HVSTERETIC RESTORING FORCE CHARACTERISTICS OF UNBONDEDPRESTRESSEDCONCRETEFRAMEDSTRUCTURE UNDER EARTHQUAKE LOAD

An analytical method, by which hysteretic restoring force characteristics of unbonded prestressed concrete framed structure can be statically pursued on the basis of material properties, is presented. The bond-slip relationship between concrete and prestressing tendon is taken into account, and thus the method covers unbonded members and bonded members. For verifying the propriety of the analytical method, the experiment is carried out on a portal frame with an unbonded prestressed concrete beam of 4.2 min length and reinforced concrete columns of 1 min height. High intensity reversed cyclic lateral loading is applied. The experimental results show a good agreement with the analytical ones in terms of load-deflection relation and the fluctuation of the tendon stress at anchorage end.


INTRODUCTION
In seismic areas, the application of unbonded prestressed concrete to primarily earthquake resistant members is prohibited due to several reasons, i.e. safety of tendon anchorage assembly against cyclic earthquake load, uncertainties with regard to the fluctuation of tendon stress, little available data on hysteretic restoring force characteristics and complexity in analysis, etc.However, unbonded prestressed concrete members can be very useful to develop the further demand for prestressed concrete structures, because of economical advantage of unbonded tendon, that is, no need for grouting at the construction site, and of practically perfect protection against corrosion comparing with the grouting which is likely to be imperfect.
In addition, the past researches reported that small amounts of additional nonprestressed reinforcement can improve the restoring force characteristics and the flexural ductility [1,4].
In this study, an analytical method, by which hysteretic restoring force characteristics of unbonded prestressed concrete members and framed structures can be statically pursued on thb basis of material properties, is presented.
In this method, a structural member is divided into several blocks along member axis and each block is further subdivided into layers, which reflect mechanical properties of the materials.The stiffness matrix of the member is derived from these segments based on the assumptions that stresses and strains are *Kyoto University, Japan.constant in a segment and the cross section of the member remain plane after loading.The bond-slip relationship between concrete and prestressing tendon is taken into account, and thus, this method covers unbonded members and bonded members.
For verifying the propriety of the analytical method, the experiment is carried out on a portal frame with an unbonded prestressed concrete beam of 4.2 min length and reinforced concrete columns of 1 min height.High-intensity reversed cyclic loading is statically applied.The experimental results are compared with analytical results in terms of lateral load -deflection relation and the fluctuation of tendon stress at the anchorage end.

GENERAL DESCRIPTION OF ANALYTICAL METHOD
For the purpose of numerical calculation, a structural member is divided into several blocks in the direction of the longitudinal member axis and each block is further subdivided into layers.This method is called "Layer Element Method" and it has been developed by many researchers.
In this study, the following are assumed: 1) Stress and strain are constant in a layer element.
2) The cross section of the member remains plane after loading, i.e., the longitudinal strain in concrete and the nonprestressed reinforcement is proportional to the distance from the neutral axis.
3) Shear deformation is not taken into consideration.Although shear deformation is undoubtedly important in the case of the column, here the bending and axial force are assumed to dominate the deformation of the member.
4) The linear bond-slip relation between concrete and prestressing tendon is assumed.

5)
Assumed stress-strain relation of concrete and reinforcement are proposed by Kent and Park [2], and Park, Kent and Sampson [5].

Bond Stress and Slip between Reinforcement and Concrete
Increment of bond stress between (j-1)-th block and j-th block (6,j) is calculated by the following equation./\ T .
where, Kj = tangential modulus of bond-slip relation.
In this study, Kj is assumed to be constant during loading.

