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MATHEMATICAL MODELING FOR DOWEL
LOAD TRANSFER SYSTEMS

Hua Guo

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

DOCTOR OF PHILOSOPHY
Department of Civil and Environmental Engineering

1992

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ii

ABSTRACT

MATHEMATICAL MODELING FOR DOWEL
LOAD TRANSFER SYSTEM,

BY

HUA GUO

Zrt has been discovered and proven that the dowel bar stiffness matrix used
lay some finite element computer programs, for simulating the mechanism of
dowel bars without looseness, fails to meet some of the basic requirements
of the finite element method. An alternative component model, consisting
of two finitely long bending beams embedded in concrete and connected by
a shear-bending beam, has been developed. The model has been proven
to be theoretically correct. The model can be installed into a
finite element program to predict the responses of the load transfer
system, including distributions of bending moment, shear force and the
bearing stress of each dowel without using the assumption of effective
length. A detailed comparison between experimental and analytical results
verified the component model can reasonably predict the responses of a

dowel bar load transfer system.

Hundreds of numerical calculations were conducted using the developed
component model to test the accuracy of existing design procedures. It
has been found that the maximum bearing stress of concrete, under the
critical dowel, can not be accurately predicted by the “effective length”
assumption which is currently used in engineering analysis. Errors in
computed values of maximum bearing stress can affect the prediction of
joint faults in pavement performance models. Three tables listing maximum
bearing stresses of concrete, for the critical corner loading cases, have

been given for dowel design in this dissertation.

A nonlinear elastic structural model has also been developed to simulate
the dowel bar looseness mechanism. The model can be used to predict
various pavement responses, including stresses, displacement distributions
and load transfer capability at different stages of pavement service life.
Numerical analyses based upon the new structural model were conducted to
investigate the effects of dowel bar looseness on critical pavement
responses. Parameters included amount of dowel looseness, configuration
and location of traffic loads, shoulder edge support effects, and dowel
bar dimensions. Many findings of the study are relevant to current rigid

pavement design and rehabilitation procedures.

DEDICATION

To my mother, my wife and all friends, who encouraged and

supported me throughout my Ph.D program.

vi

ACKNOWLEDGEMENTS

The successful completion of the research described herein required the

input and assistance of many individuals. The author gratefully

acknowledges:

Mr. James A. Sherwood of the Federal Highway Administration, who was the
major advisor of the author during the period of the Fellowship Program

sponsored by the FHWA.

Mr. Roger M. Larson of the Federal Highway Administration, who was the

first advisor of the author during the period of the Fellowship Program.

Mr. William J. Kenis of the Federal Highway Administration, who provided

advice and suggestions in various phases of the conduct of the study.

Mr. Michael R. Smith of the Federal Highway Administration, who shared his

experience in scientific paper organization.

Mr.Thomae J. Pasko,Jr. of the Federal Highway Administration, who
carefully reviewed the draft of the thesis and provided many significant

suggestions.

vii

Mr. Byron N. Lord of the Federal Highway Administration, who encouraged

the author to seek the truth without hesitation.

Mr. Hisao Tomita of Federal Aviation Administration, who reviewed a part

of the thesis and provided valuable advise to improve Chapter 9.

Dr. Robert Wen, Dr. Gilbert Baladi, Dr. Thomas Wolff, Dr. David Yen of the
Michigan State University, who provided technical guidance in the conduct

of this research effort.

Mr. Jamel Hammouda of the Computer Communication and Graphics Company, who

numerically verified some findings presented in this dissertation.

Mr. Shawn Truelove, Mrs. Linda Phillipich and Mrs. Nina McMillan of the
Michigan State University, Miss. Marcia Simon, Ms. Elke Lower and Mrs.
Jeanie LaBudie of the Federal Highway Administration, who provided

valuable and patient assistance to the author during his Ph.D program.

Special thanks are offered to Dr. Mark B. Snyder for his helpful advice,
strong support, and sincere encouragement throughout the course of this
research, and for his patience and confidence in the production of this

thesis.

Chapter One

Chapter Two

Chapter Three

Chapter Four

viii

TABLE OF CONTENTS

Introduction

Major Steps of Engineering Analysis

Joint Functions and Related Deteriorations
Research on Dowel Bar Load Transfer System
Descriptions of Existir; Problems

Research Objective and Scope

Basic Elements in Finite Element Programs

Plate Element
Bar Element
Spring Element

Modifications of the Program JSLAB-86

Needs of Modification

A Simple Analytical Model
Accuracy of the Analytical Model
Problems in JSLAB-86

Summary

Dowel Models in Finite Element Programs for
PCC Pavement Analysis

Introduction

A Direct Finite Element Method Approach

Modified Shear-Bending Beam Model

A Component Stiffness Matrix

A Proposed Shear-Bending Beam Model (4 * 4
Stiffness Matrix)

A Proposed Shear Beam Model (2 * 2
Stiffness Matrix)

Numerical Examples

Summary

Chapter Five

Chapter Six

Chapter Seven

Chapter Eight

Chapter'Nine
Reference

Appendix A

A Nonlinear Mechanistic Model for Dowel
Bar Looseness

Introduction

Load Transfer Procedure of Dowel Bars
with Looseness

Looseness Distribution and Input Data

Numerical Examples

Summary

Comparison Between Analytical and
Experimental Results

Research on Load Transfer Characteristics of
Dowels by Keeton“”"

Formulas of Bending Moment, Shear Force and Bearing
Pressure in the Dowels Embedded in Concrete.

Some Special Considerations in Input Data
Preparation

Comparison of the Results

Summary

Impact to the Dowel Design Procedure

Current Design Procedures

Some Comments

The Equivalent Effective Length
Effects on the Maximum Bearing Stress
Summary

Looseness Effects on the Pavement Responses

Introduction

Major Findings of the Responses of a
Two Slab System

Major Findings from a Four-slab System

Conclusions and Recommendations

Conclusions

Shape Functions, Strain and Stress
' Matrices of a Plate Element

Notations

Assumed Displacement Function
Shape Functions

Elements of Strain Matrix

90
90
94
101

104
117

119

119
122
124
129
143
145
145
150
153
161
167
173
173
174
183
186
188

195

201

201
201
202
203

Appendix B

Appendix C

Appendix D

Stress Matrix

Bending Moments of Unbonded and bonded
Two-layer Elements

Undonded Case
Bonded Case

Stiffness Matrix of a Plate Element

The Virtual Work Principle
Stiffness Matrix of Top Layer SW and
of Bottom Layer sum“
8“” Stiffness Matrix of Subgrade
R“ the Equivalent Nodal Force Vector Due to Loads
E, the Equivalent Nodal Force Vector Due to
Temperature Gradient

Component Model for Dowel Bar System

The Nodes of Slabs and Dowel Bars
The Stiffness Matrix of Dowel Bar

206

209

209
210
213
213
215
219
223

225

226

226
226

Fig.
Fig.

Fig.
Fig.

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

Fig.

Fig.

Fig.

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II
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3-19
3-20
3-21

xi

LIST OF FIGURES

Physical Model of a PCC Pavement
Element Models of FEM Program

Dowel Bar Before and After Deflection
Positive Directions of Bar's Displacements
and Forces

A Beam on Extensionless Elastic Base

Analysis Procedure

Comparison of Displacements (1)

Comparison of Stresses (1)

Comparison of Displacements (2)

Comparison of Stresses (2)

Displacement Distributions for Various Half
Beam Lengths (L)

Stress Distributions for Various Half Beam
Lengths (L)

The Maximum Stresses for Various Half
Beam Lengths (L)

The Uplifted Lengths (L-l) for Various half
Beam Lengths (L)

Displacement Distributions for Various
Concentrated Loads at Ends

Stresses Distributions for Various
Concentrated Loads at Ends

Comparison of Displacement Distributions

Comparison of Stress Distributions

Comparison ofem Along AB Line, Day Time,
g=3.0 OF/in

Comparison ofeg Along AB Line, Day Time,
g=3.0 °F/in

Comparison of<m Along AB Line, Night time,
g=1.5 cF/in

Comparison ofcm Along AB Line, Night Time,
g=1.S °F/in

Comparison of Day and Night Time Displacements.

Along AB Line

Comparison of Displacements By Different
Coordinate Systems

Comparison of Stresses by Different
Coordinate Systems

Notations of Displacements and Forces of
Dowel Bar Element
Elastic Beam in Elastic Medium

25
26
31
40
40
41
41
44
44
45
45
47
47
49
54
54
55
55
57
57
59

67
67

Fig.

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

Fig.

Fig.

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
FIG.
Fig.

Fig.

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XII

Configuration of Single Axle Load: 18 kips,

p=100 psi
The Maximum Displacements V.S. Dowel Lengths
Total Shear Forces Transmitted by the Dowel Bars
The Maximum Stresses V.S. Dowel Lengths
Longitudinal Stresses Along Line A-A (in Fig. 1-1)
Shear Forces of Dowel Bars
Bending Moments of Dowel Bars on the Loaded Side
Bending Moments of Dowel Bars on the Unloaded Side

Load Transfer Procedure

Dowel Behavior in Experiments

Nonlinear Model of Dowel Bars With Looseness
Looseness Distribution Assumption
Displacement Shapes (Load case one)
Displacement Distributions (Load case one)
Displacement Distributions (Load case two)
Stress Distributions (Load case one)
Stress Distributions (Load case two)

Load Transfer Properties (Load case one)
Load Transfer Properties (Load case twO)

The Cart for Application of Wheel Loads

A Static Load Acted at the Joint

Finite Element Mesh of the Test Pavement
Bending Moments of the Dowels

Shear Forces of the Dowels

Bearing Pressure of the Dowels

Longitudinal Displacements

Average Shear Forces in the Joint

Transverse Displacements of the Joint

(on the loaded side)

Transverse Displacements of the Joint

(on the unloaded side)

Measured and Calculated Dowel's Shear Forces
v.s. Load Magnitudes

Percentage of Total Load Transferred by Five Dowels

The "Effective Length" (EL) and "Equivalent

Effective Length" (EEL)

Effects of Subgrade Modulus on the Maximum

Bearing Stress

The Finite Element Mesh and Two Load Cases

Effects of Subgrade modulus on the EEL

Effects of Subgrade Modulus on the Load

Transfer Efficiency

Relationship Between the EEL and l (the Radius

of Relative Stiffness)

Effects of Slab Thickness on the Maximum
Bearing Stress

147
152
157
158
159
160

163

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

Fig.
Fig.

Fig.
Fig.

Fig.
Fig.

Fig.

Fig.

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-1

V

xiii

Effects of Subgrade Modulus on the Maximum
Bearing Stress

Effects of Dowel Diameter on the Maximum

Bearing Stress

Effects of Width of the Joint Opening

on the Maximum Bearing Stress

Effects of Dowel-concrete Interaction Coefficient
on the Maximum Bearing Stress

Effects of Concrete Elasticity modulus

on the Maximum Bearing Stress

The Effects of k Under Different Looseness
The Effects of D Under Different Looseness
The Effects of h Under Different Looseness
Finite Element Mesh of the Four-Slab System
Shear Forces of Dowels in the Transverse Joint
(corner loading)
Shear Forces of the Tie Bar Under the Edge Load
Maximum Principal Stress v.s.Looseness
Corner loading)
Maximum Edge Stress v.s. Looseness (Edge Loading)

The Positive Direction of Forces and Displacements

Unbonded Case
Equivalent Layer

An Element Partially Acted by Load

A Beam on Elastic Base

163
164
164
166
166
175
177
179
182

184
184

185
185

201

209
211

223
227

Table
Table
Table

Table

Table
Table

Table
Table
Table
Table

xiv

LIST OF TABLES

Interior Loading
Edge Loading
Corner Loading

Comparison of the Maximum Stresses cg and cy
on Top of the Slabs

Shear Force of the Dowel Bars

Bending Moments of Dowel Bars

The Maximum Bearing Stresses by Different Models
The Maximum Bearing Stress (k=50 pci)
The Maximum Bearing Stress (k=200 pci)
The Maximum Bearing Stress (k=500 pci)

51

52

85
88

151
170
171
172

CHAPTER ONE
INTRODUCTION

1 Major Steps of Engineering Analysis

Since high speed digital computers have become more available, the Finite
Element Method (FEM) has become one of the most powerful tools which are
being employed to solve a‘wide range of complex boundary value problems in
engineering. (Zienkiewicz‘m'n and Bathem‘m) Many computer programs for
jointed concrete pavement analysis based on FEM have been developed in the
past decade. (Huang‘wm, Tabatabaie‘m", Chou“"”, Ioannides“"", Majidzadeh""“,

Tayabji“”“, Hoit“”" and Nashizawafl””)
The major steps of the analysis can be summarized as follow:

(1) Physical Model. Based on some assumptions, pavement systems are
simplified into a physical theoretical model. Using the FEM, the pavement
system can be simplified as an assemblage of finite number of plates, bars
and elastic springs interconnected at structural nodes as shown in Fig. l-
1 and Fig. 1-2. During this simplification procedure, the effects of many
secondary parameters have been ignored. For example, the variation of the
slab thickness, the non-uniformldistribution of properties of concrete and
the permanent deformation features of the soil. One engineering system
can be simplified into several physical models, more or less complicated,
depending on the needs of engineers and current level of computation

techniques.

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Fig. 1-2 Element Models in FEM Program

(2) M em t Mo Based on more assumptions, the physical
theoretical model is simplified into a mathematical model which can be
described by a group of differential or algebraic equations. During this
simplification procedure, several important parameters are selected to
simulate the original system's behavior under a certain environment. For
example, the nodal displacements are taken as basic parameters based on
stiffness method of the FEM. When traffic is the only type of loads to be
considered, the traditional Winkler equation can be used to simulate the
behavior of the base and subgrade, so that the corresponding equations can
be derived. However, if temperature gradient is considered for the
responses of the pavement, the subgrade must at least be modeled by
”extensionless springs", and another type of equations should be employed.
One physical model may also be simulated by several different mathematical
models (or different types of mathematical equations) which mainly depends
on the working environment of the pavement, the level of precision

required by engineers and the mathematical tools currently available.

(3) Engingggigg_prgg§iggg The results produced in step two will be

checked with the data collected in laboratory and field tests to verify
the accuracy and validity of the simplifications made in previous two
steps. The discrepancy between the calculated results using the
mathematical model and the data from laboratory or field usually leads to
modification of the physical and/or mathematical models or calibration of
the model's input. Engineering practice is the most important criterion
to evaluate the related analytical models. Only those models which are

theoretically correct and practically verified canlbe confidently employed

in engineering practice.

The emphasis in this research was on the second step: mathematical
modeling of dowel bar load transfer system for the PCC pavement analysis.
Attention was paid to evaluate the correctness in theory for existing
mathematical models of dowelled joints, and to verify the proposed models

by theory and by available laboratory and field data.

2 Joint Functions and Related Deteriorations

Joints are widely used in portland cement concrete pavement design and

rehabilitation. The major functions of joints are:

e To provide enough space and freedom for movement of pavement slabs
due to volume change of the concrete due to the change in

temperature and moisture content

e To effectively transfer traffic loads from one to the adjacent slab
e To control the width and location of cracking
c To temporarily stop construction

Concrete pavement joints may be designed as contraction, expansion,
construction, or longitudinal joints according to their functions.
However, the joints are usually the weakest portion of PCC pavements, and
the earliest deteriorations are often discovered near the joints. These
include faulting, pumping, water bleeding and seal damage. Many types of

distresses of PCC pavement, such as transverse cracking, corner break,

longitudinal cracking, blowup, lane and/or shoulder drop off, D cracking,
and lane and shoulder separation etc., are directly or indirectly caused
by the deterioration of the joints. These distresses caused by lack of
attention to such structural weakness often occupy most of the time of the

maintenance engineers.

The most common load transfer device is the smooth steel dowel bar. The
objectives of past research on load transfer include investigating the
effects of dowel systems on the behavior of pavement structures, cost
effective design and rehabilitation procedures for dowel load transfer
systems. Many projects have been supported by the Federal, State and
local agencies to survey the joint related deteriorations of the PCC
pavement, to discover and verify the sources of the distress, and to

develop new techniques to improve the joint capabilities. (Hveeml‘m',

Darter“”", Van Ness“”“ and Smith“”m)

3 Research on Dowel Bar Load Transfer Systems

Experimental Studies

The earliest large scale field tests of dowel bar performance were
conducted by Teller“”“, in Arlington, VA. Laboratory study to
investigate the behavior of a loaded single dowel embedded in concrete was
performed by Friberquh The first experiment of dowel load transfer
across a full-scale concrete pavement slab joint, with the dowel

instrumented to observe their behavior, including the distributions of

deflection, bending moment and shear force along the dowel bars, was
conducted by Keeton“”m””h Shortly after, another significant laboratory
research project was conducted by Teller“”“. He studied the effects of
several variables which influence the structural performance of the dowel
bars used in the joints of concrete pavement. The parameters include
diameter and embedded length of the dowel bars, width of the joint
opening, thickness of the slabs, initial and developed dowel bar looseness
caused by repetitive loading. The project was evaluated by Friberg“”"l as
a research which "Fills a gap in joint design which has existed for many
years." Ciolko“”” performed lab tests to determine the relative ability
of dowel bars and starlugs and verified many of Teller's findings.
Snyder"""’°l conducted a laboratory study which involved repetitive shear
loads to dowels anchored in hole drilled in concrete specimens and
investigated the relationship between dowel deflection, looseness(the gap
or void between dowel and the concrete hole), number of the repeated loads
and geometrical and mechanical properties of the dowels. A review on the
significant experimental projects in the past decades can be found in

Snyder's Ph.D thesis“””.

Analytical Studies

As early as 19209, Westergaardl'm' started to analyze the dowel bar
behavior based on Timoshenko theory“”“. Then, Bradbury“”“ and
Friberg“°""I performed more analysis and developed their formulae for design.

All these investigations were based on the model of an infinitely long

elastic beam embedded in an elastic medium. The deflection, bending

moment, shear force and the critical bearing stress of the concrete can be
calculated if the shear force transmitted across the dowel is known.
Based on experimental results and Westergaard's work, Friberg““m proposed
a procedure to estimate the maximum force using the assumptions that load
transfer efficiency is known, the distribution of shear forces along the
joint is linear, the maximum transmitted force is located at the loading
point and the force of dowel with distance 1.8 (I) from the loading point
is zero, where l is radius of relative stiffness of the pavement system.

Kushing““m proposed an analytical procedure to approximately estimate the

forces to be transmitted by each dowel bar.

A potential for a real breakthrough in analytical procedure for doweled
joints was created in the late 70's with the introduction of the finite
element method into pavement engineering. The earliest finite element
program for PCC pavement analysis (Huang“””) employed the concept of load
transfer efficiency to calculate the responses of slab, so that the
behavior of individual dowel bars could not be predicted. Tabatabaie"””,
Chou‘m", Ioannideslm“, Tayabjil‘m‘, Hoit“"“' and Nashizawa‘m” developed
computer programs which are capable of simulating the mechanism of
individual dowel bars. Among the above programs, COMBO (Hoitl'g’)

requires linear and torsional spring coefficients to be determined by
users as input data for dowel bars, all the others require direct input of
the physical dowel properties such as dimensions, spacing and mechanical
properties. The finite element models make it possible to directly
predict the responses of dowel bars using the analytical model employed by

the program without further assumptions. The review and comparison of

different programs can be found in Heinrichs“”“ and Smith“”m.

After looseness was observed and measured by Tellerm‘“, Majizadeh‘m"

suggested input of a smaller bar diameter to the finite element analysis
to consider the looseness effects. This method considers the reduction of
stiffness of dowelled joint as a whole, but can not simulate the mechanism

of individual dowel looseness.

Field Survey

During the past decades, many projects supported by Federal, State and
local government agencies collected field data, including detailed design,
construction, maintenance, traffic, environmental and pavement distresses,
verified the available analysis procedures and design methods,
investigated the discrepancies between the originally expected and the
actual pavement performances, and intended to find the sources which had

r“”“ developed a concrete pavement

caused the discrepancies. Darte
evaluation system based on the available data” Recently, Smith“”m
documented the performance of 95 experimental or other in-service rigid

concrete pavements and described the evaluation of various design and

analysis models and the development of improved prediction models.

10
4 Descriptions of Existing Problems

Research efforts have resulted in many significant achievements, however,
some problems still remain unsolved and new problems are continuously
brought up ‘with the application of new techniques. The following

problems, closely related to the PCC joint modeling, have been discovered:

e Smith“”m found differences, between computer programs ILLISLAB
(Ioannides“”“) and JSLAB (Tayabji“”“), in computed deflections of up to 20
percent due to traffic loads, and differences of up to 100% due to thermal
gradients - even though the two finite element models were based on the

same assumptions to deal with slabs resting on a Winkler base.

e Guol'ml discovered that the dowel bar stiffness matrices used by JSLAB
and ILLISLAB do not meet some of the basic requirements of the finite
element method. The stiffness matrices represent elements that are not in
equilibrium (e.g., an assumed rigid body movement vector produces non-

zero-element forces).

e Based on Friberg's research (Friberg““”), dowels at distances greater
than 1.8 (I) from the point of application of the external load were
inactive, and 1.8 (l) was defined as "effective length" of the load
transfer system. Tabatabaie“”” and Henrichs“”” concluded that the
effective length should be 1.0 (1) according to their results produced by
program ILLISLAB. In dowel bar design, the diameter is dominated by the

maximum concrete bearing stress which is in proportional to the effective

11

length. (if the total of the loads transmitted across the joints are the
same.) The difference of assumptions 1.8 (l) and 1.0 (1) would affect the

dowel bar design significantly.

0 Faulting is one of the most critical distresses affecting the
performance of rigid concrete pavements and is directly (Henrichs“”") or
indirectly (Smithflwm) determined by the maximum concrete bearing stress
under the dowels. The discovered problems in the dowel bar stiffness
matrix used in finite element programs would influence the prediction of
pavement faulting. It would also affect predictions of other pavement

performance measures.

s Many experimental studies show that dowel bar looseness greatly affects
the load transfer efficiency which is an important index to determine the
quality and capability of the dowel load transfer system. Teller“”” also-
found 40,000 load cycles (2% of 2,000,000 total cycles in his tests)
produced about 50% of total looseness. That suggest most pavements
currently in service are working under a certain looseness. The existence
of looseness not only affects the behavior of joint, but also affects the
responses of slabs and other pavement performances. So far, however,
there exists no mechanistic model to simulate individual dowel bar

behavior under looseness.

12

5 Research Objectives and Scope
291mm

The purpose of this study is to advance the state-of—the-art of the

mathematical modeling of dowelled load transfer systems, by:

e Establishing a theoretically correct and practically adjustable dowel
bar model for finite element programs, instead of the inappropriate ones

employed by current programs

0 Developing a new’mechanistic model to simulate the looseness mechanism
of individual dowel bars and making the looseness simulation level
consistent with the dowel bar simulation level in finite element programs

for PCC pavement analysis.

e Investigating the impact of the new models on the pavement analysis,

design and performance predictions.

Scope

e All analysis and verification in this project was conducted using
available computer programs so that the reliability of the program was

extremely important for obtaining meaningful results. The first task was

13

to evaluate the reliability of the selected computer program JSLAB, and

correct all discovered errors.

e .Accumulated information from existing studies on modeling the dowelled
joint was reviewed to verify the discovered problems on dowel bar
stiffness matrices currently employed in some widely used finite element
programs. Detailed derivation was conducted to reveal the existence of
the errors. Numerical examples are given to demonstrate the inaccuracy of

the results produced by using the questionable dowel bar stiffness matrix.

e Some available experimental data are used to compare the analytical
results produced by the proposed model, including bending moment

distribution on the dowel bars.

s (A model to simulate the looseness mechanimn is proposed and the
corresponding numerical iteration procedure was designed to perform the
simulation. Numerical examples are given to compare the response of

dowelled concrete pavement with and without dowel looseness.

e Analyses were conducted to investigate the impact of the new dowel
model to the design of dowelled joints and pavement thickness designs.
Both theoretical and numerical sensitivity analyses are conducted to
determine the effective length of dowelled joints. The impact of the

findings to joint design procedures is also discussed.

l4

s An effort was made in numerical calculation to use the dowel looseness
model to quantitatively understand the effects of dowel looseness,

including: critical stress and location of the slabs, performance

prediction models of the joints, etc.

15

CHAPTER TWO
BASIC ELEMENTS IN FINITE ELEMENT PROGRAMS

The matrices for basic elements in finite element programs are given in
this chapter for evaluating the reliability of JSLAB-86 (Tayabji“”a). The
rectangular plate element in Fig. 1—2(a) which contains a pavement slab,
stabilizing base and subgrade, is modeled by top layer and bottom layer
plates, and extensionless distributed springs. The bar elements in Fig.
1-2(d) are used to model dowel bars with consideration of dowel - concrete
interaction. Aggregate interlock and keyway are represented by spring
element in Fig. l-2(c). These are the physical models of PCC pavements.

The mathematical models are given below.

1 Plate Element

The stiffness matrix of a rectangular and bending slab (Zienkiewicz“”") is
based on classical small displacement theory for thin plate with uniform
thickness. At each node of the element in Fig. l-2(a), there are three
displacement components: a vertical deflection W in z «direction, a

rotation 6,‘ about the x-axis and a rotation 6, about the y-axis in Fig. 2-

2(b).

A polynomial in terms of 12 parameters is used to define the displacement

function as follow:

16

"(x' Y) =51*azx*r33Y*atxz+asxy+asy2+avx3+agx2y

2 3 3 3 (2-1)
*agxy +amy +anx y+a12xy
or:
W = 4) a (2-2)
(1x1) (1813) (1281)
where:
¢=[l xyx2 xyyz x3 xzyxy2 Y3 X’yxy3]
(2-3)
a = [51 ‘32 as 34 a, as av as as am an amp-
At any point within the element:
W
W
e -59.! (2-4)
A = x = a}’
(3:1)
6. a
6x

Twelve simultaneous equations for a plate element with 4 nodes can be

written in matrix form:

2-5
V°==A.a ( )
where:
A1
v- = A: (2-6)
Inn) A,
A

and A is a 12x12 matrix in terms of nodal coordinates. Inverting Eq. (2-

5) to obtain:

17

a =A‘1 1"

Substituting Eq. (2-7) into Eq. (2-2) to obtain:

w=¢ a" V‘=N v'
(1x12) (12:12) (12:1) (1312) (12x1)

where N is a 1x12 vector of the shape functions. Their

(Zienkiewicz“”" and Ioannides“””) are shown in Appendix A.

A strain vector due to W at a distance 2 from the mid-plane is:

he.“
ax2

_52w =2 Kc =2 8 V'
(:31) Y dy‘z (3,,” (3112) (12x1)

”' 2 62w
\ aan)

 

 

 

where, Kcis a curvature vector and B is a 3x12 strain matrix:

xc=sv'

 

( _ (3le
6x2

_ am
dy’
am

\2—‘axayJ

 

 

 

(2'7)

(2‘3)

expressions

(2’9)

(2-10)

(2-11)

The stresses including the contribution by temperature gradient can be

expressed by strains (Przemienieski"°°"):

where: E

The moments in each layer can be defined as the following:

Integrating Eq.

where:

18

o 1 P 0 e

a . ,, a E n 1 0 e Eh

 

 

y
tun) 1-p3 1'! l-u
1&7 00 2 Y O

= C e -+ de
(383) (381) (m)

Young's modulus of the slab
Poisson's ratio of the slab
Thermal coefficient

Temperature change

 

1
Eu
1-
” O
1 u o
c= E.‘ #1 0
1-u2 1‘!
2

Mk
Us My a=fzadz
n
”8'

(2-15) over the thickness h leads to:

Iran x,+xc=p s v'+x,=x were

(2-12)

(2-13)

(2-14)

(2-15)

(2-16)

l9

1
Mt = -—B'5- 1 szdz
1-p o h

(2-17)

If the temperature variation along slab thickness h is assumed linear and
the temperature on top and bottom surfaces of the slab are T' and T"

respectively, M,in Eq. (2-17) can be obtained:

 

1 1
_ Eah’ 7“-T” 1 2 Eah3 (2'18)
‘ 12(1—u) h d 12(1-u)
where:
/ _ /
g : LIT-7: (2-19)

9 is defined as temperature gradient in pavement, daytime case (the top

surface is warmer than the bottom) is defined as positive.

