crack and deflection
TRANSCRIPT
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Crack and Deflection
Control of Pretensioned
Prestressed Beams
Edward G Nawy
Professor of Civil Engineering
Rutgers—The State University
of New Jersey
Piscataway, New Jersey
P T Huang
Senior Structural Engineer
Gibbs Hill
Consulting Engineers
New York, N.Y.
Based on a series of laboratory tests on
precast prestressed T and I beams, the
authors propose crack width and deflection
formulas for evaluating the serviceability of
such members. Several numerical examples
show the applicability of the recommended
crack width equations.
he primary objective of this investi-
gation was to study the serviceabil-
ity of pretensioned prestressed I and T
beams through an analytical and exper-
imental investigation of their flexural
cracking and deflection behavior up to
failure.
Available experimental data on crack-
ing in prestressed concrete members is
limited. 1 5
Formulas on crack width
prediction from previous researches, in
general, were based on
two
different
concepts. The first concept, proposed
by Ferry-Borges,
6
Nawy-Potyondy,2
-3
Holmberg,7
8
and CEB
9
is to relate
crack width to the stress or strain in the
reinforcement.
Another concept, originated by
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YIELD STRENGTH 8Y
0.2 SET METH OD
AREA 0.0356 IN2
ULT STRENG TH=9600 LBS
^ Eg
V V V
.V I
u.V V
STRAIN
(IN/IN)
250
20C
0
y 15C
U
w
rc
U
IOC
50
0
0.000
Abeles
° proposed the crack width to
be related to the flexural tensile stress
in the concrete. The analytical equa-
tions proposed in this investigation are
based on the first concept.
Tests on four series were conducted
on 20 simply supported beams of 9-ft
span and four two-span continuous
beams of effective 9-ft spans. The ma-
jor controlling parameters were the var-
iations in the steel reinforcement per-
centages of the prestressing tendons
and the non-prestressed reinforcement.
The prestressing tendons were
/4
in .
nominal diameter 7-wire strand, 250
ksi elements. The mild steel reinforce-
ment was either #3 or #4 deformed
high strength bars of yield strength
varying between 79 and 84 ksi and ulti-
mate strength between 100 and 110
ksi. The total steel percentage was var-
ied from 0.17 to 1.08 percent.
This paper proposes analytical ex-
pressions for evaluating the crack
widths in such members at working
and overload loading levels in terms of
the controlling parameters. The investi-
gation also correlates the deflection re-
suits with the expected computed de-
flections. Several numerical examples
are included to show the applicability
of the proposed crack width formulas.
Test Program
This section describes the properties
of the materials used in the experiment-
al program and gives the details of fab-
rication of the beam specimens and the
testing procedure.
Materials
The mix was proportioned for a nomi-
nal 28-da y comp ressive strength of approx-
imately 4000 psi (281 kg/cm
). The wa-
ter-cement ratio varied between
5.4
an d
7.1 gal. per sack of cement. The coarse
aggregate used was crushed stone of
a
/s
in .
9.35
mm) maximum size; while the fine
aggregate was natural local sand. The
slump varied between 5 and 7
/
2
in.
12 7
and 19.1 cm ) as given in Table 1.
Un coated, stress-relieved,
250 ksi seven-
wire
/4
-in, strands were used for pre-
stressing. The strand nominal area was
0,0356
sq in.
0.234 cm
and the ma-
terial had a un it elonga tion of 0.65 percent
Fig. 1. Typical stress-strain relation of prestressing steel.
PCI JOURNAL/May-June 1977
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Table 1. Properties of concrete in beam specimens.
Age at
Compressive
Tensile
Water-Cement
Avg.
Beam Test
Strength
(psi)
Splitting
Ratio
Slump
(days)
Strength
(gals/sack) (in.)
3 Days
At Test
(psi)
1 2 3
4 5
6
7
B1-64
28 3080
4780 650 5.4
5
B5-B6
29
2920 4100
545
6.2
6i
B7-B8
31
2920
4350
555
6.3
6 .
69-610
33
1850 2915
470 7 .1 9
611-612 35
3100
4740
705 5.8
5
B13-B14
27 2900 4200
650 6.4
7
615-B16 29
2670
4130
660
6.6
7
B17-B18
34
2650 4925 695 5.5
7z
619-620 34
1850 3010
470 7 .1
9
B21 37 3100 4720
730
5.8
5
B22
36 2900
4350
660
6.4
7
B23 39 2670 4150
670
6.6
7
B24
33
2655 4385 620 5.5 7% ,
at 70 percent of the ult imate. The m aterial
satisfied ASTM A-416 specifications and
had a typical stress-strain relation as show n
in Fig. 1.
Non-prestressed #3 and #4 deformed
bars were used as supplemen tary reinforce-
m ent at the tension side in al l specimens
except Beams B -1 to B-6 . A typical stress-
strain diagram for the deformed bars is
shown in Fig. 2. Table 2 gives the details
of the reinforcem ent used for the beams in
this test program , and Fig. 3 gives typical
cross sections of the test beams.
Fabrication and testing
Twenty single-span and four continuous
beams were fabricated
i
and tested. The
simply supported single-span beam s were
as follows:
(1 )
Beams B-1 to B-6 were T sections
with pretensioned prestressing tendons
only.
(2 )
Beams B-7 to B-18 were also T
sections reinforced with bo th pretensioned
prestressed tendons and non-prestressed
mild steel.
Table 2. Geometrical properties of beam specimens.
Compres.
Web
Beam Size
Longitudinal
Tensile Steel
Steel
Reinforcement
Beam
Sect.
Total
Eff.
Flange
Web No. No.
No.
As
As ,
Size
Spacing
Depth
Depth
Width Thick-
of
of
of
p
p
of
ness
1/4-in.
3
4
Bar
Tendons Bars Bars
(5 )
(5 )
(s9 (s9
(in.) (in.) (in.) (in.)
in.)
in.)
(in.)
