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Tim

1 / LIBRARY
.1 0 ob Michigan State
University

This is to certify that the
dissertation entitled

IMPACT OF THE INTERACTION BETWEEN STRUCTURAL,
ENVIRONMENTAL, AND LOADING FACTORS ON RIGID
PAVEMENT RESPONSES

presented by

KAENVIT VONGCHUSIRI

has been accepted towards fulfillment
of the requirements for the

Ph. D. degree In Civil Engineerigg

 

 

 

ﬂk/

/LVMajorP ofes or’s Signa
7; oj/we/

 

/Date

MSU is an Afﬁrmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this d1eckout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

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2/05 a/cﬁclomm.mis

IMPACT OF THE INTERACTION BETWEEN STRUCTURAL, ENVIRONMENTAL,
AND LOADING FACTORS ON RIGID PAVEIVIENT RESPONSES

By

Kaenvit Vongchusiri

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTORAL OF PHILOSOPHY
Department of Civil and Environmental Engineering

2005

ABSTRACT

IMPACT OF THE INTERACTION BETWEEN STRUCTURAL, ENVIRONMENTAL,
AND LOADING FACTORS ON RIGID PAVEMENT RESPONSES

By

Kaenvit Vongchusin'

This dissertation focused on the impact of the interaction between structural,
environmental, and loading factors on mechanistic responses of rigid pavements.
Computed through the mechanistic analysis approach, the pavement responses could be
linked to rigid pavement performance. Therefore, this study established the understanding
of the impact of the interaction between such factors on mechanistic responses, captured
through an extensive parametric study. Without sacriﬁcing the quality of the ﬁnal results,
this study employed several strategies to reduce the size of the experimental matrix to a
practical number of 43,092 ﬁnite element runs.

Based on the parametric study results, the insight into the impact of the interaction
between various parameters on pavement stresses was established. Understanding the
mechanistic behavior of the pavement provides an indirect connection to a better
comprehension of pavement performance. The increase in base/subbase thickness
resulted in a reduction in stress magnitude with diminishing effect as the slab thickness
increases. While lateral support condition had a signiﬁcant effect on loading stress
magnitude, its effect on thermal stress magnitude appeared to be insigniﬁcant as shown in
the ﬁgure. An increase in the magnitude of modulus of subgrade reaction resulted in a
reduction in stress magnitude with diminishing effect as the slab thickness increased. An

increase in the magnitude of modulus of subgrade reaction resulted in an increase in the

magnitude of thermal stress as the combined stress magnitudes were compared to the
loading stress magnitudes. When combined with thermal stress, an interactive effect
between thermal strain gradient and joint spacing on combined loading and thermal stress
was observed. The results suggested that a more complex axle group should result in a
lower pavement stress magnitude. However, the results did not account for the interaction
between axle spacing and joint spacing.

This study also included a development of interpolation schemes to predict
stresses in jointed concrete pavements subjected to trafﬁc and environmental loads. The
interpolation schemes were developed based on the various design scenarios that reﬂect
the current design practice. The interpolation schemes were found to be highly efﬁcient
in generating stresses for an unlimited number of scenarios, based on a limited number of
ﬁnite element runs. Validation of the interpolation schemes was conducted by comparing
the interpolated stresses to ﬁnite element analysis results.

Also, this study demonstrated that inﬂuence surfaces for rigid pavements could be
successfully developed via a numerical procedure based on a series of ﬁnite element
analyses and a multi-dimensional interpolation process. The extensive veriﬁcation
process used herein suggested that inﬂuence surfaces could precisely and accurately
quantify pavement stresses under various loading conditions. Versatility of the inﬂuence
surface technique for rigid pavements includes, but is not limited to, rapid pavement
stress calculation, determination of critical load location, pavement stress history, and
investigation of the interaction between load configuration and structural feature. The use
of this technique is clearly more effective and practical than the direct application of the

ﬁnite element method.

ACKNOWLEDGEMENTS

I would like to express my most sincere gratitude to Dr. Neeraj Buch, the chairman of my
dissertation committee, whom I am profoundly indebted to, for his encouragement and
his support both technically and personally throughout the course of my graduate studies.
He also served as a role model without whose guidance it would not have been possible
to complete this research. My sincere appreciation also goes to the committee members,
Dr. Gilbert Baladi, Dr. Karim Chatti, and Dr. Dennis Gilliland, for their perseverance and
assistance.

The ﬁnancial supports of the Royal Thai Government, the Department of Civil
and Environmental Engineering at Michigan State University, and the Michigan
Department of Transportation throughout my studies are greatly appreciated.

Special thanks are extended to Mr. Michael Sherry of the Writing Center for his
expertise in peer reviewing, to Dr. Rigoberto Burgueﬁo for his guidance on the inﬂuence
surface theorem, and also to my fellow colleagues, including Mr. Praveen Desaraju, Mr.
Hassan Salama, and Dr. Syed Waqar Haider for supporting this research and for truly
sharing their experience with me.

Finally, I would also like to thank my mother Lalida for always providing
guidance throughout my life and my girlfriend Sonali for being there and persevering

while I pursued this undertaking.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................... vii
LIST OF FIGURES .............................................................................. viii
CHAPTER I

INTRODUCTION .................................................................................. 1
1.1 Background .................................................................................. 1
1.2 Problem identiﬁcation ...................................................................... 2
1.3 Research objectives ......................................................................... 3
1.4 Chapter overview ........................................................................... 4
CHAPTER II

LITERATURE REVIEW .......................................................................... 7
2.1 Finite element analysis of rigid pavements .............................................. 7
2.2 Factors affecting mechanistic responses ................................................ 14
2.3 Application of inﬂuence surfaces ....................................................... 25
CHAPTER III

PARAMETRIC STUDY .......................................................................... 28
3.1 Data collection .......................................................................... 28
3.2 Experimental matrix ...................................................................... 30
3.3 Analysis process ........................................................................... 47
3.4 Documentation of analysis results ..................................................... 58
CHAPTER IV

INTERPOLATION SCHEME .................................................................... 88
4.1 Least squares criteria 8.8
4.2 Development of interpolation schemes ................................................. 90
4.3 Validation of interpolation schemes .................................................. 96
4.4 Sample of calculation .................................................................... 111
CHAPTER V

INFLUENCE SURFACE TECHNIQUE ..................................................... 115
5.1 Construction of inﬂuence surfaces .................................................. 116
5.2 Interpretation of inﬂuence surfaces ................................................... 129
5.3 Veriﬁcation of the proposed technique ............................................... 131
5.4 Potential applications .................................................................. 136
CHAPTER VI

CONCLUSIONS, RESEARCH SIGNIFICANCE AND RECOMMENDATIONS 160
6.1 Conclusions ............................................................................. 160
6.2 Research signiﬁcance .................................................................. 165

6.3 Recommendations for future research ................................................ 168

BIBLIOGRAPHY ................................................................................ 170

vi

Table 1:

Table 2:

Table 3:

Table 4:

Table 5:

Table 6:

Table 7:

Table 8:

Table 9:

Table 10:

Table 11:

LIST OF TABLES

Summary of design parameters from the 14 MDOT designs 29
Ranges of input parameters obtained from other sources ................ 29
Final experimental matrix .................................................... 47
Validation matrix .............................................................. 53
Summary of critical load locations ............................................ 57
Summary of interaction between parameters on stresses ................. 83
Example prediction matrices .................................................. 95
Summary of goodness of ﬁt — stage 1 ........................................ 99
Summary of goodness of ﬁt — stage 2 ........................................ 99
Comparison of scheme 15 and scheme 16 — stage 3 ....................... 110
Inventory properties of the sections used in the demonstration ......... 142

vii

Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:

Figure 9:

Figure 10:
Figure 11:
Figure 12:
Figure 13:

Figure 14:

Figure 15:
Figure 16:
Figure 17:
Figure 18:
Figure 19:
Figure 20:

Figure 21:

LIST OF FIGURES

Kirchhoff plate element with typical d.o.f. shown at node 3 ............... 10
Effects of temperature gradient on slab curling ............................. 16
Deﬂection proﬁle due to change in the temperature differential ......... 17
Daily and seasonal variation in pavement temperature gradient ......... 19
Effect of shrinkage gradient on slab curling ................................ 20
Shrinkage modeled using equivalent temperature gradient ............... 21
Stress distribution due to drying shrinkage ................................... 21
Effect of moisture gradient on slab warping ................................. 23
Critical loading conditions .................................................... 24
Critical loading condition for top-down stresses .......................... 24
Symbolic expression of reference point and general point ...................... 27
An overview of the development of experimental matrix ................. 30
Combining base and subbase layers 33

Comparison of variation in results for combined base/subbase and no

subbase approaches ............................................................ 34
Combining CTE and thermal gradient ......................................... 35
Load conﬁgurations considered in the study .............................. 37
Sensitivity trend due to the variation in base/subbase thickness .......... 45

Sensitivity trend due to the variation in modulus of subgrade reaction ...45

Sensitivity trend due to the variation in thermal strain gradient 46
Overview of structural model .................................................. 49
Required components for the analytical tool ............................... 50

viii

Figure 22:
Figure 23:

Figure 24:

Figure 25:

Figure 26:
Figure 27:
Figure 28:
Figure 29:
Figure 30:
Figure 31:
Figure 32:
Figure 33:
Figure 34:
Figure 35:
Figure 36:
Figure 37:
Figure 38:
Figure 39:
Figure 40:
Figure 41:
Figure 42:

Figure 43:

Procedure of determining critical load location ............................. 51
Validation and determination of critical load location ........................ 54

Example validation and determination (bottom stresses, MI-9, 177-in.

joint spacing) .................................................................... 56
Example validation and determination (bottom stresses, MI-20, l77-in.
joint spacing) .................................................................... 56
Example sensitivity plots of bottom-up stresses ............................. 61
Example sensitivity plots of top-down stresses ............................ 66
Impact of lateral support condition ........................................... 71
Slab curling due to different types of thermal gradients ................... 73
Effect of longitudinal joint AGG factor on stress magnitude ............. 8O
Interpolation process .............................................................. 93
Validation procedure ............................................................ 96
Overview of validation process .............................................. 97
Validation results — stage 1 ................................................... 100
Validation results — stage 2 .................................................. 104
Validation results - stage 3 .................................................. 108
Overall process of the proposed approach to inﬂuence surface ......... 117
Typical loading grids, reference point, and unit load ................... 120

Impact of tire contact pressure and aspect ratio on pavement stress 122

Typical inﬂuence surfaces (in psi) ......................................... 130
Multi-axle truck conﬁguration used in the veriﬁcation process ........ 131
Veriﬁcation results ........................................................... 133

Typical determination of critical load location for bottom-up stress 139

ix

Figure 44:
Figure 45:
Figure 46:
Figure 47:
Figure 48:
Figure 49:
Figure 50:
Figure 51:
Figure 52:
Figure 53:
Figure 54:

Figure 55:

Typical determination of critical load location for top-down stress 140
Impact of variations in slab and base elastic moduli ..................... 144
Typical truck conﬁguration extracted from the WIM database .......... 145
Pavement stress history plots for the DGAB sections ................... 146
Stress ratio history plots for the DGAB sections ........................... 147
Fatigue damage based on peak stresses from inﬂuence surfaces 148
Time-series transverse cracking on the DGAB sections ................... 149
Single unit truck conﬁguration used in the demonstration study 150
Interaction between axle spacing and slab length on bottom-up stress 152
Interaction between axle spacing and slab length on top-down stress 155
Impact of slab length on thermal stress .................................... 157
Combined loading and thermal stresses ..................................... 158

CHAPTER I

INTRODUCTION

This dissertation focused on the impact of the interaction between structural,
environmental, and loading factors on mechanistic responses of rigid pavements.
Computed through the mechanistic analysis approach, the pavement responses could be
linked to rigid pavement performance, which in this. study refers to load related
performance such as transverse cracking. This introductive chapter is comprised of four

parts: background, problem identiﬁcation, research objectives, and chapter overview.

1.1 Background
The ability to simulate numerous pavement conditions in an economically feasible
fashion makes the mechanistic analysis approach an attractive tool in the study of rigid
pavements. Mechanistic responses, the results obtained from the mechanistic analysis,
e.g. pavement stresses, are directly inﬂuenced by the interaction between structural,
environmental, and loading factors. Then, through appropriate transfer functions, the
computed mechanistic responses could be linked to rigid pavement performance.
Therefore, the understanding of the impact of the interaction between such factors on
mechanistic responses would also lead to the understanding of pavement performance.
Realizing the importance of such a connection, researchers in the area of rigid
pavement engineering have been devoting much of their effort to develop and enhance
the mechanistic approach to become more realistic; over the decades, the mechanistic

analysis of rigid pavements was substantially improved from overly simpliﬁed closed-

form solutions to sophisticated ﬁnite element models. In the 1920’s, the closed-form
solutions were only capable of dealing with a rigid pavement system of a single slab,
inﬁnite dimensions, a single layer, and a full contact interface. Recently developed ﬁnite
element models have the ability to cope with several issues related to a realistic model of
pavement conditions, including, but not limited to, multi-slab system, ﬁnite slab
dimensions, multi-layer system, load transfer devices, aggregate interlock, the existence
of a gap between layers, the application of thermal gradients, and the simulation of
complex load conﬁgurations. The improved analysis method has raised many more
issues; as a result, the ﬁnite element method of rigid pavements is continuously being
developed. Several issues relating the ﬁnite element models have been well emphasized;
however, the study on the effect of the interaction between the various factors on
mechanistic responses through the use of the developed ﬁnite element models still

remains relatively unestablished.

1.2 Problem identification

Fundamentally, it is crucial to establish a thorough understanding of the impact of the
interaction between structural, environmental, and loading factors on mechanistic
responses. However, studying mechanistic responses of rigid pavements under a limited
number of scenarios could yield valuable information only to a certain level as the
ﬁndings obtained from such a study only remain valid for the limited variable
combinations addressed in the study. On the other hand, to capture the interactive
responses between the variables, a parametric study must contain the entire possible

variable combinations, which requires an impractically massive experimental matrix and

is virtually unachievable under limitations of time and fund. Evidently, the success in
conducting such a parametric study can only be achieved through strategically reducing
the size of the experimental matrix without compromising the validity of the ﬁndings.
Also, for the analysis with the consideration of trafﬁc loading, issues related to
the mechanistic analysis of rigid pavements are further complicated by the variety in
characteristics of axle spacing lengths and axle weights throughout the design life. In
addition to the conﬁgurations of the trafﬁc loading, the locations of trafﬁc conﬁgurations
applied in the mechanistic analysis process are as important. With respect to the design of
rigid pavements, highway pavements generally have a maximum stress value
signiﬁcantly lower than the concrete modulus of rupture. Unlike the design of bridge
decks or airport pavements, where the design criteria is usually based on the worst case
loading scenario, the design of highway rigid pavements for cracking prevention is based
on load repetitions, which inﬂuence fatigue behavior. Already subjected to a massive
experimental matrix, a parametric study needs to extend the covered area to even more
combinations of load locations to effectively address the impact of loading factors on
mechanistic responses. Therefore, it is necessary to develop an analysis technique to
simultaneously account for the variations of load conﬁgurations, both axle spacing

lengths and axle weights, and load locations in a practical time frame.

1.3 Research objectives
This research investigated the impact of various parameters, and their interrelationship,
on mechanistic responses of rigid pavements obtained from ﬁnite element method. There

are four primary objectives achieved in this dissertation. The ﬁrst objective was to

perform a parametric study on current and anticipated rigid pavements, with
considerations of loading, climatic, material, subgrade support, and construction
parameters. It was also essential to establish a protocol for the development of a
comprehensive interpolation scheme that addressed the condition changes (e.g., traffic
spectra distribution, thermal gradient distribution, material property variation, etc.) in the
calculation of mechanistic responses. Another objective was to develop inﬂuence
surfaces for rigid pavements to address the impact of complex load conﬁgurations on
pavement responses and also the impact of lateral wander. Investigating the impact of the
interaction between structural, environmental, and loading factors on mechanistic

responses was the last objective.

1.4 Chapter overview

This dissertation - the study of the impact of the interaction between structural,
environmental, and loading factors on mechanistic responses of rigid pavements - is
outlined as follows:

Chapter 2 contains a synopsis of historic milestones of several topics involved
with this study. Numerous related published articles are categorized into three groups:
ﬁnite element analysis of rigid pavements, factors affecting mechanistic responses, and
application of inﬂuence surfaces.

Chapter 3 explores the sensitivity of mechanistic responses with the variation of
structural, loading and environmental variables through the ﬁnite element method. An
extensive parametric study was conducted on a complete factorial experimental matrix

that contains 43,092 combinations of concrete slab thickness, base/subbase thickness,

modulus of subgrade reaction, joint spacing, lateral support condition, axle
conﬁgurations, and truck conﬁgurations within the practical range for the state of
Michigan. Strategies used to reduce the size of the experimental matrix are discussed.
The analysis processes and assumptions in relation to the parametric study are also
brieﬂy discussed.

Based on the ﬁnite element results from Chapter 3, Chapter 4 presents the
potential use of an interpolation scheme as a means to predict stresses in rigid pavements.
Developed based on least-squares criteria, the interpolation scheme only requires a
limited number of ﬁnite element results as anchor results at nodal points to calculate a
least-squares coefﬁcient vector. An unlimited number of design scenarios, then, can be
analyzed with the use of the interpolation scheme in short time. This chapter also presents
the validation process of the results obtained from the interpolation scheme. This chapter
also brieﬂy discusses the potential implementation of the interpolation scheme for
developing a full catalog of mechanistic responses that may cover over a million possible
design scenarios in the case of the damage calculation process. Furthermore, an example
calculation using the interpolation scheme is presented in this chapter.

