Model-Based Design of a
Ultra High Performance Concrete
Prototype Highway Bridge Girder
By
Melvin Soh
B.S. Civil Engineering (2002)
Columbia University in the City of New York
Submitted to the Department of Civil and Environmental Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Civil and Environmental Engineering
at the
Massachusetts Institute of Technology
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
J UENO 2 20 03
June 2003
OL2ARE
D 2003 Massachusetts Institute of Technology
All rights reserved
LIBRARIJES
Signature of A uthor........................................
. .......... ....... ....................
Department of Civil and Envi nmental Engineering
May 21, 2003
C ertified b y ......................................................
...............................
Franz-J
Ulm
Associate Professor of Civil and Environmenta ngineering
Thesis Supervisor
Accepted by ...........................
..
.
;0
Oral Buyukozturk
Chairman, Departmental Committee on Graduate Studies
BARKR
Model-Based Design of a Ultra High Performance Concrete
Prototype Highway Bridge Girder
By
Melvin Soh
Submitted to the Department of Civil and Environmental Engineering
on May 9, 2003 in partial fulfillment of the
requirements for the Degree of Master of Engineering in
Civil and Environmental Engineering
Abstract
In light of the current situation of highway bridge girders today, the Federal Highway
Administration (FHWA) is constantly seeking to develop new bridges that can meet and adapt to
current and future traffic, minimize environmental impact, require low-maintenance and are
economical and easy to install. An Ultra High Performance Concrete (UHPC) bridge girder can
meet such goals. It requires less material than ordinary reinforced concrete bridges owing to its
considerable mechanical properties, is more durable and easier to install due to the presence of
short fibers randomly oriented throughout the material mitigating the need for reinforcing bars.
Already developed here at M.I.T. is the two-phase constitutive model for UHPC and the
validation thereof through shear and flexural.tests. Preliminary designs have optimized the shape
and section heights of the girders as well as their behavior in transferring pre-stressing stresses.
What remains to be done is to refine existing cross-sectional designs and investigate new crosssectional designs, especially with reference to slab-thickness and recessed web distances, before a
prototype bridge can be built, tested and monitored at the FHWA Turner-Fairbank Highway
Research Center in McLean Virginia. Thus the focus of this work is on this last aspect of the
design phase of UHPC prototype highway bridge girders.
Designs of the highway bridge girders were conducted with the aid of a finite element program
that has been verified for the material, UHPC, in question. The designs were assessed against
service and ultimate limit state crack-opening criterions. A 2-D model was first analyzed to
determine global flexural behavior of the beams, followed by a 3-D model to examine localized
strain effects. The results of this work showed that a girder, with a continuous 4-inch top-slab
satisfactorily met all performance criteria under both ultimate and service limit state, while other
girders with smaller slab thickness fail to meet all these requirements simultaneously. However, a
girder with recessed webs, passed the ultimate limit state, and failed the service limit state
marginally.
From the results of this report we can conclude that, a 4-inch top slab is sufficient to meet all
service limit state performance criteria. Should the minute cracks developed under service loads
be acceptable under the service limit state, a 2-inch top-slab between webs can be employed only
if the webs are recessed inwards to minimize transverse moments.
Thesis Supervisor: Franz-Josef Ulm
Title: Associate Professor of Civil and Environmental Engineering
Contents
C hap te r I ........................................................................................................................................................ 6
1. 1 Industrial Context ................................................................................................................................. 6
1.2 Background and Research Significance ................................................................................................ 9
1.3 Thesis Objective and Approach .......................................................................................................... 10
1.4 Outline of Report ................................................................................................................................
C h apter 2 ......................................................................................................................................................
2 .1 Intro d uctio n ........................................................................................................................................
2 .2 D im en sion s .........................................................................................................................................
2 .2 .1 G ird er I ........................................................................................................................................
2 .2 .2 G ird er 2 ........................................................................................................................................
2.2.3 Girder 3 "Closed-Section .............................................................................................................
2.2.4 Girder 4 "Open-Section ...............................................................................................................
2 .2 .5 W idth ...........................................................................................................................................
II
13
13
14
14
14
16
17
18
2 .3 L o ad s .................................................................................................................................................. 19
2.3.1 Load M odifier, )7 ........................ I................................................................................................ 19
2 .3 .2 D e ad L o ad s .................................................................................................................................. 2 0
2 .3 .3 L iv e L o ad s ................................................................................................................................... 2 0
2.3.4 Load Factors ................................................................................................................................ 23
2.3.5 Determ ination of M aximum M oments ........................................................................................ 4
2.4 Pre-Stressing Param eters .................................................................................................................... 25
2.5 1 -D Think m odel of the UHPC M aterial Behavior ............................................................................. 26
2.6 Design Criteria .................................................................................................................................... 30
2.6.1 Service Lim it State ...................................................................................................................... 30
2.6.2 Ultim ate Lim it State .................................................................................................................... 30
2.6.3 M odeling of Loads ....................................................................................................................... 31
Ch ap ter 3 ...................................................................................................................................................... 3 3
3 .1 In trod u ction ........................................................................................................................................ 3 3
3.2 2-D Section M odeling ......................................................................................................................... 34
3 .3 M e sh ................................................................................................................................................... 3 6
3.4 Results of 2-D Analysis ...................................................................................................................... 37
3 .4 .1 O v erv iew ..................................................................................................................................... 3 7
3.4.2 Flexural Resistance ...................................................................................................................... 39
3.5 Summ ary of 2-D analysis ................................................................................................................... 42
h
C ap te r 4 ...................................................................................................................................................... 4 3
4 .1 Intro d uctio n ........................................................................................................................................ 4 3
4.2 Design Requirem ents Revisited .......................................................................................................... 44
4.2.1 Dimensions .................................................................................................................................. 44
4.2.2 Boundary Conditions ................................................................................................................... 45
4 .2 .3 L o ads ........................................................................................................................................... 4 6
4 .2 .4 L im it S tate s .................................................................................................................................. 4 8
4 .2 .5 M e sh ............................................................................................................................................ 4 9
4 .3 D efl ection s .......................................................................................................................................... 5 1
4.4 Flexural Resistance ............................................................................................................................. 53
4.4.1 Service Loads .............................................................................................................................. 53
4.4.2 Ultimate Lim it State .................................................................................................................... 55
4.4.3 Crack Pattern ............................................................................................................................... 58
C hap te r 5 ...................................................................................................................................................... 6 0
5 .1 C on c lusio n .......................................................................................................................................... 6 0
5 .2 O u tlo o k ............................................................................................................................................... 6 3
3
List of Figures
7
Figure 1-1 Annual vehicle miles of highway travel, VMT(199-2000) [2] ................................................
Figure 1-2 Comparison of flexural strengths of UHPC (DUCTAL TM) and conventional concrete (HPC) [3]9
15
Figure 2-1 Cross Section G irder 1 ................................................................................................................
15
Figure 2-2 Cross - Section G irder 2 .............................................................................................................
16
Figure 2-3 Cross-Section Girder 3 "Closed-Section"................................................................................
17
Figure 2-4 Cross-Section Girder 4 "Open-Section" ..................................................................................
18
Figure 2-5 3-d view of one and a half bridge girders.................................................................................
22
Figure 2-6 HS-20 Truck Load Specifications............................................................................................
25
Figure 2-7 Axle location to corresponding to maximum moments .........................................................
27
Figure 2-8 l-d think model of a two-phase matrix-fiber composite material [5]......................................
28
Figure 2-9 Stress-strain response of the two-phase composite model [4].................................................
28
Table 2-3 Values of "effective" UHPC model input parameters..............................................................
34
Figure 3-1 Cross Section of one and a half girders...................................................................................
35
Figure 3-2 E quivalent 2-D G irder.................................................................................................................
37
F ig ure 3 -3 2 -D M esh ....................................................................................................................................
Figure 3-4 2-d Model with boundary conditions and axle loads (a) initial shape of 2-d model (b) deflected
37
shape of 2-d model, dotted lines represent initial shape ..................................................................
38
Figure 3-5 Maximum principal plastic strain contours of 2-d model at ULS ............................................
