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Flexural Behavior of an Ultrahigh-Performance Concrete I-Girder

Article in Journal of Bridge Engineering · November 2008

DOI: 10.1061/(ASCE)1084-0702(2008)13:6(602)

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 Flexural Behavior of an Ultrahigh-Performance
Concrete I-Girder
Benjamin A. Graybeal, Ph.D., P.E., M.ASCE1

Abstract: The flexural behavior of an ultrahigh-performance concrete 共UHPC兲 was investigated through the testing and related analysis
of a full-scale prestressed I-girder. A 28 ksi 共193 MPa兲 compressive strength steel fiber reinforced concrete was used to fabricate an 80 ft
共24.4 m兲 long AASHTO Type II girder containing 26 prestressing strands and no mild steel reinforcement. Intermediate and final
behaviors, including cracking, flexural stiffness, and moment capacity, were investigated. Test results are compared to predictions based
on standard analytical procedures. A relationship between tensile strain and crack spacing is developed. The uniaxial stress-strain response
of UHPC when subjected to flexural stresses in an I-girder is determined and is verified to be representative of both the stress and flexural
stiffness behaviors of the girder. A flexural design philosophy for this type of girder is proposed.
DOI: 10.1061/共ASCE兲1084-0702共2008兲13:6共602兲
CE Database subject headings: Concrete structures; Concrete beams; Prestressed concrete; Flexural strength; High-strength
concrete; Fibers; Girders.

Introduction crete of this type currently commercially available in North

America. This high cement, high silica fume content concrete has
Ultrahigh-performance concrete 共UHPC兲 is a new class of con- an extremely low water-cementitious materials ratio 共less than
crete that has been developed in recent years. When compared 0.20兲 and requires high-range water-reducing admixtures to
with high performance concrete 共HPC兲, UHPC exhibits superior achieve the correct rheology. This concrete contains no coarse
compressive and tensile mechanical behaviors, as well as excep- aggregate and is internally reinforced by 0.5 in. 共13 mm兲 long,
tional durability properties. A research program was initiated to 0.008 in. 共0.2 mm兲 diameter straight steel fibers that are included
characterize many of the behaviors relevant to the use of UHPC at a volumetric ratio of 2%. Further details on the mix composi-
in the highway bridge industry 共Graybeal 2006a,b兲. Full-scale tion can be found in Graybeal 共2007兲.
AASHTO Type II prestressed concrete girders were tested under The material characterization research program associated
flexural and shear loadings. The results of the flexural testing with the research discussed herein focused on quantifying the
program are discussed herein. mechanical and durability properties of this UHPC. Table 1 pro-
The term UHPC covers a broad range of cementitious com- vides a listing of many of the properties that were quantified using
posite materials that exhibit sets of mechanical and durability standardized tests. Of particular interest in the realm of structural
properties far advanced beyond conventional concretes. The concrete, after the application of a steam curing treatment this
French led a significant portion of the early development of these UHPC has a compressive strength of 28 ksi 共193 MPa兲, a tensile
materials in corporate, national, and academic research laborato- cracking strength of 1.3 ksi 共9 MPa兲, and a creep coefficient of
ries. This background led to the publication of the Association 0.29.
Française de Génie Civil Interim Recommendations for Ultra The use of this new class of concrete in structural applications
High Performance Fiber-Reinforced Concretes 共Association has been limited. One reason for the slow implementation is the
Française de Génie Civil 2002兲. This document states that UHPCs perceived complexity of the structural behaviors of UHPC com-
tend to have a compressive strength over 21.7 ksi 共150 MPa兲, ponents as compared to conventional concrete components as
internal fiber reinforcement to ensure nonbrittle behavior, and a well as the lack of full-scale UHPC component test results. Given
the increased material cost of UHPC as compared to conventional
high cementitious materials content. UHPCs also tend to have
concrete, practical use of UHPC structural components will likely
very low water-to-cementitious materials ratios, minimal or no
require optimization of the component to make use of the ad-
coarse aggregates, and an optimized gradation of granular
vanced mechanical and durability properties. Of primary interest,
materials.
this optimization frequently leads to the use of the tensile prop-
The UHPC studied in this research program is the only con-
erties of UHPC to carry a portion of the loads imparted on the
1
component. Without an understanding of the UHPC tensile prop-
Research Structural Engineer, Federal Highway Administration, erties as exhibited on the structural component level, the optimal
Turner-Fairbank Highway Research Center, 6300 Georgetown Pike, design of said components is not possible.
McLean, VA, 22101. E-mail: benjamin.graybeal@fhwa.dot.gov
UHPC tensile properties are distinct from those of conven-
Note. Discussion open until April 1, 2009. Separate discussions must
be submitted for individual papers. The manuscript for this paper was tional concrete due to the increased tensile cracking capacity of
submitted for review and possible publication on January 7, 2008; ap- the cementitious composite matrix and the crack-bridging behav-
proved on March 6, 2008. This paper is part of the Journal of Bridge ior of the fiber reinforcement. In contrast to fiber-reinforced con-
Engineering, Vol. 13, No. 6, November 1, 2008. ©ASCE, ISSN 1084- ventional concretes, UHPC can exhibit significant, sustained
0702/2008/6-602–610/$25.00. postcracking tensile capacity prior to crack localization, fiber

