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PPoosstt––TTeennssiioonniinnggTTeennddoonnss iinn PPaarrttiiaallllyy PPrreessttrreesssseedd BBeeaammss

A Thesis

Submitted to the Building and Construction Engineering

Department in the University of Technology in

Partial Fulfillment of the Requirements for the

Degree of Master of Science in

Structural Engineering

By

Ehab Tarik Ibrahem (B.Sc. 1996)

February 2001

Acknowledgement

In the name of ALLAH, The most compassionate the most merciful.

Praise be to ALLAH and pray and peace be on his prophet Mohammed

his relatives and companions and on all those who follow him.

First of all, thanks for ALLAH who enabled me to achieve this research.

I would like to express my sincere thanks to my supervisor Prof. Dr.

Kaiss F. Sarsam for his guide, advice, considerable help and discussion

throughout this work.

The experimental program described in this research was sponsored by

AL-Rashid Contracting Company where concrete, forms, reinforcement,

strand, strand chucks and anchorages were furnished.

Thanks are due to the Department of Building and Construction

Engineering, University of Technology, several of the staff in the Department,

Laboratories, and the libraries for there immediate help where needed.

I wish to thank Chief Engineer Abdul-Ameer for his help and assistance

when carrying out this project.

I am deeply grateful and much obliged to Mr. Kahlid, Mr. Taleb for their

helps by time and effort.

Also, grateful acknowledgement to the members of central Library,

University of Technology.

Finally, I would like to thank every helping hand that enabled me to

achieve this goal.

Synopsis An investigation of the stress in the prestressed tendons at ultimate in unbonded

partially prestressed concrete beam is described with particular emphasis on:

A- The effects of nonprestressed bonded reinforcement in tension, and type of load

application (single concentrated load at midspan, and two symmetrical third-point loads)

on ultimate stress in unbonded post tension.

B- The effects of nonprestressed bonded reinforcement in comparison on ultimate

stress in unbonded post tension under two symmetrical third-point loads.

Load tests on 10 rectangular sections of unbonded beams are reported. It is shown that

the presence of unbonded nonprestressed reinforcement in tension and type of load

application has a marked influence on the flexural behavior of unbonded beams and the

related increase of stress in the prestressing steel at ultimate. For partial prestressing ratio

PPR: 0.35, 0.54, 0.75 and 1.0, it was found that the ultimate stress increase in unbonded

tendons was about 59.2 % greater with third-point loading. While for comparison

reinforcement no significant effect on flexural behavior of unbonded beams under two

symmetrical third-point loads.

The experimental results obtained were combined with 110 test results collected from

the technical literature. These were taken from 8 different experimental investigations that

perform between 1971 and 2000 in various parts of the world. The combined experimental

results were used to conduct an evaluation survey of design parameters, by using five

statistical methods: 1) mean values and standard deviations; 2) correlation coefficient; 3)

error analysis; 4) standard error of estimate; 5) frequency distribution.

Statistical analysis of these combined experimental results has led to two new

prediction equations for computing the ultimate stress in unbonded tendons. The different

existing prediction equations were evaluated and compared to the two proposed equations.

Comparison is also made with ACI 318-89/99 code equations (18-4) and (18-5).

The evaluation of total 120 test results pointed out some of the drawbacks in existing

code equations.

Based on statistical test observations the two proposed equations are the best

statistical model for predicting value of ∆fps and fps respectively. As example applying the

proposed prediction equation led to a coefficient of correlation of 0.77 and 0.92 for ∆fps

and fps respectively. This compares favorably with other existing design equations – the

coefficient of correlation for these equations ranged between 0.02–0.73 for ∆fps and

between 0.72-0.90 for fps respectively.

List of Contents

Subject Page No.

Chapter One: Introduction

1.1 General 1

1.2 Extent of partial prestressing 2

1.3 Advantage of partial prestressing 3

1.4 Research significance 3

1.5 Layout of the study 4

Chapter Two: Literature Review

2.1 Scope 5

2.2 Introduction 5

2.3 Initial studies 6

2.4 ACI Building 318- 63/99 Code equations 7

2.4.1 Warwaruk, Sozen, and Siess Model 7

2.4.2 Mattock, Yamazaki, and Kattula 8

2.4.3 Mojtahedi and Camble 9

2.5 Other Investigation dealing with fps existing studies 10

2.5.1 Pannell model 10

2.5.2 Tam and Pannell model 11

2.5.3 Burns, Charney and Vines 12

2.5.4 Cook, Park and Yong 12

2.5.5 Elzanaty and Nilson 13

2.5.6 Tao and Du model 13

2.5.7 Chakrabarti and Whang 14

2.5.8 Campbell and Chouinard 15

2.5.9 Harajli – 1 model 15

2.5.10 Harajli and Hijazi model, (Harajli – 2) 16

2.5.11 Harajli and Kinj model, (Harajli – 3) 17

2.5.12 Naaman and AL-Khairi model 18

2.5.13 Chakrabarti, Whang, Brown, Arsad and Aezeua 19

2.5.14 Chakrabarti model 20

2.5.15 Lee, Moon and Lim model 22

2.5.16 Shdhan model 22

2.6 Other code equations for fps 24

2.6.1 North American codes 24

2.6.2 European codes 24

2.7 Summary 25

Chapter Three: Experimental Program

3.1 Scope 28

3.2 Introduction 28

3.3 Test program 29

3.4 Beam specimen cross–section 29

3.5 Shear Reinforcement details 31

3.6 Materials 32

3.6.1 Concrete 32

3.6.1.1 Cement 33

3.6.1.2 Coarse aggregate and fine aggregate 33

3.6.1.3 Superplasticizer 34

3.6.2 Prestressing steel 34

3.6.3 Unprestressed reinforcement steel 34

3.7 Concrete mixing 35

3.8 Casting and curing 35

3.9 Post–tensioning operations 35

3.10 Measurements 36

3.10.1 Strain Increase in unbonded tendons 36

3.10.2 Concrete compressive Strain 36

3.10.3 deflection 37

3.11 Testing beam specimens 37

3.12 Testing control specimens 38

3.12.1 Compressive strength of concrete 38

3.12.2 Splitting tensile strength of concrete 38

3.12.3 Modulus of rupture of concrete 38

3.13 Summary 39

Chapter Four: Test Results

4.1 Test results 40

4.2 Cracking behavior 40

4.3 Load–deflection response 40

4.4 Stress increase in unbonded tendons 41

Chapter Five: Proposed Equation

5.1 General 50

5.2 Introduction 50

5.2.1 Mean values and standard deviations, method 1 51

5.2.2 Correlation coefficient, method 2 51

5.2.3 Error analysis, method 3 53

5.2.4 Standard error of estimate, method 4 53

5.2.5 Frequency distribution, method 5 54

5.3 Proposed design equation 55

5.4 Characteristics of proposed design equation 56

Chapter Six: Statistical Analysis

6.1 Experimental data analysis 58

6.2 Evaluation of existing prediction equation 60

6.2.1 Mean values and standard deviations, method 1 60

6.2.2 Correlation coefficient, method 2 61

6.2.3 Error analysis, method 3 64

6.2.4 Standard error of estimate, method 4 70

6.2.5 Frequency distribution 92

6.3 Evaluation of proposed equations 99

Chapter Seven: Conclusions & Future research

7.1 Conclusions 100

7.2 Future research 101

References 102

Appendix A 107

List of Figures Figure No. Page No.

3.1 Beam cross – section details 30

3.2 Shear reinforcement details 31

3.3 Gradation of coarse aggregate 33

3.4 Gradation of fine aggregate 33

3.5 Measurement instrument on beam specimen 36

3.6 Type of load application details 37

4.1 Experimental observed applied midspan moment versus

midspan deflection

44

4.2 Applied midspan moment versus measured stress increase

in prestressing steel for beams A-1, A-2, B-1, B-2 and B-3

45

4.3 Applied midspan moment versus measured stress increase

in prestressing steel for beams B-4, B-5, B-6, C-2 and C-3

46

4.4 Applied midspan moment versus measured stress increased stress

in prestressing steel

47

4.5 Crack patterns at ultimate load for beams A-1, A-2, B-1, B-2,

B-3 and B- 4

48

4.6 Crack patterns at ultimate load for beams B-5, B-6,

C-2 and C-3

49

6.1 Comparison of predicted stress by predicted design Eq. (2.7): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Warwaruk et al).

73

6.2 Comparison of predicted stress by predicted design Eq. (2.9): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (ACI 318-71)

74

6.3 Comparison of predicted stress by predicted design Eq. (2.9) and

(2.10): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (ACI

318M-1999)

75

6.4 Comparison of predicted stress by predicted design Eq. (2.22): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Canadian code)

76

6.5 Comparison of predicted stress by predicted design Eq. (2.23): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (British code)

77

6.6 Comparison of predicted stress by predicted design Eq. (2.24): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Dutch code)

78

6.7 Comparison of predicted stress by predicted design Eq. (2.25):

(a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (German code)

79

6.8 Comparison of predicted stress by predicted design Eq. (2.14): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Du and Tao)

80

6.9 Comparison of predicted stress by predicted design Eq. (2.15):

(a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Harajli-1)

81

6.10 Comparison of predicted stress by predicted design Eq. (2.16): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Harajli-2)

82

6.11 Comparison of predicted stress by predicted design Eq. (2.17): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Harajli-3)

83

6.12 Comparison of predicted stress by predicted design Eq. (2.18): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Naaman and

Al-Khairi) or (AASHTO-1994).

84

6.13 Comparison of predicted stress by predicted design Eq. (2.19): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Chakrabarti)

85

6.14 Comparison of predicted stress by predicted design Eq. (2.20): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Lee at al)

86

6.15 Comparison of predicted stress by predicted design Eq. (2.21): (a)

∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Shdhan)

87

6.16 Comparison of predicted stress by proposed design [Eq. (5.7)

approach-I]: (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps

88

6.17 Comparison of predicted stress by proposed design [Eq. (5.7)

approach-II ]: (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps

89

6.18 Comparison of predicted stress by proposed design [Eq. (5.7)

approach-III]: (a) ∆fps; (b) fps; (c) error of ∆fps and error (d) of fps

90

6.19 Comparison of predicted stress by proposed design [Eq. (5.7)

approach-IV]: (a) ∆fps; (b) fps; (c) error of ∆fps and error (d) of fps

91

6.20 Combined cumulative frequency of ∆fps 95

6.21 Combined cumulative frequency of ∆fps 96

6.22 Combined cumulative frequency of fpsp 97

6.23 Combined cumulative frequency of fpsp. 98

List of Tables

Table No. Subject Page No.

2-1 Characteristics of investigation reviewed in this study 26

2-2 Parameters and their frequency in use of various design equations 27

3-1 Details of reinforcement of various beam specimens 30

3-2 Concrete mix proportions 32

3-3 Results of cement chemical and physical test 33

3-4 Testing control specimen results 39

3-5 Summary of reinforcement and strength parameter of various 39

beam Specimens

4-1 Summary of test results 43

5-1 Summary of coefficients used in the proposed Eq. (5.7) 56

6-1 Characteristics of experimental investigations considered in 59

this study

6-2 ∆fps and fps statistical data 71

6-3 ∆fps and fps error statistical analysis 72

Notations

Aps area of prestressing steel As area of nonprestressed tensile steel As

’ area of compression steel b width of the section bw web width of a flanged member c depth from concrete extreme compressive fiber to neutral axis cy depth from concrete extreme compressive fiber to neutral axis

calculated using yield strength of the prestressing steel dps depth from concrete extreme compressive fiber to centroid of the

prestressing steel ds depth from concrete extreme compressive fiber to centroid of the

nonprestressed tensile steel d’ depth from concrete extreme compressive fiber to centroid of

nonprestressed compressive steel de distance from extreme compressive fiber to centroid of tensile force

in the tensile reinforcement (effective depth) Eps modulus of elasticity of the prestressing steel f load geometry factor fc` concrete cylinder compressive strength fcu concrete compressive strength taken from cube test fpe effective prestress in prestressing steel, after all loses fps ultimate strength in the prestressing steel fpse experimental value of fps fpsp predicted value of fps fpu ultimate strength of the prestressing steel fpy yield strength of the prestressing steel fy yield strength of nonprestressed tensile steel h height of the section hf flange thickness of a flanged member

L length of span under consideration L1 length of loaded span or sum of lengths of loaded spans L2 length of tendon between end supports Le length of the tendon between anchors divided by the number of

plastic hinges required for developing a failure mechanism in the span under consideration

Mcr cracking moment Mu ultimate flexural moment no number of loaded spans in the member n total number of spans in the member r correlation coefficient S span length from anchorage to anchorage of simply supported

member, or length of the span under consideration for continuous member

S/dp span to depth ratio of member Sr sum of squares of residuals (or sum of squares about regression) Sy standard deviation Sylx standard error of estimate γ plastic hinge length ratio β1 ACI concrete compression block reduction factor ∆fps stress increase in unbonded tendons at ultimate ∆fpse experimental value of ∆fps ∆fpsp predicted value of ∆fps (∆εcps)m maximum strain increase in concrete at the level of an equivalent

amount of bonded prestressing steel beyond the effective prestress (∆εpsu)m maximum strain increase in prestressing steel beyond the effective

prestrain εce strain in concrete at the level of prestressing steel due to effective

prestress εcu strain in concrete top fiber at ultimate εpe effective prestrain in the prestressing steel

εps strain in prestressing steel at ultimate λ strain compatibility factor ρps prestressing steel reinforcing ratio ρs nonprestressing tensile steel reinforcing ratio ϖ global reinforcement index φc resistance factor for concrete φps resistance factor for prestressing steel φs resistance factor for nonprestressing steel ξ type of load application coefficient Ωυ bond reduction coefficient

Chapter One

Chapter One Introduction 1

1.1 General:

Unbonded prestressed concrete is a type of reinforced concrete in which the

steel reinforcement has been tensioned against the concrete. This tensioning

operation results in a self- equilibrating system of internal stresses (tensile

stresses in the steel and compressive stresses in the concrete) which improves

the response of the concrete to external loads. While concrete is strong and

ductile in compression it is weak and brittle in tension, and hence its response to

external loads is improved by applying a precompression (1).

In posttensioning the tendons are stressed and anchored at the ends of the

concrete member after the member has been cast and has attained sufficient

strength. Commonly, a mortar-tight plastic tube or duct (also called sheath) is

placed along the member before concrete casting. The tendons may have been

preplaced loose inside the sheath prior to casting or could be placed after

hardening of the concrete, When the duct is filled with grease instead of grout

(bonded tendon), the bond would be prevented throughout the length of the

tendon and the tendon force would apply to the concrete member only at the

anchorage, this leads to unbonded tendons (2).

The behavior of a member with unbonded tendons is different from that of

a member with bonded tendons. For the member with bonded tendon,

equilibrium and compatibility equations can be derived, assuming that tendons

and concrete behave as a body because they are bonded completely in the

member. For members with unbonded tendons, however, tendons and concrete

deform independently, except at the ends of the member. Therefore, any

analytical procedure must satisfy the global compatibility requirement rather

than the local compatibility requirement. The latter means that the tendons and

concrete elongate equally at any given section of the member. The overall

elongation of the concrete is equal to the total lengthening of the tendon. Much

research has been carried out concerning a member with unbonded tendons, on

Chapter One Introduction 2

the basis of which various design equations (such as the ACI Code equation)

have been proposed .It is difficult to say, however, that the global compatibility

requirement is appropriately satisfied in the existing design equations that have

been proposed or put into use. Furthermore, those equations are often based on

very limited parameters and experiments (3).

1.2 Extent of partial prestressing:

Partially prestressed concrete (PPC) members can be defined as those that

contain both prestressed and nonprestressed reinforcement intended to resist

similar external loads. The extent of partial prestressing can be characterized by

one of four different parameters: partial prestressing ratio (PPR), prestressing

index, degree of prestress, and global reinforcing index (ϖ ) (4).

PPR = )2/ad(fA)2/ad(fA

)2/ad(fA

sysppsps

ppsps

−+−

− …(1.1)

Prestressing index =yspyps

pyps

fAfAfA+

…(1.2)

Degree of prestress =yspyps

peps

fAfAfA+

…(1.3)

ϖ =ec

ys

ec

ys

ec

psps

db'f'f'A

db'ffA

db'ffA

−+ …(1.4)

yspsps

sysppspse fAfA

dfAdfAd

++

=

were Aps and As = area of prestressed and nonprestressed reinforcement, respectively; fpe = effective prestress applied; a = depth of equivalent compression stress block; fpy and fy = yielding strength of prestressed and nonprestressed reinforcement, respectively; de = depth from the extreme compressive fiber to the centroid of the tensile force; dp and ds = depth of the prestressing tendon and nonprestressing steel from the compression fiber, respectively

Chapter One Introduction 3

1.3 Advantages of partial prestressing:

In many prestressed concrete structures, it is unlikely that the full service

load will be applied during the life of the structure. It is therefore possible to

design prestressed structural members in such away that cracking will occur

under full service load, should it be applied. This can be achieved by the use of

partial prestressing.

The advantages of partial prestressing are:

• Reduction of prestressing force.

• Reduction of initial camber, which is of importance for some

types of precast members.

• In some instances reduction in prestressing force may allow an

increase in tendon eccentricity.

• Reduction of cracking in the end zones may be obtained.

One of the disadvantages of partial prestressing is that the reduced stiffness the

member after cracking may result in an increase in deflection, which may be

sufficient to exceed the acceptable serviceability limit (5).

1.4 Research significance:

In this study an attempt is made to make a statistical study of the accuracy

of various equations for prediction of the ultimate stress of unbonded prestressed

beams (fps) suggested by different investigators in recent years, and present the

results of statistical study performed on (15) different models, including the

ACI-1999 model, to determine which provides the best prediction of the actual

(experimental) values of fps and ∆fps. A large number of previously published

data was readily available. All of the data used in this study was from the

researcher work, preformed to develop their model to predict fps. In this study all

the data was combined to provide the largest population for statistical analysis.

Chapter One Introduction 4

1.5 Layout of the study:

• A review of existing experimental and analytical investigations dealing

with fps followed by a summary of prediction equations recommended by

various North American and European codes is presented in Chapter Two.

• Extensive details of the experimental program of this study are presented

in Chapter Three.

• Test results are given in Chapter Four with relevant discussion pertaining

to the behavior of beam specimens under applied load.

• In Chapter Five, statistical analysis of extensive experimental results has

led to a new prediction equation for computing fps in unbonded tendons.

• The different prediction equations, as well as the equations recommended

in major design codes, were evaluated and compared to the proposed equation.

