DETERMINATION OF THE IN SITU COEFFICIENT OF FRICTION AND IMPERFECTION OF PRESTRESSING CABLES

Modern reinforced concrete structure of significant spans cannot be imagined without its prestressing. Nowadays, the design established at the very beginning of such systems, with small changes introduced in the recent past, is in use worldwide. One of the basic premises is that the force in the cable at the active end during tensioning must not exceed the value given in the following formula (1):

In the production of steel and plastic materials new things which enable the production of safer structures are introduced every day. Thus, with the application of modern steel for prestressing the relaxation of cables is reduced to a minimum. In construction practice in the case of prestressed structures for tracing cables plastic corugated pipes made of new types of plastics are used very often. The effect of the application of new types of materials on the values of the coefficient of friction is often left for the designer to estimate. Most often at the design level coefficient values are taken from applicable policies and regulations that may not take into account the application of new materials in prestressing. Previous experience in the field of determining the coefficients of friction and imperfection of the execution of prestressing are mainly related to the theoretical and laboratory testing. As it is already mentioned, the coefficient of friction depends largely on the applied materials, while the coefficient of imperfection depends largely on the level of training and conscientiousness of the contractor to perform work on the prestressing [3.4]. The parameters are adopted empirically within the limits prescribed by national standards and regulations of the country where the structure is performed. Actual determination of the aforementioned parameters is a very complicated process which requires additional resources from the contractor, as well as the involvement of specific equipment. Therefore the designed parameters are usually not checked.
In post-tensioning of the cables the prestressing force and the corresponding elongation of the cable must be checked by measurements, and actual losses since the friction have to be controlled. In addition to these parameters, it would be of great use for designers and contractors of the structure to analyze the parameters related to the coefficient of friction and the coefficient of imperfection of execution of work on prestressing [5].

