AN ABSTRACT OF THE THESIS OF
Melissa J. Robelo for the degree of Master of Science in Civil Engineering
presented on July 6, 2004.
Title: Analysis of Diagonally Cracked Conventionally Reinforced Concrete
Girders in the Service Load Range
Abstract Approved:
Redacted for privacy
Christopher C.
Performance
evaluation
of conventionally
reinforced
concrete
(CRC)
bridge
superstructure elements with diagonal cracks is of interest to the bridge engineering
community. Standardized methods to predict service-level stress magnitudes in cracked
bridge girders under combined bending and shear forces are not available. An analysis
procedure was developed to determine the response of CRC bridge girders with existing
diagonal cracks. The method estimates the stirrup stress range without prior knowledge of
the previous stirrup strain history.
Modified Compression Field Theory (MCFT), a
sectional analysis procedure, is used in the current specifications to predict capacity of
CRC and prestressed concrete beams and relies on equilibrium and compatibility
conditions based on an initial uncracked section. In this research, the method was used to
predict service-level performance for cracked sections. Effective response prediction of a
previously cracked beam was achieved through alteration of the constitutive relationships
to permit softening behavior of the concrete to begin much earlier in the loading history.
Validation of the procedure was performed through comparison of analytically predicted
stirrup strains with those from full-size laboratory specimens. Empirical data were found
to correspond well with the predicted responses. The cracked section analysis procedure
was then utilized to estimate the potential for low-cycle fatigue damage of an in-service
bridge. A moment and shear interaction surface corresponding to stirrup yielding was
estimated and compared with the load effects produced from a data set of over 14,000
permitted trucks to estimate the anticipated life of the bridge based on low-cycle fatigue
damage.
©Copyright by Melissa J. Robelo
July 6, 2004
All Rights Reserved
Analysis of Diagonally Cracked Conventionally Reinforced Concrete Girders in
the Service Load Range
by
Melissa J. Robelo
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Presented July 6, 2004
Commencement June 2005
Master of Science thesis of Melissa J. Robelo presented on July 6, 2004.
APPROVED:
Redacted for privacy
Major Professor, representing Civil Engineering
Redacted for privacy
Head of the Department of Civil, Construction, and Environmental Engineering
Redacted for privacy
Dean of the Graduate School
I understand that my thesis will become part of the permanent collection of Oregon
State University libraries. My signature below authorizes release of my thesis to
any reader upon request.
Redacted for privacy
Melissa J. Robelo, Author
ACKNOWLEDGEMENTS
I would like to express my sincere appreciation to my Major Professor, Dr.
Christopher Higgins, whose encouragement and support have made this thesis possible. I
can honestly say that it has been a joy working with Dr. Higgins. I also would like to
thank my Minor Professor, Dr. Solomon Yim, and committee members Dr. Thomas
Miller, and Dr. Cherri Pancake. A special thank you is given to Dr. Pancake for being a
constant inspiration and supporting me in so many ways through the years.
I would also like to express my gratitude to my fellow graduate students on the
ODOT project. In particular I would like to thank Tanarat Potisuk, Aeyoung Lee, and
Theresa Daniels for all their help and support along the way.
Finally, I am particularly grateful to my mom, Lucia Robelo, not only for her
constant moral support and encouragement, but for being someone that I have always
looked up to and been inspired by.
TABLE OF CONTENTS
Introduction
.1
Background
.3
AnalysisApproach ......................................................................................................... 16
Amplification Factor for Yielding of the Stirrups ...................................................... 23
ReducedCracking Stress ............................................................................................ 26
Results ............................................................................................................................ 29
-t Prediction ........................................................................................................... 29
M-V Interaction Yield Surfaces ................................................................................. 41
Conclusions .................................................................................................................... 52
References ...................................................................................................................... 54
Appendix ........................................................................................................................ 56
LIST OF FIGURES
Figure
Eg
1.
Principal stress trajectories [MacGregor, 1997] .......................................................... 3
2.
Mohr'scircleofstress .................................................................................................. 4
3.
Truss analogy model for reinforced concrete members in shear [Collins and Vecchio,
1988] ............................................................................................................................ 5
4.
Average strains of a cracked element [Collins and Mitchell, 1991] ............................ 6
5.
Average stresses of a cracked element [Collins and Vecchio, 1988] ........................... 6
6.
Panel tests performed by Collins and Vecchio [Collins and Mitchell, 1991] .............. 6
7.
MCFT compressive stress-strain relationship [Collins and Mitchell, 1991] ............... 8
8.
Concrete tension before and after cracking [Collins and Vecchio, 1988] ................... 8
9.
Crack spacing parameters [Collins and Mitchell, 1991] ............................................ 12
10. Detailed analysis [Collins and Mitchell, 1991] .......................................................... 14
11. Approximate analysis [Collins and Mitchell, 1991] .................................................. 14
12. Comparison of detailed and approximate procedures [Collins and Vecchio, 1988]
15
13. Specimen T12 used for analysis correlation .............................................................. 17
14. Specimen IT 12 used for analysis correlation ............................................................. 18
15.
Specimen T24 used for analysis correlation .............................................................. 18
16. T 12 specimen laboratory setup and crack map .......................................................... 19
17. IT 12 specimen laboratory setup and crack map ......................................................... 20
18. T24 specimen laboratory setup and crack map .......................................................... 21
19. M-V capacity for specimen T12 ................................................................................ 22
20. M-V capacity for specimen ff12 ............................................................................... 22
LIST OF FIGURES (Continued)
Figure
21. M-V capacity for specimen T24 ................................................................................ 23
22.
Average versus maximum transverse strain ............................................................... 24
23.
Average versus maximum transverse strain at high M/V .......................................... 25
24.
Cracking strain, 8cr determination of specimen 2T12 north ................................... 27
25.
Cracking strain,
26.
Cracking strain, 6cr ,determination of specimen
27. V-se
28.
at north side, M/V=4 2T12 North ..................................................................... 33
at south side, M/V=4 for specimen 21T12 ........................................................ 34
at south side,
MN4 for specimen 10T24
....................................................... 35
Location of strain gage at south side, MIV=4 for specimen
37. V-es
38.
at south side, M/V=4 for specimen 2T12 ......................................................... 32
Location of strain gage at south side, M/V=4 for specimen 21T12 ........................... 34
35. V-
36.
at south side, M1V4 for specimen 3T12 ......................................................... 31
Strain gage location relative to crack at north side, M/Vz4 for specimen 2T12 ....... 33
33. V-t
34.
south .................................. 28
Strain gage location relative to crack at south side, M/V4 for specimen 2T12 ....... 32
31. V-e,
32.
21T12
Strain gage location relative to crack at south side, MIV=4 for specimen 3T12 ....... 31
29. V-t
30.
,determination of specimen 2T12 south ................................... 27
1 0T24 ...........................
