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Numerical assessment of continuous prestressed NSC and HSC members with external CFRP tendons
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Numerical assessment of continuous prestressed NSC and HSC members with external CFRP tendons Tiejiong Loua,b, Di Mina, Wei Sunb, Bo Chena,
⁎
a b
Key Laboratory of Roadway Bridge & Structure Engineering, Wuhan University of Technology, 430070 Wuhan, China Faculty of Engineering and Physical Sciences, University of Southampton, SO17 1BJ Southampton, United Kingdom
ARTICLE INFO
ABSTRACT
Keywords: CFRP tendons High-strength concrete Continuous member Structural response
Carbon fiber reinforced polymer (CFRP) is a promising material for prestressing tendons. This study assesses the structural performance of continuous prestressed members with external CFRP tendons, focusing on the effect of using high-strength concrete (HSC) instead of normal-strength concrete (NSC). A nonlinear model is introduced and numerical tests are then carried out. Three concrete cylinder compressive strengths of 40, 60 and 90 MPa are used. The area of external CFRP tendons varies from 200 to 1700 mm2. The results show that the effect of using HSC instead of NSC heavily depends on the area of external CFRP tendons. In general, HSC members with external CFRP tendons exhibits considerably different response characteristics from those of NSC ones, including cracking, deformations, tendon and rebar stresses, neutral axis depths and so on. The concrete grade influence on moment redistribution is found to be negligible, provided that the members demonstrate favorable ductility. The rules in ACI 318-14 concerning the unbonded tendon stress and the redistribution of moments are evaluated.
1. Introduction External prestressing is broadly used to rehabilitate existing structures as well as to build new bridges [1,2]. External tendons are often exposed to harsh environments as they are placed outside the structural element. As a consequence, conventional steel tendons are suffered from corrosive damage, resulting in structural deterioration of externally prestressed members. A promising solution to this problem is to substitute steel tendons with non-corrosive fiber reinforced polymer (FRP) composites, which are commonly used in civil engineering for structural retrofit or strengthening [3–5]. Carbon fiber reinforced polymer (CFRP) possesses superior mechanical properties (e.g. high tensile strength, high stiffness and excellent creep resistance) and is suitable for prestressing applications [6]. Anchorages for CFRP tendons need to be specially developed [7–9]. Extensive research has been carried out concerning simply supported external FRP tendon members [10–15]. Relatively few works on continuous external FRP tendon members were reported. Tan and Tjandra [16] tested a total of 12 two-span concrete beams externally strengthened either by steel or by CFRP tendons. The test results indicated that external CFRP tendons produce similar structural behavior to that of external steel tendons. Lou et al. [17] evaluated secondary moments of two-span externally CFRP prestressed members based on linear transformation concept. The same authors [18] also conducted a ⁎
parametric study on the redistribution of moments at ultimate in such members. The interaction between moment redistribution and secondary moment was analyzed in a more recent study [19]. External tendon members, if their second-order effect is minimized, are known to behave similarly to internal unbonded tendon members. Some research work on concrete members prestressed with internal unbonded CFRP tendons has been reported [20–22]. External prestressing is commonly applied to long-span prestressed bridges, which are often cast with high-strength concrete (HSC) so as to achieve the desired structural strength and durability. Despite its noteworthy advantages, HSC is significantly more brittle than normalstrength concrete (NSC). HSC exhibits a much steeper stress-strain curve with a smaller ultimate compressive strain when compared to NSC [23]. The brittleness of HSC may raise concerns on the structural performance of members made of HSC. A large amount of research efforts [24–29] have been made to assess performance of HSC members, in particular, the flexural ductility and moment redistribution. The previous works generally showed that reinforced HSC members exhibit favorable ductile behavior and moment redistribution capacity. A set of numerical studies [30,31] were conducted to evaluate the moment redistribution in reinforced and bonded prestressed NSC and HSC members. The studies indicated that moment redistribution of reinforced HSC members tends to be considerably higher than that of the counterparts made of NSC [30]. The redistribution difference between
Corresponding author. E-mail addresses: [email protected] (T. Lou), [email protected] (W. Sun), [email protected] (B. Chen).
https://doi.org/10.1016/j.compstruct.2019.111671 Received 30 January 2019; Accepted 3 November 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Tiejiong Lou, et al., Composite Structures, https://doi.org/10.1016/j.compstruct.2019.111671
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Fig. 1. Beam element model.