D in Eq. (2-16) is an Elastic Matrix. Two cases, fully bonded and fully
unbonded two layer systems are considered in JSLAB. Their detail
derivations are given in Appendix B. The major conclusions are listed as

below. For the unbonded case:

Dthop + Dbocton

2-20
1 p 0 ( I

Erh3 p 1 0
layer 12(1-p2) O O 1"}:
2

Where the summation includes both of the top and bottom layers.

20

For the bonded case:

%hc(hc+hb)

 

ht + _b hb
Et
a:c =% (h, + hb) -ab (2-22)
D = Drop + Dbottoe
1 u 0 (2-23)

8(12a2h+h3) P 1 0
layer 12(1-I‘2) 0 0 _El-
2

where a( and ab are distances from the mid-plane of the top and bottom

layers to the neutral axis of the equivalent cross section respectively.

.(See Fig. 8-2, Appendix B).

The general "stress" matrix (Zienkiewicz“”") in Eq. (2-16) can be written

as:

(2-24)

Eq. (2-24) offers a transfer matrix between nodal displacements and
Bending Moments for elements. R and B are given in appendix A. After

bending moment M is obtained, bending stress a can be calculated by

following expression:

12M (2-25)

 

where z is measured from the plate neutral-plane and its positive
direction is given in Fig. 1-2(b).
There are three nodal forces corresponding to the three displacements

given in Eq. (2-4):

Pr
P" FM:
pay

(2-26)

The stiffness equations of a plate element in Fig. 1-2(a) are derived by
the virtual work principle as shown in Appendix C. The results used in
computer program are exhibited as follow.

For each plate element:

(3“, + 3mm. + Sm) V‘ = P, + P. = p. (2-27)
where:

SW Stiffness matrix of the top layer

Sb”. Stiffness matrix of the bottom layer

8“. Stiffness matrix of the subgrade

Pd Equivalent nodal force vector due to external applied
loads

P. Equivalent nodal force vector due to temperature
gradient

P‘ Total equivalent nodal force vector of the element

22

Their expressions are given below:

Sta, - ffsres 8T Dc“, 8 dXdy

shun-==f 5“22Nmu-£’dxdy
RIC.

em = If“... kNTNdxdy

Pd [fun p(x,y) NT dxdy (2-28)

P:

-lf.,.. .7 ., M

where p(x,y) is intensity of the applied loads and k is modules of
pavement subgrade. All element formulae in above equations are derived

and listed in Appendix C. Thus, the stiffness matrix of plate element

are:

(2-29
‘9. = Stop + Shorten + Ssub )

23

2 Bar Element

The dowel bar system can be modeled by three segments of beams: two
bending beams embedded in concrete, Cib and jg) in Fig. 2-1(a), and one
shear-bending beam in the joint, igh in Fig. 2-1(a). Before pavement
being loaded, the nodes of slab and dowel bar, i, and i. or j, and j" are
assumed identical. However, after the pavement being loaded, they are

separated as shown in Fig. 2-1(b).

The stiffness matrix of beam igh can be written as:(Przemieniecki“””)

 

 

12 61 -12 61
8 - EI 61 (4+¢)12 -6l (Z’OIIZ
” m -12 -61‘ 12 -61
61 (2-¢)12 '61 (4+¢)12 (2_3°)
Dlx 02x -Dlx DZX
_ D2x 03x mm: 04:: . 3n 5::
-Dlx -DZx Dlx -DZx 8,1 3::
02x 04x -DZx 03)!
Where: E Elastic Modulus of the bar
I Moment of inertia of the cross section
1 Length of the bar (or width of the joint)
:1: 24(1+u)I/A‘12
A. Cross-sectional area effective in shear
u Poison ratio
on: =- mar/Paw)
02x . ear/Paw)
03x - (4+¢)EI/l(l+¢)
04X = (2-¢)EI/l(1+¢)

24

11

DIX'DZX
DZX'D3X

 

_ —Dlx 02x
-DZX'D4x

5%1=‘3§

Dlx -D2x
Sn =

-DZx 03x

The force and displacement vectors are defined as:

(2-31)
Pb = [Or ”I Q: My];

, 2-32
vg==pn 6,99 eflb ( )

Where w and 6 are vertical and rotational displacements, and Q and M are
shear force and bending moment at the bar nodes respectively. Their

positive directions are defined in Fig. 2-2.

Most dowel bar models employed in currently available computer programs
are based on Eq. 2-30 (Tabatabaie“”“, Majidzadeh“”“, Tayabji“”“ and
Nashizawa“""). Different authors modified Eq. (2-30) using different
assumptions to consider the interaction between dowel bars and concrete.

Guo“”” has found that the dowel bar stiffness matrices employed by some

25

 

 

 

 

 

 

_§L€lb Bar
,"/
"f l / .
o lily " lsgib D
_$( ........... ,2 ........... £7 .......... j 5F.
l Joint

 

 

 

 

 

 

(b) After Deflection

Fig. 2-1 Dowel Bar Before and After Deflection

26

 

S
P
.2
0

U.

Fig. 2-2 Positive Directions of Bar’s Displacements and Forces

computer programs for jointed concrete pavement analysis, including JSLAB-
86 and ILLISLAB, failed to satisfy the equilibrium condition which is one
of the basic requirements of the FEM. A component dowel bar model has
been proposed to simulate the behavior of the dowel bar load transfer

system. The detailed discussion is given in Chapter 4.

3 Spring Element

Spring elements are used to model aggregate interlock and the keyway in
which only the vertical forces are transferred across the joint. At each

node of the spring element shown in Fig. 1-2(c) , only one displacement

27

unknown, vertical displacement W in z direction, is defined. The nodal
force corresponding to the vertical displacement is vertical shear force

Q. The stiffness equation is:

0: (SP '3P)Wi _ (2-33)
[0,.) S? (t) - v

The stiffness matrix of a spring element can be written as:

.9 _(3p -3p] (2-34)
’ -sp SP

where SP, which may be defined as the equivalent spring stiffness for
aggregate interlock or keyway, can be obtained by laboratory and/or field

tests.

28

CHAPTER THREE
MODIFICATION OF PROGRAM JSLAB-86

1. Need for Modification

Many computer programs are available for analysis of jointed concrete
pavements for both design and research purposes. Because these programs
were developed independently, it is necessary to evaluate their purported
capabilities and assess their accuracies before using them in engineering
practice. Recently Smith”"" and Mueller“”” presented their evaluation

for many programs popularly used in the United States. .As stated by
Mueller“”m, some widely used programs, such as ILLISLAB (Ioannides“”")

and JSLAB (TayabjiI'm'), both based on the finite element method and

employing the same assumptions, produce results that are significantly
different. For example, the differences in predicted maximum
displacements can be as great as 20 percent, and the difference in

predicted thermal stresses can exceed 100 percent.

It can be expected that all correct computer programs employing the same
theory and assumptions should produce reasonably close results. The most
efficient means to check the accuracy of a program is to compare its
results with those produced by using a precise analytical procedure. A
simple analytical procedure is proposed to calculate the response of a
beam resting CH1 Winkler elastic base under uniformly distributed and
concentrated loads, temperature gradient or both. The major advantage of

this model is to provide solutions as precise as desired when a portion of

29

the beam is separated from the base. Separation cases are conunonly
encountered when predicting thermal stresses of concrete slabs on
elastic base. Timoshenkom‘" and Westergaard's models (Yoder‘m‘l) are
unable to treat the case accurately, since they assume that "extension
springs" still exist between the separated portion of the beam and the
Winkler base. In fact, there is no interaction between the beam and base
when they are separated fromleach other. ,An numerical iteration technique
is employed by almost all computer programs to approach the final results.
However, the correct results will never be received unless the contact
condition is appropriately defined and the iteration loops are logically
written. A precise and closed form solution, even if only one dimension,

would be very helpful to validate these computer programs.

In this chapter, the analytical model is first developed, and detailed
derivations are given. Numerical demonstrations are given to prove the
accuracy of the analytical results through comparisons between results
calculated and those by other theoretical models. (Timoshenkom‘u and

Yoder“””.) More numerical examples are presented to briefly introduce the

features of the responses due to combinations of loads and temperature
gradient. Then the proposed analytical procedure is employed to detect
problems in computer program JSLAB-86. Finally, numerical examples
indicate that the modified JSLAB, JSLAB-92, produces responses of single
slab under different traffic loading identical to those produced by
ILLISLAB. The discussion of dowel bar modeling problems discovered in

JSLAB and ILLISLAB is presented in Chapter 4.

30

2. A Simple Analytical Model

Basis model assumptions include:

(1) The material behavior of beams is linear elastic

(2) The intensity of base reaction is proportional to the deflection at
any section which is in contact with the base

(3) The intensity of the base reaction is zero at any section which is
separated from the base

(4) The temperature variation is linear along the thickness of the beam

(5) The beam, loads and temperature gradient are symmetrical to y axis

(see Fig. 3-1).

Although the beam and base are both assumed elastic, the responses of the
system (beam plus base) under loads and temperature gradient are nonlinear
since the contact length between beam and base always varies with the
temperature gradient and loads. The major difference between the defined
problem and the solved problems in many classical textbook, such as
"Strength of Materials" by Timoshenko“””, is the added third assumption.
The classical theory assumes the intensity of the interaction forces
between the beam and base are always proportional to the section
deflection, whether they are in contact or separated from each other. The
classical theory is accurate enough to predict the responses of slender
beams on elastic bases due to loads, since the weight of the beam usually
causes contact between the beam and base when additional loads are added.

However, the third assumption must be added to deal with the responses, in

 

31

 

 

 

 

 

 
 

 

M Q V Q+dQ M+dM

~-—__._--... __.,_,.-_.___,__ .__. .._.-_

t A i \p
i . ._ .. _ 3 J

__ - ,_ Ia-.v_l - 1.-.... -._-

—KY

All notations presented are positive

Fig. 3-1 A Beam on Extensionless Elastic Base

32

case the beam and base become significantly separated.

W

The coordinate system and assumed positive notations are presented in Fig.
3-1 where coordinate XOY is defined at the center horizontal line of the
beam before its weight is acted on, and coordinate XNLY} is defined at the
center horizontal line of the beam after its weight is acted on. The
following discussions are based on coordinate system XOY unless
specifically mentioned.

The basic equations for contact portion of the beam can be written as
(Timoshenko““”):

d‘y

dxd

EI

 

+Ky=-q (-1 ers 1) (3-1)

However, the equation for separated portion of the beam should be:

——4=-q (-1 > x or) x < 1) (3-2)

In Fig. 3-1 and above equations:

1 is a half of length of contact portion of the beam;
E is elastic modules of the beam material;

I is bending inertia moment of the beam section;

9 = AT/H, is temperature gradient;

H . is thickness of the beam.

K is base coefficient of the beam.

33

The beam responses due to a combination of temperature gradient and loads
can be calculated by two steps as shown in Fig. 3-2.

Where:

a is thermal coefficient of the beam material.

The first step is to add two artificial rotation constraints at each end
of the beam, and apply a temperature gradient. In this case, the beam
ends will be acted by two bending moments (M = EIag) caused by the assumed
temperature gradient as shown in Fig. 3-2(b). The second step is to add
two moments M = EIag with directions opposite to the ones in step one and
to add other loads as they are, as shown in Fig. 3-2 (c). The total
response of displacements and forces in Fig. 3-2(a) is the sum of

responses shown in Fig. 3-2(b) and (c).

W

If only a uniformly distributed load to simulate the self-weight of the
beam and symmetrically concentrated loads act on the curled area (-1 > x
or x > 1), the problem can be greatly simplified. The general solution of

Eq. (3-1) is:

y=e°"(A cosBx+3 sinpx) +e'“"(C cosBx+D sinflx) -% (3-3)

Where:

g = L_£_)°JS (3’4)

The following conditions can be used to determine the constants in

Eq. (3-3):

34

P
Load q and temperature gradient 9
A t / H=g q Q
are?
. ,"Y///// \ l 1;; JV; ”L

 

 

 

 

 

 

 

////7

(8)

added rotation constraint

 

/.

 

A

r )f
/

 

 

Fig. 3—2 Analysis Procedure

 

 

35

y<-x) = Y(x) ‘3“)
2 ' ” 3-6
M(x=1)=EI%x% (x=1) =-—g(L—1)2-§ P1 (d1-1)+EIag ( )
d3 "
le=1) = 157—de2I = qlL—l) + gr; (3-7)
I N
[o ykdx = -qL - ,2 P1 (3-8)
y(x=1) = o (3-9)

where N is the total number of concentrated loads acted on the curled
portion of half of the beam, and diis distance of the ith concentrated
load to the center of the beam (origin of the coordinate system, Fig. 3-

1(a)).

It can be easily proved that Eq. (3-7) must be satisfied if Eq. (3-8) is
satisfied. Therefore, Eq. (3-7) will not be employed in the following

discussion. Substituting Eq.(3-3) into Eq. (3-5), we obtain:

A = C (3-10)

36

B = ~13 (3-11)

Substituting Eq. (3-10) and Eq. (3-11) to Eq. (3-3) to obtain

displacement:

Y = A (el’x + 9"”) cosBx + B (6” - e"")sinflx ‘ q (3'12)

Two unknowns (A and B) are included in Eq. (3-12) and they can be solved

by substituting Eq. (3-12) into Eq. (3-6) and (3-9). Then A and B can be

derived as:

 

N
B Q'a+d[t‘-q'(L‘-1’)2-E 2P;(dI-1')] (3-13)
_= 1'1
H a2 4, d2
.4 = i - - 1_9 (3-14)
I! dlq Ha)

and 1' can be obtained by substituting Eq.(3-13) and Eq.(3-l4) into
A B "
Fll‘) = (b+c)7!+(b-c)-fi+2(L‘-1‘)q‘+; 2p; = 0 (3-15)
n

Where:

37

sin 1' (el°-'e'ld .

a =
t): sin 1‘ (e).+ e‘ld
C.= cos 1' (e). - e‘ld
c1: cos 1' (el'+ e'IW

1:431
L°=pL
.= _E.
q M!
s pa’
Pi:_“_B.

KH
d;=‘fid1
t'= “9

ZBZH

Substituting l'= Bl, received from Eq. (3-15), into Eq.(3-12) and dividing

by the beam height H obtains the non-dimensional displacement of the

38

contact portion of the beam as:

£=§Cosflx(ebx+e'px) +£Sian(efix-e-fix) “q. (3-16)

( -l < x < l )
The curled portion of the beam is statically determinate and the

displacement can be derived by fundamental beam theory as:

N
1:321“ -1! . .2. -1 s -5 -1 -1 ._
H (1 L) +3(1 I) (1 ) 2(1 I) 2;}?!

 

6 L
(3-17)

+é- 2+ -‘+"2§—12
[H(C'b)+1q(clb)](fix 1 ) r L (L ‘L)

( -l > x, or x > 1 )
Where Yn is displacement due to the ith concentrated load. If only one

concentrated load acts at end of the beam, the following formula can be

used:
ZN: Y =3P'L‘3 l2 (1-ll3-3 (1-llzll-5) +(1-3-(l3l (3-18)
bl P: 3 L L L L
where:
p. = a
kH (3-19)

The bending moment of the contact portion of the beam is:

dzy

r4: EI (3‘20)
dX

 

- EIag

39

Substituting Eq.(3-16) into Eq. (3-20) obtains the non-dimensional bending

moment as:

#2]??- [-%sinfix(e”-e”’) +gcosfixle’x+e"xl ] -t:' (3-21)

( —1 < x < l )

The curled portion of the beam is statically determinant so its bending

moment can be written directly:

M- -——q(L'X)2 - N P (d- 3 22
- 2 z; 1 1 x) ( ' )

( -l > x, or x > 1 )

Its non-dimensional form is:

N
M e e e . (3-23)
—————— = — (L — x)2- 2P (d - x)
kH/zp2 q B ,2 1 1 l5
3. Accuracy of the Analytical Model

The accuracy of the analytical model can be proved by checking its
assumptions and derivation process step by step. An alternative is to
compare its results with those obtained by some well known models

developed under the same conditions. Fig. 3-3 to Fig. 3-6 present the

DisplacemenIs (inches)

0.006
0.005
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003

-0.004
0

250

200

40

 

 

 

 

: l. a :12 Inch I W
I - I l
Foundation Cos“. = 200 cc:
_ ,1
fp=tOOIx Mo=50000b-in ! if
1.
1‘1-
1

 

 

 

 

 

 

 

 

  

 

 

 

l
p

---__ _--'.._ _.. .____....___,_.

0.72 0.4 0.6 0.8 1
r/L

 

 

:r .‘ Thuoshcnkdllllll 4" 1116 Developed 310ch l

 

Fig. 3-3 Comparison of Displacements (1)

 

 

 

150

Stresses (psi)
0
O

   

 

[——
L ‘ 5‘2 itch
Loundolion Cos". = 200 Del =
p a ‘00 s. No .-. 50000 b-n ' .l...... :

 

 

 

-50 1-,L_._._..w-___.--_-,.__,-_-_,_. - _ ...-_ ___._._

 

 

 

0 0.2 0.4 0.6 0.8 1.

x/L

 

l7—m—7EQLQJRHT9'IH’1‘fififiéfifi Model l

 

Fig. 34 Comparison of Stresses (l)

Displacements (inches)

0.008

0.006

0.

0.

41

 

 

004

002

foundation Coslt. = 200 ad

p e :00 a. lo a 30000 s-s

 

  
 

     

       

 

 

 

 

  

 

 

r/L

 

fm- nmmhenkalgm + The Developed Male ]

 

Fig. 3-5 Comparison of Displacements (2)

 

 

 

 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

160 p:
140 to
00 w ' a
-.
80 l a __.-.._’"
; foundation Coett. = 200 Del .
60 . "S -, _.
4O
20
O 3“ r T a ,
0 0.2 0.4 0.6 0.8
x/L
(i.— Y‘Xk'11975l “I’- Thr Developed Model}

 

Fig. 3-6 Comparison of Stresses (2)

42

comparison of displacements and stresses received by the developed
procedure and the models given by Timoshenko“°‘" and Westergaard (Yoder‘m’l) .
A weightless beam with half length L=512 inches is used, since the models
developed by Timoshenko and Westergaard are for infinite beams or slabs.

The input data are as follows:

Beam size: D = 7.5 inch
H = 10 inch
Material properties: E = 5000000 psi
u = 0
k = 200 pci (K = 1500 psi for beam)
a = 0.000005 1/°F
Loads: Mo = 30000 lb-in, same direction to the moment M

in Fig. 3-2 (c), acted at the ends

P = 400 lb, downward, acted at the ends

Fig. 3-3 and Fig. 3—4 show the resulting displacements and stresses. Since
the assumption of "extension springs" applies even if a portion of the
beam is separated from the base, the results by Timoshenko are smaller
than those generated by the developed analytical model. The differences
between the displacements near the beam ends are even more significant.
Fig. 3-5 also presents the displacements produced by Timoshenko model and
the model presented in this chapter, except using a shorter half length,
L=128 inch, instead of L=512 inch. As shown in Fig. 3-5, the difference

increases as the half length decreases.

43

Fig. 3-6 shows the comparison of stress predicted by Westergaard formula
(Yoderl'm') and theimodel in this chapter. A night time temperature
gradient g=1.5 °F/in is applied, and the unit weight of the beam is
assumed 0.09 pci. No additional loads act on the beam in this case. It
can be seen that Westergaard's results overestimate the stress responses
since it was also developed based on the assumption that "extension
springs" exist, even if a portion of the slab is separated from the base.
However, the responses of the location far from the edge, 'x/Ll < 0.7,are

identical.

Figs. 3—7 to Fig. 3-10 show the effects on beam length to the thermal
response of beam on an elastic base. The night time temperature gradient

is 1.5 °F/in, and the unit weight is 0.09 pci. The other parameters are

the same as listed above in this section.

Fig. 3-7 presents the deflected shape of the beams with various half beam
lengths. When the length of beam is long enough, the displacements of
interior portion of the beam approach the average vertical settlement of

the beam due to its weight. In this case:

d = ~77? = ~9'—0§—6—0‘T—1—9 = —0.0045 inch

Thus, the shorter beam would have larger settlement in the middle.

Fig. 3-8 also presents the variation of stress distribution of beams with

various half lengths. It can be seen that the effect of beam length is

44

 

 

0.015
0.01 ,g I”

 

 

 

l Lat—4 g" .4:

 

Displacements (inches)

 

 

 

 

(ti—imam in «o— esre in «~— [.2256 m ‘
We} 1:120 in ~x—- 1:64 in |

4

Fig. 3-7 Displacement Distributions for Various Half Beam
Lengths (L)

 

 

 

 

160 A RX 2:1 1;
‘ if?” 1 X 1
120 f 434.:
(WWI. 9:1.ST/h I ‘x, ’- §
80

Fmstdolion Cosllicienl = 200 pci i \ X
- .wr. _fl_ 7 .. *

0:2 0.4 0.6 0.3 1
x/L

i —m- 1:1024 m + 1:512 in ~=+'-— l.:256 in I
:43- mza m 961:0: in 4

1

 

 

 

Stresses (psi)

 

 

 

 

 

4*
o

 

Fig. 3-8 Stress Distributions for Various Half Beam Lengths (L)

115

 

 

240

200

 

 

 

O)
O

 

.wfilm. 9:1.Sf/in I

 

Stresses (psi)

N
O

 

[Foundation Coo“. :. 200 oci I

 

 

80

 

 

 

40

128 192 255 320 384 448 512
L (incn)

Fig. 3-9 The Maximum Stresses for Various Half Beam Lengths (L)

32

UPLuIED LENGIH fin)
N N N
o # m

_s
O")

—l
N

 

 

 

 

 

 

 

 

 

 

 

 

C
j . 13...... . ....,TI ,
L LI"-.. __ _, _,_____l . fi ‘
I ’ W
i I
l g
j %
‘ ./
+7 ¢_7 f, . j
0 64 128 192 255

L (inch)

Fig. 3-10 The Uplifted Lengths (L-l) for Various Half Beam

Lengths (L)

46

very significant. The stress at the middle of the beam approaches a
constant as the length of the beam increases to infinity. The constant is

just the precise interior stress given by Yoder‘m’l for infinitely large

slab with the assumption of poisson ratio u = O:

6 -6
a = Eagh ==5t10 *5*10 *1'5*1°=187.5 psi

interior 2(1_“) 2*(1-0)

 

Fig. 3-9 presents the maximum bending stresses versus half beam lengths L.
The maximum thermal stresses remain constant if the beam's half length is
larger than 256 inch (entire length of the beam is longer than 42 feet).
When the half length of the beam is approximately 176 inch, the thermal
stress of the beam is at a maximum value. It can be seen from Fig. 3-8

that the maximum stresses are not always located at the center of the

beam.

Fig. 3-10 shows the length of the uplifted portion versus beam length. It
is reasonable that the uplifted lengths will not be changed if the half
length of the beam is equal to or longer than 176 inch. The 128 inch long
beam would have maximum uplifted length when all beams meet the same

temperature gradient.

A group of displacements and stresses for a 144 inch long.beam (72 inch
half length), subjected to 3°F/in night time temperature gradient and
resting on k=300 poi elastic base, are presented in Fig. 3-11 and Fig. 3-
12.~ Load P is acted on two ends of the beam with a unit weight of 0.09
poi. The results reflect the response properties of a beam with finite

length and Subjected to a combination of uniformly distributed load,

47

 

Concentrated Load P -.

HdtBeamL th=72 inchh . ,1
Lm—_‘ 30° ‘b ' /
0.015 ‘ 5‘“le ,

 

 

O
O
N
W’At
g o
a 5'
k
_\L
I

 

 

Ttemperature Gradient = 5 F/in

  
 

 

 

 

rannotation Caett. = .500 pct

 

 

 

 

Displacements (inches)

 

 

 

 

 

 

Fig. 3— ll Displacement Distributions for Various Concentrated
Loads at Ends

 

 

 

 

 

 

 

 

350
Concentrated Load P =
f t
I t
300 gas: I
g :50 b l
I 300 b |
‘ ISO b
2 so .A 7‘ u .
\ \\\ k
r; ‘X. A ._ [Holt Boom Length = 72 inch
3.200 » y
m \‘wr X.‘ A iorrperature Gradient = 5 F/in
a -.L ‘ \ \‘ ..
m ‘x Foundation Caeit. = 300 oci
“a! 150 ~._—--—*~—~ ‘7— “-
m \-. \C:

 

 

 

 

 

 

 

\\+‘\i T\\:: \
50 \u\
‘-\\
‘.
O a r x
0 0 2 0.4 0 6 0.8 I
x/L

Fig. 3-12 Stresses Distributions for Various Concentrated Loads at Ends

48

concentrated load, and temperature gradient. The results can be used to
check any existing finite element program for analysis of concrete

pavements.

4. Problems in JSLAB-86

As mentioned in the introduction, JSLAB has been evaluated by Smith?”” and
Mueller“”°'. JSLAB was selected to conduct the research on dowel bar

mathematical modeling because it is a well organized and user friendly
software, and is not a copyright reserved program. It will be beneficial
for all users of JSLAB if its reliability can be fully studied and
improved. Furthermore, there existed disagreements in evaluations of

JSLAB. It would be helpful for all to find the appropriate answer.

8 t ice I l s s in J -86.

Fig. 3-13 presents the displacements produced by JSLAB-86 and the
analytical model. The analytical curve is one of the six shown in Fig. 3-
11. The input data used for executing JSLAB-86 are the same as mentioned
above, and Poisson's ratio was taken as O for the one dimensional problem.
The two curves are identical. However, the consistency of displacements
does not prove there exist no problems. After carefully checking
subroutine ELEM of JSLAB-86, it has been found that S(lO,8), S(ll,7),

S(ll,9) and S(12,8) in the subgrade stiffness matrix are the same

49

 

0.011

Hdt Beam Length = 72 inch ‘I
0.005 '-

 

 

 

 

 

  

 

W "II. 9 : 5 F/‘n !
A
3 Foundation Coett. = 300 pci
f2 0 .
c m
In
E
o
E.
g -0.005
‘6.
.9
O

 

 

_.a JL.

-0.015 r f I .
0 0.2 0.4 0.6 0.8
x/L

 

ii—THISPM’FR+ ISIAH fl

Fig. 3-13 Comparison of Displacement Distributions

 

400

 

 

 

 

 

 

 

Stresses (psi)

 

 

 

-200 x Q - SKX/L)  .” ,1

 

 

W ° ' 'L ”

-300 'I 7' .
[”0” 390'" LW'N' - 77 inch Ilfoundjtian Coelt. = 300 pci 6' j

—400 0 I

11
I ~I~ tSIAB-as «:51 mm 12mm Model +3- -JSI.I.B—86 ]
l l

 

 

 

 

 

 

 

 

Fig. 3-14 Comparison of Stress Distributions

 

SO

“93‘". By using

in.magnitude but different in sign to those used by Ioannides
virtual work principle as a check, it has been verified that the results

used by Ioannides are correct.

Tables 3-1, 3—2 and 3-3 show the results obtained by JSLAB-86, JSLAB-92
and ILLISLAB for interior, edge and corner loading conditions
respectively. The input data and finite element meshes are the same to
those used by Ioannides “”“, page 107, 149 and 170 respectively. Although
the results of ILLISLAB were obtained by using a mainframe computer with
double precision and those of the JSLAB-92 by 486 PC computer with single
decision mode, they are nearly identical. Therefore, the following

conclusions may be obtained:

(1) The errors in the subgrade stiffness matrix of the original JSLAB
causes small differences from the modified JSLAB and ILLISLAB for a

single slab system under traffic loads.

(2) The JSLAB-92 program produces the same results as ILLISLAB does, so
it is concluded the both are credible to predict the responses of a

single slab under traffic loads.

(3) Single precision is applicable in PC computer programs to provide

sufficiently accurate responses.

