reponen_m

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

Copyright 2006

by

Michael Reponen

This document is dedicated to my parents.

iv

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.

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, French, Stolarski 1998...................................................................8 Summary of End Cracking Reduction Recommendations ...........................................9

3 MANUFACTURER SURVEY AND 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

vi

5 RESULTS ...................................................................................................................37

Introduction.................................................................................................................37 Test Case 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 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 Debonded Lengths of 10 Strands ...............................40 Modification 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

Test Case 2..................................................................................................................42 Modification 1: Alter the 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

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 Concrete Release Strength..........................................................................51

5-5 Alter the Beam Lengths.............................................................................................52

5-6 Alter the Temperature Change ..................................................................................54

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

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

5-9 Alter the Number of Beams.......................................................................................58

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



5-13 Test Case 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

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 Concrete Bottom Flange .............................................................................34

4-5 Axial and Camber Movement....................................................................................35

4-6 Analytical Model Flow Chart ....................................................................................36

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 Prestressing 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


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





5-25 Test Case 2...............................................................................................................65

x

5-26 Alter the Friction Coefficient 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

xi

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

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 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.

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 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

2

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 (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 from zero at the end of the beam, to the full

3

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/oF (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).

4

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 (µs) multiplied by the

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

dynamic friction (µd) 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

5

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

Figure 1-3. General Effects of Friction

Free Strand

Distance along beam

Force

Reverse Transfer Length

Tension Pull

Prestressing Force

Force in Free Strand

∆

Cut Strands

Weight = W

N = W/2 N = W/2

F F

For F less than max static friction forceNo movement

F F

Weight = W

N = W/2 N = W/2

F F

For F greater than max static friction force Movement occurs

Fs = µs*N Fs = µs*N

Concrete elastic shortening and rotation

6

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

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 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

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 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

9

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

11

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.

12

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

movement occured 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

13

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 δ and the radius shrinks by an amount δ`. The ratio of the strain created by δ

to the strain created by δ` 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

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 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.

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 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 occured 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

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 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

17

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

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

18

Figure 3-4. Radial Cracking

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

Top flange vertical crack

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

20

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

Crack.

Figure 3-10. Vertical Crack

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 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.

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, 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 ∆tot is very small compared to (L-3Lb).

23

c cc

c

A Ek3L

= (4-1)

s ss

c

A EkL 3L

=−

(4-2)

totc s

Fk k

∆ =+

(4-3)

s totTP k= ∆ (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 system to regain equilibrium, Beams 1 and 3 are

forced to react according to equation 4-6.

totBeam2End1 Beam2End2 6

∆∆ = ∆ = (4-5)

Beam1End2 Beam3End1 0∆ = ∆ =

totBeam1End1 Beam3End2 3

∆∆ = ∆ = (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.

24

tot totslide slide

s2 s1

6 3L L

∆ ∆−∆ + ∆

= (4-7)

s1 s2tot

slides2 s1

L L( )6 3

L L

∆ −∆ =

+ (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, 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.

c cc

c

A Ek

L= (4-9)

s ss1 s4

s1

A Ek kL

= = s ss2 s3

s2

A Ek kL

= = (4-10)

s ssBeam2

s2

s s s s

s2 s2

A E1k 1 1 2LA E A EL L

= =+

s ssBeam1 sBeam3

s1 s2

s s s s

s1 s2

A E1k k 1 1 L LA E A EL L

= = =++

(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

25

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.

dBeam2End1 Beam2End2

c c s s

c s2

.5(F F )A E A E

L 2L

−∆ = ∆ =

+ (4-12)

FreeStrands2 FreeStrands3 Beam2End1 s2TP TP k= = ∆ (4-13)

sBeam2End1 sF F= sBeam2End2 sF F= (4-14)

Because Ls1 is much larger than Ls2, and equilibrium must be satisfied, all beam

shortening for Beam1 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.

Beam1End2 Beam3End1 0∆ = ∆ =

dBeam1End1 Beam3End2

c c s s

c s1 s2

F FA E A E

L L L

−∆ = ∆ =

++

(4-15)

FreeStrands1 FreeStrands4 Beam1End1 s1TP TP k= = ∆ (4-16)

sBeam1End1 sBeam3End2 sF F F= =

sBeam1End2 sBeam3End1 s FreeStrands2 FreeStrands1F F F (TP TP )= = − − (4-17)

26

Analytical Commentary

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

referred to as the ∆UTP “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 ∆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 the 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 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

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 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

∑F = 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 “fcalc” is the calculated

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

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 1f

≥ Vertical Crack Forms (4-18)

ciff 5

psi= (4-19)

The calculated stress in the concrete bottom flange (fcalc) 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 fcalc (See Figure 4-4).

2s s

calcbf bf bf bf bf

F *e FCRTL N*RTL*e TPf =( - - - - )*-1A I A I A

(4-20)

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 δ is the unit weight of the concrete.

6 1.5cif δE = (40000 +10 )( ) psipsi 145pcf

(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 casting and the time of detensioning. The thermal coefficient of the prestressing strands is 6.67x10-6 in/in/oF (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.

29

• 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 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 prestressing strand due to the jacking force. The variable fTP is the stress in the prestressing strand due to the tension pull.

Jci

3TransferLength=0.33f *Df

(4-22)

TPci

3ReverseTransferLength=0.33f *Df

(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

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 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

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 For 3 Beam Symmetrical System

Ls1 >> Ls2 Lb Ls2 Lb Ls2 Lb Ls1>>Ls2

PROPERTIES: Concrete: Steel Friction Ac Ec As Es Fs = µsN Fd = µdN

Beam1 Beam2 Beam3

Free Strands1 Free Strands2 Free Strands3 Free Strands4

End1 End2 End1 End2 End1 End2

L

3 BEAM SYMMETRICALLY PLACED SYSTEM GIVEN: SIMULTANEOUSLY ADD A COMPRESSION FORCE "F" TO ALL BEAM ENDS

32

Figure 4-2. Change in Acting Static Friction Force

Transferred compression From cut strands Tension Pull1

Fs Fs - ∆UTP

For ∆UTP > 0 kipFor ∆UTP < Fs

Fs |Fs - ∆UTP|

For ∆UTP > 0 kip For ∆UTP > Fs

Tension Pull2

FsFs - |∆UTP|

For ∆UTP < 0 kip For |∆UTP| < Fs

Fs|Fs - |∆UTP||

For ∆UTP < 0 kip For |∆UTP| > Fs

Transferred compressionFrom cut strands


Transferred compression From cut strands

Tension Pull1

Tension Pull1

Tension Pull1

Tension Pull2

Tension Pull2

Tension Pull2

33

Figure 4-3. Global Motion of Beam


Fs

For ∆UTP = 2*Fs Impending global motion

Tension Pull1

Fs

No Sliding

Fd

For ∆UTP > 2*Fs global motion occurs

Fd

Sliding

Fs

For |∆UTP| = 2*Fs Impending global 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




34

Figure 4-4. Stress in Concrete Bottom Flange. This figure shows the combination of

stresses a prestressed beam experiences during the detensioning process.

N = W/2 N = W/2 Fs Fs

Beam Bottom Flange Free StrandsFree 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 friction 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 the end face equal to RTL

µ*N

µ*N*e

µ*N Axial Stresses

e

Flexural Stresses

N

N

Lrt*N

e

Flexural Stresses

CRTLForce in cut strands transferred over Lrt

Axial Stresses Tension Pull:

TP Force in uncut strands transferred over Lrt

Axial Stresses

35

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.

Camber

Camber movement

CL

Axial movement

36

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.

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 cut

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 cracking tendency "CT" of prestressed beams, once data input is complete

37

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 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, 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 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).

38

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 fcalc/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 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 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 fcalc/f value for each friction coefficient is shown in bold (See

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 graphically. 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 5-

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

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


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 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 case is shown in bold (See Table 5-6).

40



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 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 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 the prestressed beams on the casting bed.

The fcalc/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 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.

41

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




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 Ls1 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 fcalc/f value for each free strand length is shown in bold (See Table 5-11).




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 Ls1 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 fcalc/f value for each free strand length is shown in bold (See Table 5-12).

42



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 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 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 fcalc/f results for each case at each beam end are

listed in Table 5-16.

43

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.

44

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

45

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.

Figure 5-1. Test Case 1

Lc1 Lc2

Beam 1 Beam 2

Ls1 Ls2 Ls3

End 1 End 1End 2 End 2

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

0 5 10 15 20

Number of Cut Strands

fcal

c / f

Figure 5-2. Test Case 1 No Alterations

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

0 5 10 15 20


fcal

c / f #PS = 30

#PS = 40#PS = 50

Figure 5-3. Alter the Number of Prestressing Strands

48

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

30 35 40 45 50

Number of Prestressing Strands

fcal

c / f

max

imum

s

Figure 5-4. Number of Prestressing Strands fcalc/f Maximums

49

Table 5-3. Alter the Friction Coefficient Number of Cut Strands µ = .15 µ = .25 µ = .35 µ = .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

0 5 10 15 20


fcal

c / f

µ = .15µ = .25µ = .35µ = .45

Figure 5-5. Alter the Friction Coefficient

50

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0.15 0.2 0.25 0.3 0.35 0.4 0.45

Friction Coefficient

fcal

c / f

max

imum

s

Figure 5-6. Friction Coefficient fcalc/f Maximums

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

0 5 10 15 20


fcal

c / f

fci = 6ksifci = 7ksifci = 8ksifci = 9ksi

Figure 5-7. Alter the Concrete Release Strength

52

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

6 6.5 7 7.5 8 8.5 9

Concrete Release Strength (ksi)

fcal

c / f

max

imum

s

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

53

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20


fcal

c / f

L = 100ftL = 120 ftL = 140ftL = 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

100 110 120 130 140 150 160

Beam Lengths (ft)

fcal

c / f

max

imum

s

Figure 5-10. Beam Lengths fcalc/f Maximums

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

0 5 10 15 20


fcal

c / f

∆F = -40∆F = -20∆F = 0∆F = 20∆F = 40

Figure 5-11. Alter the Temperature Change

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 -10 0 10 20 30 40

Temperature Change (deg F)

fcal

c / f

max

imum

s

Figure 5-12. Temperature Change fcalc/f Maximums

Table 5-7. Alter the Number 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

56

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20


fcal

c / f

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

Figure 5-13. Alter the Number of Debonded Strands

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

4 5 6 7 8 9 10

Number of Debonded Strands

fcal

c / f

max

imum

s

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

57

Table 5-8. Alter the Debonded 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

0 5 10 15 20


fcal

c / f

5 ft10 ft15 ft20 ft

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

58

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

5 7 9 11 13 15 17 19

Debonded Length for 10 Strands

fcal

c / f

max

imum

s

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

59

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20


fcal

c / 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

2 3 4

Number of Beams

fcal

c / f

max

imum

s

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

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

0 5 10 15 20


fcal

c / f

40ft50ft60ft70ft

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

61

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

25 30 35 40 45 50 55 60 65 70

Free Strand Length (ft)

fcal

c / f

max

imum

s

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

62


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20


fcal

c / f

40ft50ft60ft70ft


63

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

25 30 35 40 45 50 55 60 65 70


fcal

c / f

max

imum

s


64


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20


fcal

c / f

40ft50ft60ft70ft


65

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

25 30 35 40 45 50 55 60 65 70


fcal

c / f

max

imum

s


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


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

66

Table 5-14. Alter Friction Results Friction Coefficient E1 E2 E3 E4 E5 E6 E7 E8 µ = .45 1.256 0.702 1.643 1.626 1.626 1.643 0.702 1.256 µ = .35 1.097 0.462 1.391 1.372 1.372 1.391 0.462 1.097 µ = .25 0.938 0.499 1.147 1.128 1.128 1.147 0.499 0.938 µ = .15 0.780 0.606 0.904 0.892 0.892 0.904 0.606 0.780 µ = .05 0.621 0.526 0.653 0.633 0.633 0.653 0.526 0.621 µ = .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

0 0.1 0.2 0.3 0.4

Friction Coefficient

fcal

c / f

E1 = E8E2 = E7E3 = E6E4 = 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

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’ 23/8” Ls2 2’ 10” Lc2 139’ 23/8” Ls3 2’ 10” Lc3 139’ 23/8” Ls4 88’ 3”


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 µd 0.25 Camber B1 = 3” Estimated µs 0.30 Camber B2 = 25/8” Camber B3 = 31/4”

Beam 1 Beam 2 Beam 3

Lc1 Lc2 Lc3

Ls2Ls1 Ls3 Ls4

68

Table 5-18. End Movements for Beam 2 Bottom Strand

Left End Measured

Left End Calculated Difference

BottomStrand

Right EndMeasured

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 CalculatedShorteningDifference

0.827” 0.819” 0.008”

0

0.1

0.2

0.3

0.4

0.5

0.6

6 16 26 36


Mov

emen

t (in

)

MeasuredCalculated

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

69

0

0.1

0.2

0.3

0.4

0.5

0.6

6 16 26 36


Mov

emen

t (in

)

MeasuredPredicted

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

Table 5-19. End Movements for Right End of Beam 1 Bottom Strand

Right End Measured

Right EndCalculated 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”

70

0

0.1

0.2

0.3

0.4

0.5

0.6

4 14 24 34


Mov

emen

t (in

)

MeasuredCalculated

Figure 5-30. Beam 1 Right End Measured vs Calculated

Table 5-20. End Movements for Left End of Beam 3 Bottom Strand

Left End Measured

Left End Calculated Difference

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”

71

-0.2

-0.1

0

0.1

0.2

0.3

0.4

6 16 26 36


Mov

emen

t (in

)

MeasuredCalculated

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

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 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

73

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

74

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

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 the beams to be effective.

75

APPENDIX A SAMPLE RETURNED SURVEY FORMS

APPENDIX B VERTICAL CRACK PREDICTOR

OR

IGIN

1V

ert

ical C

rack P

redic

tor

Univ

ers

ity o

f F

lorida

2006

INP

UT

:

1)

Ch

oic

e:

1=

72

" F

lorid

a B

ulb

T

2

= 7

8"

Flo

rid

a B

ulb

T

3

= A

AS

HT

O T

yp

e 6

4

= A

AS

HT

O T

yp

e 5

5=

AA

SH

TO

Typ

e 4

6=

AA

SH

TO

Typ

e 3

7=

AA

SH

TO

Typ

e 2

8

= 4

8"

Flo

rid

a U

Be

am

9

= 5

4"

Flo

rid

a U

Be

am

10

=6

3"

Flo

rid

a U

Be

am

11

=7

2"

Flo

rid

a U

Be

am

12

= C

usto

m

Choic

e2

If C

hoic

e =

12

th

en

Sp

ecify th

e C

ross S

ectional A

rea o

f th

e B

eam

(A

beam

) and t

he A

rea o

f th

e B

ott

om

Fla

ng

e (

Ab

ott

om

fla

ng

e),

th

e d

ista

nce

from

the b

otto

m o

f th

e b

ea

m to

th

e c

en

tro

id o

f th

e c

ross s

ection "

EcS

pef"

, th

e d

ista

nce f

rom

the c

entr

oid

of

the

bo

tto

m f

lan

ge

to

th

e b

ott

om

of

the b

eam

"F

rice

Sp

ef"

, a

nd

th

e M

om

en

t of

Inert

ia o

f th

e b

ott

om

fla

nge "

IBott

om

Spef"

:

Abea

m22

5in

2A

bott

om

flan

ge

225

in2

EcS

pef

28.5

inF

rice

Spef

7.5

inIB

ott

om

Spef

9047i

n4

2)

Len

gth

of

be

am

s:

Lbea

ms

15

0155

157

152

()f

t

3)

Exp

ecte

d In

itia

l C

am

be

rIn

Cam

ber

33.1

3.2

3(

)in

4)

Nu

mb

er

of

Bo

tto

m F

lan

ge

Pre

str

es

sin

g S

tran

ds

:N

um

ber

Str

ands

49

5)

Avera

ge C

on

cre

te R

ele

as

e S

tre

ng

th a

t T

ime o

f D

ete

nsio

nin

g

(Dete

rmin

ed

fro

m C

ylin

de

r B

rea

ks

- N

OT

SP

EC

IFIE

D V

AL

UE

):

Rel

ease

Str

ength

8000psi

6)

Co

ncre

te u

nit

we

igh

t:w

c150

lbf

ft3

81

7)

Bo

tto

m F

lan

ge

Str

an

d J

ac

kin

g F

orc

e p

er

str

an

d:

Fst

rand

44kip

8)

Dis

tan

ce B

etw

ee

n B

ea

ms

an

d B

etw

een

Beam

s a

nd

the B

ulk

head

s:

Fre

eStr

and

50

33

345

()f

t

9)

Typ

e o

f p

res

tre

ss

ing

str

an

d:

1 =

.500in

Gra

de

27

0

2 =

.500in

Sp

ecia

l G

rad

e 2

70

3 =

9/1

6in

Gra

de

27

0

4 =

.600in

Gra

de

27

0

Str

aChoic

e4

10)

Sp

ecif

y lo

ca

tio

n o

f d

eb

on

de

d s

tra

nd

s (

in t

he c

utt

ing

ord

er)

an

d

asso

cia

ted

deb

on

ded

le

ng

th (

ft).

D

O N

OT

IN

CL

UD

E U

NIT

S.

DE

BO

ND

ED

ST

RA

ND

S

SH

OU

LD

BE

AT

TH

E E

ND

OF

TH

E C

UT

TIN

G O

RD

ER

. D

O N

OT

IN

CLU

DE

TO

P

ST

RA

ND

S IN

TH

E S

TR

AN

D N

UM

BE

R C

OU

NT

.

ST

RA

ND

NU

MB

ER

Deb

ond

42 5

43 5

44 5

45 5

46

10

47

10

48

15

49

15

DE

BO

ND

ED

LE

NG

TH

11)

Sp

ecif

y T

em

pe

ratu

re C

ha

ng

e (

F)

fro

m T

ime o

f B

eam

Casti

ng

to

Str

an

d D

ete

nsio

nin

g.

Positiv

e =

TE

MP

AT

ST

RA

ND

DE

TE

NS

ION

ING

IS

CO

LD

ER

TH

AN

TE

MP

AT

BE

AM

CA

ST

ING

.

Negative =

TE

MP

AT

ST

RA

ND

DE

TE

NS

ION

ING

IS

WA

RM

ER

TH

AN

TE

MP

AT

BE

AM

CA

ST

ING

.