Fig. 2. Continuous beam specimens strengthened with external CFRP tendons [16].
bonded prestressed NSC and HSC members appeared to be not very important [31]. The previous works were focused on the use of HSC instead of NSC in nonprestressed or bond prestressed members. The influence of using HSC instead of NSC on the performance of continuous externally FRP prestressed members has yet to be explored. A numerical assessment of the response characteristics of two-span NSC and HSC members prestressed with external CFRP tendons is presented herein. The main investigated variables are the concrete strength and the tendon area. Global member behavior is examined and some important findings are concluded.
2. Numerical method A two-nodal beam element is considered (see Fig. 1). The element properties are described in the local coordinate system (x, y). The axial displacement u is assumed to be a linear function, and the transverse displacement v is assumed to be a cubic polynomial. Assume that a plane section keeps plane after bending and that the shear deformation is negligible. The following element tangential equilibrium equations are obtained [32]:
dP e = KTe due = (K1e + K2e ) due
2
(1)
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Fig. 3. Comparison between numerical and experimental results. (a) load versus deflection at one-third point from the end support; (b) load versus tendon stress.
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Fig. 4. Structure and cross-section of the investigated members.
ue = {u1 v1 Pe
u2 v2
1
= { N1 V1 M1 N2 V2 M2
K1e =
V
B = [ N1
BT EBdV , K2e = N2 y
(2)
T 2}
V
N3 y N4
}T
(3)
J T JdV
(4)
N5 y
N6 y ]
J = [0 N2 N3 0 N5 N6 ]
(5) (6)
where and = element nodal displacements and equivalent nodal loads, respectively; KTe = element stiffness matrix, comprising two components representing respectively the material nonlinearity and the large displacement effect; E = tangential modulus; = stress; N1N6 = shape functions. The stiffness matrix is evaluated based on a layered technique, where the cross section is divided into a number of concrete layers and reinforcing steel layers as shown in Fig. 1. The external tendons are contributed to equivalent nodal loads. Details of the numerical model can be seen in [32]. To allow the numerical model to be applicable to NSC and HSC members, compressive behavior of concrete is simulated by [33]:
ue
c
fcm
=
k 1 + (k
Pe
2
2)
(7)
where c = concrete stress; fcm = fck + 8; = c / c0 ; k = 1.05Ec c 0/ fcm ; 0.31 c = concrete strain; c0 = strain at peak stress, and c 0 (%) = 0.7f cm ; fck = cylinder compressive strength; Ec = modulus of elasticity of concrete (in GPa). The ultimate compressive strain is calculated by: for fck < 50 MPa , fck 50 MPa , and for u (%) = 3.5; fcm )/100]4 . u (%) = 2.8 + 27[(98 The concrete in tension assumes elastic and linearly tension-stiffening behavior. The tensile strength ft for NSC and HSC is determined by [33]: for NSC, ft = 0.3fck2/3 ; and for HSC, ft = 2.12 ln(1 + fcm /10). The CFRP tendons are linear-elastic until rupture. The reinforcing steel in both tension and compression is linear-elastic before yielding, followed by perfectly plastic behavior. During the analysis, when the specified maximum strain for concrete or reinforcement is reached, the member is assumed to have collapsed. The analysis is therefore terminated. The proposed beam element can conduct the nonlinear analysis of continuous prestressed NSC and HSC beams with external FRP tendons over the entire loading ranges. 3. Comparison with experimental results Three two-span beam specimens externally strengthened with unbonded CFRP tendons are investigated. These beams were tested by Tan and Tjandra [16] and were designated as MCBC-1, MCBC-2 and MCBC3. The beams were of T-shaped cross section and had a total length of 6.0 m. The main variable was the layout of external CFRP tendons (see Fig. 2). These tendons were placed over the sagging region only for
Fig. 5. Concrete strain distribution for different concrete strengths and tendon areas.
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Fig. 6. Deformation development for different concrete strengths and tendon areas. (a) moment-curvature behavior; (b) load-deflection behavior.
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Fig. 7. Variations of Pu and δu with varying ρp for different concrete strengths. (a) Pu; (b) δu.
