Tensile Force Evaluation of Unbonded PT Bars in Precast Concrete Columns

Transcription

1 Fédération Internationale du Béton Proeedings of the 2 nd International Congress ID 2-32 Session 2 Tensile Fore Evaluation of Unbonded PT Bars in Preast Conrete Columns Tani, M., Nishiyama, M. Department of Urban and Environmental Engineering, Kyoto University, Kyoto, Japan INTRODUCTION For evaluation of flexural strengths of prestressed onrete members, it is neessary to evaluate tensile fore of post-tensioning *(PT) bars as aurately as possible. An analytial study on antilever preast onrete olumns post-tensioned by unbonded tendons was reported in the referene []. If unbonded tendons are provided straight and the member is subjeted to anti-symmetrial loading, variation of tensile fore in an unbonded PT bar is small beause the bar in ompression at one end is in tension at the other end of the member as shown in Fig.. The behavior is onsidered to differ from that of a antilever member. This paper desribes a parametri study on preast onrete olumns fousing on tensile fore evaluation of unbonded PT bars by using the analytial method reported in the referenes [] and [2]. The olumns are assumed to be subjeted to anti-symmetrial loading. The parameters are axial load level, prestressing fore level and ompressive strength of onrete. Keywords: Preast onrete olumn, post-tension, unbond, layer-element method, maro model SUMMERY OF ANALYSIS The analysis is based on the layer-element method in whih the stiffness matrix of a member is onstruted in terms of layers in the setion and longitudinal elements as shown in Fig. 2. The referene [2] dealt with antilever members, while this paper is intended for the members under anti-symmetrial loading, whih simulates earthquake loading against a member restrained at the ends. The envelope urve of onrete stress-strain relation proposed by NewRC projet [3] is used, and the hystereti rules follow a model in the referene [4]. The hystereti stress-strain idealization for mild steel is based on Ramberg-Osgood model [5] onsidering Baushinger effet. The hystereti stress-strain urves of a PT bar is based on Menegotto-Pinto model [6]. Morita and Kaku model [7] is used for bond stress-slip relation. (a) anti-symmetrial loading (b) antilever loading Fig.. Comparison between anti-symmetrial loading and antilever loading Fig. 2. Analytial modeling

2 COMPARISON OF ANALYTICAL RESULTS WITH EXPERIMENTAL RESULTS [8] To prove the validity of the analysis, the omparison between analytial results and past experimental results [8] in whih beam speimens post-tensioned by unbonded tendons is onduted. Analytial models The beam speimen of the referene [8] is shown in Fig. 3. The beam ross-setion dimensions are 6 x 2 mm and the beam length is 3 mm. Four mm diameter mild steel bars are plaed at the top and bottom of the beam setion as longitudinal reinforement, and 6 mm diameter hoops are plaed at 5 mm in enters as shear reinforement. The beam is post-tensioned with two 3 mm diameter unbonded tendons. The properties of onrete, reinforement and unbonded tendon are summarized in Tables and 2. The prestressing fore level on the beam ross-setion is η P =P/( f )=.7. As shown in Figs. and 4, the beam is divided into 5 longitudinal elements, and the setion onsists of layers. The beam has stubs at the ends. The total depth of the stub 3mm x2 is inluded in the length of the unbonded tendons. The deformation of the stubs is not taken into onsider. 3 unbonded PT bars D mild steel bars 5 Be a m φ6 hoops LC Stu b Unit : mm Fig. 4. Analytial modeling Fig. 3. Dimensions of test speimen Tab.. Properties of onrete f ε m E (N/mm 2 ) (%) ( 4 N/mm 2 ) ε m : strain at ompression strength, E : /3f seant modulus of elastiity Tab. 2. Properties of reinforement and tendon f y (N/mm 2 ) (%) ( 5 N/mm 2 ) D φ6 35* -.89 φ3(tendon) 245* -.96 f y : yield strength, σ y : strain at yield strength, E s : modulus of elastiity,*.2% offset yield strength ε y E s Analytial results omparison of moment - olumn rotation angle relations and tensile fore of PT bar - olumn rotation angle relation between the analytial and the experimental results is shown in Fig. 5. The behavior of the member and PT bar are predited with good auray. 2

