A Constitutive Model for Concrete Cylinder Confined by Steel Reinforcement and Carbon Fibre Sheet

Transcription

1 A Constitutive Model or Conrete Cylinder Conined by Steel Reinorement and Carbon Fibre Sheet By Yeou-Fong Li 1 Tsang-Sheng Fang 2 and Ching-Churn Chern 3 ABSTRACT: In this paper, we modiy the L-L model (Li et al., 22) and extend the appliation o this model to onrete ylinders onined, respetively, by steel reinorement only, by arbon ibre reinored plasti (CFRP) only, and by both steel reinorement and CFRP. Thirty-six onrete ylinders with dimensions o 3 6 m were tested to veriy the eetiveness o the Modiied L-L model. The design parameters o the onrete ylinders inlude the dierent oninement types o the steel reinorement (suh as spiral and irular hoop) and the number o layers o CFRP. The experimental test results show that dierent types o steel reinorement have a great eet on the ompressive strength o onrete ylinders onined by steel reinorement, but the dierent types o steel reinorement have very little eet on onrete ylinders onined by both steel reinorement and CFRP. Compared with the stress-strain urves o onined onrete ylinders, we an onlude that the Modiied L-L model an provide more eetive predition than Kawashima models. 1. INTRODUCTION Columns are an important strutural member, and their strength and dutility aet the seismi perormane o the entire struture signiiantly. Thereore, the seismi retroit o olumns has beome a very important issue in areas o high seismiity. Although retroitting materials and methods have been around or a long time, CFRP omposite materials applied to seismi retroit projets have beame a very popular method in the last deade. In this paper, the Modiied L-L model was proposed and veriied by the stress-strain relationships o thirty-six onrete ylinders with three dierent types o steel reinorements (irular hoop, irular lap splied hoop, and spiral) onined by CFRP. Conined onrete onstitutive models have been researhed extensively sine the early 2 th entury. 1 Assoiated Proessor, Department o Civil Engineering, National Taipei University o Tehnology, Taiwan, R.O.C. 2 Graduate student, Department o Civil Engineering, National Taiwan University, Taiwan, R.O.C. 3 Proessor, Department o Civil Engineering, National Taiwan University, Taiwan, R.O.C. 1

2 Kawashima et al. (1997, 1998, and 1999) proposed a series o stress-strain models or onrete onined by steel reinorement, CFRP, and both steel reinorement and CFRP. In the onstitutive model o onrete onined by steel reinorement proposed in 1997, the asending branh was idealized by an n th -order polynomial equation. Kawashima et al. subsequently used the regression analysis o experimental results and modiied the above-mentioned 1997 model to extend the appliation to dierently onined materials by adjusting the oeiients. The model or onrete onined with both steel reinorement and CFRP proposed by Kawashima et al. (1999) will be alled the Kawashima model in this paper. Li et al. (22) proposed an eetive onstitutive model or onrete onined with CFRP. More details about this model will be disussed in the next setion. 2. CONSTITUTIVE MODEL The onined onrete onstitutive model proposed by Li and Lin (22) (L-L model) was originally developed or onrete onined by CFRP. In this paper, we modiy the L-L model (Modiied L-L model) and extend the appliation o this model to onrete ylinders onined, respetively, by steel reinorement only, by CFRP only, and by steel reinorement and CFRP together. In the Modiied L-L model, the equation o the lateral onining stress due to steel reinorement was adopted rom the Mander model (1988), and the equation o the lateral onining stress due to CFRP was adopted rom the L-L model. In this setion, the asending branh o the Modiied L-L model stress-strain urve or onrete onined by both steel reinorement and CFRP will be introdued and detailed. Fig. 1. illustrates a steel reinorement and CFRP onining onrete ylinder. Beause the mehanism o onined onrete is similar to the mehanism o soil under tri-axial loading, the stress relationships o onined onrete an be derived rom tri-axial stress relationships. Aording to the Mohr-Coulomb ailure envelope o the soil under onined stresses ( σ 3 ) and axial stress ( σ 1) ould be expressed as ollows: 2 ( 45 + φ / 2) + σ tan ( 45 / 2) σ 1 = 2 tan 3 + φ (1) In Eq. (1), σ 1 is the axial stress, is the ohesion o the soil or rok, σ 3 is the lateral onining stress, and φ is the angle o internal rition o the material. I Eq. (1) is the tri-axial stress relationship equation or onined onrete, then σ 3 is the eetive onining stress and l = σ 3, while σ 1 is the maximum axial strength and =σ1. When σ (i.e. the unonined situation), 3 = 2