In Chapter 5, the development of inﬂuence surfaces for rigid pavements is
presented as a means to address the impact of complex loading conﬁgurations, load
locations, and structural features on mechanistic responses. An inﬂuence surface is
developed by analyzing for mechanistic responses at the midslab under the wheel path.
This midslab corresponds to a unit point load at various locations all over the pavement
surface. This chapter also elaborates the potential applications of the proposed technique.

The following four tasks in the ﬁeld of rigid pavement study were performed using the

inﬂuence surface technique: rapid pavement stress calculation, determination of critical
load location, pavement stress history, and investigation of the interaction between load
conﬁguration and structural feature.

Chapter 6 presents the conclusions of the research. This chapter also suggests

future subsequent research.

 

CHAPTER II

LITERATURE REVIEW

This chapter reviews numerous related literature to determine what previous research
have been conducted on mechanistic analysis of rigid pavements, especially with regard
to the impact of the interaction between structural, environmental, and loading factors on
mechanistic responses. Three main areas of the reviewed articles are: ﬁnite element
analysis of rigid pavements, factors affecting mechanistic responses, and application of

inﬂuence surfaces.

2.1 Finite element analysis of rigid pavements

Finite element method is basically a numerical approach to estimate the solutions to the
partial differential equations that govern the characteristics of rigid pavements. As matrix
manipulation and a series of numerical integrations are unavoidable parts of the method,
the ﬁnite element analysis of rigid pavements only became practical in the 1970’s after
the development of efﬁcient computation. The ﬁnite element application for rigid
pavements has been enhanced over the past decades to address numerous factors related
to mechanistic responses into the analysis (Huang and Wang, 1973; Tabatabaie and
Barenberg, 1978; Chou and Huang, 1981; Ozbeki et al., 1985; Davids and Mahoney,
1999). Three aspects of ﬁnite element analysis of rigid pavements are reviewed: plate

theory, pavement foundation, and the mathematical process of the ﬁnite element analysis.

 

Plate theory

In this study, a two-dimensional ﬁnite element model idealized using the Kirchhoff

theory, ISLABZOOO, was employed to address the impact of the interaction between the

multi-factors on mechanistic responses. The Kirchhoff theory is applicable to thin plates

with an assumption of no shear deformation, while thick plates with an inclusion of shear

deformation in the computation would require the Mindlin theory (Cook et al., 1989;

Reddy, 1993). Since rigid pavement thickness is signiﬁcantly less than the other two

dimensions, with the exception of airport pavement slabs, transverse shear deformation is

insigniﬁcant and can be neglected. With the consideration of the Kirchhoff theory,

therefore, all stress-strain relation terms that involved shear deformation vanish and the

reduced plane stress-strain relation matrices for an isotropic material can be shown

below.

ax E’ E’ 0 8x a-T

0'), = E' E’ O - 6y -— a-T (1)
rxy O O G 7,0, 0

where 0;; and 0' y = normal stresses in x and y directions
Ix), = shear stress
8 x and 6‘ y = normal strains in x and y directions
ny = shear strain

a = coefﬁcient of thermal expansion of concrete

,u = Poisson’s ratio of concrete

T = temperature differential between top and bottom of concrete

E'=£—=——— (2)
I” ]—'u

03—5—— (3)

2'0 +11)
Based on the stress-strain relation matrices, the stiffness matrix of concrete slab

[KP] may be derived using the following formula.
__ T .
[Kpl-jAtB] ~[Dkl-[BldA, (4) .

where

[B] = the strain-displacement matrix (will be discussed later).

 

A = area boundary of an element

F .-

D ,u-D 0

 

 

[Dkl= #0 D 0 (5)
0 0 ———(1"2‘)'D

D = ﬂexural rigidity

3
D : __E_£__2_ , (6)
12 - (1 - ﬂ )
where t represents the slab thickness.
Pavement foundation

Theoretically, in the case of slab-on-grade, rigid pavement can be approximately
considered as one elastic structure supported by a foundation model called the Winkler
foundation. There are a great many other foundation models available for rigid pavement

foundation idealization; however, the Winkler foundation is traditionally used and

considered as the most effective model. Details of characteristics, advantages, and
disadvantages of the Winkler foundation will not be discussed at this time. Another
name of the Winkler foundation is “Dense Liquid” foundation because this foundation
simulates the behavior of the subgrade or original soil under concrete slab by providing a
vertical resistant pressure equal to Bw, when w is vertical deﬂection and B is the Winkler
foundation modulus (modulus of subgrade reaction). The stiffness matrix of the

foundation is written below in matrix form.

[Kf1=jAﬁ-1N1T-[N1dA. (7)
where [N] = interpolation functions matrix (will be discussed later)
A = area boundary of an element.

Mathematical process of the finite element method

Since rigid pavement has rectangular geometry, it can be discretized using rectangular
linear FE with three degrees of freedom at each node: one vertical displacement, and two
horizontal rotations as shown in Figure 1. In other words, one FE contains twelve
degrees of freedom, meaning each element has a 12x12 stiffness matrix, a 12x1 force

vector, and a 12x1 displacement vector.

 

 

.4
{I
11’3/3 A
{x y
.a y

2

Z ----- b-/~--b ------ 3 w”‘3
W

X ’y3

 

 

Figure 1: Kirchhoff plate element with typical d.o.f. shown at node 3

10

n . $.-

 

Since Kirchhoff plate elements provide interelement continuity of vertical
displacements and rotations in the x- and y-directions, the elements can be considered Cl
elements; therefore, interpolation functions for C0 elements, like Lagrange’s interpolation
formula, may not be applied. The Herrnitian interpolation function, one of interpolation
functions for C 1 elements, can be used for this situation (thin plate elements). For an

element that has four nodes (1, 2, 3, and 4), Hermitian interpolation functions can be

- .‘np

derived using the following formulae:

 

w=N1-wl+Nx1-6x1+Ny1-0y1+N2-w2 +Nx2-6x2+Ny2-6y2

. (8)
+N3-W3+Nx3-6x3+Ny3-6y3+N4-W4+Nx4-t9x4+Ny4-0y4
where
[N1 le Ny1]=Xll——6Y— l[X—lYl X2Y2 +2X1Y2 +2Y1Y2 2bY1Y2 —2aX1X2], (9)
[N2 Nx2 Ny2]=X126Y1-[X2Y1- X1Y2 +2X1Y2 +2Y1Y2 2bY1Y2 20X1X2], (10)

[N3 Nx3Ny3]=-Xi— l6 2[X—2Y2 X1Y1+2X1Y2+2Y1Y2 -2bY1Y2 2aX1X2],(11)

Y
[N4 Nx4 Ny4]=%—2--[X1Yz—X2YI+ZX1Y2+2Y1Y2 -2bY1Y2 —2(1X1X2],(12)

X1 =1—i, (13)
(1

X2 =1+i, (14)
a
y

Y =1——, 15

ll

Y2 =1+%. (16)

Now the interpolation functions can be written in matrix form 1x12 as shown
below.
[N] = [N1 le Nyl N2 Nx2 Ny2 N3 Nx3 Ny3 N4 Nx4 Ny4] (17)

Strain-displacement matrix [B] can also be written in matrix form 3x12 as shown

‘ ’n-H

 

 

 

 

 

 

 

 

 

below.
3le 32le 32Ny1 32Ny4 :
3x2 3x2 3x2 8x2
33le dZN 1 azNyl a2Ny4
[B]=_ __ __x__ (13)
ayz ayz ayz ay2
2 2
.3le 2.221"_x1 2.3 ”1’1 29$
L dxdy dxdy dxdy dxdy

 

From the previous section, the stiffness matrix of each element [Kc] (12x12) can

be derived as follows:

[Kpl‘Il‘p}+IKf]'Il‘f}={re} , (19)
but {upl= iufl={ue}. (20>
[Ke]°{“e}={re}a (21)
[Kel=le1+1Kf1, (22)

where {up} = slab displacement vector,
{Uf} = foundation displacement vector,

{ uc} = element displacement vector (12x 1),

12

r

W1 l
0x1
0y1
"’2
0x2
65,2
“’3
9x3 .
6y3 [
W4 I;
6x4 ,
l0y4.

). (23)

{ue}=<

 

 

 

{re} = element force vector (12x1),

{re}: [AIBJT tum-mom. (24)

(25)

The global stiffness matrix and force matrix can be computed based on the
element stiffness matrix and element force matrix. The concept of generating the element
stiffness matrix and the element force vector into the global stiffness matrix and global
force vector is exactly the same as the concept of using the Boolean matrix that is
applicable for CO elements; however the method is slightly different. This is because
each node of a Kirchhoff element has 3 degrees of freedom. This means the element
stiffness matrix, which is actually 12x12, can be considered 4x4 and the element force
vector, which is actually 12x1, can be considered 4x1 in order to generate them into the

global system as shown below.

13

 

"l

 

K11(3x3) K12(3x3) K13(3x3) K14(3x3)
[K ]= K21(3x3) K22(3x3) K23(3x3) K24(3x3) (26)
e K31(3x3) K32(3x3) K33(3x3) K34(3x3)
_K41(3x3) K42(3x3) K43(3x3) K44(3x3)j
31(3x1)‘
’2(3xl)
=4 i 27
{re} ’3(3x1) ( )
f4(3xl)J

 

 

Once the global stiffness matrix and global force vector are derived, the

displacement vector of the global system can be computed.

{U}3Nx1 = [KGEIIVQ N -{F}3le (28)
where {U} = global displacement vector,
[KG] = global stiffness matrix,
{F} = global force vector,

N = number of nodes in global system

2.2 Factors affecting mechanistic responses

The interaction between structural, environmental, and loading factors affects
mechanistic responses of rigid pavements. The design of the rigid pavements mainly
governs the structural feature inputs to the ﬁnite element analysis, including, but not
limited to, layer thicknesses, slab dimensions, joint design, lateral support condition,
layer properties, and modulus of subgrade reaction. Therefore, the details of structural
inputs will not be further discussed. Without any controls over their conditions, the ﬁnite
element analysis inputs, addressing environmental and loading factors, require further

consideration.

l4

 

Environmental factors

It has been well known that environmental effects on rigid pavements can be accounted
for in term of temperature differential between the top and bottom layers of the slab.
Positive temperature gradient in the daytime (the top layer is warmer than the bottom
layer) contributes to downward curling. In the nighttime, the top layer of the slab is
cooling down, but the bottom layer still remains warm; rigid pavements will have a
negative temperature gradient (the bottom layer is warmer than the top layer),
contributing to upward curling.

Slab curling shape (concave or convex) is very important to the analysis and
design because it results in different types of stresses in pavements. For upward curling,
the top layer of the slab contracts, while the bottom layer expands with respect to the
neutral axis; however, the concrete slab weight will try to move the comers of the slab
down. Negative moment due to the slab weight will cause a tension at the top of the slab
layer and a compression at the bottom of the slab. In contrast to upward curling, the top
of the slab expands, while the bottom of the slab contracts with respect to the neutral axis
for downward curling; the comers of the slab will move down but the slab center will lift
up. Consequently, the slab weight will try to move its center down, causing tension at the
bottom and compression at the top of the slab. Figure 2 illustrates the concepts of curling
due to temperature differentials (Armaghani et al., 1987).

Note that thermal gradients across the concrete slab depth may not be linear in
reality. To effectively assess the impact of such a non-linear thermal gradient, the concept
of temperature—moment could be used to normalize the effect of the non-linear thermal

gradient to an equivalent linear thermal gradient (J anssen and Snyder, 2000).

15

.P"

 

Tt = Surface Temperature
Tb = Bottom Temperature

 

O
............................................
...........................................

 

 

 

 

 

coco... 00000....OOIOeo'leOOco'oe... . .90
.....o.09.0.0000...0.00.90.0900.0.0000".......
00009.00000009'.' °°°° "00'0’00000000 ......

Temperature Curling

Figure 2: Effects of temperature gradient on slab curling

However, several researchers (Armaghani, 1987; Byrum, 2000; Eisenmann and
Leykauf, 1990; Janssen, 1987; Poblete et al., 1988; Yu et al., 1998) observed that many
rigid pavements maintained an upward curl even when they were subjected to zero
temperature gradients (temperature at the top was the same as at the bottom of the slab).
Moreover, even in low positive temperature gradients, some of rigid pavements still

appeared to curl up as shown in Figure 3 (Arrnaghani et al., 1987).

16

WWI-”HI.

,_ T = Surface Temperature
600 AT = T, — Tb '

Tb = Bottom Temperature

400

§

200

 

 

Deﬂection (mm x 103)

§

  
    

 

 

A rox. +9°F
[/— PP _/
/

O.
// \\. — ___..—-
/ \vjrx’
( . , 35°F
100 \——6.5°F

l l 1 l l L 4 l

l L 1 l
O 30 60 90 120 150 180 210 240 270 300 330 360
Distance along undowelled joint in cm

 

Figure 3: Deﬂection profile along a joint due to change in the temperature

differential (Armaghani, 1987)

Permanent upward curl in rigid pavement can be explained by the interaction of

the following causes: temperature gradient locked into the slab during the setting period

17

(construction curl), drying shrinkage at the slab surface, and moisture expansion at the
slab bottom.

The basic concept of construction curl is that when the concrete material in rigid
pavements is setting and hardening, the temperature gradient at that time will be locked
into the slab without curling. This is because at an early age, concrete does not have a
sufﬁcient stiffness to curl its edges or comers. However, after hardening, the locked in
temperature gradient will have a very important effect on the behavior of the slab in that
it will become a temperature gradient of the opposite sign. For example, if the slab sets,
when exposed to a +15 °F temperature gradient, the effect of the curling at a zero
temperature differential after hardening will be the same as if it was exposed to a -15 °F
temperature differential, and the slab will be ﬂat again, when exposed to a +15 oF
temperature differential. Technically, the slab has a locked-in temperature differential of
-15 °F. Since the construction of pavements is usually done in the daytime, when the slab
is subjected to a positive temperature gradient, most of rigid pavements may have a
locked-in negative temperature gradient, and this can result in an upward curl of the slab,
even when exposed to a zero temperature gradient.

The opposite sign of temperature gradient at hardening time can be simply
quantiﬁed in order to consider the effect of construction curl and also can be rationally
superimposed to the temperature gradient. However, due to the seasonal and daily
variation in temperature gradient, it might not be easy to quantify locked in temperature
gradients. It would be reasonable to consider several values of locked in temperature

throughout a project as illustrated in the ﬁgure below (Yu et al., 2001).

18

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When exposed to ambient humidity (lower than 100 % relative humidity),
removal of water from concrete can cause strain associated with drying shrinkage. It was
found that the loss of moisture in concrete slab was generally concentrated within the 2
in. below the surface (Eisenmann and Leykauf, 1990; Janssen, 1994). Therefore, the
effect of drying shrinkage will be very high only for the top 2 in. of the slab. Contraction
of the pavement surface due to the high level of drying shrinkage at the top layer of the 5.
slab can cause upward curling. Figure 5 shows upward curling due to shrinkage

differential (Armaghani, 1987).

 

High Shrinkage

/——I.ow Shrinkage

OOOOIOOCOOCOOUO ......... .0...’.............

 

 

 

 

"90000'000 ooooooo 0..ooO'OOOoeoceoeeoo.a....
°‘ooOoOOocoocoOOO..."0¢OOOOOOO. ...... g...

Shrinkage Warping

Figure 5: Effect of shrinkage gradient on slab curling

With ﬁnite element application, effect of drying shrinkage can be accounted for in

terms of equivalent temperature gradient applying to only the top 2 inches of the slab as

shown in Figure 6 (Heath et al., 2001).

20

 

. it
No shrinkage-P”"""l Full shrinkage

Slab depth

b

Figure 6: Shrinkage modeled using equivalent temperature gradient

 

 

 

 

 

 

 

 

 

 

 

 

 

T I I I I I I I I I T
o _ / ﬂ
’8‘ 1 - / -
I _ —1
o 2
z
:3 3 ~— —
8 4 — STRESS DISTRIBUTION a
O' DUE TO
Z 5 - DRYING SHRINKAGE _
E
(L 6 P T
m
o 7 .. _,
8 _ a
J I I 1 1 L l l l l l
.200 (i 200 400 600 800 1000
COMPRESSION | TENSION

STRESS (PSI)

Figure 7: Stress distribution due to drying shrinkage (J anssen 1987)

21

'A—r

Shrinkage strain can be considered as thermal strain; with known material
pr0perty (coefﬁcient of thermal expansion), equivalent temperature differential at the slab
surface can be computed using the following formula.

83;, =Tsh -a (29)
Where Tsh = Equivalent temperature differential due to shrinkage

or = Coefﬁcient of thermal expansion of concrete
83h = Shrinkage strain

However, the effect of drying shrinkage on the slab curling estimated using the
above formula has to be adjusted with considerations of the shrinkage characteristic and
the effect of creep in relaxing shrinkage strain. With rising elastic modulus and
shrinkage strain, the upward deﬂection will be larger. This means the effect of drying
shrinkage on upward curling in a new pavement (low elastic modulus and low shrinkage
strain) is small and may be neglected (Eisenmann and Leykauf, 1990). For old
pavements, however, sustained stress from the slab weight due to upward curling can
result in creep relaxation that reduces stress due to shrinkage up to 50 % (Altoubat,
1999).