3 Moment [knm] Analysis Experiment Column rotation angle [%] (a) Moment Column rotation angle relations Tensile fore of PT bar [kn] Analysis Experiment Column rotation angle [%] (b) Tensile fore of PT bar Column rotation angle relations Fig. 5. Comparison between the analytial and the experimental results PARAMETRIC STUDY ON PRECAST CONCRETE COLUMNS POST-TENSIONED BY UNBONDED TENDONS Analyses for the preast onrete olumns post-tensioned by unbonded tendons under anti-symmetrial loading are arried out. The parameters are ompressive strength of onrete f, axial fore level η N (=N/ f ) and prestressing fore level η P (=P i / f ), where N: the axial load, : the olumn setion area, and P i : the effetive prestressing fore before axial load is applied. The interrelationship between the parameters and the behavior of PT bar is disussed. Using a simple maro model, tensile fores of PT bar and olumn rotation angle at the maximum strength are evaluated, and the results are ompared with layer-element analysis results. Analytial models and parameters As shown in Fig. 6, the olumn ross-setion dimensions are 4 x 4 mm and the height is 2 mm. The olumn has the stubs at its ends. They are assumed to be rigid. Four mm diameter mild steel bars are plaed as longitudinal reinforement, and mm diameter hoops are plaed at 4 mm in enters as shear reinforement. The diameters of PT bars are shown in Fig. 6 (b). As shown in Fig. 6 (a), the olumn is longitudinally divided into 6 elements and the olumn setion is onsists of layers. Analytial parameters are given in Table 3. The maximum axial load on the olumn given as η N +η P is.6. A series of the models whose f is 4 N/mm 2 and η N is. are named 4 -.Series. Yield strengths of mm diameter mild steel and PT bar are 325N/mm 2 and 35N/mm 2, respetively. (a) Analytial modeling (b) Details of olumn Fig. 6. Analytial modeling 3

4 Tab. 3. Analytial parameters f' (N/mm 2 ) 4, 6 η N,.5,.,.2,.3,.4,.5 η P.,.2 Analytial results Beause the member is under anti-symmetri deformation in terms of the beam enter, the longitudinal deformation of onrete loated at a straight PT bar is independent of the eentriity of the PT bar. If PT bars are unbonded, their eentriity does not affet their tensile fore variation. If PT bars are plaed symmetrially in terms of the entral axis of the olumn setion, their tensile fores indue no flexural moment. Therefore, their tensile fores an be regarded as the axial fore on the olumn. The main subjet of this paper is to disuss the effet of the axial fore level η N+P =η N +η P on the member behavior. Fig. 7 shows the relations between the ratio / P i and η N+P where is tensile fore of PT bars at flexural strength. When η N+P is larger than.3, is smaller than P i regardless of onrete ompressive strength, and the alloation of the axial load and the amount of prestressing fore. Fig. 8 shows the relations between the ratio / P e and η N+P. is larger than P e (the effetive prestressing fore after axial load is applied) when η N+P is small. If η N+P is larger than.45, is smaller than P e. All PT bars did not yield as shown in Fig. 9. Tensile fore variation of PT bars of large prestressing levels is smaller that of small prestressing levels, if the total axial load η N+P is the same. This indiates that axial deformation of the large prestressing level models is smaller than that of the small prestressing level models. This is beause large initial prestressing fore results in larger diameter of PT bar, and stiffness of the olumn setion inreases, and therefore, axial deformation beomes smaller. The bonded PT bars plaed in tension side are assumed to yield at the flexural strength. As desribed above, if the PT bars are unbonded, the tensile fore of PT bars an be regarded as the axial fore. The flexural strength an be alulated based on the total axial load inluding the tensile fore of unbonded PT bars. But, there is few experimental data about olumn members post-tensioned by unbonded PT bars. More experiments and investigations have to be onduted in the future. /P i Series 4-.2Series 6-.Series 6-.2Series /P e Series 4-.2Series 6-.Series 6-.2Series.9.8 Fig. 7. Ratio of PT bar tensile fore at flexural strength to initial prestressing fore.9 Fig. 8. Ratio of PT bar tensile fore at flexural strength to effetive prestressing fore 4

