Numerical evaluation of plastic rotation capacity in RC beams

Transcription

1 Numerical evaluation o plastic rotation capacity in RC beams A.L. Gamino & T.N. Bittencourt Structural and Geotechnical Engineering epartment, Polytechnical School o São Paulo University, São Paulo, Brazil ABSTRACT: The objective o the present paper is the analysis o the parameter θ pl in reinorced-conete beams by means o smeared ack approaches. The eects o the compressive strength and the size dependence in bending are re-investigated. The non-linear analyses were carried out by means o the Finite Element Method. The programs IANA, CASTEM000, and QUEBRA/FEMOOP were used in the numerical simulations. Appropriate constitutive models or conete, rebars and bond-slip interaces were implemented to represent more realistically the behavior o the structural system. Interace and reinorcement (disete and embedded approaches) inite elements were used to accomplish an explicit representation o the conetereinorcement bond. INTROUCTION Since the publication o Kani in 967, it is known that the resistant capacity o beams is highly dependent on size eects. In 966 Corley (966) advocated that size eect did not signiicantly aect the plastic rotation capacity θpl in R/C beams. However, in 988 Hillerborg (988) concluded, using racture mechanics concepts, that θpl in R/C sections is inversely proportional to the height o the structural element. Over the last decades research on ductility o beams has evolved and demonstrated the inluence o the reinorcement ratio, conete grade, stirrup spacing and inally size eects. The importance o the size eect in beams o reinorced-conete is in act that this can cause reductions in the capacities o inelastic deormation and plastic rotation as well as transition o types o brittle/ductile ruptures. These eects can appear in two orms: Modiications in the a/d relations ( a is the shear span; d is the eective beam depth) that over all aect the load capacity o the beams to the shear Walraven & Lehwalter (994); Variations in the slenderness inside or out o pure bending regions; The irst research on ductility o beams had involved to portray the inluence o the reinorced ratio Leslie, Rajagopalan & Everard (976), strength o the conete Tognon et al. (980), Ashour (000), span between stirrups Shin et al. (989) and inally size eect Alca et al. (997). This work is concentrated on the evaluation o the ductility in beams using smeared ack models in the conete. The numerical analysis is based on the inite element method implemented in CASTEM 000 developed by the épartement de Mécanique et de Technologie (MT) du Commissariat Français à l Energie Atomique (CEA). This program uses the constitutive elastoplastic perect model or the steel, the rucker-prager two parameter model or the conete and the Newton-Raphson or the solution o non-linear systems. In the QUEBRA/FEMOOP system appropriate constitutive models or conete, rebars and bondslip interaces have been implemented to represent more realistically the behavior o the structural system. Interace and reinorcement (disete and embedded approaches) inite elements have been used to accomplish an explicit representation o the conete-reinorcement bond. In the numerical simulations, the established computational program in the Finite Element Method IANA was developed by the TNO Building and Constructions Research o the epartment o Computational Mechanics (Netherlands). First, the inluence o each smeared ack model on the numerical results was presented. Later, a parametric study is carried out. An analytical model is presented in order to calculate the plastic rotation capacity "θpl" o the analyzed beams. Later the experimental results are compared with the results rom the numerical simulations.