1
2
3
4 5 6
7
8 9
10
11
12 13
14
15
B-1 T 10
8.00
8 3
3
-
-
.108
0.17 .2 2
.34
3
6
B-2
T 10
7.75 8 3
4
- -
.144
0.23
.2 2
.3 5
3
6
B-3
T
10
7.30 8
3
5 - -
.180
0.31 .2 2 .38
3
6
B-4 T
10
7.00
8
3
6
-
- .216
0.39
.2 2
.3 9
3
6
B-5 T 10
6.57
8 3
7
-
-
.252
0.48
.2 2 .42
3 4
B-6
T
10
6.25
8 3
8
- .288
0.58
.2 2 .44
3 4
B-7 T 10
8.80
8
3
3
2 - .328
0.47 .2 2 .3 1
3
6
B-8 T
10
8.80 8 3
3
2 -
.328
0.47
.2 2
.3 1
3
6
6-9
T
10
8.65 8
3
4
2
-
.364
0.53
.2 2
.3 2
3
0-10
T 10
8.65 8 3
4
2 - .364
0.53 .2 2
.3 2
3
6
B-11 T
10
8.37
8
3 5
2
-
.400
0.60
.2 2
.3 3
3
4
B-12 T 10
8.37
8 3
5
2
-
.400
0.60 .2 2
.3 3
3
4
B-13 T
10
8.46 8
3
6
-
2 .616
0.91
.2 2
.3 3
3 4
8-14 T
10
8.46
8
3
6 -
2 .616
0.91
.2 2
.3 3
3 4
B-15
T 10
8.22
8
3
7
-
2
.652
0.99
.22•
.3 3
3 4
B-16 T 10
8.22 8
3
7 -
2
.652
0.99
.2 2
.3 3
3 4
0-17 T
10
8.00 8
3
8
-
2
.688
1.08 .2 2 .34
3 4
B-18
T 10
8.00
8
3
8 2 .688
1.08
.2 2
.3 4
3
4
B-19
10
9.00
6 3
3
2
-
.328
0.61
.2 2
.41
3
B-20
10
9.00
6
3
4
2
,544
1.01
.2 2
.41
3
4
8-21
10
9.00 6
3
3
2
-
,328
0.61
.2 2
.41
3
4
B-22
10 9.00 6 3
3
2
-
.328
0.61
.2 2 .41
3
4
8-23
10
9.00
6
3
4
-
2 ,544
1.01 .2 2 .41
3
4
B-24
10
9.00
6 3
4
-
2
.544
1.01
.2 2 .41
3 4
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0)
Y
W
h
Fig. 2. Typical stress-strain relation of non-prestressing steel.
(3) Beams B-19 and B-20 were I sec-
tions with both prestressed and non-pre-
stressed reinforcement.
Beams B-21 through B-24 were I sec-
t ions cont inuous on two spans re inforced
with both pretensioned prestressed tendons
and mild steel reinforcement. Straight
strand profiles were used in all beams.
For continuous beam s, strands were in-
serted into plastic hoses in the com pression
regions to achieve zero bond between con -
crete and strands. Before testing, those
strands in the compression regions were
cut through to elimin ate the effectiveness
of strands in these zones.
All beams were 10 in. (25 cm) deep as
shown in Table 1. They were all over-de-
signed to resist diagonal tension. Deformed
closed stirrups (#3 bars) at 6 in. (15 cm)
center-to-center were used throughout the
span for beams with a low tensile steel
reinforcement percentage and 4 in. (10
(a)
T-
SE TION
(b)
I - SECTION
Fig. 3. Geometric beam cross -
sectional dimensions for T and I sections.
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cm) for beams
containing a high percent-
age of tensile reinforcem ent (see Table 4).
Pretensioning frames p rev iously designed
for
the
work reported in References 2 and.
3 were
used in this investigation. Each
frame provided an independent unit with
60,000 lb (13,6 ton s) axial capacity.
All reinforcemen t was instrumented w ith
electric strain gages at critical locations.
Readings were taken of the change in
strain at all the necessary stages of pm-
stressing and loading. In addition, de-
mo untable mechan ical gages were used to
measure the variation of strain on the con-
crete faces of
the
beams.
Mechanical dial gages having 2 in. (5
em) travel and 0.001. in, (0.025 mm) ac-
curacy were used to m easure the change in
Table 3 Measured and c lcul ted cr cking nd ultim te
moments of simply supported beam specimens
Meas.
M cr
Calc, M
M cr
Meas. M.
Calc.
M
u
Meas. M
.
Mode of
Ca-
ailure
Cal c.Mu
(in.-kips) (in.-kips) (in.-kips)
(in. -kips)
1
2
3
4
5
6 7
8
B-1
137.0
1 3 6 . 9 .
1.00
258.3
219.74
1.175
Tension
B-2 164.0 159.7 1.03
310.8
277.30
1.121
Tension
B -3
178.2 170..6
1 . . 0 4 344.4
299.73.
1 . 1 4 9 .
Tension
8
-4
189.0 183.6 7.03
390.6
330.68
1.181
Tension
B-5 1 , 8 3 . 0
186.7
0 , 9 8 .
4 . 1 7 . 9 ,
935,999;
1..240 Compression
6 -6 1 9 3 , 2 ,
103,4; 1. 03
430.5.
345.59
1.240
Compression
8-7
126.0 137.6
0.92
336.0
296.10
1.135 Tension
B-8
136.4 140.0
0 . 9 8 . 336.0
296.78
1.132
Tension
B-9
130.0
1 3 6 . . 2
0.96 336.0
329.11 1.021
Compression
B-10
126.5
138.4
0.92 327.6
328.21
0.998
Compression
6-95
979,0
7 7 7 , 3 :
0„9
1 3
445.2
37 9.62 1.17 2
Compressi©n.
6-12 173.0 177.3 0.98
443.1
380.14 1.166
I
Compression
B-13
994.0
182.3
1.06
588.0
546.79
1 . 0 7 5
Compression
B-14 195.0
187.5
1 . 0 4 .
558.6
547.94 1 ; 0 1 . 9
Compression
0-15
2 0 5 . 9 . 199.0
1.03
562.8
513.15
1.092
Compression
B
-16
201.$. 1 9 7 . . 5 . 1.02
554.4
510,36. 1,999
C o m p r e s s . i ;
Q n
B-17 2 1 6 . 0 1 215.0 1.00
600.6. 580.65
1.030
Compression
B-1@
253,0
219.1
0.97 592.2
584,90
1..015
0ompres$ s o n
B - 1 9 .