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Table 1. Mechanical and Durability Properties
Material characteristic Value
Compressive strength—ASTM C39; 28-day strength 28.0 ksi 共193 MPa兲
Modulus of elasticity—ASTM C469; 28-day modulus 7,600 ksi 共52.4 GPa兲
Tensile cracking strength—Combined result from four independent test methods 1.3 ksi 共9.0 MPa兲
Long-term creep 共Ccu兲—ASTM C512; 11.2 ksi 共77 MPa兲 sustained load 0.29
Total unrestrained shrinkage—From casting via vibrating wire gauge embedded in ASTM C157 concrete prism 850 microstrain
Coefficient of thermal expansion—AASHTO TP60-00 8.7⫻ 10−6 / ° F 共15.6⫻ 10−6 / ° C兲
Chloride ion penetrability—ASTM C1202; 28-day test 18 C
Chloride ion permeability—AASHTO T259; 0.5 in. 共12.7 mm兲 depth 0.004 lb/ ft3 共⬍0.06 kg/ m3兲
Scaling resistance—ASTM C672 No scaling
Abrasion resistance 共grams lost兲—ASTM C944 2 ⫻ weight; ground surface 0.0004 lb. 共0.17 g兲 lost
Freeze-thaw resistance 共RDM兲—ASTM C666A; 600 cycles 96%
Alkali-silica reaction—ASTM C1260; tested for 28 days Innocuous

pullout, and loss of tensile capacity. Fig. 1 shows a schematic of Experimental Program
the three distinct tensile behaviors that UHPC can exhibit: 共1兲
linear-elastic behavior before cracking; 共2兲 postcracking strain
Test Specimen
hardening behavior and dispersed discrete cracking; and 共3兲 soft-
ening behavior during strain localization across specific cracks The test specimen for this research effort was an AASHTO Type
共Habel et al. 2006兲. Behavior through the conclusion of strain II prestressed I-girder. This 36 in. 共0.91 m兲 deep girder was 80 ft
hardening involves the development of distinct cracks that are 共24.4 m兲 long and contained 26 0.5 in. 共12.7 mm兲 diameter,
closely spaced 共less than half of the fiber length兲 and have small 270 ksi 共1,860 MPa兲 low relaxation prestressing strands as indi-
widths 共generally measured in tens of micrometers兲. This portion cated in Fig. 2. The girder design was modified in three specific
of the tensile behavior of UHPC is of interest in the design of ways for this research program. First, the conventional concrete
structural components; softening behavior is of little practical in- was replaced with UHPC. Second, the girder contained no mild
terest as crack localization and fiber pullout coincide with loss of steel reinforcement. Third, the strands in the girder were only
tensile capacity and component failure unless secondary load jacked to 55% of their ultimate strength due to concern over
paths are available. potential detrimental end region behaviors resulting from the
The research effort discussed herein focused on determining large amount of prestress on this relatively small cross section.
the specific tensile behaviors that this UHPC exhibits when sub- The compressive properties of the UHPC were measured both
jected to flexure in a full-scale prestressed AASHTO bridge through the testing of cylinders cast and cured alongside the beam
girder. Of particular interest are the shapes of Parts I and II of the and through the testing of cylinders produced as part of the re-
curve shown in Fig. 1, the spacing of the dispersed discrete cracks lated material characterization study 共Graybeal 2006a兲. The re-
at various strain levels, the flexural stiffness that a UHPC I-girder lease of strands was specified to occur when the compressive
displays, and the development of a design philosophy that ac- strength of the UHPC was above 10 ksi 共69 MPa兲. Recognizing
counts for the tensile behaviors of UHPC. the rapid strength gain this UHPC exhibits as it strengthens to-
It is recognized that the girder cross-sectional geometry tested ward 15 ksi 共103 MPa兲 and the different curing conditions that
herein is not necessarily a structurally efficient form for use with
are present within the girder as compared to within a cylinder, it
UHPC. However, the AASHTO cross section provides a true in-
is assumed that the compressive strength of the girder was 12 ksi
dication of full-scale structural behaviors of this concrete. Knowl-
共83 MPa兲 at strand release. The compressive strength of the
edge of these behaviors is necessary for subsequent structural
UHPC girder at the time of flexure testing was 28 ksi 共193 MPa兲
optimizations.
based on cylinder tests, and the modulus of elasticity was
8,100 ksi 共55.8 GPa兲 based on the elastic flexural behavior of the
girder. Time-dependent prestressing losses began at stressing and
continued for the life of the girder. As such, the compressive
εt,hardening
Stress Stress strength and modulus of elasticity expressed during prestressing
ft,max losses vary. Based on the strength to modulus of elasticity rela-
ft,1st tionship presented in Graybeal 共2007兲, a modulus of elasticity of
II 5,000 ksi 共34.5 GPa兲 was assumed to be representative of the
concrete stiffness at strand release and 6,000 ksi 共41.4 GPa兲 was
assumed to be representative of the concrete stiffness during pre-
stress losses.
III
I
Strain Crack Opening
Experimental Investigation
εt,1st εt,max wt,max
The 80 ft 共24.4 m兲 long girder was tested in flexure on a 78.5 ft
Fig. 1. Tensile behavior of UHPC 共Habel et al. 2006, with 共23.9 m兲 span. The girder was loaded in four point bending, with
permission兲 the point loads each located 3 ft 共0.91 m兲 from midspan as is

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 12 in. (305 mm) 1 in. (25 mm)
3 in. (76 mm) 5 in. (127 mm) 10 in. (254 mm)

6 in. (152 mm) 15 in. (381 mm)