Figure 3-6 Graph of cumulative load vs. maximum principal plastic strain, for 2-d ULS load conditions.. 38
Figure 3-7 Graph of normalized ULS live load vs. normalized maximum plastic strain .......................... 39
Figure 3-8 Graph of girder height, H (in) vs. uniaxial stress in the longitudinal direction, axx (MPa, tension
41
+), 2-d m odel U L S loading ...................................................................................................................
44
Figure 4-1 3-d Wire-frame of half girder section being analyzed ............................................................
45
Figure 4-2 Boundary conditions applied on 3-d half girder.....................................................................
47
Figure 4-3 Location of applied truck wheel load at mid-span, load condition 1 .......................................
Figure 4-4 Location of applied truck wheel load at the stress-free edge, load condition 2........................ 48
50
Figure 4-5 3-d M esh em ployed on G irder 1 .............................................................................................
50
Figure 4-6 3-d m esh employed on G irder 4..............................................................................................
52
Figure 4-7 Deflected shape 3d girder under pre-stressing loads only........................................................
52
Figure 4-8 Deflected shape 3-d girder under SLS loads ............................................................................
Figure 4-9 Graph of Normalized SLS Live Load versus Normalized Max Plastic Strain (Load Condition 1)
..............................................................................................................................................................
54
Figure 4-10 Graph of Normalized SLS Live Load versus Normalized Max Plastic Strain (Load Condition
2 ) ...........................................................................................................................................................
54
Figure 4-11 Graph of Normalized Live Loads ULS vs. normalized max plastic strain, Girder 2, Load
C o nd itio n 1 ...........................................................................................................................................
57
Figure 4-12 Graph of normalized ULS live loads against normalized max plastic strain, Girder 4, Load
c o nd itio n 1 ............................................................................................................................................
57
Figure 4-13 m ax plastic strain location at SLS.........................................................................................
58
59
Figure 4-14 Maximum plastic strain location at ULS................................................................................
4
List of Tables
Table 2-1 Summ ary of dimensional characteristics of girders..................................................................
18
Table 2-2 Summary of Loads and Load Factors as per AASHTO-LRFD guidelines [1]........................ 24
Table 3-1 Maximum longitudinal compressive stress as a ratio of the allowable compressive stress.......... 42
5
Chapter 1
INTRODUCTION
1.1 Industrial Context
Highways are an integral component of our Nation's infrastructure. They allow goods
and services to flow freely between states and enhance our quality of life by enabling greater
distances to be reached in a shorter amount of time.
Vehicle miles of travel in the United States have been steadily increasing at a rate of
about two percent each year (Fig. 1-1) [6]. Vehicle-miles of travel and the number of vehicles
using these roads and streets have risen 80% and 39.8% respectively for the period between 19802000 [6]. However, road and street mileage for this same period has increased by only 2% [8].
This disproportionate growth in transportation demand versus supply is a result of both
economical and ecological reasons. Furthermore, 29% of the Nation's estimated 585,542 bridges
are structurally deficient or functionally obsolete [6]; and 23% of the 130,224 bridges on the
National Highway System are structurally deficient or obsolete [12]. Since there exists a strong
correlation between the Nation's Gross Domestic Product and Vehicle-Miles of Travel, these
6
inadequacies in our Nation's transportation infrastructure not only cause increases in wear-andtear of vehicles, fuel consumption and emissions but also directly affect our Nation's economy.
Failure to properly maintain and replenish our nation's supply of highways would lead to frequent
pollution
congestions, interruptions in the flow of goods and services, cause greater atmospheric
and affect public safety. In order to mitigate these effects, it was suggested that the next
generation of highway bridges address these issues in the following manner [6]:
*
Longer life spans with higher durability and less maintenance;
*
Adaptability to new traffic;
"
Improved reliability and safety;
*
Environmental friendliness;
"
Easier and faster construction;
E VMT, in billion miles
~~-~---~~-
3,1000
2,500
2,000
1,500
1,000
500
1992 1993 1994 1995 1996 1997 1998 1999 2000
Figure 1-1 Annual vehicle miles of highway travel, VMT(199-2000) [2]
7
A means of achieving the above stated objectives is to employ newly developed materials
available to the construction industry. Since concrete is the most commonly adopted material
used in the construction of bridges, 60% of all US highway bridges and 40% of deteriorated
bridges are made of concrete, it makes practical sense that an improved concrete material replace
current concrete materials as the material of choice in achieving the stated goals.
One such concrete material capable of achieving such goals is Ultra-High Performance
Concrete (UHPC). UHPC is a new generation of fiber-reinforced cementitious material, an ultra
high strength concrete reinforced with organic or steel fibers. UHPC possesses ultra high
strength, exceptional durability and ductility. It is composed of Portland cement, silica fume,
crushed quartz, sand, fibers, superplasticizers and water. Short fibers, with typical lengths of
13mm and diameters of 0.2mm, are randomly orientated and distributed throughout the material.
This has enabled UHPC to achieve considerable mechanical properties. A typical UHPC material
has a design compressive strength of
f
200MPa (29ksi) and a ductile tensile strength of
f
=
10 - 15MPa (1.5 - 2.2 ksi). Thus section sizes can be considerably reduced in comparison to
ordinary steel-reinforced concrete structures today. Figure 1-2 highlights the exceptional flexural
strength of UHPC against normal High Performance Concrete (HPC). UHPC structures are
ductile in behavior and are able to carry loads even after first cracking. This was realized through
the careful understanding of fiber-matrix interactions that enabled well-dispersed micro cracking
to occur unlike that of ordinary concrete. This ductility has allowed UHPC structures to be built
without reinforcing bars or shear-stirrups. This provides for the ease of installation and the
manufacture of UHPC structures. These characteristics, along with the inherent durable nature of
UHPC materials, allow UHPC structures to be employed economically, quickly, efficiently and
with minimum environmental impact.
8
50
-Ductaf:
-- HPC
S45
40
3>
30
40
0
m 15
---
-
Level of normal serv ice loads
~10
W .
0
0-0
DefIectlon (m m)
Tm
Figure 1-2 Comparison of flexural strengths of UHPC (DUCTAL ) and conventional concrete
(HPC) [3]
1.2 Background and Research Significance
UHPC materials have been used in various structural applications. Examples of UHPC
materials in structural applications include: the Sherbrooke Bridge in Quebec, Canada, which was
built in 1997, a pedestrian bridge of 60m (190ft) with a deck thickness of only 3cm (1.25in); The
footbridge of Peace in Seoul, Korea, built in 2002. The span of this bridge is 120m (400ft) and its
deck thickness is 3cm (1.25in).
The Federal Highway Administration (FHWA) in the United States has conducted some
research into the bridge technology, part of which focuses on implementing the use of UHPC
materials. Large-scale structural UHPC beam girder tests were performed at the FHWA TurnerFairbank Highway Research Center in McLean, Virgnia (2001-2002). These girders were found
to possess greater flexural and shear strengths than ordinary girders even in the absence of
9
reinforcing bars and shear stirrups. Having accomplished this, the FHWA intends to design and
build a highly innovative and optimized highway bridge structure using UHPC on geosynthetic
reinforced soil (GRS) abutments currently in place on the Turner-Fairbank campus in Virginia
where it will be periodically loaded and monitored for a period of five years prior to conducting
an ultimate load test.
Developed in recent years here at M.I.T. is a two-phase constitutive model for UHPC by
Chuang and Ulm [4]. This model enables model-based simulation of UHPC structures in a two
and three-dimensional environment by means of a finite element program (CESAR-LCPC).
Validation of this model has been achieved through verification with the FHWA flexure and
shear test [7,9]. This has allowed one to simulate global structural behavior as well as in-situ local
material behavior of UHPC structures without the need for laboratory testing. Model-based
simulations developed have thus provided us with an adequate engineering tool for structural
design. Using this tool, preliminary design and optimization of the highway bridge girders to be
built has been completed by Park et al. [13] along with slab-joint position and prestressing
transfer analysis.
