Crack and Deflection

8/18/2019 Crack and Deflection

1/18

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

30


2/18

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

31


3/18

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

32


4/18

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.


33


5/18

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

34


6/18

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


35


7/18

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

36


8/18

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


37


9/18

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

38


10/18

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

39


11/18

= (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.

40


12/18

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


41


13/18

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.

42


14/18

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

43


15/18

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.

44


16/18

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.


45


17/18

(b )


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 )


= 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

46


18/18

= (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

Documents

Crack and Deflection