Reliability assessment on maximum crack width of GFRPreinforced

Transcription

1 Fourth International Conerence on FRP Composites in Civil Engineering (CICE2008) 22-24July 2008, Zurich, Switzerland Reliability assessment on maximum crack width o GFRPreinorced concrete beams Z. He and F. Qiu College o Resources and Civil Engineering, Northeastern University, Shenyang, , China ABSTRACT: The suggested provisions in guideline ACI 440.1R-06 or predicting the maximum crack width o GFRP-reinorced concrete beams are assessed rom the probabilistic point o view by the Rackwitz-Fiessler method. The assessment indicates that the global average reliability index is 0.57, meeting the target reliability index o serviceability ultimate state. Eective-height-to-height ratio, d/h, width, b, and height-to-width ratio, h/b, are the irst three dominating inluencing actors on reliability level. The average reliability index is basically independent o relative FRP reinorcement ratio, η. The eect o load eect ratio, α, on reliability level is also signiicant and the index tends to converge at β=. The parametric study on the bond-dependent coeicient, k b, indicates that its mean does not put any eect on reliability level while its C.O.V. is rather signiicant. As the C.O.V. o k b increases rom 0.05 to 0.20, the average index decreases rom 0.53 to The limit o the maximum crack width, [w], is shown to be no relations with reliability level. Speciication on the limit o maximum allowable crack width is dependent only on the consideration o durability and visible tolerance. 1 INTRODUCTION The relatively low moduli o elasticity o FRP bars can cause considerable delection and crack width in lexural FRP-reinorced concrete beams even in their serviceability states. Many eorts have been devoted to develop the prediction models or those states. Such models become more and more accurate with expanding experimental database. But, most o those models have been modiied rom those corresponding to steel reinorced concrete structures, some possible potential risks should be identiied careully because o the sharp dierences in mechanical properties between reinorcing bars and FRP materials (He & Ou, 2005). In 1995, Plevris et al (Plevris et al., 1995) introduced the concept o reliability into the lexural capacity design o concrete beams strengthened with CFRP laminates. Ater this, the lexural capacity o FRP-reinorced concrete beams was assessed rom the probabilistic point o view by some researchers (Huang, 2005, He et al., 2006). He and Qiu (He & Qiu, 2007) conducted the reliability assessment on the serviceability state o FRP-reinorced concrete beams. In the assessment, the delection design provisions in guideline ACI 440.1R-06 (ACI 440, 2006) were evaluated. As the result o the assessment, the delection limit o l/120 is suggested or the short-term delection limit or GFRPreinorced concrete beams. As accompanying research work, this paper presents a reliability assessment on the provisions or predicting the maximum crack width o FRP-reinorced concrete beams designed by ACI 440.1R-06 (ACI 440, 2006). 2 LIMIT STATE FUNCTION The maximum crack width, w, or FRP-reinorced concrete members is determined by the ollowing equations, - 1 -

2 w k d 2 2 s = 2β b c + E 2 β = ( h kd) d(1 k) (2) M a = Ad1 k 3 (3) ( ) ( ) 2 k = n + n n (4) 2ρ ρ ρ where, E is the modulus o elasticity o FRP bar; d c is the thickness o cover rom tension ace to center o closest bar, d c =h-d; h and d are the height and the eective height o cross section, respectively; s is bar spacing, s=b-2d c ; b is the width o cross section; k b is the bond-dependent coeicient, or the case where k b is not known rom experimental data, is that a conservative value o 1.4 should be assumed; β is the ratio o distance between neutral axis and tension ace to distance between neutral axis and centroid o reinorcement; is the reinorcement stress o FRP bar; M a is applied moment, it consists o the moment caused by dead load, M G, and the moment caused by service live load, M Q, i.e. M a =M G +M Q ; kd is neutral axis depth; ρ is reinorcement ratio, ρ =A /bd; A is the sectional area o FRP bar; n is the modular ratio o FRP bar to concrete, n =E /E c. E c is calculated by (ACI , 2005). Ec = 4750 c (5) where, c is the compressive cylinder strength o concrete. The limit state unction, z, o lexural FRP-reinorced concrete beams is expressed as z = [ w] α ww (6) where, [w] is the limit o maximum allowable crack width. The value o [w] is implicitly suggested as 0.5mm or most cases or exterior exposure and as 0.7mm or interior exposure when FRP reinorcement is used; α w is the uncertainty actor associated with the predictions by Eq.(1). It is deined as the ratio o the actual maximum crack width o FRP-reinorced concrete beam to the predicted maximum crack width by Eq.(1). Obviously, α w is a random variable. Through a statistical analysis o a total o 50 eective o GFRP-reinorced concrete beams (Theriault & Benmokrane, 1998; Masmoudi et al., 1998; Xue & Kang, 1999; Gao et al., 2001; Wang & Shi, 2002; Toutanji & Saai, 2003; Liu, 2003; Wang & Belarbi, 2005), the statistical mean and the standard deviation o α w are and 0.139, respectively. For simplicity, the normal distribution is assumed or α w and k b in the assessment. (1) 3 RELIABILITY ANALYSIS 3.1 Design variables From Eqs.(1) to (6), we can observe that the reliability level o maximum crack width or FRPreinorced concrete beams are associated with many actors, e.g., sectional width, b, sectional height, h, eective height, d, the modulus o elasticity o concrete, E c, and FRP bar, E, FRP reinorcement ratio, ρ, load eects, M G and M Q, etc. Among these actors, b, h, d, E c and E are treated as random design variables. The statistics o b, h and d are selected rom Ellingwood et al. s report (Ellingwood et al., 1980). As we know, the modulus o elasticity o concrete, E c, is closely associated with concrete strength, c, so Eq.(5) can be used to evaluate the statistical data o E c by the statistical model o c in Nowak & Szerszen s statistical data (Nowak & Szerszen, 2003). The same method is also applied to evaluate the statistics o E through linear regression o 50 data points between E and GFRP tensile strength, u, as expressed in Eq. (7) (Qiu, 2007). E = u (GPa) (7) The statistical data o GFRP bar in Group A and Group B are abstracted rom Pilakoutas et al s paper (Pilakoutas et al., 2002) and He s report (He, 2003), respectively. The statistical data o dead load and live load are listed in Table 3 (Ellingwood et al., 1980). ρ is treated as a deterministic variable. To make the assessment more general, two extreme nominal values (A and B) are selected or each random design variable as well as twenty ive FRP relative reinorcement ratios, η=ρ /ρ b, where ρ b is the balanced FRP reinorcement ratio as speciied in ACI - 2 -