1.3 Thesis Objective and Approach
Building on the mentioned highway bridge designs, the objective of this work is to refine
the design of UHPC double-tee bridge girders for the prototype structure to be installed on the
Turner-Fairbank campus (VA). The cross-section dimensions of the double-tee bridge girders are
further optimized with changes made to slab thickness and web locations, in order to determine if
they were capable
of carrying prescribed American Association of State Highway and
Transportation Officials (AASHTO) traffic loads. This bridge is to be 21.4m (70ft) long and
10
possess a deck of not more than 10.2cm (4in) thick. Analysis of the bridge will be simulated in a
computer environment by means of a finite element program, CESAR-LCPC, verified for this
material [4].
To determine structural serviceability and adequacy a design criteria must be adopted. In
contrast to typical reinforced concrete design criteria, where a maximum material stress level
determines the limit states, UHPC design criteria must be based on a maximum allowable crack
opening criteria, below which, the UHPC material achieves full ductile tensile strength with high
confidence. In this work, we employ the crack opening criteria issued by the French Association
of Civil Engineering (AFGC') [10] to define both the service limit state, and the ultimate limit
state.
The maximum allowable stress level defined will be transformed into an allowable
maximum plastic strain that is linked with crack opening. Maximum principal plastic strains of
UHPC bridge girder sections, after they have been loaded to their corresponding limit states, will
be analyzed to determine if they measure up to the stipulated performance criteria such as "nocracking" during the service limit state (SLS), and a prescribed maximum crack width for the
ultimate limit state (ULS).
1.4 Outline of Report
This report is divided into five chapters. It begins with an overview of the design
requirements in Chapter 2; the design space under consideration, design loads applied onto the
bridge, pre-stressing levels applied and the design criteria defining both the SLS and the ultimate
'Association Frangaise de Gdnie Civil
11
limit state.
Chapter 3 then progresses to describe the 2-d simulation employed to determine
overall structural feasibility of the various sections under ultimate limit state conditions. Chapter
4 describes the 3-d simulations performed to determine both structural performance in the SLS
and the ULS. Chapter 5 summarizes the findings and structural feasibility of the sections
investigated.
12
Chapter 2
DESIGN REQUIREMENTS
2.1 Introduction
This chapter introduces the general design requirements of the bridge girders and is
applicable to both 2-d and 3-d analysis. It specifies the shape and sizes of the girder designs going
into details with regard to the varying slab thickness and overhang distance. The bridge girder
under consideration is 21.4m (70ft) long and 2.4m (8ft) wide. Three of these bridge girders,
placed side by side, provide enough width for two lanes of traffic. Thus the overall size of the
bridge is 21.4m (70ft) by 7.2m (24ft). A discussion on AASHTO [1] loading requirements on a
bridge of this type will be carefully explained followed by a description of the design criteria used
to define successful performance under service loading conditions and ultimate loading
conditions. Pre-stressing forces applied to mitigate deflections under self-weight and how this is
attained will be explained. Lastly, parameters used to define the UHPC material, DUCTALTM
will be made clear.
13
2.2 Dimensions
sections under investigation in this project all possess similar general
The double-tee
characteristics. The length 21.34m (70ft), height 0.84m (33in) and width 2.44m (8ft) of all the
girders are the same while the interior dimensions, such as top-slab thickness, differ.
2.2.1 Girder 1
Girder 1 (Fig. 2-1) is a typical double-tee section possessing a 5.1cm (2in) top-slab. Top-slab
thickness, remains the same for this section. Its height is 0.8382m (33in) and bottom flange width
is 0.305m (12in) per flange. Web thickness, T., varies along the web, from 5.1cm (2in) on the
bottom to 7.62cm (3in) at the junction between the web and the top-slab. The outer top-slab
thickness, T, increases from 5.1cm (2in) at the stress-free edge to 6.35cm (2.5in) at the junction
with the web. Likewise, bottom flange thickness, 15.2cm (6in), increases to 16.5cm (6.5in) at the
junction with the web. This design is common amongst all the other 3 sections in this project.
2.2.2Girder 2
Girder 2 (Fig. 2-2) is very similar to Girder 1 in design, however it possesses a continuous 4-inch
top-slab, T7,
and T 2, instead of 2 inches. It has similar design details with girder 1 at junctions
between the top-slab and the web and the bottom-flange and the web.
14
Ts2
Tw2
B9
-
H
p 4
H
T
Tw
B2
Bfj
Figure 2-1 Cross Section Girder 1
Tw2
T,2
B
TT
Bfl
Figure 2-2 Cross - Section Girder 2
15
2.2.3 Girder 3 "Closed-Section"
Girder 3 (Fig. 2-3) is a cross between girder one and two. The outer span of the top-slab, TI, has
thickness of 10.2cm (4in) while the inner span, T,2, has thickness of 5.1cm (2in). Design details at
the junctions are similar to that of Girder 1.
T
B
B,
H
f,
Figure 2-3 Cross-Section Girder 3 "Closed-Section"
16
2.2.4 Girder 4 "Open-Section"
Girder 4's unique characteristic (Fig. 2-4) is its recessed bottom-flange and web of
distance, R= 22.9cm (9in). The outer span of its top-slab is 10.2cm (4in) thick while the inner
span is 5.08cm (2in) thick much like Girder 3. Design details at the junctions remain similar to
that of Girder 1. Table 1 lists a summary of each girder's dimensional characteristics.
s21
9)
H
T1
B
B
R
Figure 2-4 Cross-Section Girder 4 "Open-Section"
17
Ts2
Ts1
|
0.0508
0.0508
2
2
0.1016
0.1016
4
4
0.1016
0.0508
4
2
0.1016
0.1016
Girder 1, 2-inch TopSlab thickness
Girder 2, 4-inch TopSlab thickness
Girder 3 "Closed"
Section
Girder 4 "Open"
m
in
m
in
m
in
m
Bg
2.4384
96
2.4384
96
2.4384
96
2.4384
H
0.83883
33.0
0.83883
33.0
0.83883
33.0
0.83883
Section
in
96
33.0
2
4
6
Bf
min. Tw
R
m
0.3048
Bf2
0.254
Tw2
Girder 1, 2-inch Top-
0.0762
0.0508
0
Slab thickness
Girder 2, 4-inch TopSlab thickness
Girder 3 "Closed"
Section
Girder 4 "Open"
Section
in
m
in
m
in
m
in
12
0.3048
12
0.3048
12
0.3048
12
10
0.254
10
0.254
10
0.254
10
3
0.0762
3
0.0762
3
0.0762
3
2
0.0508
2
0.0508
2
0.0508
2
0
0
0
0
0
0.2286
9
Tf
0.1524
6
0.1524
6
0.1524
6
0.1524
Table 2-1 Summary of dimensional characteristics of girders
2.2.5 Width
In 2-d analysis, the width of the concrete deck under consideration is of length B= 3.66m (12ft)
or one lane. In 3-d analysis, half a bridge girder is considered, a width of 1.22m (4ft). The width
of one bridge girder is B'= 2.44m (8ft).
L
B
\Iola
Figure 2-5 3-d view of one and a half bridge girders
18
2.3 Loads
The basic design expression in the AASHTO (1994) LRFD Bridge Specifications[1] that must be
satisfied for all limit states is:
OR,
17 7,Q,
(2.1)
where:
Qj -
force effect
R, - nominal resistance
v - load factor
#-
Resistance factor
q - load modifier
2.3.1 Load Modifier, r7
The load modifier, q takes into account the ductility, redundancy and operational importance of
the bridge. q takes on different values for the service limit state (SLS) or the ultimate strength
limit state (ULS) and is defined by AASHTO-LRFD [1]
7-y= 7rq
In Equation 2.2, the Ductility Factor,
7
1
(2.2)
D = 0.95 for ULS (ductile components and connections)
and 1.0 for SLS [11]. The Redundancy Factor is, qR =1.05 for ULS (non-redundant members)
and 1.0 for SLS. The last parameter is the operational Importance Factor, 77, = 1.0 for both SLS
and ULS. Therefore given these parameters, the load modifier is 7 = 0.9975
and SLS.