MCBC-1, over the hogging region only for MCBC-2, and over both the sagging and hogging regions for MCBC-3. Third-point loading was symmetrically applied on the beams. The CFRP tendons were provided on both sides of the beams. Their tensile strength was 1868 MPa and elastic modulus was 139 GPa. The tendon area was 111.4 mm2 and the initial prestress was approximately 50% of the tensile strength. The concrete strength fck = 30 MPa. The bottom longitudinal nonprestressed reinforcement consisted of 2 steel rebars with 16 mm in diameter each. The top one consisted of 4 and 6 steel rebars with a diameter of 10 mm each over the sagging and hogging regions, respectively. The steel yield strength was 540 MPa. According to the numerical analysis, first cracking for Beam MCBC2 appears at the critical sagging region, following by second cracking at the inner support. Reversed order of the appearance of cracking is observed for MCBC-1 and MCBC-3. Failure of all specimens is caused by concrete crushing, after the formation of plastic hinges over both the critical hogging and sagging regions. The above statements about cracking and failure of the beams are consistent with the experimental observations. The predicted load versus deflection and tendon stress characteristics are compared to experimental data in Fig. 3. The numerical analysis considerably overestimates the stress in external CFRP tendons for Beam MCBC-1. This may be attributed to the premature termination of the test of the specimen [16]. Despite some discrepancy, Numerical predictions and experimental data show satisfactory agreement.
elastic modulus of 200 GPa. 4.1. Failure and cracking modes All the members fail due to concrete crushing at one of the critical sections. Sufficient development of plastic hinges is observed in the members but the NSC member with a tendon area of 1700 mm2. At failure the external CFRP tendons are well bellow their rupture capacity. Crushing failure may take place at the midspan or inner support, depending on the concrete grade and tendon area. For fck = 40 MPa, crushing failure takes place at the midspan at a low tendon area while at the inner support at a high tendon area. For fck = 60 MPa, the midspan section is collapsed when Ap is not greater than 1325 mm2 and failure takes place at the inner support when Ap = 1700 mm2. For fck = 90 MPa, failure always happens at the midspan and the exploitation of the critical hogging region is improved as Ap increases. The concrete strain distribution at failure is shown in Fig. 5. At a low tendon area (Ap = 200 mm2), the concrete tensile strains at the midspan and inner support are very large while in the other zones the strains are small. This indicates that there are big crack widths appearing in the critical sections against small crack widths in the noncritical regions. At a low tendon area, the crack width at the midspan appears to be larger than that at the inner support, while a high tendon area results in an opposite observation. At a low tendon area (Ap = 200 mm2), NSC (i.e. fck = 40 MPa) leads to a slightly larger crack width at the critical section than HSC (i.e. fck = 60 and 90 MPa). As the tendon area increases, the maximum crack width at the critical section reduces. The reduction of the crack width is more effectively for NSC than for HSC. As a result, when the tendon area increases to 595 mm2 or above, NSC turns to produce smaller crack width than NSC. The difference between the crack widths for NSC and HSC becomes increasingly apparent with increasing area of external tendons. All the members with different concrete strengths and tendon areas exhibit almost the same crack zone.
4. Numerical investigation Fig. 4 illustrates details of continuous members with external CFRP tendons used for the study. The investigated variables are the concrete strength fck and the tendon area Ap (or tendon ratio ρp): fck = 40, 60 and 90 MPa; and Ap = 200–1700 mm2 or equivalently ρp = 0.15% to 1.29%. CFRP tendons have tensile strength of 1840 MPa and elastic modulus of 147 GPa. The initial prestress is 1104 MPa. The tensile reinforcing steel areas over sagging and hogging regions, As1 and As2, are 1600 and 800 mm2, respectively. The compressive reinforcing steel area, As3, is 360 mm2. All the steels have the same yield strength of 450 MPa and
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Fig. 8. Stress increase in external tendons for different concrete strengths and tendon areas. (a) with the applied load; (b) with the deflection.
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Fig. 9. Variation of Δσpu with varying ρp for different concrete strengths. (a) numerical predictions; (a) comparison with ACI 318-14.
