CENTER FOR INFRASTRUCTURE ENGINEERING STUDIES

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1 CENTER FOR INFRASTRUCTURE ENGINEERING STUDIES FLEXURAL BEHAVIOR OF FRP-REINFORCED CONCRETE MEMBERS By Alkhrdaji, Mettemeyer, Belarbi and Nanni University o Missouri-Rolla CIES 99-15

2 Disclaimer The contents o this report relect the views o the author(s), who are responsible or the acts and the accuracy o inormation presented herein. This document is disseminated under the sponsorship o the Center or Inrastructure Engineering Studies (CIES), University o Missouri-Rolla, in the interest o inormation exchange. CIES assumes no liability or the contents or use thereo.

3 The mission o CIES is to provide leadership in research and education or solving society's problems aecting the nation's inrastructure systems. CIES is the primary conduit or communication among those on the UMR campus interested in inrastructure studies and provides coordination or collaborative eorts. CIES activities include interdisciplinary research and development with projects tailored to address needs o ederal agencies, state agencies, and private industry as well as technology transer and continuing/distance education to the engineering community and industry. Center or Inrastructure Engineering Studies (CIES) University o Missouri-Rolla 223 Engineering Research Lab 1870 Miner Circle Rolla, MO Tel: (573) ; ax cies@umr.edu

4 1. INTRODUCTION Many structures, such as bridges and parking garages, are usually treated with deicing salts and are, thereore, subjected to an aggressive environment. For such structures, possible corrosion o reinorcing and prestressing steel may eventually lead to concrete deterioration and loss o serviceability or capacity. To control corrosion problems, proessionals have turned to alternative reinorcements such as epoxy-coated steel bars. However, such remedies were only ound to slow down, rather than eliminate corrosion problems. Recently, iber reinorced polymer (FRP) materials have emerged as an alternative to steel reinorcement. FRP materials are corrosion resistance and exhibit several properties that make them suitable as structural reinorcement (ACI Committee 440, 1996 and Japan Society o Civil Engineers (JSCE), 1997). In order to apply this new technology to practice, a better understanding o the behavior o FRP reinorced members is required. In addition, current codes provide no guidance on how to modiy the existing requirements when reinorcing with materials other than steel. Some o the main dierences o FRP reinorcement when compared to steel reinorcement are higher tensile strength, lower stiness, and elastic behavior up to ailure with no yielding (no plasticity). These dierences are relected on the lexural behavior o FRP-reinorced members. The conventional concept o under-reinorced members as a avorable design approach is not practical or FRP-reinorced members because it will result in members having lower stiness hence, larger delection and crack widths are expected. Available experimental results o FRP reinorced sections indicate that when FRP reinorcing bars ruptured (tension-controlled ailure), the ailure was sudden and led to the collapse o the member (Nanni, 1993; GangaRao and Vijay, 1997; and Theriault and Benmokrane, 1998). However, a more progressive and less catastrophic ailure was observed when the member ailed due to the crushing o concrete (compressioncontrolled ailure). This behavior results in higher deormability, which is deined as the ratio o energy absorption (area under moment-curvature curve) at ultimate to that at service level (Jaeger et al., 1997). In general, lexural design can be perormed using the principles o the ultimate strength method given in ACI , building code or according to the principles o the allowable stress method using the approach provided in Appendix A o the ACI , building code. The latter can produce more conservative results (e.g. stier sections having smaller delections). To address the eect o FRP properties on the lexural behavior, comparisons will be made or sections reinorced with the three main types o FRP reinorcement namely, glass (GFRP), aramid (AFRP), and carbon (CFRP). The properties o the three types considered hereater are given in Table 1. These are typical properties o FRP bars available in the market today. Table 1. Material Properties o FRP Reinorcement Reinorcement Type Ultimate Strength Modulus (ksi) (ksi) Ultimate Strain GFRP AFRP CFRP

5 1.1 Basic Assumptions Experimental data on concrete members reinorced with FRP bars indicates that the lexural capacity can be predicted using the same assumptions made or steel reinorced concrete members (Faza and GangaRao, 1993; Nanni, 1993; and GangaRao and Vijay, 1997). Thereore, or FRP reinorced members, it will be assumed that plane sections beore loading remain plane ater loading (i.e., no shear deormation) and that concrete and FRP strains are proportional to the distance rom the neutral axis (i.e., perect bond). The maximum usable strain at the extreme concrete compression iber is assumed to be in/in provided that the speciied ultimate design strain o FRP, ε u, is not reached irst. Concrete is assumed to resist no tension. The compressive stress distribution in the concrete at ultimate is represented by Whitney s equivalent stress block, provided that suitable stress block actors are used based on the compressive strain at the extreme concrete iber. The stress-strain relationship o FRP is linear up to ailure with the maximum stress equal to the speciied ultimate design strength, u. The recommended value or the speciied ultimate design strength can be calculated as the mean strength minus three times the standard deviation ( u = u,ave 3σ) (Mutsuyoshi et al., 1990). This speciied ultimate strength provides a 99.87% probability that the guaranteed strength is exceeded. Research on FRP bars indicates that their mechanical properties can degrade with time due to creep rupture, atigue, and aggressive environment (see section 2.5.1). Using the guaranteed strength or design will only ensure the short-term strength o the lexural member. To account or this, strength knockdown actors are introduced, which are denoted as Kd. The knockdown actors will secure the design against premature ailure due to long-term eects on the reinorcing bars. Using the knockdown actors will also result in a design that can meet serviceability and allowable stress requirements. The ultimate design strength o FRP reinorcement can thereore be taken as d = u.kd. The proposed values or Kd, given in Table 1, are consistent with the allowable stresses or the given FRP types discussed in Section Table 1. Proposed Values or the Knockdown Factor, K d. Fiber Type GFRP AFRP CFRP Stress Ratio o u DESIGN FOR FLEXURE In the ultimate design method, the primary attention is placed on the predicted strength with the serviceability limits being checked ater the design is completed. However, in many cases, serviceability considerations will control the proportioning o FRP reinorced members. The structural members is designed to have a design strength at least equal to the required strength using the load actor combinations speciied by Section 9.2 o the ACI building code. The present conception o under-reinorced and over-reinorced ailure modes or concrete members reinorced with steel is not applicable since FRP reinorcement does not yield. 2

