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LIBRARY

Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

EXPERIMENTAL INVESTIGATION OF THE BEHAVIOR OF RC
DECK SLAB WITH ADDED LATERAL RESTRAINT

presented by

Made Sukr awa

has been accepted towards fulﬁllment
of the requirements for

Ph.D. Civil Engineering

degree in

 

 

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Major professor

Date 7[24/00

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em cJCIRC/DateDuepBS-p. 15

 

EXPERIMENTAL INVESTIGATION OF THE BEHAVIOR OF RC
DECK SLAB WITH ADDED LATERAL RESTRAINT

By

Made Sukrawa

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
In Civil Engineering

Department of Civil and Environmental Engineering

College of Engineering

2000

ABSTRACT

EXPERIMENTAL INVESTIGATION OF THE BEHAVIOR OF RC DECK SLAB
WITH ADDED LATERAL RESTRAINT

By: Made Sukrawa

The present study proposes a new method of retroﬁtting an existing composite
reinforced concrete bridge deck slab by increasing its lateral restraint, by connecting the
top ﬂanges of steel girders with tie rods. To verify the method, experimental and
analytical investigations were conducted. Experimental investigation was conducted by
testing a 1/5 scale composite steel I-beam bridge. Two different deck panels were
incorporated into the construction of a three-girder simply supported bridge. One panel
was retroﬁtted with tie rods, and the other panel was used as a control panel. With a span
to thickness ratio of 12.5, the model deck slab was reinforced with 0.2% isotropic welded
wire mesh. Each slab panel was consecutively tested at four different load points with a
statically applied concentrated load. The behavior of the slab panels was measured with
dial and strain gages to determine the stiffness and strength of the panels. The load
response history of the slab, the center beam, and the tie rods was observed at load
increments of 2,224 N (500 lbs) until the slab failed.

The slab panel with tie rods was signiﬁcantly stronger and stiffer than the control
slab panel. At lower load levels, the net deﬂection of the slab with tie rods reduced by a
factor of 34%, and at higher load levels, the reduction was up to 50%. The cracked and
failure loads of the slabs with tie rods increased by factors of 30% and 13.7%,

respectively. Compared to a half axle load of 72.5 kN (16.3 kip) with an impact factor of

1.3, safety factors of 15.9 and 13.9 were obtained for the slab with and without tie rods,
respectively. Thus, with added lateral restraint the safety factor of the slab increased by a
factor of 14.4%.

Prediction of the load response history of the bridge was conducted by developing
3-D nonlinear ﬁnite element (FE) models using a commercial ﬁnite element software
ABAQUS. The models incorporated each component of the bridge including the tie rods.
The deck slab was modeled using 3-D solid elements with steel reinforcement embedded
in the concrete elements. Several models were developed for studying the sensitivity of
the models to various modeling parameters. The response of each model was compared to
the observed response. In addition to the FE analysis, strength analyses using punching
shear theory of restrained slabs and the ACI Code provisions on punching shear strength
of concrete slabs were also utilized for comparison.

The arching action in the deck slab was well predicted by 3-D solid elements,
through the development of strains in the tie rods in the vicinity of the load. The observed
load response history of the bridge was also accurately simulated in the FE models up to
a load level about 60% of the failure load. At higher load levels however, the FE models
showed stiffer response than the observed one. The responses of the FE models were
strongly dependent on the material model for the concrete.

Strength analysis using the punching shear model of restrained slabs predicted the
punching strength of the slabs with corresponding restraint factor values of 0.663 and
0.786 for the slab panel without and with tie rods, respectively. The addition of tie rods
increased the lateral restraint of the slab by a factor of 18.6%. The ACI Code provisions

gave nominal shear strengths signiﬁcantly lower than the observed values.

To:

The People of Bontihing - Bali

iv

ACKNOWLEDGEMENTS

The author wishes to express sincere gratitude to Dr. William E. Saul for his
professional encouragement, guidance, and especially his personal friendship throughout
this endeavor. The author also wishes to thank the committee members: Dr. Ronald S.
Harichandran, Dr. Frank J. Hatﬁeld, and Dr. Charles R. McCluer for their valued
technical input and suggestions throughout the course of this research.

The research was partly funded by Civil and Environmental Engineering
Department - Michigan State University, and The Great Lakes Fabricator & Erectors
Association of Michigan. The author also gratefully acknowledges the Structural Lab of
Michigan DOT for their help and the use of their lab.

Most importantly, the work would not have been possible without the ﬁnancial
support from the Engineering Education Development Project of the Department of
Education and Culture of Indonesia. The author also appreciates the ﬁnancial supports
through fellowships from the College of Engineering, Ofﬁce for International Students
and Scholars, and the Graduate School of Michigan State University. Finally, invaluable

support and understanding of the author’s wife, Putu Sumiarthi, are most appreciated.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix

LIST OF FIGURES ............................................................................................................ x
CHAPTER 1

INTRODUCTION .................................................................................................. 1

1.1 Enhancing Arching Action in RC Deck Slabs ................................................. l

1.2 Hypothesis ........................................................................................................ 3

1.3 Purpose and Scope of Research ........................................................................ 5
CHAPTER 2

LITERATURE REVIEW ....................................................................................... 8

2.1 Introduction ...................................................................................................... 8

2.2 Experimental Investigations of RC Deck Slabs for Bridges ............................. 9

2.2.1 Tests by Batchelor et a1 ...................................................................... 9

2.2.2 Test by Dorton et al., and Csagoly et a1. .......................................... . 11

2.2.3 Test by Beal ..................................................................................... 12

2.2.4 Test by Fang et al. ........................................................................... 13

2.2.5 Test by Puckette et a1. ...................................................................... 14

2.2.6 Test by Perdikaris ............................................................................ 15

2.3 External Reinforcement in Deck Slab and Steel Beam .................................. 16

2.4 Empirical Design Method of RC Deck Slabs for Bridges .............................. 18

2.5 Shear Strength of Reinforced Concrete Slabs ................................................ 22

2.5.1 Empirical Methods .......................................................................... 22

2.5.2 Rational Model of Punching Shear ................................................. 25

2.6 Finite Element Analysis of Reinforced Concrete Slab ................................... 28
CHAPTER 3

THEORY AND ANALYSIS ............................................................................... 36

3.1 Introduction .................................................................................................... 36

3.2 Rational Model of Punching Shear Strength of Restrained Slabs .................. 38

vi

3.2.1 Mechanical Model for Simply Supported Slabs .............................. 42

3.2.2 Mechanical Model of Restrained Slabs .......................................... 45

3.3 Analyses of Punching Strength of RC Slabs with Additional Lateral restraint
........................................................................................................................ 48
3.4 ACI Provision on Punching Shear Strength of Concrete Slab ....................... 53
3.5 Finite Element Analysis ................................................................................. 55
3.5.1 Geometric Model ............................................................................. 56
3.5.2 Material Models .............................................................................. 60
3.5.2.1 Concrete Model ................................................................ 61
3.5.2.2 Steel Model ...................................................................... 64
3.5.3 Input Files ............................................... 64
3.5.4 Finite Element Analyses Results ..................................................... 68

CHAPTER 4
DESIGN AND CONSTRUCTION OF MODEL BRIDGE ................................. 76
4.1 Introduction .................................................................................................... 76
4.2 Design of Prototype Bridge ............................................................................ 77
4.2.1 Reinforced Concrete'Deck Slab ...................................................... 78
4.2.2 Design of Tie Rods .......................................................................... 80
4.2.3 Design of Steel Girders ................................................................... 81
4.2.4 Design of Edge Beams ..................................................................... 81
4.2.5 Design of Shear Connectors ............................................................ 83
4.3 Design of Scaled Model Bridge ..................................................................... 83
4.4 Properties of Material for Model Bridge ........................................................ 88
4.4.1 Micro Concrete ................................................................................ 90
4.4.2 Steel Reinforcement ........................................................................ 92
4.4.3 Structural Steel ................................................................................ 93
4.5 Construction of Model Bridge ........................................................................ 93
4.5.1 Concrete Cylinders .......................................................................... 94
4.5.2 Curing .............................................................................................. 95
CHAPTER 5

EXPERIMENTAL PROGRAM .......................................................................... 97

vii

5.1 Introduction .................................................................................................... 97

5.2 Instrumentation ............................................................................................... 98
5.3 Load Testing of Model Bridge ....................................................................... 99
5.4 Recorded Data .............................................................................................. 102
5.4.1 Cylinder Test ................................................................................. 102
5.4.2 Loading Test .................................................................................. 105
CHAPTER 6
PRESENTATION AND DISCUSSION OF RESULTS ................................... 113
6.1 Introduction .................................................................................................. 113
6.2 Punching Strength of Slabs (Analyses and Test Results) ............................ 114
6.3 Load Deﬂection Diagram of Slabs (FEM and Test Results) ........................ 119
6.4 Deﬂection of Center Beam (FEA and Test Results) .................................... 127
6.5 Strain in Tie Rods (FEA and Test Results) .................................................. 131
6.6 Summary of Results ..................................................................................... 134
CHAPTER 7
CONCLUSION AND RECOMMENDATION ................................................. 137
7.1 Summary and Conclusion ....................... ° ..................................................... 137
7.2 Recommendation .......................................................................................... 141
REFERENCES ............................................................................................................... 143
APPENDICES
APPENDIX A: PUNCHING SHEAR MODEL OF RESTRAINED SLAB,
DERIVATION OF EQUATIONS ........................................... 149

APPENDD( B: FLOW DIAGRAM AND CODE LISTING TO CALCULATE
PUNCHING SHEAR STRENGTH OF RESTRAINED

SLAB ......................................................................................... 163
APPENDIX C: EXAMPLE OF FIN ITE ELEMENT INPUT AND OUTPUT
FILES ......................................................................................... 169

viii

LIST OF TABLES

Table 3.1 Punching Load of Slab (Restrained Slab Theory) ............................................ 52
Table 3.2 Shear Strength of Model Slab (ACI Formulae) ............................................... 55
Table 3.3 Description of FE Models ................................................................................ 68
Table 3.4 Summary of FEA Results ................................................................................. 69
Table 4.1 Properties of Prototype Bridge ......................................................................... 84
Table 4.2. Scale Factor ..................................................................................................... 85
Table 4.3 Properties of Prototype and 1/5 Scale Model Bridges ..................................... 89
Table 5.1 Results of Cylinder Tests ............................................................................... 103
Table 5.2 Crack Load, Failure Load, and Maximum Deﬂections of Test Slab ............. 107
Table 6.1 Ratios of FEA and Test Results ..................................................................... 117

ix

LIST OF FIGURES

Figure 1.1 Compressive Membrane Forces in a Cracked Slab ....................................... 4
Figure 1.2 Effect of Adding Tie Rod .............................................................................. 4
Figure 2.1 K and N Model of Punching Shear .............................................................. 27
Figure 2.1 Tension Stiffening Model of RC ................................................................. 31
Figure 3.1. Crack Pattern of Restrained Deck Slab (Plan View) ................................... 40
Figure 3.2. Mechanical Punching Model of Restrained Slab ......................................... 41
Figure 3.3 Sector Element (Wedge) Showing Forces and Notations ............................ 41
Figure 3.4 Geometry of Equivalent Circular Slab ......................................................... 51
Figure 3.5.a Plan View of FE model Showing Mesh of the Deck .................................. 59
Figure 3.5.b Cross Section of FE model Showing Element Connectivity ....................... 59
Figure 3.5.c Isometric View of FE model without the Deck Elements ......... . ................... 60
Figure 3.6 Predicted Load Deﬂection Diagram of Slab with Tie Rods ........................ 70
Figure 3.7 Load Deﬂection Diagram of Slab Using Different Elements ...................... 71
Figure 3.8 Predicted Load Deﬂection Diagram of Slab without Tie Rods ................... 72
Figure 3.9 Predicted Load Deﬂection Diagram of Center Beam .................................. 73
Figure 3.10 Predicted Load Strain Diagram of Tie Rods ................................................ 75
Figure 4.1 Plan View of Prototype Bridge .................................................................... 79
Figure 4.2 Longitudinal Section Showing Edge Beam and Diaphragm ....................... 82
Figure 4.3 Cross Section of Model Bridge .................................................................... 87
Figure 4.4 Detail of Tie Rod ......................................................................................... 88
Figure 5.1 Experimental Set-Up on Loading Frame ..................................................... 99
Figure 5.2 Arrangement of Dial Gages and Strain Gage Position .............................. 100
Figure 5.3 Loading, Tie Rods, and Strain Gage Positions .......................................... 101
Figure 5.4 Typical Crack Pattern ................................................................................ 106
Figure 5.5 Top View of Punching Failure at LP3 ....................................................... 108
Figure 5.6 Bottom View of Punching Failure at LP2 .................................................. 108

Figure 5.7 Load Deﬂection Diagram of Slab .............................................................. 109
Figure 5.8 Load Deﬂection Diagram of Center Beam ................................................ 110
Figure 5.9 Load Strain Diagram of Tie Rods .............................................................. 111
Figure 6.1 Analysis Results: Punching Strength of Slab ............................................. 115
Figure 6.2 Observed Load Deﬂection Diagram of Slab Loaded near Mid-span ........ 121
Figure 6.3 Observed Load Deﬂection Diagram of Slab Loaded near the Edges ........ 122
Figure 6.4 Measured and Predicted Load Deﬂection Diagram of Slab Loaded at LP3
..................................................................................................................... 124
Figure 6.5 Measured and Predicted Load Deﬂection Diagram of Slab Loaded at LP7
..................................................................................................................... 125
Figure 6.6 Measured and Predicted Load Deﬂection Diagram of Center Beam Loaded
At LP3 (Slab with Tie Rods) ...................................................................... 128
Figure 6.7 Measured and Predicted Load Deﬂection Diagram of Center Beam Loaded at
LP7 (Slab without Tie Rods) ...................................................................... 129
Figure 6.8 Measured and Predicted Load Strain Diagram of Tie Rods Directly
Underneath the Applied Load ..................................................................... 132
Figure 6.9 Measured and Predicted Load Strain Diagram of Tie Rods Adjacent to the
Applied Load ............................................................................................... 133
Figure A.1 Isometric View of Conical Shell Element for Simply Supported Slab ....... 150
Figure A.2 Cross Section of Mechanical Model Showing Notations ........................... 150
Figure A.3 Mechanical Model of Simply Supported Slab ............................................ 156
Figure A.4 Mechanical Model of Restrained Slab ........................................................ 157
Figure A.5 Boundary Forces ......................................................................................... 160
Figure C.1 Magniﬁed Displaced Shape of FE Model of the Bridge .............................. 183
Figure C.2 Contour of E33, Strain in 3-Direction ......................................................... 184
Figure C.3 Contour of S33, Stress in 3-Direction ......................................................... 185

xi

CHAPTER 1

INTRODUCTION

1.1 Enhancing Arching Action in RC Deck Slabs

Although common practice is to assume ﬂexure for the design of a reinforced
concrete (RC) deck slab, it has been repeatedly shown that, when subjected to a
concentrated wheel load, the slab fails in punching shear (AASHTO 1994). By deﬁnition,
punching shear failure is denoted by a local failure, when a plug of concrete is punched
out of the slab immediately underneath the load.

A laterally restrained deck slab exhibits internal arching action that strengthens
the slab through the conﬁnement provided by a complex three-dimensional membrane
stress around the loaded area. The lateral restraint is provided by the supporting
components, surrounding slab, and rigid appurtenances acting compositely with the slab.
As a result, the slab fails in shear, rather than in ﬂexure, with a load capacity much higher
than required to sustain a truck wheel load.

Despite the extra strength, in a composite RC slab and steel girder system, the
girders always last longer than the slab because the slab is directly impacted by trafﬁc,
de-icing chemicals, climate, and other aggressive environmental effects. Such exposure
causes the slab to readily deteriorate. Consequently, it requires replacement during the
service life of the bridge. Strengthening an existing deck slab is one way to extend its
service life. In addition to restoring its function, retroﬁtting a bridge deck is also intended

to upgrade the bridge’s status to a higher performance level, or to meet additional

capacity requirements (Xanthakos 1996). In this case, retroﬁtting is aimed at increasing
the durability of the slab.

There are a number of ways to strengthen existing RC deck slab structures. One
way is to use the concrete-thickness increasing method, where the slab is retroﬁtted by
adding another layer of concrete with reinforcement. This method is not very popular
because it adds dead load stress on the supporting girders, which may not have been
considered in the design. It also requires adjustment of the vertical alignment of the
pavement adjoining the bridge. Bonding between the old and the new concrete is another
problem that arises in this method.

The carbon-ﬁber-sheet gluing method is the most recent process employed to
strengthen concrete structures. In this method carbon ﬁber sheets are glued to the
concrete under surface by means of an epoxy resin (Masajiro et al, 1994). This method is
suitable to increase the ﬂexural strength of the slab.

In this study, an alternate method called the lateral-restraint increasing method is
proposed. The method proposed herein is based on the addition of tension members (tie
rods) to connect the top ﬂanges of the supporting steel girders beneath the slab. The
additional lateral restraint is expected to minimize the lateral movement of the girders,
thus enhancing arching action in the slab, and therefore, increasing the strength and
stiffness of the slab. The advantage of this method, relative to the others, is its ease of
application that neither entails trafﬁc interruption nor requires special materials or tools.

Extensive research on the load capacity of RC deck slabs was reported by a
number of workers, where load tests on small and full scale deck slabs were conducted

(Bachelor et al. 1978, Csagoly et al 1978, Beal 1982, Fang et al. 1986, Puckette et al.

1989, Agarwal 1990). RC slabs with various steel reinforcement ratios and arrangements,
were built and tested up to failure. Orthotropic and isotropic steel arrangements were
used; the ﬁrst represents the conventional practice, where the steel ratio in the transverse
direction of the bridge is higher than that in the longitudinal direction; the later represents
the same steel ratios in both directions of the bridge. The results of these experiments
showed that the arching action enhanced the shear strength of the slab.

Taking advantage of arching action, extensive tests on steel-free ﬁber-reinforced
concrete (SFFRC) deck slabs were reported (Mufti et al. 1993, Bakht and Mufti 1996).
Instead of conventional reinforcement, steel straps were used as external reinforcement,
where the straps were attached to the top ﬂanges of the steel girders to increase the lateral
restraint of the slab. The experiments showed that the addition of steel straps resulted in a

punching failure load twice as high as the failure load of a slab without restraining straps.

1.2 Hypothesis

From the aforementioned research, it can be concluded that the behavior of RC
deck slabs on steel beams can be improved by increasing their lateral restraint. The lateral
restraint in RC deck slabs is provided by the shear connectors, internal steel
reinforcement, and the supporting components of the deck. Adding shear connectors or
internal steel reinforcement to an existing deck slab is not practical. It is possible,
however, to increase the lateral restraint of the slab by using external reinforcement,
similar to that used for the SFFRC deck slabs. By adding tension members to connect the
underside of the top ﬂanges of the steel beams, it is hypothesized to increase the lateral

restraint of the slab, and therefore, improve the strength and stiffness of the deck slab.

The enhancement of lateral restraint through the addition of a tie rod is illustrated
in Figures 1.1 and 12. Figure 1.1 shows the formation of ﬂexural cracks and in-plane
forces corresponding to the stress distribution in a restrained slab. The compressive
membrane action produces net in-plane forces at the slab boundaries that cause the
supporting girders to move both vertically and laterally. Figure 1.2 shows the diagram of

the compressive arch tied by steel reinforcement and a tie rod.

 

 

 

Load
Compressive In-plane Membrane
_ Stress Distribution / Forces Force
{ V (N: .................. .. V IL-
. J, """"""" -
T Crack T

Figure 1.1 Compressive Membrane Forces in a Cracked Slab

Load

 

 

 

 

 

Figure 1.2 Effect of Adding Tie Rod

Adding a tie rod to the arch horizontally (connecting top ﬂanges of the girders)
will reduce the lateral movement of the arch, and hence, enhance the arching action in the
slab. The compressive membrane forces together with the external load develop three-
dimensional compression in the slab, which considerably increases the strength and
stiffness of the concrete around the loaded area.

To test the validity of this hypothesis, an experimental study was conducted.
Analytical studies based on a rational model of punching shear and ﬁnite element
analyses (FEA) were conducted to verify the experimental results. The application of the
concept of adding tie rods for retroﬁtting an existing or a newly constructed RC deck slab

is the original contribution of this research.

1.3 Purpose and Scope of Research

The main objective of this research is to study the effect of adding lateral restraint

(tie rods) to an existing composite RC deck slab and steel girder system on the behavior
of the slab subjected to a static concentrated load. For this purpose, experimental testing
and theoretical analyses were conducted. The load response history of the slab, the steel
beam, and the tie rods is used to measure the effect of adding tie rods. In addition to the
main objective, there are a number of other purposes of this research. They are:

0 To verify the adequacy of the strength and stiffness of RC deck slabs with minimum

steel reinforcement under service load and overload conditions.
0 To develop three-dimensional, nonlinear ﬁnite element (FE) models of the bridge to

predict the load response history of the slabs with and without additional tie rods.

To verify the application of the rational and empirical approaches, to predict the
punching shear strength of a restrained RC deck slab with and without additional

lateral restraint.

As a preliminary study, to prevent bias observation due to uncontrolled conditions

of the deck slabs, only a newly constructed slab (without deterioration) is considered. The

deck slab is assumed to be deﬁcient because there is a need to increase the performance

and durability of the deck. Having stated the purposes and scope (limitation), the steps of

this research are listed in the following order:

Literature Review: reviews the experimental work related to the arching action in the
deck slab for bridges, and the theoretical and numerical analyses of RC deck slabs.
Theory and Analyses: include derivation of the equations to calculate the punching
shear strength of an RC slab, and development of ﬁnite element models using ﬁnite
element (FE) software ABAQUS.

Design and Construction of Model Bridge: design the prototype and small-scale
bridge, prior to building the model bridge.

Experimental Program: includes instrumentation, and load testing of the model
bridge.

Presentation and Discussion of Results: evaluate the results from the analyses and lab
experiment, and verify the validity of the analyses methods based on the test results.
Conclusion and Recommendation: draw conclusions from the test and analyses
results, and provide recommendations related to application of the concept of adding

lateral restraint to an existing or a newly constructed deck slab.

The results of these experimental and analytical studies will be applicable to
retroﬁt RC deck slabs on steel girders for bridges, and it can be extended to other slab
structures subjected to concentrated loads. Therefore, through this research, a new

method of retroﬁtting an existing RC deck slab is to be veriﬁed.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

Experiments on punching shear failure of reinforced concrete slabs discovered
that the presence of in-plane compression due to restraint in the slab boundaries
substantially increased the shear strength of the slab. These ﬁndings have inﬂuenced
bridge researchers to review and investigate the shear strength of RC deck slabs for
bridges. For the past three decades, after the composite I-beam bridge became popular in
North America, extensive research into the behavior of RC deck slabs for bridges has
been conducted.

Research on the shearing strength of composite bridge deck slabs was initiated at
the University of Illinois in 1946. Since then, more extensive tests were conducted to
speciﬁcally investigate the behavior of deck slabs under concentrated loads. Tests at
Queen’s University were among the pioneer studies in this area (Hewitt 1972; Hewitt
and Batchelor 1975; Batchelor et al. 1978). More detail on this experimental program is
discussed in the following sections, together with a number of other related experimental
work.

Taking advantage of arching action in the slab, external reinforcement has been
used to provide additional lateral restraint. Experimental work initiated at the Technical
University of Nova Scotia, TUNS, was part of an effort to eliminate internal
reinforcement in deck slabs (Mufti et. al. 1993; Bakht and Mufti 1996; Newhook 1997).

This particular work is also reviewed in detail.

The application of these ﬁndings for bridge deck design was marked by the
inclusion of the empirical design method, for the ﬁrst time, in both the Ontario Highway
Bridge Design Code, OHBDC 1983 and the American Association of State Highway and
Transportation Ofﬁcial (AASHTO 1994) design speciﬁcations for bridges. These design
methods were reviewed in detail in consideration of the beneﬁts gained utilizing arching
action in the design as compared to the conventional method. Finally, reviews on
theories on the punching shear strength of reinforced concrete slabs and ﬁnite element

analyses of RC slabs are presented in the last parts of this chapter.

2.2 Experimental Investigations of RC Deck Slabs for Bridges

A number of experimental projects have been reported investigating the behavior
of small model and full scale RC deck slabs for bridges subjected to concentrated loads.
Among the published work are those conducted by: Batchelor et al., Dorton et al. and
Csagoly et al., Agarwal, Beal, Fang et al., Puckette et al., and Perdikaris et al. In general,
the natures of the investigations are similar, however, the focus of each differs

somewhat. Therefore, it is necessary to discuss some of them emphasizing their focus.

2.2.1 Tests by Batchelor et al.

Experimental investigations of the ultimate strength of deck slabs of composite
steel concrete bridges were conducted by testing one-eight scale models of 24.4 m (80
ft.) span prototype bridges. Three-girder and four-girder model bridges were designed

with beam spacing of 0.45 and 0.30 m (18 and 12 in.), respectively. The model bridges

were tested to failure with static concentrated loads at Queen’s University in conjunction
with the Ontario Ministry of Transportation and Communications (Hewitt 1972; Hewitt
and Batchelor 1975; Batchelor et al. 1978). The 22 mm (0.9 in.) thick model slabs were
divided into strips where each strip was reinforced with various steel ratios ranging from
O to 0.6 %. Orthotropic and isotropic steel arrangements were used to represent both the
conventional practice and the proposed application, respectively.

The results from a total of 68 failure tests showed that nearly all of the slabs
failed in punching shear. Only one panel of reinforced slab and some panels of
unreinforced slabs failed in ﬂexure. The average ultimate strength of the four-beam
model slabs ranged from 16.5 kN (3,710 lbs) for the plain concrete slab to 23.66 kN
(5,319 lbs) for the slab with 0.6% isotropic reinforcement. The slab with orthotropic
reinforcement failed at about 24 kN (5,400 lbs).

It was reported that at a load of between 25 and 50 percent of the ultimate load
visible cracking in the form of longitudinal and diagonal cracks were observed on the
underside of the slab. Subsequently, transverse cracks appeared on the upper—side of the
slab, on either side of the loaded area in the vicinity of the beams. The cracking pattern
followed that the higher the percentage of reinforcement, the more symmetrical and
closely spaced the cracking was. The deﬂection behavior of the slab followed that the
slab with lower steel ratio failed after relatively larger deﬂection. The net deﬂection of
the slab with 0.2% steel ratio, at failure load of about 18 kN (4,047 lbs), was about 7.5
mm (0.3 in.).

The conclusion of the experiment suggested that the strength of the deck was

not signiﬁcantly affected by the position of the load on the deck, previous failures of the

10

slab in adjacent panels, dead load stress and deﬂection, or the strength of the concrete. It
also recommended the use of 0.2 % isotropic steel reinforcement to account for the
temperature and shrinkage requirements, which amounts to a reduction of 66 % of the
conventional steel requirement. With the recommended minimum steel, the slab can still
be expected to have a safety factor against punching failure of about 9. In addition to the
experimental work, a theory to calculate the punching shear strength of a restrained slab
was proposed based on the theory initiated by Kinnunen and Nylander for simply
supported slabs. More detail on this theory is discussed later in this chapter and in

Chapter 3.

2.2.2 Tests by Dorton et al., and Csagoly et al.

Field testing of existing, and newly designed and built bridges was conducted by
the Ontario Ministry of Transportation and Communication, using a simulated wheel
load of 445 kN (100,000 lbs), about four to ﬁve times the actual AASHTO truck wheel
load (Dorton et al. 1976; Csagoly et al. 1978). A total of 40 bridges were tested with a
wide range of spans, slab thickness, age, bridge types, and degrees of deterioration. The
deck slabs of the existing bridges were designed according to the 1973 AASHT O
speciﬁcations, where four different bridge types were selected: Non-composite steel
beam and concrete slab; Composite steel beam and concrete slab; Monolithic concrete
beam and concrete slab; Composite AASHTO girders and concrete slab. For the
composite steel girder bridges, the slab thickness varied from 180 to 215 mm (7 to 9 in.)

with beam spacing from 1.84 to 3.51 m (8 to 12 ft.).

11

From 32 bridges examined, neither serious cracking was noticed in any of the
deck slabs, nor visible cracking observed on the top surface of the slabs in the transverse
negative moment regions. Directly under the load on the bottom surface of the slab,
hairline cracking was observed during the test. However, the cracks disappeared upon
removal of the load. For the composite steel beam bridges, the deﬂection of the slab
under the simulated load varied from 0.7 to 3.05 mm (0.02 to 0.12 in.) depending on the
beam spacing. It was concluded that even under a load 4 to 5 times larger than the
maximum half-axle weight combined with the deterioration of some decks, no failure of
restrained slabs were ever observed. Consequently, the deck slabs that were designed
according to the 1973 AASHTO Speciﬁcation were considered overly designed.

For the newly designed and built bridge, a three-span-continuous steel girder
bridge was load tested. The deck slab was designed based on the new ﬁndings from lab
tests conducted at Queen’s University, in which steel reinforcement ratio as low as 0.2%
was used. The deﬂection of the deck, subjected to the same load as that used to test the
existing bridges, was reported to vary from 1 to 3.1 mm (0.04 to 0.122 in.) for the
interior panels and 1.1 to 2.8 mm (0.044 to 0.111 in.) for the exterior panels. It was
concluded that the deck slabs could be safely designed allowing for internal arching
action, making possible a large reduction in the steel reinforcement. Agarwal (1990)

reported load tests similar to those mentioned above with comparable results.

2.2.3 Tests by Beal

Lab testing of two bridge models with a scale factor of 1 to 5.9 was conducted at

the New York State DOT (Beal 1981; Beal 1982). The prototype bridge was comprised

12

of ﬁve steel girders spaced at 2.75 m (9 ft.) center to center with 22 m (72 ft.) of span
length. In the ﬁrst model the 216 mm thick slab was heavily reinforced to represent the
standard bridge deck design according to the 1977 AASHTO Speciﬁcation. For the
second model, a 200 mm thick slab was reinforced with minimum isotropic steel
representing the new practice of design in Ontario. The strain in the steel reinforcement
was measured and used to estimate the bending moment on both uncracked and cracked
conditions of the slab. For the uncracked test, ﬁeld tests were also conducted for
comparison.

The conclusion of the tests was that, the service load stress, predicted by the
existing design procedure and method, did not develop in the slab. Under a service load
of 92.6 kN (20.8 kips), the maximum induced stress in a conventionally reinforced slab
was 57.2 MPa (8.3 ksi). A value of 80.6 MPa (11.7 ksi) was observed for the slab with
lighter steel reinforcement. The maximum ratio of the measured and theoretical induced
moment were 0.65 and 0.62 for the conventional and lightly reinforced slab,
respectively. A 100 % increase of wheel load would not overstress the steel
reinforcement and, accordingly, it was concluded that no methodology is needed to
predict induced stresses resulting from the passage of occasional overload permit
vehicles. Tests to failure resulted in punching shear capacities always larger than six
times the design wheel load, regardless of the reinforcement pattern. Thus, a 59%
reduction in total steel reinforcement did not reduce the strength of the slab below a safe

level.