4.2. Moment-curvature and load-deflection behavior
is particularly notable at a high ρp level. Fig. 9(a) shows that there is almost a linear relationship between Δσpu (ultimate tendon stress increase) and ρp. As ρp increases the value of Δσpu for NSC decreases rapidly, much quicker than that for HSC. HSC generally leads to a higher Δσpu than NSC, except at a low ρp level. ACI 318-14 [34] recommends an equation for calculating the ultimate unbonded tendon stress:
Fig. 6(a) and (b) show the moment-curvature and load-deflection curves for different concrete strengths and tendon areas, respectively. Prior to the application of live loads (i.e. for the members under selfweight only), there is initial moment consisting of self-weight induced moment and prestress secondary moment. Since external tendons are placed a bit below the concordant profile, the secondary moment produced by external prestressing is positive and the magnitude is rather small. Therefore, the initial moment is a bit lower over the inner support and a bit higher over the midspan than the self-weight induced moment. Because As1 is greater than As2, the sagging yield or ultimate moment is greater than the hogging one. The difference is notable at a low tendon area while it diminishes with increasing tendon area as a result of decreasing contribution of reinforcing steel. As expected, HSC produces higher flexural stiffness, i.e. a smaller curvature or deflection at a given moment or load, when compared to NSC. All the members exhibit favorable ductile behavior, except for the NSC member with a tendon area of 1700 mm2. The latter member does not exhibit an apparent yielding plateau as the tensile reinforcing steel at the midspan just gets to its yield strength at failure. Fig. 7 displays variations of Pu (ultimate load) and δu (ultimate deflection) with varying ρp for different concrete strengths. Fig. 7(a) demonstrates that HSC produces greater ultimate load than NSC. The load difference is slight at low ρp levels and increasingly notable with increasing ρp. At low ρp levels, the ultimate deflection for NSC (fck = 40 MPa) is obviously higher than that for HSC (fck = 60 or 90 MPa), as can be observed in Fig. 7(b). The ultimate deflection decreases as ρp increases. The decrease rate for NSC is greater than that for HSC. As a consequence, at a high ρp level of 1.29%, the ultimate deflection for NSC turns to be obviously lower than that for HSC.
pu
=
pe
+ 70 +
fck 100
p
(8)
where σpe = effective prestress. The value of Δσpu should not be greater than 420 MPa. Eq. (8) is valid for L /dp 35, in which L/dp = span-todepth ratio. Fig. 9(b) shows that for ρp greater than 0.44%, the ACI 318-14 predictions correspond well with the numerical results. Both the effects of the tendon area and concrete strength are satisfactorily reflected in ACI 318-14. However, when ρp increases up to 0.44%, the ACI code significantly overestimates the rate of decrease of the tendon stress and also fails to reflect concrete strength effect. Moreover, at ρp = 0.15%, ACI 318-14 leads to a substantial overestimate of the tendon stress, especially when HSC is used. 4.4. Neutral axis depth (NAD) The initial NAD is highly dependent on the tendon area. At a low tendon area (Ap = 200 mm2), the prestressing effect cannot resist the self-weight effect. In this case, the initial NAD at the critical section is positive, with a large value. At a medium tendon area (Ap = 950 mm2), the initial NAD is negative at the critical section. At a high tendon area (Ap = 1700 mm2), the initial hogging and sagging curvatures are considerable. In this case, the initial NAD at the critical section is also positive but its magnitude is small. Fig. 10(a) an (b) show the NAD versus the applied load and curvature relationships for different concrete strengths and tendon areas, respectively. It is seen that for specific loading levels, the inner support section exhibits smaller NAD than the midspan section. For specific curvature levels, HSC leads to lower NAD than NSC, especially at a higher tendon area.
4.3. Increase in tendon stress The stress increase in external tendons for different concrete strengths and tendon areas is shown in Fig. 8. At given load levels, HSC produces smaller stress increase than NSC. On the other hand, at a given deflection level, HSC results in a higher stress increase than NSC, which
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Fig. 10. Development of neutral axis depth for different concrete strengths and tendon areas. (a) with the applied load; (b) with the curvature.
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Fig. 11. Variation of cu with varying ρp for different concrete strengths. (a) midspan; (b) inner support.
Fig. 11 shows the relation between cu (NAD at ultimate) and ρp for different concrete strengths. The cu value increases with increasing ρp, and the rate of increase for NSC is much faster than that for HSC. In addition, at given ρp levels, HSC produces lower cu than NSC. The difference is slight for a low ρp level and increasingly pronounced with increasing ρp.
