CAMBER ANALYSIS AND DESIGN OF CONTINUOUS

PRESTRESSED CONCRETE BEAMS

by

HOWARD RICHARD HORN, JR., B.S. in C.E.

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Technological College

in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

Accepted

May, 1969

t\ • H't »^ »

T3

f\Jo.8b' COP' ^

ACKNOWLEDGMENTS

The author is deeply indebted to Dr. Kishor C. Mehta for his

direction of this thesis and to the other members of the committee.

Professor J. R. McDonald and Professor A. J. Sanger.

This thesis is dedicated to my parents, Mr. and Mrs. Howard R.

Horn, whose encouragement and inspiration made this work possible.

11

TABLE OF CONTENTS

ACKNOWLEDGMENTS i i

LIST OF FIGURES v

LIST OF SYMBOLS vi

I. INTRODUCTION 1

Background 1

Purpose and Scope 5

II. DEFLECTED SHAPE AT TRANSFER OF PRESTRESS 7

Assumption of Two Support Locations 8

Girder Moment Determination 8

Approximation of the Slope at One End of the Member 9

Evaluation of Slopes and Deflections of the Beam 10

Location of Support Points 10

Check Validity of Assumed Support Point Locations 11

Programming the Method for the Computer 11

III. DESIGN OF THE COMPOSITE BEAM 15

Method of Design 16

Section Choice 16

Cable Layout 17

Deflected Shape - Transfer Moments 18

Computer Solution 13

IV. CONCLUSIONS 21

BIBLIOGRAPHY 23

111

iv

APPENDICES

A. Constants for Symmetrical I-Sections 25

B. Example Problem 26

C. Flow Chart Symbols 36

D. Deflected Shape Computer Program Flow Chart 43

E. Design Computer Program Flow Chart 67

LIST OF FIGURES

Figure Page

1. Eccentrically Prestressed Concrete Section 2

2. Simply Supported Beam 3

3. Continuous Beam 4

4. Design Sections 6

5. Slope Approximation 10

6. Deflected Shapes 12

7. Typical Deflected Shape 13

8. Section Properties 15

LIST OF SYMBOLS

A - Cross sectional area.

A - Area of concrete; or area of precast portion.

A ' - Area of composite section.

A . - Area of steel.

c - Distance from center of gravity of concrete to extreme fiber,

c. , c. - c for bottom (top) fibers,

c. ', c.' - c for composite sections,

c.g.c. - Center of gravity of concrete section,

c.g.c' - Composite sections; c.g.c. for composite sections,

c.g.s. - Center of gravity of steel,

e. e,, ep - Steel eccentricities.

F - Total effective prestressing force after losses.

FQ - Total prestressing force at transfer,

f ' - Compressive strength of concrete,

fxf fu - Fiber stress in top (bottom) fibers,

f.', f. ' - Tensile stress in top (bottom) fibers,

h - Depth of precast beam.

I - Moment of inertia.

r - Composite sections; I for composite sections,

k. , k^ - Kern distance from c.g.c. for top (bottom).

M - Moment acting on composite section.

M ' - Negative moment acting on composite section.

Mp - Girder load moment.

M - Moment on precast portion of composite section.

vi

vil

M ' - Negative moment on precast section.

Mj - Moment due to total load.

Mj Negative " ^°^^^ negative moment.

m. , m. - Ratio of section moduli of precast portion to composite

section for bottom (top) fiber,

r - Radius of gyration, x-

S, - Length of left overhang, beam with two supports.

Sp - Distance between supports.

S-, - Length of right overhang, beam with two supports.

CHAPTER I

INTRODUCTION

Background

A prestressed concrete beam has permanent stresses induced to

improve its structural behavior. Steel cables embedded in the concrete

beam produce the permanent stresses. The cables are tensioned to

provide compressive stresses in the areas of potential tensile failure

of the concrete.

Two methods of prestressing are in common use. One method is

post-tensioning; in this case the cables are tensioned after the

concrete has hardened. The other method is pre-tensioning; that is,

the cables are tensioned before the concrete is cast. The pre-

tensioning method is employed in commercial prestressed concrete plants

to allow mass production of the prestressed beams. The production of

a pre-tensioned beam consists of tensioning the cables to the desired

amount of prestress. casting the concrete beam around the cables,

allowing the concrete to harden, and then transferring the prestress

to the concrete by cutting the cables at the ends of the beam.

The center of gravity of the steel cables is generally at some

distance from the center of gravity of the concrete. This distance is

referred to as the eccentricity of the cables. The prestressing force

created by the steel cables induces stresses as shown in Figure 1. It

is seen that the stresses produced are due to compressive force F. and

due to cable moments Fe. Normally the cable is positioned so that the

cable moments oppose the moments due to external load.

1

Eccentricity, e

Center of Gravity of ^Concrete, c.g.c.

Center of Gravity of Steel, c.g.s. t

E- - > • +

/

7

Fee ^ I

Prestress Direct Cable Moments Load Effect Effect

Figure 1. Eccentrically Prestressed Concrete Section.

The stress condition at transfer (when the cables are cut) is

critical in the design of a prestressed concrete beam. The stresses

at transfer are produced by the compressive force, the cable moments,

and the girder moments. The girder moments are caused by the dead

weight of the beam since the beam deflects due to the cable moments.

This situation can be illustrated by an example of a simply supported

prestressed beam.

The simply supported beam has positive moment throughout the beam

due to external load. Figure 2. The cables are placed such that the

cable moments, Fe, oppose the moment due to external load. At transfer,

when there is no external load, the cable moments cause the beam to

deflect as shown in Figure 2d. Since the beam lifts up from the

prestressing bed supporting itself only at the ends, girder moments

due to dead weight of the beam are created. In a simply supported

beam the stresses at transfer can be obtained easily because the cable

moments and the girder moments are easily obtained.

•External Load

(a) Beam Subject to External Load

(b) External Load Moment Diagram

ei

c.g.c.

c.g.s

Cable Placement

Cable Moment = Fe

L Rigid Prestressing Bed

(d) Deflected Shape at Transfer

Figure 2. Simply Supported Beam

However, in a continuous beam the stresses at-transfer are .diffi­

cult to determine. This difficulty can be illustrated by a three span

continuous beam. Figure 3. The cables are placed as shown in

Figure 3c so that the cable moments oppose the external load moments.

At transfer, the deflected shape of the beam is difficult to determine

because the cable moments change sign along the beam. If the deflected

shape is not known, then the location of the support points can not be

determined. The lack of known support locations prevents the evaluation

of the girder moments which, in turn, prevents the evaluation of the

stresses at transfer. A literature survey failed to uncover a method

for obtaining the deflected shape at transfer for a pre-tensioned

continuous beam. If such a method were available, the design of a pre-

tensioned continuous beam could be accomplished.

f ! f t (a) Beam Subject to External LoaJ

(b) External Load Moment Diagram

c.g.s.

L c.g.c,

(c) Cable Placement

Rigid Prestressing ^ Bed

Support Points (d) Deflected Shape at Transfer

Figure 3. Continuous Beam.

The same difficulty of determining the deflected shape at transfer

occurs 1n the case of a pre-tensioned cantilever beam with overhangs.

The cable must be placed high in the beam in the negative moment region

occurring near the interior support. This placement results in a change

in sign of the cable moments along the beam. The solution required for

this problem is the same as the solution required for any pre-tensioned

continuous beam.

Purpose and Scope

The purpose of this project is to devise a method for determining

the deflected shape of a pre-tensioned continuous beam at the transfer

of prestress. Knowledge of the deflected shape allows evaluation of

the stresses in the beam at transfer. These stresses must be known for

the design of the beam. The design is a process usually requiring many

iterations. Hence a computer program is developed to design a pre-

tensioned beam subject to loading conditions occurring in a continuous

beam.

A method to evaluate the deflected shape requiring knowledge of

only the applied moments and the section properties of the beam is

developed. The beam considered is a composite section consisting of a

symmetrical I-beam and a cast-in-plaee slab (see Figure 4), The computer

programs are written in Fortran IV language for use on the IBM 360

computer.

The method for determining the deflected shape at transfer of

prestress is presented in Chapter II. A computer program written to

execute this method is also discussed. Chapter III explains the design

of a pre-tensioned overhang beam which is also programmed. Chapter IV

discusses the limitations of the method developed and the conclusions

resulting from the project.