51

Table 3-1 Interior Loading (Displacements: inches, Stresses: psi, on top)

pc ) sp acements

*

*
O O O

*

see next 8 on.

stresses n S S ncorrect,

    

            

s gn a

Table 3-2 Edge Loading (Displacements: inches, Stresses: psi, on top)

(pc ) H( n) Max D sp acements Max Stresses
JSLAB .M-JSLAB LLIS -J

 

52

Table 3-3* Corner Loading (Displacements: inches, Stresses: psi, on top)

pc ( n ax sp acements resses
S

stresses 0

 

       

oes not ave ca ty to ca cu ate pr nc pa

The sign of stresses and the thermal stress formula

Fig. 3-14 illustrates the stress distributions obtained by JSLAB-86 and
the analytical model. The negative values of JSLAB-86 curve plus a
constant are close the results prediced by the analytical model. The zero
thermal stress at x/L = 1 was predicted by JSLAB-86 since it was
determined by boundary conditions directly rather than the thermal stress
formula. After comparing the difference, the symmetrical feature can be
clearly seen. The following notations are used for convenience. I
%(x) stress calculated by JSLAB—86

34x) stress calculated by the analytical procedure

The relation between S(x) and 34x) can be written as:

saw.) = -stx> +A ‘3'“)

Where, A is a constant of 375 psi indicated in Fig. 3-14.

53

As mentioned in section 2 of this chapter, the total response (including
displacements and stresses) should be the sum of responses given in Fig.
3-2(b) and (c). The negative sign in Eq. (3-24) indicates that the sign
of stresses calculated by JSLAB-86 was not correct. The constant A
indicates that a part of the stress was lost. Carefully checking Fig. 3-
2, the stresses due to (c) are not constant, but the stress due to (b) is

constant:

A: EIag= £0ch= 5:106:5r6t3 :10
DH2 2 2
6

=375 psi

 

Therefore, two problems have been discovered by using the model developed
in this chapter: incorrect sign of stress and a term missing in
calculating thermal stress. Fig 3—14 shows that the JSLAB-92 provides

the same results as the analytical ones.

The sagas-9; can calgulgte correct thermal stresses of slabs

 

Fig. 3-15 and Fig. 3-16 present the horizontal and vertical stress “a and
‘%) distributions along the symmetrical axis of a slab (2048 in * 2048 in)
under a daytime temperature gradient of 3 °F/in by the JSLAB-92 and
Westergaard's formula (Yoder“””). Fig 3-17 and Fig. 3-18 present the same
comparison for nighttime temperature gradient = 1.5 °F/in. As shown in

these figures the thermal stresses are practically the same. In all
cases, if and only if the slab is large enough, the calculated interior

thermal stresses are extremely identical to those obtained by Westergaard

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

*tOOjj‘ A Y
jig-150 T A B x
0
§-250«H¥
E —300
g \

)—

-350
i

 

 

 

-400
-4501__._)L\iFleriflE:TTj! I I I’ I

 

 

 

 

 

—500 . W ”l--- -___-__---_ . L
0 200 400 600 800 1000
Inches
-IP-JSLAB-92‘~_-m“~hm~*——_n__—fi

—-+- Westergaard .

 

 

12(30

Fig. 3-15 Comparison of ax Along AB Line, Day Time, g=3.0 °F/in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1260

~37O -enc-n-flm
—380 _ y
it 3 IO in. k = ?00 pa
Q-ssoi LEE-=14 We“--. \
U)
a.
v -400 A y
Vt ,____,__’
3’. ‘ I
3 -4l() - ,A e i’ : I
.t ; l——.D- E
m -420 . j t
8 ‘ i
F “430‘ a %
-440 ..__.:_ _,._.._....... .—————-. r WI
-450 .. . _- ,rm_‘ (“if Y .
‘ 0 200 400 600 800 1000
Inches
if -—s- JSLAB—92 —+— Wesrergoord

 

Fig. 3-16 Comparison Of ory Along AB Line, Day Time, g=3.0 °F/in

55

 

 

 

 

 

 

 

 

 

 

 

 

250
{N‘Ifi I I I I I
200 {f
C (I:
U, 150 ‘ I’ ‘I
3 17 I: a *
,, : ->
:71 100 wr—If L1
0- I” I . .
0 Halo“. x=200w
’— L—=I.=".--.._"—_....'--l
50 +4
II ‘
I
0 r - I 7 J
O 200 400 600 800 1000 1200
Inches

 

E-l— JSLAB-92 -+— Westergaard

 

 

Fig. 3-17 Comparison of 0x Along AB Line, Night Time, g=1.5 °F/in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

225
22() jirJl§§l-———Il ll are It: sr— II
+7

2155 if .
f: " 11
8 1'1" I .
V210? it ?* fij
33 1 +1" A 13 X
§2051 7’1“ - - 1"
g I .l !
5200'; -~~-— ““"
o. If
a I ,
" ”Is—H :22

190'?

185 . . -—~ , . r

0 200 400 600 800 1000 1200

Inches

 

 

1 -I— Modified JSLAB -+- Westergaard jI
J

I -_

 

Fig. 3-18 Comparison of 0ry Along AB Line, Night Time, g=1.5 °

56

close form formula (Yoder“””) which is based on elasticity theory. The
comparison of stresses near the edge subjected to day time thermal
gradient are better than those subjected to the nighttime temperature
gradient because the assumed "Extension springs" of Westergaard model in
the curled up portion produce greater stresses than actually exist. In
case of day time, however, the slab edge remains in contact with the base,
the uplifted portion remains a certain distance from the edge and with a
smaller amplitude. Thus, the effect of "extension springs" becomes
secondary, see Fig. 3—19. The examples are only for checking the accuracy

of existing programs.

The incgesental respgnses produced bv tmaffic load andlor tsspggggggg
gradient

In many cases, engineers need to know the responses of slabs produced by
traffic loading and/or temperature gradient only, without considerationIOf
the slab weight. The numerical example given by Tayabji“”“ is given for
the mentioned purpose. In this case, the input data of unit weight 7 is

set equal to zero, and the coordinate XfiLY,in Fig. 3-1(a) is employed for

the analysis.

Using the new coordinate system, Eq. (3-1) and Eq. (3-2) should be

replaced by:

S7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.02

0.015 +--~—-—~~~—-
A 0.011. -—-<~w—~- , a
(8 J1. LHIIO'A 11-70096
§ 0.005 T————--——-— ———Il————.
: 0~»~.=-'—a - - - 1 - *
r 1
g_0.005T_f_,:7Hzi——+ I 1 J L
g '0.01 F ‘4'
'3-0015 [L 9
.‘2 I 8 X
o I I‘

-0.02 1 1

-0.025 I ’-__.____1

-0.03 . - . - 2

0 200 400 600 800 1000 1200

Inches

 

1 Day (9:3 F/in) —1— Night (9:15 F/in) 1

#

Fig. 3-19 Comparison of Day and Night Displacements Along AB Line

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.03
I ,9
0.025 1 ms“ ""“”—"-' 173'],
{By IUSLABsSG. without sell-sign \_ / if

A 0.02 “In—am; * . s

g; \‘I. .123 /

‘5 ,, -' X /

E 0.015 D4 / /{1

7 [ Us 51.45.92 . Iitmt titlilm.\ VI , , 1;? //

‘E i x. D X

m 0.01 i \ +" xi

g :1 7“ “I

a ooos ,r- I

.% /1/ // /’ \‘\

o 0 - I'vrf, ,..-: ,5”. ii. i i c
I 73/" */-' "l ‘
CM // - \

-0.005 M“ ——;-/-.r/ .
‘1 ./-’ 1a, JSLAB-92 ma MJSLAB-BG. ~th “11.»qu '
I/
-O 01 Q ' . - e . .
0 12 24 36 48 60 72
Inches

Fig. 3-20 Comparison of Displacements By Different Coordinate
Systems

58

 

 

4
51d 1? +II(Y1 =0 (-1 sxsl) (3'25)
dXI
4
51d Y‘ = -q (--l > x or x > 1) (3'26)
dX,‘

The relation between displacement under coordinates X0! and XfimY, is:

fl = Y+l}
Y = fl : _q (3-27)
° k K

and the stresses for the two cases are same.

Fig. 3—20 shows that JSLAB-86 with the discovered errors discussed above,
and the developed analytical procedure (denoted as JSLAB-92 since their
results are very identical) predict the same displacements for the case of
unit weight y = 0.09 pci (under XOY coordinate system). However, the
displacements for the weightless case of y = 0.0 (under XflhYI coordinate
system) are significantly different. Fig. 3-21 also shows that JSLAB-86
significantly underestimates stresses in case of y=0 but the developed
model has proved that the stresses under two coordinate systems should be

same.

16C)’

140

120

100

(I!
O

O)
0

Top Stresses (psi)

40

20

0

Fig. 3—21 Comparison of Stresses by Different Coordinate Systems

59

 

L

 

 

Midis-85. .1111 3.11...sz VI

 

 

\.

 

 

 

 

; é [ mitt”; tut-01ml I
II-——-—||\\fl-\\\1-\x “R L/;
,,

 

 
 
 

 

 

 

 

 

 

  

  
    
     
   

 

 

 

 
 

I
\\\
‘\ 11:300 pci
. 'uJSLAe-ae. 1.1111001 miner ~\I\ II=Io 'n L=72 In
1 \'\\\ Night Time 9:5 i' in
? -.__._ _... -___—-_....__‘ ‘
I
I
l. . 2 .. ..-. _-_ ___--. _.. ..,__.-_- T Y fl.
0 1 O 2 O 5 O 4 O b 0 6 O 7 0
Inches

60

Instead of Eq. (3-26), JSLAB-86 uses the following equation to treat the

uplifted portion of the beam:

d 4 Y,
dx,‘

(3-28)

 

EI = O (-l > x or 1 >.x)

In other words, when the uplifted region has been determined by numerical
iteration, the weight of the slab is still set equal to zero. It can be
found that Eq. (3-25) is correct (q=0) because the weight is balanced by
the Winkler forces of the base, whereas Eq. (3-28) is incorrect since no
Winkler forces exists for the separated portion. In this case, weight of
the beam must be added back to approach the correct results. Fig. 3-20
and Fig. 3-21 also show that the JSLAB-92 produces correct results for

both displacement and stress. The mentioned concepts are valid for beams

and for slabs.

5. Summary

It would be significant to evaluate a computer program before using its
results in engineering design or research. Consistency in displacement
does not mean there exist no problems. The developed analytical procedure
has been used as a potential tool to find problems in JSLAB-86. All of
the discovered problems in JSLAB-86 dealing with single slab responses
have been corrected. The modified program is referred to as JSLAB-92.
After comparing the results of JSLAB-92 with those of the developed

analytical procedure, computer program ILLISLAB, and Westergaard's

61

equations, it can be concluded that the responses of single slab under
different types of loads are correct. The problems related to multiple

slab system are presented in Chapter 4.

62

CHAPTER FOUR
DOWEL MODELS IN FINITE ELEMENT PROGRAMS FOR

PCC PAVEMENT ANALYSIS

1. Introduction

Load transfer systems of PCC pavements have been theoretically and
experimentally investigated since the 1930's (Teller“””, Friberg“””,
Keetod”“m”" and Tellefl””U. In the application of the finite element
method for PCC pavement analysis and design many computer programs
(Tabatabaie“””, Tayabji“”“ and Ioannides“““) have modeled dowel bars as

beam elements based on classical theory (Timoshenko““” and

Przemieniecki“°‘") .

Some investigators (Tabatabaie“””, Darter“"", Ozbeki“”“ and Snyder“””)
have summarized the behavior of load transfer systems of PCC pavements
based on the mentioned classical and finite element models. Results
were recently presented by Smith””” from many of the currently available
models, both mechanistic and empirical, that simulate dowel bar
behavior. The finite element approach has been used as an powerful tool
to implement the analysis, design and evaluation of load transfer
systems. Therefore the reliability of finite element models for PCC

pavements becomes very important.

A number of errors were recently discovered in program JSLAB

(Tayabli“”“) and presented in Chapter 3. After the program was

63

modified, the new version JSLAB-92 produced the same results as the
program ILLISLAB, for the responses of a single slab system. As
mentioned by SmithP”", the difference in results from the JSLAB and
ILLISLAB for multiple slab systems with dowel bars was even greater than
for single slabs. It has been demonstrated in this chapter that part of
this difference is caused by the dowel bar stiffness matrices employed

in the two programs.

Recently Nishizawsp”” stated: "Tabatabaie, et. al., used the bar element
to present the dowel function. However the bar element can not be used
in the case where crack width is so narrow that the length of the bar
element becomes too small and thus its rigidity becomes too high. This
is caused by the assumption that the displacements at both end nodes of
the bar element are the same as those of the slab element."
Nishizawap”” presented the discrepancies between their experimental
results and the predicted results from "the bar element“
(Tabatabaie“””). However, the actual reason of the discrepancy has
never been sufficiently discussed. The "refined model of doweled joint"
presented by Nishizawsp”” contains problems which will be discussed

later in this chapter.

The "bar element" used by Tabatabaie‘m”l was modified from the standard
shear-bending stiffness matrix (Przemieniecki“”") for considering the

interaction between the dowel and concrete such that the "rigidity” of
the element has been greatly reduced. A detailed analysis is presented

in this dissertation to discuss the dowel bar models employed by

64

Tabatabaie‘m“l and Ioannides“”“. The most serious drawback in these two
programs is that the dowel bars have been modified into unequilibrium
elements. The neglect of equilibrium conditions for the stiffness
matrices employed to model dowel bars, is equivalent to the modification
of the load vector of the pavement system. Numerical examples show this
is very important to the final results for the slab critical stresses.
Thus, it is concluded that equilibrium conditions should be considered

in developing the stiffness matrix of dowel bars.

Fig. 2-1 depicts a dowel bar system before and after deflection. The
dowel bar can be modeled by three segments of a beam: two bending
segments embedded in concrete, Ci. and ij in Fig. 2-1(a), and one shear-
bending segment in the joint ii“, where subcripts b and s denote the
dowel bar and slab respectively. A rigorous model of dowel bar load
transfer systems, with minimum modification of classical theory, is
briefly introduced. The embedded length of dowel bars and the physical
properties of materials have been considered in the model so that it can
be employed to investigate optimal design for dowel systems. A 4 x 4
matrix model of a shear-bending beam element and another 2 x 2 matrix
model of a shear beam element are also developed. Numerical examples
indicate that results produced by the three models are very close to
each other for solid dowel bars popularly used in the field. The models
developed in this chapter could make a contribution by being able to

model different types of load transfer systems.

65

2. A Direct Finite Element Method Approach

If the displacements of dowel bar at i and j are equal to the

displacements of slabs at i and j respectively then the stiffness matrix

of beam ij can be written as (PrzemienieckflmaU:

'12 61 -12 61
_ EI 61 (4+¢>)12 ~61 (2-¢)12
’ 13(1.¢) —12 -61 12 -61

‘61 (2-¢)12-sl (4+¢)12

Dlx 02X -DlX DZX

 

_ D2x 03x -D2x D4x __ 5n 312 (4-1)
-Dlx -DZx Dlx -DZx 8n 3,,
02x 04x -02x 03):
Where: E Elastic Modulus of the beam
I Moment of inertia of the cross section
1 Length of the beam (or width of the joint)
o 24(1+u)I/AJP
A, Cross-sectional area effective in shear (0.9 times the

area for circular cross section)

u Poisson ratio
DIX = 1231/1’(1+¢)
02x = eel/13(1+¢)
03x = (4+¢)EI/1(l+o)
04X = (2-¢)EI/1(1+¢)

66

The force and displacement vectors are defined as:

P = [0} Mi Qj Mj]r (4-2)

v: [W e), “’1 63‘]T (4-3)

where w and 6 are vertical and rotational displacements of, and Q and M
are shear force and bending moment of the beam nodes respectively. Their

positive directions are defined in Fig. 4-1.

Fig. 2-1(b) is a dowel bar system after deflection and shows the
interaction between the steel bar and concrete. The relationship
between the two displacement vectors can be written as follows

(Timoshenko““”):

(4‘4)
26 1
k,=2[32£!l1 A]
B
50=Wb- 3
eq -eb-es “‘5’
2 TD 2%
B (451)

Where the force, moment, displacements and rotations (P, Mo, 60 and 60)

are depicted in Fig. 4-2, W is the interaction coefficient of dowel

67

 

61 M; 63 M,
wt 1 0* W: l 01

(a) Positive directions

SW1

 

 

Fig. 4-1 Notations of Displacements and Forces of Dowel Bar Element

——‘__._-..__..--_ - . .._._. “__,..-__-._.

 

 

 

 

 

Fig. 4-2 Elastic Beam in Elastic Medium

68

bar, and D is the dowel bar diameter.

'Eq. (4-4) is precise if and only if the beam is assumed to be infinitely
long, homogeneous and elastic without consideration of shear
deformation, and the concrete is assumed to be a uniform elastic medium.
In Eq.(4-5), Wb,6., and w,,6, are displacements of dowel bar and slab
respectively. If they are defined as independent unknowns for a finite
element program, a 4 X 4 stiffness matrix as follows should be added

into the global stiffness matrix of the system:

31' ’52-
"51' 51'

r

 

 

(4'5)

Stoner“”” uses Eq. (4-6) to model the dowel bars. Although this
equation models dowel bar load transfer systems with minimum assumptions
(infinitely long beam in pure elastic medium), it also has two
significant drawbacks. First the total number of unknowns has to be
greatly increased and second the bandwidth of the global stiffness
matrix will be larger, which will require much longer computation time
and will cause considerable programming difficulties. Therefore, most
investigators have established an approximate but direct relation
between dowel bar and slab displacements, instead of treating them as

independent unknowns.

69

3. Modified Shear-bending Beam Model

The computer program JSLAB (Tayabli“”“) employs the following formula to

consider the interaction between steel bars and concrete slabs:

Assuming:

then:

 

1

 

 

1

02X

___1__
1 + 1
D4X'.DcxM

 

-D2X

.____l____.
_;L_+__l__
D3X'.DcxM)

 

(4'7)

DCXM = 2,106,000 lb-in

————————— 02x —————————
_JL.+_3;. ._£_+_;L_
DIX’ DCX DlX’ DCX
DZX‘ ___—l;———— -D2X
1 + 1
03x .DCXM
35: 1 1
-_——————— -02x -————————
._£_+__£_ _;L_+_;L_
01x DCX DIX DCX
02x .————l————- -02x
._l_+__l_.
_ D4X’.DCXM
where: DCX=ZB3EI
1x304: BEI
8 = 29,000,000 psi u = 0.30
D = 1.25 inch 1 = 0.25 in
Y = 1500000 pci
I = .11984 in‘ B = 0.6060 l/in
¢ = 54.17
DCX = 1,547,000 lb/in
01x = 48,383,000 lb/in 02x =
max = 14,658,000 lb—in 04x =

6,048,000 1b

-13,146,000 lb-in

70

In this case, 8(1,1) (48,383,000 lb/in, Eq. (4-1)) is replaced by
%(l,1) (1,499,000 lb/in, Eq. (4-7)) to reduce the stiffness of dowel
bar for consideration of the interaction between steel bar and concrete.
Similar replacements were employed to modify S(l,3), 5(2,2), 8(2,4),
8(3,l), S(3,3), 8(4,2), and 8(4,4) in the JSLAB program (Tayabji“”“).
Among the eight elements, S(2,4) and S(4,2) are modified in magnitude as
well as in sign. Based on the above data, S(2,4) and 8(4,2) are changed

from -13,146,000 lb-in to 2,508,000 lb-in.

These assumptions result in two failures to satisfy equilibrium

conditions.

First, the force vector (Eq. (4-2)) will usually be a nonequilibrium

force system. For example, define a displacement vector Fig. 4-1(b):

v= [1 o o o]T (4-8)

Premultiplying Eq. (4-8) by stiffness matrix Eq. (4-7), the force vector

of the element is obtained:

P s v- ___—___1 02X ___—1 02X” (4 9)
‘ J ’ 1 + 1 1 1

__+_.__.
DlX' DCX DlX' DCX

 

 

which fails to satisfy the moment equilibrium condition, 2 M = 0.

Second, rigid body movement would produce non-zero element forces. For

instance, define a rigid body movement vector:

71

r (4’10)

 

 

Premultiplying Eq. (4-10) by Eq. (4-7). a non-zero force vector is

obtained. For example, the first element Qiis:

1 202x .
1 1 - 1 )(wi-wj) : 0 1f wjewj (4-11)

—+__
DJX' DCX

 

Q1=(

The ILLISLAB program (Tabatabaie“”” and Ioannides“”“) modifies S(l,l),
S(l,3), S(3,1) and 8(3,3) the same as JSLAB does while the other matrix
elements remain the same as Eq.(4-1). The modification causes similar

nonequilibrium problems, as analyzed above.

4. A Couponent Stiffness Matrix

Nishizawa“”fl developed a "refined model" to simulate dowel bar load
transfer systems. The entire dowel bar was divided into three segments
as shown in Fig. 4-1. The two segments embedded in concrete were
modeled by finitely long bending beams in an elastic medium
(Timoshenko“””) and the middle segment was modeled by a standard bending
beam. The stiffness matrices for each segment were derived and
assembled into a 4 x 4 final stiffness matrix for the load transfer
system. For cases of a very narrow joint, the contribution of the
middle segment was neglected. The major advantages of this model are
that:

a. The finite length of the dowel has been considered so that

the model is capable of a detailed dowel bar analysis

72

b. The number of the unknowns remains the same as other

simplified models

c. The contributions of the three segments have been involved

in the final matrix.

However, the "refined model" also has two potential sources of error in
predictions:
a. The equations 18 (a) and (b) of the reference
(Nishizawa“””) were incorrectly derived, so that the
final stiffness matrix expression is different from the one
which had been expected by the authors.
b. The middle segment of the dowel bar was inappropriately

modeled by a bending beam.

After modifying the derivation given by Nishizawa“”" and employing the

standard shear-bending beam (Eq. (4-1)) to replace the bending beam, the

following stiffness matrix is obtained:

 

 

 

 

 

 

 

-1
x -2'1 0 5 OJ — $11+T1 812 T1 0 4 12
[c] 0 T2 0 3 :321 saw, 0 T2 ( )
Where:
29(slcl+s,c1) -<sf+sf)
[T1] = L 231 , (s c -s c)
C12+C12 _(Sf+sf) 1 1 1 1

 

[3

73

 

2 ZEI 28(szc2+szc2) (322+s22)
[T3] = —L—C~2+C72 (522+522) (SZCZ-SZCZ)
‘ ‘ l3
E =
o 1

SashBL, C=chBL, s=sinBL and c=cosBL. Subscripts 1 and 2 indicate the
left and right segments respectively. B can be found in Eq. (4-5) and

Snrsnrsn and 822 can be found in Eq. (4-1). The detailed derivations are

given in Appendix D.

5. A Proposed Shear—bending Beam Model (4 x 4 Stiffness Matrix)

Mod'f'ca ion of E . 4-1

In all cases where shear deformation is more important than bending
deformation, such as a beam with relatively small length compared to
width and height, 0 in Eq. (4-1) is much greater than 1. The dowel bar
element typically fits the above condition. In the example given in
section three, ¢ = 54.17. Therefore, the assumption of l/(¢+1) = 1/¢

is substituted into Eq. (4-1) to obtain:

 

r915 G_Ag -9; 9:
1 2 1 2
s = 4-13
”1 _GA _GA GA GA ( )
7‘ “—2" “'1" ‘7
GA GA -2 __Gs E if
L2E 415 1“ 2541E+1n

 

Where f and n may be defined as two stiffness reduction coefficients for

74
considering the effects of interaction between dowel bar and concrete, f
is related to shear stiffness and n is related to bending stiffness.
The case of E = l and n = 1 corresponds to Fig. l (a) with the
assumption 1/(¢+1) = 1/0.
It can be proved that for any given displacement vector V of Eq. (4-3),
the corresponding element force vector F of Eq. (4-2) can satisfy both
vertical force and bending moment equilibrium equations. Premultiplying
any rigid body displacement vector of Eq. (4-10) by Eq. (4-13), a force

vector P is always obtained in which all elements are zero. For example:

W.~-W« WNW.

 

 

Deter-iggtion of the shear stiffness reduction coefficient g and thg

 

bending gtiffness reduction coefficient 5

If Eq. (4-4) is rewritten to obtain:

1
283151

0

(P - BMO)

 

(4-14)
8 = - 1 (p-z M
° 28251 B '3)

 

Substituting P=l,l%=0 and P=0,l%=l to obtain the vertical displacement

6 and rotational angle 6 produced by unit load P=l and Mo=1:

 

5 = 1 =._l_
263EI DCX
(4-15)
5 = __1__ = _3:_
(351' CXM
Define: AWb as displacement difference between two ends of the dowel

bar

AW,as displacement difference between two slab nodes

It can be seen in Fig. 2-1(b) that the displacement of the slab node and
the bar end on one side is identical before deflection occurs, and the
difference of the two displacements is caused by interaction between
steel bar and concrete. Based on above definition and Fig. 2-1(b), the

following geometric relation can be obtained:

Aws = Awb + 260,, “'16)

where, Q, is shear force of the dowel bar and can be written as:

(4'17)
0,, = 91,511“; = 91:3 (Aw, - 260,;

Then the relation between Qh and AW< can be derived by solving Qb in

Eq.(4-17):

= £4 1 (4-18)
0" 1 1+260A Aws
1

 

Comparing Eq. (4-17) and Eq. (4-18) defines the shear reduction

76

coefficient as follows:

 

 

E ‘ 1 _ 1 (4-19)
1+26m0 1+ 26%
1 ltDCX

Introducing a parameter K” in Eq. (4-19) to calibrate the model
predictions to field data, Eq. (4-19) can be rewritten as:

E = 1 (4-20)

+ 26%
erCX 1

where K,nmy be determined by experimental research and field survey

data. If the difference caused by the assumptions employed can be

neglected, K.== 1 may be used.

Similarly the relation between dowel bar bending moment and slab node
relative rotational angle, which is different from the rotational angle

of the dowel bar ends, can be derived as:

 

_ HI 1 __EI 1 (4-21)
A! - __“_______ ___.__________
b 1 1*265I6‘ 1 1+ 2131 6’
l ltDCXM

Then X, may be introduced to also consider the effects of these

assumptions and to rewrite Eq. (4-21) as:

77

E
Mb = 765“
(4-22)
g 1
n 1+ 251
1*DCXM 2

Numerical examples in this chapter show that responses of slabs are
relatively insensitive to the values of K. and K3, and that K; has much
less importance than K, so that Kfiflg is assumed to do numerical

analysis to reduce the number of parameters.

6. A Proposed Shear Beam Model (2 x 2 Stiffness Matrix)

It is interesting and significant to investigate whether loads are
transfered through dowel bars mostly by shear force and whether the
contribution of dowel bar bending stiffness can be neglected in
transferring the load. If bending stiffness can be neglected, an

alternative 2 x 2 stiffness matrix may be employed:

£5 -95. (4-23)
3 _ 1 f 1 E
“n ' _cm GA

‘1‘E 75

Where, e is 9:111 defined by Eq. (4-20).

Eq. (4-23) may be derived directly by assuming dowel bars to be modeled
by shear beams or obtained by eliminating the second and fourth lines
and columns in Eq. (4-13). Eq. (4-17) can still be used and the

notation "=" may be replaced by an "=" in this case.

78

7. Numerical Examples

Paveeegt pedal, finite element mesh and load configgration

The pavement model and finite element mesh are presented in Fig. 1-1.

An 18000 lb. single axle load is located at the transverse joint and one

tire is at the longitudinal edge as shown in Fig.

The load

configuration is given in Fig. 4-3.' The dowel bar system is assumed as

the only load transfer system in the pavement and the effects of

aggregate interlock are neglected.

