Citation

## Material Information

Title:
Prevention of Vertical End Cracks on Prestressed Beams during Fabrication
Creator:
REPONEN, MICHAEL ( Author, Primary )
2008

## Subjects

Subjects / Keywords:
Airfoil camber ( jstor )
Analytical models ( jstor )
Coefficient of friction ( jstor )
Concretes ( jstor )
Pipe flanges ( jstor )
Prestressing ( jstor )
Reverse transfer students ( jstor )
Static friction ( jstor )
Steels ( jstor )

## Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Michael Reponen. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
3/1/2007
Resource Identifier:
658231278 ( OCLC )

Full Text

PREVENTION OF VERTICAL END CRACKING ON PRESTRESSED BEAMS
DURING FABRICATION

By

MICHAEL REPONEN

A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING

UNIVERSITY OF FLORIDA

2006

by

Michael Reponen

This document is dedicated to my parents.

ACKNOWLEDGMENTS

This degree would not have been possible without my parents, my friends, and a

few complete strangers that helped me along the way. I would also like to thank the

FDOT, Dr. Cook, Dr. Lybas, Dr. Hamilton, Dr. Consolazio, and Gate Concrete for their

kind support of this research project.

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ................................................... vii

LIST OF FIGURES ........................................... .............................viii

ABSTRACT ............................................................................. xi

CHAPTER

1 IN T R O D U C T IO N .............................. .......................... ..................1...

Project O overview ......................................................................................1................
Prestressed C concrete B background ........................................................... ...............2...

2 END CRACKING LITERATURE REVIEW ........................................................6...

In tro d u ctio n ............................................................................. ..................... 6
Review of M irza and Taw fik 1978 ....................... ...............................................6...
R eview of K annel, French, Stolarski 1998.............................................. ...............8...
Summary of End Cracking Reduction Recommendations.....................................9...

3 MANUFACTURER SURVEY AND FIELD INSPECTIONS..................................11

M an u fact rer Su rv ey .................................................................................................. 1 1
F field Inspection Introduction .................................... ....................... ............... 11......
Field Inspection Results ...................................................................... ............ 13
F ield Inspection Sum m ary ......................................... ......................... ............... 16

4 VERTICAL CRACK ANALYTICAL MODEL ..................................................21

In tro d u c tio n ................................................................................................................. 2 1
A nalytical M odel T heory ........................................... ......................... ................ 22
G lobal M otion W without Friction..................................................... ................ 22
G lobal M otion W ith Friction.................. .................................................... 24
A nalytical C om m entary ........................................ ....................... ................ 26
A nalytical M odel A ssum options ............................................................. ................ 28
A nalytical M odel Input V ariables ......................................................... ................ 30
A nalytical M odel Flow C hart ....................................... ...................... ................ 30

5 R E S U L T S ................................................................................................................... 3 7

Introduction .................................................................................. ....................... 37
T est C ase 1 ......................................................................... ..... .........................38
Modification 1: Alter the Number of Prestressing Strands .................................38
Modification 2: Alter the Friction Coefficient ...............................................38
Modification 3: Alter the Concrete Release Strength.....................................39
M odification 4: Alter the Beam Lengths ................ .................................... 39
Modification 5: Alter the Temperature Change .............................................39
Modification 6: Alter the Number of Debonded Strands ...............................40
Modification 7: Alter the Debonded Lengths of 10 Strands ..............................40
M odification 8: Alter the Number of Beams.................................. ................ 40
Modification 9: Alter the Free Strand Length for 2 Beams...............................41
Modification 10: Alter the Free Strand Length for 3 Beams...............................41
Modification 11: Alter the Free Strand Length for 4 Beams...............................41
T est C ase 2 ...................... ......... .. ........... .. .......................................... 42
M odification 1: Alter the Friction Coefficient ............................... ................ 42
M odification 2: Alter the Beam Spacing ................ ................................... 42
Analytical M odel Conclusions ........................................................ 43
Field Data Results ..... ............... .. ........... .......................................44
Field D ata Conclusions ... ............................................................................... 44

6 CONCLUSIONS AND RECOMMENDATIONS ................................................72

APPENDIX

A SAMPLE RETURNED SURVEY FORMS.........................................................75

B VERTICAL CRACK PREDICTOR ..................................................... ................ 80

C SIMPLIFIED VERTICAL CRACK PREDICTOR...................... ...................250

D FIELD STUDY STRAND LAYOUT .......... .........................256

LIST O F R EFEREN CE S ... ................................................................... ................ 257

BIOGRAPH ICAL SK ETCH .................. .............................................................. 258

LIST OF TABLES

Table page

5-1 T est C ase 1 Input D ata ..................................................................... ................ 46

5-2 Alter the Number of Prestressing Strands ............... ....................................47

5-3 A lter the Friction C oefficient ...................................... ...................... ................ 49

5-4 A lter the Concrete R release Strength..................................................... ................ 51

5-5 A lter the B eam L engths .................................................................. ............... 52

5-6 A lter the Tem perature C change ............................................................. ................ 54

5-7 Alter the Number of Debonded Strands...............................................................55

5-8 Alter the Debonded Lengths of 10 Strands ..........................................................57

5-9 A lter the N um ber of B eam s........................................ ....................... ............... 58

5-10 Alter the Free Strand Length for 2 Beams..........................................................60

5-11 Alter the Free Strand Length for 3 Beams..........................................................62

5-12 Alter the Free Strand Length for 4 Beams..........................................................64

5-13 T est C ase 2 Input D ata ................................................................... ................ 65

5-14 A lter F riction R results .. .................................................................... ................ 66

5-15 Free Strand L engths..... .................................................................. .............. 66

5-16 B eam Spacing R results .................................................................... ................ 67

5-17 72" Florida B ulb-T Input D ata ........................................................... ................ 67

5-18 End M ovem ents for B eam 2...................................... ...................... ................ 68

5-19 End Movements for Right End of Beam 1 .........................................................69

5-20 End Movements for Left End of Beam 3 ...........................................................70

LIST OF FIGURES

Figure page_

1-1 Strand A nchorage Sy stem .......................................... ......................... ...............4...

1-2 Term inology .......................................................................................................5

1-3 G general Effects of Friction .................. ............................................................5......

2-1 Steel Bearing Plate .................... .. ........... ..................................... 10

3-1 Strand C cutting P rocess.. ..................................................................... ............... 16

3-2 Prestressed Strand C rack ................................................................... ............... 17

3-3 Bursting Forces Caused by Prestressing Strands..................................................17

3 -4 R adial C rack in g ......................................................................................................... 18

3 -5 L iftin g D ev ices .......................................................................................................... 18

3 -6 A n g u lar C rack ............................................................................................................ 19

3-7 W eb-Flange Junction C rack ....................................... ....................... ................ 19

3 -8 E d g e S p a ll ................................................................................................................ .. 1 9

3-9 Horizontal Cracks .................... .. ........... ...............................20

3-10 V vertical C rack ............... .. .................. .................. ................. ... ....... ......... ...20

4-1 Global Movement For 3 Beam Symmetrical System...........................................31

4-2 Change in A cting Static Friction Force................................................ ................ 32

4-3 G global M option of B eam .................................................................. ................ 33

4-4 Stress in C concrete B ottom Flange ........................................................ ................ 34

4-5 A xial and C am ber M ovem ent...................................... ...................... ................ 35

4-6 A nalytical M odel Flow C hart...................................... ...................... ................ 36

5-1 T est C ase 1 .................................................................................. ........................45

5-2 T est C ase 1 N o A lterations........................................ ........................ ................ 46

5-3 Alter the Number of Prestressing Strands ............... ....................................47

5-4 Number of Prestressing Stands fcalc/f Maximums ................................................48

5-5 A lter the Friction C oefficient ...................................... ...................... ................ 49

5-6 Friction Coefficient fcaic/f M axim um s .................................................. ................ 50

5-7 A lter the Concrete R release Strength..................................................... ................ 51

5-8 Concrete Release Strength fcaic/f Maximums .......................................................52

5-9 A later th e B eam L length s .............................................................................................53

5-10 B eam Lengths fcaic/f M axim um s......................................................... ............... 53

5-11 A lter the Tem perature C change ........................................................... ................ 54

5-12 Temperature Change fcal/f Maximums ..................................................55

5-13 Alter the Num ber of D ebonded Strands ............................................. ................ 56

5-14 Number of Debonded Strands fcali/f Maximums................................................56

5-15 Alter the Debonded Lengths of 10 Debonded Strands......................................57

5-16 D ebonded Lengths fcaic/f M axim um s.................................................. ................ 58

5-17 A lter the N um ber of B eam s....................................... ...................... ................ 59

5-18 N um ber of Beam s fcal/f M axim um s................................................... ................ 59

5-19 Alter the Free Strand Length for 2 Beam s.......................................... ................ 60

5-20 Free Strand Length for 2 Beams fcali/f Maximums ...........................................61

5-21 Alter the Free Strand Length for 3 Beam s.......................................... ................ 62

5-22 Free Strand Length for 3 Beams fcal/f Maximums ...........................................63

5-23 Alter the Free Strand Length for 4 Beams ................ ...................................64

5-24 Free Strand Length for 4 Beams fcali/f Maximums ...........................................65

5 -2 5 T e st C a se 2 .............................................................................................................. 6 5

5-26 Alter the Friction Coefficient for Multiple Beam Ends......................................66

5-27 72" Florida Bulb-T A rrangem ent ....................................................... ................ 67

5-28 Beam 2 Left End Measured vs Calculated .........................................................68

5-29 Beam 2 Right End Measured vs Calculated.......................................................69

5-30 Beam Right End Measured vs Calculated........................................................70

5-31 Beam 3 Left End Measured vs Calculated .........................................................71

Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering

PREVENTION OF VERTICAL END CRACKING ON PRESTRESSED BEAMS
DURING FABRICATION

By

Michael Reponen

December 2006

Chair: Ronald Cook
Major Department: Civil and Coastal Engineering

The purpose of this research project was to determine the causes and cures of the

vertical end crack found on the bottom flange of AASHTO, Florida Bulb-T, and Florida-

U prestressed beam ends. This vertical crack forms during the transfer of the prestressing

force to the concrete. This type of crack forms at the base of the beam just a few inches

from the end of the beam, and propagates vertically upward towards the web region of

the beam. According to interviewed field personnel, this type of end crack is a

maintenance issue that slows down production and also raises questions regarding loss of

bond and ingress of chlorides to the prestressing strands.

The research project began by mailing surveys to three Florida Department of

Transportation prestressed beam manufacturers in Florida to determine the extent and

types of end cracking each manufacturer was experiencing during the production process

of their AASHTO, Florida Bulb-T, and Florida U-beams. Site visits followed the surveys

to allow the researchers to observe the beam production process. The site visits also gave

the researchers the opportunity to talk to each plant's engineers and technicians about the

different types of end cracks and when, where, and how each type of crack forms. A

computer model was then created in MathSoft's MathCad Version 12 to determine the

sensitivity vertical cracking has to variations in input variables such as spacing between

the beams, friction coefficient between the beam and the casting bed, debonded lengths,

etc. The analytical model determined that the variables that have the greatest effect on

vertical cracking are atmospheric temperature change between the time of beam casting

and the time of strand detensioning, friction coefficient between the casting bed and the

bottom of the beams, concrete release strength, beam length, and number of prestressing

strands. Beam spacing, and the number of beams on the casting bed have the next

greatest effect. Beam spacing becomes more important as the number of beams on the

casting bed increases. The number of debonded strands and the lengths of the debonded

strands have a small effect on vertical cracking. The conclusion that can be drawn from

this research study is that the three most important things to do in order to reduce the

occurrence of vertical cracks are to detension the prestressing strands when the

atmospheric temperature is similar or warmer than the atmospheric temperature when the

beams were cast, to lower the coefficient of friction between the casting bed and the

bottom of the beams, and to add additional space between the beams.

CHAPTER 1
INTRODUCTION

Project Overview

The purpose of this research project was to determine the causes and potential cures

of the vertical end crack found on the bottom flange of AASHTO, Florida Bulb-T, and

Florida- U prestressed beam ends. This vertical crack forms during the transfer of the

prestressing force to the concrete. This type of crack forms at the base of the beam just a

few inches from the end of the beam, and propagates vertically upward towards the web

region of the beam. According to interviewed field personnel, this type of end crack is a

maintenance issue that slows down production and also raises questions regarding loss of

bond and ingress of chlorides to the prestressing strands.

The research project began by mailing surveys to three FDOT prestressed beam

manufacturers in Florida to determine the extent and types of end cracking each

manufacturer was experiencing during the production process of their AASHTO, Florida

Bulb-T, and Florida U-beams. Site visits followed the surveys to allow the researchers to

observe the beam production process. The site visits also gave the researchers the

opportunity to talk to some of each plant's engineers and technicians about the different

types of end cracks and when, where, and how each type of crack forms. A computer

model was then created in MathSoft's MathCad Version 12 to determine trends that

should be followed to maximize the effectiveness of any vertical crack prevention plan

for any type of AASHTO, Florida Bulb-T, or Florida U-beam. Beam end movements

were measured for three 139 ft long 72" Florida Bulb-T beams at Gate Concrete in

Jacksonville Florida and compared to the predicted values from the MathCad model. It

was determined that the analytical model could not predict the exact movements of the

prestressed beam ends in the field due to non-simultaneous cutting, and dynamic effects.

However, the analytical model was determined to be a valuable tool for determining

trends that should be followed to reduce the occurrence of a vertical crack on any type of

prestressed beam.

Prestressed Concrete Background

Prestressed beams are formed by stretching steel strands with hydraulic jacks

across a casting bed that can be as long as 800 feet (Nilson 1987, Naaman 2004). The

strands are then anchored with chucks to bulkheads (See Figure 1-1) at both ends of the

casting bed. Beams are then cast along the length of the casting bed with a single set of

prestressing strands running through all of the beams. When the concrete hardens the

prestressing strands become bonded to the concrete. The portion of the strands between

the beams that do not have concrete bonded to them is referred to as free strands (Kannel,

French & Stolarski 1998). When the compressive strength of the sample concrete

cylinders reaches the project specified release strength, the free strands are then cut one at

a time and the force within each prestressing strand transfers to the concrete beams,

placing the beams in a state of compression. This compression force causes the concrete

beams to axially shorten and camber. Unlike post-tensioned strands, prestressed strands

require a certain distance to fully transfer their force through bond to the concrete. The

distance required is known as the transfer length of the prestressing strand. ACI 318-02

defines the transfer length (It) as equal to one third the effective stress in the steel strand

(fse) multiplied by the diameter of the strand (db). The transfer length is an important

concept because the transferred force varies from zero at the end of the beam, to the full

prestress force at the transfer length (Nilson 1987). Because a transfer length is required

to transfer the prestress force to the beam, the ends of prestressed beams are vulnerable to

cracking if tension strains develop in the end region concrete. With this idea in mind, the

way to prevent the vertical end crack is to reduce the tension strains the concrete feels in

the transfer length region of the beam. Tension pull and friction are two sources of

tension strain at the end of a prestressed beam that can be controlled and reduced by both

the prestressed beam designer and manufacturer.

When some of the free strands are cut, the prestressed beams on the casting bed

axially shorten and camber (Naaman 2004). The axial shortening provides the largest

movement while the camber produces a small additional amount of end movement due to

the rotation of the end face of the beam. As the beams shorten and rotate the uncut free

strands are forced to stretch to accommodate this movement (Mirza & Tawfik 1978).

This stretch creates a tension force in the uncut free strands in addition to the prestress

force (Mirza & Tawfik 1978). This additional force is referred to as "tension pull" (See

Figure 1-2). Temperature change in the free strands between the time of beam casting

and the time of strand detensioning changes the tension pull magnitude. If the

temperature of the free strands decreases, the tension pull is increased. If the temperature

of the free strands increases, the tension pull is decreased. The thermal coefficient of the

prestressing strands is 6.67x10-6 in/in/F (Barr, Stanton & Eberhard 2005). The tension

pull transfers into the concrete beam over the transfer length required for the given

tension pull magnitude. The transfer length required for a given tension pull magnitude

is referred to as the reverse transfer length (Kannel, French & Stolarski 1998).

Friction between the bottom of the concrete beam ends and the steel casting bed is

another force that can create tension strain at the ends of a prestressed beam. The role of

friction is to reduce movements of the beam, either in the form of reducing the axial

shortening, or by reducing the amount the beam shifts on the casting bed. Static friction

force (Fs) is generally modeled as the coefficient of static friction (s) multiplied by the

normal force (N). Dynamic friction force (Fd) is generally modeled as the coefficient of

dynamic friction (.d) multiplied by the normal force (N). The static friction force must

be overcome before any movement can occur. The dynamic friction force is the friction

force a body feels while it is in motion. If at any time the force causing motion of the

body becomes less than the dynamic friction force, motion ceases. For motion to occur

again, the static friction force must once again be overcome. In the case of a prestressed

beam, the friction acts at both ends of the beam as the beam cambers (See Figure 1-3).

The normal force (N) is equal to one half of the beam's weight (W). Given a constant

coefficient of friction, the greater the beam length, the larger the friction force at the two

beam ends becomes.

A B
Figure 1-1. Strand Anchorage System A) Typical Bulkhead B) Chucks

1 A
!

Reverse Transfer Length

Cut Strands

Tension Pull
Force in
Free Strand

Prestressing Force

Distance along beam

Figure 1-2. Terminology. This figure shows how tension pull is created and how it is
transferred to the concrete beam over the Reverse Transfer Length

For F less than max static friction force
No movement

Weight = W
F F
4--- --

N = W/2

N = W/2

For F greater than max static friction force
Movement occurs

Weight W
F F

Fs = Ps*N

Fs = Ps*N

N = W/2
Figure 1-3. General Effects of Friction

Concrete elastic
shortening and
rotation

N = W/2

Force

h

CHAPTER 2
END CRACKING LITERATURE REVIEW

Introduction

The following summarizes two studies previously conducted on end cracking in

prestressed beams. The first study conducted by Mirza and Tawfik focused on vertical

end cracking on 73' AASHTO Type III beams (Mirza & Tawfik 1978). The second study

conducted by Kannel, French, and Stolarski investigated vertical, angular, and horizontal

end cracking on 45", 54", and 72" I-beams with draped strands and steel bearing plates

(Kannel, French & Stolarski 1998).

Review of Mirza and Tawfik 1978

In order to determine how to prevent vertical cracking in the AASHTO Type III

beams, Mirza and Tawfik first experimented on 45' to 50' long rectangular beams. The

goal was to determine the root cause of the vertical cracking. It was theorized that the

vertical end cracking was caused by the restraining force in the uncut strands as the beam

was being detensioned. As strands are cut, the beam shortens and the uncut strands,

because they are still attached to the beam and the bulkhead, are forced to stretch. This

stretch creates a resisting force that is transferred to the concrete beams. The magnitude

of the resisting force is dependent upon the length of the strands between the beams. The

beams were cast in three sets of two, with each set having a different length of strand

between the beam ends and the bulkheads. By attaching strain gages to the steel strands

and by using dial gages on the beam ends, the total resisting force in the uncut strands

was determined. The experiment showed that although the resisting force per strand

increases throughout the cutting process, the total resisting force reaches a maximum at a

point when approximately half the strands have been cut. This is the point when the

cracks were observed to form. It was also observed that the crack widths decreased

during the cutting of the second half of the strands. Because the cracks were within a few

inches of the beam ends it was concluded that the resisting force must be transferred to

the concrete over a short distance, and that this distance was less than the compression

transfer length of the cut strands.

To enable the researcher to determine the most important variables that cause beam

end cracking, a computer analytical model was created by idealizing the beams and the

uncut strands as bilinear springs. Using a stiffness analysis, the resisting force in the

uncut strands was determined after each strand was cut. These analytical values were

compared to experimental values and found to be on average 10 to 20 percent larger in

the middle range of the cutting order. The analytical model determined that simultaneous

release of the strands resulted in the lowest tensile stresses in the concrete beams. It also

determined that if non-simultaneous release did occur it was best to cut the longer strand

(between the bulkhead and the beam) before cutting the shorter strand (between the two

beams).

In order to combat the resisting force in the uncut strands, the AASHTO Type III

beams were modified in three ways; fifteen inch long steel bearing plates (See Figure 2-

1) were installed on the bottom of the beam ends, two three foot long Grade 40 rebars

were added in the bottom flange of the beam ends, and additional prestressing strands

were debonded for each beam. After making these modifications, vertical cracks were no

longer observed in the AASHTO Type III prestressed beams. To prevent vertical

cracking in general, Mirza and Tawfik recommended making the length of the

prestressing strands between the bulkhead and the prestressed beams at least 5% of the

bed length. If this length could not be provided, they recommended debonding additional

prestressing strands for a debonded length equal to or greater than the compression

transfer length. Debonding reduces the resisting force in the uncut strands by reducing

the average stiffness of the uncut strands. A debonded strand also helps by moving a

portion of the resisting force to an interior region of the beam where the prestress force

has been fully developed and the concrete can handle the resisting force without cracking.

Review of Kannel, French, Stolarski 1998

The study conducted by Kannel, French, and Stolarski investigated vertical,

angular, and horizontal end cracking on 45", 54", and 72" I-beams with draped strands

and steel bearing plates. An ABAQUS Finite Element model of a half beam was created

to model the stresses in the concrete at the end region of the beam during the

detensioning process. Multiple strand cutting patterns were chosen for analysis to

determine the relationship between end cracking and strand cutting pattern. The

favorable strand cutting patterns were then tested on full scale 45", 54", and 72"

prestressed I-beams.

The vertical crack in this study formed in a different way than the vertical crack in

the Mirza and Tawfik (1978) study. This vertical crack formed due to tension strains

created from the release of the draped strands. The researchers determined through

analytical and field testing that if two straight strands were cut before every six draped

strands were cut, the vertical crack would not form. The angular crack formed due to

shear stresses created from the compression forces from the cut strands and the tension

forces from the uncut strands. The researchers determined through analytical and field

testing that changing the strand cutting pattern to better balance the tensile and

compressive forces on the bottom flange cross section would reduce the occurrence of the

angular crack. The horizontal crack at the web-flange interface formed due to stress

concentrations at that location. This type of crack was shown to occur independently of

the strand cutting pattern. The researchers proposed that increasing the slope of the

flange over the first 18" would reduce the occurrence of this horizontal crack.

Kannel, French, and Stolarski (1998) determined that end cracks in general form

due to two things; the restraining force from the uncut strands, and the shear stresses

created from the strand cutting pattern. To reduce the occurrence of end cracking in

prestressed beams Kannel, French, and Stolarski suggested four things; change the strand

cutting pattern, debond additional prestressing strands, lower the coefficient of friction

between the beam and the casting bed, and provide at least 10 to 15% of the total bed

length in free strand length. Adding extra free strand length reduces the tensile forces in

the uncut strands. Lowering the coefficient of friction between the beam and the casting

bed helps balance the tensile forces at the two ends of the beam by allowing the beam

more freedom to shift on the casting bed. For beams with steel bearing plates, it was

recommended that the debonded length should be at least six inches greater than the

length of the steel plate.

Summary of End Cracking Reduction Recommendations

To reduce the occurrence of vertical cracking, Mirza and Tawfik recommended

making the length of the prestressing strands between the bulkhead and the prestressed

beams at least 5% of the bed length (Mirza & Tawfik 1978). If this length could not be

provided, they recommended debonding additional prestressing strands for a debonded

length equal to or greater than the compression transfer length (Mirza & Tawfik 1978).

10

Kannel, French, and Stolarski suggested four things; change the strand cutting pattern,

debond additional prestressing strands, lower the coefficient of friction between the beam

and the casting bed, and provide at least 10 to 15% of the total bed length in free strand

length (Kannel, French & Stolarski 1998).

Figure 2-1. Steel Bearing Plate

CHAPTER 3
MANUFACTURER SURVEY AND FIELD INSPECTIONS

Manufacturer Survey

Surveys were sent in November 2004 to three FDOT prestressed beam

manufacturers in Florida to determine the extent and types of end cracking each

manufacturer was experiencing during the production process of their AASHTO, Florida

Bulb-T, and Florida U-beams. A sample returned survey can be seen in Appendix A.

The returned surveys showed an interesting result; vertical cracks were only one of

several commonly occurring end cracks. It was also learned that multiple crack types

could occur on a single beam end.

Field Inspection Introduction

Following the surveys, site visits allowed the researchers to observe the beam

production process. Three Florida prestressed concrete manufacturers were visited from

January 2005 to February 2006. AASHTO, Florida Bulb-T, and Florida U-beams in the

manufacturer's storage areas and on the casting beds were visually inspected for end

cracking. The prestressed beams were of various lengths and consisted of various

numbers and types of prestressing strands. The detensioning process of AASHTO Types

3 and 4, and 72" Florida Bulb-T prestressed beams was also observed. The site visits

gave the researchers the opportunity to converse with plant engineers and technicians

about the different types of end cracks to obtain their opinions as to when, where, and

how each type of crack formed.

Researchers observed that during the detensioning process, most of the beam

movement occurred near the end of the strand cutting process. Not only did beams

shorten and camber as strands were cut, but beams also slid as units on the casting bed.

The beams next to the bulkheads were most likely to slide, and this sliding appeared to be

most likely the result of non-simultaneous strand cutting. For example, the strand on the

left side of the beam was cut before the strand on the right side of the beam, and the beam

slid to the right. The researcher observed an AASHTO Type III beam set into harmonic

motion after non-simultaneous strand cutting. After one cycle of motion the movement

abated. This type of motion raised questions regarding the amount tension strains in the

concrete beam ends were magnified due to dynamic effects on the casting bed.

During the detensioning process, a flagman signaled when each prestressing strand

should be cut. The workmen, standing in between every beam on the casting bed applied

their torches to the specified strand (See Figure 3-1). The researcher observed that

sometimes the strands "popped" right when the torch was applied, and at other times

cutting took ten seconds or more. Occasionally, as a torch was being removed, an

additional strand was accidentally cut. As the seven wire strands were cut, distinctive

popping sounds were heard, as each of the seven wires, in each prestressing strand, broke.

The researcher determined that simultaneous cutting was problematic and an unrealistic

assumption in design.

Manufacturers indicated that end cracking on prestressed beams was a common

occurrence. They relayed that end cracking tended to occur more frequently on larger,

longer span beams. The cracking sometimes appeared to occur randomly. For example,

the third beam on a casting bed of five cracked, yet none of the other four beams would

have any cracks. End cracking was not a completely random process despite the random

nature of material properties. Each beam end on the casting bed experienced slightly

different forces during the detensioning process due to non-simultaneous cutting,

accidental additional strand cuts, and the beam spacing arrangement on the casting bed.

To reduce the occurrence of end cracking, the root source of each type of crack must be

determined. The first step in determining the cause of each type of cracking was to

distinguish the different types of end cracks. The following information presents the

different types of end cracks found during the site visits of the three Florida prestressed

concrete manufacturers.

Field Inspection Results

Eight types of end cracks were discovered during the multiple site visits to three

Florida prestressed concrete manufacturers. The first crack type shall be referred to as a

"prestressed strand crack" (See Figure 3-2). This crack originated at a prestressing strand

and propagated toward the outer surfaces of the beam. The prestressed strand crack often

ran through multiple prestressing strands before reaching the exterior surface of the

concrete beam. The researcher proposed that this crack type was caused by two things;

Poisson's Effect and rusting of the prestressing strands.

When a load is applied to a prestressing strand, the prestressing strand elongates by

an amount 6 and the radius shrinks by an amount 6'. The ratio of the strain created by 6

to the strain created by 6' is a constant known as Poisson's ratio (Hibbler 2000). In the

transfer length region of a prestressed beam, the force within an individual cut

prestressing strand varies from zero at the end of the beam to the full prestress value at

the compression transfer length. Due to Poisson's effect, the prestressing strand wants to

expand as the force in the strand reduces to zero at the end of the beam. This expansion

effect creates a bursting force on the concrete (See Figure 3-3). This led the researcher to

propose that rust further increases the bursting force at the very end of the beam because

metal bars expand as they rust.

A second cracking type shall be referred to as "radial cracking". Radial cracking is

a fan shaped multiple crack pattern that extends the entire height of the beam (See Figure

3-4). This cracking pattern was observed on a 72" Florida Bulb-T, and a 78" Florida

Bulb-T prestressed beam. The cracks originating in the bottom flange were angled

upward, the cracks in the web were approximately horizontal, and the cracks in the top

flange were angled downward. Three or four vertical top flange cracks spaced at about

five feet along the top flange finished off the pattern. Excluding the vertical top flange

cracks, when the cracks were extended with a chalk line, the chalk lines intersected at a

point in the web region of the beam. This led the researcher to propose that radial

cracking was caused by the lifting hook arrangement/design (See Figure 3-5a) or the

lifting procedure (See Figure 3-5b).

The third type of crack was the angular crack. This crack originated in the sloped

part of the bottom flange, a few inches from the end of the beam and propagated upward

at an angle towards the web. Kannel, French, and Stolarski (1998) found angular cracks

form due to shear stresses created from the compression forces from the cut strands and

the tension forces from the uncut strands. Kannel, French, and Stolarski determined

through analytical and empirical research that changing the strand cutting pattern, to

better balance the tensile and compressive forces on the bottom flange cross section,

reduced the occurrence of the angular crack.

The fourth and fifth types of cracks shall be referred to as the "web-flange junction

crack" and "edge spelling". The web-flange junction crack crossed the end face of the

beam in the web and then proceeded downward, but did not extend past the sloped

portion of the bottom flange (See Figure 3-7). A manufacturer suggested that tension

strains created between the prestressed bottom flange and the non-prestressed web region

was the cause of the crack. This led the researcher to propose that this crack could be

prevented by adding additional horizontal mild steel in the web-flange region. Edge

spalls were a common occurrence, especially on beams with skewed ends.

A "horizontal top flange crack" (See Figure 3-9a) and a "horizontal web crack"

(See Figure 3-9b) were the next two cracks identified. The horizontal top flange crack

began at the end face of the upper flange and moved inward horizontally. Manufacturer

field personnel suggested that this crack was caused by formwork pressing against the

concrete when the beam cambered during detensioning. The manufacturer advised that

the horizontal top flange crack could be prevented by providing space between the

formwork and the concrete before detensioning began. The horizontal web crack looked

similar to the horizontal top flange crack except that the horizontal web crack occurred in

the web portion of the beam.

An eighth crack type identified was the vertical crack (See Figure 3-10). Mirza and

Tawfik's (1978) research determined that the vertical crack could be caused by the

resisting forces in the uncut strands during the detension process. The vertical crack

observed was located on a beam that did not contain any draped strands, so Kannel,

French, and Stolarski's 1998 vertical crack explanation did not apply (Kannel, French &

Stolarski 1998). The vertical crack in Figure 3-10 was the object of study, for this

research project. Manufacturer field personnel believed that reducing the coefficient of

friction between the casting bed and the bottom of the prestressed beam helped reduce the

occurrence of the vertical crack.

Field Inspection Summary

Eight types of end cracks were identified during the field survey of three Florida

prestressed concrete manufacturers; the prestressed strand crack, radial cracks, the

angular crack, the web-flange junction crack, the edge spall, the horizontal top flange

crack, the horizontal web crack, and the vertical crack. On many occasions more than

one type of crack was found on the same beam end. The focus of this research project

was the vertical end crack (See Figure 3-10) and the following chapters will focus

exclusively on the vertical end crack.

Figure 3-1. Strand Cutting Process

A B
Figure 3-2. Prestressed Strand Crack. A) This figure shows how the prestressed strand
crack propagates toward the outer surface of the beam. B) Photo of
prestressed strand crack

Bursting Forces

-T Concrete

1 Cut Prestressing Strand

CTL
-CTL CTL = Compression Transfer Length

Figure 3-3. Bursting Forces Caused by Prestressing Strands. This figure shows how
expansion due to Poisson's effect and rust creates a bursting force in the
concrete beam end.

Top flange vertical crack

ULrrJ.

A A
Figure 3-5. Lifting Devices A) Typical Lifting Hook B) Lifting Machine

Figure 3-6. Angular Crack. The angular crack shown here has been highlighted with
chalk to increase its visibility.

Junction Crack

figure 3-8. Edge spall

A B
Figure 3-9. Horizontal Cracks A) Horizontal Top Flange Crack B) Horizontal Web
Crack.

* *

* -S
p
w
4

Figure 3-10. Vertical Crack

CHAPTER 4
VERTICAL CRACK ANALYTICAL MODEL

Introduction

The occurrence of vertical cracking can be affected by many variables; length of

the free strands, modulus of elasticity of the concrete, friction coefficient between the

beam and the casting bed, temperature change, debonding lengths, number of debonded

strands, number of prestressing strands, jacking force per strand, tension strength of the

concrete, cross-sectional area of the beam, beam length, and beam spacing configuration.

Because there were so many different variables that influenced the formation of vertical

cracks, it was necessary to determine which variables had the greatest effect on vertical

crack formation. This allowed the researcher to determine the best possible solution to

vertical end cracking. To accomplish this, a MathCad 12 analytical model was created

(See Appendix B). Imputing specifications of beam number, beam length, the number of

bottom strands, the type of strand, the jacking force, the free strand length, the

temperature change at casting verses cutting time, and the coefficient of static and

dynamic friction between the bottom of the beam and the casting bed, the cracking

tendency for the specified conditions could be calculated.

The analytical model is based on four major assumptions listed below.

* The strands between all the beams on the casting bed are cut exactly at the same
time for every strand in the cutting order.

* The strand cutting pattern (See Appendix D) evenly balances the transferred
compression forces from the individual cut strands to the bottom flange of the
beam.

* The strands are all heated and cut slowly.

* The prestressing strands and the concrete beams are idealized as linear elastic
springs.

Analytical Model Theory

During detensioning the friction force, between the beam and the bottom form, is

distributed over a certain area of the bottom of the beam ends. As free strands are cut and

the beam camber increases, this distributed area shrinks until the friction force acts nearly

as a line load across the very ends of the beam. When the tension pulls at the two ends of

a beam are unequal, the acting direction of the static friction force may change and the

beam may slide as a unit on the casting bed. This phenomenon is referred to as "global

movement". When only a single beam is detensioned on a casting bed, the tension pull

on the two ends of the beam is always equal and global movement can not occur. The

following sections explain how global movement can occur for a three beam

symmetrically placed system (See Figure 4-1) with and without friction.

Global Motion Without Friction

When friction is absent from the system shown in Figure 4-1, equilibrium requires

that all three beams shorten exactly the same amount, and that tension pull in all the free

strands is equal. The system resists the imposed force "F" in two ways; the concrete

resists shortening by equation 4-1, and the steel resists stretching by equation 4-2.

Compatibility requires that the total amount of concrete shortening in the system is equal

to the total amount of steel stretching in the system. This value is equal to equation 4-3.

The tension pull created in the free strands is equal to equation 4-4. Equation 4-4

assumes that the Atot is very small compared to (L-3Lb).

k= AEc (4-1)
3L,

k,- Ass (4-2)
s L-3L,

Atot F (4-3)
kc + k,

TP= kAtot (4-4)

Beam 2 will not slide and the end movements at both ends of Beam 2 will be equal

due to symmetry of the system. Combining this fact with the fact that all the beams

shorten the same amount, equation 4-5 is derived. Because Lsi is much greater than Ls2,

equation 4-6 is derived. In order for the system to regain equilibrium, Beams 1 and 3 are

forced to react according to equation 4-6.