6 2.1. Saety Factor or Flexure ( Factor) Due to the limited data available on service and long-term perormance o FRP composites in concrete structures, a more conservative value or the φ actor or lexural design should be adopted. The Japanese recommendations or design o lexural members using FRP suggest a material reduction actor o 1/1.3 or glass FRP reinorcement and 1/1.15 or carbon and aramid FRP reinorcement (Japan Society o Civil Engineers, 1997). ACI Committee 318 suggests that a lower saety actor should be used or compression controlled sections than is used or tension-controlled section because compression controlled sections generally have less ductility and are more sensitive to variations in concrete strength. Thereore, a φ actor o 0.7 is suggested or the lexural design o FRP reinorced members. It should be recognized however that or most FRP reinorced systems, service requirements control the design and a φ actor o 0.7 is not too restrictive Stress o FRP Reinorcement The capacity o steel reinorced concrete members is calculated based on the assumption that all tension reinorcement yield at ultimate. Thereore, the tension orce is assumed to act at the centroid o the reinorcement with a magnitude equal to the area o tension reinorcement multiplied by the yield strength o steel. This assumption is valid or steel reinorcement arranged in one layer or multiple layers. FRP materials, on the other hand, have no plastic region. The stress in each layer o the FRP reinorcement will vary depending on its distance rom the neutral axis. In this case, the lexural capacity must be based on a strain compatibility approach. When using a single type o FRP reinorcement, the outermost layer controls reinorcement ailure. Similarly, i dierent types o FRP bars are used to reinorce the same member, the variation in the stress level in each bar type must be considered when calculating the lexural capacity. Unless concrete crushing controls, the layer that reaches it capacity irst controls the capacity o the member. The ailure mode controlled by the rupture o FRP bars is catastrophic and thereore undesirable. For this reason, members reinorced with FRP should be so proportioned to ensure a compression ailure (Nanni, 1993). However, or some nonrectangular sections (e.g., T sections), space limitations may not allow or the placement o suicient FRP reinorcement to over-reinorce the member. In such cases, underreinorced sections may be used i an appropriate saety actor is included and serviceability requirements are satisied. ACI Committee 318 (1995) speciies that the maximum amount o reinorcement in steel reinorced concrete sections is limited to 75 percent o the amount o reinorcement at the balanced strain condition. This limit was set to consistently ensure yielding o reinorcement prior to crushing o concrete. Analogous limit can be set when designing with FRP reinorcement to attempt to ensure the crushing o concrete (compressioncontrolled ailure). To achieve this, the amount o FRP reinorcement should be larger than the balanced amount, ρ,b. However, It should be noted that i the concrete strength is higher than the design strength (as it may occur because o production or age), FRP 3

7 rupture could become the controlling ailure mode. The saety limit can be set as 1/0.75 or 1.33 and the minimum amount o FRP reinorcement is such that: ρ,min = 1. 33ρ,b (1) where ' c E εcu ρ,b = 0.85β1 (2) E ε + in which, ε cu is the ultimate permissible strain in concrete taken as 0.003, as shown in Figure 1. Utilizing equilibrium and compatibility requirements, the relationship between the ratio o FRP reinorcement and the stress in the reinorcement can be expressed as ollows: ' c E ecu? = 0.85ß1 (3) E ecu + Dividing Equation (3) by Equation (2) yields the ollowing expression: e cu 1+? u eu = (4)?, b ecu + e Equation (4) is only valid or the purpose o veriying the short-term behavior o FRP reinorced members (e.g., laboratory experiments). For design purposes, the ultimate design strength o FRP bars should account or two actors. The irst, is the low allowable stress at service level, which can be as low as 20% o the ultimate strength o a bar (see Section 2.5.1), and the second, is satisying serviceability requirements. u u cu u u b ε cu = d N.A. c b a b =β 1 c b 0.85 c ba b A,b ε,u A,b u Figure 1. Flexural stress and strain diagrams at balanced strain condition. Generally, FRP bars have lower stiness than steel reinorcement, resulting in larger delections and wider cracks. To these eects, the strength knockdown actor can result in more practical designs that can satisy the allowable stress and meet serviceability requirements. The knockdown actor, Kd, is used to calculate a conservative ultimate design strength d, rom which the corresponding design strain e,d is determined. Examining Equation (2) indicate that the relationship between the balanced amount o reinorcement and the strength o FRP bars is not linear and the change in the balanced reinorcement ratio r,b is not proportional to the reduction in the ultimate strength. For the three types o FRP bars considered in this document (GFRP, AFRP, and CFRP) and 4