35
at north side, M/V=6 for specimen 2T12 ......................................................... 36
Strain gage location relative to crack at north side, MIV=6 for specimen 2T12 ....... 36
LIST OF FIGURES (Continued)
Figure
Page
39. V-es at south side, MIV=6 for specimen 2T12 ......................................................... 37
40. Strain gage location relative to crack at south side, MIV=6 for specimen 2T12 ....... 37
41. V-se at south side, MIV=6 for specimen 21T12 .................................................................... 38
42. Location of strain gage at south side, MJV=6 for specimen 21T12 ........................... 38
43. V-st at north side, MIV=6 for specimen 21T12 ........................................................ 39
44. Location of strain gage at north side, MIV=6 for specimen 21T 12............................ 39
45. V-se at north side, MIV=6 for specimen 1 0T24 ....................................................... 40
46. Location of strain gage at north side, MIV=6 for specimen 1 0T24 ........................... 40
47. M-V yield surface and predicted capacities for specimen Ti 2 .................................. 42
48. M-V yield surface and predicted capacities for specimen IT 12 ................................ 43
49. M-V yield surface and predicted capacities for specimen T24 .................................. 44
50. Sections A and B from McKenzie River Bridge........................................................ 45
Si. McKenzie River Bridge at section A ......................................................................... 46
52. McKenzie River Bridge at section B ......................................................................... 46
53. M-V yield curve for McKenzie River Bridge girder section A ................................. 47
54. M-V yield curve for McKenzie River Bridge girder section B ................................. 48
55. McKenzie River Bridge at section A using AASHTO LRFD LL distribution factors.
......................................................................................................... 50
56. McKenzie River Bridge at section B using AASHTO LRFD LL distribution factors.
......................................................................................................... 50
LIST OF TABLES
Table
Shear to produce yield as % of AASHTO-99 MCFT for specimen T12 ................... 42
2.
Shear to produce yield as % of AASHTO-99 MCFT for specimen ..................
1T12
43
3.
Shear to produce yield as % of AASHTO-99 MCFT for specimen T24 ................... 44
4.
Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River Bridge
girdersection A .......................................................................................................... 47
5.
Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River Bridge
girdersection B .......................................................................................................... 48
Analysis of Diagonally Cracked Conventionally Reinforced
Concrete Girders in the Service Load Range
Introduction
The AASHTO 2003 LRFD Specification contains a relatively new procedure for design
and analysis of reinforced concrete beams and beam-columns for shear. The method,
developed by Collins and Vecchio at the University of Toronto is called Modified
Compression Field Theory (MCFT). MCFT contains relationships that reasonably predict
the behavior of cracked reinforced concrete elements in combined shear and bending
[Collins and Vecchio, 1986]. In particular, MCFT relationships tend to agree well with
the softening behavior after first cracking and reasonably account for the post-cracking
strength contribution of reinforced concrete. MCFT is used in the current specifications to
predict capacity of conventionally reinforced concrete (CRC) and prestressed concrete
girders based on an initial uncracked condition. However, MCFT does not predict the
service level performance of CRC sections starting from a cracked condition. Analysis of
previously cracked sections prior to reloading requires that MCFT be adjusted to account
for the concrete material reloading at a more softened state than in the initially uncracked
condition.
Through analysis of laboratory data of multiple reinforced concrete beam specimens, in
which strain gages were used to record stirrup strains during the entire loading history, a
factor was derived to modify the value of the cracking stress, thus allowing the softening
behavior of the concrete to begin at an earlier loading stage. Using this adjustment factor
2
for the cracking stress, analysis using MCFT relationships was conducted to produce a
moment-shear (M-V) interaction curve at the onset of yielding for the stirrups. This M-V
interaction curve was then compared to the ultimate capacity M-V interaction curve
produced by
Response 2000TM
[Bentz, 2000] and the AASHTO-99 MCFT curve. The
fraction of ultimate at which yielding occurs was calculated. This relationship may be
used, based upon laboratory data, to determine if a reinforced concrete deck girder
(RCDG) bridge has experienced loading above the M-V yield surface. The number of
loadings above the yield threshold may be calculated based on historical data to produce
an estimate of the number of high magnitude events for low-cycle fatigue evaluation.
This in turn, may give a much better estimate of the remaining life for a cracked RCDG
bridge girder. To demonstrate the use of the M-V yield surface developed from MCFT
and the adjustment factor for previously cracked sections, the McKenzie River Bridge was
analyzed at the section with the lowest fi, where ,8 is the number of standard deviations
away from the mean capacity [Daniels, 2004].
3
Background
The performance of conventionally reinforced concrete (CRC) bridge superstructure
elements with diagonal cracks is of interest to bridge engineers. Diagonal cracks may
result in the premature failure of the member, if web and longitudinal reinforcement are
not sufficiently provided. This type of failure is called a shear failure and may occur
suddenly in a non-ductile fashion [Collins and Mitchell, 19911.
Cracks form perpendicular to the principal stress trajectories which are dependent on the
relative quantities of flexural, shear, and axial load applied to the section (Fig. 1).
Fig. 1 - Principal stress trajectories [MacGregor, 1997].
The principal stress angles can be found from Mohr's circle for stress and strain (Fig. 2).
Given an applied shear stress, rfl,, and normal stresses in the x and y-directions,
o, and
o,, respectively, the principal tensile stress, ai' principal compressive stress, o, and
the angle of inclination for the principal stresses, O, can be derived. Principal stresses
are calculated as [Ugural and Fenster, 1995]:
ri!
I
Fig. 2 - Mohr's circle of stress.
amax,min
l2
2
ax+ay
2
2
2
[1]
J
The angle of inclination of the principal stresses is calculated by:
ay J
[2]
Early methods to design reinforced concrete beams in shear were developed by Ritter and
Morsch, who used a truss analogy for the internal force carrying mechanisms [Ritter,
1899]. The truss analogy became the basis for the strut-and-tie model and describes the
cracked reinforced concrete beam as an equivalent truss where the web is made up of
diagonal concrete struts, the stirrups are transverse ties, and the compression zone and
flexural steel make up the compression and tension chords (Fig. 3).
[i1
Compression chord
inclined
2P
Transverse ties
(ie stirrups)
45°
Fig. 3 - Truss analogy model for reinforced concrete members in shear ICollins and
Vecchio, 19881.
The angle for the compression struts has commonly been assumed to be 45°. However,
using this steep angle conflicts with many tests and is overly conservative, especially for
beams with small amounts of web reinforcement [Collins and Vecchio, 1988]. As a
result, the AC! Building code [ACI 318-83] adopted an empirical correction factor to the
truss equations to provide better correlation with experiments [Collins and Vecchio,
1988]. However, the ACI approach is known to provide wide scatter between predicted
and empirical capacities for shear.
To better characterize the behavior of diagonally cracked reinforced concrete, Collins and
Vecchio developed the relationships for MCFT from laboratory experiments as well as
solid mechanics principles of equilibrium and material compatibility. The method relates
average strains in a cracked section and Mohr's circle for stress and strain (Fig. 4 and Fig.
5).
Relationships for stress and strain were derived from panel testing of large-sized
reinforced concrete elements loaded in various combinations of shear and normal stresses
(Fig. 6).
6
/2
j(i
\
Ot\
Fig. 4 - Average strains of a cracked element ICollins and Mitchell, 1991].
V
-"i
..fcy
vcZY\2OJ
Fig. 5 - Average stresses of a cracked element ICollins and Vecchio, 1988].
Fig. 6 - Panel tests performed by Collins and Vecchio ICollins and Mitchell, 19911.
7
The stresses and strains were measured across multiple cracks to determine average values
for stress and strain across the section. Based on these tests as well as those of other
researchers, constitutive relationships were proposed.
The average principal compressive stress in the concrete, based on a parabolic-shaped
constitutive relationship, depends on the strain magnitude in the longitudinal direction,
as well as the principal tensile strain,
The average principal compressive stress,
f2, at a point along the height of the section is:
r
f2f2max21 e,cJ
[
21
[3]
j
where
f2max
IC
C
[4]
S
which is based on experiments of maximum compressive stress for cracked concrete (Fig.
7). The maximum compressive stress, f's, is based on cylinder tests at 28 days and
the corresponding strain.
is
-
/
/
/i_____P__IIl7r(t
:
Fig. 7 - MCFT compressive stress-strain relationship [Collins and Mitchell, 1991].