ABeam2End1 ABeam2End2 -- (4-5)
6

ABeamlEnd2 A Beam3End1 0

ABeamlEnd1 A Beam3End2 --At (4-6)

Because the final tension pull in the system must be equal in all the free strands to

satisfy equilibrium, the final strain in all the free strands must also be equal. In order to

satisfy these requirements, Beams 1 and 3 are forced to experience global movement.

Beams 1 and 3 will slide exactly the same distance towards Beam 2 due to symmetry of

the system. The distance of the slide is determined from the strain compatibility equation

4-7. The solution of equation 4-7 is shown in equation 4-8.

Atot slide tot A ide
6 3 (4-7)
L L
Ls2 Lsl

A tot -- Ls2
Aside 6 3 (4-8)
Ls2 + Lsl

Global Motion With Friction

The following explanation assumes that the force "F" is greater than the static

friction force "Fs". When friction is present, equilibrium requires that Beams 1 and 3

shorten exactly the same amount. Because of friction, Beam 2 will shorten a different

amount than Beams 1 and 3. The system resists the imposed force "F" in two ways; the

concrete resists shortening by the equation 4-9, and the steel resists stretching by equation

4-10. The total steel resistance to the shortening of each beam is shown in equation 4-11.

k, = (4-9)
L

A E A E
ks1 = ks4 Es ks2 = ks3 (4-10)
Lsl Ls2

1 AsEs
ksBeam2 1 1 2L
+
AsE AsE
Ls2 Ls2

1 AE
ksBeaml =ksBeam3 1 1 L + L
++L
AsEs AsEs
Lsi Ls2

Beam 2 will not slide and the end movements at both ends of Beam 2 (shown in

equation 4-12) will be equal due to symmetry of the system. The tension pull in Free

Strands 2 and 3 is shown in equation 4-13. A positive friction force is assumed to act

away from the center of each beam. The final acting static friction force for Beam 2

(shown in equation 4-14) is determined from symmetry of the system and the

requirement of the system to regain equilibrium.

ABeam2Endl ABeam2End2 A5(F E AsE(412)

Lc 2Ls2

TPFreeStrands2 TPpFreeStrands3 ABeam2Endl ks2 (4-13)

FsBeam2Endl Fs FsBeam2End2 Fs (4-14)

Because Lsi is much larger than Ls2, and equilibrium must be satisfied, all beam

shortening for Beam occurs at End 1, and all beam shortening for Beam 3 occurs at End

2. The amount each beam end shortens is shown in equation 4-15. The tension pull

created in Free Strands 1 and 4 is shown in equation 4-16. The final acting static friction

force for Beams 1 and 3 (shown in equation 4-17) is determined from symmetry of the

system and the requirement of the system to regain equilibrium. A positive friction force

is assumed to act away from the center of each beam. Equation 4-17 assumes that beam

end movement due to an imbalance of tension pulls at the two ends of the beam is

negligible.

ABeamlEnd2 A Beam3Endl 0

F-F
BeamlEndl ABeam3End2 A d (4-1A5)

Lc Lsl +Ls2

TPFreeStrandsl TPFreeStrands4 ABeamlEndlksl (4-16)

FsBeamlEndl s FsBeam3End2 Fs

FsBeamlEnd2 FsBeam3Endl = Fs (TPFreeStrands2 TPFreeStrandsl ) (4-17)

Analytical Commentary

The difference in magnitudes of the tension pulls on the two ends of a beam is

referred to as the AUTP "unbalanced tension pull". The amount the acting static friction

force reduces on the end of the beam with the larger tension pull is approximately equal

to the unbalanced tension pull (See Figure 4-2). For Figure 4-2 and 4-3, a positive AUTP

occurs when Tension Pull2 is greater than Tension Pull1. For Figure 4-2 and 4-3, a

positive static friction force acts towards the direction of axial shortening.

If the magnitude |AUTPI is greater than two times the static friction force (2*Fs)

global movement will occur. In the case of global movement, ignoring dynamic loading

effects, the beam will slide on the casting bed until the magnitude |AUTPI becomes less

than two times the dynamic friction force (2*Fd) (See Figure 4-3).

Difficulty arises when attempting to model the concrete strains in the end region of

a prestressed beam because this is a disturbed region (MacGregor 1997). In addition,

sudden strand release known as "popping" and non-simultaneous cutting of the free

strands can result in unpredictable dynamic effects. ACI 318-02 defines a disturbed

region as "The portion of a member within a distance equal to the member height h or

depth d from a force discontinuity or geometric discontinuity". Within a disturbed

region, classical beam theory can no longer be applied because plane sections do not

remain plane. A strut and tie model is one method of designing D-zones in concrete, but

a strut and tie model can not determine the actual stresses in the concrete at a given

location (Portland Cement Association 2002). Another choice is to use a Finite Element

analysis. With a Finite Element analysis the analyzed object may have any size or shape,

any type of boundary conditions, and any type of materials (Cook, Malkus, Plesha & Witt

2002). However, when using Finite Elements the following questions arise regarding the

method for modeling the following items.

* How should the transfer length and the reverse transfer length be modeled? The
magnitude of the reverse transfer length constantly changes as free strands are cut.

* The friction force magnitude, acting location, and acting direction are constantly
changing. Before movement, the static friction force acts on the concrete. During
movement, the dynamic friction force acts on the concrete.

* The compression load transferred to the concrete from each cut strand does not
transfer instantaneously, but rather slowly as the strand is heated with a torch and
the strand yields. This gives the beams the ability to react to the forces being
developed in the neighboring beams. The ability to react to the movements of other
beams alters the amount of movement each beam end experiences during the
detensioning process.

* How should the effects of the mild steel be accounted for?

* A single beam end or even a single beam can not be analyzed individually during
the detensioning process because the movements of each beam end are dependent
upon the movements of the opposite beam end and the movements of the
neighboring beams on the casting bed.

Three things must be maintained in any structural analysis; equilibrium,

compatibility, and constitutive relationships. Equilibrium requires that Newton's 2nd law

ZF = ma be maintained at every point within the structural system. Compatibility

equations describe displacement constraints that occur at supports of a member. For

example, if the end of the concrete beam moves 0.3 inches, the free strands that are

connected to the concrete must also move 0.3 inches. Constitutive relationships refer to

the material properties, such as the stiffness, of the object of analysis. The cracking

criterion that is used in the analytical model in Appendix B is equation 4-1. A simplified

hand calculation procedure is shown in Appendix C. The variable "fcali" is the calculated

stress (psi) in the concrete at a chosen location. The variable "f' is the allowable tension

stress (psi) in the concrete bottom flange and is calculated using equation 4-2. In

equation 4-2, "foi" is the compressive strength of the concrete at the time of cutting.

fca > 1 Vertical Crack Forms (4-18)
f

f =5/f (4-19)
psi

The calculated stress in the concrete bottom flange (fcaic) is based on four factors;

the transferred prestress force, the static friction force, the bearing force, and the tension

pull. The stress in the concrete is calculated at the bottom of the beam at a distance from

the end face of the beam equal to the reverse transfer length (See Figure 4-4). Equation

4-3 is used to calculate fcaic (See Figure 4-4).

CRTL F *e2 Fs N*RTL*e TP
calc=( A I )*1 (4-20)
Abf bf Abf bf Abf

Analytical Model Assumptions

The assumptions made in the analytical model of Appendix B are listed below.

* The modulus of elasticity of the concrete is calculated using equation 4-4 (Nawny
1996). The compressive strength of the concrete at the time of cutting is fci, and 6
is the unit weight of the concrete.

E = (40000 +10)( 16 )1 5psi (4-21)
\psi 145pcf

* The unit weight of the concrete is taken as 150 pcf (Prestressed Concrete Institute
1999)

* Temperature strain is superimposed on the free strands only, for temperature
changes between the time of beam casting and the time of detensioning. The
thermal coefficient of the prestressing strands is 6.67x10-6 in/in/F (Barr, Stanton &
Eberhard 2005).

* The tension pull created in each uncut free strand set due to beam movements is
based on the average lengths of all the free strands in each set. These lengths
include any debonding lengths.

* The effects of top flange prestressing strands are ignored.

* The transfer length of a prestressing strand is modeled as shown in equation 4-5
(Abrishami & Mitchell 1993). The reverse transfer length is calculated using
equation 4-6. The variable foi (ksi) is the compressive strength of the concrete at
the time of cutting. The variable D (in) is the diameter of the prestressing strand.
The variable fj (ksi) is the stress in the prestressing strand due to the jacking force.
The variable fTp is the stress in the prestressing strand due to the tension pull.

TransferLength=0.33fj *D (4-22)

ReverseTransferLength=0.33 fp *D (4-23)

* The prestress force is assumed to linearly transfer through bond to the concrete
over the compression transfer length (American Concrete Institute Committee 318
2002). The tension pull is assumed to linearly transfer through bond to the concrete
over the reverse transfer length. For the purposes of concrete elastic shortening the
prestress force from a cut strand is assumed to act at a distance from the end face of
the beam equal to 2/3rds of the compression transfer length of the strand. For
debonded strands, the prestress force is assumed to act at a distance from the end
face of the beam equal to the debonded length plus 2/3rds of the compression
transfer length of the strand.

* Each strand cut is divided into 20 calculation steps. These calculation steps allow
for beam movements to occur as the strand is weakened during the cutting process.

* Beam movements are considered small compared to the average lengths of the free
strands.

* Dynamic beam motions are ignored.

* Strand relaxation is ignored. Maximum relaxation for low-relaxation strand is
3.5% when the strand has been loaded to 80% of the tensile strength (Nilson 1987).

* The prestressing strands and the concrete beams are assumed to be linear elastic
throughout the entire detensioning process. The elastic modulus of grade 270 low
relaxation strand is taken as 28500ksi (Portland Cement Association 2002). This
assumption is acceptable because the vertical cracks form within the first half of the
cutting order, and if the prestressing strands do become inelastic, this occurs during
the second half of the cutting order.

* Any inputted debonded length needs to be greater than the transfer length of the
fully bonded prestressing strands. This is necessary because the model assumes if a

strand is debonded that the tension pull in that strand is transferred to a region of
the beam beyond the crack-prone end region.

* The reverse transfer length is considered the critical section for the analytical model
calculations. This is the point where all of the tension pull has been transferred
through bond to the concrete.

* Camber end movement after each strand cut, is added to the axial end movement.
Camber end movement after each strand cut is calculated using the equation; (Axial
movement due to strand cut)* (Total camber movement after all strand cuts)/(Total
axial Movement after all strand cuts) (See Figure 4-5).

Analytical Model Input Variables

The first input variable for the model consist of the type of beam; BT-72, BT-78,

AASHTO 2, AASHTO 3, AASHTO 4, AASHTO 5, AASHTO 6, FUB-48, FUB54,

FUB-63, FUB-72, and a custom setting where the user can input a custom beam area.

Two or more beams can be chosen for simultaneous analysis. The beams can also be

different lengths on the same casting bed. The number of bottom strands, type of strand,

and jacking force per strand must then be specified. The choices for type of strand

consist of .500" 270ksi, 9/16" 270ksi, and .600" 270ksi strands. The free strand length

between all the beams must be specified, with the free strand length for the end beams as

the length between the beam face and the abutment. Each debonded strand and its

associated debonded length must then be specified. Temperature change in the free

strands from the time of beam casting to the time of strand cutting can also be inputted.

Finally the coefficient of static and dynamic friction between the bottom of the beams

and the casting bed must be specified.

Analytical Model Flow Chart

The solution procedure used in the analytical model shown in Appendix B is

outlined in Figure 4-5. Equations are not provided because the cracking criterion fcalc/f

solution procedure can not be hand calculated due to the high level of iteration required

for multiple beam casting beds. For a simplified hand calculation procedure of fcalc/f for

single symmetrically prestressed beam, see Appendix C.

3 BEAM SYMMETRICALLY PLACED SYSTEM

GIVEN: SIMULTANEOUSLY ADD A COMPRESSION FORCE "F" TO ALL BEAM ENDS

L

Endi End2 Endi End2 Endi End2

Lsl >> Ls2 Lb Ls2 Lb Ls2 Lb Lsi>>Ls2

__ I Beami | Beam2 Beam3

Free Strands1 Free Strands2 Free Strands3 Free Strands4

PROPERTIES:
Concrete: Steel Friction
Ac Ec As Es Fs gsN Fd -dN

Figure 4-1. Global Movement For 3 Beam Symmetrical System

For AUTP > 0 kip
For AUTP < Fs

Tension Pull1

Transferred compression
From cut strands

Tension Pull2

Fs Fs AUTP N

For AUTP > 0 kip
For AUTP > Fs

Tension Pulli
Tension Pull

Tension Pull
4-

Tension Pulli
4-

Transferred compression
From cut strands

SFs |Fs- AUTPI

For AUTP < 0 kip
For |AUTP| < Fs

Transferred compression
From cut strands

Fs IAUTPI Fs --

For AUTP < 0 kip
For |AUTPI > Fs

Transferred compression
From cut strands

Tension Pull2
-+*

Tension Pull2

Tension Pull2

No*

IF, AUTPII F, 0

Figure 4-2. Change in Acting Static Friction Force

-NO.

0

q

10

4 -

0

0 -

For AUTP > 0 kip

For AUTP = 2*Fs
Impending global motion

Tension Pull1

Transferred compression
From cut strands

Fs

For AUTP > 2*Fs
global motion occurs Sliding

Transferred compression
From cut strands

Tension Pull2

Fd -

For AUTP < 0 kip

For |AUTP| = 2*Fs
Impending global motion

Tension Pull1

Transferred compression
From cut strands

Tension Pull2
^-------

Fs Fs

For |AUTP| > 2*Fs
global motion occurs

Tension Pulli

Transferred compression
From cut strands

-- Fd

Figure 4-3. Global Motion of Beam

No Sliding

Ter

vision Pull2

Tension Pt

4-

No Sliding

Ten

sion Pull2

Fd

F, 4

-No.

--No.

Beam Bottom Flange

N W/2

F,

Free Strands

N W/2

Friction:

Axial Stresses

Flexural Stresses

Normal Force:

Flexural Stresses

Transferred Prestress:

Force in cut strands
transferred over LI

CRTL

Axial Stresses

Tension Pull:

Force in uncut -
strands transferred
over Lrt

Axial Stresses

LEGEND

W = Beam weight N = Bearing force
e = Distance from centroid of bottom flange to bottom of beam Fs = Static friction force
L, = Reverse transfer length (Eq 4-6) Abf = Area of bottom flange
Ibf = Moment of inertia of bottom flange TP = Tension pull
CRTL = Tranferred prestress force at a distance from
the end face eaual to RTL

Figure 4-4. Stress in Concrete Bottom Flange. This figure shows the combination of
stresses a prestressed beam experiences during the detensioning process.

Free Strands

51F

e T

CL

Camber

Axial movement

I I1~

Camber movement

Figure 4-5. Axial and Camber Movement. This figure shows the total axial movement
and the total camber movement for a beam after all the strands have been cut.

Analytical Model's Flow Chart
The step by step calculations used by the MathCad 12 computer model to determine the
specifications that minimize the cracking tendency "CT" of prestressed beams, once data input is
complete

START: User inputs input variables
required for analysis

"TS" is calculated A strand is cut

Tension pull due to temperature for The movement at each beam
each strand cut is calculated end is calculated

Transfer length of the prestressing The length change of each free
strand is calculated strand set is calculated

Modulus of elasticity of the The tension pull "TP" for each
concrete is calculated free strand set is calculated

The average spring stiffness of each beam The unbalanced tension pull "AUTP" for
for each strand cut is calculated each beam is calculated

The static and dynamic friction forces "Fs" The acting static friction force "AFF" at
and "Fd" for each beam are calculated each beam end is calculated

The average spring stiffness for each free A small amount of global motion is applied
strand set for each strand cut is calculated to the beam(s) with a AUTP > 2*Fs

I_ Iteration occurs until AUTP <
The effective free strand spring stiffness for 2*Fd for all beams
each beam for each strand cut is calculated
The reverse transfer length "RTL" at each
beam end is calculated
The total beam axial shortening for each
beam for each strand cut is calculated
fcalc/f at each beam end is calculated

k Calculations loop for each
strand cut

END: The maximum fcalc/f for each
beam end is calculated

Figure 4-6. Analytical Model Flow Chart. By inputting the specifications for the number
of beams, beam length, number of bottom strands, the type of strand, the
jacking force, the free strand length between all the beams, the debonded
strand lengths, temperature change at casting and cutting times, and the
coefficient of static and dynamic friction between the bottom of beam and
casting bed, the cracking tendency for specified conditions can be calculated.

CHAPTER 5
RESULTS

Introduction

Tension strain in the end region of a prestressed beam can be affected by many

things as shown in Chapter 4. For this reason it was necessary to determine which

variables had the greatest effect on tension strains in the end region, so that the most

efficient solution to vertical cracking could be determined. This was accomplished by

performing a sensitivity analysis on the MathCad 12 analytical model shown in Appendix

B. Using test beam cases, one input variable was altered at a time and the resulting

change in fcali/f was noted. Test case 1 is shown in Figure 5-1. The input data for test

case 1 is shown in Table 5-1. The fcali/f results for test case 1 are shown in Figure 5-2.

Eleven alterations are made to the test case 1 input data shown in Table 5-1 in order to

determine which input changes result in the largest fcaic/f output changes. The alterations

are the number of prestressing strands, friction coefficient, concrete release strength,

beam length, temperature change, number of debonded strands, debonded lengths for ten

debonded strands, number of beams, free strand length for two beams, free strand length

for three beams, and free strand length for four beams. For all eleven alterations of test

case 1, only the fcacl/f output for beam 1 end 1 for the first twenty strands cuts is shown

(See Figure 5-1). Test case 2 is shown in Figure 5-25. The input data for test case 2 is

shown in Table 5-13. Two alterations are made to the test case 2 input data, the friction

coefficient, and the beam spacing. For both alterations, the maximum fcacl/f output for all

beam ends is shown (See Figure 5-25).

Test Case 1

Test case 1 is a 72" Florida Bulb-T configuration (See Figure 5-1). The input data

is shown in Table 5-1. The fcalc/f results for the input data shown in Table 5-1 is shown

in Figure 5-2. Eleven alterations are made to the input data shown in Table 5-1 in order

to determine which input changes result in the largest fcaic/f output changes. The

alterations are the number of prestressing strands, friction coefficient, concrete release

strength, beam length, temperature change, number of debonded strands, debonded

lengths for ten debonded strands, number of beams, free strand length for two beams, free

strand length for three beams, and free strand length for four beams. For all eleven

alterations, only the fcalc/f output for beam 1 end 1 for the first twenty strands cuts is

shown (See Figure 5-1).

Modification 1: Alter the Number of Prestressing Strands

The first modification is the total number of prestressing strands. The fcalc/f output

is shown for 30, 40, and 50 prestressing strands (See Table 5-2). The maximum fcal/f

value for each number of prestressing strands is shown in bold (See Table 5-2). Figure

5-3 shows the fcali/f results of Table 5-2 graphically. Figure 5-4 shows the maximum

fcalc/f results of Table 5-2 graphically.

Modification 2: Alter the Friction Coefficient

The second modification is the static and dynamic friction coefficients between the

casting bed and the bottoms of the prestressed beams. The fcalc/f output is shown for

static friction coefficients of 0.15, 0.25, 0.35, and 0.45 (See Table 5-3). The dynamic

friction coefficient is assumed to be 0.05 less than the static friction coefficient in all

cases. The maximum fcal/f value for each friction coefficient is shown in bold (See

Table 5-3). Figure 5-5 shows the fcal/f results of Table 5-3 graphically. Figure 5-6

shows the maximum fcali/f results of Table 5-3 graphically.

Modification 3: Alter the Concrete Release Strength

The third modification is the concrete release strength. The fcalc/f output is shown

for concrete release strengths of 6ksi, 7ksi, 8ksi, and 9ksi (See Table 5-4). The maximum

fcaic/f value for each concrete release strength case is shown in bold (See Table 5-4).

Figure 5-7 shows the fcalc/f results of Table 5-4 graphically. Figure 5-8 shows the

maximum fcali/f results of Table 5-4 graphically.

Modification 4: Alter the Beam Lengths

The fourth modification is the lengths of the prestressed beams (See Figure 5-1).

The fcal/f output is shown for beam lengths of 100ft, 120ft, 140ft, and 160ft (See Table 5-

5). The maximum fcal/f value for each beam length case is shown in bold (See Table 5-

5). Figure 5-9 shows the fcali/f results of Table 5-5 graphically. Figure 5-10 shows the

maximum fcali/f results of Table 5-5 graphically.

Modification 5: Alter the Temperature Change

The fifth modification is the temperature change in the free strands. A positive

temperature change indicates that the temperature at the time of detensioning is lower

than the temperature at the time of beam casting. When this occurs, the free strands

attempt to shorten, but are prevented by the beams and the bulkheads. A negative

temperature change indicates that the temperature at the time of detensioning is higher

than the temperature at the time of beam casting. When this occurs, the free strands relax

an amount dependent upon the magnitude of the temperature change. The fcali/f output is

shown for temperature changes of -40F, -20F, 0F, 20F, and 40F (See Table 5-6). The

maximum fcali/f value for each temperature change case is shown in bold (See Table 5-6).

Figure 5-11 shows the fcalc/f results of Table 5-6 graphically. Figure 5-12 shows the

maximum fcal/f results of Table 5-6 graphically.

Modification 6: Alter the Number of Debonded Strands

The sixth modification is the number of debonded strands. The fcalc/f output is

shown for 4, 6, 8, and 10 debonded strands (See Table 5-7). The maximum fcal/f value

for each number of debonded strands is shown in bold (See Table 5-7). Figure 5-13

shows the fcali/f results of Table 5-7 graphically. Figure 5-14 shows the maximum fcal/f

results of Table 5-7 graphically.

Modification 7: Alter the Debonded Lengths of 10 Strands

The seventh modification is the debonded length for the case of 10 debonded

strands. The fcalc/f output is shown for debonded lengths of 5ft, 10ft, 15ft, and 20ft (See

Table 5-8). The maximum fcal/f value for each debonded length is shown in bold (See

Table 5-8). Figure 5-15 shows the fcali/f results of Table 5-8 graphically. Figure 5-16

shows the maximum fcali/f results of Table 5-8 graphically.

Modification 8: Alter the Number of Beams

The eighth modification is the number of the prestressed beams on the casting bed.

The fcali/f output is shown for 2, 3, and 4 beams present on the casting bed (See Table 5-

9). The free strand lengths between the beams are equal to Ls2 for all cases (See Figure 5-

1). The free strand lengths between the beams and the bulkheads are equal to Lsi for all

cases (See Figure 5-1). The maximum fcal/f value for each number of beams case is

shown in bold (See Table 5-9). Figure 5-17 shows the fcal/f results of Table 5-9

graphically. Figure 5-18 shows the maximum fcali/f results of Table 5-9 graphically.

Modification 9: Alter the Free Strand Length for 2 Beams

The ninth modification is the free strand length between the beams and the

bulkheads for the case of two beams present on the casting bed. The fcalc/f output is

shown for free strand lengths of 25ft, 40ft, 55ft, and 70ft (See Table 5-10). The

maximum fcali/f value for each free strand length is shown in bold (See Table 5-10).

Figure 5-19 shows the fcalc/f results of Table 5-10 graphically. Figure 5-20 shows the

maximum fcali/f results of Table 5-10 graphically.

Modification 10: Alter the Free Strand Length for 3 Beams

The tenth modification is the free strand length between the beams and the

bulkheads for the case of three beams present on the casting bed. The free strand lengths

between the beams are equal to Ls2 for all cases (See Figure 5-1). The free strand lengths

between the beams and the bulkheads are equal to Lsi for all cases (See Figure 5-1). The

fcalc/f output is shown for free strand lengths of 25ft, 40ft, 55ft, and 70ft (See Table 5-11).

The maximum fcali/f value for each free strand length is shown in bold (See Table 5-11).

Figure 5-21 shows the fcalc/f results of Table 5-11 graphically. Figure 5-22 shows the

maximum fcali/f results of Table 5-11 graphically.

Modification 11: Alter the Free Strand Length for 4 Beams

The eleventh modification is the free strand length between the beams and the

bulkheads for the case of four beams present on the casting bed. The free strand lengths

between the beams are equal to Ls2 for all cases (See Figure 5-1). The free strand lengths

between the beams and the bulkheads are equal to Lsi for all cases (See Figure 5-1). The

fcalc/f output is shown for free strand lengths of 25ft, 40ft, 55ft, and 70ft (See Table 5-12).

The maximum fcali/f value for each free strand length is shown in bold (See Table 5-12).

Figure 5-23 shows the fcalc/f results of Table 5-12 graphically. Figure 5-24 shows the

maximum fcal/f results of Table 5-12 graphically.

Test Case 2

Test case 2 is a 78" Florida Bulb-T configuration (See Figure 5-25). The input data

is shown in Table 5-13. Two alterations are made to the input data shown in Table 5-13;

the friction coefficient and the free strand lengths. For both alterations, the fcalc/f output

for all beam ends is shown (See Figure 5-25).

Modification 1: Alter the Friction Coefficient

The first modification is the static and dynamic friction coefficients between the

casting bed and the bottoms of the prestressed beams (See Figure 5-25). The fcalc/f output

is shown for static friction coefficients of 0.01, 0.05, 0.15, 0.25, 0.35, and 0.45 (See Table

5-14). The dynamic friction coefficient is assumed to be 0.05 less than the static friction

coefficient in cases except the last two cases where the static friction coefficient is equal

to 0.05 and 0.01. For these cases, the dynamic friction coefficient is 0.001. The

maximum fcali/f value for each friction coefficient case for all beam ends is shown in

Table 5-14. Figure 5-26 shows the fcalc/f results of Table 5-14 graphically.

Modification 2: Alter the Beam Spacing

The second modification is the beam spacing (See Figure 5-25). The beam spacing

is shown by the lengths of the free strands in Table 5-15. Nine spacing modifications are

listed in Table 5-15 and the maximum fcali/f results for each case at each beam end are

listed in Table 5-16.

Analytical Model Conclusions

According to the analytical model in Appendix B, the following trends have been

determined from test case 1.

* Increasing the number of prestressing strands makes the beam more likely to crack.

* Increasing the coefficient of friction between the casting bed and the bottom of the
beam make the beam more likely to crack.

* Decreasing the concrete release strength makes the beam more likely to crack.

* Increasing the beam length makes the beam more likely to crack

* A temperature reduction in the free strands from the time of beam casting to the
time of strand detensioning makes the beam more likely to crack. A temperature
increase in the free strands from the time of beam casting to the time of strand
detensioning makes the beam less likely to crack.

* Decreasing the number of debonded strands makes the beam more likely to crack.

* Decreasing the debonded length of the debonded strands makes the beam more
likely to crack.

* Increasing the number of beams on the casting bed makes the beam more likely to
crack.

* Decreasing the free strand length between the bulkhead and the beam makes the
beam more likely to crack. This effect is increased as the number of beams on the
casting bed increases.

The variables that have the greatest effect on the tension strains the end region of a

prestressed beam experiences are temperature change, friction, concrete release strength,

beam length, and number of prestressing strands. The free strand lengths and the number

of beams on the casting bed have the next greatest effect. The free strand lengths become

more important as the number of beams on the casting bed increases. The number of

debonded strands and the lengths of the debonded strands have a small effect on the

tension strains in the end region of a prestressed beam.

According to the analytical model in Appendix B, the following trends have been

determined from test case 2.

* Given a symmetrical beam placement (See Figure 5-25) on the casting bed, the
middle beams are most likely to crack when friction is present. As the friction
coefficient approaches zero, all the beams become equally as likely to crack (See
Figure 5-26).

* The beam that is farthest away from the long free strands (See Table 5-15) is most
likely to crack (See Table 5-16).

Field Data Results

In February 2006 field data was collected at Gate Concrete in Jacksonville Florida.

Beam end movements were measured for the three 139 ft long 72" Florida Bulb-T beams

on the casting bed (See Figure 5-27). Measurements of movement were made at both

ends of beam 2, the right end of beam 1, and the left end of beam 3, during the strand

cutting process (See Figure 5-27). Measurements were taken visually with a millimeter

scale from a reference mark after the desired strands were cut. The field data was then

compared to the calculated values from the analytical model in Appendix B. The input

values for the analyzed beams are listed in Table 5-17. The movements for beam 2 are

listed in Table 5-18. The movements are shown graphically in Figures 5-28 and 5-29.

The movements for the right end of beam 1 are listed in Table 5-19. The movements are

shown graphically in Figure 5-30. The movements for the left end of beam 3 are listed in

Table 5-20. The movements are shown graphically in Figure 5-31. The cutting pattern

and the locations of the debonded strands can be seen in Appendix D.

Field Data Conclusions

The first half of the field data results for beam 2 are higher than calculated on the

left side of the beam and lower than calculated on the right side of the beam. There are

many possible explanations for this, but most likely the beam experienced global motion

to the right due to non-simultaneous cutting. The existence of global motion is supported

by data point #42 on the left end of beam 2. The only possible explanation for the end

movement of a beam to remain constant or reduce in value when additional prestress

force is added to the cross section is that the beam experienced global motion. The total

calculated beam shortening for beam 2 (.827") agrees with the measured total beam

shortening (.819"). Beam 1 and beam 3 data show that global motion is a very significant

issue. Data points #10 through #38 for the left end of beam 3 either remain constant or

reduce in value from their previous points. Data points #32 through #42 on the right end

of beam 1 either remain constant or reduce in value from their previous points. The

conclusion that can be drawn from the field data is that without being able to determine

which workman will cut their strand the fastest, it is not possible to calculate the actual

movements of the beam ends in the field.

Ls1 Le Ls2 Lc2 Ls3

SEnd 1 End 2 End 1 End 2

Beam 1 Beam 2

Figure 5-1. Test Case 1

Table 5-1. Test Case 1 Input Data
Variable Value Variable Value

Type of Beam

BT-72
140ft
140ft

Number of Strands 40
Jacking Force per Strand 44k
Concrete Release Strength 8ksi
Unit Weight of Concrete 150pcf
Temperature Change 0
Static Coefficient of Friction 0.45
Dynamic Coefficient of Friction 0.40

4 0.8

0.6

Strand Type
Debonded Strands

Camber

10
Number of Cut Strands

Figure 5-2. Test Case 1 No Alterations

40ft
3ft
40ft
.600
270ksi
#37 5ft
#38 5ft
#39 5ft
#40 5ft
2.5in

Table 5-2. Alter the Number of Prestressing Strands
Number of Cut
Strands #PS = 30 #PS = 40 #PS = 50

0.763
0.822
0.875
0.923
0.964
0.999
1.029
1.052
1.069
1.080
1.084
1.082
1.074
1.060
1.038
1.011
0.977
0.936
0.889
0.837

1.6

1.4

1.2

4-- 1

4 0.8

0.6

0.4

0.763
0.836
0.900
0.958
1.011
1.059
1.101
1.137
1.168
1.194
1.213
1.227
1.235
1.237
1.233
1.223
1.207
1.185
1.157
1.123

0.771
0.849
0.923
0.991
1.054
1.113
1.166
1.215
1.258
1.296
1.329
1.357
1.379
1.395
1.407
1.412
1.412
1.407
1.395
1.378

Number of Cut Strands

Figure 5-3. Alter the Number of Prestressing Strands

- #PS = 30
- -#PS=40
- -#PS = 50

30 35 40 45
Number of Prestressing Strands

Figure 5-4. Number of Prestressing Strands fcal/fMaximums

Table 5-3. Alter the Friction Coefficient
Number of Cut
Strands i =.15 p = .25 p =.35 p =.45
1 0.308 0.461 0.614 0.763
2 0.375 0.529 0.682 0.836
3 0.438 0.592 0.746 0.900
4 0.495 0.649 0.803 0.958
5 0.546 0.701 0.856 1.011
6 0.593 0.748 0.903 1.059
7 0.634 0.789 0.945 1.101
8 0.669 0.825 0.981 1.137
9 0.699 0.855 1.011 1.168
10 0.723 0.880 1.036 1.194
11 0.741 0.898 1.055 1.213
12 0.754 0.911 1.069 1.227
13 0.760 0.919 1.076 1.235
14 0.761 0.920 1.078 1.237
15 0.756 0.915 1.074 1.233
16 0.745 0.905 1.063 1.223
17 0.728 0.889 1.047 1.207
18 0.705 0.865 1.025 1.185
19 0.675 0.836 0.996 1.157
20 0.640 0.801 0.961 1.123

1.6

1.4

1.2 15

1 0l p-= .15
0.6

0.4

0.2-
0 5 10 15 20
Number of Cut Strands

Figure 5-5. Alter the Friction Coefficient

50

1.6

1.5

1.4

1.3

1.2
1.1
o 11

4- 0.9

0.8

0.7

0.6 -
0.15 0.2 0.25 0.3 0.35 0.4 0.45
Friction Coefficient

Figure 5-6. Friction Coefficient fcaic/f Maximums

Table 5-4. Alter the Concrete Release Strength

Number of Cut
Strands
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

fci = 6ksi fci = 7ksi
0.891 0.822
0.981 0.900
1.066 0.973
1.143 1.039
1.214 1.100
1.279 1.155
1.337 1.204
1.388 1.246
1.431 1.283
1.468 1.313
1.498 1.337
1.521 1.354
1.536 1.365
1.544 1.370
1.545 1.368
1.538 1.359
1.524 1.343
1.501 1.321
1.471 1.292
1.434 1.256

- fci = 6ksi
- fci = 7ksi
- fci = 8ksi
- fci = 9ksi

10 1
Number of Cut Strands

Figure 5-7. Alter the Concrete Release Strength

fci = 8ksi
0.763
0.836
0.900
0.958
1.011
1.059
1.101
1.137
1.168
1.194
1.213
1.227
1.235
1.237
1.233
1.223
1.207
1.185
1.157
1.123

fci = 9ksi
0.722
0.783
0.840
0.892
0.939
0.981
1.018
1.050
1.077
1.099
1.115
1.126
1.132
1.132
1.128
1.117
1.101
1.080
1.053
1.020

1.6

1.5

1.4

1 1.3

E 1.2

II-
1.1
0
4 0.9

0.8

0.7

0.6
6 6.5 7 7.5 8 8.5 9
Concrete Release Strength (ksi)

Figure 5-8. Concrete Release Strength fcaic/f Maximums

Table 5-5. Alter the Beam Lengths
Number of Cut
Strands L = 100ft L = 120 ft L = 140ft L = 160ft
1 0.558 0.663 0.763 0.870
2 0.610 0.723 0.836 0.949
3 0.656 0.778 0.900 1.022
4 0.698 0.827 0.958 1.090
5 0.736 0.872 1.011 1.152
6 0.769 0.912 1.059 1.208
7 0.797 0.947 1.101 1.258
8 0.821 0.976 1.137 1.302
9 0.840 1.001 1.168 1.341
10 0.854 1.020 1.194 1.373
11 0.863 1.034 1.213 1.399
12 0.868 1.043 1.227 1.419
13 0.867 1.046 1.235 1.432
14 0.862 1.044 1.237 1.439
15 0.852 1.037 1.233 1.439
16 0.837 1.024 1.223 1.433
17 0.817 1.006 1.207 1.421
18 0.792 0.982 1.185 1.401
19 0.761 0.952 1.157 1.375
20 0.726 0.917 1.123 1.342

- L= 100ft
- -L= 120ft
- -L= 140ft
- -L = 160ft

5 10 1
Number of Cut Strands

Figure 5-9. Alter the Beam Lengths

0.6
100

i 130
Beam Lengths (ft)

Figure 5-10. Beam Lengths fcaic/f Maximums

Table 5-6. Alter the Temperature Change

Number of Cut
Strands
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

AF = -40
0.179
0.271
0.357
0.439
0.515
0.585
0.650
0.710
0.764
0.812
0.854
0.891
0.922
0.947
0.966
0.979
0.986
0.987
0.982
0.971

AF =-20 AF=(
0.473 0.763
0.554 0.836
0.629 0.900
0.698 0.958
0.763 1.011
0.822 1.059
0.875 1.101
0.923 1.137
0.966 1.168
1.003 1.194
1.034 1.213
1.059 1.227
1.079 1.235
1.092 1.237
1.100 1.233
1.101 1.223
1.097 1.207
1.086 1.185
1.070 1.157
1.047 1.123

- AF = -40
- -AF=-20
- -AF=0
- AF = 20
................................ A F = 4 0

10 1
Number of Cut Strands

Figure 5-11. Alter the Temperature Change

) AF = 20
1.061
1.119
1.171
1.218
1.259
1.295
1.326
1.351
1.371
1.384
1.393
1.395
1.391
1.382
1.367
1.345
1.318
1.285
1.245
1.199

AF = 40
1.355
1.401
1.442
1.477
1.507
1.532
1.551
1.565
1.573
1.575
1.572
1.563
1.548
1.527
1.5
1.467
1.429
1.384
1.333
1.275

0 10 20 30 40

Temperature Change (deg F)

Figure 5-12. Temperature Change fcalc/f Maximums

Table 5-7. Alter the Number of Debonded Strands

Number of Cut
Strands
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

#DS = 4
0.763
0.836
0.900
0.958
1.011
1.059
1.101
1.137
1.168
1.194
1.213
1.227
1.235
1.237
1.233
1.223
1.207
1.185
1.157
1.123

#DS =
0.766
0.831
0.891
0.946
0.996
1.040
1.078
1.111
1.139
1.161
1.177
1.187
1.192
1.191
1.184
1.171
1.152
1.127
1.095
1.058

6 #DS=
0.764
0.826
0.883
0.935
0.981
1.021
1.057
1.086
1.110
1.129
1.142
1.149
1.15
1.146
1.136
1.120
1.098
1.070
1.036
0.996

8 #DS = 10
0.763
0.822
0.875
0.923
0.966
1.003
1.035
1.062
1.082
1.098
1.107
1.111
1.110
1.102
1.089
1.070
1.045
1.015
0.979
0.937

-40 -30 -20 -10

- #DS = 4
- -#DS=6
- #DS = 8
- #DS = 10

0 5 10 15
Number of Cut Strands

Figure 5-13. Alter the Number of Debonded Strands

4 5 6 7 8
Number of Debonded Strands

Figure 5-14. Number of Debonded Strands fcalc/f Maximums

Table 5-8. Alter the Debonded Lengths of 10 Strands
Number of Cut
Strands 5 ft 10ft 15 ft 20 ft
1 0.763 0.762 0.761 0.760
2 0.822 0.818 0.815 0.811
3 0.875 0.869 0.863 0.858
4 0.923 0.914 0.907 0.899
5 0.966 0.955 0.945 0.936
6 1.003 0.990 0.978 0.967
7 1.035 1.020 1.006 0.994
8 1.062 1.044 1.029 1.015
9 1.082 1.063 1.046 1.031
10 1.098 1.077 1.059 1.042
11 1.107 1.085 1.066 1.049
12 1.111 1.088 1.068 1.050
13 1.110 1.086 1.065 1.046
14 1.102 1.078 1.057 1.038
15 1.089 1.065 1.043 1.025
16 1.070 1.046 1.025 1.007
17 1.045 1.022 1.002 0.985
18 1.015 0.993 0.974 0.958
19 0.979 0.959 0.942 0.927
20 0.937 0.919 0.905 0.893

1.6

1.4

1.2-

4-. 1 5 ft
II- 1 ,, ^.

