysis overestimated thrusts

below the points where curvature changes (NC and SC), and underestimated thrusts above

those points especially at shoulders (NH and SH). Thrusts at the crown for 1 ft cover were

estimated by the 3D analysis relatively well. The 3D analysis overestimated moments in many

cases; especially, crown moments were significantly overestimated. The 3D analysis estimated

moments at the shoulders and curvature points relatively well.


HDPE pipe culverts due to live load. These ratios are tabulated in Table 40 through Table 43.

For the HDPE pipe culverts, a ratio of the vertical displacement at the crown of the 3D analysis

to that of the field test ranged from 0.78 to 2.13. A ratio of the diametrical change at springline

of the 3D analysis to that of the field test data ranged from 0.78 to 10.03 except for a few data

points where shortening of diameter was found in the field tests. The 3D analysis slightly

underestimated displacements of the most of 1 ft cover cases, and the 3D analysis

overestimated displacements of the most of 3 ft cover cases. Thrusts at the crown were

overestimated by the 3D analysis when compared to the field test data of May 2001. Moments

at the crown were underestimated by the 3D analysis for Pipe Run 1 with the heavy truck, and

they were overestimated Pipe Run 9 with the light truck when compared to the field test data of


May 2001. However, moments were estimated by the 3D analysis relatively well especially for

Pipe Run 9.

There seems to be a tendency that the 3D analysis prediction becomes better as the response

becomes larger. The 3D analyses estimated responses of the metal arch culvert and the HDPE

pipe culvert better than those of the concrete culvert. The vertical displacements at the crown

were estimated better than the horizontal displacements in most cases. The difference in small

structural responses between the 3D analysis and the field tests may have stemmed from the

accuracy of the measurements in the field. For example, the 3D analyses estimated the

diametrical change at the springline of the HDPE pipe due to live loads to be an order of 0.01 to

0.02 in. This diametrical change translates to 0.005 to 0.01 in. of horizontal displacement at the

springline on each side. The accuracy of the LVDTs that were used in the MNDOT study was

0.005 in., which is the same order as the horizontal displacement of the springline. Therefore, it

is likely that the measured horizontal displacements were less accurate than the measured

vertical displacements.

Since the vertical crown displacements were estimated by the 3D analyses relatively well

especially for the metal arch and the HDPE pipes, we conclude that the Mohr-Coulomb model is

an appropriate soil model for backfill.


Figure 100—Ratios of 3D Analysis Results to Field Test Data for Displacements of

Concrete Arch

Figure 101—Ratios of 3D Analysis Results to Field Test Data for Displacements of Metal

Arch

0.0

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(a) Crown vertical displacement (b) Horizontal chord extension

(a) Crown vertical displacement (b) Horizontal chord extension

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Figure 102—Ratios of 3D Analysis Results to Field Test Data for Displacements of HDPE

Pipes

4.4 Comparison between the Mohr-Coulomb and Hardening-Soil Models in Three-Dimensional Analysis in PLAXIS

Initial investigation of culvert responses to live loads from 2D analyses with linear-elastic, Mohr-

Coulomb, and Hardening-soil models showed that responses from the Mohr-Coulomb and

Hardening-soil models were very close to each other whereas responses from the linear-elastic

model were significantly different from other models. As a result, the Mohr-Coulomb model was

selected to be used in the 3D analysis of field tests. Subsequent Panel comments suggested

(a) Crown vertical displacement (Heavy truck) (b) Crown vertical displacement (Light truck)

0.0

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1.5

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Position 3 (Oct-00)

Position 3 (May-01)

Position 3 (Aug-02)

Position 4 (Oct-00)

Position 4 (May-01)

Position 4 (Aug-02)

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Position 3 (Oct-00)

Position 3 (May-01)

Position 3 (Aug-02)

Position 4 (Oct-00)

Position 4 (May-01)

Position 4 (Aug-02)

-4.0

-2.0

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Position 3 (Oct-00)Position 3 (May-01)Position 3 (Aug-02)Position 4 (Oct-00)Position 4 (May-01)Position 4 (Aug-02)

-4.0

-2.0

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Position 3 (Oct-00)Position 3 (May-01)Position 3 (Aug-02)Position 4 (Oct-00)

Position 4 (May-01)Position 4 (Aug-02)

(c) Horizontal diameter change (Heavy truck) (d) Horizontal diameter change (Light truck)

Pipe Run 1: A-1 backfill and 1.4 ft cover Pipe Run 3: A-2 backfill and 1.6 ft cover Pipe Run 7: A-2 backfill and 2.8 ft cover Pipe Run 9: A-1 backfill and 2.5 ft cover


comparison of the Mohr-Coulomb and Hardening-soil models in the 3D analysis as a

confirmation of selection of an appropriate soil model. To compare culvert responses from

these two soil models, 3D analyses were performed of long-span metal arch from NCHRP

Project 12-45 (McGrath et al. 2002) and HDPE pipe from the MNDOT study (McGrath et al.