• Chapter Seven summarizes the conclusions drawn from this study, and

suggestions for future research.

• A numerical example illustrating the use of the proposed prediction

equation is provided in Appendix A.

Chapter Two

Chapter Two Literature Review 5

2.1 Scope: This chapter examines mainly some background information followed by

comprehensive study for the existing code equation and many purposed

experimental, analytical and empirical investigations dealing with the stress fps

at ultimate in unbonded tendons.

A summary of prediction equations for fps suggested by different

investigation from 1963 to 2000 were reviewed in Tables. (2.1) and (2.2).

2.2 Introduction: To perform flexural design and analysis of prestressed concrete members,

the ultimate stress in the prestressing reinforcement, referred to fps must be

known. However, this stress is a variable based on many features of the

particular beam.

For unbonded post-tensioning it is a combination of the effective

prestressing and increment stress caused by loading on the member. The stress

caused by effective prestressing is a known quantity valley, but the additional

stress that result when the ultimate moment capacity is reached is unknown.

Most of the fps predictor equations are based on the following equation

fps = fpe + ∆fps …(2.1)

where fpe is the effective prestress due to the force and dead load moment and

∆fps is the increased stress caused by additional loading in reaching the ultimate

moment condition.

Chapter Two Literature Review 6

2.3 Initial studies: Many studies have been carried out to predict the stress in unbonded

tendon at ultimate. Among the first is a study by Baker ( 6), who expressed the

strain in the prestressing steel as follows

εps = εpe + (∆ εpsu)av …(2.2)

εps = εpe + λ (∆ εcps)m …(2.3)

where εpe is the strain in the prestressing steel under effective prestress,

(∆ εcps)m is the maximum strain increase in the concrete at the level of an

equivalent amount of bonded prestressing steel beyond effective prestress,

(∆ εpsu)av is the ratio of average strain increase in the prestressing steel beyond

the effective prestress, and λ is a coefficient defined as the ratio of average

concrete stress adjacent to the steel to the maximum concrete stress adjacent to

the steel. Baker (6) suggested a value of λ = 0.1 for the ultimate limit state. If the

steel remains in the elastic range of behavior at ultimate as is mostly the case in

the practice, then the stress in the prestressing steel at ultimate can be written as

fps=Eps εps = Eps [εpe + (∆ εpsu)av] …(2.4)

fps = fpe +(∆ εpsu)av =fpe + λ Eps (∆ εcps)m …(2.5)

where Eps is the modulus of elasticity of the prestressing steel.

Gifford (7) defined a strain compatibility factor λ as the ratio of the average

effective concrete strain at the level of the prestressing steel to the concrete

strain at the section of maximum moment. He suggested an empirical safe limit

value of λ = 0.2 as typical for most cases. Janney et al (8) tested in third point

loading a number of simple beams prestressed with unbonded tendon and having

a span-depth ratio close to 13. Based on their test, they suggested a value of

neutral axis at ultimate to the prestressing steel depth, i.e., λ = c/dps

Chapter Two Literature Review 7

2.4 ACI 318- 63/99 Code models: The model that the ACI Code suggests has evolved since the 1963 version of the

code, where the prediction of fps was to simply add 105 MPa to fpe (34).

fps = fpe + 105 MPa …(2.6)

But with each new model a more accurate method of prediction fps has come.

2.4.1 Warwaruk, Sozen, and Siess model: Warwaruk et al (9) conducted an extensive investigation comprising tests

on 82 simply supported partially prestressed rectangular beams. Of these beams,

41 contained unbonded tendons. The main variables were the amount of

reinforcement, the concrete compressive strength, and the type of loading. They

reported that beams containing no supplemental reinforcement failed by

developing only one major crack, while those with supplemental reinforcement

developed multiple cracking before failure. The stress in the unbonded beams

remained in the elastic range up to failure. For prediction purposes, several

parameters were plotted against the stress in the prestressing steel. The best

correlation led to the following prediction equation

)10*'f

5.47207(ff 4

C

pspeps

ρ−+= MPa …(2.7)

fpe < 0.6 fpu

where ρps is the prestressing steel reinforcement ratio, and fc’ is the compressive

strength of concrete.

Chapter Two Literature Review 8

2.4.2 Mattock, Yamazaki, and Kattula model: Mattock et al (10), conducted an experimental study on seven simply

supported partially prestressed concrete beams and three continuous beams over

two spans. The primary variables were the presence and absence of bond, the

amount of supplementary nonprestressed reinforcement, and the use of seven-

wire strand as bonded unprestressed reinforcement. The span-depth ratio was

fixed at 33.6. The authors drew the following conclusions: 1) fps for unbonded

tendons as predicted by ACI 318-63 (34) was approximately 30 percent less than

that predicted from experiment; 2) as the ratio ρps /fc’ increase, the margin of

excess strength between predicted and observed fps decreases; 3) ACI 318-63

satisfactorily reflects the behavior of unbonded tendons for simply supported

beams; 4) the distribution and width of the cracks developed in the unbonded

beams were very similar to those developed in the bonded beams provided

additional nonprestressed reinforcement is present; 5) a minimum amount of

reinforcement bars equal 0.4 percent of the total area of the critical beam section

must be provided when unbonded tendons are used.

They also show that both the ACI Building Code equation [Eq. (2.6)] and Eq.

(2.7)] were too conservative at low reinforcement ratio and they proposed the

following equation

)100

'f4.170(ff

ps

Cpeps ρ

++= MPa …(2.8)

Eq. (2.8) was later adopted with a slight modification by the 1971 and 1977

ACI Building Codes (11),(12) , as follows

)100

'f70(ff

ps

Cpeps ρ

++= MPa …(2.9)

with the limitation that fps < fpe + 400, and fps < fpy

Chapter Two Literature Review 9

2.4.3 Mojtahedi and Gamble (13): Their study showed that the ACI 318-77 Code (12) might overestimate the

stress of unbonded tendons in the members whose span-depth ratio is high. They

used the results of existing experimental work and proposed a simple structural

model to verify their argument. Then they put emphasis on the span-depth ratio

so that the span-depth ratio is considered to be a parameter in the code.

The result was, then, reflected in the ACI 318-83 Code and has been used to the

present. The new code equation provides lower stress for a member whose span-

depth ratio is high, while preserving the existing code by employing 300 instead

of 100 as a denominator in the ACI 318-77 (12) in case when the span-depth ratio

is more than 35.

The span-depth ratio, however, is not the only difference between a shallow

slab and an ordinary beam. The amount of tendon should be considered also

since fewer tendons are usually used for slabs than beams. The tendon ratio ρps

is in the denominator in the equation of the code. Thus, the smaller the value is,

the greater the tendon stress becomes, regardless of the member depth.

There is a higher possibility of overestimating the tendon stress if a small

amount of tendon is provided. In consequence, a higher span-depth ratio may

not be the only parameter to overestimate the tendon stress. Based on their

findings, the ACI Building Code restricted in its 1983 version the use of Eq.

(2.9) to members with span-to-depth ratio < 35 and implemented a new more

conservative equation for members with span-to-depth ratio >35–Eq. (2.10)

)300

'f70(ff

ps

Cpeps ρ

++= MPa …(2.10)

with the limitation that fps < fpe + 200, and fps < fpy. Eqs. (2.9) and (2.10) have

been adopted, as Eqs. (18.4) and (18.5) respectively, in later ACI Building Code

editions – ACI 318 M-89 (14) and ACI 318 M-99 (15).

Chapter Two Literature Review 10

2.5 Other investigations dealing with fps : 2.5.1 Pannell model:

Pannell (16) conducted a comprehensive experimental and analytical

investigation to study the effect of the span-to-depth ratio on the flexural

behavior of beams prestressed with unbonded tendons. A total of 38 beams were

tested. The main variables were the span-to-depth ratio, the effective prestress,

and the amount of reinforcement. Based on the findings, he proposed an

equation for the stress in the prestressing steel at ultimate, assuming that: 1) the

stress in the prestressing steel remains in the linear elastic range, 2) the effective

prestrain in the concrete is negligible. He first assumed the following

relationship

P

CPS Ll∆

ε∆ = …(2.11)

where ∆εcps is the strain change in the concrete at the level of the prestressing

steel, Lp is the width of the plastic zone assumed to occur at ultimate, and l∆ is

the concrete elongation at the level of the prestressing steel measured within the

width of the plastic zone. Based on experimental results, Pannell (16) suggested a

value of Lp equal to 10.5c, where c is the depth of neutral axis at ultimate. Then,

using strain compatibility and equilibrium, he derived the following equation

ps

cups

'fqf

ρ= MPa …(2.12)

in which

αλλ

+

+=

1

qq eu

'

'

c

pe

ps

pse

c

pspscups

ff

d b A

q

f Ld E 5.10

=

=ερ

λ

Chapter Two Literature Review 11

where εcu is the strain in the concrete top fiber at ultimate, α = 0.85 β1 (based on

cylinder compressive strength) or α = 0.68 β1 (based on cube compressive

strength), β1 is the stress block reduction factor as defined by ACI Building

Code , ρps is the prestressing steel reinforcement ratio, and dps is the distance

from the extreme compression fiber to the centroid of the prestressed

reinforcement. The preceding equation [Eq. (2.12)] was used as the basis for the

equation recommended in British Code (17) .

2.5.2 Tam and Pannell model:

Tam and Pannell (18) tested eight partially prestressed beams with

unbonded tendons subjected to a single concentrated load at midspan. The main

variables were the amount of prestressed and nonprestressed tensile steel, the

span-to-depth ratio, and the effective prestress. They observed that all beams

developed fine cracks similar to those developed in beams containing bonded

reinforcement. Based on their observations, they modified the prediction

equation for fps presented earlier [Eq. (2.10)] to account for the effect of

additional nonprestressed tensile reinforcement. In its final form, their equation

that applies to rectangular section behavior only can be expressed as follows

ps

cups

'f qf

ρ= MPa …(2.13)

in which λα

λ

αλλ

+−

+

+= se

u

q

1

qq

'cps

yss f d b

f Aq =

where As and fy are the area and the yield strength of the additional

nonprestressed reinforcement, respectively.

Chapter Two Literature Review 12

2.5.3 Burns, Charney and Vines: Burns et al (19) conducted an experimental investigation, which included

the testing of two half-scale continuous slabs prestressed with unbonded

tendons. Their work led to modifications of ACI Building Code

recommendations related to the use of additional nonprestressed reinforcement

in these slab systems. In a more recent experimental investigation of continuous

beams, Burns et al (20) observed that the increase in stress in unbonded tendons

at ultimate depends on the number of spans being loaded and the tendon profile

is each span. They thus warned that the increase in stress might not be as

optimistic as predicted by the ACI Building Code equations [Eqs. (2.9) and

(2.10)] if only one span is loaded to failure while the others are not loaded.

2.5.4 Cooke, Park and Yong: Cooke et al (21) conducted an experimental investigation to study the effect

of the span-to-depth ratio and the amount of prestressing steel on the stress at

ultimate in unbonded tendons. They tested nine simply supported fully

prestressed one-way slabs with unbonded tendons. The slabs were subdivided

into three groups with varying span-to-depth ratios, each group having varying

amounts of prestressing steel. The following observations were made: 1) the

equation for unbonded tendon given by ACI 318-77 (12) overestimates the stress

in the prestressing steel at low reinforcing indexes by 2.4, 8.7, and 11.6 percent

for slabs having L/dps ratio of 20, 30, and 40, respectively; 2) the equations

proposed by Warwaruk et (9) [Eq. (2.7)] and Pannell (16) [Eq. (2.12)]

conservatively predict fps; 3) the fps equation for unbonded tendons given by ACI

318–77 (12) overestimates the stress in the prestressing steel at low reinforcing

indexes by 2.4, 8.7, and 11.6 percent for slabs having span–to–depth ratios of

18, 28, and 38 respectively; 4) flexural instability, which occurs in slabs

containing low amounts of prestressing steel, can be prevented by using

Chapter Two Literature Review 13

additional nonprestressed reinforcement. Since ACI 318-77 (12) does not provide

a satisfactory and complete method for predicting the stress in unbonded

tendons at ultimate, they recommend the use of the ACI 318-63 (34) equation

[Eq. (2.6)].

2.5.5 Elzanaty and Nilson:

Elzanaty and Nilson (34) studied the effect of varying the amount of the

initial prestressing force on the flexural strength of unbonded post-tensioned

partially prestressed concrete beams. They tested eight small-scale models in

two series: under-reinforced (U series) and over-reinforced (O series). They

arrived at the following conclusions: 1) beams of Series U and O showed

excellent ductility at failure; 2) increasing the level of prestress in Series O

increases the ultimate moment capacity, since ∆fps remained constant for all four

beams of the series; 3) the ACI 318-77 (12) code equation for predicting fps was

conservative for Series O and unconservative for Series U.

2.5.6 Du and Tao (22) model: The research to formulate this model of predicting fps was initiated to

include the benefits that nonprestressed steel provides by increasing fps , which

the ACI 318-77 Code (12) model ignores. The authors of this model decided to

use the combined reinforcement index qo as the method of including the

nonprestressed steel in the prediction of fps (although they do not explain their

reasoning for choosing qo ). The variable qo is defined as the combined steel

index (qo =qpe+ qs) where qpe is the prestressing steel index (qpe = Aps fps / b dp fc’)

and qs is the nonprestressing steel index (qs= As fs/ b dp fc’).

The model was then formulated by performing experiments on a total of 22

concrete beams. All beams in this test program were of the same physical

dimensions with the same loading condition applied (two-point load). These

Chapter Two Literature Review 14

beams differed by having different concrete strengths, effective prestress, and

nonprestressed steel yield strengths. The value of fps was measured and

calculated by averaging the measurement from the multiple strain gages placed

on the prestressing strand steel. From this data, liner regression was performed

to develop an equation to predict fps. This equation is very accurate in predicting

the value of fps for the 22 test specimens from which it was developed; that is, a

correlation coefficient of 0.97 was obtained (with 1 being a perfect fit and 0

being the worst fit). The model proposed by Du and Tao (22) is based on the

following set of equation

fps = fpe+( 786-1920 qo ) MPa … (2.14) fps < fpy

in which qo need not be taken more than 0.30 and 0.55 fpy < fpe < 0.65 fpy .

Du and Tao considered that qo is a rational parameter for prediction of the

stress at ultimate in unbonded tendons, however, they expressed qo as a function

of the effective prestress fpe this implies that for a given section with As , Aps and

fc’ , the value of qo remains constant, regardless of the force present in the

prestressing steel at ultimate. Therefore, the use of qo can lead to unconservative

estimates of the reinforcing index, which can also lead to unconservative

prediction of fps.

2.5.7 Chakrabarti and Whang: Chakrabarti and Whang (34) tested eight partially prestressed concrete

beams with unbonded tendons under third-point loading. The partial prestressing

PPR was varied while L/dps was fixed at 21. The following observations were

made: 1) the nonprestressed tensile steel yielded at ultimate; 2) all beams failed

by crushing of the concrete top fiber at ultimate; 3) for equal reinforcement

index the stress in the prestress steel increased when PPR decreased;

Chapter Two Literature Review 15

4) decreasing the combined reinforcing index increased ∆fps; 5) there is a need

to improve ACI 318-83 equation.

2.5.8 Campbell and Chouinard: Campbell and Chouinard (24) tested six partially prestressed concrete

beams with unbonded tendons in third-point loading. The only variable was the

amount of nonprestressed tensile steel. Five of these beams were over–

reinforced. They made an interesting observation about the strain distribution in

the concrete at the level of the prestressing steel in the constant moment region,

namely: for beams with no additional nonprestressed reinforcement, very high

strains developed near the mid-span section where only two wide cracks formed,

while the strains were uniformly distributed when nonprestressed reinforcement

was present and multiple fine cracks occurred.

2.5.9 Harajli-1 model. Harajli (25) conducted an analytical investigation in which he studied the

effect of loading type and span to depth ratio on the stress at ultimate in

unbonded tendons. He incorporated the span to depth ratio in the ACI 318-89 (14) equation to allow for a continuos transition for various span to depth ratios,

and proposed the following prediction equation to replace Eq. (18-4) and (18-5)

of the ACI 318-89 (14) .

]d

L84.0)[

100'f

70(ff

PSPS

CpePS +++=

ρ MPa …(2.15)

with the limitation that fps < fpe +414, and fps < fpy

Harajli (25) indicated that the proposed equation [Eq. (2.15)] is excessively

conservative for simply supported members loaded with third-point or uniform

loading. He thus noted that a more accurate determination of fps could be based

on strain compatibility analysis. It should be indicated that this model most

resembles the ACI Building Code [Eq. (2.8)], where the only difference is that

Chapter Two Literature Review 16

the span to depth ratio is directly utilized in the calculation of fps (instead of only

being used as a method of choosing which of two equations should be utilized).

2.5.10 Harajli and Hijazi model, (Harajli-2): Harajli and Hijazi (26) made an experimental study by using a nonlinear

analysis. In this analysis, they studied the influence of span-depth ratio and its

effect on the fps. The following observations were made: 1) the parameter ρp / fc’

which was adopted in the ACI 318-89 (14) Code is not a rational design

parameter especially in partially prestressed members; 2) the increase in ∆fps

depends mainly on the geometry of the applied load; 3) the limitation of span-

depth ratio greater or smaller than 35 which is adopted in the ACI Code for all

type of loading is unwarranted. Their design equation is written as

spu

pep

wff

fwcf

ppupssowc

fysyspupsp

popusps

pupepypspeps

ff

bbChcfI

hcforbbfC

dfAbfCfAfAfA

c

dcff

ffffff

γµ

β

β

γββµ

βγ

+=

==⇒≤

>⇒−=

+

−−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=∆

≤≤∆+=

,0

)('85.0

/'85.0'

1

5.0,

1

1

1

75.1)n/n(d/S2.125.0

75.1)n/n(d/S1.14.0

8.1)n/n(d/S

21.0

oop

s

oop

c

oop

s

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

βγ

βγ

βγ

MPa …(2.16)

for single concentrated load.

for two equal 1/3 point loads.

for uniform load.

Chapter Two Literature Review 17

If compression reinforcement is taken into account when calculating ∆fps,

then c/dp ≥ 0.25 and d’ ≤ 0.15 dp , were (no/n) is the pattern loading.

It can be seen that Eq. (2.16) accounts for the member span-depth ratio

parameter in a uniform rather than a limiting manner; hence they eliminate the

discontinuity in the stress at the span depth ratio of 35 obtained using the current

ACI 318-99 (15) Code, and this will make Eq (2.16) Superior to the ACI Code.