4 0.8 -

0.6 -

0.4 -

0.2
0

- l t
- 15ft
- 20ft

5 10 15 20
Number of Cut Strands

Figure 5-15. Alter the Debonded Lengths of 10 Debonded Strands

___100000000*40-

-

-

-

fl

1.6

1.5

1.4

w 1.3
E
E 1.2

E 1.1
4--

S0.9 -

0.8

0.7

0.6 -
5 7 9 11 13 15 17 19
Debonded Length for 10 Strands

Figure 5-16. Debonded Lengths fcaic/f Maximums

Table 5-9. Alter the Number of Beams
Number of Cut
Strands #B = 2 #B = 3 #B = 4
1 0.763 0.769 0.769
2 0.836 0.841 0.841
3 0.900 0.908 0.914
4 0.958 0.969 1.002
5 1.011 1.037 1.081
6 1.059 1.098 1.152
7 1.101 1.153 1.218
8 1.137 1.201 1.275
9 1.168 1.243 1.329
10 1.194 1.277 1.372
11 1.213 1.305 1.407
12 1.227 1.326 1.431
13 1.235 1.340 1.450
14 1.237 1.341 1.458
15 1.233 1.340 1.459
16 1.223 1.332 1.449
17 1.207 1.314 1.426
18 1.185 1.289 1.401
19 1.157 1.259 1.358
20 1.123 1.218 1.312

-#B = 2
- -#B=3
- -#B=4

0 5 10
Number of Cut Strands

Figure 5-17. Alter the Number of Beams

2 3
Number of Beams

Figure 5-18. Number of Beams fcalc/f Maximums

Table 5-10. Alter the Free Strand Length for 2 Beams
Number of Cut
Strands 40ft 50ft 60ft 70ft
1 0.763 0.762 0.759 0.756
2 0.836 0.819 0.808 0.799
3 0.900 0.872 0.853 0.838
4 0.958 0.921 0.894 0.874
5 1.011 0.965 0.932 0.907
6 1.059 1.004 0.965 0.937
7 1.101 1.039 0.995 0.962
8 1.137 1.069 1.021 0.985
9 1.168 1.095 1.042 1.004
10 1.194 1.115 1.060 1.019
11 1.213 1.131 1.074 1.031
12 1.227 1.143 1.083 1.039
13 1.235 1.149 1.088 1.043
14 1.237 1.151 1.090 1.044
15 1.233 1.147 1.087 1.042
16 1.223 1.139 1.079 1.035
17 1.207 1.126 1.068 1.025
18 1.185 1.107 1.052 1.012
19 1.157 1.084 1.032 0.994
20 1.123 1.055 1.008 0.973

1.6

1.4

1.2 ---------------

_o. I --- 50ft

cc 60ft
.- 70ft

0.6

0.4

0.2
0 5 10 15 20
Number of Cut Strands

Figure 5-19. Alter the Free Strand Length for 2 Beams

25 30 35 40 45 50 55
Free Strand Length (ft)

Figure 5-20. Free Strand Length for 2 Beams fcalc/fMaximums

60 65 70

Table 5-11. Alter the Free Strand Length for 3 Beams
Number of Cut
Strands 40ft 50ft 60ft 70ft
1 0.769 0.763 0.759 0.756
2 0.841 0.823 0.810 0.801
3 0.908 0.878 0.857 0.841
4 0.969 0.934 0.905 0.884
5 1.037 0.991 0.956 0.929
6 1.098 1.037 0.995 0.964
7 1.153 1.084 1.036 1.001
8 1.201 1.125 1.072 1.028
9 1.243 1.160 1.098 1.056
10 1.277 1.185 1.124 1.075
11 1.305 1.208 1.140 1.094
12 1.326 1.227 1.157 1.104
13 1.340 1.239 1.168 1.110
14 1.341 1.240 1.169 1.116
15 1.340 1.240 1.169 1.113
16 1.332 1.233 1.160 1.109
17 1.314 1.217 1.150 1.097
18 1.289 1.198 1.130 1.080
19 1.259 1.168 1.106 1.062
20 1.218 1.136 1.079 1.036

1.6

1.4

1.2

1- 40ft
1 I-- 5Oft
c- 60ft
S0.8
2- 70ft

0.6

0.4

0.2
0 5 10 15 20
Number of Cut Strands

Figure 5-21. Alter the Free Strand Length for 3 Beams

25 30 35 40 45 50 55
Free Strand Length (ft)

Figure 5-22. Free Strand Length for 3 Beams fcalc/fMaximums

60 65 70

Table 5-12. Alter the Free Strand Length for 4 Beams
Number of Cut
Strands 40ft 50ft 60ft 70ft
1 0.769 0.763 0.759 0.756
2 0.841 0.823 0.810 0.801
3 0.914 0.884 0.863 0.853
4 1.002 0.958 0.929 0.907
5 1.081 1.026 0.984 0.957
6 1.152 1.088 1.039 1.002
7 1.218 1.143 1.084 1.043
8 1.275 1.192 1.128 1.078
9 1.329 1.235 1.161 1.109
10 1.372 1.265 1.195 1.140
11 1.407 1.295 1.217 1.160
12 1.431 1.317 1.238 1.176
13 1.450 1.332 1.248 1.186
14 1.458 1.341 1.252 1.191
15 1.459 1.337 1.255 1.191
16 1.449 1.331 1.247 1.185
17 1.426 1.313 1.233 1.173
18 1.401 1.291 1.212 1.156
19 1.358 1.258 1.184 1.129
20 1.312 1.218 1.150 1.100

1.6

1.4

1.2

1"' --- 40ft
50ft
c 0- 60ft
S0.8
70ft

0.6

0.4

0.2
0 5 10 15 20
Number of Cut Strands

Figure 5-23. Alter the Free Strand Length for 4 Beams

0.8

0.7

0.6 -
25 30 35 40 45 50 55 60 65 70
Free Strand Length (ft)

Figure 5-24. Free Strand Length for 4 Beams fcal/fMaximums

Lsl Lel L,2 Lc2 Ls3 Le3 L,4 Lc4 L,5

LI

I. -.I

-I-I-

-I I

-I- ~J

El E2 E3 E4 E5 E6 E7 E8 -

Figure 5-25. Test Case 2

Table 5-13. Test Case 2 Input Data
Variable Value Variable Value

Type of Beam
Lcl = Lc2 = Lc3 = Lc4

BT-78 Lsi = Ls5
150ft Ls2 = Ls3 = Ls4

Number of Strands 49
Jacking Force per Strand 44k
Concrete Release Strength 8ksi
Unit Weight of Concrete 150pcf
Temperature Change 0
Static Coefficient of Friction 0.45
Dynamic Coefficient of Friction 0.40

Strand Type
Debonded Strands

Camber

60ft
3ft
.600
270ksi
#46 5ft
#47 5ft
#48 5ft
#49 5ft
3in

Table 5-14. Alter Friction Results
Friction
Coefficient El E2 E3
=i .45 1.256 0.702 1.643
=.35 1.097 0.462 1.391
S=.25 0.938 0.499 1.147
p=.15 0.780 0.606 0.904
= .05 0.621 0.526 0.653
S=.01 0.555 0.546 0.557

E4
1.626
1.372
1.128
0.892
0.633
0.558

E5
1.626
1.372
1.128
0.892
0.633
0.558

E6
1.643
1.391
1.147
0.904
0.653
0.557

E7
0.702
0.462
0.499
0.606
0.526
0.546

E8
1.256
1.097
0.938
0.780
0.621
0.555

400
4WOP
4V
4001V

OIV
lop*

,0 01

.0000000 100,

- El = E8
- E2 = E7
- E3 = E6
- E4= E5

Friction Coefficient

Figure 5-26. Alter the Friction Coefficient for Multiple Beam Ends

Table 5-15. Free Strand Lengths

Modification
#1
#2
#3
#4
#5
#6
#7
#8
#9

LsI (ft)
3
3
3
3
25.8
16.125
3
3
117

Ls2 (ft)
60
3
3
3
25.8
32.25
3
117
3

Ls3 (ft) Ls4 (ft)
3 3
60 3
3 60
60 60
25.8 25.8
32.25 32.25
117 3
3 3
3 3

Ls5 (ft)
60
60
60
3
25.8
16.125
3
3
3

Total
Length (ft)
129
129
129
129
129
129
129
129
129

Table 5-16. Beam Spacing Results

Modification El
1 1.418
2 1.875
3 2.072
4 1.717
5 1.388
6 1.392
7 1.601
8 1.246
9 1.144

E2
1.415
1.875
2.072
1.717
1.383
1.389
1.628
1.244
0.677

E3
1.420
0.749
1.303
0.738
1.389
1.393
0.727
1.247
1.530

E4
0.850
1.486
1.714
1.326
1.388
1.393
1.255
0.702
1.126

E5
1.619
1.481
0.702
1.325
1.389
1.393
1.255
1.638
1.916

E6
1.619
1.484
1.332
1.326
1.389
1.393
0.727
1.303
1.793

14 Lo IL

E7
0.726
0.771
0.886
1.322
1.383
1.389
1.628
2.011
2.333

E8
1.232
1.120
1.131
1.325
1.388
1.392
1.601
1.997
2.315

Lc3

Beam 1 Beam 2 Beam 3

LSI Ls2 Ls3 Ls4

Figure 5-27. 72" Florida Bulb-T Arrangement

Table 5-17. 72" Florida Bulb-T Input Data
Variable Value Variable Value
Type of Beam BT-72 Lsi 58' 5"
Lei 139' 23/8" Ls2 2' 10"
Lc2 139' 23/8" Ls3 2' 10"
Lc3 139' 23/8" Ls4 88' 3"

Number of Strands
Jacking Force per Strand
Concrete Release Strength
Unit Weight of Concrete
Temperature Change
Camber Bl = 3"
Camber B2 = 25/8"
Camber B3 = 31/4,

42
44k
7360psi
150pcf
NA

Strand Type
Debonded Strands

Estimated [Ld
Estimated s

.600
270ksi
4 x 5'
4 x 10'
2 x 15'
0.25
0.30

Table 5-18. End Movements for Beam 2

bottom Left End
Strand Measured
4 0
8 0.039"
12 0.079"
16 0.079"
20 0.118"
24 0.157"
28 0.276"
32 0.394"
36 0.472"
40 0.512"
42 0.276"

Total

Left End BottomFRght End eight End Difference
Calculated Difference Strand IMeasured ICalculated

0.021" -0.021"
0.034" 0.005"
0.050" 0.029"
0.071" 0.008"
0.099" 0.019"
0.134" 0.023"
0.181" 0.095"
0.230" 0.164"
0.275" 0.197"
0.298" 0.214"
0.312" -0.036"

Calculated

0
0.079"
0.079"
0.118"
0.157"
0.236"
0.354"
0.354"
0.472"
0.551"

0.050"
0.098"
0.150"
0.204"
0.257"
0.308"
0.357"
0.403"
0.449"
0.507"

-0.050"
-0.019"
-0.071
-0.086"
-0.100"
-0.072"
-0.003"
-0.049"
0.023"
0.044"

Shortening Shortening Difference
0.827" 0.819" 0.008"

*- Measured
Calculated

6 16 26 36
Number of Cut Strands

Figure 5-28. Beam 2 Left End Measured vs Calculated

0.6

0.5

" 0.4

0.3
0
o
2 0.2

0.6
*
0.5 -

S0.4 -

3 Measured
0.3 -
-Eu Predicted

I 0.2

0.1

0
6 16 26 36
Number of Cut Strands

Figure 5-29. Beam 2 Right End Measured vs Calculated

Table 5-19. End Movements for Right End of Beam 1
Bottom Right End Right End
Strand Measured Calculated Difference
4 0 0.000" 0.000"
8 0.039" 0.000" 0.039"
12 0.039" 0.000" 0.039"
16 0.079" 0.000" 0.079"
20 0.079" 0.000" 0.079"
24 0.079" 0.000" 0.079"
28 0.197" 0.000" 0.197"
32 0.394" 0.000" 0.394"
36 0.079" 0.000" 0.079"
40 0.079" 0.036" 0.043"
42 0.079" 0.052" 0.027"

0.5

- 0.4

S0.3

o
2 0.2

- Measured
Calculated

A- *~

Number of Cut Strands

Figure 5-30. Beam 1 Right End Measured vs Calculated

Table 5-20. End Movements for Left End of Beam 3
bottom Left End Left End
Strand Measured Calculated Difference

0.000"
0.039"
0.039"
0.039"
0.000"
-0.039"
-0.039"
-0.039"
-0.118"
-0.039"

-0.026
-0.058
-0.092
-0.121
-0.141
-0.151
-0.151
-0.151
-0.151
-0.142

0.026"
0.097"
0.131"
0.160"
0.141"
0.112"
0.112"
0.112"
0.033"
0.103"

S0.2

0.1 -

-0.1

-0.2
6 16 26 36
Number of Cut Strands

Figure 5-31. Beam 3 Left End Measured vs Calculated

*- Measured
Calculated

CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS

The occurrence of vertical cracking can be affected by many variables; length of

the free strands, modulus of elasticity of the concrete, friction coefficient between the

beam and the casting bed, temperature change, debonding lengths, number of debonded

strands, number of prestressing strands, jacking force per strand, tension strength of the

concrete, cross-sectional area of the beam, beam length, and beam spacing configuration.

Because there are so many different variables that influence the formation of vertical

cracks, it was necessary to determine which variables had the greatest effect on vertical

crack formation so that the best possible solution could be determined. The MathCad 12

analytical model in Appendix B was created to allow the researchers to determine the

best vertical crack solution for a given casting bed of beams. This analytical model was

not created to predict the exact stresses in the concrete beams and the steel strands

because that is not possible due to non-simultaneous cutting, dynamic effects, and the

disturbed region properties of a prestressed beam end. For this reason, no hard and fast

rule can be created to eliminate vertical cracking in prestressed beams. However, by

performing a sensitivity analysis on the analytical model (See Appendix B), trends were

developed and the variables that are most likely to cause vertical cracking were

determined.

The analytical model determined that the variables that have the greatest effect on

vertical cracking are temperature change between the time of beam casting and the time

of strand detensioning, friction coefficient between the casting bed and the bottom of the

beams, concrete release strength, beam length, and number of prestressing strands. The

free strand lengths and the number of beams on the casting bed have the next greatest

effect. The free strand lengths become more important as the number of beams on the

casting bed increases. The number of debonded strands and the lengths of the debonded

strands have a small effect on vertical cracking. The trends that were developed with the

analytical model in Appendix B are listed below.

* Increasing the number of prestressing strands increases the likelihood of vertical
cracking.

* Increasing the coefficient of friction between the casting bed and the bottom of the
beam increases the likelihood of vertical cracking.

* Decreasing the concrete release strength increases the likelihood of vertical
cracking.

* Increasing the beam length increases the likelihood of vertical cracking.

* A temperature reduction in the free strands from the time of beam casting to the
time of strand detensioning increases the likelihood of vertical cracking. A
temperature increase in the free strands from the time of beam casting to the time of
strand detensioning decreases the likelihood of vertical cracking.

* Decreasing the number of debonded strands increases the likelihood of vertical
cracking.

* Decreasing the debonded length of the debonded strands increases the likelihood of
vertical cracking.

* Increasing the number of beams on the casting bed increases the likelihood of
vertical cracking.

* Decreasing the free strand length between the bulkhead and the beam increases the
likelihood of vertical cracking. This effect is increased as the number of beams on
the casting bed increases.

The conclusion that can be drawn from this research study is that the three most

important things to do in order to reduce the occurrence of vertical cracks are to

detension the prestressing strands when the temperature of the free strands is similar or

warmer than the temperature of the free strands when the beams were cast, to lower the

coefficient of friction between the casting bed and the bottom of the beams, and to add

additional space between the beams. Lowering the coefficient of friction between the

casting bed and the bottom of the beam ends can be accomplished by smoothing the

casting bed before each new pour, adding lubricants under the beam ends, and by

installing steel bearing plates at the beam ends (See Figure 2-1). If the coefficient of

beams. If the coefficient of friction is high, the additional beam spacing must be

distributed between all of the beams to be effective.

APPENDIX A
SAMPLE RETURNED SURVEY FORMS

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APPENDIX B
VERTICAL CRACK PREDICTOR

81

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PAGE 1

PREVENTION OF VERTICAL END CRACKING ON PRESTRESSED BEAMS DURING FABRICATION By MICHAEL REPONEN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2006

PAGE 2

PAGE 3

This document is dedicated to my parents.

PAGE 4

iv ACKNOWLEDGMENTS This degree would not have been possibl e without my parents, my friends, and a few complete strangers that helped me along the way. I w ould also like to thank the FDOT, Dr. Cook, Dr. Lybas, Dr Hamilton, Dr. Consolazio, and Gate Concrete for their kind support of this research project.

PAGE 5

v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES..........................................................................................................viii ABSTRACT....................................................................................................................... .xi CHAPTER 1 INTRODUCTION........................................................................................................1 Project Overview..........................................................................................................1 Prestressed Concrete Background................................................................................2 2 END CRACKING LITERATURE REVIEW..............................................................6 Introduction................................................................................................................... 6 Review of Mirza and Tawfik 1978...............................................................................6 Review of Kannel, Fr ench, Stolarski 1998...................................................................8 Summary of End Cracking Reduction Recommendations...........................................9 3 MANUFACTURER SURVEY AN D FIELD INSPECTIONS..................................11 Manufacturer Survey..................................................................................................11 Field Inspection Introduction......................................................................................11 Field Inspection Results..............................................................................................13 Field Inspection Summary..........................................................................................16 4 VERTICAL CRACK ANALYTICAL MODEL........................................................21 Introduction.................................................................................................................21 Analytical Model Theory............................................................................................22 Global Motion Without Friction..........................................................................22 Global Motion With Friction...............................................................................24 Analytical Commentary.......................................................................................26 Analytical Model Assumptions..................................................................................28 Analytical Model Input Variables..............................................................................30 Analytical Model Flow Chart.....................................................................................30

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vi 5 RESULTS...................................................................................................................37 Introduction.................................................................................................................37 Test Case 1..................................................................................................................38 Modification 1: Alter the Numb er of Prestressing Strands.................................38 Modification 2: Alter th e Friction Coefficient....................................................38 Modification 3: Alter the C oncrete Release Strength..........................................39 Modification 4: Alter the Beam Lengths.............................................................39 Modification 5: Alter the Temperature Change..................................................39 Modification 6: Alter the Number of Debonded Strands....................................40 Modification 7: Alter the De bonded Lengths of 10 Strands...............................40 Modification 8: Alter th e Number of Beams.......................................................40 Modification 9: Alter the Free Strand Length for 2 Beams.................................41 Modification 10: Alter the Fr ee Strand Length for 3 Beams...............................41 Modification 11: Alter the Fr ee Strand Length for 4 Beams...............................41 Test Case 2..................................................................................................................42 Modification 1: Alter th e Friction Coefficient....................................................42 Modification 2: Alter the Beam Spacing.............................................................42 Analytical Model Conclusions...................................................................................43 Field Data Results.......................................................................................................44 Field Data Conclusions...............................................................................................44 6 CONCLUSIONS AND RECOMMENDATIONS.....................................................72 APPENDIX A SAMPLE RETURNED SURVEY FORMS..............................................................75 B VERTICAL CRACK PREDICTOR..........................................................................80 C SIMPLIFIED VERTICAL CRACK PREDICTOR.................................................250 D FIELD STUDY STRAND LAYOUT.....................................................................256 LIST OF REFERENCES.................................................................................................257 BIOGRAPHICAL SKETCH...........................................................................................258

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vii LIST OF TABLES Table page 5-1 Test Case 1 Input Data..............................................................................................46 5-2 Alter the Number of Prestressing Strands.................................................................47 5-3 Alter the Friction Coefficient....................................................................................49 5-4 Alter the Conc rete Release Strength..........................................................................51 5-5 Alter the Beam Lengths.............................................................................................52 5-6 Alter the Temperature Change..................................................................................54 5-7 Alter the Numb er of Debonded Strands....................................................................55 5-8 Alter the Debonde d Lengths of 10 Strands...............................................................57 5-9 Alter the Number of Beams.......................................................................................58 5-10 Alter the Free Strand Length for 2 Beams...............................................................60 5-11 Alter the Free Strand Length for 3 Beams...............................................................62 5-12 Alter the Free Strand Length for 4 Beams...............................................................64 5-13 Test Ca se 2 Input Data............................................................................................65 5-14 Alter Friction Results..............................................................................................66 5-15 Free Strand Lengths.................................................................................................66 5-16 Beam Spacing Results.............................................................................................67 5-17 72 Florida Bulb-T Input Data................................................................................67 5-18 End Movements for Beam 2....................................................................................68 5-19 End Movements for Right End of Beam 1..............................................................69 5-20 End Movements for Left End of Beam 3................................................................70

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viii LIST OF FIGURES Figure page 1-1 Strand Anchorage System...........................................................................................4 1-2 Terminology.............................................................................................................. ..5 1-3 General Effects of Friction..........................................................................................5 2-1 Steel Bearing Plate.....................................................................................................10 3-1 Strand Cutting Process...............................................................................................16 3-2 Prestressed Strand Crack...........................................................................................17 3-3 Bursting Forces Caused by Prestressing Strands.......................................................17 3-4 Radial Cracking.........................................................................................................18 3-5 Lifting Devices.......................................................................................................... 18 3-6 Angular Crack............................................................................................................ 19 3-7 Web-Flange Junction Crack......................................................................................19 3-8 Edge Spall............................................................................................................... ...19 3-9 Horizontal Cracks......................................................................................................20 3-10 Vertical Crack..........................................................................................................20 4-1 Global Movement For 3 Beam Symmetrical System................................................31 4-2 Change in Acting Static Friction Force.....................................................................32 4-3 Global Motion of Beam.............................................................................................33 4-4 Stress in C oncrete Bottom Flange.............................................................................34 4-5 Axial and Camber Movement....................................................................................35 4-6 Analytical Model Flow Chart....................................................................................36

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ix 5-1 Test Case 1.............................................................................................................. ...45 5-2 Test Case 1 No Alterations........................................................................................46 5-3 Alter the Number of Prestressing Strands.................................................................47 5-4 Number of Pr estressing Stands fcalc/f Maximums.....................................................48 5-5 Alter the Friction Coefficient....................................................................................49 5-6 Friction Coefficient fcalc/f Maximums.......................................................................50 5-7 Alter the Concrete Release Strength..........................................................................51 5-8 Concrete Release Strength fcalc/f Maximums............................................................52 5-9 Alter the Beam Lengths.............................................................................................53 5-10 Beam Lengths fcalc/f Maximums..............................................................................53 5-11 Alter the Temperature Change................................................................................54 5-12 Temperature Change fcalc/f Maximums...................................................................55 5-13 Alter the Number of Debonded Strands..................................................................56 5-14 Number of Debonded Strands fcalc/f Maximums.....................................................56 5-15 Alter the Debonded Lengths of 10 Debonded Strands............................................57 5-16 Debonded Lengths fcalc/f Maximums.......................................................................58 5-17 Alter the Number of Beams.....................................................................................59 5-18 Number of Beams fcalc/f Maximums........................................................................59 5-19 Alter the Free Strand Length for 2 Beams...............................................................60 5-20 Free Strand Length for 2 Beams fcalc/f Maximums.................................................61 5-21 Alter the Free Strand Length for 3 Beams...............................................................62 5-22 Free Strand Length for 3 Beams fcalc/f Maximums.................................................63 5-23 Alter the Free Strand Length for 4 Beams...............................................................64 5-24 Free Strand Length for 4 Beams fcalc/f Maximums.................................................65 5-25 Test Case 2............................................................................................................. ..65

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x 5-26 Alter the Friction Coeffi cient for Multiple Beam Ends...........................................66 5-27 72 Florida Bulb-T Arrangement............................................................................67 5-28 Beam 2 Left End Measured vs Calculated..............................................................68 5-29 Beam 2 Right End Measured vs Calculated............................................................69 5-30 Beam1 Right End Measured vs Calculated.............................................................70 5-31 Beam 3 Left End Measured vs Calculated..............................................................71

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xi Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering PREVENTION OF VERTICAL END CRACKING ON PRESTRESSED BEAMS DURING FABRICATION By Michael Reponen December 2006 Chair: Ronald Cook Major Department: Civil and Coastal Engineering The purpose of this research project was to determine the causes and cures of the vertical end crack found on th e bottom flange of AASHTO, Florida Bulb-T, and FloridaU prestressed beam ends. This vertical crack forms during the transfer of the prestressing force to the concrete. This type of crack form s at the base of the beam just a few inches from the end of the beam, and propagates ve rtically upward towards the web region of the beam. According to interviewed field personnel, this type of end crack is a maintenance issue that slows down production an d also raises questi ons regarding loss of bond and ingress of chlorides to the prestressing strands. The research project began by mailing surveys to three Florida Department of Transportation prestressed beam manufacturers in Florida to determine the extent and types of end cracking each manufacturer wa s experiencing during th e production process of their AASHTO, Florida Bulb-T, and Florida U-beams. Site visits followed the surveys to allow the researchers to observe the beam pr oduction process. The si te visits also gave

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xii the researchers the opportunity to talk to each plant's engineers and technicians about the different types of end cracks and when, where, and how each type of crack forms. A computer model was then created in MathSo ft's MathCad Version 12 to determine the sensitivity vertical cracking ha s to variations in input vari ables such as spacing between the beams, friction coefficient between th e beam and the casting bed, debonded lengths, etc. The analytical model determined that th e variables that have the greatest effect on vertical cracking are atmosphe ric temperature change betwee n the time of beam casting and the time of strand detensioning, friction coefficient between th e casting bed and the bottom of the beams, concrete release stre ngth, beam length, and number of prestressing strands. Beam spacing, and the number of beams on the casting bed have the next greatest effect. Beam spacing becomes more important as the numb er of beams on the casting bed increases. The number of debonde d strands and the lengths of the debonded strands have a small effect on vertical crac king. The conclusion that can be drawn from this research study is that the three most im portant things to do in order to reduce the occurrence of vertical crack s are to detension the pres tressing strands when the atmospheric temperature is similar or warmer than the atmospheric temperature when the beams were cast, to lower the coefficient of friction between the casting bed and the bottom of the beams, and to add additional space between the beams.

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1 CHAPTER 1 INTRODUCTION Project Overview The purpose of this research project was to determine the causes and potential cures of the vertical end crack found on the botto m flange of AASHTO, Florida Bulb-T, and FloridaU prestressed beam ends. This ver tical crack forms during the transfer of the prestressing force to the concrete. This type of crack forms at the base of the beam just a few inches from the end of the beam, and propagates vertically upward towards the web region of the beam. According to interviewed field personnel, this type of end crack is a maintenance issue that slows down production an d also raises questi ons regarding loss of bond and ingress of chlorides to the prestressing strands. The research project began by mailing surv eys to three FDOT prestressed beam manufacturers in Florida to determine th e extent and types of end cracking each manufacturer was experiencing during the pr oduction process of their AASHTO, Florida Bulb-T, and Florida U-beams. Site visits followed the surveys to allow the researchers to observe the beam production process. The si te visits also gave the researchers the opportunity to talk to some of each plant's engineers and technicians about the different types of end cracks and when, where, and how each type of crack forms. A computer model was then created in MathSoft's Math Cad Version 12 to determine trends that should be followed to maximize the effectiven ess of any vertical crack prevention plan for any type of AASHTO, Florida Bulb-T, or Florida U-beam. Beam end movements were measured for three 139 ft long 72" Flor ida Bulb-T beams at Gate Concrete in

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2 Jacksonville Florida and compared to the predicted values from the MathCad model. It was determined that the analytical model c ould not predict the ex act movements of the prestressed beam ends in the field due to non-simultaneous cutting, and dynamic effects. However, the analytical model was determined to be a valuable tool for determining trends that should be followed to reduce the occurrence of a vertical crack on any type of prestressed beam. Prestressed Concrete Background Prestressed beams are formed by stretc hing steel strands with hydraulic jacks across a casting bed that can be as long as 800 feet (Nilson 1987, Naaman 2004). The strands are then anchored with chucks to bul kheads (See Figure 1-1) at both ends of the casting bed. Beams are then cast along the leng th of the casting bed with a single set of prestressing strands running th rough all of the beams. When the concrete hardens the prestressing strands become bonded to the conc rete. The portion of the strands between the beams that do not have concrete bonded to th em is referred to as free strands (Kannel, French & Stolarski 1998). When the compre ssive strength of the sample concrete cylinders reaches the project sp ecified release strength, the free strands are then cut one at a time and the force within each prestressing strand transfers to the concrete beams, placing the beams in a state of compression. This compression force causes the concrete beams to axially shorten and camber. Unlik e post-tensioned strands prestressed strands require a certain distance to fully transfer their force thr ough bond to the concrete. The distance required is known as the transfer length of the prestressing strand. ACI 318-02 defines the transfer length (lt) as equal to one third the effective stress in the steel strand (fse) multiplied by the diameter of the strand (db). The transfer length is an important concept because the transferred force varies fr om zero at the end of the beam, to the full

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3 prestress force at the transfer length (Nilson 1987). Because a transfer length is required to transfer the prestress force to the beam, th e ends of prestressed beams are vulnerable to cracking if tension strains deve lop in the end region concrete. With this idea in mind, the way to prevent the vertical end crack is to re duce the tension strains the concrete feels in the transfer length region of the beam. Tension pull and friction are two sources of tension strain at the end of a prestressed beam that can be controlled and reduced by both the prestressed beam designer and manufacturer. When some of the free strands are cut, the prestressed beams on the casting bed axially shorten and camber (Naaman 2004). The axial shortening provides the largest movement while the camber produces a small additional amount of end movement due to the rotation of the end face of the beam. As the beams shorten and rotate the uncut free strands are forced to stretch to accommoda te this movement (Mirza & Tawfik 1978). This stretch creates a tension force in the unc ut free strands in addition to the prestress force (Mirza & Tawfik 1978). This additional force is referred to as "tension pull" (See Figure 1-2). Temperature cha nge in the free strands between the time of beam casting and the time of strand detensioning cha nges the tension pull magnitude. If the temperature of the free strands decreases, the tension pull is increase d. If the temperature of the free strands increases, the tension pull is de creased. The thermal coefficient of the prestressing strands is 6.67x10-6 in/in/oF (Barr, Stanton & Eberha rd 2005). The tension pull transfers into the concrete beam over the transfer length required for the given tension pull magnitude. The transfer length required for a given tension pull magnitude is referred to as the reverse transfer length (Kannel, French & Stolarski 1998).

