Numerical analysis of externally prestressed concrete beams part 1

Acknowledgements

First of all, the author would like to express his deepest gratitude and appreciation to his
advisor, Prof. Umehara Hidetaka of Department of Environmental Technology and Urban
Planning, Nagoya Institute of Technology, for his invaluable guidance, continuous motivation,
patient explanations, useful comments and understanding that have been the great source of
support throughout the course of this research. His tireless devotion, unlimited kindness even
for the private matters has earned the author’s highest respect. Only his ingenious ideas,
constructive suggestions and tireless guidance made the completion of this study possible.
The author wishes to express thanks to the members of his dissertation examining
committee, Prof. Umehara, H., Prof. ….and Prof. ….for going through the text of this thesis
painstakingly and for making enlightening suggestions and comments that helped to refine the
scope and content of this study.
The author wishes to express special thanks to Prof. Tanabe Tada-Aki of Department of
Civil Engineering, Nagoya University, for his continuous guidance and discussions,
invaluable suggestions from the days of the author’s studying under his supervision up to
present. That is in addition to his going through the text of the journal papers making
enlightening suggestions and useful comments through the course of study.
The author also wishes to express his sense appreciation to Associate Prof. Uehara Takumi
of Department of Civil Engineering, Nagoya Institute of Technology, for his assistance during
the course of study. Thanks are also expressed to Research Assistant Mr. Kimata, H., Dr. Ito,
A., of Concrete laboratory of Department of Civil engineering, Nagoya University for his help
in various kinds of matters.
Special thanks are expressed to Dr. Shahid, N, Mr. Brohi, K. and all Vietnamese friends in
Aichi Ken for their help and assistance to overcome the daily difficulties.
The author likes to express special appreciation go to Mr. Hirahara, H., Mr. Ushida, K.
and all members of Concrete laboratory, Nagoya Institute of Technology for their assist to
overcome the daily difficulties and for their friendly attitude during the whole research period.
Grateful acknowledgement is given to the Ministry of Education Science and Culture of
Japan, since it has generously provided the financial support, which has made it possible for
the author to pursue this course of study.

The author would like to express special thanks to Dr. Do Huu Tri, Doctor General of the
Research Institute for Transportation Science and Technology (RITST) of Vietnam, and to
Dr. Nguyen Xuan Dao, former Doctor General of RITST for their continuous supports in
various kinds of matter that allowing the author completes the course of study in Japan.
Grateful appreciations are also expressed to all members of Department of Bridges and
Tunnels of RITST for their help and continuous cooperation in the current research field.
Last but not the least, the author would like to express his deep sense of gratitude to his
wife for her patience, understanding and moral support, for her high limit state of endurance
over the years, which made the full completion of this dissertation a reality. About all, the
author wishes to express his deepest sense of respect, for which words are not enough, to his
mother for her tenderness, love, care, sacrifices and encouragement.


Chapter 1

INTRODUCTION

1.1

GENERAL

One of the latest developments in prestressed concrete technology has been the use of
external cables, which may be defined as a method of prestressing where major portions of
the cables are placed outside the concrete section of a structural member. The prestressing
force is transferred to the beam section through end anchorages, deviators or saddles. This
type of prestressing could be applied not only to new structures, but also to those being
strengthened. Substantial economic and construction time saving have been indicated for this
innovative type of construction.

External prestressing system was used in the bridge construction in the early days of
prestressing. However, due to a generally inadequate technology, external prestressing has
received a bad image and was almost abandoned in the 1950’s. This is because the corrosion
problem for the external cables was serious, and the internal prestressing system with the
bonded cable was emphasized. With the development of partial prestressing techniques and
protective system for the external cables, it is possible to have structures with external cables,
whose performance is as good as the structures with bonded cables. In recent years, external
prestressing revives in the construction of new structures and has a great development in the
bridge construction.
The deterioration of existing bridges due to increased traffic loading, progressive structural
aging, and reinforcement corrosion from severe weathering condition has become a major
problem around the world. The number of heavy trucks and the traffic volume on these
bridges has both risen to a level exceeding the value used at the time of their design, as a
result of which many of these bridges are suffered fatigue damage and are therefore in urgent