Capillary sorption can also cause expansion in concrete, when water is
sufﬁciently supplied to the concrete. Most of the time, pavements have a positive
moisture gradient (the top of the slab is drier than the bottom) and infrequently may have
a negative moisture gradient for a short period after a rainfall. Therefore, the bottom of
the slab will usually expand as compared to the neutral axis, and consequently this will
result in a warping (an upward curling due to a moisture gradient) in rigid pavements as

shown in Figure 8 (Armaghani, 1987).

22

W'Rmu—H

 

 

 

Figure 8: Effect of moisture gradient on slab warping

Basically, the existence of a moisture gradient causes an upward warping in rigid
pavements. To analyze the effect of a moisture gradient on warping and its interaction
with other causes using the ﬁnite element method, an equivalent negative temperature
gradient can be estimated. Fang (2001) recommended that a factor of 0.5 can be
multiplied to the daily peak positive temperature gradient in the summer and a factor of 2

for the winter to account for the effect of a moisture gradient.

Loading factors

The use of the ﬁnite element method allows the two aspects of loading factors,
conﬁguration and location, to be simultaneously addressed. Traditionally, rigid
pavements are analyzed for the edge loading condition, since it produces maximum stress
at the bottom of the slab. However, in the case of high locked-in negative temperature
gradient, caused by construction curl, drying shrinkage gradient, and moisture gradient,
the edge loading condition that had traditionally been considered as the critical loading
condition is not the most critical loading case anymore. Instead, the comer loading

condition is the critical loading condition because it magniﬁes the effect of the negative

23

 

moment over the upward curled pavement (Yu et al., 2001). Both the upward curling and
corner loading condition cause tension at the top of the slab, causing top-down cracking
in rigid pavements. As illustrated in Figure 10, with multi-axle trucks, top-down stress
situation can be magniﬁed more when axles are placed near transverse joints of a slab.
Figures 9 and 10 illustrate the various loading conditions for jointed plain concrete

pavements (Yu et al., 2001).

 

Interior loading
I

 

Comer
loading Edge adin g
E

 

 

 

Figure 9: Critical loading conditions (Y 11 et al., 2001).

 

Drive and steering
axles of a truck

 

 

 

Figure 10: Critical loading condition for top-down stresses (Y 11 et al., 2001).

24

Due to the complexity of multi-axle truck configurations, the critical loading
conditions can vary substantially, depending on axle spacing lengths, weights, and
structural parameters. While the direct application of the ﬁnite element analysis may
address the impact of loading factors in a time-consuming fashion, the use of inﬂuence

surface technique can tackle complex loading scenarios effectively and efﬁciently.

2.3 Application of inﬂuence surfaces

The technique of inﬂuence surfaces was ﬁrst introduced in the 1910’s. Although the term
inﬂuence surface was not directly mentioned in his work, Hencky may have been the ﬁrst
to develop inﬂuence surfaces for deﬂections of elastic plates by applying Maxwell’s
theorem of reciprocal deﬂections (Hencky, 1913). More than a decade later, a
comprehensive work on plate theory and the inﬂuence surfaces of plate deﬂections,
which became the basic theory of inﬂuence surfaces, was published (Nadai, 1925).
However, deﬂections of the structure are of lesser interest, from a design standpoint, as
compared to internal forces, such as the bending moment or shear force. Westergaard was
among the ﬁrst to develop inﬂuence surfaces for internal forces (Westergaard, 1930).
While Westergaard is known for being a pioneer in the mechanistic analysis of rigid
pavements, his work on inﬂuence surfaces only focused on the inﬂuence ﬁelds of bridge
decks. Since that time, many researchers have developed inﬂuence surfaces for different
internal forces, boundary conditions, and reference points; yet, application of inﬂuence
surfaces remained limited to bridge decks (Lansdown, 1966; Oran and Lin, 1973;
Dowling and Bawa, 1975; Kawama et al., 1980; Williams and du Preez, 1980; Irnbsen

and Schamber, 1982; Memari and West, 1991).

25

I‘-

 

For the ﬁrst time, in 1976, Nayak et a1. attempted to develop inﬂuence surfaces
for a plate resting on a dense liquid foundation, which is the general case for rigid
pavements, by applying a pinch load and differentiating the inﬂuence ﬁelds of deﬂections
using the ﬁnite element method (Nayak et al., 1976). However, the inﬂuence surfaces
developed by Nayak et al. are limited only to single slab-on-grade systems without
addressing the impact of boundary slabs or an elastic base layer.

Since the ﬁrst development of the inﬂuence surface technique in the early 1910’s,
inﬂuence surfaces have conventionally been obtained through an analytical approach, in
which the formulations of the surfaces are explicitly expressed. To begin this approach,
expression of the deﬂection contour of the structure with a unit load applied at the
reference point of the inﬂuence surface must be mathematically derived. Based on the
Maxwell-Betti-Reciprocal Theorem, the inﬂuence surface of deﬂections can then be
directly obtained from the deﬂection contour (Hencky, 1913). As illustrated in Figure 11,
if the unit load is applied at the coordinate A (u, v), and B (x, y) is a general point on the
plate, then the deﬂection contour can be symbolically expressed as Y (11, v, x, y).
Consequently, through the reciprocal system, the coordinate (u, v) denotes the coordinate
of the reference point of the inﬂuence surface of deﬂection, while the coordinate (x, y)
also denotes the point of the applied unit load. Subsequently, differentiating the inﬂuence
surface of deﬂections then provides the inﬂuence surfaces of internal forces, such as

shear or moment.

26

Amy

 

 

. 86:, y)

. A(u,V)

 

u,x

 

>

Figure 11: Symbolic expression of reference point and general point

27

CHAPTER III

PARAMETRIC STUDY

This chapter established the understanding of the impact of the interaction between such
factors on mechanistic responses, captured through an extensive parametric study. Data
collection procedure, development of the experimental matrix, and process involved with

the analysis are included in this chapter, as well as documentation of the results.

3.1 Data collection

The Michigan Department of Transportation (MDOT) Technology Advisory Group
(TAG) provided 14 “approved” designs for projects that were either recently constructed
or were programmed for construction in the near future. The designs provided the
structural parameters used for Michigan rigid pavements, e.g., cross-sections, pavement
features, material properties, etc. The ranges of inputs obtained from the MDOT designs
are summarized in Table 1. The details of the approved designs projects may be found in

the ﬁnal report to MDOT by Buch et al. (Buch et al., 2004).

In addition to these input parameters, the analytical model required the following
additional parameters (1) coefﬁcient of thermal expansion (CTE) of the concrete, (ii)
thermal gradients, (iii) axle and truck conﬁgurations, (iv) Poisson’s ratio and unit weight.
Based on the review of the literature (Klieger and Lamond, 1994), LTPP database, Truck
driver’s guidebook for Michigan (Michigan Center for Truck Safety, 2001), and
conversations with the TAG, ranges for these additional input parameters were

established and are summarized in Table 2.

28

Table 1: Summary of design parameters from the 14 MDOT designs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Inputs Minimum Maximum
Slab thickness 9.5 in. 12.0 in.
Base thickness 4.0 in. 16.0 in.
Subbase thickness No subbase 12.0 in.
Joint spacigg 177 in. 315 in.
Lane width 12 ft 14 ft
Lateral support condition Widened lane
, , Doweled 1.25 in. diameter at 12 in.
Jornt des1gn .
spacrng center to center
Concrete elastic modulus 4.2x106 psi
Modulus of subgrade reaction 90 psi/in. I 220 psi/in.

 

Table 2: Ranges of input parameters obtained from other sources

 

 

 

 

 

 

 

 

 

 

 

Input variables Ranges
Concrete unit weight 0.0087 rb/in.3
Concrete Poisson's ratio 0.15 - 0.20
Aggregate base unit weight 0.0061 lb/in.3
Aggregate base Poisson's ratio 0.35
Thermal gradient -4 - +4 °F/in.
Coefﬁcient of thermal expansion 3x10'6 - 9 x 1045 in./in./°F
Location of stress Top and bottom
. Sin 1e axle, tandem axle,. .. Multi-axle (8),
Load conﬁguration MI g1 MI—2,. .. MI-20

 

29

 

3.2 Experimental matrix

An experimental matrix was constructed based on the concept of complete factorial
(Fisher, 1960) for all combinations of design inputs reﬂecting MDOT practice, climatic
condition, and load conﬁgurations in Michigan. Several engineering principles and
common knowledge were applied to make the experimental matrix more concise, but still
provide the same level of information. An overview of the process is illustrated in Figure

12.

 

Data collection

|
I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Recent pavement designs LTPP database —> thermal Truck driver’s guidebook
from WOT —+ cross- gradients for Michigan and WIM data —> load
sections, pavement features, ’ conﬁgurations and axle
and material properties weights
TAG inputs Modiﬁcation of design inputs

 

 

 

 

 

 

I

Experimental matrix

 

 

 

 

Figure 12: An overview of the development of experimental matrix

30

 

An important first step in data analysis is to ensure that the project objectives can
be accomplished within the limitations of time and funds. If every combination of input
parameters is to be considered, the complete factorial experimental matrix would result in
millions of FE runs. To reduce the experimental matrix size, the preparation of the ﬁnal
matrix was achieved by carrying out the following strategies: combining variables,
considering only frequently seen load conﬁgurations, and adjusting increments for non-

discrete inputs.

Combining variables

Two variables are combined into one variable to reduce the number of input
combinations in the experimental matrix based on an assumption that the mechanistic
response computed either with one combined variable or two separate variables would be
the same or approximately the same. The variables to be combined are base thickness and

subbase thickness, which are combined into base/subbase thickness, and CTE (0t) and

thermal gradient (AT/D), which are combined into thermal strain gradient.

Figure 13 illustrates how base thickness and subbase thickness can be combined.
It is assumed that the two layers have an unbonded interface, one elastic modulus
represents the combined layer, and the Poisson’s ratios of the two layers are
approximately the same (Khazanovich and Yu, 2001). This sensitivity study of the
accuracy of the combined base/subbase thickness was conducted for the 14 MDOT
designs by comparing the mechanistic responses computed based on the two-layer system
(concrete and combined base/subbase layers on the top of the subgrade) and that based on

the three-layer system (concrete, base and subbase layers on the top of the subgrade). In

31

this sensitivity study, for the three-layer system approach, an unbonded interface
condition and Totski interface model (ERES Consultants, 1999) were considered between
base and subbase layers and between concrete and base layers, respectively. An
unbonded interface condition was considered for the two-layer system approach. It was
found that the difference in the magnitudes of stresses between the two approaches is less
than 4%. The results from the sensitivity study are illustrated in Figure 14 as compared
with the results based on no subbase for the 14 MDOT designs.

The CTE and thermal gradient are simultaneously accounted for in terms of the
product of the two variables, or(AT/D) or thermal strain gradient. Figure 15 illustrates the
sensitivity plots to validate this assumption. The sensitivity study was conducted for nine
cases by comparing the mechanistic responses computed based on two analysis
approaches. Analysis approach 1 consists of varying CTE values, while keeping a
thermal gradient constant. Analysis approach 2 consists of keeping a CTE value constant,

while varying thermal gradients.

32

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It was found that the mechanistic responses computed based on the two
approaches are identical. A statistical experiment to illustrate the validity of combining
CTE and thermal gradient was conducted by repeating this process for eight more
combinations of pavement parameters selected based on a fractional factorial (Buch et al.,
2004). It should be noted that comparison between pavements with different slab
thickness even with the same thermal strain gradient is not valid, since the pavements are
subjected to different temperature differentials. Comparison of pavement responses under

a curled slab condition, therefore, should only be made within the same slab thickness.

Considering only frequently seen load conﬁgurations

Several axle and truck conﬁgurations are contained in the Truck driver’s guidebook for
Michigan (Michigan Center for Truck Safety, 2001). Based on the TAG’s
recommendations, certain axle and truck conﬁgurations, not existent or not frequently
seen, could presumably be omitted. Only 8 axle conﬁgurations and 11 truck
conﬁgurations are selected for the experimental matrix. Figures 16 (a) and (b) illustrate

the axle and truck conﬁgurations included in the parametric study.

36

 

 

Axle Type Axle Conﬁguration

 

Single

 

! 18 kips

 

l6 kips each at
Tandem 3’6” spacing

 

 

13 kips each at

Tridem 3’6” spacing

 

 

 

l3 kips each at

uad
- Q 3'6” spacing

 

 

1‘ t i '1‘
13 kips each at
GrouP 0f 5 m 3’6” spacing

 

13 kips each at
Group of6_ , 9 , 9 9 9 3’6" spacing

 

 

l3 kips each at
Group Of 7 3’6” spacing

 

 

 

13 kips each at
Group Of 8 m 3’6” spacing

 

 

 

 

(a) Axle conﬁgurations

Figure 16: Load configurations considered in the study

 

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Adjusting increments for non-discrete inputs

Input increments need to be carefully considered for non-discrete variables; in this case,
these included base/subbase thickness, modulus of subgrade reaction (k-value), and
thermal strain gradient. The ﬁner increments can better capture trends of the mechanistic
responses, but will also result in increasing the required FE runs. Therefore, it is crucial
to capture trends of the mechanistic responses with as large increments of input
parameters as possible. Five values of each non-discrete variable were used in the
sensitivity study of input increments. Based on this “mini-analysis”, it was determined
that response trends could be adequately captured by using three values for each non-
discrete variable. These values for the base/subbase thickness, k-value, and thermal strain
gradient are 4, 16, 26 in., 30, 100, 200 psi/in, and o, :10, :20x106 in", respectively.
Positive thermal gradients are considered for analysis of stresses at the bottom of the
concrete slab, while negative thermal gradients are considered for analysis of stresses at
the top of the concrete slab, since the critical stress locations correspond with the types of
thermal gradient. Figures 17 through 19 illustrate the trends of stresses with variations of
base/subbase thickness, modulus of subgrade reaction, and thermal strain gradient,
respectively. Note that if not speciﬁed, the parameters for these sensitivity plots are lO-in.
concrete slab, l6-in. base/subbase, lOO-psi/in. modulus of subgrade reaction, concrete

shoulder, 177-in. joint spacing, 18-kips single axle, and thermal strain gradient of zero.

 

 

 

 

 

 

 

150
I
E W
g 100* —— —7 — —- —~ ”_-_. __ i. 7- - _ i a _
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i 50» A :2_— f - _ _.___._ _. _. _ ___ r
2
O T T T l I
0 5 10 15 20 25 30

Base thickness, in.

+ Transverse stress at bottom of FCC —I— Longitudinal stress at bottom of FCC

Figure 17: Sensitivity trend due to the variation in base/subbase thickness

 

 

 

 

 

 

 

 

0 I T r I
0 50 100 150 200 250

Modulus of subgrade reaction, [Bi/in.

+ Transverse stress at bottom of FCC —I— Longitudinal stress at bottom of FCC

Figure 18: Sensitivity trend due to the variation in modulus of subgrade reaction

45

250

 

 

 

 

 

 

 

 

 

 

 

zoo ._
'5
Q-
g 150 -
{I}
E
g 100 —— ~—
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so —» -—~~ —~ - - ~e~w~ ___,
o . . .
o 5 10 15 20

MAT/D), 10" in“

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Figure 19: Sensitivity trend due to the variation in thermal strain gradient

In addition to the above mentioned strategies, locations of stresses (at the bottom
and the top of the concrete slab) are also effectively selected to reduce the number of
runs. For positive thermal gradients, only stresses at the bottom of the concrete slab are
considered, while stresses at the t0p of the concrete slab are considered for negative
thermal gradients. The experimental matrix size has been reduced to 43,092 FE runs as
illustrated in Table 3. It should be noted that this ﬁnal experimental matrix addresses all
possible input parameters for all discrete variables and three levels of each non-discrete
variable. However, the combinations of non-discrete variables that are not addressed in
this ﬁnal experimental matrix are still of interest and will be obtained through the

interpolation scheme, which is to be discussed later.

46

Table 3: Final experimental matrix

 

Input variables

Number of cases

 

PCC slab thickness

7 (6, 7,... 12 in.)

 

Base/subbase thickness

3 (4, 16, 26 in.)

 

Modulus of subgrade reaction

3 (30, .100, 200 psi/in.)

 

Slab length (joint spam)

2 (177 in. and 315 in.)

 

 

 

 

 

 

 

 

 

Joint design 1
Shoulder type 3
0t.AT/D 3 (o, :10, 120,110" in")
Location of stress 2
Load conﬁguration 19
Total combinations 43,092

 

3.3 Analysis process

Based on a complete factorial of 43,092 combinations of parameters identified
previously, a preliminary parametric study is conducted by performing a series of FE
analyses using the ISLABZOOO program. The results obtained from this parametric study
are included in this section. The parametric study will be presented in four parts:
structural model, analysis process, documentation of analysis results and interpretation of
analysis results.

The pavement system for this analysis typically is comprised of three to. six
concrete slabs, depending on the length of the load conﬁguration. This is to ensure that
the ﬁrst and last concrete slabs are unloaded as recommended in Report 1-26 (NCHRP,
1990) to analyze the pavement system with extended slabs in order to reﬂect realistic
boundary conditions that all the slabs are bounded by two slabs on both directions. Two
lane widths (12 and 14 ft) and two shoulder types (untied AC and tied concrete) are
considered. The study focuses on the analysis of the mechanistic responses in the outer

lane (the truck lane), which is traditionally the design lane. Two joint spacing lengths

47

(177 and 315 in.) are considered. The structural model with two traffic lanes was not
found to result in different pavement response in the outer wheel path as compared to the
results obtained from the structural model with one traffic lane. Therefore, the second
trafﬁc lane is not included in the structural model to reduce the structure size and,
consequently, analysis time. The wheel path considered in this study is 20 in. from the
center of the outer wheel to the traffic stripe, similar to the pavement model used by
Darter et al, 1994. Mesh size of 12x12 in. is used as a standard mesh size. This mesh size
was found to achieve both satisfactory convergence and reasonable runtime. Figure 20

illustrates the typical slab structure layout as modeled using ISLABZOOO.