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4 Friction between the bottom of the concrete beam ends and the steel casting bed is another force that can create tens ion strain at the ends of a pr estressed beam. The role of friction is to reduce movements of the beam, either in the form of reducing the axial shortening, or by reducing the amount the beam shifts on the casting bed. Static friction force (Fs) is generally modeled as the co efficient of static friction (s) multiplied by the normal force (N). Dynamic friction force (Fd) is generally modeled as the coefficient of dynamic friction (d) multiplied by the normal force (N). The static friction force must be overcome before any movement can occur. The dynamic friction force is the friction force a body feels while it is in motion. If at any time the force causing motion of the body becomes less than the dynamic friction forc e, motion ceases. For motion to occur again, the static friction force must once again be overcome. In the case of a prestressed beam, the friction acts at both ends of the b eam as the beam cambers (See Figure 1-3). The normal force (N) is equal to one half of the beam's weight (W). Given a constant coefficient of friction, the greater the beam le ngth, the larger the friction force at the two beam ends becomes. A B Figure 1-1. Strand Anchorage System A) Typical Bulkhead B) Chucks

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5 Figure 1-2. Terminology. This figure shows how tension pull is cr eated and how it is transferred to the concrete beam over the Reverse Transfer Length Figure 1-3. General Effects of Friction FreeStrand Distance along beam Force Reverse Transfer Length Tension Pull Prestressing Force Force in Free Strand CutStrands Weight = W N = W/2 N = W/2 F F For F less than max static friction force No movement F F Wei g ht= W N = W/2 N = W/2 F F For F greater than max static friction force Movement occurs Fs = s*N Fs = s*N Concrete elastic shortening and rotation

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6 CHAPTER 2 END CRACKING LITERATURE REVIEW Introduction The following summarizes two studies pr eviously conducted on end cracking in prestressed beams. The first study conducted by Mirza and Tawfik focused on vertical end cracking on 73' AASHTO Type III beams (Mirza & Tawfik 1978). The second study conducted by Kannel, French, and Stolarski inve stigated vertical, a ngular, and horizontal end cracking on 45", 54", and 72" I-beams with draped strands and steel bearing plates (Kannel, French & Stolarski 1998). Review of Mirza and Tawfik 1978 In order to determine how to prevent vertical cracking in the AASHTO Type III beams, Mirza and Tawfik first experimented on 45' to 50' long rectangular beams. The goal was to determine the root cause of the ve rtical cracking. It wa s theorized that the vertical end cracking was caused by the restrain ing force in the uncut strands as the beam was being detensioned. As strands are cut, the beam shortens and the uncut strands, because they are still attached to the beam and the bulkhead, are forced to stretch. This stretch creates a resisting force that is transf erred to the concrete beams. The magnitude of the resisting force is dependent upon the le ngth of the strands be tween the beams. The beams were cast in three sets of two, with each set having a different length of strand between the beam ends and the bulkheads. By attaching strain gages to the steel strands and by using dial gages on the beam ends, th e total resisting force in the uncut strands was determined. The experiment showed that although the resisting force per strand

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7 increases throughout the cutting process, the total resisting force reaches a maximum at a point when approximately half the strands have been cut. This is the point when the cracks were observed to form. It was also observed that the cr ack widths decreased during the cutting of the second half of the stra nds. Because the cracks were within a few inches of the beam ends it was concluded that the resisting force mu st be transferred to the concrete over a short dist ance, and that this distance was less than the compression transfer length of the cut strands. To enable the researcher to determine the most important variables that cause beam end cracking, a computer analytical model was created by idealizing the beams and the uncut strands as bilinear springs. Using a s tiffness analysis, the resisting force in the uncut strands was determined after each strand was cut. These analytical values were compared to experimental values and found to be on average 10 to 20 percent larger in the middle range of the cutting order. The an alytical model determined that simultaneous release of the strands resulted in the lowest tens ile stresses in the concre te beams. It also determined that if non-simultaneous release did occur it was best to cut the longer strand (between the bulkhead and the beam) before cutting the shorter strand (between the two beams). In order to combat the resisting force in the uncut strands, the AASHTO Type III beams were modified in three ways; fifteen inch long steel bearing plates (See Figure 21) were installed on the bottom of the beam ends, two three foot long Grade 40 rebars were added in the bottom flange of the beam ends, and additional prestressing strands were debonded for each beam. After making th ese modifications, vertical cracks were no longer observed in the AASHTO Type III prestressed beams. To prevent vertical

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8 cracking in general, Mirza and Tawfik recommended making the length of the prestressing strands between the bulkhead and the prestressed beams at least 5% of the bed length. If this length could not be provided, they recommended debonding additional prestressing strands for a debonded length e qual to or greater than the compression transfer length. Debonding redu ces the resisting force in the uncut strands by reducing the average stiffness of the uncut strands A debonded strand also helps by moving a portion of the resisting force to an interior region of the beam where the prestress force has been fully developed and the concrete can handle the resisting fo rce without cracking. Review of Kannel, French, Stolarski 1998 The study conducted by Kannel, French, and Stolarski investigated vertical, angular, and horizontal end cr acking on 45", 54", and 72" I-bea ms with draped strands and steel bearing plates. An ABAQUS Finite Element model of a half beam was created to model the stresses in the concrete at the end region of the beam during the detensioning process. Multiple strand cutting patterns were chosen for analysis to determine the relationship between end cracking and strand cutting pattern. The favorable strand cutting patterns were th en tested on full scale 45", 54", and 72" prestressed I-beams. The vertical crack in this study formed in a different way than the vertical crack in the Mirza and Tawfik (1978) study. This ver tical crack formed due to tension strains created from the rel ease of the draped strands. Th e researchers determined through analytical and field testing that if two strai ght strands were cut before every six draped strands were cut, the vertical crack would not form. The angular crack formed due to shear stresses created from the compression forces from the cut strands and the tension forces from the uncut strands. The research ers determined through analytical and field

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9 testing that changing the st rand cutting pattern to bette r balance the tensile and compressive forces on the bottom flange cross section would reduce th e occurrence of the angular crack. The horizontal crack at the web-flange interface formed due to stress concentrations at that locati on. This type of crack was s hown to occur independently of the strand cutting pattern. The researchers proposed that increasing the slope of the flange over the first 18" would reduce the occurrence of this hor izontal crack. Kannel, French, and Stolarski (1998) determ ined that end cracks in general form due to two things; the restraining force from the uncut strands, a nd the shear stresses created from the strand cutting pattern. To reduce the occurrence of end cracking in prestressed beams Kannel, Fren ch, and Stolarski suggested fo ur things; change the strand cutting pattern, debond additional prestressing strands, lower the coefficient of friction between the beam and the casting bed, and provi de at least 10 to 15% of the total bed length in free strand length. Adding extra free strand length re duces the tensile forces in the uncut strands. Lowering the coefficient of friction between the beam and the casting bed helps balance the tensile forces at the two ends of the beam by allowing the beam more freedom to shift on the casting bed. Fo r beams with steel bearing plates, it was recommended that the debonded length should be at least six inches greater than the length of the steel plate. Summary of End Cracking Reduction Recommendations To reduce the occurrence of vertical cracking, Mirza and Tawfik recommended making the length of the prestressing strands between the bulkhead and the prestressed beams at least 5% of the bed length (Mirza & Tawfik 1978). If this length could not be provided, they recommended debonding additio nal prestressing strands for a debonded length equal to or greater than the compressi on transfer length (M irza & Tawfik 1978).

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10 Kannel, French, and Stolarski suggested four things; change the strand cutting pattern, debond additional prestressing strands, lower th e coefficient of friction between the beam and the casting bed, and provide at least 10 to 15% of the total bed length in free strand length (Kannel, French & Stolarski 1998). Figure 2-1. Steel Bearing Plate

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11 CHAPTER 3 MANUFACTURER SURVEY AN D FIELD INSPECTIONS Manufacturer Survey Surveys were sent in November 200 4 to three FDOT prestressed beam manufacturers in Florida to determine th e extent and types of end cracking each manufacturer was experiencing during the pr oduction process of their AASHTO, Florida Bulb-T, and Florida U-beams. A sample retu rned survey can be seen in Appendix A. The returned surveys showed an interesting result; vertical crack s were only one of several commonly occurring end cracks. It was also learned that multiple crack types could occur on a single beam end. Field Inspection Introduction Following the surveys, site visits allowed the researchers to observe the beam production process. Three Florida prestressed concrete manufacturers were visited from January 2005 to February 2006. AASHTO, Florida Bulb-T, and Florida U-beams in the manufacturer's storage areas and on the casti ng beds were visually inspected for end cracking. The prestressed beams were of various lengths and consisted of various numbers and types of prestressing strands. The detensioning pro cess of AASHTO Types 3 and 4, and 72" Florida Bulb-T prestressed be ams was also observed. The site visits gave the researchers the opport unity to converse with plan t engineers and technicians about the different types of end cracks to obt ain their opinions as to when, where, and how each type of crack formed.

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12 Researchers observed that during the dete nsioning process, most of the beam movement occured near the end of the st rand cutting process. Not only did beams shorten and camber as strands were cut, but b eams also slid as units on the casting bed. The beams next to the bulkheads were most likel y to slide, and this s liding appeared to be most likely the result of non-simultaneous stra nd cutting. For example, the strand on the left side of the beam was cut before the stra nd on the right side of the beam, and the beam slid to the right. The researcher observed an AASHTO Type III beam set into harmonic motion after non-simultaneous strand cutting. After one cycle of motion the movement abated. This type of motion raised questions regarding the amount te nsion strains in the concrete beam ends were magnified due to dynamic effects on the casting bed. During the detensioning process, a flagma n signaled when each prestressing strand should be cut. The workmen, standing in be tween every beam on the casting bed applied their torches to the specified strand (See Fi gure 3-1). The researcher observed that sometimes the strands "popped" right when the torch was applied, and at other times cutting took ten seconds or more. Occasi onally, as a torch was being removed, an additional strand was accidentally cut. As th e seven wire strands were cut, distinctive popping sounds were heard, as each of the seven wires, in each prestressing strand, broke. The researcher determined that simultaneou s cutting was problematic and an unrealistic assumption in design. Manufacturers indicated that end crac king on prestressed beams was a common occurrence. They relayed that end cracking tended to occur more frequently on larger, longer span beams. The cracking sometimes appeared to occur randomly. For example, the third beam on a casting bed of five crack ed, yet none of the other four beams would

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13 have any cracks. End cracking was not a co mpletely random process despite the random nature of material properties. Each beam end on the casting bed experienced slightly different forces during the detensioning process due to non-simultaneous cutting, accidental additional strand cuts, and the beam spacing arrangement on the casting bed. To reduce the occurrence of e nd cracking, the root source of e ach type of crack must be determined. The first step in determining the cause of each type of cracking was to distinguish the different type s of end cracks. The following information presents the different types of end cracks found during the s ite visits of the three Florida prestressed concrete manufacturers. Field Inspection Results Eight types of end cracks were discovered during the multiple site visits to three Florida prestressed concrete manufacturers. Th e first crack type shall be referred to as a prestressed strand crack (See Figure 3-2). This crack originated at a prestressing strand and propagated toward the outer surfaces of th e beam. The prestressed strand crack often ran through multiple prestressing strands befo re reaching the exterior surface of the concrete beam. The researcher proposed that this crack type was caused by two things; Poissons Effect and rusting of the pr estressing strands. When a load is applied to a prestressi ng strand, the prestressing strand elongates by an amount and the radius shrinks by an amount . The ratio of the strain created by to the strain created by  is a constant known as Poisson s ratio (Hibbler 2000). In the transfer length region of a prestressed beam, the force within an individual cut prestressing strand varies from zero at the end of the beam to the full prestress value at the compression transfer length. Due to Poisso ns effect, the prestressing strand wants to expand as the force in the strand reduces to ze ro at the end of the beam. This expansion

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14 effect creates a bursting force on the concrete (See Figure 3-3). This led the researcher to propose that rust further increases the bursting force at the very end of the beam because metal bars expand as they rust. A second cracking type shall be referred to as radial cracking. Radial cracking is a fan shaped multiple crack pattern that extends the entire height of the beam (See Figure 3-4). This cracking pattern was observed on a 72 Florida Bulb-T, and a 78 Florida Bulb-T prestressed beam. The cracks origin ating in the bottom flange were angled upward, the cracks in the web were approximate ly horizontal, and the cracks in the top flange were angled downward. Three or four vertical top fl ange cracks spaced at about five feet along the top flange finished off th e pattern. Excluding th e vertical top flange cracks, when the cracks were extended with a chalk line, the chalk lines intersected at a point in the web region of the beam. This led the researcher to propose that radial cracking was caused by the lifting hook arra ngement/design (See Figure 3-5a) or the lifting procedure (See Figure 3-5b). The third type of crack was the angular cr ack. This crack originated in the sloped part of the bottom flange, a few inches from the end of the beam and propagated upward at an angle towards the web. Kannel, Fren ch, and Stolarski (1998) found angular cracks form due to shear stresses created from th e compression forces from the cut strands and the tension forces from the uncut strands. Kannel, French, and Stolarski determined through analytical and empirical research th at changing the strand cutting pattern, to better balance the tensile and compressive forces on the bottom flange cross section, reduced the occurrence of the angular crack.

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15 The fourth and fifth types of cracks shall be referred to as the web-flange junction crack and edge spalling. The web-flange junction crack crossed the end face of the beam in the web and then proceeded dow nward, but did not extend past the sloped portion of the bottom flange (See Figure 3-7) A manufacturer s uggested that tension strains created between the prestressed botto m flange and the non-prestressed web region was the cause of the crack. This led the rese archer to propose that this crack could be prevented by adding additional horizontal mild steel in the web-flange region. Edge spalls were a common occurrence, esp ecially on beams with skewed ends. A horizontal top flange crack (See Figure 3-9a) and a horizontal web crack (See Figure 3-9b) were the next two cracks identified. The horizontal top flange crack began at the end face of the upper flange a nd moved inward horizontally. Manufacturer field personnel suggested that this crack was caused by formwork pressing against the concrete when the beam cambered during dete nsioning. The manufact urer advised that the horizontal top flange crack could be prevented by providing space between the formwork and the concrete before detensi oning began. The horizontal web crack looked similar to the horizontal top flange crack excep t that the horizontal web crack occured in the web portion of the beam. An eighth crack type identified was the vert ical crack (See Figur e 3-10). Mirza and Tawfik's (1978) research determined that the vertical crack could be caused by the resisting forces in the uncut strands during the detension pr ocess. The vertical crack observed was located on a beam that did not contain any draped strands, so Kannel, French, and Stolarskis 1998 ve rtical crack explanation did not apply (Kannel, French & Stolarski 1998). The vertical crack in Figure 3-10 was the object of study, for this

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16 research project. Manufacturer field personnel believed that reducing the coefficient of friction between the casting bed and the bottom of the prestressed beam helped reduce the occurrence of the vert ical crack. Field Inspection Summary Eight types of end cracks were identified during the fiel d survey of three Florida prestressed concrete manufacturers; the pr estressed strand crack, radial cracks, the angular crack, the web-flange junction crack, the edge spall, the horizontal top flange crack, the horizontal web crack, and the ver tical crack. On many occasions more than one type of crack was found on the same beam end. The focus of this research project was the vertical end crack (See Figure 3-10) and the following chapters will focus exclusively on the vertical end crack. Figure 3-1. Strand Cutting Process

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17 A B Figure 3-2. Prestressed Strand Crack. A) This figure shows how the prestressed strand crack propagates toward the outer su rface of the beam. B) Photo of prestressed strand crack Figure 3-3. Bursting Forces Caused by Prestressing Strands This figure shows how expansion due to Poisson's effect and rust creates a bursting force in the concrete beam end. Cut Prestressing Strand CTL CTL = Compression Transfer Length Bursting Forces Concrete

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18 Figure 3-4. Radial Cracking A B Figure 3-5. Lifting Devices A) Typical Lifting Hook B) Lifting Machine Top flange vertical crack

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19 Figure 3-6. Angular Crack. The angular crack shown here has been highlighted with chalk to increase its visibility. Figure 3-7. Web-Flange Junction Crack Figure 3-8. Edge Spall

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20 A B Figure 3-9. Horizontal Cracks A) Horizontal T op Flange Crack B) Horizontal Web Crack. Figure 3-10. Vertical Crack

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21 CHAPTER 4 VERTICAL CRACK ANALYTICAL MODEL Introduction The occurrence of vertical cracking can be affected by many variables; length of the free strands, modulus of elasticity of th e concrete, friction coefficient between the beam and the casting bed, temperature change, debonding lengths, number of debonded strands, number of prestressing strands, jacki ng force per strand, tension strength of the concrete, cross-sectional area of the beam, beam length, and beam spacing configuration. Because there were so many different variables that influenced the formation of vertical cracks, it was necessary to determine which va riables had the greatest effect on vertical crack formation. This allowed the researcher to determine the best possible solution to vertical end cracking. To accomplish this, a MathCad 12 analytical model was created (See Appendix B). Imputing specifications of beam number, beam length, the number of bottom strands, the type of strand, the j acking force, the free strand length, the temperature change at casting verses cutti ng time, and the coefficient of static and dynamic friction between the bottom of the beam and the casti ng bed, the cracking tendency for the specified condi tions could be calculated. The analytical model is based on four major assumptions listed below. The strands between all the beams on the casting bed are cut exactly at the same time for every strand in the cutting order. The strand cutting pattern (See Appendix D) evenly balances the transferred compression forces from the individual cu t strands to the bott om flange of the beam.

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22 The strands are all heated and cut slowly. The prestressing strands and the concrete beams are idealized as linear elastic springs. Analytical Model Theory During detensioning the friction force, be tween the beam and the bottom form, is distributed over a certain area of the bottom of the beam ends. As free strands are cut and the beam camber increases, this distributed area shrinks until the friction force acts nearly as a line load across the very ends of the beam When the tension pulls at the two ends of a beam are unequal, the acting direction of th e static friction force may change and the beam may slide as a unit on th e casting bed. This phenomenon is referred to as "global movement". When only a single beam is de tensioned on a casting bed, the tension pull on the two ends of the beam is always equal and global movement can not occur. The following sections explain how global m ovement can occur for a three beam symmetrically placed system (See Figur e 4-1) with and without friction. Global Motion Without Friction When friction is absent from the system shown in Figure 4-1, equilibrium requires that all three beams shorten exactly the same am ount, and that tension pull in all the free strands is equal. The system resists the imposed force "F" in two ways; the concrete resists shortening by equation 4-1, and the steel resists stretc hing by equation 4-2. Compatibility requires that the total amount of concrete shorte ning in the system is equal to the total amount of steel stre tching in the system. This value is equal to equation 4-3. The tension pull created in the free strands is equal to equation 4-4. Equation 4-4 assumes that the tot is very small compared to (L-3Lb).

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23 cc c cAE k 3L (4-1) ss s cAE k L3L (4-2) tot csF kk (4-3) stotTPk (4-4) Beam 2 will not slide and the end movements at both ends of Beam 2 will be equal due to symmetry of the system. Combining this fact with the fact that all the beams shorten the same amount, equation 4-5 is derived. Because Ls1 is much greater than Ls2, equation 4-6 is derived. In order for the sy stem to regain equilibrium, Beams 1 and 3 are forced to react according to equation 4-6. tot Beam2End1Beam2End26 (4-5) Beam1End2Beam3End10 tot Beam1End1Beam3End23 (4-6) Because the final tension pull in the system must be equal in all the free strands to satisfy equilibrium, the final strain in all the free strands must also be equal. In order to satisfy these requirements, Beams 1 and 3 ar e forced to experien ce global movement. Beams 1 and 3 will slide exactly the same di stance towards Beam 2 due to symmetry of the system. The distance of the slide is dete rmined from the strain compatibility equation 4-7. The solution of equation 4-7 is shown in equation 4-8.

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24 tottot slideslide s2s163 LL (4-7) s1s2 tot slide s2s1LL () 63 LL (4-8) Global Motion With Friction The following explanation assumes that the force "F" is greater than the static friction force "Fs". When friction is present, equi librium requires that Beams 1 and 3 shorten exactly the same amount. Because of friction, Beam 2 w ill shorten a different amount than Beams 1 and 3. The system resi sts the imposed force "F" in two ways; the concrete resists shortening by the equation 49, and the steel resist s stretching by equation 4-10. The total steel resistance to the shorteni ng of each beam is shown in equation 4-11. cc c cAE k L (4-9) ss s1s4 s1AE kk L ss s2s3 s2AE kk L (4-10) ss sBeam2 s2 ssss s2s2AE 1 k 11 2L AEAE LL ss sBeam1sBeam3 s1s2 ssss s1s2AE 1 kk 11 LL AEAE LL (4-11) Beam 2 will not slide and the end movements at both ends of Beam 2 (shown in equation 4-12) will be equal due to symmetry of the system. The tension pull in Free Strands 2 and 3 is shown in equation 4-13. A positive friction force is assumed to act

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25 away from the center of each beam. The fi nal acting static friction force for Beam 2 (shown in equation 4-14) is determined from symmetry of the system and the requirement of the system to regain equilibrium. d Beam2End1Beam2End2 ccss cs2.5(FF) AEAE L2L (4-12) FreeStrands2FreeStrands3Beam2End1s2TPTPk (4-13) sBeam2End1sFF sBeam2End2sFF (4-14) Because Ls1 is much larger than Ls2, and equilibrium must be satisfied, all beam shortening for Beam1 occurs at End 1, and a ll beam shortening for Beam 3 occurs at End 2. The amount each beam end shortens is shown in equation 4-15. The tension pull created in Free Strands 1 and 4 is shown in e quation 4-16. The final acting static friction force for Beams 1 and 3 (shown in equation 417) is determined from symmetry of the system and the requirement of the system to regain equilibrium. A positive friction force is assumed to act away from the center of each beam. Equation 4-17 assumes that beam end movement due to an imbalance of tensi on pulls at the two ends of the beam is negligible. Beam1End2Beam3End10 d Beam1End1Beam3End2 ccss cs1s2FF AEAE LLL (4-15) FreeStrands1FreeStrands4Beam1End1s1TPTPk (4-16) sBeam1End1sBeam3End2sFFF sBeam1End2sBeam3End1sFreeStrands2FreeStrands1FFF(TPTP) (4-17)

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26 Analytical Commentary The difference in magnitudes of the tensi on pulls on the two ends of a beam is referred to as the UTP unbalanced tension pull. Th e amount the acting static friction force reduces on the end of the beam with th e larger tension pull is approximately equal to the unbalanced tension pull (See Figure 4-2). For Figure 4-2 and 4-3, a positive UTP occurs when Tension Pull2 is greater than Tension Pull1. For Figure 4-2 and 4-3, a positive static friction force acts towards the direction of axial shortening. If the magnitude | UTP| is greater than two times the static friction force (2*Fs) global movement will occur. In the case of global movement, ignoring dynamic loading effects, the beam will slide on th e casting bed until the magnitude | UTP| becomes less than two times the dynamic friction force (2*Fd) (See Figure 4-3). Difficulty arises when attempting to model the concrete strains in the end region of a prestressed beam because this is a dist urbed region (MacGregor 1997). In addition, sudden strand release known as "popping" and non-simultaneous cutting of the free strands can result in unpredictable dynamic effects. ACI 318-02 defines a disturbed region as "The portion of a member within a distance equal to the member height h or depth d from a force discontinuity or geom etric discontinuity". Within a disturbed region, classical beam theory can no longer be applied because plane sections do not remain plane. A strut and tie model is one method of designing D-z ones in concrete, but a strut and tie model can not determine the ac tual stresses in the concrete at a given location (Portland Cement Association 2002). A nother choice is to use a Finite Element analysis. With a Finite Element analysis th e analyzed object may have any size or shape, any type of boundary conditions, and any type of materials (Cook, Malkus, Plesha & Witt

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27 2002). However, when using Finite Elements the following questions arise regarding the method for modeling the following items. How should the transfer length and the reve rse transfer length be modeled? The magnitude of the reverse transfer length c onstantly changes as free strands are cut. The friction force magnitude, acting locati on, and acting direc tion are constantly changing. Before movement, the static fr iction force acts on the concrete. During movement, the dynamic friction force acts on the concrete. The compression load transferred to the concrete from each cut strand does not transfer instantaneously, but rather slowly as the strand is heated with a torch and the strand yields. This gives the beams the ability to react to the forces being developed in the neighboring beams. The ab ility to react to the movements of other beams alters the amount of movement each beam end experiences during the detensioning process. How should the effects of the mild steel be accounted for? A single beam end or even a single beam can not be analyzed individually during the detensioning process because the move ments of each beam end are dependent upon the movements of the opposite beam end and the movements of the neighboring beams on the casting bed. Three things must be maintained in any structural analysis; equilibrium, compatibility, and constitutive relationship s. Equilibrium requires that Newton's 2nd law F = ma be maintained at every point with in the structural system. Compatibility equations describe displacement constraints th at occur at supports of a member. For example, if the end of the concrete beam moves 0.3 inches, the free strands that are connected to the concrete must also move 0.3 inches. Constitutive relationships refer to the material properties, such as the stiffne ss, of the object of analysis. The cracking criterion that is used in the analytical mode l in Appendix B is equation 4-1. A simplified hand calculation procedure is shown in Appendix C. The variable fcalc is the calculated stress (psi) in the concrete at a chosen locati on. The variable f is the allowable tension

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28 stress (psi) in the concrete bottom flange and is calculated using equation 4-2. In equation 4-2, fci is the compressive strength of the concrete at the time of cutting. calcf 1 f Vertical Crack Forms (4-18) cif f5 psi (4-19) The calculated stress in the concrete bottom flange (fcalc) is based on four factors; the transferred prestress force, the static frict ion force, the bearing force, and the tension pull. The stress in the concrete is calculated at the bottom of the beam at a distance from the end face of the beam equa l to the reverse transfer length (See Figure 4-4). Equation 4-3 is used to calculate fcalc (See Figure 4-4). 2 ss calc bfbfbfbfbfF*eF CRTLN*RTL*eTP f=(----)*-1 AIAIA (4-20) Analytical Model Assumptions The assumptions made in the analytical m odel of Appendix B are listed below. The modulus of elasticity of the concrete is calculated usi ng equation 4-4 (Nawny 1996). The compressive strength of the concrete at the time of cutting is fci, and is the unit weight of the concrete. 61.5 cif E = (40000+10)()psi psi145pcf (4-21) The unit weight of the concrete is taken as 150 pcf (Prestressed Concrete Institute 1999) Temperature strain is superimposed on the free strands only, for temperature changes between the time of beam casti ng and the time of detensioning. The thermal coefficient of the pr estressing strands is 6.67x10-6 in/in/oF (Barr, Stanton & Eberhard 2005). The tension pull created in each uncut fr ee strand set due to beam movements is based on the average lengths of all the fr ee strands in each set. These lengths include any debonding lengths.

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29 The effects of top flange pres tressing strands are ignored. The transfer length of a prestressing stra nd is modeled as shown in equation 4-5 (Abrishami & Mitchell 1993). The revers e transfer length is calculated using equation 4-6. The variable fci (ksi) is the compressive strength of the concrete at the time of cutting. The variable D (in) is the diameter of the prestressing strand. The variable fJ (ksi) is the stress in the prestressi ng strand due to the jacking force. The variable fTP is the stress in the prestressing strand due to the tension pull. J ci3 TransferLength=0.33f*D f (4-22) TP ci3 ReverseTransferLength=0.33f*D f (4-23) The prestress force is assumed to linearl y transfer through bond to the concrete over the compression transfer length (Am erican Concrete Institute Committee 318 2002). The tension pull is assumed to linea rly transfer through bond to the concrete over the reverse transfer length. For the purposes of concrete elastic shortening the prestress force from a cut strand is assumed to act at a distance from the end face of the beam equal to 2/3rds of the compre ssion transfer length of the strand. For debonded strands, the prestress force is assu med to act at a distance from the end face of the beam equal to the debonded length plus 2/3rds of the compression transfer length of the strand. Each strand cut is divided into 20 calculat ion steps. These calculation steps allow for beam movements to occur as the strand is weakened during the cutting process. Beam movements are consider ed small compared to the average lengths of the free strands. Dynamic beam motions are ignored. Strand relaxation is ignored. Maximum relaxation for low-relaxation strand is 3.5% when the strand has been loaded to 80% of the tensile strength (Nilson 1987). The prestressing strands and the concrete beams are assumed to be linear elastic throughout the entire detensioning process. The elastic modulus of grade 270 low relaxation strand is taken as 28500ksi (Portl and Cement Association 2002). This assumption is acceptable because the vertical cracks form within the first half of the cutting order, and if the prestressing stra nds do become inelastic, this occurs during the second half of the cutting order. Any inputted debonded length needs to be gr eater than the transfer length of the fully bonded prestressing strands. This is necessary because the model assumes if a

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30 strand is debonded that the tension pull in that strand is transferred to a region of the beam beyond the crack-prone end region. The reverse transfer length is considered th e critical section for the analytical model calculations. This is the point where all of the tension pull has been transferred through bond to the concrete. Camber end movement after each strand cut, is added to the axial end movement. Camber end movement after each strand cut is calculated using the equation; (Axial movement due to strand cut)* (Total cambe r movement after all strand cuts)/(Total axial Movement after all strand cuts) (See Figure 4-5). Analytical Model Input Variables The first input variable for the model consist of the type of beam; BT-72, BT-78, AASHTO 2, AASHTO 3, AASHTO 4, AASHTO 5, AASHTO 6, FUB-48, FUB54, FUB-63, FUB-72, and a custom setting where the user can input a custom beam area. Two or more beams can be chosen for simu ltaneous analysis. The beams can also be different lengths on the same casting bed. Th e number of bottom strands, type of strand, and jacking force per strand must then be specified. The choices for type of strand consist of .500" 270ksi, 9/16" 270ksi, and 600" 270ksi strands. The free strand length between all the beams must be specified, w ith the free strand length for the end beams as the length between the beam face and th e abutment. Each debonded strand and its associated debonded length must then be sp ecified. Temperature change in the free strands from the time of beam casting to the time of strand cutting can also be inputted. Finally the coefficient of static and dyna mic friction between the bottom of the beams and the casting bed must be specified. Analytical Model Flow Chart The solution procedure used in the analytical model shown in Appendix B is outlined in Figure 4-5. Equations are not provided because the cracking criterion fcalc/f solution procedure can not be hand calculated due to the high level of iteration required

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31 for multiple beam casting beds. For a simplified hand calculation procedure of fcalc/f for single symmetrically prestressed beam, see Appendix C. Figure 4-1. Global Movement Fo r 3 Beam Symmetrical System Ls1 >> Ls2 Lb Ls2 Lb Ls2 Lb Ls1>>Ls2 PROPERTIES : Concrete: Steel Friction Ac Ec As Es Fs = sN Fd = dN Beam1 Beam2 Beam3 Free Strands1 Free Strands2 Free Strands3 Free Strands4 End1 End2 End1 End2 End1 End2 L 3 BEAM SYMMETRICALLY PLACED SYSTEM G I V E N : S IM U LTA N E OUS LY ADD A CO MPRE SS I ON F O R C E "F" T O ALL BEAM E N D S

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32 Figure 4-2. Change in Ac ting Static Friction Force Transferred compression From cut strands Tension Pull1 Fs FsUTP For UTP > 0 kip For UTP < Fs Fs |Fs UTP| For UTP > 0 kip For UTP >Fs Tension Pull2 Fs Fs | UTP| For UTP < 0 kip For | UTP | < Fs Fs |Fs | UTP|| For UTP < 0 kip For | UTP| > Fs Transferred compression From cut strands Transferred compression From cut strands Transferred compression From cut strands Tension Pull1 Tension Pull1 Tension Pull1 Tension Pull2 Tension Pull2 Tension Pull2

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33 Figure 4-3. Global Motion of Beam Transferred compression From cut strands Fs For UTP = 2*Fs Im p endin g g lobal motion Tension Pull1 Fs No Sliding Fd For UTP > 2*Fs global motion occurs Fd Sliding Fs For | UTP| = 2*Fs Im p endin g g lobal motion Fs No Sliding Fd For | UTP| > 2*Fs global motion occurs Fd Sliding For UTP > 0 kip For UTP < 0 kip Tension Pull2 Tension Pull1 Tension Pull1 Tension Pull1 Tension Pull2 Tension Pull2 Tension Pull2 Transferred compression From cut strands Transferred compression From cut strands Transferred compression From cut strands

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34 Figure 4-4. Stress in Concrete Bottom Flange. This figure shows the combination of stresses a prestressed beam experien ces during the detensioning process. N = W/2 N = W/2 F s Fs Beam Bottom Flange Free Strands Free Strands Lrt Friction : Normal Force : Transferred Prestress : LEGEND W = Beam weight N = Bearing force e = Distance from centroid of bottom flange to bottom of beam Fs = Static fri ction force Lrt = Reverse transfer length (Eq 4-6) Abf = Area of bottom flange Ibf = Moment of inertia of bottom flange TP = Tension pull CRTL = Tranferred prestress force at a distance from th e e n d fa ce equ al t o RTL *N *N*e *N Axial Stresses e Flexural Stresses N N Lrt*N e Flexural Stresses CRTL Force in cut strands transferred over Lrt Axial Stresses Tension Pull : TP Force in uncut strands transferred over Lr t Axial Stresses

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35 Figure 4-5. Axial and Camber Movement. Th is figure shows the total axial movement and the total camber movement for a beam after all the strands have been cut. Camber Camber movement CL Axial movement

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36 Figure 4-6. Analytical Model Fl ow Chart. By inputting the specifications for the number of beams, beam length, number of bo ttom strands, the type of strand, the jacking force, the free strand length between all the beams, the debonded strand lengths, temperature change at casting and cutting times, and the coefficient of static and dynamic fric tion between the bottom of beam and casting bed, the cracking tendency for speci fied conditions can be calculated. START: User inputs input variables required for analysis Tension pull due to temperature for each strand cut is calculated Transfer length of the prestressing strand is calculated "TS" is calculated Modulus of elasticity of the concrete is calculated The average spring stiffness of each beam for each strand cut is calculated The static and dynamic friction forces "Fs" and "Fd" for each beam are calculated The average spring stiffness for each free strand set for each strand cut is calculated The effective free strand spring stiffness for each beam for each strand cut is calculated The total beam axial shortening for each beam for each strand cut is calculated A strand is cut The movement at each beam end is calculated The tension pull "TP for each free strand set is calculated The length change of each free strand set is calculated The unbalanced tension pull UTP" for each beam is calculated The acting static friction force "AFF" at each beam end is calculated A small amount of global motion is applied to the beam(s) with a UTP > 2*Fs Iteration occurs until UTP < 2*Fd for all beams The reverse transfer length "RTL" at each beam end is calculated fcalc/f at each beam end is calculated Calculations loop for each strand cu t END: The maximum fcalc/f for each beam end is calculated Analytical Model's Flow Chart The step by step calculations used by the MathCad 12 computer model to determine the specifications that minimize the cr acking tendency "CT" of prestre ssed beams, once data input is complete

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37 CHAPTER 5 RESULTS Introduction Tension strain in the end region of a pr estressed beam can be affected by many things as shown in Chapter 4. For this reason it was necessary to determine which variables had the greatest effect on tension st rains in the end region, so that the most efficient solution to vertical cracking could be determined. This was accomplished by performing a sensitivity analys is on the MathCad 12 analytical model shown in Appendix B. Using test beam cases, one input variab le was altered at a time and the resulting change in fcalc/f was noted. Test case 1 is shown in Figure 5-1. The input data for test case 1 is shown in Table 5-1. The fcalc/f results for test case 1 are shown in Figure 5-2. Eleven alterations are made to the test case 1 input data shown in Table 5-1 in order to determine which input changes result in the largest fcalc/f output changes. The alterations are the number of prestressing strands, fricti on coefficient, concrete release strength, beam length, temperature change, number of debonded strands, debonded lengths for ten debonded strands, number of beams, free strand length for two beams, free strand length for three beams, and free strand length for four beams. For all eleven alterations of test case 1, only the fcalc/f output for beam 1 end 1 for the first twenty strands cuts is shown (See Figure 5-1). Test case 2 is shown in Figure 5-25. The input data for test case 2 is shown in Table 5-13. Two alterations are made to the test case 2 input data, the friction coefficient, and the beam spacing. For both alterations, the maximum fcalc/f output for all beam ends is shown (See Figure 5-25).