The major input data are listed below:
Length of each slab
Width of slab
Thickness of slab
Elastic modulus of concrete
Poisson's ratio of concrete
Subgrade reaction k-value
Unit weight of concrete
Dowel bar diameter
Dowel bar spacing
Elastic modulus of dowel bar steel
Width of joint
Poisson's ratio of steel

Dowel-concrete interaction coefficient

180 inches
144 inches

10 inches
5,000,000 psi
0.15

200 pci

0.09 pci

1.25 inches
12 inches
29,000,000 psi
0.25 inch
0.30

1,500,000 pci

79

Physical explanation of K,

As disCussed previously, the coefficient Kg can be used to calibrate the

model to field or lab. data. The major model assumptions are that:

a. The length of dowel bars is assumed infinite, whereas it is
actually finite.

b. The dowel bars are assumed to be perfectly bound by elastic
concrete whereas there exists nonlinear behavior of interaction
between the concrete and dowel bars which could be caused by
construction procedures, dowel bar looseness, installation and
other factors.

c. The formula to determine stiffness reduction coefficients E and n
are approximate, for instance, the effects of dowel bar bending
moments are only approximately considered.

d. The dowel-concrete interaction coefficient (Y in Eq. (4-7)) is
difficult to estimate precisely and the values obtained by

different investigators cover a large span (Tabatabaie“””).

The first three factors would tend to overestimate the resistance
capability of concrete, thus, K, should be taken greater than 1. The
fourth factor could overestimate or underestimate the concrete
resistance capability. It is suggested that the fourth factor be
neglected in determining KP Thus, if the dowel-concrete interaction
coefficient Y is assumed to be correctly estimated, K¢=l would be
corresponding to the upper bound of the concrete resistance capability.

Eq. (4-21) and Eq. (4-23) indicate that the greater the value of X; the

80

96 , ‘1

 

'77

 

 

   

Fig. 4-3 Configuration of Single Axle Load: 18 kips, p=100 psi

 

 

-0.0185 T ---—~—---———~—---— I
-0.019 Wm
-0.0195 \‘s 1.1“..."
-D.02
-0.0205
-0.021 ~—
-0.0215 “WWW-W" W , - , __ ,1
-0.022 5- "W- W
-0.0225 «-——~~- g
—0.023
-O.‘0235 ‘————77"‘"
-0.024 {J
-0.0245

-0325? I r " .
1 2 3 ‘ 4 5 6 7 8 9 10 ii 12

DOWEL BAR LENGTHS. INCH

 

 

 

 

 

 

 

 

 

 

IN

 

 

 

 

 

 

 

 

 

 

 

1
1

l::bu—s’157211"???"013—651-177

 

Fig. 4-4 The Maximum Displacements v.s. Dowel Lengths

81

less the resistance capability of the concrete. So, Kfid, 2 and 3 are

selected in following numerical analysis.

The effects of embedded length of the dowel bars

Fig. 4-4, Fig. 4-5 and Fig. 4-6 present the maximum displacements of the

loaded slab at point B and F, total forces transmitted by the load

transfer system and the maximum longitudinal stresses of line A-A, B-§

(Fig. 1-1) in terms of the embedded length of dowel bars respectively by

using the component model. The following conclusions can be obtained:

a. The longer the embedded length the higher the load tranfer
capability of the dowel bar system that leads to an increase
of maximum displacement and maximum stress in the unloaded slab,
the decrease of those in the loaded slab and more total shear
force being transmitted from the loaded to the unloaded slabs.

b. Based on the example presented, when the embedded length is
longer than five times the diameter of the bar the difference of
results between the finitely and infinitely long dowel bar models
can be neglected, thus the dowel bars currently used in pavements
can be appropriately modeled by assuming the embedded length to be
infinitely long.

c. In the example presented, the maximum stresses on line A-E (the
loaded side), Fig. 1-1, are always greater than that on line F-A
(the unloaed side), the comparison of maximum stresses on line B-
B are likely on the contrary, however, the difference between the
maximum stress on the loaded slab and the unloaded slab is less

than 1% on line A-A and less than 2% on line B-S.

82

 

7.9

-F '
7.8 /

 

 

 

KPS

7.7

 

 

7.6 f

7.5

 

 

 

7 ' r

5 6 7 8 91 1o 11 12
00th BAR LENGTHS. INCH

0‘1
p

2

Fig. 4-5 Total Shear Forces Transmitted by the Dowel Bars

85

80 \.\.§!
1 , i i

HST/Kr—

7O

 

 

 

 

 

PSI

65

 

60

 

 

 

 

55

 

501 T Y r T I I I T
2 1 3 4 5 6 7 8 9 10 11 12

DOWEL BAR LENGTHS. INCH

.4
c1

 

+ A-E LINE ‘4— F—A LINE —"'— B—I LINE ‘5‘- J-B LINE

 

 

 

Fig. 4-6 The Maximum Stresses V.S. Dowel Lengths

83

 

 

Eg. (5-13) and shear beam model 801114-231

Fig. 4-7 presents the comparison of longitudinal stress distributions of
line A-A, Fig.1-1 predicted by the three models. It can be clearly seen
that for the solid dowel bar system, the three models provide very close
results. Tables 4-1 to Table 4-3 present a comparison for the maximum
stresses, shear forces and bending moments of each dowel bar. The
difference of the results predicted by the three models are pratically

negligible.

 

sla

Fig. (4-7) also shows the stress a,<distribution produced by program
JSLAB-86. As discussed in section three, JSLAB-86 employs a
nonequilibrium stiffness matrix Eq. (4-7) which predicts maximum q,iJ1
the unloaded slab about 30% greater than that of the loaded slab (Table
4-1). Also the JSLAB-86 results in the unloaded slab are greater than
all of those by modified JSLAB-86 employing equilibrium stiffness matrix
Eq. (4-12), Eq. (4-13) or Eq. (4—23). Whereas the JSLAB-86 results of

the loaded slab are smaller than those predicted with the new models.

SIGMA x (psi)

Fig.

Fig.

84

 

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-100 my 1 r T T f 1
O 45 90 135 180 225 270 315 360

X (inches)

 

I—I— JSLAB—86 ~8- Eq.(4-l2) —— Eq.(4—13) — Eq.(a-stI

 

4-7 Longitudinal Stresses Along Line A-K (in Fig. 1-1)

 

2.5

 

 

 

 

 

SHEAR FORCES (kips)

 

 

 

 

6 r18Y30742154r66V78190'102Y114126Y138
LOCATION: Y (inches)

 

In JSLAB-86 - 5414-12) 51.14-13) m 51114—23)

 

4-8 Shear Forces of Dowel Bars

85

Table 4-l Comparison of the Maximum Stresses a, and a, on Top of the

 

 

 

 

 

 

 

Slabs

cm on A8 cm, on FA am on BE 0, at H

JSLAB-86 69.4 90.6 -84.0 '84.8
Eq. (4-13), K=l 76.3 77.1 -92.95 -46.9
Eq. (4-13), x=2 80.9 73.4 -100.9 -43.9
Eq. (4-13), x=3 84.5 70.0 —104.3 -42.o
Eq.(4-23), k=l 77.5 78.2 -92.8 -43.5
Eq. (4-12) 77.0 76.3 -94.88 -46.6

 

 

 

 

 

 

 

Table 4-1 shows that the maximum c7,I on top of the loaded slab on line

AE, Fig. 1-1, predicted with JSLAB is 9.8% smaller than that by the

component model Eq. (4-12) and the maximum 0, on top of the unloaded

slab on line FA is 18.7% greater.

Table 4-2 Shear Forces of the Dowel Bars (kps)

 

 

 

 

 

 

 

Bar No.* 1 2 3 4 s 6 7 8 9 10 11 12 2

JSLAB -.09 0.08 0.30 0.66 0.96 0.85 0.45 0.17 0.15 0.45 1.12 2.07 7.17
A -.04 0.14 0.38 0.68 0.89 0.81 0.52 0.32 0.34 0.65 1.24 1.96 7.89
s 0.03 0.18 0.37 0.58 0.72 0.69 0.53 0.43 0.49 0.74 1.16 1.66 7.58
c 0.07 0.20 0.35 0.52 0.64 0.63 0.53 0.48 0.55 0.76 1.09 1.48 7.30
0 -.04 0.14 0.38 0.68 0.89 0.81 0.52 0.32 0.34 0.65 1.24 1.96 7.89
8 -.03 0.15 0.38 0.66 0.85 0.78 0.53 0.35 0.38 0.67 1.23 1.89 7.84

 

 

 

 

* No. 1 bar is located at Y = 6 inch. and N0. l2 is at Y = 138 inch in Fig. 1-1. A: Eq. (4-13), K=l.

B: Eq. (4-13), K=2. C: Eq. (4-13). K=3. D: Eq. (4-23), K=l. E: Eq. (4-12), component model.

86

The most significant difference for'cg occurs at point B by the edge
tire, where JSLAB-86's prediction is 81% greater. The critical position
of the load stress predicted by JSLAB is point H, only 10 inches away
from the joint, however, the critical position predicted by the
component model, Eq. (4-13) and Eq. (4-23) is point G which is 50 inches

away from the joint.

Fig. 4-9, Fig. 4-10 and Table 4-3 show the comparison of bending moments
of the dowel bars predicted with JSLAB-86 and with the JSLAB-92 based on
89. (4-12), Eq. (4-13) and Eq (4-23). The output of bending moments at
two ends of each dowel bar by the JSLAB-92 forms an equilibrium system
which satisfy 2 9,: 0 and z Mi= 0, whereas moments output from JSLAB-
86 fails to satisfy equilibrium, as discussed in this chapter. The
nonequilibrium forces produce significant differences in bending
moments. More seriously, it causes an incorrect sign of bending moments
on the unloaded slab, which is why JSLAB—86 overestimates the stress
responses on the unloaded slab by up to 18.7% and overestimates the

stresses on the loaded slab up to 81% in this example.

Diggegggg gglggg of 5,

Table 4-2 also shows comparison of shear forces of each dowel bar
calculated by JSLAB-92 based on Eq. (4-13) with Ksl, K=2 and K83. The
right hand column in Table 4-2 are total forces transfered by all dowel
bars. It can be clearly seen that the larger the value K is used, the
less load is transferred from the loaded slab to the unloaded slab,

therefore, the greater are the stresses produced on the loaded slab and

87

 

 

 

 

 

”effifl L‘

 

BENDING MOMENTS (kip—in)

luau Annnn H]:
hinn- ‘I-‘n -‘nnn

I-nn- I-"-
51:31!
71-11““ 1121)
11:13:“

HEIHKI
11“)!" jun-

Y1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

WALL
18 30 42 54 66 78 90102114126138
LOCATION: Y (inches)

 

 

-J5U‘9’35 EMU-'2) 5:]Eq.(4-13).K:1 I
% 5914-13). K22 5:] Eq.(4-13).K_-3 j

 

 

Fig. 4-9 Bending Moments of Dowel Bars on the Loaded Side

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12

i

I
A10 ,1
.E T ‘
I i
5 8 .
" 1 I
W
F- 6 H '11
Z 11 11
L” 11 . I 11
g a I 11 1
I 4 1 1 1 1 1 1
o I I E I I 1
z 11 i 1 11 . 1 11 1
5 2 1 1 1.1 1 11 1 1 1 1 1
z 1 : 1 : ' 1 1 1 1 1
U I 1 11 1 i I 1 1 I 1
m 0' r- r r'

-2 _r

 

6 18130'42154'66Y78'901102314426738
LOCATION: Y (inches)

 

£39233 JSLAB-86 = Eq.(4-12) [1:] Eq.(4-13). K=l
m Eq.(4-13). K=2 Eq.(4-13). K23

 

 

 

 

Fig. 4-10 Bending Moments of Dowel Bars on the Unl0aded Side

88

the smaller are produced on the unloaded slab.

Qggpggisgg gf dowel b5; shear forces

Fig. 4-8 presents the shear forces of each dowel bar calculated by
JSLAB-86, Eq. (4-12), Eq. (4-13) and Eq. (4-23) with Kean The
differences between the shear force results are less significant than
the bending moments of the dowel bars and the critical stresses in the

slabs.

Table 4-3 Bending Moments of Dowel Bars

 

1 2 3 4 5 6 7 8 5 1o 11 12

 

JSLAB .04 .88 1.96 3.57 4.87 4.53 2.96 1.95 2.06 3.57 6.70 11.03

 

-35 0.20 1.05 2.15 3.78 5.10 4.77 3.21 2.20 2.32 3.86 7.01 11.36 ‘

 

.23 .26 .32 .39 .44 .44 .41 .39 .42 .49 .60 .73

 

-.24 -.23 -.22 -.22 -.22 -.24 -.28 -.31 -.33 -.33 -.30 -.24

 

.23 .26 .32 .38 .43 .44 .41' .40 .42 .49 .60 .72

 

fi
I
H
h)
J; 3: J! P: J; .F

4-12 -.24 -.23 -.22 -.22 -.22 -.24 -.28 -.31 -.33 -.33 -.30 -.25

 

 

 

 

 

All figures and tables presented show that the results by using Eq. (4-
23) are practically the same as those by Eq. (4-12) and Eq. (4-13) with
K=1, including the maximum displacements of the slabs, critical stresses
in the slabs, shear forces and bending moments of the dowel bars. The
above comparisons lead to a significant conclusion: Eq. (4-12) and Eq.
(4-13) with K81 may be replaced by Eq. (4-23) for simulating the solid
dowel bar mechanism in FEM program for concrete pavement analysis. The
conclusion implys that the shear beam model is not only a simple but

also an applicable model in PCC pavement analysis.

89

8 Sue-ary

Based on theoretical analysis and numerical examples presented in this
chapter, it is concluded that the neglect of equilibrium condition of
the dowel bar stiffness matrix causes significant differences in
prediction of dowel bar forces and critical slab stresses. The
reference (Nishizawa“””) was found to provide a good concept to simulate
the dowel bar mechanism in detail. The error in derivation should be
corrected and the bending beam model to simulate the dowel segment in
the joint (between two slabs) should be replaced by a shear-bending beam
or a shear beam element. Three new stiffness matrices Eq. (4-12)
(component model), a 4x4 matrix Eq. (4-13) and a 2x2 matrix Eq. (4-23)
are proposed and investigated in this paper. All of them satisfy the
necessary equilibrium conditions. The identical results produced by the
three models confirm that the shear resistance capability of the solid
steel dowel bars is the dominant parameter of the load transfer system.
Furthermore Eq. (4-20) and Eq. (4-22) can be used to determine the
stiffness reduction coefficients for simulating the mechanism of
interaction between dowel bars and concrete. Numerical examples
demonstrate that the shear beam model Eq. (4-23) with Kfid is applicable
to simulate the solid steel dowel bar in finite element programs for PCC
pavement analysis. Eq. (4-12) is developed with minimum assumptions and
includes most considerations of dowel bar geometry and physical
properties so that it has potential to study dowel behavior for

analysis, design and optimization.

90

CHAPTER FIVE
A NONLINEAR MECHANISTIC MODEL FOR DOWEL

BAR LOOSENESS

1 Introduction

/

Many studies have demonstrated that the capability of load transfer]
systems significantly affect the service quality and remaining life of the:

pavement (HveemP“”, Teller“”” and Snyder“””). Dowel bars are a popularly 7

used system to transfer load from the loaded slab to the unloaded slab.

Since the Finite Element Method was introduced into the analysis of rigid I

pavements, the dowel bar mechanism has been simulated by many models which
have been discussed in chapter four. These models are valid for dowels
perfectly embedded in concrete, in other words, there exists no gap or

void between the dowel and the concrete, or no looseness.

However, some experimental studies (Kushing“””, Finney“”", Keeton“”",
Teller‘m", Ball“””, Snyder“""’’ and Reiter‘m") have shown that dowel bar
looseness greatly affects the load transfer efficiency, maximum
deflection, critical stresses, pumping, faulting and further the remaining
life of the pavement. The maximum stresses of slabs with and without
dowel bars under the same traffic loads at the joint, as calculated by the
finite element computer program, could have differences of more than 100%.

The actual responses of slabs with loose dowel bars should be in between.

i

91

Maj izadeh‘m“ summarized major findings in references published before 1984
and establishes their model to consider the dowel bar looseness through
modifying the dowel bar diameter in their computer program RISC. The

major assumptions of their model are:

(1) Looseness is uniform for all dowel bars.

(2) The loaded slab has to deflect by the amount of looseness before
dowel bars become effective in load transfer, i.e., the loaded slab
behaves as a single slab with a free edge until the deflection
exceeds the amount of looseness, at which time the free edge is
transformed into a joint with fully effective dowel bars and without
voids.

(3) A void with depth equal to the amount of looseness forms under the
joint of the loaded slab due to the high stress concentration (at
the slab-foundation interface) under an undoweled joint.

(4) All dowel bars come into contact at the same time (deflection value)

independent of the distance from the applied loads.

Finally, an effective dowel bar diameter D' = k x D, where k is a
function of looseness and always smaller or equal to one, is employed in
numerical calculation. As explained by the author, the assumptions were
made in order to analyze looseness inside a linear elastic model. The

model would be improved if:

(1) Looseness was a function of bar location, distribution of the tire

load along the transverse joint, and support conditions of the

92

slab, etc.

(2) Dowel bars came in contact one after another and the contact order
depends on the relative location of the loads and each dowel bar
(3) A nonlinear model was used because the second assumption leads to a

nonlinear load transfer procedure.

Based on the above discussion, a nonlinear elastic model to simulate the
dowel bar looseness mechanism is proposed in this paper. The model can be
used to predict responses of doweled rigid pavements with consideration of
the effects of the dowel bar looseness. If the looseness of each bar is
known, no matter whether they are obtained from field survey or assumed,
the developed model can calculate the responses of pavement with any
looseness distribution, the order of dowel bars come in contact, and the
final critical stresses. Numerical examples are presented in this chapter
for demonstrating the validity of the model and for conceptually and

quantitatively understanding the nonlinear load transfer mechanism.

Dowel Bar Stiffness Matrix

As discussed in chapter 4, the following equation can be used for dowel

bar stiffness matrix for general purpose:

where:

where:

93

 

 

 

 

'93: .63 fig .53
l 2 l 2
GA GA 51 GA GA EI
___ .__.1 -+.__ ___. ___ul -___ -
_ 2 4 E In 25 4 E 1“ (51)
' -2115 -24 25 __GA
1 2 l 2
13A (54 Elf (£4 (£4 Elf
-—— -———l -——— -___ ___1,-1___
2E 4 E 111 25 4 5 1"
Elastic Modules of the Dowel Bar
Moment of Inertia of the Cross Section
Elastic Shear Modules of the Dowel Bar
Width of the Joint
Cross Sectional Area Effective in Shear
5 = 1 (5-2)
1+. ZCEA
1*IX1X 1
TI' 1 (5-3)
+ JLEI
110C201"

Dcx = 215331

DCXM = as:

B = (Yo/4EI)°”

Y

D

Kux.

Dowel - Concrete Interaction Coefficient
Diameter of the Dowel Bar

Parameters to be determined for different dowel systems.
K, = 1 and K2 = l are suggested in chapter 4 for solid

dowel bars.

94

Due to the fact that load transfer capability is dominated by its
capability of shear resistance, Eq. (5-1) can be replaced by a simpler

matrix in the computer program:

GA .24
S - “7'5 1 i (5-4)
”’- _G’A GA

7‘ 7‘

numerical examples presented in chapter 4 indicate that Eq. (5-1), (5-4)
and the component stiffness matrix (Eq. (4-12)) provide very close results
for predicting responses of pavements with solid dowel bar system. Among
them Eq. (5-4) is the simplest one so that it is employed in this chapter
to investigate the effects of dowel bar looseness. Regardless of the
application of E9. (5-4) or (5-1), or another stiffness matrix, the
concept and idea to develop a model for dowel bar looseness analysis are

same, and suitable for any type of dowel bar stiffness matrix.

2 Load Transfer Procedure of Dowel Bars with Looseness

Fig. 5-1 presents the load transfer procedure of a dowel bar with
looseness. Fig. S-l(a) shows a cross section view of slab with the dowel
before any traffic load moves in or after it goes out. Fig. 5-l(b) and
(e) show that each slab performs as a single slab with free edge when the
relative deflection of the two slabs is smaller than the dowel looseness.
Fig. S-l(c) and (d) show that the dowel bar becomes effective since the
relative deflection between the two slabs exceeds the looseness of the

dowel bar.

Teller“”” presented excellent experimental results to study the dowel

95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

be an]

’ i
1 a.)

Ikmnflflshbnnhaamu

 

 

 

 

 

 

 

 

 

 

 

 

(c)

Fig. 5-1 Load Transfer Procedure

96

 

 

 

 

 

 

 

 

 

(d)

 

 

 

 

 

 

 

Woodedslsbsmfsceline

X 6.. 6” ~ 0
f 6* X
16.,

 

 

 

 

 

(a)

Fig. 5-1 Load Transfer Procedure (continued)

97

behavior with consideration of the initial looseness and the looseness
developed by repetitive loads. Fig. 5-2 shows relation between applied
load and relative deflection after various numbers of load cycles and
effect of repetitive loading on the development of dowel looseness (Copied

from Teller“””). The findings can be summarized as follow:

(1) During the first 5000 lb load, the dowels were in a state of
adjustment in which existing looseness was being taken up, and a
condition of full bearing was being established.

(2) When load was greater than 5000 lb, the relation between increments
of load and increments of relative deflection became constant. The
slope was independent of the number of load cycles.

(3) Initial looseness may be defined as the vertical coordinate of
the intersections between the dashed lines and the y axis, as

indicated in Fig. 5-2(a).

Based on the previous research and also because all loads are acted on the
approach slab then the leave slab evenly, the following assumptions may be

added:

(1) The void, or gap between the dowel bar and concrete is uniformly
distributed along the bar. This assumption implies that "The loaded
slab has to deflect by the amount of looseness before dowel bars
become effective in load transfer."

(2) Before any load moves in, (Fig. 5-l(a)), 6m + 6g = 5m + 6n. = Lu

which is defined as Looseness of the dowel bar (see Fig. S-2(a)).

98

 

 

 

 

 

 

 

 

 
 
     
 

 

 
 

 

 

 

 

 

0:2
MUN!!! OF
1.0110 CYCLES
9.02,
'°'° 500000
0‘:
3‘; 101,000
g 5.000
- .003 I’a/‘U
l I’ V o
g ’l’:"’ /
: V/” ”d
0.006"
U I
u.) I
o I
.. ’ 24%
> ,ooc
;; , 1V
4
.J
U
1: /
ooz
S-INGH sue DEPTH
i-mcn 01m. courts
8-011! EMBEDMENT
! -INGH JOlNT mom

   

 

O 2 4 S 8 IO l2
STATIC TEST LOAD - THOUSANDS OF POUNDS

a. Relation Between Applied Load and Relative Deflection,
After Various Numbers_of Load Cycles. (from Teller,
Fig. 5, 1958)

e003 { V
1 3
510.000 " pOUND

REPEATED LOAD
.002 *- —~-r—~— --——----— _-_ T- .-_.. -_._ - - __.

i a

l 1
s-INGH SLAB DEPTH 1
l-INCH DIAM. DOWEL
8 'DIAM. EMBEOMENT
; i—INCH JOINT WlOTH
O 1 I

O 4 8 I2 16 20

NUMBER OF LOAD CYCLES - HUNDREDS OF THOUSANDS

 

 

 

.001 ._'._-

 

INCREASE IN DOWEL
LOOSENESS - INCHES

 

 

 

 

 

 

b. Effect of Repetitive Loading on the Development of Dowel
Looseness. (from Teller, 1958, Figure. 6)

Fig. 5-2 Dowel Behavior in Experiments

99

(3) The deformation of dowel bar is assumed inside of elastic range, or
the permanent deformation of dowel bar will not be considered in the
mean time. So, when the tire moves far away from the joint, the
structural cross section view of the slabs will be the same as shown
in Fig. 5-1 (a).

(4) The surface of two slabs are in same horizontal line before any load
moves in, or, the permanent faulting is not considered in developing

dowel bar looseness model. The interactive effects between dowel

looseness and faulting will not be considered at this time.

The pavement slabs can still be modeled by elastic plates resting either

on an extensionless Winkler base (Tabatabiel‘m‘ and Tayabjil'm), or on
multiple elastic layers (Majidzadehp””), or on stress dependent layers

(Ioannides“”“).

Based on the above assumptions, the behavior of a single dowel bar with
looseness can be graphically described in Fig. 5-3(a) and (b), where P is
a load acted on a slab node and A represents the relative displacement
between the loaded and unloaded slabs. &,is the stiffness contributed by
the loaded slab, base and subgrade under the slab only, and A0 = I» is
defined by the second assumption. Before the relative displacement
between the loaded and unloaded slab nodes exceeds the defined looseness,
namely A <.Lfi, the dowel bar is not effective in load transfer. When A >
I”, the dowel bar becomes effective and Slis contributed by the two slabs
and their support system. The Force-Deflection relation is a typical

bilinear model.

100

 

 

 

 

 

 

 

\ \

 

 

 

 

jwo Slabs
(a)

Bilinear Model of Single Dowel Bar with Looseness

 

 

 

(c) (d)

Model for Multiple Dowels

Fig. 5-3 Nonlinear Model of Dowel Bars with Looseness

101

Similarly, the functional behavior of a multiple dowel bar system can be
described in Fig. S-3(c) and (d). The force-deflection curve of load P
and the relative displacement between the loaded and unloaded slabs is
given in Fig. 5-3, (d). 1m,indicates the first point of changing stiffness
of the loosed dowel system. Before the relative displacement exceeds Am
no bar is effective in load transfer, and the loaded slab performs as a
single slab without load transfer system. When the displacement is
between A0 and A”, only one dowel is in contact and effective in load
transfer. The second bar starts in contact when the displacement equal to
A]. The stiffness of entire pavement structure becomes greater since a
contribution is also provided by the unloaded slab through more effective
dowel bars. The curve shown in Fig. S-3(d) is a multi-linear model which
can be coordinated with any available computer program to calculate the

responses of pavement system step by step.
3 Looseness Distribution and Input Data

Fig. S-4(a) indicating effect of increasing the magnitude of the repeated
load on the development of dowel looseness was copied from TelLefl””L Two
significant findings can be summarized by analyzing Fig. 5-2(b) and Fig.

5-4(a):

(1) About 40000 load cycles (2% of 2,000,000 total cycles) produced
about 50% of looseness by the 2,000,000 cycles. That indicates if
magnitude of the repetitive loads remains the same, looseness is

developed quickly at the beginning and increases very slowly when

INIJIII 1)

INCHEA‘SL IN DOWEL LOOSLNLSS ‘

102

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

WfieEl path as pa

b. Tire Load Distribution

Fig. 5-4

Looseness Distribution Assumption

: :4 ,— , K
: I I I ; F
‘ I : ,
003! 4L 45r1f" .
' ]/ I_ \_Is.000-pou~01
3 ; 95951750 LOAOI
’ I
! I
002 A T I
' . ' I i i ]
I0.000-eou~0 I : .1 .
REPEATED LCADI I g 1
.00! T I 1 f
' I ~ I I 2 .
‘ I : I I I
] I ll i [I k-INCH DIAM. 00‘”ij
. I 1 .L 1
0O I 2 3 4 5 5 7 8 9 IO II
NUMBER OF LOAD CYCLES - HUNDREDS OF THOUSANDS
a. Effect of Increasing the Magnitude of the Repeated
Load on the Development of Dowel Looseness.
(from Teller, Figure. 26, 1958)
Parental
totalloads

103

number of load cycles is large.
(2) When the magnitude of the repetitive loads was replaced by a greater
one (Fig. 5-4(a)), new looseness was developed immediately, and the

looseness development procedure was still similar to the stage one.

The above findings indicate that most increased looseness of dowel bar is
caused by heavy trucks. The percentage of tire loads across a joint can
be approximately expressed by Fig. S-4(b). Although the distribution is
not uniform, the percentage of tires passing each dowel spacing could
still be more than 2%. If this is true, the experimental findings by
Teller““’“’l implies that the looseness level of different dowels should not
make a significant difference. Therefore, as a first stage of
investigating the effects of looseness, a uniform looseness distribution
is employed to conduct numerical analysis though the developed model is

capable of dealing with any type of looseness distributions.

A finite element mesh (Fig. 1-1) with two load cases is employed in
numerical analysis. The pavement analyzed contains two slabs which have
equal length and width, so that it is symmetrical in both X and Y

directions. The two load cases are:

Load case one: A concentrated 9000 lb load acted at point I in Fig. 1-1,

(Xa180 inch, on the approach slab and Y266 inch)

Load case two: An 18000 lb single axle load with four tires which

configuration is given in Fig. 4-3.

