Numerical analysis of externally prestressed concrete beams part 1

Numerical analysis of externally prestressed concrete beams

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

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

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 extra-dosed 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

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

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

Fig.1.1 Classification of prestressed concrete beams

Unbonded Prestressed Concrete BeamsPrestressed Concrete

Beams

Internally Unbonded Bonded Prestressed

Concrete Beams

Perfectly Bonded

Partially Bonded

css εε=Δ

l 0

Externally Unbonded

Assumption for computing method of cable strain

Unbonded Prestressed Concrete BeamsPrestressed Concrete

Beams

Internally Unbonded

Internally Unbonded Bonded Prestressed

Concrete Beams

Perfectly BondedPerfectly

Partially BondedPartially Bonded

css εε=Δ

l 0

1εε= ∫ΔΔ slcsdx

l 0

Externally Unbonded

Assumption for computing method of cable strain

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

Fig.1.3 Classification of externally prestressed beams

Deviators are located within the depth of cross section

Cross SectionArrangement of external cables

Deviators are located within the depth of cross section

Cross SectionArrangement of external cables

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

• 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

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

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-Saint-Georges, 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

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

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

c) Beam with external cables and folded steel web

Fig.1.4 New type of bridges using external cables (Continue)

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

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 tensioned 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

post-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

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

Fig.1.5 Difference between the deformed shapes of beams

and external cablesExternal

Applied load

Applied load

Deviator

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

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

Fig.1.6 Behavior of fully prestressed and partially prestressed beams

with external cables a) Shallow tied arch behavior

b) Flexural behaviorExternal

Non-prestressedreinforcementCrushing of

concretea) Shallow tied arch behavior

b) Flexural behaviorExternal

Non-prestressedreinforcementCrushing of

concrete

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

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 zero- friction 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-to-depth ratio will be carried out in the future

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

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

Fig.1.7 Organization of thesis Chapter 2

Chapter 2

Non-linear analysis algorithm and displacement control methodNon-linear analysis algorithm and displacement control method

Analytical methodology for externally prestressed concrete beams

Analytical methodology for externally prestressed concrete beams

Numerical analysis of conventionally prestressedconcrete beams with external cables

Numerical analysis of conventionally prestressedconcrete beams with external cables

Numerical Investigation of externally prestressed concrete beams with large eccentric cablesNumerical Investigation of externally prestressed concrete beams with large eccentric cablesParametric Study

cables, which were proposed by the previous researchers, are reviewed and discussed An equation computed the increment of cable strain, which is based on the deformation compatibility of beam and the force equilibrium at the deviators, is then proposed, in which the increment of cable strain is computed based on the change of overall deformation of the beam, friction at the deviators and cables angles

Criteria for the cable slip based on the force equilibrium at a deviator are described Before the proposed equation for the cable slip and its computing procedure, the some equations for the cable slip proposed by the others researchers are briefly reviewed Initial condition of the beam as well as the second order effects such as eccentricity variation of external cable, joint opening of precast segmental beams and is briefly presented at the end of Chapter 3

In Chapter 4, using the proposed method presented in Chapter 3 performs a comprehensive program of numerical investigations for conventionally prestressed concrete beams with external cables Simply supported beams prestressed with external cables as well as multiple span continuous beams with arbitrary cross section and loading condition tested by many researchers are considered for the analysis as numerical examples Effects of different parameters on the behavior of externally prestressed concrete beams such as loading pattern, amount of non-prestressed reinforcement, distance between two successive deviators, cable layout, casting method, etc are also studied in Chapter 4 The predicted results are presented and discussed in terms of both load vs deflection and load vs increase of cable stress curves under the light of the experimental findings Since friction exists naturally at the deviators or at the intermediated supports of continuous beams, the redistribution of cables strain though the slippage is also investigated The validity of the proposed method is verified by comparing the predicted results with the experimental observations, which are available in the technical literature Some concluding remarks drawn from the predicted results as well as the experimental observations are given at the end of Chapter 4

Chapter 5 performs a parametric study of the effect of friction at the deviators and loading arrangement on the behavior of externally prestressed concrete beams The friction effect is investigated in four different cases: 1) free slip; 2) slip with friction; 3) partially fixed; and 4) perfectly fixed The effect of loading arrangement is carried out on two span continuous beams having cables continued from one end to the other end The applied load is arranged so that it makes unbalanced loading arrangement on the both span The effect of loading arrangement is studied in five different cases in this chapter The load carrying capacity and the stress variation in the external cables are then discussed with emphasized the friction and

the loading arrangement Some concluding remarks drawn from the parametric evaluation are given at the end of this chapter