2005).


Soil-structure interaction analysis of culverts subjected to the surface live load is performed

using Plaxis 3D. Two structural models were selected as described above: (1) Long-span metal

arch, Test 2, 3 ft cover (NCHRP Project 12-45); and (2) HDPE pipe, Pipe Run 7, A-2 backfill, 2.8

ft cover (MNDOT study). These structures were analyzed with both Mohr-Coulomb and

Hardening-Soil models, and structural responses were compared. In the metal arch model,

backfill was assumed to have properties of SW85, and the soil above the crown of arch was

assumed to have properties of SW95. In the HDPE pipe model, backfill was assumed to have

properties of ML95. The interface strength was assumed to be 50% of strength of surrounding

soil..

4.4.2 Results

4.4.2.1 Metal Arch in Test 2 with 3 ft Cover

Figure 103 compares vertical crown displacements and horizontal chord extensions along the

culvert between the two cases of the metal arch analysis: the case with the Mohr-Coulomb

model and the case with the Hardening-soil model. Figure 104 compares thrusts and moments

under the wheel load between the two cases. These figures also show displacements and

forces measured in the field tests. Measured thrusts and moments in Figure 104 are average

values of measurements under the left and right wheels. Table 44 through Table 46 also

compare displacements and forces between the two case. Differences in moments and thrusts

between the two soil models were insignificant. Displacements were slightly smaller with the

Hardening-soil model than the Mohr-Coulomb model: by 9 percent for the vertical crown

displacement and by 16 percent for the horizontal displacement. Therefore, displacement

results were closer to the filed measurements with the Hardening-soil model than with the Mohr-

Coulomb soil model in this case.

4.4.2.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover

Figure 105 and Figure 106 compare vertical crown displacements and horizontal diameter

extensions along the culvert between the two cases of HDPE pipe analysis: the case with the


Mohr-Coulomb model and the case with the Hardening-soil model. These figures also show

displacements measured in the field tests. Figure 107 and Figure 108 compares thrusts and

moments under the wheel load between the two cases. Table 47 through Table 49 also

compare displacements and forces between the two case. Calculated displacements were

larger with the Hardening-soil model than with the Mohr-Coulomb model: by about 30 percent

for the vertical crown displacement and by about 55 percent for the horizontal displacement.

Therefore, displacement results from the Mohr-Coulomb model were closer to the measured

displacements in the field tests in this particular case. Due to the larger displacements,

moments and thrusts were also larger with the Hardening-soil model. Thrusts from the

Hardening-soil model were up to 20 percent higher than those from the Mohr-Coulomb model,

and moments were up to 44 percent higher. Softer soil responses obtained for the Hardening-

soil model can be explained by the lower stiffness of the Hardening soil model when compared

to that of the Duncan-Selig model. By selecting higher stiffness values for Hardening-soil

properties of ML95, it will be possible to bring force results closer to those from the Mohr-

Coulomb model.

Figure 103—Comparison of Displacements between Cases with Mohr-Coulomb and

Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover)


0.0

0.1

0.2

0.3

0.4

0.5

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0 5 10 15 20


Vert

ical

Dis

plac

emen

t at C

row

n (in

) Mohr-Coulomb Model (Test 2, 3 ft Cover)Hardening Soil Model (Test 2, 3 ft Cover)Field Test (Test 2, 3 ft Cover)

Wheel Load Location0.00

0.05

0.10

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0.50

0 5 10 15 20


Hor

iz. E

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t of 8

8 in

. (in

)

Mohr-Coulomb Model (Test 2, 3 ft Cover)Hardening Soil Model (Test 2, 3 ft Cover)Field Test (Test 2, 3 ft Cover)

Wheel Load Location


Figure 104—Comparison of Thrusts and Moments under Wheel between Cases with

Mohr-Coulomb and Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover)

Table 44—Summary of Displacements under Wheel (Metal Arch, Test 2, 3 ft Cover)

Vertical or Horizontal

Field Test (in.)

Plaxis 3D (in.) Ratio: Plaxis 3D / Field Test

Mohr-Coulomb

Hardening-Soil

Mohr-Coulomb

Hardening-Soil

Vertical crown displacement 0.45 0.79 0.72 1.74 1.58

Horizontal chord extension 0.25 0.40 0.34 1.60 1.34

Table 45—Summary of Thrusts under Wheel (Metal Arch, Test 2, 3 ft Cover)

Location Field Test Plaxis 3D (kip/ft) Ratio: Plaxis 3D/Field Test(kip/ft) Mohr- Hardening Mohr- Hardening

Coulomb Soil Coulomb SoilNS 0.69 3.12 3.54 4.54 5.15NC 1.42 4.41 4.70 3.10 3.31NH 9.07 5.92 6.27 0.65 0.69CR 3.08 7.05 7.75 2.29 2.51SH 5.84 5.80 6.18 0.99 1.06SC 2.46 4.08 4.46 1.66 1.81SS -0.34 2.75 3.25 -7.98 -9.44