-1-


need of strengthening and repair. A method for strengthening and rehabilitation of such
structures has become increasingly important.
External prestressing is considered one of the most powerful techniques used for
strengthening or rehabilitation of existing structures and has grow recently to occupy a
significant share of the construction market. The adoption of external cables has been
proposed as a very effective method for repairing and strengthening damaged structures.
Although external prestressing is a primary method for rehabilitation and strengthening of
existing structures, it is being increasingly considered for the construction of new structures,
particularly bridges. Since the external prestressing system is simpler to construct and easier
to inspect and maintain as compared with the internal prestressing system, the beams
prestressed with external cables have attracted the engineer’s attention in recent years, and it
has been proposed in the design and construction of new bridges. A large number of bridges

with monolithic or precast segmental block have been already built in the United States,
European countries and Japan by using the external prestressing technique. Recently, a new
type of structures using the external cables or combination with either bonded cables or
unbonded cables has been increasingly developed around the world such as externally
prestressed concrete bridges consisting of concrete flanges and folded steel web or extradosed bridges with a short tower.
In this chapter, the definition of post-tensioned prestressed concrete beams and
classification of beams prestressed with external cables is initially presented. The application
of external prestressing is discussed together with its advantages and disadvantages. The
historical development of external prestressing is also discussed, following by literature
reviews of the previous studies. A general overview of problem arisen from the application of
external prestressing is highlighted. The differences between internally unbonded cables and
external cables at all loading stage are also briefly presented and discussed. Finally, the
objectives and scope of the present study as well as the organization of the course of study are
defined and given at the end of this chapter.
1.1.1

Definition of post-tensioned prestressed concrete beams

An initial distinction is helpful when dealing with definition of prestressed concrete
structures. A post-tensioned prestressed concrete beams may be classified as either bonded or
unbonded. Frequently, the prestressing cable is placed inside the concrete cross section and
bonded by filling the ducts with cement grout after the desired prestressing force has been
-2-


applied; this is called as conventional prestressing or conventionally prestressed concrete
beams. On the contrary, the ducts may be left empty, or filled with grease, in this case the
bond between the concrete and the prestressing cable is eliminated, friction inside the ducts is
artificially reduced to minimum value and the cables transfer their load to the concrete beam
through

the

end

anchorages

and

the

deviators,

the

terms

“unbonded”

and

“external“ prestressing are adopted. The term “unbonded prestressing” is used if when the
cable is placed inside the cross section and friction between the duct and the cable is equal to
zero. Whereas, the term “external prestressing” is used if the cable is placed outside the cross
section and attached to the beam at some deviator points along the beam.
Prestressed concrete beams may also be classified as either fully or partially prestressed.
Fully prestressed beams contain only prestressing cables, whereas partially prestressed beams
contain bonded non-prestressed reinforcement in addition to the prestressing cables in the
tension zone.

Prestressed Concrete
Beams

Bonded Prestressed
Concrete Beams

Perfectly
Bonded

Unbonded Prestressed
Concrete Beams

Partially
Bonded

Internally
Unbonded

Externally
Unbonded

Assumption for computing method of cable strain

Δε s = Δε cs

⎛1 l
⎞
Δε s = K s ⎜ ∫ Δε cs dx − Δε cs ⎟ + Δε cs
⎝l 0
⎠

Δε s =

1 l
Δε cs dx
l ∫0

Δε s = ???

Fig.1.1 Classification of prestressed concrete beams

Depending on the extent of bondage between the concrete and the prestressing cables, all
the prestressed concrete beams can be mainly divided into two groups, namely, prestressed
concrete beams with bonded cables and prestressed concrete beams with unbonded cables.
And in each group, beams can be divided into small subgroup. For example, beams
prestressed with bonded cables may be classified either perfectly bonded or partially bonded
-3-


cables, whereas beams prestressed with unbonded cables may be classified either internally
unbonded cables or external cables. Fig.1.1 shows the classification of prestressed concrete
beams. In this figure, the equations of cable strain for each group are also presented, except
the equation for externally prestressed concrete beams, which is a main target of this study
and will be presented in Chapter 3 and Chapter 6.
1.1.2. Classification of externally prestressed concrete beams
Generally, external prestressing is defined as a prestress introduced by the high strength
cable, which is placed outside the cross-section and attached to the beam at some deviator
points along the beam. Fig.1.2 shows a typical view of a concrete box girder bridge
prestressed by external cables.