126.0
737.3 0 . 9 2 ,
308.,7
293.57
1.052
Compression.
8-08
127.3 167.9 0.75
300
„3.
454,11
0.660
Shear
Table 4 Measured stabilized crack spacirugs versus their theoretical values
Meas.
Theo.
a Meas. a
Meas.
Corre-.
a
Stab.
min.
Stab. min. Csmin.
Stab. max. St ab . max. csmax.
Stab.
sponding
Theo.
csmean
Beam
Crack
Crack te
Crack
Crack
Mean
Initial
Mean
T
a
Spacing Spacing
CSein.
Spacing Spacing
osmax.
Crack M/ M Crack
csmean
a c s
min.
ac s
min.
a c s
acs
Spacing.
n
Spacing
max. max.
a
a
csmeanea n
(in.) (in.)
(in.) ( i n . ) ,
(in.)
(in.)
2
3
4 5 6
7
8
9
1 0 . .
1 1
B-1
5.00
4.08
1 . 2 3 - 7.25 8 . 1 5 :
0 . . 8 9
6.12
0 . . 5 9
6.33
0.97
B-2
4.13
3.93 1.05:
7.63
7.86
0,97
5.90
0.55 5.34
1.11
B - 3 -
4.31 3.73
1 . 1 . 6
6.63
7 . 4 , 6 .
0.89 5.60 0.67 5.13
1.09
8-4
3 , 7 . 3
3.26
0.96 6 . 3 8 : 6.52 0.98
4.89
0.67
4 . 7 5 .
1.03
B - 5 . 2.38
2.77
0.86 5 . 0 0 ,
5 . 5 3 , 0.90 4.15
0.70
4.39 0.95
6-6
2.50
2.41 1.04
5.13
4.51 1 . . 0 7
3.61
0 . . 6 6
3.92 0.92
0-7
2 . 1 3 . 3
2.13
1.00
3.94
4 . 2 6 .
0.92
3.20
9 , . 7 5
2.97 1.08
B - 8 .
1.94.
1.98
0 . 9 3 . 4 . 0 6 . 3 . 9 4 : 7.03
2.97
0.75
2.97
1.00
B-9
2.13 1.90
1.12
3 . 7 5 .
3.80
0.97'
2.85
0.73 2.72
1.05
B-10
2.50 1.84
1.36
5.00
3 . 6 8 .
1.36 2.76
0 . 7 6 :
2.72
1.02
8-11 1.75
1 . 6 3 . 1.07
3.13 3.26
0 . . 9 6
2.44
0.66 2.67
0.91
B-12 1.75
1.65 1.06
3.38
3 . 3 0 .
1.02
2.47
0.66
2.67
0.93
0-13
1.44
1.58
0.91 3,56
3.16
1.13
2.37
0.68 2.21
1.07
0 - 1 4 . 1.38
1 . 5 1 .
0.91
3.63
3.03 1.20 2.27
0.71
2 . 2 1 .
1.03
B-15 1,25 1.39
0.90:. 3.73
2 . 7 7 -
1.13
2 . 0 8 -
0 . . 7 5 2 . 4 6 - .
0.84.
B
-16
1 . 7 5 .
1.50 1.17 3.00 3 . 0 0 .
1.00
2 . 2 5 . 0.76
2.46
0.91
8 - 1 7
1.63 1.50
1.08
3 . 2 5 . 3 . . 0 0
1.08 2.25 0.70
2.29 0.98
8-18
1 . 5 0 . 1.37
1 . 0 9 .
3 . 0 6 - .
2 . 7 5 - 1 . 1 1
2.06
0.71
2.29
0.90
0 - 1 9 .
1,88
2 . 0 9 .
0.90
4.25
4.18
1.02
3 . 7 3 -
0.72
3.52
0.89
B-20
1 . 6 9 . 1 . 8 5 .
0,91
3.75
3.70
1.03 2 . 7 8 .
0.70
2.57
1.08
Mean
1 . 0 4 :
1.03 0 . 9 9 .
5.0.
0 . 1 3 , 0 . 1 1 .
0.08
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deflection due to loading. Crack widths
were measured with illuminated 25-p ower
microscopes having a
0.05 mm accuracy.
Crack spacin gs of all the developing cracks
were also accurately recorded as well as the
crack pen etration of the principal cracks.
For most
of the
beams, eight to nine
increments of load were applied to fail-
ure. At the con clusion of each beam test,
the concrete con trol cyl inders were tested
for both com pressive and tensile failure.
Test Results
This section summarizes the test re-
sults for effective prestress force, mo-
ment capacity, flexural crack widths,
and load-deflection data.
Effective prestress and
moment capacities
A measurement of the effective pre-
stressing force is necessary to determine
the net steel stress for each loading
state. The initial prestress varied be-
tween 174.1 and 180.6 ksi (12,240 and
12,696 kg/cm
, respectively). The ef-
fective prestress correspondingly varied
from 138.0 to 148.0 ksi (9701 to 10404
kg/cm
2) after deducting anchorage,
elastic shortening and creep and shrink-
age losses.
The
measured external mom ents, Mc,.,
producing the first hair crack, is com-
pared to the calculated cracking mo-
ment in Table 3. This table also gives
the measured and calculated ultimate
moments .
Flexural cracking
Maximum crack widths were mea-
sured at the reinforcement level and at
the bottom tensile face of the concrete.
The spacings of the cracks were meas-
sured an both faces of each beam at
each loading stage. These spacings
were summarized for each test speci-
men and the mean crack spacing was
calculated.
Table 4 gives the mean stabilized
crack spacing for load ratios of 50 to
70 percent of the ultimate load. Table
5 gives the measured crack width of the
stabilized cracks at the reinforcement
levels of the steel closest to the outer
fibers for the various stress levels.
Table 5. Observed versus theoretical maximum crack width at steel level.
Net Steel Stress afs
30 ksi
40 ksi
60 ksi
80 ksi.
Beam
W obs.
W theo,
Obs.
obs, W heo,
obs.
Wis. W +eo,
W ig,
obs,
rhea,
obs.
theo.
the,,.