13

2.2.4 Tests by Fang et al.

Lab testing of a full-scale model bridge was conducted at the University of Texas
at Austin (Fang et al. 1986, Fang et al. 1990). The 190 mm (7.5 in.) slab was supported
on three steel girders spaced at 2.13 m (7 ft.). The span length of the model bridge was
14.94 m (49 ft.). Isotropic steel reinforcement amounted to 60% of the existing
AASHTO speciﬁed requirement was used. The level of stress in the deck due to design
and fatigue loading was recorded.

The research concluded that at the design load level, the overall behavior of the
deck slab, including the local stiffness of the deck at the loaded point, was satisfactory.
Under overload condition, about three times the design load, the behavior of the slab was
essentially linear. Fatigue loading did not signiﬁcantly change the behavior of the deck
due to the design load, even under the overload cited above. Another important
assessment from their conclusions was that the compressive membrane forces did not

signiﬁcantly affect the performance of the bridge at loads below cracking.

2.2.5 Tests by Puckette et al.

The University of Wyoming in conjunction with the Wyoming Highway
Department and the Federal Highway Administration (Puckette et al. 1989) conducted
tests on two noncomposite steel girder bridges. One deck slab was reinforced according
to conventional practice, and the other used minimum isotropic reinforcement. The
stresses in the longitudinal and transverse steel were measured at 133.44 kN (30,000 lbs)
applied load. The AASHTO design load was 92.52 kN (20.8 kips) which corresponds to

a 71.168 kN (16 kips) prototype wheel loads with a 1.3 impact factor.

14

The maximum observed stress in the bottom transverse steel under the 133.44 kN
(30,000le) load was 45.5 MPa (6.6 ksi) and 54.5 (7.9 ksi) for the conventional and
isotropic slabs, respectively. The stresses in the other steel reinforcement of the slab
were even lower. It was concluded that a slab with isotropic steel arrangement

performed satisfactorily under the AASHTO design load.

2.2.6 Tests by Perdikaris et al.

Perdikaris did a number of tests related to the behavior of RC deck slabs under
static and moving loads for slabs with various degrees of edge restraint. Among the test
specimens was a l to 6.6 scale model bridge tested at Case Western Reserve University,
in conjunction with the Ohio DOT and the Federal Highway Administration (Perdikaris
and Beim 1988; Petrou and Perdikaris 1990; Petrou 1991). The 15.24 m (50 ft.)
prototype bridge was supported by four non composite steel girders spaced at 2.13 m (7
ft.). The 216 mm (8.5 in.) slab with orthotropic and isotropic reinforcement was loaded
with pulsating and moving loads in addition to static loads. The pulsating load was
applied sinusoidally at a ﬁxed point at frequencies between 1 and 8 Hz. The maximum
load was about 60% of the static ultimate strength of the slab and the minimum load was
1.779 kN (400 lbs). The constant magnitude of moving load was applied at a maximum
speed of 460 mm/s (1.5 ft.lsec).

The results of the tests showed that the orthotropically reinforced slab was
stronger and stiffer than the slab with isotropic reinforcement. However, the later slab
still had a safety factor of 14 in terms of the static failure load. In term of cracking load

the safety factor was about four. At the design load of 92.6 kN (20,800 lbs), the expected

15

stress in the steel reinforcement was about 20.7 MPa (3 ksi), which is a negligible
amount compared to the yield stress of the steel. Under the pulsating load, the slab with
isotropic reinforcement showed lower fatigue strength than the one with orthotropic
reinforcement. Under moving load, however, the fatigue life (number of load passages)
of the isotropically reinforced slab was about 20 times the fatigue life of the slab with
orthotropic reinforcement. With a reduction of 43% in steel content, the slab with
isotropic reinforcement failed at a 37% lower load, but its ductility increased by about
270%, as compared to the orthotropic reinforcement.

Still other tests related to the application of the new design approach have been
done. The common conclusion of the tests was that a deck slab reinforced with as low as
0.2% isotropic reinforcement is adequate to sustain the current AASHTO design load,
including the provision for fatigue strength. The minimum speciﬁed reinforcement of
0.2% was intended to account for temperature and shrinkage induced stresses. It was
also implied that the conventional design method of slab reinforcement was

unnecessarily excessive.

2.3 External Reinforcement in Deck Slab and Steel Beam

Experiments on steel-free ﬁber reinforced concrete (SFFRC) deck slabs were
conducted at TUNS (Mufti et. al. 1993; Bakht and Mufti 1996; Newhook 1997). These
experiments avoided using steel reinforcement and took advantage of the internal
arching system, which develops in laterally restrained slabs. Fibers were used to control

the cracking due to the effects of shrinkage and temperature in the concrete. Instead of

16

internal steel reinforcement, external steel straps were used to provide lateral restraint of
the slabs.

As reported by the investigators, the initial experiment was guided by trial and
error where each experiment was conceived after studying the results of the preceding
ones. In the ﬁrst experiment, a half-scale model of a two-girder bridge was constructed.
A 100 mm (4 in.) thick steel-free ﬁber reinforced concrete slab was supported by two
W460x82 steel girders, spaced at 1,067 mm (42 in.) center to center, with a longitudinal
span of 3,500 mm (137.8 in.). Three C200x17 diaphragms were placed at the inner
portion of the girders, but none at the ends. Therefore, the free edges of the slabs were
not restrained. Pairs of 75-mm (3 in.) long shear studs, spaced at 305-mm (12 in.) center
to center, were used to connect the slabs with the girders. The deck slab contained 38-
mm (1.5 in.) long polypropylene ﬁbers with 0.88% of volume fraction (0.34% by
weight). The deck slab was tested under a central rectangular patch load measuring 257 '
by 127 mm2 (10 by 5 in.). The slab failed in ﬂexure at a load of 173 kN (38.9 kip).

In the second model, end diaphragms were added to the steel framework, with
the other dimensions the same as in the ﬁrst experiment. Interestingly enough, the slabs
failed at a load of 222 kN (49.9 kip), but not in punching shear. The ﬁrst two models led
to a third model where 64 x 10 mm (2.5 x 0.4 in.) steel straps spaced at 457 mm (18 in.)
centers were welded to the top ﬂanges of the steel girders. This model resulted in a
punching shear failure at a load of 418 kN (93.9 kip). Taking advantage of the localized
failure, the slab was tested at two other locations at distances of 0.868 and 0.438 from
the closer transverse free edge, where S is the spacing of the girders. The failure loads

for the two locations were 316 and 209 kN (71 and 47 kips), respectively, with a hybrid

17

between punching shear and ﬂexure failure modes. The decrease in load capacities near
the transverse edges was believed to be due to the lack of longitudinal restraint. In the
fourth model, three girders were used to study the behavior of the slab under pairs of
loads, one on either side of an internal girder. Testing at mid span of this model resulted
in simultaneous punching shear failures at a load of 418 kN (94 kip) for each loading
pad.

In comparison, the AASHTO Speciﬁcation requires that interstate highway
bridges be designed for a maximum load of 145 kN (32 kip), which is a maximum HL-
93 truck load (AASHTO 1994). In a full scale, the third and fourth models mentioned
above would fail at a load of 1,672 kN (376 kip), which have more than enough factor of
safety to sustain the maximum HL-93 truckload.

This experiment suggested that the transverse free edges of the slab need to be
stiffened by edge beams with their major ﬂexural rigidities in the horizontal plane and
include shear connectors to ensure an effective transfer of in-plane forces from the slabs
to the steel girders. Extensive work has been done as an extension of this experiment,
including the dynamic performance of SFFRC deck slabs (Bakht and Mufti 1996; Mufti
and Bakht 1996; Newhook 1997). These research ﬁndings have been extended to ﬁeld
veriﬁcation by the construction of a full-scale highway bridge. These results have also
been adopted by the Canadian Highway Bridge Design Code (CHBDC) into the design

provisions for steel-free deck slabs on beams.

18

2.4 Empirical Design Method of RC Deck Slabs for Bridges

In addition to the conventional method of design, where the deck slab design is
based on ﬂexure, an empirical method has been adopted for the ﬁrst time in the Ontario
Bridge Design Code (OHBDC 1983). This method is based mainly on the presumption
of signiﬁcant compressive membrane action in the deck slab. Among the prerequisites
for this method are as follows:

0 The girder spacing shall not exceed 3.7 m (12.1 feet).

0 The cantilever portion of the slab shall extend at least 1 m (3.3 feet) beyond the
centerline of the exterior girders. The curb integral with the slab may be used instead
of the 1 m overhang, provided that the combined cross-sectional area of the curb,
plus the slab beyond the centerline of the exterior girder, is not less than the cross-
sectional area of a 1 m length of the deck slab.

0 - The span length to thickness ratio of the slab shall not exceed 15. In skew slabs, the
skew span shall be used in calculating this ratio.

0 The slab thickness is not less than 225 mm (9 in.) and four layers of isotropic
reinforcement bars (p = 0.3% for each layer) shall be used with spacing not to
exceed 300 mm (12 in.).

o Diaphragms shall extend throughout the transverse cross section of the bridge
between external girders, and the maximum spacing of such diaphragms shall be 8 m
(26 feet) for steel I-beams and box girders.

Deck slabs that meet the above requirements need not be analyzed, and are
presumed to have met crack control requirements and adequate shear strength. If the

empirical method is not applicable, yield line methods rather than elastic analyses shall

19

be used to determine the ultimate resistance. The serviceability limit state of cracking

should also be checked.

The AASHTO LRFD Bridge Design Speciﬁcations 1994 (AASHTO 1994)
permits for the ﬁrst time an empirical design method for deck slabs. Excluded in the
speciﬁcation is the cantilevered overhang of the slab. Section 9.7.2 of AASHTO 1994
speciﬁes the method with a long commentary. In order to use the empirical method
certain conditions have to be satisﬁed. These conditions and the logical explanation
behind them are discussed in the following:

0 The supporting components are made of steel and /or concrete. Slabs supported by
timber beams are not allowed to be designed with this method because there is not
enough evidence to show adequate lateral shear transfer between the slab and the
relatively soft timber beams.

0 The deck is fully cast-in-place and water cured. Precast slabs are not allowed to be
designed with this method because it does not provide a good lateral shear transfer
between the deck and the beam. The development of bond strength between concrete
and steel is enhanced in water cured concrete.

0 The deck is of uniform depth, except for haunches at girder ﬂanges and other local
thickening. In the case of non-uniform thickness of the deck slab, the minimum
thickness should be used as the slab thickness in the design.

0 The ratio of effective length to design depth does not exceed 18 and is not less than
6.

0 Core depth of the slab is not less than 100 mm. Core depth is the depth without

concrete covers.

20

The effective length of slab shall be taken as the face-to-face distance (for slabs
monolithic with beams), or the distance between ﬂange tips plus the ﬂange overhang
(for slabs supported on steel or concrete girders), and does not exceed 4, 100 mm.
The minimum depth of the slabs is not less than I 75 mm, excluding a sacriﬁcial
wearing surface where applicable.

There is an overhang beyond the centerline of the outside girder of at least 5 times
the depth of the slab. This condition is satisﬁed if the overhang is at least 3 times the
depth of the slab, and a structurally continuous concrete barrier is made composite
with the overhang.

The speciﬁed 28-day strength of the concrete is not less than 28 MPa.

The deck is made composite with the supporting structural components. To satisfy
this requirement, a minimum of two shear connectors at 600 mm centers shall be
provided in the negative moment region of the continuous steel superstructures. For
concrete girders, the use of stirrups extending into the deck is sufﬁcient to satisfy the
requirement.

Having satisﬁed all of the above conditions, the slab designed using the empirical

method shall be provided with four layers of isotropic reinforcement. The minimum

amount of reinforcement shall be 0.570 mmzlmm for each bottom layer and 0.380

mmzlmm for each top layer. The reinforcing steel shall be Grade 400 or better with

spacing not to exceed 450 mm. It is also speciﬁed to use straight bars with lap splices.

Tests indicated that 0.2% reinforcement (based on effective depth) satisﬁes the

strength requirement. The speciﬁed steel area for the bottom layers corresponds to 0.3%

of the gross area for a 190 mm thick slab. The intent of the extra reinforcement is to

21

provide better crack control in the positive moment region. For the negative moment
region, ﬁeld measurements showed very low stress in the steel, consequently, 0.2%
reinforcement was speciﬁed. The lower steel percentage is also intended to prevent

spalling of the deck due to corrosion of the steel.

2.5 Shear Strength of Reinforced Concrete Slabs

There are a number of analysis methods available to calculate the shear strength
of a concrete slab. However, most of the methods are empirical based on test results of
slab structures supported on columns. Accordingly, direct application of the empirical
methods to calculate the shear strength of a deck slab supported on beams is not
appropriate. In the following discussions, two methods of analysis are reviewed, one
being the empirical method used in the ACI Code, and the other is a rational model for

calculating the punching shear capacity of a restrained concrete slab.

2.5.1 Empirical Methods

As reported by Joint ASCE-ACI Task Committee 426 (Joint ASCE—ACI 1974),
the expression for ultimate shear strength of a slab can be divided into two categories.
The ﬁrst were the expressions dependent primarily on concrete strength, and the other
were expressions dependent primarily on ﬂexural effects such as bending moment and
steel ratio. Among the expressions, none of them included the condition of the slab at the
boundary. Nonetheless, it is interesting to note the conclusion of the committee

regarding the factors that primarily affect the shear strength of a concrete slab. Among

22

the factors was the restraint (provided by the external frames) where the in-plane
compression due to the presence of restraint considerably enhances the capacity of the
slab. However, it was stated that the degree of strength enhancement is difﬁcult to
calculate.

Another signiﬁcant ﬁnding from the committee of prime interest to designers is
that the shear strength of a slab can be increased substantially by nonyielding portions of
a structure surrounding the loaded area. Therefore, it is necessary to consider the
stiffness of the slab and the rigidity against lateral movements of its supports to
determine strength changes caused by the surrounding elements of the structure.

Comparing test results and calculated values (using various empirical formulae),
the committee found that the shear capacity equation from the ACI 318-71 gives a lower
bound value of the shear strength, which is preferable for the purpose of design. The

complete form of the expression (in US. common units) is given by:

V“ = W?

.4(c+d)

 

where Vu is the ultimate shear strength of the slab with effective depth to centroid of
reinforcement, d, and column (load) side length of c. The term on the right hand side is
the limited shear stress of the slab subjected to bending in two directions, in which fc’
has unit of pound per square inch (psi). For the purpose of comparing the calculated
value to the experimental test result, the strength reduction factor, 4) is taken equal to
one.

The shear strength of nonprestressed slabs and footings according to the ACI

318-95 (ACI 1995) is given in three different expressions depending on the size and the

23

position of column, load, or reaction area. For slabs loaded (supported) at the center, its

shear capacity is limited to the following two expressions:

ACI Equation 11-36: Vc = (2 +-b4—) fc’bod and

ACI Equation 11-37: Vc = 4 fc’bod
where Vc is the shear capacity of the slab without shear reinforcement, be is the ratio of
long side to short side of concentrated load or loaded area, and ho is the perimeter of the
critical section. The critical section can be calculated at a distance d/Z from the edges of
the load or reaction area. For a value of ﬁe less than or equal to 2.0 equation 11-37
governs.

For two-way prestressed slabs and footing, the ACI Code formula for shear

strength is:
ACI Equation 11-38: Vc =(ﬂP1/fc’+0.3fpc)bod +Vp

The constant [3,, has a limiting value of 3.5, and the compressive stress (at the centroid of
the cross section), fpc is limited to a value between 125 psi and 500 psi. Vp is the vertical
component of effective prestress force at the section.

It is obvious that the ACI Code formulae will not be appropriate for slabs with
varying boundary conditions and steel reinforcement. However, for comparison purpose,
these equations can be used to obtain an estimate of the shear capacity of restrained
slabs. The compressive stress, ﬁx in ACI equation 11-38 can be viewed as being due to
the compressive membrane forces resulting from arching action in the slab.

Other groups of researchers proposed different empirical formulae to estimate the

shear strength of a deck slab. Among the groups are: J iang and Shen, and Azad et al.

24

J iang and Shen (1986) proposed a plasticity-based failure model in the form of a simple
formula to calculate the punching shear strength of a concrete slab as given by:

Vjs = 0.074 fc’7r(d0 + h)h
In this formula, do is the diameter of the loaded area and h is the thickness of the slab.
Azad et al. (1994) proposed a modiﬁcation to the ACI 318-89 formula to calculate the
shear strength of a concrete slab, in which the critical section was to be determined using
the shear plane angle equal to 30 degree instead of 45 degree. For a rectangular load area
of size a and b, the resulting equation for the perimeter of critical section is given by:

he = 2(a + b) + 6.93d
Azad et al. also compared the theoretical shear strength values given by the Jiang and
Shen model and Azad’ formulae to a limited number of test results. It was shown that
using a shear plane angle of 30 degrees to determine the perimeter of critical section in
the ACI formula yielded a shear strength value closer to the test results, on the safe side.
The Jiang and Shen formula was reported to hold true for deck slabs with lower concrete

strength.

2.5.2 Rational Model of Punching Shear

The empirical approaches have been of value in the studies of punching shear
capacity of reinforced concrete slabs. The application of each approach is limited to the
conditions upon which the approach was made. For a slab in general, a rational
mechanical model that accounts for arching action, boundary conditions, and steel

reinforcement is necessary.

25

The ﬁrst rational model to calculate the punching shear capacity of a reinforced
concrete slab was proposed by Kinnunen and Nylander (1960). The K and N model was
developed based on extensive tests of circular slabs supported on a central column. The
load was applied along the perimeter of the slab, which therefore represents a simply
supported circular slab with a concentrated load at the column. To apply this model for
the analysis of bridge deck slabs, it is necessary to introduce boundary forces along the
perimeter of the slab. For that purpose, Hewitt and Batchelor ( 1972) modiﬁed the K and
N model. The punching model of a restrained slab proposed by Hewitt and Batchelor
became the model, speciﬁcally developed for RC deck slabs. However, both models are
not as popular as the empirical approaches because they involve a great deal of analysis
and entail somewhat vague assumptions. Petrou and Perdikaris (1996) proposed an
approach, where the slab was modeled as a truss with springs along the slab boundaries.
This model involved so many constants and assumptions that it proved difﬁcult to
follow. For that reason, the model is not included in this study. In the following section,
the basic principles of the K and N, and Hewitt and Batchelor models are discussed,
while a more detail discussion is presented in Chapter 3.

The theory proposed by Kinnunen and Nylander was based on extensive
experiments conducted in the late 1950’s in Stockholm, Sweden (Kinnunen and
Nylander 1960, Kinnunen 1963). Referring to Figure 2.1, Kinnunen and Nylander
argued that the outer portion of the slab, which is bounded by the shear crack and by
radial cracks, is loaded through a compressed conical shell that develops from the

column (load) to the end of the shear crack. The conical shell is assumed to have a shape

26

and varying thickness in such a manner that the compressive stress at the intersection
with the column and the root of the shear crack are approximately equal.

The criterion of failure, as argued by Kinnunen and Nylander is as follows:
The strength and eﬂective modulus of elasticity of the slab increase due to the presence
of a three-dimensional state of compressive stress in the conical shell. The tangential
strain at the top surface of the slab at a distance (B/2 + y) from the center of the column
was taken as a measure of the supporting strength of the conical shell. Failure takes
place when the strain reaches such a high value that the strength of the conical shell is
impaired. Thus, failure is described as the failure of the conical shell in compression,

which takes place when the tangential strain reaches a characteristic value.

 

 

C P s/(zn)

 

 

 

 

 

 

Conbal shell

 

Figure 2.1 K and N Model of Punching Shear

The punching load was calculated by assuming a dimension of the conical shell
and then following a convergent iterative procedure. They found that their theory gave
values for the punching load which were in satisfactory agreement with results of their

own tests as well as those of others. For the case of a two-way reinforced slab the

27

calculated values were lower than the test results. Kinnunen, in his later paper, argued
that the higher test values were attributed to the dowel and membrane effects, and
consequently, the calculated value should be corrected by increasing the value by 20
percent.

Modiﬁcation of the K and N model was proposed by Hewitt and Batchelor
(Hewitt 1972, Hewitt and Batchelor 1975, Batchelor et al. 1978). The proposed model
incorporated boundary restraining forces and moments (ﬁxed boundary actions) into an
idealized model developed for simply supported slabs by Kinnunen and Nylander.

The compressive membrane and ﬁxed boundary action forces depend greatly on
the slab deﬂection and the degree of edge restraint. When a slab fails in punching, the
ultimate resisting moment of the slab does not necessarily develop. Although the
boundary forces could theoretically be calculated using the stress-strain relationship for
reinforcement and the concrete, the stress distribution and the resultant in-plane forces at
the instant of failure by punching is extremely difﬁcult to calculate or verify.
Consequently, an empirical factor called the “restraint factor”, Fr, was introduced to
account for the fraction of boundary forces at the instance of punching failure. The
factor, Fr, has a value between zero and one, depending on the slab’s condition at the
boundary. A value between 0.52 and 0.74 was appropriate for the deck slabs tested by
Hewitt and Batchelor.

As reported by Petrou (Petrou 1991), Bousias modiﬁed the Hewitt and Batchelor
model, by including the effect of compressive reinforcement. A rectangular compressive
block was used instead of a parabolic stress distribution to calculate the boundary forces

at failure. Such modiﬁcations resulted in a punching load about 17 to 20% higher than

28

that of the Hewitt and Batchelor model. In view of this difference, it was concluded that
the value of the restraint factor used by Hewitt and Batchelor has to be reduced, thus this
modiﬁcation resulted in a different value for the restraint factor. To limit the discussion

this model will not be included in the analysis.

2.6 Finite Element Analysis of Reinforced Concrete Slab

Application of ﬁnite element analyses, FEA, for reinforced concrete structures
has increased steadily because of the development of relatively powerful ﬁnite element
procedures over the last four decades. However, the analysis results obtained from the
FEA still vary considerably as compared to experimental results, particularly those
involving ultimate capacity of the structure. One possible cause of the problem is that the
reinforced concrete materials exhibit a complex structural response with various
nonlinearities. Interaction between concrete and the steel reinforcement embedded in the
concrete complicate the behavior of the composite concrete material. In addition, the
presence of cracks in concrete affects the overall behavior of the structure that requires
the model to include the post-cracking behavior of concrete. However, modeling the
post-elastic behavior of concrete remains a poorly understood area.

Ngo and Scordelis (1967) published the ﬁrst paper on ﬁnite element analysis of
reinforced concrete beams in 1967. This ﬁrst model included cracking and bond
simulation capability that enabled the computation of stress in the vicinity of the crack.
This led to the development of a discrete crack model. The second model tried to locate
the zones of cracking and how its development affected the overall behavior of the

structure. This second approach led to the development of a smeared crack model

29

(Schnobrich 1985). Subsequent studies have been undertaken to study the overall
behavior of special types of structures, such as RC plates and shells. Emphasis was
placed on the development of a smeared crack model within a layered plate and shell
element. Among those who worked on this approach are Hand et al. (1973), and Lin and
Scordelis (1975).

Finite element analysis of RC plate and shell structures gained its popularity
because their geometry is usually complex. The rapid development of shell elements in
the aerospace industry is another factor that improves the FEA of RC plate and shell
structures. Many investigators used a comer supported two-way RC slab for a numerical
example. This frequently referenced slab was tested by McNeice at the University of
London, England in 1967 (J ofriet and McNeice 1971). Using an effective section
approach to account for the crack development in the ﬂexural response, Jofriet and
McNeice reported two FE models of the McNeice slab that yielded load deﬂection
diagrams very similar to the test results.

Using nonlinear layered analysis, Hand et al. (1973) reported the FE model of
McNeice slab. Lin and Scordelis (1975) also analyzed the same slab, in which tension
stiffening was included in the material model for reinforced concrete. Improved results
were reported with the inclusion of tension stiffening. Another researchers (Gilbert and
Warner 1978) reported the analysis of McNeice slab utilizing different methods of
incorporating tension stiffening. In addition to modifying the stress strain relation for
concrete, Gilbert and Warner used modiﬁed steel properties (increased E values) to
assign the tension stiffening. They claimed that the second approach was as satisfactory

as the ﬁrst one, but it required less computer time.

30

The importance of including a realistic tension stiffening model was emphasized
when analyzing lightly reinforced sections or structures that are endangered by stability
failures (Schnobrich 1985). As a global property of reinforced concrete, the tension
stiffening is a consequence of the presence of steel within the concrete. Accordingly,
different models of tension stiffening have been used in the literature. One of the models
most commonly used is as shown in Figure 2.2. This model is similar to the post-
cracking response of concrete in tension when gradually unloaded. The factor ,6
characterizes the length of the descending branch of the concrete stress-strain curve.

Based on limited experimental results, Vecchio and Collins reported reasonably
high tension stiffening effects of up to one hundred times greater than the cracking
strain. Admittedly, Collins’ graph of measured tension stiffening, as reproduced by
Schnobrich (1985), was obtained from a heavily reinforced section with a reinforcement
percentage of 1.875% in at least one direction. The value of ,6 = 20 is sufﬁciently
conservative and was used by many investigators. Schnobrich suggested that the upper

bound value of ﬁe, should be equal to the yield strain of the steel reinforcement.

f, A

 

 

 

 

\
/
ﬁle:

Figure 2.2 Tension Stiffening Model for Reinforced Concrete

31

Hsu and Zhang (1996) proposed a different approach to account for tension
stiffening for an RC membrane element, in which they considered two effects
simultaneously. First, the tensile stress strain in concrete; second, the stress strain curve
in steel embedded in concrete. They argued that after the cracking of concrete, higher
steel strains are located at the cracks, while between the cracks the strains are smaller.
Therefore, the steel stress strain curve should be modiﬁed accordingly. They
demonstrated the effects of using simpliﬁed, modiﬁed, and accurate constitutive laws for
concrete and steel in tension, on the response of six full scale RC panels tested at the
University of Houston. It was shown that using the accurate constitutive laws into the
softened truss model produced an accurate prediction of load-deformation response
throughout the loading history.

The importance of incorporating tension stiffening in the concrete model was
also shown in the example manual of a commercially available FE program, ABAQUS
(H, K & S, Inc. 1996; H, K & S, Inc. c 1999). They also used McNeice slab as an
example of non-linear analysis of RC structures. Three different values of tension
stiffening: 0.0005, 0.001, and 0.002 were used to obtain three different load deﬂection
diagrams. The model with a tension stiffening value of 0.001 yielded a load deﬂection
diagram that best matched the test result. It was also shown that with a tension stiffening
value of 0.0005, the responses of the slab model became brittle and failed at a lower load
level. Using a value of 0.002 very ductile response was obtained, in which the analysis
continued until a much larger deﬂection than the maximum test value was reached.

To date shell elements are widely used for the analysis of RC slab structures.

Even to model punching shear, Polak proposed the use of layered plate and shell element

32

(Polak 1998). Huria et al. (1993) reported a nonlinear FEA of an RC slab bridge using an
isoparametric shell element. Abbasi et al. (1992) and Azad et al. (1994) also reported a
ﬁnite element analysis of a reinforced concrete deck slab using plate and shell elements.
Unfortunately, they only reported the ultimate load values instead of the load deﬂection
relationships. The later is more meaningful in studying the behavior of deck slabs under
lower load levels. Gonzales-Vidosa et al. (1988) used axisymmetric solid elements to
model symmetrical punching of RC slabs. They claimed that the FE package they were
using could be used in studying the punching problem in RC slabs with conﬁdence.
Admittedly, their package was limited for the analysis of RC elements subjected to
plane-stress and axisyrnmetrical loading conditions.

The analysis of a complex 1:6 scale model of a reactor vessel has served as a
yardstick for the FEA of general concrete structures. This structure was built and
experimentally tested in 1987 at Sandia National Laboratories (Noguchi and Schnobrich
1991; Bathe et al. 1989; H, K & S, Inc. 1996). Pretest predictions of the structure (using
axisymmetric solids) were provided by 10 organizations using six different FE programs
including ABAQUS, ADINA, and NFAP. There was signiﬁcant variation in the reported
predictions. However, the experimental results also showed considerable variation.
Therefore, the differences in the analytical predictions did not seem severe, especially
for the prediction of the global response of the structure. The broader band of analysis
values lied on the values of tension stiffening. As reported by Schnobrich, the consensus
of the investigators was that a no tension concrete model gave the best correlation with
the observation of the test. The explanation for this is that the cracking state has been

developed prior to the full pressure test. The actual failure of this containment structure

33

occurred at locations not predicted by the FEA. For this reason, post-test analyses were
undertaken using three-dimensional models to verify their ability to reproduce the local
failure behavior. Bathe et al. (1989) claimed to have a high degree of conﬁdence in the
ADINA system and the concrete model mentioned above. However, they admitted that
concrete still remains a very complex material to model numerically. They suggested
improvement of the material model through the open exchange of ideas and mutual
respect between software developers and structural analysts.

Different from the aforementioned researchers, Wegner and Mufti used three-
dimensional solid elements (instead of shell elements) to model a ﬁber reinforced
concrete deck slab for the bridge mentioned earlier in this chapter (W egner and Mufti
1994). They used a commercial program, ADINA to model every component in their
bridge to obtain as realistic a model as possible. The deck slab was modeled using 20-
node, three-dimensional isoparametric elements, while two-node beam elements were
used for the steel girders. Vertically oriented rigid links (made of beam elements with
high section properties) were used to connect the beam elements to the deck elements.
The links were capable of transmitting both vertical forces and shear forces between the
steel girders and the concrete deck, where full composite action was assumed. Channel
section diaphragms were also modeled using beam elements, and steel straps were
modeled using truss elements capable of transmitting axial forces only. A total of eleven
different models were developed and their respective results of nonlinear analysis were
reported in the form of load-deﬂection curves.

The failure loads resulting from each model were reported as being widely

scattered, which shows that the models were very sensitive to a number of modeling

34

parameters, and therefore required some experimentation to tune the numerical
procedures to a particular application. Among the models that gave results closest to the
load tests was the one incorporating the entire system (without taking advantage of
conditions of symmetry). The load was applied over a realistic area (rather than at a
single node), and the girder top ﬂanges were also modeled to represent the effect of a
shortened span for the slab. A model with a reduced value of concrete modulus of
elasticity was also developed. Interestingly enough, this model gave analysis results
closest to the experimental result. They concluded that the three-dimensional
isoparametric element used to model the deck slab produced overly stiff responses, and
suggested the need for improvement of numerical techniques to be more reliable.

There are a number of works related to the FEA of RC structures reported in the
literature including those published in foreign languages (Aoyama and Noguchi 1985;
Noguchi and Schnobrich 1991). Despite the overwhelming number of papers on the FEA
of concrete structures, very view studies have addressed a true three-dimensional
problem. Currently however, several commercial general-purpose FE programs have
incorporated three-dimensional model for concrete into their element library. Among the
programs are: ABAQUS, ADINA, and DIANA. For the present study, ABAQUS was

used to analyze the model bridge in Chapter 3.