C

\ZJ3 Precast Section

c Cast-in-Place Slab n

Composite Section

Figure 4. Design Sections

CHAPTER II

DEFLECTED SHAPE AT TRANSFER OF PRESTRESS

A method for determining the deflected shape of a pre-tensioned

prestressed beam subject to loading conditions occurring in a continuous

beam at transfer of prestress is developed in this chapter. Knowledge

of the deflected shape is necessary for determination of the girder

moments and the stresses at transfer. The deflection of the beam at

transfer occurs under the combined influence of the cable moments and

the girder moments. However, the girder moments depend on the location

of the points of support of the beam which, in turn, are influenced by

the deflected shape of the beam. Thus, the deflected shape and the

girder moments are interdependent.

A simple method for determining the deflected shape and the loca­

tion of the support points Is not available because of a lack of known

boundary conditions. Neither a value of the slope nor a value of the

deflection is known at any point on the beam. The beam has no restric­

tions such as specified support locations imposed upon it.

Three approaches were considered in seeking a closed solution:

(1) application of moment-area principles, (2) treatment as a beam on

an elastic foundation, and (3) a numerical solution to the differential

equations for the beam. A lack of definite boundary conditions prevents

a direct application of any of these approaches. A solution using an

approximate value of the slope at one end of the beam and an iterative

procedure is possible by either of approaches (1) or (3),

8

The moment-area method is chosen for the ensuing analysis because

the moments are easily obtained at given intervals. The method for

determining the deflected shape and the girder moments at transfer is

based on two assumptions. First, the value of the slope at one end of

the beam can be reasonably approximated. Second, the beam at transfer

supports itself on only two points. After assuming the two support

locations, use of an iterative procedure produces a successive correc­

tion of the location of the support points until convergence is reached.

Execution of the method is composed of the following steps:

Step 1. Assumption of two support locations.

Step 2. Determination of the girder moments.

Step 3. Approximation of the slope at one end of the member.

Step 4. Evaluation of slopes and deflections at Intervals along

the beam.

Step 5. Location of support points.

Step 6. Check validity of assumed support point locations.

Assumption of Two Support Locations

The locations of the two support points must be assumed in order

to start the iterative procedure. Convergence occurred regardless of

the accuracy of the assumed locations in the problem tested. However,

a good assumption reduced the number of iterations required for

convergence.

Girder Moment Determination

The girder moment due to the dead weight of the precast beam can

be determined for an assumed support condition. Since, by hypothesis,

there are only two supports equations of statics may be used.

Approximation of the Slope at One End of the Member

A knowledge of the slope at one end of the beam is sufficient to

permit evaluation of the deflected shape at intervals along the beam.

The slope at one end of the beam can be approximated using moment-area

principles, and some simplifying assumptions.

Initially the left end of the beam is assumed to be fixed as shown

in Figure 5. Using this assumption, the tangential deviation. BC. of

the right end resulting from girder moments and cable moments can be

determined by moment-area principles. Now. if the ends of the beam A

and C do not have a large difference in deflection from the horizontal

base line, the tangential deviation. BC. can be assumed equal to BD,

Therefore, the slope of the beam at A is

..n A - BD ^ BC ^ " ^A • Length ' Length

Thus a boundary condition, slope at the left end. Is established from

the combined girder and cable moments of the beam.

The accuracy of the value of the slope depends on the difference

in deflection of the two ends from the base line. Generally, this

difference is small because the deflections of the beam are small. For

a symmetrical deflected shape, the deflections of the two ends are the

same, hence the slope obtained using this method is exact to within

the error inherent in the assumption that the length of the beam and

its horizontal projection may be considered to be equal.

10

Beam Length = L

Base Line

Tangential Deviation

Figure 5, Slope Approximation.

Evaluation of Slopes and Deflections of tlie Beam

The slopes and deflections of the beam are calculated using the

girder moments, the cable moments, and the slope at the left end

(boundary condition). The moment-area method of evaluating the slopes

and deflections is used.

Location of Support Points

The two support points may be located at the ends of the beam, at

two interior points, or at a combination of an end point and an interior

point. The interior support point is located at a point of zero slope

11

at which the slope change is from negative to positive. By locating

the points of zero slope and examining the deflected shape the corrected

location of the support points can be found.

Check Validity of Assumed Support Point Locations

The initial assumption of support locations is checked against the

calculated support point locations. If the two sets of values do not

agree within the desired accuracy the procedure is recycled through

Step 1 using the calculated support locations. This Iteration produces

stable values of the support locations after several cycles.

Programming the Method for the Computer

The iterative procedure of assuming a support location and then

calculating a corrected value makes a computer program desirable. In

order to program the method, the following considerations were made.

First, there are four possible stable support conditions as shown

in Figure 6. A set of span lengths SI, S2, and S3 determines a

specific support condition and a set of equations for the girder

moments. The computer can determine the girder moments for the given

condition at one foot Intervals.

Second, the slopes and deflections are also calculated at intervals

of one foot. A point of zero slope at which the slope changes from

negative to positive indicates the location of an interior support.

The point of zero slope may fall between two adjacent node points. In

this case it is necessary to locate the point within the interval. If

the slope at node point K is negative while that at node point K + 1

SI I sF

A. SI = 0, S2 ?< 0, S3 ?« 0

T

12

B. SI / 0, S2 ?< 0, S3 = 0

C. SI / 0, S2 ?< 0, S3 ?< 0

D. SI = 0, S2 ?« 0, S3 = 0

E. S2 = 0 unstable

Figure 6. Deflected Shapes.

is positive, there is a point of zero slope within the interval. To

locate a point of zero slope within a one foot interval, it is assumed

that the slope variation in a one foot interval is linear. The assump­

tion has been adequate in the problems tried. Within the interval, the

distance X' from node point K to the support is

13

Y' I (Slope at K) - (1 foot) ^ " '(Slope at K + 1) - (Slope at K)

Then the distance D to the interior support is

D = X,< + X'

where Xj, is shown in Figure 7b. The error introduced by the assumption

of linear variation of slope is negligible because deflections in the

beam are small. In the computer program the deflections are referenced

from the first point of zero slope for convenience. The computer deter­

mines the interior support locations and prints them as output.

Slope at K Slope at K+1

Line of Zero Slope

End Support Point

(a)

Zero Slope (Positive to Negative)^

(b)

Interior Support (Slope Negative

to Positive)

Figure 7. Typical Deflected Shape,

14

A program for evaluating the deflected shape at transfer is

written based on the considerations above. The flow chart for such a

program is given in Appendix D. The use of this program together with

a design program given in Chapter III Is illustrated in the example

problem in Appendix B. Chapter III explains the design method in

detail, showing the function of transfer moments, resulting from the

deflected shape, in the design.

CHAPTER III

DESIGN OF THE COMPOSITE BEAM

A method for the design of a pre-tensioned prestressed beam subject

to loading conditions occurring in a continuous beam Is described. The

design method presented here may be used to design a composite section

subject to both positive and negative moments. The method Is based on

a well-known design procedure for composite sections subject to positive

moment given in Design of Prestressed Concrete Structures by T. Y.

2 1

Lin. The "Building Code Requirements for Reinforced Concrete gives

the design specifications. The composite section consists of a pre-

tensioned precast beam with a cast-in-plaee concrete slab, as illus­

trated in Figure 8. The width of the cast-in-place slab to be used in

the composite section is taken as the flange width of the precast

portion plus an overhang of eight times the slab thickness on each side

of the precast portion.

z\

I c.g.c,

c.g.c.

c.g.s.

IZ IT

IZ b !

%'

(a) Precast Pre-Tensioned Part (b) Composite Section

Figure 8. Section Properties

15

16

Basic data for the design such as the magnitude of loadings, the

cast-in-place slab thickness, the distance between adjacent beams, and

the properties of the materials are known. An assumption of the cross-

sectional area of the pre-tensioned beam is made initially. The beam

is then analyzed for various dead and live load conditions and a moment

envelope is developed. Upon completion of the analysis, the positive

and negative moments are known at any point in the beam. A composite

section is designed for the maximum positive moment of the beam and

checked for adequacy for the maximum negative moment.

Method of Pes i gn

The method of design is divided Into three parts as follows:

1. Choice of cross section.

2. Determination of cable layout.

3. Determination of the deflected shape of the precast beam and

a set of transfer girder moments.

Section Choice

The procedure for determining a cross section is based on the

2 method presented by Lin for designing a composite section for the

positive moment. The equations are rearranged for the maximum negative

moment encountered in continuity of the beam.