48

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49

The ﬂow chart in Figure 21 illustrates the required components for the FE
analysis. It can be seen that all structural and environmental factors have been addressed
in the ﬁnal experimental matrix. However, the critical load location needs to be derived
ﬁrst before the creation of the stress catalog. The critical load location is deﬁned by the

load location along the wheel path that results in the most critical mechanistic response,

the highest value of the maximum responses for each load location.

 

 

Mechanistic Responses

 

 

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l I L
Structural Environmental Loading Factors
Factors Factors
_._—.1 I
-— PCC Thickness L Thermal Gradients Load
__‘J Conﬁgurations
Basel subbase Positive Thermal
Thickness Gradients Axles
__ MOdUIUS 0f Ne ative Thermal
Subgrade Reaction gGradients Trucks
—- Joint Spacing
A Material Properties
Lateral Support
Condition
Joint Design

 

 

 

Figure 21: Required components for the analytical tool

50

Procedure of determining critical load location

The procedure for determining critical load location is illustrated in Figure 22. The
procedure involves the computation of stresses at every load location along the wheel
path for a given set of conditions. The load location that results in the most critical

(maximum) stress will be considered as the critical load location.

 

 

 

 

 

 

 

 

 

Specify load
configuration
Position the load
Specify pavement
feature and temperature
gradient \L Move load at 12”

 

 

 

 

 

 

ISLABZOOO analysis @— Increment in the
direction of traffic

 

 

 

 

Repeat this loop
over the slab length

Al Pavement responses

Vlr

Critical load location = load location with the maximum pavement response

 

 

 

 

 

 

 

Figure 22: Procedure of determining critical load location

Assumptions and validation process

The procedure for determining critical load location is a time consuming process; it is
impractical to perform the procedure for all possible combinations of input parameters in
the ﬁnal experimental matrix. It was assumed that variations in the following variables do

not affect critical load locations:

51

0 Slab thickness,

0 Base/subbase thickness,

0 k-value,

0 Lateral support condition and

0 Thermal strain gradient.

Validation of these assumptions was conducted to show that the critical load
location is constant with the variation of the ﬁve non-influential variables. The fractional
factorial design of —l3— - 35 = 9 is the method used to study the impact of variables within
a practical size of vilidation matrix. The validation matrix used for all trucks and axles is
summarized in Table 4. Fundamentally, fractional factorial design is a statistical method
that allows for fractionation of a complete experimental factorial, while still balancing the
fraction.

This process needs to be repeated for every axle and truck conﬁguration, joint
spacing, and stress location (top and bottom of the concrete slab) as these factors are
considered inﬂuential in affecting critical load locations. Critical load locations for all
eight axle conﬁgurations and 11 truck conﬁgurations are summarized in Table 5. Critical
load locations for axle conﬁgurations were found to be in the vicinity of the middle of the
slab and the transverse joint for stresses at the bottom and top of the concrete slab,
respectively. However, no typical location was found for critical load locations for truck

conﬁgurations due to the complex combinations of the axles and axle spacing lengths

within truck conﬁgurations.

52

 

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54

This example shows the determination process of the critical load location for
bottom-up stresses for MI-l6 on l77-in. joint spacing pavements. Stresses were
computed for load locations along the wheel path for the nine validation cases as
identiﬁed in Table 4. The non-inﬂuential variables were found to impact the stress
magnitude; however, the non-inﬂuential variables did not signiﬁcantly impact critical
load location. For this example, the critical load location was approximately 84 in. for all
the nine cases irrespective of the variation of the non-inﬂuential factors. Figure 23 (a)
illustrates the physical meaning of the computed critical location. An example stress
proﬁle for validation case 1 and the corresponding critical stress location are illustrated in

Figure 23 (b). More example illustrations can be seen in Figures 24 and 25.

55

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57

3.4 Documentation of analysis results

The parametric study results provide information used in investigating the impact of
structural, environmental, and loading factors, and the interaction between them on three
types of mechanistic responses: longitudinal stress at the bottom of the concrete slab,
transverse stress at the bottom of the concrete slab, and longitudinal stress at the top of

the concrete slab.

Impact of structural factors

Figures 26 (a) through (f) are example illustrations of the impact of structural features on
the longitudinal stress at the bottom of the concrete slab under various conditions as
stated in the ﬁgures. Note that these ﬁgures represent MI-l6 loading (see Figure 26 (c)
for conﬁguration), l6-in. base/subbase, lOO-psi/in. k-value, concrete shoulder, and 177-
in. joint spacing unless identiﬁed otherwise. All the ﬁgures show that slab thickness has a
signiﬁcant impact in reducing stresses. In addition, the ﬁgures show that the changes in
stresses due to changes in base/subbase thickness, k-value, and lateral support condition
appear to be less relevant as the concrete slab becomes thicker. Also, joint spacing does
not appear to have signiﬁcant impact on edge stresses. Impact of lateral support condition
will be discussed in detail later. Figures 26 (d) and (f) show an interaction of k-value and
joint spacing with thermal gradients, which is to be discussed later. Although the
magnitude of longitudinal stress at the bottom of the concrete slab was found to vary with
combinations of input parameters, similar trends were observed in sensitivity plots over
the entire experimental matrix. Similar trends were observed for the transverse stress at

the bottom of the concrete slab with the exception of the impact of joint spacing, which

58

was found to have no signiﬁcant impact on the transverse stresses, even under the
inﬂuence of a thermal gradient. An example critical location of stress is illustrated in
Figure 26 (g).

The impact of structural features on longitudinal stress at the top of the concrete
slab is illustrated in Figures 27 (a) through (f). Note that these ﬁgures represent MI-16
loading (see Figure 16 (b) for conﬁguration), 16-in. base/subbase, lOO-psi/in. k-value,
concrete shoulder, and l77-in. joint spacing, the same conditions as previous parts unless
identiﬁed otherwise. It can be seen in these ﬁgures that the magnitudes of longitudinal
stresses at the top of the concrete slab are lower than the longitudinal stresses at the
bottom of the concrete slab illustrated in the previous part. However, the trends observed
for these stresses are similar. It should be noted that negative thermal gradients are
considered in Figures 27 (d) and (f), since the critical location of stresses is at the top of
the concrete slab in these ﬁgures. An example critical location of stress is illustrated in

Figure 27 (g).

Impact of loading factors

Figures 26 (h) and 27 (h) are example illustrations of the impact of the load
conﬁgurations (axles and trucks) on the magnitude and normalized magnitude (by total
weight of the conﬁguration) of longitudinal stresses at the bottom and top of the concrete
slab, respectively. In order to compare the contribution of each axle type (carrying
different weight) on loading stress, it is necessary to express the stress as psi/kip. It can
be seen that the normalized stress magnitudes are lower as the axle conﬁgurations have

more load carrying wheels, implying that at the same stress level, a multi-axle can carry

59

heavier loads than a single or tandem axle. However, the impact of truck conﬁgurations is
not shown in these figures because each truck conﬁguration makes various numbers of
passes at the point of interest on the pavement slab. For example, the truck type MI-l6
(see Figure 16 (b)) will result in four peaks of stresses corresponding to one single axle
(driving axle), one quad axle, and two tandem axles. Hence, normalization based on total
weight is not valid. The normalization should be based on the number of passes made by
each axle group.

Impact of load lateral placement on mechanistic responses presented in Figure 28
(a), in which stresses are shown for several load locations across the lane width, suggests
that the concrete shoulder resulted in the lowest stresses among the three lateral support
conditions considered in the study for the load located along the wheel path. It was found
that the magnitudes of longitudinal stresses for AC shoulder (12-ft lane with AC
shoulder) were higher than that for widened lane (also AC shoulder but with l4-ft lane).
This could be attributed to the fact that a widened lane (14 ft.) creates a pseudo-interior
loading condition (the wheel path shifted 2 ft towards the centerline, resulting in the
reduction of stresses from edge loading). An example sensitivity plot of temperature-
induced stresses in Figure 28 (b) illustrates that lateral support condition does not have a

signiﬁcant impact on temperature-induced stress in longitudinal direction.

60

 

300

250 -
'3 200 - Nbess 6"

 

 

 

 

 

a 150 - =— -r\_ 9..
2% 100 - : : _.A 12"
50 —
o I T
4 16 26

Base/subbase thickness, in.

(a) Impact of base/subbase thickness, thermal strain gradient of 0x10'6 in.-1

 

 

 

PCC shoulder Widened lane AC shoulder
I 6" slab 9" slab D 12" slab

(b) Impact of lateral support condition, thermal strain gradient of 0x10.6 in.-1

Figure 26: Example sensitivity plots of bottom-up stresses

61

 

 

 

 

 

 

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300 ~
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m 100 - ' + ﬂ 12..

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300 _ ‘/'/‘ 12"
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100 -

O I n

30 100 200
Modulus of subgrade reaction, psi/in

Stress, psi

 

 

 

(d) Impact of modulus of subgrade reaction, thermal strain gradient of 20x10.6 in.-1

Figure 26: Example sensitivity plots of bottom-up stresses (continued)

62

 

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66

 

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67

 

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Impact of environmental factors

Environmental factors in this study are addressed in terms of thermal strain gradient (the
product of CTE with positive or negative thermal gradients). As illustrated in Figures 29
(a) and (b), a positive gradient causes a downward curling of the slab, while a negative
gradient causes an upward curling of the slab. The increase in magnitude of thermal
gradient results in the increase in the magnitude of stresses, when positive and negative
thermal gradients are considered in computation of stresses at the bottom and top of the

concrete slab, respectively.

 

(a) Downward curling (b) Upward curling

Figure 29: Slab curling due to different types of thermal gradients (Y u et al, 2004)

As observed in the previous section (Figures 26 (d) and 27 (d)), the magnitude of
the longitudinal stress appears to be impacted by the interaction between the thermal
strain gradients, k-value and pavement thickness. This interactive trend is supported by
the curling stress equations by Bradbury (Huang, 1993), where thermal curling stress is a
function of the ﬁnite slab correction factor. This factor generally increases with the
increase in the ratio of joint spacing (for longitudinal stresses) to radius of relative

stiffness. This phenomenon could also be explained by

73

Significance of parametric study results

Based on the parametric study results, the insight into the impact of the interaction
between various parameters on pavement stresses was established. Understanding the
mechanistic behavior of the pavement provides an indirect connection to a better
comprehension of pavement performance. From the mechanistic standpoint, the results
suggest various types of interactions in that several parameters were found to affect
pavement stresses. These effects include i) the interactive effect between slab thickness
and base/subbase thickness on loading stress, ii) direct effect of lateral support condition
on loading stress and combined loading and thermal stress, iii) the interactive effect
between slab thickness and modulus of subgrade reaction on loading stress, iv) the
interactive effect between thermal strain gradient and modulus of subgrade reaction on
combined loading and thermal stress, v) the interactive effect between thermal strain
gradient and joint spacing on combined loading and thermal stress.

Figures 26 (a) and 27 (a) illustrate the interactive effect between slab thickness
and base/subbase thickness on loading stress at the bottom and top of the slab,
respectively. It can be seen that the increase in base/subbase thickness results in a
reduction in stress magnitude with diminishing effect as the slab thickness increases. As
the base/subbase layer provides uniformity of support to the slab, an increase in
base/subbase thickness reduces the magnitude of loading stress. The results suggest that
from the loading stress standpoint the base/subbase thickness should also have a
substantial effect on slab cracking for a pavement system with a thin slab, especially
thinner than 10 in. However, the results should not suggest that the base/subbase

thickness has a less signiﬁcant impact for a pavement system with a thicker slab, since

74

the base/subbase layer could also affect the drainage characteristic of the pavement
system.

Figures 26 (b) and 27 (b) suggest that lateral support condition has only direct
effect on loading stress and combined loading and thermal stress. For various
surrounding conditions, it was found that AC shoulder results in the highest stress
magnitude as compared to PCC shoulder and widened lane. This could imply that from
the standpoint of load-related distress a pavement system with PCC shoulder or widened
lane should have a better performance than a pavement system with AC shoulder.

The impact of lateral support condition along with lateral wander of trafﬁc
loading was further illustrated in Figure 28 (a), suggesting that the lateral load location
directly affects the magnitude of pavement stress for all types of lateral support condition.
The results also imply the signiﬁcance of the location of trafﬁc paint stripe as it would
dictate the location of wheel path.

It should be noted that the parametric study results are based on wheel path as the
lateral load locations, which are 20 in. for PCC and AC shoulders and 44 in. for widened
lane. The impact of lateral support condition and various lateral load locations are
illustrated in Figure 28 (b). While lateral support condition has a signiﬁcant effect on
loading stress magnitude, its effect on thermal stress magnitude appears to be
insigniﬁcant as shown in the ﬁgure. This implies that the variation in lateral support
condition should not affect temperature-related performance of the pavement.

Figures 26 (c) and 27 (c) illustrate the interactive effect between slab thickness
and modulus of subgrade reaction on loading stress at the bottom and top of the slab,

respectively. It can be seen that the increase in modulus of subgrade reaction results in a

75

reduction in stress magnitude with diminishing effect as the slab thickness increases. The
results suggest that from the loading stress standpoint the modulus of subgrade reaction
should also have a substantial effect on slab cracking for a pavement system with a thin
slab, especially thinner than 10 in.

However, Figures 26 (d) and 27 ((1) illustrate an interactive effect between
thermal strain gradient and modulus of subgrade reaction on combined loading and
thermal stress. It could be suggested that an increase in the magnitude of modulus of
subgrade reaction results in an increase in the magnitude of thermal stress as the
combined stress magnitudes are compared to the loading stress magnitudes in Figures 26
(c) and 27 (c). From the mechanistic standpoint, this could imply that a roadbed with
higher modulus of subgrade reaction should result in a better load-related performance
but not for a temperature-related performance. However, the mechanistic behavior alone
may not sufﬁciently provide such a conclusion to the actual performance of the
pavement, since a roadbed with higher modulus of subgrade reaction usually also has a
better erodibility resistance and also a better drainage characteristic.

Figures 26 (e) and 27 (e) suggest that from the loading stress standpoint joint
spacing should not have a signiﬁcant effect on load-related performance of the pavement.
As stated by the Portland Cement Association’s thickness-design procedure, the presence
of joints has no effect on the pavement stress magnitude, since the load is placed adjacent
to the midslab away from the joints. However, it should be noted that the results did not
account for the interaction between axle spacing and joint spacing, which will be further

discussed in Chapter 5.

76

When combined with thermal stress, Figure 26 (f) and 27 (f) illustrate an
interactive effect between thermal strain gradient and joint spacing on combined loading
and thermal stress. This implies that an increase in joint spacing should result in a higher
level of temperature-related distress.

Figures 26 (h) and 27 (h) illustrate the impact of load conﬁguration on pavement
stress magnitude. The results imply that a more complex axle group should result in a
lower pavement stress magnitude. However, the results did not account for the interaction
between axle spacing and joint spacing. The impact of load conﬁguration and its
interaction with joint spacing will be further discussed in Chapter 5 through the use of

inﬂuence surface technique.

Justiﬁcation of the selection of AGG factor

Boundary support condition along the longitudinal joints of the slabs is characterized
through AGG factor in ISLABZOOO program. It is crucial that an appropriate value of
AGG factor is selected to represent the load transfer mechanism. The AGG factor can be
empirically estimated as follows (Crovetti, 1994):

l

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0.012
Where AGG = AGG factor
LTE = Load transfer efﬁciency, percent
6 = Radius of relative stiffness, in
k = Modulus of subgrade reaction

77

The radius of relative stiffness is defined as follows:

 

 

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12(1 — #2) - k
Where E = Radius of relative stiffness, in
E = Elastic modulus of layer 1
h = Thickness of layer 1
u = Poisson's ratio for layer 1
k = Modulus of subgrade reaction

In general, the typical values of LTE for tied concrete shoulder and untied AC

shoulder vary from 25-90% and O-40%, respectively. Based on equation 1, the ranges of

AGG/kl were calculated as 0-0.77 and 034-165 for tied concrete shoulder and untied

AC shoulder, respectively. Based on the inputs in the parametric study, the range of kt’

varies from 1188 to 8286 psi. A sensitivity study of the effect of AGG factor on
magnitude of edge stresses is conducted for ranges of AGG factor from 5 to 7,000 psi
(AC shoulder and widened lane) and from 300 to 2,500,000 psi (concrete shoulder).
Based on these results, the AGG factors of 1,000,000 psi and 1,000 psi are selected for
tied concrete shoulder and untied AC shoulder for the parametric study, respectively.
Note that this sensitivity study is conducted for 177-in. joint spacing and lS-kips single
axle at flat slab condition. Several sensitivity plots are generated as illustrated in Figures
30 (a) through (c). It can be seen that the stress magnitude is not signiﬁcantly sensitive to

AGG factor for concrete shoulder and widened lane, while for AC shoulder the variation

78

in stress magnitude could be up to 10% from the stress magnitude computed based on the
selected AGG factor (1,000 psi). Table 6 summarizes the details of the documented

interaction.