35

CHAPTER 3

THEORY AND ANALYSIS

3.1 Introduction

Extensive research into the behavior of RC deck slabs discovered that the primary
action by which these slabs resist a concentrated wheel load is not ﬂexure, as was
traditionally believed, but a complex three dimensional membrane stress state referred to
as internal arching action. Cracking of the deck in the positive moment region and the
resulting upward shift of the neutral axis in that portion of the slab make the action
possible. The arching action is sustained by in-plane membrane forces which develop as a
result of lateral conﬁnement provided by the surrounding concrete slab, rigid

appurtenances and supporting components acting compositely with the slab.

By now, the arching action phenomenon has been widely accepted among bridge
engineers, though some state DOT are still reluctant to use the empirical design method
of deck slab design that takes advantage of the arching action. Despite the overwhelming
data available through extensive experimental work, there is not yet a theoretically sound
analytical work available. The complexity of the problem has contributed to the slow
development of the analytical method. In addition, the nonlinear behavior of concrete
subjected to a complex stress state is still insufﬁciently known. Accordingly, the
available analysis methods rely on empirical coefﬁcients that are not necessarily suitable

for general application. Punching theory of a restrained slab proposed by Hewitt and

36

_,..

Batchelor is one of the rational approaches that incorporates empirical factors and
coefﬁcients. Moreover, this theory was a direct modiﬁcation of the K and N theory for
simply supported slabs for which design parameters were developed. Nevertheless, the
theory could only predict the strength and deﬂection of the deck at failure. However,
being a method developed speciﬁcally for deck slabs, the theory is helpful for
understanding the phenomenon of arching action. For this reason the theory is
analytically re-derived in this chapter. The theory is also utilized to calculate the deck

strength under different lateral restraint conditions.

Petrou and Perdikaris (1990) reported predictions of deck strengths using yield
line theory and the ACI Code formulae. The yield line theory was reported to under
estimate the punching strength of the deck slab. Kuang and Morley (1992), in their
experiment on punching shear behavior of restrained reinforced concrete, also concluded
that J ohansen’s yield line theory is not suitable for predicting the punching load capacity
of a restrained slab. In this study, however, the theory is not considered in the analysis.
The ACI formulae, on the other hand, were reported to give strength values closer to the

experimental results than those from the yield line theory.

Kuang and Morley compared the values of slab punching strength given by the
British Standard, BS 8110 and the ACI 318-89. It was reported that for a slab with lower
steel ratio the ACI formula gave punching strength values closer to the test results than
the British Standard did. Nevertheless, the ACI formulae were developed for design
purposes, which accordingly, are on the conservative side. Moreover, the formulae do not
consider the effect of restraint provided by the slab boundaries, which admittedly under

estimate the strength of a restrained slab. However, the formulae are simple and useful to

37

rapidly estimate the shear strength of the deck. For these reasons, the formulae are

included for comparison.

In this study, the behavior of a deck slab under lower load magnitudes is more
important than its load capacity because in practical cases, the deck is subjected to a
wheel load that is much lower (about 10%) than its ultimate capacity. An analytical
method capable of predicting the response of the deck under any magnitude of applied
loads would be more useful for understanding the deck behavior. More importantly, the
method should also be able to explicitly incorporate the tie rods into the analysis in order

to study their effect on the deck behavior.

Considering the successful application of numerical analyses for RC plate and
shell structures, ﬁnite element analysis (FEA) is utilized in this study to analyze the deck
slab as an integral part of a model bridge. Even though there are not many reports on the
successful applications of FEA on three-dimensional RC deck slabs, it is widely used and
readily available. The theory itself will not be discussed here because it has been widely
accepted as a standard method for stress analysis. However, the material modeling for
reinforced concrete is brieﬂy summarized since the development of the constitutive

model of the material is not yet as advanced as that of other structural materials.

3.2 Rational Model of Punching Shear Strength of Restrained Slabs

Theory of punching shear failure for simply supported RC slabs was proposed by
Kinnunen and Nylander based on extensive experimental work conducted in the late

1950’s in Stockholm, Sweden (Kinnunen and Nylander 1960, Kinnunen 1963). The

38

theory was based on the observed behavior and failure mechanism of a circular slab
supported by a central column. For a composite RC slab and steel girder system, the
conditions at the boundaries of the slab are partially restrained. In order to use the model
proposed by Kinnunen and N ylander (K and N model), some modiﬁcations of the model
were required. For this purpose, Hewitt and Batchelor proposed a punching shear model
for restrained slabs, where the boundary restraining force, Pb and the boundary restraining
moment, Mb are incorporated into the K and N model (Hewitt and Batchelor, 1975,
Batchelor et al 1978). Mb and Fb are forces per unit length of slab, acting in the plane of

slab along its boundary.

Similar to the Hewitt and Batchelor reports, Mufti and Newhook (1998) showed
that under a concentrated load, the failure mechanism of a deck slab supported on steel
girders is as shown in Figure 3.1. Under an initiating load level, radial cracks formed on
the bottom surface of the concrete slab, originating below the load. As the load increases,
the radial cracks gradually propagated to the top surface of the slab to become full depth
cracks. Circular cracks formed on the top surface of the slab at a diameter approximately
equal to the girder spacing (this circular crack pattern is the basis of the assumed
equivalent circular slab). At a load somewhat less than the failure load, an inclined shear
crack developed, originating on the bottom surface of the slab at some distance away
from the load, in a circular manner. At punching failure, the shear crack formed the upper
surface of the punch cone. This failure mechanism was similar to that of the circular slabs
tested by Kinnunen and Nylander. Accordingly, the deck slab could be analyzed as an

equivalent circular slab.

39

#1!
[A Loaded Area

 

 

Circular Crack Pattern 1
(Equivalent Circle) f

>Radial Crack T
_-_._$hser_Cra¢k,_---_-m--

-‘O’.-~--v-~w~-‘nt .-.. .......-..‘-—-.~_-.‘...- -—_---p--'n-o—n-vn-—-~.—----------—-—-- ......

Sector Element
’l—J‘.

Figure 3.1. Crack Pattern of Restrained Deck Slab (Plan View)

 
 
  

 

 

 

 

 

 

Kinnunen and Nylander argued that the portion of the slab, which is bounded by
the shear crack and radial cracks, is loaded through a compressed conical shell that
develops from the circular column (concentrated load) to the end of the shear cracks. The
imaginary conical shell is assumed to have the shape as shown in Figure 3.2, and its
thickness varies in such a manner that the compressive stresses at the intersection with
the column and the root of the shear crack are approximately equal. At the tip of the
conical shell, at the top surface of the slab, the concrete is in a state of three-dimensional
compressive stress where conﬁnement provided by these stresses signiﬁcantly increases

the strength of the concrete in that area.

The section of the slab outside the shear cracks can be divided into wedges or
sector elements of angle [3 bounded by shear cracks, radial cracks, and the outside edges
of an equivalent circular slab as shown in Figures 3.1, 3.2, and 3.3. These sector elements
are assumed to act as rigid bodies, and under failure loading, they rotate about a center of
rotation, CR, with angle w. Figure 3.3 also shows forces in a sector element used as the

basis for the derivation of equations to determine the punching strength of a restrained

40

slab (Kinnunen and N ylander 1960, Kinnunen 1963, Hewitt and Batchelor 1975). The
details of the derivation are given in Appendix A. In the following sections, the punching

shear model of a circular slab is summarized.

 

 

 

 

 

P
(‘- P Conical shell
I
\y or E T Slcar crack A
l I :T b
y
L... ................. -9..qu - - .. - ............... A. Fl,
"""" , "“" ""‘ \y‘j.
K- Co ->i p B [(210 "

 

 

T Blair)

  

 

 

 

 

 

 

P B/(21c)

Figure 3.3 Sector Element (Wedge) Showing Forces and Notations

41

3.2.1 Mechanical Model for Simply Supported Slabs

Referring to Figure 3.2 and Figure A2 in Appendix A, the load P, which is

transmitted through the conical shell is given by the following expressions:

tan a(l— tana)

 

fm) = sin acos a(l — tan a) = 1+ tan2 a Equation (1)
B y (___B+2y) .
P: h h (B+y) —--—hf,f(a,)2 Equatron(2)

The complete list of notation is given in Appendix A. The above equations are valid for

both simply supported and restrained slabs.

If ‘1’ is the angle of the rotation around the CR, then using the expression for
strain a, of reinforcing steel at a distance, r = r8 (the boundary between the yield and

elastic stressranges) from the center of the slab, the following expression is obtained:

  

:- |..‘I
ll

s W1 " 3%) Equation (3)

3

 

.9: Is

Referring to Fig. A3 in Appendix A, the resultant force in the ring reinforcement, R1 is

calculated using dAs = p h dr as follows:

 

 

For r8 2 Co: R1 = pf h2 (r—‘ - 29-) + 5‘- ln Cl 2 Equation (4.a)
” h h h t;
For r5 < Co: R1 = M0h2[%ln Cl 2] Equation (4.b)

42

a...

At failure, the steel reinforcement inside the shear crack yields, therefore the steel stress

is equal to f,,. Accordingly, the resultant force of the radial steel reinforcement, R2 can be

expressed as:
C
For r8 2 Co: R2 = e‘sEsphﬂC0 = fwhpCoﬂ = [2)“,th 719,6 Equation (5.a)
For rs < Co: R = pf ,h2 I’- ,6 Equation (5b)
2 s) h

The compressive strain in concrete has the relation: ed = w): , and the corresponding
r

resultant compressive force in concrete, R3 can be written as:

h2 B C/ 2
R3 = 34%)2 5(3): + 1) mm with a radial component of R3B.

Failure occurs when set at the top of the slab (under the load) reaches a limiting value.
Based on measurement during their experiment, Kinnunen and Nylander proposed the

following expressions for the Young’s modulus and failure ratio of concrete:

EC = 105(035 + 03%?) Equation (6)

f.

f = 235 Equation (7)

r-(BIZoy)

The term fcubc denotes the compressive strength of a lSO-mm concrete cube in kilogram
force per square centimeters. The failure ratio is expressed in terms of the ratio of the
stress of concrete in the imaginary conical shell and the tangential concrete stress at

radius r = (B/2 + y) from the center of the circular slab.

43

Hewitt and Batchelor converted the compressive strength of 6 by 12 in. cylinder

(fcyl) in psi units into fcubc using the following relation:

= fcyl
“b: 0.75 + 0.000025 f0,

 

fc Equation (8)

For slabs with two-way reinforcement, Kinnunen (1963) proposed the following

 

equations:
B B B .

For — S 2; y! = 0.0035(1— 0.22—)(l + —) Equation (9.a)
h h 2 y
B B .

For h > 2; VI = 0.00195(1+ 2—;) Equation (9.b)

B
and (if = E +1.8 Equation (10)

The expression for the stress in the conical shell, f, becomes as follows:

B B
For 0 < — S 2: f, = 825(0.35 + 03%)(1— 022—) Equation (1 La)
h 150 h
B f “be .
For 71- > 2: f, = 0.00195Ec = 460(0.35+0.3—1£5-6-) Equation (1 Lb)

Referring to Fig. A3 in Appendix A, the equilibrium condition is satisﬁed by taking

moments with respect to the point of intersection of R1 and PB/(21t). It yields:

 

lwtana 1 y l C/2
Kta —1-—————=-——1 —1—-
( y "a )(l+tan2a) 4.7(+B)"iB/z+y]
C/2-B/2
h K =————
were y (h—y/3)

Rearrangement of terms results in the second equation for punching load of a simply

supported slab:
2 R
P = 750?, + 732-) Equation ( 12)

Using equation (1) through (12) the punching load of a simply supported slab can be
determined by solving the punching load P in equations (2) and (15) through an iterative
procedure. For the case of a restrained slab, the equations for a simply supported slab are
coupled with additional equations as a result of the additional boundary forces as given in

the following section.

3.2.2 Mechanical Model of Restrained Slabs

The model for a restrained slab proposed by Hewitt and Batchelor is basically the
one proposed by Kinnunen and Nylander, with moment and force at the level of the
compression reinforcement at the boundary included. Referring to Figure A.4, taking

moments with respect to the point of intersection of the resultant of the forces R1 and

 

 

PIS/(21:) gives:
27:1?3 Mb 71C
=K — ta— ,
P ’ co P (h- y/3)

where the notations are given in Appendix A.

Assuming a circular fan failure mode, the resisting moments Mb, is given by
4112M.» Therefore, replacing Mb/P with X/(4rt), in which X is a factor of the order unity,

yields:

45

M
X = 471'?" Equation (13)

Using ftlfc, = 2.35 and substituting the expressions for P and R3 for simply supported slabs

the following equations are obtained:

XC
K =K ———- E ti 14

(l—tana)
(1+tan2a)

 

1+ B
( y/ ) [ C/Z Equation(15)

=Kt —1
4.7 “is/2”] (zana )

Veriﬁcation of these equations is obvious: if Mb = 0, then X = 0, and therefore, Kz = Ky

which is identical to that of simply supported slabs.

Referring to Fig. A5 in Appendix A, the equilibrium equation is obtained by
taking moments about the point of intersection of TB/(21t) in the conical shell and the
resultant of the forces, R3. Then the second expression for punching load of a restrained

slab is given by the following relation:

2n R C .
P=-K—Z(R, +-j+F,—§- Equatron(l6)

Again, if Fb and Mb = 0, then X = 0 and therefore, K, = Ky, In that case Equation (16) is
identical to Equation (12) for a simply supported slab. This means that the expressions

for punching load of a restrained slab are those for a simply supported slab, provided that

the boundary forces (are zero.

In order to solve for the punching strength of a restrained slab, it is necessary to
develop an expression for the boundary forces using static equilibrium at the slab

boundary. Shown in Figure A5 are resultant maximum boundary stresses and forces in

46

the slab. The maximum tensile force, Ft, per unit length of the boundary, at the level of

tensile reinforcement, is given by:
F; = hm”, Equation (17)

Likewise, the maximum compressive force per unit length of the slab boundary is given

by:
t A
PC = kfm (2 — 7;) Equation (18)

At failure, the stresses in concrete and steel reinforcement at the slab boundary,
do not necessarily reach maximum values. Accordingly, a restraint factor, Fr was
introduced to account for the lower stresses in the slab boundary. The factor Fr is purely
empirical, to make up the difference between the calculated and measured punching load.
However, it is reasonable that the actual boundary restraints at failure are less than the

maximum possible values. Accordingly, the following expressions are adopted:

13 3
M, = EMMM) = F,[F,(2h — :)— Fc(h—Et-3—2A)] Equation (19)

F, = Fanm) = F,[F, — 13] Equation (20)

Finally, assuming a small angle of rotation, w, the slab deﬂection at failure, A can be

approximately expressed as:

(C-3)
2

 

A __. 1" Equation (21)

This concludes the development of the equations necessary to solve for the

punching load, P, of a circular restrained slab with a circular concentrated load through

47

an iterative procedure. For the case of a non-circular slab with a non-circular loaded area,
it is necessary to assume equivalent circles for the slab and load by equating their

perimeters (Kinnunen and Nylander 1960).

N ylander reported that, the punching load calculated using the above mentioned
theory for simply supported slabs is about 20% less than the observed values (N ylander
1963). The reason for this difference is that the formulae were derived based on the
conditions at shear crack formation stage. The ultimate loads are substantially higher than
the loads at the ﬁrst crack stage, because when a shear crack crosses the steel
reinforcement, the crack propagation is arrested due to the bond between the steel and the
concrete. This phenomenon is known as dowel action. In addition to the dowel action, the
compressive membrane forces developed under the applied load also cause the crack to
stop. Both phenomena are logically sound and experimentally provable. However, it is
not easy to establish analytical expressions that explain the phenomena because the dowel
and membrane forces developed at failure are difﬁcult to measure or calculate. An
empirical factor of 1.2 was reported to give a corrected punching load very close to the
measured value. The factor was adopted for restrained slabs by Batchelor and Hewitt.

Accordingly, the corrected theoretical punching load, V is given by: V = 1.2P.

3.3 Analyses of Punching Strength of RC Slabs with Additional Lateral Restraint

Analyses of RC deck slabs with additional lateral restraint were conducted
utilizing an empirical approach based on the punching shear model of a restrained slab

developed by Kinnunen and Nylander, and Hewitt and Batchelor. For practical cases of

48

deck slabs, the value of the restraint factor, F, reported by Hewitt and Batchelor, ranges
from 0.52 to 0.74 (Hewitt and Batchelor 1975). In this study, the lateral restraint for a
slab with additional tie rods should be higher than that without tie rods. Consequently, the
restraint factor, F, for a slab with tie rods should be larger than that without tie rods,
which also results in a higher punching load for slab with tie rods. Unfortunately, the
exact values of these factors could not be determined until test results are obtained.
Accordingly, the punching loads are determined using restraint factors between 0.5 and
0.9, where the slab in this study is expected to have a restraint factor value within this

range.

For this analysis, a 1/5 scale model bridge is assumed, and the properties of the
slab and its supporting structures are as designed in Chapter 4, and used in the
experimental work in Chapter 5. The model bridge consists of three steel girders and a
concrete deck of 40.6 mm (1.6 in.) thick, with girder spacing of 508 mm (20 in.). The
load was applied through a 102 mm by 51 mm (4 in. by 2 in.) pad, which corresponds to
a perimeter of 305 mm (12 in.). The geometry and dimensions of the equivalent circular

slab used for the analysis are shown in Figure 3.4.

The steps required to determine the punching load capacity of a restrained slab

with known value of the restraint factor are listed as follows:

1. Increment a value for deﬂection, A;

2. Calculate. F, and F, from Equations (17) and (18), respectively;
3. Calculate Mb and Fb from Equations ( 19) and (20), respectively;

4. Increment a value for X;

49

10.

11.

12.

I3.

14.

15.

Increment 3 value for y/h;

Calculate f, from Equation (1 La) or ( 1 Lb), and tan or for for from Equation (15) and
(1);

Calculate P from Equation (2);

Calculate w , rs and Co from Equations (9.a) or (9.b), (3) and (10);

Calculate R, from Equations (4.a) or (4.b), and R2/B from Equation (5.a) or (5.b);
Calculate A and P from Equations (21) and (16);

Compare the values of P obtained from step 7 and step 10. If the difference is

signiﬁcant go to step 5;
Calculate X from Equation (13);

Compare the value of X in step 4 with that calculated in step 12. If the difference is

signiﬁcant go to step 4;
Calculate F, from Equation (18) using the new value of A;

Compare the values of F, calculated in step 2 and 14. If the difference is signiﬁcant

go to step 1;

16. Determine the corrected punching load, V = 1.2 P.

A computer program that implements the above steps was developed. The

ﬂowchart and the complete listing of the code are given in Appendix B, Flow Diagram

and Code Listing to Calculate Punching Shear Strength of Restrained Slabs. Upon

completion of the experimental tests, the restraint factors for the model slabs could be

determined by equating the calculated and the experimentally observed values.

50

 

 

P
Eqivabnt Load Diarmter,
9i B = 97 rrm(3.82 in)
I

 

 

 

 

 

 

T ,_ T
'1‘: 41mm H: 33m
(1. 6 in) Isotropi: Steel, Rho = 0.2% (1.3 in)
I e e e__ 4-
, Eqivalent Slab Diarmter,
C =508 rrm(20 in) >i

 

Figure 3.4 Geometry of Equivalent Circular Slab

The complete dimensions and properties of the slab, used as the input of the

program, are as follows (notations are adopted from the code listing):

0 Slab thickness, T = 40.6 mm (1.6 in.), slab effective depth, H = 33 mm (1.3 in.).
0 Concrete compressive strength, FCCYL = 27.58 MPa (4.0 ksi).

0 Steel reinforcement ratio, RHO = 0.2%.

o Restraint factor, FR is varied from 0.5 to 0.9.

0 Equivalent diameter of slab (equal to beam spacing), C = 508 mm (20 in.).

0 Equivalent diameter of load, B = 97 mm (3.82 in.).

0 Yield stress of steel reinforcement, FSY = 500 MPa (72.5 ksi).

0 Steel Young’s modulus, ES = 200 GPa (29,000 ksi).

51

The program was run using the above input values and different values of the
restraint factor. The punching load of the deck, for each value of the restraint factor is
shown in Table 3.1. The net deﬂection of the slabs is also listed in the table along with

the corrected punching load due to the dowel and compressive membrane forces effects.

Table 3.1 Punching Load of Slab (Restrained Slab Theory)

 

 

 

 

 

 

 

 

 

 

Restraint Punching Load Deﬂection Corrected Punching Load

Factor, FR P (N) P (lbs) (m) V (N) V (lbs)
0.50 35,719 8,030 1.76 42,863 9,636
0.55 38,290 8,608 1.67 45,948 10,330
0.60 40,843 9,182 1.59 49,012 11,018
0.65 43,370 9,750 1.52 52,044 11,700
0.70 45,861 10,310 1.46 55,033 12,372
0.75 48,316 10,862 1.40 57,980 13,034
0.80 50,727 1 1,404 1.35 60,873 13,685
0.85 53,094 11,936 1.31 63,712 14,323
0.90 55,416 12,458 1.26 66,499 14,950

 

From the table, it is apparent that the punching strength of the model slab is

directly proportional to the value of restraint factor, the higher the restraint factor the

higher the punching load of the deck. The magnitude of the values, however, is not

meaningful unless it is compared to a reference value such as wheel load used to design

 

the bridge deck slab. In a full-scale bridge the load capacity of a 1/5-scale slab is 25 times
the load capacity of the model slab. Accordingly, the theoretical load capacity of the
prototype slab ranges from 1,071.57 to 1,662.52 kN (240.90 and 373.73 kip). Using an
impact factor of 1.3 the concentrated load due to a truck load of 145 kN (32.597 kips) is
94.25 kN (21.188 kip). The corresponding factors of safety provided by these slabs range
from 11.4 to 17.6. This is in good agreement with the values reported in the literature
(Hewitt 1972, Beal 1982, Fang et al. 1986, Petrou and Perdikaris 1990). The numbers in
Table 3.1 will be compared to the experimental test results to determine the actual value

of restraint factors of the slabs with and without additional lateral restraint.

3.4 ACI Provision on Punching Shear Strength of Concrete Slab

The model deck slab previousily analyzed using the restrained slab theory will
now be analyzed using the ACI Code provision on punching shear strength of concrete
slabs (Section 11.122 of the ACI 318-95 Code). Using a loaded area with ratio of long
side to short side dimensions, 13, equal to 2, the expression for the nominal punching
shear strength of a reinforced concrete slab without shear reinforcement is given in US

customary units as:
Vc = 4,/fc’bod (ACI: 11-37)

In SI units the above equation becomes (Hassouri, 1998):

Vc = $7,”, ’bod (ACI-SI: 11-37)

53

V, is the nominal shear strength in pounds (lbs) and newton (N) for the US. customary
and SI units, respectively. The concrete compressive strength, f,’, is 27.58 MPa (4,000
psi). The empirical nature of the formula requires that f,’ be provided in the units shown;
the square root of f,’ has the same unit as f,’. The perimeter of the critical section, ho, is
437 mm (17.2 in), and the effective depth of the slab, d, is 33 mm (1.3 in).

Due to the arching action, the top portion of the slab around the loaded area is
under compression, similar to that of a prestressed concrete slab. Consequently, the ACI
formula for a prestressed concrete slab could be applied to restrained RC deck slabs.
Since the magnitude of the stress is not known an assumed value can be used. For this
purpose, the ACI Code formula for shear strength of a two-way prestressed concrete slab
is also included in this analysis. Omitting the vertical component of effective prestress

force, VP, the expression of shear strength in US. customary units is given as:

Vc = (ﬂP,/fc’+0.3fpc )bod ' (ACI: 11-38)
The Code limits the values of [3p and the compressive stress f,, to 3.5 and 500 psi,
respectively. In this case, a maximum value is assumed. Using these ACI formulae the

ultimate shear strength of the model slab is calculated and the result is shown in Table

3.2.

It is apparent that the ACI Code formulae do not take into consideration the span
length, the boundary conditions, or the ﬂexural reinforcement of the slab. They also
neglect the dowel aetion due to the presence of steel reinforcement. However, using the
formula for a prestressed concrete slab (assuming the compressive membrane stress as

the stress due to the prestressing), the shear strength of the slab increases considerably. In

54

J.’

the actual case, the magnitude of the membrane stress at failure could be higher than 500

psi, for which accordingly, the strength of the slab would be higher.

Table 3.2 Shear Strength of Model Slab (ACI Formulae)

 

 

 

 

ACI Formulae / Maximum Shear Strength
Equations N Lbs
ACI: 11-37 25,162 5,657
ACI: 11-38 36,936 8,304

 

 

 

 

 

3.5 Finite Element Analysis

The ﬁnite element modeling (FEM) of the model bridge was conducted to
calculate the deck response under a full range of loads, from zero to failure load. Ideally,
the FE model should also be able to verify the experimental test results. In this study, the
FE model was designed to take into consideration the effect of each component in the
bridge to mimic the load deﬂection curve observed during the experiment up to failure.
For this purpose, it is necessary to model every component in the bridge, taking into
consideration the non-linearity of the concrete material. In addition, previous test results
showed that the deck slabs underwent large deformation. Therefore, geometric non-

linearity should be included in the model.

ABAQUS was among the commercial ﬁnite element software available at the

Division of Engineering and Computing Services (DECS) at MSU that has nonlinear

55

capability. It also has a special feature to model the inelastic behavior of a concrete

material. For this reason ABAQUS/Standard Version 5.8 was used in the present study.

To study the effect of adding tie rods, two distinct types of models were
developed. One type for the slab panel with tie rods, and the other for the control slab
panel without tie rods. For each type, several computer runs were necessary in order to
study the sensitivity of the material model. From observations made during the
experiment it was apparent that only the concrete slab underwent inelastic deformation,
while the steel beams remained elastic up to the failure of the deck. Accordingly, special
care was given in modeling the concrete material. Mesh sensitivity was beyond the scope
of this study; however, three different element types and arrangements were incorporated
in the FE models. Agreement between the FEA and the experimental test results was

expected in this study.

3.5.1 Geometric Model

The unsymmetrical loading positions on the bridge required the FE model to
incorporate the entire bridge. The model included the deck slab, steel girders, edge
beams, diaphragms, and tie rods. The deck slab was modeled with a three-dimensional
solid element, which in ABAQUS is referred to C3D type. The steel girders were divided
into ﬂanges and web, in which a 9-node shell element of type S9R5 was used for both the
ﬂanges and the web. The edge beams and the diaphragms were modeled using a beam

element of type B31. Truss element of type T3D2 was used to model the tie rods.

56

During the development of the models, a shell element was used for the deck slab.
However, the stress in the tie rods underneath the load was held to zero, because the shell
element was meant to undergo ﬂexure without in-plane displacement. Therefore, shell
elements could not be utilized in this study because they could not model the arching
action in the deck slab. Accordingly, a solid element was the only choice for the RC slab

model.

There are a number of solid element types available in ABAQUS. Among the
element types are: an 8-node linear, a 20-node quadratic, and a 27-node high order
element. In this study, the 8-node element C3D8 was used as the reference element for
the deck slab. Either one or two elements were used along the thickness of the slab. A
model with four solid elements along the depth of the deck was also investigated.
However, using four elements resulted in so large a model that it could not be solved
within a reasonable period of time. For that reason, the model with four elements wasnot
included in this study. For the purpose of comparison, a 27-node solid element with
reduced integration (type C3D27R) was also used to model the RC slab. Some of its
nodes were omitted to reduce the size of the model. The steel reinforcement was modeled
as layers of rebar embedded in the concrete elements. The interactions between concrete
and rebar, such as bond slip and dowel action, were approximately modeled in the

material model for concrete.

The ﬂanges and the webs of the steel beams were modeled using type S9R5
element, which is a 9-node shell element with reduced integration. This type of element
was meant for a thin shell, which is appropriate to model the effect of bending of the top

ﬂanges on which the concrete slab was supported. The width of the ﬂange and the depth

57

of the web dictated the mesh sizes. A mesh of 50 x 100 mm. (2 x 4 in.) was used for the
ﬂanges and 100 x 150 mm. (4 x 6 in.) mesh were used for the web. The thickness of the
ﬂange and web elements was 4.34 mm and 2.90 mm, respectively. No evidence of bond
failure between the ﬂanges of the steel girders and the concrete was found during the
demolition of the test specimen. Accordingly, no attempt was made to model slip
between the top ﬂanges of the steel beam and the concrete slab. In other words, a full

composite action was assumed in the FE model.

The tie rods were modeled with type T3D2, a 2-node truss element. The cross
sectional area of the rod was 161 mm2 (0.25 inz), which is the area of the square bar on
which the strain gage was mounted. The truss element was chosen because the tie rod
should only be subjected to axial tension due to the arching action in the slab. The
diaphragms and the edge beams were modeled with a type B31 element, a linear three-
dimensional beam element. The moment of inertia for the beam element were 395,420
mm4 ( 0.95 in4) with respect to the weak axis and 1,147,966 mm4 (2.758 in‘) with respect
to the strong axis. Based on these properties, the equivalent dimensions of the beam
element could be determined. The composite action between the edge beams and the
concrete slab was not included in the FE model since its effect is negligible if the load is
applied at a distance away from the edge of the slab. The geometric model is shown in

Figures 3.5.a, 3.5.b, and 3.5.c.

58

1,422 mm (56 imﬂ

 

A

L
P

2,438 mm (96 in)

Figure 3.5.a Plan View of FE Model Showing Mesh of the Deck (Using 8-node Solid
Elements), Tie Rods, and Loaded Area.

 

 

 

 

 

 

 

 

Solid 51- C3D8 (Slab) Truss El. T3D2 (Tierod) 41 mm (1.6 in)
| | l J l I l l l I l [T
Shell E1. S9R5 152 mm
(Web & Flanges) ‘ (6 in)
—;am E1. B31 (Map;—
L 508 mm 508 mm (20 in)

Figure 3.5.b Cross Section of FE model Showing Elements Connectivity.

59

Main Girder Diaphragm Tie Rod

    

Diaphragm

Figure 3.5.c Isometric View of FE model without the Deck Elements

3.5.2 Material Models

The reinforced concrete modeling in ABAQUS was accomplished by combining
plain concrete elements and rebar elements. These elements were Stiperimposed on the
mesh of plain concrete elements, and were used with standard metal plasticity models
that describe the behavior of the rebar material. A one-dimensional strain theory was used
for the rebar element. With this modeling approach, the concrete behavior was considered
independently of the rebar. The effect associated with the rebar/concrete interface, such
as bond slip and dowel action, was modeled approximately by introducing tension
stiffening into the concrete model. The purpose of the tension stiffening is to simulate the
load transfer across cracks through the rebar. More detail on the reinforced concrete

model is explained in the following sections.