W theo .
then
2 3
4
5 --- 6 7 8 9
10
11
12
13
B-1
.0089 .0089
1.00
.0122
,0118
1,03
.0198 .0177
1.12
.0285
.0236
1.21
B-2 .0071
.0075
0.95
.0104
.0100
1.04
.0172 .0150
1.15
.0242
.0198
1.22
B-3 .0062
.0072 0.86
.0090
.0096
0.94
:,0149
.0144
1.03 .0212
.0193
1,10
0-4
,0053
,0067
0.79
.0075
.0089
0.84
.0127
.0133 0.95
.0184 .0178 1.03
6-5
.0047
,0065
0.72
.0067
.0087 0.77 ,0108
.0130
0.83
.0160 .0174
0.92
B-6
.0042
.0058
0.72
,0058
.0078
0.74
.0095
.0116
0,82
.0140 .0155
0.90
B-7
.0040 .0041
0,98
.0055
.0055
1.00
,0095 .0082 1.16
,0135
,0109 1.25
B-8
.0039
0041
0.95
.0054
.0055
0.98 ,0089
.0082
1.09
.0128
.0109
1.17
B-9
,0077
,0035
1.06
.0050
.0047
1,06
.0081
.0070 1.16 10123
.0094
1.31
B-10
.0041 .0035
1.17
.0056
.0047
1.19 .0090
,0070
1.29 .0132
.0094
1.40
B-11 .0036 .0043
0.84
.0050 .0058
0.86
.0080 .0087
0.92
.0123
.0115
1.07
B-12 .0035 ,0043
0.81
,0052
.0058
0,90 .0086
.0087
0.99
,0130
.0115
1.13
8-13
.0030
.0034
0.86
.0045
.0046
0.98
.0077
.0088
1.13
.0112
.0091
1.23
B-14
,0032 .0034
0,94 ,0047
.0046
1.02 .0080
.0068
1.16
.9117
,0091
1.28
B-15
.0025
.0040
0.63
.0042
.0053
0.79
.0077
.0080
0.96
.0117
.0107
1.09
8-16
.0026 .0040
0.65
.0039
,0053 0.74 .0067
.0080
57.84
.9355
.0107
0.98
B-17
.0027
.0437
0.73
.0042
.0049
0.86
.0074
.0074 1.00
.0112
.0098
1.14
B-18
.0021 ,0037
0.57
.0031
.0049
0.63 .0063
.0074
0.85
,0106
.0098
3.08
B-19
.0041
.0049 0.84
.9957
,0065
0.88
.0090
.0097
0.93
.0132
.0130
1.02
B-20
.0026 .0036
0.72
,0039 .0047
0.83 .0070
.0071
0.99
.0110
.0096
1.16
Mean
0,84
0.90
1.02
1.13
S.
0. 0,18 0.14
0.14
0.13
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a I
O
J
0.0
0.3
0.6
0.9
1.2
1.5
1. 8
2.1
DEFLECTION (IN)
Fig. 4. Typical load
-
deflection relation for Beams B-3 and B-4.
Deflection
Deflection behavior of prestressed
concrete structural elements differs
from that of reinforced concrete sys-
tems. Initial reverse deflection due to
prestressing, namely camber, has to be
considered. Typical composite net
load-deflection relations due to external
load are shown in Fig. 4.
These curves essentially show a tri-
linear relation. The first stage repre-
sents the precracking stage, essentially
elastic, the second stage, namely, the
post-cracking stage where the deflection
of the beam increases faster as more
cracks develop, and the third stage de-
notes the behavior prior to failure.
a) FOR EVEN DISTRIBUTION OF
b) FOR NON-EVEN DISTRIBUTION OF
REINFORCEMENT IN CONCRETE
R E I N FO R C E M E N T I N C O N C R E T E
Fig 5 E ffective area in tension for distribution of reinforcement
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Mathematical Model for
Evaluating Serviceability
This section describes the mathemati-
cal model used to evaluate serviceabil-
ity based on crack spacing, crack width,
and deflection.
Crack spacing
Primary cracks form in the region of
maximum bending moment when the
external load reaches the cracking load.
As loading is increased, additional
cracks will form and the number of
cracks will be stabilized when the stress
in the concrete no longer exceeds its
tensile strength at further locations re-
gardless of load increase.
This condition is important as it es-
sentially produces the absolute mini-
mum crack spacing which can occur at
high steel stresses, to be termed the
stabilized minimum crack spacing. The
maximum possible crack spacing un-
der this stabilized condition is twice
the minimum, to be termed the stabil-
ized maximum crack spacing. Hence,
the stabilized mean crack spacing
ad 8
is shown to be the mean value of the
two extremes.
The total tensile force T transferred
from the steel to the concrete over the
stabilized mean crack spacing can be
def ined as:
T
= f a,, u, , Io
(1)
where
f =
a factor reflecting the distribu-
tion of bond stress
urn
= maximum bond stress which is
a function of
\ fa
Jo = sum of the circumferences of
the reinforcing elements
The resistance R of the concrete
area in tension
A
t
an be defined as:
R
= A
t f t
(2)
By equating Eqs. (1) and (2), the
following expression for
a
C3 is obtained,
where c is a constant to be developed
from the tests:
a0 8
\
(3)
The concrete stretched area, namely,
the concrete area in tension A
t
for both
the evenly distributed and non-evenly
distributed reinforcing elements is
shown in Fig. 5.
With a mean value of
f tVL = 7.95
in this investigation, a regression analy-
sis of the test data resulted in the fol-
lowing expression for the mean sta-
bilized crack spacing:
aC8
=
1.20 A
t/so
(4)
Fig. 6 gives the basic regression
analysis plot for
a C 8
Crack width
If Afg is he net stress in the pre-
stressed tendon or the magnitude of
the tensile stress in the normal steel at
any crack width load level in which
the decompression load (decompression
here means f
c
= 0 at the level of the
reinforcing steel) is taken as the refer-
ence point,
3 5
then for the prestressed
tendon:
ifs = fn.t — fd
(5)
where
f ^ t
= stress in the prestressing steel
at any load level beyond the
decompression load
f,
t stress in the prestressing steel
corresponding to the decom-
pression load
The unit strain e,
= Of
8
/E
3
ince it
is reasonable to disregard the unit
strains in the concrete due to the ef-
fects of temperature, shrinkage and
elastic shortening.
4 5
Hence, the maxi-
mum crack width can be defined as:
2U
mx
= k
a
0 a
(6)
where
k
and
a
re constants to be
established by tests.