104

The major input data are listed as follow:

Elastic Modules of the Concrete 5,000,000 psi
Thickness of the Slab 10 inch

Poison Ratio of the Concrete 0.15

Subgrade Modules 200 pci

Dowel Bar Diameter 1.25 inch
Elastic Modules of Dowel Bar 29,000,000 psi
Width of the Joint 0.25 inch
Poison Ratio of the Steel 0.3
Dowel-Concrete Interaction Coefficient 1,500,000 pci
Dowel Bar Looseness As indicated

4 Numerical Examples

W

Fig. 5-5(a)-(e) show ‘theI displacement shapes in terms of different
looseness levels under load case one. The figures indicate that the more
serious the looseness, the less percentage of total loads is transmitted
from the loaded slab to the unloaded slab. When all dowels are
ineffective, the calculated displacements equal to the responses of a

single slab without dowels.

Fig. 5-6(a)-(c) present the displacement distributions along Line B-B, E-E
and F-F (Fig. 1-1) under load case one. Fig. S-7(a)-(d) present the
displacement distributions along Line A-A, B-B, E-E and F-F under load
case two (Fig. 4-3). These figures quantitatively show the effects of

looseness level to the displacement responses. For instance, Fig. 5-7(b)

105

  

A /" 2
// f1/
\f/ \. \‘~/// \A/\
.. \ ‘ , ‘ \\ /.\‘ (I, "\ ~ /
‘\ \ \ ‘ ‘7 \I a X , '1“ :(f \\‘ \. I // z
1. \ ‘ \ , . ~ ’ \ z‘..// \‘ <2
\\ 1‘ \ .* J‘?_ /\ K/ I; 1’ fl
_“ \_ , .1 .
' v.3»;ir
\\.\/ I / C; 3
\\~ 1, I 1.",
a. Looseness

\,/

0.0, the Maximum Displacement

0.0086 in

 

 

b.

0.0101 in
,/ \. ‘1
/‘/(r I" ‘\ 2 \
,’/ . /,./ '. ~ ‘.
J 1 / (V I 3‘ 1 \.
\ ‘ ,.\ '\ ‘ ‘ _ ‘2 /
W I ‘

Looseness = 0.003 in, the Maximum Displacement -
/\/\\
<\ \ ‘

 

,2 1 ,}‘£é;;::::;7
x" . \\ ‘\l ‘ \(l ,
I 1 2' \ 1 .A"
oi. ‘~ ‘2 ‘ \\ X, i
‘ \ x 0/
\\ ,1, ,
\\
/; \ ’4
~\. \\//,
\ .1 '7
C e

Looseness a 0.006 in, the Maximum Displacement
Fige 5'5

Displacement Shapes (load case one)

0.0115 in

44"

106

 

d. Looseness = 0.009 in, the Maximum Displacement = 0.0129 in

 

e. Looseness = 0.015 in, the Maximum Displacement = 0.0154 in

Fig. 5-5 Displacement Shapes (continued, load case one)

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
  

 

 

 

 

 

0.005
0
-0.005
(I)
g —o.01
Z
—0.015
—0.02
4.025 1 v r v v v v Y Y T v
0 30 60 90 120 150 180 210 240 270 300 330 360
lNCl-ES
+ Lo=.000 —'- Lo=.005 + Lo=.012
‘9' Lo=.018 + Lo=.024 + Lo=.030
(a) UISDLHCEMENTS OF LINE B~B
-0.002
-0.004
-0.008
-0.008
E -0.010
-0.012
-0.014
—0.016
0 24 48 ‘72 96 120 144
mm:
+ [0=.000 '3- [0=.003 -N- [0:006
7-0— [0:009 -0- [0:012 — [0:015
(b) DISPLACEMENT 0F LINE E-E
0.000
-0.00[
—0.002
£3 -0.003
5
a -0.004

-0.005
-0.006

     

 

-0.007
0 24 48 72 98 120 144
m
-- [0:000 —e—— [0:003 + [0:006
—0— [0:009 —0— [0=.0[2 — [0--.015

 

 

 

(C) DISPIACEMENT OF LINE F-F

Fig. 5—6 Displacement Distributions (load case one)

108

 

0.005

 

 

 

-0.005

—0.01 k JOINT

 

INCHES

 

-0.015

 

-0.02

 

 

 

_00025 V ' V T I I T v I v I
0 30 60 90 120 150 180 210 240 270 300 330 360
INCHES

 

+ Lo=.000 —*— Lo=.006 + Lo=.012
—E— Lo=.018 —>‘— Lo=.024 + Lo=.030

 

 

 

(o) DISDLHCEMENTS OF LINE 8.3

0.005

-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
—-0.035

INCHES

 

0 30 60 90 120 150 180 210 240 270 300 330 360
INCHES

 

+ Lo=.000 —*— Lo=.006 + Lo=.012
' *3“- Lo=.018 + Lo=.024 + Lo=.030

 

 

 

(b) DISDLHCEMENTS OF LINE H-H

Fig. 5-7 Displacement Distributions (load case two)

109

 

 

-0.005

 

-0.01

 

 

-0.015

 

INCHES

-0.02

— . 5 ,
if; m:
<\‘HI

-0.0_35 . , . I
0 24 48 72 96 120 144

INCHES

 

 

 

 

 

 

+ L0=.OOO —+— Lo=.006 + Let-.012
*5"— L0=.018 + Lo=.024 + Lo=.030

 

 

 

(c) DISPLACEMENTS OF LINE E—E

0.002

-0.002
-0.004
-0.006
-0.008

-0.01
—0.012
—0.014
—0.016

INCHES

 

0 24 48 72 96 120 144
INCHES

 

+ Lo=.000 —+— Lo=.006 + Lo=.012
—a— Lo=.018 + Lo=.024 + Lo=.030

 

 

 

(d) DISDLHCEMENTS OF LINE F_p

Fig. 5-7 Displacement Distributions (continued,
load case two)

110

indicates that a 0.006 in looseness increases the displacement at point E

from 0.0187 in to 0.0216 in, etc.

W15

Fig. 5-8(a)-(c) quantitatively present the stress distributions along Line
B-B, E-E and F-F under load case one and Fig. 5-9(a)-(d) present the
similar results along A-A, B-B, E-E and F-F under load case two. For
instance, Fig. 5-9(a) and (b) indicate that the full looseness of the
dowel bars may cause the maximum longitudinal stress even 100% more than
those without looseness, and a 0.006 in looseness may increase the
maximum longitudinal stress more than 20%. In load case two, each 0.003
inch looseness could cause approximately 10% in crease in the maximum
stress of line A-A, 14% increase of the maximum stress of line 8-3. When
temperature gradient is considered, the mentioned increase of maximum
longitudinal stress would become more critical. For PCC pavement
thickness design procedures based on fatigue criterion, effect of

additional stress due to the dowel bar looseness could be significant.

Where is the most critical point in pavement? Which stress is more
critical, corner stress or edge stress? Many investigators have concluded
that the edge stress is most critical at the middle between two joints.
However, all these research results were obtained based on assumption of
no looseness. The serious looseness significantly increases the maximum
corner stress whereas the maximum edge stress is not sensitive to the
looseness level because the location of the maximum edge stress is far

away from the joint. The above analysis plus the consideration of dynamic

111

90
80
70
60
50
40‘
30
20
[0

    

                    

30 60 90 [ l 210 240

 

-.- [0:000 -.- [1:003 + [0:000
-0— [0:009 -0—' [0:012 — [03.016

 

 

 

(o) SIGMA-X or 1th 8-3

 

 

-400
24 48 72 96 [20 [44
[-a— 19:00:: —.— [moon —- [0:015 ]

 

(b) SIGMA-Y or LINE E-E

 

 

24 48 72 96 [20 [44
mm
-.- [0:000 -B- [0:003 -- [0:008
-0- [0:000 —¢— [0:012 — [0:015

 

 

 

(c) SIGMA-Y or LINE 5-1:

Fig. 5-8 Stress Distributions (load case one)

112

 

120

10° {32"‘5‘ 4 JOINT I
so 4" 1.... 3‘
m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
_ 601 // \i;
E 40 //_‘.‘u\\
201 /;’;‘
o _
_20 r 1' I t I r 1 r y 1 f v 1
0 30 60 90 120 150 180 210 240 270 300 330 360
INCHES
+ Lo=.000 + Lo=.006 + Lo=.012
‘5" Lo=.018 + Lo=.024 + Lo=.030
(a) SIGMH-X or LINE 8-8
150 ~
\ ":3
f» “x /IJOINT I
100 f I
Z 50- I ‘-
O
O 30 60 90 120 150 180 210 240 270 300 330 350
INCHES

 

+ Lo=.000 —*— Lo=.006 + Lo=.012
fi— Lo=.018 + Lo=.024 + Lo=.030

 

 

(b) SIGMA-X or LINE e—e

Fig. 5-9 Stress Distributions (load case two)

113

 

150

100 fl

 

 

., 52m \
m \ / E91

Y

 

 

 

 

 

 

'"150 “r Y Y 1* r
O 24 48 72 96 120 144

INCHES

 

i... Lo=,ooo _._ Lo=.005 + Lo=.030 1

g

(c) SIGMA-Y OF LINE E—E

 

 

 

 

 

 

 

 

 

 

 

 

o 24 45 7E 96 110 144
INCHES

 

+ Lo=.OOO —+— Lo=.006 + Lo=.012
‘5— Lo=.018 + Lo=.024 + Lo=.030

 

 

 

(d) SIGMfl-Y OF LINE E—E

Fig. 5-9 Stress Distributions (load case two,continued)

114

coeeficient due to the interaction between truck and pavement near slab
corner being greater than that at slab edge, the critical position needs

further study.

Fig.5-8(b) and Fig.5-9(c) demonstrate that the transverse stress
distributions of line E-i are not sensitive to the dowel bar looseness,
and the maximum transverse stress (370 psi) presented in Fig. S-8(b) does
not make sense in rigid pavement design because there exists no
concentrated load on the pavement as used in the calculation. Fig. 5-8
presented herein is for numerical comparison only. The results shown in
Fig. S-8(c) and Fig. 5-9(d) look sensitive to the looseness level,
however, the effects of looseness are to reduce the maximum stress
responses in the unloaded slab. Therefore, the effects are not

significzat in pavement thickness design.

Loa e ca abi it

Fig. 5-10(a) and Fig. 5-11(a) present the variation of dowel bar shear
forces due to the increase of assumed looseness under load case one and
load case two respectively. In load case one, Fig. 5-10(a) indicates that
numbers of the effective dowels are reduced from 12 for looseness a O, to
5 for looseness 8 0.003 in and 0.006 in and to 3 for looseness = 0.009 in.
The corresponding results in load case two are presented in Fig. S-ll(a):
12 for zero looseness, 9 for looseness a 0.006 in. Fig. 5-10(b) and Fig.
5-11(b) show the decrease of the total forces (the value without looseness

is taken as 100%) transmitted from the loaded to unloaded slabs due to the

115

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.40
1.20 .
1.00
0.30 1
g 0.60
0.40 11 1““
0.20
0.00 U n . fl .. U
‘0'20 Tl‘2'3T4I5f6'7r8'9'10111112
N0.0FBA16
[:1 [0:000 - [0:003 Lo=.006
[0:009 [0:012 #235312; ID=.015
(o) SHEAR FORCES 0F DOWEL BARS
100
90 g
o 80
II 70
60 g
g 50 § § 0
D:
8 40
30 0 0 0 0 0
C
r 20
10 § §

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 '0003 '0.006 10.009 '0012 0015
10085113515

(b) TOTAL SHEAR FORCES TRANSMI'I'I‘ED BY BARS

Fig. 5-10 Load Transfer Properties (load case one)

116

 

 

HKHR
lexxl’]

 

A.
XXUI XIXXX IX
xux‘ v

 

u
nlnnn unnnn nunnn In“,

 

 

 

nun"): nunnn “I

“VHHHH Rm
1’! 1111111 Ill

 

 

 

 

 

 

 

 

VIIILXIIXI
qu‘

11-
1.5 HELL

 

 

 

5 ’18‘30 42 54 66 78 90102114126138
Y(INCHES)

 

3 ”732 Lo=.000 - Lo=.006 &\\\\‘§ Lo=.012
@ Lo=.018 m Lo=.024 @ Lo=.030

 

 

 

 

(o) SHEQQ FOQCE‘S OF THE DONEL BFIQS

100% FOR LOOSENESS = 0

 

O 0.006 0.012 0.018 0.024 0.03
LOOSENESS. INCHES

(b) TOTAL FOQCES TQHNSFEQQED BY DONEL 8008

Fig. 5-11 Load Transfer Properties (load case two)

 

117

.increase of looseness level. The 0.003 inch looseness causes 22% loss of
the load transfer capability for load case one, and approximately 14% loss
for load case two. As mentioned by, Majidzadehl‘m', .003 - .006 inch
looseness is often observed in pavement in service, even the initial dowel
bar looseness of new pavement could be greater than .003 inch which still

affects load transfer capability significantly.
5 Summary

A nonlinear elastic model to simulate the dowel bar looseness mechanism is
proposed in this study. For any given looseness of the dowel bars, the
model can prediCt the responses of the pavement, including the final
stress and displacement distributions and load transfer capability. The
looseness level depends on the construction quality, pavement service
life, accumulate traffic loads, environmental condition of the pavement,
etc. Once the looseness level and distribution are measured or estimated
by an appropriate model, the responses of pavement at different stages of
its service life may be predicted by the developed model. The findings in

this chapter are summarized as follow:

(1) Based on the numerical example presented in this paper (load case
two), each increase of 0.003 inch looseness could cause a 10%

increase in the»maximum stress at the edge, 14% increase of the maximum
stress in the middle of the slab, and causes 14% loss of the load
transfer capability.

(2) The maximum longitudinal stress in pavement is sensitive to the

(3)

(4)

118

looseness level. It would.become1more critical to transverse cracks
when temperature gradient is considered. This finding could have
a significant effect to any PCC pavement thickness design procedure
based on fatigue criterion.

0.003 inch dowel bar looseness might change the stress distribution
quite significantly. That explains why the quality of dowel bar
installation is very important to its service quality and life.
Further more, any coating material to be used for protecting dowel
bars against corrosion should be carefully verified to ensure it is
thin and strong enough, and will not produce effect similar to a
serious looseness.

The consideration of looseness causes some significant changes in
pavement response. More numerical analysis corresponding to
pavements with different service periods and under different
environmental conditions shouldIbe conducted to study the effects of

looseness to some existing research conclusions.

119

CHAPTER SIX
COMPARISON BETWEEN ANALYTICAL AND
EXPERINIENTAL RESULTS

1 Research on Load Transfer Characteristics of Dowels by Keeton“”"

A very significant experimental study on the load transfer
characteristics of dowels used in airfield pavement expansion joints was
conducted by the 0.8. Naval Civil Engineering Research and Evaluation
Laboratory in the 1950's. Not only the test results, but also the
structural, material and environmental data were presented in the
literature, hence, it is possible to use the experimental results to
compare the results produced by the analytical models presented in

Chapter 4.

The primary objective of the experimental research was the development
of a realistic evaluation procedure for load transfer devices. The
interrelationships among deflection, moment, shear and bearing pressure
during load transfer in an airport pavements, were studied by
constructing a full-size concrete slab with instrumented dowels across
an expansion joint and imposing upon the slab loads of the magnitude of
those resulting from the use of modern aircraft. (Tire load varied from

10,000 lb to 100,000 lb)

The test slab was 10 in. thick, 15 ft. wide and 50 ft. long, consisting
of two 25—foot sections jointed by dowels across 0.75 in. expansion

joint. A transverse weakened plane joint was provided at the center of

120

 

Fig. 6-1 The Cart for Application of Wheel Loads
(from Keeton‘m")

 

Fig. 6-2 A Static Load Acted at the Joint (from
Keetonmm)

122

applied at the center of the slab with the tire print tangent to one

face of the joint as shown in Fig. 6-2.

2 Formulas of Bending Moment, Shear Force and Bearing Pressure in

Dowels Embedded in Concrete

The bending moment and shear forces at two ends of a dowel in the joint
(the intersections between the two surfaces of slabs and the center line
of the dowel bar) can be calculated using the formulae given in Chapter
4, the detailed derivation can be found in Appendix 4. The reaction
moments and shear forces acting on the ends of two segments of the dowel
embedded in concrete must be the same as their original moments and
forces. Taking the segment embedded in the leave slab as an example and

considering Fig. D-1,1% and E,have been obtained and the following

expressions can be written.

Relative displacement of the dowel:

0 (x) =A’cthcosBx+B’Cthsian+C’sthcosBx
(5'1)

+ D’shfixsinfix

bending moment:

M(x) = -2 WEI [D’cthcos Bx-C’cthsian+B’sthcosBx

‘ 6-2
— A’shflxsinflx] ( )

123

Shear force:

Q(x) =ZB3EI[ (C’-B’) cthcosBx+ (A’+D’) chfixsian

 

 

 

 

+ (A’—D’)sthcosfix + (B’+C’)sthsian] (6'3)
Bearing pressure:
p(x) = 6(x) D ‘1' (5'4)
Where:
A’ - 60
B’ = .1. (.29. P0
2 B 231133
C, = 1139. . Po (6-5)
2 [3 ZEIB3
DI _. M0
25102
Y = interaction coefficient between dowel and concrete
0 = diameter of the dowel
l3 = MID/431)“15
E = Elastic modulus of teh dowel
I = Moment of inertia of the cross section of the beam
50 and o0 can be solved from the following equation:
Po 25192 25605") 32+52 5o (6-6)
= ‘ 2 2 SC-sc
M0 ch32 S +s (pa

13

124

For convenience of presentation, the positive moment is defined as the

bottom surface of the dowel in extension.

3 Sane Special Considerations in Input Data Preparation

Modulgg of subgrade reaction

Keeton“”" concluded: "Tests have indicated the presence of an air void
at the center of the slab at the joint amounting to about 0.04 in."
When responding to the discussion by B. F. Friberg, on the measurement
of the modulus of subgrade reaction, Keeton described: "The modulus of
subgrade reaction was measured at the joint two days before the
construction of the slab and was found to be about 200 psi per in. If
the slab were in intimate contact with the subgrade, the measured slab
deflections would indicate a maximum subgrade pressure of 17.8 psi based
on k3200 pci. This pressure is not likely to cause subgrade failure to
the extent of 0.03 in." For verifying the above statement, Keaton also
stated: "Over 400 load applications (50,000 lb) were made on the slab.
From beginning to end, the test results did not reflect any drastic
changes in the subgrade such as would be evident in event of a subgrade
failure." From above statement, k = 200 pci is used in most numerical
calculations without considering the variation of the modulus of
subgrade reaction under the 50,000 lb load or the reduction of the k
value due to the air gap. However, it is believed that the complicated
slab deflection measuring system embedded under the slabs might have
significant effects to the subgrade modulus. A few numerical results

are also presented by using k=50 pci for comparison. The presented

125

results in this chapter are obtained using k=200 pci except those

specifically mentioned.

Simulation of the air gap under the slab

Keeton also concluded: ‘"It is most probable that the void beneath the
center of the slab at the joint is the result of slab warping during the
curing." However, he also described the test environment as: "The slab
is inside a building and is therefore subject to minimum ambient
temperature changes and is not exposed to the direct rays of the sun.”
The air void beneath the slab was found by the authors and can also be
verified by the presented displacements versus the magnitude of the
loads (Table 5 in the reference, Keeton“”"). Any measured response was
the difference of the responses corresponding to the states before and
after 50,000 lb load being applied on the slab so that the initial state
should be defined as the one before the 50,000 lb load moved on. The
determination of the initial state becomes extremely important in
comparing the experimental and analytical results. The only
significant related information provided by the authors is the existence
of air gap beneath the slab. It is assumed here that: the initial
shape of the slab was the one with about average 0.04 inch gap at the
joint and caused by temperature gradient. By numerical tests, it has
been found that 3.2° F/in night-time temperature gradient (g=-3.2) would
cause average 0.0405 inch curled-up deflection at the joint. Therefore,
the analytical results for comparing the experimental ones are the

difference between responses due to 50,000 lb plus g=-3.2° F/in and the

responses due to g=-3.2 °F/in temperature gradient only. The assumption

126

is acceptable because the initial shape of the slab is more important in

the analysis than the source which had produced the initial shape.

Sisul tion of the ex ansion 'oint and the weakened la oin

The 0.75 inch width expansion joint is simulated by using the component
model as a dowelled joint and the weakened plan joint at the center of
each 25 ft slab is assumed as a 0.1 inch width joint with aggregate
interlock only. The existence of the interlocked narrow joint has
secondary effects on the response of the dowels and the slab near the
expansion joint under the tire load, however, it has significant effects

to determine the initial state of the slabs as discussed above.

Det ' tion 0 ast' it mod 1 s of e c crete

Keeton reported that the concrete had a compressive strength fl.of 6160
psi based on 28-day specimens. The modulus of elasticity E can be
predicted with reasonable accuracy from the empirical equation found in

ACI Code:
5' = 33 y1-5 fl”; (6-7)

Carrasquillo“”” and Martinez“”“ reported that for compressive strength
in the range from 6000 to 12000 psi, the ACI Code equation overestimates
E for both normal weight and lightweight material by as much as 20%.
Based on their research, the following equation is recommended for

normal density concretes with fl.in the range from 3000 to 12000 psi:

127

E = (40,000 fl; + 1,000,000) <_1%_5)Ls (5'3)

The units used in the equation are pounds/irfi for strength and pounds/ft3
for density 7. Eq. 6-8 was employed to determine the modulus of

elasticity of concrete in this analysis.

Input data used in numerical analysis

The major input data used in the analysis are listed below:

Length of each section of the slab 25 ft.

Width of slab 15 ft.
Thickness of slab 10 in.
Elastic modulus of concrete 4,140,000psi
Poisson ratio of concrete 0.15
Subgrade reaction k value 200 pci

Unit weight of concrete 145 pcf
Dowel bar diameter 1.125 in.
Dowel bar spacing 12 in.

Elastic modulus of dowel steel

29000000 psi

Width of joint 0.75 in.
Poisson ratio of steel 0.30
Dowel-concrete interaction coefficient 1,500,000pci

The interlock spring modulus

The finite element mesh of the slab is given in Fig.

6-3 0

100,000 psi

128

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129

4 Comparison of The Results

The comparison of bending moments of the four dowels just under and with
distance 1, 2 and 3 ft from the load center are presented in Figs. 6-4,
(a) to (d). The maximum stress of the dowel is 18.1 kpsi (experiment)
and 28.0 kpsi (analysis), both are relative high. As discussed in
Chapter 4, the critical responses of dowels and slabs affected by the
dowel moments usually are not as sensitive as by the shear forces.
However, The test slab is only 10 inch thick and the single tire load is
very high (50,000 lb), so thatthe maximum bending stress of the dowel
under the load becomes relative high. For a 9,000 lb single tire load,
the maximum bending stress of the dowel would only be about 5,000 psi.
In another word, the 10 inch thickness is not sufficient for airport

pavement to withstand very heavy tire load.

The measured shear forces for two symmetrically located dowels on the
two sides of the load were different, so Figs. 6-5 (a) to (d) only show
the comparison of shear forces of four dowels on the same side of the
load. The differences between the measured and the analytical shear
forces are no more than 20%. The comparison of bearing pressure on the
four dowels are presented in Fig. 6-6. The "measured" maximum bearing
pressure (18,100 psi) is much higher than the measured compressive
strength of the concrete (6160 psi). It is not clear, however, how and

what device was used to measure the bearing pressure in the experiment.

The shear forces of five dowel bars at the joint surface are given in

130

 

 

 

 

 

 

 

 

 

 

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-10—9 -8 :7 leis 41-3 32-10 12f3'is'é 7' 0 S110
DISTANCE FROM THE JOINT CENTER. inch

 

1 —-— TEST —+— ANALYSIS 1

 

(a) Under the load

 

a
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DISTANCE FROM THE JOINT CENTER. inch

 

2 —-— TEST + ANALYSIS

 

(b) 1 ft. from the load

Fig. 6-4 _ Bending Moments of the Dowels

131

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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2000 l f i \\\ CENTERor JONl
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; —-— TEST —+-— ANALYSIS?
(c) 2 ft. from the load
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DISTANCE FROM THE JOINT CENTER. inch

 

fl

1 + TEST _..._ ANALYSIS I

 

(d) ~ 3 ft. from the load

Fig. 6-4 Bending Moments of the Dowels (continued)

132

30001
1
2000

 
 
 
 

1000
O

-1000
.J

3
C
8
°'_ZOOOI
-3000

-4000
-5000

-6000
—10-9-8-7-6-5-4-3-2-1012 3 4 5 6 7 8 910
DISTANCE FROM THE JOINT CENTER, inch

I-«r—TEST —+—-ANALYSSI

4

 

(a) Under the load

50001 |
4
200041 T 7
10002 A l J \
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0 I

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4 .

 

-30001:

 

-40001:

—5000'

.(

 

 

 

-6000 Y Y Y Y Y Y Y Y Y Y Y Y Y Y
—10-9-8-7-6-5—4-3-2-1 O 1 2 3 4 5 6 7 8 910
DISTANCE FROM THE JOINT CENTER. inch

 

 

2 a

l-irrTEST '4—-ANALYSS I

 

(b) 1 it. from the load

Fig. 6-5. Shear Forces of the Dowels

133

3000
2000
1000
0
-1000
42000
-3000
-4000

In

'0
C
3
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Q

-5000

 

-6000
—10-9-8-7—6-S-4-3-2-10 12 3 4 5 6 7 8 910
DISTANCE FROM THE JOINT CENTER. inch

 

v1

I-IF-TEST ——+— ANALYSS J

 

(c) 2 ft. from the joint

3000
2000

 

.4

 

4

1000

W " *‘x 1 QM
—1000 1. ..

4

 

 

 

 

 

if

 

n
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c
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-4000'

 

 

 

.4

 

-5000

4

I

I

—6000 T T f T T V Y V V ‘Jf j V Y Y Y Y Y r r

-10-9-8-7-6-S-4-3-2-1012 3 4 5 6 7 8 910
DISTANCE FROM THE JOINT CENTER. inch

 

 

 

 

2+ TEST —+— ANALYSIS I

 

(d) 3 ft. from the joint

Fig. 6-5 Shear Forces of the Dowels (continued)

134

 

 

 

 

 

 

 

 

 

 

 

 

 

12T
1 \
82 - x\\
1| A \ CENTER OF JOINT
4f ,2
‘2 02——-4.-‘F‘-IFI.E:’/={)‘ -:jp—1Fql-1r—Ih1I—lk——~
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-IO—9-8—7—6—5-4-3-2-10 12 3 4 5 6 7 8 910

(a)

DISTANCE FROM THE JOINT CENTER.
—-o—- ANALYSIS I

I—I— TEST

I

Under the load

inch

 

>0

 

 

\‘
,_4

 

 

‘K\

 

‘ \CENTER OF JOINT

 

Ib/In
(Thousands)

-6?

.
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-10

 

—12

 

 

 

 

-“ T T T fiT T T T T T T T T T T T r T
-10-9-8-7-6-5-4-3-2-1012 3 4 5 6 7 8 910

DISTANCE FROM THE JOINT CENTER.
—+— ANALYSIS 2

I+ TEST

1*

inch

(b) 1 ft. from the load

Fig. 6-6

Bearing Pressure of the Dowels

135

I2
l0
8 CENTEROFJOINT
6
4
’g 2
ES 0
\ In
2 3 -2
é -4
-6
-8
-I0
-I2

 

- -10—9—8—7-6—S-4-3-2-l o I 2 3 4 s 6 7 8 910
DISTANCE FROM THE JOINT CENTER. inch

I -I— TEST --0— ANALYSIS }

 

 

(c) 2 ft. from the load

12
10

8 CENTER OF JONT

ONkO)

-2
-4
-5
-8
-l0
-I 2

Ib/Hn
(Thousands)

 

—lO-9-8-7—6-—5-4-3-2_-l o I 2 3 4 5 6 7 8 910
DISTANCE FROM THE JOINT CENTER, inch

 

[—I— TEST + ANALYSIS l

(d) 3 ft. from the load

Fig. 6-6 Bearing Pressure of the Dowels (continued)

136

Fig. 6-8. The calculated shear forces must be symmetrical to the load
center and the same when the load applied at the two sides of the joint
if the structural and support conditions of the two slabs were the same.
However, the measured ones were not symmetrically distributed when the
load was applied on the two sides of the joint. For comparison, the
average of the four measured values (shear forces on two symmetrical
dowels when the load acted at two sides of the joint) is used. The
difference between the maximum experimental and analytical results is
less than 10%. Fig. 6-8 suggests that the experimental and analytical

shear forces have good agreement.