60

3.5.2.1 Concrete Model

The concrete material was modeled by specifying its Young’s modulus, Poisson’s
ratio, compressive behavior, tension behavior, uniaxial and multiaxial behavior, post-
failure stress strain relation, and cracked shear retention. The Young’s modulus of
concrete was used for the elastic part of the stress strain relation, while the compressive
behavior was necessary to model the non-linear behavior of the material. The uniaxial
tension, uniaxial compression, and multiaxial compression behavior of the concrete were
used to determine the failure ratios in tension and compression. The postfailure stress
strain relation was required to introduce the tension stiffening behavior of the concrete
element. Finally, cracked shear retention was used to model the diminishing shear

stiffness of the cracked concrete.

In the present study, cylinder tests were conducted to obtain the compressive and
tensile strength of the concrete. The detail of the tests is presented in Chapter 5. The
remaining concrete parameters required to model the concrete material were determined
utilizing well known relationships. The value of Poisson’s ratio of concrete is between
0.15 and 0.2 with a higher value for lower strength concrete (Mehta and Monteiro 1996).
In this study a value of 0.2 was used for all models. Different values of Poisson’s ratio
were tried in some models but there were no signiﬁcant differences in the results. For the
compressive stress strain relation, a hypothetical model was developed based on test data
conducted by Kupfer and Gerstle (Kupfer et a1 1973). The model is incorporated in the

material model input for the concrete element.

For the basic FE model, the value of Young’s modulus used was 25,000 MPa.

(3,600 ksi), which is about one eight of the value for steel. Yielding of concrete was

61

assumed to start at a stress of about 12.5% of its maximum stress, and the total strain
corresponding to the maximum stress of 27.58 MPa (4,000 psi) is assumed to be about
0.003. For some models, the above values were arbitrarily assumed to study the
sensitivity of the model with respect to each modeling parameter. A total strain of 0.005
was used for some models to include a hardening behavior. This behavior was made

possible by the presence of compressive membrane action in the deck slab.

The tension behavior of the concrete material model included tension stiffening in
order to model the strain softening behavior of the cracked concrete. The stress strain
relation was assumed to be linear up to failure. After failure, the strain softening reduces
the stress linearly to zero. The strain at which the stress is equal to zero for a heavily
reinforced member was reported to be as high as one hundred times its failure strain
(Schnobrich 1985). Assuming linear stress strain behavior under tension, the failure strain
of the model concrete is about 0.0001. Accordingly, the maximum strain for the tension

stiffening models could be as high as 0.01.

Knowing that the yield strain of a 500 MPa (72.5 ksi) steel is about 0.0025, then a
reasonable value for the tension strain of concrete is equal to the yield strain of steel. At
failure however, the steel strain could be much higher than 0.01. Accordingly, a tension
stiffening value higher than 0.0025 is possible. Tension stiffening values between 0.0005
and 0.002 were used in ABAQUS for the analyses of a one way slab tested by Jain and
Kennedy, and a two way slab tested by McNiece (H, K & S, Inc. c 1998; Jain and
Kennedy 1974). The model using tension stiffening of 0.001 yields a load deﬂection
curve that ﬁts the experimental result best. Admittedly, this parameter should be

calibrated to a particular case. It was also shown that, the higher the tension stiffening the

62

more ductile the model. In this study, a value of 0.0025 was used as a standard value for
tension stiffening. In some models, a tension stiffening value as high as 0.0075 was used

to obtain longer solutions.

In addition to the above mentioned properties, ABAQUS requires additional input
values to determine the failure surface of a concrete element which are termed failure
ratios. The failure ratios include the ratio of the ultimate biaxial compressive stress to the
ultimate uniaxial compressive stress, and the absolute value of the ratio of uniaxial tensile
stress at failure to the ultimate uniaxial compressive stress. Kupﬂer et al. (1969) showed
that the maximum biaxial strength is about 25% higher than the uniaxial compressive
strength and occurs at 0', = f,, where fl) is the prism compressive strength of concrete.
Under 61 = 0'2, the biaxial compressive strength of concrete is about 16% greater than the
uniaxial strength. The ratio of uniaxial tensile stress to uniaxial compressive stress was
between 7% and 11% with higher values for lower strength concrete (Mehta and
Monteiro, 1996). Based on the above data, values of 1.25 and 0.1 were used as reference
values. In some models, values of 1.3 and 0.11 were used for the failure ratio in

compression and tension, respectively.

The cracked shear retention feature in ABAQUS was meant to account for the
reduction in shear modulus as the concrete cracks. The shear retention option assumes
that the shear stiffness of open cracks reduces linearly to zero as the crack opening
increases. If the option is omitted in the material data block, ABAQUS will assume full
shear retention, where the shear response is not affected by the cracked element. In this
study, the option was omitted, which means that there is no shear stiffness loss in the

planes of crack, once they have formed. The reason for this assumption is due to the fact

63

that the deck slab undergoes a three-dimensional state of stress, in which the response is
controlled by the material behavior normal to the crack planes, whereas the material
behavior in the plane of the crack is not important. Accordingly, the choice of full shear
retention not only has no inﬂuence in the results but also provides a more efﬁcient

numerical solution (H, K & S, Inc. b 1998).

3.5.2.2 Steel Model

The material model for steel was obtained by specifying its Young’s modulus and
Poisson’s ratio for the elastic part, and the yield stress and plastic strain for the plastic
portion. For the Young’s modulus a value of 200,000 MPa (29,000 ksi) was used with a
Poisson’s ratio of 0.3. The yield stress for rebar was 500 MPa (72.5 ksi) and 250 MPa (36
ksi) for the structural steel. No plastic strain was included in the models since all of the

structural steel is expected to be within its elastic stress range.

3.5.3 Input Files

ABAQUS/Standard version 5.8 runs as a batch application that requires an input
ﬁle to be prepared prior to running the software. General pre-processor software such as
Hyperrnesh can also be used to prepare the input ﬁle. However, for a three—dimensional
model, such as the subject bridge model, it is easier to write the input ﬁle than to use a -

graphical pre-processor.

The input ﬁle contains model data and history data. Model data deﬁne the ﬁnite

element model: the nodes, elements, element properties, material deﬁnitions, and any

other data that specify the model itself. History data deﬁne what happens to the model,

the sequence of events or loading for which the response of the model is sought.

ABAQUS has some useful features for simplifying the writing of the input ﬁle. Among

the features used in this model are node and element sets, node and element generations,

and node copy. The complete features are described in the three volumes of

ABAQUS/Standard User’s Manual (H, K & S, Inc. b 1998).

For the purpose of modeling the bridge in this study, the input ﬁle was prepared in

the following sequences. An example of input ﬁle is given in Appendix C.

1.

3.

Mesh Deﬁnition: includes deﬁnition of nodes and elements. Nodes are deﬁned by
number and coordinates, and elements are deﬁned by number, type, and the node

numbers that form the element.

Material Deﬁnition: speciﬁes the name and the properties of the material. For an
elastic material such as steel, only its Young’s modulus is required. For elastic plastic
materials such as concrete, the material deﬁnition is lengthy and complex, especially
when properties beyond the elastic range are to be considered. Among the necessary
data to deﬁne a concrete material are the compressive stress, Young’s modulus,
Poisson’s ration, plastic strain, and failure ratios. The two-way reinforcement is
modeled using the REBAR option where the area and the location of the steel

reinforcement along the thickness of the element are speciﬁed.

Boundary Conditions: deﬁne the prescribed values of displacement at the boundaries.

In this FE model, one end of the center beam is assumed to resist lateral, longitudinal,

65

and vertical displacements, while the other end is free to displace longitudinally. The

other two beams are free to displace, except in the vertical direction.

4. Step Deﬁnition: speciﬁes the history data such as the loading type and its magnitude,
solution method, etc. For this model, a nonlinear static solution is expected, therefore,
an incremented load is speciﬁed. Since considerable nonlinearity is expected in the
response, including possible unstable regimes, the modiﬁed Riks method is used with
automatic incrementation of the load. The option to specify this method in ABAQUS

is STATIC, RIKS.

5. Output Requests: list all the output data to be created by ABAQUS. Unless
speciﬁcally deﬁned, by default ABAQUS will create a data ﬁle that lists the model
deﬁnition and all of the output results that may cause memory problems. For problem
of this type, a status ﬁle that contains information about the progress of the analysis is
necessary. The status ﬁle is created using the MONITOR option, where the value of
one degree of freedom of one node in the model is written (to the status ﬁle) at the
end of each increment. Therefore, the status ﬁle can be examined as the analysis job
is executing, which is very useful for large size problems with hours of running time.
The other important output ﬁle is the restart ﬁle, which can be used for graphical
post-processing using ABAQUS/Post. This ﬁle is created using the RESTART,
WRITE options. With this restart ﬁle hard copies of the model can be created using

ABAQUS/Plot.

During execution, ABAQUS, like other FE software, creates a number of sizable
temporary scratch ﬁles. To manage these ﬁles, it is necessary to create an environment

ﬁle that speciﬁes how and where the scratch ﬁles should be written and saved.

66

Depending on the size of the problem, the size of scratch ﬁles could be more than a

gigabyte, which accordingly, must be properly managed to avoid memory problem.

In the present study, a number of FE models were created to obtain a model that
best mimics the experimental test results. The models were numbered and named, and the
variables included in the models are summarized in Table 3.3. All of the models use one
8-node element along the depth of the deck, except for model m82 and m27. Model m82

uses two 8-node elements, and m27 represents a model with one 27-node element.

In addition to the input ﬁle, a sample status ﬁle is also given in Appendix C
together with the output from ABAQUS/Plot as shown in Figures C1, C2, and C3. In
Figure C1, a magniﬁed deformed shape of the bridge model is shown. Figures C2 and C3
show the contour plot of strain and stress in 3—direction (vertical direction), respectively.
The load response history of the bridge model is more important in studying the effect of
adding tie rods to the deck slab. Accordingly, the FEA results were presented in the form
of load deﬂection diagrams for the slab and the beam, and load strain diagram for the tie

rods.

Each FE model shown in Table 3.3 uses a different model for the concrete
material in order to study the effect of varying material properties on the response of the
FE model. Model m6 uses a concrete model that represents the properties used for
example problems in ABAQUS/Standard Example Manual (H, .K & S, Inc. c 1998).
However, these properties do not necessarily produce the best model for the subject case.

The results of the FEA using varying concrete models are given in the following sections.

67

Table 3.3 Description of FE Models

 

 

 

 

 

 

 

 

 

 

 

 

 

FE Model Concrete Failure Ratio Tension
No Name Ec/Eco f,y / f, 3,, Comp. Tension Stiff'

1 m1 100% 12.5% 0.0030 1.25 0.10 0.0025
2 mla 100% 10.0% 0.0015 1.16 0.10 0.0020
3 mlb 100% 12.5% 0.0030 1.16 0.10 0.0040
4 mlc 100% 12.5% 0.0030 1.16 0.10 0.0060
5 m2 100% 12.5% 0.0050 1.30 0.1 1 0.0075
6 m3* 50% 12.5% 0.0050 1.30 0.1 1 0.0075
7 m4 100% 12.5 % 0.0030 1.20 0.10 0.0100
8 m5 100% 50% 0.0030 1.20 0.10 0.0075
9 m6 100% 50% 0.0015 1.16 0.10 0.0020
10 m82 100% 12.5% 0.0050 1:30 0.1 1 0.007 5
1 1 m27 100% 12.5% 0.0050 1.30 0.11 0.0075

 

 

 

 

 

 

 

 

 

 

Note: * Model m3 uses reduced Young’s modulus. Eco = 24,822 MPa
(3.6 E6 psi), and f, = 27.58 MPa (4000 psi).

3.5.4 Finite Element Analysis Results

The results of the FEA are presented in tables and diagrams. Table 3.4 shows the
summary of results of the analyses using various models. The complete description of
each model is shown in Table 3.3. It is apparent from the table that the response of each
model varies considerably. The models with tie rods fail at a higher load than those
without tie rods. These results show that the addition of tie rods improve the strength of

the slab. It also shows that the FEA can model the arching action in the slab. However,

68

the results are not meaningful unless they are compared to the test results. Therefore,

further discussion on these results is given in Chapter 6 together with the results from the

experimental program presented in Chapter 5.

Table 3.4 Summary of FEA Results

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model Slab with Tie Rods Slab without Tie Rods
No Name P-max. A-max. P—max. A-max.
N lbs. Mm in. N lbs. Mm in.
1 M1 33,949 7,632 4.17 0.164 32,952 7,408 4.50 0.177
2 M 1 a 30,959 6,960 3.43 0.135
3 Mlb 38,504 8,656 4.80 0.189
4 mlc 39,144 8,800 4.90 0.193
5 m2 46,048 10,352 7.24 0.285 43,770 9,840 7.95 0.313
6 m3 60,851 13,680 11.91 0.469 42,418 9,536 8.59 0.338
7 m4 43,913 9,872 5.77 0.227 37,934 8,528 5.31 0.209
8 m5 56,368 12,672 5.89 0.232
9 m6 30,817 6,928 3.02 0.1 19
10 m82 15,088 3,392 1.45 0.057
1 1 m27 15,871 3,568 1.61 0.063

 

Figures 3.6, 3.7, and 3.8 show the load-deﬂection relationship of the deck slab

with and without tie rods loaded near the mid-span of the bridge. Figure 3.7 shows the

effect of using different element types and conﬁgurations, where three diagrams are

69

 

shown. In Figure 3.8, the load deﬂection diagram for the model without tie rods is shown,
where the name with extension “nr” is associated with this model. In this case, the tie
rods were removed from the model. It is apparent from the ﬁgures that the responses of

the models vary according to the variables used to model the concrete material.

 

Load Deﬂection Diagram of Slab With Tie Rods

 

 

 

 

70,000
-x—m1(fr:1.25,
ts:0.0025)
60,000 + m1a(fr:1.16, “
ts:0.002)
+ mlb(fr:1.l6,
50,000 ts:0.004) I

+m1c(fr:1.2,

40,000 .- . .. ts:0.006)
x —-)(-— m2(fr:l .3,
. iv o ts:0.0075)

 

 

 

 

 

:2: .—J
'O
8
.1 30,000 - v -e—m3(50%Ec) r1
’5 ' ’1
l/ 4(f 12
. +111 1': .
20,000 a 7’ ’
ts:0.01)

 

in + m5(fcy:0.5fc)

10,000 ~
/ I_ § + m6
0 l l l l T T l l T

0 2 4 6 8 10 12 14 16 18 20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Total Deﬂection (mm)

 

 

 

Figure 3.6 Predicted Load Deﬂection Diagram of Slab with Tie Rods

Models m5 and m6 show the stiffest response, in which the load deﬂection

relation is nearly linear. Model m5 with a higher tension stiffening value provides a

70

longer response than model m6. This shows that the tension stiffening value used in the
ABAQUS example problem is not appropriate for the model in this study. Model m3
gives the longest response, which results in the highest failure load and the largest
deﬂection of the slab. Some models are not included in the ﬁgures because they are on

top of each other. The responses for these models can be seen in Table 3.4.

 

Load Deﬂection Diagram of Slab With Tie Rods
(Using Different Concrete Elements)

50,000
45,000
40,000
35 ,000
30,000

25,000

Load (N)

20,000

15,000 C3D8

10,000

—0— 2C3D8

5,000 + C3D27
0

 

0 2 4 6 8 10

Total Deﬂection (mm)

 

 

 

 

Figure 3.7 Load Deﬂection Diagram of Slab Using Different Elements

71

Figure 3.7 shows the effect of using different element types and conﬁgurations.
The model using one 8-node element along the thickness of the slab provides the longest
response. Using two 8-node elements and one 27-node element along the slab thickness,
the models fail at a load much lower than those using one 8-node element. The load

deﬂection curves however, are closely tied together.

 

Load Deﬂection Diagram of Slab Without Tie Rod

50,000
45 ,000
40,000
35,000
30,000

25 .000

Load (N)

20,000
1 5 .000
1 0,000

5 ,000

 

0
0 2 4 6 8 10 12

Total Deﬂection (mm)

 

 

 

 

Figure 3.8 Predicted Load Deﬂection Diagram of Slab Without Tie Rods

72

The load deﬂection diagram for the FE models without tie rods is shown in Figure
3.8. Four FE models were developed to show the effect of material properties used for the
models. Model m3nr, which uses a reduced value of Young’s modulus, gives the largest
deﬂection while models mlnr and m4nr give stiffer responses. The responses of model
m2nr and m3nr are similar except that model m2nr is stiffer than model m3nr. These two

responses show that the effect of modulus of elasticity is properly modeled in the FEA.

 

Load Deﬂection Diagram of Center Beam
(FE Prediction)

70,000

60,000

50,000

40,000

Load (N)

30,000

20,000
+ml --A-— m2 + m3

10,000

+mlnr —e— m2nr —e— m3nr

 

Deﬂection (mm)

 

 

 

Figure 3.9 Predicted Load Deﬂection Diagram of Center Beam

73

‘0

.7»:

Figure 3.9 shows the load-deﬂection curve for the center beam adjacent to the
loaded slab. Similar to the result of slab deﬂection, the deﬂection of the center beam is
signiﬁcantly affected by the presence of tie rods. It can be seen from the response of
model m1 and model mlnr that with tie rods, model m1 gives stiffer response than model
mlnr without tie rods. Similar responses can be observed from the other models. It is also

interesting to note that the load deﬂection relation in the center beam is obviously linear.

In addition to deﬂection of the center beam and the slab, the strain developed in
the tie rods at a distance nearest to the load point is also recorded from the FEA. This
strain is an important evidence of the presence of arching action in the slab that causes
the supporting steel girders to move laterally. During the deve10pment of the FE models,
shell elements were used to model the deck slab. However, no strain was developed in
the tie rods, which meant that shell elements could not model the arching action

developed in the deck-slab.

The load strain relation in the tie rods underneath (S 5) and immediately adjacent
to the applied load (S4) are given in Figure 3.10. It is apparent from the ﬁgure that the
higher the load the higher the strain in the tie rods, with strain rate higher than the rate of
the load. It is also interesting to note that the development of strain in the tie rods started

at a very low load level, under which bridge structures operate.

In general, the FEA results show that the effect of tie rods in the deck slab is
properly modeled The response of the FE models with and without tie rods show that the
addition of tie rods improves the behavior of the slab through deﬂection reduction and
strength enhancement in the slab. These results however, remain to be veriﬁed through

the experimental test.

74

 

 

Load (N)

Load Strain Diagram in Tie Rod
(FE Prediction)

70.000

60.000

50,000

40,000

 

30,000

 

20,000 +S4(m1) —e—SS(ml)
—+— S4(m2) + SS(m2)
10,000
—I— S4(m3) —a- SS(m3)
0
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012
Strain

 

Figure 3.10 Predicted Load Strain Diagram of Tie Rods

75

 

CHAPTER 4

DESIGN AND CONSTRUCTION OF MODEL BRIDGE

4.1 Introduction

To study the effect of adding lateral restraint (i.e. tie rods) to an RC deck slab on
its behavior under static concentrated load, laboratory load testing up to failure was
decided upon. For that purpose, two different types of slab panels were planned: one
panel without tie rods was used as a control panel, and the other panel with added tie rods
served as a treated panel. In this study, both types of slab were incorporated into the
construction of one bridge in order to obtain two slab panels with comparable properties;
A bridge with three steel girders was adopted to obtain two similar slab panels.
Therefore, the effect of adding tie rods could be observed by testing two similar rows of
slab panels in the same bridge, in which one of the panels was treated by connecting the
underside of the top ﬂanges of the girders with tie rods.

Testing a full-scale prototype structure is ideal for accuracy as well as easy to
design the structure. However, to build a full scale bridge is not only expensive but also
time-consuming, particularly since the bridge will be tested up to failure. Therefore, a
reduced scale model is more appropriate for the purpose of destructive testing.

The ACI Manual (1998) deﬁnes a small scale model as “any structural element or
assembly of structural elements built to a reduced scale (in comparison with full size
structures) which is to be tested, and for which laws of similitude must be employed to

interpret test results.” There are two commonly used approaches in designing a reduced

76

scale model structure. The ﬁrst approach is using scaled loads to design a model structure
of the form of the prototype. The second approach is to scale the dimensions and
sectional properties of a prototype structure to obtain a model structure. Previous
researchers have used both approaches and obtained results with comparable accuracy. In
this study, the second approach was adopted. For this purpose, a hypothetical prototype
bridge was designed according to the AASHTO Speciﬁcation, and then, each component
of the bridge was scaled to obtain the model bridge. This way, the size of the model
bridge could be adjusted according to the experimental condition. Following the design
was the construction of the model-bridge. More detail on the design and construction of

the model bridge is given in the following sections.

4.2 Design of Prototype Bridge

The hypothetical prototype bridge was designed according to AASHTO LRFD

Bridge Design Speciﬁcation, SI Units (Appendix B — Basic Steps for Steel Bridge
Superstructures of AASHTO 1994). Three steel I-beams of Grade 250 MPa (36 ksi) were
used with beam spacing of 2,540 mm (8 ft. - 4 in.). The simple span bridge was designed
with a span length of 12,190 mm (40 ft) and a total width of 7,112 mm (23 ft.- 4 in.), that
represents a one lane bridge at an exit ramp. The steel beams and the RC deck slab were
designed to act as a composite beam utilizing shear connectors. Figure 4.1 shows a plan
view of the hypothetical prototype bridge showing deck panels with and without tie rods.
The design is summarized and presented in this section, while the details of the design of

the prototype-bridge are not included in this thesis.

77

4.2.1 Reinforced Concrete Deck Slab

The behavior of RC deck slabs with and without additional lateral restraint is the
primary interest of this study. For that reason, the deck was carefully designed by
considering the test results from previous studies in addition to the AASTHO
Speciﬁcation (AASHTO 1994). The thickness of the RC deck slab was designed to be
203 mm (8 in), using a concrete compressive strength of about 28 MPa (4,000 psi). For
the purpose of rebar development, the deck overhang was designed to be about 1,016 mm
(40 in.), which is about ﬁve times the deck thickness.

Those numbers are within the AASTHO requirements for empirical design of RC
deck slabs. The minimum concrete strength required by AASHTO was used for this
project so that the experimental results could be used to interpret the punching strength of
deck slabs with a wider range of compressive strengths. The thickness of the slab,
however, was made thicker than the minimum requirement of 175 mm (7 in.), to
represent common bridge deck design practice. No sacriﬁcial wearing surface was
included in the deck thickness for this design purpose. Accordingly, the steel
reinforcement was placed symmetrically along the depth of the slab (using the same top
and bottom concrete covers).

For the reinforcing steel, a smaller amount than that required by the Code was
used in order to amplify the effect of adding lateral restraint (tie rods), provided such an
effect could be shown to exist. It was also intended to verify the results of previous tests
reported in the literature. The AASHTO minimum requirements for steel reinforcement

are 0.57 mmzlmm for each bottom layer, and 0.38 mmzlmm for each top layer. With the

78

slab thickness of 203 mm, the above numbers correspond to steel ratios of 0.28 % and

0.19 % for the top and bottom layers, respectively.

1,016 /— Main Girder rs Diaphragm

l i ......................................... /

2,540 [Control Panel /

 

 

 

 

 

 

: /-Edge
7,110mm l rTreated Panel if Beam
(280 in.) ----------- - .

I I I I I I I J I I I I I g
2,540 I I I I I I I g

 

 

 

 

 

 

I lillr—Tie
t .---l ..... 1 ..... I ...... I ..... 1 ..... 1--..--l ..... 1 ..... 1 ..... I ..... I ...... 6. Rod

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

 

 

1.016 m <—-—— 6,095 mm (240 in.) —->l<— 6.095 mm (240 in.) 6*
(40 m') 12,190 m (480 in.)

 

 

 

 

Figure 4.1 Plan View of Prototype Bridge

In this study, a steel area of 0.40 mmzlmm, which corresponds to a steel ratio of
about 0.2 %, was used in all four layers with isotropic arrangement. Number 13 bars with
12.7 mm (0.5 in.) diameter and 305 mm (12 in.) spacing provides a steel area of 0.415
mmzlmm. The minimum yield strength of steel reinforcement required by the code is 400
MPa (60 ksi). Steel with higher yield strengths could also be used as long as the
minimum area is satisﬁed.

Using 0.2% isotropic steel results in lower steel ratio for the bottom layers but
higher steel ratio for the t0p layers. The choice of isotropic steel arrangement was based
on the possible beneﬁt gained by using prefabricated steel such as welded wire fabric or

wire mesh. Previous research also showed that an RC deck slab reinforced with 0.2 %

79

steel provides more than adequate strength to resist a concentrated load from a truck
wheel (Hewitt 1972, Beal 1982). Therefore, the same steel ratio was adopted in this
study, not only to amplify the effect of adding tie rods, but also to verify the adequacy of

using minimum steel reinforcement in an RC bridge deck slab.

4.2.2 Design of Tie Rods

The design of steel tie rods was based upon previous research found in the
literature (Newhook, 1997). The steel strap that is used in a ﬁber reinforced concrete deck
slab, also known as a steel-free deck slab, is meant for a new construction where the strap
was installed prior to the pouring of the concrete deck. In this study, the tie rods are
intended to be installed after the concrete slab is constructed, because it is meant to
improve the performance of an existing deck slab. Consequently, direct application of the
strap design used for the steel-free deck slab is similar, but sufﬁciently different for the
design of the tie rods. However, it was used as a nearest available guideline.

Considering their replaceability and ease of installation, the shape of the tie rods
could be round or square with threaded ends to facilitate bolted connections. With bolted
connections, the rods can be pre-tensioned by tightening the bolts prior to loading. The
size of tie rods was determined following the Canadian Highway Bridge Design Code
(CI-IBDC) requirements for ﬁber reinforced concrete deck slabs.

Another important issue in designing the tie rods was that their spacing should be
small. Otherwise, they may not be effective if the slab is loaded at a distance away from
the rods. The CHBDC requires that the maximum spacing of straps, S] is 1.5 meter. For a

slab thickness of 200 mm with a beam spacing of 2,500 mm, the minimum area of the

80

strap required by CHBDC is about 938 mm2 (Newhook, 1997). In this study tie rods with
40-mm (1.575 in.) diameter, spaced at 1 m (40 in) was adopted. The area of the rod was
1,257 mm2, which is about 34% larger than the minimum area required by the CHBDC
for a steel-free deck slab. A total of 12 rods were used and numbered R1 to R12 as shown

in Figure 4.1.

4.2.3 Design of Steel Girders

The three main girders were designed using rolled steel wide ﬂange sections. The
section used for the prototype bridge was W760x160 (W 30x108) of M270 Grade 250
MPa (36 ksi). The wide ﬂange sections were designed as composite I-beams with a span
length of 12,190 m (40 ft. - 0 in.) and beam spacing of 2,540 mm (8 ft.—4 in.) from
center to center. Shear studs of 3 x 19 mm (0.75 in.) diameter, spaced at 160 mm (6.3 in)
center to center were provided for that purpose. As required by the Code, the bridge was
designed to have three diaphragms, one at mid-span and two at the ends of the bridge.
Channel sections of MC450x86 (MC 18x58), M270, Grade 250 MPa (36 ksi) were used

for these diaphragms.

4.2.4 Design of Edge Beams

At the ends of the bridge, the top portions of the girders were connected together
by means of steel channels termed the edge beams. These beams were also provided with
shear connectors to obtain composite action between the beams and the deck slab. The

strong axis of the channels was placed in the horizontal direction to provide greater

81

lateral restraint to the slab near the longitudinal edges. The geometry of the edge beam is

 

 

 

 

 

 

 

shown in Figure 4.2.
A'q X-°::.:AV:°:AV ’1‘
0° 0 p 0 9° a 13° 203 mm
°<I A’oAZ°a:i°<i Z°o>A°qXJI
Edge Beam (K
MC450 " 86 Main Girder
, W760 x 160
Diaphragm
MC450 x 86

 

 

 

 

 

 

 

 

Figure 4.2 Longitudinal Section Showing Edge Beam and Diaphragm

The idea of using edge beams was based on previous research, in which it was
shown that by adding lateral stiffeners at the ends of the slab, the strength of the slab
become uniform along the slab (Bakht and Mufti, 1996). Without edge beams, the
strength of the slabs near the edge were lower than that at a distance from the supports.

The CHBDC requirement for the ﬂexural rigidity of an edge beam, E1 is 3.5 L,4
in MN -m2 unit, where L, is the unsupported length of the edge beam which is equal to the
span of the slab. For an L, of 2.233 meter (beam spacing less the ﬂange width), the
required ﬂexural rigidity E1 was 87 MN -m2. The channel section used for edge beam was
the same as that used for diaphragms, which is MC450x86 (MC18x58) of M270 Grade
250 MPa (36 ksi). Considering composite action between the edge beam and the concrete

deck slab, the moment of inertia of the edge beam was 4.837 x 10'4 m4 (1,162 in’). With

82

steel elastic modulus of 200,000 MPa (29,000 ksi), the ﬂexural rigidity of the edge beam

became 96.7 MN-mz, that is about 11% higher than that required by CHBDC.

4.2.4 Design of Shear Connectors

The shear connectors for the steel girders were designed based on strength and
fatigue requirements. The design was governed by fatigue requirement of the AASHTO
Speciﬁcation. Three studs of l9-mm (3/4 in.) diameter were used with spacing of 160-
mm (6.3 in.) center to center. The properties of the shear studs and the other components

of the prototype bridge are summarized in Table 4.1.

4.3 Design of Scaled Model Bridge

The two basic dimensions required in testing a model-structure for static loading
are force and length, denoted F and L, respectively. For a practical true model, the scale
factors of stress, strain, and the modulus of elasticity for both concrete and steel are all
unity. Since the dimension of stress is FL'2 = 1, then the scale factor of force is the square
of the scale factor of length.

Ideally, a length scale factor as close as possible to unity is preferable to avoid
possible size effects and dimensional inaccuracy. In this study, however, to minimize the
cost of construction, a length scale factor, 3;, of 5 was adopted. A strength or realistic
model, which is a direct model made of materials similar to that of the prototype
materials, was used in order to predict the prototype behavior up to failure (Sabnis et a1

1983, Aldridge and Breen 1970, and Harris et al. 1970).

83

Table 4.1 Properties of Prototype Bridge

 

No Component Pr0perties

 

l Deck Slab 203 mm (8 in.) thick of 28 MPa (4000 psi) concrete
Isotropic steel ratio of 0.2 %.

 

2 Steel Girders W760x160 (W 30x108) of M270 Grade 250 MPa (36 ksi)
Span length: 12,190 m (40 ft.)

Beam Spacing: 2,540 mm (8 ft. — 4 in.)

 

3 Diaphragm MC450x86 (MC18x58) of M270 Grade 250 MPa (36 ksi).

 

4 Edge Beam MC450x86 (MC18x58) of M270 Grade 250 MPa (36 ksi).

 

5 Shear Connectors (3) l9-mm (3/ ”) diameter studs

160-mm (6.3”) spacing.

 

6 Tie Rods 40 mm (1.575 in.) diameter

1015-mm (3’ - 6”) spacing.