5 .8 4-.Series 4-.2Series 6-.Series 6-.2Series.6 /P y.4.2 Fig. 9. Ratio of PT bar tensile fore at flexural strength to yield strength of PT bar Evaluation of flexural strength by maro model Tensile fore of a bonded PT bar an be alulated by a plain setion analysis. A plain setion analysis with the strain ompatibility fator F an be applied to unbonded members. In this paper, a simple maro model is proposed to obtain PT bar tensile fores and olumn rotation angle at flexural strength more easily than the layer-element analysis desribed earlier. The model onsists of two parts: flexural deformation zones at the ends and a rigid zone in the middle as shown in Fig.. Flexural and axial deformations are attributed to the flexural deformation zones. In the rigid zone axial deformation only is onsidered. Using ACI equivalent retangular stress blok in the flexural deformation zone, neutral axis depth x n is alulated by Eq.(). The elongation of the flexural deformation zone and rigid zone ( L, L 2 ) are alulated by Eq.(2) and Eq.(3). Substitution of Eq.() into Eq.(2) gives longitudinal elongation of the member in terms of. Based on the deformation ompatibility of the member and the PT bar, Eq.(4) is obtained. The relational expression about derived from these equations is a quadrati form, from whih an be alulated. The member rotation angle R u is defined by Eq.(5). x n C =.85 f ' b N + Pu + atσ y aσ y =.85 f ' b D L = ( 2) ε u L' x n N + Pu L2 = ( L 2L') E A + E a L + L2 ε u R u = L' x n p s p s Pu Pi = l E a p () (2) (3) (4) (5) where N : axial fore, P i : initial prestressing fore, a t,a : ross setional areas of tensile and ompressive ordinary reinforement, σ y : yield strength of reinforement, f : onrete ompressive strength, b : olumn width, D : olumn depth, L : length of flexural deformation zone, ε u : ultimate strain of onrete(=.3), E, E p, E s : modulus of elastiities of onrete, PT bar and ordinary reinforement, respetively,, a p, a s : ross setional areas of onrete, PT bar and reinforement, respetively, L : olumn total height, l p : length of PT bar 5

6 εu.85f xn C=a s σy C M Ts=atσy.42xn N+Pu Fig.. Flexural deformation zone and rigid zone Fig.. Strain and stress distribution of flexural deformation zone at flexural strength The relationship between the tensile fore of PT bar at flexural strength obtained from the layer-element analysis (referred to analytial value ) and that from the maro-model (referred to theoretial value ) is shown in Fig. 2. The squares and irles plotted in Fig.2 and Fig.3 indiate the analytial results, and the lines with small squares or irles show the theoretial results with the same prestressing level. The theoretial results agree well to the analytial results, although the theoretial results are slightly smaller than the analytial ones when the axial load level is small. P i an be derived from the quadrati equation, but the equation is so ompliated that simplifiation may be needed for pratial appliation. The analytial and theoretial values of the rotation angle at flexural strength are plotted against the axial fore level in Fig.3. The theoretial results are smaller than the analytial ones when the axial load level is small. The olumn rotation angles at flexural strength derease as the axial fore level inreases. The theoretial results agree well to the analytial ones. In this paper, the length of the flexural deformation zone L is assumed equal to the element length of the layer-element analysis, 2mm (=.5D). As shown in Eq.(5), the olumn rotation angle is in proportion to the length of the flexural deformation zone. Therefore, an appropriate evaluation of the length is neessary [kn] Series(analysis) 4-.Series(theory) Series(analysis) 4-.2Series(theory) [kn] Series(analysis) 6-.Series(theory) 6-.2Series(analysis) 6-.2Series(theory) (a) f = 4 Series (b) f = 6 Series Fig. 2. Comparison of PT bar tensile fore at flexural strength obtained from layer-element analysis and maro model 6