3 440.1R-06 (ACI 440, 2006). Thus, it deines a broad design space with a total o =800 design cases. The Rackwitz-Fiessler method (Rackwitz & Fiessler, 1978) is applied to calculate reliability index. Five dierent load eect ratios, i.e. α=m Q /M G =0.5, 1.0, 1.5, 2.0 and 2.5, are selected to evaluate the eect o α upon reliability level. So, there are 800 5=4,000 indexes. Assume that both M Q and M G are caused by uniorm distribution loads in the reliability analysis. k b is taken as 1.40 in the analysis. Table 1 Statistical data o design variables Design variable Nominal (Group A) b (mm) 200 h (mm) 1.5b d (mm) 0.8h E c (MPa) E (GPa) Mean μ, Standard deviation, σ μ=b+2.54 σ=3.66 μ=h-3.05 σ=6.35 μ=d-4.70 σ=12.70 μ=25.12 σ=1.17 μ=43.84 σ=0.77 Nominal (Group B) b 0.95h Mean μ, Standard deviation, σ μ=b+2.54 σ=3.66 μ=h-3.05 σ=6.35 μ=d-4.70 σ=12.70 μ=32.27 σ=0.576 μ=51.65 σ=0.86 Probability distribution Table 2 Statistical data o dead load and live load Load pattern Mean/Nominal, C.O.V. a Probabilistic, δ κ distribution Load actor Dead Live b Extreme I 1.0 a : C.O.V.= coeicient o variance. b : 50-year maximum 3.2 Eects o design variables Figure 1 shows the eects o all design variables associated with sectional resistance on average reliability level o crack width o GFRP-reinorced concrete beams. The average reliability index or each design variable is obtained by dividing all the indexes into two groups according to the nominal value o the variable considered and then to calculate the average index in each group. It can be seen that the eect o eective-height-to-height ratio, d/h, is the most signiicant inluencing actor, accompanied by width, b, and height-to-width ratio, h/b. As d/h increases rom 0.8 to 0.95, the average reliability index increases up to 31%. A change o b rom 200mm to 500mm could result in an increase o 26.8% in average reliability level. I h/b increases rom 1.5 to 3.0, the average index would increase up to 10%. The eects o the modulus o elasticity o concrete and GFRP bar, i.e. E c and E, are relatively insigniicant in contrast with d/h, b and h/b. A decrease o 4.4% in average reliability index could be expected i E increases rom 41.83GPa to 49.08GPa. As or E c, the decrease would be 2.3% i it increases rom 21.60GPa to 30.54GPa. The horizontal straight line in Figure 1 shows global the average reliability index o all indexes. As suggested in the General principles on reliability or structures (ISO/DIS 2394, 1998) or the target reliability index o serviceability ultimate state, the target reliability index could be 0 or reversible limit state and 1.5 or irreversible limit state. For partial reversible limit state, the target index could be determined between 0 and 1.5, depending on the degree o reversibility. The study on the lexural behavior o GFRP-reinorced concrete beams has indicated that those beams could exhibit excellent reversibility o crack width i applied load is removed (He, 2003). So, the global average reliability o 0.57 is acceptable as a whole. Figures 2-3 illustrate the eects o load eect ratio, α, and FRP relative reinorcement ratio, η, on the average level. Unlike the signiicant eect o η on the average reliability level o delection o GFRP-reinorced concrete beams (He and Qiu, 2007), it seems that η does not put any eect on reliability level o maximum crack width o GFRP-reinorced concrete beams. As we expected, the eect o load eect ratio, α, is obvious, especially or α is less than 1.5. The - 3 -