The System of Linear Equations of Increments of Slip
The method used here was proposed by Kosaka et al. [3].Dividing the member in the longitudinal direction into block elements i _ = 1, 2, ... , j , . . . of length lj , as shown in Fig. 1 From the equilibrium of forces, we get the relationship between the stress of reinforcement and slip as below, at the end 1 of member, as shown in Fig. 2; ( J-1 )-TH BLOCK J-TH BLOCK LJ-1 where, E 5 j and As= tangential modulus and cross sectional area of reinforcement, pKl = -6sP1/1:>.ps1 , i\ 0 K2 = AsP2/6pS2 , 6sP1 and AsP2 = force increment of the reinforcement, i\pSl and ApS2 = slip increment of reinforcement at the ends of the member.
Although pKl and pK2 should be determined by some calculation, in this study because of simplification it is assumed that they are constant and large enough to restrain the tendon from coming out from the joint.This could be justified because the prestressing tendon was anchored almost perfectly on the steel plate.
These equations (7-9) represent the system of linear equation of ASj .
Increments of slip (LSj) can be determined by given AcE:j (strain increment of concrete element located at the reinforcement level).
Substituting the calculated values of Sj into Eq.5, we get AbEj Stress increment of reinforcement Lsoj are calculated by

Derivation of the Stiffness Matrix of the Cross Section
The cross section of the member is subdivided in the z-direction into layers i 1,2, ... ,m of cr?ss-sectional areas Ati and centroidal coordinates Zi• .Centroiaal coordinates of longitudina1 steel reinforcement in j-th block is denoted as sZij, (i = 1,2, ... ,n}.
The longitudinal nominal strain increment of concrete at any point of the cross section is A E: ..
in which, E:oj = normal strain at the member axis, x;¢j = curvature of j-th block.
Bending moment LM,c and normal force AN,c in concrete are e~pressed by the followirig equations.
where, Aij = area of concrete in i-th layer of j-th element, and VcOij = normal stress in the concrete in the longitudinal direction in i-th layer of j-th block element.Iij = moment of inertia of i-th layer about centroidal axis of its layer.
Similar expressions for reinforcements can be obtained as follows: On the other hand, normal stresses in concrete 2 0 2 0 . . . .LI"' .
The total forces l'iN• and 6Mj in the cross section are obtainetl by summing up the contributions of concrete and steel.This yields 6N.
/'iM_c + 6M.s = s.6EOj + y j 6¢j i=l s iJ s iJ s iJ ( 26) Eqs. 23 and 24 represent the force-deformation relations for a cross-section of the beam.When Eqs. 23 and 24 are rewritten in the matrix form, this provides ( 27)

Consideration of Prestress
From the equilibrium of forces shown in Fig. 3, the increments of axial force and bending moment in j-th block, l'iNj and 6Mj are obtained as follows. I\N.
where, l'iN1, l'iQ1 and I\M1 = axial force and moment at the end 1 of the member, and 6MjP = moment induced by the prestress transfer, given by I\Np•ej, where /\NP= effective prestressing force, ej = eccentricity of tendon measured from the centroidal axis of the section.That is, prestress, which is divided into some loading steps, is assumed to be applied to each block as external loads /'iN~ and 6MjP The stiffness matrices of the member and the structure were constituted by the procedure presented by Kosaka et al. [3] and Tani et al. [6].The effect of prestress is included in the terms due to bondslip and nonlinearity of material properties.

Computational Algorithm
The computational algorithm in each loading step may proceed as follows: 1.For all elements, assume that 6Eoj and /'i¢j are the same as in the previous step.2. Set up the bond stiffness equation in which the bond stiffness and the tangential modulus of reinforcement are assumed to be the same as in the previous step.3. Solve the equation above to get the slip between concrete and reinforcement in each block.4. Set up the stiffness matrix of each block and transfer it into the flexibility matrix.5. Combine the flexibility matrices of the sections so that the member flexibility matrices can be obtained.6. Transfer the member flexibility matrices into the stiffness ones and constitute the stiffness matrix of the structure.7. Solve the equation above under some support conditions and get the unknown displacements and forces.8. Calculate ~EOj and ~¢j again by using the displacement and force increments obtained above.9. Compare the assumed values and calculated values of ~EOj and ~¢j , and return to step 1 if the difference between them is larger than one percentage of the assumed values.If not, go next.10.Calculate the stresses and tangential modulus of concrete and reinforcement according to the strains obtained by using ~EOi and ~¢j .If the tangential modulus ifi this step differs from the assumed value, unequivalent stress is calculated and released in the next step.