Fig.1.2 Typical view of a prestressed concrete box girder bridge with external cables

Location of deviators

Conventionally prestressed beams
with external cables

Externally prestressed beams
with large eccentric cables

Deviators are located within the depth
of cross section

Deviators are located outside the depth
of cross section

Cable
Cable

Cross Section

Deviator

Arrangement of external cables

Cross Section

Arrangement of external cables

Fig.1.3 Classification of externally prestressed beams

-4-

Deviator


In an external prestressing system, depending on the location of deviators, there are two
kinds of the beams. Deviators are placed within the depth of cross section, the term
“conventionally prestressed beam with external cables“ is used, and otherwise the name
“beam with large eccentric cables“ is adopted (see Fig.1.3).
1.1.3

Advantages and disadvantages of external prestressing

External prestressing, initially developed for bridge strengthening, is now used for new
bridges, particularly for precast or cast-in-place concrete segmental bridges. New design
concepts and prestressing techniques have been developed to implement external prestressing,
especially in France, the United States and Japan. These efforts were undertaken because of
the following advantages of external prestressing:
•

External prestressing leads to simple cable layouts, with very limited angular deviations,
reduce friction losses and improve the concreting conditions by eliminating ducts from
webs.

•

Concreting of new structures is improved because there are no cables inside the section
(external prestress only) or there are fewer cables (internal combined with external
prestress).

•

Dimensions of the concrete cross section, especially, the webs can be reduced due to the
partial or full elimination of internal cables (deadweight reduction).

•

Profile of external cable is simpler and easier to check during and after the installation.

•

Grouting is improved because of a better visual control of the operation and therefore, a
better protection of prestressing cable is obtained. It is also possible to easily inspect the
cable during the entire life of the structure.

•

External cable can be removed and replaced if the corrosion protection of the external
cable allows the release of the prestressing force (for instance, cables made of
individually lubricated sheathed strand.

•

Friction losses are significantly reduced because external cables are contacted to the
structure only at the some deviator points and anchorages.

•

The main construction operations, concreting and prestressing are more independent of

one another. Therefore, the influence of workmanship on the overall quality of the
structure is reduced.
-5-


Though the above-mentioned advantages are attractive, some shortcomings are
nevertheless encountered due to external prestressing, which are as follows:
•

Because the external cables are located outside the cross section of the beam, they have
to be protected from corrosion by the high density polyethylene (HDPE) ducts, which
results in a higher initial material cost for the prestressing system over that of the
internal cables.

•

External cables do not participate in the local crack control.

•

The strain difference between the cable and the concrete may lead to movements of the
cable over the deviators and thus to friction corrosion.

•

The external prestressing system is transferred the force to the beam via the anchorages
and the deviator points along the beam. Therefore, the anchorages and the deviator
points must carry the high concentrated forces under the applied load. Consequently,
they become the critical regions of the structures. They must be designed to support
large longitudinal or transverse forces, and their connection to the cross section usually

introduces shear transfer in the form of concentrated load acting on the cross section.
These elements should be carefully detailed and adequately reinforced.

•

In the deviation zones, the high transverse pressures are acting on the prestressing cable.
The saddle inside the deviation zones made of metal tubes or sleeves should be
precisely installed to reduce friction as much as possible and to avoid damage to the
prestressing cable, which could lead to the strength reduction.

•

The behavior of anchor head of the external cable is more critical. Failure of the anchor
head of an external cable means a complete loss prestress in that cable. Therefore, the
anchor head should be carefully protected against corrosion.

•

At the ultimate state, failure with a little warning due to insufficient ductility is a major
concern for externally prestressed structures.

•

Under the ultimate bending condition, more prestressing force is required to generate
the ultimate strength similar to that of internally bonded cables.

•

External cables are subjected to vibrations and, therefore, their free length should be
limited.