Alternatively, the average principal compressive stress over the whole section being
analyzed, as opposed to at each point along the height of the section, was derived from
Mohr's circle as:
f2=(tan9+cot9)
bjd
fi
[5]
where 0 is principal angle of strain, V is the shear force,
b,,
is width of the web, jd
is
shear depth of the section, andf1 is principal tensile stress. The recommended relationship
between the average tensile stress and the average tensile strain is illustrated in
Fig. 8.
'C
'C
'C,
Cr
C
cr
Fig. 8 - Concrete tension before and after cracking ICollins and Vecchio, 19881.
The average tensile stress-strain relationship is linear until the principal tensile strain
reaches the cracking strain, 5cr Therefore, if
then:
fi=ki
[6]
where E is the modulus of elasticity of the concrete and
5cr is:
fcr
[7]
5cr
An estimate for the stress at which cracking
= 42J7
where ,%
psi
occurs,fcr,
is [Collins and Mitchell, 1991]:
[8]
is a factor that accounts for the density of the concrete [Collins and Mitchell,
1991]:
2
1.00 normal-weight concrete
10
2 = 0.85 for sand-lightweight concrete
2 = 0.75 for all-lightweight concrete
After cracking, i >
the average tensile stress-strain relationship is a function of the
reinforcing bond characteristics, the principal tensile strain, and the cracking stress.
However, average tensile stresses will differ with variations of stress along the section,
mostly at crack locations. Tensile concrete stresses at the crack approach zero, while the
tensile stress in reinforcement crossing the crack increases. The ability of the member to
transmit forces across the crack may limit the shear capacity. As a result, a limit on the
principal tensile stress is required to maintain equilibrium [Collins and Mitchell, 1991].
The relationship for principal tensile stress after cracking is:
a1a2,.
fi 1+500s1
vcitanO+(fvy
f)
Sb
[9]
Where ai is a factor that accounts for the type of reinforcement bond.
ai = 1.0 for deformed reinforcing bars
ai
= 0.7 for plain bars, wires, or bonded strands
ai
= 0 for unbonded reinforcement
The parameter a accounts for the type of loading.
a
= 1.0 for short-term monotonic loading
a
= 0.7 for sustained and/or repeated loads
The limiting value of shear stress that can be transmitted along the crack is a function of
the concrete compressive strength,f', crack width, w, and the maximum aggregate size, a.
V1
2.16%j7
24w
0.3 1+
a+0.63
psi and in.
[10]
11
where the crack width is derived as a function of principal tensile strain,
and the
maximum diagonal crack spacing.
[11]
W =
The maximum diagonal crack spacing, s1, is a function of the maximum longitudinal
crack spacing, Smx, the maximum transverse crack spacing, Smv, and the angle of
principal strain, 0:
SmO=.
[12]
Smx
Smv
The maximum longitudinal and transverse crack spacing indicates the ability of the
longitudinal and transverse reinforcement to control cracking as:
Smx =
2(CX
+
10
) + 0.25k1
Px
[13]
where c, is the maximum distance to the reinforcement, s is the spacing of the
longitudinal reinforcement, and
db
is the diameter of the longitudinal reinforcement (Fig.
9). The adjustment factor for bond, k1, is 0.4 for deformed bars or 0.8 for plain bars or
bonded strands. The longitudinal reinforcement ratio, Px' is:
Px
where
[14]
is the area of the longitudinal reinforcing steel and A is the gross area of the
concrete.
Similarly, the maximum transverse crack spacing is computed as:
Smv
2(c +)+o.25k1
10
[15]
12
where c, is the maximum distance to the transverse steel, s is the stirrup spacing, db is the
diameter of the stirrups, and Pv is the transverse reinforcement ratio:
Pv =
A
[16]
bws
where A. is the area cross-sectional area of the stirrups.
SI
;.
I
-:
I_I,
Fig. 9 - Crack spacing parameters ICollins and Mitchell, 1991].
The resulting shear force is expressed in terms of the concrete and steel contribution for
any angle of principal strain, 0 as:
V=(Avfv +fib)jdcot0
S
The principal compressive strain is derived from Eqn. [1] as:
[17]
13
52'cI1/1_
f2
f2max)
[18]
Longitudinal and transverse strains are derived based on Wagner's expression for the
angle of inclination for the diagonal compression and Mohr' s circle [Wagner, 1929].
tan2O=2
[19]
S
From Mohr's circle:
[20]
From these two equations, the longitudinal and transverse strain can be found. The
longitudinal strain is described by:
tan2 0+
[21]
l+tanO
and the transverse strain is described by:
6=
2 tan2 0+ei
[22]
1+tan2G
Stirrup stress is computed assuming a bi-linear, elastic perfectly-plastic stressstrain relationship. Therefore, the stress in the stirrups,f, is linear until yielding
occurs. After reaching the yield strain,j
f:=EStfy
,
the stress remains constant at yield:
[23]
The MCFT can be used in two different ways. Either a detailed dual-sectional analysis or
an approximate analysis may be conducted [Collins and Vecchio, 1988]. The detailed
analysis (Fig. 10) involves dividing the specimen into sub layers, for which each sublayer
must satisfy static equilibrium. For this type of analysis, the principal angle of stress,
14
which in MCFT is assumed to be equal to the principal angle of strain, is found at each
sublayer. As a result, the principal angle of strain varies throughout the section height.
The advantage of the detailed sectional analysis is that the results better predict the
capacity and behavior, maintaining the compatibility and equilibrium conditions, but also
requires more computational resources.
LET-
-
Th>
cross section shear stresses
tongitudina
strains
principal
compressive
Stress traiectorses
Fig. 10 - Detailed analysis [Coffins and Mitchell, 19911.
The approximate method of sectional analysis using MCFT sets the angle of principal
strain to be constant throughout the height of the cross-section.
Thus every sublayer
utilizes the same angle of principal strain in this approximate analysis (Fig. 11).
di
:T
hUH fl
cross section shear stresses longitudinal
strains
sectional torces
Fig. 11 - Approximate analysis [Collins and Mitchell, 19911.
Collins and Vecchio (1988) demonstrated that the approximate method using a constant
angle of principal strain provides reasonable results compared with the detailed sectional
analysis. Example moment and shear interaction diagrams comparing the approximate
with the detailed analysis are shown in Fig. 12.
15
1000
Parabolic
shear strain
800
a
z
Dual secti
800
400
0
200
0
o
200
400
800
MOMENT (kN-m)
800
i
Fig. 12 - Comparison of detailed and approximate procedures [Collins and Vecchio,
19881.
Several computer programs have been developed based on MCFT for analysis of
reinforced concrete sections. Response 2000TM is the most well-known program and was
developed by Evan Bentz under the direction of Michael Collins at the University of
Toronto
Bentz, 2000].
Previously developed programs include Shear (1990) and
Response (1990). Shear calculates the resulting shear force, stress in the stirrups, and the
principal stresses based on a value of principal strain and concrete section properties.
Response does this but also incorporates a greater number of options for sectional
analysis.
The applied loading is provided as input and the program iterates for the
resulting average stresses, average strains, and corresponding average principal angle. It
is important to note that these are averages over the entire section as compared to results
from Response 2000 TM
Response 2000TM uses a detailed analysis approach while
Response uses the approximate sectional analysis. However, as described previously, the
approximate analysis approach has been shown to correspond well with the detailed
analysis approach.