19
1I for both ULS
2.3.2 Dead Loads
Dead loads of the structural components in this project comprise only the self-weight of the
UHPC bridge girder. This can be simply evaluated by multiplying the density of the UHPC
material by the volume of the bridge girder and gravitational acceleration. The density of UHPC
is, p = 2500kg / m3 = 0.0911b /in 3
Thus the unit weight of the bridge girder,
g, = pg = 24.5kN / m 3 (0.091b / in3 )
(2.3)
In addition to the self-weight of the girder, the dead weight of a possible future-wearing surface,
of asphalt or concrete, must be considered. Though this load is difficult to determine because it is
difficult to determine how layers and associated thickness of wearing surfaces may be applied in
the course of its service life, it is necessary to account for it. This future-wearing surface rests on
top of the slab. A typical value for its mass per square meter is 122kg/M 2 (251b/ft 2 ). Therefore the
pressure applied to the top surface, A, of the slab by the wearing surface is,
g2= 122 x 9.81=1.2kPa (0.17psi)
(2.4)
2.3.3 Live Loads
The live loads applied in this project are consistent with HL-93 in AASHTO (1994) LRFD
Bridge Specifications [1]. Under this specification, design truckload, HS-20, or a design tandem
load, is superimposed onto a design lane load to obtain maximum loading for any conditions.
20
Design Truck Load HS-20
The design truckload is a model that resembles the semi-trailer truck (Fig. 2-6). The front axle
carries a load of 2P,,. Located 4.3m (14ft) behind the front axle, is the drive axle of load 2P, 2.
Finally, of varying distance between 4.3~9m (14-30ft) is the rear trailer axle carrying another
load of 2P, 2 . This is better illustrated in fig 2-6. The variable spacing is to allow the designer to
determine the critical load effect. Therefore, the total load is,
Pwi =17.8kN (4kips)
(2.5)
P,2 = 71.2kN (16kips)
(2.6)
P, = 2P 1 + 2 x (2P 2 )kN = 320.2kN (72kips)
(2.7)
AASHTO Standard Specifications provides for a "tire contact area" that is b,=0.51 m (20in) wide
and l = 0.25m (10in) long. Wheel loads are assumed to be uniformly distributed over this tire
contact area. Thus, after dividing by two, the wheel loads can be obtained from the axle loads and
subsequently the wheel surface pressure applied onto the deck of the girder can be found.
Front Wheel Tire Pressure: p I =
Rear Wheel Tire Pressure: p
Pw
I = 148kPa (20psi)
_ .2 _ 148kPa (20psi)
(2.8)
(2.9)
Where,
AW = bw x l
= 0.1275m 2 (200in 2 )
(2.10)
21
AASHTO specified design tandem loads, as an alternative to the design truck loads, produce
force effects much smaller than that of the design truck loads and is not further investigated in
this bridge design.
Dynamic Impact Factor
The design wheel load has to be multiplied by the dynamic impact factor, 6 = 1.33, in calculating
factored loads.
Design Lane Load
The design lane load consists of a uniformly distributed load of 951 kg/m (6401b/ft) per one lane,
B = 3.66m (12ft), of traffic. This is equivalent to a surface pressure of
951 x 9.81
P2
=
(0.37psi)
32.6kPa
3.66
2
L-
HS20-44
HS15-44
8.000 LBS.
6.000 LBS.
II
I
32.000 LBS.
24,000 LBS.
32000 LBS.
24,000 LBS.
Figure 2-6 HS-20 Truck Load Specifications
22
(2.11)
2.3.4 Load Factors
Service Limit State
The third service limit state, service III from AASHTO LRFD Bridge Design Specifications [1],
is considered here. This service limit state refers to the load combinations relating only to tension
in pre-stressed concrete structures with the objective of crack control. The design specifications
suggest the following load combinations for SLS:
*
DC - Dead load of structural components and non-structural attachments
*
DW - Dead load of wearing surface and utilities
"
LL - Vehicular Live Load
FSLS = DC+ DW +.8LL
=1.0 x (g, + g 2 )+0.8 x ((5x p + p 2 )
(2.12)
Ultimate Strength Limit State
The limit state Strength I of the AASHTO LRFD Bridge Design Specifications is considered
here. This strength limit state is the basic load combination relating to the normal vehicular use of
the building without wind. This limit state represents the maximum possible load that the
structure is designed to sustain. The suggested load factors are:
FULS = 1.25DC+1.5DW +1.75LL =1.25 x g, +1.5
23
xg
2
+1.75 x (( x p, + p 2 )
(2.13)
Load Factors
|ULS
ISLS
Loads
1
Self-weight, g1
1
Future Wearing Surface, g2
0.8
Traffic Lane Load, p
0.8
Front wheel of truck, P1
0.8
Rear wheels of truck, P2
I Nominal Loads
24.5 kN/mA3
1.25
1.20 kPa
1.5
2.55 kPa
1.75
1.75
35.60 kN/axle
142.39 kN/axle
1.75
(0.09
(0.17
(0.37
(8
(32
lb/inA3)
psi)
psi)
kips)
kips)
Load due to the wheels of the truck need to be multiplied by the Dynamic Impact Factor
Dynamic Impact Factor
1.33
Table 2-2 Summary of Loads and Load Factors as per AASHTO-LRFD guidelines [11
Table 2-2 lists a summary of the loads and load factors as per AASHTO-LRFD guidelines [1] for
highway bridge design.
2.3.5 Determination of Maximum Moments
In order to produce maximum
moments along the bridge
under the proposed HL-93
specifications, the truck wheel load locations have to be varied along the length of the bridge.
Maximum moment was found to occur when the middle axle is located 0.71m off the center of
the bridge in the direction of the lighter front axle, as illustrated in Fig. 2-7, which places the
weight of the truck as close to the mid-span of the bridge girder as possible.
24
L/2
Z
V.I Im yM
427 m (14 ft)
)
1.27m
DI
(24ft
Pi
P2
L
g, = pgdV
\ -
L
Figure 2-7 Axle location to corresponding to maximum moments
2.4 Pre-Stressing Parameters
The number of pre-stressing tendons, N, is restricted by two conditions:
1.
N must be greater than or equal to the minimum number of tendons required to attain a
specific total pre-stressing of a section.
2.
N must be smaller than or equal to the maximum number of tendons that can fit into the
bottom flange.
These conditions can be mathematically expressed by:
= Nmin
N
Nmax
=
f TAstendon
XiTxK
(2.14)
C
with:
P = pA1 ;
CT
P =Y
f)
sotal
Sjtotal
Ac,element
25
(2.15)
cT
=
xsendon
(2.16)
where,
P is the total pre-stressing force on the girder;
A,
tendon
= 1.26677 /10' m2 (0.5 in diameter tendons) cross sectional area of tendons;
y= 0.8 the (effective) level of pre-stressing;
f T=
1870MPa is the strength of the tendons;
<a> denotes the greatest non-negative integer that satisfies <a> _<a;
c = 5.1cm (2in) minimum distance between concrete and clear cover;
Ac,eement
CT =
is the cross sectional area of the structural element where the tendons are located;
2.2% is the reinforcement ratio.
With these parameters, a bottom flange cross-sectional area of 0.90 x 0.15 m 2 (36 x 6 in 2 ) allows
one to place a rough total of 30 tendons in the flange. This allows us to achieve the desired prestress force level of pooo = 4.45MN (1,000kips) or equivalently, an effective pre-stress pressure
ofp ooo = 33.OMPa (4.79ksi).
2.5 1-D Think model of the UHPC Material Behavior
The 1-d think model, proposed by Chuang and Ulm [5],
used to capture the various
characteristics of a composite UHPC material is illustrated in Figure 2-8. The elastic brittleplastic behavior of the composite matrix is modeled with an elastic spring of stiffness CM, and a
brittle-plastic crack device (crack strengthf, frictional strength km). An elasto-plastic material law
governs the composite fiber behavior. It is described by an elastic spring (stiffness CF) placed in
series with a friction element (strengthfy). The irreversible matrix behavior (strain c' ) is linked
with the irreversible reinforcement behavior (strain
coupled by an elastic spring of stiffness M.
26
e') by means of two parallel elements
E
Cm
Cf
1k
f
ft
M
p
fy
Figure 2-8
1-d think model of a two-phase matrix-fiber composite material [5]
These 6 model parameters (CM, CF, M f, kM, fy) govern the composite material behavior. The
total macroscopic stress, Z, is composed of the stresses acting on the crack element (cM) and the
friction device
(UF).