In Chapter 6, problems of externally prestressed concrete beams with large eccentric cables are initially given Since the deformation compatibility of beam, which is suitable for the analysis of conventionally prestressed beams with external cables, is not appropriated for the analysis of externally prestressed beams with large eccentric cables, a new equation for the calculation of cable strain based on the deformation compatibility of cable and the force equilibrium at the deviators is developed A non-linear analysis of beams prestressed with large eccentric cables is performed with three main aims: 1) to verify the validity of the proposed equation for the computing cable strain in the analysis of beams prestressed with large eccentric cables; 2) to verify the applicability of the developed equation in the analysis of conventionally prestressed beams with external cables; 3) to show the effect of cable eccentricity on the behavior of externally prestressed concrete beams by comparing the results obtained from the numerical investigations as well as from the experimental observations of the both kinds of the beams with external cables

The predicted results in terms of both load vs deflection and load vs increase of cable stress curves are presented and discussed to better understanding of the findings from the experiments The accuracy of the proposed method of analysis is then verified by comparing the predicted results with the experimental observations

Chapter 7 presents summary and conclusions from the course of study and gives some recommendations for the future research work

In general, the non-linear analysis is carried out in a step-by-step manner to trace the deformational history of structure based on the applied load, material characteristics, and geometrical deformations Although the load control method can capture the behavior of structure until the maximum load capacity of the beam, however, the displacement control method is usually chosen to trace the softening behavior in practice of structural analysis The displacement method is analogous to the force method except that nodal displacements are considered as the unknown instead of the redundant forces or moment Since the nodal displacements represent the freedom to move or rotate, the term “degree of freedom” represents the nodal displacements As well known Newton-Raphson algorithm with controlled displacement is used for solving the non-linear system of equations With this algorithm, the load increments at each calculation step depend on a single displacement, which is given the displacement controlled point The convergence is controlled through unbalanced forces between the internal force and the external force

In this chapter non-linear algorithm together with displacement control method is briefly

presented Solution for the beam element with six degree of freedom is then described in detail Application of Newton-Raphson iterative technique based on the displacement control method is shortly reviewed Finally, flowchart of non-linear analysis is presented It should always keep in mind that the efficient integration technique should be developed to reduce the computation time

2.2 NONLINEAR ANALYSIS ALGORITHM AND ITS SOLUTIONS

2.2.1 Application of displacement control method

Assume that a single force ΔF is applied at a displacement controlled point, at which the

incremental displacement ΔUk is given, and external forces, ΔF1, ΔF2, , ΔFn are applied at the points, at which the displacements ΔUu are unknown (Fig.2.1) These external forces can be expressed in terms of the proportional forces as λ1ΔF, λ2ΔF, , ΔF, λnΔF where λ1, λ2, …, λn

are coefficients defined by λi=ΔFi/ΔF The reactions ΔF1u, ΔF2u, …, ΔFnu are applied to the

fixed points, at which the displacements are equal to zero The proportional force vector is

expressed as (λ1ΔF,λ2ΔF, ,λnΔF,ΔF,ΔF1u,ΔF2u, ,ΔFnu)Tand rewrite in general form (λkΔF, ΔF, ΔFu)T, correspondingly the incremental displacement vector is (ΔUu, ΔUk, 0)T

From the force-displacement relationship can be written as:

(2.1)

And rewriting again

Δ=

2.2.2 Solving matrix for beam element with six degree of freedom

From the well-known load-displacement relationship, the following equation can be written as:

where [K] is the total stiffness matrix; {ΔU} is the nodal displacement vector; {ΔF} is the

vector of the applied load

Depending on the boundary condition of the beam, the displacements at the nodal points will be either zero or nonzero value, i.e., the displacements compose two components; one is a non-zero-value component; and the other is a zero-value component From Eq.(2.5), the nodal displacement vector {ΔU}, therefore, can be separately arranged into two components, and

then rewritten as in the following form:

(2.6)

where K11 and K22 are the stiffness matrices of free nodes and fixed nodes, respectively; K12and K21 are resulted stiffness matrices, when the total stiffness matrix has been separated; ΔF1is a vector of the applied load at free nodes corresponding to nonzero-displacement vector

ΔU1; ΔF2 is a vector of forces at the fixed nodes (reactions at supports) corresponding to the zero-displacement vector ΔU2

From Eq.(2.6) can be obtained:

The displacement vector ΔU2 at the fixed nodes is always equal to zero Therefore, Eq.(2.7) and Eq.(2.8) can be rewritten as:

Now, considers that a simply supported beam is subjected to several loads as shown in Fig.2.2 The vector of the applied load {ΔF1} can be written in terms of the proportional forces as:

(2.11)

in which the proportional coefficients of applied load bi are defined by the following equation:

where ΔF is an applied load at the displacement control point, and is being taken as a

referenced load; ΔFi is the applied load at the node (i)

The same way, we can write the nodal displacement vector for the free nodes,{ΔU1}

(2.13)

in which the proportional coefficients of displacement are defined by the following equation:

ΔΔ

where ΔUk is a given displacement at the displacement control point

It should be noted that the vector {b} of the proportional coefficients of applied load is

always known in advance before the calculation In the case of displacement control method, the incremental displacement ΔUk is given at the displacement control point before the calculation of each step, the unknown displacements at the other points will be then calculated by Eq.(2.4) Here, there is only an unknown value of the applied load ΔF at the displacement

control point To find the value of the applied load at the displacement control point by substituting Eq.(2.11) and Eq.(2.13) into Eq.(2.9), we can write as:

the simplest ways to determine the value of the applied load ΔF at the displacement control

point will be presented hereinafter

For the displacement control method, among the loading points, a single point is commonly chosen as a displacement control point Therefore, from Eq.(2.16) can write as:

(2.17)

Therefore, the value of the applied load ΔF at the displacement control point is defined as:

FcUk = dcp Δ

cUF= Δ

Once, the value of the applied load ΔF at the displacement control point is obtained, the

unknown displacement vector {ΔU1} can be defined by Eq.(2.16) The vector of reactions ΔF2at the fixed nodes (at supports) can be then determined by substituting {ΔU1} into Eq.(2.10)

To repeat this procedure until the final stage of loading, the entire load-displacement response of structure can be investigated

2.2.3 Application of Newton-Raphson iteration method

In this section, the displacement control method based on Newton-Raphson procedure will be reviewed, briefly

Assuming that the calculation is convergence for the last load increment, the load will be increased by {ΔF}, the tangential stiffness matrix at this stage is [K], and the residual force remains for the last load increment is {R} (although {R} is rather small to satisfy the convergent criteria, {R} is not equal to zero), {ΔU} is the increment of displacement The

nonlinear equation can be expressed as:

Eq.(2.20) can be rearranged into two components as in Eq.(2.6), and given at:

(2.21)

Solving the Eq.(2.21), we can obtain:

{ }[ ]({ } { }11 [ ]21{ 2})

111

finished at this stage It should be continuously checked that whether or not the residual force satisfies the convergent criterion, i.e., the external force is approximately equal to the internal force The residual force is determined as:

(2.25)

where (2) means the second iteration and {ΔU2}=0 Repeat this procedure until the residual

force becomes so small that it can satisfy the convergent criterion For the nth iteration, the equation of common form can be expressed as:

ni

ni

21

{ΔF}; {ΔR} is the residual force And Fig.2.4 shows the flowchart of the iterative procedure

for the displacement control method

2.2.4 Computing program and analytical flowchart

A finite element method has been firmly established as a powerful and popular analytical tool The conventional finite element method often approximates a deformed shape of beam element with interpolation functions such as a cubic polynomial function for transverse displacement and a linear function for longitudinal displacement The cubic function implies a linear variation of curvature along the element However, the analysis of unbonded members in general or the analysis of beams prestressed by external cables in particular necessitates an accurate evaluation of curvature variations since the compatibility equation should be formulated with the values of concrete strain at the lever of cable Thus, a large number of short elements are necessary for an adequate evaluation of cable strain

A non-linear finite element program together with the displacement control method has been developed to obtain the entire behavior of the structures up to the ultimate limit state

Fig.2.4 Iterative procedure by Newton-Raphson method

= i

= i

0=Δuo

The program uses a stepwise analysis and deformation control to trace the nonlinear response of prestressed concrete beams with external cables, which may be simply supported or continuous beams, and subjected to either concentrated with one loading point or more, or subjected to uniformly distributed loading This program is capable of accounting for not only the flexural deformation, but also for the shear deformation, friction at the deviators, location of deviator located within or outside the depth of cross section, and external cables with different configuration (straight or polygonal profile) In the analysis, the beam is represented by a set of beam elements connected together by nodes located at the either end Each node has three degrees of freedom, namely, horizontal displacement, vertical displacement and rotation A cable stress equal to the effective stress after all losses is taken as an initial value in the analysis Cross section of the beam is divided into layers, in which each layer might have different material properties but its properties are assume to be constant over the layer thickness Based on the effective stress of cable, the concrete strain of each layer for every beam element is determined, and appears to be taken as the initial condition of the beam

Fig.2.5 Flowchart of analytical method

ucs εε<

Discrete beam element, and layers of cross section

Establishment of the total stiffness matrix, matrix for the concrete strain,

and matrix for the cable strainEstablishment of vector of the applied

load, and given displacement at a displacement control point

Calculation of nodal displacement vector and vector of the applied load.