16
Analysis Approach
The procedure outlined by Collins and Mitchell (1991) for the programs Shear and
Response was used to create a spreadsheet for implementation of MCFT. The spreadsheet
results are referred to here as MR. This allowed for a greater flexibility in iterating for
solutions using the MCFT relationships. The procedure outlined for Shear was used to
calculate the shear force as well as the average stresses and strains across the section by
varying the principal tensile strain. The Shear program's output included average stresses
and strains including the transverse strain, e, the longitudinal strain, s, the stress in the
stirrups, f, principal compressive stress,
f2,
principal compressive strain, 62, principal
tensile stress, f1, and any stresses and strains in prestressing tendons if included. Axial
force equilibrium was obtained by iteratively changing the angle of the principal strain.
Moment equilibrium was added according to suggestions in Appendix A of Collins and
Mitchell (1991).
A plane-section analysis method was conducted by numerically
integrating concrete sublayers. Any concrete in tension was neglected and concrete in
compression was analyzed using the parabolic model (Eqn.
3).
Step-by-step
implementation of the methodology is illustrated in the appendix.
The output for this spreadsheet was compared with output from Response and with
instrumented behaviors for three laboratory specimens: a T-beam with 12 in. stirrup
spacing, T12 (Fig. 13 and Fig. 16), an inverted T-beam with 12 in. stirrup spacing, 1T12
(Fig. 14 and Fig. 17), and a T-beam with 24 in. stirrup spacing, T24 (Fig. 15 and Fig. 18).
The spreadsheet results were found to correspond well with the Response 2000TM capacity
17
curve, the AASHTO-99 MCFT capacity curve, and the Response capacity curve (Figs. 19
-21).
In general, MR provides results above Response and below Response 2000TM. This was
likely due to improved convergence with the spreadsheet implementation, whereby the
user, to a greater extent than with Response, could control the degree of precision of
convergence.
The analysis performed at low MIV ratios has been found to provide
unreliable results. In addition, the procedure is only valid for monotonic loading and does
not allow for unloading and reloading of cracked specimens.
I in Clear_6in
6TV',
1.5 in Clear cover
I
#4 Grade 60 L=25ft-9in
#4Grade4O
42in
3.75
/ Spacing 12 in.
#llGrade6O
°
14 in.
1.5 in Clear cover
Fig. 13 Specimen T12 used for analysis correlation.
18
1 .5 in Clear cover
#11 Grade 60
#4 Grade 40
Spacing 12 in.
42 in
--! I
6in I/° __J
1 in Clear COVer
Fig. 14
'#11 Grade 60 L=25ft-9in
II
0
0
5jn
_
1.5 in Clear cover
Specimen 1T12 used for analysis correlation.
1.5 in Clear cover
n Clear cover
Grade 60 L25ft-9in
42 in
rade 40
24n o.c.
1 Grade 60 L24ft-6in
Grade 60 L=25ft-9in
lear cover
]4 In
Fig. 15 Specimen T24 used for analysis correlation.
3(I114
P12
i0©12 in
L,,
North End
P/2
4@6n
I
13 5 in.
44 ri
44
I--- 'I
44
'1
fl
21
Wi
-
'1'
-I-
.-
x=8liri.
18
1I][J
y42in
x:945,i.
i=3O.5in.
y1
l2in
144 in
4 in. dssp. sensor
I
<) Added stralo gage
0.5 in. disp. sensor
0 Strain gage (embedded)
5 in. string pot (on both sides of beam)
0
TIt sensor
Note All equipment is on west side of beam unless otherwise noted.
0.5 in. disp. sensors located from nearest corner of stern.
Fig. 16 T12 specimen laboratory setup and crack map.
144 in.
/ /7/ /7
Faikire mode: Shear Compression
Peak Load: 378 kips.
Widest aad; 0.109 ii.
North End
P/2
7 in 14
1.1+
135,n
I
J
1O©12 in.
4
44ri
+
I
44wi
1-r-1---r--r-j-r
r
44in.
1
-t
2
-i
r
I
I
I
P/2
I
1O©12 in.
14 In
k +
7 ii
in.
r
r
I
I
cfiFrr3r
L1
48in.
x62.5in.
x93.5ri.
yl9 75in
y2Oin.
y22.25.
x:78ifl.
y 16 5in
IJEI
;
x=88.5in iLl
// //
x57.75in
y=l5in.
/
//
x=49.75in.
yrOri.
12 ft.
4 in disp sensor
0.5 in. disp. sensor
0 Strain gage (embedded)
12 ft
() Added strain gage
wi. string pot (on both sides of beam)
Failure mode: Shear Tension/Anchorage
Peak Load: 358 kips
lit sensor
Note: All equipment is on west side of beam unless otherwise noted.
0.5 Ni. disp. sensors located from nearest corner of stem
tJ
0
Fig. 17 IT1 2 specimen Jaboratory setup and crack map.
P/2
++
P/2
4O6m
North End
6in__________________________
in.
xr66.51I
y20 75
_J[_J
;
,//i////
l2in
x-85n
yr12 5m
xrlO7.75in
=24
y=30 75
X661n
ysi 6tn
x52 5ri
y1 3 5
I
144r
I
-
-------
///4/ /
144in
__________________________
- 4in.ClipGage
4 in. disp sensor
(> Added strain gage
Failure mode; shear-compression
0.5 in. asp. sensor
0
Strain ge (embedded)
5 in. stnng pot (on both sides ci beam)
Pe* load: 241 ks
Widesi cradi. 0.09Th.
Tit sensor
Note: M equipment is on west side ci beau unless otherwise noted
Eidemal strain gages located on east side of beam
I')
Fig. 18 T24 specimen laboratory setup and crack map.
22
M-V Capacity Surface
2T 12
stirrup spacing= 12
220
200
180
160
140
120
100
80
- Ultimate Capacity from Response
2000TM
Ultimate Capacity from AASHTO-99 MCFT
60
*-* T12 Failure M/V
S. Ultimate Capacity from MR
40
A-A Ultimate Capacity from Response
20
0
250
500
750
1,000 1,250 1,500 1,750 2,000 2,250 2,500 2,750 3,000
M (k-ft)
Fig. 19 - M-V capacity for specimen T12.
M-V Capacity Surface
21T12
stirrup spacingl2"
L
2
60
40
cn
20
00
Ultimate Capacity from Response 2000w
-+ Ultimate Capacity from AASHTO-99 MCFT
80
** lTl2 Failure M/V
60
S. Ultimate Capacity from MR
40
A-A Ultimate Capacity from Response
20
0
250
500
750
1,000 1,250 1,500 1,750 2,000 2,250 2,500 2,750 3,000
M (k-ft)
Fig. 20 - M-V capacity for specimen 1T12.
23
M-V Capacity Surface
10T24
stirrup spacing24"
160
140
120
iE.II
0.
.
80
>
60
- Ultimate Capacity from Response
2000w
e-. Ultimate Capacity from AASHTO-99 MCFT
40
*-* T12 Failure MN
20
£4
Ultimate Capacity from MR
Ultimate Capacity from Response
0
0
250
500
750
1,000
1,250
1,500
1,750
2,000 2,250 2,500 2,750
M (k-ft)
Fig. 21 - M-V capacity for specimen T24.
Amplijication Factor for Yielding of the Stirrups
Predicting the moments and shears that cause yielding of stirrups is required to determine
the potential for low-cycle fatigue on the stirrups. Steel is fatigued when subjected to
repeated yielding. Therefore, it is important to be able to predict the moments and shears
that may cause yielding of stirrups in a reinforced concrete girder.
To estimate the onset of yielding in the transverse reinforcement, an amplification factor
was used to relate the average transverse strains over the section to the maximum stirrup
strain at a diagonal crack location (Fig.
22).
24
erage Transverse Strain
Maximum Transverse Strain
Fig. 22 - Average versus maximum transverse strain.