-5

25

kip/ft


-30

60

kip-in/ft


Table 46—Summary of Moments under Wheel (Metal Arch, Test 2, 3 ft Cover)

Location Field Test Plaxis 3D (kip-in./ft) Ratio: Plaxis 3D/Field Test(kip-in./ft) Mohr- Hardening Mohr- Hardening

Coulomb Soil Coulomb SoilNS -0.03 -3.14 -3.61 94.73 108.90NC -0.38 -2.75 -2.13 7.27 5.63NH -5.24 -5.75 -5.42 1.10 1.03CR 2.04 13.71 13.38 6.74 6.57SH -3.28 -6.43 -6.26 1.96 1.91SC -1.97 -2.62 -1.78 1.33 0.91SS 0.07 -2.66 -3.31 -35.61 -44.31

Figure 105—Comparison of Crown Vertical Displacements between Cases with Mohr-

Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.) Mohr-Coulomb Model (Heavy P3)

Mohr-Coulomb Model (Heavy P4)Hardening Soil Model (Heavy P3)Hardening Soil Model (Heavy P4)Heavy P3 (Oct-00)Heavy P3 (May-01)Heavy P3 (Aug-02)Heavy P4 (Oct-00)Heavy P4 (May-01)Heavy P4 (Aug-02)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.) Mohr-Coulomb Model (Light P3)

Mohr-Coulomb Model (Light P4)Hardening Soil Model (Light P3)Hardening Soil Model (Light P4)Light P3 (Oct-00)Light P3 (May-01)Light P3 (Aug-02)Light P4 (Oct-00)Light P4 (May-01)Light P4 (Aug-02)


0.00

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0 2 4 6 8 10 12


Vert

ical

Dis

plac

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t at C

row

n (in

.)

Mohr-Coulomb Model (Heavy P3)Mohr-Coulomb Model (Heavy P4)Hardening Soil Model (Heavy P3)Hardening Soil Model (Heavy P4)Heavy P3 (Oct-00)Heavy P3 (May-01)Heavy P3 (Aug-02)Heavy P4 (Oct-00)Heavy P4 (May-01)Heavy P4 (Aug-02)

0.00

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0.04

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0.08

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0.12

0 2 4 6 8 10 12


Vert

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Dis

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t at C

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n (in

.)Mohr-Coulomb Model (Light P3)Mohr-Coulomb Model (Light P4)Hardening Soil Model (Light P3)Hardening Soil Model (Light P4)Light P3 (Oct-00)Light P3 (May-01)Light P3 (Aug-02)Light P4 (Oct-00)Light P4 (May-01)Light P4 (Aug-02)


Figure 106—Comparison of Horizontal Diameter Changes between Cases with Mohr-Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)

Figure 107—Comparison of Thrusts between Cases with Mohr-Coulomb and Hardening-

Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)

Figure 108—Comparison of Moments between Cases with Mohr-Coulomb and Hardening-

Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)

Table 47—Summary of Vertical Displacements under Wheel (HDPE Pipe, A2 Soil, 2.8 ft Cover)

Truck Position Field Test (in.) Plaxis 3D (in.) Ratio: Plaxis 3D / Field TestMohr- Hardening Mohr-Coulomb Model Hardening Soil Model

Oct-00 May-01 Aug-02 Coulomb Soil Oct-00 May-01 Aug-02 Oct-00 May-01 Aug-02Heavy 3 0.061 0.036 0.039 0.069 0.089 1.13 1.91 1.76 1.47 2.48 2.29

4 0.065 0.040 0.039 0.076 0.103 1.18 1.91 1.96 1.59 2.58 2.65Light 3 0.049 0.018 0.022 0.053 0.068 1.07 2.92 2.39 1.38 3.75 3.07

4 0.049 0.026 0.026 0.058 0.077 1.18 2.23 2.23 1.58 2.98 2.98


-20

-15

-10

-5

0

5

10

15

20

25


Ben

ding

Mom

ent (

lb-in

./in.

)

Mohr-Coulomb Model(Heavy P3)Mohr-Coulomb Model(Heavy P4)Hardening Soil Model(Heavy P3)Hardening Soil Model(Heavy P4)

-20

-15

-10

-5

0

5

10

15

20

25


Ben

ding

Mom

ent (

lb-in

./in.

)

Mohr-Coulomb Model(Light P3)Mohr-Coulomb Model(Light P4)Hardening Soil Model(Light P3)Hardening Soil Model(Light P4)


0

5

10

15

20

25

30

35

40

45


Thru

st (l

b/in

.)

Mohr-Coulomb Model (Heavy P3)Mohr-Coulomb Model (Heavy P4)Hardening Soil Model (Heavy P3)Hardening Soil Model (Heavy P4)

0

5

10

15

20

25

30

35

40

45


Thru

st (l

b/in

.)