Alternatively:
w.
mam
= k
a
as(Of,)
a
(6a)
Eq. (6a)
is
rewritten in terms of
Of8
so that an analysis of the test data of
all the simply supported test beams
leads
to
the following expression at the
PCI JOURNAL/May-June 1977
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6
d
to
a)
3
C. ,
rc
C. ,
z
a2
w
0
w
M
0
0
o
o
o
f At
a ^ g
=0.I5I
0
0
9
0
STAND ERROR OF EST. 0.2698
IV
LV
ov
•sv
2
Fig.
6.
Stabilized mean
crack spacing versus
A t
/V?,
o•
l a
t
Wmax.5.85xt9" (Mo ^Dfs
12
.-
10
*40%
.. _______
• -40%
1 8
-
W4
/
•
STAND. ERROR
OF EST,
0.00125'"
)o
0
'SC)
Ivy
13v
Gvv
c:v
avv
aav
A)A6
fig. 7, Linearized maximum crack width versus net steel stress.
0.1
0 ,
-0.,
z
• 0.
E
0.
x
a c
g 0
rc
U
a1
Q 0
0
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reinforeement level::
= J.4 X W
'i
4
g
S
Of4)1..31
7).
A 25 percent band of sc3tt-cv en-
velopes all the data for the expression.
ir> Eq.. (7) for Af
20 to 80 ksi.
Linearizing Eq. (7) for easier use by
the design engineer leads to the fol-
lowing simplified expression of the
maximur crack width at the reinforc-
ing steel level:
w.
= 5.85 X 10-
5
A7
Io) Ofs
(8a)
The maximum crack width (in.) at
the tensile face of the concrete is given
by.:,
tp Max
=
5.85>< 1
0-5
(R
4
)(
A
t/o)^s (8b)
where R
is a distance ratio as defined
in the notation.
A plot of the data and the best fit
expression for Eq (8a) is given in Fig. 7
with a 40 percent spread which is rea-
sonable in view of the randomness of
crack development and the lineariza-
tion of the original expression of Eq.
(7).
Tables 5 and 6 give the relation be-
tween the observed and theoretical
crack widths at the reinforcing steel
level as well as at the tensile concrete
faces of the beams at net stress levels
t f s
of 30, 40, 60, and 80 ksi.
Deflection
Deflection computations under ser-
vice load conditions will
usually
be
necessary
to ensure deflection service-
ability in addition to crack control
ser-
viceability. Since the deflection which
concerns the design engineer most is a
service load condition, both the nn-
cracked and cracked section properties
are needed
1 1 1 2
for the computations.
An estimate of the magnitude of de-
flection can be made from the follow-
ing equation:
Sc
M
¢
L ?
(9 )
where c
is a
constant depending
on
the
loading and support conditions.
Branson's generally accepted11,13,14
expression for the effective moment of
inertia I
e
is as follows:
Table 6 Observed versus theoretical maximum
cr k width at tensile face of beam.
let
Steel Stress.4fs.
39 ksi.
40 ksi 6 ks
0ks
Beam
W
obs.thea.
W
obs.
W obs-
Wibeo.
W
obS.
W e b s . W t he O.
Wobs.
Web s
W eb
s
c
Wtheo.
Wtheo.
Wtheo.
1
2
3
4
5
6
7
8
9 10 1 1
1 2
13
B 1
.0111 .0731 0.847
.0151
.0175. 0,865 .0261
..0262
1.000
.0400
.0349:. 1.1.45.
6-2
.0127
.0110
1.079
.0204
.0157 1.299
.0275
. `9236.
1.108
.0409 .0313 1.309
B-3
.0131
.0128 1.022
.0166: .0172
0.971
.0304
.0256, 1.166 .0382 .0344
1.112
B-4
.0097 .0130
0.742
.0158
.0174
0.910
.0226
.0259 :
0.871
.0304 .0347
0.876
B-5 .0091
.0147
0.619
.0117
.0197.
0.595 .0205
.0294
0.698
.0320
.0393,
0.814
6-6
.0124 .0148.
0.835
.0181
.0199
0.906
.0213 .0297: 0.717 .0364
.0397
0.917
B-7
.0052.
.0051 1..015
.0068
.0069
0.989 .0117 .0103:
1.141
.0188 .0137
1.380
B-8
.0049 .0051 0,956 .0061 .0069 0.887
.3111 .01.03 1.Q63
.0146
.0137
1.072
6-9 .0051 .0045
1.130
.0064 .0 0 61
1 .0 56,
.0107
.0090 1..185:
.0165
.0121
1.361
B-10
.0058
.0045 1.285
.0082
.0061 1.352
.0134
.0090
1,484
.0185. .0121
1.526
8-11
.0054 .0059 0,917
.0069 .0079
0.868
.0112
.01.19
0.940
.0172.
.0158 1.092
B-12
.0048 .0059
0,815 .0076 .0079
0.956
.0134 .0119
1.124
.0192 .0158
1.219
B-13
.0043 .0046 0.937
.0058
.0062
0.934
.0105 .0092
1 144
.0138 .0123
1.1.23
B-14
.0052 .0046 1.133 .0059 .0062
0.950
.0103 .0092
1.122 .0145
.0123
1.180
B-15 .0039 .0057
0.682 .0 0 61 .0076 0.805
.011.5
.0114 1.005 .0181 .0153
1.183
B-16
.0038 .0057. 0.664
.0057
.0076
0.752
.0093
.0114 0.81.3 .0160 .0153
1.046
B-17
.0039 .0056
0.698
.0060 .0074 0.811
.0098
.0112 0.877 .0159 .0148 1.074
B-18 .0030 .0056
0.537
.0045 .0074
0,608
.0086 .0112 0,770 .0147 .0148
0.993
B-19
.0057 .0061 0.931.
.0085
.0081.
1.046. .0129
.0121
1.064 .0202
.0163
1.243
B-20
.0034 .0045 0.750 .0045: .0059,
0.760
.0089
.0089
1.000
.0139 .0119
1.161
Mean
0.880-
0.916
1.017
1.141
S.D.
0.196.