E = CM +
where,
L
0-
"
=
0-F
CM
K
CF
Here, E refers to the total strain, C.
cracking and 16
- CM
rCM+CF
-
(CM + M)
M
(2.17)
-F
F
CF
M
'E
EP
-(CF + M)), E
I
(2.18)
is the permanent strain associated with the composite matrix
is the permanent strain associated with the permanent composite fiber
deformation.
27
f
E
Figure 2-9 Stress-strain response of the two-phase composite model [4]
From the model above, the stress-strain response of the two-phase composite model is illustrated
in Figure 2-9. Ko and K, represent the composite stiffness of the material before and after
cracking respectively. The 1-d UHPC model extends to three-dimensions by replacing the 1-d
scalar quantities by their tensorial 3-d equivalents. This is dealt with in greater detail by Chuang
and Ulm [5].
Model
Parameter
UHPC Only
SI
[
IU
UHPC-1,000
SI
|
iU
CM
53.9 GPa (7,820 ksi) 53.9 GPa (7,820 ksi)
CF
M
v
0
0
1.65 GPa (240 ksi)
0.17
f,
0.7 MPa
(0.1 ksi)
0.7 MPa
(0.1 ksi)
km
UMC
6.9 MPa
190 MPa
(1.0 ksi)
(28 ksi)
U-At,
220 MPa
(32 ksi)
6.9 MPa
190 MPa
220 MPa
(1.0 ksi)
(28 ksi)
(32 ksi)
fy
4.6 MPa
(0.67 ksi)
12.7 MPa
(1.85 ksi)
10 MPa
(1.5 ksi)
30 MPa
(4.4 ksi)
-Fc
4.4 GPa (640 ksi)
1.65 GPa , (240 ksi)
0.17
Table 2-3 Values of "effective" UHPC model input parameters
28
Table 2-3 lists the model input parameters for the UHPC material on its own provided by Lafarge
for DUCTALIM. am,, and
cub
represent the initial compressive strength of the matrix and initial
biaxial compressive strength of the matrix respectively. All parameters (CM, CF, M, kM) are known
parameters of the UHPC material.
Table 2-3 also provides the values for the effective stiffness and strength of the UHPC material
reinforced by pre-stressing tendons. Indeed, if there is perfect adhesion between the tendons and
surrounding UHPC, the effect of pre-stressing tendons serves to enhance composite stiffness. The
stiffness of the bottom flange, CB , becomes,
CB =C
+CT(E -CF
(2.19)
Likewise, the effective strength UHPC and the tendons realize at failure is also enhanced by the
presence of pre-stressing tendons:
if fy +
,
where
CT
is the reinforcement ratio,
fj
T[(1 -,V)f
(2.20)
y
y+CT
-
fy2.0
is the tendon strength, and
f,
is the ductile tensile
strength of the composite fiber phase. Equations (2.19) and (2.20) were used to determine the
effective composite properties of the pre-stressed bottom flange and are noted UHPC-1,000 in
Table 2-3.
29
2.6 Design Criteria
Since the strength of UHPC materials depend on the development of cracks, the right-hand side
of Equation (2.1), denoting the reduced strength, should be substituted for an expression denoting
this limitation; thus:
(pRn = F(max[[w]])
(2.21)
2.6.1 Service Limit State
The service limit state relates to how the bridge performs under everyday loading during the
course of its life in service. In order to define this limit state for UHPC structures, a crack opening
criteria is used. A "no-cracking" criteria is used in this work to define the service limit state. This
criteria will serve to safe-guard the bridge from long-term durability problems and fatigue crack
opening from cyclic loading:
max [[w (FsLs)]]
where
e,
0 or max (c
P
(X)) = 0
(2.22)
stands for the principal plastic strain associated with matrix cracking.
2.6.2 Ultimate Limit State
The ultimate limit state is the state where the structure is subjected to the maximum loads that it
can sustain. Surpassing this limit state leads to failure of the structure. This limit state ensures the
mechanical composite performance of the UHPC material. UHPC-guidelines by the French
30
Association of Civil Engineering [10] recommend a crack opening width as a design criterion for
this limit state.
[[w]]"
max[w( FULs
m""
for unreinforced sections
= 0.3mm
( 2.23)
Lf. h
IW IIim - min( - ' -)
for reinforced sections
4 '100
where [[w]]Iim represents the total cumulative crack opening measured over a characteristic
length, l = (2/3) h. Lf = 13mm, is the length of fibers employed in the specific UHPC material,
DUCTALT M , and h is the height of the girder or plate structure.
Thus in dimensionless form,
[I'm""
max(E (FULS ))
m
p- r e
m
=
=
1.5[[w]]""/ h
for unreinforced sections
(2.24)
m
8h(--;8h 200
for reinforced sections
2.6.3 Modeling of Loads
Due to the non-linear nature of the stress-strain relation of UHPC material, the loads discussed
above have to be applied gradually to the model until the desired load level is reached. First stage
of this gradual loading consists of the material's self-weight and pre-stressing. This stage of
loading would be likened to the loads experienced by the structure upon initial installation.
Secondly, load due to the future-wearing surface, dead loads, is applied. Lastly, live loads due to
the design lane load and the design truckloads are applied incrementally onto the structure. These
loads are increased by 10% at each stage until the desired load level has been reached. This load
scheme is illustrated in Figure 2-8.
31
(a)
0.8 (JPI + P2
129
L.0g 2
1.0 g +p
Time Step
(b)
1.75 (&P1 + P2)
1.5 g 2
1.25 g, +p
Time Step
Figure 2-8 Load Scheme (a) FSLS load scheme (b) FULS load scheme
(8=1.33 is the dynamic amplification factor)
32
Chapter 3
2-D MODEL BASED DESIGN
3.1 Introduction
In this chapter, the girders under investigation are modeled in a 2-d form and loaded to
the ultimate limit state where the stresses and strains developed throughout the section will be
analyzed to determine its feasibility under such conditions. The chapter begins with a description
of the model used to conduct this analysis and how real 3-d bridge girder has been simplified into
a 2-d form. The boundary conditions and the mesh employed are described thereafter and the
results are shown. The 2-d model is then checked if it meets the ultimate limit state criteria for
admissible cracks developed on the bottom edge and maximum compressive stress developed on
the top edge.
33
3.2 2-D Section Modeling
Finite Element simulations were conducted to examine the behavior of these girders in two
dimensions. These simulations were conducted using the UHPC model in CESAR-LCPC.
Analysis was first performed in two dimensions to determine the overall adequacy of the doubletee girders in resisting the applied moments. In order to model the double-tee girders in two
dimensions, the cross section of the double tee girders have to be collapsed into an equivalent
cross section.
12
B
4
t
14
4-
T
-+
:f
Figure 3-1 Cross Section of one and a half girders
One and a half bridge girders, corresponding to one traffic lane, will be modeled in two
dimensions. As depicted in the Figure 3-1, the intricate designs of the double-tee girders have to
be collapsed into a girder of equivalent cross sectional dimensions. Actual double-tee girder will
be broken down into sections of consistent height. The widths of the equivalent cross section will
then simply be the sum of the widths of the actual double-tee girder consistent with their section.
34
Thus, in this manner, the actual double-tee girder will be collapsed into an equivalent girder
depicted in Figure 3-2.
4B
3B
Figure 3-2 Equivalent 2-D Girder
The thicker portion of the top-slab at its stress free corner is modeled as TI- T,2. Since one and a
half girders were modeled, the web thickness and bottom flange width of the equivalent cross
section is simply three times that of the original double-tee girder. Thus, in this manner, we have
broken down the complex design of the double-tee girder into an equivalent section that is readily
analyzable in 2-d.
35
3.3 Mesh
The meshes used in the finite element models contain between 6,400-6,800 elements per bridge
girder. 16 layers of elements were used to model the entire depth of the bridge with at least 4
layers used to model the top-slab. Four node quadrilaterals were used to define the mesh. The
girder was partitioned along its height to model the different sections of the actual girder. The
equivalent widths of each section were taken into account in the finite element model. An
example of the mesh used is illustrated in Figure 3-3. The bridge is simply supported at its ends:
One end of the bridge is pivoted while the other end rests on a roller.