•

External cables might be susceptible to fire damage.
-6-


1.2. LITERATURE REVIEW
1.2.1

Historical development and application of external prestressing

External prestressing was a mode of construction in the early days of prestressing. Several
bridges were built for example in Germany first, with the Adolf Hitler Bridge at Aue in 1936,
designed by Franz Dischinger. In Belgium then, under the influence of Magnel with the
Sclayn bridge in 1950. And in France between 1950 and 1952, the bridge at Villeneuve-SaintGeorges, designed by Lossier, the bridge at Vaux-Sur-Seine and port a Binson, built by
Coignet, and bridge at Can Bia. These first attempts, however, did not produce excellent
results. Most of these externally prestressed structures suffered from corrosion. This
experience gave a poor image of external prestressing, and very few externally prestressed
concrete bridges were built in the sixties and in the seventies except for a series of road
bridges in Belgium were built between 1960 and 1970, and in England the Bournemouth
Bridge and the Exe and Exminster viaducts.
After lying dormant for some time, external prestressing has been rediscovered as an
attractive application of prestressing. Under the influence of French engineers-Jean Muller in
the United States and SETRA (Service Technique des Routes et Autoroutes) in France,
external prestressing has been made possible by the development of the prestressing
technology, and numerous structures have been designed and built with external prestressing
around the world, especially, in Europe and the United States. One of the recent projects is the
construction of the second stage expressway system in Bangkok, Thailand, which commenced
in 1989 where external cables and dry jointed precast segmental desk were used1). The

Shigenobu river bridge was the first externally prestressed segmental type bridge in Japan.
And from the experience of that bridge, numerous bridges have been designed and
constructed with external prestressing in Japan up to present.
The development of high capacity cables has resulted in a reduction in the number of
external cables, which eases design and construction. And above all, the experience of
strengthening some classical prestressed concrete bridges, in which the initial prestressing
forces were not great enough, has made it possible to put into use protective systems adapted
to ensure resistance to corrosion of external cables.
Furthermore, experience in strengthening these bridges, which perforce had to be placed
the cables outside the concrete section, made designers aware of the advantages of external
prestressing. This led them to consider its use in building new bridges. The principal
-7-


advantages are the considerable simplification of the cable layout and the large reduction in
losses of prestress due to friction.
In recent years, the external prestressing technology is widely used in the construction of
concrete bridges. Highways and elevated railways are being constructed using the external
prestressing with precast segments. Another application of external prestressing is the
strengthening or rehabilitation of existing concrete structures, which is restored for
economical of legal reasons instead of being demolished. Furthermore, the application of this
technology have paved way to many innovative structures. Extra-dosed bridge is one such
example where the cable is placed above the girder over the supports in continuous bridges,
similar to the cable stayed bridge, but with a short tower. External prestressing has been
applied also in composite bridges such as steel beams with a concrete top slabs, or other

a) Extra-dosed bridge with a short tower

b) Bridge with large eccentric cables
Fig.1.4 New type of bridges using external cables

-8-


c) Beam with external cables and folded steel web
Fig.1.4 New type of bridges using external cables (Continue)

combinations of steel elements and concrete slabs. Recently, external prestressing is
increasingly used in composite structure consisting of concrete flanges and folded steel plates
as web such as T-beams or box girder bridges. By this method the self-weight of the structure
is greatly reduced and span length can be increased as compared to classical concrete girders.
Fig.1.4 shows several types of bridges using external cables, which have been recently built in
Japan.
1.2.2

Previous investigations

After the recent revival of interest in the external prestressing, Virlogeux, M.2, 3) was one of
the earlier authors to explain the entire important factor involved in the analysis of externally
prestressed concrete beams and to propose a method of analysis to take into account all these
parameters. For the service stage of analysis, the cable length variation between the two
successive deviators was obtained from the displacement of the deviators by assuming that the
beam was uncracked, remained linearly elastic and no any movement at these deviators. At
the ultimate state, the author proposed a plastic hinge concept for predicting the cable
elongation. For considering the friction at the deviators, the author has made use of the
Cooley formulation applied to a cable with discrete deviators.
Muller, J. and Gauthier, Y.4) have developed a finite element program with 3-D element for
the analysis of precast segmental box girders. The program was designed to predict the
complete moment versus curvature response of simply supported and continuous beams
beyond joint opening up to the ultimate limit state. They have considered only elastic material