Cross-section 20 in. (508 mm)
36 in. (914 mm)

18 in. (457 mm) Load Load

South Elevation T T

South Face
Bottom Face
North Face

3 ft (0.91 m) 3 ft (0.91 m)
78.5 ft (23.9 m)
Potentiometer
Strain Gage Note: Figure not drawn to scale.
T Tilt Meter

Fig. 2. Cross section, loading configuration, and instrumentation plan

shown in Fig. 2. The girder was instrumented with load cells, with 共1,920 kN m兲, corresponding to a load of approximately 78 kips
2 in. 共51 mm兲 gauge length surface-bonded electrical resistance 共350 kN兲. Also note the stability of the neutral axis depth during
strain gauges, with potentiometers, and with tilt meters. The ma- the unloading and reloading sequences.
jority of strain gauges were placed on the midspan cross section Significant audible cracking—which started when the load
in order to capture the strain profile and the neutral axis location reached 73 kips 共325 kN兲 and continued throughout the remain-
throughout the test. Midspan strain gauges were located on the der of the test—emanated from the girder. However, these crack-
top extreme fiber, bottom extreme fiber, and at 1, 3, 5, 10, 15, and ing sounds could not be individually correlated to cracks on the
20 in. 共25, 76, 127, 254, 381, and 508 mm兲 down from the top surface of the girder. In fact, at loads below approximately
extreme fiber. The potentiometers measured vertical deflection at 160 kips 共700 kN兲 the cracks were not visible with the naked eye.
midspan, at each load point, and at each quarter point. One tilt The test was halted overnight just after a peak load of 140 kips
meter measured the rotation of the mid-depth of the girder at each 共620 kN兲 and a deflection of 12 in. 共0.30 m兲 was reached. Neither
bearing location. inspection of the girder nor the instrumentation results showed
The girder was initially loaded in 4 kip 共17.8 kN兲 load steps. that any appreciable changes occurred overnight. Prior to resum-
After the girder began to sustain inelastic damage and exhibit ing the test the cracks on the bottom of the bottom flange were
reduced flexural stiffness, the test was incremented on midspan mapped. Fig. 5 shows photographic results of this mapping at six
deflections of 0.2 in. 共5 mm兲 and then 0.5 in. 共12.7 mm兲 until points along the length of the girder. The crack spacing near mid-
failure. Periodically throughout the test the load level was de- span was approximately 0.2 in. 共5 mm兲. This spacing had in-
creased to approximately 75% of the maximum load previously creased to 0.4 in. 共10 mm兲 at 10 ft 共3 m兲 from midspan, 1 in.
achieved. The load was then increased and the residual stiffness 共25 mm兲 at 16 ft 共4.9 m兲, and 5 in. 共127 mm兲 at 22 ft 共6.7 m兲.
of the girder was measured. The loading of the girder continued The peak applied moment prior to failure was 3,225 k ft
until failure, which was defined as pullout of fiber reinforcement 共4,370 kN m兲, corresponding to a combined applied plus self-
crossing a dominant crack and the subsequent rupturing of the weight moment of 3,540 k ft 共4,800 kN m兲. After the deflection
prestressing strands. increment bringing the applied load to 178 kips 共790 kN兲 was
Fig. 3 shows the midspan load-deflection response of the reached, the midspan deflection continued to increase as the hy-
girder from initial load application through the peak load of
178 kips 共790 kN兲 occurring at a deflection of 18.5 in. 共0.47 m兲.
The load-deflection response shows that the girder behavior Midspan Deflection (mm)
0 100 200 300 400 500
began to soften at 75 kips 共330 kN兲 of applied load and a deflec- 200
tion of approximately 3 in. 共76 mm兲. The girder exhibited signifi- 175 800
Total Applied Load (kips)

Total Applied Load (kN)

cant additional load-carrying capacity after initial cracking, due in 150

700
part to the fiber reinforcement. 125
600
The 16 strain gauges located at midspan were used to create a 100
500
400
strain profile over the depth of the girder and to identify the 75
300
location of the neutral axis. Individual gauges were used until 50 200
their readings became unreliable due to cracking of the underly- 25 100
ing concrete. The data from both sides of the girder correlated 0 0
well. Fig. 4 shows the recorded midspan top flange strain, along 0 5 10 15 20
with the interpolated neutral axis depth values and extrapolated Midspan Deflection (in.)

bottom flange strain values. The figure shows that the neutral axis
began to rise at an applied moment of approximately 1,415 k ft Fig. 3. Flexural behavior of AASHTO Type II UHPC girder

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 14000 6 共Graybeal 2006b兲. Thus the overall effective prestressing force
(152)
12000 Neutral Axis Depth was 455 kips 共2,020 kN兲 at an eccentricity of 9.2 in. 共0.23 m兲
10000 12 below the elastic neutral axis.
(305)

(in. (mm) from top))

A rectangular stress block approximation for the compressive

Neutral Axis Depth

8000
behavior of concrete in flexure allows for a simplified means to
Microstrain