DO

NO

T I

NC

LU

DE

TH

E U

NIT

S

Tem

pC

han

ge

30

12)

Sta

tic F

ric

tio

n C

oe

ffic

ien

t B

etw

ee

n B

ott

om

of

Pre

str

essed

Beam

En

ds a

nd

th

e

Casti

ng

Bed

( is

oft

en

be

twe

en

.3

an

d .

45):

.3

D.2

513)

Dyn

am

ic F

ric

tio

n C

oe

ffic

ien

t B

etw

een

Bo

tto

m o

f P

restr

essed

Beam

En

ds a

nd

th

e

Casti

ng

Be

d (

D is

alw

ay

s le

ss

th

an

):

82

Calc

ula

tions

:B

eam

Cro

ss S

ectional A

rea

Num

bC

alcs

20

Full

Len

gth

2A

875

in2

Choic

e1

=if

1105in

2C

hoic

e2

=if

1125in

2C

hoic

e3

=if

1053in

2C

hoic

e4

=if

789

in2

Choic

e5

=if

559.5

in2

Choic

e6

=if

369

in2

Choic

e7

=if

1146in

2C

hoic

e8

=if

1212in

2C

hoic

e9

=if

1311in

2C

hoic

e10

=if

1410in

2C

hoic

e11

=if

Abea

mC

hoic

e12

=if

"err

or"

oth

erw

ise

Ecc

ent

33.9

5in

Choic

e1

=if

40.3

9in

Choic

e2

=if

36.3

8in

Choic

e3

=if

31.9

6in

Choic

e4

=if

24.7

3in

Choic

e5

=if

20.2

7in

Choic

e6

=if

16.3

8in

Choic

e7

=if

19.6

7in

Choic

e8

=if

22.2

3in

Choic

e9

=if

27.0

4in

Choic

e10

=if

30.1

6in

Choic

e11

=if

EcS

pef

Choic

e12

=if

"err

or"

oth

erw

ise

Pre

str

essin

g S

tra

nd

Are

a

Aps

0.1

53

in2

Str

aChoic

e1

=if

0.1

67

in2

Str

aChoic

e2

=if

0.1

92

in2

Str

aChoic

e3

=if

0.2

192

in2

Str

aChoic

e4

=if

"err

or"

oth

erw

ise

Aps

0.2

192

in2

Modulu

s o

f ela

sticity o

f p

restr

essin

g s

tra

nds

(can't

be c

han

ge

d)

Eps

28500

ksi

Weig

ht

of

the

be

am

(kip

/ft)

wt

wc

A

wt

1.1

51042

kip ft

A1105

in2

83

Eccentr

icitie

s a

nd

Mo

me

nts

of In

ert

ia o

f th

e b

ott

om

fla

nge:

Fri

ce5.5

73

inC

hoic

e1

=if

7.5

09

inC

hoic

e2

=if

7.5

97

inC

hoic

e3

=if

7.5

97

inC

hoic

e4

=if

7.2

66

inC

hoic

e5

=if

6.2

33

inC

hoic

e6

=if

5.2

inC

hoic

e7

=if

5in

Choic

e8

=if

5in

Choic

e9

=if

5in

Choic

e10

=if

5in

Choic

e11

=if

Fri

ceS

pef

Choic

e12

=if

"err

or"

oth

erw

ise

IBott

om

3697in

4C

hoic

e1

=if

8766in

4C

hoic

e2

=if

9047in

4C

hoic

e3

=if

9047in

4C

hoic

e4

=if

7280in

4C

hoic

e5

=if

3873in

4C

hoic

e6

=if

1829in

4C

hoic

e7

=if

4667in

4C

hoic

e8

=if

4667in

4C

hoic

e9

=if

4667in

4C

hoic

e10

=if

4667in

4C

hoic

e11

=if

IBott

om

Spef

Choic

e12

=if

"err

or"

oth

erw

ise

Fri

ce7.5

09

in

IBott

om

8766

in4

84

Are

a o

f B

otto

m F

lan

ge

Direct

Tensio

n S

trength

of

Concre

te "

TS

"

Abf

325

in2

Choic

e1

=if

399

in2

Choic

e2

=if

404

in2

Choic

e3

=if

404

in2

Choic

e4

=if

361

in2

Choic

e5

=if

262.7

5in

2C

hoic

e6

=if

180

in2

Choic

e7

=if

560

in2

Choic

e8

=if

560

in2

Choic

e9

=if

560

in2

Choic

e10

=if

560

in2

Choic

e11

=if

Abott

om

flan

ge

Choic

e12

=if

ConcT

ensS

tren

gth

5R

elea

seS

tren

gth

psi

psi

ConcT

ensS

tren

gth

447.2

13595

psi

Abf

399

in2

Ten

sionA

rea

Abf

Ten

sionA

rea

399

in2

ConcA

llow

able

Ten

sion

Ten

sionA

rea

ConcT

ensS

tren

gth

ConcA

llow

able

Ten

sion

178.4

38225

kip

Dia

mete

r of

pre

str

essin

g s

trand

D.5

inS

traC

hoic

e1

=if

.5in

Str

aChoic

e2

=if

.5625

inS

traC

hoic

e3

=if

.6in

Str

aChoic

e4

=if

"err

or"

oth

erw

ise

D0.6

in

85

Assem

ble

s d

eb

on

de

d le

ng

ths in

to a

ma

trix

Deb

ondL

ength

out g

0

g1

Num

ber

Str

ands

for out D

ebond

1h

Deb

ond

2h

h1

cols

Deb

ond

()

for

out

out

ft

out

Deb

ondL

ength

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 0

ft

Deb

on

dL

eng

th

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

15

ft

Initia

l conditio

ns in the p

restr

essin

g

str

ands

Tem

pS

trai

n.0

00

00

66

7T

emp

Ch

ang

eT

empS

trai

n0.0

002

Tem

pS

tres

sT

empS

trai

nE

ps

Tem

pS

tres

s5.7

0285

ksi

Ori

gS

tres

sStr

and

Fst

ran

d

Ap

sO

rig

Str

essS

tran

d2

00

.72

99

27

ksi

Ori

gF

orc

eStr

and

Ori

gS

tres

sStr

and

Aps

Num

ber

Str

ands

Ori

gF

orc

eStr

and

21

56

kip

Str

essS

tran

dO

rigS

tres

sStr

and

Tem

pS

tres

sS

tres

sStr

and

195.0

27077

ksi

Init

ialS

trai

nS

tran

ds

Str

essS

tran

d

Ep

sIn

itia

lStr

ainS

tran

ds

0.0

0684

86

Adds d

ebon

din

g le

ng

ths to

fre

e s

tra

nd

le

ngth

s

All

Str

andL

ength

s

out p

1F

reeS

tran

d1

1D

ebondL

ength

p

out p

2F

reeS

tran

d1

2D

ebondL

ength

p

p1

Num

ber

Str

ands

for

cols

Fre

eStr

and

()

2=

if

out i

jF

reeS

tran

d1

j2

Deb

ondL

ength

i

i1

Num

ber

Str

ands

forj

1co

lsF

reeS

tran

d(

)fo

r

out k

1out k

1D

ebondL

ength

k

out k

cols

Lbea

ms

()

1out k

cols

Lbea

ms

()

1D

ebondL

ength

k

k1

Num

ber

Str

ands

for

cols

Fre

eStr

and

()

2if out

Adds u

p a

ll th

e s

tra

nd

le

ng

ths s

o th

at th

e a

vera

ges f

or

each s

trand c

ut

can b

e d

ete

rmin

ed

NS

Num

ber

Str

ands

1N

S48

TotS

tran

dL

ength

s

out N

um

ber

Str

ands

wA

llS

tran

dL

ength

s Num

ber

Str

ands

w

out z

wout z

1w

All

Str

andL

ength

s zw

zN

S1

for

w1

cols

All

Str

andL

ength

s(

)fo

r

out

87

All

Str

andL

ength

s

12

34

5

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

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

ftA

llS

tran

dL

ength

s

12

34

5

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

50

33

345

55

13

13

13

50

55

13

13

13

50

55

13

13

13

50

55

13

13

13

50

60

23

23

23

55

60

23

23

23

55

65

33

33

33

60

65

33

33

33

60

ft

88

TotS

tran

dL

ength

s

12

34

5

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

2520

287

287

287

2275

2470

284

284

284

2230

2420

281

281

281

2185

2370

278

278

278

2140

2320

275

275

275

2095

2270

272

272

272

2050

2220

269

269

269

2005

2170

266

266

266

1960

2120

263

263

263

1915

2070

260

260

260

1870

2020

257

257

257

1825

1970

254

254

254

1780

1920

251

251

251

1735

1870

248

248

248

1690

1820

245

245

245

1645

1770

242

242

242

1600

1720

239

239

239

1555

1670

236

236

236

1510

1620

233

233

233

1465

1570

230

230

230

1420

1520

227

227

227

1375

1470

224

224

224

1330

1420

221

221

221

1285

1370

218

218

218

1240

1320

215

215

215

1195

1270

212

212

212

1150

1220

209

209

209

1105

1170

206

206

206

1060

1120

203

203

203

1015

ftT

otS

tran

dL

eng

ths

12

34

5

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

1370

218

218

2181240

1320

215

215

2151195

1270

212

212

2121150

1220

209

209

2091105

1170

206

206

2061060

1120

203

203

2031015

1070

200

200

200

970

1020

197

197

197

925

970

194

194

194

880

920

191

191

191

835

870

188

188

188

790

820

185

185

185

745

770

182

182

182

700

720

179

179

179

655

670

176

176

176

610

620

173

173

173

565

570

170

170

170

520

520

167

167

167

475

470

164

164

164

430

415

151

151

151

380

360

138

138

138

330

305

125

125

125

280

250

112

112

112

230

190

89

89

89

175

130

66

66

66

120

65

33

33

33

60

ft

89

Avera

ge S

tra

nd

le

ng

ths in

ea

ch

fre

e s

tra

nd s

et

aft

er

each s

trand

is c

ut

Concre

te M

odulu

s o

f E

lasticity

AvgS

tran

dL

ength

s

inte

r1w

TotS

tran

dL

ength

s zw

inte

r2w

Index

Sz

out z

w

inte

r1w

inte

r2w

z1

Num

ber

Str

ands

forw

1co

lsA

llS

tran

dL

ength

s(

)fo

r out N

um

ber

Str

ands

1w

w0

ft

ww

1co

lsA

llS

tran

dL

ength

s(

)fo

r

out

E40000

Rel

ease

Str

eng

th

psi

10

6w

c

145

lbf

ft3

1.5

psi

E4816.5

16412

ksi

90

AvgS

tran

dL

ength

s

12

34

5

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

51.429

5.857

5.857

5.85746.429

51.458

5.917

5.917

5.91746.458

51.489

5.979

5.979

5.97946.489

51.522

6.043

6.043

6.04346.522

51.556

6.111

6.111

6.11146.556

51.591

6.182

6.182

6.18246.591

51.628

6.256

6.256

6.25646.628

51.667

6.333

6.333

6.33346.667

51.707

6.415

6.415

6.41546.707

51.75

6.5

6.5

6.5

46.75

51.795

6.59

6.59

6.5946.795

51.842

6.684

6.684

6.68446.842

51.892

6.784

6.784

6.78446.892

51.944

6.889

6.889

6.88946.944

52

77

747

52.059

7.118

7.118

7.11847.059

52.121

7.242

7.242

7.24247.121

52.188

7.375

7.375

7.37547.188

52.258

7.516

7.516

7.51647.258

52.333

7.667

7.667

7.66747.333

52.414

7.828

7.828

7.82847.414

52.5

88

847.5

52.593

8.185

8.185

8.18547.593

52.692

8.385

8.385

8.38547.692

52.8

8.6

8.6

8.6

47.8

52.917

8.833

8.833

8.83347.917

53.043

9.087

9.087

9.08748.043

53.182

9.364

9.364

9.36448.182

ft

91

AvgS

tran

dL

ength

s

12

34

5

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

52.5

88

847.5

52.593

8.185

8.185

8.185

47.593

52.692

8.385

8.385

8.385

47.692

52.8

8.6

8.6

8.6

47.8

52.917

8.833

8.833

8.833

47.917

53.043

9.087

9.087

9.087

48.043

53.182

9.364

9.364

9.364

48.182

53.333

9.667

9.667

9.667

48.333

53.5

10

10

10

48.5

53.68410.368

10.368

10.368

48.684

53.88910.778

10.778

10.778

48.889

54.11811.235

11.235

11.235

49.118

54.375

11.75

11.75

11.75

49.375

54.66712.333

12.333

12.333

49.667

55

13

13

13

50

55.38513.769

13.769

13.769

50.385

55.83314.667

14.667

14.667

50.833

56.36415.727

15.727

15.727

51.364

57

17

17

17

52

57.77818.556

18.556

18.556

52.778

58.75

20.5

20.5

20.5

53.75

59.28621.571

21.571

21.571

54.286

60

23

23

23

55

61

25

25

25

56

62.5

28

28

28

57.5

63.33329.667

29.667

29.667

58.333

65

33

33

33

60

65

33

33

33

60

00

00

0

ft

92

Cre

ate

s m

atr

ix w

ith

ja

ckin

g fo

rce

fo

r e

ach s

trand

Pre

stre

ssT

ransf

er

out g

Fst

rand

g1

Num

ber

Str

ands

for

out

Adds u

p t

ota

l p

restr

ess tra

nsfe

rre

d to

th

e b

eam

TotP

rest

ress

Tra

nsf

er

out q

Fst

rand

q

q1

Num

ber

Str

ands

for

out

Adds u

p t

ota

l p

restr

ess tra

nsfe

rre

d to

en

d o

f beam

only

pre

str

ess in d

eb

on

de

d s

tra

nd

s is n

ot in

clu

ded

TotP

rest

ress

Tra

nsf

erE

nd

out q

TotP

rest

ress

Tra

nsf

erq

q1

4fo

r

out r

out r

1F

stra

nd

Deb

ondL

ength

r0

ft=

if

out r

out r

1D

ebondL

ength

r0

ftif

r5

Num

ber

Str

ands

for

out

93

Pre

stre

ssT

ransf

er

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

44

kip

Pre

stre

ssT

ran

sfer

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

44

kip

94

TotP

rest

ress

Tra

nsf

er

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

1276

kip

To

tPre

stre

ssT

ran

sfer

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

2156

kip

95

TotP

rest

ress

Tra

nsf

erE

nd

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

1232

kip

TotP

rest

ress

Tra

nsf

erE

nd

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

1804

kip

96

Calc

ula

tes c

om

pre

ssio

n tra

nsfe

r le

ng

th o

f th

e p

restr

essin

g s

trands

MP

a145.0

37738

psi

Com

pT

ransL

ength

0.3

3

6.9

Ori

gS

tres

sStr

and

MP

a

D mm

20.7

Rel

ease

Str

ength

MP

a

mm

Com

pT

ransL

ength

24.3

29204

in

Calc

ula

tes f

rictio

n fo

rce

s o

n b

ea

m e

nd

s

Bea

ring

wt

Lbea

ms

2B

eari

ng

86.3

28125

89.2

05729

90.3

56771

87.4

79167

()

kip

FR

fB

eari

ng

FR

fDyn

DB

eari

ng

FR

f25.8

98438

26.7

61719

27.1

07031

26.2

4375

()

kip

FR

fDyn

21.5

82031

22.3

01432

22.5

89193

21.8

69792

()

kip

Convert

s s

tatic frictio

n to

a la

rge

r m

atr

ix

FR

fw

out 1

1F

Rf

out 1

2F

Rf

cols

Fre

eStr

and

()

2=

if

out 1

2g

1F

Rf

1g

out 1

2g

FR

f1

g

g1

cols

FR

f(

)fo

rco

lsF

reeS

tran

d(

)2

if

ou

t

Bea

ringw

out 1

2g

1B

eari

ng

1g

out 1

2g

Bea

ring

1g

g1

cols

Bea

ring

()

for

out

97

The t

ota

l are

a o

f p

restr

essin

g th

at h

as y

et

to b

e c

ut

at

each s

tep in t

he c

utt

ing o

rder

ApsU

ncu

tout N

um

ber

Str

ands

0in

2

out N

um

ber

Str

ands

1A

ps

out j

Num

ber

Str

ands

j(

)A

ps

jN

um

ber

Str

ands

21

for

out

ApsU

ncu

t

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

5.042

in2

ApsU

ncu

t

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 0

in2

98

The t

ota

l are

a o

f p

restr

essin

g th

at h

as y

et

to b

e c

ut

at

each s

tep in t

he c

utt

ing o

rder

not

inclu

din

g th

e d

eb

on

de

d s

tra

nd

s

ApsU

ncu

tEnd

out 1

ApsU

ncu

t 1co

lsD

ebond

()

Aps

out q

out q

1A

ps

Deb

ondL

ength

q0

ft=

if

out q

out q

1D

ebondL

ength

q0

ftif

q2

Num

ber

Str

ands

for

out

ApsU

ncu

tEnd

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 0

in2

ApsU

ncu

tEnd

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

4.603

in2

99

Spring s

tiff

ne

sse

s fo

r p

restr

essin

g s

tra

nd

s

kS

teel

out q

f

28500

ksi

AvgS

tran

dL

ength

s q1

f

ApsU

ncu

t q

f1

cols

Fre

eStr

and

()

forq

1N

um

ber

Str

ands

1fo

r

out

Eff

ective s

trand s

tiff

ness f

or

each b

eam

Convert

s k

Ste

el to

a la

rge

r m

atr

ix

kS

teel

w

out q

1kS

teel

q1

out q

2kS

teel

q2

cols

Fre

eStr

and

()

2=

if

out q

2co

lsL

bea

ms

()

kS

teel

qco

lskS

teel

()

out q

2c

2kS

teel

qc

out q

2c

1kS

teel

qc

c2

cols

kS

teel

()

1fo

r

cols

Fre

eStr

and

()

2ifq

1ro

ws

kS

teel

()

for

out

kef

fSte

el

out q

1

1

1

kS

teel

q1

1

kS

teel

q2

cols

Fre

eStr

and

()

2=

if

out q

f

1

1

kS

teel

qf

1

kS

teel

qf

1

f1

cols

Lbea

ms

()

for

cols

Fre

eStr

and

()