2 NUMERICAL EVALUATION OF PLASTIC ROTATION CAPACITY. Conete Constitutive Model in CASTEM 000 The conete constitutive model used was the rucker-prager two parameters model (derived rom the Ottosen our parameters model). This model was ormulated in 95 and can be seen as a simple modiication o the iterion o Von Mises, including the inluence o the hydrostatic pressure, according to Equation (). α p Figure. Thoreneldt hardening curve. p α ( σ ) = αi + J k 0 () dp = where α and kdp are the material constants, I and J the invariants that depend on the normal stress on a body. Figure brings the graphical representation o the surace o plasticity in the plan σ-σ where t is the tensile strength, c is the compressive strength and bc is the biaxial strength in compression. bc c t c σ bc t σ.. Sotening Eect For the corresponding curve to the behavior o the conete under traction the desibed bilinear model or the code (CEB-Fip, 993) was used in some analyses in agreement with to Equation 3. _ ( ctm, ) σ =.0,05 + ctm (3) _ 0,05 ε In other simulations the non-linear curve o Hordijk was used. This model presented in Figure 3 uses an exponential relation between the normal tensile stress and the deormations, with c and c assuming the values respectively o 3,0 and 6,93. σ t σ ( ε ) ε = ( + c 3 ) exp( c ) t εult 3 ε ε + c exp c... ε ult ε ult 0,or ε < ult ε <,or 0 ε < εult < Figure. rucker-prager yield iteria.. Conete Constitutive Models in IANA Three smeared ack models were available in.. Hardening Eect this program: ixed, multi-directional and rotating. For the conete under compression the model desibed in the code (CEB-Fip, 993) was used in.3 Conete Constitutive Model in FEMOOP some modeling represented in the Equation. A our parameter model o Ottosen (977) was used Eci εc εc to represent an unacked material structure. The σ c Ec 0,% 0,% = () rupture surace is given by: ( cm E ci ε + c E c 0,% J J I In other modeling the non-linear hardening model F = A + λ + B = 0 (4) o Thoreneldt was used. This model presented in cm cm cm Figure uses an equation between compressive A linear sotening curve law was utilized to represent the acked region with a rotating smeared- stress and the deormations based in the adoption o diverse parameters. ack model. G /h ε ult ε Figure 3. Hordijk sotening curve.

3 The contact and truss inite elements were implemented (disete and embedded approach in agreement with Elwi & Hrudey (989) ormulations) or the numerical representation o rebars and interaces. where σ sr is the reinorced stress in the ack, ε s and ε sy respectively are the yield deormations in steel reinorcements and acks. The Figure 5 represents the process o the plastic rotating capacity using an equivalent beam..4 Steel Reinorcements In all programs the reinorcements were modeled with the Von Mises plasticity model. L/ Q b o.5 Steel Bonding Model in FEMOOP The Homayoun & Mitchell (996) multilinear bondslip model desibed in Figure 4 was implemented or the representation o interaces (conete/steel) behavior. The ollowing parameters are given: τ s is the interace bond strength, τ sr is the interace residual bond strength, s is the interace slip at the peak stress, s r is the residual interace slip, E b is the prepeak bond modulus and E d is the post-peak bond modulus. 0,5.Q R sa υ l / l / B R sb Rsy a y Rs max R c V=0 R s max τ s Bond Stress Figure 5. Equivalent beam. By using the Figure 5 can be calculated the conete and steel orces: τ sr E d Rc = M / z + N.zs / z + 0,5 V..cot υ Rs = M / z + N. zs / z + 0,5 V..cot ( ) υ (7) E b s Figure 4. Homayoun bond-slip curve. s r Slip 3 THEORETICAL ANALYSIS OF PLASTIC ROTATING CAPACITY Using the MC-90 the plastic rotation capacity can be deined as: l pl θ pl = d x a = 0 [ ε sm ( a) ε smy ] ( a) (5) where ε sm (a) is the reinorced strain in the member plastiication place, ε smy is the reinorced deormation or ack tension equal to yk. Using the deinition and applying the model σs(εsm) by Kreller (989): l pl 0,8 σ θ = sr pl d x a= 0 yk da ( ε ε ) s sy dx (6) where υ is the inclination angle o the compression ield. The tensile orces can be derived rom the ollowing orm: Rs Rs ( x) = R 4. ( R R )(. x / l ) x l / s,max R s,max R ( x) = R + sb sa.( L x) L / x l / sa lb (8) And rom this orm is ound the length o the plasticized zone: l pl Rs max Rsy a y = = 0,5.l. Rsy > RsB Rs max RB (9) l pl Rsy RsA ay = = 0,5. L lb Rsy RsB RB R sa Solving the integral o Equation (6) the plastic rotation capacity can be calculated: θ pl. δ = 3 ( d x) σ sr l yk AsEs sb ( R R ) s max sy 3 Rs max RsB