-9-


properties, but considered the opening of joints between segments in the analysis. However,
their analysis model is limited by several drawbacks, which include: 1) the model requires
information regarding the moment versus curvature or moment versus the joint rotation
relationship of each element; 2) the model does not account material non-linearity; 3) no
verification was made regarding the similarity between the beams prestressed with internally
unbonded cables, and those with external cables (eccentricity variations or second-order
effects were not isolated). They concluded that behavior of beams prestressed with either
internal or external cables is essentially the same way at all loading stages up to ultimate.
However, the results of some experiments, which were lately conducted by the other
investigators, contradicted their conclusions.
Ramos, G. and Aparicio, A.C.5, 6) developed a nonlinear analysis using the finite element
method including the nonlinear behavior of material and geometrical non-linearity for the
prediction of load-displacement response of the monolithic or segmental beams with internal
or external cables. Two extreme cases of bond condition (free slip and perfectly fixed) at the
deviator points were considered in the analysis. The analysis takes into account second-order
effects to evaluate the possible loss of the eccentricity of the external cables at the midspan
section.
Pisani, M.A.7) developed a method to evaluate the behavior of singly supported beams with
external cables and symmetrical loading condition. The algorithm based on the finite different
method includes second-order effects and large displacement. The analysis takes into account
two extreme cases of bond condition of cable at the deviators, namely, free slip and perfectly
fixed. However, the precast segmental beams, especially, the beams with dry joints are
excluded from the proposed method of analysis.
Kreger, M.E. et al.8) modeled segmental structures with external cables using the finite
element method. However, they arrested cable movement at the deviator points, and their
main aim to examine the effect of dry joints on the strength and the ductility of box girder
construction.

El-Habr, K.C.9) developed a nonlinear analysis algorithm based on the finite element
method for the prediction of the moment versus deflection response of externally prestressed
bridge girders composed of precast elements. The purpose of investigation was to determine
several important limit states, namely, cracking of concrete, opening of the joints between
segments, yielding of unbonded cables and ultimate nominal capacity. The analysis was taken
into account two nonlinear effects, namely, nonlinear material behavior and opening of the
-10-


joints at the interface of the precast segments. The model is limited because it does not
consider slipping of the cable at the deviators and the stiffness of the joint element is obtained
from a parametric study to avoid ill conditioned stiffness matrices, losing any physical
meaning.
Alkhairi, F.M, and Naaman, A.E.10) presented an analytical procedure for unbonded
prestressed concrete beams with internal or external cables using the moment curvature
relationships. They considered material non-linearity, span to depth ration and the effect of
eccentricity variation. The model has been applied for simply supported beams and must be
extended for continuous beam. The proposed model does not accept segmental construction,
and no slipping of the cable at the deviators is allowed.

1.3 GENERAL OVERVIEW OF PROBLEM
1.3.1

Problem of externally prestressed concrete beams

External prestressing was at first found very convenient as technique for repair of posttensioned concrete beams. It is now currently used in construction of new bridges. A
significant number of monolithic or precast segmental prestressed concrete box girder bridges
with external cables have already been constructed. Substantial economic and construction
time saving have been indicated for this type of construction. However, relatively little
analytical investigation has been undertaken to evaluate the behavior of such bridges,

incorporating the new developments for all range of loads. Nevertheless for the analytical
purpose still exists a general problem of computing tool, which has to account for the best
new aspects arisen from a specific structural behavior.
Unlike the analysis of beams prestressed with either internally unbonded or external cables,
the analysis of beams prestressed with bonded cables is well understood and documented in
the technical literature11~19). This is attributed to the perfect bond assumption that exists
between the prestressing cable and the surrounding concrete. This assumption leads to a
relatively simple section-analysis at the section of maximum moment. That is the stress in the
prestressing cable in a bonded member is a section-dependent and may be determined by the
strain compatibility approach applied to the failure section.
The case is quite different for beams prestressed with either internally unbonded or
external cables, where the perfect bond assumption between the prestressing cable and