8 considering a 5000 psi concrete, introducing Kd values o 0.35, 0.5, and 0.65 (given in Table 2) will cause r,b to increase 5.0, 2.6, and 1.6 times, respectively. The larger value obtained or glass FRP bars relects the inerior long-term properties and lower stiness o this type o reinorcement. Figure (2) shows the eect o the knockdown actor on the relationship between /u and r/r,b or over-reinorced members. Similarly, the relation between /,d and r/r,b is obtained by substituted or,u and e,u by the corresponding design values o,d and e,d in Equation (4), which yields the ollowing ecu 1+? d ed expression: = (5)?, b ecu + e Equations (4) and (5) are plotted in Figure 3 or the three dierent types o FRP bars namely, glass, aramid and carbon. Also plotted in Figure 3 is the relation between s/y and rs/rb or steel bars with y o 60 ksi. The igure clearly indicates that or typical crosssections, imposing a reinorcement ratio o 1.33r,b will limit the tensile stress in the FRP reinorcement at ultimate to a value o 0.86u or Equation (4) and 0.85d or Equation (5) while or steel, a reinorcement ratio o 1.33rb will correspond to a stress o 0.8y. The resulting stress limits in FRP bars relect the saety actor used to prevent the rupture o the bars. The curve or steel alls below those or FRP due to the higher stiness o steel compare to that o FRP bars. The igure also reveals that or a given ratio o r/r,b, the type o FRP reinorcement has very small inluence on the stress ratio, /,u. In addition, Equations (4) and (5) indicate that the stress level in FRP reinorcement is not inluenced by the concrete strength. d d 5

9 /,u or /,d r /r,b = 1.33 Glass Aramid Carbon Glass (Kd) Aramid (Kd) Carbon (Kd) Steel, y=60 ksi r /r,b Figure (3). Normalized relationship between stress level and amount o reinorcement or over-reinorced sections. Although a ratio o 1.33r,b may seem high, or a typical rectangular RC member, the minimum amount o FRP reinorcement is much smaller than the balanced amount o steel reinorcement and can be even smaller than the minimum amount o steel reinorcement. For example, or a rectangular RC cross section with Grade 60 steel and c o 5,000 psi, the minimum amount o steel reinorcement as speciied by the ACI-318 building code is and the balanced ratio o steel reinorcement is The minimum ratios o GFRP, AFRP and CFRP reinorcement determined or,d using Equation (1) are , , and , respectively. These minimum amounts o FRP reinorcement are smaller than the maximum allowable ratio o steel reinorcement. For a typical rectangular cross-section with b d equal to 8 in. 12 in., the calculated ratios correspond to 5#7, 2#6, and 3#4 o GFRP, AFRP and CFRP bars, respectively. FRP reinorcement has a compressive strength that is highly variable and signiicantly lower than its tensile strength (Kobayashi and Fujisaki, 1995 and Japan Society o Civil Engineers, 1997). Except when large amount is provided, the inluence o FRP reinorcement in compression on the lexural capacity o the member is minimal (Almusallam et al., 1997). The strength o any FRP bar in compression can thereore be ignored when calculating the lexural capacity o FRP reinorced members (Japan Society o Civil Engineers, 1997). 6

10 2.3. Minimum Reinorcement Area at Cracking Condition A minimum amount o reinorcement should be used such that the actored nominal capacity o the member, φm n, exceeds the cracking moment o the section, M cr. The current ACI equations or minimum steel reinorcement are based on this concept and, with some modiications, can be equally applied to FRP reinorced members. The modiications result rom two sources. The irst is the choice o a dierent φ actor (i.e., 0.7 instead o 0.9); and the second is the limitation o stress in the FRP to avoid its rupture. The suggested value or stress limit is 0.8 d (see Figure 2). The minimum reinorcement area or FRP reinorced members can be thereore obtained by multiplying existing ACI equations or steel limits by a constant equal to , which yields the ollowing expression: i.e., ' c A,cr = b w d (6) d but not less than: 320 A,cr = b w d (7) d In analysis, Equations (6) and (7) should be used as a check when the reinorcement ratio is less than the balanced ratio (ρ < ρ,b ) Ultimate Strength Method Design The design o an FRP-reinorced member according to this method is similar to the conventional over-reinorced case o a steel-reinorced member in which concrete crushes beore the yielding o steel reinorcement. For compression-controlled ailure (see Figure 4) and using equilibrium and compatibility conditions, the ollowing two equations can be derived: M ρ 2 n = ρ ( ) bd (8) ' c (E ε ) 2 cu 0.85β1 ' c = + E εcu 0.5E εcu (9) 4 ρ Equations (8) and (9) are only applicable to rectangular sections with one layer o FRP reinorcement. 7

11 b ε cu = d N.A. c a b =β 1 c 0.85 c ba A ε < ε,d A Figure 4. Stress and strain distribution o FRP reinorced sections at ultimate The design can be achieved by solving Equations (8) and (9) simultaneously. For preliminary design, the designer can use Equations (4) instead o (9) and assume an appropriate value or / u (utilizing Figure 2, say / u =0.7). In all cases, a inal check should be made to ensure that the stress in FRP reinorcement at service level will not exceed the allowable stresses Analysis In analysis, the controlling mode can be determined by comparing the reinorcement ratio, ρ, to the reinorcement ratio at the balanced strain condition, ρ,b. Two modes o ailure are considered in this process: Compression ailure. When ρ ρ,b. In this case, the stress in FRP reinorcement can be determined rom Equation (9). The capacity is then calculated using Equation (8). Tension ailure. When ρ ρ,b, the section ails by rupture o the FRP bars without crushing o concrete, which indicates that concrete did not reach its ultimate strain. Calculations are complicated because the strain in concrete at ultimate is less than and hence, the depth o neutral axis, c, and the lever arm, jd, are unknown. In addition, the rectangular stress block actors, α 1c and β 1c corresponding to the concrete strain ε c are also unknown. The calculations can be simpliied i appropriate values are assumed or the depth o neutral axis and the lever arm. The ultimate strength o FRP under-reinorced rectangular member can be expressed as ollows: ß1c c M ne = A d (d ) (10) 2 where the subscript e in M ne reers to the exact analysis. The parameters α 1c and β 1c or Whitney s equivalent stress block corresponding to any compressive strain in concrete extreme ibers can be expressed as ollows: 4ε co εc β 1c = (11) 6ε 2ε co c 8