Calculation of concrete strain & stress,internal force and cable strain & stress

Calculation of total displacement at all

ucs εε<

Discrete beam element, and layers of cross section

Establishment of the total stiffness matrix, matrix for the concrete strain,

and matrix for the cable strainEstablishment of vector of the applied

load, and given displacement at a displacement control point

Calculation of nodal displacement vector and vector of the applied load.

Calculation of concrete strain & stress,internal force and cable strain & stress

Calculation of total displacement at all

UU<

Detail of formulation for the beam element and the correlation between the deformation of concrete element and cable element will be presented in Chapter 3

It is important to note that a single displacement control point, which can be arbitrarily chosen among the points of the applied load, is applied in the analysis It is truly said that application of a single displacement control point is in a general form, and can be solved for the arbitrary loading arrangement Finally, the analytical flowchart is presented in Fig.2.5

This chapter presents non-linear analysis procedure for the beam element based on the displacement control method together with Newton-Raphson iterative technique In the analysis, a single displacement control point is applied to capture the entire behavior of the beams No particular difficulties arise from different loading scheme or from the different constraint system, such as in simply supported or continuous beams subjected to one or more loading points, which are either symmetrically or non-symmetrically located from the center line of the beams

One of the major problems concerning the beams prestressed with external cables is in calculating the cable stress beyond the effective prestress In the case of beams prestressed with bonded cables, since the cable strain is assumed to be the same as the concrete strain at the cable level, the calculation of cable strain under the applied load is a problem related only to a section of maximum moment, i.e., the increase of cable strain is a section-dependent This is totally different in the case of beams prestressed with external cables Since the cable is unbonded, the cable freely moves in the relative change of the beam deformation Therefore, the cable strain is basically different from the concrete strain at every cross section, i.e., the cable strain cannot be determined from the local strain compatibility between the concrete and the cable For the calculation of cable strain, it is necessary to formulate the global deformation compatibility of beam between the extreme ends This makes the analysis of a

beam with external cables more complicated, and a proper modeling of the overall deformation of beam becomes necessary

When behavior of externally prestressed concrete beams was investigated, many researchers attempted to calculate the increase of cable stress beyond the effective prestress either by using their formulations with some parameters involved for certain cases21, 28~31, 38), or by using equations, which are provided in the codes for unbonded beams33~35) Since there is no crack under the service loads, the stress increase in the cable is extremely small, it can be negligible As a result, the cable stress at ultimate could be computed by using the strain compatibility with the strain reduction coefficient39~40) Whereas Lu, Z., et al.41) assumed that after cracking, the total elongation of a cable is equal to the total crack widths of concrete surrounding the prestressing cable Some researchers20~24) assumed that the strain variation in a cable is uniform over its entire length, and tried to calculate the cable strain by adopting the assumption, which states that the total elongation of a cable must be equal to the total elongation of concrete at the cable level Since the prestressing force is transferred to the concrete beam through the deviator points and anchorage ends, the cable friction obviously exists at the deviator points, resulting in a different level of strain increase between the two successive cable segments Due to the complicated calculation of cable strain, almost analytical approaches, however, did not consider friction at the deviators because of its unknown extent For the purpose of simplicity, some researchers consider only two extreme cases, namely, free slip and perfectly fixed at the deviator when the cable strain is computed5~7) And they also quoted that the measured increase in cable stress of a tested beam is always between those obtained from the numerical models considering free slip and perfectly fixed at the deviators This implies that there are some influences of friction on the increase of cables stress, which did not properly consider so far in many studies

In previous studies, test performed by Chouinard, K.L., et al.42)

on unbonded, partially prestressed concrete beams have indicated that the stress in the prestressing cable is greater than that computed from the compatibility of deformation based on flexural theory This discrepancy has been attributed to the effect of deformation due to the shear force not being taken into account in flexural theory43) Consequently, to accurately predict the stress in the unbonded cable, it is necessary to consider the effect of shear force in computing the deformation of concrete at the level of the prestressing cable along the length of the beam