The MCFT analysis uses the average strains over a section height, which was taken to be
the section of concrete between the compression and flexural reinforcement, referred to
here as DV. The transverse strain distribution would be better represented by a parabolic
shape over the actual shear depth,
geometry and calculus.
h.
The amplification factor is derived from simple
The relationship between the average strain over the whole
section and the maximum strain of a corresponding parabolic distribution was found to be:
8parabola_max =
where
h,
3 DV
2h
8average
[24]
assumed to extend from the flexural tensile reinforcement to the edge of the
deck/stem interface for a T-section is:
h = d
tdeck
[25]
and h for an inverted T-section, which was assumed to be the distance between the
flexural tensile reinforcement and the edge of the compression zone, a, is:
h=da
[26]
25
The depth of the compression zone, a, is computed based on an equivalent rectangular
stress block as:
a=
fA
[27]
O.85f' b
where j is the stress in the reinforcement, and A is the cross-sectional area of the
reinforcement.
A potential limitation of using this strain amplification approach arises in high moment to
shear (MIV) ratios. At high M/V ratios, the strain in the flexural steel at the bottom of the
section may increase significantly permitting larger transverse strains at the level of the
reinforcement. This may distort the strain distribution. The new shape, containing an
offset at the bottom where the flexural strains have increased, causes the real strain
distribution to approach the average strain distribution. Therefore, for high MN ratios,
the amplification factor may be less reliable (Fig. 23).
High Strain Due
to Flexure
Fig. 23 - Average versus maximum strain at high M/V.
26
Reduced Cracking Stress
The stress at first cracking used by
Response
was calculated using Eqn. 8. This is the
stress at initial cracking, when concrete material that was previously uncracked begins to
crack and thus soften. However, this value of cracking stress no longer functions for the
onset of material softening for a concrete member that already contains diagonal cracks.
Reloading of a cracked specimen induces material softening behavior once crack surfaces
have decompressed. The beam load increases without an increase in strain until a new,
reduced, value
Ofjr
is reached. In order to account for this phenomenon, laboratory data
from specimens that are typical of Oregon's 1950's vintage reinforced concrete deck
girder (RCDG) bridges were analyzed. The load versus strain data collected from strain
gages placed on stirrups during the tests to failure were used to assess concrete softening
behavior. For the laboratory specimens, the value for se,. (the onset of material softening
behavior on the reloading branch) tended to be in the range between 33 percent and 50
percent of the original er,. value with 33 percent being the more typical for strain gages
near diagonal cracks.
Therefore, cracking stress of 1/3 the initial value was used to
analyze the cracked reinforced concrete sections for reloading response prediction (Figs.
24-26).
27
Shear vs. Stirrup Strain @ 4' from Support
Specimen 2T12, stirrup spacing=12"
North Side
200
Strain Gage
First ()acking @ V=100 kips
Softened R)st-Q'acking @ V=33 kips
175
150
: 125
gioo
200
0
400
600
800
1,000
1,200
1,400
1,600
1,800
Stirrup Strain (nc)
Fig. 24 - Cracking strain,
cr,
determination of specimen 2T12 north.
Shear vs. Stirrup Strain @ 4 from Support
Specimen 2T1 2, stirrup spacing=1 2"
South Side
200
175
Strain Gage
- First Cracking @ V80 kips
- Softened Rst-Cracking @ V=40 kips
150
125
100
50
25
(1
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
Stirrup Strain (sic)
Fig. 25 - Cracking strain, Ecr, determination of specimen 2T12 south,
28
Shear vs. Stirrup Strain @ 4 from Support
Specimen 21T12, stirrup spacingl2
South Side
200
175
150
125
Strain Gage
- First cracking © V108 kips
- Softened Fkst-Cracking © V33 kips
I
200
400
600
800
1,000
1,200
1,400
1,600
1,800
Stirrup Strain (nc)
Fig. 26 - Cracking strain,
Ecr,
determination of specimen 21T12 south.
Results
V-es Prediction
Laboratory data were employed to verify the validity of using reduced cracking stress for
analyzing the effects of reloading a cracked reinforced concrete section. Three laboratory
specimens were evaluated that were characteristic of vintage cracked 1950's RCDG
Oregon bridges. Two T-beams with stirrup spacings of 12 in. and 24 in. as well as one
inverted T-beam (IT) with stirrups spaced at 12 in were investigated. The softened
cracking stress was taken as one-third of the original cracking stress, as discussed
previously. The results of this analysis corresponded well with the reloading branches of
the laboratory specimens demonstrating that the reduced cracking stress provided a
suitable modification to model the reloading behavior of cracked concrete specimens (Fig.
27
46). The model for reloading based on MCFT, adjusted with a reduced cracking
stress, can be seen as the dashed curves on each plot. In some cases, the curve has been
shown both at the origin and offset to a distance where softening behavior is more readily
evident.
In addition, the amplification factor from Eqn. 24 was applied to the model based on
MCFT (spreadsheet MR) and compared with the laboratory data. In general, the results
showed that the amplified models corresponded more reasonably to the initial loading
branch, while the unamplified model provided a better correlation with the softened
reloading branches after cracking. A possible explanation is that the stirrups, on initial
loading, are fully bonded. Therefore, the strains vary locally throughout the height of the
30
stirrup, reaching maximum magnitude at the crack location. This corresponds well with
the purpose of the amplification factor, which is to modify the average transverse strain to
account for the maximum transverse strain over the section. On the reloading branch after
cracking, the concrete softening results in diminished bond to the steel. With less bond to
cause local variations in strain, the strain over the stirrup height approaches the average.
Stirrup strains for a fully unbonded stirrup are constant (average) over the stirrup height.
Consequently, the strain in the stirrup for the reloading branch is predicted more
realistically by the average strain over the section height (the unamplified model) due to
reduced bond between the softened concrete and the stirrup.
One additional factor affecting the degree of correlation between the data and the MCFT
model (spreadsheet MR) is the location of the strain gage relative to the crack(s). The
model assumes that the strain is measured at the crack location. Comparing the model
predictions with a strain gage located at the crack produces good results. Conversely, the
comparison of the model predictions with a strain gage located a distance away from the
nearest crack shows results that are not as well correlated. An example of this can be seen
in Fig. 33.
31
Shear-Stirrup Strain © M/V4 and S12"
3T12 Precrack
South Side
20C
190
180
170
160
150
140
- -
Cl)
a.
G'
LL.
c
0
120
110
100
90
80
.
U)
60
50
40
30
20
10
n
0
300
600
900
1,200
1,500
1,800
2,100
2,400
2,700
3,000
Stirrup Strain (J.Le)
*
*
Strain Gage
yield strain=1 760
rvR
rvR
j.IE
c arrplified by 1.5
arrplified by 3*/(2h)_1 .63
VR&rlI3cr_original
Rr=l/3Ecr_origjn offset (for derronstration)
Fig. 27 - V- at south side, M/V=4 for specimen 3T12.
WAW
Fig. 28 - Strain gage location relative to crack at south side, M/V=4 for specimen
3T12.
32
Shear-Stirrup Strain @ M/V4 and S=12'
2T1 2
South Side
L
1
1
1
(0
0.
ai
0
U.
I-
G)
U)
'I
I
0
300
600
900
1,200
1,500
1,800
Stirrup Strain (t)
* *
Strain Gage
yield strain=1 760 p
fflMR
MR arTplif led by 1.5
MR c arrpllf led by 3*DV/(2h)....1 .63
- MR Ir1'3Ecr_originaI
Fig. 29 - V-
at south side, M/V=4 for specimen 2T12.
IIIrI:
Fig. 30 - Strain gage location relative to crack at south side, M/V=4 for specimen
2T12.