The cross-sectional area and the prestressing force, F, are deter­

mined for the maximum positive moment. For the negative moment, the

prestressing force is held constant and only the cross-sectional area

is determined. In this determination, the composite section is assumed

17

to carry its negative moment capacity. If the area required for the

negative moment is larger than the area designed for the positive

moment, the design is revised by increasing the area.

Cable Layout

The cable layout is obtained from the known moment envelope, pre­

stressing force, and cross section, A cable zone, specifying the

region in which the cable may be placed is determined at discrete

points along the beam.

There are four conditions which determine the cable zone, two for

positive moment and two for negative moment. Since both positive and

negative moments are possible at the same point, all four conditions

must be checked at each increment along the beam. The four conditions

are as follows:

1. Positive moment at working load requires that the cable be low

enough to prevent tensile failure on the bottom of the precast

beam (upper limit),

2. In a positive moment region the cable must be high enough so

that it will not cause a tensile failure on the top of the

precast beam (lower limit).

3. Negative moment at working load requires that the cable be

high enough to prevent tensile failure on the top of the pre­

cast beam (lower limit).

4. In a negative moment region, the cable must be low enough so

that it will not cause a tensile failure on the bottom of the

precast beam (upper limit).

18 2

The method presented by Lin is used to obtain the cable zone. It

must be remembered that the va^ue of the girder moment. Mg. necessary

for evaluation of the cable zone is the moment at transfer of prestress.

In the first cycle of the design the girder moment at transfer is not

known since it is dependent upon the deflected shape of the beam.

Therefore. In the first cycle the girder moment of the pre-tensioned

beam in place is used. In the successive cycles the girder moments

obtained from the deflected shape at transfer are used. Close attention

to signs and due regard to the negative moment regions must be observed

in determination of the cable layout zone.

Deflected Shape - Transfer Moments

In order to complete the design the transfer girder moments of the

precast beam must be found. This is the problem considered in Chapter

II. A method is developed whereby the support points for the beam at

transfer can be obtained. This allows evaluation of the girder moments

at transfer which can be used to revise the design. Through a recycling

process a final design 1s obtained. The design process is repeated as

many times as necessary using the updated values of moments at transfer

to obtain the final design.

Computer Solution

A program is developed to complete one cycle of design. The pro­

gram is written with one foot increments of beam length chosen for

accuracy. A flow chart is presented in Appendix E while an illustrative

problem is solved in Appendix B.

19

An explanation of some of the procedures of programming the method

are listed below:

1. The analysis portion of the flow chart is for a beam with two

supports and one overhango This portion may be replaced by

an analysis for any beam configuration desired.

2. The analysis of the precast beam in place is written as a

subroutine so that it may be revised easily as the area of

concrete changes.

3. It being necessary to specify the shape of the cross section,

this is accomplished by using charts such as the one illus­

trated in Appendix A.

4. Since at least one dimension of the cross section must be

specified, the depth is chosen as the specified dimension.

5. The location of the cable within the cable zone must be

specified. The arbitrary choice is the upper limit of the

cable zone,

6. A check for any points of zero cable zone must be made. If

this condition occurs the cable is placed at the point of zero

cable zone to complete the problem. The output of the problem

makes note of these points.

7. For the first cycle of design, an estimate of the support

locations of the precast beam is desired to enable the designer

to make good estimates for use in the deflection program. This

is accomplished by using the portion of the deflection program

which evaluates the deflected shape at transfer at the end of

20

the design. A moment diagram consisting only of the moment due

to cable eccentricity is used. Explanation of the method used

in evaluating support locations 1s given in Chapter II.

At the conclusion of one run of the design program, the designer

ma y use the deflection program to find the deflected shape and girder

moments of the pre-tensioned beam at transfer. The design may then be

revised for these moments. The example problem in Appendix B illus­

trates the step-by-step procedure in the combined use of the design

program and the deflection program. The use of the two programs brought

forth their limitations. These limitations along with the conclusions

of the project are given in Chapter IV.

CHAPTER IV

CONCLUSIONS

Two methods are presented in this paper, one for evaluating trans­

fer deflections in a pre-tensioned beam subject to loading conditions

occurring in a continuous beam and the other for designing a pre-

tensioned overhang beam of composite section. The conclusions of the

study are:

1. It is possible to determine the deflected shape of a pre-

tensioned beam subject to loading conditions occurring in a

continuous beam at transfer by approximating the value of the

slope at one end of the beam.

2. The deflection computer program written for the method is an

independent program which can be applied to additional

problems.

3c A final design of a pre-tensioned overhang beam can be

obtained by using the design computer program (see Appendix

B). The computer time required for the example problem is

less than fifteen minutes on the IBM 360 computer.

The computer programs developed in this study are subject to the

following limitations:

1. The deflection program assumes that the deflection at transfer

(camber) will produce only two support points.

2. The structural analysis part of the design computer program is

valid for a simply supported beam with overhangs. This

21

22

limitation may be eliminated by incorporating a continuous

beam analysis.

3. If the maximum negative moment is much larger (about 25% or

greater) than the maximum positive moment, the cross section

obtained from the design program will be uneconomically large.

BIBLIOGRAPHY

1. ACI Committee 318. "Building Code Requirements for Reinforced Concrete," (ACI 318-63). American Concrete Institute Publication. June. 1963.

2. Lin, T. Y. Design of Prestressed Concrete Structures. Second Edition. Jonn IJTley and Sons, Inc.. New York. 1963.

3. Morice. P. B. "Analysis of Prestressed Concrete Structures and Application of Recent Research." Institution of Civil Engineers. Proceedings, v. 6, March, 1957.

4. Saint John, A. D. "Computer Design of Prestressed Concrete," PCI Journal. v. 8, No. 4. August. 1963. pp. 31-34.

5. Stubbs, I. "Prestressed Concrete Design Using Electronic Digital Computers." PCI Journal, v. 7, No. 2, April, 1962. pp. 69-81

23

APPENDICES

A. Constants for Symmetrical I-Sections 25

B. Example Problem 26

C. Flow Chart Symbols 36

D. Deflected Shape Computer Program Flow Chart 43

E. Design Computer Program Flow Chart 67

24

APPENDIX A

TABLE 1*

CONSTANTS FOR SYMMETRICAL I-SECTIONS

t ?

±

U± ^ ^ Sec- b' t tion F F

K. K.

a 0.1 0.1 0.28bh O.SOOh O.SOOh 0.0449bh'^ 0.160h^ 0.320h 0.320h

b 0.1 0.2 0.46 0.500 0.500 0.0671

c 0.1 0.3 0.64 0.500 0.500 0.0785

d 0.2 0.1 0.36 0.500 0.500 0.0492

e 0.2 0.2 0.52 0.500 0.500 0.0689

f 0.2 0.3 0.68 0.500 0.500 0.0791

g 0.3 0.1 0.44

h 0.3 0.2 0.58

0.500 0.500 0.0535

0.500 0.500 0.0707

i 0.3 0.3 0.72 0.500 0.500 0.0796

j 0.4 0.1 0.52

k 0.4 0.2 0.64

0.500 0.500 0.0577

0.500 0.500 0.0725

1 0.4 0.3 0.76 0.500 0.500 0.0801

0.146 0.292 0.292

0.123 0.246 0.246

0.137 0.274 0,274

0.132

0.117

0.121

0.122

0.111

0.111

0.113

0.105

0.264

0.234

0.243

0.244

0.222

0.222

0.226

0.211

0.264

0.234

0.243

0.244

0,222

0.222

0.226

0.211

*Taken from Reference 3 of Appendix C

25

APPENDIX B

EXAMPLE PROBLEM

1. The example problem is the design of a prestressed composite

beam having a single overhang. The fixed geometry is shown

in Figure 1.

2. The cable moment sign convention adopted considers eccentricity

below the center of gravity of concrete positive.