12 2 3ε coε c εc α 1c = 2 (12) 3β1cεco Equations (11) and (12) are derived based on parabolic stress-strain relationship o concrete using numerical integration. The parameter ε co is the concrete strain corresponding to peak concrete stress, c, and is deined as ollows (Todeschini, 1982): 1.71 c ε co = (13) E c Using compatibility requirement and knowing that the stress in FRP reinorcement will reach its ultimate design stress at ailure, u,, the depth o neutral axis can be expressed as ollows: εc c = d (14) ε + ε c Using equilibrium requirement and utilizing Equation (14) compatibility, the relationship between the FRP reinorcement ratio, ρ, and the strain in concrete extreme ibers, ε c, can be expressed as ollows: ' c ec? = α1c ß1c (15) d ec + εd Substituting Equation (14) in Equation (10) yields the ollowing: ß1c εc M ne = A d d(1 ) (16) 2 ε c + εd In general, as the ratio o FRP reinorcement, ρ, increases the compressive strain in concrete extreme ibers, ε c, at ailure will increase and will reach ε cu = when ρ = ρ,b. The value o β 1c, on the other hand, will decrease as ε c increases. Hence the product o β 1c c given in Equation (10) will remain relatively constant and can be approximated by their product at balanced strain condition given as β 1 c b. Where β 1 is calculated rom equation (11) or ε c = ε cu = or using the β 1 value given in the ACI building code. Based on this approach, the approximate ultimate capacity o under-reinorced members can be expressed as ollows: ß1 cb M na = A d (d ) (17) 2 In which the depth o neutral axis at balanced strain condition is calculated using the ollowing expression: εcu c b = d (18) εcu + εd Substituting Equation (18) in Equation (17) yields the ollowing: ß1 εcu M na = A d d(1 ) (19) 2 εcu + εd The ratio o M na /M ne is then obtained by dividing Equation (19) by Equation (16) as ollows: d 9

13 ß1 εcu 1 M na 2 εcu + εd jb = = (20) M ß ne 1c εc j 1 2 ε c + εd Where j b and j are the ratios o the lever arm to reinorcement depth, d, at balanced strain condition and at any under-reinorced case, respectively. In addition, the ratio o ρ /ρ,b is obtained by dividing Equation (15) by Equation (2) as ollows: ρ α1cß 1c εc εcu + εd = (21) ρ α ß ε ε + ε, b 1 1 The relationship between M na /M ne and ρ /ρ,b is plotted in Figure 5 or the three types o FRP bars using two dierent concrete strengths, 4 and 8 ksi. Relationships are terminated when the reinorcement ratio o ρ is smaller than the minimum amount given by equation (6). The igure clearly indicates that the simpliied approach, expressed by Equation (17) gives a slightly conservative approximation o the ultimate strength o under-reinorced rectangular members. It was mentioned earlier that the ailure o FRP under-reinorced member is sudden and can occur without warning. Thereore, an additional actor o saety should be included to secure against this ailure mode. Similar consideration was addressed earlier or over-reinorced members where the stress in FRP reinorcement was limited to 0.86 u indirectly by imposing a reinorcement ratio limit o 1.33ρ,b. Since the ailure o underreinorced members is catastrophic, the authors recommend that the stress in FRP underreinorced members be limited to 0.8 u. Hence, the simpliied equation or calculation o the nominal capacity o under-reinorced sections can be expressed as ollows: β1cb M n = 0.8A d (d ) (22) 2 In all cases, the provided area o reinorcement should be lager than the minimum amount, A,cr, speciied by Equations (6) and (7) Allowable Stress Method FRP reinorced concrete can be designed according to the allowable stress method outlined in Appendix A o the ACI , building code. Using the allowable stress method usually results in stier sections having smaller delections. The design o concrete members according to the allowable stress method ollows the assumptions that concrete compressive stress and tensile stress in FRP do not exceed the allowable stresses, the strain distribution in the concrete is linear and proportional to the distance rom the neutral axis, and the concrete stresses can be calculated rom the strain using Hooke s Law. The stress and strain conditions at service level are shown in Figure 7. The ollowing discussion is limited to rectangular sections with one layer o FRP reinorcement. cu c d 10

14 Mn a /Mn e GFRP AFRP CFRP d (ksi) E (ksi) GFRP, 'c=4 AFRP, 'c=4 CFRP, 'c=4 GFRP, 'c=8 AFRP, 'c=8 CFRP, 'c= r /r,b Figure (5). Accuracy o the proposed equation or under-reinorced sections or dierent values o r /r b. j GFRP, 'c=4 AFRP, 'c=4 CFRP, 'c=4 GFRP, 'c=8 AFRP, 'c=8 CFRP, 'c= GFRP AFRP CFRP d (ksi) E (ksi) r /r,b Figure (6). Variation o parameter j with respect to r /r b. 11