2,100
33
Shear-Stirrup Strain @ M/V4 and S12"
2112
North Side
2"
U
2 25
2 10
95
80
65
(j
3:
Wi
'-1
0
1I
0
.
Cl)
0
250
500
750
1,000
1,250
Stirrup Strain
1,500
1,750
2,000
2,250
(l.L)
* *
Strain Gage
yield strain=1 760
(>
MRcarrplified by 1.5
UIffiMR
<
MRcarrplified by 3*Dv/(2h)163
MR r'13cr_orIginaI
MREcrl/3cr_originai offset (for derronstration)
Fig. 31 - V-
at north side, M/V=4 2T12 North
I(
I
I
Fig. 32 - Strain gage location relative to crack at north side, M/V=4 for specimen
2T12.
2,500
34
Shear-Stirrup Strain @ MN=4 and S12"
2 ITI 2
South Side
-'U
1
70
1
30
1
50
1
40
1
30
'
0.
0
U(5
w
C/)
501
f:
I;'
0
150
300
450
600
750
900
1,050
1,200
Stirrup Strain (.tc)
Strain Gage
yield strain=1 760
rvRarrplifled by 1.5
rvRc arrplified by 3*DV/(2h)....1 44
- IVR Ccrl /3Ccr_orig inal
FvR CCr=l I&cr_orig inal
offset (derronstration)
Fig. 33 - V-E at south side, M/V=4 for specimen 21T12.
I -------I
I
I
I
I
C:..,
'.
Fig. 34 - Location of strain gage at south side, M/V=4 for specimen 21T12.
1,350
35
Shear-Stirrup Strain @ MN=4 and S24"
I 0T24
South Side
1 DI
14
13
12
11'
j 10'
0
a
0'
0
U-
J
3
2
1
0
300
600
900
1,200
1,500
Stirrup Strain
* *
mm
1,800
2,100
2,400
(j.t)
Strain Gage
yield strain
rvREan-pliIied by 1.5
rvR c an-plified by 3*DV/(2h).1 .63
- IVREcrlI3rcr_originai
FvR Cr=lI3Ecr_originaI
offset (for derronstration)
Fig. 35 - V-E at south side, M/V=4 for specimen 10T24.
wAfl\
Fig. 36 - Location of strain gage at south side, M/V4 for specimen 10T24.
2,700
36
Shear-Stirrup Strain © M/V6 and S12"
2T1 2
North Side
22C
200
180
160
-'I--.
140
. d.-
120
I-
0
u.
100
I-
80
Cl)
60
40
20
C
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
Stirrup Strain ()
* *
K)
Strain Gage
yield strainl 760
E
K) MR amplified by 1.5
MRcarrplified by 3*/(2h).....163
MRrl/3cr_origina
MRrl/3cr_originai
(for derronstration)
Fig. 37 - V-E at north side, MIV=6 for specimen 2T12.
I
Fig. 38 - Strain gage location relative to crack at north side, M/V6 for specimen
2T12.
2,000
37
Shear-Stirrup Strain © M/V6 and S12"
2T1 2
South Side
'U
I
1
30
I
1
30
1
50
1
U)
a.
I
30
20
10
I
)0
1
1
w
0
UI-
5
0
30
U)
30
50
30
20
10
0
300
600
900
1,200
1,500
1,800
2,100
Stirrup Strain ()
* *
mm
yieId=1760
O 0 IVR c arrplified by 1.5
X X
R c arrplif led by 3*DV/(2h)1 .63
Strain Gage
R Ccr'3Cr_OriginaI
-M
Fig. 39 - V-
grl/3Ecr_originaI
offset (for derronstration)
at south side, MIV=6 for specimen 2T12.
I
Fig. 40 - Strain gage location relative to crack at south side, MIV=6 for specimen
2T12.
Shear-Stirrup Strain @ MN6 and S12"
2 ITI 2
South Side
U'-)
1
90
I 80
I
1
I
1
C',
0.
0
U.
I-
w
.c
Cl)
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
Stirrup Strain ()
* *
Strain Gage
yield strain=1 760 pz
MRcarrplified by 1.5
FvRcarrphuied by 3*/(2h)...144
- rvR &r=l/&cr_originaI
VRrl/3Fr_originaI
Fig. 41 - V-
offset (for derronstration)
at south side, MIV=6 for specimen 21T12.
Fig. 42 - Location of strain gage at south side, M/V=6 for specimen 21T12.
2,000
39
Shear-Stirrup Strain @ MN=6 and S12"
21T1 2
North Side
J,.J
1
1
1
(ii
e
0
1
30
70
50
50
40
30
20
10
DO
U-
.z
U)
ic
n
0
400
800
1,200
1,600
2,000
2,400
2,800
Stirrup Strain ()
* *
Strain Gage
yield strain=1 760 pz
fflmR
O 0 rVRcarrphlied by 1.5
FvRcarrplif led by 3*[/(2h)144
cr
I3r
ofig irial
RrlI3r_originai offset (for derronstration)
Fig. 43 - V-
at north side, M/V=6 for specimen 21T12.
Fig. 44 - Location of strain gage at north side, MIV=6 for specimen 21T12.
3,200
40
Shear-Stirrup Strain @ MN6 and S=24"
I 0T24
North Side
U')
1
40
I 30
1
1
U)
.x
e
0
U-
I-
20
V
10
70
60
C,
50
40,..
f
30
20
10
n
0
400
800
1,200
1,600
2,000
2,400
2,800
Stirrup Strain (.tc)
Strain Gage
yield strain
IvRarrplified by 1.5
rvR c ariplified by 3*DV/(2h)....1 .63
- FVR Erlt3Ecr_onginaI
'Rrl/3Ccr_originai
Fig. 45 -
V-E
offset (for derronstration)
at north side, MJV=6 for specimen 10T24.
Fig. 46 - Location of strain gage at north side, MIV6 for specimen 10T24.
3,200
41
M- V Interaction Yield Surfaces
Low-cycle fatigue of reinforced concrete girders may occur when load effects (M and V)
combine to produce yielding of stirrups. The moment-shear (M-V) interaction yield
surface is needed to perform a low-cycle fatigue evaluation. Using the MCFT with the
adjustment factor for cracking stress, as well as an amplification factor to achieve the
maximum strain in the section based on the average strains, an M-V interaction curve at
yielding of the stirrups was developed.
The yield M-V interaction diagrams were
developed for the three laboratory specimens and two critical sections of the McKenzie
River Bridge, located in Lane County, Oregon. The ratio between yielding and capacity
values were calculated with respect to the AASHTO-99 MCFT capacity curve. Yielding
was conservatively based on amplified strains corresponding to well bonded stirrups. The
amplified yield curve exhibited lower moments and shears to produce the yield condition
than that based on average strains.
Through analysis of the M-V interaction yield surfaces, it was found that for higher M/V
ratios, there is a point at which the stirrups do not yield before capacity is reached for the
section. As a result, sections at high M/V ratios do not appear to be prone to low-cycle
fatigue.
Laboratory Specimens
The three laboratory specimens analyzed were the T12, IT 12, and T24. On average, the shear and
corresponding moment to cause yielding in the stirrup was approximately 67% of the ultimate
capacity as predicted by the AASHTO-99 MCFT for the T12 specimen for MJV ratios of 3 to 14,
approximately 69% for 1T12 for MJV ratios of 3 to 10, and for T24 was about 63% for MN ratios
42
from 3 through 20. These results are displayed in Tables 1 to 3, and graphically displayed in Figs.
47 to 49.
M-V Yield Surface
T12
stirrup spacingl2"
I
1
I
1
I
0.