3. Output is shown in the form obtained from the computer.

417'

50' T 20' • * - H — >

I. T

Slab

V X

20' - ^ > •

Figure 1. Example Problem

Step J_ - Design Program

Input:

Span 1 = 0 feet

Span 2 = 50 feet

Span 3 = 20 feet

Beam Spacing = 20 feet

Slab Thickness = .417 feet

Beam Depth =3.5 feet

26

27

Output:

Initial Concrete Area = 2.2 feet

Weight of Concrete = 150 pounds per foot

^ci " ^c' " ^°°° P ' Live Load = 80 psf

Dead Load = 10 psf

Allowable Steel Stress = 250 ksi

Constants for Section h from Table 1^, Appendix A

FLANGE WIDTKt-0.9522

AREAt 1.9330

MAX +M*t 885.95

WEB THICKNESSt 0.2857

COMP WIDTKt-7.6242

MAX -M*+ -676.19

FLANGE DEPTHt 0.7000

COMP AREAt 5.1123

DEPTHt 3.5000

AREA OF STEELt 1.7575

PRESTRESSING FORCEt INITIAL FORCEt PRECAST INERTIAt COMPOSITE INERTIAt 263.6228 307,5598 2.8864 7.5448

eet

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

LOWER BOUND

0,7032 0.6817 0.6612 0.6419 0.6237 0.6066 0.5906 0.5758 0.5620 0.5493 0.5377 0.5272 0.5178 0.5096 0.5024 0.4963

MOD ELASTICITYt 550000,0

UPPER BOUND CABLE LOCATION

2.7388 2.5408 2,3522 2.1729 2,0029 1.8423 1.6910 1.5491 1.4165 1.2933 1.1794 1.0749 0.9797 0.8939 0.8174 0,7502

2.7388 2,5408 2,3522 2.1729 2.0029 1.8423 1,6910 1.5491 1,4165 1.2933 1,1794 1.0749 0.9797 0.8939 0.8174 0.7502

CABLE MOMENT

-304.125 -243.229 -185,208 -130,061 -77.788 -28,390 18,134 61,783 102,558 140.459 175,485 207.636 236.914 263,316 286.844 307.499

28

EET

17 18 19 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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

LOWER BOUND

0.4914 0.4875 0.4847 0.4831 0.4825 0.4831 0.4847 0.4875 0.4914 0.4963 0.5024 0.5096 0.5178 0.5272 0.5377 0.5493 0.5620 0.5758 0.5906 0.6066 0.6237 0.6419 0.6612 0.6817 0.7032 0.7258 0.7495 0.7743 0.8002 0.8377 1.0342 1.2375 1.4476 1.6644 1.4144 1.1771 0.9527 0.8670 0.8499 0.8339 0.8190 0.8052 0.7925 0.7809 0.7705 0.7611 0.7528 0.7456

UPPER BOUND

0.6924 0.6439 0.6048 0.5750 0.5546 0.5435 0.5418 0.5494 0.5664 0.5927 0.6283 0.6733 0.7277 0.7914 0,8644 0,9468 1.0385 1.1396 1,2500 1.3698 1.4989 1.6373 1,7851 1.9423 2,1087 2.2846 2.4698 2.6643 2.8487 2.8757 2.9038 2.9331 2.9634 2,9948 2,9733 2.9529 2.9336 2.9154 2,8983 2.8823 2,8674 2.8536 2.8410 2.8294 2.8189 2.8095 2.8012 2.7941

CABLE LOCATION

0.6924 0.6439

^ 0,6048 0.5750 0.5546 0.5435 0.5418 0.5494 0.5664 0.5927 0.6283 0.6733 0.7277 0.7914 0.8644 0.9468 1.0385 1.1396 1,2500 1,3698 1.4989 1.6373 1.7851 1,9423 2.1087 2.2846 2.4698 2.6643 2.8487 2.8757 2.9038 2.9331 2.9634 2.9948 2,9733 2,9529 2,9336 2.9154 2.8983 2,8823 2.8674 2,8536 2,8410 2,8294 2.8189 2.8095 2.8012 2.7941

CABLE MOMENT

325.279 340.184 352,214 361.370 367.653 371.060 371.593 369.251 364.035 355,945 344,981 331,142 314.429 294,840 272.379 247,043 218,830 187,745 153,783 116.949 77.241 34.659 -10.801 -59.130 -110.337 -164.422 -221.375 -281.204 -337.910 -346.222 -354.874 -363.865 -373.195 -382.865 -376.250 -369.973 -364,035 -358.437 -353,178 -348.259 -343,678 -339,437 -335,535 -331,973 -328.750 -325.865 -323,321 -321,115

EET LOWER BOUND

65 0.7396 66 0.7346 67 0.7307 68 0.7280 69 0.7263 70 0.7258

SLOPE CHANGES SIGN TO -0.00017 0.00009

UPPER BOI

2.7880 2.7830 2.7792 2.7764 2.7748 2.7742

AT 22 55

MAXIMUM VALUES FOR ENTIRE BEAM

29

E LOCATION

2.7880 2.7830 2.7792 2.7764 2.7748 2.7742

CABLE MOMENT

-319.249 -317.723 -316,535 -315.687 -315.178 -315.008

FOR USE IN ESTIMATING SUPPORT LOCATIONS. CHANGE TO POSITIVE SLOPE INDICATES POTENTIAL SUPPORT.

tALL UNITS IN FEET AND KIPS EXCEPT AREA OF STEEL IN INCHES

30

Step 2 - Deflection Program

Input:

Span 1 • 0 feet

Span 2 • 55 feet

Span 3 « 15 feet

Cable Moments. Weight of Concrete. Area of Concrete,

Modulus of Elasticity, Precast Moment of Inertia,

from Output of Step 1

Output:

FEET

0 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 30

SLOPE

0.00042 0.00059 0.00074 0.00085 0.00094 0.00100 0.00102 0.00103 0.00101 0.00097 0.00091 0.00083 0.00073 0.00062 0.00050 0.00036 0.00021 0.00006 -0.00011 -0.00028 -0.00045 -0.00062 -0.00080 -0.00097 -0.00114 -0.00131 -0.00147 -0.00163 -0.00177 -0.00191

DEFLECTION

-0.01292 -0.01256 -0.01205 -0.01139 -0.01059 -0.00969 -0.00873 -0.00772 -0.00669 -0.00567 -0.00468 -0.00374 -0.00287 -0.00209 -0.00142 -0.00086 -0.00043 -0.00014 -0.00001 -0.00004 -0.00023 -0.00059 -0.00113 -0.00184 -0.00272 -0.00378 -0.00501 -0.00640 -0.00795 -0.00966 -0.01150

step 2...Output...Continued

31

FEET SLOPE DEFLECTION

SLOPE

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

CHANGES SIGN TO -0.00011 0.00015 SUPPORT

51.33356

-0.00203 -0.00214 -0.00223 -0.00230 -0.00236 -0.00239 -0.00241 -0.00240 -0.00236 -0.00230 -0.00221 -0.00208 -0.00193 -0.00174 -0.00152 -0.00128 -0.00105 -0.00080 -0.00056 -0.00032 -0.00008 0.00015 0.00038 0.00059 0,00080 0,00100 0.00120 0.00140 0.00160 0.00180 0,00200 0.00220 0.00240 0.00260 0.00280 0.00300 0.00319 0.00339 0,00359 0.00379

AT 19 52

-0.01346 -0.01555 -0.01773 -0.02000 -0.02233 -0.02470 -0.02711 -0.02951 -0.03189 -0.03422 -0.03647 -0.03862 -0.04063 -0.04246 -0.04410 -0.04550 -0.04667 -0.04759 -0.04828 -0.04872 -0,04892 -0.04888 -0.04861 -0.04813 -0,04743 -0,04653 -0,04543 -0.04413 -0.04263 -0,04093 -0.03903 -0,03693 -0.03463 -0,03213 -0,02944 -0.02654 -0,02345 -0.02015 -0.01666 -0,01297

DEFLECTIONS MEASURED FROM FIRST POINT OF ZERO SLOPE

32

Step 2 - Deflection Program

Input:

Same as Step 2 except

Span 1 = 0 feet

Span 2 = 52 feet

Output:

Span 3 = 18 feet

EET

0 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 30 31 32 33

GIRDER MOMENT

6.480 12.671 18.573 24.184 29.507 34.540 39.283 43.737 47.901 51.776 55,361 58,657 61.663 64.380 66.808 68.945 70.794 72.353 73.622 74.602 75.292 75,693 75.804 75.626 75.159 74.401 73.355 72.019 70.393 68.478 66.273 63.779 60.995

SLOPE

0.00074 0.00092 0.00107 0.00118 0.00126 0.00132 0.99134 0.00134 0.00132 0.00127 0.00121 0.00112 0.00102 0.00090 0.00077 0.00063 0.00047 0.00031 0.00014 -0.00004 -0.00022 -0.00041 -0.00059 -0.00078 -0.00096 -0.00114 -0,00132 -0.00148 -0.00164 -0,00179 -0,00192 -0,00205 -0,00215