_I.
H,
X:Tf
1.
Number of ncdes
Numt~er of element.
Number oF group :
- il)
36
6817
:
400
4
Figure 3-3 2-D Mesh
3.4 Results of 2-D Analysis
3.4.1 Overview
Figure 3-4a illustrates a typical 2-d section with the applied loads and boundary conditions. Fig 34b shows the deflected shape of the section under ultimate load conditions. Fig 3-5 displays the
principal plastic strains, as iso-contours, present in the composite matrix at the ultimate limit
state. Note that principal plastic strains develop on the bottom flange of the 2-d section.
Pwi
(a)
Fw2
4 4
~w2
0
z
P.]
P'2
'P'2
(b)
Figure 3-4 2-d Model with boundary conditions and axle loads (a) initial shape of 2-d model (b)
deflected shape of 2-d model, dotted lines represent initial shape
37
Figure 3-5 Maximum principal plastic strain contours of 2-d model at ULS
-4.50
2. 00
1.80 -4.00
1.60 -,-"35
z1.40 -
3.50
00
.
t
3.00 <
o 1.20
T 2.50
1.00
f
0
2.00 -
>-e 0.80 -girder
-
E 0.60 -
-
-
1
girder 2
Z
1.50 M
girder 3
o
s
nirdmr 4
1.00
0.40
0.50
0.20
-_
0.00
0
0.002
-0.00
0.006
0.004
0.008
0.01
Absolute Strain (1)
Figure 3-6 Graph of cumulative load vs. maximum principal plastic strain, for 2-d ULS load
conditions
38
E
0
3.4.2 Flexural Resistance
Fig 3-6 shows the total load, including self-weight, applied on the section versus
maximum strain. Maximum strain in the 2-d model occurred on the bottom flange. It can be
readily inferred from the graph that the performance of girders 1, 3 and 4 are very similar, while
the performance of girder 2, with the 4-inch thick top-slab, exceeds that of the other girders. It
can resist greater moments, due to greater loading, for the same plastic strain occurring in the
bottom flange.
1.20
1.00
0
-j
0.80
C,)
J 0.60
.N
--
0.40
-
E
girder 1
-
-
girder 2
girder 3
o 0.20
girder 4
n 0n
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Normalized max plastic strain
Figure 3-7 Graph of normalized ULS live load vs. normalized maximum plastic strain
39
Fig 3-7 displays the relationship between normalized ULS live loads and normalized strains for
sections under 2-d analysis subjected to loads in concordance with the ultimate limit state.
Normalized live loads, FULS, were determined by dividing the applied live load at the given
load-step with the total live load demanded by the ultimate limit state criteria, i.e. from:
FULS =1.25 x g, +1.5
xg
2
+
FULS x 1.75(5 x p, + p 2 )
(3.1)
Normalized strains were obtained by taking the maximum observed plastic strain values at each
load-step and dividing it by the maximum allowable plastic strain for reinforced concrete sections
prescribed by the French Association of Civil Engineers (AFGC) for UHPC materials under
ultimate limit state loads (refer to Eqn. 2.24)1. From the graph, we can infer, since maximum
allowable strains are similar for all the bridge girders, that girder 2 can withstand the greatest load
for any given strain value. All girder sections satisfy the ultimate limit state criteria at slightly
greater than 30% of the allowed maximum strain values. Thus all the girders are expected to
avoid excessive cracking of the bottom-flange causing a loss in moment carrying capacity.
Recall
Pre = min(
""m
3L,-
3
) = 0.0058159
8h '200
-
40
--
-4
-5
L.
-6
7
0.90
--
-2
'-3
_
f
-1
0
12
11
3 (ksi)
35.00
0.80
30.00
0.70
25.00
0.60
20.00
0.50
IM
Z.
0.40
15.00
0.30
-
-
-
Girder 1
Girder 2
10.00
Girder 3
0.20
Girder 4
5.00
0.10
I
0.00
-5 0.00
I
I
-40.00
-30.00
I.i
-20.00
Stress,
ar,
-10.00
0.00
10.00
20.00
A
AA
U.U
30.00
(MPa, tension+)
Figure 3-8 Graph of girder height, H (m) vs. uniaxial stress in the longitudinal direction,
tension +), 2-d model ULS loading
acxx (MPa,
Fig 3-8 illustrates the variation of longitudinal stress, a, in the longitudinal direction, along the
height of the section. Positive stress values correspond to tension while negative stress values
correspond to compression. Maximum compressive stress developed at the top of the 2-d sections
is 40.4MPa for girder 1, 26.OMPa for girder 2, 37.9MPa for girder 3 and 36.7MPa for girder 4.
The maximum allowable compressive stress, as stipulated by the manufacturer of the UHPC
material, is
-uc= 190MPa (refer Table 2-3). Thus all the girders possess maximum compressive
stress at the ultimate limit state under the allowable maximum compressive stress. They are
unlikely to fail in compression at the ultimate limit state. We tabulate the ratio of maximum
uniaxial longitudinal stress developed over allowable compressive stress, a; in Table 3-1.
41
Girder
Girder
Girder
Girder
1
2
3
4
0.212632
0.136941
0.199345
0.193007
Table 3-1 Maximum longitudinal compressive stress as a ratio of the allowable compressive stress
3.5 Summary of 2-D analysis
All double-tee bridge girder sections satisfy the ultimate limit state criteria, limited cracking, set
forth. Thus the overall dimension of the bridges appears to resist any of the applied moments due
to dead and live loads. It is unlikely that the bridge structure will fail due to excessive strain on
the bottom flange of the bridge. Furthermore, compressive stresses developed at the top of the
section is significantly less than the maximum allowable compressive stress of the material. Thus
the section in its flexural state is unlikely to fail due to compressive or tensile stresses developed.
Girder 1 performs best on a per weight basis. Adding additional concrete to the top-slab does not
improve the overall stiffness of the double-tee section because the added self-weight increases the
applied moment in excess of the stiffness provided by the additional UHPC.
42
Chapter 4
3-D MODEL BASED DESIGN
4.1 Introduction
This Chapter examines the 3-d analysis performed for the various bridge girder designs. The
service limit state is analyzed here to determine the adequacy of top-slab thickness. The ultimate
limit state is then analyzed to determine detailed structural feasibility of selected girders. This
section begins with describing the geometry of the problem and the various assumptions used to
simplify it. Loading conditions, limit states and the mesh adopted in the finite element analysis
will also be discussed. Results from the finite element analysis and the interpretation of it will
conclude the chapter.
43
4.2 Design Requirements Revisited
4.2.1 Dimensions
In the ensuing 3-d analysis, all four girders from the 2-d analysis above are examined. However,
making use of transversal symmetry conditions, only half the cross-sections is modeled, thus the
width of the top slab is 1.2m (4ft). Adequate boundary conditions are prescribed to simulate
continuity of the top-slab of the girders. The length of the girders examined is 21.4m (70ft).
Figure 4-1 provides a 3-d wire-frame view of a girder being analyzed.
z
Figure 4-1 3-d Wire-frame of half girder section being analyzed
44
4.2.2 Boundary Conditions
In order to simulate continuity of the top-slab, at mid-span of the girder axis, the slimmer edge of
the top-slab in Figure 4-2 below was allowed to translate freely in the y-direction but not allowed
to rotate about the z-axis. To accomplish this in CESAR-LCPC, the two parallel lines running
along the length of the top-slab, shown connected to a fixed-roller in Figure 4-2, were prevented
from translating in the x-direction. This provided the necessary restraining torque and accurately
models the continuity of moments present in the top-slab while allowing the mid-span of the
girder to deflect downwards. The bottom flange of the cross-section is simply supported. It posses
a roller at one end, along the outermost edge of the bottom flange, locking it in the y-direction,
and is pinned at the opposite end, locking it in the y and z directions.
yJ
x
Figure 4-2 Boundary conditions applied on 3-d half girder
45
4.2.3 Loads
Loads congruent with the service limit state were imposed on the modeled 3-d girders. We recall
that there are, in the SLS:
FsLs= DC + DW+ 0.8LL = 1.0 x (g, + g 2)+ 0.8 x (5xp] +p2)
(4.1)
and in the ULS:
FULS
1.25DC + 1.5DW+ 1.75LL= 1.25 x g, + 1.5 x g 2 + 1.75 x (Sx p +p 2)
(4.2)
where:
3- Dynamic Impact Factor 1.33
DC - Dead load of structural components and non-structural attachments
DW- Dead load of wearing surface and utilities
LL - Vehicular Live Load
Loads are incrementally applied onto the bridge as per Figure 2-8: The design truckload was also
applied onto the 3-d bridge girder to generate maximum moments. This was done in a similar
fashion to the 2-d analysis, where a HS-20 truck's mid-axle was positioned 0.7m off the center of
the bridge girder as per Figure 2-7. The truck wheel load pressure is distributed over the number
of nodes present at the location where it is to be applied; they are subsequently applied as point
loads to all node locations within the wheel load patch.