The actual load-moment curves differ from the elastic curves after cracking (Fig. 14(a)), indicating that the moment redistribution does not occur until cracking (Fig. 14(b)). As the applied load increases, the deviation tends to more and more pronounced (Fig. 14(a)), indicating increasing redistribution of moments (Fig. 14(b)). In addition, the deviation at the inner support appears to be more pronounced (Fig. 14(a)), indicating higher moment redistribution in comparison with the midspan (Fig. 14(b)). Moreover, a higher tendon area leads to a smaller deviation between the actual and elastic values (Fig. 14(a)), indicating lower moment redistribution (Fig. 14(b)). The load-moment relationships for different concrete strengths are very similar (Fig. 14(a)), indicating the impact of concrete grade on moment redistribution is not important (Fig. 14(b)). Fig. 15 shows the relation between βu (ultimate moment redistribution) and ρp for different concrete strengths. The βu value diminishes with increasing ρp. At given ρp levels, NSC and HSC members exhibit almost the same moment redistribution, except at a high ρp level of 1.29%. For ρp = 1.29%, NSC produces substantially lower moment redistribution than HSC due to lacking of ductility of the NSC member with external tendons. Therefore, it may be concluded that the concrete grade has negligible influence on moment redistribution in these members, provided that the members have sufficient plastic rotational capacities. In ACI 318-14 [34], the allowable redistribution of moments is determined by
4.5. Stress and strain in reinforcing steel The load versus stresses in reinforcing steels curves for different concrete strengths and tendon areas are illustrated in Fig. 12. It is seen that concrete cracking affects remarkably the stress evolution in tensile reinforcing steel, but its influence on the compressive reinforcing steel is not apparent. The stress increase in the tensile reinforcing steel is slow at first while it turns to be rapid after cracking. The tensile reinforcing steel at the inner support reaches its yield strength firstly, followed by yielding at the midspan. For HSC members, all the compressive reinforcing steels do not get to their yield strength at ultimate. For NSC members, on the other hand, the compressive reinforcing steel may reach its yield strength at a high tendon area. In general, a high tendon area is more likely to cause the yielding of compressive steels at the critical sections. At given load levels, NSC leads to a higher tensile or compressive steel strain (stress also if not having yielded) than HSC. Fig. 13 demonstrates the relation between εsu (strain in reinforcing steel at ultimate) and ρp for different concrete strengths. As ρp increases, the tensile steel strain decreases while the compressive steel strain increases. The midspan section exhibits larger strain in tensile steel at low ρp levels while smaller one at high ρp levels than the inner support section. In addition, the midspan section generally produces higher strain in compressive steel than the inner support section, except for NSC at a high ρp level of 1.29%. NSC generally leads to a lower strain in tensile steel and a higher strain in compressive steel than HSC
u
= (1000 t )%
(9)
where εt = net strain in extreme tensile steel at ultimate. The value of βu should not be greater than 20%. A comparison between the βu-ρp relationships for NSC and HSC members with external tendons by ACI 318-14 and numerical analysis is illustrated Fig. 16(a). It is seen that the influence of the tendon area and concrete strength is satisfactorily reflected in ACI 318-14. However, ACI 318-14 substantially underestimates the moment redistribution, especially at a low ρp level. Based on an extensive parametric study, Lou et al. [18] proposed a simplified equation to calculate βu in externally CFRP prestressed
4.6. Moment redistribution Fig. 14(a) and (b) show the development of moments (actual M and elastic Me) and moment redistribution ( = 1 M /Me ), respectively.
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Fig. 12. Development of the stress in reinforcing steel for different concrete strengths and tendon areas. (a) midspan; (b) inner support.
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Fig. 13. Variation of εsu with varying ρp for different concrete strengths. (a) tensile reinforcing steel; (b) compressive reinforcing steel.
• HSC generally produces higher ultimate stress in external tendons
members. The equation is a modification of Eq. (9) by including a parameter λ: (10)
= [ (1000 t )]%
u
where λ is a parameter related to As2/As1, i.e. when As2 /As1 1,
= 0.65
•
(11a)
1.2 ln(As2 / As1 )
•
and when As2 /As1 > 1
= 0.65 + 0.67 ln(As2 /As1 )
2.76ln2 (As2 /As1 )
(11b)
A comparison of the simplified equations (i.e. ACI 318-14 and modified ACI 318-14) against the numerical predictions for βu of the prestressed NSC and HSC members with external CFRP tendons investigated is illustrated in Fig. 16(b). It is seen that by introducing the structure-related factor As2/As1, the modified ACI 318-14 equation, i.e. Eq. (10), demonstrates better correlation than ACI 318-14.
•
5. Conclusions The results of analysis of two-span continuous prestressed NSC and HSC members with external CFRP tendons lead to the following conclusions:
Acknowledgments The work has been supported by the Fundamental Research Funds for the Central Universities under Grant No. 2018ⅣA006 and by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 751921.
• NSC members with a high tendon area may fail to exhibit favorable •
than NSC. In calculating the ultimate tendon stress, ACI 318-14 satisfactorily reflects the impact of concrete grade and tendon area for ρp greater than 0.44%. However, ACI 318-14 is non-conservative at a low ρp level, especially when HSC is used. HSC produces a lower cu value than NSC, and the difference is increasingly significant with increasing ρp. The cu value for the inner support is lower than that for the midspan. HSC leads to higher ultimate strain in tensile reinforcing steel than NSC, except at low ρp levels. The ultimate strain in compressive reinforcing steel for HSC is lower than that for NSC. The inner support section exhibits smaller strain in tensile reinforcing steel that the midspan section at low ρp levels, while the observation is opposite at high ρp levels. The concrete strength has negligible impact on moment redistribution in these members provided that the members have sufficient plastic rotational capacities. The predictive model of Eq. (10) is more accurate than ACI 318-14 when predicting moment redistribution in this structural typology.
ductile behavior, while the counterpart members made of HSC are able to develop sufficient rotation of plastic hinges. HSC produces larger crack widths at the critical sections than NSC except at a low ρp level. HSC leads to higher ultimate load-carrying capacity than NSC, particularly obvious at high ρp levels. HSC produces smaller ultimate deflection at low ρp levels but larger one at high ρp levels than NSC.