Mohr-Coulomb Model (Light P3)Mohr-Coulomb Model (Light P4)Hardening Soil Model (Light P3)Hardening Soil Model (Light P4)


Table 48—Summary of Horizontal Chord Extensions under Wheel (HDPE Pipe, A2 Soil, 2.8 ft Cover)

Truck Position Field Test (in.) Plaxis 3D (in.) Ratio: Plaxis 3D / Field TestMohr- Hardening Mohr-Coulomb Model Hardening Soil Model

Oct-00 May-01 Aug-02 Coulomb Soil Oct-00 May-01 Aug-02 Oct-00 May-01 Aug-02Heavy 3 0.009 0.004 0.005 0.016 0.024 1.74 3.91 3.13 2.69 6.06 4.85

4 0.016 0.008 0.006 0.019 0.031 1.21 2.43 3.24 1.95 3.90 5.20Light 3 0.005 0.003 0.003 0.012 0.018 2.40 4.00 4.00 3.67 6.11 6.11

4 0.007 0.006 0.004 0.015 0.023 2.11 2.46 3.69 3.35 3.91 5.87

Table 49—Summary of Force Results (HDPE Pipe, A2 Soil, 2.8 ft Cover)

Truck Position Thrust (lb/in.) Moment (lb-in./in.) Crown Peak Peak Positive Peak Negative

MC HS MC HS MC HS MC HSHeavy 3 19.5 24.6 28.7 32.1 15.7 20.5 -11.1 -15.4

4 24.5 32.8 33.2 39.8 11.7 14.7 -11.5 -16.5Light 3 14.9 18.8 22.0 24.3 12.0 15.1 -8.6 -11.7

4 18.5 24.9 25.2 30.2 8.8 10.5 -8.7 -12.4

4.4.3 Conclusion

With the soil properties reported used, there was not a consistent trend in structural responses

when the soil model was changed from the Mohr-Coulomb model to the Hardening-soil model.

However, for both structures (the metal arch and the HDPE pipe), changing the soil constitutive

model from the Mohr-Coulomb model to the Hardening-soil model affected displacement results

more than moments and thrusts.

We will examine appropriateness of the Mohr-Coulomb model by simulating the field tests in

ABAQUS as described in the next section.

4.5 Three-Dimensional Analysis of Field Tests in ABAQUS

4.5.1 Introduction

In Task 2 of NCHRP Project 15-29, SGH performed 3D analyses of field live load tests (NCHRP

Project 12-45 and MNDOT study) using PLAXIS 3D with the Mohr-Coulomb constitutive model

for backfill. A few panel members showed their concern about appropriateness of the Mohr-

Coulomb model because calculated structural responses were not close to the measured

responses in some cases. SGH believes that there are two reasons for the deviation of


calculated responses from the measured responses: (1) orthotropic section properties for metal

arch and HDPE pipe and (2) level of backfill compaction.

Corrugated metal plate arch and HDPE pipe have different stiffnesses in the circumferential and

longitudinal directions due to their corrugation or plate profile. Since orthotropic properties

cannot be assigned to plate elements in PLAXIS 3D, we inserted strips of very thin elements to

match circumferential and longitudinal stiffnesses. However, due to these thin elements,

effective shear stiffness of the metal arch or HDPE pipe becomes lower than the actual shear

stiffness, which resulted in a concentration of displacement near the wheel loads.

In the cases analyzed for the metal arch in PLAXIS 3D, crown vertical displacements in Test 1

were predicted well by the analysis whereas those calculated for Test 2 were significantly

greater than measured displacements. In Test 1, backfill soil was compacted for a target

compaction of 95 percent of the maximum standard proctor density and resulted in 92 percent.

In Test 2, backfill soils above and below the crown were compacted for target compaction of 95

percent and 85 percent, respectively, and resulted in 96 percent and 87 percent. SGH believes

that the backfill soil below the crown was further compacted when the backfill soil above the

crown was compacted to 96 percent of the standard proctor density in Test 2. The PLAXIS 3D

analysis reported in the earlier did not consider this additional compaction effort, and soil

properties for SW85 were used for the backfill soil below the crown.

To examine the two items discussed above, SGH performed two cases of 3D soil-structure

interaction analysis in ABAQUS: (1) long-span metal arch, Test 2, 3 ft cover (NCHRP Project

12-45); and (2) HDPE pipe, Pipe Run 7, A-2 backfill, 2.8 ft cover (MNDOT study).


Soil-structure interaction analysis of culverts subjected to the surface live load was performed

using ABAQUS. Two structural types were selected as described above: (1) Long-span metal

arch (Test 1 and Test 2 with 3 ft cover in NCHRP Project 12-45); and (2) HDPE pipe (Pipe Run

7 with A-2 backfill and 2.8 ft cover and Pipe Run 3 with A-2 backfill and 1.6 ft cover in the

MNDOT study).