2if

q1

Num

ber

Str

ands

1fo

r

out

100

kS

teel

12

34

5

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

485.612

4223.459

4223.459

4223.459

537.876

475.209

4092.546

4092.546

4092.546

526.318

464.806

3962.553

3962.553

3962.553

514.761

454.403

3833.509

3833.509

3833.509

503.205

444.001

3705.447

3705.447

3705.447

491.65

433.599

3578.399

3578.399

3578.399

480.094

423.197

3452.4

3452.4

3452.4

468.54

412.797

3327.485

3327.485

3327.485

456.986

402.396

3203.692

3203.692

3203.692

445.433

391.996

3081.061

3081.061

3081.061

433.881

381.597

2959.631

2959.631

2959.631

422.329

371.199

2839.448

2839.448

2839.448

410.779

360.801

2720.555

2720.555

2720.555

399.229

350.404

2603

2603

2603

387.681

340.008

2486.833

2486.833

2486.833

376.133

329.612

2372.106

2372.106

2372.106

364.587

319.218

2258.875

2258.875

2258.875

353.043

308.825

2147.196

2147.196

2147.196

341.499

298.433

2037.13

2037.13

2037.13

329.958

288.042

1928.743

1928.743

1928.743

318.418

277.653

1822.1

1822.1

1822.1

306.88

267.266

1717.273

1717.273

1717.273

295.344

256.88

1614.338

1614.338

1614.338

283.811

246.496

1513.372

1513.372

1513.372

272.28

236.115

1414.46

1414.46

1414.46

260.753

225.736

1317.691

1317.691

1317.691

249.228

215.359

1223.157

1223.157

1223.157

237.708

204.986

1130.959

1130.959

1130.959

226.192

kip in

101

kS

teel

12

34

5

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

277.653

1822.1

1822.1

1822.1

306.88

267.266

1717.273

1717.273

1717.273

295.344

256.88

1614.338

1614.338

1614.338

283.811

246.496

1513.372

1513.372

1513.372

272.28

236.115

1414.46

1414.46

1414.46

260.753

225.736

1317.691

1317.691

1317.691

249.228

215.359

1223.157

1223.157

1223.157

237.708

204.986

1130.959

1130.959

1130.959

226.192

194.617

1041.2

1041.2

1041.2

214.68

184.252

953.993

953.993

953.993

203.175

173.891

869.456

869.456

869.456

191.675

163.536

787.714

787.714

787.714

180.184

153.188

708.902

708.902

708.902

168.701

142.848

633.162

633.162

633.162

157.228

132.516

560.646

560.646

560.646

145.768

122.196

491.516

491.516

491.516

134.323

111.89

425.945

425.945

425.945

122.896

101.601

364.119

364.119

364.119

111.491

91.333

306.235

306.235

306.235

100.115

81.093

252.507

252.507

252.507

88.776

70.89

203.161

203.161

203.161

77.485

61.468

168.936

168.936

168.936

67.13

52.06

135.809

135.809

135.809

56.793

42.672

104.12

104.12

104.12

46.482

33.318

74.371

74.371

74.371

36.216

24.66

52.645

52.645

52.645

26.774

16.018

31.552

31.552

31.552

17.353

8.009

15.776

15.776

15.776

8.677

kip in

102

kS

teel

w

12

34

56

78

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

485.612

4223.459

4223.459

4223.459

4223.459

4223.459

4223.459

537.876

475.209

4092.546

4092.546

4092.546

4092.546

4092.546

4092.546

526.318

464.806

3962.553

3962.553

3962.553

3962.553

3962.553

3962.553

514.761

454.403

3833.509

3833.509

3833.509

3833.509

3833.509

3833.509

503.205

444.001

3705.447

3705.447

3705.447

3705.447

3705.447

3705.447

491.65

433.599

3578.399

3578.399

3578.399

3578.399

3578.399

3578.399

480.094

423.197

3452.4

3452.4

3452.4

3452.4

3452.4

3452.4

468.54

412.797

3327.485

3327.485

3327.485

3327.485

3327.485

3327.485

456.986

402.396

3203.692

3203.692

3203.692

3203.692

3203.692

3203.692

445.433

391.996

3081.061

3081.061

3081.061

3081.061

3081.061

3081.061

433.881

381.597

2959.631

2959.631

2959.631

2959.631

2959.631

2959.631

422.329

371.199

2839.448

2839.448

2839.448

2839.448

2839.448

2839.448

410.779

360.801

2720.555

2720.555

2720.555

2720.555

2720.555

2720.555

399.229

350.404

2603

2603

2603

2603

2603

2603

387.681

340.008

2486.833

2486.833

2486.833

2486.833

2486.833

2486.833

376.133

329.612

2372.106

2372.106

2372.106

2372.106

2372.106

2372.106

364.587

319.218

2258.875

2258.875

2258.875

2258.875

2258.875

2258.875

353.043

308.825

2147.196

2147.196

2147.196

2147.196

2147.196

2147.196

341.499

298.433

2037.13

2037.13

2037.13

2037.13

2037.13

2037.13

329.958

288.042

1928.743

1928.743

1928.743

1928.743

1928.743

1928.743

318.418

277.653

1822.1

1822.1

1822.1

1822.1

1822.1

1822.1

306.88

267.266

1717.273

1717.273

1717.273

1717.273

1717.273

1717.273

295.344

256.88

1614.338

1614.338

1614.338

1614.338

1614.338

1614.338

283.811

246.496

1513.372

1513.372

1513.372

1513.372

1513.372

1513.372

272.28

236.115

1414.46

1414.46

1414.46

1414.46

1414.46

1414.46

260.753

225.736

1317.691

1317.691

1317.691

1317.691

1317.691

1317.691

249.228

215.359

1223.157

1223.157

1223.157

1223.157

1223.157

1223.157

237.708

204.986

1130.959

1130.959

1130.959

1130.959

1130.959

1130.959

226.192

194.617

1041.2

1041.2

1041.2

1041.2

1041.2

1041.2

214.68

kip in

103

kS

teel

w

12

34

56

78

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

267.266

1717.273

1717.273

1717.273

1717.273

1717.273

1717.273

295.344

256.88

1614.338

1614.338

1614.338

1614.338

1614.338

1614.338

283.811

246.496

1513.372

1513.372

1513.372

1513.372

1513.372

1513.372

272.28

236.115

1414.46

1414.46

1414.46

1414.46

1414.46

1414.46

260.753

225.736

1317.691

1317.691

1317.691

1317.691

1317.691

1317.691

249.228

215.359

1223.157

1223.157

1223.157

1223.157

1223.157

1223.157

237.708

204.986

1130.959

1130.959

1130.959

1130.959

1130.959

1130.959

226.192

194.617

1041.2

1041.2

1041.2

1041.2

1041.2

1041.2

214.68

184.252

953.993

953.993

953.993

953.993

953.993

953.993

203.175

173.891

869.456

869.456

869.456

869.456

869.456

869.456

191.675

163.536

787.714

787.714

787.714

787.714

787.714

787.714

180.184

153.188

708.902

708.902

708.902

708.902

708.902

708.902

168.701

142.848

633.162

633.162

633.162

633.162

633.162

633.162

157.228

132.516

560.646

560.646

560.646

560.646

560.646

560.646

145.768

122.196

491.516

491.516

491.516

491.516

491.516

491.516

134.323

111.89

425.945

425.945

425.945

425.945

425.945

425.945

122.896

101.601

364.119

364.119

364.119

364.119

364.119

364.119

111.491

91.333

306.235

306.235

306.235

306.235

306.235

306.235

100.115

81.093

252.507

252.507

252.507

252.507

252.507

252.507

88.776

70.89

203.161

203.161

203.161

203.161

203.161

203.161

77.485

61.468

168.936

168.936

168.936

168.936

168.936

168.936

67.13

52.06

135.809

135.809

135.809

135.809

135.809

135.809

56.793

42.672

104.12

104.12

104.12

104.12

104.12

104.12

46.482

33.318

74.371

74.371

74.371

74.371

74.371

74.371

36.216

24.66

52.645

52.645

52.645

52.645

52.645

52.645

26.774

16.018

31.552

31.552

31.552

31.552

31.552

31.552

17.353

8.009

15.776

15.776

15.776

15.776

15.776

15.776

8.677

kip in

104

kef

fSte

el

12

34

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

435.535

2111.73

2111.73

477.113

425.77

2046.273

2046.273

466.344

416.008

1981.276

1981.276

455.579

406.249

1916.755

1916.755

444.816

396.492

1852.724

1852.724

434.058

386.737

1789.2

1789.2

423.302

376.986

1726.2

1726.2

412.551

367.238

1663.743

1663.743

401.804

357.494

1601.846

1601.846

391.061

347.753

1540.53

1540.53

380.323

338.015

1479.816

1479.816

369.59

328.283

1419.724

1419.724

358.863

318.554

1360.277

1360.277

348.141

308.831

1301.5

1301.5

337.426

299.112

1243.417

1243.417

326.717

289.399

1186.053

1186.053

316.016

279.693

1129.437

1129.437

305.323

269.993

1073.598

1073.598

294.639

260.3

1018.565

1018.565

283.964

250.615

964.371

964.371

273.299

240.939

911.05

911.05

262.645

231.272

858.637

858.637

252.004

221.616

807.169

807.169

241.376

211.971

756.686

756.686

230.762

202.338

707.23

707.23

220.166

192.72

658.845

658.845

209.587

183.118

611.579

611.579

199.029

173.533

565.479

565.479

188.493

163.969

520.6

520.6

177.983

kip in

105

kef

fSte

el

12

34

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

240.939

911.05

911.05

262.645

231.272

858.637

858.637

252.004

221.616

807.169

807.169

241.376

211.971

756.686

756.686

230.762

202.338

707.23

707.23

220.166

192.72

658.845

658.845

209.587

183.118

611.579

611.579

199.029

173.533

565.479

565.479

188.493

163.969

520.6

520.6

177.983

154.426

476.996

476.996

167.501

144.909

434.728

434.728

157.053

135.422

393.857

393.857

146.641

125.967

354.451

354.451

136.272

116.552

316.581

316.581

125.952

107.182

280.323

280.323

115.689

97.866

245.758

245.758

105.493

88.613

212.973

212.973

95.377

79.436

182.06

182.06

85.356

70.351

153.118

153.118

75.449

61.381

126.253

126.253

65.683

52.553

101.58

101.58

56.092

45.07

84.468

84.468

48.04

37.634

67.904

67.904

40.046

30.267

52.06

52.06

32.136

23.01

37.186

37.186

24.356

16.794

26.322

26.322

17.748

10.624

15.776

15.776

11.196

5.312

7.888

7.888

5.598

kip in

106

Calc

ula

tes th

e s

tiffn

ess o

f o

nly

th

e n

on

de

bonded s

trands

kS

teel

End

out q

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28500

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AvgS

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eng

ths q

1f

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q

f1

cols

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Makes k

Ste

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End

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()

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out q

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ms

()

kS

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qco

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out q

2c

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teel

End

qc

out q

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1kS

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End

qc

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cols

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End

()

1fo

r

cols

Fre

eStr

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kS

teel

End

()

for

out

107

kS

teel

End

12

34

5

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

404.677

3519.549

3519.549

3519.549

448.23

394.322

3395.942

3395.942

3395.942

436.732

383.97

3273.413

3273.413

3273.413

425.238

373.62

3151.996

3151.996

3151.996

413.747

363.273

3031.729

3031.729

3031.729

402.259

352.929

2912.651

2912.651

2912.651

390.775

342.588

2794.8

2794.8

2794.8

379.294

332.251

2678.22

2678.22

2678.22

367.818

321.917

2562.954

2562.954

2562.954

356.347

311.587

2449.048

2449.048

2449.048

344.88

301.261

2336.551

2336.551

2336.551

333.418

290.939

2225.513

2225.513

2225.513

321.962

280.623

2115.987

2115.987

2115.987

310.512

270.312

2008.029

2008.029

2008.029

299.068

260.006

1901.696

1901.696

1901.696

287.632

249.706

1797.05

1797.05

1797.05

276.203

239.414

1694.156

1694.156

1694.156

264.782

229.128

1593.081

1593.081

1593.081

253.371

218.851

1493.896

1493.896

1493.896

241.969

208.583

1396.676

1396.676

1396.676

230.578

198.324

1301.5

1301.5

1301.5

219.2

188.076

1208.452

1208.452

1208.452

207.835

177.84

1117.618

1117.618

1117.618

196.485

167.617

1029.093

1029.093

1029.093

185.151

157.41

942.974

942.974

942.974

173.835

147.219

859.364

859.364

859.364

162.54

137.047

778.373

778.373

778.373

151.269

126.896

700.117

700.117

700.117

140.023

kip in

108

kS

teel

End

12

34

5

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

177.84

1117.618

1117.618

1117.618

196.485

167.617

1029.093

1029.093

1029.093

185.151

157.41

942.974

942.974

942.974

173.835

147.219

859.364

859.364

859.364

162.54

137.047

778.373

778.373

778.373

151.269

126.896

700.117

700.117

700.117

140.023

116.77

624.72

624.72

624.72

128.808

106.672

552.312

552.312

552.312

117.627

96.606

483.031

483.031

483.031

106.486

86.578

417.025

417.025

417.025

95.391

76.594

354.451

354.451

354.451

84.35

66.662

295.476

295.476

295.476

73.373

56.793

240.277

240.277

240.277

62.472

46.999

189.045

189.045

189.045

51.663

37.297

141.982

141.982

141.982

40.965

27.709

99.305

99.305

99.305

30.407

18.267

61.247

61.247

61.247

20.023

9.01

28.056

28.056

28.056

9.864

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

kip in

109

kS

teel

Endw

12

34

56

78

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

404.68

3519.55

3519.55

3519.55

3519.55

3519.55

3519.55

448.23

394.32

3395.94

3395.94

3395.94

3395.94

3395.94

3395.94

436.73

383.97

3273.41

3273.41

3273.41

3273.41

3273.41

3273.41

425.24

373.62

3152

3152

3152

3152

3152

3152

413.75

363.27

3031.73

3031.73

3031.73

3031.73

3031.73

3031.73

402.26

352.93

2912.65

2912.65

2912.65

2912.65

2912.65

2912.65

390.77

342.59

2794.8

2794.8

2794.8

2794.8

2794.8

2794.8

379.29

332.25

2678.22

2678.22

2678.22

2678.22

2678.22

2678.22

367.82

321.92

2562.95

2562.95

2562.95

2562.95

2562.95

2562.95

356.35

311.59

2449.05

2449.05

2449.05

2449.05

2449.05

2449.05

344.88

301.26

2336.55

2336.55

2336.55

2336.55

2336.55

2336.55

333.42

290.94

2225.51

2225.51

2225.51

2225.51

2225.51

2225.51

321.96

280.62

2115.99

2115.99

2115.99

2115.99

2115.99

2115.99

310.51

270.31

2008.03

2008.03

2008.03

2008.03

2008.03

2008.03

299.07

260.01

1901.7

1901.7

1901.7

1901.7

1901.7

1901.7

287.63

249.71

1797.05

1797.05

1797.05

1797.05

1797.05

1797.05

276.2

239.41

1694.16

1694.16

1694.16

1694.16

1694.16

1694.16

264.78

229.13

1593.08

1593.08

1593.08

1593.08

1593.08

1593.08

253.37

218.85

1493.9

1493.9

1493.9

1493.9

1493.9

1493.9

241.97

208.58

1396.68

1396.68

1396.68

1396.68

1396.68

1396.68

230.58

198.32

1301.5

1301.5

1301.5

1301.5

1301.5

1301.5

219.2

188.08

1208.45

1208.45

1208.45

1208.45

1208.45

1208.45

207.83

177.84

1117.62

1117.62

1117.62

1117.62

1117.62

1117.62

196.48

167.62

1029.09

1029.09

1029.09

1029.09

1029.09

1029.09

185.15

157.41

942.97

942.97

942.97

942.97

942.97

942.97

173.84

147.22

859.36

859.36

859.36

859.36

859.36

859.36

162.54

137.05

778.37

778.37

778.37

778.37

778.37

778.37

151.27

126.9

700.12

700.12

700.12

700.12

700.12

700.12

140.02

kip in

110

kS

teel

Endw

12

34

56

78

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

198.32

1301.5

1301.5

1301.5

1301.5

1301.5

1301.5

219.2

188.08

1208.45

1208.45

1208.45

1208.45

1208.45

1208.45

207.83

177.84

1117.62

1117.62

1117.62

1117.62

1117.62

1117.62

196.48

167.62

1029.09

1029.09

1029.09

1029.09

1029.09

1029.09

185.15

157.41

942.97

942.97

942.97

942.97

942.97

942.97

173.84

147.22

859.36

859.36

859.36

859.36

859.36

859.36

162.54

137.05

778.37

778.37

778.37

778.37

778.37

778.37

151.27

126.9

700.12

700.12

700.12

700.12

700.12

700.12

140.02

116.77

624.72

624.72

624.72

624.72

624.72

624.72

128.81

106.67

552.31

552.31

552.31

552.31

552.31

552.31

117.63

96.61

483.03

483.03

483.03

483.03

483.03

483.03

106.49

86.58

417.03

417.03

417.03

417.03

417.03

417.03

95.39

76.59

354.45

354.45

354.45

354.45

354.45

354.45

84.35

66.66

295.48

295.48

295.48

295.48

295.48

295.48

73.37

56.79

240.28

240.28

240.28

240.28

240.28

240.28

62.47

47

189.04

189.04

189.04

189.04

189.04

189.04

51.66

37.3

141.98

141.98

141.98

141.98

141.98

141.98

40.97

27.71

99.31

99.31

99.31

99.31

99.31

99.31

30.41

18.27

61.25

61.25

61.25

61.25

61.25

61.25

20.02

9.01

28.06

28.06

28.06

28.06

28.06

28.06

9.86

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

kip in

111

Calc

ula

tes th

e le

ng

th o

f th

e c

on

cre

te (

for

all

str

ands)

used f

or

ela

stic s

hort

enin

g c

alc

ula

tions

Eff

ConcL

eng

th

out h

1L

bea

ms

4 3C

om

pT

ransL

ength

2D

ebondL

ength

h

out h

1L

bea

ms

Full

Len

gth

1=

if

h1

Num

ber

Str

ands

for

cols

Fre

eStr

and

()

2=

if

out g

wL

bea

ms 1

w

4 3C

om

pT

ransL

ength

2D

ebondL

ength

g

out g

wL

bea

ms 1

wF

ull

Len

gth

1=

if

g1

Num

ber

Str

ands

forw

1co

lsL

bea

ms

()

for

cols

Fre

eStr

and

()