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38 Test Case 1 Test case 1 is a 72 Florida Bulb-T confi guration (See Figure 5-1). The input data is shown in Table 5-1. The fcalc/f results for the input data shown in Table 5-1 is shown in Figure 5-2. Eleven alterations are made to the input data shown in Table 5-1 in order to determine which input changes result in the largest fcalc/f output changes. The alterations are the number of prestressing stra nds, friction coefficient, concrete release strength, beam length, temperature ch ange, number of debonded strands, debonded lengths for ten debonded strands, number of b eams, free strand length for two beams, free strand length for three beams, and free strand length for four beams. For all eleven alterations, only the fcalc/f output for beam 1 end 1 for th e first twenty strands cuts is shown (See Figure 5-1). Modification 1: Alter the Number of Prestressing Strands The first modification is the total nu mber of prestressing strands. The fcalc/f output is shown for 30, 40, and 50 prestressing strands (See Table 5-2). The maximum fcalc/f value for each number of prestressing strands is shown in bold (See Table 5-2). Figure 5-3 shows the fcalc/f results of Table 5-2 graphica lly. Figure 5-4 shows the maximum fcalc/f results of Table 5-2 graphically. Modification 2: Alter the Friction Coefficient The second modification is the static and dynamic friction coefficients between the casting bed and the bottoms of the prestressed beams. The fcalc/f output is shown for static friction coefficients of 0.15, 0.25, 0.35, and 0.45 (See Table 5-3). The dynamic friction coefficient is assumed to be 0.05 less than the static friction coefficient in all cases. The maximum fcalc/f value for each friction coefficient is shown in bold (See

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39 Table 5-3). Figure 5-5 shows the fcalc/f results of Table 5-3 graphically. Figure 5-6 shows the maximum fcalc/f results of Table 5-3 graphically. Modification 3: Alter the Concrete Release Strength The third modification is the concrete release strength. The fcalc/f output is shown for concrete release strengths of 6ksi, 7ksi, 8ksi, and 9ksi (See Table 5-4). The maximum fcalc/f value for each concrete release strength case is shown in bold (See Table 5-4). Figure 5-7 shows the fcalc/f results of Table 5-4 gra phically. Figure 5-8 shows the maximum fcalc/f results of Table 5-4 graphically. Modification 4: Alter the Beam Lengths The fourth modification is the lengths of the prestressed beams (See Figure 5-1). The fcalc/f output is shown for beam lengths of 100ft, 120ft, 140ft, and 160ft (See Table 55). The maximum fcalc/f value for each beam length cas e is shown in bold (See Table 55). Figure 5-9 shows the fcalc/f results of Table 5-5 graphically. Figure 5-10 shows the maximum fcalc/f results of Table 5-5 graphically. Modification 5: Alter the Temperature Change The fifth modification is the temperatur e change in the free strands. A positive temperature change indicates that the temper ature at the time of detensioning is lower than the temperature at the time of beam cas ting. When this occurs, the free strands attempt to shorten, but are prevented by the beams and the bulkheads. A negative temperature change indicates that the temper ature at the time of detensioning is higher than the temperature at the time of beam casti ng. When this occurs, the free strands relax an amount dependent upon the magnitude of the temperature change. The fcalc/f output is shown for temperature changes of -40oF, -20oF, 0oF, 20oF, and 40oF (See Table 5-6). The maximum fcalc/f value for each temperature change cas e is shown in bold (See Table 5-6).

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40 Figure 5-11 shows the fcalc/f results of Table 5-6 graphi cally. Figure 5-12 shows the maximum fcalc/f results of Table 5-6 graphically. Modification 6: Alter the Number of Debonded Strands The sixth modification is the number of debonded strands. The fcalc/f output is shown for 4, 6, 8, and 10 debonded strands (See Table 5-7). The maximum fcalc/f value for each number of debonded strands is shown in bold (See Table 5-7). Figure 5-13 shows the fcalc/f results of Table 5-7 graphically Figure 5-14 shows the maximum fcalc/f results of Table 5-7 graphically. Modification 7: Alter the Debonded Lengths of 10 Strands The seventh modification is the debon ded length for the case of 10 debonded strands. The fcalc/f output is shown for debonded lengths of 5ft, 10ft, 15ft, and 20ft (See Table 5-8). The maximum fcalc/f value for each debonded length is shown in bold (See Table 5-8). Figure 5-15 shows the fcalc/f results of Table 5-8 graphically. Figure 5-16 shows the maximum fcalc/f results of Table 5-8 graphically. Modification 8: Alter the Number of Beams The eighth modification is the number of th e prestressed beams on the casting bed. The fcalc/f output is shown for 2, 3, and 4 b eams present on the casting bed (See Table 59). The free strand lengths be tween the beams are equal to Ls2 for all cases (See Figure 51). The free strand lengths between th e beams and the bulkheads are equal to Ls1 for all cases (See Figure 5-1). The maximum fcalc/f value for each number of beams case is shown in bold (See Table 5-9). Figure 5-17 shows the fcalc/f results of Table 5-9 graphically. Figure 5-18 shows the maximum fcalc/f results of Table 5-9 graphically.

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42 Figure 5-23 shows the fcalc/f results of Table 5-12 gra phically. Figure 5-24 shows the maximum fcalc/f results of Table 5-12 graphically. Test Case 2 Test case 2 is a 78 Florida Bulb-T confi guration (See Figure 5-25) The input data is shown in Table 5-13. Two alterations are made to the i nput data shown in Table 5-13; the friction coefficient and the free stra nd lengths. For both alterations, the fcalc/f output for all beam ends is shown (See Figure 5-25). Modification 1: Alter the Friction Coefficient The first modification is the static and dynamic friction coefficients between the casting bed and the bottoms of the prestr essed beams (See Figure 5-25). The fcalc/f output is shown for static friction coefficients of 0.01, 0.05, 0.15, 0.25, 0.35, and 0.45 (See Table 5-14). The dynamic friction coeffi cient is assumed to be 0.05 less than the static friction coefficient in cases except the last two cases where the static friction coefficient is equal to 0.05 and 0.01. For thes e cases, the dynamic friction co efficient is 0.001. The maximum fcalc/f value for each friction coefficient case for all beam ends is shown in Table 5-14. Figure 5-26 shows the fcalc/f results of Table 5-14 graphically. Modification 2: Alter the Beam Spacing The second modification is the beam spaci ng (See Figure 5-25). The beam spacing is shown by the lengths of the free strands in Table 5-15. Nine sp acing modifications are listed in Table 5-15 and the maximum fcalc/f results for each case at each beam end are listed in Table 5-16.

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43 Analytical Model Conclusions According to the analytical model in A ppendix B, the following trends have been determined from test case 1. Increasing the number of prestressing strands makes the beam more likely to crack. Increasing the coefficient of friction betw een the casting bed and the bottom of the beam make the beam more likely to crack. Decreasing the concrete release strength makes the beam more likely to crack. Increasing the beam length makes the beam more likely to crack A temperature reduction in the free strands from the time of beam casting to the time of strand detensioning ma kes the beam more likely to crack. A temperature increase in the free strands from the ti me of beam casting to the time of strand detensioning makes the b eam less likely to crack. Decreasing the number of debonded strands makes the beam more likely to crack. Decreasing the debonded length of the de bonded strands makes the beam more likely to crack. Increasing the number of beams on the casting bed makes the beam more likely to crack. Decreasing the free strand length between the bulkhead and the beam makes the beam more likely to crack. This effect is increased as the number of beams on the casting bed increases. The variables that have the greatest effect on the tensi on strains the end region of a prestressed beam experiences are temperature change, friction, concrete release strength, beam length, and number of prestressing stra nds. The free strand lengths and the number of beams on the casting bed have the next grea test effect. The free strand lengths become more important as the number of beams on the casting bed increases. The number of debonded strands and the lengths of the de bonded strands have a small effect on the tension strains in the end region of a prestressed beam.

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44 According to the analytical model in A ppendix B, the following trends have been determined from test case 2. Given a symmetrical beam placement (See Figure 5-25) on the casting bed, the middle beams are most likely to crack when friction is present. As the friction coefficient approaches zero, all the beam s become equally as likely to crack (See Figure 5-26). The beam that is farthest away from th e long free strands (See Table 5-15) is most likely to crack (See Table 5-16). Field Data Results In February 2006 field data was collected at Gate Concrete in Jacksonville Florida. Beam end movements were measured for the three 139 ft long 72" Florida Bulb-T beams on the casting bed (See Figure 527). Measurements of movement were made at both ends of beam 2, the right end of beam 1, and the left end of beam 3, during the strand cutting process (See Figure 5-27). Measuremen ts were taken visually with a millimeter scale from a reference mark after the desired strands were cut. The field data was then compared to the calculated values from the analytical model in Appendix B. The input values for the analyzed beams are listed in Table 5-17. The movements for beam 2 are listed in Table 5-18. The movements are show n graphically in Figur es 5-28 and 5-29. The movements for the right end of beam 1 ar e listed in Table 5-19. The movements are shown graphically in Figure 5-30. The movement s for the left end of beam 3 are listed in Table 5-20. The movements are shown graphi cally in Figure 5-31. The cutting pattern and the locations of the debonded strands can be seen in Appendix D. Field Data Conclusions The first half of the field data results for beam 2 are higher than calculated on the left side of the beam and lower than calculate d on the right side of the beam. There are many possible explanations for this, but most likely the beam experienced global motion

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45 to the right due to non-simultaneous cutting. The existence of globa l motion is supported by data point #42 on the left end of beam 2. The only possible explanation for the end movement of a beam to remain constant or reduce in value when additional prestress force is added to the cross section is that the beam experienced gl obal motion. The total calculated beam shortening for beam 2 (.827 ) agrees with the m easured total beam shortening (.819). Beam 1 and beam 3 data sh ow that global motion is a very significant issue. Data points #10 through #38 for the left end of beam 3 either remain constant or reduce in value from their previous points Data points #32 thr ough #42 on the right end of beam 1 either remain constant or reduce in value from their previous points. The conclusion that can be drawn from the field da ta is that without be ing able to determine which workman will cut their strand the fastest, it is not possible to calculate the actual movements of the beam ends in the field. Figure 5-1. Test Case 1 Lc1 Lc2 Beam1 Beam 2 Ls1 Ls2 Ls3 End1 End1 End 2 End 2

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46 Table 5-1. Test Case 1 Input Data Variable Value Variable Value Type of Beam BT-72 Ls1 40ft Lc1 140ft Ls2 3ft Lc2 140ft Ls3 40ft Number of Strands 40 Strand Type .600 270ksi Jacking Force per Strand 44k Debonded Strands #37 5ft Concrete Release Strength 8ksi #38 5ft Unit Weight of Concrete 150pcf #39 5ft Temperature Change 0 #40 5ft Static Coefficient of Friction 0.45 Camber 2.5in Dynamic Coefficient of Friction 0.40 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f Figure 5-2. Test Ca se 1 No Alterations

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47 Table 5-2. Alter the Number of Prestressing Strands Number of Cut Strands #PS = 30 #PS = 40 #PS = 50 1 0.763 0.763 0.771 2 0.822 0.836 0.849 3 0.875 0.900 0.923 4 0.923 0.958 0.991 5 0.964 1.011 1.054 6 0.999 1.059 1.113 7 1.029 1.101 1.166 8 1.052 1.137 1.215 9 1.069 1.168 1.258 10 1.080 1.194 1.296 11 1.084 1.213 1.329 12 1.082 1.227 1.357 13 1.074 1.235 1.379 14 1.060 1.237 1.395 15 1.038 1.233 1.407 16 1.011 1.223 1.412 17 0.977 1.207 1.412 18 0.936 1.185 1.407 19 0.889 1.157 1.395 20 0.837 1.123 1.378 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f #PS = 30 #PS = 40 #PS = 50 Figure 5-3. Alter the Number of Prestressing Strands

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48 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 3035404550 Number of Prestressing Strandsfcalc / f maximums Figure 5-4. Number of Prestressing Strands fcalc/f Maximums

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49 Table 5-3. Alter the Friction Coefficient Number of Cut Strands = .15 = .25 = .35 = .45 1 0.308 0.461 0.614 0.763 2 0.375 0.529 0.682 0.836 3 0.438 0.592 0.746 0.900 4 0.495 0.649 0.803 0.958 5 0.546 0.701 0.856 1.011 6 0.593 0.748 0.903 1.059 7 0.634 0.789 0.945 1.101 8 0.669 0.825 0.981 1.137 9 0.699 0.855 1.011 1.168 10 0.723 0.880 1.036 1.194 11 0.741 0.898 1.055 1.213 12 0.754 0.911 1.069 1.227 13 0.760 0.919 1.076 1.235 14 0.761 0.920 1.078 1.237 15 0.756 0.915 1.074 1.233 16 0.745 0.905 1.063 1.223 17 0.728 0.889 1.047 1.207 18 0.705 0.865 1.025 1.185 19 0.675 0.836 0.996 1.157 20 0.640 0.801 0.961 1.123 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f = .15 = .25 = .35 = .45 Figure 5-5. Alter the Friction Coefficient

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50 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 0.150.20.250.30.350.40.45 Friction Coefficientfcalc / f maximums Figure 5-6. Friction Coefficient fcalc/f Maximums

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51 Table 5-4. Alter the Concrete Release Strength Number of Cut Strands fci = 6ksi fci = 7ksi fci = 8ksi fci = 9ksi 1 0.891 0.822 0.763 0.722 2 0.981 0.900 0.836 0.783 3 1.066 0.973 0.900 0.840 4 1.143 1.039 0.958 0.892 5 1.214 1.100 1.011 0.939 6 1.279 1.155 1.059 0.981 7 1.337 1.204 1.101 1.018 8 1.388 1.246 1.137 1.050 9 1.431 1.283 1.168 1.077 10 1.468 1.313 1.194 1.099 11 1.498 1.337 1.213 1.115 12 1.521 1.354 1.227 1.126 13 1.536 1.365 1.235 1.132 14 1.544 1.370 1.237 1.132 15 1.545 1.368 1.233 1.128 16 1.538 1.359 1.223 1.117 17 1.524 1.343 1.207 1.101 18 1.501 1.321 1.185 1.080 19 1.471 1.292 1.157 1.053 20 1.434 1.256 1.123 1.020 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f fci = 6ksi fci = 7ksi fci = 8ksi fci = 9ksi Figure 5-7. Alter the Concrete Release Strength

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52 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 66.577.588.59 Concrete Release Strength (ksi)fcalc / f maximums Figure 5-8. Concrete Release Strength fcalc/f Maximums Table 5-5. Alter the Beam Lengths Number of Cut Strands L = 100ft L = 120 ft L = 140ft L = 160ft 1 0.558 0.663 0.763 0.870 2 0.610 0.723 0.836 0.949 3 0.656 0.778 0.900 1.022 4 0.698 0.827 0.958 1.090 5 0.736 0.872 1.011 1.152 6 0.769 0.912 1.059 1.208 7 0.797 0.947 1.101 1.258 8 0.821 0.976 1.137 1.302 9 0.840 1.001 1.168 1.341 10 0.854 1.020 1.194 1.373 11 0.863 1.034 1.213 1.399 12 0.868 1.043 1.227 1.419 13 0.867 1.046 1.235 1.432 14 0.862 1.044 1.237 1.439 15 0.852 1.037 1.233 1.439 16 0.837 1.024 1.223 1.433 17 0.817 1.006 1.207 1.421 18 0.792 0.982 1.185 1.401 19 0.761 0.952 1.157 1.375 20 0.726 0.917 1.123 1.342

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53 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f L = 100ft L = 120 ft L = 140ft L = 160ft Figure 5-9. Alter the Beam Lengths 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 100110120130140150160 Beam Lengths (ft)fcalc / f maximums Figure 5-10. Beam Lengths fcalc/f Maximums

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54 Table 5-6. Alter the Temperature Change Number of Cut Strands F = -40 F = -20 F = 0 F = 20 F = 40 1 0.179 0.473 0.763 1.061 1.355 2 0.271 0.554 0.836 1.119 1.401 3 0.357 0.629 0.900 1.171 1.442 4 0.439 0.698 0.958 1.218 1.477 5 0.515 0.763 1.011 1.259 1.507 6 0.585 0.822 1.059 1.295 1.532 7 0.650 0.875 1.101 1.326 1.551 8 0.710 0.923 1.137 1.351 1.565 9 0.764 0.966 1.168 1.371 1.573 10 0.812 1.003 1.194 1.384 1.575 11 0.854 1.034 1.213 1.393 1.572 12 0.891 1.059 1.227 1.395 1.563 13 0.922 1.079 1.235 1.391 1.548 14 0.947 1.092 1.237 1.382 1.527 15 0.966 1.100 1.233 1.367 1.5 16 0.979 1.101 1.223 1.345 1.467 17 0.986 1.097 1.207 1.318 1.429 18 0.987 1.086 1.185 1.285 1.384 19 0.982 1.070 1.157 1.245 1.333 20 0.971 1.047 1.123 1.199 1.275 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f F = -40 F = -20 F = 0 F = 20 F = 40 Figure 5-11. Alter the Temperature Change

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55 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 -40-30-20-10010203040 Temperature Change (deg F)fcalc / f maximums Figure 5-12. Temperature Change fcalc/f Maximums Table 5-7. Alter the Numb er of Debonded Strands Number of Cut Strands #DS = 4 #DS = 6 #DS = 8 #DS = 10 1 0.763 0.766 0.764 0.763 2 0.836 0.831 0.826 0.822 3 0.900 0.891 0.883 0.875 4 0.958 0.946 0.935 0.923 5 1.011 0.996 0.981 0.966 6 1.059 1.040 1.021 1.003 7 1.101 1.078 1.057 1.035 8 1.137 1.111 1.086 1.062 9 1.168 1.139 1.110 1.082 10 1.194 1.161 1.129 1.098 11 1.213 1.177 1.142 1.107 12 1.227 1.187 1.149 1.111 13 1.235 1.192 1.15 1.110 14 1.237 1.191 1.146 1.102 15 1.233 1.184 1.136 1.089 16 1.223 1.171 1.120 1.070 17 1.207 1.152 1.098 1.045 18 1.185 1.127 1.070 1.015 19 1.157 1.095 1.036 0.979 20 1.123 1.058 0.996 0.937

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56 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f #DS = 4 #DS = 6 #DS = 8 #DS = 10 Figure 5-13. Alter the Nu mber of Debonded Strands 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 45678910 Number of Debonded Strandsfcalc / f maximums Figure 5-14. Number of Debonded Strands fcalc/f Maximums

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57 Table 5-8. Alter the Debonde d Lengths of 10 Strands Number of Cut Strands 5 ft 10 ft 15 ft 20 ft 1 0.763 0.762 0.761 0.760 2 0.822 0.818 0.815 0.811 3 0.875 0.869 0.863 0.858 4 0.923 0.914 0.907 0.899 5 0.966 0.955 0.945 0.936 6 1.003 0.990 0.978 0.967 7 1.035 1.020 1.006 0.994 8 1.062 1.044 1.029 1.015 9 1.082 1.063 1.046 1.031 10 1.098 1.077 1.059 1.042 11 1.107 1.085 1.066 1.049 12 1.111 1.088 1.068 1.050 13 1.110 1.086 1.065 1.046 14 1.102 1.078 1.057 1.038 15 1.089 1.065 1.043 1.025 16 1.070 1.046 1.025 1.007 17 1.045 1.022 1.002 0.985 18 1.015 0.993 0.974 0.958 19 0.979 0.959 0.942 0.927 20 0.937 0.919 0.905 0.893 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f 5 ft 10 ft 15 ft 20 ft Figure 5-15. Alter the Debonded Lengths of 10 Debonded Strands

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58 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 5791113151719 Debonded Length for 10 Strandsfcalc / f maximums Figure 5-16. Debonded Lengths fcalc/f Maximums Table 5-9. Alter the Number of Beams Number of Cut Strands #B = 2 #B = 3 #B = 4 1 0.763 0.769 0.769 2 0.836 0.841 0.841 3 0.900 0.908 0.914 4 0.958 0.969 1.002 5 1.011 1.037 1.081 6 1.059 1.098 1.152 7 1.101 1.153 1.218 8 1.137 1.201 1.275 9 1.168 1.243 1.329 10 1.194 1.277 1.372 11 1.213 1.305 1.407 12 1.227 1.326 1.431 13 1.235 1.340 1.450 14 1.237 1.341 1.458 15 1.233 1.340 1.459 16 1.223 1.332 1.449 17 1.207 1.314 1.426 18 1.185 1.289 1.401 19 1.157 1.259 1.358 20 1.123 1.218 1.312

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59 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f #B = 2 #B = 3 #B = 4 Figure 5-17. Alter the Number of Beams 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 234 Number of Beamsfcalc / f maximums Figure 5-18. Number of Beams fcalc/f Maximums

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60 Table 5-10. Alter the Free Strand Length for 2 Beams Number of Cut Strands 40ft 50ft 60ft 70ft 1 0.763 0.762 0.759 0.756 2 0.836 0.819 0.808 0.799 3 0.900 0.872 0.853 0.838 4 0.958 0.921 0.894 0.874 5 1.011 0.965 0.932 0.907 6 1.059 1.004 0.965 0.937 7 1.101 1.039 0.995 0.962 8 1.137 1.069 1.021 0.985 9 1.168 1.095 1.042 1.004 10 1.194 1.115 1.060 1.019 11 1.213 1.131 1.074 1.031 12 1.227 1.143 1.083 1.039 13 1.235 1.149 1.088 1.043 14 1.237 1.151 1.090 1.044 15 1.233 1.147 1.087 1.042 16 1.223 1.139 1.079 1.035 17 1.207 1.126 1.068 1.025 18 1.185 1.107 1.052 1.012 19 1.157 1.084 1.032 0.994 20 1.123 1.055 1.008 0.973 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f 40ft 50ft 60ft 70ft Figure 5-19. Alter the Free Strand Length for 2 Beams

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61 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 25303540455055606570 Free Strand Length (ft)fcalc / f maximums Figure 5-20. Free Strand Length for 2 Beams fcalc/f Maximums

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62 Table 5-11. Alter the Free Strand Length for 3 Beams Number of Cut Strands 40ft 50ft 60ft 70ft 1 0.769 0.763 0.759 0.756 2 0.841 0.823 0.810 0.801 3 0.908 0.878 0.857 0.841 4 0.969 0.934 0.905 0.884 5 1.037 0.991 0.956 0.929 6 1.098 1.037 0.995 0.964 7 1.153 1.084 1.036 1.001 8 1.201 1.125 1.072 1.028 9 1.243 1.160 1.098 1.056 10 1.277 1.185 1.124 1.075 11 1.305 1.208 1.140 1.094 12 1.326 1.227 1.157 1.104 13 1.340 1.239 1.168 1.110 14 1.341 1.240 1.169 1.116 15 1.340 1.240 1.169 1.113 16 1.332 1.233 1.160 1.109 17 1.314 1.217 1.150 1.097 18 1.289 1.198 1.130 1.080 19 1.259 1.168 1.106 1.062 20 1.218 1.136 1.079 1.036 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f 40ft 50ft 60ft 70ft Figure 5-21. Alter the Free Strand Length for 3 Beams

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63 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 25303540455055606570 Free Strand Length (ft)fcalc / f maximums Figure 5-22. Free Strand Length for 3 Beams fcalc/f Maximums

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64 Table 5-12. Alter the Free Strand Length for 4 Beams Number of Cut Strands 40ft 50ft 60ft 70ft 1 0.769 0.763 0.759 0.756 2 0.841 0.823 0.810 0.801 3 0.914 0.884 0.863 0.853 4 1.002 0.958 0.929 0.907 5 1.081 1.026 0.984 0.957 6 1.152 1.088 1.039 1.002 7 1.218 1.143 1.084 1.043 8 1.275 1.192 1.128 1.078 9 1.329 1.235 1.161 1.109 10 1.372 1.265 1.195 1.140 11 1.407 1.295 1.217 1.160 12 1.431 1.317 1.238 1.176 13 1.450 1.332 1.248 1.186 14 1.458 1.341 1.252 1.191 15 1.459 1.337 1.255 1.191 16 1.449 1.331 1.247 1.185 17 1.426 1.313 1.233 1.173 18 1.401 1.291 1.212 1.156 19 1.358 1.258 1.184 1.129 20 1.312 1.218 1.150 1.100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 05101520 Number of Cut Strandsfcalc / f 40ft 50ft 60ft 70ft Figure 5-23. Alter the Free Strand Length for 4 Beams

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65 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 25303540455055606570 Free Strand Length (ft)fcalc / f maximums Figure 5-24. Free Strand Length for 4 Beams fcalc/f Maximums Figure 5-25. Test Case 2 Table 5-13. Test Case 2 Input Data Variable Value Variable Value Type of Beam BT-78 Ls1 = Ls5 60ft Lc1 = Lc2 = Lc3 = Lc4 150ft Ls2 = Ls3 = Ls4 3ft Number of Strands 49 Strand Type .600 270ksi Jacking Force per Strand 44k Debonded Strands #46 5ft Concrete Release Strength 8ksi #47 5ft Unit Weight of Concrete 150pcf #48 5ft Temperature Change 0 #49 5ft Static Coefficient of Friction 0.45 Camber 3in Dynamic Coefficient of Friction 0.40 Lc1 Lc2 Ls1 Ls2 Ls3 E1 E3 E2 E4 E5 E6 E7 E8 Ls4 Ls5 Lc3 Lc4

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66 Table 5-14. Alter Friction Results Friction Coefficient E1 E2 E3 E4 E5 E6 E7 E8 = .45 1.256 0.702 1.643 1.626 1.626 1.643 0.702 1.256 = .35 1.097 0.462 1.391 1.372 1.372 1.391 0.462 1.097 = .25 0.938 0.499 1.147 1.128 1.128 1.147 0.499 0.938 = .15 0.780 0.606 0.904 0.892 0.892 0.904 0.606 0.780 = .05 0.621 0.526 0.653 0.633 0.633 0.653 0.526 0.621 = .01 0.555 0.546 0.557 0.558 0.558 0.557 0.546 0.555 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 00.10.20.30.4 Friction Coefficientfcalc / f E1 = E8 E2 = E7 E3 = E6 E4 = E5 Figure 5-26. Alter the Friction Coefficient for Multiple Beam Ends Table 5-15. Free Strand Lengths Modification Ls1 (ft) Ls2 (ft) Ls3 (ft) Ls4 (ft) Ls5 (ft) Total Length (ft) #1 3 60 3 3 60 129 #2 3 3 60 3 60 129 #3 3 3 3 60 60 129 #4 3 3 60 60 3 129 #5 25.8 25.8 25.8 25.8 25.8 129 #6 16.125 32.25 32.25 32.25 16.125 129 #7 3 3 117 3 3 129 #8 3 117 3 3 3 129 #9 117 3 3 3 3 129

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67 Table 5-16. Beam Spacing Results Modification E1 E2 E3 E4 E5 E6 E7 E8 1 1.418 1.415 1.420 0.850 1.619 1.619 0.726 1.232 2 1.875 1.875 0.749 1.486 1.481 1.484 0.771 1.120 3 2.072 2.072 1.303 1.714 0.702 1.332 0.886 1.131 4 1.717 1.717 0.738 1.326 1.325 1.326 1.322 1.325 5 1.388 1.383 1.389 1.388 1.389 1.389 1.383 1.388 6 1.392 1.389 1.393 1.393 1.393 1.393 1.389 1.392 7 1.601 1.628 0.727 1.255 1.255 0.727 1.628 1.601 8 1.246 1.244 1.247 0.702 1.638 1.303 2.011 1.997 9 1.144 0.677 1.530 1.126 1.916 1.793 2.333 2.315 Figure 5-27. 72 Florida Bulb-T Arrangement Table 5-17. 72 Florida Bulb-T Input Data Variable Value Variable Value Type of Beam BT-72 Ls1 58 5 Lc1 139 23/8Ls2 2 10 Lc2 139 23/8Ls3 2 10 Lc3 139 23/8Ls4 88 3 Number of Strands 42 Strand Type .600 270ksi Jacking Force per Strand 44k Debonded Strands 4 x 5 Concrete Release Strength 7360psi 4 x 10 Unit Weight of Concrete 150pcf 2 x 15 Temperature Change NA Estimated d 0.25 Camber B1 = 3 Estimated s 0.30 Camber B2 = 25/8 Camber B3 = 31/4 Beam 1 Beam 2Beam 3 Lc1 Lc2 Lc3 Ls2 Ls1 Ls3 Ls4

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68 Table 5-18. End Movements for Beam 2 Bottom Strand Left End Measured Left End CalculatedDifference Bottom Strand Right End Measured Right End Calculated Difference 4 0 0.021 -0.021 6 0 0.050 -0.050 8 0.039 0.034 0.005 10 0.079 0.098 -0.019 12 0.079 0.050 0.029 14 0.079 0.150 -0.071 16 0.079 0.071 0.008 18 0.118 0.204 -0.086 20 0.118 0.099 0.019 22 0.157 0.257 -0.100 24 0.157" 0.134 0.023 26 0.236" 0.308 -0.072 28 0.276" 0.181 0.095 30 0.354" 0.357 -0.003 32 0.394" 0.230 0.164 34 0.354" 0.403 -0.049 36 0.472" 0.275 0.197 38 0.472" 0.449 0.023 40 0.512" 0.298 0.214 42 0.551" 0.507 0.044 42 0.276" 0.312 -0.036 Total Shortening Calculated Shortening Difference 0.827 0.819 0.008 0 0.1 0.2 0.3 0.4 0.5 0.6 6162636 Number of Cut StrandsMovement (in) Measured Calculated Figure 5-28. Beam 2 Left E nd Measured vs Calculated

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69 0 0.1 0.2 0.3 0.4 0.5 0.6 6162636 Number of Cut StrandsMovement (in) Measured Predicted Figure 5-29. Beam 2 Right E nd Measured vs Calculated Table 5-19. End Movements for Right End of Beam 1 Bottom Strand Right End Measured Right End CalculatedDifference 4 0 0.000 0.000 8 0.039" 0.000 0.039 12 0.039" 0.000 0.039 16 0.079" 0.000 0.079 20 0.079" 0.000 0.079 24 0.079" 0.000 0.079 28 0.197" 0.000 0.197 32 0.394" 0.000 0.394 36 0.079" 0.000 0.079 40 0.079" 0.036 0.043 42 0.079" 0.052 0.027

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70 0 0.1 0.2 0.3 0.4 0.5 0.6 4142434 Number of Cut StrandsMovement (in) Measured Calculated Figure 5-30. Beam 1 Right E nd Measured vs Calculated Table 5-20. End Movements for Left End of Beam 3 Bottom Strand Left End Measured Left End CalculatedDifference 6 0.000" -0.026 0.026 10 0.039" -0.058 0.097 14 0.039" -0.092 0.131 18 0.039" -0.121 0.160 22 0.000" -0.141 0.141 26 -0.039" -0.151 0.112 30 -0.039" -0.151 0.112 34 -0.039" -0.151 0.112 38 -0.118" -0.151 0.033 42 -0.039" -0.142 0.103

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71 -0.2 -0.1 0 0.1 0.2 0.3 0.4 6162636 Number of Cut StrandsMovement (in) Measured Calculated Figure 5-31. Beam 3 Left E nd Measured vs Calculated

PAGE 84

72 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS The occurrence of vertical cracking can be affected by many variables; length of the free strands, modulus of elasticity of th e concrete, friction coefficient between the beam and the casting bed, temperature change, debonding lengths, number of debonded strands, number of prestressing strands, jacki ng force per strand, tension strength of the concrete, cross-sectional area of the beam, beam length, and beam spacing configuration. Because there are so many different variables that influence the formation of vertical cracks, it was necessary to determine which va riables had the greatest effect on vertical crack formation so that the best possible so lution could be determined. The MathCad 12 analytical model in Appendix B was created to allow the researchers to determine the best vertical crack solution fo r a given casting bed of beams. This analytical model was not created to predict the exact stresses in the concrete beams and the steel strands because that is not possible due to non-si multaneous cutting, dynamic effects, and the disturbed region properties of a prestressed beam end. For th is reason, no hard and fast rule can be created to eliminate vertical cracking in prestressed beams. However, by performing a sensitivity analys is on the analytical model (See Appendix B), trends were developed and the variables that are most likely to cause vertical cracking were determined. The analytical model determined that the va riables that have the greatest effect on vertical cracking are temperat ure change between the time of beam casting and the time of strand detensioning, friction coefficient between the casting bed and the bottom of the

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73 beams, concrete release strength, beam length, and number of prestr essing strands. The free strand lengths and the number of beams on the casting bed have the next greatest effect. The free strand lengths become more important as the number of beams on the casting bed increases. The number of debonde d strands and the lengths of the debonded strands have a small effect on vertical cracking. The trends that we re developed with the analytical model in Appendix B are listed below. Increasing the number of prestressing stra nds increases the likelihood of vertical cracking. Increasing the coefficient of friction betw een the casting bed and the bottom of the beam increases the likeli hood of vertical cracking. Decreasing the concrete rele ase strength increases the likelihood of vertical cracking. Increasing the beam length increases th e likelihood of vertical cracking. A temperature reduction in the free strands from the time of beam casting to the time of strand detensioning increases the likelihood of vertic al cracking. A temperature increase in the free strands from the time of beam casting to the time of strand detensioning decreases the likelihood of vert ical cracking. Decreasing the number of debonded strands increases the like lihood of vertical cracking. Decreasing the debonded length of the debonde d strands increase s the likelihood of vertical cracking. Increasing the number of beams on the cas ting bed increases the likelihood of vertical cracking. Decreasing the free strand length between the bulkhead and the beam increases the likelihood of vertical cracking. This effect is increased as the number of beams on the casting bed increases. The conclusion that can be drawn from this research study is that the three most important things to do in or der to reduce the occurrence of vertical cracks are to detension the prestressing strands when the te mperature of the free strands is similar or

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74 warmer than the temperature of the free stra nds when the beams were cast, to lower the coefficient of friction between the casting be d and the bottom of the beams, and to add additional space between the beams. Lowering the coefficient of friction between the casting bed and the bottom of the beam ends can be accomplished by smoothing the casting bed before each new pour, adding l ubricants under the beam ends, and by installing steel bearing plates at the beam e nds (See Figure 2-1). If the coefficient of friction is low, the additional beam spacing can be added between the bulkheads and the beams. If the coefficient of friction is high, the additional beam spacing must be distributed between all of th e beams to be effective.