6000 18
4000
(457) predict the flexural behavior of prestressed concrete girders. This
Bottom Flange Strain
empirical relationship has been found to accurately approximate
2000 24
(610) the nonlinear stress-strain behavior of conventional concrete and
0
is thus widely used in flexural provisions of reinforced concrete
-2000 Top Flange Strain 30 design specifications. However, two of the assumptions inherent
(762)
-4000 in United States design codes 共AASHTO 2007; ACI 2005兲 that
-6000 36 use this analytical technique are violated by the behaviors of this
(914)
0 20 40 60 80 100 120 UHPC. First, this UHPC exhibits a compressive stress-strain re-
Load Step sponse that is nearly linear to high stress levels and more closely
resembles a triangular stress distribution than the familiar para-
Fig. 4. Strain profile results for midspan of girder bolic distribution of conventional concrete 共Graybeal 2007兲. Al-
though this behavior could be accounted for through a
modification of the parameters of the rectangular stress block,
draulically actuated load decreased slightly. After this load step these design codes currently do not contain provisions allowing
was reached and just prior to failure, a single large crack was the proper modifications to occur. Second, this UHPC exhibits a
observed in the bottom flange and web below the west load point. sustained tensile capacity after cracking to high tensile strain lev-
Unlike any other cracks in the girder, this crack was of sufficient els 共Graybeal 2006b兲. Code-based ultimate flexural capacity cal-
width to be clearly visible to observers from a distance of 16 ft culations assume that the concrete carries no tensile force, thus
共5 m兲 and was indicative of fiber pullout occurring at this loca- these calculations may be significantly in error if applied to
tion. The fiber pullout caused a local stress increase in the strands UHPC. For both of these reasons, the use of the rectangular stress
and precipitated strand rupture, with the girder separating into block approximation and associated analytical methods to predict
two unconnected pieces. the behavior of UHPC girders may not be warranted.
To illustrate this fact, the following presents an analysis of the
UHPC girder based on the standard code-based analytical
Analysis of Test Results method. This result is compared to the experimentally determined
flexural capacity of the UHPC girder and to the capacity that this
code-based method would predict for a conventional HPC pre-
Predictions of Girder Flexural Capacity stressed girder. Recall that the peak load applied to the girder was
The initial, elastic behavior of the girder was analyzed through a 178 kips 共790 kN兲, corresponding to an applied moment of
strain compatibility analysis. This analysis provided the state of 3,225 k ft 共4,370 kN m兲.
stress in the girder at the initiation of live load application and A girder geometrically equivalent to that tested in this study
thus is the starting point for all subsequent flexural analyses. The was analyzed assuming f ⬘c = 8 ksi 共55 MPa兲, ␣ = 0.85, and ␤1
time-dependent stress losses in the strands resulting from concrete = 0.65. These calculations resulted in a calculated flexural capac-
creep, concrete shrinkage, and strand relaxation were approxi- ity of 1,700 k ft 共2,300 kN m兲, a neutral axis depth of 16.4 in.,
mated to total 14%. This loss value is based on specific material and a bottom fiber tensile strain at failure of 0.0036. Given that
behaviors observed in the associated study of this UHPC 共Gray- the rectangular stress block model is widely accepted as being
beal 2006a兲 and its derivation is presented in detail elsewhere representative of the compressive flexural behavior of conven-

Fig. 5. Crack spacing on bottom flange of girder after 12 in. 共305 mm兲 of midspan deflection

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tional and high-performance concretes, this calculation demon- 5000

Bottom Flange Tensile Strain

Note: 1 in. = 25.4 mm
strates that UHPC I-girders can display significantly increased 4000
flexural capacity as compared to similar HPC girders.

(microstrain)
The analysis was then completed again with two more sets of 3000
parameters: 共1兲 f ⬘c = 28 ksi 共193 MPa兲, ␣ = 0.85, ␤1 = 0.65; and 共2兲 Equation (1)
2000
f ⬘c = 28 ksi 共193 MPa兲, ␣ = 0.75, ␤1 = 0.667. The first of these is
equivalent to the analysis above with only the compressive 1000
strength being modified. The second allows the rectangular stress
0
block to mimic the compressive behavior of a concrete that be- 0.1 1.0 10.0
haves linear elastically through failure. In both cases the calcu- Crack Spacing (in.)
lated flexural capacity is approximately 73% of the observed
capacity. However, to reach this capacity both of these calcula- Fig. 6. Tensile strain related to flexural crack spacing
tions predict that the extreme tensile fiber strain would be greater
than 0.016 and the neutral axis depth would be approximately
5 in. 共130 mm兲 below the top of the girder. This strain is well midspan of the test girder are presented in Fig. 4. At the load step
beyond the experimentally observed strain at which the fiber re- where crack spacing observations were recorded the extreme ten-
inforcement pulled out and the prestressing strands ruptured in the sile fiber strains along the length of the girder were then deter-
physical girder test. This neutral axis is also much higher than mined. The prestress forces, the girder self-weight, and the
was observed in the laboratory test just prior to girder failure. In applied loads are accounted for within these calculations.
total, these results indicate that common design code-based cal- The crack spacing versus strain results are presented in Fig. 6.
culation methods do not provide an accurate representation of the This figure, presented in semilog format, also shows a curve rep-
flexural behaviors observed. resenting the best-fit equation for the results. Eq. 共1兲 defines this
Finally, a direct comparison of the flexural capacity results was curve with the strain, ␧, in microstrain calculated from the crack
completed between the girder tested in this study and a girder spacing, scr, in inches or millimeters. This equation has an
tested by Russell and Burns 共1993兲. Russell and Burns tested a R-squared value of 0.952
46 in. 共1.17 m兲 deep decked I-girder. This girder had a 6 ksi
共41 MPa兲, 72 in. 共1.83 m兲 wide top flange, and was prestressed 500 40
with 28 0.5 in. 共12.7 mm兲 strands. This 28% deeper girder dis- ␧ = 450 + + with scr in inches
played a similar moment capacity to the UHPC girder tested in
冑scr 2
scr
the present study. Achieving a similar flexural capacity with a
significantly shallower girder demonstrates just one of the poten- 2,520 25,800
tial applications for which UHPCs may find use. ␧ = 450 + + with scr in millimeters 共1兲
冑scr 2
scr