3 1 = 2 tan 45 + φ / 2 = the plain onrete strength an be expressed as ( ) σ. By using the above physial-based onstitutive model or onined onrete, the ompressive strength o onined onrete ( ) an be alulated as ollows: = + tan 2 φ l 45 + (2) 2 In Eq. (2), is the ompressive strength o the unonined onrete and l is the eetive lateral onining strength. The eetive lateral onining strength might ome rom steel reinorement, CFRP, or both steel reinorement and CFRP together. The Modiied L-L model an be alulated as ollows: ( + ) tan 2 φ 45 + = + l1 l 2 (3) 2 In the above equation, is the peak ompressive strength o the onined onrete, and is the ompressive strength o the unonined onrete. In Eq. (3), as ollows: l1,, and φ an be represented l 2 l1 1 = ke ρ s yh (From Mander s model) (4) 2 k n t E ε l = 2 2 (5) D φ = + (6) 36 1 In Eq. (4), l1 is the eetive lateral onining strength due to steel reinorement, k e is the oninement eetiveness oeiient, and k e depends on the type o lateral steel reinorement, shown as ollows: 3

4 k e = A A e s 1 2d = 1 ρ s 2 (For irular hoop) (7) k e s 1 2d = 1 ρ s (For irular spiral) (8) Also in Eq. (4), ρ s is the ratio o the volume o transverse onining steel to the volume o onined onrete ore, and yh is the yield strength o the transverse reinorement. In Eq. (5), k is the oeiient o setion shape (Priestley et al., 1996), n is the jaket layer o CFRP, t is the thikness o CFRP per layer, E is the elasti modulus o CFRP, D is the diameter o the ylinder, andε is the ultimate strain o CFRP. In Eq. (6), φ is the angle o internal rition o onrete, whih is in proportion with the ompressive strength o onrete, and usually varies rom 36 o to 45 o (Goodman, 1989). The angle o internal rition φ an be expressed as a untion o onrete strength (Li et al., 22) as shown in Eq. (6). In Eq. (7) and Eq. (8), A e is the area o the eetively onined ore onrete, A is the area o the ore o olumn setion within enter lines o perimeter spiral, s is the lear spaing between spiral or hoop bars, d is the diameter o spiral, and ρ is the ratio o the area o axial steel to the area o the ore o setion. s When the axial stress reahes the peak ompressive strength, the CFRP breaks and its strain reahes the ultimate strain ε. For the ompatibility ondition o onrete ylinder and CFRP deormation, the ultimate strain is mainly ontrolled by the strength o CFRP omposite materials. Thereore, ε an be expressed as the ollowing equation: ε α tan 45 φ + 2 l 2 = ε (9) 4

5 In Eq. (9), ε is the strain at the ompressive strength o the unonined onrete ( ), usually set at ε =.2. Parameter α is related to the material properties o oninement material; α equals to 2.24 (Li et al., 22) in this paper and its orresponding CFRP material properties are listed in Table 1. As the strain ε alls between to ε, the asending branh stress-strain relation an be simulated by using the seond-order paraboli equation. Substituting three boundary onditions = (atε = ), = (atε = ), and d dε = (atε = ε ε ) into the seond-order paraboli urve, the stress-strain relation o onined onrete are shown as ollows: ε ε ε + 2 ε = 2 (1) where and ε are alulated rom Eq. (2) and Eq. (9). 5