79

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87

CHAPTER IV

INTERPOLATION SCHEME

Interpolation is a statistical procedure used to approximate unknown values (non-nodal
points) in the vicinity of known values (nodal points). An interpolation scheme in this
study is used because it is required to obtain mechanistic responses for all the
combinations of the non-discrete inputs, not addressed in the ﬁnal experimental matrix.
The experimental matrix includes all possibilities of all the discrete design inputs: slab
thickness, joint spacing, lateral support condition, and load conﬁguration. However, only
three values were speciﬁed for each of these non-discrete inputs in the ﬁnal experimental

matrix:

0 Modulus of subgrade reaction (30, 100, 200 psi/in.),

0 Base/subbase thickness (4, 16, 26 in.),

. Thermal strain gradients (0, :10x10'6, :20x10’6inf1).

The interpolation process in this study is used to approximate the results that are
not directly analyzed by the ﬁnite element model across ranges of the three non-discrete
input parameters: modulus of subgrade reaction, base/subbase thickness, thermal strain

gradient, this interpolation scheme is a three-dimensional process.

4.1 Least-squares criteria
The statistical method of least-squares approximation, proposed by Carl Friedrich Gauss

in 1795 (Rassias, 1991), is applied to develop and evaluate interpolation schemes in this

88

study. The method is unbiased and algebraically provides an approximation to a
dependent variable Y, with minimal variance. With a linear model, coefﬁcients Bj for the
least-squares solution satisfy the normal equations:
n
8[ Z eiz ]
i=1 =o,j=0,l,2,...,m (32)

3,6}-

where $7,: is the value of variable i7 at point i,

37,- = ,60 +,B1 Gil-1 + ,82 361-2 + ,63 ~36? + ...+ﬂm 56;" is the predicted value at point i,

~ ~ ~m
1x2,xi3,...xi

i , z are independent

e, = )7: — 3",. is the predicted value’s error, and 35

(predictor) variables evaluated at point i, i = 1, 2, ..., n.

The matrix formulation of the solution in the nonsingular case is:

 

 

y,-=ﬂ0+ﬂ1~2i1+ﬂ2~§i2+ﬂ3-3c'i3+...+ﬂmoiim=33? (33)
lﬂo‘
ﬂl
__ —1 .
ﬂ=,ﬂzl=[xT.x] .xT.y (35)
.ﬂm.
,_, r ~1 ~2 ~m‘
x1 1 x1 1 ' "1
X 132 1 36% 3% 3'5" (36)
=<x3>= ~l ~ ~m
: ll 1c.3 x3 x3
~"J (1 35,; 7:3 33"}

 

 

 

 

89

(37)

~<>

II
551

II

N
w

 

 

4.2 Development of interpolation scheme

First, a sensitivity study was conducted to investigate the impact of the three non-discrete
input parameters. The impact of modulus of subgrade reaction and base/subbase
thickness on the magnitude of stresses were found to be highly non-linear as the change
in the slope of the relationship was observed. On the other hand, initial trials showed the
impact of thermal strain gradient to have little curvature. Therefore, the interpolation
process is divided into two steps: (i) two-dimensional interpolation based on known
anchor results obtained from the ﬁnite element model across the ranges of the
base/subbase thickness and the modulus of subgrade reaction at each level of the thermal
strain gradient, and (ii) one-dimensional interpolation based on the interpolated results
from step 1 across the range of the thermal strain gradient. The overview of the
interpolation process is illustrated in Figure 31. Using the least-squares criteria, several
interpolation schemes were developed and compared as discussed later. The prototype of

the interpolation scheme is explained below in matrix form.

a(H*,k*,a,-)= Y *-B (38)

9O

Where 0'(H*,k*,ai ) is mechanistic response for the target combination of

base/subbase thickness and modulus of subgrade reaction at level a, of
thermal strain gradient

)7 * is the vector of predictor variables

 

_ ... *2
X* = {I H * H *2 ln(k*) H *-ln(k*) H *2 -ln(k*) -1— -H— H } (39)
k * k * k *
Where H* is target base/subbase thickness

k* is target modulus of subgrade reaction

a, is anchor value 0 in.”1 of thermal strain gradient
a2 is anchor value :10x10’6 in." of thermal strain gradient

a3 is anchor value :20x10'6 in.’1 of thermal strain gradient

,6 is least—squares coefficient vector

 

$333

ﬁ)
1!

[34L = [XT-X]—l-XT-o“’ (40)
ﬂs
,56

 

 

.ﬂsl

91

 

 

 

 

2 2 1 H1 ”1
1HH 1k lkHlkH ———
11“(1)11(1)1n(1)1k1k1kl
2 2 1 H1 ”12
1H H 1 k 1 k H 1 k H _—
1 111(2)n(2)1n(2)1k2 k2 k2
2
._ . 2 2 L in 5’;
{{(HL/q) 1H1 Hl ln(k3) ln(k3)H1 ln(k3)Hl k3 k3 k3
X(Hl’k2) 2 2 1 H H
Y(H1,k3) 1 H2 H2 ln(k1) ln(k1)H2 ln(k1)H2 ? -—2- —2
— 1 k1 k1
{(HZJCI) H H2
X=<.{(H2,k2)i= 1 H2 H5- ln(k2) ln(k2)H2 ln(k2)H22 1:1— 7;; -k—7-— (41)
{(11213) 2 2 H3
{(H3’kl) 1 H2 H22 ln(k3) ln(k3)H2 ln(k3)H§ i 512— i
{(H3,k2) k3 k3 k3
LX(H3J<3)1 2 2 1 H3 ”3
1H H 1k lkH lkH -———
3 3 n(1)n(1) 3 n(1) 3 k1k1k1
2
2 2 1 H3 H3
1HH 1k lkHlkH __—
3 3 n(2) n(2) 3 n(2) 3 k2 k2 Q
2 2 1 H3 ”3
1H H lk lkH lkH ———
_ 3 3 n(3)n(3) 3 n(3) 3 k3 k3 k3
Where H1, H2 and H3 are base/subbase thicknesses 4, l6 and 26 in.
k1 , kg and k3 are k-values 30, 100 and 200 psi/in.
.011.
012
013
0‘21
0"=<0'22i (42)
023
031
032
1033.

 

 

Where 0,7 is known anchor value stress from finite element analysis at H, and kj

92

(43)
Where 0'(H*,k*,a*)is mechanistic response for the target combination of
base/subbase thickness, modulus of subgrade reaction, and product of

oc(AT/D)

£7 * is the vector of predictor variables based on 01(AT/D)

 

5*:ﬁ a’l‘ a*2} (44)
-—1
Fl a1 a2
yo 1 a(H*,k*,a1)
?= 71 =1 a2 0% - 0(H*,k*,6¥2) (45)
L 3-

 

   
 

Final interpolated result

 

 

Step 1 Step 2

Figure 31: Interpolation process

93

 

Several interpolation schemes were developed following this prototype with
different terms used in the prediction vectors (39) in step 1 and (44) in step 2. Examples
of prediction vectors used in some of the schemes developed in this study are given in
Table 7. It should be noted that the natural logarithm of modulus of subgrade reaction and
the interaction terms with base/subbase thickness in the prediction matrices for schemes
15 and 16 are similar to terms suggested in the Westergaard’s closed form stress
equations (Huang, 1993). A signiﬁcant drop in error due to the use of these terms was
observed. Comparing the interpolated results with ﬁnite element results at non-nodal
points validates these two interpolation schemes. Also note that the solutions to the
normal equations for schemes 15 and 16 produce perfect ﬁts at the nine nodal points
corresponding to each level of the product 01(AT/D). Several more schemes have also

been investigated. Most of these schemes that contain high order interaction term(s) in

2 2 3 3 . . .
“step 1”, e.g. H* k* , H*k* , H* k*, were found to result in low predictive power.

94

 

 

 

 

 

 

 

 

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95

4.3 Validation of interpolation schemes

The validation process is illustrated in Figures 32 and 33. This process involves obtaining
ﬁnite element results at non-nodal points that were not used in developing interpolation
schemes. Error is deﬁned as the difference between the interpolated result and the ﬁnite

element result directly obtained from the ISLABZOOO.

 

Identify statistical
terms to be used in
interpolation scheme

i/

Construct an interpolation
Scheme based on the identiﬁed
statistical terms using the analyzed
stresses from the ﬁnalized matrix

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘1’ Identify variable
Stress prediction é combinations for an > ISLAB2000 analysis
l experimental matrix

 

Interpolated stress results
(for variable combinations Actual stress results
in the experimental matrix)

 

 

 

 

 

 

 

 

 

I g Comparison of results from
7 the two approaches —-> MSE

 

 

 

Figure 32: Validation procedure

96

 

 

Generate 20 interpolation schemes based on
various prediction matrices in steps 1 & 2

 

 

based on limited 20 variable combinations

Select best four interpolation schemes
for second-stage validation

irL

First-stage test: test all interpolation schemes ]

 

 

Second-stage test: 12,348 variable combinations
(focus on single axle, tandem axle, tridem axle)

 

 

for third-stage validation

 

 

Third-stage test: 500 random combinations
(focus on all axle conﬁgurations)

111

Select best two interpolation schemes 1

 

Figure 33: Overview of validation process

More than 12,000 non-nodal ﬁnite element results have been obtained and used to
validate and select from interpolation schemes. The three stages of the validation process

are as follows:

Validation stage 1: In the first stage, all interpolation schemes that were developed are
validated with a limited number of non-nodal points. The validation matrix covers 20
non-nodal points with variations of all three non-discrete variables for a ﬁxed
combination of discrete variables (lO-in. SLAB thickness, l6-in. base/subbase thickness,
177-in. joint spacing, concrete shoulder, and single axle edge loading). Non-nodal points
at the middle between the anchor values are considered in this validation stage. These
non-nodal points are believed to result in large magnitudes of errors since they are far
from the anchor values. Mean square of errors (MSE), bias, and variance are the

measures of the goodness of ﬁt of the interpolation schemes considered in this study.

97

 

These values were calculated for the errors (difference between the ﬁnite element results
and interpolated results) obtained from the validation process. Table 8 and Figures 34 (a)
through (d) illustrate the validation results at the ﬁrst stage for the six most promising
interpolation schemes. The comparison between ﬁnite element and interpolated results
illustrated in Figure 34 (a) suggests that all these schemes have high predictive power.
However, based on MSE, bias, and variance in Table 8 and Figures 34 (b) through (d),
schemes 5, 6, 15, and 16 appear to be the best four performing interpolation schemes, and

consequently are selected for the next stage of validation.

Validation stage 2: The validation matrix for this stage consists of 12,348 non-nodal
points (midpoints between nodal points). The experimental matrix of “validation stage 2”
is a complete factorial of all discrete variable and ﬁve values of each of the three non-
discrete variables (including two midpoints). The process focuses on single, tandem, and
tridem axles for all non-discrete and discrete variables. The middle points between nodal
points are also used for this validation stage. The validation results are illustrated in Table
9 and Figures 35 (a) through (d). Based on the validation results, the two best performing

schemes are 15 and 16.

Validation stage 3: Instead of using the middle points between nodal points in the
validation process, this validation stage considers non-nodal points that are randomly
selected. This validation stage is based on 300 cases for single through tridem axles and
200 cases for quad through multi-axle (8). The validation results illustrated in Figures 36

(a) and (b) and Table 10 suggest that scheme 16 is the best performing interpolation

98

 

scheme. It should be noted that the only difference between schemes 15 and 16 is the
prediction matrix in step 2. The values of MSE, bias, and variance obtained from this
validation stage were found to be larger than those obtained from the other stages. Since

the values for all three non-discrete variables are randomly selected, this validation stage

should produce a more realistic result compared to the other stages.

Table 8: Summary of goodness of fit - stage 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Goodness of Fit Scheme 5 Scheme 6 Scheme 9 Scheme 10 Scheme 15 Scheme 16
MSE. PSi2 6.34 32.29 202.31 267.70 1.24 1.22
Variance, psi2 6.23 13.64 69.31 120.71 1.08 1.07
Absolute bias, psi 0.33 4.32 11.53 12.12 0.40 0.39
Table 9: Summary of goodness of fit — stage 2
Goodness of Fit Scheme 5 Scheme 6 Scheme 15 Scheme 16

MSE, psi2 16.47 41.43 4.15 3.11

Variance, psi2 16.40 28.39 4.14 3.11

Absolute bias, psi 0.25 3.61 0.11 0.01

 

 

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110

MSE represents the overall measure of goodness of fit, estimated by the average
of square of errors (difference between actual and interpolated values). The MSE can be
decomposed into two parts: square of bias, and variance. Bias is the average value of
errors, while variance is the average of squared deviation of errors from average error.
Based on the results from validation stage 3, scheme 16 was found to be most promising.
Figures 36 (a) and (b) provide for a comparison between actual and interpolated values
based on these schemes. These ﬁgures suggest that the interpolation schemes can be a
reliable alternative for approximating mechanistic responses. Table 10 also shows that the
interpolated results for single through tridem axles are exceptionally accurate and precise.

The biases and variances associated with the longitudinal stress at the bottom of the

concrete slab for scheme 16 are 0.0 psi and 3.38 psiz, respectively. Overall maximum

absolute biases based on this scheme are 0.6 psi and 2.1 psi for single through tridem
axles and quad through multi-axle (8), respectively.

As the validation process has been completed, the interpolation scheme is used to
generate a catalog of stresses by assigning a series of sets of design inputs that are not
addressed in the experimental matrix. The catalog of stresses can be found in the MDOT

report.

4.4 Sample of calculation

Interpolation schemes can simply be implemented by carrying out the mathematical
expressions as described earlier. For example, the longitudinal stress is estimated at the
bottom of the slab. The pavement cross-section includes a 275-mm (l l-in.) concrete slab,

500-mm (20—in.) base/subbase thickness, 40.7-kPa/mm (ISO-psi/in.) modulus of subgrade

lll

 

reaction, 8.0-m (27-ft) joint spacing, tied concrete shoulder, thermal strain gradient of

62110-7 mrn-l (15x10-6in.-1), l42-kN (32-kips) tandem axle.

Step 1: Interpolation in 2-D space across the ranges of base/subbase thickness and

modulus of subgrade reaction

 

 

 

P-
Prediction vector was computed based on H* and k* at the target point (equation 39) 5;
- 2 2 1 500 5002 .
X* = 1 500 500 1n(40.7) 500- ln(40.7) 500 ~ln(40.7) -
40.7 40.7 40.7

A nine by nine matrix was computed based on H, and kj at nodal points (equation 41)

l

 

"X’(100,8.13) '1 100 10000 2.10 210 20956 0.1230 12.30 1230‘
£000,271) 1 100 10000 3.30 330 32995 0.0369 3.69 369
f(100,54.2) 1 100 10000 3.99 399 39927 0.0185 1.85 185
f(400,8.13) 1 400 160000 2.10 838 335290 0.1230 49.20 19680
X: 2?(400,27.1) =1 400 160000 3.30 1320 527925 0.0369 14.76 5904
2?(400,54.2) 1 400 160000 3.99 1597 638829 0.0185 7.38 2952
1?(650,8.13) 1 650 422500 2.10 1362 885374 0.1230 79.95 51968
X‘(650,27.1) 1 650 422500 3.30 2145 1394053 0.0369 23.99 15590
_Y(650,54.2)_ _1 650 422500 3.99 2595 1686908 0.0185 11.99 7795‘

 

 

 

Anchor stresses were obtained from ﬁnite element analysis at Hi and kj for 01:0, 4 and

8x10.7 mm.l (equation 42)

112

’1092.6‘ ’2192.8‘ ’3293.1‘
752.2 2312.6 3857.2
619.3 2320.8 3989.4
1074.0 2182.2 3290.4
6a=0 =1 738.9 rkPa 36.4 =4 2298.5 rkPa 6a=8 = 3857.1 » kPa
608.0 2305.9 4035.2
1017.1 2150.9 3284.7
698.3 2255.9 3813.5
. 573.7 J 2266.3 3969.3,

 

 

 

 

 

 

 

Then, stresses at target H* and k* corresponding to the three levels of a were computed

(equations 38 and 40)

_ —1
a(500,40.7,0)=x*-[[XT-x] .XT...-0]-647.5 I...
— T ‘1 T
a(500,40.7,4)=X*- [X x] .x -&a.__4 =22925 kPa

_ —1
a(500,40.7,8) = X * [[x T x] - X T 60.8] = 3954.2 kPa

Step 2: Interpolation in 1-D across the range of thermal strain gradient

Prediction vector was computed based on 01* at the target point (equation 44)
58*:{1 6 62}
A least-squares coefficient vector was computed based on oq at nodal points and

computed stresses obtained from step 1 (equation 45)

0 "'1 647.5 647.5
4 16 - 2292.5 = 409.1
8

64 3954.2 0.526

113

Then, the target stress at H*, k* and 0t* was computed (equation 43)

647.5
a(H*,k*,a*)={1 6 36}- 409.1 =3121.2 kPa
0.526

The stress computed using interpolation scheme is 3121.2 kPa (452.353 psi),
while the result directly obtained from ﬁnite element analysis is 3121.8 kPa (452.436

psi). The error of interpolated result in this example is 0.6 kPa (0.1 psi) or 0.02%.

114

 

CHAPTER V

INFLUENCE SURFACE TECHNIQUE

The use of inﬂuence surfaces is considered the most effective and efﬁcient approach for
the analysis of structures that carry moving loads, and for the derivation of critical
loading scenarios for such structures. However, in practice the inﬂuence surface
technique has mostly been limited to the analysis of bridge decks. Logically, the analysis
of rigid pavements should also beneﬁt from the versatility of inﬂuence surfaces, since
they also carry moving loads.