Fig. 6-7 presents the comparison of longitudinal displacements under and
1, 2 and 3 ft. from the load center. The maximum displacement obtained
by using k=200 pci in analysis is 38% lower than the test one.

However, if the R value is assumed as 100 pci for considering the
effects of the deflection measuring system under the slab, the
temperature gradient needed to produce 0.04 inch average joint
displacement on the loaded side should be about -3.5 °F/in. The
calculated transverse displacements on the two sides of the expansion
joint are presented in Fig. 6-9 and Fig. 6-10. The results from using
ksZOO pci and g=-3.2°F/in are also given in the same figures for
comparison. The results with the assumption of kalOO pci are much

closer to the test results.

Apart from the responses of the slab and joint under 50,000 pounds at
the center of the joint, Keeton also presented shear forces measured at

the dowel under the load, 1 and 2 ft from the load and the maximum

Fig. 6-7

inch

inch

137

0.04
0.02

CENTER (I JONT

-0.02
-0.04
-0.06
-0.08

—0.1
-0.l2
-0.l4
-0.16
-0.18

 

.2
-8-7-6-5-4-3-2-lOl2345678
DISTANCE FROM THE JOINT CENTER. H

[-I— TEST __._ ANALYSIS J

 

 

(3) Under the load

0.04
0.02

 

-0.02
-0.04
-0.06
-0.08

-0.1
-0.12
-0.l4
-0.16
-0.18

-0.2

 

-8—7-6-5-4-3-2-1012 3 4 5 6 7 8
DISTANCE FROM THE JOINT CENTER. H

F.— TEST + ANALYSIS

 

 

(b) 2 it. from the load

Longitudinal Displacements

138

0.04
0.02

 

—o.oz
-o.04
-o.os

E -0.08
-o.1
—o.12
-0.l4
-O.16
—0.18
-o.2

 

-8—7-6-5-4-3-2-10l2 3 4 5 6 7 8
DISTANCE FROM THE JOINT CENTER. fl

 

F—u— TEST -+- ANALYSFI

 

LONGITUDINAL DISPLACEMENTS 4 fi FROM THE LOAD

(c) 4 ft. from the load

0.04
0.02

 

-o.02
-o.o4
—o.oe

E -o.os

-o.I
-0.l2
-0.l4
—o.is
-0.l8

-o.2

 

-8-7—6-5-4-3-2-l0 12 3 4 5 6 7 8
DISTANCE FROM THE JOINT CENTER. H

 

lI—n— TEST -+- ANALYSIS j]

(d) 6 ft. from the load

Fig. 6—7 Longitudinal Displacements(continued)

139

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o I 2 ' 3 4
DISTANCE mom LOAD CENTER. TI

I TEST m ANALYSIS I

 

 

Fig. 6-8 Average Shear Forces in the Joint

0.04

0.02 k=100 9d. $4.5 r

ITIZOO -.32 F is
-o.oz "a " /

-o.04
-o.os

E -o.oa
-0.l
-0.12
-0.14
-O.16
-O.18

 

-8-7-5-5-4-3-2-1012 3 4 5 6 7 8
DISTANCE FROM THE JONT CENTER. fI

Fig. 6-9 Transverse Displacements of the Joint (on the loaded
side)

140

0.04
0.02

kxloo. g=- F/h

-0.02
-0.04
-0.06
-0.08

-0.1
-0.12
-0.14
-0.16
-0.18

-0.2

inch

 

-8-7-6-5—4-3-2-1012 3 4 5 6 7 8
DISTANCE FROM TFE JOINT CENTER. 11

Fig. 6-10 Transverse Displacements of the Joint (on the unloaded
side)

 

10000

D

‘Kfl

8000 X/
5000

‘ ”MUN“

k=100 ocl. g=-3.5 Us
4000
AN‘LYSTS

.1 k=200 pci. g=-3. 2 F/in

2000 //

O T T T T T T T v
10 20 30 40 50 60 70 80 90 100
TOTAL LOAD. THOUSAND POUNDS

 

 

POUNDS

 

 

 

 

(a) Under the Load Center
Fig. 6-11 Measured and Calculated Dowel’s Shear Forces v.s.
Load Magnitudes

141

ANALYSB ‘
A=Ioo pci. 9:3.2 F/in E
\ .
ANALYSIS
u=2oo pci. g=-3.2 F/in
.;:;/
L/S'L

 

8000

 

 

6000

 

 

POUNDS
h
C)
<3
C)

 
  

 

2000

 

 

 

 

10 2D 30 40 5'0 50 7D 80 90 100
TOTAL LOAD. THOUSAND POUNDS

(b) 1 ft from the Load Center

 

ANALYSIS ‘

k=100 ed. 95-15 F/h 4///)',////1r

 

  
     

4000‘
I ANALYSIS
m I k=200 9g. g=-3.2 r/an P
9 J \
8
Q \
\ TEST
2000

 

 

 

 

I0 20 30 4b 50 60 7b 810 9}) 100
TOTAL LOAD. THOUSAND POUNDS

(c) 2 ft from the Load Center
Fig. 6-11 Measured and Calculated Dowel’s Shear Forces v.s.
Load Magnitudes (continued)

142

 

100

 

80

 

60

 

40

 

20

PERCENT OF APPLPIED LOAD

 

 

 

O T T T T T r T n
10 20 30 40 50 60 70 80 90 100

TOTAL LOAD. THOUSAND POUNDS

 

+ TEST ‘4— ANALYSIS

 

 

 

Fig. 6-12 Percentage of Total Load Transferred by Five Dowels

143

displacement of the slab under the load varying from 10,000 to 100,000
pounds. The comparison between the measured and analyzed shear forces
are given in Fig. 6-11. The results by using subgrade modulus k-lOO pci
and g--3.S °F/in are also presented for comparison. It is interesting
to point out that the analytical model underestimates almost all shear
forces on the dowel just under the load center and overestimates almost
all shear forces on the dowel 2 ft away from the load center. The
dowel shear forces are not very sensitive to the variation of subgrade
modulus k and tempreture gradient g. Fig. 6-12 shows good agreement
between the measured and analyzed percentage of total load transferred

by the five dowels nearest the load center.

The analytical results under the combination of load and temperature
gradient were obtained by using nonlinear iteration procedure, in other
words, the nonlinear behavior of the system was considered. The
relationship between the shear force and the applied load in Fig. 6-11
indicates that a linear relation exists when temperature gradient

remains constant and the magnitudes of load are large enough.

The comparison verifies that the analytical model simulates the load
transfer characteristics quite well. All the input data used in the

analysis are obtained from the paper published by Keeton“”". The only

assumption employed in this chapter is the initial state of the slab

144

shape was produced by temperature gradient.

The shear forces cf the dowels obtained by the analysis are in good
agreement with those obtained by the experiment. The shapes of all
compared responses, including the bending moment, shear force
distribution of the dowels, displacement of the slab and the bearing
pressure on the concrete, are identical to those of the experiment. The
analytical results are closer to the experimental ones, when based on
the average values of the measurement. Unfortunately, the distributions
of bending moment and shear force on the dowels symmetrical to those
with the presented measurement are not available in the reference
(Keeton“”"), hence, the corresponding comparison can only be conducted
by using the results on one side of the longitudinal symmetrical line of

the slabs.

The modulus of subgrade reaction k and the dowel-concrete interaction
coefficient Y can be adjusted to produce analytical results very
identical to the measured ones. The essential objective of the
comparison is to simulate the load transfer characteristics rather than
to simulate the measured results. Therefore, no effort was made to

adjust the parameters at this time.

145

CHAPTER SEVEN
IMPACT TO THE DOWEL DESIGN PROCEDURE

1 Current Design Procedures

Smooth round dowel bars have been employed as a load transfer device in
jointed concrete pavements for a long time. Many experimental and
analytical researches have been conducted to develop and improve the
design procedure of the dowels. Before the 60’s, the most influential
analytical models were developed by Timoshenko‘m", and FribergI'mm‘” etc.,
and some significant experimental research studies for dowelled jointed
slabs were conducted by Tellerl‘mll'm', Kushing‘m" and Keetonm‘" etc. The
complete review of these studies and the application in engineering design

can be found in Snyder“”” and Heinrichs“””.

The conclusion has been obtained that the maximum concrete bearing stress
is the most important parameter to be determined in PCC pavement joint
design. Currently, the maximum bearing stresses of concrete under dowels
are required to be equal to or smaller than the concrete bearing strength.
Furthermore, the level of the bearing stress has direct effects on the
accumulation of joint faulting which is one of theb most important
parameter to evaluate the performance of PCC pavements.(Darter“"" and

Heinrichs“””)

The Friberg procedure can be generally divided into two steps to determine I

the maximum bearing stress. The first step is to predict the maximum

146

shear force acting on the critical dowel bar. The second step is to

calculate the maximum bearing stress of the concrete under the critical

bar by using the maximum shear force obtained in the first step.

The first step is based on three assumptions:

Where

A certain percentage of the total load is transferred by the
dowelled joint. The range of percent varies from 0% to about 50%,
depending on the quality of the joint, pavement structural
parameters and the load type. 50% would be a conservative
estimation, Henrichs“”” suggests using 45%.

The dowel shear forces are linearly distributed along the joint, see
Fig. 7-1.

An ”Effective Load Transfer Length" L, was assumed as 1.8 l by
Fribergw”” and all dowels located farther than L, from the load
center are assumed to not contribute in transferring load. Where 1
is radius of relative stiffness of the slab to be determined by the
following formula:

_av3__-i ' <7-1)
12(1-p2) k

E is elasticity modulus of concrete
u is poisson ratio of the concrete
h is thickness of the concrete slab

k is modulus of the subgrade

 

Fig. 7-1

147

Unear Distributlon assumed
Based on 'Effectlve Length'.

L1 - 1.8lbyFT1berg
L1 - 1.01by Tabatabaie

n-5

 

L18 'Equivalent Effective Length'

Determined by the 'Actusl Maximum
Shear Force' P m Calculated by

Finite Element Program.

 

n - 3

ShearForossottheDowsls

/ /

 

/.
I

\7;
\

 

The "Effective Length"(EL) and "Equivalent Effective

Length"(EEL)

148

Using the above assumptions, the maximum shear force acted on the critical

dowel bar can be calculated. As mentioned above, LI 2 1.8 l was proposed

by Friberg based on Westergaard's theory.

In the second step, the shear force of the dowel is assumed known.
Timoshenko“”” model gives a procedure to predict the behavior of a steel
bar embedded in "pure elastic" concrete. Based on the Timoshenko theory,

Friberg derived the maximum bearing stress formula:

 

7-2
omax = ‘I’ 6O ( )
where:
P1IZ+B JO) (7-3)
0
493591
in which:
P| is the maximum shear force acting on the dowel,

predicted in the first step

JO width of the joint opening
E, modulus of elasticity of the dowel bar
I moment of inertia of dowel bar cross-section,

=0.25 n (D/2)fl
D Diameter of the dowel bar
[3 (Y D/4E,I)°~‘-’

?(PSI) Dowel-concrete interaction coefficient

149

After conducting 3-D finite element analysis for a dowel embedded in
elastic concrete space, Tabatabaie‘m"I proposed to use the following formula

to determine the maximum bearing stress directly:

(800+0.068E)
4
3

 

(1+0.355J0)P1 (7-4)
D

where, Pm is the maximum shear force acting on the critical dowel and was

determined by Tabatabaie by using ILLISLAB program as follow:

P1 = a s p, (7'5)

in which:

a = 0.0091, for edge load

a = 0.0116, for protected corner load

a = 0.0163, for unprotected corner load

S = dowel spacing

R = total load
Based on the results produced by using the finite element program
ILLISLAB, Tabatabaie“”” concluded: "only the dowels within a distance 1.0
1 from the center of the load are effective in transferring the major part
of the load." It is obvious that the Tabatabaie’s assumption is more
conservative than the Friberg's. Henrichs“”” proposed to use the L,-I1J0
1 instead of 1.8 l in the first step to predict the maximum shear force
acting on the critical dowel, then to use the Friberg model (Eq. (7-3)) to

determine the maximum bearing stress.

150

2 Some Comments

On the effective length

As summarized above, the determination of the maximum bearing stress can
be divided into two steps, the "Effective Length" was introduced to
predict the maximum shear force acting on the critical dowel. However,
the only purpose of the first step is to determine the maximum shear force
on the dowel. If the "Effective Length" assumption is good, the maximum
shear force Plpmedicted by using the assumed "Effective Length" should be
identical or close to the actual maximum shear force Pm. Fig. 7-1
indicates that when the distribution of the dowel shear forces is strongly

nonlinear, the mentioned procedure could bring significant error.( P,<:PIn

in most cases)

Compggigon of a few numerical examples

Table 7-1 presents the maximum bearing stresses calculated by using
Friberg's model (Eq. 7-2 and Eq. 7-3, with assumption effective length Ll
= 1.8 l and 1..1 = 1.0 l), Tabatabaie's model (Eq. 7-4), and the component

dowel bar model developed in Chapter 4.

The parameters are: h = 10 in, Y = 1,500,000 pci, E = 4,500,000 psi, J.O.

= 0.25 in.

151

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 7-1 The Calculated flaximum Bearing Stresses(psi)
by Different Models

Friberg's model, L = 1.8 1

k \ D 0.75 in 1.25 in 1.75 in

50 pci 2387 945 516

-200 pci 3208 1270 694

500 pci 3868 1532 837
Friberg's model, L = 1.0 l I

k \ D 0.75 in 1.25 in 1.75 in

50 pci 3907 1547 845

200 pci 5143 2037 1113

500 pci 5962 2361 1290
Component dowel bar model P

k \ D 0.75 in 1.25 in 1.75 in

50 pci 5158 2815 1964

200 pci 4459 2478 1692

500 pci 3846 2229 1524
Tabatabai's model

D 0.75 in 1.25 in 1.75 in

k=50,200,500pci 3111 1574 1005

 

 

 

 

 

 

Table 7-1 demonstrates that the results received by using different models

are very different.

On the effects of the subgrade modulus

Fig. 7-2 was copied from Fig. 44, Henrichs“””. It can be seen that the
maximum bearing stress increases when the subgrade modulus k increases.

the same conclusion can also be obtained by using Eq. (7-2) and (7-3) (see
Table 7-1). However, the conclusion is difficult to understand. It seems

not logical that the stronger the subgrade support is and the stronger

l-
m
h)

10 Bearing Stress (ksi)

 

Load - 9000 lbs
Load transferred - 452
Slab thickness - 10 in

8 L- Dowel spacing - 12 in

6 Dowel Dlameter 0.75 In
I-

4 I- , 1.0 In

 

2 _E//'T , 1.25 In

 

 

 

o l l l l 1
50 150 250 350 450 550 650

+ Subgrade k-value (pci)

Fig. 7-2 Effects of Subgrade Modulus on the Maximum Bearing
Stress (from Henrichsms”)

153

capability of withstanding load the loaded side slab has, the greater
maximum shear force needs to be transferred across the critical dowel near

the load.

0n thg effects of concrete modulus E

Eq. 7-4 indicates that the maximum bearing stress increases as the
concrete modulus increases when the other parameters remains the same.
This conclusion does not agree with the results from Friberg's model. As
discussed above, the higher concrete modulus means the loaded side has a
stronger load resistance capability, so that the total load and the
maximum load transferred by the critical dowel should be reduced, but not

increased as suggested by Eq. 7—4.
Since developed in 1940's, the "Effective Length" concept has been widely

used in PCC pavement design for a half century. It is worth to re-

investigate the concept again and improve the design procedure.

3 The Equivalent Effective Length(EEL)

Based on the "Effective Length"(BL) assumption, if the total load is known
and the percent of the total load transferred is assumed, the maximum

shear force can be calculated by the following formulas:

When the load is located at the edge of the joint:

154

 

 

13 C
P‘- g ( + (7-6)
2n+1-—£—2—£L
L
when the load is located at the unprotected corner of the joint:
I3 C
p = .- (7'7)
n+1_n(n+1)s
2L
where, P represents the maximum shear force,
n the number of the effective dowels on one side of the
bar under the load,
S the dowel spacing.
C percent of the total load transferred across the joint
1 radius of the relative stiffness of the slab
P, Total load

Some of the above parameters are shown in Fig. 7-1.

Since the most important parameter in the first step of the current design
procedure is the Maximum Shear Force, the BEL may be defined as a length
L which can be determined by Eq. (7-6) or Eq. (7-7) depending upon the
edge or corner loading case. The maximum shear force and the percent of
the total load transferred can be calculated by an appropriate finite
element program as discussed in Chapter 4. Under the above definition, n
is not the total number of the effective dowels on one side of the bar
under the load, it. is only the number of the dowels which can
significantly transfer load. As shown in Fig. 7-1, n is five under the EL

definition, but is only 3 under the EEL definition.

155

The EEL formulas can be obtained by solving L in Eq. (7-6) and Eq. (7-7)

as follow:

when the load is located at the edge of the joint:

L _ thz(n+1) S (7-3)

1 _ l [(2n+1) Pm - PCC]

 

when the load is located at the unprotected corner of the joint:

L Pmn (n+1) S (7-9)

 

1 ' 2 1 [(n+1) pm- ptcl
In any case the following formula must be satisfied:

n s-g 5 07+ 1) (7-10)

There exists significant difference between the concept of the EL and of
the EEL. The BL is an assumed one to predict the maximum shear force
which must different from the actual one more or less, whereas the EEL is
the one calculated by using the "actual maximum shear force" predicted by
the finite element method so that when it is substituted back to Eq. (7-6)
or (7-7), the "predicted" maximum shear force must be equal to the "actual

force" calculated by the finite element program.

Friberg proposed Ll/l = 1.8 'and Tabatabaie suggested L,/l = 1.0, both

assumed L/l constant. However, Eq. (7-8) and Eq. (7-9) indicate that L/l,
where L is corresponding to the "actual" maximum shear force of the

dowels, is a function of n, S, Pm, P” C as well as the radius of relative

stiffness l.

156

Some characteristics of the EEL

JSLAB-92 was employed to calculate the maximum shear force of the dowel
bar. The finite element mesh is given in Fig. 7-3. The numerical.
analyses were conducted for two loading cases. The first is a 9000 lb
load with tire pressure 50 psi acting at the node I in Fig. 7-3. The
loading area is 12 x 15 id% and the case is defined as edge loading. The
second is the same type of load acting at the node J in Fig. 7-3 and the
case is defined as corner loading. Using the calculated maximum shear
forces and the percentage of total load transferred across the joint, L/l

may be calculated by using Eq.(7-8) to Eq.(7-10).

By using the BL assumption, the higher subgrade modulus always reduces 1
value (Eq. 7-1), and then reduces LI (Ll = 1.8 l or LI = 1.0 l) and the
number of effective dowels n (Bq. (7-10). Since the percent of total load
transferred is assumed constant, the maximum shear force and bearing
stress will be always increased as shown in Table 7-1. However, Fig. 7-4
indicates that L/l increases when the k value increases. Fig. 7-5 shows
that the total load transferred decreases as the R value increases. The
two figures explain that the increase of subgrade modulus does not have to

increase the maximum bearing stress.

Rglatiog between EEL and 1

Fig. 7-6 plots the twelve examples with the same dowel-concrete

interaction coefficient(1.5 x lU’pci), dowel diameter(1.25 in) and joint

opening(0.25 in), but with different slab thickness and subgrade modulus

mommo coo.— ozfi. ocm zoos. EoEoE SEE 2:. MN .9“.
u w

 

157

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.2: 8m 3mm gm 8m liaise... 2m 8m owes.“ a: a: as as am .3 E. a
x -- a
.T ma
I 1.. T om
. III IIILTI I; 3
I. TIT. I em
m. I m 3
I ITL S 2
I I I om
IIII Now
III ,, III III: U:
I -I T I . wma
< VIIIIIIII7 IE- - . -. I - - w; < mm

IIWIIIIIIIIIIIII . I: II .IIIII,.IIIH.II IlIIII II. I. I- III‘ .III. “a.
_ Z. :2. lil. I 7: am: :.I. _;

158

 

 

 

 

0.0. = 0.25 in, E=4.5 mpsi

 

 

 

 

 

 

 

 

So 100 150 200 250 500 350 400 450 500 550
k. pci

 

I+h=8 in -+-n=10 in +h=12 in-B-h=‘l4 in I

 

(1) Load at the Corner

 

 

 

 

 

 

 

 

0.7 % PSlzi 5 mpciv 13:11!) in y
. ITO. = 0.25 In, 5:4.5 mpsi

o so 100 150 200 250 300 550 400 450 500 550
k. pci

 

 

 

 

I . . . . fl
I+h=8 In —+—h=10 In +h=12 In -B-h=l4 In I

J

(2) Load at the Edge

Fig. 74 Effects of Subgrade Modulus on the EEL

159

 

 

 

 

PSI=1.5 mpci, 0:1.25 in
0.0. = 0.25 in. E=4.5 mpsi

 

 

 

O 50 100 ISO 200 250 300 350 400 450 500 550
k. pci

 

I+h:3 in —o—h:10 in --h:12 in -B-h=14 in I

 

(1) Load at the Corner

 

    
 
 
 
 

 

’_.

1' 51:1.5 mpci. 0:115 in
.0. = 0.25 in. E=4.5 mpsi

 

 

 

. I
‘ I
I

I

as: - - , a - . . a 4
0 50 100 150 200 250 300 350 400 450 500 550

k. pci

 

I+h=8 in -0—h:10 in file-11:12 in -Ev—h=14 in I

 

(2) Load at the Edge
Fig. 7-5 Effects of Subgrade Modulus on the Load Transfer

Efficiency

160

 

 

 

 

 

 

 

 

 

 

 

 

 

 

45 f T .
I & /
I - \ /
I ,- \ x/
I ”I" .. \ '
I /'I K - ‘, \ ///-
I 1‘ \' \. \ /
’ “I \ ’// ' 0'8] \ /
_ K/ ; Y
\ 35 fl x], '\\\ A: 1.0 . [V f
‘ 1‘ '\. /.
I / \ \ \/
I 1 In /'
i /
J /‘\. \y
\\ \
I \ \ -'
I. I p \. PSI-:15 moci. 0:1.25 in
f ‘ q/J.o. = 0.25 in. E=4.s mpsi
;._I e \.,« n=8. 10. 12. 14 in, use. 200. 500 pci
' . ,
25 a \ . . ' - ' ,
25 30 35 4O 45 50 55 60 65 70
inchs
(1) Load at the Corner
70 : T . S.
J I/ .I '\\//
. n. ,/
: / \‘V/l '
1 "I I
60 T “ .\ T T 7|
3 1 ‘ ; I 3 ,
.\ I
I ‘/ .g '
50 f - ‘ 4‘9
‘I K I. p'SI:1.£> mpci. 021.25 in
I" ‘I ' I 10. = 0.25 in. E:4.5 mpsi
i(- - , 3 h=8. 10. 12. 14 in. k=50. 200. 500 pci
40 I 7 r Y I
25 3O 35 40 45 50 55 60 65 70

u IUChS

(2) Load at the Edge
Fig. 7-6 Relationship between the EEL and l (the Radius of Relative
Stiffness)

161

for both edge and corner loading cases. For the same radius of the
relative stiffness value 1, the L values could be very different. The
central line of the shaded area indicates that L/l a 1.3 for the edge
loading and L/l z 0.87 for the corner loading case. Therefore, the rather
wide bandwidth suggests that the assumptions of L/l = 1.8 or 1.0 might not
be an appropriate assumption to accurately predict the maximum shear
forces on the critical dowel and the maximum bearing stress of the

concrete.
4 Effects on the Maximum Bearing Stress

When Friberg developed the dowel bar analytical model in-l940's, it was
impossible to analytically predict the maximum shear force acting on the
critical dowel precisely, hence, he proposed the approximate but simple
procedure for the dowel bar design. Since the finite element method was
developed and the application of high speed computers has been very
popular, more options become available in the analysis of load transfer
mechanism. For example, it is not necessary to divide the entire analysis
procedure into two steps as summarized in section one this chapter. As
discussed in Chapter 4, the component model of dowel bar can be installed
into a finite element program to calculate the responses of each dowel,
including the distribution of bending moments, shear forces, the relative
displacements of the beam and the bearing stresses of the concrete. The
results are calculated with comprehensive consideration of all inputs
simultaneously and without more assumptions such as effective length and

percent of total load transferred. In this section, more numerical

162

examples will be given to analyze the effects of different parameters on

the maximum bearing stress of the critical dowel.

Effects of slab thickness and subgrade modulus

Fig. 7-7 shows that the bearing stress decreases when the slab thickness
increases. Four curves of the maximum stress v.s. subgrade modulus are
presented in Fig. 7-8 which indicates that the maximum stress decreases
when the subgrade modulus (k value) increases. This conclusion is
different from the results presented by Tabatabaie‘m‘" and Henrichs‘m”, (also
see Fig. 7-2 and Table 7-1 in this chapter). It is believed that the
discrepancy was caused by using the "Effective Length” assumption which
sometime can not accurately describe the maximum bearing stress

characteristics.

Effects of dowel diameter and width of the joint gpening

Fig. 7-9 indicates that the maximum bearing stress of the concrete is very
sensitive to the dowel's diameter D which might be the most sensitive
parameter among the all. Smaller diameter can cause dramatic increase of
the maximum stress. This finding qualitatively has good agreement with
Friberg and Tabatabaie's Formulas. (Eq. (7-2),(7-3) and (7-4)). Both Fig.
7-9 and Fig. 7-10 indicate the insensitivity of the maximum bearing stress
due to the variation of width of the joint opening. Using Eq. (7-4) to
predict the maximum bearing stress, the increase of joint opening from

0.25 inch to 0.75 inch provides about 16% increase of the maximum bearing

163

 

4000 r

 

 

PSI=L5 mpci. 0:1.25 in
J.O. = 0.25 in. E=4.5 mpsi

 

/

//

3000 ‘5

 

 

 

9 1 l 13 15
n. inchs

 

Ta—x :50 mi —+—:< =200 psi+K=500 psiI

 

Fig. 7-7 Effects of Slab Thickness on the Maximum Bearing Stress

 

 

I’Slzl.5 mpci. 0:1.25 in
9.0. = 0.25 in. E=4.5 mpsi

 

 

 

 

100 200 300 400 500 600
x. pci

 

I+h=8in —9—h=10in+h:12in-B-h=l4in7

#

Fig. 7-8 Effects of Subgrade Modulus on the Maximum Bearing
Stress

164

 

 

 

       

 

 

 

 

4500 v; ;991=1 ‘1 mgr-“iv [1:1 95 in
I §k=200 pci. h=10 in
. =4.5 mpsi
3500 If
.3 ,
2500 if
1500+ ~ r . .
c.s 0.75 1 1.25 1.5 1.75 2

0. inchs

 

-I-J.O.=0.I in -+—J.0.=0.253n-¢-.'.0:0.S:n -e-J.o,=o7s an I

 

Fig. 7-9 Effects of Dowel Diameter on the Maximum Bearing

 

 

 

 

 

 

 

 

 

 

 

 

Stress

4500 *L -_________._..‘__ = = J

I I PSl=1.5 mpci. D=1.25 in I

I I k=200 pci. h=10 in. E:t.5 mpsi I

3500 I
.5 i v '
0- ”" i
I

I

2500 =F — ‘ *fi
}_ l3? east is: I

7'_ g: z "71 I

1500 +— . f . 1 . . , I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

J.0.. inchs

 

g—¢I-o=.7ssn +D=I.Oin -—o=1.25h7
f-S-o=l.53n +0=L7san

 

Fig. 7-10 Effects of Width of Joint Opening on the Maximum
Bearing Stress

165

stress (0.355 x (.75-.25)/(1+.355 x .25)) x 100 = 16.3). However, by
using the dowel bar component model in JSLAB-92, the increase is only 1.2%
for the D = 0.75 inch dowel, 3.3% for the D = 1.25 inch dowel and 2.7% for

the D = 1.75 inch dowel.