2if

out

Calc

ula

tes th

e a

ve

rag

e c

on

cre

te le

ng

ths (

from

above)

at

each s

trand c

ut

Abso

lute

Va

lue

fu

nctio

n

AvgE

ffC

oncL

ength

out i

w

1

i

n

Eff

ConcL

ength

nw

i

iN

um

ber

Str

ands

1fo

rw1

cols

Eff

ConcL

ength

()

for

out

absV

alout

0if

out

10

if

out

112

Eff

ConcL

eng

th

12

34

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

147.297152.297154.297149.297

ftE

ffC

oncL

ength

12

34

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

117.297122.297124.297119.297

ft

113

AvgE

ffC

oncL

ength

12

34

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

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

147.297

152.297

154.297

149.297

ftA

vgE

ffC

oncL

ength

12

34

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

144.44

149.44

151.44

146.44

ft

114

Equiv

ale

nt

sp

rin

g s

tiffn

ess fo

r b

ea

m

kC

onc

out q

z

AE

AvgE

ffC

oncL

ength

qz

z1

cols

AvgE

ffC

oncL

ength

()

forq

1N

um

ber

Str

ands

for

out

kC

onc

12

34

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

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3015.95

2916.77

2878.91

2975.48

3020.61

2921.14

2883.16

2980.02

3025.07

2925.31

2887.22

2984.36

3029.35

2929.31

2891.12

2988.53

3037.97

2937.37

2898.97

2996.92

3046.27

2945.13

2906.53

3004.99

3058.66

2956.71

2917.8

3017.04

3070.63

2967.89

2928.7

3028.69

kip in

kC

onc

12

34

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

3011.07

2912.21

2874.47

2970.73

kip in

115

Calc

ula

tes th

e to

tal p

restr

ess fo

rce

le

ft to

short

en t

he b

eam

and

str

etc

h t

he u

ncu

t str

an

ds a

fte

r fr

ictio

n h

as b

een o

verc

om

e

TotP

Taf

terF

ric

out q

0kip

TotP

rest

ress

Tra

nsf

erq

FR

fif

out q

TotP

rest

ress

Tra

nsf

erq

FR

fDyn

TotP

rest

ress

Tra

nsf

erq

FR

fif

cols

Fre

eStr

and

()

2=

if

out q

b0

kip

TotP

rest

ress

Tra

nsf

erq

FR

f1

bif

out q

bT

otP

rest

ress

Tra

nsf

erq

FR

fDyn

1b

TotP

rest

ress

Tra

nsf

erq

FR

f1

bif

b1

cols

kC

onc

()

for

cols

Fre

eStr

and

()

2if

q1

Num

ber

Str

ands

for

out

116

TotP

Taf

terF

ric

12

34

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

22.42

21.7

21.41

22.13

66.42

65.7

65.41

66.13

110.42

109.7

109.41

110.13

154.42

153.7

153.41

154.13

198.42

197.7

197.41

198.13

242.42

241.7

241.41

242.13

286.42

285.7

285.41

286.13

330.42

329.7

329.41

330.13

374.42

373.7

373.41

374.13

418.42

417.7

417.41

418.13

462.42

461.7

461.41

462.13

506.42

505.7

505.41

506.13

550.42

549.7

549.41

550.13

594.42

593.7

593.41

594.13

638.42

637.7

637.41

638.13

682.42

681.7

681.41

682.13

726.42

725.7

725.41

726.13

770.42

769.7

769.41

770.13

814.42

813.7

813.41

814.13

858.42

857.7

857.41

858.13

902.42

901.7

901.41

902.13

946.42

945.7

945.41

946.13

990.42

989.7

989.41

990.13

1034.42

1033.7

1033.41

1034.13

1078.42

1077.7

1077.41

1078.13

1122.42

1121.7

1121.41

1122.13

1166.42

1165.7

1165.41

1166.13

kip

TotP

Taf

terF

ric

12

34

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

1034.42

1033.7

1033.41

1034.13

1078.42

1077.7

1077.41

1078.13

1122.42

1121.7

1121.41

1122.13

1166.42

1165.7

1165.41

1166.13

1210.42

1209.7

1209.41

1210.13

1254.42

1253.7

1253.41

1254.13

1298.42

1297.7

1297.41

1298.13

1342.42

1341.7

1341.41

1342.13

1386.42

1385.7

1385.41

1386.13

1430.42

1429.7

1429.41

1430.13

1474.42

1473.7

1473.41

1474.13

1518.42

1517.7

1517.41

1518.13

1562.42

1561.7

1561.41

1562.13

1606.42

1605.7

1605.41

1606.13

1650.42

1649.7

1649.41

1650.13

1694.42

1693.7

1693.41

1694.13

1738.42

1737.7

1737.41

1738.13

1782.42

1781.7

1781.41

1782.13

1826.42

1825.7

1825.41

1826.13

1870.42

1869.7

1869.41

1870.13

1914.42

1913.7

1913.41

1914.13

1958.42

1957.7

1957.41

1958.13

2002.42

2001.7

2001.41

2002.13

2046.42

2045.7

2045.41

2046.13

2090.42

2089.7

2089.41

2090.13

2134.42

2133.7

2133.41

2134.13

kip

117

InC

amber

33.1

3.2

3(

)in

Tota

l axia

l sh

ort

en

ing

of th

e b

ea

ms

Lbea

ms

150

155

157

152

()

ft

Xto

t

out q

b

TotP

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b

kC

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b

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()

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1N

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Str

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um

ber

Str

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v

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um

ber

Str

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v

kC

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ber

Str

ands

v

v1

cols

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Taf

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()

for

out

Ecc

ent

40.3

9in

Cam

bM

ov

out 1

j

InC

amber

1j

Ecc

ent

Lbea

ms 1

j

2

j1

cols

Lbea

ms

()

for

out

Cam

bM

ov

0.1

34633

0.1

34633

0.1

37206

0.1

32862

()

in

Tota

l axia

l sh

ort

en

ing

plu

s c

am

be

r m

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ment

of

the b

eam

s

contr

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d b

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ach

str

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ut

Xto

tInd

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ccX

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cc

cc1

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Xto

t(

)fo

r

out q

cX

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cX

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Xto

t(

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ws

Xto

t(

)2

for

out

Xto

tIndiv

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out j

k

Xto

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jk

Xto

t Num

ber

Str

and

sk

Cam

bM

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1k

Xto

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jk

k1

cols

Lbea

ms

()

forj

1N

um

ber

Str

ands

for

out

118

Xto

t

12

34

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

0.007

0.004

0.004

0.006

0.019

0.013

0.013

0.019

0.032

0.022

0.023

0.032

0.045

0.032

0.032

0.045

0.058

0.041

0.042

0.058

0.071

0.051

0.052

0.071

0.085

0.062

0.062

0.085

0.098

0.072

0.073

0.098

0.111

0.083

0.083

0.111

0.125

0.094

0.095

0.125

0.138

0.105

0.106

0.138

0.152

0.117

0.118

0.152

0.165

0.129

0.13

0.166

0.179

0.141

0.142

0.18

0.193

0.153

0.155

0.194

0.207

0.166

0.168

0.208

0.221

0.18

0.181

0.222

0.235

0.193

0.195

0.236

0.249

0.207

0.209

0.25

0.263

0.221

0.223

0.265

0.277

0.236

0.238

0.279

0.292

0.251

0.253

0.294

0.306

0.266

0.269

0.308

0.321

0.282

0.285

0.323

0.336

0.298

0.301

0.338

0.35

0.314

0.317

0.353

0.365

0.331

0.334

0.368

inX

tot

12

34

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

0.365

0.331

0.334

0.368

0.38

0.348

0.352

0.383

0.395

0.365

0.369

0.398

0.41

0.383

0.387

0.414

0.425

0.401

0.405

0.429

0.441

0.419

0.424

0.445

0.456

0.438

0.443

0.46

0.471

0.456

0.462

0.476

0.487

0.475

0.481

0.492

0.503

0.495

0.5

0.508

0.518

0.514

0.52

0.524

0.534

0.533

0.54

0.54

0.55

0.553

0.559

0.556

0.566

0.572

0.579

0.572

0.582

0.591

0.599

0.589

0.597

0.608

0.616

0.604

0.612

0.626

0.633

0.619

0.627

0.643

0.651

0.635

0.642

0.66

0.668

0.65

0.656

0.675

0.684

0.664

0.669

0.691

0.7

0.678

0.682

0.705

0.714

0.691

0.695

0.719

0.728

0.705

in

119

Xto

tIndiv

idual

12

34

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

0.007764

0.005128

0.005103

0.007629

0.015304

0.010603

0.010694

0.015239

0.015391

0.010884

0.01098

0.015335

0.015479

0.011174

0.011274

0.015432

0.015568

0.011471

0.011576

0.015529

0.015657

0.011777

0.011888

0.015627

0.015747

0.012091

0.012208

0.015727

0.015838

0.012414

0.012536

0.015827

0.01593

0.012746

0.012874

0.015928

0.016022

0.013086

0.01322

0.01603

0.016115

0.013434

0.013575

0.016133

0.016209

0.013791

0.013939

0.016236

0.016303

0.014155

0.014311

0.016341

0.016398

0.014528

0.014691

0.016446

0.016494

0.014909

0.01508

0.016553

0.016591

0.015297

0.015476

0.01666

0.016689

0.015692

0.015879

0.016768

0.016787

0.016093

0.016289

0.016877

0.016886

0.0165

0.016705

0.016987

0.016985

0.016912

0.017126

0.017098

0.017085

0.017327

0.017551

0.01721

0.017186

0.017746

0.017979

0.017322

0.017288

0.018166

0.018409

0.017436

0.01739

0.018586

0.01884

0.01755

0.017493

0.019004

0.019268

0.017664

0.017596

0.019418

0.019694

0.01778

0.0177

0.019827

0.020113

0.017895

0.017804

0.020228

0.020524

0.018012

in

120

Xto

tIndiv

idual

12

34

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

0.01739

0.018586

0.01884

0.01755

0.017493

0.019004

0.019268

0.017664

0.017596

0.019418

0.019694

0.01778

0.0177

0.019827

0.020113

0.017895

0.017804

0.020228

0.020524

0.018012

0.017909

0.020617

0.020924

0.018128

0.018014

0.020992

0.021309

0.018245

0.018119

0.021349

0.021677

0.018362

0.018223

0.021685

0.022022

0.018479

0.018328

0.021995

0.022341

0.018596

0.018432

0.022274

0.022629

0.018711

0.018534

0.022518

0.02288

0.018825

0.018636

0.02272

0.023089

0.018938

0.018735

0.022875

0.02325

0.019048

0.018832

0.022977

0.023356

0.019154

0.018924

0.023019

0.023401

0.019255

0.019011

0.022993

0.023376

0.019349

0.01909

0.022892

0.023274

0.019434

0.01775

0.020342

0.020686

0.018062

0.01782

0.020426

0.020773

0.018138

0.017884

0.020456

0.020805

0.018207

0.017937

0.020407

0.020756

0.018263

0.016591

0.018368

0.018692

0.016891

0.016636

0.018398

0.018724

0.016939

0.015297

0.016601

0.01691

0.01558

0.015338

0.016671

0.016982

0.015625

in

121

Calc

ula

tes a

re

fere

nce

to

tal b

ea

m s

ho

rte

nin

g f

or

com

parison t

o m

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sults.

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su

lts fo

r a

xia

l sh

ort

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ing

sh

ould

alw

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han t

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um

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out 1

w

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w

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for

cols

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()

2if

out

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lref

out 1

jA

xia

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ambM

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1j

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()

for

out

Axia

lrr

0.7

29165

0.7

53471

0.7

63193

0.7

38887

()

in

Axia

lref

0.8

63799

0.8

88104

0.9

00399

0.8

71749

()

in

Cam

bM

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0.1

34633

0.1

34633

0.1

37206

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in

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the f

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122

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Cre

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ma

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SF

q1

g

kS

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um

bq

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kS

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um

bq

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1

q2

g1

Xto

tIndiv

idual

Num

bq

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g0

in

q2

g1

Xto

tIndiv

idual

Num

bq

g

q2

g0

inif

q2

g1

0in

ifq1

ifg1

cols

Lbea

ms

()

for

cols

Fre

eStr

and

()

2if

129

q2

g1

0in

q2

gX

totI

ndiv

idual

Num

bq

g

qg

XX

q1

q1

XX

q2

q2

cols

Fre

eStr

and

()

2=

if

XX

qco

lsL

bea

ms

()

1q

cols

Lbea

ms

()

2

XX

qw

q2

w2

q2

w1

w2

cols

Lbea

ms

()

for

cols

Fre

eStr

and

()

2ifq

1=

if

XX

q1

XX

q1

1q

1

XX

q2

XX

q1

2q

2co

lsF

reeS

tran

d(

)2

=if

XX

qco

lsL

bea

ms

()

1X

Xq

1co

lsL

bea

ms

()

1q

cols

Lbea

ms

()

2

XX

qw

XX

q1

wq

2w

2q

2w

1

w2

cols

Lbea

ms

()

for

cols

Fre

eStr

and

()

2ifq

1if

Sen

dG

lobal

Const

Pac

kq

XX

Tq

T

in

XX

Tq

1T

in

Rec

eiveG

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Glo

bal

Sen

dG

lobal

() 1

1Tin

XX

Ri

Gl

bl

we

1co

lsF

reeS

tran

d(

)fo

r

cols

Fre

eStr

and

()

2if

q1

if

130

XX

qw

eR

ecei

veG

lobal

1w

e

Rec

eiveS

lide

Glo

bal

Sen

dG

lobal

() 1

3Tin

Sli

deq

qss

Rec

eiveS

lide

1ss

Tota

lSli

deq

qss

Tota

lSli

deq

q1

ssS

lideq

qss

ss1

cols

Lbea

ms

()

for

Ten

sionP

ull

qh

kS

teel

Num

bq

hX

Xq

hT

empS

tres

sA

psU

ncu

tNum

bq

Ten

sionP

ull

End

qh

kS

teel

EndN

um

bq

hX

Xq

hT

empS

tres

sA

psU

ncu

tEndN

um

bq

RT

Lq

h

0.3

3

6.9

Ten

sionP

ull

qh

ApsU

ncu

tNum

bq

MP

a

D mm

20.7

Rel

ease

Str

ength

MP

a

mm

Deb

ondL

ength

Num

bq

0ft

=if

RT

Lq

h0

mm

Deb

ondL

ength

Num

bq

0ft

if

h1

cols

Lbea

ms

()

1fo

rco

lsF

reeS

tran

d(

)2

if

Ten

sionP

ull

wq

1T

ensi

onP

ull

q1

Ten

sionP

ull

Endw

q1

Ten

sionP

ull

End

q1

RT

Lw

q1

RT

Lq

1

Ten

sionP

ull

wq

2T

ensi

onP

ull

q2

Ten

sionP

ull

Endw

q2

Ten

sionP

ull

End

q2

RT

Lw

q2

RT

Lq

2

cols

Fre

eStr

and

()

2=

if

Ten

sionP

ull

wq

2co

lsL

bea

ms

()

Ten

sionP

ull

qco

lsL

bea

ms

()

1

Ti

Pll

Ed

cols

Fre

eStr

and

()

2if

131

Ten

sionP

ull

Endw

q2

cols

Lbea

ms

()

Ten

sionP

ull

End

qco

lsL

bea

ms

()

1

RT

Lw

q2

cols

Lbea

ms

()

RT

Lq

cols

Lbea

ms

()

1

Ten

sionP

ull

Endw

q2

aa2

Ten

sionP

ull

End

qaa

Ten

sionP

ull

Endw

q2

aa1

Ten

sionP

ull

End

qaa

Ten

sionP

ull

wq

2aa

2T

ensi

onP

ull

qaa

Ten

sionP

ull

wq

2aa

1T

ensi

onP

ull

qaa

RT

Lw

q2

aa2

RT

Lq

aa

RT

Lw

q2

aa1

RT

Lq

aa

aa2

cols

Lbea

ms

()

for

CR

TL

wq

bb

RT

Lw

qbb

TotP

rest

ress

Tra

nsf

erE

ndN

um

bq

Com

pT

ransL

ength

VC

Tq

bb

CR

TL

wq

bb

ConcA

llow

able

Ten

sion

Ten

sionP

ull

Endw

qbb

FR

fw1

bb

Deb

ondL

ength

Num

bq

0ft

=if

CR

TL

wq

bb

CR

TL

wq

1bb

VC

Tq

bb

VC

Tq

1bb

Deb

ondL

ength

Num

bq

0ft

if Fri

ctio

nF

orc

eq

bb

FR

fw1

bb

bb

12

for US

Fq

Ten

sionP

ull

wq

2T

ensi

onP

ull

wq

1

cols

Fre

eStr

and

()

2=

if

bb

12

cols

Lb

eam

s(

)fo

r

cols

Fre

eStr

and

()