0.187 0.190 0.174
PC JOURNALMay June 1977
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= (Mcr
)
1,
+
[i_(Mcr)]1̂ ^
10
The cracking moment
M
0
of a pre-
stressed concrete beam can be com-
puted from the following expression
for evaluating
le :
M
c
.=Fe-}
AI9
frl.9
(11)
9
y
t
yt
where
I
er
and I9
can be calculated from
the properties of the section. The con-
crete modulus of rupture,
f
s equal to
7.5a/f.
.
Eqs. (10) and (11) were applied to
the deflection test results of 19 pre-
stressed beams, 13 of which had addi-
tional non-prestressed tension steel. The
range of application was from the
cracking load level to 90 percent of
the ultimate load.
Fig. 8 gives a plot of the computed
versus measured deflections for the
short-term loadings applied in this in-
vestigation. It is noted that the degree
of scatter is within a 20 percent band
which can be considered fully ade-
quate.
Discussion of
Test Results
It is observed from this investigation
that the initial flexural cracks started at
a relatively low net steel stress level
between 3 and 8 ksi. These initial
cracks formed in a rather random man-
ner and with an irregular spacing. All
major cracks usually developed at a net
steel stress level of 25 to 30 ksi.
At higher stresses the existing cracks
widened and new cracks of narrow
width usually formed between major
cracks. A visibile stabilized cracking
space condition was generally reached
z
0
0
t
U
W
J
W
W
O
O
W
F-
a.
a
U
MEASURED DEFLECTION IN.)
Fig. 8. Computed versus measured deflection.
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8.0
2
E 6.0
z
U
a
'' 4.0
U
U
V
Z
W
2.0
2
0.0
5.0
15 0
25.0
35.0
45.0
55.0
65.0
75.0
STEEL STRESS ofy KS I)
Fig. 9. Mean crack spacing versus net steel stress.
at 0.5 to 0.7 of the ultimate load.
This investigation established that
the maximum crack spacings after sta-
bilization were close to twice the mini-
mum possible spacings having a mean
value of 2.02 and a standard deviation
of 0.29. The effect of the variation of
percentages of the non-prestressed
steel was significant both on the crack
spacing and the crack width.
For beams with non-prestressed steel,
the number of flexural cracks was al-
most twice as many as those with no
mild steel. These cracks were more
evenly distributed, with considerably
less spacing and finer widths. This be-
havior can be attributed to the fact
that the bond of the mild steel to the
surrounding concrete played a pro-
nounced role in crack control.
A typical plot of the effect of the
various steel percentages on the crack
spacing at the various stress levels Of,
is given in Fig. 9. It is seen from this
plot that the crack spacing stabilized at
a net stress level of 36 to 40 ksi. The
influence of the various parameters,
particularly, the variation in the steel
reinforcement percentages of the pre-
stressing tendons and the non-pre-
stressed steel on cracking and deflection
of prestressed concrete T beams is giv-
en in Table 7.
It is also observed that it is advanta-
geous to locate the non-prestressed steel
below the prestressed tendons. This is
due to the fact that mild steel has larg-
er diameters than the prestressing rein-
forcement, hence a larger bond area of
contact with the surrounding concrete.
Also, by placing the mild steel close to
the tensile concrete face, cracks will be
more evenly distributed, hence crack
spacing and consequently crack width
will be smaller.
The effect of the spacing of the stir-
rups on the crack spacing was not pro-
nounced. It was found that the final
crack spacing and crack pattern did not
necessarily follow the vertical shear
reinforcement. Even though the first
few cracks usually started at the stir-
rups, the vertical legs of the stirrups
served only as initial weak areas of
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Table 7 Influence of various parameters on cracking
and c
eftection of prestressed concrete T beams
B e a m
Tenon
Non-
Prestressed
Steel
Percentage
of Tension
Steel
C.G. of steel*
(in.
from Top
Fiber)
Crack
Spac ing
M ax. Crack W idth
at Level
of
Stee l
at
f
= 60 ks i
N o .
of
Flexural
Cracks
Midspan
Deflection
at P = lOk
B
1
3-1/4'
N o n e 0.17 8.52
6.12
.0198
5
.950
8- 7
3-1/4
2
4
3
0.47
8.75
3.20 .0095
9
.400
64
31/4
-2=0
9.47
8,75
2,37
.0089
1 0
.395
6=2
4.1/4
588'e
0:23
8,43 5,90
.0172 5
.390
8'-9
4-1/4
2=#3
0,53
8,71
2.85
.0081
1 0
.342
8 = 1 0
4-1/4
2-#3 0.53
8.71
2.76
.0090
1
.350
B- 3
5
-1/4 N o n e 0.31
8,22 5.60
.0149
6 .320
B 1 1
5-1/4"
2=93
0,60
8.63
2.44
,00$0
1 1 .299
8-12
5 -1/4' 2-03 0.60 8.63
2.47
,0286
1 1 .288
B-4
6-1/4"
6689
0 .39 8.06
4,89
.0127
7
.255
8 1 3 6-1/4
2 -#4 0.91
8.65
2,37
.0077
1 1 .228
8-14
6-1/4'
2 -#4
0.91
8.65
2,27
.0080
1 2
.247
8- 5
7
-1/4 None
0.48 7.74
4.15
.0108 8
.234
B-15
7-1/4 2 -# 4 0 .99
8,57 2.08
.0077 1 2
.211
8
-16
7=1/4" 2 -# 4 0 .99 8.57
2.25
.0067 1 2
.207
8 -6
0-1/4
N on
0.58
7,44
3.61 .0095 8 .258
B-17
8-1/4 2-
64
1.08
8.49
2.25
.0074 1 2
.192
8-18
8-1/4 2-#4 1.08
8,49
2,06
.0093
1 3
,176
Distance from compression fiber to the center of lAytr of mild reinforcing steel closest to tension fiber.
stress concentration. In most cases, the
stabilized mean crack spacings were
sm aller than the spacings of st irrups.
Based on these observations and the
analytical results of this investigation
it can be said that the proposed equa-
tions for crack and deflection control
can be reasonably applied by the de-
sign engineer for maintaining the ser-
viceability of pretensioned partially
prestressed beams and girders under
working load and overload conditions.
Once the allowable crack width is es-
tablished for the prevailing environ-
mental conditions, the proper percent-
age of non-prestressing reinforcement
can be determined to ensure service-
able behavior. Four design examples
are given in the Appendix to demon-
strate the applicability of the proposed
crack width equations.