Data availability statement The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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Fig. 14. Development of moments and moment redistribution for different concrete strengths and tendon areas; (a) load-moment behavior; (b) load versus moment redistribution.
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Fig. 15. Variation of βu with varying ρp for different concrete strengths. (a) midspan; (b) inner support.
Fig. 16. Comparison of βu values. (a) ACI 318-14 and numerical predictions; (b) simplified equations against numerical analysis.
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beams. Struct Eng Mech 2018;66(2):195–201. [30] Lou T, Lopes SMR, Lopes AV. Evaluation of moment redistribution in normalstrength and high-strength reinforced concrete beams. ASCE J Struct Eng 2014;14(10):04014072. [31] Lou T, Liu M, Lopes SMR, Lopes AV. Moment redistribution in two-span prestressed NSC and HSC beams. Mater Struct 2017;50:246. [32] Lou T, Xiang Y. Finite element modeling of concrete beams prestressed with external tendons. Eng Struct 2006;28(14):1919–26. [33] CEN. Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings. EN 1992-1-1, European Committee for Standardization, Brussels, Belgium; 2004. [34] ACI Committee 318. Building code requirements for structural concrete (ACI 31814) and commentary (ACI 318R-14). American Concrete Institute, Farmington Hills, MI; 2014.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Numerical assessment of continuous prestressed NSC and HSC members with external CFRP tendons Tiejiong Loua,b, Di Mina, Wei Sunb, Bo Chena,
⁎
a b
Key Laboratory of Roadway Bridge & Structure Engineering, Wuhan University of Technology, 430070 Wuhan, China Faculty of Engineering and Physical Sciences, University of Southampton, SO17 1BJ Southampton, United Kingdom
ARTICLE INFO
ABSTRACT
Keywords: CFRP tendons High-strength concrete Continuous member Structural response
Carbon fiber reinforced polymer (CFRP) is a promising material for prestressing tendons. This study assesses the structural performance of continuous prestressed members with external CFRP tendons, focusing on the effect of using high-strength concrete (HSC) instead of normal-strength concrete (NSC). A nonlinear model is introduced and numerical tests are then carried out. Three concrete cylinder compressive strengths of 40, 60 and 90 MPa are used. The area of external CFRP tendons varies from 200 to 1700 mm2. The results show that the effect of using HSC instead of NSC heavily depends on the area of external CFRP tendons. In general, HSC members with external CFRP tendons exhibits considerably different response characteristics from those of NSC ones, including cracking, deformations, tendon and rebar stresses, neutral axis depths and so on. The concrete grade influence on moment redistribution is found to be negligible, provided that the members demonstrate favorable ductility. The rules in ACI 318-14 concerning the unbonded tendon stress and the redistribution of moments are evaluated.