4 . δ σ = sr l l θ pl d x yk AsEs 3 Rsy R sb (.R 3.R + R ) s max Rsy R sb l + B sb (0) ( R R ) ( R ) sb RA Making the width o the support plate (b o =0) and leaving the eect o the shear orces (cot υ=0) arrived the ollowing expressions are ound: sy sb sy lpl L M y ay = = Mu Mu My θ pl = ay. = pl L ( EI ). ( EI ) pl ( M M ) u y Mu () β=0, 4 ANALYSIS OF THE SMEARE CRACK APPROACH Three reinorced-conete beams were analyzed in the computational program IANA, analyzed in the experimental program o Barbosa (998) every 3,60m o span, width o 5cm and height o 8,3cm, with stirrups o 6mm a diameter spaced in 8cm and with applied loads in /3 and /3 o span. The longitudinal bottom reinorcements consist o two bars o 6mm a diameter and the longitudinal top reinorcements consist o two bars o 8mm a diameter. The mechanical properties o these beams is shown in Table. β=0,9 Figure 6. Crack pattern or ixed-ack approach. Table. Material properties. Beam c (MPa) y (MPa) E c (GPa) E s (GPa) , , 0 In the modeling the three models o smeared acks (ixed, rotating and multi-directional) without sotening (brittle model) and hardening model in agreement to proposal o Thoreneldt or two distinct values o the shear retention actor (β). In the ixed model (Figure 6) the use o a smaller shear retention actor implied in a more extended acking; the variations in β had not modiied the maximum values o the ack deormations (ε nn ). In the multidirectional model (Figure 7) the use o a smaller shear retention actor implied in a more extended acking mainly in the region o pure bending (or β=0, notices eight wider acks in this region were noticed; or β=0,9 six wider acks in the same region were noticed); the variations in β had modiied the maximum values o the ack deormations ( β is inversely proportional to ε nn in this model). β=0, β=0,9 Figure 7. Crack pattern or multi-directional approach. In the rotating model (Figure 8) the use o a smaller shear retention actor implied in a more extended acking mainly is o the region o pure

5 bending; variations in β had very aected the values o the deormations in the acks. The ixed, multi-directional and rotating models predicted respectively six, eight and our wider acks (or β=0,) in the region o pure bending. The experimental ack pattern (Figure 9) shows our wider acks similar to the ones ound with the rotating smeared ack model. than the experimental ones, whose inluence can be observed in the values o ound or the plastic rotation capacity: bigger values or the numerical simulation in the IANA and smaller values or CASTEM. The results ound with FEMOOP were located between the IANA and the CASTEM results (closer to the experimental results). Table. Plastic rotation capacity values. β=0, Experiment CASTEM IANA FEMOOP Beam a y (cm) θ pl (mrad) 0,63,69 0 3,7 5, ,3 33,93 0,87,0 0 7,48 8,3 03 6,6 38,90 0 4,3,93 0,3,9 03 6,4 8,30 0 3,36,66 0 9,66 4,9 03 7,77 9,77 Table 3 illustrates the displacements at the steel yield stress (δy), displacements in the rupture (δu) and global ductility ratio (μ=δu/δy) rom the experimental and numerical analysis. Table 3. uctility ratio values. β=0,9 Figure 8. Crack pattern or rotating model. Figure 9. Experimental ack pattern. 5 ANALYSIS OF THE PLASTIC ROTATION CAPACITY Experiment CASTEM IANA FEMOOP Beam δ u (mm) δ y (mm) μ d = δ u /δ y 0 9,5,3, ,53,87 3, ,94 3,55 5,30 0 9,8 5,6,9 0 70,5,89 3, ,3,30 3,0 0 0,5 3,30, ,60, ,7,9,09 0 6,8 4,6, ,,6, ,3 8,3,69 The plastic rotation capacity or the orcedisplacement curves was evaluated or experimental and numerical modeling in IANA, CASTEM and QUEBRA/FEMOOP programs or the three previous beams. Each "θpl" was calculated using the Equation (). The calculated values were disposed in Table. In general, the results rom the IANA were a little more rigid than the experimental results and the results in platorm CASTEM were a little less rigid The results point to a magniying o the inelastic deormation capacity o beam with regard to beam when it extended concomitantly with the compressive strength and reinorced yield tension. Another magniying o this ductility can be observed in beam 3 compared to beam where the compressive strength o the conete only extended itsel.