Conclusions
1. The maximum crack width (in.)
at the level of reinforcement closest to
the tensile face can be predicted from
the expression:
w = 5.85
x 10-5(A
2/I9)f8
The crack width at the outer face of
the concrete is
W
'?vax =
tom Rt
where R4
is the distance ratio.
2.
Initial flexural cracks randomly
form at irregular spacings at low net
steel stresses of 3 to 8 ksi. All major
cracks usually develop at a net stress of
25 to 30 ks i .
3 .
Visible crack spacing stabilizes at
50 to 70 percent of the ultimate load.
4.
The presence of non-prestressed
steel in the prestressed members has a
significant effect on crack control such
that the cracks become more evenly dis-
tributed and the crack spacings and
widths become smaller.
5. An increase in the percentage of
reinforcement decreases substantially
the crack spacings and width in the
part ially prestressed beam s.
6 .
For loads above the first cracking
load, the deflections are smaller for
prestressed concrete beams containing
non-prestressing reinforcement than for
a similar beam without non-prestressed
reinforcement.
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R F R N S
1. Huang, P. T., Serviceabi li ty Behavior
and Crack C ontrol in Pretension ed Pre-
stressed Concrete I- and T-Beams,
PhD Th esis under the direct ion of the
first autho r, Rutgers Un iversity, Jun e
1975, 33 8 pp.
2.
Nawy, E. C., and Potyondy, J. C.,
Moment Rotation, Cracking and De-
flection o f Spirally Boun d, Pretensioned
Prestressed Beams, Engineering Re-
search Bulletin No. 51, Rutgers Uni-
versity, Novem ber 1970, 96 pp.
3 .
Nawy, E. G., and Potyondy, J. C.,
Flexural Cracking Behavior of Pre-
tensioned, Prestressed Con crete I- an d
T-Beams,
ACI Journal,
Proceedings
V. 68, No. 5, May 1971, pp. 355-3 60.
4.
Nawy, E. G., Crack Control in Rein-
forced Con crete Structures, ACI
Jour-
nal,
Proceedings V. 65, October 1968,
pp. 825-836.
5.
ACI Committee 224, Control of
Cracking inConcrete Structures,
ACI
Journal,
Proceedings V. 69, No. 12,
December 1972, pp. 717-752.
6.
Ferry-Borges, J. N., Preliminary Re-
port, Comite Europeen du Beton,
Comm ission IV a—Cracking.
7.
Holmberg, A., and Lindgern, S.,
Crack Spacing and Crack W idth due
to Normal Force or Bending Mo-
ment, Document D2:1970, National
Swedish Building Research.
8.
Holmberg, A., Flexural Crack
Width, Nordisk Betong, 1970.
9.
CEB-FIP, International Recommen-
dations for the Design and Construc-
tion of Concrete Structures, Comite
Europeen du B eton/Federation Inter-
nationale de la Precontrainte, Paris
(English Edition Published by the Ce-
ment and Concrete Association, Lon-
don), 1970, pp. 1-80, Appendix V,
pp. 1-47.
10.
Abeles, P. W., Design of Partially
Prestressed Concrete Beams,
ACI
Journal,
Proceedings V. 64, No. 10,
October 1967, pp. 669-677.
11.
ACI Committee 318, Building Code
Requirements for Reinforced Con-
crete (ACI 318-71), American Con-
crete Institute, Detroit, 1971, 78 pp.
12.
ACI Comm ittee 3 18, Comm entary on
the Building Code Requirements for
Reinforced Concrete (ACI 318-71C),
American Concrete Institute, Detroit,
1971, 96 pp.
13 .
Branson, D. E., Instantaneous and
Time-Dependen t Def lections of Simple
and Continuous Reinforced Concrete
Beam s, Part 1, Report No. 7, Alabama
Highway Research Report, Bureau of
Public Roads, August 1963.
14. PCI Design Handbook—Precast and
Prestressed Concrete
Prestressed Con -
crete Institute, Chicago, 1971, pp. 1-1
to 12-8.
OT
Four numerical examples showing the appli-
cability of the recommended crack width
equations and a notation section, summariz-
ing in alphabetical order the meaning of each
mathematical symbol, appear on the next four
pages.
PCI JOURN AL/May-June 1977
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APPENDIX
ESIGN EXAMPLES
The following design examples are
prepared solely to illustrate the appli-
cation of the preceding recommended
crack width equations. In these exam-
ples it is assumed that the member
geometry, load condition, stresses in
tendons and mild steel have been de-
fined. Consequently, the detailed mo-
ment and stress calculations are
omitted.
Example Al
A prestressed concrete beam has a T
section as shown in Fig. Al. It is pre-
stressed with fifteen 7/16-in. diameter,
7-wire strand of 270-kip grade. The loca-
tions of neutral axis and cen ter of gravity
of steel are shown in the figure. f 's, = 5000
psi E
= 57 ,000V f ', ,
E = 28,000 psi.
Find the m ean stabi l ized crack spacing
and the crack widths at the steel level as
well as at the tensile face of the beam at
Ofa = 30 ksi. Assume that no failure in
shear or bond takes place.
(a )
Mean stabi lized crack spacing
A, = 7 X 14 = 98 sq in.
Eo=15,rD
=
15i-(7/16)
= 20.62 in.
at ,
=1.2(A:/yo)
= 1.2(98/20.62)
=5.7in.
(b )
Maximum crack width at steel level
w ,
n = 5.85 x
10-s(A ,
/lo)Ofs
= 5.85
x
10-6(98/20.62)30
= 834.1
x
10
-6
in .
-^ 0.0083 in.
(c )
Maximum crack width at tensile face
of beam
5 - 10.36
R̀
25-10.36-3.5
= 1 .3 1
w „ ax = wm a.c
R
= 0.0083 X 1.31
= 0.011 in.
Example A2
For the prestressed beam in Example
Al except that three additional #6 non-
prestressed mild steel bars are added as
shown in Fig. A2.
Find the crack spacing and crack widths
at
Af,
= 30 ksi.
(a )
Mean stabilized crack spacing
A, = 14(3
x
1.75
+
/ X
hs+
1 3 a
= 14
x
6.84
= 95.8 sq in.