2.5.11 Harajli and Kanj model, (Harajli-3): Harajli and Kanj (27) conducted an experimental investigation, which

included the testing of 26 partially prestressed concrete beams with unbonded

tendons. The main variables were the reinforcing index, and member span-to-

depth ratio. They concluded that: 1) the effect of the length of the plastic hinge

at ultimate is as important as the effect of span-to-depth ratio on the stress in the

prestressing steel; 2) the increasing span–to–depth ratio from 8 to 20 resulted in

a drop in the measured stress increase in unbonded tendons by about 35 percent;

3) the parameter 'f CPSρ , which is the basis of Eqs. (18.4) and (18.5) of the

ACI 318-83 [Eqs. (2.9) and (2.10)], is not a rational design parameter. Harajli

and Kanji (27) proposed a design equation written as

fps = fpe + γo fpu [1.0-3.0 ω] < fpy MPa … (2.17)

ω = ρp /c

se

ff + ρs

p/

c

sy

dfdf

γo = (no/n1)[0.12+2.5/(L/dp)]

in which ρp and ρs are prestressing and reinforcing ratio, respectively; no is the

length of loaded span(s); and n1 is the length of the tendon between the

anchorage ends. The proposed equation includes various influential parameters

such as the partial prestressing ratio, the span-depth ratio (L/dp), and the pattern

loading (no/n1). In applying Eq. (2.17) the term ω is not be taken more than

Chapter Two Literature Review 18

0.23. It should be noted that Eq. (2.14) takes a form similar to Eq. (18.3) in the

ACI Code for the calculation of bonded tendon stress. It neglects the effects of

the loading type, however, which is controversial, depending on the researcher.

2.5.12 Naaman and Al-Khairi model: Naaman and Al-Khairi (28) proposed a model based on deficiencies they

found in model from a study of current and proposed equation to predict fps. This

model was then developed from experimental results from 143 beams. The

model attempts to account for most of the variables found important in the

prediction of fps. This model proposed by Naaman and AL-Khairi is based on

the following set of equations

2

1pscupsupeps L

L)1

cd

(Eff −+= εΩ MPa …(2.18)

fps < 0.94 fpy

)

dL(

5.1

PS

u =Ω for one-point loading

)

dL(

0.3

PS

u =Ω for third-point or uniform loading

c = 1

1111

A2CA4BB −+−

1A = 1w/

c bf85.0 β

B1= fw/

cys/

y/spe

2

1cupsups h)bb(f85.0fAfA)f)

LL

(E(A −+−+−εΩ

C1 = )LL

(dEA2

1pscupsups εΩ−

Chapter Two Literature Review 19

for rectangular sections or rectangular section behavior of flanged section, use

bw = b; (L1/L2) is the ratio of the loaded span length to the summation of lengths

of all spans.

It can be seen that this model introduces a bond reduction coefficient (Ω)

that accounts for the variances of strain between bonded and unbonded tendon.

This coefficient reduces the analysis of beams prestressed with unbonded

tendons to that of beams prestressed with bonded tendons. This model also

accounts for the modulus of elasticity of the prestressing steel, the strain in the

concrete compressive fiber at ultimate, the depth of the neutral axis, and any

nonprestressed steel that is in the beam, including compression steel. This

proposed model was compared to experimental results, and good agreement was

obtained. Later, the equation was adopted in the AASHTO LRFD Code (1994) (33).

It can be seen that [Eq. (2.18)], however, is somewhat complex to be used

for design purposes because the neutral axis depth should be computed in

advance. Because the tendon stress and the neutral axis depth are coupled

together in the computational process, they cannot be computed independently.

Thus, a lengthy computational procedure with a quadratic equation is needed to

get the values.

2.5.13 Chakrabarti, Whang, Brown, Arsad and Aezeua: Chakrabarti et al (29) tested 33 beams post-tensioned with unbonded tendons.

In this project, four groups of beams were tested with the following variables

taken into account: different mixes of reinforcing and prestressing steel, T-

beams and rectangular beams, normal and high-strength concrete, low and high

ratios (L/d), and different initial stresses in the tendons. They also describe

general precracking and post-cracking behavior observed in the testing. Based

on test observations and load-deflection plots, they concluded that: 1) Beams

with a moderate reinforcing index and moderate partial prestressing ratio

Chapter Two Literature Review 20

exhibited better ductility; 2) Maintaining the reinforcing index at the same level,

gradual reduction of PPR resulted in improvement in overall beam behavior in

terms of crack control, deflection control, and ductility in the post cracking

range; 3) Both in T-beams and rectangular beams, some improvement in

strength and deflection control was observed using high-strength concrete when

reinforcing index and PPR were maintained within an optimum range ; 4) When

(L/d) exceeded 35, cracking behavior and deflection control were greatly

improved with a small amount of additional reinforcement ; 5) the effect of

change of initial stress in the tendon on the overall beam behavior was not that

pronounced. However, as the values of fpe were increase, the values of ∆fps were reduced.

2.5.14 Chakrabarti model (30):

This model has evolved over many years of development and

experimentation. This model was formulated from a desire to include the benefit

of nonprestressed steel in the calculation of fps , but with many additional

considerations. Not only the amounts of nonprestressed steel but also the yield

strength and locations of the nonprestressed steel are taken into considerations.

The beam configurations were tested to develop this model including

rectangular beams, T-beams, and slabs. Of all the models included in this study,

this model has more variables taken within the equations.

The model proposed by Chakrabarti (30) is based on the following set of

equations

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−

+++= sef

)B1(

A70pefksefpsf MPa …(2.19 a)

in which 138 )025.0

1(f

414dd

100'fA s

ys

ps

s

C ≤+=ρ

ρ …(2.19 b)

Chapter Two Literature Review 21

33dL for 65.0k,8.0r

33dL for 1.0 k , 0.1r

25.0 f 100

'f rB

ps

ps

PePS

C

>==

≤==

≤=ρ

where ρs is the nonprestressed tensile steel reinforcing ratio, and ds is the depth

from concrete extreme compressive fiber to centroid of the nonprestressed

tensile steel. However, for thin prestressed members without any bonded

reinforcement, the calculated value of fps given by Eq. (2.19a) shall be further

reduced to fps (modified) as given in Eq. (2.19f)

( )pePSpeps f-(2.13a)] .Eq[f65.0f)ified(modf += …(2.19 f)

The maximum value of fps, calculated by Eq. (2.19 a) or Eq. (2.19 b), shall

not exceed the following limits

33dL where 276ff

33dL where 414ff

ps

peps

ps

peps

>+=

≤+=

It is somewhat unclear as to how Chakrabarti’s (30) equation determined

the coefficient of each parameter. For example, the limit of span-depth ratio is

ambiguously 33 in contrast to the ACI Code equation. Also, Eq. (2.19 e) uses

the ratio of 0.65 to account for the effect of high span-depth ratio. Further, it is

hard to forecast the effects of many variables with the equation. For example, ρs

and fse are used in the denominator and the numerator simultaneously. Thus,

designers can hardly expect their effects when an adjustment in prestressing

force has to be made during the design stage.

…(2.19 c)

…(2.19 e)

…(2.19 d)

Chapter Two Literature Review 22

2.5.15 Lee, Moon and Lim model (3): In this study, a design equation for computation of the unbonded tendon

stress at the member flexural failure was proposed in such away that main

parameters and their combination were obtained from theoretical study, they

made these conclusions: 1) The unbonded tendon stress at the member flexural

failure can be influenced by the level of the effective stress of the tendons; 2)

The stress increases of unbonded tendon may be proportional to the square root

of fc’/ρp , unlike the ACI Code; 3) The span-depth ratio has to be considered

together with the loading type since those are dependent upon the plastic hinge

length. Their design equation is written as

pypsse

pp

c

p

s

ps

ysseps

ff70fwhere3f10f

)20.2...(d/L

1f1'f

dd80

Af)A'A(

151f8.070f

≤≤+==

⎥⎥⎦

⎤

⎢⎢⎣

⎡++

−++=

ρ

2.5.16 Shdhan model: Shdhan (31) conducted an experimental study on two simply supported fully

prestressed concrete beams. The primary variables were the type of load

application, the first beam under one point load and the second under two point

load. The span-depth ratio was fixed at 10. Shdhan (31) drew the following

conclusions: 1) beams loaded under one point loading mobilize the least ∆fps compared with two point loading; 2) very high compressive strain developed in

the section for a fully prestressed beams compared with more uniformly

distributed strain when nonprestressed reinforcement are present; 3) The 1999

ACI Building Code (15) equation [Eq. (18.4) and (18.5)] give results that are

reasonable on the safe side, and this leads to a large scatter in the comparison

…(2.20) MPa

for single concentrated load.

for two equal 1/3 point loads.

Chapter Two Literature Review 23

between predicted and experimental observed results; 4)the design equation

proposed by British(17) and Canadian Code(32) gave a negative value; 5)

Naaman and Al-Khairi (28) have developed a rational methodology with strong

theoretical basis. Shdhan (31) proposed the following equation

pypspeps f95.0 fff ≤+= ∆ MPa …(2.21)

in which 2

1

ps

y

5.0

PS

pups LL)

dc

75.11()

dL(

ff −=ξ∆

and ξ = 1.0 for one-point loading

ξ = 1.5 for third-point or uniform loading.

where L1/L2 is the ratio of the loaded span length to the summation of lengths of

all spans-thus, for simply supported members, the ratio L1/L2 is equal to one,

where fpu is the ultimate stress of the prestressing steel, L/dps is the member

span-to-depth ratio, and ξ is a coefficient representing the influence of the type

of load application on fps. The term cy/dps, evaluated at the section into

consideration, need not be taken more than 0.41, where cy is the depth from the

extreme compression fiber to the neutral axis calculated assuming a stress of fpy

in the tendon. Shdhan (31) concluded that applying the proposed design equation

[Eq. (2.21)] on evaluation of total 116 experimental results led to a coefficient of

correlation of 0.93 in comparison with different investigations, (North American

and European codes). The coefficient of correlation for these design equations

ranged between 0.65-0.89. The proposed equation [Eq. (2.21)] is simpler than

the Harajli and Hijazi (26) equation [Eq. (2.16)]. Shdhan (31) simplified the

computing procedure for span-depth ratio with a little modification.

Chapter Two Literature Review 24

2.6 Other code equations for fps:

It is useful to review other codes for the purpose of comparison and

evaluation, starting first with North American codes.

2.6.1 North American codes: The ACI Building Code equations for predicting fps at ultimate for

unbonded tendons were reviewed previously [Eqs. (2.9) through (2.10)].

The Canadian code(32) recommends a stress increase in unbonded

tendons of flexural members obtained from the following equation

pyypspeps f )cd(Le

5000ff ≤−+= MPa …(2.22)

b'f85.0fAfA

cC1c

yssypspd

y βφφφ +

=

φps =0.9, φs, =0.85, φc =0.6

where φps , φs, and φc are resistance factors for the prestressing steel,

nonprestressed steel, and concrete, respectively.

2.6.2 European codes: The method recommended in the British code (17) is based on the studies

of Tam and Pannell (18), as shown in Eq. (2.13). The British code (17)

recommends the following equation

)d bfAf7.1

1()d/L(

7000ffpscu

pspu

pspeps −+= MPa …(2.23)

pups f7.0f ≤

Chapter Two Literature Review 25

where fcu is the strength of concrete taken from cube tests, and the length of the

plastic zone at ultimate is assumed equal to 10c, where c is the depth of the

neutral axis.

To predict the stress in the prestressing steel at ultimate in unbonded

tendons, the 1984 issue of the Dutch code (34) recommends the following

relation

peps f05.1f = MPa …(2.24)

It seems that in the Dutch code, a value of fps at ultimate five percent larger than

that of fpe is generally assumed.

The 1980 issue of the German code (34) recommends a stress increase in

unbonded tendons of flexural members obtained from the following equation

pypspeps f )LL(Eff <+=

∆ MPa …(2.25)

where ∆L = dps / 17, and L is the length o the tendon between end anchorages.

For simply supported beams, L is equal to the span length.

The German code(34) method is also recommended by the 1984 issue of

the CEB–FIP (28) for the design of flat slabs using unbonded tendon.

2.7 Summary: The investigations reviewed in this study summarized in Table. (2.1),

where a general look of equations proposed by researchers from 1963 to 2000

are reviewed. Also in Table. (2.2), important parameters and their frequency in

use of the equations are summarized where recent equations have a trend to take

into account partial prestressing effect, loading type, the span- depth ratio, and

the pattern loading.

Chapter Two Literature Review 5

Reference Date Authors ±

Type of study •

fps equation proposed

Type of prestressing construction used in

experimental ÷

Type of loading≠

Span to depth ratio rang (L/dps)

Tendon profile+

No. of simply

supported beams tested

9 1962 Warwaruk, Sozen, and Siess B Yes B S,T 13.8 – 15.2 S 41 16 1969 Pannell B Yes F S 12 - 40 S 34

10 1971 Mattock, Yamazaki, and Kattula B Yes P U 34 P 6

18 1976 Tam, and Pannell B Yes P S 18 - 43 S 8 19 1978 Burns B No P U 53 S 6 21 1981 Cooke, Park, and Yong B No F T 18 - 38 S 9

34 1982 Elzanaty and Nilson B No P T 21 S, M (P,S) 8

22 1985 Du and Tao E Yes P T 19 S 20 34 1989 Chakrabarti and Whang E No B T 21 DD 8 28 1990 Naaman and AL-Khairi T Yes - - - - - 25 1990 Harajli T Yes - - - - - 26 1991 Harajli and Hijazi T Yes - - - - - 27 1991 Harajli and Kanj B Yes B S,T 8 - 20 S 26 24 1991 Campbell and Chouinard E No B T 15 S 6

29 1994 Chakrabarti, Whang, Brown, Arsad, and Amezeua E No B T 17.1 – 55.2 S 33

30 1995 Chakrabarti T Yes - - - - -

35 1997 Ament, Chakrabarti and Putcha T No - - - - -

3 1999 Lee, Moon and Lim T Yes - - - - - 31 2000 Shdhan E Yes F S,T 10 S 2 2001 Current B Yes B S,T 9 S 10

± In addition to the above studies, fps was evaluated using the ACI, British, AASHTO, Canadian, German Codes, and the Dutch practice. • T = theoretical; E = experimental; B = both. ÷ F = full prestressing with unbonded tendons; P = partially prestressing with unbonded tendons, B= both. ≠ S = single concentrated load at midspan; T = tow-point loading; U = uniform distributed load.

+ M = mixed; DD = double draping point; P = parabolic; S = straight.

Table. 2.1- Characteristics of investigation reviewed in this study

Chapter Tw

o Literature

Review

Chapter Two Literature Review 5

Parameters

Prestressing effects Partial prestressing Member geometry and loading type Design equation

fse ± fse + Aps fpy fpu As fy A’s fc

’ L/dps f ÷ P ≠ U.S (ACI-1963) 3 - - - - - - - - - - - U.S (ACI-1977) 3 - 3 - - - - - 3 - - - U.S (ACI-1999) 3 - 3 - - - - - 3 3 - - Canadian (1984) 3 - 3 3 - 3 3 - 3 3 - - Dutch (1990) 3 - - - - - - - - - - - German (1980) 3 - - - - - - - - 3 - - British (1985) 3 - 3 - 3 - - - 3 3 - - Warwaruk et al. (1962) Eq.(2.7) 3 - 3 - - - - - 3 - - - Pannell (1969) Eq.(2.12) 3 - 3 - - - - - 3 3 - - Mattock (1971) Eq.(2.8) 3 - 3 - - - - - 3 - - - Tam, and Pannell (1976) Eq.(2.13) 3 - 3 - - 3 3 - 3 3 - - Du and Tao (1985) Eq.(2.14) - 3 3 - - 3 3 - 3 - - - Harajli (1990) Eq.(2.15) 3 - 3 - - 3 3 - 3 3 - - Harajli and Hijazi (1991) Eq.(2.16) - 3 3 - 3 3 3 3 3 3 3 3 Harajli and Kanj (1991) Eq.(2.17) - 3 3 - - 3 3 - 3 3 3 - Naaman and AL-Khairi (1990) Eq.(2.18) - 3 3 - - 3 3 3 3 3 3 3

Chakrabarti (1995) Eq.(2.19) - 3 3 - - 3 3 - 3 3 - - Lee, Moon and Lim (1999) Eq.(2.20) - 3 3 - - 3 3 3 3 3 3 3

Shdhan (2000) Eq.(2.21) 3 - 3 3 3 3 3 - 3 3 3 3 Total frequency in use 13 6 16 2 3 10 10 16 3 13 5 14

± = ∆fps not influenced by fse. + = ∆fps influenced by fse. ÷ f = loading type. ≠ P = pattern loading.

Table. 2.2-Parameters and their frequency in use of various design equations

Chapter Tw

o Literature

Review

Chapter Three

Chapter Three Experimental Program

28

3.1 Scope: The tests of ten concrete beam specimens prestressed with unbonded

tendon and reinforced with and without ordinary reinforcing steel are described,

two parameters and their effect on the magnitude of stress in the prestressing

steel fps at nominal flexural strength of the members were examined, these two

parameters are: (1 area of ordinary reinforcement steel and; (2 type of load

application. The stress increase in unbonded tendon ∆fps was measured by using

mechanical (Left–Right) dial gage and mechanical (Demec) strain gage at

various stages of loading. Midspan deflection and laboratory testing was also

measured.

3.2 Introduction: The main factors that may affect the behavior of unbounded partially

prestressed concrete beams are:

1- Amount of prestressed reinforcement.

2- Amount of nonprestressed reinforcement in tension and compression.

3- Material properties.

4- Effective prestressed in tendon immediately before testing.

5- Span / depth ratio.

6- Initial tendon profile.

7- Form of loading.

8- Friction between tendon and duct.

9- The degree of confinement of the concrete in the compression zone.

Only items 2 and 7 were investigated, studying these two parameters will be

superior to Eq. (18-4) and (18-5) of the ACI Building Code (1999) (15), which

despite the effect of ordinary nonprestressed reinforcement and type of load

application.

Chapter Three Experimental Program

29

3.3 Test program:

A total of ten simply supported beam specimens with rectangular cross

section were tested. The beams were divided into three tests Groups (A, B and

C). For each set of Groups A and B, two beams were tested, one beam under

single concentrated load at midspan, and the other beam under two symmetrical

third-point loads. The beams of Group C were tested under two symmetrical

third point loads.

Test procedure and beam specimens were essentially in agreement with the

recommendations of ACI 318M-99 (15).