6 6 CONCLUSIONS For the ixed rotation ack model the use o a smaller shear retention actor implicates wider acks; the variations in β don t modiy the maximum values o the ack deormations (ε nn ). For the multi-directional ack model the use o a smaller shear retention actor implicates also wider acking mainly in the region o pure bending (or β=0, eight wider acks were noticed in this region; or β=0,9 six wider acks in the same region were noticed); the variations in β modiy the maximum values o the ack deormations ( β is inversely proportional to ε nn in this model). For the rotating ack model the use o a smaller shear retention actor implicates more extended acking in the region o pure bending; variations in β aect the deormation values in the acks. The ixed, multi-directional and rotating models predict respectively six, eight and our wider acks (or β=0,) in the region o pure bending. The experimental ack pattern (Figure 9) shows our main acks in agreement with the rotating smeared ack model. The results can be observed in the values o the plastic rotation capacity parameters: larger values or the numerical simulation in the IANA and smaller values or the CASTEM. The results obtained with the FEMOOP were ound to be between the ones obtained with the IANA and the CASTEM (closer to the experimental results). The results point to a magniication o the inelastic deormation capacity o beam with regard to beam when the compressive strength and steel yield stress are ineased. Another magniication o ductility can be observed in beam 3 compared to beam where the compressive strength o the conete only is ineased. REFERENCES Alca, N., Alexander, S..B. & MacGregor, J.G Eect o size on lexural behavior o high-strength conete beams, ACI Structural Journal, 94(), Ashour, S. A Eect o Compressive Strength and Tensile Reinorcement Ratio on Flexural Behavior o High S- trength Conete Beams. Elsevier Science Ltd., Engineering Structures, pp Barbosa, M.P An Experimental and Numerical Contribution on High Reinorced-conete Structures: Study o the Anchorage and the Behavior o Bending Beams. Engineering School, São Paulo State University, FEIS, Unesp, 74p. Comité Euro International du Béton Model Code or Conete Structures. CEB FIP MC 90, 437 p. Corley, G.W Rotational capacity o reinorced-conete beams, ASCE, 9(5), -46. Elwi, A.E. & Hrudey, T.M Finite element model or curved embedded reinorcement. Journal o Engineering Mechanics, v.5, n.4, pp Hillerborg, A Rotational capacity o reinorced-conete beams, Norwegian Conete Research, publication n.7, -34. Homayoun, H.A. & Mitchell, Analysis o bond stress distributions in pullout specimens. Journal o Structural Engineering, v., n.3, pp Kani, G.N.J How sae are our large reinorced-conete beams, ACI Journal, 64(3), 8-4. Kreller, H Zum nicht-linearen Trag- und Verormungsverhalten von Stahlbetonstabtragwerken unter Last- und Zwangeinwirkung. issertation, Universitat Stuttgart. Leslie, K.E., Rajagopalan, K.S. & Everard, N.J Flexural behavior o high strength conete beams, ACI Structural Journal, 73(9), Ottosen, N.S A ailure iterion or conete. Journal o the Engineering Mechanics ivision, v.03, n.4, pp Shin, S.W., Gosh, S.K. & Moreno, J Flexural ductility o ultra high strength conete members, ACI Structural Journal, 86(4), Tognon, G., Ursella, P. & Coppetti, G esign and properties o conetes with stregth over 500kg/cm. ACI S- tructural Journal, 77(3), Walraven, J. & Lehwalter, N Size eects in short beams loaded in shear, ACI Structural Journal, 9(5),