1. Introduction External prestressing is broadly used to rehabilitate existing structures as well as to build new bridges [1,2]. External tendons are often exposed to harsh environments as they are placed outside the structural element. As a consequence, conventional steel tendons are suffered from corrosive damage, resulting in structural deterioration of externally prestressed members. A promising solution to this problem is to substitute steel tendons with non-corrosive fiber reinforced polymer (FRP) composites, which are commonly used in civil engineering for structural retrofit or strengthening [3–5]. Carbon fiber reinforced polymer (CFRP) possesses superior mechanical properties (e.g. high tensile strength, high stiffness and excellent creep resistance) and is suitable for prestressing applications [6]. Anchorages for CFRP tendons need to be specially developed [7–9]. Extensive research has been carried out concerning simply supported external FRP tendon members [10–15]. Relatively few works on continuous external FRP tendon members were reported. Tan and Tjandra [16] tested a total of 12 two-span concrete beams externally strengthened either by steel or by CFRP tendons. The test results indicated that external CFRP tendons produce similar structural behavior to that of external steel tendons. Lou et al. [17] evaluated secondary moments of two-span externally CFRP prestressed members based on linear transformation concept. The same authors [18] also conducted a ⁎
parametric study on the redistribution of moments at ultimate in such members. The interaction between moment redistribution and secondary moment was analyzed in a more recent study [19]. External tendon members, if their second-order effect is minimized, are known to behave similarly to internal unbonded tendon members. Some research work on concrete members prestressed with internal unbonded CFRP tendons has been reported [20–22]. External prestressing is commonly applied to long-span prestressed bridges, which are often cast with high-strength concrete (HSC) so as to achieve the desired structural strength and durability. Despite its noteworthy advantages, HSC is significantly more brittle than normalstrength concrete (NSC). HSC exhibits a much steeper stress-strain curve with a smaller ultimate compressive strain when compared to NSC [23]. The brittleness of HSC may raise concerns on the structural performance of members made of HSC. A large amount of research efforts [24–29] have been made to assess performance of HSC members, in particular, the flexural ductility and moment redistribution. The previous works generally showed that reinforced HSC members exhibit favorable ductile behavior and moment redistribution capacity. A set of numerical studies [30,31] were conducted to evaluate the moment redistribution in reinforced and bonded prestressed NSC and HSC members. The studies indicated that moment redistribution of reinforced HSC members tends to be considerably higher than that of the counterparts made of NSC [30]. The redistribution difference between
Corresponding author. E-mail addresses: [email protected] (T. Lou), [email protected] (W. Sun), [email protected] (B. Chen).
https://doi.org/10.1016/j.compstruct.2019.111671 Received 30 January 2019; Accepted 3 November 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Tiejiong Lou, et al., Composite Structures, https://doi.org/10.1016/j.compstruct.2019.111671
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Fig. 1. Beam element model.
Fig. 2. Continuous beam specimens strengthened with external CFRP tendons [16].
bonded prestressed NSC and HSC members appeared to be not very important [31]. The previous works were focused on the use of HSC instead of NSC in nonprestressed or bond prestressed members. The influence of using HSC instead of NSC on the performance of continuous externally FRP prestressed members has yet to be explored. A numerical assessment of the response characteristics of two-span NSC and HSC members prestressed with external CFRP tendons is presented herein. The main investigated variables are the concrete strength and the tendon area. Global member behavior is examined and some important findings are concluded.
2. Numerical method A two-nodal beam element is considered (see Fig. 1). The element properties are described in the local coordinate system (x, y). The axial displacement u is assumed to be a linear function, and the transverse displacement v is assumed to be a cubic polynomial. Assume that a plane section keeps plane after bending and that the shear deformation is negligible. The following element tangential equilibrium equations are obtained [32]:
dP e = KTe due = (K1e + K2e ) due
2
(1)
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Fig. 3. Comparison between numerical and experimental results. (a) load versus deflection at one-third point from the end support; (b) load versus tendon stress.
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Fig. 4. Structure and cross-section of the investigated members.
ue = {u1 v1 Pe
u2 v2
1
= { N1 V1 M1 N2 V2 M2
K1e =
V
B = [ N1
BT EBdV , K2e = N2 y
(2)
T 2}
V
N3 y N4
}T
(3)
J T JdV
(4)
N5 y
N6 y ]
J = [0 N2 N3 0 N5 N6 ]
(5) (6)
where and = element nodal displacements and equivalent nodal loads, respectively; KTe = element stiffness matrix, comprising two components representing respectively the material nonlinearity and the large displacement effect; E = tangential modulus; = stress; N1N6 = shape functions. The stiffness matrix is evaluated based on a layered technique, where the cross section is divided into a number of concrete layers and reinforcing steel layers as shown in Fig. 1. The external tendons are contributed to equivalent nodal loads. Details of the numerical model can be seen in [32]. To allow the numerical model to be applicable to NSC and HSC members, compressive behavior of concrete is simulated by [33]:
ue
c
fcm
=
k 1 + (k
Pe
2
2)
(7)
where c = concrete stress; fcm = fck + 8; = c / c0 ; k = 1.05Ec c 0/ fcm ; 0.31 c = concrete strain; c0 = strain at peak stress, and c 0 (%) = 0.7f cm ; fck = cylinder compressive strength; Ec = modulus of elasticity of concrete (in GPa). The ultimate compressive strain is calculated by: for fck < 50 MPa , fck 50 MPa , and for u (%) = 3.5; fcm )/100]4 . u (%) = 2.8 + 27[(98 The concrete in tension assumes elastic and linearly tension-stiffening behavior. The tensile strength ft for NSC and HSC is determined by [33]: for NSC, ft = 0.3fck2/3 ; and for HSC, ft = 2.12 ln(1 + fcm /10). The CFRP tendons are linear-elastic until rupture. The reinforcing steel in both tension and compression is linear-elastic before yielding, followed by perfectly plastic behavior. During the analysis, when the specified maximum strain for concrete or reinforcement is reached, the member is assumed to have collapsed. The analysis is therefore terminated. The proposed beam element can conduct the nonlinear analysis of continuous prestressed NSC and HSC beams with external FRP tendons over the entire loading ranges. 3. Comparison with experimental results Three two-span beam specimens externally strengthened with unbonded CFRP tendons are investigated. These beams were tested by Tan and Tjandra [16] and were designated as MCBC-1, MCBC-2 and MCBC3. The beams were of T-shaped cross section and had a total length of 6.0 m. The main variable was the layout of external CFRP tendons (see Fig. 2). These tendons were placed over the sagging region only for
Fig. 5. Concrete strain distribution for different concrete strengths and tendon areas.