DEFLECTION

-0.01865 -0,01795 -0,01712 -0,01613 -0.01500 -0.01378 -0,01250 -0.01117 -0.00983 -0.00850 -0.00720 -0.00596 -0.00479 -0.00372 -0,00276 -0.00192 -0.00122 -0.00067 -0.00028 -0.00005 -0.00000 -0.00014 -0.00045 -0.00095 -0.00164 -0,00251 -0,00356 -0.00479 -0.00619 -0.00776 -0.00947 -0.01133 -0.01331 -0,01542

Step 3...Continued...Output

33

FEET

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

SLOPE CHANGES -

GIRDER MOMENT

57.922 54.559 50.907 46.966 42.735 38.214 33.404 28.304 22.915 17.236 11.268 5.010

-1.536 -8.373 -15.499 -22.915 -30.620 -38.615 -46.899 -41.833 -37.056 -32.569 -28.371 -24.463 -20.844 -17,515 -14.475 -11.725 -9.264 -7.094 -5.211 -3.618 -2.316 -1.303 -0.579 -0.144 -0.000

SIGN TO 0.00004 0.00005

SLOPE

-0,00225 -0.00232 -0.00237 -0.00240 -0.00241 -0.00239 -0.00234 -0.00227 -0.00217 -0.00203 -0.00187 -0.00167 -0.00145 -0.00123 -0,00102 -0.00080 -0.00057 -0.00036 -0.00015 0.00005 0.00026 0.00046 0.00066 0.00086 0.00106 0.00126 0,00146 0.00166 0.00186 0.00206 0.00226 0.00246 0.00265 0.00285 0.00305 0.00325 0.00345

AT 20 53

SUPPORT Span 1 = Span 2 = Span 3 =

DEFLECTION

-0.01762 -0.01990 -0.02224 -0.02463 -0.02703 -0.02943 -0.03180 -0.03410 -0.03632 -0.03843 -0.04038 -0.04215 -0.04371 -0.04505 -0.04618 -0.04708 -0.04777 -0.04823 -0.04849 -0.04854 -0.04838 -0.04803 -0,04747 -0.04671 -0.04575 -0.04459 -0.04323 -0.04167 -0.03991 -0.03795 -0.03579 -0.03344 -0.03088 -0.02813 -0.02518 -0.02203 -0.01868

STABILIZE 1 -- 0 feet = 52 feet = 18 feet

AT

34

Step £ - Design Program

Input:

Same as Step 1 except girder moments from Step 3 are read

in before evaluating the cable zone.

Output:

Same as Step 1 except for cable zone and cable layout shown

below.

EET

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 30 31 32 33 34

LOWER BOUND

0.6984 0.6721 0.6469 0,6228 0.5997 0.5778 0.5570 0.5373 0.5187 0.5012 0.4847 0.4694 0.4552 0.4421 0.4300 0.4191 0.4903 0.4005 0.3929 0.3850 0.3809 0.3766 0.3733 0.3712 0.3701 0.3702 0.3713 0.3736 0.3769 0.3814 0.3869 0.3935 0.4013 0.4101

UPPER BOUND

2.7344 2.5319 2.3388 2.1551 1.9806 1.8156 1.6598 1.5134 1.3763 1.2486 1.1302 1,0212 0,9215 0.8312 0.7501 0.6785 0.6161 0.5632 0.5195 0.4839 0.4602 0.4446 0.4383 0.4414 0.4538 0,4755 0.5066 0.5470 0.5968 0.6559 0.7243 0,8021 0.8893 0.9857

CABLE LOCATION

2.7344 2.5319 2.3388 2.1551 1.9806 1.8156 1.6598 1.5134 1.3763 1,2486 1,1302 1,0212 0.9215 0.8312 0.7501 0,6785 0.6161 0.5632 0.5195 0.4839 0.4602 0.4446 0.4383 0.4414 0.4538 0.4755 0.5066 0.5470 0.5968 0.6559 0.7243 0.8021 0.8893 0.9857

CABLE MOMENT

-302.758 -240,492 -181.100 -124.580 -70.933 -20,161 27,740 72.768 114.922 154.203 190.612 224.147 254,809 282.598 307.514 329.558 348.729 365.026 378.451 389.393 396,681 401,486 403,418 402.478 398.664 391.978 382.420 369.986 354,682 336,502 315,452 291.528 264.729 235.060

35

EET

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

LOWER BOUND

0.4200 0.4311 0.4432 0.4564 0.4708 0.4862 0.5027 0.5203 0.5390 0.5589 0.6479 0.8377 1.0342 1.2375 1.4476 1.6644 1.4144 1.1771 0.9527 0.8174 0.8493 0.8334 0.8186 0.8048 0.7922 0.7807 0.7703 0.7609 0.7527 0.7455 0.7395 0.7346 0.7307 0.7280 0.7263 0.7258

UPPER BOUND

1.0915 1.2067 1.3312 1.4650 1.6082 1.7607 1.9226 2.0938 2.2743 2.4642 2.6282 2.6502 2.6733 2.6975 2.7229 2.7493 2.7768 2.8054 2,8351 2.8659 2.8978 2.8818 2.8670 2.8533 2.8407 2.8291 2.8187 2.8094 2.8011 2.7940 2.7879 2.7830 2.7792 2.7764 2.7748 2.7742

CABLE LOCATION

1.0915 1.2067 1.3312 1.4650 1.6082 1.7607 1.9226 2.0938 2.2743 2.4642 2.6282 2.6502 2.6733 2.6975 2.7229 2.7493 2,7768 2.8054 2.8351 2.8659 2.8978 2.8818 2.8670 2,8533 2.8407 2.8291 2.8187 2.8094 2.8011 2.7940 2.7879 2.7830 2.7792 2.7764 2.7748 2.7742

CABLE MOM

202.514 167.098 128.809 87.648 43.610 -3,296

-53.076 -105.735 -161.260 -219.659 -270.103 -276.873 -283.982 -291.427 -299.210 -307.332 -315.791 -324.588 -333.723 -343.195 -353.005 -348.108 -343.549 -339,326 -335,443 -331.896 -328.688 -325.816 -323.284 -321,088 -319,230 -317.710 -316,530 -315.684 -315.177 -315,008

BEAM DESIGN FROM FIRST RUN CHECKS. NO MAJOR CHANGE IN CABLE ZONE OR LOCATION INDICATES DESIGN ADEQUATE.

APPENDIX C

FLOW CHART SYMBOLS

Block Diagram Symbols

Processing Operation Input

Output Change of Control

^

Offpage Connector

A=(l,1,100)

If (A-B). go to sign of (A-B)

Iteration, Initial A=l. Increment =1. Terminate at 100

All statements used in flow charts use standard Fortran symbols such as ** for exponentiation.

36

37

Program Variable Names

AC - A^, Area of concrete; or area of precast portion,

ACF - Area of concrete required to prevent a compressive failure at the

bottom of the beam.

ACFF - Larger of ACF and ACPF.

ACPF - Area of concrete required to prevent a compressive failure at the

top of the beam.

AC2 - A ', Area of composite portion^

AMC - M , Moment acting on the composite section.

AMCNG - Maximum allowable value of negative moment on composite secion,

AMG - Maximum positive value of Mg.

AMP - M , Moment on precast portion of composite section,

AMPNG - Total negative moment applied to a section less AMCNG.

AMTNM - Maximum negative value of BN(K),

AMTPM - Maximum positive value of BP(K),

AST - Area of steel,

AVGM - Average value of moment between a point of zero slope and the

nearest increment.

B - Width of beam or its flange.

BA - A , Area of concrete; or area of precast portion.

BASE - Slope of a line connecting beam ends.

BH3 - I, Moment of inertia

BH32 - r. Moment of inertia of composite section.

BH32A, BH32B, BH32C. BH32D - Components of BH32,

BK - k. . Kern distance for precast section.

38

BK2, TK2 - Kern distances for composite section.

BN(K) - Maximum negative bending moment at increment K.

BP(K) - Maximum positive bending moment at increment K.

BPP - Width of web of beam.

CAC, CBPB, CBH3, CCB, CCT, CKB, CKT, CR2. CTH - Constants as defined in

Appendix B.

CB - c. , Distance from c.g.c. to bottom fiber.

CB2 - c. ', Distance from c.g.c. to bottom fiber of composite section.

CB2P - Component of CB2.

CGSY - Distance from the extreme fiber of the precast beam to the c.g.s.

CM - Defined locally as moment due to eccentricity or as total moment

acting on precast beam at transfer.

CT - c^, Distance from c.g.c, to top fiber.