 

 

 

 

 

For RC structures, the similitude requirements are expressed in term of the scale
factor, 3, deﬁned by

i

'U

5.:

He

where iP and im refer to prototype and model quantities, respectively. Using a length scale
factor of 5 results in a scale factor of 25 for force. Therefore, a 1/5 scale deck slab that

fails at an ultimate load of 1 kN, for example, will fail at a load of 25 kN in the full scale

84

prototype structure. Accordingly, the scale factors for the other similitude requirements

could be calculated as shown in Table 4.2.

 

 

 

 

 

 

 

 

 

 

 

Table 4.2. Scale Factor

Quantity Dimension Scale Factor
Stress FL'2 1
Strain LI;1 1
Modulus of elasticity FL’2 1
Length L 5
Force F 25
Deﬂection L 5
Area , L2 25
Section modulus L3 125
Moment of inertia , L4 625
Density FL'3 US

 

 

 

 

 

The problem with a reduced scale model is related to the dead load induced
stresses. The density or unit weight of a structural material is expressed in force per cubic
length, FL'3. By scaling the length, while keeping the stresses in prototype and model
structures constant, it is required to scale the unit weight of the model structure by a

factor of 1/5. However, in concrete structures the density of the material will affect their

85

properties. Accordingly, in this study, the density of the concrete was not scaled in order
to get materials of comparable properties for both prototype and model structures.

It was assumed that the stress caused by the self-weight of the structure is
relatively small compared to the live load induced stress, because punching shear
capacity is related to a concentrated load instead of the distributed load due to self-
weight. Therefore, the dead load induced stress was omitted in this study.

By not scaling the mass density of the model structure, the model becomes
deﬁcient as long as dead load stresses are of concern. If the dead load effects were to be
properly scaled, a supplementary load equal to (sL — 1) times the actual model weight
would have to be applied to the model structure. However, an earlier study showed that
incorporating the effect of self weight by applying dead load compensation has little, if
any, effect to the punching shear capacity of the deck slab (Hewitt, 1972). Accordingly,
for practical purposes, it was recommended that any effect due to differential dead load
stresses or deﬂection could be ignored.

Based on the above assumptions, the model structure was designed by dividing
the properties of the prototype structure by an appropriate scale factor. The ratio between
the scaled prototype and model structures should be as close as possible to unity.
However, due to the availability of materials, some ratios could not be kept to unity.

Other difﬁculties were encountered in selecting the rolled steel sections that
match the scaled properties of prototype structural component. Unless a built-up section
was used, it was not possible to obtain rolled sections that matched all dimensions and
sectional properties of the prototype section. Sections that match the cross sectional area

may not match the moment of inertia and vise versa. However, the important or

86

controlling factor was the moment of inertia, so that was used for the basis in the
selection of the section. The same procedure has been used in bridge design practice,
where the choice of a rolled section is dictated by the availability of the section in the
market. In this study, however, special care was given to the modeling of the deck slab
thickness, which proportionally affects the moment of inertia of the composite section.
Figure 4.3 shows the cross section of the model bridge, while the complete dimensions of

the prototype and model-bridges are given in Table 4.3.

WWM No 4 @ 152 Tie Rod

       
  
  

40.6
(1.6 in)

152
(6 in)

  

e e
e e
C76x6.1
508 (20 in)

  

203

(8 in) a .M150x6.5

     

 

 

1,422 mm
(56 in)

Figure 4.3 Cross Section of Model Bridge

For the model tie rods, a square bar was used to facilitate installation of strain
gages in the rods. The ends of the bar were threaded with a net area equal to that required
for the scaled rods. The threaded ends in the rods allowed the use of nuts to tighten the
rod connection prior to load testing the slab. The detail of the tie rods is shown in Figure
4.4. Some difﬁculties related to the detail of the rod connection were anticipated mainly
because of the limited space available. The ﬂange width of the model girder was so small
that an additional plate was required to facilitate the rod connection. This condition was

not beneﬁcial because the eccentricity caused by the additional plate would be the source

87

of additional stress in the connection. In a full scale bridge, however, such problems
would not exist because the ﬂange width of the girder is usually large enough to

accommodate the connection for the tie rods.

Bottom of Slab

   
   

   

   
   

12.7 x 12.7 bar
8 mm threaded rod

Strain Gage

Figure 4.4 Detail of Tie Rods

4.4 Properties of Materials for Model Bridge

In-order to obtain a true model, the properties of the materials in both prototype
and model structures should be the same. In concrete structures, however, there are size
effects that can not be avoided. It is generally believed that micro concrete exhibits
higher strength but is more prone to shrinkage cracking than prototype concrete. In
practice, modeling only the compressive strength of the concrete is considered acceptable
for the purpose of experimental study. Special consideration for curing of micro concrete
was recommended in order to prevent excessive cracking due to shrinkage. One way to
control the shrinkage cracking problem is to cure the wet concrete under water as long as
possible. The procedure of water curing is required by the AASHTO Speciﬁcation for a

prototype deck slab that is designed using the empirical method.

88

Table 4.3 Properties of Prototype and 1/5 Scale Model Bridge

 

 

 

 

 

 

 

 

 

 

 

 

Component Prototype Scaled MODEL Ratio
1. Deck Slab (fc’ 28 MPa)
Thickness (mm) 203 40.64 40.64 1.00
2. Steel Reinforcement fy 400 MPa fy 500 MPa
N013 @317 No4@152
Area (mmzlmm) 0.400 0.080 0.083 1.04
3. Girders (fy 240) W760 x 160 M150x6.5
Area (mmz) 20452 818 832 1.02
Ix (mm4) 1.86 E9 2,976,000 2,996,866 1.01
Iy (mrn‘) 60.77 E6 97,232 70,759 0.73
4. Diaphragms and Edge
Beams (fy 240) MC450x86 C75x6.l
Area (mm2) 11,032 441 781 1.77
Ix (mm4) 2,813,724,444 450,196 690,944 1.53
Iy (mm4) 7,408,919 11,854 83,246 7.02
5. Shear Stud (fy 240) (3) o 19 mm (1) ¢ 8 mm
Spacing (mm) 160 50.8
Area (mmz/mm) 5.316 1.06 0.99 0.93
6. Tie Rods (fy 240) o 40 mm o 8 mm
Spacing (mm) 1000 203
Area (mmzlmm) 1.24 0.257 0.25 0.97

 

For a steel structure the mechanical properties of small and large size structures

are considered the same. However, the availability of rolled sections in small sizes is

limited. Consequently, slight distortions of a model structure can not be avoided. As long

89

 

as the model does not signiﬁcantly affect the behavior of the deck slab, the effect of
adding tie rods could be measured properly. In this case, a slight distortion in the model
for steel girders will not affect the test results as much as the distortion of the deck model.
Accordingly, modeling the deck slab as accurately as possible is necessary.

As shown in Table 4.3, the ratio between the scaled prototype and model bridge
components, are not equal to unity except that for the theoretical deck thickness. The
exact thickness of the slab may vary due to imperfections during the construction.
However, such variability is also found in the ﬁeld, thus, with proper care in construction,
it need not be considered in the design of the model. Since the effect of adding tie rods is
of main concern, as long as the material properties are the same for both control and
treated slab panels, the effect of the treatment could be measured properly. Further
discussion of the material properties used for the model-bridge is given in the following

sections.

4.4.1 Micro Concrete

The target value of compressive strength for the micro concrete was about 28
MPa (4000 psi), which is the minimum strength allowed by the AASHTO empirical
design for deck slabs (AASHTO 1994). The compressive strength was the only concrete
property to be modeled considering that the other properties such as tensile strength, the
modulus of elasticity, and Poisson’s ratio are related to the compressive strength. The
mix design of regular (prototype) concrete recommended by ACI was used to design the

mix proportion of the micro concrete, which in SI units proceeds as follows.

90

The maximum aggregate size for the prototype concrete was about 38 mm (1.5
in.). For the micro concrete, the maximum size of aggregate was scaled to about 7.6 mm
(0.3 in.). In this study, a natural aggregate, so called “pea gravel”, with maximum size of
9.5 m (3/8 in.) was used. Only the maximum size of the aggregate was scaled in
obtaining the mix design of micro concrete. Direct scaling of coarse and ﬁne aggregate is
not necessary because it may result in a large proportion of ﬁnes that require more
absorption water. For this reason, regular sand with ﬁneness modulus of 2.94 was
utilized.

The micro concrete was intended to have a slump value of 100 — 125 mm (4 — 5
in.). Workability was another concern for the micro concrete because the slab was so thin
that a special effort to compact the concrete to avoid segregation problem was required.
Consequently, it was necessary to use highly workable, high slump micro concrete to
assure good consolidation. With maximum aggregate size of 9.5 mm, the approximate
mixing water required for the mix was about 230 kg/m3 of concrete. To achieve a
compressive strength of about 28 MPa (4000 psi), the maximum water cement ratio, w/c
was about 0.54 as given by AC1211, Table A1.5.3 (ACI 1994). Using ﬁne aggregate
with a ﬁneness modulus of 2.94, the required volume of coarse aggregate per unit volume
of concrete was 0.48 as shown in Table A1.5.3.6 of ACI211.

The ﬁnal weight of each ingredient for one cubic meter of concrete was; water:
230 kg, cement: 426 kg, ﬁne aggregate: 986 kg, coarse aggregate: 768 kg. The ﬁnal mix
proportion by weight of water: cement: aggregatewas 0.5 4: 1: 4.1. This mix proportion is
very similar to that used by Hewitt (Hewitt, 1972), in which using this proportion a

compressive strength of least 28 MPa (4,000 psi) was reported. Accordingly, in this

91

study, the mix proportion was directly used for the design of the micro concrete, no trial

mix was conducted prior to using the mix design for the actual micro concrete.

4.4.2 Steel Reinforcement

Welded Wire Mesh (WWM) was used for the reinforcement of the model slab.
The diameter of the wire was 4 mm (0.16 in.) with spacing of 152 mm (6 in.) in both
directions. The wire that was readily available in the market had a minimum yield
strength of 500 MPa (72.5 ksi). No attempt was made to anneal the wire as the process is
complex and may introduce new variability in the properties of the wire. Bridge
engineers have been reluctant to use welded wire mesh in highway structures. Their main
concern is poor fatigue strength due to welded intersections. Nevertheless, there are many
beneﬁts to be gained from using WWM in bridge decks. Among the beneﬁts are
considerable savings of construction time, improved bonding and anchorage, small
probability of error, and ease of placement.

A previous experimental testing program (Al-Mutairi 1989) showed that a deck
slab reinforced with WWM performed well with the exception of a simply supported slab
in the ﬂexure test, where the slab had less ductility than an equivalent slab reinforced
with conventional reinforcement. However, the welded wire mesh reinforced slab
deformed sufﬁciently and showed clear signs of distress before failure, a behavior that
can provide visual warning of imminent failure. In addition, knowing that a deck slab has
a high reserve capacity compared to the maximum possible applied load, the ductility of
the structure is not a concern. Csagoly et al. (1978) also recommended the use of welded

wire mesh for the reinforcement of deck slabs. Finally, test results showed that the strain

92

in a steel rebar induced by a truck load is so low that the steel is not required as far as
strength is concerned (Beal 1982, Puckette et al. 1989). Thus, WWM can be used, with

proper engineering judgment, as reinforcement of bridge decks.

4.4.3 Structural Steel

The structural steel for girders, diaphragms, edge beams, and tie rods, the
properties of the material in both the model and prototype were the same. In this case, all
of the steel used in the model bridge had a yield strength of 240 MPa (36 ksi). Therefore
the material model for the structural steel was not distorted provided that the variability

of the material properties are the same for large and small sections.

4.5 Construction of the Model Bridge

The model bridge was constructed in two phases, the steel framing phase and the
concrete pouring phase. In the ﬁrst phase, all steel framing for the girders, diaphragms,
edge beams, and tie rods were prepared in the shop. The shear connectors were welded to
the girders and edge beams prior to the frame assembly. The girders were cut longer than
the designed span length and protruded beyond the bridge deck to facilitate supporting
the bridge at the designed support points. Special structures for the supports were built
high enough to provide clearance (working space) underneath the bridge. After all the
dimensions were checked and veriﬁed, and formwork was prepared, the steel frame was

transported in pieces to the Structural Lab of the Michigan Department of Transportation.

93

In the lab, the frame was re-assembled without the tie rods. The formwork and the
reinforcing steel were put in place for pouring phase. The tie rods were not put in position
prior to pouring of the concrete because it was meant to be added to an existing structure
instead of being part of a new structure. During the pouring, two batches of concrete mix
were required for the entire bridge and the associated preparation of concrete test
cylinders. To avoid unwanted variability between the control slab panel and the treated
one, pouring was started from the support end instead of from the side end of the bridge.
One batch of concrete mix was enough for half the length of the bridge. In this way,

similar concrete can be expected for both panels of the slab.

4.5.1 Concrete Cylinders

Scaled concrete cylinders are commonly used to determine the mechanical
properties of the micro concrete. However, it is generally accepted that the strength of
concrete cylinder in either compression or tension bears little relationship to the strength
of the same concrete in a structure. The more accurate way to test the strength of concrete
in a structure is by cutting cylinders out of the structure. In this study, however, since the
thickness of the slab was only 40 mm (1.6 in.), it was not possible to cut a cylinder out of
the slab. Accordingly, to determine the strength of the micro concrete, small concrete
cylinders were cast from the same batch of concrete used for the deck slab. The suggested
diameter of cylinder is about the least dimension'of the structure, which in this case the
thickness of the slab. However, a very small cylinder may not be effective for measuring

the strength of the concrete (Aldridge and Breen 1970).

94

In this study, the size of the concrete cylinders used for compression and split-
cylinder tests was 75 mm diameter by 150 mm (3 by 6 in.). The cylinders were cast using
plastic molds, following the procedures suggested by the American Standard for Testing
and Materials, ASTM C469 and ASTM C496 (ASTM 1996). The cylinders were cast at
the time of pouring the concrete for the deck slab. Five cylinders were cast for each of the
two batches. A total of ten cylinders were obtained, six for compression test and four for
split-cylinder test.

The consistency of the mix was measured with a slump test according to the
ASTM C143 (ASTM 1996). The value of the slump measured during the test was about
175 mm (7 in), which is higher than the expected value of about 100 to 125 mm (4 to 5
in). The higher slump value was partly attributed to the small maximum size of the coarse
aggregate and the water content of the sand. For micro concrete, however, higher slump
concrete is beneﬁcial for workability, compaction, and ﬁnish without harmful

segregation. It also indicates lower compressive strength.

45.2 Curing

Previous research on structural concrete modeling showed that curing is one of
the most important factors affecting the properties of micro concrete. Extensive shrinkage
cracking was reported for micro concrete cured under wet burlap or using a chemical
compound (Hewitt, 1.975). To avoid the same problem, it is necessary to cure the micro
concrete well in the sense that the amount of lost water should be limited as far as
possible. For the concrete cylinders, de-molding were done after 24 hours of casting. The

cylinders were then cured in the curing room for 28 days. Curing of the model slab was

95

done by ponding. For this purpose, the edges of the slab were made about 50 mm thicker
to hold the ponding water. Ponding was continued for 28 days, until the removal of the
formwork when the slab was ready for testing. In this way, the concrete cylinders and
model slab were cured in a similar manner, and therefore, a better relationship can be

expected between the concrete properties of the slab and the cylinders.

96

CHAPTER 5

EXPERIMENTAL PROGRAM

5.1 Introduction

The main purpose of the experimental program was to investigate the behavior
and punching shear capacity of RC deck slabs with and without tie rods. For this purpose,
a one-ﬁfth-scale model bridge that incorporated two slab panels (with and without tie
rods) was built and tested in the Structural Lab of the Michigan Department of
Transportation. The behavior of deck slabs under varying magnitude of loads is of
primary interest in this experiment. Moreover, the deﬂection of the beam and the strain
developed in the tie rods were also investigated. The model bridge was instrumented
accordingly.

A shored construction was adopted in this study to minimize the dead load
induced stresses because neither density scaling, nor dead load compensation was
planned for this experiment. The choice of shored construction was also intended to
minimize dead load deﬂection, therefore, no carnbering of the steel girder was necessary.
Even though unshored construction was used, the dead load induced stress and deﬂection
could still be disregarded for the purpose of testing a small-scale model structure (Hewitt
1972).

The instrumentation was designed to verify the focus of the experiment that was
aimed at the effect of adding tie rods. Consequently, the instrumentation was directed

toward measurements of the load deﬂection relationship in the slab and steel girder as

97

well as the stress/strain developed in the tie rods. In this case, the quantities to be
measured during the load test were the magnitude of applied loads, the deﬂection of the
slab and the girder corresponding to the applied load, and the strain in the tie rods. More

detail on the experimental program is explained in the following sections.

5.2 Instrumentation

Measurement of the load and the corresponding deﬂection of the slabs in the lab
was done by applying a concentrated load generated incrementally by means of an
instrumented ram (a hydraulic jack) reacting with a steel loading frame. The concentrated
load was applied to the deck slab through a 50 by 100 mm (2 by 4 in.) neoprene pad of 13
m (1/2 in.) thick, that corresponds to the nominal prototype wheel-print of a truck that
is about 250 by 500 mm (10 by 20 in). The load was generated and recorded using a
computer-based loading device that was capable of controlling the intended load
magnitudes. A sketch of the experimental set up on the loading frame is shown in
Figure 5.1.

The vertical deﬂection was measured using 0.025 mm. (0.001 in.) dial deﬂection
gauges. The net deﬂection of the slab relative to the beam was measured by mounting the
gage on the center beam adjacent to the load. When the beam and the slab deﬂect the
same amount, then the gage reading will be zero. Likewise, if the beam and slab
deﬂections are different then the gage reading will be the difference between the two
deﬂections. A typical arrangement of the dial gauges to measure deﬂection, and the strain

gage position in a tie rod are shown in Figure 5.2.

98

The measurement of strain in the tie rods was done using general-purpose strain
gages, type CEA-06-250UW-120. The active length of the gage was 250 mils (0.25 in. or
6.35 mm) with a resistance of 120 Ohms and a gage factor of 2.065 at 24 °C. A total of
15 strain gages were mounted to 12 tie rods. To simplify the reading of the strains
manually, each wire from the strain gage was connected to a low resistance switch box
that can accommodate as many as 32 strain gages. The readings of the strain for each
gage were obtained from a digital strain reader that was connected to the switch box by

pressing the appropriate buttons that correspond to the strain gage.

Load Cell

  

 

 

 

 

 

 

 

 

 

Model Brid '
ge W 50x100 Neoprene Pad

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.1 Experimental Set-Up on Loading Frame

5.3 Load Testing of the Model Bridge

Eight different load points, four for each slab panel, were planned for the loading
test. The loading device could only be moved in one direction along the bridge.

Therefore, to test the other panel of the slab, it was necessary to move the bridge. The

99

diagram of the loading position showing tie rods and the strain gage positions on the tie
rods are shown in Figure 5.3. A total of eight tests on eight loading positions, numbered
LPl to LP8, were conducted. The numbers that follow the two letters for the loading
positions corresponded to the order of testing. The strain gages were also labeled 8] to

S 15 for identiﬁcation purposes. As shown in Figure 5.3, two strain gages were mounted
into each of the three tie rods located near the middle span of the bridge. The extra gages

were stand by, meant to be used if some of the gages did not work properly.

 

 

 

 

 

Concentrated Load
Appl'nd on 50x100 Neoprene Pad
1‘
40.6
152

    
  

in Tie Rod

Note: Diaphragm is not shown
for chrity.

 

Figure 5.2 Arrangement of Dial Gages and Strain Gage Position

The order of the testing at the various points of the slab was initially designed in
order to minimize the inﬂuence of previous tests. The testing was started by applying the
load near the end of the control slab panel (LPl) and continued on the other end of the

strengthened slab panel (LP2). After three tests, however, it became apparent that such

100

inﬂuence was so minimal that special precaution was unnecessary. Accordingly, the
order of testing was rearranged to minimize the movement of the model bridge.

Using a load increment of 2.224 kN (5001bs.), each test was continued until the
slab failed. In each increment the load was applied for about two minutes to allow for the
manual reading of deﬂections of the slab and the center beam, and strains in the tie rods
(for the treated slab panel). Additional time was required for visual inspection of cracking
in the slab as the load increased. The instance at which the ﬁrst crack became visible was
not easy to observe, however, with special precaution the load that caused the ﬁrst visible

crack could be approximately recognized.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2,515 mm (99 in)
305mm; 2.635mm_)<_635mm_,<_635mm_) (.305mm
(12 in) (25 in) (25 in) (25 in) (12 in)
7?
:,-:.-. .................. -. -2: .2::::::::::::'.:::' .ZZZZZZZIZZZZZZZZZT- -
Controlled
Panel LPl I LP6 I LP7 LP8 I
, -- ---- ---. -.'.'.":11":1:'."'.'.'.'."'.':.:"'.:'.'. ---- -- 1,422 mm
Panel I LP4I I aLPS I ELP3 LP2! I
:ﬁ::!:::::!::::.l:::::':::::!:::::'::::::l::::::!::::.J:::::!::::!--“.1"--.

 

I l I l I . I l I l
I 32 I 54 i867 ISIO’III 313 I 515
S1 s3 85 S89 812 814

Figure 5.3 Loading, Tie Rods, and Strain Gage Positions

For the eight tests conducted, each slab failed in punching shear and failures were

marked by the formation of a punching cone with the top size slightly larger than the size

101

of the load pad. The bottom size of the cone, however, was signiﬁcantly larger than the
top size, which means the angle of the shear crack was greater than 45 degrees. Failure,

in each case, was sudden with large slab deﬂection.

5.4 Recorded Data

Among the data recorded during the experiment were the load and the
corresponding deﬂection of the slab and the center beam, and the strains in the tie rods.
The data from the cylinder tests were also recorded to determine the strength of the micro

concrete. The detail of the recorded data is summarized in the following sections.

5.4.1 Cylinder Test

The preparation of the test cylinders was discussed in chapter four where the
cylinders were cast and cured following the ASTM procedures. Tests on the concrete
cylinders were performed at the structural lab of Michigan State University prior to the
load testing of the model slab. The original size of the cylinder was 76 m diameter by 152
mm (3 by 6 in). Prior to testing, the cylinders were cut to square their ends.
Unfortunately, the surfaces of the cylinder ends could not be made perfectly square
(parallel). Capping both ends of the cylinder, using a sulphur capping compound, would
have been better than cutting them. Six cylinders were used for compression tests (ASTM
C469), and four cylinders were used for split-cylinder tests (ASTM C496). The results of

the tests are given in Table 5.1.

102

The cross sectional area of the cylinder was 4,560 mm2 (7.069 in2). With an
average failure load of 116.00 kN (25,8821bs.), the average compressive strength of the
cylinder became 25,249 kPa (3,662 psi). For the split cylinder test, the average failure
load was 51.79 kN (11,555 lbs). The average length of the cylinders, after they were cut,
was 150 mm (5.91 in.). The relation between the failure load and the split tensile
strength, fsp (in psi unit) is given by

2P

(far

Where P is the failure load in lbs, L and D are the length and the diameter of the cylinder

in inches, respectively. The corresponding average split tensile stress of the cylinder was

 

 

 

 

 

 

 

 

 

 

2,861 kPa (415 psi).
Table 5.1 Results of Cylinder Tests
Failure Load, kN (lbs)
No. Compression Test Split-Cylinder Test
1 121.01 (27,000) 46.84 (10,450)
2 114.65 (25,580) 50.20 (11,200)
3 114.07 (25,450) 55.89 (12,470)
4 118.55 (26,450) 54.23 (12,100)
5 114.47 (25,540)
6 113.26 (25,270) -—
Average 116.00 (25,882) 51.79 (11,555)

 

 

 

 

 

103

In general, the lower the compressive strength of the concrete, the higher it’s
relative tensile strength. The value of the splitting strength, fSp of a normal weight
concrete can be related to its compressive strength using a formula proposed by Mirza et

a1 (Mirza et al, 1979) as given, in psi units, by

f, = 64¢?

For a split tensile strength of 2,861 kPa (415 psi), the corresponding compressive
strength of the concrete would be 28.993 MPa (4205 psi).

The concrete strength obtained from the cylinder tests was 25.249 MPa (3662
psi). The result from compression tests was about 14.8% lower than the value calculated
from the split-cylinder test. The main cause of the difference was suspected to be the
cylinder ends that were not perfectly square. In this case, the result from the splitting test
was more reliable than that from the uniaxial compressive test because the splitting test is
not affected by the end surface condition of the cylinder. Accordingly, the result from the
splitting test was used to determine the compressive strength of the model concrete.

The low value of compressive strength was also attributed to the high water
cement ratio, which was meant to increase the workability of the micro concrete. For the
purpose of the analysis, however, a value of 27.58 MPa (4000 psi) was assumed for the
compressive strength of the model concrete. The sensitivity of the value of the
compressive strength to the punching shear capacity of the concrete slab is low because

the shear strength is related to the square root of the compressive strength of the concrete.

104

5.4.2 Loading Test

The data recorded during the testing of the slab included the load, the net
deﬂection of the slab underneath the load application, the deﬂection of the center beam
adjacent to the load, and the strain in the tie rods for the treated slab panel. The crack
pattern of the slab was also observed. The cracking load, however, was very difﬁcult to
be recognized through visual inspection of the crack. Therefore, the crack load could not
be determined accurately.

The mode of failure, for both types of slab panels, was punching shear, in which
the failure surface was localized under the load in a circular manner. Prior to failure,
ﬂexural cracks developed on both surfaces of the loaded slab. On the bottom surface, the
cracks originated from the center of the load and formed radial cracks. On the top
surface, the ﬂexural cracks started at the beam lines adjacent to the applied load. At a
loadonear the failure load, shear cracks developed, originating on the bottom surface of
the slab at a distance away from the load. As the load increased, the shear cracks grew to
the top surface of the slab and formed the upper surface of the punch cone. Figures 5.4
(a) and (b) show the typical crack pattern on both surfaces of the slab.

The failure of the slabs was so sudden that it was difﬁcult to observe the growth
of the shear crack. After failure, however, the crack pattern was still visible even though
the cracks closed as the load was removed. The load at which the slab started to crack
was difﬁcult to determine accurately because the initial crack was not visible. However,
with some approximation and reference to the load deﬂection relationship, the cracking
load could be recognized and shown in Table 5.2. The table also includes the load and

maximum deﬂection at failure of the control and treated slab panels.

105

 

Flexural Crack Pu h C
—-\ N /—— nc one

..........................................................................

..............................

 

 

L.—. ___ .-- __ _- \

IL Steel Beam \

Load Pad

 

 

 

(a) Top Surface

/— Shear Crack

/

AV

WAY

L Steel Beam \
(X

Radial Crack

 

 

 

Z

N

 

 

 

 

 

(b) Bottom Surface

Figure 5.4 Typical Crack Pattern

Sample photographs of the top and bottom views of punched slab are shown in
Figures 5.5 and 5.6. As can be seen from Figure 5.5, the top view of the failed slab
loaded at LP3 shows the size of the upper cone (about the size of the load pad) that
denotes a localized failure. Figure 5.6 shows the bottom part of the slab loaded at LPl ,

which shows a more extensive failure. In this case, however, the damage was caused by

106

the penetration of the load pad into the slab due to delayed release of the hydraulic jack
after failure. Nevertheless, the failure shows the radial and circular crack patterns similar

to those reported in the literature.

Table 5.2 Crack Load, Failure Load, and Maximum Deﬂections of Test Slab

 

 

 

 

 

 

 

 

 

 

 

Load Position Crack Load Failure Load Max. Deﬂection, mm (in)
(Slab Type) kN (lbs) kN (lbs) Beam Slab
LP 1 (Control) 31.14 (7,000) 53.38 (12,000) 1.2 (0.049) 10.4 (0.409)
LP 6 (Control) 28.91 (6,500) 51.15 (11,500) 4.6 (0.182) 11.4 (0.450)
LP 7 (Control) 26.69 (6,000) 53.38 (12,000) 4.7 (0.186) 12.7 (0.500)
LP 8 (Control) 24.47 (5,500) 53.38 (12,000) 1.2 (0.046) 11.9 (0.468)
LP 2 (Treated) 40.03 (9,000) 60.05 (13,500) 1.4 (0.054) 5.7 (0.264)
LP 3 (Treated) 33.36 (7,500) 60.05 (13,500) 4.6 (0.180) 9.4 (0.370)
LP 4 (Treated) 31.14 (7,000) 60.05 (13,500) 0.9 (0.034) 9.2 (0.362)
LP 5 (Treated) 40.03 (9,000) 60.05 (13,500) 4.3 (0.170) 9.0 (0.356)

 

 

 

 

 

 

The load deﬂection relation of the slab and center beam, for each of the test point
sets is given in Figures 5.7 and 5.8. It is apparent from Table 5.2 and Figure 5.7 that the

slab with the additional tie rods cracked and failed at higher loads, and deﬂected less than
the slab without tie rods. Comparing the load deﬂection curves of LPl and LP2, and LP8

and LP4, it is apparent that the slab panel with added tie rods was signiﬁcantly stronger

107

and stiffer than that without tie rods. The same observation is obvious by comparing the

curves for LP3, and LPS and LP6, and LP7.

rd!

 

Figure 5.6 Bottom View of Punching Failure at LPl

108

The deﬂection of the center beam was also signiﬁcantly affected by the addition
of tie rods. From Figure 5.8, it is apparent that, when loaded near the supports, the
deﬂection of the beam was less than that when loaded near the mid-span of the bridge.
With added tie rods, the beam deﬂections were signiﬁcantly lower than that without tie

rods.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Load - Deflection Diagram of Slab
70,000
60,000 ﬁg
50,000
3 40,000 —
'O
N
3 30,000 -
20,000 - + LP1 + LP6 E
+LP7 +LP8
10,000 —)I(- LP2 —a— LP3 -
-°- LP4 —l— LPS
0 'r i i
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
Net Deﬂection (mm)

 

 

 

Figure 5.7 Load Deﬂection Diagram of Slab

109

Typical behavior of the rods is presented in the form of the load-strain curves
given in Figure 5.9. When the load was applied at LP2 and LP3, most of the connection
in the tie rods adjacent to the load failed (at a load of about 20 kN). This observation
indicates that the welded connection was the weakest point in the structures. Therefore,
prior to the next test, the welds were strengthened by adding another layer of weld. When
loads were applied at LP4 and LPS, the strain developed in the tie rods was higher than

that of the previous tests. Nevertheless, some of the welds also failed before the failure of

 

Load - Deﬂection Diagram of Center Beam

 

70,000

60,000 [2%
50,000 . %_3

640,000 7

'8
3 30,000 — Z

k

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20,000 7 + LP1 + LP6 *
+ LP7 —)6- LP8
10,000 ‘ + LP2 + LP3 7
—+— LP4 + LPS
0 I i
0.0 1.0 2.0 3.0 4.0 5.0

Deﬂection (mm)

 

 

 

Figure 5.8 Load Deﬂection Diagram of Center Beam

110

the slab, but at a higher load level than those applied at LP2 and LP3. The failure of the
weld, however, was a strong indication that the tie rods were under a high magnitude of

stress due to the arching action developed in the slabs.