Although longitudinal and circumferential stiffnesses for axial force and bending are known, we

can match only three of the four stiffnesses exactly because shell elements have uniform

thickness and only orthotropic material properties are defined. We calculated moduli of

elasticity of the longitudinal and circumferential directions and thickness of the shell element to


match longitudinal and circumferential bending stiffnesses and circumferential axial stiffness.

Shear modulus in the orthotropic properties used in the analysis was determined by multiplying

actual shear modulus by a ratio of the actual plate thickness to the effective plate thickness

used in the analysis. Table 50 shows orthotropic properties used in ABAQUS analyses, and

Table 51 compares stiffnesses calculated from the orthotropic properties given in Table 50 with

actual stiffnesses. Although axial stiffness of the metal arch used in ABAQUS turned out to be

close to the actual stiffness, axial stiffness of the HDPE pipe used in ABAQUS was significantly

smaller than the actual stiffness.

Table 50—Orthotropic Properties Used in ABAQUS Analyses

Structure Modulus of elasticity (psi) Poisson’s

ratio Shear

modulus (psi) Shell element thickness (in.) Circumferential Longitudinal

Metal arch 3,234,000 17,670 0.3 1,001,000 2.390 HDPE pipe 12,680 162 0.35 2,865 4.227

Table 51—Orthotropic Stiffness Properties

Structure Actual or ABAQUS

Axial, EA, (lb/in.) Bending, EI, (lb-in.2


Longitudinal Circumferential Longitudinal

Metal arch Actual 7,731,000 47,917 3,681,000 20,106

Used in ABAQUS 7,731,000 42,231 3,681,000 20,106

HDPE pipe Actual 53,600 32,700 79,800 1,019

Used in ABAQUS 53,600 684 79,800 1,019

Analyses in ABAQUS were performed with the Mohr-Coulomb model, and structural responses

were compared with those measured in the field tests. In the metal arch model for Test 2, two

sets of backfill properties were examined for backfill below the crown of arch: SW85 and SW90.

Properties of SW95 were used for the soil above the crown in Test 1 and Test 2 and for backfill

in Test 1. For the HDPE pipe, two sets of backfill properties were used: ML90 and ML95. Also,

soft haunch and void areas were modeled by using very soft soil properties as shown in Figure

109. These areas were not modeled in PLAXIS 3D. Heavy live load truck was positioned in a

symmetric manner over crown (Position P4) for the HDPE pipe. The interface was introduced

between the structure and surrounding soil using contact elements. Coefficient of friction was

assumed to be 50 percent of angle of friction of surrounding soil. Structures, live load tests, and

soil properties of SW85, SW95, ML90, and ML95 were described in detail earlier. Figure 110

and Figure 111 show ABAQUS models of the metal arch and the HDPE pipe.


60 in.Nominal

AASHTO Backfill(A-2)

30 ft

In-Situ Soil

6 in. Bedding

Cover Depth(varies)

12 in.


8 ft 8 in. 84 in.

Soft Haunch

Void

Figure 109—Cross Section of Finite Element Model for HDPE Pipe in ABAQUS

Table 52—Soil Porperties Used for Soft Haunch and Void Areas

Area Modulus of Elasticity

E

Poisson’s Ratio ν

Angle of Friction φ

Dilatation Angle ψ

Cohesion c

(psi) (deg) (deg) (psi) Soft haunch 1,000 0.35 28 0 1.0

Void 50 0.30 23 0 2.5


Figure 110—ABAQUS Metal Arch Model with 3 ft Cover

Figure 111—ABAQUS HDPE Pipe Model with 3 ft Cover

4.5.3 Validation of ABAQUS Model

Before performing analyses with orthotropic material properties for structure in ABAQUS, we a

performed analysis of the metal arch in ABAQUS with strips of thin elements to examine

whether ABAQUS can produce the same results as PLAXIS 3D did. The same material

properties were used in both models. However, finite element meshes in the cross section were

not the same because 15-noded wedge elements were used for soil in PLAXIS 3D while 8-

noded brick elements were used in ABAQUS. Figure 112 and Figure 113 compare

displacement and force results from PLAXIS 3D and ABAQUS, respectively. Vertical crown

displacement and horizontal chord extension under wheel in ABAQUS differ from those of

PLAXIS 3D by only 2 percent and 7 percent, respectively. Maximum thrust and moment in

ABAQUS differ from those of PLAXIS 3D by only 4 percent and 3 percent, respectively. We

concluded that ABAQUS can produce essentially the same results as PLAXIS 3D when the

same problem is analyzed.