Cracking Behavior of UHPC

Tensile Stress-Strain Behavior of UHPC
The tensile cracking behavior of this prestressed UHPC girder
was observed to be significantly different than would be expected The effective structural use of UHPC requires knowledge of its
in a conventional concrete girder. As shown in Fig. 5, the extreme uniaxial stress-strain behavior. An analytical derivation of the
tensile fiber of this prestressed girder displayed a very high crack uniaxial stress-strain response of the UHPC was obtained from
density. Recall that the crack spacing observations were recorded the experimental results of the flexural testing of this girder. One
when the total applied load on the girder was 140 kips 共620 kN兲. principal component of this derivation was the strain profile on
The crack spacing at any cross section along the span was ob- the midspan cross section of the girder. Another important experi-
served to be inversely proportional to the maximum tensile strain mental result was the moment applied to the midspan of the girder
experienced by the bottom fiber as a result of flexural forces throughout the test. Finally, both an approximation of the com-
applied to the girder. Thus, the load and strain data collected pressive stress-strain response of UHPC and an approximation of
during the testing of the girder, along with the crack spacing the tensile stress-strain response of 270 ksi low relaxation pre-
density observations, allow for the derivation of a relationship stressing strand were necessary and were referenced from Gray-
between the curvature-based UHPC extreme tensile fiber strain beal 共2007兲 and Caltrans 共2006兲, respectively.
and the observed crack spacing on the extreme tensile fiber of the Results from the previously discussed strain compatibility
prestressed girder. This relationship could prove useful for two analysis were used here again to define the state of strain and
reasons. First, crack spacing could provide an indication of maxi- stress in the girder prior to the initiation of flexural testing. From
mum tensile strain experienced by a UHPC structural component this analysis, the compressive stresses in the concrete at the ini-
whose load history is unknown. Second, the crack spacing could tiation of the flexure test were calculated to be 1.0 ksi 共6.9 MPa兲
provide an indication of whether a UHPC laboratory component at the top of the midspan cross section and 1.5 ksi 共10.3 MPa兲 at
exhibited the expected tightly spaced tensile cracking behavior the bottom of the midspan cross section. The calculated stresses
necessary for significant postcracking tensile capacity. in the strands ranged from 122 ksi 共843 MPa兲 in the top flange
The basic assumption required to initiate the derivation of this strands to 119 ksi 共820 MPa兲 in the lowest bottom flange strands.
relationship is that the girder is undergoing pure flexural behavior These results provide the basis for the moment-curvature
with plane sections remaining plane. Under pure flexural behav- analysis that was completed to determine the full uniaxial stress-
ior, the observed neutral axis location and curvature of the mid- strain behavior of UHPC in this prestressed I-girder. In contrast to
span cross section can be used to calculate the midspan effective a standard moment-curvature analysis wherein known concrete
strain in the extreme tensile fiber of the girder at any applied and steel stress-strain behaviors are applied to a cross section
moment. These calculated strains in the extreme tensile fiber at whose curvature is adjusted until internal and external moments

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 5 (34)

0
-4000 -2000 0 2000 4000 6000 8000 10000
Microstrain
-5 (-34)

4 (-28)
-10 (-69) Transitional Behavior

Stress (ksi (MPa))

3 (-21)

Stress (ksi (MPa))

-15 (-103)
2

-20 (-138) 1
Post Cracking Behavior
-25 (-172) 0
-500 0 500 1000 1500 2000
-1 (-7) Microstrain
-30 (-207)
-2 (-14)

Fig. 7. Analytically derived uniaxial stress-strain behavior of UHPC in prestressed I-girder