Since the width of the AASHTO designated design truck, HS-20, is 1.83m (6ft) and the modeled
3-d girders only possess widths of 1.2m (4ft), only one set of wheels, either the left or right side
46
of the truck, can exist on the half girder being modeled at any time. Thus in order to examine the
full-spectrum of behavior of loads being applied to the bridge, the truck wheel loads were applied
first to the mid-span of the girder (fig 4-3), load condition 1, followed by the stress-free edge of
the girder (fig 4-4), load condition 2, in separate analysis. Load condition 2 is analogous to the
situation of a joint between slabs. Should there be perfect continuity at this slab-joint location,
load condition 1 (mid-span) would be expected to govern the design; however should there be a
discontinuity at this location, load condition 2 might govern the design. In order to account for
this possible discontinuity at the slab-joint location, a zero stress boundary condition was applied
to this location to simulate cantilever action. This is the worse possible situation should there be
no continuity at all.
yt
X
Figure 4-3 Location of applied truck wheel load at mid-span, load condition 1
47
P
x
Figure 4-4 Location of applied truck wheel load at the stress-free edge, load condition 2
4.2.4 Limit States
The service limit state will be used to determine the adequacy of performance of the top-slab. A
"no-cracking" criteria will be imposed on the service limit state, (see section 2.6.1):
max[[w(FsLs )]]
0 -> max(eCf
(x))
where, CP, (x) is the principal plastic strain of the UHPC matrix.
48
0
(4.3)
For dimensional consistency, the obtained strain values will be normalized against AFGC
recommended values for maximum crack opening at the ultimate limit state, given by Equation
(2.24). These are the maximum crack openings for an unreinforced section of a 2 and 4 inch slab.
4.2.5 Mesh
Four-node quadrilaterals were employed to create the mesh. There exists a minimum of 4 layers
of elements on the top-slab (see Figs. 4-5 and 4-6) to accurately model the local strains due to the
design truckload. Four layers of elements were modeled into the bottom flange to capture plastic
strains due to overall
deflection of the girder. The mesh is repeated every 0.3~0.4m
(0.091~0.122ft) throughout the length of the girder, 21.4m (70ft).
49
- .1............
..............
I
[
Figure 4-5 3-d Mesh employed on Girder 1
L
_____
iLE
=_4
Figure 4-6 3-d mesh employed on Girder 4
50
=__=_
4.3 Deflections
When the bridge is initially installed, only its self-weight and pre-stressing stress exist. Thus it
deflects upwards as depicted in Fig 4-7. Upon gradual application of the dead load of the
structural components and the applied traffic loadings, the deflection of the bridge gradually
decreases till the bridge deflects downwards at application of full service loads. Fig 4-8 displays
the deflected position of the bridge at service loads when the truck wheel loads are at the stressfree edge (load condition 2). The largest deflections recorded under service loading conditions
occurred in girder 1. Girder 1 deflected -0.0156m (0.61in) downwards under service loading
conditions. This is within acceptable limits of deflection in service conditions under AASHTOLRFD guidelines [1]; as allowable deflection under service loading conditions is L/800
0.02667m (1.05)> 0.0156m (0.61 in).
51
=
Figure 4-7 Deflected shape 3d girder under pre-stressing loads only
Figure 4-8 Deflected shape 3-d girder under SLS loads
52
4.4 Flexural Resistance
4.4.1 Service Loads
Load Condition 1
Figure 4-9 below depicts how, for load condition 1, normalized service limit state loads vary with
strain values normalized against ultimate limit state requirements for all four bridge girder
designs. The strain values represent principal maximum plastic strains located on the top-slab due
to service truckloads. These strain values were normalized against maximum strain values for the
ultimate limit state (Eq. (2.27)) in order to be consistent in comparisons. The maximum strain
values normalized against correspond accordingly to the thickness of the top-slab of the bridge
girder, i.e. 2 or 4 inches. Normalized loads were obtained simply by dividing the total applied live
load on the girder with the live load required by the service limit state.
It can be readily inferred from Figure 4-9 below that only girder 2 meets the required "nocracking" criteria for UHPC bridge girders subject to service limit state conditions. Girder 2
possesses a 4-inch thick top-slab greatly adding to its moment carrying capability. Girders 1 and
3, which both possess a 2-inch top-slab fail to meet the "no-cracking" criteria, at the service load,
they develop normalized (transversal) maximum plastic strains of 6.3% and 5.3% of the ULS
value respectively, which should exclude them from further consideration. Girder 4 possesses a 2inch thick top-slab but the distance to mid-span from the web is 9 inches shorter than that of the
other girders. This reduces transverse moments and thus causes less strain at the bottom edge of
the top-slab. However, girder 4 cracks before the service limit state is obtained, at the service
limit state, it develops normalized maximum plastic strains of 1.35% which corresponds to [[w]]=
0.0135x30ptm =4pm. This is a very small value and is much smaller than typical shrinkage
cracks on the order of 100pm, and could therefore be eventually accepted as crack opening for
the service limit state.
53
1.60
0
1.40
0
1.20
0
1.00
0.80
girder 1
an 0.60
- - - girder 2
0.40
girder 3
-J
N
E
0.20
0
-
girder 4
Z
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Normalized Max Plastic Strain [1]
Figure 4-9 Graph of Normalized SLS Live Load versus Normalized Max Plastic Strain (Load
Condition 1)
1.60
1.40
0
1.20
0
-j
4)
1.00
0.80
N
--
girder 1
- - - girder 2
0.60
E
0
Z 0.40
g irder 3
0.20
--
girder 4
0.00
0.00
0.01
0.01
0.02
0.02
0.03
0.03
Normalized Strain [1]
Figure 4-10 Graph of Normalized SLS Live Load versus Normalized Max Plastic Strain (Load
Condition 2)
54
Load Condition 2
Load condition 2, where the truck axle load is applied at the cantilevered overhang, is examined
next. Figure 4-10 describes how normalized loads applied under service limit conditions vary
with normalized strain developed on the top surface of the top-slab. Normalized loads were
determined simply by dividing the absolute value of the applied live load with the absolute value
of the live loads required under service limit conditions. Normalized strains were determined by
dividing the obtained strain values on the top surface of the top-slab by the corresponding
maximum strain values for the respective slab thickness under ultimate loading conditions.
All of the bridge girder designs met the "no-cracking" criteria under service load conditions.
Girders 1,2 and 3 all developed no cracks under service loading conditions and even above
service loading conditions. Girder 4 only starts to develop cracks at above 1.08 times the applied
service load. At 1.16 times the service load, maximum normalized plastic cracks developed were
1.0% of the maximum plastic strain admissible under ultimate loading conditions. Therefore we
can infer that this load condition does not govern the slab design at all, because under the same
load, load condition 1 starts cracking earlier and is therefore more critical. Load condition 2 will
not be further investigated in the ensuing ultimate limit state analysis.