Figure 112—Comparison of Vertical and Horizontal Displacements between PLAXIS 3D

and ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover)

Figure 113—Comparison of Thrusts and Moments under Wheel between PLAXIS 3D and

ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover)

4.5.4 Results

4.5.4.1 Metal Arch with 3 ft Cover

Figure 114 and Figure 115 show displacement and force results from ABAQUS analyses with

orthotropic properties for the metal arch. Table 53 compares displacement results between

ABAQUS analyses and field test data. By comparing displacements between the ABAQUS

model with orthotropic properties and SW85 properties and the PLAXIS model, it is obvious that

both vertical and horizontal displacements were significantly reduced in the ABAQUS model,

and that horizontal displacements were affected more than vertical displacements by modeling

the metal arch with orthotropic properties. These results confirm that modeling technique used

in PLAXIS 3D for the corrugated metal arch caused unrealistically large horizontal

displacements. In Figure 114, we can also see that using SW90 properties for backfill in Test 2

case brought calculated displacements much closer to the measured data, which shows that the

backfill soil in Test 2 that was compacted to 87 percent of the maximum standard proctor


Test 2, 3 ft Cover (PLAXIS, SW85)

Test 2, 3 ft Cover (Field Test)

Test 2, 3 ft Cover (ABAQUS, SW85)

-2.5

12.5

kip/ft




-15

30

kip-in/ft


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20


Vert

ical

Dis

plac

emen

t at C

row

n (in

) Test 2, 3 ft Cover (PLAXIS, SW85)



Wheel Load Location

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 5 10 15 20


Hor

iz. E

xten

sion

of C

hord

at H

eigh

t of 8

8 in

. (in

)




Wheel Load Location


density was compacted to about 90 percent of the maximum standard proctor density when the

soil above the crown was compacted.

Figure 114—Vertical and Horizontal Displacements from ABAQUS Analyses with

Orthotropic Properties (Metal Arch, 3 ft Cover)

Figure 115 –Thrusts and Moments under Wheel from ABAQUS Analyses with Orthotropic

Properties (Metal Arch, 3 ft Cover)

Table 53—Summary of Displacements from ABAQUS Analyses with Orthotropic Properties (Metal Arch, 3 ft Cover)

Field Test (in.) ABAQUS (in.) Ratio: ABAQUS / Field Test

Vertical Horizontal Vertical Horizontal Vertical Horizontal Test 1 (SW95 in ABAQUS) 0.413 0.083 0.397 0.163 0.961 1.970

Test 2 (SW90 in ABAQUS) 0.453 0.252 0.489 0.208 1.080 0.824


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20


Vert

ical

Dis

plac

emen

t at C

row

n (in

) Test 1, 3 ft Cover (Field Test)Test 1, 3 ft Cover (ABAQUS, SW95)Test 2, 3 ft Cover (Field Test)Test 2, 3 ft Cover (ABAQUS, SW85)Test 2, 3 ft Cover (ABAQUS, SW90)Test 1, 3 ft Cover (PLAXIS, SW95)Test 2, 3 ft Cover (PLAXIS, SW85)

Wheel Load Location

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 5 10 15 20

Distance from Symmetry Line of Tandem Axles (ft)H

oriz

. Ext

ensi

on o

f Cho

rd a

t Hei

ght o

f 88

in.

(in)

Test 1, 3 ft Cover (Field Test)Test 1, 3 ft Cover (ABAQUS, SW95)Test 2, 3 ft Cover (Field Test)Test 2, 3 ft Cover (ABAQUS, SW85)Test 2, 3 ft Cover (ABAQUS, SW90)Test 1, 3 ft Cover (PLAXIS, SW95)Test 2, 3 ft Cover (PLAXIS, SW85)

Wheel Load Location


Test 1, 3 ft Cover (Field Test)Test 1, 3 ft Cover (ABAQUS, SW95)Test 2, 3 ft Cover (Field Test)Test 2, 3 ft Cover (ABAQUS, SW85)Test 2, 3 ft Cover (ABAQUS, SW90)

-2.5

12.5

kip/ft

Test 1, 3 ft Cover (Field Test)Test 1, 3 ft Cover (ABAQUS, SW95)Test 2, 3 ft Cover (Field Test)Test 2, 3 ft Cover (ABAQUS, SW85)Test 2, 3 ft Cover (ABAQUS, SW90)

-15

30

kip-in/ft


4.5.4.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover

Figure 116 and Figure 117 show displacement and force results from ABAQUS analyses with

orthotropic properties for the HDPE pipe. Table 54 compares displacement results between

ABAQUS analyses and field test data measured in October 2000, which were the first

measurements after the pipe installation. In the case of 2.8 ft of cover, calculated

displacements with ML 95 properties were greater than the data measured in October 2000 by

18 percent and 25 percent for vertical and horizontal displacements, respectively, while in the

case of 1.6 ft of cover, calculated displacements with ML95 properties were smaller than the

data measured in October 2000 by 27 percent and 7 percent for vertical and horizontal

displacements, respectively. These results show that displacements calculated in ABAQUS are

in good agreement with those measured in the first live load tests after the pipe installation.