In general, the same logic applied to an inﬂuence line for a one-dimensional
member is applicable to an inﬂuence surface for a two-dimensional structure. An
inﬂuence surface for a plate or shell structure is usually represented pictorially as a three-
dimensional plot (e.g., a surface plot or contour plot) of mechanistic responses at a
reference point due to a unit load at various locations over the entire surface of the
structure. When placing the unit load at a location on the structure, the values displayed
on an inﬂuence surface represent the magnitudes of the mechanistic response at the
reference point. In this study, a series of inﬂuence surfaces have been generated as the
inﬂuence ﬁeld data for various reference points, types .of mechanistic responses, and
boundary conditions stored in an extensive computer database. These are then directly
used to compute magnitudes of stresses for design purposes and determine critical
loading scenarios. The inﬂuence ﬁeld data are also used to draw three-dimensional plots

of inﬂuence surfaces to provide graphical interfaces for the application.

115

The use of inﬂuence surfaces for rigid pavements did not gain popularity since
rigid pavements were designed empirically, where the mechanistic responses of the
pavements were not essential to the design process. With the recent development of the
mechanistic-based design procedure, and its forthcoming adoption, the application of the
inﬂuence surface technique will be useful for the analysis and design of rigid pavements.
This chapter establishes the application of the inﬂuence surface technique in rigid
pavement study by accomplishing three goals: (1) to propose an approach for developing
inﬂuence surfaces for rigid pavements, (ii) to validate the proposed approach, and (iii) to

demonstrate potential applications of inﬂuence surfaces for further rigid pavement study.

5.1 Construction of inﬂuence surfaces

While the analytical approach to obtain inﬂuence surfaces has been well established with
successes documented by many researchers, the process requires the structural analysis to
be expressed mathematically. This alone can be a daunting task for a large and complex
structure like a rigid pavement system. Therefore, a numerical process to obtain inﬂuence
surfaces is proposed. The overall process of constructing inﬂuence surfaces is presented

in Figure 37.

116

 

 

Establish the experimental
matrix for the inﬂuence surfaces

\V

 

 

 

 

Apply a set of structural feature
inputs from the experimental matrix
into a ﬁnite element analysis

 

 

 

 

 

r

W

 

 

 

Apply a unit-load on the
structure surface at a loading = Perform
grid analys1s

the ﬁnite element

 

 

 

 

 

A Structural analysis process is to be
' repeated for all loading grids

k

 

v

 

 

 

 

 

 

Store the results and Record the results at the
corresponding structural and . reference points from the ﬁnite
loading inputs into database element analysis

 

 

 

 

 

 

 

Repeat these steps for all sets of
structural feature inputs from the

 

 

 

 

experimental matrix

 

W

 

Identify the required
structural feature inputs,
iresponse type and the
reference point of the
inﬂuence surface

 

 

 

 

variable combinations in the experimental matrix

Obtain an extensive database of inﬂuence surfaces for all

 

 

W

 

 

Apply multi-dimensional interpolation schemes K——

 

 

\l/

 

Obtain the on-demand inﬂuence surface

 

 

 

Figure 37: Overall process of the proposed approach to inﬂuence surface

The structural analysis part of this approach relies on an available ﬁnite element

program specially developed for a multi-plate pavement system supported by a dense

liquid foundation (Tabatabaie and Barenberg, 1978). It should be noted that the proposed

ll7

 

 

approach is also applicable to other conventional models for analysis of rigid pavements
(Huang and Wang, 1973; Chou and Huang, 1981; Ozbeki et al., 1985; Davids and
Mahoney, 1999). Through the determination of inﬂuence ﬁelds provided by a series of
ﬁnite element analyses, inﬂuence surfaces for rigid pavement stresses are numerically
assembled. Numerous ﬁnite element analyses are performed for a unit load by placing it
all over the entire pavement slab surface one point at a time, while mechanistic responses
at selected reference points are recorded and stored in an extensive computer database.
This process is then repeated for other design variables. With the use of an interpolation
scheme based on least-squares criteria, inﬂuence surfaces can be numerically generated
on demand.

In general, limitations of the proposed approach should not be different from the
limitations of the ﬁnite element method, on which the construction of the inﬂuence
surfaces are based. While geometric non-linearity can possibly be handled by the ﬁnite
element method (Davids and Mahoney, 1999), it is absolutely unacceptable to violate the
principle of superposition, as it is the foundation of the inﬂuence surface technique. As a
result, the proposed approach may not be applicable to heavily curled rigid pavements
where the pavement structure is likely to exhibit geometric non-linearity due to the gap

between slab and base layer.

Experimental matrix
As described in the overall process, the initial step towards the numerical approach of
inﬂuence surfaces is to establish an experimental matrix containing ranges of structural

feature inputs to be varied. Practically, variation of structural feature inputs of rigid

118

 

 

pavements is unlimited, including, but not limited to, layer thickness, layer material
properties, joint designs, lateral support conditions, slab dimensions, skew orientations,
and roadbed moduli. While the proposed technique should logically be applicable to all
types and ranges of structural feature inputs, it is not an objective of this study to include
all possible combinations of the aforementioned variables, but to propose the process and
demonstrate the potential of the numerical approach of inﬂuence surfaces for rigid

pavements. Concrete elastic modulus, Poisson’s ratio, and unit weight are kept constant

at 4x106 psi, 0.15, and 150 lb/ft3, respectively. Aggregate elastic modulus, Poisson’s

ratio, and unit weight are kept constant at 30x103 psi, 0.35, and 105 lb/ft3, respectively.

Joint design is kept constant at 1.5 in. dowel diameter with 12 in. spacing center to center.
This study does not include shoulder to offer signiﬁcant lateral support to the slabs,
assembling the general characteristics of an asphalt concrete shoulder. The key inputs
selected as part of the experimental matrix in this study include concrete slab thickness
ranging from 8 to 12 in., dense-graded aggregate base thickness ranging from 4 to 18 in.,
modulus of subgrade reaction ranging from 50 to 250 psi/in., slab length ranging from 15
to 20 ft, and slab width ranging from 12 to 14 ft. Altogether, there are 270 input
combinations, including two levels of slab width, three levels of slab length, ﬁve levels of
slab thickness, three levels of aggregate base thickness, and three levels of modulus of
subgrade reaction.

In addition to these structural feature inputs, the construction of inﬂuence surfaces
also involves various general points and reference points, as illustrated in Figure 11. Each
combination of inputs requires a series of ﬁnite element analyses to be performed for a

unit load positioned at every loading grid all over the pavement surface. As illustrated in

119

Figure 38, loading grids are spaced at 12 in., covering the entire area of three-slab surface
of the pavement model. The number of loading grids on the pavement surface range from
540 to 840 grids, depending on the dimension of the slabs. However, with the advantage
of the structural symmetric property, only a quarter of the surface area will need to be

covered.

12 ft

0ft

 

Unit load

Figure 38: Typical loading grids, reference point, and unit load

An inﬂuence surface for a pavement system is a three-dimensional plot of a
reference point’s mechanistic re5ponses to a unit load at various locations over the
surface of the pavements; to be meaningful, the reference points should be the points
where the mechanistic responses are pertaining to the design criteria. To relate the use of
inﬂuence surfaces to the transverse cracking of the pavements, various mid-slab locations
across the width of the slab were selected for reference points where critical values of
longitudinal stress are expected. Figure 38 also illustrates a typical reference point in this
study. The reference point shown in the ﬁgure is located at the mid-slab 12 in. away from
the edge.

Since tire pressure, tire contact area, and tire aspect ratio have an impact on the

mechanistic responses of rigid pavements obtained from the ﬁnite element method

120

 

(Ioannides, 1985), wheel load may not be treated as a discrete point load. Thus, in the
formation of inﬂuence surfaces, it is of importance to appropriately select contact
pressure, contact area, and aspect ratio for the unit load. Figure 39 presents a series of
ﬁnite element analyses conducted to examine the impact of tire pressure and aspect ratio

on pavement stress. In Figure 39 (a), the wheel load and tire aspect ratio are maintained at

9,000 lb and 1:1, respectively, while varying the tire contact area from 25 to 144 in.2. The

tire contact pressures ranged from 63 to 360 psi. Similarly, in Figure 39 (b), the wheel

load and tire contact area are maintained at 9,000 lb and 100 in.2, respectively, while

varying the tire aspect ratio (width: length) from 2:1 to 1:3. As revealed in the ﬁgures, it
is clear that both tire contact pressure and tire aspect ratio have an insignificant effect on
pavement stress when the variations are within practical limits. As a result, a 10 in. by 10
in. square load of 9,000 lb is selected for the standard unit load. When compared to other
tire pressures and aspect ratios, the results in Figure 39 suggest that the errors associated
with the use of the standard unit load, instead of the actual wheel, are less than two

percent within reasonable ranges.

121

 

 

 

Pavement Stress (psi)
Percent Difference from
Standard Unit Load

 

 

 

Tire Contact Pressure (psi)

 

 

- Pavement Stress —0— Percent Difference as Compared to Standard Unit Load

 

a) Impact of tire contact pressure on pavement stress

 

 

 

 

 

 

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Tire Aspect Ratio (width: bngth)
- Pavement Stress +Pement Difference as Conpared to Standard Unit Load

 

 

b) Impact of tire aspect ratio on pavement stress

Figure 39: Impact of tire contact pressure and aspect ratio on pavement stress
(Details: three-slab system of dimensions 12 ft by 15 ft with doweled joint, slab thickness of 10 in.,
aggregate base thickness of 4 in., modulus of subgrade reaction of 100 psi/in., wheel path at 24 in. from

slab edge, mid-slab longitudinal stress underneath wheel path)

122

 

Interpolation process

As described in the overall process in Figure 37, the next step is to apply a series of
interpolation schemes containing mechanistic responses only at loading grids and only
for the structural features identiﬁed in the experimental matrix to the database. The
interpolation process is applied only to the non-discrete variables, which are continuous
in nature. Such variables include slab thicknesses, aggregate base thicknesses, moduli of
subgrade reaction, and coordinates of general points and reference points on inﬂuence
surfaces. While theoretically possible to also include slab widths and slab lengths in the
interpolation process as they are numerical in their characteristic, they are actually
limited to only a small number of design alternatives in practice. Consequently, slab
widths and slab lengths are treated as discrete variables in this study. Only applicable
across the ranges of non-discrete variables, the interpolation process thus requires
repetitions for each of the combinations of discrete variables.

The interpolation process, through the least-squares method, involves a series of
interpolation schemes. To disentangle the numerical complexity, the interpolation process
is divided into three fundamental steps: (i) one-dimensional interpolation for a reference
point, (ii) two-dimensional interpolation for a general point, and (iii) three-dimensional
interpolation for structural feature inputs. Generally, with a function of two coordinates

(as illustrated in Figure 11), reference point A (u, v) in this study could simply be viewed

as a function of only one variable A (“mid-slab, v), since the location for the reference

points in this study is narrowed down to the various locations across the width of the slab

at the mid-slab, where “mid-slab is the ﬁxed u-coordinate at mid-slab. As a result, the

response on the inﬂuence surface in this study is adjusted to a six-dimensional function

123

17 (v,x, y,D,H ,k), incorporating reference point (v), general point (x, y), slab thickness
(D), aggregate base thickness (H), and modulus of subgrade reaction (k). The prototype

of the interpolation schemes in matrix form is explained below.

M One-dimensional interpolation for a reference point

In the ﬁrst step, a one-dimensional cubic interpolation scheme is selected. Cubic
interpolation is preferred over linear and quadratic schemes since its error is signiﬁcantly
less. While more complex, the fourth degree interpolation scheme was not found to
substantially improve the interpolated results. Note that each interpolated result obtained
from a one-dimensional cubic interpolation scheme requires four mechanistic responses
from the database. As the beginning of the interpolation process, this step directly

retrieves the mechanistic responses at the anchor inputs from the database,

“vi ,x j , yk , DI , H m ,kn ). Repeated at all anchor points (xj, yk, D1, Hm, kn), this step of

the interpolation process is to propagate results at the desired reference point

y'(v*,xj,yk,Dl,Hm,kn).

y'(v*,xj,)’k.D[,Hm,kn):21(v*).,81(xj,yk,Dl,Hm,kn) (46)
Where 52'(v*, x j , y k , D] , H m ,kn )is interpolated result for the targeted reference

point at level xJ', yk ,Dl , Hm,and kn , and
fl (v *) is the vector of predictor for step 1

Yr(v*)={1 v* v*2 v*3} (47)

Where v* is the targeted reference point,

124

.—
...r—.

 

(Xj, yk) are the sixteen most adjacent anchor general points,

D. are anchor values of slab thickness where D1, D2,. .., D4 are the four

most adjacent anchor values of slab thickness,

Hm are anchor values of aggregate base thickness where H1, H2 and H3
are 4, 10, and 18 in., respectively,

kn are anchor value of modulus of subgrade reaction where k1, kg and k3

are 13.6, 40.7, and 67.8 kPa/mm, respectively, and

 

A

,61 (x j , y k , D] , H m , kn ) is the least-squares coefﬁcient vector for step 1

 

 

—1
A T A
ﬂ1(xj,yk,Dz,Hm,kn)=[XrT - X1] ~Xr 'Yl(xjv)’kal,Hm,kn) (48)
2 3
1 v1 v12 v5
1 v v v
X1: 2 g g (49)
1 V3 v3 v3
2 3
L1 V4 v4 v4 _
Where v1, v2,. . ., V4 are the four most adjacent anchor reference points

—
~

Y(v1,xj,)’k,Dl,Hmrkn)-

A fv2,x-,yk,Dl,H ,k
Y1(x,-,yk.Dz.Hm,kn)= 1 m " (50)

Y V3,xj,yk,Dl,Hm,kn

 

 

_Y V4,xj,yk,D1,Hm,kn _

Step 2: Two-dimensional interpolation for a general point
Based on the results at the speciﬁc reference point obtained in the ﬁrst step, the second

step further interpolates the results to specified general points on inﬂuence surfaces,

125

which is then repeated for all anchor values of structural inputs. To cubically interpolate
the results for general points on a two-dimensional surface, each interpolation in this step
requires 16 interpolated results from the previous step, which now serve as responses at

anchor inputs. Repeated for all anchor structural inputs, the second step subsequently

feeds results at the desired general points 33(v*,x*,y*,D1,Hm,kn)into the last step of
the interpolation process.

3'1"": x*,y*,D1 ’Hm’kn)= 22(x’ka)’ *)'/§2(V*’Dl ’Hmikn) (51)
Where 55(v*, x*, y*, D], H m , kn )is interpolated result for the targeted reference

point and general point at level DI , Hm , and kn , and

552 (x*, y *)is the vector of predictor for step 2
_. 3
X2(X*,y*)={x*3 x*2.y* x*,y*2 y* } (52)

Where (x*,y*) is targeted general point, and

5’2 (v*, D1, H m , kn )is the least-squares coefﬁcient vector for step 2

A —1 A
52(v*,Dl,Hm,kn)=[X2T -X2] -X2T -Y2(v*,D[,Hm,kn) (53)
3 2 2 3‘
Fxl x1 'YI xl'ylz Ya
3 2
X2 = x1 x1 9’2 xl'yz )12 (54)

2. . 2 .3
ini x4'Y4 x4'Y4 Y4

 

 

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Y(V*’x1’)’l,Dlva’kn)—
y("*’xl,)’2’:Dlva’kn) (55)

 

 

_§(V*,x4,)’4’DlaHm’kn )_

126

Step 3: Three-dimensional interpolation for structural feature input
The last step, which is a three-dimensional interpolation process, is divided into two
parts: (i) one-dimensional interpolation across the range of slab thicknesses (D) to obtain

results at the desired slab thickness y'(v*,x*, y*,D*,Hm,kn) and (ii) two-dimensional

interpolation across the ranges of aggregate base thickness (H) and moduli of subgrade

reaction (k) to ultimately obtain the result at the desired slab thickness |
?(V*,x*, y*,D*,H*,k *).

i
5"(V*,x*,y*,D*,Hm,kn)=Y3a(D*)-,33a(v*,x*,y*,Hchn) (56) !
Where §(v*,x*, y*, D*, H m ,kn )is the interpolated result for the targeted i

reference point, general point, and slab thickness at level H m , and kn , and

)7 3a (D *) is the predictor vector for the ﬁrst part of step 3

x3a(D*)={r 0* 0*2 W3} (57)
Where D* is targeted slab thickness, and
33a (v*, x*, y*, H m , kn )is the least-squares coefﬁcient vector for the ﬁrst

part of step 3

A -1 ..
ﬂ3a(V*,x*’y*’Hmikn)=|:X3aT 'X3a] 'X3aT 'Y3a(V*vx*’)’*,Hmikn) (58)

1 Dl 012 013
2 3
l 02 D D
X 3a — 3 :2, (59)
1 D3 D3 D3
1 D4 0} 02

 

 

127

 

 

y(v*,x*,y*,,01 Hm,k,,)
,. y(v* ,,x*,y* D2, Hm,k n)
Y v*,x*,y*,H ,k = ~ (60)
3a( m n) y(v*,,*x*y H03 Hm,k n)
J(V* ”35*,” D4 Hm’k n)
y(v*,x*,y*,D*,H*,k *) : f3b(H*'k *)-33b(v*,x*,y*,D*) (61)

Where 32’(v*,x*, y* , D*, H *,k *) is the ﬁnal interpolated result, and

)7 3b (H *, k *) is the vector of predictor for the second part of step 3

2
H* H*
7.? } (62>

 

i
k*

Y3b(H*,k*)={l H* 11*2 ln(k*) H*ln(k*) H*21n(k*) k*

Where H* is targeted aggregate base thickness,

k* is targeted modulus of subgrade reaction , and

1831) (v*, x*, y*, D *)is the least-squares coefﬁcient vector for the second

part of step3
* T ‘1 T ~
ﬂ3b (v*,x’k’ y*,D *): [X3b . X3b:] . X3b . Y3b (v*,x’l" y*,D *) (63)
- 2 2 1 H1 H12 _
1H1 H1 ln(k1)H11n(k1) H1 ln(k1) — _. __
k k1 k1
2
1 H1H121n(k2) Hlln(k2) 3121,10,?) .1. £11 ”_1
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3b 1 H1H121n(k3) Hlln(k3) H121n(k3) i El EL.
k3 k3 k3
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1H3 H3 ln(k3) H3ln(k3) H3 ln(k3) — __ _
_ k3 k3 k3 _

 

 

128

 

 

'i(v*.x*.y*.D*.Hr.k1)'
33 V*,X*,y*.D*.H1,k2;

Y3b(V*,X*,y*,D*)= ? V*.X*,y*.D*.H1,k3 (65)

 

 

L9(v*,x*, y*,.D*,H3,k3 )_

5.2 Interpretation of inﬂuence surfaces

The details of each step in the interpolation process presented in the previous section can
be implemented with the aid of any mathematical software or spreadsheet program. As
the inﬂuence ﬁeld database has been developed via the ﬁnite element method, an
inﬂuence surface plot can be generated by feeding the targeted variables into the
formulations described above. Three typical inﬂuence surfaces are illustrated in Figures
40 (a) through (c) in contour plot format through the use of spreadsheet program.