CT2 - c^', Distance from c.g.c, to top fiber of composite section.

DCM - Moment due to eccentricity.

DEFAPP. DEFIP - Deflections of the precast beam,

DEF(K) - Deflection of the precast beam at K.

DLOAD - Dead load of the precast beam.

DT - Fixed point number defined locally.

E - Defined locally as total allowable eccentricity or modulus of

elasticity of concrete.

EA, EB, EC, ED - Allowable eccentricities under various loading

conditions.

El - Eccentricity allowable due to allowable tensile stress.

E2 - Eccentricity allowable due to girder moment.

39

FB - fj , Fiber stress at bottom fibers.

FBC - Allowable compressive stress in concrete at transfer.

FCI - Ultimate unit stress in concrete, at transfer.

FCP - f^'. Compressive strength of concrete.

FO - F Q . Total prestressing force at transfer.

FRSTM. FRSTMP - First moment of cable moment diagram.

FSP - Ultimate unit stress in steel.

FTC - f^. Fiber stress at bottom fibers.

FTP - Allowable tensile stress in concrete at transfer.

GIRDM(BA) - Subroutine for finding grider moments.

GM. GMX - G| . Girder load moment.

H - h. Depth of precast beam.

I. IS2P - Control parameters for program execution.

ISl, IS2, IS3 - S(l), S(2), S(3) respectively.

K, KKK. KL - Control parameters for program execution.

L, LAB, LAC, LAE, LAX, LTH, LTHTB, LTHTC - Control parameters for

program execution.

LTHT - Total length of precast beam.

NUMB - Control parameter for program execution. 2

R2 - r , radius of gyration squared.

R22 - Radius of gyration squared, composite section.

RATIO - Ratio of ACFF to AC.

RDEFl, RDEF2, RDEF3, RDEF4 - Distance beam deflects below line

connecting assumed supports.

RL - Reaction at left support under maximum positive moment loading.

40

RLN - Reaction at left support under maximum negative moment loading.

RR - Reaction at right support under maximum positive moment loading.

RRN - Reaction at right support under maximum negative moment loading.

S(l), S(2), S(3) - SI, S2, S3 respectively.

SAT - Ultimate tensile fiber stress under working load.

SINT - Sine of angle between horizontal and a line connecting left end

of the beam with tangential deviation of right end.

SLO(K) - Slope of precast beam at K.

SLOPE - Slope of a line connecting the two support points.

SLOPEI - Initial slope of left end of precast beam,

SLP(K) - Slope of precast beam at K.

ST - Thickness of floor slab.

T - Thickness of beam flange.

TANDEV - Tangential deviation of right end of the beam with respect to

left.

THETA - Arc sine of SINT.

TK - k^. Kern distance from c.g.c. for top.

TMB - m.. Ratio of section moduli for top fibers.

TMT - m, . Ratio of section moduli for bottom fibers.

TS - Thickness of floor slab.

WC - Weight of concrete.

WD - Total dead load on precast beam.

WDPSF - Weight of dead load on slab, psf.

WIDTH - Defined locally as spacing between adjacent beams or as width

of slab in composite beam.

41

WL - Total live load on precast beam.

WLPSF - Weight of live load on slab. psf.

X - Denotes a location on the beam.

XACF - Area of concrete required to prevent a compressive failure at

top of beam, negative moment condition.

XACPF - Area of concrete required to prevent a compressive failure at

bottom of beam, negative moment condition.

XAL, XI. XIS2P. XLAC. XX, XXAL. XXX - Locally defined, used in program

execution.

XAMG - Maximum negative value of Mg.

XBK - k., Top Kern distance from c.g.c.

XBK2 - k^'. Top Kern distance from c.g.c' for composite section.

XCB - c.. Distance from c.g.c. to top fiber.

XCB2 - c.'. Distance from c.g.c' to top fiber.

XCT - c. . Distance from c.g.c to bottom fiber.

XCT2 - Cu', Distance from c.g.c' to bottom fiber.

XLTHT - Total length of beam.

XSUP - Distance locating a support point.

XSUPA - Distance locating a left interior support point.

XSUPB - Distance locating a right interior support point.

XTK - k. . Bottom Kern distance from c.g.c. b

XTK2 - ku'. Bottom Kern distance from c.g.c' for composite section, D

XTMB - m.. Ratio of section moduli for top fiber,

XTMT - m. , Ratio of section moduli for bottom fiber. D

XXACF - Area of concrete required to prevent tensile failure at design

load due to negative moment.

42

XXO - Distance from left end of the beam.

XZERO - Distance from the left end of the beam to point of zero

deflection (first point of zero slope).

Yl - Deflection at the left end of beam.

Y2 - Deflection at the right end of beam.

YCGSYA - Lower limit of allowable cable zone.

YCGSYB - Upper limit of allowable cable zone.

YC6SYF - Final location of cg.s.

YCGSYl. YCGSY2, YC6SY3. YCGSY4 - Location of c.g.s. under various

loading conditions.

YXSUP - Deflection of a support point,

YXSUPA - Deflection of left interior support point.

YXSUPB - Deflection of right interior support point.

YXSUPE - Deflection of left end of a beam with both supports interior.

ZACFF - Larger XACF and XACPF.

ZDEF - Zero deflection.

APPENDIX D

DEFLECTED SHAPE COMPUTER PROGRAM FLOW CHART

The input data to the computer program includes the beam section

properties, the material properties, and the cable moments. The cable

moments are determined by the cable layout obtained in design. The

flow chart for this program follows. The block diagram symbols of the

flow chart are explained in Appendix C, Appendix B includes an example

problem solved in part by this program.

43

44

EVALUATION OF MOMENT DIAG

[ AC,S1,S2,S3,WC,E, BH3

LTHT- S1+S2+S3

FCK4000.

BA AC

DL0AD^BA*WC/1000

RL-(DL0AD*(Sl+S2)*((Sl+S2)/2.)-(DL0AD*S3**2.)/S2,)/S2

RR^DL0AD*(S1+S2+S3)-RL

K l

O e X K

GM(K)^-(DL0AD*X**2,)/2,

K^K+1

G X K

GM(K)^RL*(X-S1)-(DL0AD*X**2.)/2,

K^K+1

X

0

6 M-

45

e 6M(K)^RL*(X-S1)+RR*(X-Sl-S2)-(DL0AD*X**2.)/2,

K^K+1

0

TITLE: FEET, CABLE MOMENT, GIRDER MOMENT

(^DCM(K),K^1,LTHT

- ^

K^(1,1,LTHT) )

CM(K)^DCM(K)-GM(K)

K,DCM(K),GM(K),CM(K)

^

46

EVALUATION OF SLOPES AND DEFLECTIONS

A2 E^(33.*1830./1.8)*SQRT(FCI)*(144./1000.)

K l

FRSTM-(CM(l ) /2 . )* ( (S( l )+S(2)+S(3)) - ( .5) )

K- 1

LAD^LTHT-1

- <

I 1^(1.1.LAD)

ZZZ^K

FRSTMP^(CM(K)+CM(K+1))/2.)*((S(1)+S(2)+S(3))-(ZZZ+. 5))

FRSTM FRSTM+FRSTMP

K^K+1

LTHT^(S(1)+S(2)+S(3))

TANDEV^FRSTM/(E*BH3)

SINT-^TANDEV/(S(1)+S(2)+S(3))

THETA^ARSIN(SINT)

SLOPEI^TAN(THETA)

KKK^l

K^l

I TITLE: SLOPE CHANGES TO. AT

47

A3 DLS(1)^(CM(1))/(2.*E*BH3)

SLP(1)^SL0PEI-DLS(1)

241 I K^Ktl

48

DLS(K)-(CM(K)+CM(K-1))/(2.*E*BH3)

SLP(K)^SLP(K-1)-DLS(K)

0 SLO(KK)^SLP(K)

XXO(KKK) K

KKK KKK+1

'SLP(K).K

K^K+1

DLS(K)^(CM(K)+CM(K-1))/(2.*E*BH3)

^

49

A5 $LP(K)^SLP(K-1)-SLS(K)

SLO(KKK)^SLP(K)

XXO(KKK)^K

KKK^KKK+1

SLP(K).K

25

251 TITLE: SLOPE. K

^ K ^ ( I . I . L T H " T ^

^ ^

50

A 6 | SLP(K),K

TITLE: DE FLECTION,K |

LTH^XXO(l)

ZDEF-HO.