2if

132

CR

TL

wq

bb

RT

Lw

qbb

TotP

rest

ress

Tra

nsf

erE

ndN

um

bq

Com

pT

ransL

ength

Deb

ondL

ength

Num

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0ft

=if

CR

TL

wq

bb

CR

TL

wq

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Deb

ondL

ength

Num

bq

0ft

if

US

Fq

ccT

ensi

onP

ull

wq

2cc

Ten

sionP

ull

wq

2cc

1

cc1

cols

Lbea

ms

()

for

VC

Tq

2i

1C

RT

Lw

q2

i1

ConcA

llow

able

Ten

sion

Ten

sionP

ull

Endw

q2

i1

FR

fw1

2i

1

VC

Tq

2i

CR

TL

wq

2i

ConcA

llow

able

Ten

sion

Ten

sionP

ull

Endw

q2

iF

Rfw

12

iU

SF

qi

Deb

ondL

eng

thN

um

bq

0ft

=if

VC

Tq

2i

1V

CT

q1

2i

1

VC

Tq

2i

VC

Tq

12

i

Deb

ondL

eng

thN

um

bq

0ft

if Fri

ctio

nF

orc

eq

2i

1F

Rfw

12

i1

Fri

ctio

nF

orc

eq

2i

FR

fw1

2i

US

Fq

i

US

Fq

i0

kip

if

VC

Tq

2i

1C

RT

Lw

q2

i1

ConcA

llow

able

Ten

sion

Ten

sionP

ull

Endw

q2

i1

FR

fw1

2i

1ab

sVal

US

Fq

kip

VC

Tq

2i

CR

TL

wq

2i

ConcA

llow

able

Ten

sion

Ten

sionP

ull

Endw

q2

iF

Rfw

12

i

Deb

ondL

eng

thN

um

bq

0ft

=if

VC

T2

i1

VC

T1

2i

1

Deb

ondL

eng

thN

um

bq

0ft

ifUS

Fq

i0

kip

ifi1

cols

Lbea

ms

()

for

133

VC

Tq

2i

1V

CT

q1

2i

1

VC

Tq

2i

VC

Tq

12

i

Fri

ctio

nF

orc

eq

2i

1F

Rfw

12

i1

absV

alU

SF

qi

kip

kip

Fri

ctio

nF

orc

eq

2i

FR

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2i

out

in

Ten

sionP

ull

kip

VC

T

kip

RT

Lw

in

XX in

Ten

sionP

ull

End

kip US

F

kip

Fri

ctio

nF

orc

e

kip

in

RT

L

in

Ten

sionP

ull

w

kip

Tota

lSli

deq

in

in

CR

TL

w

kip

Ten

sionP

ull

Endw

kip

RT

Lw

in

EndM

ovx

Equat

ion

11

inS

tran

dM

ovX

XE

quat

ion

12

in

Ten

sPull

Equat

ion

21

kip

Ten

sPull

End

Equat

ion

22

kip

Rev

erse

Tra

nsL

ength

Equat

ion

23

in

Cra

ckP

redic

tor

Equat

ion

31

kip

Unbal

ance

Forc

eE

quat

ion

32

kip

Ten

sPull

wE

quat

ion

33

kip

Ten

sPull

Endw

Equat

ion

34

kip

Rev

erse

Tra

nsL

ength

wE

quat

ion

41

in

Fri

ctio

nV

alu

eE

quat

ion

42

kip

Bea

mS

lide

Equat

ion

43

inC

om

pA

tRT

LE

quat

ion

24

kip

134

All

the f

ollo

win

g fu

nctio

ns ta

ke

extr

em

ely

larg

e m

atr

ices

and m

ake t

he

m s

ma

ll m

atr

ice

s (

the

on

es s

how

n in t

he r

esults)

TotE

ndM

ovx

out 1

cE

ndM

ovx

1c

c1

cols

EndM

ovx

()

for

out q

ccout q

1cc

EndM

ovx

qcc

cc1

cols

EndM

ovx

()

forq

2ro

ws

EndM

ovx

()

for

out

EndM

ovxP

erS

tran

d

out q

ceT

otE

ndM

ovx

qN

um

bC

alcs

ce

ce1

cols

TotE

ndM

ovx

()

forq

1ro

ws

TotE

ndM

ovx

()

Nu

mbC

alcs

for

out N

um

ber

Str

and

s2

c1

Xto

tIndiv

idual

Num

ber

Str

ands

c

2out N

um

ber

Str

ands

12

c1

out N

um

ber

Str

and

s2

c

Xto

tIndiv

idual

Num

ber

Str

ands

c

2out N

um

ber

Str

ands

12

c

c1

cols

Xto

tIndiv

idual

()

for

out

135

Ten

sPull

Per

Str

and

out q

ceT

ensP

ull

qN

um

bC

alcs

ce

ce1

cols

Ten

sPull

()

forq

1ro

ws

Ten

sPull

()

Num

bC

alcs

for

out

Ten

sPull

EndP

erS

tran

d

out q

ceT

ensP

ull

End

qN

um

bC

alcs

ce

ce1

cols

Ten

sPull

End

()

forq

1ro

ws

Ten

sPull

End

()

Num

bC

alcs

for

out

Rev

erse

Tra

nsL

ength

Per

Str

and

out q

ceR

ever

seT

ransL

ength

qN

um

bC

alcs

ce

ce1

cols

Rev

erse

Tra

nsL

ength

()

forq

1ro

ws

Rev

erse

Tra

nsL

ength

()

Num

bC

alcs

for

out

Com

pA

tRT

LP

erS

tran

d

out q

ceC

om

pA

tRT

Lq

Num

bC

alcs

ce

ce1

cols

Com

pA

tRT

L(

)fo

rq1

row

sC

om

pA

tRT

L(

)

Num

bC

alcs

for

out

Str

andM

ovX

XP

erS

tran

d

out q

ceS

tran

dM

ovX

Xq

Num

bC

alcs

ce

ce1

cols

Str

andM

ovX

X(

)fo

rq1

row

sS

tran

dM

ovX

X(

)

Num

bC

alcs

for

out

Unbal

ance

Forc

ePer

Str

and

out q

ceU

nbal

ance

Forc

e qN

um

bC

alcs

ce

ce1

cols

Unbal

ance

Forc

e(

)fo

rq1

row

sU

nbal

ance

Forc

e(

)

Num

bC

alcs

for

out

136

Fri

ctio

nV

alu

ePer

Str

and

out q

ceF

rict

ionV

alue q

Num

bC

alcs

ce

ce1

cols

Fri

ctio

nV

alue

()

forq

1ro

ws

Fri

ctio

nV

alue

()

Num

bC

alcs

for

out

Bea

mS

lideP

erS

tran

d

out q

ceB

eam

Sli

de q

Num

bC

alcs

ce

ce1

cols

Bea

mS

lide

()

forq

1ro

ws

Bea

mS

lide

()

Num

bC

alcs

for

out

Cra

ckP

redic

torP

erS

tran

d

out q

ceC

rack

Pre

dic

tor q

Num

bC

alcs

ce

ce1

cols

Cra

ckP

redic

tor

()

forq

1ro

ws

Cra

ckP

redic

tor

()

Num

bC

alcs

for

out

NS

NN

um

ber

Str

ands

Num

bC

alcs

137

EN

D M

OV

EM

EN

T R

ES

UL

TS

:

Tota

l E

nd M

ove

me

nt fo

r E

ach

Be

am

En

d a

fter

each s

trand c

ut

(this

does N

OT

inclu

de g

lobal m

otion)

In o

rder

to d

ete

rmin

e th

e to

tal m

otio

n o

f each b

eam

end t

his

valu

e m

ust

be a

dded t

o t

he s

lidin

g v

alu

e

EndM

ovxP

erS

tran

d

12

34

56

78

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

0.00772

0.00004

0.00337

0.00176

0.00174

0.00337

0.00004

0.00759

0.02303

0.00004

0.01044

0.0053

0.00536

0.01044

0.00004

0.02282

0.03842

0.00004

0.01772

0.00889

0.00905

0.01772

0.00004

0.03816

0.0539

0.00004

0.02867

0.00912

0.01161

0.02644

0.00004

0.05359

0.06947

0.00004

0.03953

0.00973

0.01464

0.03499

0.00004

0.06912

0.08512

0.00004

0.05067

0.01037

0.01748

0.04404

0.00004

0.08475

0.10087

0.00004

0.06184

0.01129

0.02057

0.05315

0.00004

0.10047

0.11671

0.00004

0.07339

0.01215

0.02319

0.06307

0.00004

0.1163

0.13264

0.00004

0.08586

0.01243

0.0262

0.07293

0.00004

0.13223

0.14866

0.00004

0.09774

0.01364

0.02943

0.08292

0.00004

0.14826

0.16478

0.00004

0.11048

0.01432

0.03336

0.09257

0.00004

0.16439

0.18098

0.00004

0.12311

0.01549

0.03614

0.10373

0.00004

0.18063

0.19729

0.00004

0.13674

0.01601

0.04007

0.1141

0.00004

0.19697

0.21369

0.00004

0.14972

0.01756

0.04389

0.12498

0.00004

0.21342

0.23018

0.00004

0.16281

0.01939

0.04657

0.13738

0.00004

0.22997

0.24677

0.00004

0.17738

0.02011

0.05148

0.14794

0.00004

0.24663

0.26346

0.00004

0.19106

0.02213

0.05457

0.16073

0.00004

0.2634

0.28025

0.00004

0.2063

0.02298

0.06006

0.17153

0.00004

0.28027

0.29713

0.00004

0.222

0.02378

0.06529

0.18301

0.00004

0.29726

0.31412

0.00004

0.23655

0.02613

0.06864

0.19678

0.00004

0.31436

0.3312

0.00004

0.25124

0.02877

0.07334

0.20964

0.00004

0.33157

0.34839

0.00004

0.26748

0.03027

0.07908

0.22187

0.00004

0.34889

0.36568

0.00004

0.28514

0.03078

0.08528

0.23408

0.00004

0.36633

0.38307

0.00004

0.30009

0.03442

0.08998

0.24822

0.00004

0.38388

0.40056

0.00004

0.31538

0.03813

0.09533

0.26213

0.00004

0.40154

in

138

min

EndM

ovx

()

0in

EndM

ovxP

erS

tran

d

12

34

56

78

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

0.40056

0.00004

0.31538

0.03813

0.09533

0.26213

0.00004

0.40154

0.41816

0.00004

0.33043

0.0425

0.10095

0.27621

0.00004

0.41932

0.43586

0.00004

0.34521

0.04755

0.10682

0.29045

0.00004

0.43722

0.45366

0.00004

0.36

0.05298

0.11124

0.30656

0.00004

0.45523

0.47157

0.00004

0.37882

0.05478

0.1208

0.31792

0.00004

0.47336

0.48958

0.00004

0.39361

0.06099

0.12819

0.33184

0.00004

0.4916

0.5077

0.00004

0.41285

0.0631

0.13815

0.34356

0.00004

0.50996

0.52593

0.00004

0.42742

0.07021

0.1456

0.35813

0.00004

0.52844

0.54425

0.00004

0.44219

0.07743

0.15317

0.37291

0.00004

0.54704

0.56269

0.00004

0.45716

0.08474

0.16083

0.38787

0.00004

0.56575

0.58122

0.00004

0.47229

0.09213

0.16858

0.403

0.00004

0.58458

0.59986

0.00004

0.48756

0.09958

0.1764

0.41827

0.00004

0.60351

0.61859

0.00004

0.50293

0.10708

0.18427

0.43365

0.00004

0.62256

0.63742

0.00004

0.51838

0.11461

0.19219

0.44909

0.00004

0.64171

0.65635

0.00004

0.53385

0.12216

0.20011

0.46456

0.00004

0.66097

0.67536

0.00004

0.54931

0.12969

0.20803

0.48002

0.00004

0.68032

0.69445

0.00004

0.56469

0.1372

0.21592

0.49541

0.00004

0.69975

0.7122

0.00004

0.57839

0.14385

0.22292

0.5091

0.00004

0.71781

0.73002

0.00004

0.59212

0.15054

0.22996

0.52283

0.00004

0.73595

0.74387

0.00408

0.60625

0.15686

0.24181

0.53178

0.00927

0.74493

0.74476

0.02112

0.61459

0.16894

0.25452

0.53983

0.02662

0.74584

0.74476

0.03771

0.62151

0.18038

0.26659

0.54645

0.04351

0.74584

0.74476

0.05434

0.62844

0.19185

0.27869

0.55308

0.06045

0.74584

0.74907

0.06534

0.63648

0.20041

0.28988

0.5588

0.07406

0.74781

0.75674

0.073

0.64481

0.20875

0.29837

0.56729

0.08187

0.75563

in

Axia

lref

0.8

63799

0.8

88104

0.9

00399

0.8

71749

()

in

139

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Tota

l E

nd M

ovem

ent

B1 E

nd1

Str

and N

um

ber

Total Movement (in)

140

05

10

15

20

25

30

35

40

45

0

0.0

5

0.1

0.1

5

Tota

l E

nd M

ovem

ent

B1 E

nd2

Str

and N

um

ber

Total Movement (in)

141

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Tota

l E

nd M

ovem

ent

B2 E

nd1

Str

and N

um

ber

Total Movement (in)

142

05

10

15

20

25

30

35

40

45

0

0.0

5

0.1

0.1

5

0.2

0.2

5

Tota

l E

nd M

ovem

ent

B2 E

nd2

Str

and N

um

ber

Total Movement (in)

143

05

10

15

20

25

30

35

40

45

0

0.0

5

0.1

0.1

5

0.2

0.2

5

0.3

0.3

5

Tota

l E

nd M

ovem

ent

B3 E

nd1

Str

and N

um

ber

Total Movement (in)

144

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Tota

l E

nd M

ovem

ent

B3 E

nd2

Str

and N

um

ber

Total Movement (in)

145

05

10

15

20

25

30

35

40

45

0

0.0

5

0.1

0.1

5

0.2

Tota

l E

nd M

ovem

ent

B4 E

nd1

Str

and N

um

ber

Total Movement (in)

146

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Tota

l E

nd M

ovem

ent

B4 E

nd2

Str

and N

um

ber

Total Movement (in)

147

End M

ovem

en

t E

ach

Str

an

d C

on

trib

ute

s

Does N

OT

in

clu

de

glo

ba

l m

otio

n

EndM

ov

out 1

cE

ndM

ovxP

erS

tran

d1

c

c1

cols

EndM

ovxP

erS

tran

d(

)fo

r

out q

ccE

ndM

ovxP

erS

tran

dq

ccE

ndM

ovxP

erS

tran

dq

1cc

cc1

cols

EndM

ovxP

erS

tran

d(

)fo

rq2

Num

ber

Str

ands

for

out

EndM

ov

12

34

56

78

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

0.007724

0.00004

0.00337

0.001758

0.001737

0.003366

0.000043

0.007586

0.015304

00.007065

0.003538

0.003623

0.007071

00.015239

0.015391

00.007286

0.003598

0.003693

0.007287

00.015335

0.015479

00.010947

0.000226

0.00256

0.008714

00.015432

0.015568

00.010862

0.000609

0.003028

0.008548

00.015529

0.015657

00.011134

0.000643

0.002835

0.009053

00.015627

0.015747

00.011171

0.00092

0.003098

0.00911

00.015727

0.015838

00.011556

0.000858

0.002616

0.00992

00.015827

0.01593

00.012464

0.000282

0.003012

0.009862

00.015928

0.016022

00.011881

0.001205

0.003228

0.009992

00.01603

0.016115

00.012748

0.000686

0.003927

0.009648

00.016133

0.016209

00.01262

0.00117

0.002781

0.011158

00.016236

0.016303

00.013634

0.000521

0.003937

0.010374

00.016341

0.016398

00.012981

0.001547

0.003816

0.010875

00.016446

0.016494

00.013085

0.001824

0.002675

0.012404

00.016553

0.016591

00.014574

0.000722

0.004914

0.010562

00.01666

0.016689

00.013675

0.002017

0.003093

0.012786

00.016768

0.016787

00.015242

0.000851

0.005492

0.010797

00.016877

0.016886

00.015699

0.000801

0.005224

0.011481

00.016987

in

148

EndM

ov

12

34

56

78

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

0.016985

00.014557

0.002354

0.003352

0.013774

00.017098

0.017085

00.014684

0.002644

0.004696

0.012855

00.01721

0.017186

00.016247

0.001499

0.005746

0.012233

00.017322

0.017288

00.017661

0.000505

0.006199

0.01221

00.017436

0.01739

00.014948

0.003637

0.004701

0.014139

00.01755

0.017493

00.015292

0.003712

0.005352

0.013916

00.017664

0.017596

00.01505

0.004369

0.005614

0.01408

00.01778

0.0177

00.014773

0.005055

0.005876

0.014237

00.017895

0.017804

00.014795

0.005432

0.004415

0.016109

00.018012

0.017909

00.018816

0.001801

0.009562

0.011362

00.018128

0.018014

00.014791

0.006201

0.007387

0.013922

00.018245

0.018119

00.019238

0.002111

0.009957

0.01172

00.018362

0.018223

00.01457

0.007115

0.007457

0.014565

00.018479

0.018328

00.014777

0.007218

0.007564

0.014777

00.018596

0.018432

00.014966

0.007308

0.007663

0.014966

00.018711

0.018534

00.015131

0.007386

0.007749

0.015131

00.018825

0.018636

00.015268

0.007451

0.007821

0.015269

00.018938

0.018735

00.015374

0.007501

0.007876

0.015374

00.019048

0.018832

00.015444

0.007533

0.007912

0.015444

00.019154

0.018924

00.015473

0.007546

0.007928

0.015473

00.019255

0.019011

00.015456

0.007537

0.007919

0.015456

00.019349

0.01909

00.015389

0.007503

0.007885

0.015389

00.019434

0.01775

00.013691

0.006651

0.006997

0.013689

00.018062

0.01782

00.013732

0.006693

0.007041

0.013732

00.018138

0.013848

0.004035

0.014136

0.00632

0.011855

0.00895

0.009226

0.00898

0.000897

0.01704

0.008332

0.012075

0.012706

0.00805

0.01735

0.000913

00.016591

0.006921

0.011446

0.012075

0.006617

0.016891

0

00.016636

0.00693

0.011468

0.012097

0.006627

0.016939

0

0.004304

0.010993

0.00804

0.008561

0.011188

0.005722

0.013611

0.001969

0.007669

0.007669

0.008336

0.008336

0.008491

0.008491

0.007812

0.007812

in

149

05

10

15

20

25

30

35

40

45

0

0.0

05

0.0

1

0.0

15

0.0

2

0.0

25

End M

ovem

ent

Bea

m 1

End 1

Str

and N

um

ber

Movement (in)

150

05

10

15

20

25

30

35

40

45

0

0.0

05

0.0

1

0.0

15

End M

ovem

ent

Bea

m 1

End 2

Str

and N

um

ber

Movement (in)

151

05

10

15

20

25

30

35

40

45

0

0.0

05

0.0

1

0.0

15

0.0

2

End M

ovem

ent

Bea

m 2

End 1

Str

and N

um

ber

Movement (in)

152

05

10

15

20

25

30

35

40

45

0

0.0

02

0.0

04

0.0

06

0.0

08

0.0

1

0.0

12

0.0

14

End M

ovem

ent

Bea

m 2

End 2

Str

and N

um

ber

Movement (in)

153

05

10

15

20

25

30

35

40

45

0

0.0

02

0.0

04

0.0

06

0.0

08

0.0

1

0.0

12

0.0

14

End M

ovem

ent

Bea

m 3

End 1

Str

and N

um

ber

Movement (in)

154

05

10

15

20

25

30

35

40

45

0

0.0

02

0.0

04

0.0

06

0.0

08

0.0

1

0.0

12

0.0

14

0.0

16

0.0

18

End M

ovem

ent

Bea

m 3

End 2

Str

and N

um

ber

Movement (in)

155

05

10

15

20

25

30

35

40

45

0

0.0

05

0.0

1

0.0

15

End M

ovem

ent

Bea

m 4

End 1

Str

and N

um

ber

Movement (in)