Eo=20.62+3X2.36
= 27.70 in.
a,,
= 1.2(At
/mo)
= 1.2(95.8/27.7)
= 4.15 in.
(b ) Maximum crack width at steel level
wm az = 5.85 X 10-5(A,/^o)\f,
= 5.85
x
10-5(95.8/27.7)30
= 606.9 X 10-5
- 0.0061 in.
(c ) Maximum crack width at tensile face
of beam
25 - 10.6
R̀=25-10.6-2.75
= 1.24
w x
= wma:
= 0.0061 X 1.24
= 0.007 in.
Example A3
A prestressed concrete beam has an I
section as shown in Fig. A3. It is pre-
stressed with twenty-four 7/16-in. diam-
eter, 7-wire strand of 270-kip grade. The
locations of neutral axis and center of
gravity of steel are shown in the figure.
f , = 5000 psi,
E,
= 57,000-v
f
Es = 28,-
000,000 psi.
Find the m ean stabi l ized crack spacing
and the crack widths at the steel level as
well as the tensile face of the beam at
-Af,
= 20 ksi. Assume that no failure in
shear or bond takes place.
(a) Mean stabilized crack spacing
A =7X 18
=
126 sq in.
Eo = 24irD
= 247r(7/16)
= 32.99 in.
ass =1.2(A:/
xo)
= 1.2(126/32.99)
=4.58 in.
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o
N N
^ r
Î
+ ._ _– C.̂ S.
M
1
C..Ci.S_N
I5Ø
1 5
7-WIRE S TRAND
ĵ
7-WIRE S TRAND
l
14"
f
3-s
14'—
FIG. At
FIG. A2
r
_
5.5'
o
5.5'
O1
N .A .
N.A.
^
24
6 D
20
6
7 -WIRE STRAN D
r-
7- W IR E S T R A N D
F-
-7 —
FIG. A 3 FIG. A 4
Fig. A. Geometric details of sections in Examples Al to A4.
PCI JOURNAL/May-June 1977
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(b )
Maximum crack width at steel level
w..a,
= 5.85 X 10-5 At/Eo)ofs
= 5.85
x 10-5(126/32.99)20
= 446 9
x
10 -
== 0.0045 in.
(c )
Maximum crack width at tensile face
of beam
6 — 18.72
R̀-
36-18.72— 3.5
= 1.25
w'mz = w„ ar R,
= 0 .0045 X 1 .25
= 0.006 in.
Example A
For the prestressed beam in Example
A3
except that four 7
/1 6
-in. diameter,
-wire strand of
270-kip grade at the
bottom row are replaced by four #7 non-
prestressed mild steel bars as shown in
Fig.
A4.
Find the crack spacing and crack widths
at
\f s
= 20 ksi.
(a) Mean stabilized crack spacing
At=18X 3X1.75+
/
X 7/16+1 6)
= 122.06 sq in.
= 20^D + 4 X 2.75
=207rX7/6+4x2.75
= 38.49 in .
a
g
= 1.2 At/2o)
1.2(122.06/38.49)
= 3.8
in .
(b )
Maximum crack width at steel level
= 5.85 X 10-5(A, /Xo)Of,
= 5.85
x
10`5(122.06/38.49)20
= 371 0
x
10 -5
0.0037 in.
(c ) Maximum crack width at tensile face
of beam
_
3 6 — 1 9 .2 3
36-19.23-2.79
= 1.2
w' as = w,naz R,
=0.0037>< 1.2
= 0 . 004
in .
NOTE: From the comparison of
crack width values at the tensile face
of the concrete, it is noted that even
at the high net stress p f
s
= 30 ksi, it is
possible to reduce the crack width con-
siderably for partially prestressed or
overloaded beams with the addition of
a few mild steel reinforcing bars.
For normal levels of net stress
A f 8
under working load conditions up to
15 to 20 ksi, the crack width level that
can develop at the tensile face using
the criteria developed in this paper
can become negligible.
N O T A T I O N
A
g
= gross area of concrete, sq in.
f
=
a factor reflecting distribution
A
s
= area
of
steel
reinforcement,
of bond stress
sq in.
= compressive stress in concrete,
A t
= concrete stretched area, name-
psi
ly, concrete area in tension,
f 0 =cylinder compressive strength
sq in.
of concrete, psi
a
s
=
stabilized mean crack spacing,
f t =
tensile
splitting
strength
of
in .
concrete, psi
E,
= modulus of concrete, psi
fa
=
stress in prestressing steel cor-
= 57,000
V f ,_ psi
for normal
responding to decompression
weight concrete
load; ksi
E
= Young's modulus of steel, ksi
ft
=
stress in prestressing steel at
F
=, prestressing force in tendon,
any load level beyond decom-
kips, lb
pression load, ksi
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= (f, - J,)
= net stress in pre-
stressing steel, or magnitude
of tensile stress in non-pre-
stressing steel at any load level
Jr
= modulus of rupture of con-
crete, psi, ksi
Icr
cracked moment of inertia, in.4
= effective moment of inertia,
in.4
'U
= gross moment of inertia,
in 4
= effective beam span, ft, in.
a
= maximum service load moment
in span, in.-lb, ft-kips
Ur
= cracking moment, in.-lb, ft-
kips
1
0
= sum of reinforcing element cir-
cumferences
Ri = ratio of distance from neutral
axis of beam to concrete out-
side tension face
h
) to dis-
tance from neutral axis to steel
reinforcement centroid (h1);
value has a range in this in-
vestigation from 1.25 to 2.56,
ma,dmum bond stress as a
function of \ff; psi
maximum flexural crack width
at steel level, in,
maximum flexural crack width
at tensile face of concrete, in,
=
unit strain in reinforcement, in.
per in,
deflection, in.
Acknowledgment
This investigation is part of a continu-
ing research program on the behavior of
prestressed concrete beams and slabs, ini-
tiated by the f irst author and conducted at
the Con crete Research Laboratories of the
Department of Civil and Environmental
Engineering of Rutgers University—The
State University of New Jersey. The p aper
is based in part on the PhD thesis of the
second author under the direction of the
first.
Discussion of this paper is invited.
Please forward your comments to
PCI Headquarters by November 1, 1977.
47