15 d b N.A.. ε c ε c,a kd 0.5 c bkd A ε ε,a A Figure 7. Flexural stress and strain conditions at service load Allowable Stresses The recommended allowable compressive stress in concrete, c,a, can be taken as 0.45 c (Faza, 1991 and Nanni, 1993), which is similar to the limitation o the ACI building code. Creep-rupture, atigue, and durability (i.e., resistance to environmental conditions such as temperature, moisture, and alkalinity) are major material properties that should be considered when setting the limits or the allowable stress in FRP reinorcement. Research on creep-rupture has indicated that this material property depends on iber type, stress level, and temperature o the surrounding environment (Ando et al., 1997 and Dolan et al., 1997). Table 3 shows recommended stress limits as a ratio o u to prevent the creep-rupture o FRP reinorcement. Recommended values are based on the test results available in literature (Ando et al., 1997; Dolan et al., 1997; Seki et al., 1997; and Yamaguchi et al., 1997). Table 3. Stress Limits as a Ratio o Ultimate Strength to Prevent the Creep Rupture Failure o FRP Reinorcement. Service Lie (years) GFRP AFRP CFRP > The atigue characteristics o FRP bars depend on FRP types, stress levels, stress ratios, number o cycles, loading requency, and durability. Table 4 shows the recommended stress limits as a ratio o u to prevent the atigue ailure o dierent FRP types. Recommended values were based on the available data or tests conducted at temperatures o 68 ± 5 F and a maximum to minimum stress ratio o 10. (Rahman and Kingsley, 1996; Rahman et al., 1997; Adimi et al., 1998; and Hayes et al., 1998). 12

16 Table 4. Stress Limits as a Ratio o Ultimate Strength to Prevent the Fatigue Failure o FRP Reinorcement. Fiber Type GFRP AFRP CFRP Stress Ratio o u The allowable stresses can be derived by multiplying the lowest stress limit given in Tables 3 and 4 or each type o FRP by a reduction actor o 0.8. This knock down actor is introduced to relect the environmental eects on material characteristics. The proposed allowable stresses in FRP reinorcement at service level are shown in Table 5. Table 5. Allowable Stress Ratios or FRP Reinorcement at Service Level. Fiber Type GFRP AFRP CFRP Allowable Stress,,a as a Ratio o u Design The ratio o FRP reinorcement at the balanced allowable stress condition, ρ,bs, can be expressed as ollows: c,a n c,a ρ =,bs (23) 2,a n c,a +,a where,a and c,a are the allowable stresses in FRP reinorcement and the concrete, respectively and n is the ratio o the elastic modulus o the FRP bar to that o concrete. A parameter k b is deined as the ratio o the depth o the neutral axis to reinorcement depth at the balanced allowable stress condition and is expressed as ollows: n c,a εc,a k b = = (24) n c,a +,a εc,a + ε,a The service moment capacity o a rectangular section at service loads level, M s, can be expressed as ollows: k 2 Ms = ρ (1 ) bd (25) 3 Two conditions may be encountered in design. The irst, when the stress in FRP reinorcement,, reaches its allowable stress,,a, while stress in concrete, c, is still below its allowable stress, c,a ( =,a and c < c,a ). Considering the state o stress given in Figure 6 and satisying compatibility requirements, the ratio o FRP reinorcement can be expressed as ollows: c εc ρ,s = (26) 2 ε + ε,a c,a 13

17 The second condition occur when the stress in concrete, c, reaches its allowable, c,a, while the stress in FRP reinorcement,, is still below its allowable stress,,a ( c = c,a and <,a ). In this case, the ratio o FRP reinorcement can be expressed as ollows: c,a εc,a ρ =,s (27) 2 ε + ε Dividing Equation (26) by Equation (23) yield the ollowing relationship: ε,a 2 1+ ρ,s ε c c,a = ρ,bs c,a ε c,a + ε c,a Figure 8 shows the variation o c / c,a with respect to ρ /ρ,bs. It is clear that the ratio o ρ /ρ,bs is less sensitive to the material properties. Mainly because the ratio o ε,a/ ε c,a given in Equation (28) has a value o 10 or most cases. The igure also indicates that the irst stress condition ( =,a and c < c,a ) will only occur when ρ /ρ,bs is less than 1.0. Because FRP exhibits high tensile strength and low modulus o elasticity, this condition requires minimal amount o FRP reinorcement. However, design based on this stress condition result in a section or which the cracking moment may exceed the moment capacity o the reinorced section. To investigate this behavior urther, consider the minimum amount o FRP reinorcement required to prevent ailure upon cracking as given by Equations (6). The minimum reinorcement ratio can be expressed as ollows: 5 ' c ρ,cr = (29) u Dividing Equation (23) by Equation (29) and substituting c,a = 0.45 c yields the ollowing: ρ,bs ' n u c,a = 1.42 c (30) ρ,cr,a n c,a +,a Equation (26) is plotted in Figure 9 or dierent values o c or the three types o FRP reinorcement. The igure indicates that the ρ,bs /ρ,cr ratio is inluenced by the type o FRP reinorcement (e.g., material properties) and by concrete strength. Also, or a practical range o c, the ratio o ρ,bs /ρ,cr is very close to or less than 1.0. This clearly indicates that a member designed or the irst stress condition would ail upon cracking. More conservative and practical approach can be employed by using the second stress condition ( c = c,a and <,a ). Stress condition or this approach is investigated by dividing Equation (27) by equation (23), which yields the ollowing expression: c,a c,a (28) 14

18 c/c,a Glass, 'c= GFRP AFRP CFRP Glass, 'c= u (ksi) E (ksi) Aramid, 'c=8 'c given in ksi units Carbon, 'c= r /r,bs Aramid, 'c=4 Carbon, 'c=4 Figure (8). Variation o concrete stress with respect to r /r,bs r,bs/rmin,cr GFRP AFRP CFRP u (ksi) E (ksi) 6500 'c given in ksi units Glass Aramid Carbon ' c (ksi) Figure (9). Eect o FRP properties and concrete strength on r,bs /r min,cr. 15