156

05

10

15

20

25

30

35

40

45

0

0.0

05

0.0

1

0.0

15

0.0

2

0.0

25

End M

ovem

ent

Bea

m 4

End 2

Str

and N

um

ber

Movement (in)

157

Glo

bal m

otion

of b

ea

ms-

refe

ren

ce

fra

me

= initia

l beam

positio

ns a

nd location o

f bulk

heads

Positiv

e n

um

be

r in

dic

ate

s b

ea

m s

lide

s r

ight,

negative n

um

ber

indic

ate

s b

eam

slid

es left

Bea

mS

lideP

erS

tran

d

12

34

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

00

00

00

00

0.003113

00

0

0.009488

00

-0.0059

0.017108

00

-0.011946

0.024921

00

-0.018146

0.031791

00

-0.024507

0.040019

00

-0.027773

0.047263

00

-0.034484

0.058434

00

-0.041384

0.066096

00

-0.048485

0.073985

00

-0.052141

0.082115

00

-0.05968

0.090501

00

-0.067457

0.099159

00

-0.071474

0.108108

00

-0.079778

0.117368

00

-0.084076

0.12696

00

-0.088529

0.13691

00

-0.097771

0.142077

00

-0.102573

0.152826

00

-0.107569

0.164024

00

-0.112776

0.169867

00

-0.118212

0.175976

00

-0.123898

0.182376

00

-0.129858

0.189095

00

-0.136119

in

158

Bea

mS

lideP

erS

tran

d

12

34

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

0.152826

00

-0.107569

0.164024

00

-0.112776

0.169867

00

-0.118212

0.175976

00

-0.123898

0.182376

00

-0.129858

0.189095

00

-0.136119

0.196167

00

-0.142712

0.203629

00

-0.142712

0.203629

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

0.212014

00

-0.142712

in

159

05

10

15

20

25

30

35

40

45

0

0.0

5

0.1

0.1

5

0.2

0.2

5

Tota

l B

eam

1 S

lide

Str

and N

um

ber

Movement (in)

160

05

10

15

20

25

30

35

40

45

0

0.2

0.4

0.6

0.8

Tota

l B

eam

2 S

lide

Str

and N

um

ber

Movement (in)

161

05

10

15

20

25

30

35

40

45

0

0.2

0.4

0.6

0.8

Tota

l B

eam

3 S

lide

Str

and N

um

ber

Movement (in)

162

05

10

15

20

25

30

35

40

45

0.1

4

0.1

2

0.1

0.0

8

0.0

6

0.0

4

0.0

20

Tota

l B

eam

4 S

lide

Str

and N

um

ber

Movement (in)

163

BE

AM

RE

SU

LT

S:

Tota

l A

xia

l S

ho

rte

nin

g o

f B

ea

ms a

fte

r e

ach s

trand c

ut

Bea

mS

hort

en

out q

cE

ndM

ovxP

erS

tran

dq

2c

1E

ndM

ovxP

erS

tran

dq

2c

c1

cols

EndM

ovxP

erS

tran

d(

)

2fo

rq1

row

sE

ndM

ovxP

erS

tran

d(

)fo

r

out

Bea

mS

hort

en

12

34

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

0.008

0.005

0.005

0.008

0.023

0.016

0.016

0.023

0.038

0.027

0.027

0.038

0.054

0.038

0.038

0.054

0.07

0.049

0.05

0.069

0.085

0.061

0.062

0.085

0.101

0.073

0.074

0.101

0.117

0.086

0.086

0.116

0.133

0.098

0.099

0.132

0.149

0.111

0.112

0.148

0.165

0.125

0.126

0.164

0.181

0.139

0.14

0.181

0.197

0.153

0.154

0.197

0.214

0.167

0.169

0.213

0.23

0.182

0.184

0.23

0.247

0.197

0.199

0.247

0.263

0.213

0.215

0.263

0.28

0.229

0.232

0.28

in

164

Bea

mS

hort

en

12

34

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

0.314

0.263

0.265

0.314

0.331

0.28

0.283

0.332

0.348

0.298

0.301

0.349

0.366

0.316

0.319

0.366

0.383

0.335

0.338

0.384

0.401

0.354

0.357

0.402

0.418

0.373

0.377

0.419

0.436

0.393

0.397

0.437

0.454

0.413

0.418

0.455

0.472

0.434

0.439

0.473

0.49

0.455

0.46

0.492

0.508

0.476

0.482

0.51

0.526

0.498

0.504

0.528

0.544

0.52

0.526

0.547

0.563

0.542

0.549

0.566

0.581

0.564

0.572

0.585

0.6

0.587

0.595

0.604

0.619

0.61

0.618

0.623

0.637

0.633

0.641

0.642

0.656

0.656

0.665

0.661

0.675

0.679

0.688

0.68

0.694

0.702

0.711

0.7

0.712

0.722

0.732

0.718

0.73

0.743

0.753

0.736

0.748

0.763

0.774

0.754

0.766

0.784

0.794

0.772

0.782

0.802

0.813

0.789

0.799

0.82

0.832

0.806

0.814

0.837

0.849

0.822

0.83

0.854

0.866

0.837

in

Ax

ialr

ef0

.86

37

99

0.8

88

10

40

.90

03

99

0.8

71

74

9(

)in

165

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axia

l S

hort

enin

g o

f B

eam

1

Str

and N

um

ber

Axial Shortening (in)

166

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axia

l S

hort

enin

g o

f B

eam

2

Str

and N

um

ber


167

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axia

l S

hort

enin

g o

f B

eam

3

Str

and N

um

ber


168

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axia

l S

hort

enin

g o

f B

eam

4

Str

and N

um

ber


169

Beam

Length

- a

cco

un

ts fo

r b

ea

m a

xia

l short

enin

g

Bea

mL

ength

out q

Lbea

ms

Bea

mS

hort

enq

cols

Fre

eStr

and

()

2=

if

out q

cL

bea

ms 1

cB

eam

Short

enq

c

c1

cols

Bea

mS

hort

en(

)fo

rco

lsF

reeS

tran

d(

)2

if

q1

row

sB

eam

Short

en(

)fo

r

out

Bea

mL

ength

12

34

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

149.999

155

157

151.999

149.998

154.999

156.999

151.998

149.997

154.998

156.998

151.997

149.996

154.997

156.997

151.996

149.994

154.996

156.996

151.994

149.993

154.995

156.995

151.993

149.992

154.994

156.994

151.992

149.99

154.993

156.993

151.99

149.989

154.992

156.992

151.989

149.988

154.991

156.991

151.988

149.986

154.99

156.99

151.986

149.985

154.988

156.988

151.985

149.984

154.987

156.987

151.984

149.982

154.986

156.986

151.982

149.981

154.985

156.985

151.981

149.979

154.984

156.983

151.979

149.978

154.982

156.982

151.978

149.977

154.981

156.981

151.977

149.975

154.98

156.979

151.975

149.974

154.978

156.978

151.974

ft

170

Bea

mL

ength

12

34

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

149.974

154.978

156.978

151.974

149.972

154.977

156.976

151.972

149.971

154.975

156.975

151.971

149.97

154.974

156.973

151.969

149.968

154.972

156.972

151.968

149.967

154.971

156.97

151.967

149.965

154.969

156.969

151.965

149.964

154.967

156.967

151.964

149.962

154.966

156.965

151.962

149.961

154.964

156.963

151.961

149.959

154.962

156.962

151.959

149.958

154.96

156.96

151.957

149.956

154.959

156.958

151.956

149.955

154.957

156.956

151.954

149.953

154.955

156.954

151.953

149.952

154.953

156.952

151.951

149.95

154.951

156.95

151.95

149.948

154.949

156.949

151.948

149.947

154.947

156.947

151.947

149.945

154.945

156.945

151.945

149.944

154.943

156.943

151.943

149.942

154.942

156.941

151.942

149.941

154.94

156.939

151.94

149.939

154.938

156.937

151.939

149.938

154.936

156.936

151.937

149.936

154.935

156.934

151.936

149.935

154.933

156.932

151.934

149.933

154.932

156.931

151.933

149.932

154.93

156.929

151.932

149.931

154.929

156.928

151.93

ft

171

05

10

15

20

25

30

35

40

45

149.9

2

149.9

4

149.9

6

149.9

8

15

0

Bea

m 1

Len

gth

Str

and N

um

ber

Beam Length (ft)

172

05

10

15

20

25

30

35

40

45

154.9

2

154.9

4

154.9

6

154.9

8

155

Bea

m 2

Len

gth

Str

and N

um

ber

Beam Length (ft)

173

05

10

15

20

25

30

35

40

45

156.9

2

156.9

4

156.9

6

156.9

8

157

Bea

m 3

Len

gth

Str

and N

um

ber

Beam Length (ft)

174

05

10

15

20

25

30

35

40

45

151.9

2

151.9

4

151.9

6

151.9

8

152

Bea

m 4

Len

gth

Str

and N

um

ber

Beam Length (ft)

175

ST

RA

ND

RE

SU

LT

S:

The a

mount e

ach

fre

e s

tra

nd

se

t str

ech

es a

fter

each s

trand c

ut

Str

andM

ovX

XP

erS

tran

d

12

34

5

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

0.007724

0.00341

0.003495

0.003409

0.007586

0.023028

0.010475

0.010655

0.01048

0.022825

0.041532

0.014649

0.017946

0.017767

0.03816

0.063387

0.019221

0.020733

0.020581

0.059492

0.086575

0.022463

0.024369

0.023083

0.081067

0.110045

0.025784

0.027847

0.025936

0.102894

0.132662

0.030085

0.031865

0.028685

0.124982

0.156728

0.033414

0.035339

0.035339

0.144075

0.179902

0.038633

0.038633

0.03849

0.166713

0.207095

0.039343

0.043066

0.041582

0.189644

0.230872

0.044429

0.047679

0.044129

0.212877

0.254969

0.04916

0.05163

0.05163

0.23277

0.279403

0.054665

0.056088

0.054465

0.256649

0.304187

0.05926

0.061451

0.057563

0.280873

0.32934

0.063686

0.065951

0.065951

0.301442

0.35488

0.069312

0.071587

0.068208

0.326407

0.380828

0.073727

0.076697

0.076697

0.347473

0.407207

0.079377

0.08304

0.08304

0.368804

0.434042

0.085126

0.089064

0.085279

0.395033

0.456194

0.094517

0.094771

0.094251

0.416933

0.484029

0.098451

0.10211

0.10211

0.439139

0.512413

0.1035

0.109356

0.109137

0.461668

0.535544

0.115318

0.116059

0.115911

0.48454

0.559044

0.124157

0.124398

0.124363

0.507776

0.582936

0.133049

0.133462

0.132319

0.5314

0.607252

0.141379

0.143445

0.140138

0.55544

0.632024

0.149081

0.154375

0.147782

0.579928

in

176

Str

andM

ovX

XP

erS

tran

d

12

34

5

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

0.535544

0.115318

0.116059

0.115911

0.48454

0.559044

0.124157

0.124398

0.124363

0.507776

0.582936

0.133049

0.133462

0.132319

0.5314

0.607252

0.141379

0.143445

0.140138

0.55544

0.632024

0.149081

0.154375

0.147782

0.579928

0.65729

0.156414

0.164223

0.163891

0.59794

0.675199

0.175229

0.175586

0.175253

0.616069

0.701599

0.181635

0.189175

0.189175

0.634314

0.719717

0.200873

0.201243

0.200895

0.652676

0.73794

0.215443

0.215815

0.21546

0.671155

0.756268

0.230219

0.230597

0.230237

0.689751

0.7747

0.245185

0.245568

0.245203

0.708462

0.793234

0.260316

0.260704

0.260334

0.727287

0.81187

0.275585

0.275976

0.275603

0.746225

0.830605

0.290959

0.291353

0.290978

0.765273

0.849437

0.306403

0.306798

0.306422

0.784426

0.868361

0.321875

0.322272

0.321895

0.803681

0.887372

0.337332

0.337728

0.337351

0.82303

0.906462

0.352721

0.353115

0.35274

0.842464

0.924212

0.366412

0.366763

0.366429

0.860526

0.942032

0.380144

0.380496

0.380162

0.878663

0.95588

0.398316

0.398672

0.398338

0.887644

0.956777

0.423688

0.423453

0.423738

0.888557

0.956777

0.4472

0.446974

0.447246

0.888557

0.956777

0.470766

0.470539

0.470813

0.888557

0.961081

0.489799

0.490288

0.490146

0.890526

in

177

05

10

15

20

25

30

35

40

45

0

0.2

0.4

0.6

0.8

Aver

age

Fre

e S

tran

d S

et 1

Str

etch

Str

and N

um

ber

Average Stretch (in)

178

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

Aver

age

Fre

e S

tran

d S

et 2

Str

etch

Str

and N

um

ber


179

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

Aver

age

Fre

e S

tran

d S

et 3

Str

etch

Str

and N

um

ber


180

05

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

Aver

age

Fre

e S

tran

d S

et 4

Str

etch

Str

and N

um

ber


181

05

10

15

20

25

30

35

40

45

0

0.2

0.4

0.6

0.8

Aver

age

Fre

e S

tran

d S

et 5

Str

etch

Str

and N

um

ber


182

Avera

ge len

gth

s o

f e

ach

Fre

e S

tra

nd

Se

t (inclu

des s

tretc

hin

g)

Str

aLen

gth

out q

cA

vgS

tran

dL

ength

s q1

cS

tran

dM

ovX

XP

erS

tran

dq

c

c1

cols

Str

andM

ovX

XP

erS

tran

d(

)fo

rq1

row

sS

tran

dM

ovX

XP

erS

tran

d(

)fo

r

out

Str

aLen

gth

12

34

5

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

51.459

5.917

5.917

5.917

46.459

51.491

5.98

5.98

5.98

46.491

51.525

6.045

6.045

6.045

46.525

51.561

6.113

6.113

6.113

46.561

51.598

6.184

6.184

6.184

46.598

51.637

6.258

6.258

6.258

46.636

51.678

6.336

6.336

6.336

46.677

51.72

6.417

6.418

6.418

46.719

51.765

6.503

6.503

6.503

46.764

51.812

6.593

6.593

6.593

46.811

51.861

6.688

6.688

6.688

46.86

51.913

6.788

6.788

6.788

46.911

51.968

6.893

6.894

6.893

46.966

52.025

7.005

7.005

7.005

47.023

52.086

7.123

7.123

7.123

47.084

52.151

7.248

7.248

7.248

47.148

52.219

7.381

7.381

7.381

47.216

52.292

7.523

7.523

7.523

47.289

52.37

7.674

7.674

7.674

47.366

52.452

7.835

7.835

7.835

47.449

52.54

8.008

8.009

8.009

47.537

52.635

8.194

8.194

8.194

47.631

ft

183

Str

aLen

gth

12

34

5

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

52.452

7.835

7.835

7.835

47.449

52.54

8.008

8.009

8.009

47.537

52.635

8.194

8.194

8.194

47.631

52.737

8.394

8.394

8.394

47.733

52.847

8.61

8.61

8.61

47.842

52.965

8.844

8.844

8.844

47.961

53.094

9.099

9.099

9.099

48.09

53.234

9.376

9.377

9.376

48.23

53.388

9.68

9.68

9.68

48.383

53.556

10.015

10.015

10.015

48.551

53.743

10.384

10.384

10.384

48.737

53.949

10.795

10.795

10.795

48.943

54.179

11.253

11.253

11.253

49.174

54.438

11.769

11.769

11.769

49.432

54.731

12.354

12.354

12.354

49.726

55.066

13.022

13.022

13.022

50.061

55.452

13.792

13.792

13.792

50.447

55.903

14.691

14.691

14.691

50.897

56.434

15.753

15.753

15.753

51.429

57.072

17.027

17.027

17.027

52.067

57.852

18.584

18.584

18.584

52.846

58.826

20.529

20.529

20.529

53.82

59.363

21.602

21.602

21.602

54.357

60.079

23.032

23.032

23.032

55.073

61.08

25.033

25.033

25.033

56.074

62.58

28.035

28.035

28.035

57.574

63.413

29.704

29.704

29.704

58.407

65.08

33.039

33.039

33.039

60.074

65.08

33.041

33.041

33.041

60.074

ft

184

05

10

15

20

25

30

35

40

45

30

40

50

60

70

Aver

age

Fre

e S

tran

d S

et 1

Len

gth

Str

and N

um

ber

Average Strand Length (ft)

185

05

10

15

20

25

30

35

40

45

5

10

15

20

25

30

35

40

Aver

age

Fre

e S

tran

d S

et 2

Len

gth

Str

and N

um

ber


186

05

10

15

20

25

30

35

40

45

5

10

15

20

25

30

35

40

Aver

age

Fre

e S

tran

d S

et 3

Len

gth

Str

and N

um

ber


187

05

10

15

20

25

30

35

40

45

5

10

15

20

25

30

35

40

Aver

age

Fre

e S

tran

d S

et 4

Len

gth

Str

and N

um

ber


188

05

10

15

20

25

30

35

40

45

20

30

40

50

60

70

Aver

age

Fre

e S

tran

d S

et 5

Len

gth

Str

and N

um

ber


189

Revers

e T

ran

sfe

r L

en

gth

"R

TL

"T

ensT

ransf

erL

ength

Rev

erse

Tra

nsL

ength

Per

Str

and

Ten

sTra

nsf

erL

ength

12

34

5

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

-0.459

0.007

0.164

0.155

-0.455

-0.337

0.214

0.285

0.278

-0.323

-0.208

0.355

0.444

0.384

-0.19

-0.078

0.495

0.59

0.502

-0.056

0.048

0.676

0.757

0.613

0.08

0.181

0.808

0.895

0.895

0.197

0.309

1.02

1.02

1.013

0.335

0.46

1.027

1.19

1.125

0.475

0.591

1.222

1.362

1.209

0.617

0.723

1.395

1.5

1.5

0.738

0.857

1.593

1.652

1.585

0.883

0.993

1.746

1.836

1.676

1.029

1.13

1.884

1.976

1.976

1.153

1.269

2.064

2.154

2.02

1.303

1.409

2.186

2.302

2.302

1.428

1.552

2.349

2.489

2.489

1.555

1.696

2.505

2.653

2.511

1.711

1.814

2.785

2.794

2.775

1.84

1.963

2.851

2.983

2.983

1.97

2.113

2.949

3.155

3.147

2.101

2.234

3.268

3.293

3.288

2.233

2.357

3.465

3.473

3.471

2.367

2.48

3.645

3.658

3.621

2.501

2.604

3.787

3.853

3.748

2.637

2.73

3.892

4.055

3.852

2.774

in

190

Ten

sTra

nsf

erL

ength

12

34

5

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

1.814

2.785

2.794

2.775

1.84

1.963

2.851

2.983

2.983

1.97

2.113

2.949

3.155

3.147

2.101

2.234

3.268

3.293

3.288

2.233

2.357

3.465

3.473

3.471

2.367

2.48

3.645

3.658

3.621

2.501

2.604

3.787

3.853

3.748

2.637

2.73

3.892

4.055

3.852

2.774

2.856

3.967

4.199

4.189

2.87

2.942

4.353

4.363

4.354

2.965

3.071

4.352

4.561

4.561

3.059

3.153

4.674

4.684

4.674

3.152

3.234

4.829

4.838

4.829

3.242

3.312

4.949

4.958

4.949

3.33

3.388

5.031

5.04

5.032

3.415

3.46

5.073

5.082

5.073

3.496

3.528

5.07

5.078

5.071

3.572

3.591

5.019

5.027

5.02

3.642

3.647

4.917

4.924

4.917

3.705

3.694

4.759

4.766

4.759

3.758

3.73

4.542

4.548

4.542

3.798

3.75

4.262

4.267

4.262

3.821

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

in

191

05

10

15

20

25

30

35

40

45

0

0.51

1.52

2.53

3.54

Rev

erse

Tra

nsf

er L

ength

Str

and S

et 1

Str

and N

um

ber

Reverse Transfer Length (in)