3.4 Beam specimen cross-section: All test beams (see Fig. 3.1) were approximately (250×350) mm in cross-

section, the span length of the beams were 2320 mm and overall length of the

beams were 2520 mm. The effective depth of the tendon was 260 mm; this gave

a span/depth ratio of 9.

Each beam has one straight unbounded tendon. In addition to unbounded

tendon, Groups B and C contained bonded nonprestressed deformed bars (i.e.,

partially prestressed beams). With ds = 300 mm. Group C also contained

unprestressed compression reinforcement with ds’ = 35mm, this will lead to yield

of compression reinforcement at failure.

This reinforcement was selected on the basis that the beams at failure

would fall into three categories with nonprestressed steel carrying about 20,30 or

50 percent of the total ultimate load. Thus, it was expected that the influence of

the bonded steel on the ultimate stress in unbounded tendons might also be

observed.

Chapter Three Experimental Program

30

Table 3.1- Details of reinforcement of various beam specimens.

Group

± Beam

designation

Type of

loading

÷

Type of

prestressing

Construction

used •

Prestressing

strand ≠

Total bonded

bottom

reinforcement

Total bonded

top

reinforcement

A A-1

A-2

T

S

F

F

1-strand

1-strand

-

-

-

-

B-1

B-2

T

S

PP1

PP1

1-strand

1-strand

2-8φ

2-8φ

-

-

B-3,C-1

B-4

T

S

PP2

PP2

1-strand

1-strand

2-12φ

2-12φ

-

- B

B-5

B-6

T

S

PP3

PP3

1-strand

1-strand

2-16φ

2-16φ

-

-

C

C-2

C-3

T

T

PP2

PP2

1-strand

1-strand

2-12φ

2-12φ

2-8φ

2-12φ

± Group A and B contained 2(6-mm) plain top reinforcement to support shear stirrups. • F = Full prestressing, PP. = Partially prestressing. ÷ S = Single concentrated load at mid span, T= Two point loading. ≠ All Strands were ½- in, (12.7mm), passing Grad 250ksi.

260 mm350 mm

250 mm

300 mm

Fig. 3.1-Beam cross-section details.

Chapter Three Experimental Program

31

3.5 Shear reinforcement details: In order to ensure that flexural failure could occur before shear failure,

shear strength was calculated for all beam specimens and shear reinforcement

was provided in excess of that required.

Shear reinforcement consisted of closed stirrups of 8 mm deformed bars

[cross-sectional area = 50 mm2] with yield tensile strength of 414MPa. Shear

reinforcement details are shown in Fig. 3.2.

Fig. 3.2- Shear reinforcement details.

2320 mm

23-8 mm @ 110 mm spacing

260 mm

2320 mm

12-8 mm @ 90 mm spacing

260 mm

12-8 mm @ 90 mm spacing

(a) Beam under one point load.

(b) Beam under two-point load.

Chapter Three Experimental Program

32

3.6 Materials: Prestressed concrete utilizes high-quality materials (1), namely high-strength

steel and concrete. Some of the most important design characteristics of

materials used will be explained.

3.6.1 Concrete: It was decided to adopt concrete mix produced by AL-Rashid Contracting

Company (R.C.Co). Table (3.2) illustrates composition of concrete mix used in

casting the beams, where all beams were cast in the same day.

Table 3.2 -Concrete mix proportions.

Components Quantities

Cement type I

Sand (0-5 mm)

Aggregate (5-12.7mm)

Water

Superplasticizer (BVD) (0.45% of cement weight)

550 kg/m3

950 kg/m3

1000 kg/m3

200 L /m3

3 L /m3

Water /cement ratio ≈ 0.36

Slump = 60mm

Cylinder strength (150×300) mm (fc’)* = 27 MPa at 7 days

= 38 MPa at 28 days

= 42.5 MPa at 60 days

Density = 2397 kg/m3

* Average of two tests cylinders (150 × 300) mm.

Chapter Three Experimental Program

33

3.6.1.1 Cement: Cement used was Ordinary Portland Cement (type I cement) conforming to

ASTM-C150 (41) and produced in Kufa. Table (3.3) shows the results of cement chemical and physical test done by National Center for Construction Laboratories (NCCL).

Table 3.3- Results of cement chemical and physical test.

Chemical test Physical test

Composition % Test Results

CaO

SiO2

AL2O3

Fe2O3

SO3

MgO

60.4

21.6

5.3

3.0

2.3

3.2

Loss On ignition

Insoluble residuals

Blaine Fineness

Soundness

Initial setting

Final setting

2.3

1.2

285 m2/kg

0.34%

130 min

205 mm

3.6.1.2 Coarse and fine aggregate: Gravel of (12.7mm) maximum size was used, small values of maximum

size of rang of (5-12.7mm) would result in higher compressive strength for high cement content concrete with low water to cement ratio (42). Fineness of sand of 2.5was used. Gradation of coarse and fine aggregate conformed to requirement of ASTM-C33 (41) , the sieve analyses test was done by (NCCL).

Sieve size (mm)

0

20

40

60

80

100

% P

assi

ng

ASTM - C33

0.15 0.30 0.60 1.18 2.36 7.75 9.50

Sieve size (mm)

0

20

40

60

80

100

% P

assi

ng

ASTM - C33

2.36 4.75 9.50 12.50

Fig. 3.3- Gradation of coarse aggregate Fig. 3.4- Gradation of fine aggregate

ASTM–C33 ASTM–C33

Pass

ing

% Pa

ssin

g %

Sieve size (mm) Sieve size (mm)

Chapter Three Experimental Program

34

3.6.1.3 Superplasticizer: Locally available Superplasticizer (BVD) (42) was used, conforming to

ASTM-C494 (41), It was used to have workable concrete with low water cement

ratio.

3.6.2 Prestressing steel: The prestressing strand were all cut from the same ½-in (12.7mm) diameter

coil, laboratory testing of three prestressing specimen was under taken by the

(NCCL) based on ASTM-A416 (41),

The stress-strain behavior of the prestressing strand cannot be idealized as

elastoplastic material (1), so yielding of prestressing strand is not well defined,

Tadros (43) indicates that fpy is close to 0.93(fpu) rather than 0.85 or 0.9(fpu), and

suggest that ASTM-A416 (41) should consider revising the relevant standard to

reflect available experimental data (i.e. ASTM-A416 (41) did not take the full

advantage of available steel capacity), ASTM-A416 (41) standards specify that

the yield stress of prestressing strand should correspond to total strain of one

percent.

The yield, ultimate strength and modulus of elasticity of prestressing strand

were, 1750 MPa, 1882MPa and 192 GPa, respectively. The presstressing strand

were passing Grade (250ksi) Stress-Relieved (41), these stresses are based on the

cross-section area of 92.9 mm2.

3.6.3 Unprestressed reinforcement steel: Deformed bars were used as ordinary nonprestressed reinforcement,

laboratory testing of three specimens for each diameter was under taken by the

(NCCL), the diameter, yield and ultimate strength were respectively; 8 mm,

(414 MPa), (788.2 MPa); 12 mm, (410.8 MPa), (622.3Mpa); 16 mm, (497.1

MPa), (765.5 MPa). All bars had modulus of elasticity about 200 GPa, tests

were conforming to ASTM-A615(41).

Chapter Three Experimental Program

35

3.7 Concrete mixing: The following steps were followed in concrete mixing. Coarse and fine

aggregate were mixed first, then half of total water (without Superplasticizer)

was added, followed by adding cement with continuos mixing, the other half of

water, which was mixed with Superplasticizer from the start was finally added,

concrete mixing continued until a homogenous mix was obtained (41).

3.8 Casting and curing: Nine standard cylinder (150×300) mm and two standard prisms

(100×100×400) mm were cast in steel forms and molds respectively. These

were treated with oil before putting the reinforcement cage or casting control

specimens. Beam specimens casting was done in two lifts. Each lift was

compacted by internal vibration with an electrical vibrator. Beams and control

specimens were removed from their forms and molds, respectively with in 24

hours and continuously wet-cured under water proof polyethylene covers to

keep concrete damp enough during curing. Beams and control specimens were

allowed to dry for at least 15 days before testing. The casting and curing was

carried out at (R.C.Co).

3.9 Post-tensioning operations: The prestressing strand was located in a greased duct, which consisted of

polyethylene tubing of (14.5 mm) internal diameter, complete greasing of the

duct ensured that the post-tensioning process did not have significant frictional

losses, i.e. the tendon force was practically uniform along the beam length (27 ) ,

the applied force at the jacking end was about 0.645 fpu at 112.77 kN (15) this

was identical for all beam specimens, the operation was carried out at (R.C.Co).

Chapter Three Experimental Program

36

3.10 Measurement: 3.10.1 Strain Increase in unbonded tendons:

The tendon force was practically uniform along the beam length (27) , thus

the average strain increase in the prestressing tendon at any load level could be

determined from the elongation of the tendon between the anchorage ends.

Elongation of the prestressing tendon was measured using mechanical (Demec)

strain gages. Demec gage points were placed at spacing of 100 mm from one

end to the other on one side of the beam at the level of the prestressing tendon.

In addition to the strain gages, two dial gages with travel distance of 25 mm and

accuracy of 0.002mm were put on the two ends of the beams, touching the

anchorage plate at the level of the prestressing tendon. This method was

examined to give an approximate value for elongation of the prestressing

tendon.

Fig. 3.5-Measurement instrumentation on beam specimen.

3.10.2 Concrete compression strain: The longitudinal concrete compression strain at the extreme top fiber was

measured using (Demec) strain gage. Demec gage points were placed at a

spacing of 100mm along the length of all beam specimens.

3.10.3 Deflection: Deflection of the beam specimens was at midspan using a dial gage with

travel distance of 30mm and accuracy of 0.01mm.Since the beam specimens are of short span the camber value of all beams was insignificant.

1160 mm 1160 mm

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . 100 mm

Deflection dial gage

90 mm260

Demec gage point

Left dial gage

Right dial gage

Deflection dial gage

Chapter Three Experimental Program

37

3.11 Testing beam specimens: Beam specimens were tested as simply supported beams over 2320 mm

span in 200 kN capacity (Avery) hydraulic machine. Each beam specimen was

supported and loaded by rollers. Forces were distributed through steel bearing

plate 250 mm in length to cover the beam width, to observe crack development,

beams specimens were painted white with emulsion paint before testing. Cracks

were traced by pencil.

Tests started with the application of 10 kN load to set and check dial gages, then

unloading to zero. At zero loading, initi al reading of dials gages and

mechanical strain gages were obtained. The load was applied in

10 to 15 stages. At each loading stage all the dials and strain gage reading were

taken. The interval between two consecutive stages was roughly 5 to 15 minutes.

The overall testing time took on the average 1 to 1.5 hours, depending on the

deformation capacity of the beam tested. The load was continued until failure

(defined as the highest capacity beyond which loading dropped). On removal of

load the beam reverted to near its original undeflected position.

Fig. 3.6 -Type of load application details.

1160 mm 1160 mm

P

773 mm 773 mm 773 mm

2P

2P

Chapter Three Experimental Program

38

3.12 Testing control specimens: Test procedures were essentially in agreement with ASTM requirements.

Results obtained from the test are summarized in Table (3.4).

3.12.1 Compressive strength of concrete: Six standard cylinders (300×150) mm were tested to determine concrete

compressive strength. Tests were according to ASTM-C39 (41). Cylinders were

capped conforming to requirements of ASTM-C617 (41). Failures of cylinders

were sudden and explosive. The maximum concrete compressive strength stress

fc’ was reached in 2 to 3 minutes. The static modulus of elasticity Ec , was

estimated from the Eq. Recommended by ACI Code, which is based on the work

of Pauw,A (1)

Ec= Wc1.5 0.043'

cf MPa …(3.1)

were Wc = Density of concrete mix, in kg/m3

3.12.2 Splitting tensile strength of concrete: Splitting tensile strength was determined by testing three standard cylinders

(300×150) mm conforming to ASTM-C496 (41). These cylinders failed when a

longitudinal crack broke them in two halves. Failure surface was plane and

course aggregates were broken.

3.12.3 Modulus of rupture of concrete: The modulus of rupture is the computed flexural tensile stress at which a

test beam of plain concrete fractures. Two standard prisms (100×100×400) mm

were tested with two point loading to determine modulus rupture of concrete.

Test was according to ASTM-C78 (41). Failure surface was plane and occurred in

the middle third.

Chapter Three Experimental Program

39

Table 3-4 Testing control specimen results.

Specimen fc’

MPa

Ec

GPa

fct

MPa

fr

MPa

Groups A.B and C 42.5 32.897 3.47 3.52

3.13 Summary: Summaries of beam designation and reinforcement areas are given for the

Groups A, B, and C in Table (3.5). This table also includes information relevant

to the experiment such as the effective prestressing steel. And yield stress fy of

the reinforcing steel in tension and compression.

Table 3.5- Summary of reinforcement and strength parameter of various beam

specimens.

Beam

No.

fc’

MPa

Aps

mm2 ρp* 10-3

fpe

MPa ∂

As

mm2

fy

MPa

As’

mm2

fy

MPa

Type of

loading •

A-1 42.5 92.9 1.429 980 - - - - T

A-2 42.5 92.9 1.429 980 - - - - S

B-1 42.5 92.9 1.429 980 98.2 414 - - T

B-2 42.5 92.9 1.429 980 98.2 414 - - S

B-3, C-1 42.5 92.9 1.429 980 233.2 410.8 - - T

B-4 42.5 92.9 1.429 980 233.2 410.8 - - S

B-5 42.5 92.9 1.429 980 407.6 497 - - T

B-6 42.5 92.9 1.429 980 407.6 497 - - S

C-2 42.5 92.9 1.429 980 233.2 410.8 89.2 414 T

C-3 42.5 92.9 1.429 980 233.2 410.8 233.2 410.8 T

∂ The total losses (233 MPa) calculated by using the lump sum method, taken from Reference 44. • T = Two point load, S = Single concentrated load.

Chapter Four

Chapter Four Test Results

40

4.1 Test results: Results obtained from the tests on ten beams are presented in Table 4.1 and

in Fig. 4.1 through 4.4.

4.2 Cracking behavior: At the cracking load, fully prestressed specimen loaded with two third-point

loads developed several simultaneous cracks spread mainly inside the flexural

span. However, as the load increased, only one crack or occasionally two cracks

out of several cracks formed were observed to increase significantly in width and

to propagate upward to the compression zone of the member leading to a

response commonly known in the technical literature as tied arch behavior (10).

A similar behavior was also observed in the fully prestressed specimens tested

under single concentrated load. In these members, the crack that widened most

was exclusive of the first crack formed at or very close to midspan. Because of

the presence of deformed reinforcing bars, the cracks developing in the partially

prestressed specimens increased consistently in width with no sign of

deformation concentrating at a single crack location, as occurred in the fully

prestressed beam, where the area of the ordinary reinforcement in the partially

prestressed specimens varied between a minimum and maximum of 0.002A and

0.009A (Group B), which are cover the minimum of 0.004A specified in the ACI

Building Code (A is the area of the part of the cross section between the flexural

tension face and center of gravity of the gross section)(15). Fig. (4.5) shows the

patterns of flexural cracks after beam failure.

4.3 Load-deflection response: The response of the applied midspan moment versus deflection of beam

specimens is shown in Fig. 4.1. All beams were underreinforced and, hence

Chapter Four Test Results

41

showed signs of yielding before failure. Yielding of the partially prestressed

specimens occurred due to yielding of reinforcing bars. In the fully prestressed

specimens, the yielding behavior was observed partly due to the cracking of the

specimen and partly due to the yielding of the 2(6-mm) plain reinforcement.

Because of the widening of single crack and its fast progression to the member

compression zone as mentioned earlier, the yielding of the fully prestressed

specimens was followed by considerable reduction in stiffness and significant

increase in deflection with increasing load. Despite the formation of a single

major crack, fully prestressed members mobilized a significant amount of post-

elastic deformation prior to failure. However, the trend of decreasing ductility

with increasing amount of tensile reinforcement commonly known in flexural

concrete member was not as obvious in the fully prestressed specimens compared

to the partially prestressed ones. In Fig. 4.1 no significant effect of compression

reinforcement on the behavior of moment-deflection is noticed.

4.4 Stress increase in unbonded tendons: In general there are four methods used to test the prestressing strand

stresses: (1-mechanical (Demec) strain gage; (2 -displacement transducers; (3 -

electric strain gage (Strand force) and (4 -mechanical (Left-Right) strain gage or

(Tendon elongation), methods 1 and 4 are used.

The measured strain and stresses in the unbonded prestressed steel were

below yield for all beam specimens. A summary of the observed ∆f ps results

measured using mechanical (Demec) strain gages and mechanical (left-right) dial

gages (tendon elongation) is given in Table. (4.1). Typical results showing the

variation of stress increases in the prestressing steel with applied midspan

moment are shown in Fig. (4.2) and (4.3) for all beams.

Before cracking, the stress in the prestressing steel showed only a slight

increase with applied load. After cracking, the stresses tend to increase

Chapter Four Test Results

42

significantly at a rate depending on the content of tension reinforcement. In

general, the rate increased as the reinforcing index decreased. It should be

indicated that the stresses in the prestressing steel measured from tendon

elongation being larger than the stresses measured using mechanical (Demec)

strain gages, as shown in Fig. 4.2 and 4.3, two reasons could be observed:

(1-gradual seating of the anchorage that normally accompanies the increase in

stress in the prestressing steel with increasing applied load, where the seating of

the anchorages, particularly for short-span members of the type used in this

investigation, increased the rate of increase of tendon elongation while

simultaneously decreasing the rate of increase of strain measured from strain

gages with applied load, and this was observed by Harajli and Kinj (27);

(2- profile change of the tendon as the beam deflected during testing, and this

was observed by Campbell and Chouinard (24). However, despite some

discrepancies, both methods of stress measurements were constant throughout the

loading history for most beam specimens. In Table (4.1) ∆fps proposed to be the

average of the two readings, (Demec) and (Strand elongation).

One of the primary objectives of the current experimental investigation was

to determine the effect of type of load application on the magnitude of fps at

nominal flexural strength of unbonded members. The results seem to indicate that

the type of load application has significant influence on fps. It can be observed in

Table.(4.1) that the magnitude of ∆fps for members tested under single

concentrated load are not of the same order of magnitude as their sister

specimens tested under two third-point loads i.e. (both types of loading gave ∆fps

results slightly higher than those predicted using a single concentrated load). The

results of this investigation are in contradiction with the analytical observations

made by Harajli and Kanj (27).

* See Comments by HarajlI, M, H., and Hijazi, S, A., PCI JOURNAL. V. 36. No. 5, September-October 1991, pp. 91-95.