 

Load - Strain Diagram of Tie Rods Under LPS

 

70,000

 

60,000

50,000 KEEN ;
g 40,, l 57
3.0,... f

20,000 _ . —e—Strain Gage 3
+ Strain Gage 4
+ Strain Gage 5
—x— Strain Gage 6
—0— Strain Gage 7

O I I T T T T IT WW Tj If

0 200 400 600

Micro Strain

 

 

 

 

 

 

 

 

 

 

 

 

10,000 ‘

 

 

 

 

 

 

 

Figure 5.9 Load Strain Diagram of Tie Rods

As shown in Figure 5.9, the strain in tie rod 5 (SS in Figure 5.3), and 6,7 (86,7)
dropped at a load about 50,000 N (11,240 lbs) because of the connection failure. Tie rods

3 and 4 (S3, S4) took over the tensile force until the slab failed. Strain developed in tie

111

rod 3 (S3), which was two spaces away from the load, was so small compared to that of
tie rod 5 (SS). The maximum strain in tie rod 5 (S5) was about 620 micro strain, which
corresponds to about 50% of the yield strain of a 250 MPa (36 ksi) steel. At lower load
levels, the strain in the tie rods was signiﬁcantly lower.

Additional observations were made after the tests to inspect the steel
reinforcement, shear connectors, and steel beams for stress development. The steel
reinforcement around the failed slab was visually observed and yielding in the steel was
detected. During the demolition of the concrete slab, there was no sign of yielding of the
steel bars out side the load sites in the slab.

The 8 mm shear connectors, attached to the main girders and edge beams, were
also carefully examined during the demolition of the slab. It was found that the
connectors were perfectly bonded in the concrete and there was no sign of overstress in
the connector or in the concrete around it. 'The bottom ﬂanges of the steel beams were
also observed for signs of overstress. However, no such sign was found which means that
during the load testing the steel beams were within their elastic limit. Further discussion

on the experimental results is given in Chapter 6 along with the results from the analyses.

112

CHAPTER 6

PRESENTATION AND DISCUSSION OF RESULTS

6.1 Introduction

The purpose of this chapter is to present and discuss the results from the analysis
and experimental program presented in Chapter Three and Chapter Five. Analyses of the
punching shear strength of RC deck slabs were conducted using the mechanical punching
model of restrained slabs and the ACI Code provisions on punching shear strength of RC
slabs. In addition, FEA were conducted to obtain the response of the deck under a full
range of loads, from zero up to failure. Expectedly, the FEA would simulate the slab
response observed during the experiment in the laboratories, which includes: the ultimate
load and deﬂection of slabs, the load deﬂection relationships for the slab and the center
beam, and the load strain relationship for the tie rods.

The results from the analyses are discussed and compared to the experimental
results observed during the test. Direct comparison of the analyses and test results was
possible because the actual dimensions of the model bridge, obtained from the
measurements during the experimental program, were used in the analyses. Results from
the FEA however, were discussed in more detail since these analyses provided more
comprehensive responses of the structure. For the purpose of comparison between the test
and FEA results, only the slab panels loaded near the mid-span of the bridge were
considered. More detail on the results of the experiment and the analyses are presented

and discussed in the following sections.

113

6.2 Punching Strength of Slabs (Analyses and Test Results)

The punching shear strength of the model slab, according to the punching theory
of a restrained slab, was graphed for various values of restraint factor as shown in Figure
6.1. Inserted in the graph are the average values from the experimental program for the
control and treated slab panels, and the results from the ACI Code formulae. The average
test results were 60,050 N (13,500 lbs) and 52,822 N (11,875 lbs) for slab with and
without tie rods, respectively. The ACI Code formulae gave nominal shear strengths of
25,162 N (5,657 lbs) for the RC slab, and 36,936 N (8,304 lbs) for the prestressed
concrete slab.

From the graphs, it is apparent that with restraint factor values of 0.50 to 0.90 the
strength of the slabs linearly increases from about 42.863 kN (9.636 kip) to 66.499 kN
(14.95 kips). The corresponding uncorrected deﬂections of the slab with restraint factors
of 0.5 and 0.9 were 1.76 mm (0.0694 in) and 1.26 mm (0.0498 in), respectively. The
observed values for the deﬂections were 13.65 mm (0.537 in.) and 16.7 mm (0.657 in.)
for the slab with and without tie rods, respectively. Comparing the analysis results to the
observed values, it is apparent that the punching theory of a restrained slab could predict
the punching load, but could not predict the deﬂection of the slab.

The plotted lines (Figure 6.1) for the observed strength of the slabs with and
without tie rods cross the theoretical line at two different points that correspond to two
different values of the restraint factor. Using linear interpolation, the failure loads of the
test slab panels corresponded to a restraint factor of about 0.663 for the control slab
panel, and 0.786 for the slab panel with tie rods (dashed area in Figure 6.1). The restraint

factor values of deck slabs tested by Hewitt and Batchelor were between 0.52 and 0.74

114

(1975). Apparently, the restraint factor value of 0.663 for the control slab panels is in
agreement with those reported in the literature. The 0.786 value for the slab with tie rods
also in agreement with the Hewitt and Batchelor report, in which, with additional tie rods,

the restraint factor of the slab becomes higher.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Punching Shear Strength of Restrained Slab
80.000 r t
5 —e—Exp.
70.000 r
60.000 ——Exp. : I ->K-Exr>-(nr)
Exp.(nr)
A 50.000 . +v:1.2 p
é .
.3 40.000 *
g x % x -e—P:Theory
"’ 30.000 ' '
ir—h—ﬁﬁ—F-b—
20.000 % +ACI'1 1'
I 38
10.000 + ACI:1 1-
i ‘ : 37
0.000 1 '
0.40 0.50 0.60 0.70 0.80 0.90 1.00
Restraint Factor

 

 

 

Figure 6.1 Analysis Results: Punching Strength of Slab

Referring to the test values, the addition of tie rods increased the lateral restraint
of the slab by a factor of 18.2%. The corresponding increase in the strength of the slab

was about 13.7%. The slight increase in restraint factor is attributed to the failure of the

115

tie rod connections prior to failure of the slabs. Were the connections stronger, the slab
with added tie rods would fail at a higher load, and therefore, the restraint factor of this
slab will be higher.

The nominal punching shear strength of RC slabs, given by the ACI Code formula
(ACI: 11-37), is substantially lower than that obtained from the punching shear theory of
restrained slabs. Incorporating a compressive stress of 3.448 MPa (500 psi) into the ACI
formula for prestressed concrete slabs (ACI: 11-38), however, increases the strength of
the slab from 25.162 to 36.936 kN (from 5.657 to 8.304 kips), which is a 47% increase.
ACI Equation 11-37 yields a strength value of about 48% of the measured strength of the
control slab. ACI Equation 11-38 on the other hand, yields a value about 71% of the test
value. This increase implies enhancement of shear strength due to the presence of
compressive membrane stress.

The ACI formulae on punching shear strength of RC slabs are developed mainly
for the purpose of designing the shear capacity of slabs loaded or supported by a column,
such as in ﬂat slab structures or in column footings. These formulae do not take into
account any shear strength provided by the ﬂexural reinforcement. They also neglect any
conditions at the slab boundaries, so that is not appropriate for a restrained slab.
Accordingly, the formulae give shear strength values lower than that obtained from the

punching theory of a restrained slab.

FEA and Test Results

Data regarding the failure loads and deﬂections obtained from the FEA and test

results are summarized in Table 6.1. It is apparent from the table that the FEA could not

116

 

Table 6.1 Data from FEA Compared to Test Results

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FE Model Slab With Tie Slab Without
Rods Tie Rods
Name f,y / f, 3,, f. ratio t. stiff. P/ Fe. A/Ae. P/Penr A/Aenr

M1 12.5% 0.0030 1.25 0.0025 57% 30% 63% 27%
mla 10.0% 0.0015 1.16 0.0020 52% 25%
mlb 12.5% 0.0030 1.16 0.0040 64% 35%
mlc 12.5% 0.0030 1.16 0.0060 65% 36%

m2 12.5% 0.0050 1.30 0.0075 77% 61% 84% 47%

m3* 12.5% 0.0050 1.30 0.0075 101% 87% 81% 51%

m4 12.5% 0.0030 1.20 0.0100 73% 42% 73% 32%
m5 50% 0.0030 1.20 0.0075 94% 43%
m6 50% 0.0015 1.16 0.0020 51% 22%
m82 12.5% 0.0050 1.30 0.0075 25% 11%
m27 12.5% 0.0050 1.30 0.0075 26% 12%

 

* This model uses reduced Young’s modulus.

Fe is the average experimental load = 60,050 N (13,500 lbs),

Ae is the average experimental deﬂection = 13.65 mm (0.538 in.),

Penr is the average experimental load for control slab = 52,266 N (11,750 lbs),

Aenr is the average experimental deﬂection for control slab = 16.7 mm (0.659 in.).

predict the experimentally observed values at failure, in which the ratios between the

analysis and the measured values are less than unity. Model m3 however, matched the

measured load value well, but overestimated the slab stiffness at failure by 13%.

However, in this model, the value for the concrete elastic modulus was arbitrarily

117

 

reduced to 50% of the commonly used value of 24,822 MPa (3.6E6 psi) for a 27.58 MPa
(4,000 psi) concrete, in an attempt to study the effect of varying value of elastic modulus.

Models ml, mla, mlb, and mlc show the effect of varying values of tension
stiffening on the models’ responses. In model m1, using a tension stiffening value of
0.0025 the analysis (therefore the solution) terminated at a load of 57% of the observed
failure load. The strain at which the 500 MPa (72.5 ksi) steel reinforcement yields is
about 0.0025. The value of 0.0025 also corresponds to a maximum tensile strain of 25
times the crack strain of plain concrete, which is about 0.0001. Therefore, model ml was
meant for the reference model to which the other models are compared.

Referring to the results of model ml, using tension stiffening of as much as the
yield strain for the steel reinforcement does not seem to be appropriate for this problem.
Accordingly, to study the effect of tension stiffening, values of 0.002, 0.004, and 0.006
were used for mla, mlb, and mlc, respectively. The responses of these models were not
distinguishable from each other in a load deﬂection diagram. The model with a higher
tension stiffening value however, gave a longer (higher load and deﬂection magnitudes)
response than the model with a lower value of tension stiffening.

In model mlc, using a tension stiffening value of 0.006, the ratios for the load and
the deﬂection between the FEA and test results were only 65% and 36%. In model m4
using tension stiffening value of 0.01 results in load and deﬂection ratios of 73% and
42%, respectively. The tension stiffening value of 0.01 is the maximum value reported
for heavily reinforced concrete (Schnobrich 1986). For the slab in the present study a
lower value for tension stiffening is more appropriate. However, higher failure ratio is

possible due to the conﬁnement provided by the membrane action in the vicinity of the

118

 

loaded slab area. Accordingly, a value as high as 1.3 was used for the failure ratio of
concrete in compression.

In model m2, a tension stiffening value of 0.0075 was used with a failure ratio in
compression of 1.3. The load and deﬂection ratios for this model were 77% and 53%, and
84% and 47% for the slab with and without tie rods, respectively. In models m5 and m6
the ratio of elastic stress limit and the ultimate compressive stress was set to 50%; means
that the stress strain diagram of the concrete becomes nonlinear at a stress 50% of its
compressive strength. Model m5, with tension stiffening value of 0.007 5, gave a failure
load ratio of 0.94, but a deﬂection ratio of 0.43. Model m6 on the other hand, gave ratios
of 51% and 22% for the load and deﬂection, respectively. The material model used for
model m6 is similar to that for model mla, where the value for the tension stiffening was
0.002 with failure ratio of 1.16, and plastic strain 5,, of 0.0015. With higher elastic stress
limit, model m6 failed at a lower load and deﬂection than model mla.

Models m82 and m27 gave very short responses compared to the observed
response. The analysis for these models terminated at early stage of load increment
because the system could not reach equilibrium conditions during the iteration process.
The probable cause of this problem is the large size of the models that cause unstable
analysis. The load deﬂection curves for these models however, ﬁt the curve of model m2

very well, only they were shorter (see Figure 3.7 in chapter three).

6.3 Load Deﬂection Diagram of Slabs (FEM and Test Results)

The global behavior of the slab under varying load level could not be judged only

from the failure load or deﬂection. The load deﬂection diagram of the slab would be a

119

 

more appropriate measure of the effect of adding tie rods to the behavior of the deck slab.
For this reason, the recorded data from the experimental program were analyzed and
presented in Figures 6.2 and 6.3. In these diagrams, the total slab deﬂection was used
instead of the net deﬂection to enable direct comparison to the diagram obtained from the
FEA. During the experiment the net deﬂection of the slab (deﬂection of slab relative to
the deﬂection of the center beam adjacent to the applied load) was measured. The total
slab deﬂection was calculated as the sum of the net slab deﬂection and the center beam
deﬂection. The consequence of this procedure is the introduction of a possible source of
additional error in the test results.

In Figure 6.2, the observed load deﬂection diagrams of the slab loaded near the
mid-span are presented. It is apparent that the slab panels with additional lateral restraint
(tie rods), as shown by LP3 and LPS, are stronger and stiffer than the control slab without
tie rods (LP6 and LP7). This statément also holds true for the slab panels loaded near the '
supports, as shown in Figure 6.3. Accordingly, from the experimental results, it is clear
that the addition of tie rods increased the failure load, and at the same time, it
signiﬁcantly reduced the deﬂection of the slab. Therefore, additional tie rods improved
the global behavior of the slab.

The diagrams for the slab panels near the mid-span of the bridge showed a good
correlation where the two lines from two different tests were very close together. The test
results for the slab panels near the support, however, were not as good as those near the
middle span. For the slab panel loaded at LP1, the slab deﬂection is less than that when
loaded at LP8. Theoretically, they should be the same since the slab panels are similar.

One possible reason for the difference is the order of testing. The slab panel loaded at

120

 

LP1 was tested ﬁrst while the later was tested last. When loaded at LP8, the slab panels
adjacent to the tested slab panel were already tested and failed. Variability in the material
properties may also contribute to this difference along with possible measurement error
during the test. Despite the difference between the two, the lines for the slab panels with
tie rods clearly deviated from those of slab panels without tie rods, which indicated an

improvement in the slab behavior due to additional tie rods.

 

Load - Deﬂection Diagram of Slab
Loaded Near Mid Span

70,000
60,000
50,000

40,000

Load (N)

30,000

20,000

—a— lp3 —-l— lp5
10,000

+lp6 +1p7

 

0.00 5.00 10.00 15.00 20.00

Total Deﬂection (mm)

 

 

 

 

 

Figure 6.2 Observed Load Deﬂection Diagram of Slab Loaded near Mid-span

121

Figures 6.2 and 6.3 also showed that the slab panels with tie rods exhibited stiffer
responses than those without tie rods. The deﬂections of the slab panels with tie rods
were considerably lower than those of the control slab panels. These small deﬂections
indicate smaller strains, and accordingly, lower stresses develop in the slab. From these
ﬁgures, the effect of tie rods can be seen at a load less than 4,000 N (900 lbs), which in a

prototype slab, the magnitude of the load would be less than 100,000 N (22,4811bs).

 

Load - Deﬂection Diagram of Slab
Loaded Near the Edges

70,000
60,000
50,000

40,000

Load (N)

30,000

b4 b2
20,000

+ lp2 —o— lp4

10,000
—e— lpl + lp8

 

0
0.00 5.00 10.00 15.00 20.00

Total Deﬂection (mm)

 

 

 

 

Figure 6.3 Observed Load Deflection Diagram of Slab Loaded Near the Edges

122

 

(e

Knowing that the AASHTO wheel load (with 1.3 impact allowance) is about 95
kN (21 kip), then the tie rods would actually improve the behavior of the slab under
service load. At higher load levels the improvement in the slab stiffness is more
pronounced although such effect would be more desirable at a lower load level.

It is also apparent from Figures 6.2 and 6.3 that the load deﬂection relations of the
tested slabs are highly nonlinear, starting at load levels below 10,000 N (2,248 lbs),
which is about 10 to 15% of the failure loads. This is in agreement with the stress strain
diagram of concrete material, where at lower stress levels the material behaves (nearly)
linearly, and at higher stress levels the rate of strain becomes higher than the rate of
stress.

In the FE models, the load was applied near the mid-span of the bridge for the
following two reasons. Firstly, to avoid the effect of edge beams near the support.
Secondly, to take advantage of the better test results near the mid-span’in which the band
of the data is narrower. The larger slab deﬂection near the mid-span of the bridge would
also make comparison of he test and analysis results easier.

The FEA results are presented in Figure 6.4 for the slab with tie rods, and in
Figure 6.5 for the slab without tie rods. Included in the ﬁgures are the test results of the
slab panels loaded at LP3 and LP7 for the slab panels with and without tie rods,
respectively. In Figure 6.4 only ﬁve models (out of eleven models listed in Table 6.1)
were included. The rest of the models overly predicted the stiffness of the test slab, and
are not included in the diagram.

From the ﬁve models, all but model m3 predicted the actual response of the slab

at lower load levels very accurately. At higher load levels, however, the diagrams

123

deviated from the observed response variously. From Figure 6.4 it is apparent that the
model with higher failure ratio (m2) provided a response that best ﬁts the observed
response up to a load of about 35,000 N (7,868 lbs). Model m3, with failure ratio and

tension stiffening values equal to those for n12, provided the best response at higher load

 

levels.
Load Deﬂection Diagram of Slab
(With Tie Rods)

70,000

60,000

50,000
g 40,000
'8
,3 30 000 —<>-m1(fr:1.25, ts:0.0025)

+ m2(fr:l.3, ts:0.0075)
20,000 + m3(50%Ec)

—+— m4(fr:1.2, ts:0.01)
10,000 —+—- m5(fcy:0.5fc)

-lh-EXP.

 

0 2 4 6 8 10 12 14 16 18 20

Total Deﬂection (mm)

 

 

 

Figure 6.4 Measured and Predicted Load Deﬂection Diagram of Slab Loaded at LP3

124

The models with higher tension stiffening values gave solutions in which the
models fail at higher load and deﬂection values. With a low value of tension stiffening,
the analysis terminated at a lower load increment. The reason for this early termination of

the analysis is due to a failure of convergence achieved during the iteration process.

 

Load Deﬂection Diagram of Slab
(Without Tie Rod)

60.000

50,000

40,000

30,000

Load (N)

20,000

1 0.000

 

0 5 10 15 20

Total Deﬂection (mm)

 

 

 

Figure 6.5 Measured and Predicted Load Deﬂection Diagram of Slab Loaded at LP7

The FEA results for the control slab panels shown in Figure 6.5 are not as

satisfactory as those for the slab panels with tie rods. Among the four models, m3nr with

125

a reduced modulus of elasticity ﬁtted the experimental curve better than the other models.
However, m2nr also yielded response reasonably close to the observed response. From
the responses of models m2 and m2nr shown in Figure 6.4 and 6.5, it is apparent that the
FE models appeared to fail at a load and deﬂection signiﬁcantly lower than the observed
values. Model m2 for the slab with tie rods ceased computing at a load of 77% of the
observed failure load, and a maximum deﬂection of 52% of the observed value. In model
m2nr, these ratios were 84% and 47% for the load and deﬂection, respectively. The
models with a reduced Young’s modulus gave better load and deﬂection ratios: 101%
and 87% for m3, and 81% and 51% for m3nr. Despite the better load and deﬂection
ratios, m3 underestimates the stiffness of the slab at lower load levels.

Judging from the response of m2 and m2nr, the FEA could not predict the load
deﬂection diagram of the tested slab up to failure. The models failed at an average load
about 20% below the observed failure load with a corresponding average deﬂection of
46% less than the observed value (see Table 6.1). During the tests however, the concrete
slabs actually have failed at a load much lower than the ultimate load through the
formation of shear cracks, which were not easy to observe. The dowel action between the
steel reinforcement and the concrete made it possible for the slab to sustain a higher load
after the cracks formed, as was marked by the excessive deformation in the slab prior to
failure. In the FEA, failure was also determined by the failure of concrete. The dowel
action was modeled in tension stiffening, and failure of concrete was modeled in failure
ratios. As soon as the concrete stress reaches its limiting value the slab will not take

additional load, and therefore the analysis terminates, which means that the slab has

126

 

failed. These arguments explain the reason the FE models fail at a signiﬁcantly lower
load than the observed value.

The model with a lower Young’s modulus value gave a higher response because a
smaller load increment is sufﬁcient to cause additional deﬂection that helps the iteration
process reaches convergence. Using higher values of the plastic strain and failure ratio
also allows the analysis to continue to higher values.

During the test, the tie rod connections failed before the slab, which possibly
caused larger deformation than if the tie rod connection were stronger. In the FEA the tie
rods were not allowed to fail, so the model’s response was stiffer than the observed
response. This contributed to the difference between the FEA and measured responses at
higher load levels (after the connections have failed).

Despite the differences between the observed and the FEA results, the FEA are
more meaningful for studying the global behavior of the model slab up to relatively high
load levels. Further discussion on the correlation between the observed and the FEA
results on beam deﬂections and strains development in tie rods are continued in the next

sections.

6.4 Deﬂection of Center Beam (FEA and Test Results)

The deﬂection of the center beam adjacent to the load was also recorded and its
load deﬂection diagram was developed and is presented in Figures 6.6 and 6.7. Included
in these diagrams are the results from the FEA. As expected, the deﬂection of the center
beam was linearly proportional to the load up to failure. The reason for this linearity is

that the beams were conservatively designed taking into consideration the failure load of

127

the slab. Furthermore, the addition of tie rods signiﬁcantly affected the deﬂection of the
center beam. One possible reason for this effect is that with tie rods, the two beams share

the applied better than without tie rods.

 

Load Deﬂection Diagram of Center Beam
(With Tie Rods)

 

70,000

 

60,000

 

 

50,000

 

 

40,000

 

 

Load (N)

30,000

 

 

 

20.000

 

 

10,000

 

 

 

 

 

 

 

0

Deﬂection (mm)

 

 

 

Figure 6.6 Measured and Predicted Load Deﬂection Diagram of Center Beam

Loaded at LP3 (Slab with Tie Rods)

From Figure 6.6 it is apparent that FE models m1 and m2 mimic the load
deﬂection diagram of the center beam very well. These models, however, failed at a load
signiﬁcantly lower than the actual failure load, in which failure was determined by the

failure of the concrete slab. Model m3 matched the actual failure load, but slightly

128

I.

underestimated the actual stiffness of the beam. From Figure 6.6 on the other hand, FE
models m2nr, and m3nr predicted the load deﬂection curve of the center beam better than
does mlnr. Model m2nr ﬁtted the observed curve the best. These results agreed with the

results for the slab deﬂection where model m2 gave responses better than the other

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

models.
Load Deﬂection Diagram of Center Beam
(Without Tie Rod)
60,000
50.000 , 4:.— EXP°("’) a
40,000
8
'3 30,000
O
.4
20,000 + Exp.(nr) e,
—e— mlnr
10,000 —0— m2nr H
+ m3nr
0
0 2 4 6

Deﬂection (mm)

 

 

 

Figure 6.7 Measured and Predicted Load Deﬂection Diagram of Center Beam

Loaded at LP7 (Slab without Tie Rods)

Comparing the curves for the slab and beam deﬂections, it is obvious that the

FEA mimic the beam response much better than that of the slab. The explanation to this

129

phenomenon lies on the variables associated with the modeling of the concrete material,
which is not as simple as that for the steel material model. Steel on one hand is a
relatively homogeneous material that behaves linearly up to a high stress level; in the
present study the steel beams were within their elastic limit until failure of the slab.
Concrete on the other hand, is a nonlinear material starting at a relatively low stress level.
In addition, research on FEA of concrete structures is not yet as advanced as that of other
structural materials because the application of concrete structures does not usually require
analysis in great precision. In addition, FEA of RC structures involving true 3-D models
are not yet as popular as applications that use shell or axisymmetric elements.

The observed behavior of the center beam loaded at LP3, LP5, LP6, and LP7 is
presented in Figure 5.8 in chapter ﬁve. From this ﬁgure, the deﬂection of the beam when
the load was applied to the slab with tie rods near the mid-span of the beam (at LP3 and
LP5), was less than those when the load was' applied at LP6 and LP7 of the control slab.
The probable cause of the difference is the order of the testing; when the loads were
applied at LP6 and LP7 the slab at LP3 and LPS had already been tested and failed, that T.
inﬂuenced the properties of the composite beam section, decreasing its stiffness. At load

points near the support (LP1, LP2, LP4, and LP8), however, such effect was so small that ’

 

it was not apparent.

In the FE models, the effect of testing order did not exist because the slab was
loaded at one location at a time. Any differences between the observed and the FEA
results could also be attributed to this factor. Nevertheless, the global behavior of the
center beam was accurately modeled in the FEA. The model using a reduced Young’s

modulus (m3nr) produced larger beam deﬂection than did the other models, which means

130

 

’7' ' *

o‘»
.

J.

that the FEA correctly modeled the composite action between the concrete deck and the

steel beam.

6.5 Strain in Tie Rods (FEA and Test Results)

Another important parameter to study the effect of adding tie rods to a deck slab
is the strain development in the tie rods. The effect is determined by the magnitude of
strains at lower load levels under which the bridge slab operates. Figures 6.8 and 6.9
show a typical load strain diagram in the tie rods underneath and adjacent to the load
when the load was applied at LP5.

The strains in the tie rods underneath the applied load started to develop at a very
low load level, which means that the arching action in the slab is well developed under
the service load. This veriﬁes that the skepticism whether the arching action takes effect
at low load levels is unjustiﬁed.

From the experimental program presented in chapter ﬁve, the average maximum
strain in the tie rods in the vicinity of the load was about 0.00035. With steel Young’s
modulus of 200,000 MPa (29,000 ksi) the above strain corresponded to an average tensile
force of 11.287 kN (2,538 lbs) in the 12.7 mm (0.5 in.) square bar. The corresponding
stress in the 8 mm (0.315 in.) tie rods are about 224.6 MPa (32.6 ksi), which is within the
elastic limit of the steel. Accordingly, the size of the tie rods used in the model-bridge is
adequate even if the slab is overly loaded.

As shown in Figure 6.8, models m1, and m2 ﬁtted the test curve better than model
m3. It is also clear that at higher load levels the observed response was much stiffer than

the model’s curves, because the tie rod connection failed during the test. Were the

131

 

connection stronger, the observed and FEA results would be better correlated up to a

 

 

higher load level.
Load Strain Diagram in Tie Rod
(FE Prediction and Test Results)
70,000
60,000
50,000
:2: 40,000
«a
8
.1 30,000
. 20,000
10,000 —A—-Exp-SS —e—SS(m1)
-o— SS(m2) —a— SS(m3)
0
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012
Strain

 

 

 

Figure 6.8 Measured and Predicted Load Strain Diagram of Tie Rods Directly

Underneath the Applied Load

With 250 MPa (36 ksi) steel, the yield strain of the tie rods is about 0.0012.
Figure 6.8 shows a nonlinear relation between the load and the strain in the tie rods, even

at a very low strain value. The higher the load, the greater the strain rate in the tie rods.

132

 

This means that the magnitude of the arching action increases as the load increases as

noted by the high strain rate developed in the tie rod.

 

 

 

 

 

 

Load Strain Diagram in Tie Rod
(FE Prediction and Test Results)
70,000
60,000
50.000
2 40,000
To“
8
.4 30,000
20,000
+ Exp-S4 -o— S4(m 1)
10,000
-+— S4(m2) —e— S4(m3)
0
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012
Strain
Figure 6.9 Measured and Predicted Load Strain Diagram of Tie Rods Adjacent to I
the Applied Load

The same argument holds true for the tie rods adjacent to the loaded area, as
shown in Figure 6.9. At higher load levels, the test results showed less strain than did the

FE models because during the test the tie rod connection failed before failure of the slab.

133

Model m3 gave a maximum strain value very close to the yield strain. However, under a
load level of about 6 times the wheel load, the tie rods were still within its elastic limit.
The tie rods, at a distance from the loaded area, were not strained as much as that
closer to the applied load. The strains diminished in tie rods at a distance two (tie rods)
spaces away from the load (see Figure 5.9). This observation was also well modeled in
the FEA. Therefore, the arching action is a highly localized phenomenon that causes
enhancement of the concrete strength around the loaded area. This also suggests that the
tie rods spacing should be small. A spacing of 1,000 mm in a prototype slab is sufﬁcient,

as shown in the test results.

6.6 Summary of Results

The behavior of deck slab panels with and without tie rods were experimentally
observed and theoretically and numerically analyzed. The results were presented in the
form of tables and diagrams and are summarized in the followings.

1. Experimental results showed that the slab panels with added tie rods were
signiﬁcantly stronger and stiffer than those without tie rods. A lower slab deformation
means a lower stress magnitude, which accordingly, increases the fatigue resistance
of the slab. As a result, addition of tie rods could improve the durability of the slab.

2. The effect of adding tie rods, observed through the development of strain in the tie
rods, showed that the arching action developed in the slab starting at a load level
lower than the service load. This means that it could improve the behavior of the slab
under the service load, though the improvement is more pronounced at higher load

levels.

134

 

."

3. Theoretical analysis using a punching shear model of a restrained slab predicted the
punching load of the slabs with corresponding restraint factor values of 0.663 and
0.786 for the slab panels without and with tie rods, respectively. This theory however,
could not predict the observed deﬂection of the slabs.

4. The ACI Code provision on punching shear strength of RC and prestressed concrete
slabs gave signiﬁcantly lower strengths (48% and 71%) than the observed values for
the control slab. The formula for the prestressed concrete slab gave higher punching
load than the formula for the RC slabs. This implies that the presence of compressive
stress increase the punching shear capacity of the slab.

5. The ﬁnite element analyses of the model bridge showed that the response of the
models were strongly dependent of the material model for the concrete. Among the
variables affecting the FEA response were the tension stiffening, failure ratio, plastic
strain, yield stress, and modulus of elasticity. The model with higher values for
tension stiffening, failure ratio, and plastic strain gave a higher magnitude response
than the model using the lower values. The model with higher value of yield stress
gives a stiffer response. Using lower value of Young’s modulus, the model provides
higher response with less stiffness.

6. The FEA predicted the load deﬂection behavior of the slab well at the lower load
levels (up to 60% of the observed failure load). At higher load levels however, the
models showed stiffer response than the observed response. The model with tie rods
was stronger and stiffer than the model without tie rods. This conﬁrmed the
experimental observation that the addition of tie rods improved the behavior of the

slab.

135

 

7. The FEM mimic the observed load deﬂection diagram of the center beam very well
until the model fails due to failure of the concrete slab. The model using a lower
value of concrete elastic modulus gave a more ﬂexible response, which means that
the composite action between the concrete deck and the steel beam was properly
modeled.

8. The strain developed in the tie rods was also properly modeled in the FEA. The
observed response was stiffer than the model response at higher load levels, because
during the test, the connection of the tie rods failed prior to the failure of the slab.