 

Effects of concrete elasticitv and the dowel-concrete inggggggigg

M25222.

Fig. 7—11 presents the maximum stress curves v.s. dowel-concrete
interaction coefficient Y. As summarized by Finney“”“, the values of Y
measured by different investigators varied from 0.3 x 106 to 8.6 x 10‘ pci.
In practice, 1.5 x 10" pci is often used. However, ! significantly
depends on the concrete properties, dowel bar diameter, slab thickness,
dowel length, dowel looseness and etc. Fig. 7-11 indicates the
sensitivity of Y. The higher Y is corresponding to the higher maximum
bearing stress. That means, the dowel in deteriorated joint or with

significant looseness would have smaller maximum bearing stress.

Fig. 7-12 shows that the maximum bearing stress decreases when the.
concrete elasticity modulus increases. As discussed above this is
understandable because the higher E value indicates that the stronger
loaded slab can withstand heavier load and leave less load transferred
across the dowels to the unloaded slab. The role of higher E value is
similar to the higher R value of the subgrade, both should reduce the
quantity' of load ‘transferred. across the joint. However; Eq. (7-4)

indicates that the higher E value would cause higher maximum bearing

166

 

4000 f

 

\\

 

      

 

 

 

 

 

3000;
I
'6 a
Q I
I
2000 I
I
I 0:1.25 in. J.0. = 0.25 in.
I k=200 pcl. h:10 h
I
1000 I . ea 4T, . .
O 0.5 1 1.5 2 2.5 3 3.5
PSI. million pci

 

1+ [=35 million psi ~+- [=45 miiloin psi ‘5‘— E=5.5 million psi I

Fig. 7-11 Effects of Dowel-concrete Interaction Coefficient on the
Maximum Bearing Stress

r Sl=1.5 mpci. 0:1.25 in
J.0. = 0.25 in

 

 

4000 I
I
I
1
I

 

 

 

 

 

 

 

 

‘7, \: ‘
2000 I s 3
IOOOI - , . .
3.5 4 4.5 5 5.5 6

E. milIion psi

 

fur-mic in, «:50 —+—h=Io in. uzzoo -v-—n=i4 in, k=50 -e—n=u in, k=500

 

Fig. 7-12 Effects of Concrete Elasticity Modulus on the MaXimum
Bearing Stress

167

stress. The discrepancy might be partially caused by the application of
Eq. (4-7) in ILLISLAB and Eq. (4-12) in this‘study. The former one
produces nonequilibrium element and might lead to unreasonable results
sometime.

For convenience to review, the maximum bearing stress of dowel, for h - 8,
10, 12, 14 inch, k=50, 200, 500 pci, D = 0.75, 1.25 and 1.75 inch, 3.0. 8

0.25, 0.5, 0.75 inch, E = 3.5x10‘, 4.5x106, 5.5x“ psi are given in Table 7-

2' 7-3' and 7-4e

General principle has been found by analyzing the results of hundreds of
numerical examples: the higher values of dowel diameter, slab thickness,
Iconcrete modulus, and subgrade modulus can reduce the maximum bearing
stress of the concrete under the critical dowel. The maximum bearing
stress is not sensitive to the width of joint opening but is very
sensitive to the dowel-concrete interaction behavior, though which is
difficult to control. Three tables are presented for obtaining the

maximum bearing stresses of concrete under the critical dowel bar.

The discovery of error in the stiffness matrix of dowel bar used in some
finite element programs, and of the inappropriate utilization of the joint
effective length made it necessary to re-evaluate some design procedures

for the dowel system. The major findings in this chapter are:

168

e Equation (7-4) could provide some questionable results. The maximum
bearing stress does not proportionally increases as the concrete
elasticity E increases and it is also not so sensitive to the width of

joint opening, as indicated in the equation

e The maximum bearing stress on the critical dowel increases as the
subgrade modulus decreases which is different from the conclusion
presented in some literatures. This finding indicates that the most
critical season in a year for the maximum bearing stress is spring for the
thawing reduces the subgrade modulus, rather than winter in the wet-frozen
region. And the thawing effect could cause 10 to 20% difference inthe

maximum bearing stress.

e The utilization of "Effective Length" (EL) assumed in 1940's and
modified at end of 1980's underestimates the maximum bearing stress in
some cases. The "Equivalent Effective Length" (EEL) concept has been

developed to prove that the EL assumption needs more studies.

e The most critical dowel is the one under a tire load nearest to the
unprotected corner. The maximum bearing stress could be two times even

higher than that of the critical dowel under the tire load at the edge of

the joint.

169

e Due to the significant difference between the existing and the
developed models in predicting the maximum bearing stress, it is
suggested that all empirical models which use Eq (7-4) to calculate the
maximum bearing stress and then to predict the joint faulting should be

checked before being employed in engineering projects.

170

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 7-2 The uaximum Bearing Stresses (k = 50 pci)
L m
a J.0. D h (in)
(10°psi) (in) (in)
8 10 12 14
3.5 0.25 0.75 6271 5446 4833 4390
0.25 3439 2973 2635 2391
0.75 2439 2084 1813 1656
0.5 0.75 6463 5596 4957 4510
1.25 3507 3134 2687 2438
1.75 2464 2108 1854 1673
0.75 0.75 6612 5708 5049 4603
1.25 3572 3090 2737 2482
1.75 2488 2131 1876 1693
4.5 0.25 0.75 5948 5158 4584 4176
1.25 3256 2815 2498 2265
1.75 2298 1964 1733 1567
0.5 0.75 6127 5299 4703 4293
1.25 3323 2872 2548 2311
1.75 2323 1989 1754 1583
0.75 0.75 6259 5398 4795 4386
1.25 3384 2924 2594 2347
1.75 2347 2012 1775 1601
5.5 0.25 0.75 5707 4937 4395 4018
1.25 3113 2693 2394 2169
1.75 2193 1874 1656 1500 .
0.5 0.75 5867 5064 4510 4137
1.25 3181 2748 2440 2217
1.75 2218 1897 1678 1515
0.75 0.75 5989 5160 4602. 4237
1.25 3239 2765 2482 2259
é$4flfi &%_§§=‘L—=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.171

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 7-3 The Maximum Bearing stresses (k 8 200 pci)
E J.O. D h (in)
(10°psi) (in) (in) E
8 10 12 14
3.5 0.25 0.75 5434 4721 4157 3717
1.25 2973 2611 2324 2097
1.75 2041 1785 1586 1429
0.5 0.75 5536 4783 4191 3737
1.25 3034 2661 2365 2130
1.75 2069 1810 1608 1450
0.75 0.75 5587 4799 4187 3722
1.25 3087 2703 2398 2157
1.75 2096 1833 - 1629 1467
4.5 0.25 0.75 5165 4459 3917 3497
1.25 2836 2478 2201 1980
1.75 1944 1692 1501 1350
0.5 0.75 5251 4507 3944 3510
1.25 2893 2524 2238 2011
1.75 1971 1716 1522 1369
0.75 0.75 5289 4513 3932 3492
1.25 2942 2561 2266 2034
1.75 1996 1738 1541 1385
5.5 0.25 0.75 4951 4254 3728 3334
1.25 2728 2374 2102 1893
1.75 1867 1620 1433 1290
0.5 0.75 5025 4292 3747 3347
1.25 2781 2417 2136 1920
1.75 1893 1643 1453 1308
0.75 0.75 5052 4291 3732 3325
1.25 2827 2451 2162 1941
$=éflandh

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

172

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 7-4 The Maximum Bearing Stresses (k = 500 pci)
E J.O. D h (in) H
(10°psi) (in) (in)

8 10 12 14

3.5 0.25 0.75 4688 4076 3579 3178

1.25 2638 2342 2096 1892

1.75 1796 1599 1436 1302

0.5 0.75 4705 4061 3544 3132

1.25 2683 2375 2120 1909

1.75 1822 1621 1455 1318

0.75 0.75 4673 4004 3474 3055

1.25 2717 2399 2135 1918

1.75 1844 1640 1470 1330

4.5 0.25 0.75 4460 3846 3357 2970

1.25 2528 2229 1984 1783

1.75 1723 1524 1362 1231

0.5 0.75 4464 3821 3316 2920

1.25 2569 2258 2004 1798

1.75 1747 1544 1379 1245

0.75 0.75 4421 3757 3241 2840

1.25 2599 2277 2016 1803

1.75 1768 1561 1393 1256

5.5 0.25 0.75 4276 3664 3186 2813

1.25 2440 2138 1896 1700

1.75 1664 1464 1305 1176

0.5 0.75 4271 3633 3140 2760

1.25 2477 2164 1913 1712

1.75 1687 1483 1320 1189

0.75 0.75 4221 3563 3063 2680

1.25 2504 2181 1922 1716
1AA? 1499 gig—.422.—

 

 

 

173

CHAPTER EIGHT
LOOSENESS EFFECTS ON THE PAVEMENT RESPONSES

1 Introduction

A nonlinear elastic model was recently developed to simulate the dowel bar
looseness mechanism. For any given dowel bar looseness, this model can
predict the pavement responses, including the. stress and displacement
distributions and load transfer capability. A previous experimental
study““”””” found that the measured initial looseness was typically about
0.003 inch, and that the looseness was approximately doubled after 2
million load cycles. The numerical results presented in Chapter 5 are
limited to the responses of a two-slab system (Fig. 1-1) acted upon by two
load cases: a concentrated 9000-lb load acting at the center of the joint

and an 18000-1b single axle load with four tires acting at the joint

(Fig. 4-3).

Additional numerical results are presented in this chapter to investigate
the interactive effects between dowel looseness and several design
parameters, such as the.subgrade modulus, k, the dowel diameter, D, and
the slab thickness, h. The effects of dowel looseness on critical dowel
shear forces, maximum principal stress, maximum slab displacements and
load transfer efficiency are shown for various looseness levels.
Responses of a four-slab system with transverse and longitudinal joints
under edge and corner loading cases are also presented in this chapter to

further illustrate the effects of dowel bar and tie bar looseness.

174

2 Major Findings of the Responses of a Two Slab System

A finite element mesh for two-slab system is given in Fig. 1-1. The major

example input data are listed as follow:

Elastic Modulus of the Concrete, Ec 4,500,000 psi
Poisson's Ratio of the concrete, uc 0.15

Elastic Modulus of Dowel Bar, E, 29,000,000 psi
Width of Joint, J0 0.25 in
Poisson's Ratio of Steel, u, 0.3
Dowel-concrete Interaction Coefficient,PSI 1,500,000 psi/in
Subgrade Modulus, k 200 psi/in

Slab thickness, h 10 inch

Dowel diameter, D 1.25 inch
The variation of values of k, h, D and looseness are indicated in each

figure.

Figures. 8-1 through 8-3 illustrate some responses (the maximum shear
force of the top dowel in Fig. 1-1, and the maximum displacement and
maximum principal stress of the loaded slab) and the percent of load
transferred across the joint versus k, D and h for different dowel bar

looseness. The main findings can be summarized as below:

e Increased dowel looseness increases the maximum principal stress of
the loaded slab. For k = 50 pci, 0.003 in of looseness can produce
a 3.2% stress increase, and 0.006 of looseness can produce a 5.5%

increase in the maximum principal stress. For k=500 psi/in,

175

 

 

 

 

 

 

 

 

 

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Fig. 8-1 The Effects of k for Varying Dowel Looseness

176

 

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Fig. 8-1 The Effects of k for Varying Dowel Looseness (continued)

177

 

 

 

 

 

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Fig. 8-2 The Effects of D for Varying Dowel Looseness

178

 

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Fig. 8-2 The Effects of D Under for Varying Dowel Looseness(continued)

179

 

 

 

 

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(b) Maximum Displacement vs. Slab Thickness
Fig. 8-3 The Effects of Slab Thickness for Varying Dowel
Looseness

180

 

 

 

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SLAB THICKNESS h(inches)

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(d) Percent of Total Load Transferred vs. Slab Thickness
Fig. 8-3 The Effects of h for Varying Dowel Looseness(continued)

181

the corresponding increases are 11.8% and 24.3% respectively.
However, the maximum principal stress of the loaded slab resting on
a stronger subgrade is always lower than that of the slab resting on

a weaker subgrade if all other factors are held constant.

Increased dowel looseness increases the displacement magnitude of
the loaded slab and decreases the displacement magnitude of the
unloaded slab. The increases due to 0.006 in looseness for the-
loaded slab ranged from 6.1% to 25% for k = 50 psi/in to 500 psi/in,
from 13% to 15% for D = 0.75 to 1.5 inch, and from 12.0% to 19.6%

for h = 8 to 14 inches.

Increased dowel looseness decreases the amount of load that an be
transferred across the joint. Numerical results indicate that the
number of dowels which are active in transferring load across the

joint decreases when the looseness uniformly increases.

Increased dowel looseness decreases the maximum shear force on the
critical dowel (the top bar in Fig. l-l). This shear force decrease
is caused by the decrease of load transfer efficiency. As discussed
in Chapter 7, the maximum bearing stress of concrete is proportional
to the shear force. Therefore, dowel looseness does not cause an

increase of the maximum bearing stress of concrete.

11'-0"

11 spaces @ 12 inches

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9 spaces @ 12 inches

182

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183

3 Major Findings from a Four-slab System

A four-slab system with one traffic lane and a shoulder was employed to
investigate the effects of looseness of the tie bars which often connect
the traffic lane and the shoulder. The system includes 1.25-in diameter
dowel spaced 12 inches in the transverse joints, and 0.625-in tie bars on
an 30-in centers in the lane-shoulder joint. The plane view of the four-
slab system, is shown in Fig. 8-4. Because the system is a symmetric one,
only one half of the entire system is given. Two loading cases have been
considered: a corner loading case with a single axle load at the
transverse joint and a tire at the corner, and an edge loading case with
the single axle positioned 75 inch away from the transverse joint and with
one tire at the edge of the longitudinal joint. Both loading cases are

presented in Fig. 8-4.

Fig. 8-5 shows the shear force distributions of the dowel bars along the
transverse joint. A comparison between Fig. 8-5 and Fig. S-ll(a)
indicates that the effects of dowel looseness on the two-slab and four-
slab systems are similar. Fig. 8-5 shows that the shear force on the
critical bar increases slightly when the looseness increases from zero to
0.0015 inch, and then decreases when the looseness continue to increase.
The increase due to the 0.0015 inch looseness is 6.2%. TellerIm'I and
Snyder“”” indicated that the initial looseness of dowel could be as much
as 0.003 inch. This finding implies that after the dowels are installed
in the joint, looseness always decreases the shear forces on the critical

dowel bar. Fig. 8-6 shows a similar effect for tie bar looseness.

184

 

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186

Fig. 8-7 shows that increasing the looseness of the dowel bars and tie
bars always increases the maximum principal stress for the corner loading
case. The 0.003-inch looseness can produce a 13.1% stress increase, and
the 0.006-inch looseness can produce a 21.1% stress increase. Fig. 8-8
shows that the maximum edge stress decreases when the looseness increases
from zero to 0.0015 inch . When the looseness continues to increase, the
maximum edge stress continuously increases. The maximum edge stress for
the pavement without tie bars between the traffic lane and shoulder is
234.8 psi, which is 10.7% and 15.3% higher than the results for looseness
= 0.0 inch and looseness = 0.0015 inch. As mentioned above, the
existence of initial looseness leads to the conclusion that looseness

increases the maximum edge stress.
4 Conclusions and Recommendations

The numerical analyses presented in this chapter and in Chapter 5 verify
that the effects of dowel bar looseness should be considered in
mechanistic design procedures for jointed rigid pavements. 'The looseness
generally increases the pavement responses of the loaded slab, including
the maximum displacement, the maximum corner stress and edge stress. Any
mechanistic design procedure based on fatigue analysis could be
significantly affected if the increases of slab stress due to the

looseness of dowels and tie bars are considered.

The numerical results presented in this chapter and Chapter 5 were
obtained by static analysis and without consideration of the combination
of traffic loads and temperature gradients. If the dynamic effects of
traffic loads are considered, the dynamic factor (ratio of the maximum

dynamic loading and the static loading) could be between 1.3 to 2.0

187

(GillespieII99II, and Stoner“°9”). In this case, the combined effects of
dynamic loads and looseness would be even more significant on the maximum
stress responses of the slab. When the temperature gradient and moisture
variation are considered, the combination of looseness and temperature
gradient could produce higher corner stresses during the night and higher
edge stresses during the day. The analysis will be more complicated
because ‘two 'types of nonlinear’ behaviors should. be considered: 'the
nonlinear behavior of the loose dowel bars, and the nonlinear support
provided to the uplifted slab as it comes in contact with the subgrade
gradually under increasing load. The interaction between dowel looseness
and dynamic loading, and between dowel looseness and temperature and/or

moisture gradients needs further study.

188

CHAPTER NINE
CONCLUSIONS AND RECOMMENDATIONS

Dowel bars are widely used in rigid concrete pavements to transfer loads
across joints and thus prolong the service life of the loaded slabs. It
has been shown by many experimental studies that initial looseness exists
to some degree in all dowel bars. The looseness is a. function of
construction quality, accumulation of traffic loads, and exposure to the
field environment. In current design procedures dowel bars with small
looseness are treated the same as without looseness. Somel design
procedures consider the effects of looseness by reducing the stiffness of
each dowel bar. Both models with or without consideration of the dowel
bar looseness are installed in finite element based programs to predict

pavement responses.

Recently, some widely used models of dowels without consideration of
looseness were checked by basic theory and evaluated by numerical
analysis. Some errors in the dowel bar models and in the computer program
using the model have been discovered and corrected. A component dowel bar
model has been developed to simulate the load transfer mechanism. The
impacts of the discovered errors were also discussed. A nonlinear
mechanistic model has also been developed to simulate the dowel bar
looseness mechanism. The model can be used to predict various pavement
responses at different stages of the pavement service life, including

stresses, displacement distributions and load transfer capacity.

189

The ma'o conclus'ons of this researc are s d be ow:

e The dowel bar stiffness matrix in some finite element programs for
jointed concrete pavements (e.g. JSLAB and ILLISLAB) fails to meet some of
the basic requirements of the finite element method. The largest source
of error results from this failure to satisfy dowel bar equilibrium
conditions. This dissertation presents proof that the stiffness matrices
employed by JSLAB and ILLISLAB represent elements that are not in
equilibrium. This problem appears to be caused by an inappropriate
modification of the shear-bending beam element that is found in.many basic

structural engineering texts.

e A numerical sensitivity analysis showed that ignoring dowel bar
equilibrium requirements can produce significant errors in the predicted
responses of concrete slabs and dowel bar load transfer systems. Based on
the numerical analysis, it was found that the non-equilibrium stiffness
matrix used in JSLAB overestimates the stress responses on the unloaded
slab by up to 18.7% and underestimates the stress at edge of the loaded
slab by up to 9.8%. The maximum bending moments at two ends of the dowels
calculated by the non-equilibrium model was 10 times higher than those

calculated by the equilibrium component model.

e ' A complete comparison between experimental data, published Iby
Keeton“”“, and the analytical results from the component model
(including distributions of bending moments and shear forces of the

dowels, and bearing stresses of the concrete under the dowels) indicated

190

that trends of all predicted and measured responses are the same. The
comparison verified that the component model can reasonably predict the
load transfer characteristics of the dowel bar system and has potential to
predict responses of slab and dowel bars by calibrating the subgrade

modulus and dowel-concrete interaction coefficient.

e The maximum bearing stress on the concrete under the critical dowel
has been found to be one of the most influential parameters in FCC
pavement joint design. The magnitude of maximum stresses have direct
effects on the accumulation of joint faulting which is a key indicator to
evaluate the PCC pavement performance. The sensitivity of the maximum
bearing stress has been analyzed using the component model. It was found
that higher values of dowel diameter, slab thickness, concreteImodulus and
subgrade modulus can reduce maximum bearing stress of concrete under the
critical dowel. Maximum bearing stress is insensitive to joint opening,
but very sensitive to the dowel-concrete interaction coefficient.
Unfortunately the interaction coefficient is difficult to estimate. ‘Three
tables of the critical maximum bearing stress in terms of slab thickness,
subgrade modulus, concrete elasticity modulus, width of joint opening and

diameter of dowels are presented in this dissertation. as a design

reference.

e Existing models for predicting the maximum bearing stress have been
compared with the developed component model. Results obtained with these
models are different from those obtained by the component model. For

example, based on Friberg'sII‘mI and Tabatabaie'sII‘m' model the maximum stress

191

is much more sensitive to the width of joint opening than in the component
model. Friberg concluded the higher value of subgrade modulus increases
the maximum bearing stress, and Tabatabaie concluded the higher concrete
elasticity modulus increases the maximum bearing stress. The source of
discrepancy is the effective length assumption developed by Friberg and
modified by Tabatabaie. The component model should provide more practical
conclusions, since the results are directly obtained using the model

installed in a finite element program without additional assumptions.

e Due to the significant difference in bearing stress predictions
between existing models and the one developed in this research, pavement
performance models which have been developed based on the maximum bearing
stress must be evaluated. :n: is suggested these models be evaluated

before being employed in PCC pavement design and rehabilitation.

e A mechanistic nonlinear model to simulate the mechanism of dowel bar
looseness was developed. The model estimates pavement responses to load,
including stress, displacement distributions and load transfer capability,
with consideration of dowel looseness at different stages of the pavement

service life.

e Numerical analysis were conducted to investigate the effects of
dowel bar looseness on critical pavement responses. Parameters included
magnitude of dowel looseness, configuration and location of traffic loads,

shoulder edge support effects, variation of subgrade modulus, dowel

192

diameter and slab thickness. the numerical examples indicate that an
increase in dowel looseness of 0.003 inch could produce a corresponding
10% increase in pavement edge stress, a 14% increase in maximum interior
stress, and a 14% loss of joint load transfer efficiency. Previous
studies have found that this magnitude of looseness exists initially in
many doweled full—depth replacement joints. Previous experimental studies
have shown that after two million load applications, the amount of
looseness had doubled from initial levels, which implies that the
resulting critical stress may be 20 to 28% higher than those obtained by

if dowel looseness is neglected.

e Numerical examples for a four-slab system using the nonlinear model
also show that the looseness of tie bars across the traffic lane-shoulder
joint significantly affect the ability of the shoulder to withstand edge
loads. The location of critical stresses in the concrete slab depends on
the load transfer efficiency of the lane-shoulder ties as well as upon

traffic load configuration and location, and transverse joint load

transfer efficiency.

e Dowel bar looseness of 0.003 inches can change rigid pavement stress
distributions quite significantly. This emphasizes the need for
construction quality during dowel bar installation to insure maximum
performance life. Furthermore, dowel coatings used for corrosion
protection or to prevent bonding to the concrete should be carefully

evaluated to insure that they do not allow vertical dowel movement.

193

e The maximum longitudinal stress and principal stress in rigid
pavements are sensitive to dowel looseness. This becomes even more
critical to the development of transverse cracks when temperature and
moisture gradients (curling and warping stress) are considered. This
could significantly affect PCC thickness design procedures that are based

on fatigue criteria.

Some recommendations for future resegrch are given below:

0 Joints are necessary for PCC pavements, however, they are also the
weakest portion in PCC pavements. The joint deterioration is caused by
many sources. The understanding of the interactive effects among these
source parameters is very important, such as interaction between the loss
of subgrade support and dowel bar looseness, dynamic loading, looseness
and faulting. The understanding can be reached by experimental studies
through field survey and can also be reached by' using appropriate
mechanistic models. The lack of mechanistic model‘to simulate the
deteriorated pavement suggests that more efforts should be made for
developing appropriate mechanistic models to simulate the behavior of

deteriorated PCC pavement.

e WhenImore parameters are considered.in.mechanistic analysis, optimal
design of FCC pavement will become a desired goal. Objective function
should be determined based on long term benifit, constrain conditions

might be selected based on Federal and State requirements and local needs.

194

The incorporation of mechanistic and empirical models may simplify the
mathematical problem and make it possible to select the design parameters

to minimize the objective function.

e Efforts should be made to develop a technique to measure the
looseness of dowel bars by non-destructive test. The information is
needed not only for pavement response prediction, but also for the

pavement performance prediction.

195

REFERENCES

AASHTO, 1986.
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Ball,C.G., et al., 1975.
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Bathe, K J., 1982.
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Beeson, M C., et al., 1981.
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Ciolko, A.T., et al., 1979.
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196

FAA, 1978.
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Finney, E.A., et al., 1947.
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Dowels," Proceedings, HRB, Vol. 27, 1947.

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197

Ioannides, A.M. et al., 1984.
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Design of Load Transfer Loints in Concrete Pavements." Proceedings of the

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1984.

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Nashizawa,T., et al., 1989.
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198

Nasim, M A., et al., 1991
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199

Tajabji, S.D., et.a1., 1986.
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"The Finite Element Method," McGraw-Hill Book Company Limited, London,
England, 1977.