-


surrounding concrete is no longer valid. Generally, in an unbonded member where relative
slip occurs between the prestressing cable and the adjacent concrete, the compatibility of
deformation in the prestressing cable and adjacent concrete over the entire length of the
member must be considered in determining the increase of strain in the prestressing cable at
ultimate. In this case, a section-analysis based on the strain compatibility along the section is
not sufficient to provide a complete solution as in the case of the beams with bonded cables;
the stress increase in the unbonded cables beyond the effective prestress due to the applied
load is member-dependent instead of being section-dependent. Rather, the stress in the cable
at any loading level during the response history depends on the total change in length of the
concrete at the cable level between the end anchorages. This assumption is generally
appropriated for the analysis of conventionally beams prestressed with external cables as
proven so far in many previous studies20~24). Since the cable portions are almost placed
outside the depth of cross section as in the case of the beams prestressed with large eccentric
cables, the concrete at the cable level, therefore, does not exist. As a result, the overall

deformation of concrete at the cable level does not appropriate to apply for the analysis of the
beams prestressed with large eccentric cables. This makes complicated computing procedure
for the strain variation in an external cable. Thus, there is an increasing need to look more
closely at the analytical procedure.
Because there is no compatibility between the strain in the prestressing cable and the
concrete at every cross section, the increment of stress at ultimate must be evaluated by taking
into account the whole structure, rather than performing the calculation at each section,
independently. This changes the principles for the structural analysis, which cannot be
maintained as a section-analysis. Therefore, it is necessary to formulate the global
deformation compatibility between the end anchorages. This means that the strain change in
the cable is member-dependent and is influenced by initial cable profile, span-to-depth ratio,
deflected shape of the beam, boundary condition of the end of beam, amount of initial
prestress, etc. This makes the analysis of beam with external cables more complicated, and a
proper modeling of the overall beam deformation becomes necessary.
An analytical method for externally prestressed concrete beams can, in principle, be the
same as that of prestressed concrete beams with unbonded cables. There are, however, two
specific problems, which commonly arise concerning the behavior of prestressed beams using
external cables at the ultimate state. Firstly, the increase of cable stress is a function of the
total deformation of the beam between the extreme ends, and depends also on the slip at the
-12-


contacted points between the concrete beam and the cable, at which the frictional resistance
always exists. Secondly, second order effects appear due to the fact that the cable remains
rectilinear between two successive deviators or anchorages in the process of the beam
deformation. Since the bond between the concrete and the prestressing cables is eliminated, as
a result the friction inside the ducts is artificially reduced to minimal, the stress variation in an
unbonded cable is assumed to be uniform over its entire length. When compared with
internally unbonded cables, the cable stress calculation is more involved in the case of
externally prestressed concrete beams due to the shift of cable eccentricity and the possible

frictional resistance at the deviation points.
1.3.2

Differences between internally unbonded and external cables

In an external prestressing system, the prestressing cables are not bonded to the
surrounding concrete. Hence, beams prestressed with external cables can be treated as
unbonded prestressing member. Therefore, the same parameters that are known to influence
the behavior of the beams with internally unbonded cables are expected to influence beams
with external cables. Although the behavior of externally prestressed concrete beams is
conceptually similar to that of beams with internally unbonded cables, the main difference
between the internally unbonded and externally unbonded prestressing cable lies in the
deflected shape of the beam and the cable. When beams with internally unbonded cables are
subjected to an applied load, the deflected shape of the internal cable usually follows the
deflected shape of beam itself throughout the entire span, the position of the cable relative to
the axis of the beam remain practically unchanged with increasing the beam deformation. On
the other hand, the external cable does not follow the beam deflection, except at the deviator
points, i.e., the external cables are free to move relative to the axis of the beams between the
anchorages or between the deviator points and anchorages (see Fig.1.5). This leads to a
Applied
load

External
cable

Deviator
Fig.1.5 Difference between the deformed shapes of beams
and external cables