192

05

10

15

20

25

30

35

40

45

0123456

Rev

erse

Tra

nsf

er L

ength

Str

and S

et 2

Str

and N

um

ber


193

05

10

15

20

25

30

35

40

45

0123456

Rev

erse

Tra

nsf

er L

ength

Str

and S

et 3

Str

and N

um

ber


194

05

10

15

20

25

30

35

40

45

0123456

Rev

erse

Tra

nsf

er L

ength

Str

and S

et 4

Str

and N

um

ber


195

05

10

15

20

25

30

35

40

45

0

0.51

1.52

2.53

3.54

Rev

erse

Tra

nsf

er L

ength

Str

and S

et 5

Str

and N

um

ber


196

FO

RC

E R

ES

UL

TS

:

Unbala

nced T

en

sio

n P

ull

"U

TP

":

Positiv

e n

um

be

r in

dic

ate

s b

ea

m w

an

ts to

move t

o t

he r

ight,

negative n

um

ber

indic

ate

s b

eam

wants

to

mo

ve

to

th

e le

ft

Unbal

ance

Forc

ePer

Str

and

12

34

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

10.652

0.357

-0.363

-10.317

31.928

0.736

-0.717

-30.877

38.743

13.066

-0.713

-50.758

44.882

5.794

-0.583

-48.959

44.797

7.064

-4.767

-45.676

44.551

7.383

-6.839

-43.411

47.724

6.146

-10.981

-40.472

46.487

6.407

0-51.751

51.377

0-0.457

-49.052

40.036

11.473

-4.574

-45.833

43.393

9.619

-10.508

-40.7

44.944

7.012

0-50.984

47.909

3.872

-4.414

-45.714

47.665

5.705

-10.123

-40.946

46.4

5.63

0-50.626

47.442

5.396

-8.014

-42.793

44.973

6.708

0-50.575

44.683

7.864

0-52.356

43.881

8.022

-7.712

-43.38

50.895

0.49

-1.003

-49.027

44.996

6.667

0-51.293

40.787

10.056

-0.376

-51.067

48.591

1.197

-0.24

-49.601

50.094

0.364

-0.052

-49.95

50.553

0.584

-1.617

-48.596

49.216

2.721

-4.357

-46.227

kip

197

Unbal

ance

Forc

ePer

Str

and

12

34

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

50.094

0.364

-0.052

-49.95

50.553

0.584

-1.617

-48.596

49.216

2.721

-4.357

-46.227

46.237

6.476

-8.064

-42.907

42.162

8.832

-0.376

-50.105

51.044

0.371

-0.347

-50.216

44.008

7.193

0-51.595

49.498

0.321

-0.302

-49.567

49.027

0.294

-0.28

-48.789

47.352

0.268

-0.256

-46.854

44.578

0.243

-0.231

-43.863

40.829

0.217

-0.207

-39.94

36.247

0.192

-0.183

-35.228

30.996

0.168

-0.16

-29.892

25.263

0.144

-0.137

-24.117

19.259

0.121

-0.116

-18.115

13.218

0.1

-0.095

-12.118

7.4

0.08

-0.076

-6.385

5.09

0.059

-0.056

-4.136

2.585

0.048

-0.045

-1.728

0.683

0.037

-0.035

-0.215

-0.368

-0.017

0.021

0.666

-0.051

-0.012

0.014

0.245

-0.473

-0.007

0.009

0.565

0.029

0.008

-0.002

-0.006

kip

198

05

10

15

20

25

30

35

40

45

0

10

20

30

40

50

60

Unbal

ance

d T

ensi

on P

ull

on B

eam

1

Str

and N

um

ber

Unbalanced Tension Pull (kip)

199

05

10

15

20

25

30

35

40

45

02468

10

12

14

Unbal

ance

d T

ensi

on P

ull

on B

eam

2

Str

and N

um

ber


200

05

10

15

20

25

30

35

40

45

12

1086420

Unbal

ance

d T

ensi

on P

ull

on B

eam

3

Str

and N

um

ber


201

05

10

15

20

25

30

35

40

45

60

50

40

30

20

100

Unbal

ance

d T

ensi

on P

ull

on B

eam

4

Str

and N

um

ber

Unbalanced Tension Pull (kip

202

Acting S

tatic F

rictio

n F

orc

e -

ne

ga

tive

va

lue indic

ate

s t

hat

fric

tion is a

cting in t

he s

am

e d

irection a

s t

he f

rictio

n a

t th

e o

pp

osite

be

am

en

d

Any n

um

ber

oth

er

tha

n th

at e

qu

al to

Fs is indic

ating t

he t

he f

riction is f

orc

ed t

o c

hange t

o p

revent

glo

ba

l m

otio

n o

f th

e b

ea

m

Fri

ctio

nV

alu

ePer

Str

and

12

34

56

78

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

25.9

15.2

26.8

26.4

26.7

27.1

15.9

26.2

25.9

-626.8

26

26.4

27.1

-4.6

26.2

25.9

-12.8

26.8

13.7

26.4

27.1

-24.5

26.2

25.9

-19

26.8

21

26.5

27.1

-22.7

26.2

25.9

-18.9

26.8

19.7

22.3

27.1

-19.4

26.2

25.9

-18.7

26.8

19.4

20.3

27.1

-17.2

26.2

25.9

-21.8

26.8

20.6

16.1

27.1

-14.2

26.2

25.9

-20.6

26.8

20.4

27.1

27.1

-25.5

26.2

25.9

-25.5

26.8

26.8

26.6

27.1

-22.8

26.2

25.9

-14.1

26.8

15.3

22.5

27.1

-19.6

26.2

25.9

-17.5

26.8

17.1

16.6

27.1

-14.5

26.2

25.9

-19

26.8

19.7

27.1

27.1

-24.7

26.2

25.9

-22

26.8

22.9

22.7

27.1

-19.5

26.2

25.9

-21.8

26.8

21.1

17

27.1

-14.7

26.2

25.9

-20.5

26.8

21.1

27.1

27.1

-24.4

26.2

25.9

-21.5

26.8

21.4

19.1

27.1

-16.5

26.2

25.9

-19.1

26.8

20.1

27.1

27.1

-24.3

26.2

25.9

-18.8

26.8

18.9

27.1

27.1

-26.1

26.2

25.9

-18

26.8

18.7

19.4

27.1

-17.1

26.2

25.9

-25

26.8

26.3

26.1

27.1

-22.8

26.2

25.9

-19.1

26.8

20.1

27.1

27.1

-25

26.2

25.9

-14.9

26.8

16.7

26.7

27.1

-24.8

26.2

25.9

-22.7

26.8

25.6

26.9

27.1

-23.4

26.2

25.9

-24.2

26.8

26.4

27.1

27.1

-23.7

26.2

25.9

-24.7

26.8

26.2

25.5

27.1

-22.4

26.2

25.9

-23.3

26.8

24

22.7

27.1

-20

26.2

25.9

-20.3

26.8

20.3

19

27.1

-16.7

26.2

kip

203

Fri

ctio

nV

alueP

erS

tran

d

12

34

56

78

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

-24.7

26.8

26.2

25.5

27.1

-22.4

26.2

25.9

-23.3

26.8

24

22.7

27.1

-20

26.2

25.9

-20.3

26.8

20.3

19

27.1

-16.7

26.2

25.9

-16.3

26.8

17.9

26.7

27.1

-23.9

26.2

25.9

-25.1

26.8

26.4

26.8

27.1

-24

26.2

25.9

-18.1

26.8

19.6

27.1

27.1

-25.4

26.2

25.9

-23.6

26.8

26.4

26.8

27.1

-23.3

26.2

25.9

-23.1

26.8

26.5

26.8

27.1

-22.5

26.2

25.9

-21.5

26.8

26.5

26.9

27.1

-20.6

26.2

25.9

-18.7

26.8

26.5

26.9

27.1

-17.6

26.2

25.9

-14.9

26.8

26.5

26.9

27.1

-13.7

26.2

25.9

-10.3

26.8

26.6

26.9

27.1

-926.2

25.9

-5.1

26.8

26.6

26.9

27.1

-3.6

26.2

25.9

0.6

26.8

26.6

27

27.1

2.1

26.2

25.9

6.6

26.8

26.6

27

27.1

8.1

26.2

25.9

12.7

26.8

26.7

27

27.1

14.1

26.2

25.9

18.5

26.8

26.7

27

27.1

19.9

26.2

25.9

20.8

26.8

26.7

27.1

27.1

22.1

26.2

25.9

23.3

26.8

26.7

27.1

27.1

24.5

26.2

25.9

25.2

26.8

26.7

27.1

27.1

26

26.2

25.5

25.9

26.7

26.8

27.1

27.1

26.2

25.6

25.8

25.9

26.7

26.8

27.1

27.1

26.2

26

25.4

25.9

26.8

26.8

27.1

27.1

26.2

25.7

25.9

25.9

26.8

26.8

27.1

27.1

26.2

26.2

kip

Ma

xim

um

Sta

tic f

riction f

orc

e F

s -

(show

n a

s a

refe

rence)

FR

fw25.8

98

25.8

98

26.7

62

26.7

62

27.1

07

27.1

07

26.2

44

26.2

44

()

kip

204

05

10

15

20

25

30

35

40

45

25

25.2

25.4

25.6

25.826

26.2

26.4

Act

ing S

tati

c F

rict

ion F

orc

e B

1 E

nd 1

Str

and N

um

ber

Acting Static Friction Force (kip)

205

05

10

15

20

25

30

35

40

45

30

20

100

10

20

30

Act

ing S

tati

c F

rict

ion F

orc

e B

1 E

nd 2

Str

and N

um

ber


206

05

10

15

20

25

30

35

40

45

26

26.2

26.4

26.6

26.827

27.2

27.4

Act

ing S

tati

c F

rict

ion F

orc

e B

2 E

nd 1

Str

and N

um

ber


207

05

10

15

20

25

30

35

40

45

16

18

20

22

24

26

28

30

Act

ing S

tati

c F

rict

ion F

orc

e B

2 E

nd 2

Str

and N

um

ber


208

05

10

15

20

25

30

35

40

45

16

18

20

22

24

26

28

Act

ing S

tati

c F

rict

ion F

orc

e B

3 E

nd 1

Str

and N

um

ber


209

05

10

15

20

25

30

35

40

45

26.6

26.827

27.2

27.4

Act

ing S

tati

c F

rict

ion F

orc

e B

3 E

nd 2

Str

and N

um

ber


210

05

10

15

20

25

30

35

40

45

20

100

10

20

30

Act

ing S

tati

c F

rict

ion F

orc

e B

4 E

nd 1

Str

and N

um

ber


211

05

10

15

20

25

30

35

40

45

25.6

25.826

26.2

26.4

Act

ing S

tati

c F

rict

ion F

orc

e B

4 E

nd 2

Str

and N

um

ber


212

Tota

l T

ensio

n P

ull

Fo

rce

in

Ea

ch

Fre

e S

trand S

et

Ten

sPull

Per

Str

and

12

34

5

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

-38.2

0.54

13.61

12.9

-37.86

-27.45

17.43

23.23

22.64

-26.32

-16.56

28.23

35.3

30.53

-15.15

-6.04

38.51

45.9

39.06

-4.35

3.64

51.36

57.51

46.53

6.06

13.44

59.93

66.34

66.34

14.59

22.39

73.77

73.77

73.31

24.26

32.43

72.46

83.94

79.36

33.53

40.6

83.99

93.61

83.1

42.4

48.39

93.34

100.35

100.35

49.36

55.81

103.72

107.59

103.17

57.46

62.84

110.5

116.21

106.08

65.14

69.48

115.88

121.51

121.51

70.88

75.72

123.16

128.56

120.54

77.75

81.57

126.54

133.25

133.25

82.67

87

131.69

139.55

139.55

87.19

92.03

135.91

143.93

136.22

92.84

95.15

146.05

146.54

145.53

96.51

99.39

144.39

151.05

151.05

99.76

103.2

143.99

154.04

153.67

102.6

105.07

153.66

154.86

154.62

105.02

106.55

156.64

157.01

156.96

107.01

107.64

158.19

158.78

157.16

108.56

108.33

157.54

160.26

155.91

109.68

108.61

154.85

161.32

153.26

110.35

kip

213

Ten

sPull

Per

Str

and

12

34

5

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

106.551

156.644

157.008

156.956

107.006

107.638

158.191

158.775

157.158

108.562

108.327

157.543

160.264

155.907

109.68

108.611

154.848

161.324

153.259

110.352

108.484

150.646

159.478

159.102

108.998

106.404

157.448

157.819

157.472

107.257

105.519

149.527

156.72

156.72

105.125

102.651

152.149

152.47

152.168

102.601

99.429

148.456

148.75

148.469

99.68

95.85

143.202

143.47

143.214

96.36

91.913

136.491

136.734

136.502

92.639

87.616

128.444

128.662

128.455

88.514

82.957

119.203

119.396

119.213

83.984

77.936

108.932

109.1

108.94

79.048

72.553

97.816

97.96

97.823

73.706

66.81

86.069

86.19

86.075

67.96

60.709

73.928

74.028

73.933

61.815

54.259

61.659

61.739

61.662

55.278

48.059

53.15

53.209

53.153

49.017

41.542

44.127

44.174

44.129

42.401

34.539

35.222

35.259

35.225

35.009

26.878

26.51

26.493

26.514

27.179

19.844

19.793

19.781

19.795

20.04

12.826

12.353

12.346

12.355

12.919

6.447

6.477

6.485

6.482

6.477

kip

214

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

Ten

sion P

ull

Str

and S

et 1

Str

and N

um

ber

Tension Pull (kip)

215

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

140

160

Ten

sion P

ull

Str

and S

et 2

Str

and N

um

ber

Tension Pull (kip)

216

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

140

160

180

Ten

sion P

ull

Str

and S

et 3

Str

and N

um

ber

Tension Pull (kip)

217

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

140

160

Ten

sion P

ull

Str

and S

et 4

Str

and N

um

ber

Tension Pull (kip)

218

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

Ten

sion P

ull

Str

and S

et 5

Str

and N

um

ber

Tension Pull (kip)

219

Tensio

n P

ull

in fre

e s

tra

nd

se

ts "

TP

"

(By S

trand S

et)

Ten

sPull

EndP

erS

tran

d

12

34

5

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

-31.555

0.45

11.243

10.655

-31.275

-22.57

14.333

19.097

18.617

-21.638

-13.552

23.1

28.879

24.978

-12.393

-4.914

31.348

37.357

31.791

-3.544

2.946

41.58

46.555

37.666

4.903

10.821

48.237

53.395

53.395

11.741

17.911

59.013

59.013

58.647

19.406

25.776

57.6

66.719

63.084

26.652

32.051

66.309

73.903

65.607

33.475

37.929

73.155

78.651

78.651

38.691

43.405

80.668

83.68

80.246

44.691

48.473

85.244

89.645

81.836

50.248

53.129

88.611

92.916

92.916

54.203

57.364

93.305

97.393

91.322

58.903

61.174

94.904

99.934

99.934

62.003

64.551

97.703

103.537

103.537

64.693

67.489

99.669

105.552

99.896

68.084

68.903

105.758

106.113

105.387

69.884

70.993

103.133

107.895

107.895

71.258

72.621

101.323

108.4

108.135

72.199

72.74

106.38

107.209

107.043

72.703

72.454

106.518

106.766

106.73

72.764

71.759

105.461

105.85

104.772

72.375

70.648

102.745

104.52

101.678

71.53

69.116

98.539

102.66

97.529

70.224

kip

220

Ten

sPull

EndP

erS

tran

d

12

34

5

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

72.74

106.38

107.209

107.043

72.703

72.454

106.518

106.766

106.73

72.764

71.759

105.461

105.85

104.772

72.375

70.648

102.745

104.52

101.678

71.53

69.116

98.539

102.66

97.529

70.224

67.157

93.257

98.725

98.492

67.475

63.842

94.469

94.691

94.483

64.354

61.09

86.568

90.733

90.733

60.862

57.028

84.527

84.706

84.538

57

52.639

78.594

78.75

78.601

52.772

47.925

71.601

71.735

71.607

48.18

42.893

63.696

63.809

63.701

43.232

37.55

55.048

55.141

55.052

37.935

31.906

45.847

45.921

45.851

32.302

25.979

36.311

36.367

36.313

26.349

19.787

26.677

26.716

26.679

20.102

13.362

17.214

17.238

17.215

13.592

6.745

8.214

8.225

8.215

6.868

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

00

00

0

kip

221

05

10

15

20

25

30

35

40

45

0

10

20

30

40

50

60

70

80

Bea

m E

nd T

ensi

on P

ull

Str

and S

et 1

Str

and N

um

ber

Beam End Tension Pull (kip)

222

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

Bea

m E

nd T

ensi

on P

ull

Str

and S

et 2

Str

and N

um

ber


223

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

Bea

m E

nd T

ensi

on P

ull

Str

and S

et 3

Str

and N

um

ber


224

05

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

Bea

m E

nd T

ensi

on P

ull

Str

and S

et 4

Str

and N

um

ber


225

05

10

15

20

25

30

35

40

45

0

10

20

30

40

50

60

70

80

Bea

m E

nd T

ensi

on P

ull

Str

and S

et 5

Str

and N

um

ber


226

Tra

nsfe

rred P

restr

ess F

orc

e L

ine

arly In

terp

ola

ted a

t R

evers

e T

ransfe

r Length

"C

RT

L"