Since external cables are unbonded with the concrete, the conventional analysis of flexural members based on the compatibility between strains in the cables and the concrete at a

particular section is not applicable to externally prestressed concrete beams An analytical model for externally prestressed concrete beams cannot be developed without considering the total compatibility requirement that the total elongation of a cable must be equal to the integrated value of concrete deformation at the cable level between end anchorages That is, strain avariation in an external cable depends on the deformation of every point of the beam This impllies that the adequate evaluation of cable strain depends on the accuracy in the calculation of concrete strain at the cable level Therefore, the beam should be neccesarily divided into a large number of short elements by using the finite element method

Therefore, this chapter will firstly present formulation of stiffness matrix for the concrete element based on the finite element method including the shear deformation And the computing equation for the cable strain in externally prestressed concrete beams on the basis of the force equilibrium at the deviators and the deformation compatibility between the concrete beam and the prestresing cable is then developed The process of calculating cable slip at the deviators is also explored in a general form for all kinds of cable configurations and multiple deviator points Finally, second-order effects such as eccentricity variation of external cable, joint opening for precast segmental beams, second moment due to prestressing and initial condition are also presented in this chapter

3.2 FORMULATION OF BEAM ELEMENT BY FINITE ELEMENT METHOD

3.2.1 Stiffness matrix of concrete element

In an X, Y coordinate system, a beam element with six degree of freedom is subjected to an axial force P, a shear force Q and a moment M (see Fig.3.1) Generally, displacements of the beam in the directions x and y are denoted as ux and vy, respectively These displacements can be expressed as the following:

Fig.3.1 Beam element

Y

where u is the increment of displacement caused by an axial force; vb is the increment of

displacement caused by a moment; vs is the increment of displacement caused by a shear force

The increment of axial strain εx and the increment of shear strain γxy can be obtained by differential with neglecting the high order terms as:

From Eq.(3.4), the displacement νs can be obtained as:

where EI is the flexural stiffness; GA is the shear stiffness and K=EI/GA is a ratio obtained by

dividing the flexural stiffness per the shear stiffness

The conventional finite element method often approximates a deformed shape of beam element with interpolation function such as cubic polynomial function for transverse displacement and a linear function for longitudinal displacement From finite element theory44) we can find polynomial representing displacement pattern for the beam element caused by moment and expressed as the following:

where {C}T={C1 C2 C3 C4} are four unknown coefficients of polynomial function order differential of Eq.(3.6) is performed, then substituting the result into Eq.(3.5), and arrives at:

Second-[ kKx]{ }Cx

(3.8)

Since rotation angle is defined as the first order differential of displacement, and from Eq.(3.6) can be obtained as:

y = = 2 +2 3 +3 4 2 = 0 1 2 3 2

Thus, from Eq.(3.8) and Eq.(3.9), the displacement vector can be written in Eq.(3.10)

() (){ }C [fxy ]{ }C

For a beam element with two nodes 1 and 2 as shown in Fig.3.1, the coordinate of each

node in the x direction is x1=0 and x2=L, respectively The nodal displacement vector can be

then written as:

[ ]H{ }dev

Substitute Eq.(3.11) into Eq.(3.6), Eq.(3.7) and Eq.(3.8) and arrives at:

vvbe

[ ][ ]{ }euue

u= 1⎢⎣⎡ − ⎥⎦⎤ =where

(3.13)

(3.14)

(3.15)

vbNNNNN =

[ ][011022]

vsNNNNN =

[ ][011022]

vyNNNNN =

{ }dT =[u1 ν1 θ1 u2 ν2 θ2]

These matrices have [1x6] elements with subscripts (1) and (2) indicated the nodal number

as shown in Fig.3.1 Differential is performed with respect to x for each displacement compo-

nent and arrive at:

= 1; 0; 0; 1; 0; 0'

(3.19)

where the matrices [ ][ e'e'']

[ ]=⎢⎣⎡0;−12 3;− 6 2 ;0; 12 3;− 6 2⎥⎦⎤

LKTs =+

Using the principle of virtual work, the incremental displacement can be obtained as:

[ ]K [ ] [ ]AEAdV [ ] [ ]BTGBdV

dV

By using the layer model, the concrete element is divided M, N layers in the X and Y

directions, respectively (see Fig.3.2) Each layer may have different material properties, but its properties are assumed to be constant over the layer thickness Then Eq.(3.22) becomes:

(3.23)

where Ejk,,Gjk are the Young’s modulus and shearing modulus of layer jk respectively, bjk is

the width of layer jk The matrix [Kc] is a symmetry matrix with [6x6] elements The displacement relationship can be expressed as:

load-[ ]K { }du

(3.24)

Every element of matrix [Kc] can be defined as below equations:

= =∑∑

[ 3()3()]

1114 ccKK =−

XNeutral axis

qkqk+1