7 θ u [%] Series(analysis) 4-.Series(theory) 4-.2Series(analysis) 4-.2Series(theory) θ u [%] Series(analysis) 6-.Series(theory) 6-.2Series(analysis) 6-.2Series(theory).2.2 (a) f =4 Series (b) f =6 Series Fig. 3. Comparison of olumn rotation angle at flexural strength obtained from layer-element analysis and maro model One the tensile fores of PT bars at flexural strength are evaluated, the stress distribution of the flexural deformation zone beomes lear. Therefore, moment apaity is given as follows, M = C ( D / 2.42x ) + ( a σ + a σ ) d' (6) n t p p where d is an effetive depth of PT bar. The ratio M theory / M analysis is plotted against η N+P in Fig. 4. M theory is smaller than M analysis by 5~2%. The smaller the axial load level is, the smaller results are obtained from the maro model. However, the tensile fores of PT bar alulated from the layer-element method and the maro model show a good agreement as shown in Fig.2. The reason may be attributed to the depth of the resultant ompression fore of onrete, whih is larger in the maro model than in the layer-element method. Beause the authors have not yet onduted loading tests on preast onrete olumns post-tensioned by unbonded tendons under earthquake indued loading, suh as anti-symmetrial loading, the analytial results from the layer-element analysis was used in this paper instead of an experimental result. The authors are now planning to ondut loading tests on preast onrete olumns (/3 sale) post-tensioned by unbonded tendons. After the experiments the two analytial methods will be ompared with test results. M theory /M analysis Series 4-.2Series Series 6-.2Series.6 Fig. 4. Relations between M theory / M analysis and axial load level 7

8 CONCLUSIONS Layer-element analyses predited the behavior of the beam speimens post-tensioned by unbonded tendons tested under anti-symmetrial loading with a good auray. If η N+P is larger than.3, PT bar tensile fore at flexural strength is smaller than the initial prestressing fore. If η N+P is larger than.45, the one at flexural strength is smaller than the effetive prestressing fore, regardless of onrete ompressive strength and the alloation of η N and η P. A simple maro model was proposed and employed to obtain flexural strength and tensile fores in PT bars, and olumn rotation angle at flexural strength. Tensile fores in PT bars at flexural strength obtained by the maro model agreed well to the results obtained from the layer-element method. A setion analysis using ACI onrete stress blok and plain setion assumption gave 5~2% smaller flexural strengths than the above analyses. REFERENCES. Masanori Tani, Minehiro Nishiyama, Ihizo Kishimoto. Estimation of Post-Tensioning Fore in Preast Prestressed Conrete Members under Cyli Loading. Proeedings of the Japan Conrete Institute, Vol.27, No.2, pp , Hiroshi Maeda, Minehiro Nishiyama, Ihizo Kishimoto. Analysis Method of Hystereti Behavior of Preast PC Members Considering Bond Charateristis. Proeedings of the Japan Conrete Institute, Vol.26, No.2, pp.79-74, Kenji Sakino. Stress-Strain Curve of Conrete Confined by Retilinear Hoop. Journal of Strutural and Constrution Engineering, Arhitetural Institute of Japan, No.46, pp.95-4, F.Watanabe, J.Y.Lee, M.Nishiyama. Strutual Performane of Reinfored Conrete Columns with Different Grade Longitudinal Bars. ACI Strutual Journal, vol.92, No.4, pp.42-48, Hiroyuki Tagawa, Minehiro Nishiyama. The Idealization of Stress-Strain Prestressing Steel Bar. Summaries of Tehnial Papers of Annual Meeting, Arhitetural Institute of Japan, pp , Hidefumi Enomoto. Analytial study on Hystereti Property of Prestressed Reinfored Conrete Beams. Master s thesis of Osaka University, Shiro Morita, Tetsuzo Kaku. Loal Bond Stress Slip Relationship under Repeated Loading. Journal of Arhitetural Institute of Japan, No.229, Hiroshi Muguruma, Fumio Watanabe, Mihehiro Nishiyama. Mehanial Property of Hyperstati Beams Post-Tensioned by Unbonded Tendons. Journal of Prestressed Conrete, Japan Prestressed Conrete Engineering Assoiation, Vol.27, No.2, pp.66-73,