Com

pA

tRT

LP

erS

tran

d

12

34

56

78

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

-2.491

0.036

0.036

0.888

0.888

0.841

0.841

-2.469

-2.44

1.55

1.55

2.065

2.065

2.013

2.013

-2.339

-1.882

3.208

3.208

4.011

4.011

3.469

3.469

-1.721

-0.842

5.374

5.374

6.404

6.404

5.45

5.45

-0.608

0.607

8.561

8.561

9.585

9.585

7.755

7.755

1.009

2.623

11.694

11.694

12.944

12.944

12.944

12.944

2.846

5.038

16.597

16.597

16.597

16.597

16.494

16.494

5.458

8.315

18.581

18.581

21.522

21.522

20.35

20.35

8.597

11.752

24.313

24.313

27.098

27.098

24.056

24.056

12.274

15.695

30.271

30.271

32.545

32.545

32.545

32.545

16.01

20.152

37.453

37.453

38.851

38.851

37.257

37.257

20.749

25.134

44.2

44.2

46.482

46.482

42.433

42.433

26.055

30.651

51.122

51.122

53.606

53.606

53.606

53.606

31.271

36.713

59.715

59.715

62.332

62.332

58.446

58.446

37.698

43.331

67.223

67.223

70.787

70.787

70.787

70.787

43.919

50.518

76.463

76.463

81.029

81.029

81.029

81.029

50.629

58.286

86.077

86.077

91.158

91.158

86.274

86.274

58.8

65.622

100.722

100.722

101.06

101.06

100.369

100.369

66.557

74.543

108.29

108.29

113.29

113.29

113.29

113.29

74.821

84.088

117.322

117.322

125.515

125.515

125.209

125.209

83.599

92.946

135.93

135.93

136.989

136.989

136.777

136.777

92.899

102.288

150.378

150.378

150.728

150.728

150.678

150.678

102.725

112.123

164.783

164.783

165.391

165.391

163.707

163.707

113.086

122.456

178.092

178.092

181.168

181.168

176.243

176.243

123.986

133.295

190.04

190.04

197.988

197.988

188.091

188.091

135.432

kip

227

Com

pA

tRT

LP

erS

tran

d

12

34

56

78

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

102.288

150.378

150.378

150.728

150.728

150.678

150.678

102.725

112.123

164.783

164.783

165.391

165.391

163.707

163.707

113.086

122.456

178.092

178.092

181.168

181.168

176.243

176.243

123.986

133.295

190.04

190.04

197.988

197.988

188.091

188.091

135.432

144.646

200.862

200.862

212.637

212.637

212.136

212.136

145.33

154.286

228.299

228.299

228.838

228.838

228.334

228.334

155.522

166.61

236.096

236.096

247.453

247.453

247.453

247.453

165.987

176.788

262.035

262.035

262.588

262.588

262.067

262.067

176.701

187.16

279.447

279.447

279.999

279.999

279.472

279.472

187.633

197.691

295.354

295.354

295.907

295.907

295.379

295.379

198.743

208.336

309.38

309.38

309.929

309.929

309.405

309.405

209.982

219.039

321.111

321.111

321.654

321.654

321.136

321.136

221.286

229.726

330.102

330.102

330.634

330.634

330.127

330.127

232.572

240.302

335.873

335.873

336.39

336.39

335.898

335.898

243.731

250.637

337.911

337.911

338.409

338.409

337.935

337.935

254.621

260.558

335.669

335.669

336.142

336.142

335.692

335.692

265.045

269.82

328.568

328.568

329.013

329.013

328.59

328.59

274.732

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

278.076

316

316

316.411

316.411

316.02

316.02

283.297

kip

228

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

Tra

nsf

erre

d P

rest

ress

at

RT

L B

1 E

nd 1

Str

and N

um

ber

Transferred Prestress Force (kip)

229

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

350

Tra

nsf

erre

d P

rest

ress

at

RT

L B

1 E

nd 2

Str

and N

um

ber


230

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

350

Tra

nsf

erre

d P

rest

ress

at

RT

L B

2 E

nd 1

Str

and N

um

ber


231

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

350

Tra

nsf

erre

d P

rest

ress

at

RT

L B

2 E

nd 2

Str

and N

um

ber


232

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

350

Tra

nsf

erre

d P

rest

ress

at

RT

L B

3 E

nd 1

Str

and N

um

ber


233

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

350

Tra

nsf

erre

d P

rest

ress

at

RT

L B

3 E

nd 2

Str

and N

um

ber


234

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

350

Tra

nsf

erre

d P

rest

ress

at

RT

L B

4 E

nd 1

Str

and N

um

ber


235

05

10

15

20

25

30

35

40

45

0

50

100

150

200

250

300

Tra

nsf

erre

d P

rest

ress

at

RT

L B

4 E

nd 2

Str

and N

um

ber


236

Inte

rmedia

te c

alc

ula

tio

ns to

de

term

ine

cra

ckin

g p

ote

ntial of

each b

eam

end:

CT

NoT

S

out j

k

Cra

ckP

redic

torP

erS

tran

dj

kC

oncA

llow

able

Ten

sion

Abf

j1

row

sC

rack

Pre

dic

torP

erS

tran

d(

)fo

rk1

cols

Cra

ckP

redic

torP

erS

tran

d(

)fo

r

out

Accounts

fo

r e

cce

ntr

icity o

f fr

ictio

n fo

rce

CT

NoT

S2

out j

kC

TN

oT

Sj

k

Fri

ctio

nV

alueP

erS

tran

dj

kF

rice

2

IBott

om

j1

row

sC

TN

oT

S(

)fo

rk1

cols

CT

NoT

S(

)fo

r

out

237

Incre

ases t

he

siz

e o

f th

e m

atr

ix

Ten

sTra

nsf

erL

ength

w

out q

1T

ensT

ransf

erL

ength

q1

out q

2co

lsT

ensT

ransf

erL

ength

()

2T

ensT

ransf

erL

ength

qco

lsT

ensT

ransf

erL

ength

()

Deb

ondL

ength

q1

0ft

=if

out q

1out q

11

out q

2co

lsT

ensT

ransf

erL

ength

()

2out q

12

cols

Ten

sTra

nsf

erL

ength

()

2

Deb

ondL

ength

q1

0ft

ifq1

row

sT

ensT

ransf

erL

ength

()

for

out z

2j

2T

ensT

ransf

erL

ength

zj

out z

2j

1T

ensT

ransf

erL

ength

zj

Deb

ondL

ength

z1

0ft

=if

out z

2j

2out z

12

j2

out z

2j

1out z

12

j1

Deb

ondL

ength

z1

0ft

ifz1

row

sT

ensT

ransf

erL

ength

()

forj

2co

lsT

ensT

ransf

erL

ength

()

1fo

r

out

238

Accounts

for

ecce

ntr

icity o

f b

ea

rin

g fo

rce

(assum

ing t

hat

it a

cts

at

the v

ery

end o

f th

e b

eam

)

CT

NoT

S3

out j

kC

TN

oT

S2

jk

Bea

ringw

1k

Ten

sTra

nsf

erL

ength

wj

kF

rice

IBott

om

j1

row

sC

TN

oT

S2

()

fork

1co

lsC

TN

oT

S2

()

for

out

Cra

ckin

g P

ote

ntia

l o

f e

ach

be

am

en

d b

ottom

fla

nge =

Actu

al S

tresses/

Allo

wable

Tensio

n S

tress

Cra

ckin

gP

ote

nti

al

out j

k

CT

NoT

S3

jk

1

ConcT

ensS

tren

gth

j1

row

sC

TN

oT

S2

()

fork

1co

lsC

TN

oT

S2

()

for

out

239

Cra

ckin

g C

rite

rio

n: A

bo

ve

1.0

in

dic

ate

s v

ert

ical cra

ck p

robable

CP

= A

ctu

al S

tre

ss/ A

llow

ab

le S

tre

ss

Cra

ckin

gP

ote

nti

al

12

34

56

78

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

0.1543

0.0102

0.2375

0.2327

0.2383

0.2432

0.0226

0.162

0.2137

-0.2215

0.4329

0.4229

0.4298

0.4395

-0.1939

0.2262

0.2788

-0.2533

0.5383

0.3597

0.6139

0.6236

-0.409

0.2868

0.3491

-0.2724

0.6431

0.5633

0.675

0.683

-0.3143

0.3622

0.4178

-0.2076

0.707

0.6089

0.6626

0.7287

-0.2036

0.4328

0.482

-0.1454

0.7651

0.6616

0.6807

0.7763

-0.1114

0.4987

0.5387

-0.1394

0.8355

0.7486

0.6605

0.8154

-0.0141

0.5597

0.5936

-0.0731

0.8778

0.7864

0.9233

0.9233

-0.1332

0.6073

0.641

-0.1029

0.9468

0.9468

0.9468

0.9534

-0.0498

0.6589

0.6915

0.106

0.9291

0.7622

0.9096

0.976

0.0365

0.7054

0.7291

0.0878

0.9791

0.8377

0.8298

0.9839

0.1465

0.7467

0.7618

0.0903

1.0136

0.9094

1.0597

1.0597

0.0152

0.7753

0.7897

0.0657

1.0493

0.9911

0.9908

1.057

0.1173

0.8066

0.8126

0.0836

1.0632

0.9764

0.8991

1.0527

0.2078

0.8325

0.8304

0.1119

1.067

0.9803

1.1041

1.1041

0.0641

0.8462

0.8432

0.0989

1.0757

0.9916

0.9509

1.0756

0.1919

0.8617

0.8507

0.1354

1.0636

0.9576

1.1036

1.1036

0.0628

0.8652

0.8529

0.132

1.0553

0.9292

1.0987

1.0987

0.0213

0.864

0.8497

0.131

1.0391

0.9085

0.9275

1.0527

0.1545

0.8633

0.836

-0.0109

1.0389

1.0308

1.0336

1.0502

0.0377

0.8515

0.8223

0.0609

0.9932

0.8811

1.0278

1.0278

-0.0311

0.8347

0.8028

0.1003

0.9491

0.777

0.9843

0.9908

-0.0645

0.8127

0.7739

-0.0788

0.9277

0.9068

0.9401

0.9443

-0.0825

0.7856

0.7401

-0.1565

0.8811

0.8746

0.8954

0.8963

-0.1384

0.7532

0.7015

-0.2226

0.8252

0.8146

0.8089

0.8382

-0.1703

0.7155

0.6579

-0.262

0.7598

0.7093

0.692

0.7726

-0.1892

0.6724

240

Cra

ckin

gP

ote

nti

al

12

34

56

78

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

-0.0788

0.9277

0.9068

0.9401

0.9443

-0.0825

0.7856

0.7401

-0.1565

0.8811

0.8746

0.8954

0.8963

-0.1384

0.7532

0.7015

-0.2226

0.8252

0.8146

0.8089

0.8382

-0.1703

0.7155

0.6579

-0.262

0.7598

0.7093

0.692

0.7726

-0.1892

0.6724

0.6093

-0.2758

0.6871

0.564

0.5481

0.7009

-0.1952

0.6238

0.5557

-0.2722

0.6096

0.4375

0.6226

0.6299

-0.4118

0.5691

0.4972

-0.5328

0.5287

0.5212

0.5382

0.5452

-0.4998

0.5105

0.4341

-0.4804

0.4404

0.2921

0.4529

0.4529

-0.6208

0.448

0.3679

-0.6936

0.3387

0.3319

0.3495

0.3559

-0.6778

0.3818

0.2985

-0.7895

0.2343

0.2279

0.2457

0.2519

-0.7672

0.312

0.226

-0.8644

0.1265

0.1204

0.1385

0.1443

-0.8367

0.2387

0.1507

-0.9182

0.0177

0.012

0.0302

0.0357

-0.886

0.1622

0.0727

-0.9506

-0.0894

-0.0948

-0.0765

-0.0713

-0.9148

0.0828

-0.0075

-0.9615

-0.1918

-0.1968

-0.1785

-0.1738

-0.9231

0.0007

-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

241

05

10

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30

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40

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0.50

0.51

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fact/ft

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10

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05

10

15

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0.51

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fact/ft

245

05

10

15

20

25

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35

40

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0.50

0.51

1.5

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Bea

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fact/ft

246

05

10

15

20

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40

45

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0.51

1.5

CP

Bea

m 3

End 2

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and N

um

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fact/ft

247

05

10

15

20

25

30

35

40

45

0.50

0.51

1.5

CP

Bea

m 4

End 1

Str

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um

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fact/ft

248

05

10

15

20

25

30

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0.51

1.5

CP

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249

APPENDIX C SIMPLIFIED VERTICAL CRACK PREDICTOR

ORIGIN 1≡

A simplified procedure and example for hand calculation of the cracking criterion ( ξ = fcalc/f) for thecase of a single symmetrically placed beam on the casting bed.

Lc

Ls Ls

Prestressed Beam

Given:

Prestress Values:

Total Number of uncut prestressing strands NumUncutStrands 20:=

Total Number of cut prestressing strands NumCutStrands 10:=

Jacking Force per prestressing strand JackingForce 44:= (kip)

Area of prestressing strand Aps .2192:= (in^2)

Modulus of Elasticity of prestressing strand Es 28500:= (ksi)

Temperature strain in free strands εps .00000667:= (in/in/F)

Diameter of prestressing strands D 0.600:= (in)

Length of free strands L 15:= (ft)

Concrete Values:

Concrete strength at the time of strand cutting fci 6000:= (psi)

Cross sectional area of prestressed beam A 789:= (in^2)

Bottom flange area of prestressed beam Abf 361:= (in^2)

Unit weight of the concrete beam δ 150:= (pcf)

Beam Length Lc 120:= (ft)

251

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

TempChange 30:= (F)

Static coefficient of friction between bottom of the beam and the casting bed

µs .40:=

Dynamic coefficient of friction between bottomof the beam and the casting bed

µd .35:=

Expected initial camber CamberIn 3:= (in)

Distance from centroid of cross section to bottomof beam

Dcentroid 24.73:= (in)

Distance from centroid of bottom flange to bottomof beam

DcentroidBF 7.266:= (in)

Moment of inertia of bottom flange IbottomF 7280:= (in^4)

Solution:

Step 1: Calculate the tension pull due to temperature change "TPtemp"

TPtemp TempChange εps⋅ Es⋅ Aps⋅ NumUncutStrands⋅:=

TPtemp 25.001= (kip)

Step 2: Calculate the free strand spring stiffness "ks"

ksAps Es⋅

L 12⋅NumUncutStrands⋅:=

ks 694.13= (kip/in)

Step 3: Calculate the compression transfer length of the prestressing strands "lt"

lt .33JackingForce

Aps⎛⎜⎝

⎞⎠

⋅ D⋅3

fci

1000

⋅:=

lt 28.104= (in)

252

Step 4: Calculate the concrete modulus of elasticity "E"

E40000 fci 106

+( ) δ

145⎛⎜⎝

⎞⎠

1.5

1000:=

E 4312.189= (ksi)

Step 5: Calculate the beam spring stiffness "kc"

kcA E⋅

Lc 12⋅43

lt⋅−

:=

kc 2425.845= (kip/in)

Step 6: Calculate friction forces "Fs" and "Fd":

Fsµs A⋅ δ⋅ Lc⋅

2 144⋅ 1000⋅:=

Fs 19.725= (kip)

Fdµd A⋅ δ⋅ Lc⋅

2 144⋅ 1000⋅:=

Fd 17.259= (kip)

Step 7: Calculate beam movement "∆":

check to see if static friction force has been overcome

JackingForce NumCutStrands⋅ 440= (kip) Fs 19.725= kip( )

if this number is less than Fs the beam movement ∆1 = 0otherwise

∆1JackingForce NumCutStrands⋅ Fd−

2 kc ks+( ):=

∆1 0.068= (in)

253

∆AtotNumUncutStrands NumCutStrands+( ) JackingForce⋅ Lc⋅ 12⋅

A E⋅:=

∆Atot 0.559= (in)

∆CtotCamberIn Dcentroid⋅

Lc2

12⋅

:=

∆Ctot 0.103= (in)

∆ ∆1∆1

∆Atot∆Ctot⋅+:=

∆ 0.08= in( )

Step 8: Calculate the tension pull due to beam movement "TPm"

TPm ∆ ks⋅:=

TPm 55.699= (kip)

Step 9: Calculate the total tension pull "TP"

TP TPtemp TPm+:=

TP 80.7= (kip)

Step 10: Calculate the reverse transfer length "RTL"

RTL .33TP

NumUncutStrands Aps⋅⎛⎜⎝

⎞⎠

⋅ D⋅3

fci

1000

⋅:=

RTL 2.577= (in)

Step 11: Calculate the prestress transferred to the concrete linearly interpolated at the RTL "CRTL"

CRTL JackingForceRTL

lt⎛⎜⎝

⎞⎠

⋅ NumCutStrands⋅:=

CRTL 40.35= (kip)

254

Step 12: Calculate the beam tension strength "TS"

TS5 fci1000

:=

TS 0.387= (ksi)

Step 13: Calculate the cracking criterion ξ

ξ

1−CRTL TP− Fs−

A

A δ⋅ Lc⋅

2 144⋅ 1000⋅RTL⋅ DcentroidBF⋅

IbottomF−

Fs DcentroidBF2⋅

IbottomF−

⎛⎜⎜⎝

⎞

⎠⋅

TS:=

ξ 0.893=

This procedure should be repeated for other numbers of cut strands to determine the maximum ξ.The maximum usually occurs when approximately one third of the strands have been cut. If ξ isgreater than 1.0 the section is assumed to have vertically cracked.

255

256

APPENDIX D FIELD STUDY STRAND LAYOUT

Debonded Strands:

Triangle = 15’

Square = 10’

Circle = 5’

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, Upper Saddle River, 2000.

Kannel, Jeffery J., and French, Catherine E., and Stolarski, Henryk K., “Release Methodology of Prestressing Strands,” Minnesota 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 Cracking 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.

258

BIOGRAPHICAL SKETCH

I was born in Winter Park, Florida, to Beverly and Daniel Reponen. I enjoy

swimming, saltwater fishing, canoeing, windsurfing, weightlifting, 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.

reponen_m

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