EXPERIMENT OF UNBONDED PRESTRESSED CONCRETE PORTAL FRAME
For verifying the propriety of the analytical method above, the experiment was carried out on two portal frames with an unbonded prestressed concrete beam 4.2 m long and reinforced concrete columns 1 m high, shown in Fig. 4.These two frames were so designed as to have the same lateral load resistance.The mechanical properties of concrete and reinforcement are listed in Tables 1 and 2. Specified 0.2% offset yield strength and tensile strength of prestressing tendon are 1078 MPa and 1225 MPa, respectively.The moment capacity of the column is about 1.5 times that of the beam, so that the plastic hinges are intended to be located in beam ends and column bases.One frame, 'FR35', was consisted of the beam where the eccentricity of prestressing tendon is 35 mm (D/6:D indicates the whole depth) and the other, 'FR60', has the prestressing tendon whose eccentricity is 60 mm (D/3.5).Effective prestresses are 107.8kN for FR35 and 63.7 kN for FR60.Prestress was transferred to the beam while one column base is supported by pin and the other is supported by roller.Therefore, the columns were free from the moments, shears and axial forces produced by prestressing.After prestress transfer, the frame was fixed to the floor.This procedure was followed in the calculation, but the friction of tendon during prestress transfer is ignored.
Fig. 5 shows loading set-up.Reversed cyclic horizontal load was statically applied to the mid-span of the beam by hydraulic jack.Besides the horizontal load, the vertical load was also applied at the midspan of the beam, so that the bending moments at the beam ends and the mid-span due to the prestressing were offset at the beginning of the test.This vertical load was kept constant during the test.
The frames were subjected to several slow load reversals simulating very severe earthquake loading.The 'first yield' displacement of the frames were found when all the tension reinforcements in expected hinging regions had yielded.The first loading cycle consisting of ten cycles was followed by a series of deflection controlled cycles in the inelastic range, also comprising ten full cycles to each of the displacement ductility factors of ±2, ±3, ±4, and sometimes higher, as illustrated in Fig. 6.
In the calculation bond stiffness assumed for prestressing tendon is 9.8 x 10 3 N/rnm 3 • This is about 1/1000 used for nonprestressed deformed bar.Fig. 7 shows layer elements assumed in the calculation.

EXPERIMENTAL RESULTS AND COMPARISON WITH THE ANALYTICAL RESULTS
Figs. 8 and 9 show the first cycle in each series of deflection cycles of the measured horizontal load versus horizontal deflection at the midspan of the beam.These figures also show the calculated loaddeflection curves.The deterioration due to load cycles in concrete and bond is not considered in this calculation, so that each loading cycle comprises only one full cycle.When the experimental results are compared with analytical results, fairly good agreement can be observed.3. Solve the equation above to get the slip between concrete and reinforcement in each block.4. Set up the stiffness matrix of each block and transfer it into the flexibility matrix.5. Combine the flexibility matrices of the sections so that the member flexibility matrices can be obtained.6. Transfer the member flexibility matrices into the stiffness ones and constitute the stiffness matrix of the structure.7. Solve the equation above under some support conditions and get the unknown displacements and forces.8. Calculate tEoj and t¢j again by using the displacement and force increments obtained above.9. Compare the assumed values and calculated values of tEoj and 6¢j , and return to step 1 if the difference between them is larger than one percentage of the assumed values.If not, go next.10.Calculate the stresses and tangential modulus of concrete and reinforcement according to the strains obtained by using tEQi and 6¢j .If the tangential modulus ifi this step differs from the assumed value, unequivalent stress is calculated and released in the next step.