It should be noted that the pavement system in this study contains three slabs,
where the reference point is always located on the second slab. However, the three-slab
system sufﬁciently covers the entire non-zero area on the surface; as illustrated in the
ﬁgures, all general points appear to have zero values near both ends of the three-slab
system. Furthermore, the responses on the plot are required to be normalized to the unit
load 9,000 lb to quantify the response for each wheel load. Then, through superposition,

the response due to the entire load conﬁguration at the reference point can be calculated.

129

 

 

 

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c) Reference point at 40 in.

Figure 40: Typical inﬂuence surfaces (in psi)

(Details: slab dimensions of 12 ft by 15 ft, slab thickness of 8 in., aggregate base thickness of 4 in., and

modulus of subgrade reaction of 50 psi/in.)

130

5.3 Verification of the proposed technique

To validate the proposed technique, an extensive veriﬁcation process is conducted
involving a number of independent cases analyzed directly by the ﬁnite element method
and then by the proposed technique of inﬂuence surface. Shown in Figure 41 is the multi-
axle truck conﬁguration used in the veriﬁcation process. This truck is applied to a total of
14,850 combinations of slab dimensions, reference points, general points, and structural

features. These are all features excluded from the experimental matrix.

 

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l3 kips 13 kips 18 kips l6 kips l6 kips 15.4 kips

Figure 41: Multi-axle truck configuration used in the verification process

Goodness of fits

Comparing results of the two approaches (illustrated in Figure 42 (a)) generally suggests
that the proposed technique can accurately and precisely quantify pavement stresses.
Here we are comparing the proposed approach to the results obtained directly from the

ﬁnite element method, which is also conﬁrmed to have exceptionally low values of

statistical ﬁt indicators such as a mean of squares of error (MSE) of 0.303 psiz, maximum

error of 2.263 psi, and average absolute value of errors of 0.348 psi. More importantly,

131

the distribution of percent error in Figure 42 (b) illustrates the reliability of the results
produced through the proposed approach. As can be seen, a majority of cases are
associated with less than one percent of error and the maximum percent error observed is

only 3.3 percent.

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Sources of errors

Through an analytical approach, the results obtained from inﬂuence surfaces should be
identical to those obtained directly from the ﬁnite element method. However, for a
numerical approach, such as the one proposed in this study, errors are naturally
anticipated. Basically, since several assumptions have been made to accelerate and
simplify the numerical process, calculation errors are the result of violations of the
assumptions. The level of the errors depends on the degree of violation. As shown in the
veriﬁcation process, the level of errors appears to be acceptable; it is nevertheless
important to future researchers to comprehend the sources of the errors.

One unavoidable assumption as a part the proposed approach is the unaddressed
impact of tire contact pressure and aspect ratio. Deviations of both parameters away from
the standard unit load used in the construction of inﬂuence surfaces, even within practical
limits, violate the assumption and consequently result in errors. While this violation
appears to have rrrinimal consequences (as illustrated in Figure 39), it is still responsible
for a signiﬁcant portion of the overall half percent error. Another source of error comes

from the use of power series in the interpolation process, on which the vectors of

predictor X1, X2, and X3a are based. A certain level of error is always associated with the

use of power series, depending on the degree level of the series. As the fourth degree
series was not found to substantially improve the interpolated results, the third degree

series was chosen.

135

 

 

 

5.4 Potential applications

Like the ﬁnite element method, the inﬂuence surface technique has unlimited potential as
a tool for the analysis and design of rigid pavements, but in a more time—efﬁcient fashion.
Thus, the use of the technique makes some of the most tedious tasks, if handled directly
with ﬁnite element method, more effective and practical. To elaborate the potential
applications of the proposed technique, the following four tasks in the ﬁeld of rigid
pavement study are performed using the inﬂuence surface technique: (i) rapid pavement
stress calculation, (ii) determination of critical load location, (iii) pavement stress history,

and (iv) investigation of interaction between load conﬁguration and structural feature.

Rapid pavement stress calculation

Finite element analysis is a time—consuming approach to calculate pavement stresses,
while the interpolated stresses can instantly be obtained via mathematical software or
spreadsheet programming. The proposed approach is an attractive alternative to the ﬁnite
element method.

To demonstrate the calculation process, a standard 32-kips tandem axle is placed
on the inﬂuence surface shown in Figure 40 (b). The longitudinal stress values on the
inﬂuence surface at each wheel load are 91.1 psi for the two wheels on the wheel path
and 27.3 psi for the two wheels away from the wheel path. As the applied wheel loads are
8 kips apiece, the inﬂuence surface values are then normalized to the unit load, resulting
in a value of 80.9 psi for the two wheels along the wheel path and 24.3 psi for the two
wheels away from the wheel path. Note that these values are obtained through step 2 of

the interpolation process, while the estimate values could also be obtained by reading

136

 

directly from the inﬂuence surface plot. Ultimately, via the superposition principle, it can
be calculated that the response at the reference point due to the entire load conﬁguration
is 210.5 psi. In addition, it should also be noted that the use of the inﬂuence surface
allows the pavements to be analyzed through more common mathematical software (e. g.,
MathCad®, Mathematica®, MATLAB®) or spreadsheet programs (e. g., Microsoft

Excel®) at places where access to ﬁnite element software is not available.

Determination of critical load location

In general, for highway rigid pavements, stresses have a maximum value signiﬁcantly
lower than the concrete modulus of rupture. Therefore, unlike the design of bridge decks
or airport pavements, where the design criteria is usually based on the worst case loading
scenario, the design of highway rigid pavements for cracking prevention is based on load
repetitions, which effect fatigue behavior. However, the worst loading scenario and the
maximum value of pavement stress may become signiﬁcant to the design of rigid
pavements when actual wheel loads are much heavier. In such a situation, it is of interest
to pavement engineers whether or not the pavement stresses exceed the modulus of
rupture.

While the process to determine the critical load locations is time-consuming if
done directly using the finite element method (Buch et al., 2004), they can be obtained
almost instantaneously via the application of the proposed technique. Due to the
swiftness of the inﬂuence surface technique in determining each pavement stress, with an
extra line of loop command or macro applied to inﬂuence surfaces, pavement stresses for

every load position required can be quantified in a short time frame. From the location

137

 

the driving axle approaches the pavement system to the location the last axle of the truck
leaves the pavement system, critical load location is the load location corresponding to
maximum stress. Based on the truck conﬁguration in Figure 41, an example of the
determination of critical load location is illustrated in Figure 43 for bottom-up stress at
the reference point. While it is conventionally believed that the critical load location for
bottom-up stress is the position where the heaviest axle is located at the reference point,
this example proves that such general belief may not necessarily be true. The critical load
location in this instance was found to be where the lighter tandem axle is located at the
reference point. The impact of adjacent axles must be accounted for in the analysis, as it
could either intensify or reduce the pavement stress at the reference point, depending on
the spacing between axles and the length of the slab. In the same way, Figure 44 shows

the determination of critical load location for top-down stress at the reference point.

138

 

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Pavement stress history

Addressed through Miner’s hypothesis, the fatigue damage of rigid pavements is
conventionally calculated based on moduli of rupture, peak stress values, and the number
of associated repetitions. With the Weigh-in-Motion (W IM) data, pavement stress history
can be provided via the proposed approach. For each truck conﬁguration and its traveling
speed imported from the WIM data into inﬂuence surface, pavement stress history can be
presented in the same fashion as illustrated in Figures 43 (a) and 44 (a). The use of the
proposed approach should have an advantage over considering each axle group and
weight group separately. This is because the interaction from the adjacent axles is also
addressed in the calculation. However, as WIM data do not generally contain the lateral
placements of vehicles, additional study of vehicle wander may need to be conducted to
obtain the proper y-coordinate of the general point for each truck wander and the
appropriate reference point for the location of wheel path. Furthermore, in the future,
once the fatigue analysis is improved to address the impacts of off-peak stress values and
the sequence of the associated repetitions in the damage calculation, the use of the
inﬂuence surface technique can be employed as a perfect ﬁt to this puzzle.

To further elaborate this application of the inﬂuence surface technique, several
pavement stress history plots were produced based on actual field data. The structural
inputs were extracted from the Long-Term Pavement Performance (LTPP) Speciﬁc
Pavement Study-2 (SPS-Z) sections in Michigan. Constructed in 1993 along US-23 (N)
near the Ohio-Michigan border, the experiment contains four sections with dense-graded
aggregate base (DGAB), the same type of material as used in the developed inﬂuence

surfaces. The inventory properties of the DGAB sections are shown in Table 11. The

141

 

other eight sections were not eligible in this demonstration as they were constructed with
treated bases. Theoretically, this technique is applicable to all types of elastic base layer;
however, it is not an objective of this study to capture the impact of all base types. Based
on the LTPP database (Release 17), the backcalculated modulus of subgrade reaction has
an average value of 225 and 300 psi/in. between May and June and between November

and December, respectively.

Table 11: Inventory properties of the sections used in the demonstration

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Section
Inventory properties
26-0213 26-0214 26-0215 26-0216
Lane width, ft 14 12 12 14
Slab length, ft 15 15 15 15
Dowel diameter, in. 1.25 1.25 1.50 1.50
Dowel spacing, in. 12 12 12 12
Slab thickness, in. 8.0 8.1 10.7 11.1
Base thickness, in. 6.1 5.8 6.2 5.9
Backcalcula‘ed Slat? 5,570,000 ‘ 5,030,000 5,480,000 5,360,000
elastic modulus, pSl
B “1‘93””th has? 37,100 33,500 36,500 35,700
elastic modulus, psr

 

The main issues pertaining to applying the developed inﬂuence surfaces to the
ﬁeld data are the difference between the input values used in the inﬂuence surface
analysis and the input values in actuality. These inputs are load transfer devices and layer
elastic moduli. The SPS—Z sections employed both 1.25-in. and 1.5-in. dowel diameters,
while only a 1.25-in. dowel diameter was addressed in the construction of the inﬂuence
surfaces. The slab and base elastic moduli used in devel0ping inﬂuence surfaces are

4,000 ksi and 30 ksi, respectively, different from the actual layer elastic moduli. Despite

142

 

the fact that the variations in load transfer device and layer elastic moduli were not
included in the development of inﬂuence surfaces in this study, such variations
insigniﬁcantly affect the pavement stress magnitudes. As reported in the MDOT study,
the variations in load transfer device, while signiﬁcantly affecting slab deﬂections
adjacent to the transverse joint, were found to be insensitive to the pavement stress
magnitudes near the midslab. Figures 45 (a) and (b) suggest that the variations in slab and
base elastic moduli within reasonable ranges have an insigniﬁcant effect on the pavement
stress magnitudes; only 5% change in stress magnitudes is contributed to the increase in

slab elastic moduli from 4,000 to 6,000 ksi.

143

 

 

 

Stress, psi

 

 

4,000 ksi 4,500 ksi 5,000 ksi 5,500 ksi 6,000 ksi
Concrete elastic rmdulus

I 25 ksi I 30 ksi El 35 ksi D 40 ksi
Base elastic modulus

 

 

 

(a) Pavement stress magnitudes

 

 

 

 

 

Percent Error

 

 

 

4,000 ksi 4,500 ksi 5,000 ksi 5,500 ksi 6,000 ksi
Concrete elastic modulus

I 25 ksi I 30 ksi El 35 ksi El 40 ksi
Base elastic modulus

 

 

 

(b) Percent error as compared to those used in developing inﬂuence surfaces
Figure 45 : Impact of variations in slab and base elastic moduli
(Details: 10-in. slab thickness, 6-in. aggregate base thickness, 100-psi/in. modulus of

subgrade reaction, slab dimensions of 12 ft by 15 ft, 9-kips midslab loading)

144

 

Extracted from the WIM database at the SPS-2 site, the truck conﬁguration shown
in Figure 46 traveled through the WIM station between 8 pm and 9 pm of July 11, 2000
at the speed of 63 mi/hour. This truck was applied to the inﬂuence surfaces for this
demonstration. i

130 in. 370 in. 51 in. 236 in.
\l \1 \1

7
7
.7
7
7

 

28,440 lb II—II
28,2201b II—II
27.34011» Il——ll
26,2401b II——II
10.3601b I————I

‘ 3...?

Figure 46: Typical truck configuration extracted from the WIM database

Based on the aforementioned parameters, Figure 47 illustrates the stress history
plots, produced through the use of the inﬂuence surface technique. To account for the
difference in the modulus of rupture between the sections, the pavement stress history
plots were modiﬁed to the pavement stress ratio history plots as illustrated in Figure 48.
Note that sections 26-0213 and 26-0215 have a targeted 550-psi for the l4-day modulus
of rupture, while sections 26-0214 and 26-0216 have a targeted 900-psi for the 14-day
modulus of rupture. Based on the testing results, the 365-day moduli of rupture were 915
psi for sections 26-0213 and 26-0215 and 1000 psi for section 26-0214, while the testing
results were not available for section 26-0216. For the purpose of demonstration, the

plots in Figure 48 employed the targeted moduli of rupture.

145

 

 

 

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147

To translate the computed stress ratio into transverse cracking, application of the
transfer function, such as the fatigue model suggested in NCHRP 1-37 A, was used to
calculate fatigue damage due to the passing of this truck conﬁguration as shown in Figure
49. Equation 66 is the fatigue model used in this demonstration (NCHRP, 2004). Note
that the fatigue damage is the summation of the reciprocal of the allowable load

repetitions (l/N) due to all peak stress ratios caused by the passing of the truck.

LogloN = 2 . R‘1-22 + 0.4371 (66)
Where N is the allowable load repetitions

R is the stress ratio

 

 

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Fatigue damage

 

 

26-0213 26—0214 26-0215 26-0216

 

 

 

Figure 49: Fatigue damage based on peak stresses from influence surfaces

148

 

 

Figure 50 illustrates the time-series transverse cracking on the four DGAB section
based on the LTPP database (Release 17). The lO-year distress data suggested that
section 26-0216 remained free of transverse cracking, while section 26-0213 was the ﬁrst
section to exhibit transverse cracking, followed by sections 26-0215 and 26-0214,
respectively. The trend of transverse cracking coincides with the trend observed in the

calculated fatigue damage in Figure 49.

 

 

 

 

 

 

 

 

 

6

3

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0

E4

0

5,3 A

5 / /
‘82 /

as / /
'51

= JL/
20%] VV X" v

 

 

Pavement age, years

 

 

—<>— 26-0213 -I— 26-0214 + 26-0215 —)6- 26—0216

 

Figure 50: Time-series transverse cracking on the DGAB sections

Investigation of interaction between load configuration and pavement structure

The proposed technique can effectively be used as a tool to address the impact of the
interaction between trafﬁc load and pavement structure, such as the interaction between
load conﬁguration and slab length — a task requiring a strenuous effort if directly

performed using the ﬁnite element method. The truck conﬁguration in Figure 51

149

 

 

 

represents the typical image of a single unit truck, whose spacing between the load-
carrying axles may vary substantially. As the axle spacing up to 25 ft has been
documented in practice (Michigan Center for Truck Safety, 2001), this sensitivity study is

conducted within a valid range.

 

   

 

   

Axle spacing varies” 9 ft ___, i A
I“ from 9 ~ 22 ft I I

18 kips 18 kips 15.4 kips

Figure 51: Single unit truck configuration used in the demonstration study

It is important to note that variation in critical load locations is part of the
interaction, since the critical load locations are also affected by variations in axle spacing
and slab length. In this demonstration, three slab lengths (15, 18, and 20 ft) and various
axle spacing lengths ranging from 9 to 22 ft, in increments of 1 ft, are considered. Since
the process of determining critical load locations Can be accomplished in a short time
frame using inﬂuence surfaces, repeating this process for all combinations of axle
spacing lengths and slab lengths can be accomplished in a reasonable time frame.