Orthotropic Properties (HDPE Pipe, A2 Backfill, 2.8 ft Cover)


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)

Heavy P4 (Oct-00)

Heavy P4 (May-01)

Heavy P4 (Aug-02)

Heavy P4 (ABAQUS, ML90)


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)

Heavy P4 (Oct-00)

Heavy P4 (May-01)

Heavy P4 (Aug-02)





Orthotropic Properties (HDPE Pipe, A2 Backfill, 1.6 ft Cover)

Table 54—Summary of Displacements from ABAQUS Analyses with Orthotropic Properties (HDPE Pipe, A2 Backfill)

Field Test on Oct. 00

(in.) ABAQUS (in.) Ratio: ABAQUS / Field Test

Vertical Horizontal Vertical Horizontal Vertical Horizontal 2.8 ft cover (ML95

in ABAQUS) 0.065 0.016 0.076 0.020 1.175 1.248

1.6 ft cover (ML95 in ABAQUS) 0.137 0.027 0.099 0.025 0.725 0.927

4.5.5 Conclusion

It is important to model orthotropic properties of structures to accurately calculate structural

response. The modeling technique that SGH used in PLAXIS 3D underestimated shear

stiffness of the corrugated plate and resulted in concentrated displacements under wheel loads.

To accurately simulate field tests, the actual achieved density of backfill should be estimated,

and backfill properties for the corresponding density should be used in the analysis.

Because structural responses from ABAQUS analyses with the Mohr-Coulomb soil model are in

good agreement with those measured in the field tests for the cases examined in this study, we

conclude that the Mohr-Coulomb model can be an appropriate model to produce structural

response of culvert subjected to live loads with sufficient accuracy.


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 2 4 6 8 10 12


Vert

ical

Dis

plac

emen

t at C

row

n (in

.)

Heavy P4 (Oct-00)Heavy P4 (May-01)Heavy P4 (Aug-02)Heavy P4 (ABAQUS, ML90)Heavy P4 (ABAQUS, ML95)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 2 4 6 8 10 12


Dia

met

rical

Cha

nge

betw

een

Sprin

glin

es (i

n.)

Heavy P4 (Oct-00)Heavy P4 (May-01)Heavy P4 (Aug-02)Heavy P4 (ABAQUS, ML90)Heavy P4 (ABAQUS, ML95)


Since the Mohr-Coulomb model is the soil model of our choice and orthotropic properties of

corrugated plates were found to be important, ABAQUS is the better software to use for the

problem we have than PLAXIS 3D.

5. DISCUSSION

The preliminary 2D analyses showed that a soil-structure interaction analysis of buried culverts

subjected to live loads with the linear-elastic soil model could produce significantly different

structural response than that with the Mohr-Coulomb soil model and the Hardening-Soil model.

Since a difference between the linear-elastic model and the Mohr-Coulomb model is whether or

not soil failure is modeled by plasticity, plasticity is one of the key aspects of soil models that are

suitable for this project.

The analyses presented here indicate some of the difficulties in predicting structural response of

buried culverts subjected to live loads. The soil parameters currently used in design appear to

yield soil behavior that is softer than achieved in the field tests. As noted above, given the

variability of real world soils and in field compaction effort, this conservatism is justified in

design. Soil parameters could be developed just for the current study that match the soil test

data (Section 2), which in turn produce better estimates of live load response of buried culverts.

However, the same question will arise, that is how should the parameters be modified for design

of actual structures which will experience all of the variability noted. Given the success of the

Duncan-Selig model and the Selig (1988) properties, it is appropriate to continue with design

parameters that are conservative.

In addition to the soil parameters of a specific soil type, there are other uncertainties in the field

tests, which made matching field data in the analysis difficult. Backfill densities are reported as

the density measured at the time of backfilling, however, there is considerable activity over the

pipes after the backfilling is completed, and this activity likely densifies the soil. For example:

• In the long span study, the soil surface was compacted with a large vibratory roller prior to the live load tests to assure that the surface soil could carry the heavily loaded truck without significant rutting. This likely densified the clean gravel backfill.

• In the MN/DOT study, the backfill was overlaid with 8 in. of gravel, and 4 in. of pavement. Thus again the backfill over the top crown of the test pipes was likely densified prior to live load testing.

• In the MN/DOT study, after the construction was complete, there was still considerable variability in the data as a result of seasonal differences, temperature variations and perhaps other parameters.


6. CONCLUSIONS AND RECOMMENDATIONS

This report documents the investigation of soil models for analysis of live load effects on buried

structures. We recommend that the additional studies carried out in this project be conducted

with a linearly-elastic, perfectly-plastic model with a Mohr-Coulomb failure criterion. This

selection offers the best mix of capturing the important aspects of soil behavior in transmitting

live loads to structures and of offering simplicity in modeling that will allow the team to complete

the most analyses in the least amount of time. In implementing this soil model, we do

recommend that the elastic soil properties be selected based on depth of fill as in this study.