reach equilibrium, in this analysis the tensile portion of the con- crack widths as compared to full-scale prestressed structural com-
crete stress-strain response is the unknown variable. ponents. The behavior of small-scale tension specimens is more
The midspan cross section was discretized into 72 concrete indicative of lower bound tensile behaviors than of the behaviors
slices and five prestressing strand slices parallel to the neutral that could be expected if secondary tensile load paths 共i.e., strands
axis. The measured strain values from the strain gauges on the or mild steel reinforcement兲 were present as is the case in the
midspan cross section were used to define the state of strain in tension flange of a prestressed girder. Tension stiffening as exhib-
each slice over the depth of the midspan cross section at each load ited by fiber reinforced concretes also contributes to the girder’s
step. The tensile portion of the uniaxial stress-strain response for increased tensile capacity 共Abrishami and Mitchell 1997; Bis-
the UHPC was then assumed and the forces on the cross section choff 2003; Chao et al. 2007兲.
at each load step were determined. The summation of the forces To complete this analysis, the unloading and reloading behav-
in the concrete and the strands were compared with one another to iors of UHPC were considered. After UHPC undergoes inelastic
ensure that the internal forces on the cross section were in equi- deformation, subsequent unloading and reloading of the concrete
librium at each load step. After many trial UHPC tensile stress- is modeled as linear elastic behavior with a reduced effective
strain responses, the response shown in Fig. 7 was selected as modulus of elasticity. This behavior continues until the previous
being acceptably representative of the UHPC’s behavior through- maximum strain in the load history is exceeded. In this analysis,
out the loading history. On all loading steps, the absolute values UHPC not subjected to peak tensile or compressive strains was
of the concrete and strand forces are within 7% of one another. assumed to behave linear elastically with a stiffness equal to the
Fig. 7 can be considered to be composed of four separate secant modulus of elasticity of the largest strain ever attained.
curves, each defining a portion of the behavior. In compression, This model was used in both tension and compression. For the
the relationship defined elsewhere for UHPC compressive behav- postcracking tensile behaviors, the origin for the secant modulus
ior 共Graybeal 2007兲 is used, with a compressive strength of 28 ksi was shifted to the location where the dotted curve intersects the
共193 MPa兲 and a modulus of elasticity of 8,100 ksi 共55.8 GPa兲. abscissa in Fig. 7. This location was chosen based on the
The initial tensile behavior of UHPC is considered to be linear moment-curvature analysis and is consistent with the fact that
elastic with an 8,100 ksi 共55.8 GPa兲 modulus of elasticity; thus it cracking 共and the resulting load transfer, partial fiber debonding,
is coincident with the initial slope of the compression behavior. and fiber bending兲 necessarily causes an irrecoverable displace-
The postcracking behavior curve for UHPC is indicated in Fig. 7 ment to occur across a crack. In the tensile cracking transition
with an initial modulus of elasticity of 5,000 ksi 共34.5 GPa兲. The region, the secant moduli were calculated just before and after the
intermediate portion of the behavior is the tensile transition zone transition region. The secant values in the transition region were
during the initial cracking of the concrete. This transition allows then linearly interpolated from these bounding values.
the UHPC to transform from an uncracked, elastic material into a A comparison of the analytical and experimental results is
significantly cracked material that is still capable of carrying ten- shown in Fig. 8. The derived internal forces on the cross section
sile loads. were used to determine the internal moment on the girder mid-
Note that according to this model the UHPC carries 3.0 ksi span at each load step. The load cell readings captured throughout
共20.7 MPa兲 of tensile stress before the cross section becomes suf- the loading history were used to determine the external moments
ficiently cracked to cause a change in the slope of the tensile on the midspan cross section. The experimental and analytical
stress-strain response. This value is well above the UHPC tensile results compare well until the final load steps of the test. Toward
cracking strength values ranging from 1.0 to 1.5 ksi 共7 – 10 MPa兲 the end of the loading history the model tends to overestimate the
that have been experimentally determined elsewhere 共Chanvillard internal moment, which is explainable by two reasons. First,
and Rigaud 2003; Rossi 1997; Li and Lepech 2004; Graybeal given that only the strain gauges bonded to uncracked concrete
2006a兲. Other studies within this research program have shown provided reliable results, the experimentally determined strain
that tensile behaviors in small-scale specimens without discrete profile on the cross section became less accurate at high loads
reinforcement tend to occur with less crack density and larger when the cross section was significantly cracked and the neutral

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 4000 Applied Moment (kN-m)
5000 0 1000 2000 3000 4000 5000
3500
60000

Midspan Moment (kN-m)m

3000
Midspan Moment (k-ft)m

4000

Moment of Inertia (mm )

4
Moment of Inertia (in )
50000

4
2.0E+10
2500 40000
3000 1.5E+10
2000 30000
1.0E+10
1500 2000 20000
External (Virtual Work Analysis) 5.0E+09
1000 10000
Internal (Moment-Curvature Analysis)
1000
500 Experimental Result (External Moment) 0 0.0E+00
Stress-Strain Model Result (Internal Moment) 0 1000 2000 3000 4000
0 0 Applied Moment (k-ft)
0 20 40 60 80 100 120
Load Step Fig. 9. Externally and internally determined effective moment of
inertia
Fig. 8. External and internal moments on midspan cross section

used here as this model accounts for cracking of UHPC through a

decrease in stress capacity, thereby smearing the concentrated
axis had moved closer to the top flange. Second, the analytical larger forces carried by the fibers, instead of through decreasing
method used to determine the uniaxial stress-strain response was the effective area of the cross section and maintaining the higher
more robust at lower strains as this portion of the stress-strain stress levels that actually occur in the fibers. The resulting flexural
response was utilized during the calculations performed at every stiffness of the girder cross section is plotted as a function of the
load step, while higher strain portions of the response were only applied moment in Fig. 9. Note that the plot assumes a constant
engaged at higher applied load levels. Therefore, greater certainty modulus of elasticity at 8,100 ksi 共55.8 GPa兲.
is built into the lower tensile and compressive strain ranges than The check on this result is provided through a comparison
can be expected in the higher strain ranges. with the overall flexural stiffness exhibited by the girder through-
out the test. A virtual work analysis was performed to determine
Flexural Stiffness the flexural stiffness of any cross section along the length of the
girder as a function of the self-weight and applied moment on the
The constitutive model for the stress-strain response of this girder at that location. In this analytical technique, the moment on
UHPC was also analyzed in terms of the flexural stiffness of the the girder is multiplied by the moment generated in the equilib-
girder. Although a UHPC cross section is generally modeled as rium system by a “dummy” load at the location where the deflec-
carrying tensile forces across tensile cracks, in reality only the tion is desired. Eq. 共2兲 provides the basis for this analysis. The
internal steel fiber reinforcement that comprises a small fraction integration, or in this case the summation, of nearly 1,000 discrete
of the volume of the UHPC carries any significant tensile forces cross-sectional slices along the length of the girder provides the
across cracks. Additionally, as discussed earlier, the flexural ten- solution. Note that shear deformations were excluded from the
sile cracks in UHPC tend to be of very small width and very analysis as preliminary calculations demonstrated that they were
closely spaced. As such, the postcracking tensile flexural behavior of minimal consequence
of UHPC is actually comprised of elastic straining of uncracked