EXPERIMENT OF UNBONDED PRESTRESSED CONCRETE PORTAL FRAME
For verifying the propriety of the analytical method above, the experiment was carried out on two portal frames with an unbonded prestressed concrete beam 4.2 m long and reinforced concrete columns l m high, shown in Fig. 4.These two frames were so designed as to have the same lateral load resistance.The mechanical properties of concrete and reinforcement are listed in Tables 1 and 2. Specified 0.2% offset yield strength and tensile strength of prestressing tendon are 1078 MPa and 1225 MPa, respectively.The moment capacity of the column is about 1.5 times that of the beam, so that the plastic hinges are intended to be located in beam ends and column bases.One frame, 'FR35', was consisted of the beam where the eccentricity of prestressing tendon is 35 mm (D/6:D indicates the whole depth) and the other, 'FR60', has the prestressing tendon whose eccentricity is 60 mm (D/3.5).Effective prestresses are 107.8kN for FR35 and 63.7 kN for FR60.Prestress was transferred to the beam while one column base is supported by pin and the other is supported by roller.Therefore, the columns were free from the moments, shears and axial forces produced by prestressing.After prestress transfer, the frame was fixed to the floor.This procedure was followed in the calculation, but the friction of tendon during prestress transfer is ignored.Fig. 5 shows loading set-up.Reversed cyclic horizontal load was statically applied to the mid-span of the beam by hydraulic jack.Besides the horizontal load, the vertical load was also applied at the midspan of the beam, so that the bending moments at the beam ends and the mid-span due to the prestressing were offset at the beginning of the test.This vertical load was kept constant during the test.
The frames were subjected to several slow load reversals simulating very severe earthquake loading.The 'first yield' displacement of the frames were found when all the tension reinforcements in expected hinging regions had yielded.The first loading cycle consisting of ten cycles was followed by a series of deflection controlled cycles in the inelastic range, also comprising ten full cycles to each of the displacement ductility factors of ±2, ±3, ±4, and sometimes higher, as illustrated in Fig. 6.
In the calculation bond stiffness assumed for prestressing tendon is 9.8 x 10 3 N/mm 3 • This is about 1/1000 used for nonprestressed deformed bar.Fig. 7 shows layer elements assumed in the calculation.

EXPERIMENTAL RESULTS AND COMPARISON WITH THE ANALYTICAL RESULTS
Figs. 8 and 9 show the first cycle in each series of deflection cycles of the measured horizontal load versus horizontal deflection at the midspan of the beam.These figures also show the calculated loaddeflection curves.The deterioration due to load cycles in concrete and bond is not considered in this calculation, so that each• loading cycle comprises only one full cycle.When the experimental results are compared with analytical results, fairly good agreement can be observed.

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However, the larger the deformation became, the larger difference could be observed.It •_s mainly because the shear deformation was not taken into consideration in the calculation.The shear deformation, especially •in the column, dominates the whole deformation of the frame in the loading cycles to high ductility values, and some pinching of the load-deflection loops was noticeable in the experiment.Just before the failure, where the ductility factor is almost +4, the shear deformation of the column base occupies a large portion of the whole deformation, while the deformation of the beam remains small.Therefore, the frame moves like a rigid body connected to the foundation at the column base.
In Figs. 10 and 11, the first cycle in each series of deflection cycles of the measured tendon force increments versus horizontal deflections at the midspan of the beam are shown.The calculated results are also shown in the same figures.As described before, the shear deformation at the column bases results in imposing not so large rotation on the beam ends even in the inelastic range.Therefore, the calculated results are larger than the experimental results.
In addition, in the calculation, the tendon force increment continues to increase almost linearly with the deflection of the frame because the rotations at the beam ends have a linear relationship with the lateral deflection of the frame.
The maximum tendon force increment measured in the test was up to 196 N/mm 2 • From analytical and experimental results on the portal frame, the tendon force measured at the anchorage ends is not so large.It may be not necessary to consider any risk of tendon fracture even in the inelastic range.The tendon force increment measured in the test showed good agreement with the predicted value obtained from ACI and NZS.

CONCLUSIONS
The analytical method, by which hysteretic restoring force characteristics of unbonded prestressed concrete framed structures can be statically pursued on the basis of material properties, was presented.The analytical results were compared with the experimental results of unbonded prestressed concrete portal frame in terms of lateral load versus deflection and increment of tendon stress versus deflection relation.Fairly good agreement can be observed.However, the larger difference could be found in the loading cycles to high ductility values because the shear deformation became dominant.Therefore the method has to be so improved as to take inelastic shear deformation into consideration.
The analytical and experimental results showed that the tendon force was not so large because the rotation of the beam ends was not so large while the large inelastic deformation was imposed on the frame.Therefore, it may be not necessary to consider any risk of tendon fracture and the unbonded tendon will be successfully used in primarily earthquake resistant members.

FIGURE 4 -
FIGURE 4 -DIMENSIONS AND REINFORCING DETAILS OF SPECIMEN

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FIGURE 10 -INCREMENT OF TENDON STRESS -DEFLECTION RELATION IN FR 60 , the longitudinal strain increment