4.4.2 Ultimate Limit State
Fig 4-11 depicts the variation of normalized ULS live loads versus the normalized maximum
plastic strain obtained for girder 2 (continuous 4-inch slab) under load condition 1. It graphs
plastic strain values for two locations, one at the top-slab under the wheel load and the second at
the bottom-flange. Plastic strains occurring on the top-slab were normalized against allowable
plastic strain values for unreinforced sections and plastic strains on the bottom flange were
normalized against allowable plastic strains for reinforced sections (refer Eq. (2.27)). As can be
inferred from the graph, under ultimate loading conditions, bottom flange cracking quickly
55
becomes more predominant and hence control the performance of the girder under ultimate
loading conditions. At the ultimate load, the top-slab develops transversal plastic strains, 40% of
the maximum allowable value under ultimate loading conditions. In turn, the bottom flange
develops longitudinal plastic strains, which are 90% of the maximum allowable value under
ultimate loading conditions. This seemingly high crack development appears to stand in
contradiction to our 2-d analysis (see Fig. 3-7); but one notes that in 2-d analysis, 1.5 girders
support two wheel loads as in reality. In this 3-d analysis, half a girder supports one wheel load,
as opposed to two-thirds of a wheel load if in proper ratio. The analysis conducted here is more
useful in determining top-slab performance under truck wheel loads for various limit states.
Performance of the bottom-flange can be more accurately assessed in 2-d analysis.
Figure 4-12 depicts how normalized ULS live load varies against normalized strain values
obtained for the bottom flange and top-slab of girder 4. We note that both locations perform
satisfactorily under ultimate loading conditions. Thus if the very small cracks developed under
service loading conditions for girder 4 were to be admissible under the service limit state criteria,
girder 4 would be a possible design solution as it satisfies the ultimate limit state criteria.
56
1.20
0
0
-J
cn
1.00
ON=
I
0.80
-
0.60
-
. '-
i
Top-
-.N
(D 0.40
Slab
-
-
E
0 0.20 -
Bottom
Flange
Z
0.00
0.00
0.20
0.40
0.60
0.80
1.20
1.00
Normalized max plastic Strain [1]
Figure 4-11 Graph of Normalized Live Loads ULS vs. normalized max plastic strain, Girder 2, Load
Condition 1
1.20
(U
0
-j
1.00
0.80 -
..J
C)
-j 0.60 -0
N
0.40
-
TopSlab
Bottom
0
0.20 -
Flange
Z
0.00 -p
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Normalized Strain [1]
Figure 4-12 Graph of normalized ULS live loads against normalized max plastic strain, Girder 4,
Load condition 1
57
4.4.3 Crack Pattern
Figure 4-13 max plastic strain location at SLS (load condition 1)
Figure 4-13 shows the maximum plastic strain location when the bridge girders are subjected to
service limit state loads applied at the mid-span of the girder (load condition 1). Maximum plastic
strain occurs under the wheel load applied at the top-slab when subjected to service limit state
loads.
58
Figure 4-14 Maximum plastic strain location at ULS (load conditions 1)
Figure 4-14 shows the location of maximum plastic strain when the bridge girders are subjected
to ultimate limit state loads. The location of maximum plastic strain under ultimate limit state
conditions is the bottom flange of the girder.
59
Chapter 5
CONCLUSION AND OUTLOOK
5.1 Conclusion
The motivation of this work is the need for alternative highway bridge girder design solutions to
meet tomorrow's challenges of the Nation's infrastructure. To this end, a highway bridge girder
made of UHPC material was proposed as a design solution to meet these challenges. This work
contributes to this overall effort through a refined study of the cross-sectional design, to
determine the optimal slab and web configurations for a prototype bridge to be built, monitored
and tested at the FHWA Turner-Fairbank campus in Virginia.
Four different cross-sections were examined, girder one possesses a 2-inch top-slab, girder two a
4-inch top-slab. Girder 3 is similar to girders 1 and 2 except that its top-slab is 4-inches thick at
the stress-free edge and 2-inches thick at the mid-span. Girder 4 possesses the same design detail
as girder 3 but has recessed webs to allow for maintenance and inspection.
Two-dimensional analysis conducted on the bridge girder sections indicate a sufficient overall
flexural performance of the bridge girders when loaded to the ultimate limit state. It was found
that when loaded to the ultimate limit state, all the girders satisfied the crack opening criterion set
60
forth by the AFGC. Furthermore, compressive stresses developed at the top-edge of the bridge
were no more than 20% of the prescribed allowable values by the manufacturer. This leads us to
conclude that the bridge girders possess the required flexural strength to hold up statistically
significant loads that it is expected to experience in its operational use.
The three-dimensional analysis reveals more of the local effects due to applied loads on the
bridge girder sections. The maximum plastic strains developed in the top-slab were analyzed for
feasibility under the service load. It was found that when the truck wheel loads were applied at
the stress free (cantilevered) edge of the girder, no plastic strains developed on the top-slab under
service loading. That is, it satisfies the "no-cracking" criteria required under service load. Thus
the top-slab performs adequately should there be no continuity at all at the slab-joint location.
Subsequently, analyses of the bridge girders were performed when truckloads were applied at the
mid-span (between the webs) of the bridge girders. It was found that girders 1 and 3, which both
possess top-slab thickness of 2 inches at the girder mid-span, developed cracks at the bottom of
the top-slab prior to the required service load. At the required service load, girders 1 and 3
develop cumulative crack widths of 19pm and 16pm respectively. Girder 2 (with a continuous 4
inch slab) does not develop any cracks at the service load. Girder 4 (recessed web) develops
cracks underneath the top-slab at mid-span just prior to attaining full service load, but exhibits a
very small crack opening of 4pm. This crack width is much smaller than other (inevitable)
"service state" cracks, such as shrinkage cracks, which occurs in all concrete structures on the
order of 100ptm; one could therefore argue that girder 4's cracks are indeed miniscule and be
admissible for the service limit state. Mid-span deflection obtained for all girders under the
service limit state is minor. Thus for the service limit state, only girder 2 surpasses the prescribed
design service limit state requirements strictly while girder 4 is a border-line case. Both girders
satisfy the ULS as 3-d simulations showed: Girder 2 (continuous 4-inch slab) was found to satisfy
ultimate limit state admissible cracking criteria for both cracking at the bottom flange as well as
61
underneath the top-slab at the wheel load. Bottom flange cracks quickly govern the performance
of the girder under ultimate limit state. Girder 4 (with recessed webs) was also examined under
ultimate limit state conditions. It was found that it too satisfied ultimate limit state admissible
cracking criteria for both cracking at the top-slab and the bottom-flange.
From the results obtained, we can draw a number of conclusions:
1.
A 4-inch top-slab (girder 2) is sufficient to carry the necessary AASHTO loads required
for a bridge girder of this size. It performs adequately under both service and limit state
conditions and is a feasible design solution
2.
Girder 4 (with recessed webs) performs satisfactorily in the ultimate limit state. However,
it develops miniscule cracks at the service limit state prior to the desired load. Should
these miniscule cracks be allowed under a revised service limit state criterion, girder 4 is
a possible design solution as well.
3.
For the considered load, increasing the thickness of the outer top-slab, T,2, provides no
advantages as the joint-slab position does not govern the design. However, as the webs
are recessed inwards, outer top-slab thickness should correspondingly increase as well
due to larger lever arm actions. Furthermore, a thicker outer top-slab may also be
necessary for other load situations, such as the one relating to a rail barrier, which was
not considered in this study.
These conclusions allow us to adopt girders 1 and possibly girder 4 as a feasible UHPC prototype
bridge design in tests to be conducted at the Turner-Fairbank campus in Virginia. The successful
outcome of which achieves FHWA stated goals for innovative bridge designs for tomorrow.
62
5.2 Outlook
Our findings indicate that a UHPC highway bridge with a cross-sectional design of girder 2 can
be successfully implemented. Girder 4, on the other hand, can be successfully implemented with
minor changes in the design detail, such as increasing top-slab thickness to 7.6cm (3in) or
increasing overall height to 0.91m (3ft), and further analysis should be pursued in this direction in
the interest of employing a cross-section that is more economical, materially and financially, than
girder 2.
Since this work focuses mainly on the flexural design of the UHPC bridge girder, other design
considerations have to be made before it can be functionally employed. Design considerations
such as local strains developed during pre-stressing, shear effects and natural frequency analysis
have to be taken into account. It is in this direction that the design of UHPC highway bridge
girders will be further pursued.
63
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64