19 ρ ρ,s,bs =,a ε 1+ ε The variation o /,a with respect to ρ /ρ,bs is plotted in Figure 10. The igure shows that the second stress condition requires a value o ρ /ρ,bs that is larger than 1.0. This approach is recommended or design since the larger required amount o FRP reinorcement will insure a capacity higher than the cracking moment in most cases. It is also recommended to limit the stress ratio o /,a to 0.8. This limit will provide a margin o saety that accounts or the variation o concrete strength.,a c,a,a ε + ε c,a,a (31) Glass Aramid Carbon Eq. Series4 (32) /,a Glass (8) Aramid (8) Carbon (8) GFRP AFRP CFRP u (ksi) E (ksi) r /r,bs Figure (10). Variation o FRP stress with respect to r /r,bs Typically, the ratio ε c,a /ε,a is in the order o 0.1. Thereore, Equation (31) can be simpliied urther by substituting ε c,a /ε,a = 0, which yields the ollowing: ρ ρ,s,bs = Equation (32) can be used or preliminary design o the lexural member.,a 2 (32) 16

20 Equation (32), which is plotted in Figure 10, overestimates the required amount o FRP reinorcement. For example, when ρ / ρ,bs is about 2.5, the equation will overestimate the required amount o FRP by about 8%. When the cross sectional dimensions are unknown, the design is initiated by selecting an FRP type and imposing /,a ratio o 0.8 or ρ / ρ,bs ratio o 1.6. The design is achieved using equation (25) and equation (31) where the parameter k is determined as ollows: n c,a εc,a k = = (33) n c,a + εc,a + ε In the case where the cross sectional dimensions are given, dierent approach should be employed. Because FRP bars exhibit high tensile strength and small modular ratio, n, the controlling design parameter at service level is the compressive stress in concrete. Thereore, the design solution should be obtained by assuming that c is equal to c,a. The nominal capacity at the service level can be expressed as ollows: 1 2 k M s = c,abd k (1 ) (34) 2 3 From Equation (34) the parameter k can be derived as: 6 M s k = (35) 2 bd In addition, the stresses in reinorcement and at the extreme compression iber can be related as ollows: 1 k = n c, a (36) k I is less than 0.8,a, then the required amount o reinorcement can be determined using the ollowing equation: M s A = (37) k d (1 ) 3 On the other hand, i is larger than 0.8,a then should be set equal to 0.8,a and Equations (36) and (37) are solved simultaneously to obtain the values o k and c. A inal check should be made to ensure that A is larger than the minimum amount required to prevent ailure upon cracking Analysis Two cases may be encountered in the analysis o concrete members according to the allowable stress method. I provided ratio o, ρ, is less than the balanced ratio at service, ρ,bs, then the stress in FRP reinorcement is equal to,a while the stress in concrete can be determined using Equation (28). Once the stresses in concrete and FRP are determined, the parameter k can be calculated using the ollowing Equation: n c ε c k = = (38) n + ε + ε c,a c c,a,a 17

21 The capacity is then calculated using the ollowing Equation: 1 2 k Ms = cbd k(1 ) (39) 2 3 When the provided ρ is larger than ρ,bs, then the stress in FRP reinorcement is less than,a while the stress in concrete is equal to the allowable stress c,a. In this case, the stress in FRP reinorcement is determined using Equation (32) and should not exceed 0.8,a, the parameter k is determined using Equation (33), and the capacity using Equation (25). A more simpliied approach can be achieved by taking the nominal moment capacity to be the smaller o the two values calculated using Equation (34) and Equation (25), or which =0.8,a. The value o k is calculated using the ollowing expression: k 2 = 2ρ n + ( ρ n ) ρ n (40) 3. FLEXURAL SERVICEABILITY Whether the design o a lexural member is achieved using the ultimate strength method or the allowable stress method, crack widths and delections should be checked. The current expressions and requirements o the ACI building code or crack width and delection need to be modiied to relect the dierence between material properties o steel and FRP reinorcement. In general, serviceability requirements o crack width and delection may become the controlling actors in the design o FRP reinorced members. Compared to steel reinorced members having the same cross sectional dimensions sand lexural capacity, larger delections and crack widths may be expected under service loads as a result o the lower lexural rigidity o some FRP reinorcement Cracking Crack widths in FRP reinorced members are expected to be larger than those in steel reinorced member. Experimental and theoretical research on crack width (Faza and GangaRao, 1993; Masmoudi et al., 1996; and Gao et al., 1998) has indicated that the well-known Gregerly-Lutz equation can be modiied to give a reasonable estimate o the crack width o FRP reinorced members. The original Gregerly-Lutz equation can be expressed as ollows: w = β(e 3 s εs) dc A (41) where w is the crack width in units o inch, β is the ratio o the distance rom the neutral axis to the extreme tension iber to the distance rom the neutral axis to the center o tensile reinorcement, d c is the thickness o the concrete cover measured to the center o the irst layer o reinorcement, and A is the eective tension area o concrete In reality, the crack width is proportional to the strain in tensile reinorcement rather than the stress (Wang and Salmon, 1992). Hence, Equation (41) can be applied to predict the crack width o FRP reinorced lexural members i the steel strain, ε s, is replaced by an equivalent FRP strain, ε = / E, which results in: 18