Chapter Four Test Results

43

∆fps MPa ω PPR εu

Mu MPa

Mcr kN.m

fps MPa

fse

MPa Average Tendon

elongation Strain gage

(Demec)

Beam No.

0.055 1 0.0019 50.049 25.78 1660.23 980 680.23 713.3 647.16 A-1

0.046 1 0.0030 51.181 31.27 1385.52 980 405.52 488.22 322.82 A-2

0.067 0.761 0.0030 65.216 30.33 1622.29 980 642.29 692.51 592.07 B-1

0.058 0.728 0.0014 56.877 32.17 1364.77 980 384.77 461.88 307.66 B-2

0.082 0.567 0.0030 83.794 37.21 1577.56 980 597.56 644.67 550.45 B-3

0.074 0.524 0.0030 84.174 34.12 1334.36 980 354.36 424.36 284.36 B-4

0.113 0.368 0.0030 95.542 38.75 1491.50 980 511.50 572.5 450.5 B-5

0.106 0.333 0.0030 105.209 39.27 1277.26 980 297.26 377.26 217.26 B-6

0.070 0.580 0.0030 76.211 22.75 1613.21 980 633.21 673.21 593.21 C-2

0.052 0.600 0.0030 74783 18.95 1654.51 980 674.51 754.51 594.51 C-3

Fig. 4.1-Summary of test results

Chapter F

our Test R

esults

Fig. 4.1-Summary of test results.

Chapter Four Test Results 44

Fig 4.1-Experimental observed applied midspan moment versus midspan deflection.

Mid

span

mom

ent (

kN.m

)

0

20

40

60

80

100

0 10 20 30 40 50

A - 1B - 1B - 3B - 5

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

A - 2B - 2B - 4B - 6

0

20

40

60

80

100

0 10 20 30 40 50

C- 1C - 2C- 3

0

20

40

60

80

100

120

0 10 20 30 40 50Midspan deflection (mm)

A - 1B - 1B - 3B - 5A - 2B - 2B - 4B - 6

Chapter Four Test Results 45

Fig. 4.2- Applied midspan moment versus measured stress increase in prestressing steel for beams A-1, A-2, B-1, B-2 and B-3.

Beam A-1

0

10

20

30

40

50

60

-200 0 200 400 600 800

Stress (MPa)

Mom

ent (

kN.m

)

L - RBemec

Beam B-1

0

10

20

30

40

50

60

70

-100 100 300 500 700 900Stress (MPa)

Mom

ent (

kN.m

)

L - RDemec

Beam B-2

0

10

20

30

40

50

60

-200 0 200 400 600Stress (MPa)

Mom

ent (

kN.m

)

L -RDemec

Beam B-3

0

10

20

30

40

50

60

70

80

90

-200 0 200 400 600 800

Stress (MPa)

Mom

ent (

kN.m

)

L -RDemec

Beam A-2

0

10

20

30

40

50

60

0 200 400 600

Stress (MPa)

Mom

ent (

kN.m

)

L - RDemec

Chapter Four Test Results 46

Fig. 4.3- Applied midspan moment versus measured stress increase in prestressing steel for beams B-4, B-5, B-6, C-2 and C-3.

beam B-5

0

20

40

60

80

100

120

-200 0 200 400 600 800

Stress (MPa)

Mom

ent (

kN.m

)

L - RDemec

Beam B-4

0

10

20

30

40

50

60

70

80

90

-200 0 200 400 600

Stress (MPa)

Mom

ent (

kN.m

)

L -RDemec

Beam B-6

0

20

40

60

80

100

120

-200 0 200 400

Stress (MPa)

Mom

ent (

kN.m

)

L- RDemec

Beam C-2

0

10

20

30

40

50

60

70

80

90

-200 0 200 400 600 800

Stress (MPa)

Mom

ent (

kN.m

)

L - RDemec

Beam C-3

0

10

20

30

40

50

60

70

80

0 200 400 600 800

Stress (MPa)

Mom

ent (

kN.m

)

L - RDemec

Chapter Four Test Results 47

0

20

40

60

80

100

120

-100 0 100 200 300 400 500 600

A - 2B - 2B - 4B - 6

0

20

40

60

80

100

120

-200 0 200 400 600 800 1000

A - 1B - 1B - 3B - 5

0

20

40

60

80

100

120

-100 100 300 500 700 900

Stress (MPa)

A - 1B - 1B - 3B - 5A - 2B - 2B - 4B - 6

0102030405060708090

-200 0 200 400 600 800

C - 1C - 2C - 3

Mid

span

mom

ent (

kN.m

)

Fig. 4.4-Applied midspan moment versus measured stress increased stress in

prestressing steel.

Chapter Four Test Results 48

]

A-1

A-2

B-1

B-2

B-3

B-4

Fig. 4.5-Crack patterns at ultimate load for beams A-1, A-2, B-1, B-2, B-3 and B-4.

Chapter Four Test Results 49

B-5

B-6

C-2

C-3

Fig. 4.6-Crack patterns at ultimate load for beams B-5, B-6, C-2 and C-3.

Chapter Five

Chapter Five Proposed Equation 50

5.1 Scope:

In this Chapter a statistical analysis is performed that compares

experimental value of ∆fps and fps to predicted values of ∆fps and fps. The work

reported in this Chapter is along the same lines as that performed by researchers

in the area of probabilistic analysis reported elsewhere (35), (40).

5.2 Introduction: Many statistical parameters that are calculated contain both the

experimental and predicted values of ∆fps and fps. The following notation is

utilized throughout this study.

∆fpse = experimental value of ∆fps

∆fpsp = predicted value of ∆fps

∆fps = general ∆fps (not specifically experimental or predicted)

fpse = experimental value of fps

fpsp = predicted value of fps

fps = general fps (not specifically experimental or predicted)

To perform this statistical study a collection of experimental data that

include the measured values of ∆fps and fps was required. It was also required

that as many data samples as possible from as many different sources as possible

be found and included in the study. For this effect a wide range of concrete

member characteristics are taken into account, which include rectangular and T-

beams, slabs with various sizes, lengths, amounts, locations, and strength of

prestressed and mild steel, concrete strength, span-to-depth ratios, and partial

Chapter Five Proposed Equation 51

and full prestressing. The following methods are used to compare the models

discussed previously.

5.2.1 Mean values and standard deviations, method 1: A first and simplest step to making a comparison of data samples of

predicted values is to compare mean values and standard deviations. The mean

value represents the average or central tendency of all the data samples, and the

standard deviation represents the spread or concentration of the population of

the data samples (38). The standard deviation of the experimental values is

referred to as the uncertainty prior to regression (36).

Both the mean value (Y ), comparing the model’s central tendency to the

experimental central tendency, and the standard deviation ( Sy ), comparing the

spread of data, calculated for both predictor and experimental data (39), are

calculated for both predicted and experimental data.

5.2.2 Correlation coefficient, method 2: The correlation coefficient is a statistical quantity that indicates the

“goodness” of the fit of an equation. This is done by comparing the sum of

squares about regression Sr , also referred to as the sum of squares of the

residuals, and the sum of squares about the mean St .The correlation coefficient

r is given by (38):

r = [( St – Sr )/ St ]1/2 …(5.1)

for ∆fps r = [Σ ( ∆fpspi - ∆fpse )2 / Σ ( ∆fpsei - ∆fpse )2 ]1/2

Sr = Σ ( ∆fpsei - ∆fpspi )2 …(5.2.a)

St = Σ ( ∆fpsei - ∆fpse )2 …(5.2.b)

Chapter Five Proposed Equation 52

for fps r = [Σ ( fpspi - fpse )2 / Σ ( fpsei - fpse )2 ]1/2

Sr = Σ ( fpsei - fpspi )2 …(5.3.a)

St = Σ ( fpsei - fpse )2 …(5.3.b)

where

∆fpse and fpse = the mean of the experimental values of ∆fps and fps respectively.

∆fpsei and fpsei = individual experimental value of ∆fps and fps respectively.

∆fpspi and ,fpspi = individual predicted value of ∆fps and fps respectively.

An alternative formulation for r that is more convenient for computer

implementation is:

2222 )()())((

iiii

iiii

yynxxnyxyxnr

∑−∑∑−∑

∑∑−∑= ... (5.4)

For a perfect fit, that is a model that predicts every value with exact correctness,

∆fpsei=∆fpspi or fpsei=fpspi, therefore Sr =0, then r= (St/ St)1/2 = 1. This

indicates that the model explains 100 percent of the variability of the data. The

value of the correlation coefficient is typically between 0 and 1. However, If Sr

is greater than St the correlation coefficient is undefined due to the square root

of a negative number. If this happens. It indicates that the model has more

variability than the experimental data or the model explains less than 0 percent

of the variability (36), (37) .

Chapter Five Proposed Equation 53

5.2.3 Error analysis, method 3: The error or residual is the absolute amount that the predicated value is off

from the actual value. For a perfect model, the error, at all predictions, should be

zero. The error is given by (39):

Error of ∆fps = ∆fpse-∆ fpsp … (5.5.a)

Error of fps = fpse - fpsp … (5.5.b)

From this type of analysis a comparison of the mean errors can be made to

indicate which model is most accurate by yielding the smallest mean error. Plots

of the predicted value of ∆fps and fps versus the error can be utilized to indicate

any trends in the error with respect to the predicted value, and the model could

be made to better correct this trend in error (39). This graphical representation

also indicates the absolute errors at each of the (120) data points and can

visually indicate the predictor equation with the least error at all points.

5.2.4 Standard error of estimate, method 4: The next statistic to be measured is the standard error of estimate, which

is given by: (36), (37)

Sy/x = [ Sr / (n – 2 ) ] 1/2 …(5.6)

for ∆fps = Σ( ∆fpsei - ∆fpse )2 / (n – 2 ) 1/2

for fps = Σ( fpsei - fpse )2 / (n – 2 ) 1/2

the subscript notation “y/x” designates that the error is for predicted

value of y corresponding to a particular value of x.

Chapter Five Proposed Equation 54

This statistic is the spread, or standard deviation, about the regression

line defined by the model being analyzed. This value is also referred to as the

uncertainty that remains after regression (36). This value should be as small as

possible, and should be smaller than standard deviation of the experimental data.

The reason for comparing the standard error of estimate to the standard

deviation of the experimental data is to insure that there is less uncertainty after

regression than before, which indicates that the model provides an improvement

in the fit of the data (39). Much like the error analysis that provided a graphical

indication of the spread of individual data points, the standard error of estimate

provides a single numeric value indicating the spread of the entire population.

5.2.5 Frequency distribution, method 5: In the frequency distribution analysis a grouping of the data samples is

made and the number of samples in each group is plotted. This data is plotted as

a cumulative frequency graph, which indicates the percentage of samples that

are in the particular grouping or less (39). Form this type of analysis a

determination of over/underestimation can be made. A prediction of the

probability of a value less than a certain presaged value can be also made. Both

of these indications have to be made by comparing the predictor model plots

against the experimental data plots.

Chapter Five Proposed Equation 55

5.3 Proposed design equation: Statistical analysis of combined experimental results has led to proposed

Eq. (5.7). This equation is intended to present a prediction of the tendon stress at

ultimate fps

fps= ∆fps + fpe

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

++−=ps

ys's

4pu3pe21

'c

p1pups A

fAAff

fff ααα

βρ

αγ∆ MPa …(5.7)

⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

nn

d/S1

f1 o

p

γ

with the limitation that fps < fpe + 650, and fps < 0.95 fpy

by regression analysis with previous test data, the coefficients in Eq.(5.7) were

determined as in Table (5.1).

Table 5.1 Summary of coefficients used in the proposed Eq. (5.7).

f

Approach 1-point

loading

Tow-point loading

or uniform loading 1α 2α 3α 4α

I ∞ 32 1.23 1.59 0.44 0.64

II ∞ 64 1.20 1.50 0.75 1.50

III ∞ 64 1.15 1.44 0.72 1.44

IV ∞ 32 0.95 1.6 0.43 0.82

were γ is a coefficient representing the combined influence of the type of load application with the span-depth ratio on fps,(plastic hinge length ratio); f = load geometry factor; (no/n) represent the ratio of the length of loaded spans to the total length of the member between the anchorage ends, for simply supported members no/n =1; pρ = ratio of prestressing steel (Aps/b dp); 1β = ACI concrete compression block reduction factor; 1α , 2α , 3α and 4α are constant coefficients; Aps, As and As

’ = area of prestressed, nonprestressed tension and compression

reinforcement, respectively; fpe = effective prestress applied; fpu = measured value of ultimate strength in the tendons; fpy and fy = yielding strength of prestressed and nonprestressed reinforcement, respectively.

Chapter Five Proposed Equation 56

5.4 Characteristics of proposed design equation: The proposed design equation has five characteristics:

First, the proposed design equation considers the effective prestress as a

parameter, compared to many other equations such as ACI 318M-1999 equation

which does not recognize the effect of fse on the tendon stress increment. It

means that the higher the effective stresses, the higher the ultimate stress.

Previous test results, however, showed that ∆fps decreases if the value of fse

increases.(3),(30).

Second, the proposed equation can accurately compute the fps of the

unbonded tendon in the partially prestressed member. The ACI 318M-1999 as

example cannot take into account the bonded reinforcement. Also the moment

equilibrium equation illustrates that the area of bonded reinforcements affects

the equilibrium state of the section. It, in turn, affects the stress of the tendon.

Third, the plastic hinge length ratio (γ ) is considered to be an important

parameter, the loading type and the span-depth ratio together influence it, and in

form of square root. Also the parameter (γ ) accounts for the member span ratio

in uniform instead of independently, as in the ACI 318M-1999 code; hence, it

eliminates the discontinuity in the stress level at limiting span-depth ratio (15),(30).

Fourth, the proposed equation extends to include both simple and

continuous members by using (no/n) factor, the proposed equation can predict it

with the concept of plastic hinge length (26).

Finally, It can be observed in Table (5.1) that four Approaches were

proposed to predict the characteristic coefficients ( 321 ,,,, αααγ f and 4α ) used in

Eq. (5.7). Using these approaches is for statistical purpose. In general the

existing equations are designed to lead to safe results with overestimation of ∆fps

and fps equal to zero. However, these equations usually gave a poor correlation

with very large scatter between predicted and experimental data, and this is

Chapter Five Proposed Equation 57

statistically undesirable (35), (38), (39). Naaman and Al-Khairi (28) and Shdhan (31)

recognized this behavior, the question was raised as to whether balancing

between safe results with a small overestimation and good correlation with small

scatter between predicted and experimental data could be found. These four

approaches were designed to answer that question, and as follow:

Approach I- the concentration in this approach was on correlation

coefficient, using this approach will make Eq. (5.7) give the largest correlation

coefficient between the predicted values (∆fpsp) and (fpsp) to the experimental

values (∆fpse) and (fpse) respectively than any other existing equation.

Approach II- the concentration in this approach was to make Eq. (5.7) to

have error of ∆fps ; i.e. ( ∆fpse - ∆fpsp ≈ 0 ) and of fps ; i.e. ( fpse - fpsp ≈ 0 )

respectively and at the same time the mean of predicted values ( ∆fpsp ) and ( fpsp )

provide closest estimate of the mean of experimental values ( ∆fpse ) and ( fpse )

respectively.

Approach III- This approach like approach II with a little modification,

where more attention was provided to make Eq. (5.7) have a direct image of the

experimental cumulative frequency distribution than any other existing equation.

Approach IV- a general balancing between safe results with a small

overestimation and good correlation between predicted and experimental data

are provided, statistically this is very good and wanted (38), (39).

The four approaches will be examined statistically and compared to the

other design equations, in Chapter 6.

Chapter Six

Chapter Six Statistical Analysis 58

6.1 Experimental data analysis: The experimental results obtained in this study were combined with 110

test results collected from technical literature. These were taken from eight

different experimental investigations that perform between 1971 and 2000. All

these members were flexure critical - i.e., failing in flexure. The combined

experimental results include those of simply supported beams, reinforced with

and without additional bonded nonprestressed reinforcement. These have a wide

range of member span-to-depth ratios varying between 8 and 55, which covers

current practical ranges for most beams and slabs. Thus, the combined

experimental results truly represent a comprehensive and representative sample.

Table (6.1) indicates some details of 120 specimens considered in this work.

Data from the investigations listed in Table (6.1) were extracted, and stored in a

database. Particular attention was paid to storing information related to the

observed stress at ultimate in the unbonded tendons. In collecting data from the

available literature, care was taken in interpreting the experimental results. For

example, the data collected from Reference (14) was analyzed assuming that the

cylinder strength is equal to 0.82 of the cube strength (29), also value for ε cu and

Eps equal to 0.003 and 193060 MPa, respectively, was assumed when no

information was given by authors regarding these parameters (29), In addition,

limitations on fps using the various models presented were all taken into account.

In particular, any beam having a value of fpe < 0.5 fpu was excluded from the

analysis of the ACI Building Code equations. Furthermore, the limits imposed

on all models were taken into account to insure fairness when comparing the

proposed equations to the others.

Chapter Six Statistical Analysis 59

No.

of

mem

bers

te

sted

6 8 9 20

6 26

33

2 10

Tend

on

prof

ile

Para

bolic

Stra

ight

Stra

ight

Stra

ight

Stra

ight

Stra

ight

Dou

ble

harp

Stra

ight

Stra

ight

Type

of

pres

tress

ing

cons

truct

ion

Parti

al

Parti

al

Fully

Parti

al

Mix

ed

Parti

al

Mix

ed

Fully

Parti

al

Type

of l

oad

appl

icat

ion

Four

–po

int

One

–po

int

Third

–po

int

Third

–po

int

Third

–po

int

Third

–po

int

Third

–po

int

Mix

ed

Mix

ed

L/d p

s

Ran

ge

34

18 –

43

18 –

38

19

15

8 –2

0

18 –

55

10

9

Mem

ber

Con

tinui

ty

Mix

ed

Sim

ple

Sim

ple

Sim

ple

Sim

ple

Sim

ple

Sim

ple

Sim

ple

Sim

ple

Type

of

mem

ber

Bea

m

Bea

m

Slab

Bea

m

Bea

m

Bea

m

Mix

ed

Bea

m

Bea

m

Ref

eren

ce

Mat

tock

et a

l (6)

Tam

and

Pan

nell (1

4)

Coo

k et

al (1

7)

Du

and

Tao (1

8)

Cam

pbel

l and

Cho

uina

rd (1

9)

Har

ajli

and

Kan

ji (20)

Cha

Kra

barti

(21)

Shd

han

(31)

Cur

rent

Dat

e

1971

1976

1981

1985

1991

1991

1995

2000

2001

Tabl

e 6.