>
0
0
* *
Ultimate Capacity from Response 2000TM
Ultimate Capacity from AASHTO-99 MCFT
T12 Failure MN
A A MR cr=1/3cronginaI
30
MR Er=l'&cr onginal AMPLIFIED
15
0
0
300
600
900
1,200
1,500
1,800
2,100
2,400
2,700
M (k-ft)
Fig. 47 - M-V yield surface and predicted capacities for specimen T12.
Table 1 - Shear to produce yield as % of AASHTO-99 MCFT for specimen T12.
MN
1
2
3
4
6
8
10
12
14
Moment at V,,, (k-ft)
V, kips
% V by AASHTO-99 MCFT
44%*
80
55%*
200
100
400
133.3
73%
500
125
70%
625
104
61%
800
100
61%
1000
100
64%
1200
100
68%
1350
96
69%
*Low MN values not well predicted by method.
80
43
M-V Yield Surface
IT 12
stirrup spacing=12"
lu
1
95
1
1
1
1
35
.1
20
>
90
/'
05
60
0
0
Ultimate Capacity from Response 2000
Ultimate Capacity from AASHTO-99 MCFT
* * 1T12 Failure M/V
A A MR Fr=lI3F.tronginaI
45
30
MR cr=l/3Ccr onginal AMPLIFIED
15
0
0
300
600
900
1,200
1,500
1,800
2,100
2,400
2,700
M (kip-ft)
Fig. 48 - M-V yield surface and predicted capacities for specimen 1T12.
Table 2 - Shear to produce yield as % of AASHTO-99 MCFT for specimen 1T12.
MAT
Moment at V, (k-ft)
Vs,, kips
% V, by AASHTO-99 MCFT
1
120
2
300
400
500
700
900
120
150
133.3
125
116.7
112.5
110
65%*
81%*
3
4
6
8
10
1100
*Low MN values not well predicted by method.
72%
69%
68%
68%
70%
44
M-V Yield Surface
T24
stirrup
spacing=24"
.1 rc
1
40 ____________
1
30
* *
1
1
1
Il
.
>
80
70
60
50
40
0
30
0
* *
20
Ultimate Capacity from Response 2000TM
Ultimate Capacity from AASHTO-99 MCFT
T24 Failure M/V
MR Fcr1/3Ecronginal
MR FcrlI3Ecr original AMPLIFIED
10
0
0
300
600
1,500
1,800
2,100
2,400
M (k-ft)
MR with original cracking strain (r original) reach ultimate before yield.
Therefore, ultimates are plotted for rorigIna.
900
1,200
2,700
Fig. 49 - M-V yield surface and predicted capacities for specimen T24.
Table 3 Shear to produce yield as % of AASHTO-99 MCFT for specimen T24.
M/V
3
4
6
8
10
12
14
20
Moment at
Vs,,
250
350
500
625
740
875
1000
1400
(k-ft)
Vs,,
kips
83.3
87.5
83.3
78.1
74
72.9
71.4
70
% V, by AASHTO-99 MCFT
60%
64%
63%
62%
61%
62%
63%
69%
45
McKenzie River Bridge
The two sections of the McKenzie River Bridge analyzed were an IT and T beam, sections
A and B (Fig. 50) with stirrups spaced at 9 in. and 12 in., respectively (Fig. 51 and Fig.
52). The McKenzie River Bridge section tended to yield at higher percentages of the
AASHTO-99 MCFT ultimate capacity than the laboratory specimens. For M!V between 1
and 4, the McKenzie River Bridge section A yielded on average approximately 81% of
AASHTO-99 MCFT capacity while for the same range of MIV ratios, section B on
average yielded about 86% of ultimate capacity predicted by AASHTO-99 MCFT. This
data can be seen in Tables 4 and 5, and is plotted in Figs. 53 and 54.
A
42ft
5Oft
B
112.5ft
'
5Oft
5Oft
Fig. 50 Sections A and B from McKenzie River Bridge.
46
3 #11
#4
50.0
9.00 in
4- #11
Fig. 51 - McKenzie River Bridge at section A.
108.0
4- #11
r
03
t
#4
12.00 in
3- #11
4i4
Fig. 52 - McKenzie River Bridge at section B.
47
M-V Yield Surface
Mckenzie Bridge, Section A
stirrup spacing=9"
uU
180
160
140
1
>
0
o
0
Ultimate Capacity from Response 2000
Ultimate Capacity from AASHTO-99 MC
MR ftrl/3Ecr_originap
MR Fcrl/3Ecr orlgina! AMPLIFIED
20
0
0
150
450
300
600
900
750
1,050
1,200
M (kip-ft)
Fig. 53 - M-V yield curve for McKenzie River Bridge girder section A.
Table 4 - Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River
Bridge girder section A.
M/V
Moment at V,,, (k-fl)
V,,, kips
1
115
228
115
340
440
113.3
110
2
3
4
114
% V11
by AASHTO-99 MCFT
81%
80%
80%
84%
M-VYieId Surface
Mckenzie Bridge at Section B
stirrup spacing=12"
1 8C
1 6C
1 4C
1 2C
i bc
.
>
8C
6C
4c
o
o
Ultimate Capacity from Response 2(
Ultimate Capacity from AASHTO-99
* * MR
0 0 MR
2C
0
100
rl'3Ecr_originaI
cr=lI&cr original
200
300
AMPLIFIED
400
500
600
700
800
900
M (kip-ft)
Fig. 54 M-V yield curve for McKenzie River Bridge girder section B.
Table 5 - Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River
Bridge girder section B.
MN
Moment at Vs,, (k-ft)
1
95
180
2
3
4
260
340
V, kips % V by AASHTO-99 MCFT
95
90
86.7
85
89%
85%
82%
86%
As can be seen from comparing the yielding shear level with the capacity, the laboratory
specimens yielded between approximately 60% and 73% of the nominal AASHTO-99
MCFT capacity, while the sections analyzed for the McKenzie River Bridge yielded
between about 80% and 89% of the nominal AASHTO-99 MCFT capacity.
49
The load effects produced by service level trucks combined with dead load at the sections
of interest were determined and compared with the predicted yield shear force and
moment. To estimate service level truck loads at the McKenzie River bridge, weigh-inmotion data was used which had been collected for southbound truck traffic on interstate
highway I-S during the period of one year, that being 2003. The truck data was collected
at the Wilbur weigh station, located just North of Roseburg, OR. In total, 900,000 trucks
passed through Wilbur weigh station going southbound in 2003 [Daniels, 2004]. Of
those, approximately 14,000 trucks were classified by ODOT as being permit table 3, 4,
and 5 vehicles [Daniels, 2004], plotted in Fig. 55 and Fig. 56. These are vehicles that
carry loads which exceed the legal limits. Load effects calculated using AASHTO LRFD
live load distribution factors combined with AASHTO impact factors [Daniels, 2004]
caused some trucks to induce yielding of the stirrups. For the location of section A,
approximately 5 of the 14,000 permit table 3, 4, and 5 trucks caused yielding in the
stirrups.
50
1
1
1
ti)
I65
Cl)
E
E
(5
0
100
200
300
400
500
700
600
800
900
1000
Moment (kip-ft)
Fig. 55- McKenzie River Bridge at section A using AASHTO LRFD LL distribution
factors.
1
1
- - - - - - -
10
-
AASHTO-LRFD 98):
DFv= 0 884
1
U)
0.
.)
*
30
Dead
ad Moment àidShei:
-2 Q3 ft-kips,V01= 22.42
'5
w
70
MDL =
Cl)
DO
d=41 2 in, b'430 in'
.