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75 APPENDIX A SAMPLE RETURNED SURVEY FORMS

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APPENDIX B VERTICAL CRACK PREDICTOR

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ORIGIN1 Vertical Crack Predictor University of Florida 2006 INPUT :1) Choice : 1= 72" Florida Bulb T 2= 78" Florida Bulb T 3= AASHTO Type 6 4= AASHTO Type 5 5= AASHTO Type 4 6= AASHTO Type 3 7= AASHTO Type 2 8= 48" Florida U Beam 9= 54" Florida U Beam 10= 63" Florida U Beam 11= 72" Florida U Beam 12= Custom Choice2 If Choice = 12 then Specify the Cross Sectional Area of the Beam (Abeam) and the Area of the Bottom Flange (Abottomflange), the distance from the bottom of the beam to the centroid of the cross section "EcSpef", the distance from the centroid of the bottom flange to the bottom of the beam "FriceSpef", and the Moment of Inertia of the bottom flange "IBottomSpef": Abeam225in2 Abottomflange225in2 EcSpef28.5in FriceSpef7.5in IBottomSpef9047in4 2) Length of beams : Lbeams150155157152 ()ft 3) Expected Initial Camber InCamber33.13.23 ()in 4) Number of Bottom Flange Prestressing Strands : NumberStrands49 5) Average Concrete Release Strength at Time of Detensioning (Determined from Cylinder Breaks NOT SPECIFIED VALUE) : ReleaseStrength8000psi 6) Concrete unit weight : w c 150 lbf ft3

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7) Bottom Flange Strand Jacking Force per strand: Fstrand44kip 8) Distance Between Beams and Between Beams and the Bulkheads: FreeStrand50 33345 ()ft 9) Type of prestressing strand : 1 = .500in Grade 270 2 = .500in Special Grade 270 3 = 9/16in Grade 270 4 = .600in Grade 270 StraChoice4 10) Specify location of debonded strands (in the cutting order) and associated debonded length (ft) DO NOT INCLUDE UNITS. DEBONDED STRANDS SHOULD BE AT THE END OF THE CUTTING ORDER. DO NOT INCLUDE TOP STRANDS IN THE STRAND NUMBER COUNT. STRAND NUMBER Debond 42 5 43 5 44 5 45 5 46 10 47 10 48 15 49 15 DEBONDED LENGTH 11) Specify Temperature Change (F) from Time of Beam Casting to Strand Detensioning. Positive = TEMP AT STRAND DETENSIONING IS COLDER THAN TEMP AT BEAM CASTING. Negative = TEMP AT STRAND DETENSIONING IS WARMER THAN TEMP AT BEAM CASTING. DO NOT INCLUDE THE UNITS TempChange30 12) Static Friction Coefficient Between Bottom of Prestressed Beam Ends and the Casting Bed ( is often between .3 and .45) : .3 D .25 13) Dynamic Friction Coefficient Between Bottom of Prestressed Beam Ends and the Casting Bed ( D is always less than ) :

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Calculations :Beam Cross Sectional Area NumbCalcs20 FullLength2 A 875in2Choice1 = if 1105in2Choice2 = if 1125in2Choice3 = if 1053in2Choice4 = if 789in2Choice5 = if 559.5in2Choice6 = if 369in2Choice7 = if 1146in2Choice8 = if 1212in2Choice9 = if 1311in2Choice10 = if 1410in2Choice11 = if AbeamChoice12 = if "error"otherwise Eccent33.95inChoice1 = if 40.39inChoice2 = if 36.38inChoice3 = if 31.96inChoice4 = if 24.73inChoice5 = if 20.27inChoice6 = if 16.38inChoice7 = if 19.67inChoice8 = if 22.23inChoice9 = if 27.04inChoice10 = if 30.16inChoice11 = if EcSpefChoice12 = if "error"otherwise Prestressing Strand Area Aps0.153in2StraChoice1 = if 0.167in2StraChoice2 = if 0.192in2StraChoice3 = if 0.2192in2StraChoice4 = if "error"otherwise Aps0.2192in2 Modulus of elasticity of prestressing strands (can't be changed) Eps28500ksi Weight of the beam (kip/ft) wtw c A wt1.151042 kip ft A1105in2

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Eccentricities and Moments of Inertia of the bottom flange: Frice5.573inChoice1 = if 7.509inChoice2 = if 7.597inChoice3 = if 7.597inChoice4 = if 7.266inChoice5 = if 6.233inChoice6 = if 5.2inChoice7 = if 5inChoice8 = if 5inChoice9 = if 5inChoice10 = if 5inChoice11 = if FriceSpefChoice12 = if "error"otherwise IBottom3697in4Choice1 = if 8766in4Choice2 = if 9047in4Choice3 = if 9047in4Choice4 = if 7280in4Choice5 = if 3873in4Choice6 = if 1829in4Choice7 = if 4667in4Choice8 = if 4667in4Choice9 = if 4667in4Choice10 = if 4667in4Choice11 = if IBottomSpefChoice12 = if "error"otherwise Frice7.509in IBottom8766in4

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Area of Bottom FlangeDirect Tension Strength of Concrete "TS" Abf325in2Choice1 = if 399in2Choice2 = if 404in2Choice3 = if 404in2Choice4 = if 361in2Choice5 = if 262.75in2Choice6 = if 180in2Choice7 = if 560in2Choice8 = if 560in2Choice9 = if 560in2Choice10 = if 560in2Choice11 = if AbottomflangeChoice12 = if ConcTensStrength5 ReleaseStrength psi psi ConcTensStrength447.213595psi Abf399in2 TensionAreaAbf TensionArea399in2 ConcAllowableTensionTensionAreaConcTensStrength ConcAllowableTension178.438225kip Diameter of prestressing strand D.5inStraChoice1 = if .5inStraChoice2 = if .5625inStraChoice3 = if .6inStraChoice4 = if "error"otherwise D0.6in

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Assembles debonded lengths into a matrix DebondLength outg0 g1NumberStrands for outDebond1h Debond2h h1colsDebond () for outoutft out DebondLength 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0ft DebondLength 1 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 10 10 15 15ft Initial conditions in the prestressing strands TempStrain.00000667TempChange TempStrain0.0002 TempStressTempStrainEps TempStress5.70285 ksi OrigStressStrand Fstrand Aps OrigStressStrand200.729927ksi OrigForceStrandOrigStressStrandAps NumberStrands OrigForceStrand2156kip StressStrandOrigStressStrandTempStress StressStrand195.027077ksi InitialStrainStrands StressStrand Eps InitialStrainStrands0.00684

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Adds debonding lengths to free strand lengths AllStrandLengths outp1 FreeStrand11 DebondLengthp outp2 FreeStrand12 DebondLengthp p1NumberStrands forcolsFreeStrand ()2 = if outij FreeStrand1j 2DebondLengthi i1NumberStrands for j1colsFreeStrand () for outk1 outk1 DebondLengthk outkcolsLbeams ()1 outkcolsLbeams ()1 DebondLengthk k1NumberStrands for colsFreeStrand ()2 if out Adds up all the strand lengths so that the averages for each strand cut can be determined NSNumberStrands1 NS48 TotStrandLengths outNumberStrandsw AllStrandLengthsNumberStrandsw outzw outz1 w AllStrandLengthszw zNS1 for w1colsAllStrandLengths () for out

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AllStrandLengths 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345ft AllStrandLengths 12345 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5033345 5513131350 5513131350 5513131350 5513131350 6023232355 6023232355 6533333360 6533333360ft

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TotStrandLengths 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 25202872872872275 24702842842842230 24202812812812185 23702782782782140 23202752752752095 22702722722722050 22202692692692005 21702662662661960 21202632632631915 20702602602601870 20202572572571825 19702542542541780 19202512512511735 18702482482481690 18202452452451645 17702422422421600 17202392392391555 16702362362361510 16202332332331465 15702302302301420 15202272272271375 14702242242241330 14202212212211285 13702182182181240 13202152152151195 12702122122121150 12202092092091105 11702062062061060 11202032032031015ft TotStrandLengths 12345 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 13702182182181240 13202152152151195 12702122122121150 12202092092091105 11702062062061060 11202032032031015 1070200200200970 1020197197197925 970194194194880 920191191191835 870188188188790 820185185185745 770182182182700 720179179179655 670176176176610 620173173173565 570170170170520 520167167167475 470164164164430 415151151151380 360138138138330 305125125125280 250112112112230 190898989175 130666666120 6533333360ft

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Average Strand lengths in each free strand set after each strand is cut Concrete Modulus of Elasticity AvgStrandLengths inter1wTotStrandLengthszw inter2wIndexSz outzw inter1winter2w z1NumberStrands for w1colsAllStrandLengths () for outNumberStrands1 ww 0ft ww1colsAllStrandLengths () for out E40000 ReleaseStrength psi 106 w c 145 lbf ft3 1.5 psi E4816.516412ksi

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AvgStrandLengths 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 51.4295.8575.8575.85746.429 51.4585.9175.9175.91746.458 51.4895.9795.9795.97946.489 51.5226.0436.0436.04346.522 51.5566.1116.1116.11146.556 51.5916.1826.1826.18246.591 51.6286.2566.2566.25646.628 51.6676.3336.3336.33346.667 51.7076.4156.4156.41546.707 51.756.56.56.546.75 51.7956.596.596.5946.795 51.8426.6846.6846.68446.842 51.8926.7846.7846.78446.892 51.9446.8896.8896.88946.944 5277747 52.0597.1187.1187.11847.059 52.1217.2427.2427.24247.121 52.1887.3757.3757.37547.188 52.2587.5167.5167.51647.258 52.3337.6677.6677.66747.333 52.4147.8287.8287.82847.414 52.588847.5 52.5938.1858.1858.18547.593 52.6928.3858.3858.38547.692 52.88.68.68.647.8 52.9178.8338.8338.83347.917 53.0439.0879.0879.08748.043 53.1829.3649.3649.36448.182ft

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AvgStrandLengths 12345 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 52.588847.5 52.5938.1858.1858.18547.593 52.6928.3858.3858.38547.692 52.88.68.68.647.8 52.9178.8338.8338.83347.917 53.0439.0879.0879.08748.043 53.1829.3649.3649.36448.182 53.3339.6679.6679.66748.333 53.510101048.5 53.68410.36810.36810.36848.684 53.88910.77810.77810.77848.889 54.11811.23511.23511.23549.118 54.37511.7511.7511.7549.375 54.66712.33312.33312.33349.667 5513131350 55.38513.76913.76913.76950.385 55.83314.66714.66714.66750.833 56.36415.72715.72715.72751.364 5717171752 57.77818.55618.55618.55652.778 58.7520.520.520.553.75 59.28621.57121.57121.57154.286 6023232355 6125252556 62.528282857.5 63.33329.66729.66729.66758.333 6533333360 6533333360 00000ft

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Creates matrix with jacking force for each strand PrestressTransfer outgFstrand g1NumberStrands for out Adds up total prestress transferred to the beam TotPrestressTransfer outqFstrandq q1NumberStrands for out Adds up total prestress transferred to end of beam only prestress in debonded strands is not included TotPrestressTransferEnd outqTotPrestressTransferq q14 for outroutr1 Fstrand DebondLengthr0ft = if outroutr1 DebondLengthr0ft if r5NumberStrands for out

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PrestressTransfer 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44kip PrestressTransfer 1 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44kip

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TotPrestressTransfer 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 44 88 132 176 220 264 308 352 396 440 484 528 572 616 660 704 748 792 836 880 924 968 1012 1056 1100 1144 1188 1232 1276kip TotPrestressTransfer 1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1056 1100 1144 1188 1232 1276 1320 1364 1408 1452 1496 1540 1584 1628 1672 1716 1760 1804 1848 1892 1936 1980 2024 2068 2112 2156kip

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TotPrestressTransferEnd 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 44 88 132 176 220 264 308 352 396 440 484 528 572 616 660 704 748 792 836 880 924 968 1012 1056 1100 1144 1188 1232kip TotPrestressTransferEnd 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 968 1012 1056 1100 1144 1188 1232 1276 1320 1364 1408 1452 1496 1540 1584 1628 1672 1716 1760 1804 1804 1804 1804 1804 1804 1804 1804 1804kip

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Calculates compression transfer length of the prestressing strands MPa145.037738psi CompTransLength 0.33 6.9 OrigStressStrand MPa D mm 20.7ReleaseStrength MPa mm CompTransLength24.329204in Calculates friction forces on beam ends Bearing wtLbeams 2 Bearing86.32812589.20572990.35677187.479167 ()kip FRf Bearing FRfDyn D Bearing FRf25.89843826.76171927.10703126.24375 ()kip FRfDyn21.58203122.30143222.58919321.869792 ()kip Converts static friction to a larger matrix FRfw out11 FRf out12 FRf colsFreeStrand ()2 = if out12g 1 FRf1g out12g FRf1g g1colsFRf () forcolsFreeStrand ()2 if out Bearingw out12g 1 Bearing1g out12g Bearing1g g1colsBearing () for out

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The total area of prestressing that has yet to be cut at each step in the cutting order ApsUncutoutNumberStrands0in2 outNumberStrands1 Aps outjNumberStrandsj ()Aps jNumberStrands2 1 for out ApsUncut 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 10.522 10.302 10.083 9.864 9.645 9.426 9.206 8.987 8.768 8.549 8.33 8.11 7.891 7.672 7.453 7.234 7.014 6.795 6.576 6.357 6.138 5.918 5.699 5.48 5.261 5.042in2 ApsUncut 1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5.48 5.261 5.042 4.822 4.603 4.384 4.165 3.946 3.726 3.507 3.288 3.069 2.85 2.63 2.411 2.192 1.973 1.754 1.534 1.315 1.096 0.877 0.658 0.438 0.219 0in2

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The total area of prestressing that has yet to be cut at each step in the cutting order not including the debonded strands ApsUncutEndout1ApsUncut1colsDebond ()Aps outqoutq1 Aps DebondLengthq0ft = if outqoutq1 DebondLengthq0ft if q2NumberStrands for out ApsUncutEnd 1 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 4.384 4.165 3.946 3.726 3.507 3.288 3.069 2.85 2.63 2.411 2.192 1.973 1.754 1.534 1.315 1.096 0.877 0.658 0.438 0.219 0 0 0 0 0 0 0 0 0in2 ApsUncutEnd 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 8.768 8.549 8.33 8.11 7.891 7.672 7.453 7.234 7.014 6.795 6.576 6.357 6.138 5.918 5.699 5.48 5.261 5.042 4.822 4.603in2

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Spring stiffnesses for prestressing strands kSteel outqf 28500ksi AvgStrandLengthsq1 f ApsUncutq f1colsFreeStrand () for q1NumberStrands1 for out Effective strand stiffness for each beam Converts kSteel to a larger matrix kSteelw outq1 kSteelq1 outq2 kSteelq2 colsFreeStrand ()2 = if outq2colsLbeams () kSteelqcolskSteel () outq2c 2 kSteelqc outq2c 1 kSteelqc c2colskSteel ()1 for colsFreeStrand ()2 if q1rowskSteel () for out keffSteel outq1 1 1 kSteelq1 1 kSteelq2 colsFreeStrand ()2 = if outqf 1 1 kSteelqf 1 kSteelqf1 f1colsLbeams () forcolsFreeStrand ()2 if q1NumberStrands1 for out

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kSteel 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 485.6124223.4594223.4594223.459537.876 475.2094092.5464092.5464092.546526.318 464.8063962.5533962.5533962.553514.761 454.4033833.5093833.5093833.509503.205 444.0013705.4473705.4473705.447491.65 433.5993578.3993578.3993578.399480.094 423.1973452.43452.43452.4468.54 412.7973327.4853327.4853327.485456.986 402.3963203.6923203.6923203.692445.433 391.9963081.0613081.0613081.061433.881 381.5972959.6312959.6312959.631422.329 371.1992839.4482839.4482839.448410.779 360.8012720.5552720.5552720.555399.229 350.404260326032603387.681 340.0082486.8332486.8332486.833376.133 329.6122372.1062372.1062372.106364.587 319.2182258.8752258.8752258.875353.043 308.8252147.1962147.1962147.196341.499 298.4332037.132037.132037.13329.958 288.0421928.7431928.7431928.743318.418 277.6531822.11822.11822.1306.88 267.2661717.2731717.2731717.273295.344 256.881614.3381614.3381614.338283.811 246.4961513.3721513.3721513.372272.28 236.1151414.461414.461414.46260.753 225.7361317.6911317.6911317.691249.228 215.3591223.1571223.1571223.157237.708 204.9861130.9591130.9591130.959226.192kip in

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kSteel 12345 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 277.6531822.11822.11822.1306.88 267.2661717.2731717.2731717.273295.344 256.881614.3381614.3381614.338283.811 246.4961513.3721513.3721513.372272.28 236.1151414.461414.461414.46260.753 225.7361317.6911317.6911317.691249.228 215.3591223.1571223.1571223.157237.708 204.9861130.9591130.9591130.959226.192 194.6171041.21041.21041.2214.68 184.252953.993953.993953.993203.175 173.891869.456869.456869.456191.675 163.536787.714787.714787.714180.184 153.188708.902708.902708.902168.701 142.848633.162633.162633.162157.228 132.516560.646560.646560.646145.768 122.196491.516491.516491.516134.323 111.89425.945425.945425.945122.896 101.601364.119364.119364.119111.491 91.333306.235306.235306.235100.115 81.093252.507252.507252.50788.776 70.89203.161203.161203.16177.485 61.468168.936168.936168.93667.13 52.06135.809135.809135.80956.793 42.672104.12104.12104.1246.482 33.31874.37174.37174.37136.216 24.6652.64552.64552.64526.774 16.01831.55231.55231.55217.353 8.00915.77615.77615.7768.677kip in

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kSteelw 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 485.6124223.4594223.4594223.4594223.4594223.4594223.459537.876 475.2094092.5464092.5464092.5464092.5464092.5464092.546526.318 464.8063962.5533962.5533962.5533962.5533962.5533962.553514.761 454.4033833.5093833.5093833.5093833.5093833.5093833.509503.205 444.0013705.4473705.4473705.4473705.4473705.4473705.447491.65 433.5993578.3993578.3993578.3993578.3993578.3993578.399480.094 423.1973452.43452.43452.43452.43452.43452.4468.54 412.7973327.4853327.4853327.4853327.4853327.4853327.485456.986 402.3963203.6923203.6923203.6923203.6923203.6923203.692445.433 391.9963081.0613081.0613081.0613081.0613081.0613081.061433.881 381.5972959.6312959.6312959.6312959.6312959.6312959.631422.329 371.1992839.4482839.4482839.4482839.4482839.4482839.448410.779 360.8012720.5552720.5552720.5552720.5552720.5552720.555399.229 350.404260326032603260326032603387.681 340.0082486.8332486.8332486.8332486.8332486.8332486.833376.133 329.6122372.1062372.1062372.1062372.1062372.1062372.106364.587 319.2182258.8752258.8752258.8752258.8752258.8752258.875353.043 308.8252147.1962147.1962147.1962147.1962147.1962147.196341.499 298.4332037.132037.132037.132037.132037.132037.13329.958 288.0421928.7431928.7431928.7431928.7431928.7431928.743318.418 277.6531822.11822.11822.11822.11822.11822.1306.88 267.2661717.2731717.2731717.2731717.2731717.2731717.273295.344 256.881614.3381614.3381614.3381614.3381614.3381614.338283.811 246.4961513.3721513.3721513.3721513.3721513.3721513.372272.28 236.1151414.461414.461414.461414.461414.461414.46260.753 225.7361317.6911317.6911317.6911317.6911317.6911317.691249.228 215.3591223.1571223.1571223.1571223.1571223.1571223.157237.708 204.9861130.9591130.9591130.9591130.9591130.9591130.959226.192 194.6171041.21041.21041.21041.21041.21041.2214.68kip in

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kSteelw 12345678 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 267.2661717.2731717.2731717.2731717.2731717.2731717.273295.344 256.881614.3381614.3381614.3381614.3381614.3381614.338283.811 246.4961513.3721513.3721513.3721513.3721513.3721513.372272.28 236.1151414.461414.461414.461414.461414.461414.46260.753 225.7361317.6911317.6911317.6911317.6911317.6911317.691249.228 215.3591223.1571223.1571223.1571223.1571223.1571223.157237.708 204.9861130.9591130.9591130.9591130.9591130.9591130.959226.192 194.6171041.21041.21041.21041.21041.21041.2214.68 184.252953.993953.993953.993953.993953.993953.993203.175 173.891869.456869.456869.456869.456869.456869.456191.675 163.536787.714787.714787.714787.714787.714787.714180.184 153.188708.902708.902708.902708.902708.902708.902168.701 142.848633.162633.162633.162633.162633.162633.162157.228 132.516560.646560.646560.646560.646560.646560.646145.768 122.196491.516491.516491.516491.516491.516491.516134.323 111.89425.945425.945425.945425.945425.945425.945122.896 101.601364.119364.119364.119364.119364.119364.119111.491 91.333306.235306.235306.235306.235306.235306.235100.115 81.093252.507252.507252.507252.507252.507252.50788.776 70.89203.161203.161203.161203.161203.161203.16177.485 61.468168.936168.936168.936168.936168.936168.93667.13 52.06135.809135.809135.809135.809135.809135.80956.793 42.672104.12104.12104.12104.12104.12104.1246.482 33.31874.37174.37174.37174.37174.37174.37136.216 24.6652.64552.64552.64552.64552.64552.64526.774 16.01831.55231.55231.55231.55231.55231.55217.353 8.00915.77615.77615.77615.77615.77615.7768.677kip in

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keffSteel 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 435.5352111.732111.73477.113 425.772046.2732046.273466.344 416.0081981.2761981.276455.579 406.2491916.7551916.755444.816 396.4921852.7241852.724434.058 386.7371789.21789.2423.302 376.9861726.21726.2412.551 367.2381663.7431663.743401.804 357.4941601.8461601.846391.061 347.7531540.531540.53380.323 338.0151479.8161479.816369.59 328.2831419.7241419.724358.863 318.5541360.2771360.277348.141 308.8311301.51301.5337.426 299.1121243.4171243.417326.717 289.3991186.0531186.053316.016 279.6931129.4371129.437305.323 269.9931073.5981073.598294.639 260.31018.5651018.565283.964 250.615964.371964.371273.299 240.939911.05911.05262.645 231.272858.637858.637252.004 221.616807.169807.169241.376 211.971756.686756.686230.762 202.338707.23707.23220.166 192.72658.845658.845209.587 183.118611.579611.579199.029 173.533565.479565.479188.493 163.969520.6520.6177.983kip in

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keffSteel 1234 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 240.939911.05911.05262.645 231.272858.637858.637252.004 221.616807.169807.169241.376 211.971756.686756.686230.762 202.338707.23707.23220.166 192.72658.845658.845209.587 183.118611.579611.579199.029 173.533565.479565.479188.493 163.969520.6520.6177.983 154.426476.996476.996167.501 144.909434.728434.728157.053 135.422393.857393.857146.641 125.967354.451354.451136.272 116.552316.581316.581125.952 107.182280.323280.323115.689 97.866245.758245.758105.493 88.613212.973212.97395.377 79.436182.06182.0685.356 70.351153.118153.11875.449 61.381126.253126.25365.683 52.553101.58101.5856.092 45.0784.46884.46848.04 37.63467.90467.90440.046 30.26752.0652.0632.136 23.0137.18637.18624.356 16.79426.32226.32217.748 10.62415.77615.77611.196 5.3127.8887.8885.598kip in

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Calculates the stiffness of only the nondebonded strands kSteelEnd outqf 28500ksi AvgStrandLengthsq1 f ApsUncutEndq f1colsFreeStrand () for q1NumberStrands1 for out Makes kSteelEnd a larger matrix kSteelEndw outq1 kSteelEndq1 outq2 kSteelEndq2 colsFreeStrand ()2 = if outq2colsLbeams () kSteelEndqcolskSteelEnd () outq2c 2 kSteelEndqc outq2c 1 kSteelEndqc c2colskSteelEnd ()1 for colsFreeStrand ()2 if q1rowskSteelEnd () for out

PAGE 120

kSteelEnd 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 404.6773519.5493519.5493519.549448.23 394.3223395.9423395.9423395.942436.732 383.973273.4133273.4133273.413425.238 373.623151.9963151.9963151.996413.747 363.2733031.7293031.7293031.729402.259 352.9292912.6512912.6512912.651390.775 342.5882794.82794.82794.8379.294 332.2512678.222678.222678.22367.818 321.9172562.9542562.9542562.954356.347 311.5872449.0482449.0482449.048344.88 301.2612336.5512336.5512336.551333.418 290.9392225.5132225.5132225.513321.962 280.6232115.9872115.9872115.987310.512 270.3122008.0292008.0292008.029299.068 260.0061901.6961901.6961901.696287.632 249.7061797.051797.051797.05276.203 239.4141694.1561694.1561694.156264.782 229.1281593.0811593.0811593.081253.371 218.8511493.8961493.8961493.896241.969 208.5831396.6761396.6761396.676230.578 198.3241301.51301.51301.5219.2 188.0761208.4521208.4521208.452207.835 177.841117.6181117.6181117.618196.485 167.6171029.0931029.0931029.093185.151 157.41942.974942.974942.974173.835 147.219859.364859.364859.364162.54 137.047778.373778.373778.373151.269 126.896700.117700.117700.117140.023kip in

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kSteelEnd 12345 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 177.841117.6181117.6181117.618196.485 167.6171029.0931029.0931029.093185.151 157.41942.974942.974942.974173.835 147.219859.364859.364859.364162.54 137.047778.373778.373778.373151.269 126.896700.117700.117700.117140.023 116.77624.72624.72624.72128.808 106.672552.312552.312552.312117.627 96.606483.031483.031483.031106.486 86.578417.025417.025417.02595.391 76.594354.451354.451354.45184.35 66.662295.476295.476295.47673.373 56.793240.277240.277240.27762.472 46.999189.045189.045189.04551.663 37.297141.982141.982141.98240.965 27.70999.30599.30599.30530.407 18.26761.24761.24761.24720.023 9.0128.05628.05628.0569.864 00000 00000 00000 00000 00000 00000 00000 00000kip in

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kSteelEndw 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 404.683519.553519.553519.553519.553519.553519.55448.23 394.323395.943395.943395.943395.943395.943395.94436.73 383.973273.413273.413273.413273.413273.413273.41425.24 373.62315231523152315231523152413.75 363.273031.733031.733031.733031.733031.733031.73402.26 352.932912.652912.652912.652912.652912.652912.65390.77 342.592794.82794.82794.82794.82794.82794.8379.29 332.252678.222678.222678.222678.222678.222678.22367.82 321.922562.952562.952562.952562.952562.952562.95356.35 311.592449.052449.052449.052449.052449.052449.05344.88 301.262336.552336.552336.552336.552336.552336.55333.42 290.942225.512225.512225.512225.512225.512225.51321.96 280.622115.992115.992115.992115.992115.992115.99310.51 270.312008.032008.032008.032008.032008.032008.03299.07 260.011901.71901.71901.71901.71901.71901.7287.63 249.711797.051797.051797.051797.051797.051797.05276.2 239.411694.161694.161694.161694.161694.161694.16264.78 229.131593.081593.081593.081593.081593.081593.08253.37 218.851493.91493.91493.91493.91493.91493.9241.97 208.581396.681396.681396.681396.681396.681396.68230.58 198.321301.51301.51301.51301.51301.51301.5219.2 188.081208.451208.451208.451208.451208.451208.45207.83 177.841117.621117.621117.621117.621117.621117.62196.48 167.621029.091029.091029.091029.091029.091029.09185.15 157.41942.97942.97942.97942.97942.97942.97173.84 147.22859.36859.36859.36859.36859.36859.36162.54 137.05778.37778.37778.37778.37778.37778.37151.27 126.9700.12700.12700.12700.12700.12700.12140.02kip in

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kSteelEndw 12345678 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 198.321301.51301.51301.51301.51301.51301.5219.2 188.081208.451208.451208.451208.451208.451208.45207.83 177.841117.621117.621117.621117.621117.621117.62196.48 167.621029.091029.091029.091029.091029.091029.09185.15 157.41942.97942.97942.97942.97942.97942.97173.84 147.22859.36859.36859.36859.36859.36859.36162.54 137.05778.37778.37778.37778.37778.37778.37151.27 126.9700.12700.12700.12700.12700.12700.12140.02 116.77624.72624.72624.72624.72624.72624.72128.81 106.67552.31552.31552.31552.31552.31552.31117.63 96.61483.03483.03483.03483.03483.03483.03106.49 86.58417.03417.03417.03417.03417.03417.0395.39 76.59354.45354.45354.45354.45354.45354.4584.35 66.66295.48295.48295.48295.48295.48295.4873.37 56.79240.28240.28240.28240.28240.28240.2862.47 47189.04189.04189.04189.04189.04189.0451.66 37.3141.98141.98141.98141.98141.98141.9840.97 27.7199.3199.3199.3199.3199.3199.3130.41 18.2761.2561.2561.2561.2561.2561.2520.02 9.0128.0628.0628.0628.0628.0628.069.86 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000kip in

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Calculates the length of the concrete (for all strands) used for elastic shortening calculations EffConcLength outh1 Lbeams 4 3 CompTransLength 2DebondLengthh outh1 Lbeams FullLength1 = if h1NumberStrands forcolsFreeStrand ()2 = if outgw Lbeams1w 4 3 CompTransLength 2DebondLengthg outgw Lbeams1w FullLength1 = if g1NumberStrands for w1colsLbeams () forcolsFreeStrand ()2 if out Calculates the average concrete lengths (from above) at each strand cut Absolute Value function AvgEffConcLength outiw 1 i nEffConcLengthnw i iNumberStrands1 for w1colsEffConcLength () for out absVal out 0 if out 1 0 if out

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EffConcLength 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297ft EffConcLength 1234 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 137.297142.297144.297139.297 137.297142.297144.297139.297 137.297142.297144.297139.297 137.297142.297144.297139.297 127.297132.297134.297129.297 127.297132.297134.297129.297 117.297122.297124.297119.297 117.297122.297124.297119.297ft

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AvgEffConcLength 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297ft AvgEffConcLength 1234 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.297152.297154.297149.297 147.059152.059154.059149.059 146.832151.832153.832148.832 146.615151.615153.615148.615 146.408151.408153.408148.408 145.992150.992152.992147.992 145.595150.595152.595147.595 145.005150.005152.005147.005 144.44149.44151.44146.44ft

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Equivalent spring stiffness for beam kConc outqz AE AvgEffConcLengthqz z1colsAvgEffConcLength () for q1NumberStrands for out kConc 1234 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3015.952916.772878.912975.48 3020.612921.142883.162980.02 3025.072925.312887.222984.36 3029.352929.312891.122988.53 3037.972937.372898.972996.92 3046.272945.132906.533004.99 3058.662956.712917.83017.04 3070.632967.892928.73028.69kip in kConc 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73 3011.072912.212874.472970.73kip in

PAGE 128

Calculates the total prestress force left to shorten the beam and stretch the uncut strands after friction has been overcome TotPTafterFric outq0kip TotPrestressTransferqFRf if outqTotPrestressTransferqFRfDyn TotPrestressTransferqFRf if colsFreeStrand ()2 = if outqb 0kip TotPrestressTransferqFRf1b if outqb TotPrestressTransferqFRfDyn1b TotPrestressTransferqFRf1b if b1colskConc () for colsFreeStrand ()2 if q1NumberStrands for out

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TotPTafterFric 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 22.4221.721.4122.13 66.4265.765.4166.13 110.42109.7109.41110.13 154.42153.7153.41154.13 198.42197.7197.41198.13 242.42241.7241.41242.13 286.42285.7285.41286.13 330.42329.7329.41330.13 374.42373.7373.41374.13 418.42417.7417.41418.13 462.42461.7461.41462.13 506.42505.7505.41506.13 550.42549.7549.41550.13 594.42593.7593.41594.13 638.42637.7637.41638.13 682.42681.7681.41682.13 726.42725.7725.41726.13 770.42769.7769.41770.13 814.42813.7813.41814.13 858.42857.7857.41858.13 902.42901.7901.41902.13 946.42945.7945.41946.13 990.42989.7989.41990.13 1034.421033.71033.411034.13 1078.421077.71077.411078.13 1122.421121.71121.411122.13 1166.421165.71165.411166.13kip TotPTafterFric 1234 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1034.421033.71033.411034.13 1078.421077.71077.411078.13 1122.421121.71121.411122.13 1166.421165.71165.411166.13 1210.421209.71209.411210.13 1254.421253.71253.411254.13 1298.421297.71297.411298.13 1342.421341.71341.411342.13 1386.421385.71385.411386.13 1430.421429.71429.411430.13 1474.421473.71473.411474.13 1518.421517.71517.411518.13 1562.421561.71561.411562.13 1606.421605.71605.411606.13 1650.421649.71649.411650.13 1694.421693.71693.411694.13 1738.421737.71737.411738.13 1782.421781.71781.411782.13 1826.421825.71825.411826.13 1870.421869.71869.411870.13 1914.421913.71913.411914.13 1958.421957.71957.411958.13 2002.422001.72001.412002.13 2046.422045.72045.412046.13 2090.422089.72089.412090.13 2134.422133.72133.412134.13kip

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InCamber33.13.23 ()in Total axial shortening of the beams Lbeams150155157152 ()ft Xtot outqb TotPTafterFricqb kConcqb keffSteelqb b1colsTotPTafterFric () for q1NumberStrands1 for outNumberStrandsv TotPTafterFricNumberStrandsv kConcNumberStrandsv v1colsTotPTafterFric () for out Eccent40.39in CambMov out1j InCamber1j Eccent Lbeams1j 2 j1colsLbeams () for out CambMov0.1346330.1346330.1372060.132862 ()in Total axial shortening plus camber movement of the beams contributed by each strand cut XtotInd out1cc Xtot1cc cc1colsXtot () for outqc Xtotqc Xtotq1 c c1colsXtot () for qrowsXtot ()2 for out XtotIndividual outjk XtotIndjk XtotNumberStrandsk CambMov1k XtotIndjk k1colsLbeams () for j1NumberStrands for out

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Xtot 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 0.0070.0040.0040.006 0.0190.0130.0130.019 0.0320.0220.0230.032 0.0450.0320.0320.045 0.0580.0410.0420.058 0.0710.0510.0520.071 0.0850.0620.0620.085 0.0980.0720.0730.098 0.1110.0830.0830.111 0.1250.0940.0950.125 0.1380.1050.1060.138 0.1520.1170.1180.152 0.1650.1290.130.166 0.1790.1410.1420.18 0.1930.1530.1550.194 0.2070.1660.1680.208 0.2210.180.1810.222 0.2350.1930.1950.236 0.2490.2070.2090.25 0.2630.2210.2230.265 0.2770.2360.2380.279 0.2920.2510.2530.294 0.3060.2660.2690.308 0.3210.2820.2850.323 0.3360.2980.3010.338 0.350.3140.3170.353 0.3650.3310.3340.368in Xtot 1234 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0.3650.3310.3340.368 0.380.3480.3520.383 0.3950.3650.3690.398 0.410.3830.3870.414 0.4250.4010.4050.429 0.4410.4190.4240.445 0.4560.4380.4430.46 0.4710.4560.4620.476 0.4870.4750.4810.492 0.5030.4950.50.508 0.5180.5140.520.524 0.5340.5330.540.54 0.550.5530.5590.556 0.5660.5720.5790.572 0.5820.5910.5990.589 0.5970.6080.6160.604 0.6120.6260.6330.619 0.6270.6430.6510.635 0.6420.660.6680.65 0.6560.6750.6840.664 0.6690.6910.70.678 0.6820.7050.7140.691 0.6950.7190.7280.705in