1 C

hara

cter

istic

s of e

xper

imen

tal i

nves

tigat

ions

con

side

red

in th

is st

udy

Chapter Six Statistical Analysis 60

6.2 Evaluation of existing prediction equation:

The accuracy of different existing prediction equations, as well as the

equations recommended in major design codes to predict fps in unbonded

tendons at ultimate, was evaluated and compared to the proposed prediction

equations. This was achieved, for each prediction equation, by comparing

experimental and predicted results of ∆fps and fps, and computing the

corresponding coefficient of correlation.

6.2.1 Mean values and standard deviations, method 1:

A comparison of the mean values and standard deviations of ∆fps and fps

listed in Table (6.2), calculated by the predictor equations (∆fpsp) and (fpsp) to the

experimental values (∆fpse) and (fpse) respectively, indicates that the German

code (34), Du/Tao (22), Harajli/Hijazi (26), Naaman/AL-Khairi(28) and the

proposed equation [Eq.(5.7) approach I ] equations tends to overestimate the

value of ∆fps by (337.95, 157.66, 170.41 ,197.65 and 96.22 MPa) respectively

and fps by (110.01, 107.60, 63.24, 40.844 and 64.39 MPa) respectively, while the

other equations are extremely conservative and underestimate the values of ∆fps

and fps [see Table.(6.2)]. The proposed equation [Eq. (5.7) approach II ], while

also conservative, provides the closet estimate of ∆fps and fps with an

underestimation of (2.016 and 9.864 MPa) respectively.

The standard deviations of all predictor equations are very similar to the

standard deviation of the experimental data. This indicates that all 18 predictor

equations are estimating the data with the same spread of values as those

provided by the experimental data. Therefore, all mean values generated and

compared previously seem reasonable and can be accepted as valid measures of

the central tendencies of the predictor equations.

Chapter Six Statistical Analysis 61

6.2.2 Correlation coefficient, method 2: To check the accuracy of each prediction equation reviewed, two graphs

were prepared for each equation: one compared the predicted value of fps at

ultimate with the experimentally observed value of fps, and the other compared

the predicted change in stress ∆fps beyond the effective prestress with

experimentally observed change in stress. The line of perfect correlation was

also plotted. Note that the second graph is much more representative of the

accuracy of a given equation, since the value of ∆fps is generally small when

compared to the value of fpe, and the value of fpe is assumed given. Thus, the

influence of ∆fps will appear minimized when only fps is analyzed. The correlation coefficient for each predicted equation is listed in Table 6.2.

Fig. (6.1.a) and (6.1.b) describes the results obtained in evaluating

Warwaruk et al (9) [Eq. (2.7)]. It can be observed that while the prediction of

∆fps and fps is generally on the safe side, the results show a poor correlation,

Fig. (6.2.a) and (6.2.b) describes the results obtained in evaluating ACI

318-77 (12) [Eq. (2.9)]. It can be indeed observed that while the correlation

between predicted and observed values of fps at ultimate is reasonable, the

correlation for ∆fps is quite poor.

Fig. (6.3.a) and (6.3.b) describes the results obtained in evaluating ACI

318M-99 (15) [Eq. (2.9) and (2.10)]. Here also the correlation between predicted

and observed values of fps at ultimate is reasonable; the correlation for ∆fps is

quite poor.

Fig. (6.4.a) and (6.4.b) describes the results obtained in evaluating

Canadian code (32) [Eq. (2.22)]. While the predictions of fps are generally on the

safe side, not only large scatter but also inconsistent results may be obtained for

∆fps, where negative values could be predicted.

Chapter Six Statistical Analysis 62

Fig. (6.5.a) and (6.5.b) describes the results obtained in evaluating British

code (17) [Eq. (2.23)]. While there is a large scatter when experimental and

predicted values of ∆fps are compared, the correlation between predicted and

observed values of fps at ultimate is approximately reasonable.

Fig. (6.6.a) illustrates how conservative the Dutch code (34) [Eq. (2.24)] can

be. Fig (6.6.b) is also is quite poor for predictions fps at ultimate.

Fig. (6.7.a) and (6.7.b) show the results obtained in evaluating German

code (34) [Eq. (2.25)]. Here, not only the correlation between predicted and

observed results is poor, but also the predicted values of ∆fps and fps are generally

larger than experimentally observed result, thus on the unsafe side.

Fig. (6.8.a) and (6.8.b) show the results obtained in evaluating Du/Tao (22)

[Eq. (2.14)]. Here not only poor correlation but also unsafe predictions are

observed.

Fig. (6.9.a) and (6.9.b) show the results obtained in evaluating Harajli-1 (25)

[Eq. (2.15)]. It can be observed that while the correlation between predicted and

observed values of fps at ultimate is reasonable, the correlation for ∆fps is quite

poor with a large scatter.

Fig. (6.10.a) and (6.10.b) show the results obtained in evaluating Harajli-2

(26) [Eq. (2.16)]. While the correlation between predicted and observed values of

∆fps and fps are quite good, most of them are on the unsafe side.

Fig. (6.11.a) and (6.11.b) show the results obtained in evaluating Harajli-3

(26) [Eq. (2.16)]. While most of the predicted and observed values of ∆fps and fps

are on the safe side, poor correlation between predicted and observed values are

observed. It can be seen that Harajli-3 (26) model provides better estimates than

Harajli-1 (25).

Fig. (6.12.a) and (6.12.b) describes the results obtained in evaluating

Naaman/AL-Khairi (28) [Eq. (2.18)]. It can be observed that while the

correlation between predicted and observed values of fps at ultimate is

Chapter Six Statistical Analysis 63

reasonable, the correlation for ∆fps is quite poor with large scatter and unsafe

prediction.

Fig. (6.13.a) and (6.13.b) describes the results obtained in evaluating

Chakrabarti (29) [Eq. (2.19)]. It can be indeed observed that while the

correlation between predicted and observed values of fps at ultimate is

reasonable, the correlation for ∆fps is quite poor, also a very large scatter is

observed.

Fig. (6.14.a) and (6.14.b) describes the results obtained in evaluating Lee

at el (3) [Eq. (2.20)]. The correlation between predicted and observed values of

fps at ultimate is reasonable, the correlation for ∆fps is quite poor, with a large

scatter.

Fig. (6.15.a) and (6.15.b) describes the results obtained in evaluating

Shdhan (31) [Eq. (2.19)]. The correlation between predicted and observed values

of fps at ultimate is reasonable with a good accuracy; the correlation for ∆fps is

quite poor, with a very large scatter.

Fig. (6.16.a) and (6.16.b) describes the results obtained in evaluating

[Eq.(5.7) approach I ]. While The correlation between predicted and observed

values of fps and ∆fps provides the smallest scatter in the comparison when

compared to any other existing method, most of the predicted and observed

values of fps and ∆fps are in the unsafe side.

Fig. (6.17.a) and (6.17.b) describes the results obtained in evaluating

[Eq.(5.7) approach II ]. Here a good correlation but also unsafe predictions are

observed.

Fig. (6.18.a) and (6.18.b) describes the results obtained in evaluating

[Eq.(5.7) approach III ]. It is easy to see that a general balancing between a

good correlation and safe prediction is provided.

Fig. (6.19.a) and (6.19.b) describes the results obtained in evaluating

[Eq.(5.7) approach IV ]. This approach like approach III except to note that

Chapter Six Statistical Analysis 64

more attention was provided to increase the correlation coefficient and percent

of the safe prediction. Compared to the previous equation this model is

reasonable with a good accuracy value to represent the experimental data.

In summary, this statistic indicates that the two proposed [Eq. (5.7)

approach III and IV ] models is the best fit of the experimental data.

6.2.3 Error analysis, method 3: The figure of Warwaruk at el (9) (Fig. 6.1.c) indicates that only 7 of 120

data values were overestimated, with the largest overestimate being 84.79 MPa

and a mean value of the overestimates being 31.93 MPa, this graph does not

indicate any trend in the data. (Fig. 6.1.d) indicates that only 5 of 120 data

values were overestimated, with the largest overestimate being 84.79 MPa and a

mean value of the overestimates being 37.06 MPa. Also this graph does not

indicate any trend in the data. In general this equation has middle conservatism

compared with other equations.

The two figures of ACI 318-77 (12) [Fig. (6.2.c) and (6.2.d)] indicates that

18 of 120 data values were overestimated, with the largest overestimate being

146 MPa and a mean value of the overestimates being 57.886 MP. Compared to

the Warwaruk at el(9) [Fig.(6.1.c) and (6.1.d)] a small improvement can be

noted, where the mean of ∆fps and fps rose from 171.6 to 250 MPa and from

1150.8 to 1267.94 MPa provides closest estimate to ∆fpse and fpse respectively.

The figure of ACI 318M-1999 (15) (Fig. 6.3.c) where the effect of the

span-depth ratio considered as parameter in [Eq. (2.9) and (2.10)] indicates that

32 of 120 data values were overestimated, with the largest overestimate being

309 MPa and a mean value of the overestimates being 90.96 MPa, . (Fig. 6.3.d)

indicates that 43 of 120 data values were overestimated, with the largest

overestimate being 674 MPa and a mean value of the overestimates being

238.52 MPa. Compared to the ACI 318-77(12) [Fig.(6.2.c) and (6.2.d)] the mean

Chapter Six Statistical Analysis 65

of ∆fps and fps decreased from 250.30 to 241.76 MPa and from 1267.94 to 1259.4

MPa respectively, also the mean value of the overestimates of ∆fps and fps

increased from 57.88 to 90.96 MPa and from 57.163 to 238.53 MPa

respectively. This means that no improvement can be noted. This result was

expected (26).

The figure of Canadian code (32) (Fig. 6.4.c) indicates that only 8 of 120

data values were overestimated, with the largest overestimate being 296.87 MPa

and a mean value of the overestimates being 117.66 MPa. Inconsistent results

may be obtained for ∆fpsp, where negative values could be predicted. (Fig. 6.4.d)

indicates that only 8 of 120 data values were overestimated, with the largest

overestimate being 201 MPa and a mean value of the overestimates being 88.43

MPa, verifying that this equation is very like the ACI 318M-1999 (15). Also

largest overestimation appear at higher values of ∆fpsp and fpsp respectively.

The figure of British code (17) (Fig. 6.5.c) indicates that 53 of 120 data

values were overestimated, with the largest overestimate being 466.58 MPa and

a mean value of the overestimates being 110.26 MPa, the error seems slightly

smaller at lower values of ∆fpsp. (Fig. 6.5.d) indicates that 53 of 120 data values

were overestimated, with the largest overestimate being 96.62 MPa and a mean

value of the overestimates being 35.624 MPa. This graph does not indicate any

trend in the data. In general this equation is very conservative like the Canadian

code (32).

The figures of Dutch code (34) [Fig. (6.6.c) and (6.6.d)] indicate no sign of

overestimation, with all the predicted values on the safe side, verifying that this

equation is extremely conservative. The two figures does not indicate any trend

in the data.

The figure of German code (34) (Fig. 6.7.c) indicates that 106 of 120 data

values were overestimated, with the largest overestimate being 1827.79 MPa

and a mean value of the overestimates being 391.6 MPa. Inconsistent results,

Chapter Six Statistical Analysis 66

(Fig. 6.7.d) indicates that 100 of 120 data values were overestimated, with the

largest overestimate being 468 MPa and a mean value of the overestimates being

146.55 MPa.

This is not acceptable because these 100 concrete members have reached

their ultimate strength before the predicted ultimate stress in the tendon has been

reached. Therefore, designing with this prediction would tend to make an

underdesigned member and could cause failures. In (Fig. 6.7.c) the error seems

slightly smaller at lower values of ∆fpsp, (Fig. 6.7.d) does not indicates any trend

in the data.

The figures of Du and Tao (22) [Fig. (6.8.c) and (6.8.d)] like the German

code (34) [Fig. (6.7.c) and (6.7.d)] have the same problem. (Fig. 6.8.c) indicates

that 101 of 120 data values were overestimated, with the largest overestimate

being 535.84 MPa and a mean value of the overestimates being 194.41 MPa,

(Fig. 6.8.d) indicates that 94 of 120 data values were overestimated, with the

largest overestimate being 464.83 MPa and a mean value of the overestimates

being 157.7 MPa. The two figures does not indicates any trend in the data.

The figure of Harajli-1 (25) (Fig. 6.9.c) indicates that only 14 of 120 data

values were overestimated, with the largest overestimate being 148 MPa and a

mean value of the overestimates being 62.07 MPa. This graph does not indicate

any trend in the data. (Fig. 6.9.d) indicates that only 13 of 120 data values were

overestimated, with the largest overestimate being 144.37 MPa and a mean

value of the overestimates being 58.46 MPa, Also this graph does not indicate

any trend in the data. In general this equation like the Warwaruk at el (9)

equation was middle conservative.

The figure of Harajli-2 (26) (Fig. 6.10.c) indicates that 94 of 120 data

values were overestimated, with the largest overestimate being 543.55 MPa and

a mean value of the overestimates being 229.63 MPa. As far as data trend, a

slight trend can be seen; that is, the error slightly smaller at lower values of ∆fpsp.

This indicates that this equation is better at predicting the value of ∆fps when

Chapter Six Statistical Analysis 67

∆fpsp is between 58 and 370 MPa than it is when ∆fpsp is over 380 MPa. (Fig.

6.10.d) indicates that 82 of 120 data values were overestimated, with the largest

overestimate being 411 MPa and a mean value of the overestimates being

113.44 MPa. Also as far as data trend, a slight trend can be seen; that is, the

error slightly smaller at lower values of fpsp. This indicates that this equation is

better at predicting the value of fps when fpsp is between 725 and 1535 MPa.

The figure of Harajli-3 (27) (Fig. 6.11.c) indicates that 30 of 120 data values

were overestimated, with the largest overestimate being 150.55 MPa and a mean

value of the overestimates being 49.54 MPa. (Fig. 6.11.d) indicates that only 30

of 120 data values were overestimated, with the largest overestimate being

150.55 MPa and a mean value of the overestimates being 49.54 MPa. This

equation is not as conservative as the Harajli-1 (25) and Harajli-2 (26)

respectively. There is no classic trend in the data except to note that most

overestimations occur at higher values of ∆fpsp and fpsp respectively.

The figure of Naaman/AL-Khairi (28) (Fig. 6.12.c) indicates that 90 of

120 data values were overestimated, with the largest overestimate being 1304.7

MPa and a mean value of the overestimates being 299.23 MPa, inconsistent

results may be obtained for ∆fpsp. It should be noted that most overestimation

occurs at higher values of ∆fpsp, and increased gradually. (Fig. 6.12.d) indicates

that 82 of 120 data values were overestimated, with the largest overestimate

being 355 MPa and a mean value of the overestimates being 104.21 MPa. The

figure does not indicates any trend in the data.

The figure of Chakrabarti (29) (Fig. 6.13.c) indicates that 41 of 120 data

values were overestimated, with the largest overestimate being 190 MPa and a

mean value of the overestimates being 77.76 MPa. (Fig. 6.13.d) indicates that 40

of 120 data values were overestimated, with the largest overestimate being 190

MPa and a mean value of the overestimates being 75.9 MPa. The two figures do

not indicate any trend in the data.

Chapter Six Statistical Analysis 68

The figure of Lee at el (3) (Fig. 6.14.c) indicates that only 20 of 120 data

values were overestimated, with the largest overestimate being 133 MPa and a

mean value of the overestimates being 52.53 MPa. (Fig. 6.14.d) indicates that

only 20 of 120 data values were overestimated, with the largest overestimate

being 133 MPa and a mean value of the overestimates being 52.53 MPa. This

equation is not conservative as some other models in its estimation of ∆fpsp and

fpsp respectively. As far as data trend, a slight trend can be seen; that is, the error

is slightly smaller at higher values of ∆fpsp and fpsp respectively.

The figure of Shdhan (31) (Fig. 6.15.c) indicates that 38 of 120 data values

were overestimated, with the largest overestimate being 300.2 MPa and a mean

value of the overestimates being 95.5 MPa. (Fig. 6.15.d) indicates that 40 of 120

data values were overestimated, with the largest overestimate being 190 MPa

and a mean value of the overestimates being 75.9 MPa. It should be noted that

most overestimation occurs at higher values of ∆fpsp and fpsp respectively.

The figure of proposed [Eq.(5.7) approach I ]; (Fig. 6.16.c) indicates that

101 of 120 data values were overestimated, with the largest overestimate being

345.7 MPa and a mean value of the overestimates being 120.98 MPa. (Fig.

6.16.d) indicates that 92 of 120 data values were overestimated, with the largest

overestimate being 297.6 MPa and a mean value of the overestimates being

95.927 MPa. It can be seen that this proposed equation is very conservative, the

design with this prediction will lead to under design members and could cause

failures. The two figures do not indicate any trend in the data.

The figure of proposed [Eq. (5.7) approach II ]; (Fig. 6.17.c) indicates

that 58 of 120 data values were overestimated, with the largest overestimate

being 258.64 MPa and a mean value of the overestimates being 81.91 MPa.

(Fig. 6.17.d) indicates that 56 of 120 data values were overestimated, with the

largest overestimate being 209.1 MPa and a mean value of the overestimates

being 66.77 MPa. Compared to the approach I a small improvement can be

noted, with the mean of ∆fps and fps reduced from 474.6 to 368.9 MPa and from

Chapter Six Statistical Analysis 69

1453.09 to 1378.84 MPa provides closest estimate to ∆fpse and fpse respectively.

Also the mean value of the overestimates of ∆fps and fps reduced from 120.98 to

81.91 MPa and from 95.927 to 66.77 MPa respectively making this equation not

as conservative as the approach I equation. There is no classic trend in the data.

The figure of proposed [Eq. (5.7) approach III ];(Fig. 6.18.c) indicates

that 52 of 120 data values were overestimated, with the largest overestimate

being 236.41 MPa and a mean value of the overestimates being 72.268 MPa.

(Fig. 6.18.d) indicates that 38 of 120 data values were overestimated, with the

largest overestimate being 190.25 MPa and a mean value of the overestimates

being 44.725 MPa. Compared to the approach II an improvement can be noted,

with mean value of the overestimates of ∆fps and fps reduced from 81.91 to 72.26

MPa and from 66.77 to 44.725 MPa respectively making this equation not as

conservative as the approach II equation. Statistically this is good, but as

previously discussed overestimation of ∆fps and fps should not occur. There is no

classic trend in the data.