THEORETICAL PREMISES
Mean value of prestressing force Pm,t(x) at a given time t, at a distance x from the active end of the cable is equal to the maximum force Pmax which tension the cable on the active end, reduced for current losses and losses that depend on the time. All losses are considered in absolute values.
The value of the initial prestressing force Pm0(x ) for t=t 0, where the concrete is exposed immediately after the tensioning and anchoring of the cables, or after the transmission of the prestressing to concrete, is obtained when the prestressing force Pmax is reduced by the current loss ∆Pi (x), and should not exceed the value obtained by the equation (2): gde je σpm0(x) napon u kablu neposredno posle utezanja ili prenošenja sile kao manja vrednost od k 7·fpk i k 8·fp0,1. Preporučene vrednosti za koeficijente jesu k 7=0,75 i k 8=0.85.
where σpm0(x) is the stress in the cable immediately after the tensioning or transmitting the force as a smaller value of k 7·fpk and k 8·fp0.1. The recommended values for the coefficients are k 7=0.75 and k 8=0.85.
Gubitak sile prednaprezanja javlja se u slučaju naknadno prednapregnutih elemenata usled trenja između kablova i zaštitnih cevi u betonu. Veličina ovog gubitka jeste funkcija oblika trase kabla -efekat zakrivljenosti i odstupanja prilikom montaže trase kablova -ugaono odstupanje. Vrednosti koeficijenata koji definišu gubitke sile često se preciziraju dok se The prestressing force is usually not constant along the cable due to the friction of the cable on the walls of the channels through which it passes and friction due to the changes in the direction of the cable. In addition, the tensioning force is reduced over time due to relaxation of steel, shrinkage and creeping of concrete.
Accordingly, in a cross-section of the cable there is the initial force Pm0 in the moment of prestressing, force Pm,t in some time "t" and permanent force for t ∞ → , Pm∞. For the permanent force the changes of the force due to the effect of permanent and moving load after prestressing should also be taken into account. Losses that are computed are presented in standard SRPS EN:1992-1-1:2004 [6].
When determining the current t=t 0 losses ∆Pi(x ) the following current impacts which lead to losses in tensioning are considered, depending on which of these impacts is relevant: − losses at anchorage -due to insertion of the wedge -∆Psl, − losses due to elastic deformation of concrete -∆Pel, − losses due to the short-term effects of relaxation -∆Pr , − losses due to friction -∆P µ (x). Mean value of prestressing force Pm,t(x ) at time t > t 0 is determined depending on the method of tensioning. In addition to these current losses, the losses of prestressing which depend on the time ∆Pc+s+r (x), which are the result of creeping and shrinkage of concrete and long-lasting relaxation of steel for prestressing should be taken into account, so that: Initial force in cable (Ppo) is less than the initial force on the press (Pp) due to the insertion of wedges during the transmission of force to the anchor. During the insertion of wedges simultaneously at approximately the same size, cables are also drawn, leading to a drop in force in the cable due to the lost elongation. Insertion of wedges occurs on the side of the active and passive anchor. Insertion of wedges in the passive anchor need not to affect the reduction of the initial force if total elongation at press is increased by this amount. Insertion of wedges on the side of the press, active anchor, leads to a reduction in the initial force in the cable for the size of the lost elongation, which can have significant value for short cables. Insertion of wedges of active and passive anchor is approximately the same and is taken at the level of 3-5mm. There are ways to partially or completely eliminate this loss with the use of special procedures.
Elastic deformation of structures, leads to a total loss of the initial force of the structure if more than one cable is being tensioned. The loss is significant with highly prestressed elements with small cross-section: hangers, piles and the like. Then the loss of force in the cable due to deformation of concrete, as well as dependence from the order of tensioning cables (according to EN: 1992-1-1: 2004) have to be taken into account.
Loss of tensioning force occurs in post-prestressed elements due to friction between the cables and protective pipes in concrete. The value of this loss is a function of the form of the route of the cable-effect of obavlja priprema projekta izborom različitih tipova i oblika trase kablova. Pošto je efekat zakrivljenosti unapred određen, ugaono odstupanje jeste rezultat slučajnog ili neizbežnog odstupanja, pošto ne postoji mogućnost da se trasa zaštitnih cevi idealno montira [7].
Gubici usled trenja ∆P µ (x ), pri utezanju kablova, mogu da se procene prema obrascu (3): curvature and deviation during the installation of cable route-angular deviation. The values of coefficients that define the losses of force is often specified while the preparation of the project by selecting different types and shapes of cable route is being done. As the effect of the curvature is predetermined, angular deviation is the result of accidental or unavoidable deviation, as there is no possibility for the trace of protective pipes to be ideally instaled [7].
It should be kept into account that the maximum value of the loss of force due to friction will be on the other end of the element, if it is tension from one end. Therefore, loss due to friction varies linearly along the element span and can be interpolated for specific positions if there is a need for greater accuracy.
In previous formula θ represents the sum of the diverting angles at a distance x expressed in rad; µ is the coefficient of friction between the cable and pipes in which it is placed, expressed in rad -1 ; k determines accidental diverting angle of internal cables expressed in rad/m; and x is the distance expressed in m` along the cable from the point where the force in the cable is equal to Pmax i.e. the force on the active end of the cable during tension time.
For the friction coefficient µ it should be emphasized that it is highly dependent on surface characteristics of cables and pipes, in particular on the degree of corrosion of contact surfaces. Amout of friction also depends on the type of pipes used and the type of cable used. There are two mechanisms that produce friction. The first is the curvature of the cable, and the other is a random difference between the center of gravity of the cables and the pipe.