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XtotIndividual 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0.0077640.0051280.0051030.007629 0.0153040.0106030.0106940.015239 0.0153910.0108840.010980.015335 0.0154790.0111740.0112740.015432 0.0155680.0114710.0115760.015529 0.0156570.0117770.0118880.015627 0.0157470.0120910.0122080.015727 0.0158380.0124140.0125360.015827 0.015930.0127460.0128740.015928 0.0160220.0130860.013220.01603 0.0161150.0134340.0135750.016133 0.0162090.0137910.0139390.016236 0.0163030.0141550.0143110.016341 0.0163980.0145280.0146910.016446 0.0164940.0149090.015080.016553 0.0165910.0152970.0154760.01666 0.0166890.0156920.0158790.016768 0.0167870.0160930.0162890.016877 0.0168860.01650.0167050.016987 0.0169850.0169120.0171260.017098 0.0170850.0173270.0175510.01721 0.0171860.0177460.0179790.017322 0.0172880.0181660.0184090.017436 0.017390.0185860.018840.01755 0.0174930.0190040.0192680.017664 0.0175960.0194180.0196940.01778 0.01770.0198270.0201130.017895 0.0178040.0202280.0205240.018012in

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XtotIndividual 1234 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0.017390.0185860.018840.01755 0.0174930.0190040.0192680.017664 0.0175960.0194180.0196940.01778 0.01770.0198270.0201130.017895 0.0178040.0202280.0205240.018012 0.0179090.0206170.0209240.018128 0.0180140.0209920.0213090.018245 0.0181190.0213490.0216770.018362 0.0182230.0216850.0220220.018479 0.0183280.0219950.0223410.018596 0.0184320.0222740.0226290.018711 0.0185340.0225180.022880.018825 0.0186360.022720.0230890.018938 0.0187350.0228750.023250.019048 0.0188320.0229770.0233560.019154 0.0189240.0230190.0234010.019255 0.0190110.0229930.0233760.019349 0.019090.0228920.0232740.019434 0.017750.0203420.0206860.018062 0.017820.0204260.0207730.018138 0.0178840.0204560.0208050.018207 0.0179370.0204070.0207560.018263 0.0165910.0183680.0186920.016891 0.0166360.0183980.0187240.016939 0.0152970.0166010.016910.01558 0.0153380.0166710.0169820.015625in

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Calculates a reference total beam shortening for comparison to model results. The model results for axial shortening should always be less than these reference numbers. Axialrrout NumberStrandsFstrand ()Lbeams AE colsFreeStrand ()2 = if out1w NumberStrandsFstrand ()Lbeams1w AE w1colsLbeams () forcolsFreeStrand ()2 if out Axialref out1j Axialrr1j CambMov1j j1colsCambMov () for out Axialrr0.7291650.7534710.7631930.738887 ()in Axialref0.8637990.8881040.9003990.871749 ()in CambMov0.1346330.1346330.1372060.132862 ()in All the following functions with "Numb" at the end transform previously created matrices and make them much larger. If NumbCalcs = 20 then the matrices get 20 times larger kSteelwNumb outNumbCalcsq amp 1 c kSteelwqc ampNumbCalcs1 for c1colskSteelw () for q1rowskSteelw () for out kSteelNumb outNumbCalcsq amp 1 c kSteelqc ampNumbCalcs1 for c1colskSteel () for q1rowskSteel () for out

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XtotIndividualNumb outNumbCalcsq amp 1 c XtotIndividualqc NumbCalcs ampNumbCalcs1 for c1colsXtotIndividual () for q1rowsXtotIndividual () for out RefSlideNumb outNumbCalcsq amp 1 c RefSlideqc NumbCalcs ampNumbCalcs1 for c1colsRefSlide () for q1rowsRefSlide () for out kSteelEndNumb outNumbCalcsq amp 1 c kSteelEndqc ampNumbCalcs1 for c1colskSteelEnd () for q1rowskSteelEnd () for out AvgStrandLengthsNumb outNumbCalcsq amp 1 c AvgStrandLengthsqc ampNumbCalcs1 for c1colsAvgStrandLengths () for q1rowsAvgStrandLengths () for out

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ApsUncutNumb outNumbCalcsq amp 1 c ApsUncutqc ampNumbCalcs1 for c1colsApsUncut () for q1rowsApsUncut () for out DebondLengthNumb outNumbCalcsq amp 1 c DebondLengthqc ampNumbCalcs1 for c1colsDebondLength () for q1rowsDebondLength () for out ApsUncutEndNumb outNumbCalcsq amp 1 c ApsUncutEndqc ampNumbCalcs1 for c1colsApsUncutEnd () for q1rowsApsUncutEnd () for out TotPrestressTransferEndNumb outNumbCalcsq amp 1 c TotPrestressTransferEndqc ampNumbCalcs1 for c1colsTotPrestressTransferEnd () for q1rowsTotPrestressTransferEnd () for out

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Creates a matrix of zeros CreateZeros outzg 0 g1colsFreeStrand () for z13 for out CreateZeros 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Takes three variables and places them into a matrix so they can be transferred around as one variable ConstPackqqXXX XXXp ()outCreateZeros out11 qq out2g XXX1g out3g XXXp1g g1colsCreateZeros () for out

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Determines global motion and tension pull GlobalPackIn () SlideIndicatorss1 SlideValuess0in SlideTotperq1ss 0in ss1colsLbeams () for qGPackIn11 XXGyyPackIn2yy in XXprevGyyPackIn3yy in yy1colsFreeStrand () for XXGkkXXGkkSlideValuekk XXGkk1 XXGkk1 SlideValuekk SlideTotperqiteratekk SlideTotperqiterate1 kk SlideValuekk USFGkk0kip if XXGkkXXGkkSlideValuekk XXGkk1 XXGkk1 SlideValuekk SlideTotperqiteratekk SlideTotperqiterate1 kk SlideValuekk USFGkk0kip if SlideIndicatorkk2 = if kk1colsLbeams () for iterate1 if iterate11000 for

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TensionPullGhXXGhkSteelNumbqGh TempStressApsUncutNumbqG h1colsLbeams ()1 for TensionPullwG1TensionPullG1 TensionPullwG2colsLbeams () TensionPullGcolsLbeams ()1 TensionPullwG2aa 2 TensionPullGaa TensionPullwG2aa 1 TensionPullGaa aa2colsLbeams () for USFGccTensionPullwG2cc TensionPullwG2cc 1 USFGItiteratecc USFGcc SlideIndicatorcc2 absVal USFGcckip kip 2FRf1cc if SlideValueccRefSlideNumbqGcc SlideIndicatorcc1 absVal USFGcckip kip 2FRfDyn1cc ifSlideIndicatorcc2 = if SlideTotperqGccSlideTotperqiteratecc cc1colsLbeams () for XXoutuuXXGuu uu1colsLbeams ()1 for breakmaxSlideIndicator ()1 = if out XXout in USFGIt kip SlideTotperqG in out

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Determines movement at each of the beam ends Determines fcalc/f values for all beam ends Equation Slideqq0in TotalSlideqq0in q2 kSteelwNumbq1 XtotIndividualNumbq kSteelwNumbq2 kSteelwNumbq1 q1 XtotIndividualNumbqkSteelwNumbq1 XtotIndividualNumbq kSteelwNumbq2 kSteelwNumbq1 q1 = if q2 kSteelwNumbq1 XtotIndividualNumbq USFq1 kSteelwNumbq2 kSteelwNumbq1 q1 XtotIndividualNumbqq2 q2 0in q1 XtotIndividualNumbq q2 0in if q1 0in q2 XtotIndividualNumbq q1 0in if q1 if colsFreeStrand ()2 = if q1NumberStrands1 ()NumbCalcs for

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Slideqqg 0in TotalSlideqqg 0in q2g kSteelwNumbq2g 1 XtotIndividualNumbqg kSteelwNumbq2g kSteelwNumbq2g 1 q2g 1 XtotIndividualNumbqg kSteelwNumbq2g 1 XtotIndividualNumbqg kSteelwNumbq2g kSteelwNumbq2g 1 q2g 0in q2g 1 XtotIndividualNumbqg q2g 0in if q2g 1 0in q2g XtotIndividualNumbqg q2g 1 0in if q1 = if q2g kSteelwNumbq2g 1 XtotIndividualNumbqg USFq1 g kSteelwNumbq2g kSteelwNumbq2g 1 q2g 1 XtotIndividualNumbqg q2g q2g 0in q2g 1 XtotIndividualNumbqg q2g 0in if q 2 g 1 0in if q1 if g1colsLbeams () for colsFreeStrand ()2 if

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q2g 1 0in q2g XtotIndividualNumbqg q g XXq1 q1 XXq2 q2 colsFreeStrand ()2 = if XXqcolsLbeams ()1 qcolsLbeams ()2 XXqw q2w 2 q2w 1 w2colsLbeams () for colsFreeStrand ()2 if q1 = if XXq1 XXq1 1 q1 XXq2 XXq1 2 q2 colsFreeStrand ()2 = if XXqcolsLbeams ()1 XXq1 colsLbeams ()1 qcolsLbeams ()2 XXqw XXq1 w q2w 2 q2w 1 w2colsLbeams () for colsFreeStrand ()2 if q1 if SendGlobalConstPackq XXTqTin XXTq1 Tin ReceiveGlobalGlobalSendGlobal ()11 Tin XX RiGlbl we1colsFreeStrand () for colsFreeStrand ()2 ifq1 if

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XX qwe R ece i ve Gl o b a l 1we ReceiveSlideGlobalSendGlobal ()13 Tin Slideqqss ReceiveSlide1ss TotalSlideqqss TotalSlideqq1 ss Slideqqss ss1colsLbeams () for TensionPullqh kSteelNumbqh XXqh TempStressApsUncutNumbq TensionPullEndqh kSteelEndNumbqh XXqh TempStressApsUncutEndNumbq RTLqh 0.33 6.9 TensionPullqh ApsUncutNumbq MPa D mm 20.7ReleaseStrength MPa mm DebondLengthNumbq0ft = if RTLqh 0mm DebondLengthNumbq0ft if h1colsLbeams ()1 for colsFreeStrand ()2 if TensionPullwq1 TensionPullq1 TensionPullEndwq1 TensionPullEndq1 RTLwq1 RTLq1 TensionPullwq2 TensionPullq2 TensionPullEndwq2 TensionPullEndq2 RTLwq2 RTLq2 colsFreeStrand ()2 = if TensionPullwq2colsLbeams () TensionPullqcolsLbeams ()1 TiPllEd colsFreeStrand ()2 if

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T ens i on P u llE n d wq2colsLbeams () T ens i on P u llE n d qcolsLbeams ()1 RTLwq2colsLbeams () RTLqcolsLbeams ()1 TensionPullEndwq2aa 2 TensionPullEndqaa TensionPullEndwq2aa 1 TensionPullEndqaa TensionPullwq2aa 2 TensionPullqaa TensionPullwq2aa 1 TensionPullqaa RTLwq2aa 2 RTLqaa RTLwq2aa 1 RTLqaa aa2colsLbeams () for CRTLwqbb RTLwqbb TotPrestressTransferEndNumbq CompTransLength VCTqbb CRTLwqbb ConcAllowableTension TensionPullEndwqbb FRfw1bb DebondLengthNumbq0ft = if CRTLwqbb CRTLwq1 bb VCTqbb VCTq1 bb DebondLengthNumbq0ft if FrictionForceqbb FRfw1bb bb12 for USFqTensionPullwq2 TensionPullwq1 colsFreeStrand ()2 = if bb12colsLbeams () for colsFreeStrand ()2 if

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CRTLwqbb RTLwqbb TotPrestressTransferEndNumbq CompTransLength DebondLengthNumbq0ft = if CRTLwqbb CRTLwq1 bb DebondLengthNumbq0ft if USFqcc TensionPullwq2cc TensionPullwq2cc 1 cc1colsLbeams () for VCTq2i 1 CRTLwq2i 1 ConcAllowableTension TensionPullEndwq2i 1 FRfw12i 1 VCTq2i CRTLwq2i ConcAllowableTension TensionPullEndwq2i FRfw12i USFqi DebondLengthNumbq0ft = if VCTq2i 1 VCTq1 2i 1 VCTq2i VCTq1 2i DebondLengthNumbq0ft if FrictionForceq2i 1 FRfw12i 1 FrictionForceq2i FRfw12i USFqi USFqi 0kip if VCTq2i 1 CRTLwq2i 1 ConcAllowableTension TensionPullEndwq2i 1 FRfw12i 1 absVal USFq kip VCTq2i CRTLwq2i ConcAllowableTension TensionPullEndwq2i FRfw12i DebondLengthNumbq0ft = if VCT 2 i 1 VCT 1 2 i 1 DebondLengthNumbq0ft if USFqi 0kip if i1colsLbeams () for

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VCT q 2 i 1 VCT q 1 2 i 1 VCTq2i VCTq1 2i FrictionForceq2i 1 FRfw12i 1 absVal USFqi kip kip FrictionForceq2i FRfw12i out in TensionPull kip VCT kip RTLw in XX in TensionPullEnd kip USF kip FrictionForce kip in RTL in TensionPullw kip TotalSlideq in in CRTLw kip TensionPullEndw kip RTLw in EndMovxEquation11 in StrandMovXXEquation12 in TensPullEquation21 kip TensPullEndEquation22 kip ReverseTransLengthEquation23 in CrackPredictorEquation31 kip UnbalanceForceEquation32 kip TensPullwEquation33 kip TensPullEndwEquation34 kip ReverseTransLengthwEquation41 in FrictionValueEquation42 kip BeamSlideEquation43 in CompAtRTLEquation24 kip

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All the following functions take extremely large matrices and make them small matrices (the ones shown in the results) TotEndMovx out1c EndMovx1c c1colsEndMovx () for outqcc outq1 cc EndMovxqcc cc1colsEndMovx () for q2rowsEndMovx () for out EndMovxPerStrand outqce TotEndMovxqNumbCalcs ce ce1colsTotEndMovx () for q1 rowsTotEndMovx () NumbCalcs for outNumberStrands2c 1 XtotIndividualNumberStrandsc 2 outNumberStrands1 2c 1 outNumberStrands2c XtotIndividualNumberStrandsc 2 outNumberStrands1 2c c1colsXtotIndividual () for out

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TensPullPerStrand outqce TensPullqNumbCalcs ce ce1colsTensPull () for q1 rowsTensPull () NumbCalcs for out TensPullEndPerStrand outqce TensPullEndqNumbCalcs ce ce1colsTensPullEnd () for q1 rowsTensPullEnd () NumbCalcs for out ReverseTransLengthPerStrand outqce ReverseTransLengthqNumbCalcs ce ce1colsReverseTransLength () for q1 rowsReverseTransLength () NumbCalcs for out CompAtRTLPerStrand outqce CompAtRTLqNumbCalcs ce ce1colsCompAtRTL () for q1 rowsCompAtRTL () NumbCalcs for out StrandMovXXPerStrand outqce StrandMovXXqNumbCalcs ce ce1colsStrandMovXX () for q1 rowsStrandMovXX () NumbCalcs for out UnbalanceForcePerStrand outqce UnbalanceForceqNumbCalcs ce ce1colsUnbalanceForce () for q1 rowsUnbalanceForce () NumbCalcs for out

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FrictionValuePerStrand outqce FrictionValueqNumbCalcs ce ce1colsFrictionValue () for q1 rowsFrictionValue () NumbCalcs for out BeamSlidePerStrand outqce BeamSlideqNumbCalcs ce ce1colsBeamSlide () for q1 rowsBeamSlide () NumbCalcs for out CrackPredictorPerStrand outqce CrackPredictorqNumbCalcs ce ce1colsCrackPredictor () for q1 rowsCrackPredictor () NumbCalcs for out NSNNumberStrandsNumbCalcs

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END MOVEMENT RESULTS:Total End Movement for Each Beam End after each strand cut (this does NOT include global motion) In order to determine the total motion of each beam end this value must be added to the sliding value EndMovxPerStrand 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0.007720.000040.003370.001760.001740.003370.000040.00759 0.023030.000040.010440.00530.005360.010440.000040.02282 0.038420.000040.017720.008890.009050.017720.000040.03816 0.05390.000040.028670.009120.011610.026440.000040.05359 0.069470.000040.039530.009730.014640.034990.000040.06912 0.085120.000040.050670.010370.017480.044040.000040.08475 0.100870.000040.061840.011290.020570.053150.000040.10047 0.116710.000040.073390.012150.023190.063070.000040.1163 0.132640.000040.085860.012430.02620.072930.000040.13223 0.148660.000040.097740.013640.029430.082920.000040.14826 0.164780.000040.110480.014320.033360.092570.000040.16439 0.180980.000040.123110.015490.036140.103730.000040.18063 0.197290.000040.136740.016010.040070.11410.000040.19697 0.213690.000040.149720.017560.043890.124980.000040.21342 0.230180.000040.162810.019390.046570.137380.000040.22997 0.246770.000040.177380.020110.051480.147940.000040.24663 0.263460.000040.191060.022130.054570.160730.000040.2634 0.280250.000040.20630.022980.060060.171530.000040.28027 0.297130.000040.2220.023780.065290.183010.000040.29726 0.314120.000040.236550.026130.068640.196780.000040.31436 0.33120.000040.251240.028770.073340.209640.000040.33157 0.348390.000040.267480.030270.079080.221870.000040.34889 0.365680.000040.285140.030780.085280.234080.000040.36633 0.383070.000040.300090.034420.089980.248220.000040.38388 0.400560.000040.315380.038130.095330.262130.000040.40154in

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minEndMovx ()0in EndMovxPerStrand 12345678 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0.400560.000040.315380.038130.095330.262130.000040.40154 0.418160.000040.330430.04250.100950.276210.000040.41932 0.435860.000040.345210.047550.106820.290450.000040.43722 0.453660.000040.360.052980.111240.306560.000040.45523 0.471570.000040.378820.054780.12080.317920.000040.47336 0.489580.000040.393610.060990.128190.331840.000040.4916 0.50770.000040.412850.06310.138150.343560.000040.50996 0.525930.000040.427420.070210.14560.358130.000040.52844 0.544250.000040.442190.077430.153170.372910.000040.54704 0.562690.000040.457160.084740.160830.387870.000040.56575 0.581220.000040.472290.092130.168580.4030.000040.58458 0.599860.000040.487560.099580.17640.418270.000040.60351 0.618590.000040.502930.107080.184270.433650.000040.62256 0.637420.000040.518380.114610.192190.449090.000040.64171 0.656350.000040.533850.122160.200110.464560.000040.66097 0.675360.000040.549310.129690.208030.480020.000040.68032 0.694450.000040.564690.13720.215920.495410.000040.69975 0.71220.000040.578390.143850.222920.50910.000040.71781 0.730020.000040.592120.150540.229960.522830.000040.73595 0.743870.004080.606250.156860.241810.531780.009270.74493 0.744760.021120.614590.168940.254520.539830.026620.74584 0.744760.037710.621510.180380.266590.546450.043510.74584 0.744760.054340.628440.191850.278690.553080.060450.74584 0.749070.065340.636480.200410.289880.55880.074060.74781 0.756740.0730.644810.208750.298370.567290.081870.75563in Axialref0.8637990.8881040.9003990.871749 ()in

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051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Total End Movement B1 End1Strand NumberTotal Movement (in)

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051015202530354045 0 0.05 0.1 0.15 Total End Movement B1 End2Strand NumberTotal Movement (in)

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051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Total End Movement B2 End1Strand NumberTotal Movement (in)

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051015202530354045 0 0.05 0.1 0.15 0.2 0.25 Total End Movement B2 End2Strand NumberTotal Movement (in)

PAGE 156

051015202530354045 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Total End Movement B3 End1Strand NumberTotal Movement (in)

PAGE 157

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Total End Movement B3 End2Strand NumberTotal Movement (in)

PAGE 158

051015202530354045 0 0.05 0.1 0.15 0.2 Total End Movement B4 End1Strand NumberTotal Movement (in)

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051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Total End Movement B4 End2Strand NumberTotal Movement (in)

PAGE 160

End Movement Each Strand Contributes Does NOT include global motion EndMov out1c EndMovxPerStrand1c c1colsEndMovxPerStrand () for outqcc EndMovxPerStrandqcc EndMovxPerStrandq1 cc cc1colsEndMovxPerStrand () for q2NumberStrands for out EndMov 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0.0077240.000040.003370.0017580.0017370.0033660.0000430.007586 0.01530400.0070650.0035380.0036230.00707100.015239 0.01539100.0072860.0035980.0036930.00728700.015335 0.01547900.0109470.0002260.002560.00871400.015432 0.01556800.0108620.0006090.0030280.00854800.015529 0.01565700.0111340.0006430.0028350.00905300.015627 0.01574700.0111710.000920.0030980.0091100.015727 0.01583800.0115560.0008580.0026160.0099200.015827 0.0159300.0124640.0002820.0030120.00986200.015928 0.01602200.0118810.0012050.0032280.00999200.01603 0.01611500.0127480.0006860.0039270.00964800.016133 0.01620900.012620.001170.0027810.01115800.016236 0.01630300.0136340.0005210.0039370.01037400.016341 0.01639800.0129810.0015470.0038160.01087500.016446 0.01649400.0130850.0018240.0026750.01240400.016553 0.01659100.0145740.0007220.0049140.01056200.01666 0.01668900.0136750.0020170.0030930.01278600.016768 0.01678700.0152420.0008510.0054920.01079700.016877 0.01688600.0156990.0008010.0052240.01148100.016987in

PAGE 161

EndMov 12345678 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0.01698500.0145570.0023540.0033520.01377400.017098 0.01708500.0146840.0026440.0046960.01285500.01721 0.01718600.0162470.0014990.0057460.01223300.017322 0.01728800.0176610.0005050.0061990.0122100.017436 0.0173900.0149480.0036370.0047010.01413900.01755 0.01749300.0152920.0037120.0053520.01391600.017664 0.01759600.015050.0043690.0056140.0140800.01778 0.017700.0147730.0050550.0058760.01423700.017895 0.01780400.0147950.0054320.0044150.01610900.018012 0.01790900.0188160.0018010.0095620.01136200.018128 0.01801400.0147910.0062010.0073870.01392200.018245 0.01811900.0192380.0021110.0099570.0117200.018362 0.01822300.014570.0071150.0074570.01456500.018479 0.01832800.0147770.0072180.0075640.01477700.018596 0.01843200.0149660.0073080.0076630.01496600.018711 0.01853400.0151310.0073860.0077490.01513100.018825 0.01863600.0152680.0074510.0078210.01526900.018938 0.01873500.0153740.0075010.0078760.01537400.019048 0.01883200.0154440.0075330.0079120.01544400.019154 0.01892400.0154730.0075460.0079280.01547300.019255 0.01901100.0154560.0075370.0079190.01545600.019349 0.0190900.0153890.0075030.0078850.01538900.019434 0.0177500.0136910.0066510.0069970.01368900.018062 0.0178200.0137320.0066930.0070410.01373200.018138 0.0138480.0040350.0141360.006320.0118550.008950.0092260.00898 0.0008970.017040.0083320.0120750.0127060.008050.017350.000913 00.0165910.0069210.0114460.0120750.0066170.0168910 00.0166360.006930.0114680.0120970.0066270.0169390 0.0043040.0109930.008040.0085610.0111880.0057220.0136110.001969 0.0076690.0076690.0083360.0083360.0084910.0084910.0078120.007812in

PAGE 162

051015202530354045 0 0.005 0.01 0.015 0.02 0.025 End Movement Beam 1 End 1Strand NumberMovement (in)

PAGE 163

051015202530354045 0 0.005 0.01 0.015 End Movement Beam 1 End 2Strand NumberMovement (in)

PAGE 164

051015202530354045 0 0.005 0.01 0.015 0.02 End Movement Beam 2 End 1Strand NumberMovement (in)

PAGE 165

051015202530354045 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 End Movement Beam 2 End 2Strand NumberMovement (in)

PAGE 166

051015202530354045 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 End Movement Beam 3 End 1Strand NumberMovement (in)

PAGE 167

051015202530354045 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 End Movement Beam 3 End 2Strand NumberMovement (in)

PAGE 168

051015202530354045 0 0.005 0.01 0.015 End Movement Beam 4 End 1Strand NumberMovement (in)

PAGE 169

051015202530354045 0 0.005 0.01 0.015 0.02 0.025 End Movement Beam 4 End 2Strand NumberMovement (in)

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Global motion of beamsreference frame = initial beam positions and location of bulkheads Positive number indicates beam slides right, negative number indicates beam slides left BeamSlidePerStrand 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0000 0000 0.003113000 0.00948800-0.0059 0.01710800-0.011946 0.02492100-0.018146 0.03179100-0.024507 0.04001900-0.027773 0.04726300-0.034484 0.05843400-0.041384 0.06609600-0.048485 0.07398500-0.052141 0.08211500-0.05968 0.09050100-0.067457 0.09915900-0.071474 0.10810800-0.079778 0.11736800-0.084076 0.1269600-0.088529 0.1369100-0.097771 0.14207700-0.102573 0.15282600-0.107569 0.16402400-0.112776 0.16986700-0.118212 0.17597600-0.123898 0.18237600-0.129858 0.18909500-0.136119in

PAGE 171

BeamSlidePerStrand 1234 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 0.15282600-0.107569 0.16402400-0.112776 0.16986700-0.118212 0.17597600-0.123898 0.18237600-0.129858 0.18909500-0.136119 0.19616700-0.142712 0.20362900-0.142712 0.20362900-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712 0.21201400-0.142712in

PAGE 172

051015202530354045 0 0.05 0.1 0.15 0.2 0.25 Total Beam 1 SlideStrand NumberMovement (in)

PAGE 173

051015202530354045 0 0.2 0.4 0.6 0.8 Total Beam 2 SlideStrand NumberMovement (in)

PAGE 174

051015202530354045 0 0.2 0.4 0.6 0.8 Total Beam 3 SlideStrand NumberMovement (in)

PAGE 175

051015202530354045 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Total Beam 4 SlideStrand NumberMovement (in)

PAGE 176

BEAM RESULTS:Total Axial Shortening of Beams after each strand cut BeamShorten outqc EndMovxPerStrandq2c 1 EndMovxPerStrandq2c c1 colsEndMovxPerStrand () 2 for q1rowsEndMovxPerStrand () for out BeamShorten 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0.0080.0050.0050.008 0.0230.0160.0160.023 0.0380.0270.0270.038 0.0540.0380.0380.054 0.070.0490.050.069 0.0850.0610.0620.085 0.1010.0730.0740.101 0.1170.0860.0860.116 0.1330.0980.0990.132 0.1490.1110.1120.148 0.1650.1250.1260.164 0.1810.1390.140.181 0.1970.1530.1540.197 0.2140.1670.1690.213 0.230.1820.1840.23 0.2470.1970.1990.247 0.2630.2130.2150.263 0.280.2290.2320.28in

PAGE 177

BeamShorten 1234 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0.3140.2630.2650.314 0.3310.280.2830.332 0.3480.2980.3010.349 0.3660.3160.3190.366 0.3830.3350.3380.384 0.4010.3540.3570.402 0.4180.3730.3770.419 0.4360.3930.3970.437 0.4540.4130.4180.455 0.4720.4340.4390.473 0.490.4550.460.492 0.5080.4760.4820.51 0.5260.4980.5040.528 0.5440.520.5260.547 0.5630.5420.5490.566 0.5810.5640.5720.585 0.60.5870.5950.604 0.6190.610.6180.623 0.6370.6330.6410.642 0.6560.6560.6650.661 0.6750.6790.6880.68 0.6940.7020.7110.7 0.7120.7220.7320.718 0.730.7430.7530.736 0.7480.7630.7740.754 0.7660.7840.7940.772 0.7820.8020.8130.789 0.7990.820.8320.806 0.8140.8370.8490.822 0.830.8540.8660.837in Axialref0.8637990.8881040.9003990.871749 ()in

PAGE 178

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Axial Shortening of Beam 1Strand NumberAxial Shortening (in)

PAGE 179

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Axial Shortening of Beam 2Strand NumberAxial Shortening (in)

PAGE 180

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Axial Shortening of Beam 3Strand NumberAxial Shortening (in)

PAGE 181

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Axial Shortening of Beam 4Strand NumberAxial Shortening (in)

PAGE 182

Beam Length accounts for beam axial shortening BeamLength outqLbeamsBeamShortenq colsFreeStrand ()2 = if outqc Lbeams1c BeamShortenqc c1colsBeamShorten () forcolsFreeStrand ()2 if q1rowsBeamShorten () for out BeamLength 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 149.999155157151.999 149.998154.999156.999151.998 149.997154.998156.998151.997 149.996154.997156.997151.996 149.994154.996156.996151.994 149.993154.995156.995151.993 149.992154.994156.994151.992 149.99154.993156.993151.99 149.989154.992156.992151.989 149.988154.991156.991151.988 149.986154.99156.99151.986 149.985154.988156.988151.985 149.984154.987156.987151.984 149.982154.986156.986151.982 149.981154.985156.985151.981 149.979154.984156.983151.979 149.978154.982156.982151.978 149.977154.981156.981151.977 149.975154.98156.979151.975 149.974154.978156.978151.974ft

PAGE 183

BeamLength 1234 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 149.974154.978156.978151.974 149.972154.977156.976151.972 149.971154.975156.975151.971 149.97154.974156.973151.969 149.968154.972156.972151.968 149.967154.971156.97151.967 149.965154.969156.969151.965 149.964154.967156.967151.964 149.962154.966156.965151.962 149.961154.964156.963151.961 149.959154.962156.962151.959 149.958154.96156.96151.957 149.956154.959156.958151.956 149.955154.957156.956151.954 149.953154.955156.954151.953 149.952154.953156.952151.951 149.95154.951156.95151.95 149.948154.949156.949151.948 149.947154.947156.947151.947 149.945154.945156.945151.945 149.944154.943156.943151.943 149.942154.942156.941151.942 149.941154.94156.939151.94 149.939154.938156.937151.939 149.938154.936156.936151.937 149.936154.935156.934151.936 149.935154.933156.932151.934 149.933154.932156.931151.933 149.932154.93156.929151.932 149.931154.929156.928151.93ft

PAGE 184

051015202530354045 149.92 149.94 149.96 149.98 150 Beam 1 LengthStrand NumberBeam Length (ft)

PAGE 185

051015202530354045 154.92 154.94 154.96 154.98 155 Beam 2 LengthStrand NumberBeam Length (ft)

PAGE 186

051015202530354045 156.92 156.94 156.96 156.98 157 Beam 3 LengthStrand NumberBeam Length (ft)

PAGE 187

051015202530354045 151.92 151.94 151.96 151.98 152 Beam 4 LengthStrand NumberBeam Length (ft)

PAGE 188

STRAND RESULTS:The amount each free strand set streches after each strand cut StrandMovXXPerStrand 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 0.0077240.003410.0034950.0034090.007586 0.0230280.0104750.0106550.010480.022825 0.0415320.0146490.0179460.0177670.03816 0.0633870.0192210.0207330.0205810.059492 0.0865750.0224630.0243690.0230830.081067 0.1100450.0257840.0278470.0259360.102894 0.1326620.0300850.0318650.0286850.124982 0.1567280.0334140.0353390.0353390.144075 0.1799020.0386330.0386330.038490.166713 0.2070950.0393430.0430660.0415820.189644 0.2308720.0444290.0476790.0441290.212877 0.2549690.049160.051630.051630.23277 0.2794030.0546650.0560880.0544650.256649 0.3041870.059260.0614510.0575630.280873 0.329340.0636860.0659510.0659510.301442 0.354880.0693120.0715870.0682080.326407 0.3808280.0737270.0766970.0766970.347473 0.4072070.0793770.083040.083040.368804 0.4340420.0851260.0890640.0852790.395033 0.4561940.0945170.0947710.0942510.416933 0.4840290.0984510.102110.102110.439139 0.5124130.10350.1093560.1091370.461668 0.5355440.1153180.1160590.1159110.48454 0.5590440.1241570.1243980.1243630.507776 0.5829360.1330490.1334620.1323190.5314 0.6072520.1413790.1434450.1401380.55544 0.6320240.1490810.1543750.1477820.579928in

PAGE 189

StrandMovXXPerStrand 12345 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 0.5355440.1153180.1160590.1159110.48454 0.5590440.1241570.1243980.1243630.507776 0.5829360.1330490.1334620.1323190.5314 0.6072520.1413790.1434450.1401380.55544 0.6320240.1490810.1543750.1477820.579928 0.657290.1564140.1642230.1638910.59794 0.6751990.1752290.1755860.1752530.616069 0.7015990.1816350.1891750.1891750.634314 0.7197170.2008730.2012430.2008950.652676 0.737940.2154430.2158150.215460.671155 0.7562680.2302190.2305970.2302370.689751 0.77470.2451850.2455680.2452030.708462 0.7932340.2603160.2607040.2603340.727287 0.811870.2755850.2759760.2756030.746225 0.8306050.2909590.2913530.2909780.765273 0.8494370.3064030.3067980.3064220.784426 0.8683610.3218750.3222720.3218950.803681 0.8873720.3373320.3377280.3373510.82303 0.9064620.3527210.3531150.352740.842464 0.9242120.3664120.3667630.3664290.860526 0.9420320.3801440.3804960.3801620.878663 0.955880.3983160.3986720.3983380.887644 0.9567770.4236880.4234530.4237380.888557 0.9567770.44720.4469740.4472460.888557 0.9567770.4707660.4705390.4708130.888557 0.9610810.4897990.4902880.4901460.890526in

PAGE 190

051015202530354045 0 0.2 0.4 0.6 0.8 Average Free Strand Set 1 StretchStrand NumberAverage Stretch (in)

PAGE 191

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 Average Free Strand Set 2 StretchStrand NumberAverage Stretch (in)

PAGE 192

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 Average Free Strand Set 3 StretchStrand NumberAverage Stretch (in)

PAGE 193

051015202530354045 0 0.1 0.2 0.3 0.4 0.5 0.6 Average Free Strand Set 4 StretchStrand NumberAverage Stretch (in)

PAGE 194

051015202530354045 0 0.2 0.4 0.6 0.8 Average Free Strand Set 5 StretchStrand NumberAverage Stretch (in)

