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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
'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