6 3. EXPERIMENTAL PROGRAM Thirty-six onrete ylinders with a dimensions o 3 6 m were designed and tested to veriy the eetiveness o the Modiied L-L model. In this setion, the design and abriation o onrete ylinders and the related uni-axial test programs will be disussed. 3.1 Design o Conrete Cylinders The thirty-six onrete ylinders were divided into 4 groups, and eah group was applied with no, one or 2 layers o CFRP omposite material. For eah design parameter, three onrete ylinders were needed. Groups A, B, C, and D represent dierent types o steel reinorement, suh as irular hoop, two C-shaped lap-splie hoops, irular spiral, and without steel reinorement, respetively. The illustration oniguration o the onrete ylinder is shown in Fig. 2. Table 2 introdues the nomenlature priniples used or the thirty-six onrete ylinders. The irst letter means the type o steel reinorement. A means irular hoop, B means two C-shaped lap-splie hoops, C means irular spiral, and D means no steel reinorement. The number ollowing the irst letter means the number o layers o CFRP applied on the onrete ylinder. The seond number means the serial number o the onrete ylinder. For example, A-1-2 represents the speimen onined by irular hoop with 1-layer CFRP, and its serial number is 2. The design onrete strength was 17.2 MPa (175 kg/m 2 ), and the design onrete slump is 12 m. The steel reinorement used in the onrete ylinders is No. 3 steel with a yield strength o MPa. The steel reinorement was plaed at 1 m spaings, and the onrete over thikness was 2.5 m. For the irular hoop and two C-shaped lap-splie hoops, the lap length o steel reinorement was 11.5 m. The longitudinal rebars are used to hold the horizontal steel reinorements in position. A total o thirty-six plasti pipes, with an internal diameter o 3 m and a height o 6 m, were used as the ormwork or the onrete ylinders. Premixed onrete is used in the experiment. 3.2 Instrumentation o the Compression Test This test program was undertaken using a 49 kn universal-testing mahine, whih is load ontrolled, at the strutural laboratory o the National Taipei University o Tehnology. The experimental equipment inludes load ells, a linear voltage displaement transormer, an analog/digital onverter with a signal ampliier, and a personal omputer. 6

7 3.3 al Observations The ailure mode shape o the onrete ylinders onined by CFRP is approximately onial. The ailure mehanism o ylinders onined by steel reinorement and 2-layer CFRP is desribed as ollows. The onrete between CFRP and steel reinorement spalled, and the CFRP was broken in the enter o the ylinder. This shows that the onrete inside the steel reinorement was still onined by steel reinorement when the CFRP was broken. The experimental observations o the rest o the speimens onined with CFRP and with or without steel reinorement are desribed as ollows. When the stress o the atuator reahed the peak strength o the onined onrete, breaking sounds o the CFRP was heard ontinuously, the CFRP subsequently broke in the middle o the ylinder and the onrete was rushed. The breaking position o the CFRP was not neessarily at the overlaying position. The relationships o the average peak ompressive strength o onined onrete and the number o layers o CFRP are drawn in Fig. 3. As seen rom Fig. 3, we an obtain the ollowing onlusions: 1. When the onrete ylinder is onined by steel reinorement (Groups A, B, and C), its ompressive strength is higher than those without steel reinorement (Group D) and the ompressive strength is highly dependent on the types o steel reinorement. The ompressive strength due to spiral is larger than the 2-C-shaped lap-splie and irular hoop reinorement. 2. When onrete ylinders are onined by steel reinorement and CFRP together (Groups A, B, and C), their ompressive strengths are very lose to eah other. This indiates that the ompressive strength o onrete ylinders onined by CFRP is irrelevant to the types o steel reinorement. The reason is that when CFRP reahes its ultimate strain (usually.15), the strains o steel reinorement o Groups A, B, and C are still within yielding and ultimate strains. The stresses o steel reinorement o Groups A, B, and C are all at the yielding stresses. 4. DISCUSSION ON THE THEORETICAL/EXPERIMENTAL RESULTS In this setion, the theoretial (alulated rom dierent models) and experimental results o the peak strength o onined onrete and the stress-strain urve are ompared. 4.1 The Comparison o the Peak Strength The Modiied L-L model The peak strengths o Group D speimens are ompared with the peak strengths alulated by the Modiied L-L model, and the error analysis o the peak strengths are listed in Table 3. Sine the unonined onrete strength is the peak strength o the speimens, the average error o the D- 7