9. In general, the FEM could mimic very well the global behavior of the model bridge at
lower load levels. Accordingly, the FEM is more appropriate for a non-destructive
evaluation, in which the structure is not loaded up to failure. To predict the slab
response up to failure, considerable efforts would be necessary to improve the
material model as well as the mechanical constitutive models for concrete subjected
to high magnitude three-dimensional stress states. Nevertheless, if properly
understood, the FE method is one of the most powerful tools available for stress F

analyses.

 

 

136

 

CHAPTER 7

CONCLUSION AND RECOMMENDATION

7 .1 Summary and Conclusion

Experimental investigation of the behavior of a reinforced concrete deck slab with
additional lateral restraint (tie rods) was conducted by testing two different slab panels,
which were incorporated into the construction of a 1/5 scale model bridge. The deck slab
was reinforced with a 0.2% isotropic wire mesh arrangement that represented the
minimum steel requirement for the empirical method of slab design in the AASTHO
LRFD Bridge Design Speciﬁcations (1994). One slab panel was strengthened by
connecting the top ﬂanges'of the steel girders with steel tie rods, immediately underneath
the slab. The other slab panel served as a control slab without tie rods. Each slab panel
was consecutively tested at four different load points with a full range static concentrated
load. The load response history of the slab, the center beam, and the tie rods was
observed under a load increment of 2,224 N (500 lbs). The effect of adding tie rods was
measured from the stiffness and strength of each slab panel, as well as the strain
developed in the tie rods.

To simulate the load response histories observed during the experiment, nonlinear
three-dimensional ﬁnite element models were developed using a commercial ﬁnite
element program, ABAQUS. The models incorporated each component of the bridge,
including the deck, steel girders, edge beams, diaphragms, and the tie rods. The deck was

modeled using 3-D solid elements where the steel reinforcement was modeled as layers

137

of rebar embedded in the concrete elements. The ﬁnite element analyses involved several

models to study the sensitivity of the models to various modeling parameters for concrete

material.

In addition to the FEA, a rational punching shear theory of restrained slabs and
the ACI Code provisions on punching shear strength of concrete slabs were also utilized
to analyze the ultimate shear capacity of the slab. In the punching shear theory, the deck
slab was transformed into an equivalent circular slab with diameter equal to the center to
center spacing of the steel girders. For the ACI Code provision, two equations were
utilized, one equation for RC slabs and the other for prestressed concrete slabs. The
results obtained from the experiment and the analyses were carefully discussed toward
the effect of adding tie rods on the behavior of the deck slab and the veriﬁcation of the
analysis methods.

From the experimental results, the following conclusions can be drawn:

1. The slab panel with tie rods was signiﬁcantly stronger and stiffer than the slab panel
without tie rods. The signiﬁcant reduction in the slab deﬂection indicates a lower
tensile strain in the slab, which accordingly, improves the crack resistance of the
deck. Therefore, additional tie rods could improve the durability of an existing deck
slab.

2. The strain developed in the tie rods is a strong indication of increased lateral restraint,
which enhances the arching action in the deck slab and consequently, increases the

stiffness and the strength of the deck.

138

 

. With the addition of tie rods, the net deﬂection of the slab was reduced by a factor of
34% at lower load levels and 50% at higher load levels. The cracked and failure loads
of the slabs with tie rods increased by factors of 30% and 13.7%, respectively.
. The strength and stiffness enhancement in the slab with tie rods was observed starting
at a load magnitude lower than the standard truck wheel load, which means that tie
rods can improve the performance of a slab under service load conditions.
. The present study veriﬁed the earlier ﬁndings that under static concentrated load, a
deck slab reinforced with 0.2% steel exhibited a punching shear strength much higher
than that required to support a standard truck loading. In a prototype bridge, the
failure load of the slab with and without tie rods would be 1,501.268 kN (337.5 kip)
and 1,306.659 kN (293.75 kip), respectively. Compared to a standard wheel load of
7 2.5 kN (16.3 kip) with an impact factor of 1.3, the expected safety factors of the
slabs with and without tie rods are 15.9 and 13.9, respectively.

From the ﬁnite element analyses the following conclusions are drawn:
. The observed nonlinear load response history of the model bridge was well simulated
in the ﬁnite element models up to about 60% of the failure load. At higher load levels
however, the FE models showed stiffer responses than the observed ones.
. Solid 3-D elements could properly model the arching action in the RC slab through
the development of strain in the tie rods. The shell elements in ABAQUS, however,
were conﬁned to ﬂexural strains only and so the strain in the tie rods did not develop.
. The responses of the ﬁnite element models were strongly dependent on the material

model for the concrete slab. Among the variables that affect the model’s responses

139

 

are tension stiffening, failure ratio, plastic strain, yield stress, and elastic modulus of
concrete.

4. At higher load levels, the ﬁnite element models mimic the observed load deﬂection
diagram of the center beam better than that of the concrete slab. This result indicates
the needs for the improvement in modeling the post-cracking behavior of the
reinforced concrete element.

5. The model using a lower value of concrete elastic modulus allowed larger deﬂection
in the center beam, which means that the composite action between the concrete deck
and the steel beam was properly modeled by the FEA.

6. In general, the ﬁnite element models could mimic very well the global behavior of the
slab at lower load levels. To predict the slab response at higher load levels,
considerable efforts are necessary to improve the material model as well as the
mechanical constitutive models for the concrete. Accordingly, the FEA of RC
structures is more appropriate for a non-destructive evaluation, in which the structure
is not loaded up to failure.

From the rational punching shear theory of restrained slabs and the ACI Codes
provision on punching shear strength of concrete slabs, the following conclusions can be
drawn:

1. Theoretical analysis using the punching shear model of restrained slabs predicted the
punching strength of the slabs with restraint factor values of 0.663 and 0.786 for the
slab panel without and with tie rods, respectively. This theory however, gave a slab
deﬂection about 10% of the observed value. For the purpose of design, a restraint

factor value of 0.6 could be assumed for a deck slab of comparable design conditions.

140

 

 

2. The ACI Code provisions on punching shear strength of RC and prestressed concrete
slabs gave punching shear strength of 48% and 71% of the observed strength of the
control slab. These formulae however, are readily available for the purpose of design,
or to rapidly estimate the punching shear strength of concrete slabs. Higher punching
load for the prestressed concrete slab implies that the presence of compressive stress

increases the punching shear capacity of the slab.

7 .2 Recommendation

The present study showed that the newly constructed deck slab with additional tie
rods exhibited better behavior than that without tie rods. Considering the extraneous
possible beneﬁts gained from the concept of adding tie rods to an existing deck slab,
further research to study the long-term performance of slabs with varying degrees of
deterioration is recommended. In addition, further study on the cost effectiveness of the

concept is necessary. The effect of applying varying magnitudes of tension forces in the

 

tie rods, to prestress the slab, would also be worth studying. For a newly constructed I:
bridge, the concept could be applied with much less effort by prefabricating the rod

connections. In addition, minimum steel reinforcement should be used to minimize the

risk of corrosion-induced deterioration of the deck slabs. E

For the purpose of applying the concept of adding tie rods to deck slabs of an
existing bridge, the following notes are worth considering:
0 The spacing of the tie rods should be kept small because the distribution of the stress
in the tie rods is localized in the vicinity of the load. A 40 mm diameter tie rods with

a 1,000 mm spacing would be sufﬁcient for a full scale bridge.

141

o The tie rod connection should be designed to accommodate possible replacement and
applying tension force in the tie rods. Fatigue resistance of the connection should also
be considered.

0 In the negative moment region, where the top ﬂanges of the girders are in tension,
bolted connections could be used to avoid welded connections.

Related to the finite element analysis, the present study veriﬁed the earlier
ﬁndings that a true 3-D ﬁnite element modeling of RC structures could not predict the
response of the structure up to failure. Accordingly, interpretation of the model response
should be limited to the lower load levels. In addition, the application of modeling
parameters used in the present study should be limited to untested models of similar
conﬁguration. Finally, development of material and constitutive models for the reinforced
concrete element still remains to be investigated, taking advantage of test results reported

in the literature.

142

B.
I

 

 

I In" " '.' '92:..-

 

 

REFERENCES

AASHTO (1994). LRFD Bridge Design Speciﬁcations, SI Units, First edition, American
Association of State Highway and Transportation Ofﬁcials, Washington, DC.

Abbasi, M.S., Baluch, M.H., Azad, A.K., Abdel-Rahrnan, H.H. (1992). “Nonlinear Finite
Element Modeling of Failure Modes in R/C Slabs,” Computer & Structures, Vol. 42, No.
5, 1992, PP. 815-823.

ACI Committee 318 (1995). Building Code Requirements for Structural Concrete (ACI
318-95) and Commentary (ACI 318R-95), American Concrete Institute, PO. Box 9094
Farmington Hills, MI 48333.

Agarwal, AC. (1990). “Load Testing of New Concrete Bridge Deck Slabs,” Third Int.
Conference on Short and Medium Span Bridges, Toronto, Canada, pp. 277-289.

Aldridge, W.W. and Breen, LE. (1970). “Useful Techniques in Direct Modeling of
Reinforced Concrete Structures,” Models for Concrete Structures, ACI, SP 24-5.

Al-Mutairi, NM. (1989). “The Structural Feasibility of Using Welded Steel Mesh in
Bridge Decks,” Ph.D. Thesis, University of Maryland College Park, 1989.

Aoyama, H., and Noguchi, H. (1986). “Future Prospect for Finite Element Analysis of
Reinforced Concrete Structures,” Finite Element Analysis of Reinforced Concrete
Structures, Proceeding of the International Workshop, ASCE 345 East 47th Street New
York, New York 10017-2398.

 

ASCE-ACI Task Joint Committee 426 on Shear and Diagonal Tension of the Committee
on Masonry and Reinforced Concrete of the Structural Division (1974). “The Shear
Strength of Reinforced Concrete Members - Slabs,“ ASCE Journal of the Structural
Division, Vol. 100, No. ST8, pp. 1543-1591.

ASTM (1996). Annual Book of ASTM Standards, Section 4 Construction, Volume 04.02
Concrete and Aggregates, American Society for Testing and Materials, 100 Barr Harbor
Drive, West Conshohocken, PA 19428.

Azad, A.K., Baluch, M.H., Abbasi, M.S., Kareem, K., (1994). “Punching Capacity of
Deck Slabs in Girder-Slab Bridges,” ACI Structural Journal, Vol. 91, No. 6, November-
December 1994, pp. 656-662.

Bakht, B., and Mufti, AA. (1996). “FRC Deck Slabs Without Tensile Reinforcement,”
Concrete International, Vol. 18, No. 2, pp. 50—55.

143

 

Barker, R.M., and Puckett, IA. (1997). Design of Highway Bridges Based on AASHTO
LRFD Bridge Design Speciﬁcations, John Wiley & Sons, Inc.

Batchelor, B. deV., Hewitt, B.E., Csagoly, P., and Holowka, M. (1978). “Investigation of
the Ultimate Strength of Deck Slabs of Composite Steel/Concrete Bridges,” TRR Record
No. 664 Transportation Research Board, pp. 162-170.

Bathe, K-J., Walczak, J ., Welch, A., and Mistry, N. (1989). “Nonlinear Analysis of
Concrete Structures,” Computers & Structures, Vol. 32, No.3 /4, pp. 563-590.

Beal, DB. (1981). “Strength of Concrete Bridge Decks,” Research Report 89,
Engineering Research and Development Bureau, New York State Department of
Transportation.

Beal, DB. (1982). “Load Capacity of Concrete Bridge Decks,” ASCE Journal of the
Structural Division, Vol. 108, No. ST4, April 1982, pp. 814-832.

Cook, R.D., Malkus, D.S., and Plesha, ME. (1989). Concept and Applications of Finite
Element Analysis, Third Edition, Wiley, New York.

Csagoly, P. (1997). “Testing of a Composite Steel-Concrete Bridge Deck,” Concrete
International, February 1997.

Csagoly, P., Holowka, M., and Dorton, R. (1978). “The True Behavior of Thin Concrete
° Bridge Slabs,” TRR Record No. 664 Transportation Research Board, pp. 171-179.

Dorton, R.A., Holowka, M., and King, J.P.C. (1977). “The Conestogo River Bridge —
Design and Testing,” Canadian Journal of Civil Engineers, Vol. 4 No. 18, 1977, pp. 18-
39.

 

pt
Fang, I.-K., Tsui, C.K.T., Burns, NH, and Klingner, RE. (1990). “Load Capacity of 3
Isotropically Reinforced, Cast-in-Place and Precast Panel Bridge Decks,” PCI Journal, :1
July-August 1990. 1
Fang, I.-K., Worley, J .A., Burns, NH, and Klingner, RE. (1986). “Behavior of Ontario-

Type Bridge Decks on Steel Girders,” Research Report 350-1 , Center for Transportation

Research, University of Texas at Austin. ..

Gilbert, R.I., and Warner, RF. (1978). “Tension Stiffening in Reinforced Concrete
Slabs,” Journal of the Structural Division, Proceeding of the ASCE, Vol. 104, No. ST12,
December 1978.

Gonzales-Vidosa, F., Kotsovos, M.D., and Palvovic, MN. (1988). “Symmetrical

Punching of Reinforced Concrete Slabs: an Analytical Investigation Based on Nonlinear
Finite Element Modeling,” ACI Structural Journal, May-June 1988.

144

 

Hand, F.R., Pecknold, D.A., and Schnobrich, WC. (1973). “Nonlinear Layered Analysis
of RC Plates and Shells,” Journal of the Structural Division, Proceeding of the ASCE,
Vol. 99, No. 817, July, 1973.

Harris, H.G., Sabnis, G.M., and White, RN. (1970). “Reinforcement for Small Scale
Direct Models of Concrete Structures,” Models for Concrete Structures, ACI, SP 24-6.

Hassoun, MN. (1998). Structural Concrete, Theory and Design, Addison-Wesley.

Hewitt, BE. (1972). “An Investigation of the Punching Strength of Restrained Slabs with
Particular Preference to the Deck Slabs of Composite I-Beam Bridges,” Ph.D. Thesis,
Department of Civil Engineering, Queen’s University at Kingston, Canada.

Hewitt, BE, and Batchelor, B. deV. (1975). “Punching Shear Strength of Restrained
Slabs,” ASCE Journal of Structural Division, Vol. 101, No. 8T9, pp. 1837-1853.

H, K & S, Inc. a (1998). Abaqus Theory Manual, Version 5.8, Hibbit, Karlsson &
Sorensen, Inc. 1080 Main Street Pawtucket, RI. 02860-4847.

H, K & S, Inc., b (1998). Abaqus/Standard User’s Manual, Version 5.8, Hibbit, Karlsson
& Sorensen, Inc. 1080 Main Street Pawtucket, RI. 02860-4847.

H, K & S, Inc., c (1998). Abaqus/Standard Example Manual, Version 5.8, Hibbit,
Karlsson & Sorensen, Inc. 1080 Main Street Pawtucket, RI. 02860-4847.

H, K & S, Inc., (1 (1998). Abaqus/Post Manual, Version 5.8, Hibbit, Karlsson &
Sorensen, Inc. 1080 Main Street Pawtucket, RI. 02860-4847.

H, K & S, Inc. (1996). Analysis of Concrete Structures with Abaqus, Karlsson &
Sorensen, Inc. 1080 Main Street Pawtucket, RI. 02860-4847. "

Hsu, T.T.C. and Zhang, L. (1996). “Tension Stiffening in Reinforced Concrete
Membrane Elements,” ACI Structural Journal, Vol. 93, No. l January-February 1996. .'

Huria, V., Lee, K., and Aktan, AB. (1993). “Nonlinear Finite Element Analysis of RC
Slab Bridge,” Journal of Structural Engineering, ASCE Vol. 119, No. 1, January 1993. E

 

Jain, S.C., and Kennedy, J .B. (1974). “Yield Criterion for Reinforced Concrete Slabs,”
Journal of the Structural Division, Proceeding of the ASCE, Vol. 100, No. 8T3, March,
1974.

Jiang Da-Hua and Shen Jing-Hua (1986). “Strength of Concrete Slabs in Punching

Shear,” Journal of Structural Engineering, ASCE, Vol. 112, No. 12, December 1986, pp.
2578-2591.

145

Jofriet, J .C, and McNeice, G.M. (1971). “Finite Element Analysis of Reinforced Concrete
Slabs,” Journal of the Structural Division, Proceeding of the ASCE, Vol. 97, No. ST3,
March 1071.

Kinnunen, S. (1963). “Punching of Concrete Slabs with Two-way Reinforcement,”
Transaction of the Royal Institute of Technology, Stockholm, Sweden, No. 198.

Kinnunen, S., and Nylander, H. (1960). “Punching of Concrete Slabs without Shear
Reinforcement,” Transaction of the Royal Institute of Technology, Stockholm, Sweden,
No. 158.

Kotsovos, M.D., and Pavlovic, MN. (1995). Structural Concrete F inite-Element Analysis
for Limit-State Design, Thomas Telford Services Ltd., Thomas Telford House, 1 Heron
Quay, London E14 4JD.

Kuang, J .S., and Morley, CT. (1992). “Punching Shear Behavior of Restrained
Reinforced Concrete Slabs,” ACI Structural Journal, Vol. 89, No. 1, January-February
1992, PP. 13-19.

Kupfer, H.B., and Gerstle, K.H. (1973). “Behavior of Concrete under Biaxial Stresses,”
AS CE Journal of the Engineering Mechanics Division, Proceedings of the American
Society of Civil Engineers, Vol. 99, N0. EM4, August 1973, pp. 853-866.

Lin, GS, and Scordelis, AC. (1975). “Nonlinear Analysis of RC Shells of General
Form,” Journal of the Structural Division, Proceeding of the ASCE, Vol. 101, No. ST3,
March, 1975.

Masajiro, K., Hiroshi, U., Masahito, U., and Takitaro, H. (1994). “Rehabilitation and
Strengthening of Concrete Structures with Carbon Fiber,” Proceeding of 10”I US-Japan
Bridge Engineering Workshop, Lake Tahoe - Nevada, pp. 304-319.

Mehta, PK. and Monteiro, P.J.M. (1996) Concrete Structure, Properties, And Materials,
Second Ed., Prentice Hall Inc., Englewood Cliffs, New Jersey 07632.

Mufti, A.A., and Bakht, B. (1996). “Design Provisions for Steel-free Deck Slabs on
Beams,” Proceedings of the Sixth International Colloquium on Concrete in Developing

Countries, pp. 35-44.

Mufti, A.A., Jaeger, L.G., Bakht, B., and Wegner, L.D. (1993). “Experimental
Investigation of Fiber-reinforced Concrete Deck Slabs without Internal Steel
Reinforcement,” Canadian Journal of Civil Engineer, Vol. 20, pp. 398-406.

Mufti, A.A., and Newhook, J .P. (1998). “Punching Shear Strength of Restrained

Concrete Bridge Deck Slabs,” ACI Structural Journal, Vol. 95, No. 4, July-August 1998,
pp. 375-38 1.

146

Newhook, J .P. (1997). “The Behavior of Steel-Free Concrete Bridge Deck Slabs under
Static Loading Condition,” Ph.D. Thesis, Faculty of Engineering, Technical University of
Nova Scotia, Halifax, Canada.

Ngo, D., and Scordelis, AC. (1967). “Finite Element Analysis of Reinforced Concrete
Beams,” ACI Journal / March 1967.

Noguchi, H., and Schnobrich, W.C. (1991). “Finite Element Analysis of Reinforced
Concrete in Japan and US,” Finite Element Analysis of Reinforced Concrete Structures II,
Proceeding of the International Workshop, ASCE, 345 East 47‘h Street New York, NY
10017-2398.

Perdikaris, PC, and Beim, S. (1988). “RC Bridge Decks Under Pulsating and Moving
Load,” ASCE Journal of Structural Engineering, Vol. 114, No. 3, pp. 591-607.

Petrou, M.F. (1991). “A Model Study on Highway Concrete Bridge Decks Subjected to
Concentrated Load (Ontario VS. AASHTO Design),” MS Thesis, Department of Civil
Engineering, Case Western University May 1991.

Petrou, M.F. and Perdikaris, RC. (1990). “Small Scale Model Tests: Arching Action in
Reinforced Concrete Bridge Decks,” Third Int. Conference on Short and Medium Span
Bridges, Toronto, Canada, pp. 317-326.

Petrou, M.F. and Perdikaris, PC. (1996). “Punching Shear Failure in Concrete Decks as
Snap-Through Instability,” AS CE Journal of Structural Engineering, Vol. 122, No. 9,
September 1996, pp. 998-1005.

Polak, MA. (1998). “Modeling Punching Shear of Reinforced Concrete Slabs Using
Layered Finite Element,” ACI Structural Journal, Vol. 95, No. 1, J anuary-February 1998.

Puckett, J.A., Naiknavare, RD, and Lohrer, JD. (1988). “Evaluation of Bridge Deck
Utilizing Ontario Bridge Deck Design Method,” Final Report, Project No. 38717,
Department of Civil Engineering, University of Wyoming, Laramie, WY, January 1988.

Sabnis, G.M., Harris, H.G., White, RN, and Mirza, MS. (1983). Structural Modeling
and Experimental Techniques. Prentice-Hall, Inc., Eanglewood Cliffs, NJ. 07632.

Schnobrich, W.C. (1986). “The Role of Finite Element Analysis of Reinforced Concrete
Structures,” Finite Element Analysis of Reinforced Concrete Structures, Proceeding of
the International Workshop. ASCE 345 East 47th Street New York, New York 10017-
2398.

Wegner, L.D., Mufti, AA. (1994). “Finite Element Investigation of Fiber Reinforced

Concrete Deck Slabs without Internal Steel Reinforcement,” Canadian Journal of Civil
Engineer, Vol. 21, pp. 231-136.

147

 

 

Youn, S-G., and Chang, S-P. (1998). “Behavior of Composite Bridge Decks Subjected to
Static and Fatigue Loading,” ACI Structural Journal, Vol. 95, No. 3, May-June 1998.

Xanthakos, RP. (1996). Bridge Strengthening and Rehabilitation, Prentice-Hall, Inc.,
Upper Saddle River, NJ. 07458.

148

 

t
..-

‘.

APPENDIX A
PUNCHING SHEAR MODEL OF RESTRAINED SLAB

DERIVATION OF EQUATIONS

1. Mechanical Model for Simply Supported Slabs

Kinnunen and Nylander, based on observation made in their tests, argued that the
outer portion of the slab, which is bounded by the shear crack and by radial cracks, is
loaded through a compressed conical shell that develops from the column (load) to the
end of the shear crack. The element shown in Figure A.1 is acted upon by reactions
PB/(Ztt) and by the following forces which are caused by rotation:

o The oblique compression force TB/(Zn) in the compressed conical shell.

- Tangential forces in the reinforcement at right angles to the radial cracks, with
resultant R1.

0 Radial forces in the reinforcement cutting across to the radial cracks, with resultant
R2.

0 Tangential compressive forces in the concrete under the center of rotation, with
resultant R3.

The conical shell was assumed to have the shape of Figure A2, and its thickness
varies in such a manner that the compressive stress at the intersection with the column

and the root of the shear crack are approximately equal, as given by:

149

B B
2n3(2c,)f, 5 2m; + y)(2c1 )f,

 

 

 

 

 

 

 

P B/(Zn)

Figure A.l Isometric View of Conical Shell Element for Simply Supported
Slab

B/2 P
BIZ - chota + y/3cot0t
<—y ,‘ .
Zr It a “3"”)
VH"

A -
,,""
x” 2C2
"24III.- ll

Z1 C.R.

 

y/3

 

 

 

 

 

Rebar

 

 

VI
r=Co ’l

 

 

 

Figure A.2 Cross Section of Mechanical Model Showing Notations

In this equation, ft is the concrete compressive stress in the imaginary conical shell, 2C2

and 2c1 are the thickness of the conical shell as given by:

2-3/2+y_1+ﬂ
c,” 3/2 B

150

Similarly, the following relation holds:

From the geometry in Figure A.2 it can be seen that:
Therefore,

2y y(1—tan a)

—)Z' = 2(1+y/B)

Over the cross section 2c] we have:

2cl = 22, cos a
The assumed compressive force, T, along the lower bounding surface of the cone can be
approximated by:

T = 27t(B/2+ y)2c1f,

The force, P, which is transmitted through the conical shell is given by:

 

1— tana
P = Tsina = 27¢,(B/ 2+ y)Z(-(1—;y—/E)lsinacosa
Deﬁne:
_ . _tana(1—tana) ,
fm, — srn acos a(1 tan a) — 1+ tan2 a Equatron (1)
Then P can be rewritten as
_B_ _y_ (__B+2y) .
P— - h h (B+ y) ——-hf,f(,,,)2 Equatron (2)

If \y is the angle of the rotation around the C.R., then the strain 8, of reinforcing steel at a

distance, r from the center of the slab is given by:

151

_ 91.22-}: _2
6‘,-V’ r —rV/(1 h)

The corresponding steel stress is given by:
f: k e E: E": — w(1-—)

In this case k1 is a coefﬁcient greater than one (1), expresses the fact that the stress of
steel varies between the radial cracks, and reaches a maximum value at the crack. Here k1
is assumed as unity. At the boundary between the yield and elastic stress ranges, r = rs,
this expression holds:

ES

‘<

 

        

 

 

 

 

7' .
7;. = {—(1 __. h) = f Equatron (3)
In general, the resultant force in steel ring is given by:
r=r r=C/2A
R1 = f ”[g-co Z A + r 2'1"]
r
Using dAs = p h (it, then
r, C12
For r8 2 Co: R1: pfnhl:J'dr+ r, I i] = pf0h[(rs — C0)+ rs 1n Cl 2]
Co r, r s
R1 = ,Q‘ h2 (i — £9) + L'- In Cl 2 Equation (4.a)
" h h h S
For r8 < Co: R1 = pfnh2|:%ln C/ 2] Equation (4.b)
’2

The stress in steel inside the shear crack is fsy and correspondingly

For rs- > CozR — -8 E phﬂCo= fnhpCoﬂ = pfth % ﬂ Equation (5.a)

152

 

Is.

For r8 < Co: R2 = pfth h

,6 Equation (5.b)

The Compressive Stress in Concrete

When the sector element outside the shear crack undergoes angular rotation around the
C.R., compressive stress occurs in a tangential direction under a horizontal plane passing
through the C.R. These stresses increase linearly as the distance from the above

mentioned plane becomes greater. The strain is given by:

5a=W

‘I"~<

The corresponding concrete stress in the tangential direction on the top surface of the

slab, fat is

y
fct = 1‘23”: EC

where k2 is a coefﬁcient to reﬂect the stress variation which is again assumed as unity.

The total compressive force acting on a radial crack is

C/2 y C/2 2
dr 1 dr y C/2
R =E.w ( ydy)—=E.w -y2—=E.w—ln——
3 Bl23I+y I.)- r BIL-y 2 r 2 Bl 2 + y
h2 y C/2
R =E — — 21 ——
3 ”W299 n13/2+y

Substituting the expression of:

153

 

B/2+
u/Ec = f“ —;— = fa __7__y_ into the above equations, R3 can then be written as:
C/2
R3 =___ _—
h;-( chd 1nB/2+y

The radial component of R3 is R3B.

Criteria of Failure

Failure occurs when cm at the top of the slab (under the load) reaches a limiting
value. The following expressions give an essential agreement with the test results

reported by K and N:

B I}! B
F _S . — — —: . _ . —
or O<h 2,cc,— l/lr= 1+B/2y 00035(1 022’!)

For 5>2;s,, =—3"—=o.00195
h 1+B/2y

The following empirical expressions were also reported by K and N:

fcube

E=105 5+0
(0.3 .31—5--'--O

) Equation (6)

f,

-f—--— = 2.35 Equation (7)

Here font,e is the 150 mm cube strength of concrete with unit of kilogram force per square
centimeters.

The following expression was used by Hewitt and Batchelor to convert the
compressive strength of a 6 in. diameter by 12 in. cylinder (fcyl) in units of pounds per

square inch into fcube.

154

 

 

fot

 

= t' 8
fad" O.75+0.000025f0,, 5““ 1°“ )

Accordingly, the expression for ‘1’ becomes:

B B B .
For — S 2; y! = 0.0035(1 - O.22—)(l + —) Equatron (9.a)

h h 2 y

B B .
For 71- > 2;![1 = 0.001950 + -2—y) Equatron (9.b)

The last two equations were derived on the basis of measurements of the concrete strain
in a radial direction on the top surfaces of a slab with ring reinforcement. Kinnunen later

proposed the following equation for slabs with two-way reinforcement.

 

Co B .
And h — 2h+1.8 Equatron(lO)

The expression for the stress in the conical shell, ft become as follows:

B ' B
For 0 < 71- s 2: f, = 0.0035(1— 0.227));

B
f, = 825(035 + 03%)(1 — 022;) Equation (11.a)
B
For h > 2: f, = 0.00195Ec = 460(0.35 + 0.3%) Equation (11.b)

Condition of Equilibrium

Taking moments with respect to the point of intersection of R1 and PIS/(21:) gives:

27:123(h—y/3)+-£—(h—y/3)-P(C/2-B/2)=0
tana

155

 

 

 

22m, C/2-B/2

 

=— t +——-——=K— t
P 00a (h-y/3) y COG
C/2-B/2
H K =——————
ere y (h-y/3)

 

 

 

 

 

 

 

 

 

 

 

 

y/3
Rsi3 (I,
t h T
Z
I
k——C/2 1
P B/(21t) P B/(Zn)

Figure A.3 Mechanical Model of Simply Supported Slab

Substituting the expression for P, R3, and ft/fct = 2.35 yields:

2
y B C/2 B C/2
—— —+1ln——— —+11 ——
2hf“(2y ) [rs/2”] y(2y “133/2”
27: 3: =Ky

B+2
2mm 3+;

 

 

 

—cota

 

fa
2353 B+2 —
( ”3+3,

Rearrangement of terms yields

tana(1—tana)

 

 

_1_ (B + 2y)(B+ y)1n[—§—Q—]=(Ky _ cot 60f“ = (K, — cot a)

4.7 B(B+2y) B/2+y 1+tan2a
Therefore,
l-tana 1 y C/2
Ktan -1———=—-1+-—1 _—
(Y 0’ )(1+tan2a 4.7( B)n[B/2+y]

156

 

 

The second equation of equilibrium for the sector element was obtained by taking
moments about the point of intersection of TB/(Ztt) in the conical shell and the resultant

of the forces R3 (see Figure A.3).

g(C/2—B/2)-(R,B+R2)(h-y/3)=0

 

P,B (Cl 2 - B / 2) ,B

R + R = — = P— K
(‘ﬁ 2) 27: (h—y/3) 2:: y
Therefore the second expression for P is given by:

P=2Kl:'(R. +-%—) Equation(12)

II Mechanical Model for Restrained Slabs

Hewitt and Batchelor, as shown in Figure A.4 proposed an idealized mechanical
model of a restrained slab at punching failure. The model is basically that proposed by
Kinnunen and Nylander, except a moment and force at the bottom compression
reinforcement at the boundary have been included. Mb and Fb are respectively moment n
and force per unit length of slab acting in the plane of the slab. H _'

Taking moments with respect to the point of intersection of the resultant of the

 

forces R1 and PIS/(2n) gives:

27093(h—y/3)+-t—L(h-y/3)—-P(C/2—B/2)+Mb7rC=0
ana .