This technique does not offer all of the benefits of the Duncan-Selig/Hardening Soil models in

capturing stress-dependent stiffness behavior of soil, but for the purposes of a live load study, it

appears to provide sufficient accuracy.

Parameters for the soil model should be those reported above based on the Selig 1988 and

1990 properties. The bulk modulus values in Selig 1990 should be considered suitable for

analysis when justified by data, but may not be a lower bound. The proposed properties will

prove to be somewhat conservative relative to field data, but will represent a lower bound of

behavior.

7. REFERENCES

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Brinkgreve, R.B.J. and Broere, W., Eds. (2004), Plaxis 3D Tunnel Version 2, Plaxis,

Netherlands.

Duncan, J.M., Byrne, P., Wong, K.S., and Mabry, P. (1980), Strength, Stress-Strain and Bulk

Modulus Parameters for Finite Element Analysis of Stress and Movements in Soil Masses,

Report No. UCB/GT/80-01, University of California, Berkeley, Berkeley, CA.

Fernando, N. S. M. and Carter, J. P. (1998), “Elastic Analysis of Buried Pipes under Surface

Patch Loadings,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 124,

No. 8, pp. 720-728.

Heger, F. J., Liepins, A. A., and Selig, E. T. (1985), “SPIDA: Analysis and Design System for

Buried Concrete Pipe,” Advances in Underground Pipeline Engineering, Proceedings of

the International Conference, American Society of Civil Engineers, pp. 143-154.


Howard, A. K. (1977), “Modulus of Soil Reaction Values for Buried Flexible Pipe,” Journal of the

Geotechnical Engineering, Vol. 103, No. GT1, New York, NY.

Jaky, J. (1944), “The Coefficient of Earth Pressure at Rest,” Journal of The Society of Hungarian

Architects and Engineers, Budapest, pp. 355-358 (in Hungarian).

Lade, P. V. (2005), “Overview of Constitutive Models for Soils,” Soil Constitutive Models—

Evaluation, Selection, and Calibration, J. A. Yamamuro and V. N. Kaliakin, Eds., ASCE,

Reston, VA, pp. 1-34.

Lin, R. D. (1987), Direct Determination of Bulk Modulus of Partially Saturated Soils, Master

Thesis, University of Massachusetts, Amherst, MA.

McGrath, T. J. (1998), “Replacing E′ with the Constrained Modulus in Flexible Pipe Design,”

Pipelines in the Constructed Environment, Proceedings of the 1998 Pipeline Division

Conference, American Society of Civil Engineers, pp. 28-40.

McGrath, T.J. and Beaver J.L. (2005), Performance of Thermoplastic Pipe Under Highway

Vehicle Loading, Project Report Prepared for Minnesota Department of Transportation,

Simpson Gumpertz & Heger Inc., Waltham, MA.

McGrath, T.J., DelloRusso, S.J., and Boynton, J. (2002), “Performance of Thermoplastic Culvert

Pipe Under Highway Vehcle Loading,” Pipelines 2002, G. Kurz, Ed., American Society of

Civil Engineers.

McGrath. T.J., Moore, I.D., Selig, E.T., Webb, M. C., and Taleb, B. (2002), Recommended

Specifications for Large-Span Culverts, NCHRP Report 473, Transportation Research

Board, Washington, D.C.

Moore, I. D. and Brachman, R. W. (1994), “Three-Dimensional Analyses of Flexible Circular

Culverts,” Journal of Geotechnical Engineering, Vol. 120, No. 10, pp. 1829-1844.

Musser, S. C. (1989), CANDE-89 User Manual, FHWA-RD-89-169, Federal Highway

Administration, McLean, VA.

Pang, S. (1999), “Discussion: Elastic Analysis of Buried Pipes under Surface Patch Loadings by

N. S. M. Fernando and J. P. Carter,” Journal of Geotechnical and Geoenvironmental

Engineering, Vol. 125, No. 12, p. 1104.


Selig, E.T. (1988), “Soil Parameters for Design of Buried Pipelines,” Pipeline Infrastructure, B.

A. Bennett, Ed., ASCE, New York, NY, pp. 99-116.

Selig, E.T. (1990), “Soil Properties for Plastic Pipe Installations,” Buried Plastic Pipe

Technology, STP1093, G.S. Buczala and M.J. Cassady, Eds., ASTM, Philadelphia, PA,

pp. 141-158.

Taleb, B. (2000), Behavior of Large-Span Metal and Reinforced Concrete Culverts under Earth

and Live Loadings, Ph.D. Dissertation, University of Massachusetts Amherst, Amherst,

MA.

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Dissertation, University of Massachusetts Amherst, Amherst, MA.

Documents

Appendix A Investigation of Suitable Soil Constitutive Models for 3-D Finite Element Studies of Live