The figure of proposed [Eq. (5.7) approach IV ];(Fig. 6.19.c) indicates that

37 of 120 data values were overestimated, with the largest overestimate being

200.06 MPa and a mean value of the overestimates being 68.63 MPa. (Fig.

6.19.d) indicates that 25 of 120 data values were overestimated, with the largest

overestimate being 149.45 MPa and a mean value of the overestimates being

40.451 MPa. Compared to the approach III an improvement can be noted, with

mean value of the overestimates of ∆fps and fps reduced from 72.26 to 68.63 MPa

and from 44.725 to 40.45 MPa respectively making this equation not as

conservative as the approach II equation.

In summary, this statistic indicates that the two proposed [Eq. (5.7)

approach III and IV ] models are the best fit of the experimental data.

Chapter Six Statistical Analysis 70

6.2.4 Standard error of estimate, method 4: A comparison of the standard error of estimate and standard deviations

of ∆fps and fps listed in Table (6.2). When looking at the standard error of

estimate of ∆fps the German code (34) and Naaman/AL-Khairi (28) models

provides no improvement over the spread of the experimental data with respect

to the mean (S y/x>Sy), while the other models has (S y/x<Sy) providing an

improvement. When looking at the standard error of estimate of fps all the

models has (S y/x<Sy) providing an improvement.

Chapter Six Statistical Analysis 73

fps ∆fps

Standard deviation

MPa

Standard error of estimate

MPa

Correlation coefficient

Delta mean fps

MPa Mean fps

MPa Standard deviation

MPa

Standard error of estimate

MPa

Correlation coefficient

Delta mean ∆fps MPa

Mean ∆fps MPa

Sy Sy/x r fpse - fpsp fps Sy Sy/x r ∆fpse -∆fpsp∆fps

Design equation

224.049 1388.705 154.391 370.95 Experimental 222.562 112.719 0.836 120.763 1267.942 140.824 75.164 0.593 120.49 250.30 U.S (ACI-1977) 221.126 110.005 0.8347 116.842 1259.402 144.624 79.644 0.596 131.49 241.76 U.S (ACI-1999) 225.386 115.493 0.784 -185.798 1202.907 127.661 118.23 0.519 185.83 187.12 Canadian (1984) 253.912 115.005 0.724 320.071 1068.635 193.958 7.946 0.026 320.07 50.887 Dutch (1990) 204.914 93.555 0.830 -110.019 1498.725 337.093 369.40 0.279 -337.95 708.91 German (1980) 207.366 64.197 0.870 195.497 1193.208 165.739 155.20 0.475 30.544 340.41 British (1985) 216.121 68.227 0.833 237.879 1150.826 148.599 17.085 0.621 199.35 171.60 Warwaruk et al. (1962) Eq.(2.7) 220.038 131.739 0.763 -107.605 1496.313 174.361 135.33 0.514 -157.66 528.62 Du and Tao (1985) Eq.(2.14) 223.785 111.016 0.829 148.453 1240.252 153.141 89.718 0.588 148.05 237.88 Harajli (1990) Eq.(2.15) 239.175 104.176 0.910 -63.242 1451.948 214.938 165.83 0.705 -170.41 541.37 Harajli and Hijazi (1991) Eq. (2.16) 221.459 108.086 0.860 85.663 1303.045 152.151 99.820 0.691 85.66 285.29 Harajli and Kanj (1991) Eq.(2.17)

233.873 126.642 0.854 -40.884 1429.551 296.088 304.33 0.554 -197.65 568.61 Naaman and AL-Khairi (1990) Eq.(2.18), and AASHTO (1994)

206.542 102.611 0.829 61.727 1326.978 130.996 75.860 0.587 61.727 309.23 Chakrabarti (1995) Eq.(2.19)

239.960 124.413 0.860 119.288 1269.417 176.158 121.03 0.732 119.28 251.66 Lee, Moon and Lim (1999) Eq.(2.20)

223.667 95.797 0.900 67.636 1321.069 157.762 116.47 0.682 43.914 327.04 Shdhan (2000) Eq.(2.21) 214.772 73.572 0.931 -64.392 1453.092 147.261 77.456 0.801 -96.224 474.67 Proposed Eq.(5.7) approach I 221.066 94.215 0.903 9.864 1378.842 149.545 96.423 0.751 2.016 368.94 Proposed Eq.(5.7) approach II 213.765 82.671 0.911 51.619 1337.090 147.724 93.304 0.750 16.008 355.48 Proposed Eq.(5.7) approach III 209.943 73.559 0.922 74.998 1312.666 138.084 74.963 0.778 40.457 329.39 Proposed Eq.(5.7) approach IV

Table 6.2- ∆fps and fps Statistical data.

Chapter Six S

tatistical An

alysis

Chapter Six Statistical Analysis 74

∆fps fps

Design equation Average of

overestimation MPa

Maximum overestimation

MPa

Percent of overestimation

% ∗

Average of overestimation

MPa

Maximum overestimation

MPa

Percent of overestimation

% ∗

U.S (ACI-1977) -57.886 -146.01 15.12 -57.16 -146.00 15.126

U.S (ACI-1999) -90.968 -309.02 26.89 -238.5 -674.00 36.134

Canadian (1984) -117.66 -296.87 6.722 -88.43 -201.00 6.722

Dutch (1990) - - - - - -

German (1980) -391.60 -1827.8 89.07 -146.5 -468.00 84.033

British (1985) -110.26 -466.58 44.53 -35.62 -96.623 10.084

Warwaruk et al. (1962) Eq.(2.7) -31.636 -84.790 5.880 -37.06 -84.790 4.201

Du and Tao (1985) Eq.(2.14) -194.41 -535.84 84.87 -157.7 -464.83 10.924

Harajli (1990) Eq.(2.15) -62.078 -148.01 11.76 -58.46 -144.37 68.907

Harajli and Hijazi (1991) Eq. (2.16) -229.63 -543.55 79.00 -113.4 -411.00 25.210

Harajli and Kanj (1991) Eq.(2.17) -49.541 -150.55 25.21 -49.54 -150.55 68.907 Naaman and AL-Khairi (1990) Eq.(2.18), and AASHTO (1994) -299.23 -1304.7 75.63 -104.2 -355.01 68.907

Chakrabarti (1995) Eq.(2.19) -77.460 -190.11 34.45 -75.90 -190.01 33.613

Lee, Moon and Lim (1999) Eq.(2.20) -52.534 -133.01 16.80 -52.53 -133.00 16.806

Shdhan (2000) Eq.(2.21) -95.501 -300.25 31.93 -52.47 -205.44 23.531

Proposed Eq.(5.7) approach III -120.98 -345.75 84.87 -95.92 -297.60 77.311

Proposed Eq.(5.7) approach III -81.913 -258.64 49.57 -66.77 -209.12 47.058

Proposed Eq.(5.7) approach IIII -72.268 -236.41 43.69 -44.725 -190.25 31.932

Proposed Eq.(5.7) approach IVI -68.634 -200.06 31.09 -40.451 -149.451 21.008

∗ The percent is from 120 experimental data tested.

Table. 6.3- ∆fps and fps error statistical analysis.

Chapter Six S

tatistical An

alysis

Chapter Six Statistical Analysis 73

Fig. 6.1-Comparison of predicted stress by predicted design Eq. (2.7): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Warwaruk et al).

0

200

400

600

800

0 200 400 600 800

∆ fpse (MPa)

∆fps

p

(M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

fps

p

(MPa

)

-600

-400

-200

0

200

400

600

110 126 143 147 157 161 163 166 171 176 179 180 182 183 188 191 191 191 195 197

∆f psp (MPa)

Err or (

fps

e-f

psp

)(MPa

)

-600

-400

-200

0

200

400

600

818 974 1006 1046 1053 1074 1146 1165 1171 1193 1221 1266 1291 1298 1311

f psp (MPa)

Error (

fps e

-fps

p

)(MPa

)

R =0.833 R = 0.621

Delta Mean 199.355 MPa

Delta Mean237.879 MPa

(b)

(c)

(a)

(d)

Chapter Six Statistical Analysis 74

0

200

400

600

800

0 200 400 600 800∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

119 128 144 149 165 172 178 187 201 222 241 245 259 268 325 361 367 367 400 400

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

831 983 1019 1056 1076 1133 1181 1222 1240 1286 1322 1347 1386 1410 1499 1564 1604

f psp (MPa)

Erro

r (f p

se-f

psp)

(MPa

)

R = 0.593 R = 0.862

Delta Mean120.654 MPa

Delta Mean120. 663 MPa

(b)

(c)

(a)

(d)

Fig. 6.2-Comparison of predicted stress by predicted design Eq. (2.9): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (ACI 318-71).

Chapter Six Statistical Analysis 75

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(M

Pa)

-600

-400

-200

0

200

400

600

87.7 119 129 144 159 172 179 190 215 241 243 259 301 353 367 374 400

∆ f psp (MPa)

Erro

r ( ∆

f pse

- ∆f p

sp) (

MPa

)

-800

-400

0

400

800

761 969 1015 1068 1097 1173 1194 1233 1247 1292 1339 1352 1396 1416 1494 1543 1603

f psp (MPa)

Erro

r (f p

se-f

psp)

(MPa

)

R = 0.596 R = 0.847

Delta Mean131.498 MPa

Delta Mean116.842 MPa

(a) (b)

(c)

(d)

Fig. 6.3-Comparison of predicted stress by predicted design Eq. (2.9) and (2.10): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (ACI 318M-1999).

Chapter Six Statistical Analysis 76

-200

0

200

400

600

800

-200 0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

fpse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

-76 28 38 47 78 106 113 129 135 151 166 176 187 202 216 247 312 346 422 448

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

699 896 1005 1048 1100 1141 1158 1192 1237 1285 1322 1349 1390 1413 1450

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.519 R = 0.784

Delta Mean183.833 MPa

Delta Mean185.798 MPa

(c)

(b)

(d)

(a)

Fig. 6.4-Comparison of predicted stress by predicted design Eq. (2.22): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Canadian code).

Chapter Six Statistical Analysis 77

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

94 109 140 175 192 243 268 280 294 296 313 317 327 337 362 387 453 557 709 709

∆ f psp (Mpa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

778 1001 1036 1036 1066 1148 1214 1232 1236 1288 1302 1302 1302 1317 1317

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

Delta Mean30.544 MPa

Delta Mean195.497 MPa

R = 0.475 R = 0.87

(a) (b)

(c)

(d)

Fig. 6.5-Comparison of predicted stress by predicted design Eq. (2.23): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (British code).

Chapter Six Statistical Analysis 78

0

200

400

600

800

0 200 400 600 800

∆ f pse ( MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-800

-400

0

400

800

33 39 41 42 43 43 44 44 45 45 47 47 48 49 49 49 50 50 52 53 54 56 57 58 59 61 63 64 65 65∆ f psp (MPa)

Erro

r ( ∆

f pse

- ∆f p

sp) (

Mpa

)

-800

-400

0

400

800

689 861 897 926 941 978 1011 1029 1042 1100 1140 1202 1231 1317 1361

f psp (MPa)

Erro

r (f p

se-f

psp)

(MPa

)

R = 0.026 R = 0.724

Delta Mean320.07 MPa

Delta Mean320.07 MPa

(a) (b)

(c)

(d)

Fig. 6.6-Comparison of predicted stress by predicted design Eq. (2.24): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Dutch code).

Chapter Six Statistical Analysis 79

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(M

Pa)

-2000

-1000

0

1000

2000

206 307 338 415 535 535 616 632 639 639 647 757 1010 1217 1566

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-800

-400

0

400

800

930 1290 1290 1325 1360 1427 1486 1500 1549 1582 1603 1635 1655 1674 1750

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.279 R = 0.83

Delta Mean-337.6 Mpa

Delta Mean-110.019 MPa

(a) (b)

(d)

(c)

Fig. 6.7-Comparison of predicted stress by predicted design Eq. (2.25): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (German code).

Chapter Six Statistical Analysis 80

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

213 213 304 337 355 402 443 503 527 555 584 593 603 625 645 659 676 697 702 739

∆ f psp (MPa)

Erro

r (∆

f pse

−∆

f psp

) (M

Pa)

-600

-400

-200

0

200

400

600

1097 1175 1221 1302 1377 1415 1448 1479 1520 1591 1634 1659 1683 1744 1790

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.514 R = 0.763

Delta Mean-157.663 MPa

Delta Mean-107.605 MPa

(d)

(c)

(b)(a)

Fig. 6.8-Comparison of predicted stress by predicted design Eq. (2.14): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Du and Tao).

Chapter Six Statistical Analysis 81

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

73.9 92.4 117 119 135 146 153 172 192 196 214 224 263 312 401 414 414

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

757 961 1010 1044 1098 1158 1193 1224 1251 1316 1371 1394 1416 1480 1560

fpsp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.588 R = 0.829

Delta Mean148.058 MPa

Delta Mean 148.453 MPa

(b)(a)

(c)

(d)

Fig. 6.9-Comparison of predicted stress by predicted design Eq. (2.15): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Harajli-1).

Chapter Six Statistical Analysis 82

0

200

400

600

800

1000

0 200 400 600 800 1000

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

58.1 135 213 227 321 393 508 555 594 609 667 700 720 742 761 768 804

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

725 1011 1100 1184 1290 1302 1360 1479 1528 1557 1588 1645 1674 1674 1674 1674 1750

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.705 R = 0.911

Delta Mean-170.412 MPa

Delta Mean-63.242 MPa

(a) (b)

(c)

(d)

Fig. 6.10-Comparison of predicted stress by predicted design Eq. (2.16): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Harajli-2).

Chapter Six Statistical Analysis 83

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

94.5 111 136 142 156 203 229 246 263 285 306 321 342 362 384 444 597

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

783 1004 1059 1115 1166 1193 1228 1252 1333 1384 1421 1483 1515 1577 1617

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.691 R = 0.86

Delta Mean 85.663 MPa

Delta Mean85.663 MPa

(b)(a)

(c)

(d)

Fig. 6.11-Comparison of predicted stress by predicted design Eq. (2.17): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Harajli-3).

Chapter Six Statistical Analysis 84

0

400

800

1200

1600

0 400 800 1200 1600

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-1500

-1000

-500

0

500

1000

1500

7 96 212 267 316 387 430 503 552 627 688 756 879 987 1321

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

676 960 1173 1277 1290 1290 1360 1400 1441 1471 1508 1588 1636 1645 1674 1674 1750f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.554 R =0.854

Delta Mean-197.658 MPa

Delta Mean-40.844 MPa

(d)

(c)

(a) (b)

Fig. 6.12-Comparison of predicted stress by predicted design Eq. (2.18): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Naaman and Al-Khairi) or

(AASHTO-1994).

Chapter Six Statistical Analysis 85

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-300

0

300

600

117 183 193 202 211 227 238 252 276 276 294 338 366 414 414 414 414 414 414 414

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-300

0

300

600

888 1043 1084 1118 1190 1246 1276 1292 1304 1365 1394 1394 1422 1482 1537 1568 1586

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.587 R = 0.829

Delta Mean61.727 MPa

Delta Mean61.727 MPa

(c)

(c)

(a) (b)

Fig. 6.13-Comparison of predicted stress by predicted design Eq. (2.19): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Chakrabarti).

Chapter Six Statistical Analysis 86

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

70 70 70 70 70 87.7 108 142 215 238 276 320 382 410 430 482 575

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

798 918 978 1038 1071 1138 1180 1215 1238 1275 1295 1343 1377 1416 1549 1621 1674

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R =0.732 R = 0.86

Delta Mean119.288 mPa

Delta Mean120.722 MPa

(d)

(a) (b)

(c)

Fig. 6.14-Comparison of predicted stress by predicted design Eq. (2.20): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Lee at al).

Chapter Six Statistical Analysis 87

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-300

0

300

600

68.3 106 141 174 186 196 258 290 303 334 353 395 422 456 468 522 591

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-300

0

300

600

724 988 1035 1114 1168 1210 1226 1263 1287 1349 1403 1454 1477 1509 1578 1590 1663

f psp (MPa)

Erro

r (f p

se-f

psp

) (M

Pa)

R = 0.682 R = 0.9

Delta Mean43.914 MPa

Delta Mean67.636 MPa

(a) (b)

(c)

(d)

Fig. 6.15-Comparison of predicted stress by predicted design Eq. (2.21): (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps, by (Shdhan).

Chapter Six Statistical Analysis 88

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

161 258 280 346 367 390 413 427 447 465 490 507 521 535 540 551 574 607 644 650

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

869 1138 1192 1274 1290 1304 1380 1422 1459 1500 1556 1588 1630 1630 1674 1674 1674

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.801 R = 0.931

Delta Mean-96.293 MPa

Delta Mean-64.390 MPa

(a) (b)

(c)

(d)

Fig. 6.16-Comparison of predicted stress by proposed design [Eq. (5.7) approach-I]: (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps.

Chapter Six Statistical Analysis 89

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (

MPa

)

600

1000

1400

1800

600 1000 1400 1800

f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

7 135 174 210 232 277 312 321 354 367 389 401 409 432 439 461 475 512 556 650

∆ f psp (MPa)

Erro

r (∆

f pse

- ∆f p

sp) (

MPa

)

-600

-400

-200

0

200

400

600

811 1028 1147 1209 1255 1286 1310 1341 1404 1479 1533 1585 1619 1630 1674

f psp (MPa)

Erro

r (f p

se-f

psp) (

MPa

)

R = 0.903

(a) (b)

R = 0.75

Delta Mean2.016 MPa

(c)

Delta Mean9.864 Mpa

(d)

Fig. 6.17-Comparison of predicted stress by proposed design [Eq. (5.7) approach-II ]: (a) ∆fps; (b) fps; (c) error of ∆fps and (d) error of fps.

Chapter Six Statistical Analysis 90

Fig. 6.18-Comparison of predicted stress by proposed design [Eq. (5.7) approach-III ]: (a) ∆fps; (b) fps; (c) error of ∆fps and error (d) of fps.