 

27m,
P

b ”C
P (h—y/3)

 

 

=Ky—cota—

157

 

 

 

 

 

 

 

-Rsﬁ t

 

 

 

 

 

 

T _—
t h
F}, ’ /
I
Mb
é—-— C/2
P B/(Zn)

 

P B/(Zn)

Figure A.4 Mechanical Model of Restrained Slab

For convenience, M.,/P is replaced by X/(41t), in which X is a factor of the order unity

(assuming a circular fan failure mode the resisting moments M.,, is given by 41th).

M
X=47r—’L
P

Thus

Equation ( 13)

Then again assuming ft/fct = 2.35 and substituting equations for P and for R3, the

following equations are obtained:

2

y B C/2
-—-— -—+l 1n
B+2y
B+y

2n

 

27¢.faBy

C/2

4.7 B/2+y

1 B
+y/ ml:
(l—tana)

[(K (1+tan2a)

__L—tan _1
’ 4(h-y/3)’ 0’ 1

Deﬁne:

158

B/2+y]
=Ky—cota—X

:l=(K), —cota-

__.C__
4(h-y/3)

XC

———4(h_y/3))fa=

Equation (14)

 

 

 

Then,

(l—tana)

(1+y/B)ln C/2
(1+tan2a)

= K tana—l
4.7 B/2+y:l ( z )

Equation (15)

When M., = 0, X = 0, and K2 = Ky, then Equation (15) is identical to that for a simply
supported slab.
Taking moments about the point of intersection of TB/(21t) in the conical shell and

the resultant of the forces R3, gives

P_/3 E _1_ E-
2ﬂ(C/2—B/2)—(R,,B+R2+Fbﬂ2)(h 3) M,,62_0

13.1.31 _ _ g_ E -1
2”(C/2 B/2) XPﬂgﬂ—(R,B+R,+Fbﬂz)(h 3)

PB C/2-B/2 XC C PB
— — = R +R +F — =—K
27t( h—y/3 4(h—y/3)) (‘ﬂ 2 “62) 27: ‘

 

+ F — Equation (16)

If Fb = 0 and M., = 0, X = 0 and K, = K,, then Equation (16) is identical to Equation (12)

for a simply supported slab.

Restraint Factor

In a practical situation, the boundary restraining forces on a slab loaded to
punching failure are usually not known and would be difﬁcult to either calculate or
measure accurately, even under laboratory conditions. For these reasons a single factor, to
be termed the ‘restraint factor’, was proposed by Batchelor and Hewitt. The factor ranges

from zero for a simply supported slab to 1 (one) for a fully restrained slab.

159

 

 

 

The resultant maximum boundary stresses and forces are given in Figure A5. The
maximum tensile force, Ft, per unit length of the boundary, at the level of tensile
reinforcement is given by:

F; = hpf:y Equation (17)

If the slab is not reinforced at the boundary, F; is zero. The maximum compressive force

per unit length of the boundary in the concrete is given by
t A .
F, = k-fmax (-2- — 4) Equatron (18)

where k is the ratio of the average stress to the maximum stress in the concrete, fw and
depends on the stress distribution, and A is the central deﬂection of the slab at failure.

Assuming a parabolic distribution of stress, k = 2/3.

Ft= h p fsy
F=c k fmax (t/2- A/4)

3/8(t/2- A/4)

  

! h T
t/2- A/4

Figure A.5 Boundary Forces

Using a maximum concrete stress of 0.85 fc’, the idealized maximum boundary restraints

arethengivenby
13 3
Mb =F2h— F -— ——
“m" ‘( t) (h 16’ 32A)
Fb(unx)=F;:_Fi

160

 

 

 

 

Realizing that the maximum boundary restraints would rarely occur at punching

failure, the restraint factor, Fr was introduced. The actual boundary restraints at failure are

then given by
13 3 .
Mb = FrMMm, = F, F,(2h — t) — Fc(h — ﬁt - 33A) Equatron (19)
F, = FINN) = F,[FC — F,] Equation (20)

The slab deﬂection, A, can be approximately expressed as

C—B
=,_<___>

A
2

Equation (21)

The uncorrected theoretical punching load, P, for a restrained slab is determined by
iteration using Equations (1) through (21).
Using correction factor for dowel and membrane effects of 1.2, the following
equation was adopted:
V =1.2P
V is the corrected ultimate load of slab, and P is the load calculated without consideration

of dowel or tensile membrane effects.

List of symbol and notations

0 0t - is the angel between the horizontal plane and TB/(21t)

0 B - is the angel of sector element of the slab under the load.
0 B - is the diameter of load (column).

0 C - is the diameter of slab.

0 2c] — is the thickness of conical shell at its lower boundary.

161

 

 

 

 

2C2 — is the thickness of conical shell at its upper boundary.

Co - is the distance from the center of slab to the shear crack.

8c. - is the concrete strain at top of slab in tangential direction.

t:s - is the strain in reinforcing steel at a distance r from the center of the slab.

fc. - is the compressive stress at the top of slab in a tangential direction.

fem,e - is the 150 mm cube strength of concrete with unit of kilogram force per square
centimeters.

ft - is the compressive stress in the imaginary conical shell.

Ft, - is the boundary force at the boundary of restrained slab.

Fe - is the concrete compressive force at the slab boundary.

Ft - is the tensile force at the slab boundary.

Mb - is the boundary moment at the boundary of a restrained slab.

P - is the load (force) which is transmitted through the conical shell.

R1 - is resultant force in the ring reinforcement

R2 - is resultant force in the radial reinforcement acts at the section r = C0

R3 - is resultant force in the concrete acting in a radial crack

rs - is the radius of slab area in which the yield point stress is reached under the failure
load.

T - is oblique compressive force acting at the conical shell.

y - is the distance from the top surface of the slab to the root of shear crack at failure.
\lt - is the angle of rotation in a radial vertical plane of the slab portion adjacent to the

load outside shear crack.

162

 

 

APPENDIX B

FLOW DIAGRAM AND CODE LISTING TO CALCULATE
PUNCHING SHEAR STRENGTH OF RESTRAINED SLAB

1. Flow Diagram to Calculate Punching Strength of Restrained Slabs

 

 

 

Estimate A j

U

Calculate: Ft, Fc, Mb, and Fb

U

Estimate X ’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Estimate y/h 5 2T.

U

3 Calculate: ft, tan 0:, Pl, w, rs, and Co

 

 

 

 

 

 

 

Calculate: R1, R2, A, and P2

 

 

 

 

 

 

 

 

Calculate: X and Fe

 

 

NO

 

    

NO

 

 

Fc = Fc-old ?

 

 

 

V= 1.2P

 

163

 

11. Code Listing to Calculate the Punching Strength of a Restrained Slab

{******* PROGRAM To CALCULATE PUNCHING SHEAR STRENGTH *******}
{********* OF A RESTRAINED SLAB WITH KNOWN VALUE OF FR **********}
{***************** USING BORLAND DELPI—II VERSION 3o ****************}

{***VARIABLES DECLARATION*** }
VAR
T, H, B, C, FCCYL, FCCUB, RHO, FSY, ES, RF, BF : double;
X, PU, DEFL, PDIFF, XDIFF, FT, FC, V : double;
BM, BMPF, YONH, HOLDl, HOLD2, HOLD3, HOLD4, HOLDS : double;
HOLD6, KY, KZ, TANA, FALFA, F, SIGT, P1, P2, PSI : double;
RS, CO, Y, PERRl, PERR2, PDHOLD, YHOLD, XCHECK, XDHOLD : double;
CCHECK, R1, R2, BONH, CDIFF, PERR3 : double;
cERR1,cERR2,cERR3: integer;

{*** PROCEDURE TO CALCULATE BOUNDARY RESTRAINING FORCES ***}
Procedure CachRes;
begin
FT:=RF*FSY*H*RHO;
FC:=RF*((T/2)-(DEFI/4))*FCCYL* 1 .7/3;
BF:=FC-FT;
BM:=F'I‘*(2*H-T)-FC*(H-T*13/16-DEFL*3/32);
BMPF:=12.568*BM/1000;

end;

{*PROCEDURE TO CALCULATE THE FIRST VALUE OF PUNCHING LOAD*}
Procedure CalcPl;
begin
Y:=YONH*H;

164

HOLD] :=(1/4.7)*( l+YlB)*Ln(C/(B+2*Y));
KY:=(C-B)/(2*(H—Y/3));
KZ:=KY-X*C/(4*(H-Y/3));
HOLD2:=KZ+HOLD1 ;
HOLD3:=-KZ-1;
HOLD4:=HOLD1+1;
HOLDS:=(HOLD3*HOLD3-4*HOLD2*HOLD4);
IF HOLDS < 0 THEN TANA:=-HOLD3/(2*HOLD2)
ELSE
TAN A:=(-HOLD3-SQRT(HOLD5))/(2*HOLD2);
FALFA:=TANA*(1-TANA)/(1+TANA*TANA);
BONH:=B/H;
F:=14.22;
IF BONH < 2 THEN SIGT:=825*(0.35+FCCUB/(F*500))*(1-0.22*BONH)*F
ELSE
SIGT:=460*(0.35+FCCUB/(F*500))*F;
P1:=3. l42*BONH*YONH*(B+2*Y)*SIGT*FALFA*H*H/(B+Y);

end;

 

{*PROCEDURE TO CALCULATE THE SECOND VALUE OF PUNCHING LOAD*}
Procedure CALCP2;
begin
IF BONH < 2 THEN PSI:=0.0035*(1-0.22*BONH)*(1+0.5*B/Y)
ELSE
PSI:=0.00195*(1+0.5*B/Y);
RS:=H*ES*PSI*(1-YONH)/FSY;
CO:=0.5*B+1.8*H;
HOLD6:=RHO*FSY*H;
IF RS <= CO THEN
BEGIN

165

 

R1:=HOLD6*RS*Ln(0.5*C/CO);
R2:=HOLD6*RS;
END
ELSE
IF RS > 0.5*C THEN RS:=0.5*C
ELSE
BEGIN
R1:=HOLD6*((RS-CO)+RS*Ln(0.5*C/RS));
R2:=HOLD6*CO;
END;
DEFL:=PSI*(C-B)/2;
P2:=6.284*(R1+R2+BF*C*(H-Y/3-DEFL)/(2*(H-Y/3.0)))/KZ;
END;

{***** THE MAIN PROGRAM *at***}
_ Begin

T := StrToFloat(edT.Text);

H := StrToFloat(edB.Text);

B := StrToFloat(edB.Text);

C := StrToFloat(edC.Text);

FCCYL := StrToFloat(edFCCYL.Text);
RHO := StrToFloat(edRHO.Text);

FSY := StrToFloat(edFSY.Text);

ES := StrToFloat(edES.Text);

RF := StrToFloat(edRF.Text);
FCCUB:=FCCYIJ(0.75+0.000025*FCCYL);
DEFL:=T/4;

PU:=0;

V:=0;

X:=1.0;

166

 

YONH:=0.5;
cERRl := 0;
cERR2 := 0;
cERR3 := 0;
PERR3 := 1.1;
WHILE PERR3 > 0.1 DO
BEGIN
Inc(cERR3);
CachRes;
PERR2 := 1.1;
WHILE PERR2 > 0.1 DO
BEGIN
Inc(cERR2);
PERRI := 1.1;
WHILE PERRl > 0.1 DO
BEGIN
Inc(cERRl);
CALCPl; {*CALCULATE Pl*}
CALCP2; {*CALCULATE P2*}
{**** COMPARE P1 & P2 ****}
PDIFF:=ABS(Pl-P2);
PERR1:=100*PDIFF/P1;
{*** ITERATION WITH Y/H ***}
YONH:=YONH*0.5*(1+P2/Pl);
END;
PU:=(P1+P2)/2000;
V:=1.2 * PU;
XCI-IECK:=BMPF/PU;
{*** COMPARE X & XCI-IECK ***}
XDIFF:=ABS(X-XCHECK);

167

 

PERR2:=ABS(100*XDIFF/X);

X:=(X+XCHECK)/2;

END;
{*** CHECK COMPRESSIVE BOUNDARY FORCE ***}
CCHECK:=RF*((T/2)-(DEFU4))*FCCYL* 1 .7/3;
CDIFF:=ABS(FC-CCHECK);
PERR3:=100*CDIFF/FC;
END;

{aunt PRINT OUTPUT ***}
le.Caption := FloatToStr(PU);
le.Caption := FloatToStr(V);
Labell 1.Caption := IntToStr(cERRl);
Labe112.Caption := IntToStr(cERR2);
Labell3.Caption := IntToStr(cERR3);
Labell4.Caption := FloatToStr(YONH);
Labe115.Caption := FloatToStr(X);

end;

end.

168

 

Ir—

APPENDIX C

EXAMPLE OF FINITE ELEMENT INPUT AND OUTPUT FILES

I. Input File (ABAQUS/Standard Version 5.8)

*HEADING
M 1: Basic Model Using 8-node Elements (psi units).
**Node Deﬁnition
*N ODE
l,0.0,0.,0.
l9,36.,0.,0.
23,40.,0.,0.
51,96.,0.,0.
201 ,0.0,7.,0.
219,36.,7.,0.
223,40.,7.,0.
251,96.,7.,0.
401 ,0.0,9.,O.
419,36.,9.,0.
423,40.,9.,0.
451,96.,9.,0.
801 ,0.0, 16.,0.
819,36.,16.,0.
823,40.,16.,0.
851,96.,16.,0.
1001,0.0,20.,0.
1019,36.,20.,0.
1023 ,40.,20.,0.
1051,96.,20.,0.
1401,0.0,27.,0.

169

I!

1419,36.,27.,0.
1423,40.,27.,0.
1451,96.,27.,0.
1601,0.0,29.,0.
16l9,36.,29.,0.
1623,40.,29.,0.

165 1,96.,29.,0.

2401 ,0.0,47.,0.
2419,36.,47.,0.
2423,40.,47.,0.
2451,96.,47.,0.

2601 ,0.0,49.,0.
2619,36.,49.,0.
2623,40.,49.,0.
2651,96.,49.,0.
2801,0.0,56.,0.
2819,36.,56.,0.
2823,40.,56.,0.
2851,96.,56.,0.
*NGEN, NSET=NL1
1,201,100
*NGEN, N SET=NL2
201,401,100
*NGEN, NSET=N L3
401,801,100
*NGEN, NSET=NL4
801,100] , 100
*NGEN, NSET=NL5
1001, 1401 , 100
*NGEN, NSET=NL6
1401,1601 ,100

170

 

*NGEN, NSET=NL7
1601,2401,100

*NGEN, N SET=NL8
2401,2601,100

*NGEN, NSET=NL9
2601,2801,100

*NSET, NSET=NL
NL1,NL2,NL3,NL4,NL5 ,NL6,NL7,NL8,NL9

*NGEN, NSET=NMIDI
19,219,100

*NGEN, NSET=NMID2
219,419,100

*NGEN, NSET=NMID3
419,819,100

*NGEN, NSET=NMID4
819,1019,100

' *NGEN, NSET=NMIDS
1019,1419,100

*NGEN, NSET=NMID6
1419,1619,100

*NGEN, NSET=NMID7
1619,2419,100

*NGEN, N SET=NMIDS
2419,2619,100

*NGEN, NSET=NMHD9
2619,2819,100

*NSET, NSET=NMIDL

NMIDl, NMIDZ, NMID3, NMID4, NMIDS, NMID6, NMID7, NMID8, NMI9

*NGEN, NSET=NMIR1
23,223,100

*NGEN, NSET=NMIR2

171

 

223,423,100
*NGEN, NSET=NMIR3
423,823,100
*NGEN, NSET=NMIR4
823,1023, 100
*NGEN, NS ET=NMIR5
1023,1423,100
*NGEN, NSET=NMIR6
1423,1623,100
*NGEN, NSET=NMIR7
1623,2423, 100
*NGEN, NSET=NMIR8
2423,2623,100
*NGEN, N SET=NMIR9
2623,2823,100
*NSET, NSET=NMIDR
NMIR 1,NMIR2,NMIR3,NMIR4,NMIR5,NMIR6,NMIR7,NMIR8,NMI9
*NGEN, NSET=NR1
5 1,251 , 100
*NGEN, NS ET=NR2
25 1 ,45 1,100
*NGEN, NSET=NR3
45 1,85 1,100
*NGEN, N SET=NR4
85 1 ,105 1, 100
*N GEN, NSET=NR5
1051,1451,100
*NGEN, NSET=NR6
1451,1651,100
*NGEN, NS ET=NR7
1651,2451,100

172

 

f

"‘5 Out-

*N GEN, NSET=NR8
2451,2651,100

*NGEN, NSET=NR9
2651,2851,100

*NSET, NSET=NR

NR1 ,NR2,NR3,NR4,NR5,NR6,NR7,NR8,NR9
*NFILL, NSET=SLEFI‘
NL,NMIDL, 18,1
*NFILL, NSET=SMID
NMIDL,NMIDR,4,1
*NFILL, NSET=SRIGHT
NMIDR,NR,28,1
*NSET, NSET=STOP
SLEFT,SMID,SRIGHT
*NODE

50301 ,0.0,8.,-4.6
50319,36.,8.,-4.6 ’
50323,40.,8.,46
50351,96.,8.,-4.6
51501,0.0,28.,-4.6
51519,36.,28.,—4.6
51523,40.,28.,-4.6
51551,96.,28.,-4.6
52501 ,0.0,48.,-4.6
52519,36.,48.,-4.6
52523,40.,48.,46
52551,96.,48.,-4.6
*NGEN, NSET=WEBL1
50301,50319,1

*NGEN, N SET=WMID1
50319,50323,1

173

 

*NGEN, NSET=WEBR1

50323,5035 1,1
*NGEN, NSET=WEBL2

5]501,51519,l
*NGEN, NSET=WMID2

51519,51523,1

*N GEN, NSET=WEBR2

51523,51551,1

*NGEN, NSET=WEBL3

52501,52519,1

*NGEN, NSET=WMID3

52519,52523,1

*NGEN, NSET=WEBR3

52523,52551,1

*NCOPY, CHANGE NUMBER=40000, OLD SET=STOP, NEW SET=SBOT,
REFLECT=MIRROR

0.,0.,-0.8,0.,1.,-0.8

1.,0.,-0.8

*N SET, NSET=FTOPL
40201,40301,40401,41401,41501,41601,42401,42501,42601
*NSET, NSET=FI'OPR
40251,40351,40451,41451,41551,41651,42451,42551,42651
*NCOPY, CHANGE NUMBER=20000, OLD SET=FTOPL, NEW SET=FB OTL,
REFLECT=MIRROR

0.,0.,-4.6,0.,1.,-4.6

l.,O.,-4.6
*NCOPY, CHANGE NUMBER=20000, OLD SET=FTOPR, NEW SET=FB OTR,
REFLECT=MIRROR

0.,0.,-4.6,0.,] .,-4.6

1.,0.,-4.6
*NFILL,NSET=FB OT

174

 

FBOTL,FBOTR,50,]
**End of Node Deﬁnition
**Start Element Deﬁnition

**Reinforced Concrete Element for Slab
*ELEMENT, TYPE=C3D8, ELSET=SLABTOP
1,40001,40002,40102,40101,1,2,102,101
*ELGEN, ELSET=SLABTOP

l,50,1,1,28,100,100
*SOLID SECTION, ELSET=SLABTOP, MATERIAI;RC
*MATERIAL, NAME=RC
*ELASTIC

3.6E6,0.2
*CONCRETE

500.,0.

4000.,0.003
*FAILURE RATIOS

1.25,0.l
*TENSION STIFFENING

1.,0.

0.,0.0025
*REBAR, ELEMENT=CONTINUUM, MATERIAI;WWM,
GEOMETRY=ISOPARAMETRIC, NAME=YYBOT
SLABTOP,.00333,].,0.,0.1875,4,2
*REBAR, ELEMENT=CONTINUUM, MATERIAI;WWM,
GEOMETRY=ISOPARAMETRIC, NAME=XXB OT
SLABTOP,.00333,].,0.,0.1875,1,l
*REBAR, ELEMENT=CONTINUUM, MATERIAI;WWM,
GEOMETRY=ISOPARAMETRIC, NAME=YYTOP
SLABTOP,.00333,].,0.,0.8125,4,2

175

*REBAR, ELEMENT=CONTINUUM, MATERIAIFWWM,
GEOMETRY=ISOPARA1VIETRIC, NAME=XXT OP
SLABTOP,.00333,].,0.,O.8125,1,1

*MATERIAL, NAME=WWM

*ELASTIC

29.E6

*PLASTIC

72.5E3

**Shell Element for Steel Girders

*ELEMENT, TYPE=S9R5, ELSET=TOPFL1

4000] ,40201 ,40203 ,40403,4040 1 ,40202,40303 ,40402,40301 ,40302
*ELGEN, ELSET=TOPFL1

40001,25,2,1,

*SHELL SECTION, ELSET=TOPFL1, MATERIAI.=A36ST1

0. l7 1 ,

*MATERIAL, NAME=A36ST1

*ELASTIC

29.E6

*ELEMENT, TYPE-=S9R5, ELSET=TOPFL2
40101,41401,41403,41603,41601,4]402,41503,41602,41501,41502
*ELGEN, ELSET=TOPFL2

40101,25,2,1,2,1000,100

*SHELL SECTION, ELSET=TOPFL2, MATERIAL=A36ST2
0.171,

*MATERIAL, NAME=A36ST2

*ELASTIC

29.E6

*ELEMENT, TYPE=S9R5, ELSET=BOTFL1
60001,60201,60203,60403,60401,60202,60303,60402,60301,60302
*ELGEN, ELSET=BOTFL1

176

 

60001,25,2,1,

*SHELL SECTION, ELSET=BOTFL1, MATERIAL=A36ST3
0. 171 ,

*MATERIAL, NAME=A36ST3

*ELASTIC

29.E6

*ELEMENT, TYPE=S9R5, ELSET=BOTFL2
60101,6140],61403,61603,61601,61402,61503,61602,6]501,61502
*ELGEN, ELSET=B OTFL2

60101,25,2,1,2,1000,100

*SHELL SECTION, ELSET=BOTFL2, MATERIAL=A36ST4
0. 171,

*MATERIAL, NAME=A36ST4

*ELASTIC

29.E6

*ELEMENT, TYPE=S9R5, ELSET=WEB]
41001,60301,60303,40303,40301,60302,50303,40302,50301,50302
*ELGEN, ELSET=WEB]

41001,25,2,1,

*SHELL SECTION, ELSET=WEB 1, MATERIAL=A36ST5
0.1 14,

*MATERIAL, NAME=A36ST5

*ELASTIC

29.E6

*ELEMENT, TYPE=S9R5, ELSET=WEB2
41101,61501,61503,41503,41501,61502,51503,41502,51501,51502
*ELGEN, ELSET=WEB2

41101,25,2,1,2,1000,100

*SHELL SECTION, ELSET=WEB2, MATERIAL=A36ST6
0.] 14,

*MATERIAL, NANIE=A36ST6

 

177

 

*ELASTIC
29.E6

**Beam Element for Edge Beams
*ELEMENT, TYPE=B3 l , ELSET=EDGEBM
50001 ,40401 ,40501

*ELGEN, ELSET=EDGEBM
50001,2,50,100,18,100,1

*BEAM SECTION, SECTION=RECT AN GULAR, ELSET=EDGEBM,
MATERIAL=STEDGEBM

2.758,0.95

*MATERIAL, NAME=STEDGEBM
*ELASTIC

29.E6

**Truss Element for Tie Rods

*ELEMENT, TYPE=T3D2, ELSET=TIEROD]

5 1001 ,40403,41403

*ELGEN, ELSET=TIEROD1

5 1001,5,4,1

*SOLID SECTION, ELSET=TIEROD], MATERIAL=STROD1
0.25

*MATERIAL, NAME=STROD1

*ELASTIC . (
29.E6

*ELEMENT, TYPE=T3D2, ELSET=TIEROD2
51101,40425,41425

*ELGEN, ELSET=TIEROD2

5] 101,7 ,4,l

*SOLID SECTION, ELSET=TIEROD2, MATERIAL=STROD2
0.25

 

178

 

*MATERIAL, NAME=STROD2
*ELASTIC

29.E6

*ELSET, ELSET=TIERODS
TIEROD], TIEROD2

**Beam Element for Diaphragms

*ELEMENT, TYPE=B31, ELSET=DIAPH1

52001,50301,51501

*BEAM SECTION, SECTION=RECTANGULAR, ELSET=DIAPH1,
MATERIAL=STDIAP]

0.95,2.758

*MATERIAL, NAME=STDIAP1

*ELASTIC

29.E6

*ELEMENT, TYPE=B3], ELSET=DIAPH2

52002,51501,52501

*BEAM SECTION, SECTION=RECTANGULAR, ELSET=DIAPH2,
MATERIAL=STDIAP2

0.95,2.758

*MATERIAL, NAME=STDIAP2

*ELASTIC

29.E6

*ELEMENT, TYPE=B3], ELSET=DIAPH3

52003,50327,51527

*ELGEN, ELSET=DIAPH3

52003,2,24,1,

*BEAM SECTION, SECT ION=RECT AN GULAR, ELSET=DIAPH3,
MATERIAL=STDIAP3

0.95,2.758

*MATERIAL, NAME=STDIAP3

179

 

*ELASTIC

29.E6

*ELEMENT, TYPE=B3 1, ELSET=DIAPH4
52005,51527,52527

*ELGEN, ELSET=DIAPH4

52005,2,24,1,

*BEAM SECTION, SECTION=RECT AN GULAR, ELSET=DIAPH4,
MATERIAL=STDIAP4

0.95,2.758

*MATERIAL, NAME=STDIAP4
*ELASTIC

29.E6

**Boundary Conditions
*BOUNDARY
6020],3
60301,3
6040],3
61401,3

6150] ,1,3
61601,3
62401,3
62501,3
62601,3
60251,3
60351,3
60451.,3
61451,3
61551,2,3
61651,3
62451.3

180

 

62551,3
62651,3

**Step Deﬁnition and Output Request
*RESTART, WRITE

*STEP, INC=100

*STATIC, RIKS
0.1,l.,,,,40921,3,-0.55

**Load Deﬁnition

*ELSET, ELSET=LOADEDEL
819,820,919,920

*DLOAD

LOADEDEL,P2,2000.

*EL PRINT, ELSET=TIERODS

E

*NSET, NSET=NODALDISP

4092 1 ,6152 l

*NODE PRINT, NSET=NODALDISP
U

*MONITOR, NODE=40921, DOF=3
*END STEP

181

 

 

II. ABAQUS/Standard Status File

SUMMARY OF JOB INFORMATION:

MONITOR NODE: 40921 DOF: 3

STEP INC ATT SEVERE EQUIL TOTAL STEP INC OF DOF IF
DISCON ITERS ITERS TIME/LPF TIME/LPF MONITOR RIKS

 

ITERS
1 1 l 0 3 3 0.0999 0.09995 -0.0264 R
1 2 1 0 6 6 0.198 0.09815 -0.0553 R
1 3 1 0 6 6 0.292 0.09364 -0.0883 R
1 4 1 0 5 5 0.381 0.08966 -0.124 R
1 5 2 0 5 5 0.403 0.02197 -0.133 R
1 6 1 0 5 5 0.425 0.02177 -0. 142 R
1 7 1 0 5 5 0.447 0.02160 -0.151 R
1 8 2 0 3 3 0.452 0.005430 -0.154 R
1 9 1 0 4 4 0.458 0.005385 -0.156 R
1 10 2 0 4 4 0.461 0.003041 -0.157 R
1 1] -1 0 5 5 0.465 0.004513 -0.159 R
1 12 2 0 3 3 0.466 0.001 149 -0. 160 R
l 13 1 0 3 3 0.467 0.001 141 -0.160 R
1 14 ] 0 3 3 0.469 0.001694 -0.161 R
1 15 2 0 3 3 0.470 0.0009607 -0. 161 R
1 l6 1 0 4 4 0.471 0.001427 -0.162 R
1 17 2 0 3 3 0.472 0.0008104 -0.162 R
1 l8 1 0 5 5 0.473 0.001204 -0.163 R
1 19 2 0 3 3 0.474 0.0003035 -0.163 R
1 20 l 0 3 3 0.474 0.0003035 -0.163 R
1 21 1 0 3 3 0.475 0.0004545 -0. 163 R
1 22 1 0 4 4 0.475 0.0006827 -0.164 R
1 23 2 0 3 3 0.476 0.0003859 -0.164 R
1 24 2 0 3 3 0.476 0.0002182 -0.164 R
1 25 1 0 3 3 0.476 0.0003228 -0.164 R
1 26 2 0 3 3 0.476 0.0001816 -0. 164 R
1 27 1 0 4 4 0.477 0.0002716 -0.164 R
1 28 2 0 3 3 0.477 0.0001533 -0.164 R
1 29 l 0 4 4 0.477 0.000231 1 -0. 164 R

182

 

IH. ABAQUS/Plot Output

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algae—666% t t t....t.t....t.t.

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tuttiilltllrll

183

Figure C.1 Ma '
grnfied Displaced Shape of FE Model f th
0 e Bridge

wvumvumo “NEH? OOONISIVN "N93 Hum.m 3&0ng mboa
ﬂmv.o NEHB QEB‘ADSDUU‘ A169”... magic QHBM “Hm—h. 2H DEAﬁSOU M.,—AH?
NH BSD—4H .n Aﬂam NE n HAHW BEE“:

H

 

nooﬂnn.wb
noeﬂhu.ve
«oeunn.no
doenuh.u¢
nooMuw.at

noeﬂmm.mr
novﬂnm.hl
noqﬂmv.MI
n94MvH.HI
nooMVM.HI
naoﬂnm.«:
noelnh.an
no;ﬂ«m.«n
NDA<>

nooﬂnu.nv!

 

 

 

Figure C.2 Contour of S33, Stress in 3-direction

184

mvnmv.mo an. OOONISIGN .MB‘O Him...“ .ZOHmdg mbog

 

ﬂmv.o mud-H.H. DEBEEDUU‘ #5608 nmv.o mam MHIB 2H Bahama-OD MIHB
NH EOE H mam NE n NAHh BN‘BMNM H

 

mo:lom.ne
nolﬂvn.no
Hannah.«e
novlvv a. !
nail-o.ﬁv
vo:ﬂau.he «
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votmvn.nl I
corn-dtwl
noIﬂvo.ﬁI

norHo- «I
mollmh.nt
n0:l0d.«t

Hadl> nnﬂ

 

 

 

Figure C.3 Contour of E33, Strain in 3-direction

185

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

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