4 average index increases at a slowing rate as α increases. The average index would increase about 7.6% i α increases rom 0.5 to 2.5, tending to converge at β= b 0.57 h/b d/h Design variables E c Figure 1 Eects o design variables on β α=0.5 α=1.0 α=1.5 Group A Group B Relative reinorcement ratio η E α=2.0 α=2.5 Figure 2 The relationships between β, η and α Load eect ratio α Figure 3 Eect o α on β 4 PARAMETRIC STUDY As shown in Figure 2, the average reliability index is basically independent o relative FRP reinorcement ratio, η. So, only single value o η is selected, i.e. η=2.0 in the ollowing parametric study o the eects o design variables. Figures 4-6 show the relationships between average reliability index and b, h/b and d/h. As b increases gradually rom 150mm to 500mm, the average reliability index increases signiicantly and nonlinearly, exhibiting a close interrelationship between average reliability level and sectional width. The relationship between average reliability level and height-to-width ratio, h/b, can be approximated by a straight line. Figure 6 shows that average reliability index would increase slowly and nearly linearly as eective-height-toheight ratio, d/h, increases rom 0.5 to 0.8, and that i d/h>0.8, the index would increase very rapidly as d/h increases up The statistics o six values o concrete strength (Nowak & Szerszen, 2003), c, are selected to evaluate the statistics o the modulus o elasticity o concrete, E c, by Eq.(5). The relationship between average reliability index and E c is illustrated in Figure 7 rom which we can observe that and the mean and the standard deviation o E c have some eects on average reliability level. But those eects are insigniicant in contrast with those o sectional design variables. In a general sense, the average index decreases as the nominal value o E c increases. The eect o bond-dependent coeicient, k b, on reliability level is also obvious. For FRP bars having bond behavior inerior to steel, k b is larger than 1.0, and or FRP bars having bond behavior superior to steel, k b is smaller than 1.0. The average k b values vary widely rom to 1.72, with a mean o 1.10 (ACI 440, 2006). To make urther understanding about the eect o k b on reliability level, k b is treated as a random variable here with assumed normal probability distribution. Its mean ranges rom 0.6 to 1.7 and our values o C.O.V. are also taken into consideration, i.e. δ=0.05, 0.10, 0.15 and The analytical results are illustrated in Figure 8, showing that the mean o k b does not put any eect on reliability level. The eect o the C.O.V. o k b is shown to be rather signiicant. As the C.O.V. o k b increases rom 0.05 to 0.20, the average index decreases rom 0.53 to Since the limit or maximum crack width, [w], is a constant - 4 -

5 as speciied in ACI 440 guideline (ACI 440, 2006), the reliability level has no relations with [w] which is conirmed by Figure 9. So, rom the probabilistic point o view, any value o [w] can be speciied or the design o the serviceability state o FRP-reinorced concrete beams. Speciication on the limit o maximum allowable crack width is dependent only on the consideration o durability and visible tolerance. Average Reliability Index β Width b (mm) Figure 4 Eect o b on β Eective depth/height d/h Figure 6 Eect o d/h on β δ=0.05 δ=0.10 δ=0.15 δ= Mean o bond-dependent coeicient k b Figure 8 Eect o k b on β Average Reliability Index β Height/width h/b Figure 5 Eect o h/b on β Modulus o elasticity o concrete E c (GPa) 0.54 Figure 7 Eect o E c on β Crack width limit [w] Figure 9 Eect o [w] on β 5 CONCLUSIONS The reliability assessment indicates that the global average reliability index is 0.57, meeting the requirement o the target reliability index or serviceability ultimate state. Eective-height-toheight ratio, d/h, width, b, and height-to-width ratio, h/b, are the dominating inluencing actors on reliability level. The average reliability index is basically independent o relative FRP reinorcement ratio, η. As load eect ratio, α, increases, the average reliability level also increases but a slowing rate. The parametric study on the bond-dependent coeicient, k b, indicates that its mean does not put any inluence on reliability level while its C.O.V. is rather distinct. The limit o the maximum crack width, [w], is shown to be no relations with reliability level. Speciication on the limit o maximum allowable crack width is dependent only on the consideration o durability and visible tolerance

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