XZERO^XXO(1)-ABS((SLP(LTH))/(SLP(LTH)-SLP(LTH-1)))

DEF1P^(CM(1) / (2 .*E*BH3))* .5

DEFAPP^DEFIP

DT^.5

L-1

LAX^LTH-1

253

\

/LAB^(I,I,LAX)W-

DT-^DT+l.O

DEFIPP^((CM(L)+CM(L+1))/(2.*E*BH3))*DT

DEFAPP^DEFAPP+DEFIPP

L^L+1

^

253 W

XAL^LAX

i

51

A7 AVGM^CM(LAX)+(CM(LTH)-CM(LAX))*((XZER0-XAL)/2.)

XXAL^LTH-LAB

DEFAPP^EFAPP+(XZER0-XAL)*AVGM*(XXAL+((XZERO-XAL)/2.)) /(E*BH3)

DEF(LAB)^(-DEFAPP)

LAE-LAB-1

I I DEF(LAB),L"AE

[ L LAB

DEFAPP^.

DT^-.5

DEF(LTH)^(AVGM*(XZERO-XAL)*( (XZERO-XAL) /2 . ) / (E*BH3) )

I DEF(LTH) .LAX

I ZDEF,XZERO 1

1 AVGM-CM(LTH)-(CM(LTH)-CM(LAX))*((XX0(l)-XZER0)/2.)

D E F ( L T H + 1 ) ^ ( ( X X 0 ( 1 ) - X Z E R 0 ) * A V G M * ( ( X X 0 ( 1 ) - X Z E R 0 ) / 2 . ) / ( E * B H 3 ) )

LTHH-HLTH+1

52

A8| LTHB-^LTHT-1 I

( LAC^(LTH.1,LTHTB)

258

h--DEFAPP O

L LAC

DT-0 .5

DT^DT+1.0

DEFIPP-((CM(L)+CM(L+1))/(2.*E*BH3))*(DT)

DEFAPP-HDEFAPP+DEFIPP

L^L-1

XLAC LAC

^

AVGM*<:M(LTH)-(CM(LTH)-CM(LAX))*((XXO(I)-XZERO)/2.)

DEFAPP^DEFAPP+(XX0(l)-XZER0)*AVGM*(((XLAC+l.)-XX0(l))+((XX0(l)-XZER0/2))

/(E*BH3)

I DEF(LAC+2.)^-DEFAPP

53

A9 LAfrHLAC+l

LAGfrHLAC+2

i DEF(LAGG).LAG

^ CHECK ASSUMED SUPPORTS

YKDEF(l)

Y2^DEF(LTHT+1)

XLTHT^LTHT

BASE^(Y2-Y1)/XLTHT

XXX^O.

0 - > ^ 1^(1.1,LTHT ) ^

54

Aiofx^

CONTINUE

I RDEF1-DEF(I+1)-(DEF(1)+(BASE*X))

XXX=1.

0

55

56

20 S(3)^XLTHT-S(2)

XSUP^S(2)

IS2^S(2)

YXSUP^DEF(IS2+1)

22 XSUP-S(2)-ABS((SLP(IS2)/(SLP(IS2)-SLP(IS2-1))))

YXSUP-DEF(IS2)+SLP(IS2-1)*.5*(XSUP-(S(2)-1.))

24 XSUP-S(2)+ABS((SLP(IS2)/(SLP(IS2)-SLP(IS2+1))))

YXSUP^DEF(IS2+1)+SLP(IS2)*.5*(XSUP-S(2))

23 SL0PE^(YXSUP-DEF(1))/XSUP

TITLE: SUPPORT, DEFLECTION, SLOPE

L XSUP.YXSUP,SLOPE.

57

A11(I^ (2.1,LTHTC h

28 CONTINUE

27

RDEF2^DEF(I)-DEF(1)+SL0PE*XI)

XXX=1.

CONTINUE

15 W

XLTHT^LTHT -<-

-0

58

31 S(1)-(S1)-1.

ISUS(l)

8

0

0

0

0 0

0 0

59

30 S(2)-XLTHT-S(1)

XSUP^S(l)

I S l - S ( l )

YXSUP^DEF(ISl-H)

33 XSUP^S(1)-ABS((SLP(IS1)/SLP(IS1)-SLP(IS1-1))))

YXSUP-DEF(IS1)+SLP(IS1-1)*.5*(XSUP-(S(1)-1.))

34 XSUP^S(1)+ABS((SLP(IS1)/(SLP(IS1)-SLP(IS1+1))))

YXSUP-DEF(IS1+1)+SLP(IS1)*.5*(XSUP-S(1))

36 SLOPE-(DEF(LTHTn)-YXSUP)/(XLTHT-XSUP)

60

A14 TITLE: SUPPORT, DEFLECTION. SLOPE

I XSUP. YXSUP.SLOPE

I XXO^^O.

LTHTC^LTHT+1

? - > - ^ M 1.1. LTHT )^

3 ^

^ ^ ^ CONTINUE h-^

RDEF3^DEF(I)-(DEF(LTHTC)-SL0PE*(XLTHT-XI))

XXX^l.

38 I — CONTINUE

0 -0

61

0

62

0* 0*

0 0

40 S(2)^XLTHT-S(1)-S(3)

XSUPA^S(l)

ISUS(l)

YXSUPA^DEF(IS1+1)

43| XSUPA-S(1)-ABS((SLP(IS1)/(SLP(IS1)-SLP(IS1-1))))

YXSUPA-DEF(IS1)+SLP(IS1-1)*,5*(XSUPA-(S(1)-1,))

44 XSUPA^S(1)+ABS((SLP(IS1)/(SLP(IS1)-SLP(IS1+1))))

YXSUPA-DEF(IS1+1)+SLP(IS1)*,5*(XSUPA-S(1))

[ 54 XSUPA.YXSUPA

63

0 0

0

0

0

0 0

64

47 XSUPB^IS2P

YXSUPB^DEF(IS2P+1)

50 XIS2P^IS2P

XSUPB^XIS2P-ABS((SLP(IS2P)/(SLP(IS2P)-SLP(IS2P-1))))

YXSUPB^DEF(IS2P)+SLP(IS2P-1)*.5*(XSUPB-(XIS2P-1,))

51 XIS2P^IS2P

XSUPB^XIS2P+ABS((SLP(IS2P)/(SLP(IS2P)-SLP(IS2P+1))))

YXSUPB^DEF(IS2P+1)+SLP(IS2P)*,5*(XSUPB-XIS2P)

I 53 XSUPB.YXSUPB

0 0 SLOPE^(YXSUPB-YXSUPA)/(XSUPB-XSUPA)

YXSUPE^YXSUPA-SLOPE*XSUPA

XXX^O,

t ^

65

A17 LTHTC^LTHT+1

(TITLE) ASSUMED BASIC SHAPE C O R R E c O ^

66

100 (TITLE) BEAM DEFLECTS BELOW SUPPORTS

99 (TITLE) ZERO SLOPE SUPPORT OUTSIDE RANGE

© 60 END

APPENDIX E

DESIGN COMPUTER PROGRAM FLOW CHART

67

68

W 1 0 0 3

AMG^O.

COMMON WC,S,K,GMX,AMG.XAMG.GM TO SUBROUTINE GIRDM(BA)

fs(l).S(2),S(3),WIDTH,WC.BA.WLPSF.WDPSF.ST

LTHT^S(1)+S(2)+S(3)

WL^WLPSF*WIDTH

WD-(WDPSF*WIDTH)+(BA*WC)+(ST*WIDTH*WC)

XLTHT^LTHT

RR-<(WL+WD)*(S(2)**2.)/2.-»-(WD*S(3))*(S(2)+S(3)/2.))/S(2)

RRN-((WD*S(2)**2,)/2.+(lJL+WD)*S(3)*(S(2)+S(3)/2,))/S(2)

RL-WD*(S(2)+S(3))+WL*S(2)-RR

RLN-WD*(S(2)+S(3))+WL*S(3)-RRN

K^l

TITLE: FEET,MAX+M,MAX-M.

69

•HI BP(K) o!

0 BN(K)^0.|

I4J K,BP(K),BN(K)

K^K+1

0 40 X K

14

BP(K)^(-WD*((XLTHT-X)**2.)/2.)/1000.

BN(K)^(-WL+WD)*((XLTHT-X)**2.)/2.)/1000.

f

70

0

BP(K)^0.