22 w E s = β 3 dc A (42) E where is the stress in the FRP reinorcement given in ksi. When used with FRP deormed bars having bond strength similar to that o steel, Equation (42) can estimates crack width accurately (Faza and GangaRao, 1993). However, this equation may overestimate crack width when applied to a bar with a higher bond strength than that o steel, and underestimate crack width when applied to a bar with a lower bond strength than that o steel. In order to make Equation (41) more generic, it is necessary to introduce another coeicient that accounts or the bond behavior between the bar and the surrounding concrete. For FRP reinorced members, crack width calculation should be as ollows: w E s = β k 3 b d c A (43) E where k b is a bond-dependant coeicient. For steel reinorcement or FRP bars having similar bond behavior to steel, k b can be taken as 1.0. For FRP bars having inerior bond behavior, k b can be larger than 1.0, and or FRP bars having superior bond behavior, k b can be smaller than 1.0. It should be noted, however, that the value o k b may be inluenced by many other actors such as bond strength, bar diameter, and bar surace condition. The parameters aecting k b are yet to be determined through experiments. Using the test results rom Gao et al. (1998), the calculated values o k b or three types o GFRP rods were ound to be 0.71, 1.00, and These values indicate that bond characteristics o GFRP bars can vary rom superior to inerior to that o steel and can have dierent values or the dierent products even though the same types o ibers are used. The value o k b should be provided by the manuacturer o the FRP reinorcement. Further research is needed to veriy the eect o bond strength on the crack width. is not known, a value o 1.3 is suggested or FRP deormed rods Short-Term Delections Branson (1977) derived an equation to express the transition rom I g to I cr. His equation was adopted by the ACI in the ollowing ormat: 3 3 M cr M cr Ie = I g 1 I cr I g M + a M (44) a The ACI ormula given by Equation (44) was based on the behavior o steelreinorced beams in the elastic range. Since FRP bars have a linear behavior up to ailure, they provide ideal conditions or the ACI equation (Zhao et al., 1997). Research on delection o FRP-reinorced beams (Benmokrane et al., 1996 and Brown and Bartholomew, 1996) indicates that experimental delection curves o simply supported beams are parallel to the theoretical curves predicted by the ACI ormula. Equation (44) may overestimate the eective moment o inertia o FRP reinorced beams (Benmokrane et al., 1996). In addition, bond characteristics o dierent FRP bar types also aect the delection behavior o a member. Gao et al. (1998a) proposed a modiied expression or the eective moment o as ollows: I k b 19

23 3 3 M cr M cr Ie = I g 1 I cr I g M β + a M (45) a where β is a reduction coeicient estimated as ollows: E β = α + 1 (46) Es where α is the bond-dependant coeicient. According to test results o simply supported beams, the value o α or a given GFRP bar was ound to be 0.5, which is the same as that or steel bars (Gao et al. 1998a). This approach is very appropriate since it does not deviate rom the ACI approach or calculating the delection. However, an extensive research program is required to determine the values o α and β and to ensure that this approach is valid or the dierent types o FRP bars Long-Term Delections Available data on long-term delections o FRP reinorced beams (Kage et al., 1995 and Brown, 1997) indicates that creep behavior in FRP-reinorced beams is similar to that o steel-reinorced concrete beams. The time-delection curves o FRP-reinorced and steel-reinorced beams have the same shape indicating that, with proper modiication, the approach used to predict the long-term delection o steel-reinorced beams can be used or FRP reinorced beams. Tests indicate that ater one year, FRP-reinorced members delected 1.2 to 1.8 times more than steel-reinorced members, depending on the type o the FRP bar (Kage et al., 1995). According to the ACI , Section , the additional long-term delection due to creep and shrinkage, (cp+sh), can be computed using the ollowing equations: + = λ ( (47) ( cp sh) i ) sus ξ λ = (48) ρ where λ is a multiplier or additional long-term delection, ξ is a time-dependant actor and ( i ) sus is the immediate delection under sustained load. Equation (47) can be used or the case o FRP reinorcement with slight modiications to account or the dierences in concrete compressive stress levels, and lower elastic modulus and dierent bond characteristics o FRP bars. It was mentioned earlier that compression reinorcement is not considered in the case o FRP reinorced members (ρ = 0), thereore, λ is equal to ξ. Using available data (Kage et al., 1995 and Brown, 1997), the calculated ratio o ξ FRP /ξ steel was ound to vary rom 0.46 or AFRP and GFRP to 0.53 or CFRP. Based on the above results, a conservative modiication actor o 0.6 is proposed. The long-term delection o FRP-reinorced members can thereore be expressed as ollows: + = 0. ξ (49) ( cp sh) 6 ( i ) sus where ξ is the time-dependant actor given in the ACI Further parametric studies and experimental work are necessary to validate this approach. 20

24 4. COMPARISON TO EXPERIMENTAL RESULTS The experimental lexural strength o 29 FRP reinorced concrete beams ound in the literature are compared to their theoretical values using the proposed design procedures or both over-reinorced and under-reinorced sections. Theoretical predictions were based on the actual material properties as reported by the investigators with no consideration to the proposed knockdown actor. Specimens considered or comparison have rectangular sections with one layer o tension FRP reinorcement. O the considered specimens, 20 specimen were ound to be over-reinorced and 9 were ound to be under-reinorced. Description o the properties o the specimens and predicted mode o ailure or the over-reinorced and under-reinorced beams are given in Tables 5 and 6, respectively. Specimens having more than one layer o tension reinorcement or compression reinorcement were excluded rom this comparison. In Tables 5, the test results or the 20 over-reinorced beams are compared to the predicted lexural strength. The mean M exp /M theo ratio is 1.04 with a standard deviation o The test results or the 9 under-reinorced beams are compared to the predicted lexural strength in Tables 6. The mean M exp /M theo ratio is 0.96 with a standard deviation o These results show good agreement between the proposed procedure and the experimental test results o the considered specimens. Comparison o the M exp /M theo ratios reveals that the proposed procedure tends to slightly under-estimate the lexural capacity o over-reinorced sections and slightly over-estimate the lexural capacity o over-reinorced sections. This behavior is related to the act that the capacity o the overreinorced specimens is based on the crushing o concrete at a maximum strain o This value is relatively conservative and represents the lower bound o maximum strain attainable in concrete. On the other hand, the ailure o the under-reinorced specimens is based on the experimental tensile strength o FRP reinorcement, which usually is widely variable and is calculated or a limited number o specimens. Thereore it is likely that the tensile strength o concrete embedded FRP reinorcement be lower than the value based on tests. In act, Table 6 reveal that or almost all the considered under-reinorced members, the actual tensile strengths o FRP reinorcement were slightly lower than that those based on tensile tests. However, an FRP tensile strength based on the ormula ( u - 3σ) should provide more conservative results. 21