-

gradual change in their eccentricity with increasing beam deformation giving rise to what is
known as second-order effects. Even though these effects may not be so significant at the
service load stage, they can have a considerable effect at the ultimate load stage depending on
the span-to-depth ratio, cable profile and position, and spacing of deviators, etc. Consequently,
the load-deflection response and ultimate strength characteristics of beams prestressed with
external cables are somewhat different from those of beams prestressed with internally
unbonded cables.
As Virlogeux, M.25~27) quoted that in the service limit state, there is no serious difference
between internally unbonded cables and external cables. The same specifications can apply
for tension limitations. There is no reason to reduce stresses in the external cables. The lack of
bond could be considered as a drawback, but on the other hand, it allows for the replacement
of cables if necessary, and the stress variations produced by the live load are more limited
than in the internal cable as shown for the ultimate limit states. Finally, the external cables are
completely independent from the concrete beam, and have not to suffer from the effects of
limited cracks on the prestressing cable durability.
The great difference in the structural behavior of internally unbonded and external cables
comes with the ultimate limit state. Since the unbonded cables have no friction with the
surrounded concrete, the stress variation is uniform over the cable length, and beams with
unbonded cables exhibit like a flexural member. The situation is completely different with
external cables. If there is no friction between the external cables and the concrete at the
deviator points, the tension is uniform in each external cable from one anchorage to the other.
The applied load can only produce an elongation of external cables, which corresponds to the
global deformation on the structure between two extreme anchorages. If the second-order
effects can be neglected, this elongation corresponds to the average deformation in the
concrete beam at the external cable level along it length. In this situation, the stress variations
are limited in the external cables, and the yield point cannot be reached excepted if the
deflections can become extremely large.
In a case the external cables have a greatly free length, the second-order effect cannot be

neglected because the reduction in cable eccentricity becomes more pronouncedly as the
applied load increases. When the crushing strain reaches in the concrete at the midspan
section, the second-order effect becomes large, leading to premature failure as compared with
the beams with unbonded cables. Sometime beams prestressed with external cables at ultimate
exhibit like a shallow tied arch member rather than a flexural member. This is obvious for the
-14-


External
cable

Crushing of
concrete

a) Shallow tied arch behavior

External
cable

Crushing of
concrete

b) Flexural behavior

Non-prestressed
reinforcement

Fig.1.6 Behavior of fully prestressed and partially prestressed beams
with external cables

case of fully prestressed beams with external cables, in which the cables have a straight
configuration as shown in Fig.1.6.
Muller, J. and Gauthier, Y.4) also quoted that when the beam is close to failure, a slight
increase of the eccentricity magnifies the beam deflection. The increased deflection induces a
loss of post tensioning cable efficiency if the cable does not follow the concrete deflection.

1.4 OBJECTIVES AND RESEARCH SCOPE OF THESIS
1.4.1

Objectives of thesis

Although an extensive body of experimental studies has been conducted to understand the
behavior of externally prestressed concrete beams, a method of prediction, which gives results
in close agreement with the experimental observations, is still in the research process.
Nevertheless, the characteristic behavior of externally prestressed concrete beams at the
ultimate state is a research topic, which has yet to be well understood in any depth. The
demand for a better understanding of the experimental observations has been an analytical
research need.
From the review of the previous studies, it is apparently found that although several
researchers attempted to discuss in detail in their analytical methods the influence of more or
less parameters involved in the analysis of beams prestressed with external cables, there have
been only few attempts to include all the parameters in the same method of analysis.
-


Moreover, it can be noted that apart from the development of a method, not many researchers
have attempted a systematic investigation on the influence or relative importance of the
various parameters. To investigate the stress variation in an external cable, most analytical
models neglected the effect of friction at the deviators because of its unknown extent.
Although several researchers investigated the strain variation in the external cables on the

basis of total compatibility requirement, most of them calculated the increment of cable stress
beyond the effective prestress by the imperial equations with some parameters involved for
certain cases21, 28~32) or by using the equations given in the codes for unbonded cables33~35). To
the best of author’s knowledge, none of researchers have attempted to incorporate the
deformation compatibility of beam (for the case of conventionally prestressed concrete beams
with external cables) or the deformation compatibility of cable (for the case of beams with
large eccentric cables) with the cable friction at the deviators in order to examine the increase
of cable strain of each segment under the applied load.
As mentioned above, there are some limitations of available analytical methods for the
beams prestressed with external cables, which are more or less related to computing method
of stress variation in the external cables at ultimate. Therefore, the main objectives of this
study are to develop a numerical method of analysis for beams prestressed with external
cables, which can overcome these limitations such as friction at the deviators, cable
eccentricity, etc. Also the proposed method should be appropriated for the numerical
investigations of all kind of beams prestressed with external cables such as simply supported
or multiple span continuous beams with any cable configuration, beams with or without
deviators and beams with large eccentric cables placed above the top or under the bottom of
cross section. Since the strain variation in an external cable depends on the overall
deformation of the beam, an equation for computing the cable strain should be developed in
the relative change of deformation of the beam, i.e., the total compatibility requirement of the
beam should be satisfied. The distributions of cable strain through the slippage and computing
procedure for the cable slip are also considered in this study.
1.4.2