201

111EP1€TVZIIJ(14.
.SlqkllfifiiIVZZAWCXTTZ)FWSL ETIYQALIDVI4JVHDlSEIYQIESE?JQIAEIYQIIZIES'(717

14 IQZQ4JIYE'IZIQEflllliAWI’

1. Notations

 

 

 

 

 

 

 

 

 

The jth nodal Mx
displacement
vector is: h l 1 j
W l
AJ= ex
6, j .6
l
JZ,vv

Fig. A-l The Positive Direction of Forces

and Displacements

2 Assumed Displacement Function

W(x, y) =a1+a2x+a3y+a,x2+a5xy+a6y2+a7x3+a8x2y+a9xy2
3 3 3
+a10y +a11x y+a12xy

202

Shape Functions

 

 

 

 

N1 = 1_ 3x2_ xy _3y‘2+ x3 +3ny+ 3x372+ y3 _ x3y _ xy3
4A2 4A3 432 4A3 8.2123 8AB2 433 82133 8AB3
N2 = 7.32.2.2- XYZ- Y’ . X3”

 

2A B ZAB 4B?- SABZ

x-.£i-_x_x. X3 + XZY- Xiy
A 28 4A7- 2A8 8AZB

_zi- xyz _ y’ + 20!"
23 4A8 432 BABZ

N = _XX_X2Y+ X3},
5 23 2A3 8,423

N _ 32:2,, xy - x3 _3x2y_ 3xy2+ X33,+ xy3

7 — 4A2 4A3 4A3 8.2123 8.2132 8.1133 8AB3

N = __X_J_’+XY2- X5"
8 2A 2A3 81432

2 3 2 3
x+x +xy_xy

N =-
9 2A 4A7- 4A3 8A2B

N = _ «‘0’ + 3x2y+3xy2_ X3}! - xy3
1° 4A3 BAZB 81-132 81138 8AB3

N ___ xy2_xy3
1‘ 4A8 8A3?-

203

4 Elements of Strain Matrix

B(1,1) ___ 3 _ 3x_ 3y +3X}!
2A2 2A3 4.428 4.1138

B(2,1) = 3 _ 3X _ 3y+ 3xy
232 4.1132 283 4.1133

1+3x 3y_3x2_3y2

 

 

 

 

 

 

 

 

 

 

B(3,1) = - +
2A3 2.428 2le2 4A3B 4.1183
B(1,2) = 0
39,2) = -3.+_£+ 3Y- 3XY
3 AB 282 4.2182
2
B(3,2) - 3-3’5 35’
.A 2U? 4LABZ
3(1,3) = _2.— 3" —_5L+ 3):}!
A 2A2 AB 4AZB
B(2,3) - o
2
B(3,3) _ —_]_‘+£§— 3x
B AB 412123
8(114) = By " 3X},
4A2B 4A3B
3(2’4) =_ 3 + 3x +3y_3xy
232 4.2182 233 4.1133
2
BOA) = 1 _ 3x_ 3y+3x +3y2

 

 

2A3 ZAZB 24le2 4A3B 4AB3

B(1,5) = O

204

 

 

 

 

 

 

 

 

 

1 X By 3xy
= -—+ + -
8(2'5) B 2A8 23?- 41132
3(3 5) = -_X.+ 3Y2
' AB 4.432
3X}’
8(1'6) = 1;.
AB 4.423
8(216) = 0
1 2x 3x2
B '6 = —-—+
(3 ) B AB 4,2123
8(1,7)=- 3 + 3X. By - My
223?- 2A3 4AZB 42138
22182 4.21.133
2 2
3(317)= 1-3x-3y+3x+3y
2A3 ZAZB 22132 4.433 4.111133
B(l,8) = o
B(2.8) = -_X_+ 3XY
AB 4.432
2
B(3.8) = 14.2.55- 3y

B(1.9)

B(2,9)

I
O

x 3x2
B 3 , 9 - _
< ) AB 4.2123

 

204

 

 

 

 

 

 

 

 

B(3,5) = -—2;+ 3Y2
AB 4AB2
3(116) = l—M—
AB 4AZB
B(216) = 0
B(3,6) = 3-33.8. 3X2
B AB 4AZB
3(117) = - 3 + BX... 3y _ BXY
2A2 2A3 4,423 4.433
B(2r7) = 3X " 3”
2.452 4.4133
B(3,7) = 1 - 3" - 35’ + 3X2 + 3y2
2A8 2.2123 2.432 42133 4.4133
8(118) " O
B(218) = -_§L+ 3XY’
AB 4.2132
B(3,8) — RH- 3y2

B(1.9) i- 3X- Y + 320/

B(2,9)

ll
0

2
B(3.9) X 3"

205

 

 

B(1,10) = - 3” + BXY
4AZB 4A3B
B(2,10) = - 3x + 3’”
4213’- 4.21133

2 2

B(3,10)= 1 + 3x + By _ 3x __ 3y

 

 

2A3 ZAZB ZAB2 4.433 42183

B(1,11) =0

I
N

+
w
e

£3C2.11J

2
3(3.11) -.- _Z.-_3_L

 

 

1U? ILABZ
3(1,12) = y — 3’”
2A8 4,423
B(2,12) = O

2
8(3'12) = —_)_(_+ 3X
AB 4AZB

Substituting

x=y=0 (defined as node 1) into above element formulae, the

strain matrix for node 1 can be obtained:

 

 

 

 

 

/ 3 2 3 1 l

o — o o o - o — o o 0

2A2 A 2A2 A
3 2 3 1
= —_ o — -— o o o 0 0

B1 23?- B 2132 B O O

- 1 _l _tl .43. o .l _J£_ —~$ 0 --—$—- 0 O

\ ZAB A B 2AB B ZAB A 2113 J

 

Similarly, for the other three nodes:

206

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/ o o o 3 o 3 o o o - 3 1‘
2.1.1.2 A 2A2 A
3 1 3 2
= — _ o _ o o o o o o o
3’ 2B2 B 282 B
_ 1 o _l _L .1 l _1_ __1_ __1_ 0
\ZAB BZABABZAB ZAB ZAJ
f- 3 o __1_ o o o 3 o —3 o o 0‘
2A2 A 2A2 A
3 2 3 1
= o o o o o o —— o - -— o
B” 23?- B 282 B
_ 1 _1; O l O ___1 —_£ —_1. -___l O l
\ 2213 A 2A8 ZAB A B 2218 B)
K o o o - 3 o -l o o o 3 o -3‘
2A2 A 2A2 A
3 1 3 2
= o o o o o o - — o — o
B‘ 252 B 252 B
\ 2A3 2A8 A 2118 B 2A3 A 131'
5 Stress Matrix
Let:
_ A
p B
1.)x D1 o
D: D1 Dy 0
o o ny

The partitioned stress matrices can be written as:

n,=[nflnj,]=osj (j=1,2,3,4)

where:

207

 

 

6P'1Dx+6pD1 -8AD1 BBDx -6le —4.ADl
1 -
Rn: 4A3 6p 101%pr -8ADy 83131 45pr -z1ADy
'2ny 4:3ny “411ny Zny O
-6p'1Dx o 430x 0 o o
_ 1 -
R13 - m '6}? 1D1 0 4BD1 0
20,0, 43ny o -2ny o o
-6pD1 4AD1 o 6p’1Dx+6pD1 8A1)1 83D
_. 1 -
Ian-m 43pr 4.1113y 0 6p 1131%pr 8A0y 83131
-2ny 0 -4Any 20,0, 4Bny 4.11ny
o o o -6p'1DX o 430x
1 _
Rn=m o OO-6p1D1 0 413191
2ny 0 0 'Zny "4Bny 0
—6p-1Dx 0 ~4BDX 0 O 0
Rn = fig -6p'1D1 o -4BD1 o o o
-2ny 413ny o 2ny o o
61.9413;qu -8AD1 -8BDx -6pD1 -4AD1
_ 1 -
Rn- 4A3 6p 101%pr -8ADy -8.BD1 -6pr -4A1)y
2ny 43ny -4Any -2ny o
o o o -6p'1Dx o ~4BDX
_ 1 _
R‘1 — m 0 0 0 -6p 1D1 0 '4301
-2ny o o 2ny 413ny o

208

-6pD1 4A1)1 o 6P'1Dx+6pD1 8A1)1 -BBDX
-6pr 4ADY O 6p'1D1+6pr BAD), -BBD1

2ny o -4ADxy -2ny 43ny 4.11ny

 

_ 1
R" 4A8

209
APPENDIX B

BENDING MOMENTS 0F UNBONDED AND BONDED TWO-
LA YER ELEMENTS
The general form of bending moment of unit length is:

Al==j; z 0 d2’

where a'is the bending stress, 2 is the distance between the stress point
and the neutral axis of the cross section area, h is thickness of the

cross section. ,

1. Unbonded case

 

 

 

 

4’4?

 

 

 

 

 

 

 

”if”?
\
./

"3 > “W,

 

CROSS SECTIGI STRESS DISTRIBUTIGI

Fig. B-l Unbonded Case

210

M=um+umm=ora+u¢

Where
ht
—2' 3 123
um = ( f 22c“, dz ) r, m, = i c“, x, mt
ht
-?
Similarly
hi
Mboetal 1—2 Chocta- Kc
So
123 11,3,
D =1—2 ctop + ficbotcon
aDan,+-lhmmw_

.E1h3 p 1. 0
layer 12(1-H2) 0 0 1;E

2. Bonded case

(1) An Equivalent Layer

(3'1)

(3'2)

(3'3)

(3’4)

NL?

3
”13' ”hr Ivu?

211

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

K———— 1'————fi K—-—- 1 -———fl
r
E, e,
_i
mid-plain '1 mutual-plane M
1 l
E “b
mfl- r
9"" _L Eb
Eb
L 1
l<—— Eh. __94
. . E! stress
original . distribution
cross section equivalent

cross section

Fig B-2 Equivalent Layer

 

 

E h E 122
I(ht+ b )=ht(hb+—£)+—£ __b
Etha 2 E: 2
(3'5)
2 2
E. + ht hb + £13 £2
_ 2 2 E:
‘ht+ fig hb

 

l
hb 3h (hc+hb)
"“’=I_—2'= E
hc+_2hb (3-6)
E
C

 

(3'7)
(2) Bending Moment for Two-Layer Element
435%) ayfl’
M= ( [ht z2 cm dz 4» Ihbz2 Chou” dz )Kc + Me
'(Cc‘?> “13-7
= ( Dtop + Dboctan ) Kc + MI:
1 p. 0
2 3
_ E(12a h+2h) ll 1 0 Kg + M:
layer 12(1-‘1 ) 0 0 l‘E
2 (B-e)

Drawn:

213

APPENDIX C
ST IFFNESS MAT RICX OF A PLATE ELEMENT

1 The Virtual Mork Principle

The internal virtual work of layered pavement can be written as:
Swim = f or; M dxdy “'1’
area

where 6Kc is virtual curvature vector corresponding to the virtual
displacement vector 6V', and

6Kc = 8 6V“

6KcT = (”)1' 3T

M = D Kc-Flt = D B V°-+lt
Substituting the above expressions into Eq. (C-l) to obtain:

ow,“ = ov'TUfBT D B dxdy) v' + ov'TffsT Mt dxdy
Employing:
D = Dcap + Dace-ca-

to obtain:
owingov" (”(BT Du,p B+BT 0mm 8 ) dxdy )V‘

+ av'Tf 13* Mt dxdy “3'2,

=5v“(sm + swam)!” - 6V“ P,

Stop and 5m...“ in Eq. (C-Z) are the stiffness matrices of the top and bottom

layers of the pavement respectively. Pt is an equivalent nodal force

214
vector due to temperature variation. The expressions are given in Eq. (C-

6).

The external virtual work consists two portions, one is contributed by
load p(x,y), the other is contributed by the subgrade reaction forces.

Winkler model is employed in the study, so that: 1

Q(X.y) = -k(x,y) W(x,y) (0-3)

where k(x,y) is the modules of subgrade reaction. The external virtual

work can be written as:
aw“, = ff 5W[p(x,y> + q<x,y>] dxdy
8198

By using Eq. (2-8), the virtual displacement vector can be written as:
SW = NSV' = 6v“ NT
Thus:

5W.“ = avfffma p(x, y) NT dxdy

- 5V'TUam k N' N dxdy V‘ (c-n

= 6V" [Pd - SM 1"]

Where, Pd is equivalent nodal force vector due to applied loads and Ssub is

215
the stiffness matrix of the subgrade. They are given in Eq. (C-G).

Equating the internal and external virtual works of each element to lead:

(Stop 1' Sbottal + stub) V. = Pd 1' Pt (C-S)
where:
Sbotton = f "“37 Dbocto- B dXde
=11”. k N: N am
Pd = [fared p(x,y) N’r dxdy ' (C-G)
So:
5. = Stop 1' Sbacco- + snub (C-7)

2 Stiffness Matrix of Top Layer S“, and Bottom Layer S

bottom

The following notations are used in computer program:
S(i,j) The element of stiffness matrix at the ith row and the jth
column. A
2A Length of the element in x direction
28 Length of the element in y direction
E Modules of elasticity

216

u Poisson’s ratio
h Thickness of each layer
R =Eh3 / [60AB x 12(1-u2)] for unbounded case

=E(12a2h+h3) / [60AB x 12(l-u2)] for bonded case

a Distance between the mid-plane of each layer and the neutral

axis of the equivalent cross section

0 =(1—u) / 2
AB = A x B
BS = (B / A)2
A5 = (A / B)2
A54 = 4 x A2
BS4 = 4 x 32

The elements of stiffness matrix are listed as follow:
S(l,l) a R x (60 x BS + 60 x AS + 30 x u + 84 x U)
S(2,l)
S(2,2)
S(3,1) = R x (30 x BS + 15 x u + 6 x U) x 2A

R x (-30 x AS - 15 x u - 6 x U) x 28

R x (20 x AS + 8 x U) x BS4

S(3,2) = R x (-15 x u) x 4AB
S(3,3) = R x (20 x BS + 8 x U) x AS4
S(4,1) = R x (30 x BS - 60 x AS - 30 x u - 84 x U)

S(4,2) = R x (30 x A5 + 5 x u ) x 23

S(4,3) = R x (15 x BS 15 x u - 6 x U) x 2A
5(4,4) = 5(1,1)
S(5,1) = -S(4,2)
S(5,2) R x (10 x AS

S(5,3)

2 x U) x BS4

0

5(5,4)
5(5,5)
S(6,l)
S(6,2)
S(6,3)
S(6,4)
S(6,5)
S(6,6)
5(7,1)
5(7,2)
5(7,3)
5(7,4)
5(7,5)
S(7,6)
5(7,7)
S(8,1)
S(8,2)
S(8,3)
S(8,4)
S(8,5)
S(8,6)
S(8,7)
S(8,8)
5(9,1)
5(9,2)
5(9,3)

217
-S(2,1)
S(2,2)
S(4,3)
0
R x (10 x BS - 8 X U) x A54

S(3,1)

-S(3,2)

S(3,3)

R x (-60 X B5 + 30 x AS - 30 x u - 84 x U)
R x (-15 x AS + 15 x u + 6 x U) x 28

R x (-30 x BS - 6 x u) x 2A

R x (-30 x BS - 30 x AS + 30 x u + 84 x U)
R x (-15 x AS + 6 x U) x 28

R x (-15 x BS + 6 x U) x 2A

S(1,1)

S(7,2)

R x (10 x AS - 8 x U) x BS4
o

-S(7,5)

R x (5 x AS + 2 x U) x BS4
o

5(2,1)

'S(2,2)

-S(7,3)
0 .
R x (10 x BS - 2 x U) x A54

5(9,4)
5(9,5)
S(9,6)
3(9,7)
S(9,8)
5(9,9)
S(10,1)
5(10,2)
5(10,3)
5(10,4)
5(10,5)
S(10,6)
. 5(10,7)
S(10,8)
5(10,9)

218

-S(7,6) '
0
R x (5 x BS + 2 x U) x AS4
-S(3,1)

-S(3,2)
S(3,3)

S(7,4)

-S(7,5)

S(7,6)

S(7,l)

-S(7,2)

S(7,3)

S(4,1)

S(4,2)

-S(4,3)

5(10,10) = 5(1,1)
5(11,1) = S(7,5)

5(11,2) = S(8,5)
5(11,3) = o
5(11,4) = -5(7,2)
5(11,5) - 5(3,2)
5(11,5) - o
5(11,7) - 5(5,1)
5(11,3) - 5(5,2)
S(ll,9) a o

S(ll,10) = -S(2,1)

219
S(ll,11) = S(9,4)

S(12,1) = S(9,9)
S(12,2) = 0
S(12,3) = S(9,6)
S(12,4) = -S(7,3)
S(12,5) = 0
S(12,6) = S(9,3)
S(12,7) = -S(6,1)
S(12,8) = 0
S(12,9) = S(6,3)

S(12,10) = -S(3,1)
S(12,11) = S(3,2)
S(12,12)= S(3,3)

Since the stiffness matrix is symmetric, only the lower triangular portion
is given. S(i,j) = S(j,i) can be used to evaluate the upper triangular

portion of the matrix.

3 Sui. Stiffness Matrix of Subgrade

If the modules of subgrade is uniform, k(x,y) = k = constant, the
following formulae can be derived:

0 a (k x A x B) / 44100

Q1 = A x Q

02 - B x Q

Q3 - A x A x Q

Q4 - B x B x Q

II
>
X

05

S(1,l)
5(2,1)
5(2,2) -
S(3,1) =
S(3,2) =
S(3,3) =
5(4,1)
5(4,2)
S(4,3) =
5(4,4) =
S(5,l) =-
S(5,2) =-
S(S,3) =
5(5,4) -

S(5,5) -
S(6,l) =
S(6,2) =
S(6,3) =
S(6,4) =
S(6,5) =
S(6,6) =
S(7,l) =
S(7,2) =
S(7,3) =

220
B x Q

24178 x Q
-6454 x Q2
2240 x Q4
6454 x Q1
-1764 x 05
2240 x Q3
8582 x Q
-3836 x 02
2786 x Q1
S(1,l)
-S(4,2)
-1680 x Q4
1176 x Q5
-S(2,l)
S(2,2)
S(4,3)
-S(5,3)
1120 x Q3
S(3,1)
-S(3,2)
S(3,3)
S(4,1)
-2786 x Q2
3836 x Q1

 

221

5(7,4) . 2758 x Q
S(7,5) = 1624 x 02
S(7,6) = 1524 x 01
S(7,7) = S(l,l)
S(8,l) = S(7,2)
S(8,2) = 1120 x Q4

S(B,3) = -S(5,3)
S(8,4) = -S(7,5)
S(8,5) - -340 x Q4
S(8,6) - —784 x 05
S(8,7) = 5(2,1)
S(8,8) = 5(2,2)
5(9,1) = -5(7,3)
S(9,2) = S(5,3)
S(9,3) = -l680 x Q3

S(9,4) = -S(7,6)
S(9,5) . S(8,6)
S(9,6) = -840 x 03
S(9,7) = -S(3,1)

5(9,3) - -5(3,2)
S(9,9) - S(3,3)
S(10,1) = S(7,4)
5(10,2) . -5(7,5)
S(10,3) = S(7,6)
5(10,4) S(7,l)
S(10,5) -S(7,2)

S(10,6)
S(10,7)
S(10,8)
S(10,9)

S(7,3)
S(4.1)

= S(492)

-S(4,3)

S(10,10) = S(l,1)

S(11,l)
5(11,2)
S(ll,3)
S(ll,4)
S(11,5)
S(ll,6)
S(ll,7)
S(ll,8)
S(ll,9)

S(11,10) = -5(2,1)

S(7,5)
S(8,5)
-S(9,5)
-S(7,2)
S(8,2)
S(9,2)
S(5,1)
S(5,2)
-S(5,3)

S(11,11) = S(2,2)

S(12,l)
S(12,2)
S(12,3)
S(12,4)
S(12,5)
S(12,6)
S(12,7)
S(12,8)
S(12,9)

S(12,10) = -5(3,1)

S(9,4)
-S(9,5)
S(9,6)
-S(7,3)
S(8,3)
S(9,3)
-S(6,1)
-S(6,2)

= S(6,3)

222

223

S(12,11) = S(3,2)

S(12,12)

S(3,3)

4 Pd, The Equivalent Nodal Force Vector Due to Loads

Fig. C-l
distributed load acted
on a part of element.
The following formulae
are valid only for the

constant load

Y 1

presents a 2 4

 

 

 

’1

1

1,.
1

 

 

 

intensity p.

Fig. C-l A Element partially acted by load

Using following notations:

F1
F2
F3
F4
F5
F6
F7
F8
F9

F10 = p x (x2

pX (X2 ‘ X1) X (Y2 ‘ Y1)
p x (X2 x X2 - X1 x X1) / 2

X

p (xz'x1)X(Y2XY2‘Y1XY1)/2
p X (X23 ‘ X13) X (Y2 ’ Y1) / 3
p X (X22 " X12) X (Yzz ‘ le) /4

pX (X2 " X1) X (Y23 ' Y13) /3

p X (X24 ' X14) X (Y2 ' Y1) / 4
P X (X23 ' X13) X (Y22 ' le) / 5
p X ”22 ' X12) X (Y23 ‘ Y13) / 5

X1) X (Y24 - Y14) / 4

224
F11 = p x (x; - x1“) x (Y: - Y3) / 8

F12 = p x (X: - X12) x (Y: 41‘) / 8

A2 - A2

82 . 82

AB = A x 8
A3 = A3

83 = 83

A82 = A x 82
A28 = A2 x 8
A83 = A x 83
A38 = A3 x 8

The elements of the equivalent force vector are listed as follow:
Pd(l) = F1 - 0.75 x F4 /A2 - 0.25 x F5 / AB - 0.75 x F6 / 82
+ 0.25 x F7 / A3 + 0.375 x F8 / A28 + 0.375 x F9 / A82
+ 0.25 x F10 / 83 - 0.125 x F11 / A38 - 0.125 x F12 / A83
Pd(2) = -F3 + 0.5 x F5 / A + F6 / 8 -0.5 x F9/A8 - 0.25 x F10 / 82
+ 0.125 x F12 / A82
Pd(3) = F2 - F4 / A - 0.5 x F5 / 8 +_0.25 x F7 / A2 + 0.5 x F8 / AB
- 0.125 x F11 / A28
0.25 x F5 / A8 + 0.75 x F6 / 82 - 0.375 x F8 / A28

Pd(4)
-0.375 x F9 / A82 - 0.25 x F10 / 83 + 0.125 x F11 / A3B
+ 0.125 x F12 / A83

Pd(5) = 0.5 x F6 / B - 0.25 x F9 / AB - 0.25 x F10 / B2

+ 0.125 x F12 / A82

225

Pd(6) = 0.5 x F5 / B - 0.5 x F8 / A8 + 0.125 x F11 / A28
Pd(7) = 0.75 x F4 / A2 + 0.25 x F5 / A8 - 0.25 x F7 / A3
-o.375 x F8 / A28 - 0.375 x F9 / A82 + 0.125 x F11 / A38
+ 0.125 x F12 / A83
9,,(8) - -o.5 x F5 / A + 0.5 x F9 / A8 - 0.125 x F12 / A82
Pd(9) = -o.5 x F4,/ A + 0.25 x F7 /A2 + 0.25 x F8 / A8
- 0.125 x F11 / A28
Pd(10) = -o.25 x F5 / A8 + 0.375 x F8 / A28 + 0.375 x F9 / A82
- 0.125 x F11 /A38 - 0.125 x F12 / A83 .
Pd(11) . 0.25 x F9 /AB - 0.125 x F12 / A82
Pd(12) = -o.25 x F8 /A8 + 0.125 x F11 / A28

5 P0 The Equivalent Nodal Force Vector Due to Temperature Gradient

If the temperature variation along slab thickness is linear, M, is a

constant vector as shown in Eq. (2-18), the equivalent thermal nodal force

vector can be written as:

.Pe=fi) ahg -lum,(1 -aAg -lflm,() aAg tMQ,() -ahg ZMQJT

.Eutfi
M:—
° 12(1-u) g

226

APPENDIX D
COMPONENT MODEL FOR DOWEL BAR SYSTEM

1 The Modes of Slabs and Dowel Bars

A dowel bar can be divided into three segments as shown in Fig. 2-1(a).
Segment Ci and jD are embedded in concrete and the segment ij is in
between the two slabs. i, and j, are denoted the slab nodes and ib and jb
the dowel bar nodes. Before any load being acted on the system, i,,ib and
j,, jb are assumed to be identical respectively. However, after the loads
being moved on, the slabs are deflected and ib and jb are separated from
is and 3', respectively. 6, and 6,- are defined as the relative deflection
between the dowel bar nodes ib, jb and the slab nodes i, and j,
respectively. Similarly, 6i and 0,- are defined as the relative rotation
angles between the slabs and the dowel bar. The relative deformation,

including deflection and rotation angle, can be analyzed by a beam resting

on elastic foundation.

2 The Stiffness Matrix of Dowel Bar
For seqment ii (Fig. 2-1)

The stiffness equation of segment ij is:

 

£1] = [4.] (0-11
1% b Sb (1'8

227

 

where:
12 61 —121 61
_‘911312_ 131' 6.1 (41111))12 '61 (2‘¢)12
"3,18,, 13(1+¢) —12 -61 12 -6.l (D 2)
61 (2-(1>)l2 -6l (4+<i>)12
E Elastic modules of the dowel bar
I Moment of the cross-sectional inertia of the bar
l Width of the joint Opening, length of the element
«p 24(1+ )I/Aelz, shear deformation factor
u Poisson’s ratio of the dowel bar
A Cross-sectional area effective in shear of the bar

d

[Q M]T Force vector of the bar’s node

[w BlT Displacement vector of the bar’s node

Their positive direction are shown in Fig. 2-2.

For seqment Ci and 10 (Fig. 2-1)

The positive notations of relative deflections, rotation angles and forces

are shown in Fig. D-l.

Where:

6=W8‘Wb

= . - b T— "“11 g <§< {1&1:_FL
w, and wb are >;:::> ::::>k

<0 9

Po PL

 

 

 

 

 

 

:///X/>//////////'///

 

displacements at
the nodes of the

slab

 

Z\)‘:—— X——>1

Fig. D-l A Beam on Elastic Base

228

and dowel bars. The beam element equilibrium eguation can be written

as(Timoshenko“94”):

81% + k6 (x) = o (o s x s L) (”'3’
dX4

where k is base coefficient of the beam, 1.8. the product of interaction

coefficient of bar and concrete and the diameter of the bar. The general

solution of Eq. (0-3) is:

6 (X) = A’ Cth cosBx + B’ cth sian

(0-4)
+ C’ sth cosBx + D’ shflx sian
where B = (k/4EI)°25
A,8,C and D can be determined by the boundary conditions at x = 0:
5(0) ==50
4(0) = %"—‘)—l.. 6,
_ d26()d _
M(O) - EI'—C?X-2—|X=O " MO (”_5)
_ d36(x)
0(0) EI dx3 |x=0 P0

The solutions are:

229

 

 

 

A’ = 60

BI = i (.92 —- PO
2 i3 25183

C, — .1; (.99 + Po (0'6)
2 l3 28183

D’ = M0
28182

Substituting Eq. (D-6) into Eq. (D-4) and using the boundary conditions:

 

 

 

M(X=L) = 0 and Q(x=L) = 0 (0’7)
db and ¢b can be expressed by Mo and Q0 as fol?
' -S
(50] ._. 1 [go—9 32*3" [Po] (D-a)
(b0 231132 632—32) 82+s2 -20 (SC+sc) M"
In which:
_ _ _ - - (ll-9)
S-shBL, C-chBL, s-81nBL and c-cosBL
Similarly, let:
M(x=0) = o and Q(x=0) = o (”‘10)
6L and ¢l can be obtained:
6 SC-SC 2 2 P
[J = 21. 2 B 5+3 [) (0-11)
4’1. 2515 (5 'S) 52+52 25(SC'+SC') ML

230
Comparing the positive directions of the nodal forces and displacements in
Fig. 2-2 and Fig. D-l, it is obvious that Eq. (D-ll) can be directly used
for segment Ci in Fig. 2-1. After P0 and MO replaced by -P0 and -Mo, Eq.

(D-8) can be used for segment jD in Fig. 2-1.

Notice: Aw = 6, A9 = 42, AP = P and AM = M, the following eqations at node

i of the segment Ci can be written:

S C'—s C
(A w) _ 1 —1—1,3—1—1 551:3? (AP) (1)-12)
A9 2.8182 (Sf-sf) AM

 

Sf+sf 28 (s,c,+s,c,)

where 3,, s,, C, and c, are obtained by substituting L=l, in Eq. (0-9).

At node j of the segment jD in Fig. 2-1:

S'C'-S C
(Aw) = 1 #B—Z—i ‘(522+522) (AP) (1)-13)
A9 25182 (Si-$3) AM

 

-(S§+s§) 28(szcz+szc2)

where 52, 52, C2 and c2 are obtained by Substituting L=l2 in EQ. (D-9).

AP and.AM can be solved, for node i of the segment Ci in Fig.2-1:

(AP) _T(Aw) - (1)-14)
AM ‘ 1 A8

for node j of the segment jD in Fig. 2-1:

231

 

(AP) _T(Aw) (1)-15)
AM 2A6
where:
28 (31C1+s,c,) - (Sf+sf)
T1 = M (SC-SC) (0-16)
Cf+cf -CSf+sf) 1 19 1 1

213(32C2+52C2) (s§+s§)

 

 

2 2E1 (D-17)
T = J— -
2 C22+c§ (S§+s§) (SZCZBSZCZ)
Using the following notations for any node n:
Qn -
1". ___ ] (o 18)
Mn
“% (D-19)

and notice the geometric relations between relative displacements and
global displacements, Eq. (0-14) and Eq. (D-lS) can be rewritten as:

for nodes is and ib:

’1. = T1 4’1 dx. (o-zo)
PH, ‘13 13 9%,

 

for nodes js and jb:

F1: _ T2 -T2 d1: (0-21)
’1. ' -T, T, djb

 

232
Component Stiffness Matrix for the Entire Dowel Bar
Indeed, a dowel bar is the assemblege of three segments. The assembled
stiffness matrix of the dowel bar can be obtained by taking summation of

the three stiffness matrices as follow:

 

 

 

{F1} / T, -T, o o (<11)
F16 _ ’Ti S11+T1 512 0 d1» (0-22)
’1. ’ o 3,, saw, Jr, ‘11.
TF1.) \ O o ’1'“ T3 Adj.)

 

 

Since there are not nodal forces acted at the bar’s nodes, the following

equations can be written:

F11) = Fjb = 0 (0-23)
Thus:
d1» = [31144.1 312 ]-1[T1 0] d1. (II-24)
d3. 55, ,S;,+1g (J 1; <1“

Substituting Eq. (0-24) into the first and the fourth equations in Eq. (0-
22), the stiffness equation for the dowel bar can be written in terms of

the nodal forces and displacements of the slabs:

F1.

db.
3,, = 1(a) (D-25)

thus, the component stiffness matrix 56 becomes:

(E 0) 311+T1 S12 -1T1 O (”'26)
0 E - 521 32214,: 0 T2

where, E is a 2 x 2 unit matrix.

 

 

1'10
S:
c o T,

 

STRTE UNIV. L

13111111 llllllllll11111111711311“

 

93009 25