25

26 Table 5. Sectional Properties and Comparison o Flexural Capacities o Over-reinorced Concrete Specimens. Reerence Designation H (in) d (in) ' c A F u E (ksi) (in 2 ) (ksi) (ksi) e u r /r,p Failure theor M theor M exp M exp/m theor Over-rein Faza & Gan., 1991 H Over-rein c Over-rein II Over-rein Al-Salloum et al., III Over-rein IV Over-rein V Over-rein Benmokrane et al., ISO Over-rein ISO Over-rein Almusallam et al., 1997 I Over-rein GB Over-rein Duranovic et al., 1997 GB Over-rein GB Over-rein Grace et al., 1998 cb-st Over-rein CB2B Over-rein CB2B Over-rein Gao et al., 1998 IS Over-rein IS Over-rein KD Over-rein KD Over-rein Average 1.04 Standard Deviation

27 Table 6. Sectional Properties and Comparison o Flexural Capacities o under-reinorced Concrete Specimens. Reerence Designation H (in) d (in) ' c A F u E M (ksi) (in 2 e ) (ksi) (ksi) u r /r,p Failure theor M theor M exp /M th exp eor Benmokrane et al, ISO Under-rein ISO Under-rein IS2B Under-rein IS2B Under-rein KD Under-rein Gao et al., 1998 IS Under-rein IS Under-rein KD Under-rein KD Under-rein Average 0.96 Standard Deviation

28 4. CONCLUSION Design and analysis o lexural members reinorced with FRP reinorcement can be achieved using the same principal assumptions used or steel reinorced members. Design can be achieved using ultimate strength or allowable stress method. For both methods, appropriate reduction actors and stress limitations should be used to account or the variability o material properties and the unclear long-term behavior o FRP reinorcement. Design and analysis approaches addressing both methods are proposed. In the ultimate strength method, tension-controlled ailure o FRP reinorced sections is not desirable since it can be more catastrophic than compression-controlled ailure. Ater cracking, FRP reinorced members have lower stiness than steel reinorced members with similar cross sectional dimensions and capacity. Hence, FRP reinorced members will generally have larger delections and crack widths. Designing or a compressioncontrolled ailure is thereore practical since it requires larger amount o FRP reinorcement resulting in stier members that are likely to pass serviceability requirements. Experimental results o over-reinorced and under-reinorced members in lexure are in good agreement with predicted capacities based on proposed procedures. Design based on the allowable stress method also results in stier sections that are compression-controlled. Design based on the allowable stress method should be such that the concrete reaches its allowable stress beore the FRP reinorcement. This is because FRP exhibits high tensile strength and low modulus o elasticity. Thereore, design based on the concrete reaching its allowable stress beore the FRP reinorcement results in a section minimal amount o FRP reinorcement, or which the cracking moment may exceed the moment capacity o the reinorced section. Crack width and short-term and long term delection can be predicted using the same equations or steel reinorced members with proper modiications. Stiness o FRP reinorcement and bond properties with concrete should be accounted or in modiying existing equations. Some o the assumptions that were made herein are yet to be proven by experiments. 25

29 5. NOTATION A = area o FRP reinorcement, in. 2 A,cr = minimum amount o FRP reinorcement needed to prevent ailure o lexural members upon cracking, in. 2 b = width o a rectangular cross-section, in. c = distance rom extreme compression iber to the neutral axis, in. c b = distance rom extreme compression iber to neutral axis at balanced strain condition, in. d = distance rom extreme compression iber to centroid o tension reinorcement, in. d c = thickness o the concrete cover measured to the center o the closest layer o longitudinal reinorcement, in. E c = modulus o elasticity o concrete, ksi E = modulus o elasticity o FRP, ksi E s = modulus o elasticity o steel, ksi c = compressive stress in concrete, ksi c = speciied compressive strength o concrete, ksi ' c = square root o speciied compressive strength o concrete, psi c,a = allowable compressive stress in concrete, ksi d = ultimate design strength o FRP reinorcement in tension, ksi = stress in the FRP reinorcement in tension, ksi,a = allowable tensile stress in FRP reinorcement, ksi u = ultimate strength o FRP, ksi I e = eective moment o inertia, in. 4 I g = gross moment o inertia, in. 4 k = ratio o the depth o the neutral axis to the reinorcement depth k b = ratio o the depth o the neutral axis to the reinorcement depth at balanced stress condition k b = bond-dependant coeicient K d = knockdown actor M cr = cracking moment M n = nominal moment capacity M na = approximate nominal moment capacity o under-reinorced sections M na = exact nominal moment capacity o under-reinorced sections M s = moment capacity at service stress level M u = ultimate moment based on actored loads n = ratio o the modulus o elasticity o FRP bars to the modulus o elasticity o concrete w = crack width in units o in. α = bond dependant coeicient used in calculation o delection, taken as 0.5 α 1 = ratio o the average stress o the equivalent rectangular stress block to c α 1c = ratio o the average stress o the equivalent rectangular stress block to c β = ratio o the distance rom the neutral axis to the center o extreme tension iber to the distance rom the neutral axis to the center o tensile reinorcement β = reduction coeicient used in the calculation o delection 26