The interaction impact between axle spacing and slab length for bottom-up stress
at the reference point for the single unit truck conﬁguration shown in Figure 51 is

illustrated in Figure 52 (a). The results reveal that bottom-up stresses at the reference

150

point are intensiﬁed by an increase in axle spacing. The increase in axle spacing, after
exceeding the slab length, however, appears to have only a marginal effect on the stress
magnitude. The observed trend is explicable, when considering inﬂuence surface plot
along with the corresponding critical load location, as shown in Figure 52 (b). The ﬁgure
reveals that while one of the load—carrying axles of this particular truck conﬁguration,
placed at the reference point, has a positive impact on the bottom-stress, the other load-
carrying axle would have a negative impact on the stress. It can be seen that the negative
impact of the adjacent axle diminishes as the spacing between the two load-carrying axles

begins to exceed the slab length.

151

 

 

 

 

 

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153

Figure 53 (a) illustrates the impact of the interaction between axle spacing lengths
and slab lengths for top-down stresses at the reference point when also considering the
truck conﬁguration shown in Figure 51. It can be seen that the top-down stresses reach
their peak values when the axle spacing lengths near the corresponding slab lengths. The
typical load location for top-down stress illustrated in Figure 53 (b) can be used to
explain the observed trend. The ﬁgure shows that the critical load location for top-down
stress concurs with the load location suggested in NCHRP l-37-A where the two load-
carrying axles are placed at each end of the slab (NCHRP, 2004). At these locations, the
inﬂuence surface for top—down stress displays its highest values.

This limited-scope study suggests that for this particular truck conﬁguration both
bottom-up and top-down stresses reach peak values when axle spacing length is slightly
shorter than slab length. While this ﬁnding may be useful to a study of the impact of the
interaction between load conﬁguration and pavement structure if applicable to all
scenarios, it is solely limited to the particular truck conﬁguration and the slab lengths
considered in this study. However, more importantly, this study indicates that the use of
the proposed technique offers a promising approach towards a detailed study to gain
insight into the interaction between load conﬁguration and structural features of rigid

pavements.

154

 

 

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156

The inﬂuence surface technique is not generally used to address the impact of
thermal gradient on stresses as temperature commonly affects the entire surface of
structures. The impact of the interaction between slab length and axle spacing, shown in
Figures 52 and 53, addresses the pavement stresses due to load only; therefore, the results
disregard the interaction between slab length and temperature. Figure 54 illustrates the
pavement stresses at the reference point due to temperature and the pavement weight for
the same cases in this demonstration. It can be seen that the longer slab lengths results in
higher thermal stress magnitudes in all cases, while this trend was not observed for the
impact of the slab lengths on loading stress magnitudes. Figure 55 shows the combined
loading and temperature stress magnitudes for this demonstration. The domination of the
thermal gradient impact was observed in the ﬁgure. Note that a positive thermal strain
gradient was used in the analysis for the bottom-up stresses, while a negative thermal

strain gradient was used for the top-down stresses.

 

 

 

 

 

 

200
g 150 -.
g 100 «
0
g, 50—
[—
0 _
15 it 18 it 2011
Slab length

 

 

 

Figure 54: Impact of slab length on thermal stress

(Details: slab width of 12 ft, slab thickness of 10 in., aggregate base thickness of 4 in., and modulus of

subgrade reaction of 100 psi/in., reference point at 18 in., thermal strain gradient of 10x10" in.")

157

 

 

 

Pavement Stress (psi)

 

350 _ _ _
300- F _ H l P

2500
200*
150-
1000
50«

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

910111213141516171819 20 2122
AxleSpacing(ft)

I 15-ft El l8-ﬁ El 20-ﬁ

 

 

a) Bottom-up stress (thermal strain gradient of +10x10'6 in.'l)

 

 

Pavement Stress (psi)

 

300 1
250

0 ! l 7

1
14 15 16 17 19 20 21
AxleSpacing(ﬁ)

aéae

I lS-ﬁ E1 l8-ﬂ Cl 20-ft

 

 

 

b) Top-down stress (thermal strain gradient of -10x1045 inil)

Figure 55: Combined loading and thermal stresses

(Details: slab width of 12 ft. slab thickness of 10 in., aggregate base thickness of 4 in., and modulus of

subgrade reaction of 100 psi/in., and reference point at 18 in.)

158

 

 

As the results from Figures 52 (a) and 53 (a) suggest that axle spacing has a
signiﬁcant impact on pavement stresses, the axle spacing should also be accounted for in
the design of the pavement. Based on this demonstration, longer joint spacing resulted in
higher bottom-up stress magnitude with diminishing effect as the axle spacing approach
15-16 ft. From the standpoint of loading stress, the selection of joint spacing should not
significantly affect the design to prevent bottom-up cracking. However, selection of
appropriate joint spacing should be a crucial part of the design to prevent top—down
cracking. This is because the top-down stress magnitudes had their peak values when the
length of axle spacing reaches the length of joint spacing. This should further warrant the
necessity to synthesize the WIM database as this would also provide a source of axle

spacing database to be considered during the selection of appropriate joint spacing.

159

 

CHAPTER VI

CONCLUSIONS, RESEARCH SIGNIFICANCE AND RECOMMENDATIONS

6.1 Conclusions
To gain an insight into the impact of various parameters and their interrelationship on
mechanistic responses of rigid pavements, this dissertation had four primary objectives:
to perform a parametric study on current and anticipated rigid pavements, with
considerations of loading, climatic, material, subgrade support, and construction
parameters, to establish a protocol for the development of a comprehensive interpolation
scheme that addresses the condition changes into mechanistic responses, to develop
inﬂuence surfaces for rigid pavements to address the impact of complex load
conﬁgurations on pavement responses and also the impact of lateral wander, and to
investigate the impact of the interaction between structural, environmental, and loading
factors on mechanistic responses. To systematically accomplish these objectives and to
provide a better understanding of each phase of the study, the research plan was divided
into three major tasks: performing of a parametric study, development of an interpolation
scheme, and application of an influence surface.

1. By conducting a parametric study based on a complete factorial experimental matrix,
the impact of the interaction between structural, environmental, and loading factors
over mechanistic responses of rigid pavements was thoroughly investigated. The
experimental matrix for the parametric study was constructed based on the data
collected from three sources: the current-practice designs of rigid pavement structure,

ﬁeld thermal gradients, and actual Michigan multi-axle truck conﬁgurations. Based

160

 

on the results from a preliminary sensitivity study, several strategies were employed
to reduce the size of the experimental matrix to 43,096 ﬁnite element runs, a practical
number, without sacrificing to the quality of ﬁnal results. These strategies include (i)
combining base and subbase layers into one layer, (ii) merging thermal gradient and
coefﬁcient of thermal expansion into a thermal strain gradient, (iii) discarding the
unfamiliar truck conﬁgurations, and (iv) selecting appropriate intervals for non-
discrete variables. Additionally, to conduct a ﬁnite element analysis, the parametric
study also requires the derivation of the critical load locations of the load
conﬁgurations, not supplied by the experimental matrix. The critical load location is
deﬁned by the load location along the wheel path that results in the most critical
mechanistic response. A time-consuming process, the procedure of determining a
critical load location was not practical to repeat for all combinations of inputs in the
experimental matrix. Hypothesized as non-inﬂuential variables to the critical load
location, ﬁve variables, including slab thickness, base/subbase thickness, modulus of

subgrade reaction, lateral support condition, and thermal strain gradient, were tested

based on a fractional factorial design of —13—-35 = 9; these variables were found to
3

have a signiﬁcant impact on the pavement stress magnitudes but appeared to be
inconsequential to the critical load location. The critical load location is inﬂuenced
only by joint spacing and truck or axle conﬁguration.

Based on the parametric study results, it was found that a change in slab thickness
from 9 to 12 in. resulted in a reduction in stress by approximately 35% for a ﬂat slab
condition. In spite of a constant thermal strain gradient, pavements constructed with

different slab thicknesses will not have a constant temperature differential, and

161

m'ﬂﬂ—
.'. ﬁ‘ 0

 

therefore, the impact of slab thickness on pavement responses, in such a case, could
not be compared. A change in base/subbase thickness from 4 to 26 in. was found to
result in about 5-30% lower stresses for a ﬂat slab condition; as the slab thickness
increases, the impact of base/subbase thickness was found to become less signiﬁcant.
Pavements with 27 feet joint spacing resulted in about 33% higher longitudinal

stresses as compared to pavements with 15 feet joint spacing for curled slab

. . . . -6 . -1 .
conditions at a thermal strain gradient value of +10x10 1n. . The severity depends

on the level of thermal curling or thermal strain gradient. For the load located along
the wheel path, pavements constructed with tied-concrete shoulders resulted in the
lowest stresses among the three lateral support conditions considered in the study.
Although the pavements were constructed with the same untied-asphalt shoulder, the
magnitudes of longitudinal stresses for pavements with a 12-ft lane (standard lane)
were higher than that for pavements with a l4-ft lane (widened lane). As the wheel
path shifted 2 ft towards the centerline for pavements with a widened lane, a pseudo-
interior loading condition was created, resulting in the reduction of stresses from edge
loading. Pavements with an untied-asphalt shoulders (12-ft lane) resulted in about
13% and 9% higher longitudinal stress values than pavements constructed with a tied-
concrete shoulder (12-ft lane) and widened lane (l4-ft lane with an untied-asphalt
shoulder), respectively. Lateral placement of trafﬁc load resulted in about 10% and
30% higher edge stresses as the load moves from the wheel path towards the
longitudinal joint (lane/shoulder joint) for a tied-concrete shoulder and an untied-

asphalt shoulder, respectively.

162

 

3. In this study, an interpolation scheme was used to obtain mechanistic responses for all
the combinations of the non-discrete inputs, excluded from the ﬁnal experimental
matrix. Via the least-squares criteria, several interpolation schemes were created with
various forms of prediction vectors. With more than 12,000 independent ﬁnite
element runs (cases not addressed in the experimental matrix), three stages of a
validation process suggested that “Scheme 16” was the best performing scheme. This
scheme contains the natural logarithm of the modulus of subgrade reaction and the
interaction terms with the base/subbase thickness, similar to the terms suggested in

the Westergaard’s closed form stress equations.

4. When including all axle types, the bias, variance, and MSE were 0.51 psi, 8.63 psi2,

and 8.89 psiz, respectively, indicating that the interpolation scheme was highly

accurate and precise in quantifying pavement responses matching with the direct

ﬁnite element results. If considering only the three major axle groups (single, tandem,

and tridem), the bias, variance, and MSE were found to be 0.02 psi, 3.38 psiz, and

.2 . . .
3.38 ps1 , respectively, an even more promrsrng performance.

5. Due to its swiftness and accuracy in quantifying pavement responses, the use of
interpolation scheme offers an attractive alternative for the analysis of rigid
pavements. With a limited number of load conﬁgurations included in the
experimental matrix, the developed interpolation scheme faces a critical issue of the
inability to address the variations in axle spacing and axle weight within the load

conﬁgurations, deviating from the conﬁgurations speciﬁed in the experimental

163

 

matrix. Extension of the interpolation scheme ability to effectively and efficiently
address such an issue was achieved in study through the inﬂuence surface technique.
The last part of the dissertation demonstrated that inﬂuence surfaces for rigid
pavements can be developed successfully through a numerical process based on a
series of ﬁnite element analyses and the multi-dimensional interpolation process.
While only a limited number of ﬁnite element results are required for the
combinations of anchor levels of each non-discrete variable in the interpolation
process, the inﬂuence surfaces can be applied to an unlimited combination of
structural features and loading scenarios through the application of multi-dimensional
interpolation schemes. The details of each step in the interpolation process are
presented in a format that can be implemented with the aid of any mathematical
software or spreadsheet program. Conﬁdence in the proposed approach was gained
through an extensive veriﬁcation process. A number of independent cases were
analyzed directly by the ﬁnite element method and then by the proposed inﬂuence
surface technique.

Based on more than 14,000 independent ﬁnite element runs, the veriﬁcation results
suggest that the inﬂuence surfaces can precisely and accurately quantify pavement
stresses as compared to the results directly obtained from the ﬁnite element method.
Versatilities of the inﬂuence surface technique for rigid pavements include, but are
not limited to, rapid pavement stress calculation, determination of critical load
location, pavement stress history, and investigation of the interaction between load
conﬁguration and structural feature. The use of this technique is clearly more

effective and practical than the direct application of the ﬁnite element method.

164

 

6.2 Research significance

1.

2.

3.

Strategies used in this research to reduce the size of the experimental matrix are
useful for future researchers.

This study introduced of the multi-dimensional interpolation scheme technique to
rigid pavement analysis.

The extensive trends observed in the parametric study better the understanding of the
interaction between structural, environmental, and loading factors on pavement
responses.

a. The increase in base/subbase thickness results in a reduction in stress
magnitude with diminishing effect as the slab thickness increases. As the
base/subbase layer provides uniformity of support to the slab, an increase
in base/subbase thickness reduces the magnitude of loading stress. The
results suggest that from the loading stress standpoint the base/subbase
thickness should also have a substantial effect on slab cracking for a
pavement system with a thin slab, especially thinner than 10in. However,
the results should not suggest that the base/subbase thickness has a less
signiﬁcant impact for a pavement system with a thicker slab, since the
base/subbase layer could also affect the drainage characteristic of the
pavement system.

b. AC shoulder results in the highest stress magnitude as compared to PCC
shoulder and widened lane. This could imply that from the standpoint of

load-related distress a pavement system with PCC shoulder or widened

165

 

lane should have a better performance than a pavement system with AC
shoulder.

Lateral load location directly affects the magnitude of pavement stress for
all types of lateral support condition. The results also imply the
signiﬁcance of the location of trafﬁc paint stripe as it would dictate the
location of wheel path.

. While lateral support condition has a signiﬁcant effect on loading stress
magnitude, its effect on thermal stress magnitude appears to be
insigniﬁcant as shown in the ﬁgure. This implies that the variation in
lateral support condition should not affect temperature-related
performance of the pavement.

An increase in the magnitude of modulus of subgrade reaction results in a
reduction in stress magnitude with diminishing effect as the slab thickness
increases. The results suggest that from the loading stress standpoint the
modulus of subgrade reaction should also have a substantial effect on slab
cracking for a pavement system with a thin slab, especially thinner than 10
in.

An increase in the magnitude of modulus of subgrade reaction results in an
increase in the magnitude of thermal stress as the combined stress
magnitudes are compared to the loading stress magnitudes. From the
mechanistic standpoint, this could imply that a roadbed with higher
modulus of subgrade reaction should result in a better load-related

performance but not for a temperature-related performance. However, the

166

 

mechanistic behavior alone may not sufficiently provide such a conclusion
to the actual performance of the pavement, since a roadbed with higher
modulus of subgrade reaction usually also has a better erodibility
resistance and also a better drainage characteristic.

g. From the loading stress standpoint joint spacing should not have a
signiﬁcant effect on load-related performance of the pavement. As stated
by the Portland Cement Association’s thickness-design procedure, the
presence of joints has no effect on the pavement stress magnitude, since
the load is placed adjacent to the midslab away from the joints. However,
it should be noted that the results did not account for the interaction
between axle spacing and joint spacing.

h. When combined with thermal stress, an interactive effect between thermal
strain gradient and joint spacing on combined loading and thermal stress
was observed. This implies that an increase in joint spacing should result
in a higher level of temperature-related distress.

i. The results suggest that a more complex axle group should result in a
lower pavement stress magnitude. However, the results did not account for

the interaction between axle spacing and joint spacing.

4. The numerical approach to the inﬂuence surface technique was proposed in this study

5.

along with the detailed formulations, which can be further studied by future research.
The inﬂuence surface technique can effectively and efﬁciently address several aspects
of the loading factors, including axle weight, axle spacing, lateral placement, and

critical load location.

167

 

6. The inﬂuence surface technique offers an effective approach to determine critical
locations for complex truck conﬁgurations.

7. Pavement stress history can be produced using the inﬂuence surface technique in a
short time frame. The pavement stress history can be used as the inputs to transfer

function to predict pavement performance.

6.3 Recommendations for future research

This research study focuses on pavement responses and several factors that affect them.
Although pavement response plays a signiﬁcant role in the mechanistic-empirical design
process, it is necessary to integrate the pavement responses with several other
components in order for it to become practical. Pavement responses need to be used as
inputs to transfer function, which relate responses to performance. However, the transfer
function coefﬁcients need to be localized and therefore it is important to ensure the
constants reﬂect the local climatic and loading conditions. The calibration process also
needs to take place to ensure the quality in the calculation process. The following
research topics are recommended: WIM data synthesis, development and calibration of
transfer functions, and cataloging of coefﬁcient of thermal expansion.

An extensive trafﬁc database, e.g. a WIM database, should be synthesized and
made available for the pavement network as hourly axle spectra is a key input for damage
computations. The hourly axle spectra allow for calculation of pavement responses that
account for daily and seasonal conditions of climate, roadbed and material. The axle
repetitions from the axle spectra and the corresponding pavement responses are the inputs

to the cumulative damage calculation.

168

 

 

 

Along with the WIM synthesis, the study of the trafﬁc lateral placement should
also be conducted. To eliminate the assumption of load only being placed at the wheel
path, the study will provide appropriate values of lateral wander or y-coordinate in the
analysis of rigid pavements using the inﬂuence surface technique.

Development and calibration of transfer functions should be conducted for key
rigid pavement distresses that reﬂect the engineering practice. The process involves
statistical correlation of the cumulative damage to the measured distresses to obtain a
calibrated model that can be used for current rigid pavement designs.

Lastly, the coefﬁcient of thermal expansion values for concrete mixes and also
aggregate (as concrete making material) used in paving need to be determined and
cataloged, since coefﬁcient of thermal expansion plays a critical role in the thermal
analysis of jointed concrete pavements. The slab movement and joint opening are also

inﬂuenced by coefﬁcient of thermal expansion of concrete.

169

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