8 speimens is zero. As or D-1 and D-2 speimens, the errors are 1.5% and 4.4% respetively. The average error o the Modiied L-L model is 2.95%. It an thereore be said that the Modiied L-L model an predit the peak strengths o onrete onined by CFRP very well. The Modiied L-L and the Kawashima models In the last entury, most o the onined onstitutive models were proposed speiially or onrete olumns onined by either steel reinorement or CFRP. Not until 1999 did Kawashima et al. (1999) propose a onstitutive model or onrete onined by both steel reinorement and CFRP *. The peak strengths o A-1, A-2, B-1, B-2, C-1, and C-2 speimens are ompared with the peak strengths alulated by the Modiied L-L model and the Kawashima model, and the error analysis o the peak strengths are listed in Table 4. As seen in Table 4, the average absolute errors o the Modiied L-L model and the Kawashima model are 2.8% and 1.1% respetively. We an onlude that the Modiied L-L model is more aurate than the Kawashima model in prediting the peak strength. 4.2 The Comparison o the Stress-Strain Curve The Modiied L-L model The stress-strain urves o the experiment results o D-, D-1, and D-2 speimens are ompared with the stress-strain urves alulated by the Modiied L-L model, shown in Fig. 4, Fig. 5, and Fig. 6, respetively. As seen in the above igures, we an onlude that the Modiied L-L model an simulate the experimental results very well. The Modiied L-L and the Kawashima models The stress-strain urves o the experiment results o the A-1, B-1 and C-1 speimens, and the Modiied L-L model and the Kawashima model are shown in Fig. 7~Fig. 9 respetively. Similarly, the stress-strain urves o the experimental results o the A-2, B-2 and C-2 speimens, and the Modiied L-L model and the Kawashima model are shown in Fig. 1 to Fig. 12 respetively. As seen in Fig. 7 to Fig. 12, the stress-strain urves o the Modiied L-L model an it the experimental stress-strain urves very well. As or the Kawashima model, the predition o stress is aeptable, but the predition o strain is greatly overestimated. From the observations and disussions on the experimental results o stress-strain urves among the three models, we an onlude that the Modiied L-L model is more eetive than the Kawashima model in the predition o the peak stress, strain at the peak stress, and the stress-strain urves. * In the papers by Kawashima et al. (1997, 1998, 1999), the diameter to height ratio o the irular ylinder (olumn) is 1:3. But, the diameter to height ratio o the irular onrete ylinder used in this paper is 1:2. Thereore, we justiy the peak strength proposed by the Kawashima model by raising it 8%. 8

9 5. CONCLUSIONS From the observation o the experimental results and by omparing the experimental results to the onstitutive models, we an arrive at the ollowing onlusions: 1. The peak stress ormula o the Modiied L-L model is a theoretial equation, and it is derived rom the Mohr-Coulomb ailure envelope theory, whih onorms to the undamental theory o plastiity. The ormula an be used in dierent levels o onining stress. 2. When onrete ylinders are onined by dierent types o steel reinorement, the ompressive strength is highly dependent on the types o steel reinorement. 3. When onrete ylinders are onined by CFRP and dierent types o steel reinorement, their ompressive strengths are very lose to eah other. This indiates that the ompressive strength o onrete ylinder onined by CFRP is irrelevant to the types o steel reinorement. 4. Compared to the test results o the 36 onrete ylinders, the average absolute errors o the peak strength estimation o the Modiied L-L model are less than 3 %, with the exlusion o the ylinders onined by steel reinorement only (suh as the A-, and C- series). As or Kawashima s models, its average absolute errors are about 2 %. 5. Comparing the stress-strain urves o the experimental results with those o the Modiied L-L and Kawashima s models, we an onlude that the Modiied L-L model is more eetive than Kawashima s models. 6. The Modiied L-L model an be applied to onrete ylinders onined by steel reinorement only, by CFRP only, and by both steel reinorement and CFRP. ACKNOWLEDGEMENTS The authors would like to thank Mr. C.-C. Fang or his support in helping to make the onrete ylinders. The valuable omments o Pro. C.-C. Chern at National Taiwan University is also greatly appreiated. REFERENCES Caltrans, Division o Strutures, The Northridge Earthquake-Post Earthquake Investigation Report (1994). Goodman, R. E., Introdution to Rok Mehanis, John Wiley & Sons,