冕
Length
concrete, elastic straining of steel fibers bridging cracks, and to a M共x兲
lesser extent plastic bending of nonperpendicular fibers bridging Deflection = m共x兲 dx 共2兲
0 EI共x兲
cracks and partial debonding of fibers at their interface with the
UHPC matrix. The occurrence of these primarily elastic behaviors In this analysis the UHPC modulus of elasticity was held constant
throughout a flexural member, even after cracking, necessitate and the moment of inertia was allowed to vary, with their product
that the constitutive stress-strain model be consistent with defining the overall flexural stiffness along the length of the
the deformations observed. As will be demonstrated below, the girder, EI共x兲.
flexural deformations of this girder can be modeled through a The full pre- and postcracking behavior of the girder was then
rational, mechanics-based methodology. determined. First, a potential flexural stiffness, EI, was proposed
The midspan cross-sectional stiffness that results from the for each self-weight plus applied load moment, M, on the girder.
uniaxial stress-strain behavior derived previously and presented in The virtual work analysis was performed to determine the result-
Fig. 7 was calculated for each load step. As with normal concrete, ing deflections and rotations at each of the potentiometers and tilt
UHPC flexural stiffness in compression and in tension prior to meters throughout the test. The analytical results were then com-
cracking can be determined through a straightforward calculation pared with the experimental observations at the potentiometers
based on the uncracked moment of inertia and the modulus of and tilt meters, and the EI versus M relationship was revised
elasticity of the concrete at the strain level that it is experiencing. accordingly. The analyses were repeated until a sufficiently accu-
After cracking, tensile behavior of UHPC is very different from rate flexural stiffness was determined. Fig. 9 graphically presents
conventional concrete, with the UHPC carrying stress and exhib- the relationship between the moment on any cross section of the
iting axial stiffness. The postcracking deformation behaviors of girder and the girder’s observed flexural stiffness at that cross
this concrete were modeled through a combination of the un- section as determined via this virtual work analysis.
cracked moment of inertia, the uniaxial stress-strain response, and In terms of the flexural stiffness of UHPC, the result generated
the secant modulus of elasticity at the peak tensile strain level that by the moment-curvature model from the midspan cross section is
the UHPC had experienced. The uncracked moment of inertia is consistent with the result generated by the virtual work model of

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the girder’s overall behavior. The internal flexural stiffness result 5 (34)
x
is always within 10% of the external result. The comparability of 0
these results shows that the UHPC flexural stiffness can be ad- -0.004 -0.002 0 0.002 0.004 0.006 0.008
-5 (-34)
dressed directly through knowledge of the constitutive stress- Strain
strain relationship. -10 (-69)

Stress (ksi (MPa))

-15 (-103)

-20 (-138)
Flexural Design of UHPC I-Girders

x
-25 (-172)