E
E
'5
=
,
= YDL
$0
'S
30
*
AASHTO-MCFT 99 M-V Interaction
Yield Points (amplified)
WIM Vehicles (Permit Tables 3 4 5)
100
150
20
10
0
50
400 450
Moment (kip-ft)
200 250 300 350
I
500 550
600 650
700
Fig. 56- McKenzie River Bridge at section B using AASHTO LRFD LL distribution
factors.
51
This procedure may be used to determine the yield moments and shears for any bridge.
However, the iterative process of finding these moment and shear combinations may be
bypassed if conservatively one were to choose a percentage of yielding to use as an initial
screening for possible low-cycle fatigue damage accumulation. Additional analyses are
needed for other bridge spans and indeterminacies to develop a reliable recommendation,
but based on the analyses performed, yielding occurs between 60% and 89% of ultimate.
A percentage within this range may be acceptable for roughly estimating low-cycle fatigue
potential.
52
Conclusions
The Modified Compression Field Theory (MCFT) is currently used in the AASHTO
LRFD specification to compute the shear capacity of reinforced concrete and prestressed
beams and beam-columns. It uses strain compatibility and equilibrium to predict the
capacity of sections under combined flexure and shear. Currently, the method is used to
analyze beams assuming an initial uncracked condition. However, analysis of cracked
sections is needed to evaluate potential for LCF in bridge girders.
MCFT was used to predict the behavior of girders with existing diagonal cracks. The
constitutive relationships were modified to realistically predict stirrup strains for girders in
the previously cracked condition. Analyses were performed for laboratory specimens and
the method was applied to an existing bridge with diagonal cracks. Based on these
analyses, the following conclusions are presented:
Stirrup strain behavior for reloading of previously cracked girders can be
reasonably predicted by MCFT and incorporating an adjustment factor to the
cracking stress, fcr
Transverse strains in the cross-section were amplified to account for higher
stresses in the stirrups at diagonal cracks to provide a better correlation with
experimental results.
For high MIV ratios, there is a limit at which the stirrups do not yield when the
section reaches capacity. There is no risk for low-cycle fatigue at these sections.
For very low M/V ratios, the MCFT does not predict the shear and moment
capacity very well.
53
.
Assessment of low-cycle fatigue for a bridge requires comparison of the load
effects from service level trucks and dead load with the yielding MIV interaction
curve of the section. The analysis technique described provides a method to
predict this threshold.
54
References
AASHTO. (2003). "LRFD Bridge Design Specifications."
Ed., Washington, D.C.
ACI Committee 318-83. (1983). Building Code Requirements for Structural Concrete.
American Concrete Institute, Detroit, Michigan.
ACI Committee 318-02. (2002). Building Code Requirements for Structural Concrete.
American Concrete Institute, Farmington Hills, Michigan.
Bentz, E. (2000). "Sectional Analysis of Reinforced Concrete Members." Ph.D. Thesis,
Department of Civil Engineering, University of Toronto, Toronto, Canada.
Collins, M.P. and Mitchell, D. (1991). Prestressed Concrete Structures. Prentice Hall,
Englewood Cliffs, New Jersey.
Collins, M.P. and Vecchio, F.J. (1982). "The Response of Reinforced Concrete to InPlane Shear and Normal Stresses." Publication No. 82-03, Department of Civil
Engineering, University of Toronto, Mar., 332 pp.
Collins, M.P. and Vecchio, F.J. (1986). "The Modified Compression Field Theory for
Reinforced Concrete Elements Subjected to Shear." ACT Journal, Vol. 83, No. 2,
Mar-Apr., pp 219-23 1.
Collins, M.P. and Vecchio, F.J. (1988). "Predicting the Response of Reinforced Concrete
Beams Subjected to Shear Using Modified Compression Field Theory." ACI
Structural Journal, May-June, pp. 258-268.
Daniels, T. (2004). "Reliability Based Bridge Assessment using Modified Compression
Field Theory and Oregon Specific Truck Loading." M.S. Thesis, Department of
Civil, Construction, and Environmental Engineering, Oregon State University,
Corvallis, OR.
Felber, A.J. (1990). "Response: A program to determine the load-deformation response
of reinforced concrete sections." M.S. Thesis, Department of Civil Engineering,
University of Toronto, Toronto, Canada.
Kani, G.N.J. (1964). "The Riddle of Shear Failure and Its Solution." ACI Journal, Vol.
61, Apr., pp. 44 1-467.
MacGregor, J.G. (1997). Reinforced Concrete Mechanics and Design. 3rd Ed. Prentice
Hall, Upper Saddle River, New Jersey.
Response. (Version 1.0) [Computer software]. (1990).
Shear. (Version 1.0) [Computer software]. (1990).
55
Ritter, W.
(1899).
"Die Bauweise Hennebique." (Construction Techniques of
Hennebique). Schweizerische Bauzeitung, ZUrich, Feb.
Ugural, A.C. and Fenster, S.K. (1995). Advanced Strength and Applied Elasticity. 3 Ed.
Prentice Hall, Upper Saddle River, New Jersey.
Wagner, H. (1929). "Ebene Blechwandträger mit sehr dUnnem Stegblech" (Metal Beams
with Very Thing Webs). Zeitschrfi für Flugtechnik und MotorIuftschfJahr, Vol.
20, Nos. 8 to 12.
56
Appendix
MCFT Procedure
57
Following is the algorithm for shear, moment, and axial load as outlined by Collins and
Mitchell in Prestressed Concrete Structures (1991) for the program Shear (1990).
LOOP FOR ITERATING ON
Step 1: Choose a value of e at which to perform the calculations.
LOOP FOR ITERATING ON 0
Step I: Estimate 0
Step H: Calculate the crack width, w, and the shear along
the
crack, v1
LOOP FOR ITERATING ON VERTICAL EQUILIBRIUM
Step a: Estimate f
Step b: Calculate the principal stress f1
Step
C:
Calculate the shear, V, as a resultant of the
shear
stress.
Step d: Calculate the average compressive stress, f2
Step e: Calculate fimax, the maximum compressive
stress
Step f: Check that f2 f2. If f2> f2m,x, the concrete
crushes
and the solution is not possible. Return to Step 1 and
choose a
58
smaller 61.
Step g: Calculate compressive strain,
62
Step b: Calculate the average longitudinal and
transverse
strains, e and s.
Step i: Calculate the stress in the stirrup utilizing the
transverse strain.
Step j: Check if the stirrup stress matches the initial
guess for the stirrup stress. If the values do not
match, return to Step a with an improved estimate
of the stress in the stirrup.
END LOOP FOR ITERATING ON VERTICAL EQUILIBRIUM
Step III: Calculate the axial force, N, on the section
resulting from
the concrete stresses in the effective shear area, bjd, and
the
moment caused by N.
tan 0
M = Ny
where y is the distance from the center of the effective shear
area to the
moment axis.
[All
[A2]
59
Step IV: Using a plane-section analysis procedure, find the
stress
resultants for the section corresponding to the longitudinal
strain,
and a curvature, b, that corresponds with the applied
moment
and longitudinal strain.
N = f(&,Ø) = Ng,.oss Nbjd
= f(s,
q) =
[A3]
Mbjd
{A4J
Step V: Calculate the axial resultant and moments
N=N
+
[A5]
MM+M
[A6]
Step VI: Change the principal angle of strain 0 until the
axial
force resultant equals the applied axial load.
END LOOP FOR ITERATING ON 0
Step VII: Check that the steel demand (right side of inequality)
has not exceeded its strength (left side of inequality).
(i
i=1
+
i=1
(, fi)4 f1bJd+[fi
(f
bjd
_fV)Jtan2O
[A7}
60
Step 2: Verify that the applied shear force equals the resultant
shear force. If the two are not close enough, revise the estimate of
Si.
END LOOP FOR ITERATING ON