10 Hoshikuma, J., K. Kawashima, K. Nagaya, and A. W. Taylor, Stress-Strain Model or Conined Reinored Conrete in Bridge Piers, Journal o Strutural Engineering, ASCE, Vol. 123, pp (1997). Hosotani, M., K. Kawashima, and J. Hoshikuma, A Stress-Strain Model or Conrete Cylinders Conined by Carbon Fiber Sheets, Civil Engineering, JSCE, 39(592), pp (1998) (in Japanese). Hosotani, M., and K. Kawashima, A Stress-Strain Model or Conrete Cylinders Conined by Both Carbon Fiber Sheets and Hoop Reinorement, Civil Engineering, JSCE, 43(62), pp (1999) (in Japanese). Li, Y.-F., C.-T. Lin, and Y.-Y. Sung, A Constitutive Model or Conrete Conined with Carbon Fiber Reinored Plastis, to appear in Mehanis o Materials (22). Mander, J. B., M. J. N. Priestley, and R. Park, Theoretial Stress-Strain Model or Conined Conrete, Journal o the Strutural Division, ASCE, Vol. 114, pp (1988). 1

11 Table 1. Material properties o CFRP Material Speiiation FAW 2 (g/m 2 ) Young s Modulus, E 23,535MPa ( kg/m 2 ) Tensile Strength 412.2MPa (42 kg/m 2 ) Thikness.11 m/layer Ultimate Strain.18 Table 2. The naming o onrete ylinders Group Speimens CFRP layers A- A A-1 1 A-2 2 B- B B-1 1 B-2 2 C- C C-1 1 C-2 2 D- D D-1 1 D-2 2 Type o steel reinorement Cirular hoop Lap-splie Spiral Nil Table 3. Error analyses o the peak strengths o the Modiied L-L model Speimens Modiied L-L Error MPa (kg/m 2 ) MPa (kg/m 2 ) (%) D (17.3) (17.3) D (26.11) 25.9 (263.97) 1.5 D (342.88) (357.94) 4.4 Average absolute error (%)=

12 Table 4. Error analyses o the peak strengths o the Modiied L-L and Kawashima models Speimen MPa (kg/m 2 ) Modiied L-L MPa (kg/m 2 ) Error (%) Kawashima MPa (kg/m 2 ) Error (%) A (328.94) 3.82 (314.15) (36.35) 9.5 A (46.58) 4.4 (48.12) (455.63) 12.1 B (324.62) 3.82 (314.15) (36.35) 11. B (416.94) 4.4 (48.12) (455.63) 9.3 C (337.74) (324.73) (36.35) 6.7 C (46.45) 41.8 (418.71) (455.63) 12.1 Average absolute error (%)=

13 Spiral N-layer CFRP Fig. 1. The illustration o steel reinorement and CFRP onining onrete ylinder 3 Plasti pipe Longitudinal reinorement Transverse reinorement Unit:m Fig. 2. The illustration onigurations o the onrete ylinder 13

14 5 (MPa) 25 Cirular hoop Lap-splie Spiral Nil Number o the layers o CFRP ( n) Fig. 3. The relationships o the average peak ompressive strengths o onined onrete and numbers o layers o CFRP 6 4 (MPa) 2 Modiied L-L ε Fig. 4. The stress-strain urves o D- speimens and the M-L-L model 14

15 6 4 (MPa) 2 Modiied L-L Fig. 5. The stress-strain urves o D-1 speimens and the M-L-L model ε 6 4 (MPa) 2 Modiied L-L Fig. 6. The stress-strain urves o D-2 speimens and the M-L-L model ε 15

16 6 4 (MPa) 2 Modiied L-L Kawashima Fig. 7. The stress-strain urves o A-1 speimens, the Kawashima, and the M-L-L models ε 6 4 (MPa) 2 Modiied L-L Kawashima Fig. 8. The stress-strain urves o B-1 speimens, the Kawashima, and the M-L-L models ε 16

17 6 4 (MPa) 2 Modiied L-L Kawashima Fig. 9. The stress-strain urves o C-1 speimens, the Kawashima, and the M-L-L models ε 6 4 (MPa) 2 Modiied L-L Kawashima Fig. 1. The stress-strain urves o A-2 speimens, the Kawashima, and the M-L-L models ε 17

18 6 4 (MPa) 2 Modiied L-L Kawashima Fig. 11. The stress-strain urves o B-2 speimens, the Kawashima, and the M-L-L models ε 6 4 (MPa) 2 Modiied L-L Kawashima Fig. 12. The stress-strain urves o C-2 speimens, the Kawashima, and the M-L-L models ε 18