20 K,BP(K),BN(Kn

TITLE: MAX+M,MAX-M"1-<^

[I AMTPM^BP(l)

ANTNM^BN(l)

I

^

- ^ K ^ ( 2.1.LTHT ) ^

71

Qosy^

[P(K) ^-(zoe)

205 :^PM-BP(7]>— —1

(202J

202A - 206

203 W- 202

AMTNM-HBN(K)

203 I CONTINUE

72

A4 AMTPM.AMTNM

CALL SUBROUTINE GIRDER MOMENT

fCBPB.CTH.CAC.CCB.CCT.CBH3.CR2,CKT,CKB

19

(^H.FCP.FSP.FCI

F^AMTPM/ ( . 65 *H)

F0^7.*F/6.

FB^.45*FCP

A C ^ 1 , 5 * F 0 / F B

B^AC/(CAC*H)

BPP^CBPB*B

T^CTH*H

AC^CAC*B*H

CB^CCB*H

CT^CCT*H

BH3^CBH3*B*(H**3.)

R2^CR2*(H**2,)

TK^CTK*H

BK^CKB*H

TS^ST

T

73

A5 WIDTH^(16.*TS)+B

AC2^AC+(TS*WIDTH)

CB2P^T*B*T/2.+BPP*(H-2.*T)*(T+(H-2.*T)/2.)

CB2^(CB2P+T*B*(H-T/2.+WIDTH*TS*(H+TS/2.))/AC2

BH32A^B*T*(T**2./12.+(CB2-T/2.)**2.)

BH32B-BPP*(H.2.*T)*((H-2.*T)**2./12+(ABS(CB2-T-(H-2.n)/2.))**2.)

BH32C^B*T*(T**2./12.+(ABS(H-T/2.-CB2))**2.)

BH32D^WIDTH*TS*(TS**2./12+(ABS(H+TS/2.-CB2))**2.)

BH32^BH32A+BH32B+BH32C+BH32D

CT2^H+TS-CB2

R22^BH32/AC2

TK2^R22/CB2

BK2^R22/CT2

TMT-H(BH3/CT)/(BH32/(CT2-TS)

TMB^(BH3/CB)/(BH32/CB2)

FTP^3.*SQRT(FCI)*(144./1000.)

EKFTP*BH3/(CT*F0)

BA^AC

CALL SUBROUTINE GIRDER MOMENT

I E2-HAMG/F0

6

74

A6 E-BK+EUE2

CGSY^CB-E

AMP (AC2/AC)*AMG

AMC AMTPM-AMP

F^((AMP+TMB*AMC)-2.*FTP*TK*AC)/(E+TK)

F0^7.*F/6.

EKFTP*BH3/(CT*F0)

E2^AMG/F0

E^BK+E1+E2

CGSY^CB-E

F^((AMP+TMB*AMC)-2.*FTP*TK*AC)/(E+TK)

F0^7.*F/6.

FBC^.6*FCI*(144./1000J

ACF^(FO+(FO*E-AMG)/TK)/FBC

ACPF^(F+((AMP+TMT*AMC)-F*E)/BK)/FB

0

75

25

0 0

ACFF^ACPF

RATIO^ACFF/AC

27 AC^ACFF

-0

29| XCB^CT

XCT^B

XTK^BK

XBK^TK

XCB2-^:T2

XCT2-HCB2

XTK2^BK2

XBK2^TK2

XTMT^TMB

XTMB^TMT

E1-FTP*BH3/(XCT*F0)

T

76

A7 E2^-XAMG/F0

E-XBK+E1+E2

CGSY-HXCB-E

AMCNfrH(2.*FTP*BH32)/XCB2

AMPNG -AMTNM-AMCNG

FTC-KFB

XXACF^(AMPNG+XTMB*AMCNG-F*(E+XTK))/(2.*FTP*XTK)

XACF^(FO+(FO*E+XAMG)/XTK)/FBC

XACPF-(F+((AMPNG+XTMT*AMCNG)-F*E)/XBK)/FTC

^ WFP M;L

©

0

12 ZACFF XACPF

77

- > -

15 AC ZACFF

16 B-AC/(CAC*H)

BPP CBPB*B

T-CTH*H

CB-KCCB*H

CT^CT*H

BH3^CBH3*B*(H**3.)

R2^R2*(H**2.)

TK^CKT*H

BK-HCKB*H

WIDTH-^(16.*TS)+B

AC2^AC+(TS*WIDTH)

CB2P^T*B*T/2.+BPP*(H-2.*T)*(T+(H-2.*T)/2,)

CB2-(CB2P+T*B*(H-T/2.)+WIDTH*TS*(H+TS/2.))/AC2

BH32A^*T*(T**2./12.t(CB2-T/2.)**2.

^

78

A8 BH32B^BPP*(H-2.*T)*((H-2.*T**2./12.+(ABS(CB2-T-(H-2.*T)/2.))**2.)

BH32C^B*T*(T**2./12.+(ABS(H-T/2.-CB2))**2.

BH32D^WIDTH*TS*(TS**2./12.+(ABS(H+TS/2.-CB2))**2.)

BH32^BH32A+BH32B+BH32C+BH32D

CT2^H+TS-CB2

R22^BH32/AC2

TK2^R22/CB2

BK2^R22/CT2

TMT-(BH3/CT)/(BH32/(CT2-TS))

TMB*(BH3/CB)/(BH32/CB2)

XTK^BK

XTMB^TMT

AST-F/(.6*FSP)

E^(33.*1830.*SQRT(FCI)*144,)/1000.

K^l

LTHT-S(1)+S(2)+S(3)

(TITLE) FLANGE WIDTH, WEB THICKNESS. FLANGE DEPTH. DEPTH.

AREA, COMP WIDTH, COMP AREA,

B.BPP,T,H,AC WIDTH. AC2.

^

79

A9| (TITLE) AREA OF STEEL. PRESTRESSING FORCE. INITIAL FORcF

(TITLE) PRECAST INERTIA. COMPOSITE INERTIA. MOD ELASTICITYI

I BH3,BH32,E |

E I

TITLE):. LOWER BOUND, UPPER BOUND, CABLE LOCATION, CABLE MOMENT

CP - <

I KL^(1,1.LTHT)

K-HKL

SAT^6.*SQRT(FCI)*(144./1000.)

AMP^(AC2/AC)*GM(K)

AMC^BP(K)-AMP

AMCNG^(2.*FTP*BH32)/XCB2

AMPNG^-BN(K)-AMCNG

EA^(AMPNG+XTMB*AMCNG-2.*FTP*XTK*AC-F*XTK)/F

YCGSYKCB+EA

EB^(FTP+F/AC+(GM(K)*CT/BH3))*(BH3/(F*CT))

YCGSY2^CB-EB

80

AlO EC^(FTP+F/AC-(GM(K)*CB/BH3))*(BH3/(F*CB))

YCGSY3^CB+EC

ED^(AMP+TMB*AMC-2.*FTP*TK*AC-F*TK)/F

YCGSY4^CB-ED

YCGSY2-^,1667

YCGSY3^H-.1667

>-( 221

-M225

^

81

All YCGSY4^(H-.1667) |

YCGSYA YCGSY2 H 1

YCGSYA^YCGSYl

82

YCGSYB-^YCGSY4]

TITLE: ZERO CABLE ZONE. UPPER, LOWER M

I K.YCGSYB.YCGSY

239 YCGSYF^YCGSYB

CM(K)^FO*(CB-YCGSYF)

T I K,YCGSYA,YCGSYB,YCGSYF.CM(K) |

2250 CONTINUE i ^

Y START DEFLECTED SHAPE FLOW CHART

83

SUBROUTINE GIRDM

COMMON WC,S.K.GMX.AMG.XAMG.GM

LTHT-S(1)+S(2)+S(3)

K^l

DLOAD^BA*WC

XLTHT-LTHT

RR^(DL0AD*((S(2)+S(3)**2.)/2.)/S(2)

RL^DL0AD*(S(2)+S(3))-RR

- » ^ K - ( 1.1.LTHT )\

^

L > ^ GMX(K)^RL*XX-(DL0AD*XX**2.)/2

84

302

307

GMX(K)—(DL0AD*(XLTHT-XX)**2.)/2.

GMX(K)^GMX(K)/1000.

GM(K)-GMX(K)

I ^ K^(2.1.LTHT)}

AMG^MX(K)

KK K

308 CONTINUE!

I XAMG^MX (1)

KKK l

85

CONTINUE I

TITLE: FEET. MAX^^GM. FEET, MAX-GM |

I KK,AMG,KKK,XAMG |

END