PAGE 195

Average lengths of each Free Strand Set (includes stretching) StraLength outqc AvgStrandLengthsq1 c StrandMovXXPerStrandqc c1colsStrandMovXXPerStrand () for q1rowsStrandMovXXPerStrand () for out StraLength 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 51.4595.9175.9175.91746.459 51.4915.985.985.9846.491 51.5256.0456.0456.04546.525 51.5616.1136.1136.11346.561 51.5986.1846.1846.18446.598 51.6376.2586.2586.25846.636 51.6786.3366.3366.33646.677 51.726.4176.4186.41846.719 51.7656.5036.5036.50346.764 51.8126.5936.5936.59346.811 51.8616.6886.6886.68846.86 51.9136.7886.7886.78846.911 51.9686.8936.8946.89346.966 52.0257.0057.0057.00547.023 52.0867.1237.1237.12347.084 52.1517.2487.2487.24847.148 52.2197.3817.3817.38147.216 52.2927.5237.5237.52347.289 52.377.6747.6747.67447.366 52.4527.8357.8357.83547.449 52.548.0088.0098.00947.537 52.6358.1948.1948.19447.631ft

PAGE 196

StraLength 12345 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 52.4527.8357.8357.83547.449 52.548.0088.0098.00947.537 52.6358.1948.1948.19447.631 52.7378.3948.3948.39447.733 52.8478.618.618.6147.842 52.9658.8448.8448.84447.961 53.0949.0999.0999.09948.09 53.2349.3769.3779.37648.23 53.3889.689.689.6848.383 53.55610.01510.01510.01548.551 53.74310.38410.38410.38448.737 53.94910.79510.79510.79548.943 54.17911.25311.25311.25349.174 54.43811.76911.76911.76949.432 54.73112.35412.35412.35449.726 55.06613.02213.02213.02250.061 55.45213.79213.79213.79250.447 55.90314.69114.69114.69150.897 56.43415.75315.75315.75351.429 57.07217.02717.02717.02752.067 57.85218.58418.58418.58452.846 58.82620.52920.52920.52953.82 59.36321.60221.60221.60254.357 60.07923.03223.03223.03255.073 61.0825.03325.03325.03356.074 62.5828.03528.03528.03557.574 63.41329.70429.70429.70458.407 65.0833.03933.03933.03960.074 65.0833.04133.04133.04160.074ft

PAGE 197

051015202530354045 30 40 50 60 70 Average Free Strand Set 1 LengthStrand NumberAverage Strand Length (ft)

PAGE 198

051015202530354045 5 10 15 20 25 30 35 40 Average Free Strand Set 2 LengthStrand NumberAverage Strand Length (ft)

PAGE 199

051015202530354045 5 10 15 20 25 30 35 40 Average Free Strand Set 3 LengthStrand NumberAverage Strand Length (ft)

PAGE 200

051015202530354045 5 10 15 20 25 30 35 40 Average Free Strand Set 4 LengthStrand NumberAverage Strand Length (ft)

PAGE 201

051015202530354045 20 30 40 50 60 70 Average Free Strand Set 5 LengthStrand NumberAverage Strand Length (ft)

PAGE 202

Reverse Transfer Length "RTL" TensTransferLengthReverseTransLengthPerStrand TensTransferLength 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 -0.648-0.525-0.521-0.525-0.644 -0.562-0.187-0.178-0.187-0.55 -0.4590.0070.1640.155-0.455 -0.3370.2140.2850.278-0.323 -0.2080.3550.4440.384-0.19 -0.0780.4950.590.502-0.056 0.0480.6760.7570.6130.08 0.1810.8080.8950.8950.197 0.3091.021.021.0130.335 0.461.0271.191.1250.475 0.5911.2221.3621.2090.617 0.7231.3951.51.50.738 0.8571.5931.6521.5850.883 0.9931.7461.8361.6761.029 1.131.8841.9761.9761.153 1.2692.0642.1542.021.303 1.4092.1862.3022.3021.428 1.5522.3492.4892.4891.555 1.6962.5052.6532.5111.711 1.8142.7852.7942.7751.84 1.9632.8512.9832.9831.97 2.1132.9493.1553.1472.101 2.2343.2683.2933.2882.233 2.3573.4653.4733.4712.367 2.483.6453.6583.6212.501 2.6043.7873.8533.7482.637 2.733.8924.0553.8522.774in

PAGE 203

TensTransferLength 12345 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 1.8142.7852.7942.7751.84 1.9632.8512.9832.9831.97 2.1132.9493.1553.1472.101 2.2343.2683.2933.2882.233 2.3573.4653.4733.4712.367 2.483.6453.6583.6212.501 2.6043.7873.8533.7482.637 2.733.8924.0553.8522.774 2.8563.9674.1994.1892.87 2.9424.3534.3634.3542.965 3.0714.3524.5614.5613.059 3.1534.6744.6844.6743.152 3.2344.8294.8384.8293.242 3.3124.9494.9584.9493.33 3.3885.0315.045.0323.415 3.465.0735.0825.0733.496 3.5285.075.0785.0713.572 3.5915.0195.0275.023.642 3.6474.9174.9244.9173.705 3.6944.7594.7664.7593.758 3.734.5424.5484.5423.798 3.754.2624.2674.2623.821 00000 00000 00000 00000 00000 00000 00000in

PAGE 204

051015202530354045 0 0.5 1 1.5 2 2.5 3 3.5 4 Reverse Transfer Length Strand Set 1Strand NumberReverse Transfer Length (in)

PAGE 205

051015202530354045 0 1 2 3 4 5 6 Reverse Transfer Length Strand Set 2Strand NumberReverse Transfer Length (in)

PAGE 206

051015202530354045 0 1 2 3 4 5 6 Reverse Transfer Length Strand Set 3Strand NumberReverse Transfer Length (in)

PAGE 207

051015202530354045 0 1 2 3 4 5 6 Reverse Transfer Length Strand Set 4Strand NumberReverse Transfer Length (in)

PAGE 208

051015202530354045 0 0.5 1 1.5 2 2.5 3 3.5 4 Reverse Transfer Length Strand Set 5Strand NumberReverse Transfer Length (in)

PAGE 209

FORCE RESULTS:Unbalanced Tension Pull UTP": Positive number indicates beam wants to move to the right, negative number indicates beam wants to move to the left UnbalanceForcePerStrand 1234 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 10.6520.357-0.363-10.317 31.9280.736-0.717-30.877 38.74313.066-0.713-50.758 44.8825.794-0.583-48.959 44.7977.064-4.767-45.676 44.5517.383-6.839-43.411 47.7246.146-10.981-40.472 46.4876.4070-51.751 51.3770-0.457-49.052 40.03611.473-4.574-45.833 43.3939.619-10.508-40.7 44.9447.0120-50.984 47.9093.872-4.414-45.714 47.6655.705-10.123-40.946 46.45.630-50.626 47.4425.396-8.014-42.793 44.9736.7080-50.575 44.6837.8640-52.356 43.8818.022-7.712-43.38 50.8950.49-1.003-49.027 44.9966.6670-51.293 40.78710.056-0.376-51.067 48.5911.197-0.24-49.601 50.0940.364-0.052-49.95 50.5530.584-1.617-48.596 49.2162.721-4.357-46.227kip

PAGE 210

UnbalanceForcePerStrand 1234 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 50.0940.364-0.052-49.95 50.5530.584-1.617-48.596 49.2162.721-4.357-46.227 46.2376.476-8.064-42.907 42.1628.832-0.376-50.105 51.0440.371-0.347-50.216 44.0087.1930-51.595 49.4980.321-0.302-49.567 49.0270.294-0.28-48.789 47.3520.268-0.256-46.854 44.5780.243-0.231-43.863 40.8290.217-0.207-39.94 36.2470.192-0.183-35.228 30.9960.168-0.16-29.892 25.2630.144-0.137-24.117 19.2590.121-0.116-18.115 13.2180.1-0.095-12.118 7.40.08-0.076-6.385 5.090.059-0.056-4.136 2.5850.048-0.045-1.728 0.6830.037-0.035-0.215 -0.368-0.0170.0210.666 -0.051-0.0120.0140.245 -0.473-0.0070.0090.565 0.0290.008-0.002-0.006kip

PAGE 211

051015202530354045 0 10 20 30 40 50 60 Unbalanced Tension Pull on Beam 1Strand NumberUnbalanced Tension Pull (kip)

PAGE 212

051015202530354045 0 2 4 6 8 10 12 14 Unbalanced Tension Pull on Beam 2Strand NumberUnbalanced Tension Pull (kip)

PAGE 213

051015202530354045 12 10 8 6 4 2 0 Unbalanced Tension Pull on Beam 3Strand NumberUnbalanced Tension Pull (kip)

PAGE 214

051015202530354045 60 50 40 30 20 10 0 Unbalanced Tension Pull on Beam 4Strand NumberUnbalanced Tension Pull (kip

PAGE 215

Acting Static Friction Force negative value indicates that friction is acting in the same direction as the friction at the op posite beam end Any number other than that equal to Fs is indicating the the friction is forced to change to prevent global motion of the beam FrictionValuePerStrand 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 25.915.226.826.426.727.115.926.2 25.9-626.82626.427.1-4.626.2 25.9-12.826.813.726.427.1-24.526.2 25.9-1926.82126.527.1-22.726.2 25.9-18.926.819.722.327.1-19.426.2 25.9-18.726.819.420.327.1-17.226.2 25.9-21.826.820.616.127.1-14.226.2 25.9-20.626.820.427.127.1-25.526.2 25.9-25.526.826.826.627.1-22.826.2 25.9-14.126.815.322.527.1-19.626.2 25.9-17.526.817.116.627.1-14.526.2 25.9-1926.819.727.127.1-24.726.2 25.9-2226.822.922.727.1-19.526.2 25.9-21.826.821.11727.1-14.726.2 25.9-20.526.821.127.127.1-24.426.2 25.9-21.526.821.419.127.1-16.526.2 25.9-19.126.820.127.127.1-24.326.2 25.9-18.826.818.927.127.1-26.126.2 25.9-1826.818.719.427.1-17.126.2 25.9-2526.826.326.127.1-22.826.2 25.9-19.126.820.127.127.1-2526.2 25.9-14.926.816.726.727.1-24.826.2 25.9-22.726.825.626.927.1-23.426.2 25.9-24.226.826.427.127.1-23.726.2 25.9-24.726.826.225.527.1-22.426.2 25.9-23.326.82422.727.1-2026.2 25.9-20.326.820.31927.1-16.726.2kip

PAGE 216

FrictionValuePerStrand 12345678 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 25.9-24.726.826.225.527.1-22.426.2 25.9-23.326.82422.727.1-2026.2 25.9-20.326.820.31927.1-16.726.2 25.9-16.326.817.926.727.1-23.926.2 25.9-25.126.826.426.827.1-2426.2 25.9-18.126.819.627.127.1-25.426.2 25.9-23.626.826.426.827.1-23.326.2 25.9-23.126.826.526.827.1-22.526.2 25.9-21.526.826.526.927.1-20.626.2 25.9-18.726.826.526.927.1-17.626.2 25.9-14.926.826.526.927.1-13.726.2 25.9-10.326.826.626.927.1-926.2 25.9-5.126.826.626.927.1-3.626.2 25.90.626.826.62727.12.126.2 25.96.626.826.62727.18.126.2 25.912.726.826.72727.114.126.2 25.918.526.826.72727.119.926.2 25.920.826.826.727.127.122.126.2 25.923.326.826.727.127.124.526.2 25.925.226.826.727.127.12626.2 25.525.926.726.827.127.126.225.6 25.825.926.726.827.127.126.226 25.425.926.826.827.127.126.225.7 25.925.926.826.827.127.126.226.2kip Maximum Static friction force Fs (shown as a reference) FRfw25.89825.89826.76226.76227.10727.10726.24426.244 ()kip

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051015202530354045 25 25.2 25.4 25.6 25.8 26 26.2 26.4 Acting Static Friction Force B 1 End 1Strand NumberActing Static Friction Force (kip)

PAGE 218

051015202530354045 30 20 10 0 10 20 30 Acting Static Friction Force B 1 End 2Strand NumberActing Static Friction Force (kip)

PAGE 219

051015202530354045 26 26.2 26.4 26.6 26.8 27 27.2 27.4 Acting Static Friction Force B 2 End 1Strand NumberActing Static Friction Force (kip)

PAGE 220

051015202530354045 16 18 20 22 24 26 28 30 Acting Static Friction Force B 2 End 2Strand NumberActing Static Friction Force (kip)

PAGE 221

051015202530354045 16 18 20 22 24 26 28 Acting Static Friction Force B 3 End 1Strand NumberActing Static Friction Force (kip)

PAGE 222

051015202530354045 26.6 26.8 27 27.2 27.4 Acting Static Friction Force B 3 End 2Strand NumberActing Static Friction Force (kip)

PAGE 223

051015202530354045 20 10 0 10 20 30 Acting Static Friction Force B 4 End 1Strand NumberActing Static Friction Force (kip)

PAGE 224

051015202530354045 25.6 25.8 26 26.2 26.4 Acting Static Friction Force B 4 End 2Strand NumberActing Static Friction Force (kip)

PAGE 225

Total Tension Pull Force in Each Free Strand Set TensPullPerStrand 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 -56.25-45.6-45.24-45.61-55.92 -47.81-15.88-15.15-15.86-46.74 -38.20.5413.6112.9-37.86 -27.4517.4323.2322.64-26.32 -16.5628.2335.330.53-15.15 -6.0438.5145.939.06-4.35 3.6451.3657.5146.536.06 13.4459.9366.3466.3414.59 22.3973.7773.7773.3124.26 32.4372.4683.9479.3633.53 40.683.9993.6183.142.4 48.3993.34100.35100.3549.36 55.81103.72107.59103.1757.46 62.84110.5116.21106.0865.14 69.48115.88121.51121.5170.88 75.72123.16128.56120.5477.75 81.57126.54133.25133.2582.67 87131.69139.55139.5587.19 92.03135.91143.93136.2292.84 95.15146.05146.54145.5396.51 99.39144.39151.05151.0599.76 103.2143.99154.04153.67102.6 105.07153.66154.86154.62105.02 106.55156.64157.01156.96107.01 107.64158.19158.78157.16108.56 108.33157.54160.26155.91109.68 108.61154.85161.32153.26110.35kip

PAGE 226

TensPullPerStrand 12345 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 106.551156.644157.008156.956107.006 107.638158.191158.775157.158108.562 108.327157.543160.264155.907109.68 108.611154.848161.324153.259110.352 108.484150.646159.478159.102108.998 106.404157.448157.819157.472107.257 105.519149.527156.72156.72105.125 102.651152.149152.47152.168102.601 99.429148.456148.75148.46999.68 95.85143.202143.47143.21496.36 91.913136.491136.734136.50292.639 87.616128.444128.662128.45588.514 82.957119.203119.396119.21383.984 77.936108.932109.1108.9479.048 72.55397.81697.9697.82373.706 66.8186.06986.1986.07567.96 60.70973.92874.02873.93361.815 54.25961.65961.73961.66255.278 48.05953.1553.20953.15349.017 41.54244.12744.17444.12942.401 34.53935.22235.25935.22535.009 26.87826.5126.49326.51427.179 19.84419.79319.78119.79520.04 12.82612.35312.34612.35512.919 6.4476.4776.4856.4826.477kip

PAGE 227

051015202530354045 0 20 40 60 80 100 120 Tension Pull Strand Set 1Strand NumberTension Pull (kip)

PAGE 228

051015202530354045 0 20 40 60 80 100 120 140 160 Tension Pull Strand Set 2Strand NumberTension Pull (kip)

PAGE 229

051015202530354045 0 20 40 60 80 100 120 140 160 180 Tension Pull Strand Set 3Strand NumberTension Pull (kip)

PAGE 230

051015202530354045 0 20 40 60 80 100 120 140 160 Tension Pull Strand Set 4Strand NumberTension Pull (kip)

PAGE 231

051015202530354045 0 20 40 60 80 100 120 Tension Pull Strand Set 5Strand NumberTension Pull (kip)

PAGE 232

Tension Pull in free strand sets "TP" (By Strand Set) TensPullEndPerStrand 12345 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 -46.877-38-37.703-38.005-46.602 -39.672-13.178-12.568-13.163-38.784 -31.5550.4511.24310.655-31.275 -22.5714.33319.09718.617-21.638 -13.55223.128.87924.978-12.393 -4.91431.34837.35731.791-3.544 2.94641.5846.55537.6664.903 10.82148.23753.39553.39511.741 17.91159.01359.01358.64719.406 25.77657.666.71963.08426.652 32.05166.30973.90365.60733.475 37.92973.15578.65178.65138.691 43.40580.66883.6880.24644.691 48.47385.24489.64581.83650.248 53.12988.61192.91692.91654.203 57.36493.30597.39391.32258.903 61.17494.90499.93499.93462.003 64.55197.703103.537103.53764.693 67.48999.669105.55299.89668.084 68.903105.758106.113105.38769.884 70.993103.133107.895107.89571.258 72.621101.323108.4108.13572.199 72.74106.38107.209107.04372.703 72.454106.518106.766106.7372.764 71.759105.461105.85104.77272.375 70.648102.745104.52101.67871.53 69.11698.539102.6697.52970.224kip

PAGE 233

TensPullEndPerStrand 12345 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 72.74106.38107.209107.04372.703 72.454106.518106.766106.7372.764 71.759105.461105.85104.77272.375 70.648102.745104.52101.67871.53 69.11698.539102.6697.52970.224 67.15793.25798.72598.49267.475 63.84294.46994.69194.48364.354 61.0986.56890.73390.73360.862 57.02884.52784.70684.53857 52.63978.59478.7578.60152.772 47.92571.60171.73571.60748.18 42.89363.69663.80963.70143.232 37.5555.04855.14155.05237.935 31.90645.84745.92145.85132.302 25.97936.31136.36736.31326.349 19.78726.67726.71626.67920.102 13.36217.21417.23817.21513.592 6.7458.2148.2258.2156.868 00000 00000 00000 00000 00000 00000 00000 00000kip

PAGE 234

051015202530354045 0 10 20 30 40 50 60 70 80 Beam End Tension Pull Strand Set 1Strand NumberBeam End Tension Pull (kip)

PAGE 235

051015202530354045 0 20 40 60 80 100 120 Beam End Tension Pull Strand Set 2Strand NumberBeam End Tension Pull (kip)

PAGE 236

051015202530354045 0 20 40 60 80 100 120 Beam End Tension Pull Strand Set 3Strand NumberBeam End Tension Pull (kip)

PAGE 237

051015202530354045 0 20 40 60 80 100 120 Beam End Tension Pull Strand Set 4Strand NumberBeam End Tension Pull (kip)

PAGE 238

051015202530354045 0 10 20 30 40 50 60 70 80 Beam End Tension Pull Strand Set 5Strand NumberBeam End Tension Pull (kip)

PAGE 239

Transferred Prestress Force Linearly Interpolated at Reverse Transfer Length "CRTL" CompAtRTLPerStrand 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 -1.172-0.95-0.95-0.943-0.943-0.95-0.95-1.165 -2.034-0.676-0.676-0.645-0.645-0.675-0.675-1.989 -2.4910.0360.0360.8880.8880.8410.841-2.469 -2.441.551.552.0652.0652.0132.013-2.339 -1.8823.2083.2084.0114.0113.4693.469-1.721 -0.8425.3745.3746.4046.4045.455.45-0.608 0.6078.5618.5619.5859.5857.7557.7551.009 2.62311.69411.69412.94412.94412.94412.9442.846 5.03816.59716.59716.59716.59716.49416.4945.458 8.31518.58118.58121.52221.52220.3520.358.597 11.75224.31324.31327.09827.09824.05624.05612.274 15.69530.27130.27132.54532.54532.54532.54516.01 20.15237.45337.45338.85138.85137.25737.25720.749 25.13444.244.246.48246.48242.43342.43326.055 30.65151.12251.12253.60653.60653.60653.60631.271 36.71359.71559.71562.33262.33258.44658.44637.698 43.33167.22367.22370.78770.78770.78770.78743.919 50.51876.46376.46381.02981.02981.02981.02950.629 58.28686.07786.07791.15891.15886.27486.27458.8 65.622100.722100.722101.06101.06100.369100.36966.557 74.543108.29108.29113.29113.29113.29113.2974.821 84.088117.322117.322125.515125.515125.209125.20983.599 92.946135.93135.93136.989136.989136.777136.77792.899 102.288150.378150.378150.728150.728150.678150.678102.725 112.123164.783164.783165.391165.391163.707163.707113.086 122.456178.092178.092181.168181.168176.243176.243123.986 133.295190.04190.04197.988197.988188.091188.091135.432kip

PAGE 240

CompAtRTLPerStrand 12345678 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 102.288150.378150.378150.728150.728150.678150.678102.725 112.123164.783164.783165.391165.391163.707163.707113.086 122.456178.092178.092181.168181.168176.243176.243123.986 133.295190.04190.04197.988197.988188.091188.091135.432 144.646200.862200.862212.637212.637212.136212.136145.33 154.286228.299228.299228.838228.838228.334228.334155.522 166.61236.096236.096247.453247.453247.453247.453165.987 176.788262.035262.035262.588262.588262.067262.067176.701 187.16279.447279.447279.999279.999279.472279.472187.633 197.691295.354295.354295.907295.907295.379295.379198.743 208.336309.38309.38309.929309.929309.405309.405209.982 219.039321.111321.111321.654321.654321.136321.136221.286 229.726330.102330.102330.634330.634330.127330.127232.572 240.302335.873335.873336.39336.39335.898335.898243.731 250.637337.911337.911338.409338.409337.935337.935254.621 260.558335.669335.669336.142336.142335.692335.692265.045 269.82328.568328.568329.013329.013328.59328.59274.732 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297 278.076316316316.411316.411316.02316.02283.297kip

PAGE 241

051015202530354045 0 50 100 150 200 250 300 Transferred Prestress at RTL B 1 End 1Strand NumberTransferred Prestress Force (kip)

PAGE 242

051015202530354045 0 50 100 150 200 250 300 350 Transferred Prestress at RTL B 1 End 2Strand NumberTransferred Prestress Force (kip)

PAGE 243

051015202530354045 0 50 100 150 200 250 300 350 Transferred Prestress at RTL B 2 End 1Strand NumberTransferred Prestress Force (kip)

PAGE 244

051015202530354045 0 50 100 150 200 250 300 350 Transferred Prestress at RTL B 2 End 2Strand NumberTransferred Prestress Force (kip)

PAGE 245

051015202530354045 0 50 100 150 200 250 300 350 Transferred Prestress at RTL B 3 End 1Strand NumberTransferred Prestress Force (kip)

PAGE 246

051015202530354045 0 50 100 150 200 250 300 350 Transferred Prestress at RTL B 3 End 2Strand NumberTransferred Prestress Force (kip)

PAGE 247

051015202530354045 0 50 100 150 200 250 300 350 Transferred Prestress at RTL B 4 End 1Strand NumberTransferred Prestress Force (kip)

PAGE 248

051015202530354045 0 50 100 150 200 250 300 Transferred Prestress at RTL B 4 End 2Strand NumberTransferred Prestress Force (kip)

PAGE 249

Intermediate calculations to determine cracking potential of each beam end: CTNoTS outjk CrackPredictorPerStrandjk ConcAllowableTension Abf j1rowsCrackPredictorPerStrand () for k1colsCrackPredictorPerStrand () for out Accounts for eccentricity of friction force CTNoTS2 outjk CTNoTSjk FrictionValuePerStrandjk Frice2 IBottom j1rowsCTNoTS () for k1colsCTNoTS () for out

PAGE 250

Increases the size of the matrix TensTransferLengthw outq1 TensTransferLengthq1 outq2colsTensTransferLength ()2 TensTransferLengthqcolsTensTransferLength () DebondLengthq1 0ft = if outq1 outq1 1 outq2colsTensTransferLength ()2 outq1 2colsTensTransferLength ()2 DebondLengthq1 0ft if q1rowsTensTransferLength () for outz2j 2 TensTransferLengthzj outz2j 1 TensTransferLengthzj DebondLengthz1 0ft = if outz2j 2 outz1 2j 2 outz2j 1 outz1 2j 1 DebondLengthz1 0ft if z1rowsTensTransferLength () for j2colsTensTransferLength ()1 for out

PAGE 251

Accounts for eccentricity of bearing force (assuming that it acts at the very end of the beam) CTNoTS3 outjk CTNoTS2jk Bearingw1k TensTransferLengthwjk Frice IBottom j1rowsCTNoTS2 () for k1colsCTNoTS2 () for out Cracking Potential of each beam end bottom flange = Actual Stresses/ Allowable Tension Stress CrackingPotential outjk CTNoTS3jk 1 ConcTensStrength j1rowsCTNoTS2 () for k1colsCTNoTS2 () for out

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Cracking Criterion: Above 1.0 indicates vertical crack probable CP = Actual Stress/ Allowable Stress CrackingPotential 12345678 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0.15430.01020.23750.23270.23830.24320.02260.162 0.2137-0.22150.43290.42290.42980.4395-0.19390.2262 0.2788-0.25330.53830.35970.61390.6236-0.4090.2868 0.3491-0.27240.64310.56330.6750.683-0.31430.3622 0.4178-0.20760.7070.60890.66260.7287-0.20360.4328 0.482-0.14540.76510.66160.68070.7763-0.11140.4987 0.5387-0.13940.83550.74860.66050.8154-0.01410.5597 0.5936-0.07310.87780.78640.92330.9233-0.13320.6073 0.641-0.10290.94680.94680.94680.9534-0.04980.6589 0.69150.1060.92910.76220.90960.9760.03650.7054 0.72910.08780.97910.83770.82980.98390.14650.7467 0.76180.09031.01360.90941.05971.05970.01520.7753 0.78970.06571.04930.99110.99081.0570.11730.8066 0.81260.08361.06320.97640.89911.05270.20780.8325 0.83040.11191.0670.98031.10411.10410.06410.8462 0.84320.09891.07570.99160.95091.07560.19190.8617 0.85070.13541.06360.95761.10361.10360.06280.8652 0.85290.1321.05530.92921.09871.09870.02130.864 0.84970.1311.03910.90850.92751.05270.15450.8633 0.836-0.01091.03891.03081.03361.05020.03770.8515 0.82230.06090.99320.88111.02781.0278-0.03110.8347 0.80280.10030.94910.7770.98430.9908-0.06450.8127 0.7739-0.07880.92770.90680.94010.9443-0.08250.7856 0.7401-0.15650.88110.87460.89540.8963-0.13840.7532 0.7015-0.22260.82520.81460.80890.8382-0.17030.7155 0.6579-0.2620.75980.70930.6920.7726-0.18920.6724

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CrackingPotential 12345678 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 0.7739-0.07880.92770.90680.94010.9443-0.08250.7856 0.7401-0.15650.88110.87460.89540.8963-0.13840.7532 0.7015-0.22260.82520.81460.80890.8382-0.17030.7155 0.6579-0.2620.75980.70930.6920.7726-0.18920.6724 0.6093-0.27580.68710.5640.54810.7009-0.19520.6238 0.5557-0.27220.60960.43750.62260.6299-0.41180.5691 0.4972-0.53280.52870.52120.53820.5452-0.49980.5105 0.4341-0.48040.44040.29210.45290.4529-0.62080.448 0.3679-0.69360.33870.33190.34950.3559-0.67780.3818 0.2985-0.78950.23430.22790.24570.2519-0.76720.312 0.226-0.86440.12650.12040.13850.1443-0.83670.2387 0.1507-0.91820.01770.0120.03020.0357-0.8860.1622 0.0727-0.9506-0.0894-0.0948-0.0765-0.0713-0.91480.0828 -0.0075-0.9615-0.1918-0.1968-0.1785-0.1738-0.92310.0007 -0.0897-0.9507-0.2863-0.2909-0.2727-0.2684-0.9107-0.0834 -0.173-0.9185-0.3692-0.3734-0.3555-0.3515-0.8779-0.1689 -0.2568-0.865-0.4366-0.4404-0.4229-0.4193-0.8248-0.255 -0.3399-0.7909-0.4844-0.4877-0.4707-0.4675-0.752-0.3403 -0.4206-0.6965-0.5079-0.5108-0.4944-0.4916-0.66-0.4229 -0.4206-0.6633-0.5079-0.5105-0.4941-0.4916-0.6276-0.4229 -0.4206-0.6272-0.5079-0.5103-0.494-0.4916-0.593-0.4229 -0.4206-0.5999-0.5079-0.5102-0.4938-0.4916-0.5712-0.4229 -0.4259-0.5901-0.5081-0.5097-0.4933-0.4919-0.5682-0.4325 -0.4214-0.5901-0.508-0.5097-0.4933-0.4918-0.5682-0.4264 -0.4274-0.5901-0.508-0.5097-0.4933-0.4917-0.5682-0.431 -0.4206-0.5905-0.5079-0.5098-0.4934-0.4916-0.5682-0.4229

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 1 End 1Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 1 End 2Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 2 End 1Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 2 End 2Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 3 End 1Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 3 End 2Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 4 End 1Strand Numberfact/ft

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051015202530354045 0.5 0 0.5 1 1.5 CP Beam 4 End 2Strand Numberfact/ft

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APPENDIX C SIMPLIFIED VERTICAL CRACK PREDICTOR

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ORIGIN1 A simplified procedure and example for hand calculation of the cracking criterion ( = fcalc/f) for the case of a single symmetrically placed beam on the casting bed. Lc Ls Ls Prestressed Beam Given: Prestress Values : Total Number of uncut prestressing strands NumUncutStrands20 Total Number of cut prestressing strands NumCutStrands10 Jacking Force per prestressing strand JackingForce44 (kip) Area of prestressing strand Aps.2192 (in^2) Modulus of Elasticity of prestressing strand Es28500 (ksi) Temperature strain in free strands ps.00000667 (in/in/F) Diameter of prestressing strands D0.600 (in) Length of free strands L 15 (ft) Concrete Values : Concrete strength at the time of strand cutting fci6000 (psi) Cross sectional area of prestressed beam A 789 (in^2) Bottom flange area of prestressed beam Abf361 (in^2) Unit weight of the concrete beam 150 (pcf) Beam Length Lc120 (ft)

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Other Values : Temperature Change between the time of beam casting and the time of strand detensioning where a positive number indicates that the temperature at the time of cutting is lower than the temperature at the time of casting TempChange30 (F) Static coefficient of friction between bottom of the beam and the casting bed s .40 Dynamic coefficient of friction between bottom of the beam and the casting bed d.35 Expected initial camber CamberIn3 (in) Distance from centroid of cross section to bottom of beam Dcentroid24.73 (in) Distance from centroid of bottom flange to bottom of beam DcentroidBF7.266 (in) Moment of inertia of bottom flange IbottomF7280 (in^4) Solution: Step 1 : Calculate the tension pull due to temperature change "TPtemp" TPtempTempChange ps Es Aps NumUncutStrands TPtemp25.001 (kip) Step 2 : Calculate the free strand spring stiffness "ks" ks ApsEs L12 NumUncutStrands ks694.13 (kip/in) Step 3 : Calculate the compression transfer length of the prestressing strands "lt" lt.33 JackingForce Aps D 3fci 1000 lt28.104 (in)

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Step 4 : Calculate the concrete modulus of elasticity "E" E 40000fci 106 145 1.51000 E4312.189 (ksi) Step 5 : Calculate the beam spring stiffness "kc" kc AE Lc12 4 3 lt kc2425.845 (kip/in) Step 6 : Calculate friction forces "Fs" and "Fd": Fs sA Lc 2144 1000 Fs19.725 (kip) Fd dA Lc 2144 1000 Fd17.259 (kip) Step 7 : Calculate beam movement ": check to see if static friction force has been overcome JackingForceNumCutStrands 440 (kip) Fs19.725 kip () if this number is less than Fs the beam movement = 0 otherwise 1 JackingForceNumCutStrands Fd 2kcks () 10.068 (in)

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Atot NumUncutStrandsNumCutStrands ()JackingForce Lc 12 AE Atot0.559 (in) Ctot CamberInDcentroid Lc 2 12 Ctot0.103 (in) 1 1 Atot Ctot 0.08 in () Step 8 : Calculate the tension pull due to beam movement "TPm" TPm ks TPm55.699 (kip) Step 9 : Calculate the total tension pull "TP" TPTPtempTPm TP80.7 (kip) Step 10 : Calculate the reverse transfer length "RTL" RTL.33 TP NumUncutStrandsAps D 3fci 1000 RTL2.577 (in) Step 11 : Calculate the prestress transferred to the concrete linearly interpolated at the RTL "CRTL" CRTLJackingForce RTL lt NumCutStrands CRTL40.35 (kip)

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Step 12 : Calculate the beam tension strength "TS" TS 5fci 1000 TS0.387 (ksi) Step 13 : Calculate the cracking criterion 1 CRTLTP Fs A A Lc 2144 1000 RTL DcentroidBF IbottomF FsDcentroidBF2 IbottomF TS 0.893 This procedure should be repeated for other numbers of cut strands to determine the maximum The maximum usually occurs when approximately one third of the strands have been cut. If is greater than 1.0 the section is assumed to have vertically cracked.

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256 APPENDIX D FIELD STUDY STRAND LAYOUT Debonded Strands: Triangle = 15 Square = 10 Circle = 5

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257 LIST OF REFERENCES Abrishami, Homayoun G., Mitchell, Denis, "Bond Characteristics of Pretensioned Strand," ACI Materials Journal, Vol. 90, No. 3, May 1993, pp. 228-235. American Concrete Institute Committee 318, Building Code Requirements for Structural Concrete and Commentary, American Concrete Institute, Farmington Hills, 2002. Barr, P. J., Stanton, J. F., Eberhard, M. O ., "Effects of Temperature Variations on Precast, Prestressed Concrete Bridge Girders," Journal of Bridge Engineering, Vol. 10, No. 2, March/April 2005, pg. 189. Cook, Robert D., Malkus, David S., Plesha, Michael E., Witt, Robert J., Concepts and Applications of Finite Element Analysis 4th ed, John Wiley & Sons, Inc, New York, 2002. Hibbler, R.C., Mechanics of Materials 4th ed, Prentice Hall, Uppe r Saddle River, 2000. Kannel, Jeffery J., and French, Catherin e E., and Stolarski, Henryk K., Release Methodology of Prestressing Strands, Mi nnesota Department of Transportation, Minneapolis, May 1998. MacGregor, James G., Reinforced Concrete : Mechanics and Design 3rd ed, Prentice Hall, Upper Saddle River, 1997. Mirza, J.F., and Tawfik, M.E., End Cr acking in Prestressed Members during Detensioning, PCI Journal, Vol. 23, No. 2, March/April 1978, pp. 66-78. Naaman, Antoine E., Prestressed Concrete Analysis and Design 2nd ed, Techno Press 3000, Ann Arbor, 2004. Nilson, Arthur H., Design of Prestressed Concrete, 2nd ed, John Wiley & Sons, New York, 1987. Portland Cement Association, Notes on ACI 318-02 Building Code Requirements for Structural Concrete with Design Applications, Portland Cement Association, Skokie, 2002. Prestressed Concrete Institute, PCI Design Handbook: Precast and Prestressed Concrete 5th ed, Prestressed Concrete Institute, Chicago, 1999.

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258 BIOGRAPHICAL SKETCH I was born in Winter Park, Florida, to Beverly and Daniel Reponen. I enjoy swimming, saltwater fishing, canoeing, windsurfing, weightli fting, softball, and other physical activities. After graduation I plan to move to the east coast of Florida and save up money to purchase a beach condo. I have had a fun and educational college career and I will always miss being a student at the University of Florida.