Scope of thesis

Since this is the first stage of development of analytical method, the applications of the
proposed method do not perform for all kind of the beams prestressed with external cables in
the current study. This is because the results from the experimental observations of such kind
of beams like composite beams or slabs are not available in the technical literature and limited

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time of the course of study. Therefore, the applications of the analytical method concentrate
only on prestressed concrete beams with external cables with arbitrary cross section and
loading condition. Prestressed concrete beams with unbonded cables are also applied by the
proposed method. By setting a large number of “fictitious deviator” along the beam and zerofriction at these deviators, no difficulties are found when beams with unbonded cables are
being analyzed. The proposed method is only applied for the analysis of externally prestressed
concrete beams subjected to monotonic loading. Of course analysis of beams prestressed with
external cables subjected to cyclic loading excludes from this study. It should be noted that all
the predicted results are interpreted in the light of the experimental findings.
At present, a 2-D model for beam prestressed with external cables based on the finite
element method with isoparametric elements is underway to develop for the next research
purposes. Shear failure and post-peak behaviors of beams prestressed with external cables
such as snap-back behavior are also interesting topics for the further development. And more
applications for composite beams prestressed with external cables such box girder with folded
steel web, bridges with a short tower like extra-dosed bridges or slabs with a great span-todepth ratio will be carried out in the future.

1.5 ORGANIZATION OF THESIS
This study presents intensively numerical investigations on the behavior of beams
prestressed with external cables up to the ultimate state. A non-linear analysis is performed on
the behavior of beam prestressed with external cables, which include either simply supported
or continuous beams with or without deviators, beams prestressed by cables with either
straight or polygonal configuration, beams subjected to one or more loading points with
arbitrary loading condition. The present study may be organized in seven chapters (see
Fig.1.7), the content of each chapter is given below:
Chapter 1 presents the introduction and general overview of problem of externally
prestressed concrete beams. Some numerical methods proposed by the other researchers for
externally prestressed concrete beams are shortly reviewed and discussed. The advantages and
disadvantages of external prestressing are given. Historical development of external

prestressing system and its applications in the past are briefly described. Finally, the
objectives and research scope as well as the organization of thesis are defined and given at the
end of this chapter.
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Organization of thesis

Chapter 1

Introduction

Chapter 2

Non-linear analysis algorithm and
displacement control method

Chapter 3

Analytical methodology for externally prestressed
concrete beams

Chapter 4

Numerical analysis of conventionally prestressed
concrete beams with external cables

Chapter 5

Chapter 6

Chapter 7

Parametric Study

Numerical Investigation of externally prestressed
concrete beams with large eccentric cables

Summary and conclusions

Fig.1.7 Organization of thesis

In Chapter 2, non-linear algorithms together with displacement control method are
described. The displacement control method and its solutions for structures with arbitrary
loading condition are presented in detail. And the Newton-Raphson iterative procedure for
capturing the non-linear behavior of structure is also presented, briefly. General solutions
dealing with the beam element with six degree of freedom are shown. It should be noted that a
single displacement control point, which can be chosen among the load points, is applied in
the analysis. Finally, a flowchart of a stepwise analysis and computing program for beams
prestressed with external cables from the zero loading stage up to the ultimate loading stage is
presented.
In Chapter 3, a non-linear analysis procedure for externally prestressed concrete beams
with consideration of coupled effects of shear deformation and friction at the deviators is
presented. A complete procedure of formulation for the beam element as well as the cable
element by using finite element method is described in detail. This includes matrices for the
calculation of concrete strain and cable strain, stiffness matrix for a beam element, stiffness
matrix for a cable element. Some equations for computing strain increase in the external
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