Based on the experimentally observed behaviors and the analyti- -30 (-207)
cal results, a flexural design philosophy for UHPC I-girders has
been developed. The design philosophy detailed herein is similar Fig. 10. Simplified uniaxial stress-strain behavior for I-girder flexural
to portions of some UHPC structural design procedures detailed design
elsewhere 共Association Française de Génie Civil 2002; Casanova
and Rossi 1996; Gowripalan and Gilbert 2000; Uchida et al.
2005兲.
The design of a prestressed UHPC I-girder for flexure requires 0.85 times the observed steam-treated compressive strength
two things. First, a conservative approximation of the UHPC’s of this concrete;
uniaxial stress-strain response must be applied to the cross sec- 2. Tensile capacity of 1.5 ksi 共10.3 MPa兲, corresponding to 0.5
tion. Second, the occurrence of the expected flexural behaviors times the pre- and postcracking uniaxial tensile capacity de-
must be ensured. Of primary importance, sufficient prestressing rived from the girders response;
strands or mild steel reinforcement must exist in the primary flex- 3. Modulus of elasticity of 7,600 ksi 共52.4 GPa兲, corresponding
ural tensile regions so that cracks in the UHPC exhibit small to the elastic modulus that this UHPC exhibits as determined
widths and close spacings. Without sufficient gross reinforcement via compression tests on cylinders; and
restraining the tensile flexural regions, individual cracks will 4. Limiting tensile strain of 0.007 corresponding to 70% of the
begin to widen as the fibers pull out and tightly spaced cracks tensile strain observed in the extreme tensile fiber of the
such as that shown in Fig. 5 will not occur. girder just prior to gross cracking, strain localization, and
Determining a sufficiently conservative approximation of girder failure.
UHPC uniaxial stress-strain behavior depends on the intended Fig. 10 graphically presents this simplified stress-strain behavior.
structural application and on the prescribed design limits. In a Using this stress-strain response, the predicted ultimate ap-
situation where cracking of the girder is not allowed at service plied moment capacity of the UHPC girder discussed herein is
loads, the girder can be designed using typical prestressed con- 2,440 k ft 共3,310 kN m兲, 76% of the experimentally determined
crete design procedures with the service loads limited to a per- applied moment capacity. As compared to the conventional meth-
centage of the first-cracking moment of the UHPC. In a situation ods for analyzing prestressed concrete girders discussed earlier in
where a minimal amount of cracking will be allowed at service this paper, the method presented here is admittedly significantly
loads, a postcracking uniform tensile stress capacity will need to more complex. However, it is also rational in that it conserva-
be assumed. Finally, for the strength limit states, a full compres- tively addresses the actual behaviors exhibited by this UHPC
sive and tensile stress-strain response will be required. This re- I-girder. The fact that this moment capacity of 2,440 k ft
sponse will include an effective tensile strain that causes fiber 共3310 kN m兲 is only 4% greater than the value calculated via the
pullout, a limiting tensile stress capacity relevant to strains below method presented in United States design codes is of little conse-
the fiber pullout strain, and a limiting compressive strength. quence; the specifications in these design codes do not address the
Until a significant number of full-scale flexure tests are com- actual behaviors of UHPC and thus would be unlikely to consis-
pleted, it will not be possible to present a calibrated set of con- tently predict capacities corresponding to a particular percentage
servative parameters for use in the flexural design of UHPC of the true ultimate capacity.
prestressed girders. Nonetheless, a suggested uniaxial stress-strain
response for UHPC based on the results detailed earlier in this
paper could be described as follows. First, the UHPC could be Conclusions
assumed to behave linear elastically in compression up to 0.85
times the compressive strength. This is reasonable since this Based on the results of this investigation, including the experi-
UHPC has been demonstrated to remain within 5% of linear elas- mental testing to flexural failure of a UHPC I-girder, the follow-
tic behavior up to over 90% of its compressive strength 共Graybeal ing conclusions are drawn:
2006a兲. Second, UHPC subjected to tensile strains below the ten- 1. UHPC I-girders will display flexural capacities larger than
sile pullout strain could be assumed to behave in an elastic- those of conventional concrete girders with similar cross-
perfectly plastic fashion at a conservative percentage of the sectional geometry;
postcracking tensile capacity. This is reasonable since it allows 2. The interaction of the fiber reinforcement and the UHPC
the designer to utilize some tensile capacity while staying within matrix allows small width, closely spaced cracks to occur
the envelope defined by experimentally observed tensile stress- and allows the UHPC to carry tensile loads after cracking;
strain behaviors. Finally, a limiting tensile strain capacity must be 3. The crack spacing in the tension flange of a UHPC I-girder is
defined as fiber pullout is likely to occur prior to strand rupture inversely proportional to the maximum tensile strain ob-
and may lead to girder failure. served in said cracked region;
For the UHPC discussed herein, a uniaxial stress-strain re- 4. A full uniaxial stress-strain response of UHPC subjected to
sponse for use in design could include the following: flexural loading in a prestressed I-girder was determined.
1. Compressive strength of 24 ksi 共165 MPa兲 corresponding to This rationally determined stress-strain response correlated

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 well with the girder’s experimentally recorded load and de- Abrishami, H., and Mitchell, D. 共1997兲. “Influence of steel fibers on
flection behavior; and tension stiffening.” ACI Struct. J., 94共6兲, 769–776.
5. The flexural design of UHPC I-girders can be completed in a American Concrete Institute 共ACI兲. 共2005兲. “Building code requirement
rational manner through the use of a conservative approxi- for structural concrete.” ACI-318–05, Detroit.
mation of the stress-strain behavior. Association Française de Génie Civil. 共2002兲. Interim recommendations
for ultra high performance fibre-reinforced concretes, Paris, France.
Acknowledgments Bischoff, P. 共2003兲. “Tension stiffening and cracking of steel fiber-
reinforced concrete.” J. Mater. Civ. Eng., 15共2兲, 174–182.
The research which is the subject of this paper was funded by the California Department of Transportation 共Caltrans兲. 共2006兲. “Seismic de-
Federal Highway Administration. The writer gratefully acknowl- sign criteria.” June.
Casanova, P., and Rossi, P. 共1996兲. “Analysis of metallic fibre-reinforced
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sarily indicate approval or endorsement of the findings, opinions,
Chanvillard, G., and Rigaud, S. 共2003兲. “Complete characterization of
conclusions, or recommendations either inferred or specifically
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expressed herein by the Federal Highway Administration or the
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Mich., Springer, The Netherlands.
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“Large scale tensile tests of high performance fiber reinforced cement
The following symbols are used in this paper: composites.” Proc., 5th Int. RILEM Workshop on High Performance
EI共x兲 ⫽ flexural stiffness of girder at any location x; Fiber Reinforced Cement Composites (HPFRCC5), Mainz, Germany.
f ⬘c ⫽ compressive strength of concrete; Gowripalan, N., and Gilbert, R. I. 共2000兲. Design guidelines for RPC
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f t,1st ⫽ concrete tensile stress at first cracking; Graybeal, B. A. 共2006a兲. “Material property characterization of ultra-high
M共x兲 ⫽ moment on girder from self-weight plus performance concrete.” Rep. No. FHWA-HRT-06-103, Federal High-
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scr ⫽ crack spacing; Highway Administration, Washington, D.C.
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uniformly stressed compression zone has correct
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␤1 ⫽ ratio of depth of equivalent uniformly
Li, V., and Lepech, M. 共2004兲. “Crack resistant concrete material for
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transportation construction.” Proc., Transportation Research Board
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