0

200

400

600

800

0 200 400 600 800∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800f pse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

804 1020 1136 1187 1187 1228 1270 1304 1380 1456 1490 1540 1540 1540 1610f psp (MPa)

Erro

r (f p

se-f

psp)

(MPa

)

-600

-400

-200

0

200

400

600

7 130 167 201 223 266 300 309 340 352 373 385 393 414 422 443 456 491 533 642

∆ f psp (MPa)

Erro

r ( ∆

f pse

- ∆f p

sp) (

MPa

)

(b)(a)

(d)

(c)

R = 0.750 R = 0.911

Delta Mean16.008 MPa

Delta Mean51.619 MPa

Chapter Six Statistical Analysis 91

0

200

400

600

800

0 200 400 600 800

∆ f pse (MPa)

∆f p

sp (M

Pa)

600

1000

1400

1800

600 1000 1400 1800

fpse (MPa)

f psp

(MPa

)

-600

-400

-200

0

200

400

600

76.8 136 176 217 245 268 295 315 325 352 365 378 391 405 425 465 527

∆ fpsp (MPa)

Erro

r ( ∆

f pse

- ∆f p

sp)

(MPa

)

-600

-400

-200

0

200

400

600

794 1023 1116 1156 1161 1224 1260 1311 1351 1424 1481 1507 1507 1507 1575

f pse (MPa)

Erro

r (f p

se-f

psp)

(M

Pa)

(d)

(c)

(a) (b)

Fig. 6.19-Comparison of predicted stress by proposed design [Eq. (5.7) approach-IV ]: (a) ∆fps; (b) fps; (c) error of ∆fps and error (d) of fps.

Chapter Six Statistical Analysis 92

6.2.5 Frequency distribution, method 5: In this test more accuracy and sensitivity to the experimental results can be

provided. All models are presented in figure (6.20) through (6.23). The models

that provide lines to the left of the experimental line are providing

underestimates, while the lines to the right indicate overestimates.

The cumulative frequency Fig. (6.20.a) indicates that four of the five

models offer conservative of ∆fps with no overestimations. These four models

are Dutch (34), Canadian (32), ACI 318 M-99 (15) and British code (17), listed

from most conservative to least conservative, except to note that the British

code (17) overestimate the groups from (580 to 760) MPa, also the Canadian

code (32) gave negative value with a minimum (–76 MPa) approximately, while

the German code (34) model is consistently overestimating in all groups.

The cumulative frequency Fig. (6.20.b) indicates that the four models offer

conservative estimates of ∆fps with no overestimations. These four models are

Warwaruk et al (9), Harajli-1 (25), ACI 318 M-99 code (15) and ACI 318-77

code (12), listed from most conservative to least conservative. It should be noted

that the four models used the parameter ρps /fc’ in represent their models,

previously this parameter is not a rational design parameter especially in

partially prestressing. A slight improvement at higher values can be shown in

the Harajli-1 (25) models where the effect of span-depth ratio has been

implicated.

The cumulative frequency Fig. (6.20.c) indicates that two of the three

models offer conservative of ∆fps with no overestimations. These two models are

Harajli-1 (25) and Harajli-3 (27), while the Harajli-2 (26) model is consistently

overestimating in all groups. However the Harajli-2 (26) model takes into

account all possible design parameters compared to Harajli-3 (27) model where

the effect of span-depth ratio is ignored, inconsistent behavior can be seen.

Chapter Six Statistical Analysis 93

The cumulative frequency Fig. (6.21.a) indicates that five of the seven

models offer conservative of ∆fps with no overestimations. These five models are

ACI 318 M-99 code (15), Chakrabarti (29), Harajli-3 (27), Lee at el (3) and

Shdhan (31) models, listed from most conservative to least conservative, except

to note that the Chakrabarti (29) and Shdhan (31) overestimate the groups from

(112 to 200) and from (675 to 795) MPa respectively, while the Du /Tao (22) and

Naaman/AL-Khairi (28) models are consistently overestimating in all groups.

The cumulative frequency Fig. (6.21.b) indicates that three of the four

proposed models offer conservative model of ∆fps with no overestimations.

These three proposed models are Eq. (5.7) approach II, IV and III

respectively, listed from most conservative to least conservative, while the Eq.

(5.7) approach I are consistently overestimating in all groups except for the 650

MPa.

General comparisons between most accurate models are presented in Fig.

(6.21.c). While non of the predicted models are a direct image of the

experimental data, it can be seen that the proposed Eq. (5.7) approach III and

IV models provides the closet, but still conservative, estimate of ∆fps.

The cumulative frequency Fig. (6.22.a) indicates that the four of the five

models offer conservative of fps with no overestimations. These four models are

Dutch (34), Canadian (32), ACI 318 M-99 (15) and British (17) code, listed from

most conservative to least conservative, while the German code (34) model are

consistently overestimating in all groups.

The cumulative frequency Fig. (6.22.b) indicates that the four models offer

conservative models of fps with no overestimations. These four models are

Warwaruk et al (9), Harajli-1 (25), ACI 318 M-99 code (15) and ACI 318-77

code (12).it should be indicate that no significant improvement between the ACI

318 M-99 code (15) and ACI 318-77 code (12).

Chapter Six Statistical Analysis 94

The cumulative frequency Fig. (6.22.c) indicates that two from three

models offer conservative of fps with no overestimations. These two models are

Harajli-1 (25) and Harajli-3 (27), while the Harajli-2 (26) model have an

overestimating group between 1100 and 1700 MPa.

The cumulative frequency Fig. (6.23.a) indicates that five of the seven

models offer conservative of ∆fps with no overestimations. These five models are

ACI 318 M-99 code (15), Lee at el (3), Harajli-3 (27), Shdhan (31) and

Chakrabarti (29), models, listed from most conservative to least conservative,

while the Du/Tao (22) and Naaman/AL-Khairi (28) models are consistently

overestimating in all groups, except the Naaman/AL-Khairi (28) model

underestimate the 1750 MPa group.

The cumulative frequency Fig. (6.23.b) indicates that three of the four

proposed models offer conservative models of ∆fps with no overestimations.

These three proposed models are Eq. (5.7) approach II, IV and III

respectively, listed from most conservative to least conservative, while the Eq.

(5.7) approach I is consistently overestimating in all groups except for the 1750

MPa.

General comparisons between most accurate models are presented in Fig.

(6.23.c). While non of the predicted models are a direct image of the

experimental data, it can be seen that the proposed Eq. (5.7) approach III

model provide the closet, but still conservative, estimate of ∆fps.

In summary, this statistic indicates that the two proposed [Eq. (5.7)

approach III and IV ] models is the best fit of the experimental data.

Chapter Six Statistical Analysis 95

0

20

40

60

80

100

0 200 400 600 800 1000

Exper.

Harajli-1.

Harajli-3.

Harajli-2.

0

20

40

60

80

100

0 100 200 300 400 500 600 700 800

Exper. ACI-77 code. ACI-99 code. Warwaruk et al. Harajli-1.

0

20

40

60

80

100

-200 200 600 1000 1400 1800 2200

Exper. Canadian code. British code. Dutch code. German code. ACI-99 code.

Fig. 6.20-Combined cumulative frequency of ∆fps.

Cum

ulat

ive

freq

uenc

y.

∆fpsp (MPa).

∆fpsp (MPa).

∆fpsp (MPa).

Experimental

(b)

(c)

(a)

Chapter Six Statistical Analysis 96

Fig. 6.21-Combined cumulative frequency of ∆fps.

Cum

ulat

ive

Freq

uenc

y.

∆fpsp (MPa).

∆fpsp (MPa).

∆fpsp (MPa).

Experimental

0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600 1800

Naaman. Chakrabarti. Shdhan. Lee at al. Harajli-3. ACI-99 code. Exper. Du/Tao.

0

20

40

60

80

100

0 100 200 300 400 500 600 700 800

Exper. Approach I. Approach II. Approach III. Approach IV

0

20

40

60

80

100

0 200 400 600 800 1000

Exper. Chakrabarti. Shdhan. Lee at al. Approach III. Approach II. Approach IV

∆fpsp (MPa)

∆fpsp (MPa)

Experimental

(a)

(b)

(c)

Cum

ulat

ive

freq

uenc

y.

Chapter Six Statistical Analysis 97

0

20

40

60

80

100

600 800 1000 1200 1400 1600 1800 2000

Exper. Canadian code. British code. Dutch code. German code. ACI-99 code.

0

20

40

60

80

100

600 800 1000 1200 1400 1600 1800 2000

Exper. ACI-77 code. ACI-99 code. Warwaruk et al.

Harajli-1.

0

20

40

60

80

100

600 800 1000 1200 1400 1600 1800 2000

Exper.

Harajli-Harajli-2. Harajli-3.

Fig. 6.22-Combined cumulative frequency of fpsp.

Cum

ulat

ive

freq

uenc

y.

fpsp (MPa).

fpsp (MPa).

fpsp (MPa).

Experimental

(b)

(c)

(a)

Chapter Six Statistical Analysis 98

Fig. 6.23-Combined cumulative frequency of fps.

Cum

ulat

ive

Freq

uenc

y.

fpsp (MPa).

fpsp (MPa).

fpsp (MPa).

Experimental

0

20

40

60

80

100

600 800 1000 1200 1400 1600 1800 2000

Exper. Approach I. Approach III. Approach II.

Approach IV.

0

20

40

60

80

100

600 800 1000 1200 1400 1600 1800 2000

Exper. Naaman. Chakrabarti. Shdhan. Lee at le. Harajli-3. ACI-99 code. Du/Tao.

0

20

40

60

80

100

600 800 1000 1200 1400 1600 1800 2000

Exper. Chakrabarti. Shdhan. Lee at le. Approch III.

Approach IV.

fpsp (MPa)

fpsp (MPa)

Experimental

(a)

(b)

(c)

Cum

ulat

ive

freq

uenc

y.

Chapter Six Statistical Analysis 99

6.3 Evaluation of proposed equations: A lesson is learnt from the preceding discussions. The accuracy of any

proposed design equations to represent the experimental results in rational

parameter is influenced by the fowling parameter (35), (38), (39).

1. The mean of ∆fpsp and fpsp should be as close as possible to ∆fpse and fpse,

not equal or more.

2. The error of ∆fpsp and fpsp (delta mean) should be as close as possible to

zero, with a positive value.

3. The correlation coefficient ( r ) should be as close as possible to 1.

4. The standard error of estimate (Sy/x ) should be as small as possible, and

should be smaller than the standard deviation ( Sy ).

5. The average of overestimation of ∆fpsp and fpsp should be as close as

possible to zero, with a small percent.

Disregarding of any previous parameters will lead to an undesirable

statistically results. From the statistical data listed in Table 6.2 and 6.3 it easy to

see that the two proposed [Eq. (5.7) approach III and IV ] models provide the

best fitting to represent the experimental results more than any other models

where a general balancing between predicted and experimental data are

provided.

Chapter Seven

Chapter seven Conclusions & Future research 100

7.1 Conclusions: From the statistical analysis performed in this study, following conclusions can be

drawn:

1. Formulating and /or analyzing a model should be performed utilizing a wide

selection of data samples, with as much variation as possible, to insure the

best results.

2. The type of load application has significant effect on fps, where unbonded

beams loaded under one-point loading mobilize the least ∆fps at their nominal

flexural resistance.

3. A very high value of PPR caused sudden large cracking in the tension zone

of the beams, and moderate value of global reinforcing index (ϖ ≈ 0.1)

caused crushing in the compression zone of the specimens.

4. The stress in prestressing steel at ultimate decreases with increasing amount

of bonded nonprestressing reinforcement.

5. A design equation for the unbonded tendon has to take into account the

partial prestressing effects that come from the bonded reinforcements.

6. The Naaman and AL-Khairi (28) mobilizes εcu, the extreme concrete fibers

stress at ultimate that is an unknown value. Therefore, to utilize this model

one has to assume a reasonable value for εcu.

7. This study indicates that the ACI-1999 (15) code equations (18-4) and (18-5)

need to be exchanged for more rational equations.

8. It seems that the models that include the effect of unprestressed steel

typically provide more accurate predictions of fps. This is best seen by

comparing the two models by Harajli (3 and 1); the one that includes the

unprestressed steel has better statistical success than the other. However, the

other factors included in the models also have some effect.

9. The Harajli-2 (26), Naaman and AL-Khairi (28), Du and Tao (22), Dutch code

(34) and German code (34) models consistently overestimate ∆fps. While the

Chapter seven Conclusions & Future research 101

Chakrabarti (29), Lee at el (3), Shdhan (31) and Harajli-3 (27) models are

medium conservative models of ∆fps and fps respectively.

10. The Canadian code (32) model may give inconsistent results for ∆fps, where

negative values could be predicted.

11. The level of the effective stress of the tendons can influence the unbonded

tendon stress at the member flexural failure.

12. The span-depth ratio has to be considered together with the loading type

since those are dependent upon plastic hinge length.

13. The results of all the tests combined indicate that the proposed [Eq. (5.7)

approach III and IV] are the best statistical model for predicting the value

of ∆fps and fps respectively. This is shown from the tests performed in

Methods 1,2,3,4 and 5 and summarized in Table 6.2 and 6.3, where a

general balancing between predicted and experimental data are provided.

14. Although the proposed [Eq. (5.7) approach III and IV] involves more detailed

requirements, it should cause no problems with the present universal case of

computers in design.

7.2 Future research: 1. Further research is needed on continuous beams and slabs with different

reinforcement ratios, location, boundary condition and material properties,

where high strength and light weight concrete are being used in prestressed

concrete structures.

2. Extend the proposed design equation to external prestressing tendons. There

is no sufficient accuracy and simplicity to be recommended for code

implementation, as example (most European codes allow no increase of

external prestressing steel at ultimate) (33).

3. More statistical tests on the same line of this research to extend the proposed

design equation of bonded prestressing tendons.

Reference 102

References

1. Collins, Michael P.; and Mitchell, Denis: “Prestressed Concrete Structure,”

New Jersey, 1991, 766 pp.

2. Naaman, Antoine E: “Prestressed Concrete Analysis and Design ,”

McGraw-Hill Co., 1982, 670 pp.

3. Lee, L, H., Moon, J.-H., and Lim, J ,-H., “Proposed Methodology for

Computing of Unbonded Tendon Stress at Flexural Failure,” ACI

Structural Journal, V. 96, No. 6, Nov. – Dec. 1999, pp. 1040 – 1049.

4. Naaman, Antoine E, “Partially Prestressed Concrete: Review and

Recommendations,” PCI Journal, V.23, No. 6, Nov. – Dec. 1985, pp. 30 –

71.

5. Rao, S, V, and, Dilger, W, H, “Evaluation of Short-Term deflection of

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Appendix A

A-1

APPENDIX A -- NUMERICAL EXAMPLE

TWO-SPAN CONTINUOUS BEAM The data for this example is taken from Reference 26. The example is

given next to illustrate the use of the proposed predictions [Eq. (5.7) approach III, and IV)] summarized in Table A-1 for computing fps and Mn in unbonded members. Section dimensions for the example beam is given in Fig. A. For consistency in comparison using various prediction methods, the area of ordinary bonded reinforcement is taken as the minimum specified in the ACI Code (As ≈ 0.004A). Results are compared with the results obtained from 1999 ACI Building Code. Material and reinforcement properties:

fc’ = 34.47 MPa

β1 = 0.81 fpu = 1861.65 MPa fpy = 1585.8 MPa fpe = 1103.2 MPa fy = 413.7 MPa

Fig A-1: Typical beam cross section

Given: Aps = 690.36 mm2 (seven ½ in. 7-wire strands); dp =850mm; ρp = 0.0016; As = 1135.55 mm2 (four No. 6 bars); ds = 939.8 mm; span-depth S/dp = 25. Required: Nominal flexural strength Mn at (A) in the vicinity of midspan (B) interior support. Approach III: Case (A): The maximum applied moment in the vicinity of midspan is obtained by loading one single span (left); hence: no/n = ½.

From Table 5.1: For one point loading f1= ∞ α1 = 1.15, α2 = 1.44, α3 = 0.72, α4 = 1.44

1016 mm

850 mm

508 mm

Appendix A

A-2

From Eq. (5.7):

⎟⎠

⎞⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

nn

dSfo

p/11γ = ( ) =⎥

⎦

⎤⎢⎣

⎡+

∝5.0

2511 0.1

∆fps( )

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧ −×+×+×

×−×=

3.6907.4135.1135044.16.186072.02.110344.1

4.3481.00016.015.16.18611.0

= 191.15 MPa < 650 MPa. … O.K

fps = fpe + ∆fps

= 1103.2+191.15 = 1294.35 MPa < 0.95 fpy = 1506.55 MPa. … O.K

∴fps = 1294.35 MPa.

Calculate the depth of the compression block [Eq. (A-1)], and then the nominal

flexural strength Mn [Eq. (A-2)].

mm 130

5103585.041412001501987

b'f85.0fAfA

ac

ySPSPS

=××

×+×=

+=

)2ad(fA)

2ad(fAM SySPPSPSn −+−= …(A-2)

= 1125.83 2.25113.8939 .7413.551135

2113.25850.9 1297.41690.36 =⎟

⎠⎞

⎜⎝⎛ −×+⎟

⎠⎞

⎜⎝⎛ −×

Case (B): The maximum applied moment in the interior support is obtained by loading the two spans simultaneously; hence: no/n = 1.

From Table 5.1: For one point loading f1= ∞ (similar to case A) α1 = 1.15, α2 = 1.44, α3 = 0.72, α4 = 1.44 (similar to case A) From Eq. (5.7): γ = 0.2 ∆fps = 388.42 MPa < 650 MPa. … O.K fps = fpe + ∆fps = 1103.2+388.42 = 1491.626 MPa < 0.95 fpy = 1506.55 MPa. …O.K

∴fps = 1491.626 MPa.

kN.m

= 113.25 mm .50847.3481.085.0

7.41355.113541.129736.690×××

×+×=

…(A-1) 0.85 1β fc’ b

Appendix A

A-3

From Eq. (A-1): a = 127.379 mm. From Eq. (A-2): Mn = 1222.213 kN.m.

ACI 318-99: For both case (A) and (B):

S/dp = 25 < 35. From Eq. (2.9): ∆fps = 371.612 MPa < 400 … O.K fps = fpe + ∆fps = 1103.2+371.61 = 1474.812 MPa < fpy = 1585.85 MPa … O.K

∴fps = 1474.812 MPa. From Eq. (A-1): a = 123.41 mm. From Eq. (A-2): Mn = 1216.024 kN.m.

Table A-1: Summary of results derived from the numerical example

Proposed Approach III Approach IV

ACI 318M-99 (15) Type of load

application ∆fps

MPa

fps

MPa

Mn

kN.m

∆fps

MPa

fps

MPa

Mn

kN.m

ACIn

Pron

)(M)(M

CASE (A) One-point loading

191.1 155.4

1294.3 1258.6

1125.8 1105.7

0.92 0.91

CASE (A) Uniform loading

227.9 207.4

1331.1 1310.6

1142.6 1132.2

0.94 0.93

CASE (B) One-point loading

388.4 310.8

1491.6 1414.0

1222.2 1184.5

1.00 0.97

CASE (B) Uniform loading

455.9 414.9

1559.1 1518.1

1257.2 1236.7

371.6 1474.8 1216.0

1.03 1.01