EXPERIMENTAL WORK IN SITU
The complete experiment was performed on construction site during the construction of the bridge Zemun-Borca. The contractor was the CRBC ( China Road and Bridge Corporation). The construction of the bridge outside the water had two systems. One of the system included the making of prestressed girders with static length 26 and 36m. The experiment used the girder with the length of 26 meters. I n the design tensioning of the girder with 4 cables of 7 ropes each with the diameter of 15,2mm was planned, while for the trace of cables corrugated plastic pipe was used. A typical cross-sections and longitudinal-section of the girder with traces of two tested cables are shown in Figure 1. Eksperiment je obuhvatio određivanje koeficijenta trenja i koeficijenta imperfekcije, s tim što nisu uzimani u obzir efekti zaklinjavanja i elastične deformacije betona. Efekti zaklinjavanja nisu se imali u vidu, jer su procedurama i modifikacijama IMS sistema prednaprezanja uticaji svedeni na minimum. Presa za utezanje modifikovana je tako da se zaklinjavanje vrši pre otpuštanja kablova. Poštujući proceduru upravljanja presom, efekat zaklinjavanja se može zanemariti. Projektom je predviđeno utezanje kompletnog nosača sa četiri kabla, a u eksperimentu je planirano utezanje s jednim kablom do ~90% projektovane sile. Efekat elastične deformacije betona na ovom nosaču i pri usvojenoj dispoziciji ispitivanja može se zanemariti, jer se vrši utezanje samo s jednim kablom koji se otpušta pre utezanja sledećeg kabla.
The experiment included the determination of the coefficient of friction and the coefficient of imperfection, except that the effects of wedging and elastic deformation of concrete were not taken into account. The effects of wedging were not taken into account because the procedures and modifications of the IMS system of presstresing minimized the impact. Press for tensioning has been modified so that wedge done before releasing the cables. By respecting procedures of using press wedging effect can be ignored. Design is anticipated tensioning of the complete girder with 4 cables, while the experiment tensioning with one cable up to ~90% designed force is planned. Effect of elastic deformation of concrete at such girders in adopted disposition tests can be ignored becouse it performs tensioning with only one cable which is released before tensioning second cable.
It was planned for cable N1 to be tensioned first and up to ~90% of the designed force in the cables. Then wedging was carried out and the force was monitored for the next 24 hours. While tensioning the cable measurements were conducted for every 20% of the maximum designed force. Practically force measurements were conducted at 20, 40, 60 and 85-95% of the maximum designed force in the cables. Measurements were also carried out after 24 hours. Then the released of tensioned cable N1 was performed and tensioning of cable N2 was started. All measurements were performed the same as for cable N1. After reaching the planned force wedging was performed and the force was followed for the next 24 hours. At the end released of cable N2 was performed. The results of the tensioning force measurements on the active and passive end are given in table 1.  1304kN  1302kN  1276kN  1180kN  1179kN

DISCUSION AND CONCLUSIONS
Test results processing was done in two ways. Coefficients of friction and coefficients of imperfection on the basis of tension forces in two cables as well as an evaluation of the comparison of the coefficient of friction for different levels of tension cables.
Calculation of the coefficient of friction and the coefficient of imperfection is reduced to solving a system of equations ( (4) and (5)) with two unknowns.
Total deviation angle θ1 in the cable N1 was 15.346°, and θ2 in the cable N2 amounted to 12.136°. Length at which the measured value of the reduction in force is the end of the cable x=26.097m. Based on the input data and solving the system of equations with two unknowns, the values for the coefficient of friction of cables μ=0.22 and imperfection coefficient k=0.004 were obtained. After solving the system of equations a calculation of the coefficient of friction of cables was made separately for different levels of cables tensioning. The calculation was done separately for each cable and imperfection coefficient ranging from 0.001 to 0.01 with step 0.001. The obtained results are shown in Figures 4 and 5. By joint processing of the test results for two cables the obtained results are comparable with the standard and recommendations defined by the values from Table  2. As it can be seen, the value obtained for plastic corrugated pipes, the friction coefficient μ=0.22 is slightly higher, while the coefficient of imperfection k=0.004 is in the suggested limits. The coefficient of i mperfections depends greatly on the contractor and its ability to instal the cord as good as possible. This significantly reduces the imperfections and the impact can be brought down to a minimum. The coefficient of friction cannot be influenced and its value depends the most on the materials used for tracing the cable. Razmatrajući rezultate po nivoima unošenja sile u kablove, jasno je da je na manjim silama utezanja koeficijent trenja daleko veći. Prilikom postavljanja kabla i manjih sila u kablovima, prvo dolazi do nameštanja kablova i gubici od aktivnog do pasivnog kraja su znatni. Kasnije, kada se kabl namesti u konačnu poziciju, trenje se smanjuje i dolazi na vrednosti koje se mogu pretpostaviti, u zavisnosti od materijala cevi za vođenje kablova.

Tabela 2. Preporučene vrednosti proizvođača za koeficijente trenja i imperfekcije
Considering the results according to the levels of force application in the cables, it is clear that for lower tension forces coefficient of friction is far greater. When setting up the cable and lower forces in the cables, the cables are first being installed and losses from the active to the passive end are considerable. Later, when the cable is installed in the final position the friction decreased and the values that can be assumed depending on the material of the pipes for cables are obtained.