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Fig. 6. Deformation development for different concrete strengths and tendon areas. (a) moment-curvature behavior; (b) load-deflection behavior.
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Fig. 7. Variations of Pu and δu with varying ρp for different concrete strengths. (a) Pu; (b) δu.
MCBC-1, over the hogging region only for MCBC-2, and over both the sagging and hogging regions for MCBC-3. Third-point loading was symmetrically applied on the beams. The CFRP tendons were provided on both sides of the beams. Their tensile strength was 1868 MPa and elastic modulus was 139 GPa. The tendon area was 111.4 mm2 and the initial prestress was approximately 50% of the tensile strength. The concrete strength fck = 30 MPa. The bottom longitudinal nonprestressed reinforcement consisted of 2 steel rebars with 16 mm in diameter each. The top one consisted of 4 and 6 steel rebars with a diameter of 10 mm each over the sagging and hogging regions, respectively. The steel yield strength was 540 MPa. According to the numerical analysis, first cracking for Beam MCBC2 appears at the critical sagging region, following by second cracking at the inner support. Reversed order of the appearance of cracking is observed for MCBC-1 and MCBC-3. Failure of all specimens is caused by concrete crushing, after the formation of plastic hinges over both the critical hogging and sagging regions. The above statements about cracking and failure of the beams are consistent with the experimental observations. The predicted load versus deflection and tendon stress characteristics are compared to experimental data in Fig. 3. The numerical analysis considerably overestimates the stress in external CFRP tendons for Beam MCBC-1. This may be attributed to the premature termination of the test of the specimen [16]. Despite some discrepancy, Numerical predictions and experimental data show satisfactory agreement.
elastic modulus of 200 GPa. 4.1. Failure and cracking modes All the members fail due to concrete crushing at one of the critical sections. Sufficient development of plastic hinges is observed in the members but the NSC member with a tendon area of 1700 mm2. At failure the external CFRP tendons are well bellow their rupture capacity. Crushing failure may take place at the midspan or inner support, depending on the concrete grade and tendon area. For fck = 40 MPa, crushing failure takes place at the midspan at a low tendon area while at the inner support at a high tendon area. For fck = 60 MPa, the midspan section is collapsed when Ap is not greater than 1325 mm2 and failure takes place at the inner support when Ap = 1700 mm2. For fck = 90 MPa, failure always happens at the midspan and the exploitation of the critical hogging region is improved as Ap increases. The concrete strain distribution at failure is shown in Fig. 5. At a low tendon area (Ap = 200 mm2), the concrete tensile strains at the midspan and inner support are very large while in the other zones the strains are small. This indicates that there are big crack widths appearing in the critical sections against small crack widths in the noncritical regions. At a low tendon area, the crack width at the midspan appears to be larger than that at the inner support, while a high tendon area results in an opposite observation. At a low tendon area (Ap = 200 mm2), NSC (i.e. fck = 40 MPa) leads to a slightly larger crack width at the critical section than HSC (i.e. fck = 60 and 90 MPa). As the tendon area increases, the maximum crack width at the critical section reduces. The reduction of the crack width is more effectively for NSC than for HSC. As a result, when the tendon area increases to 595 mm2 or above, NSC turns to produce smaller crack width than NSC. The difference between the crack widths for NSC and HSC becomes increasingly apparent with increasing area of external tendons. All the members with different concrete strengths and tendon areas exhibit almost the same crack zone.
4. Numerical investigation Fig. 4 illustrates details of continuous members with external CFRP tendons used for the study. The investigated variables are the concrete strength fck and the tendon area Ap (or tendon ratio ρp): fck = 40, 60 and 90 MPa; and Ap = 200–1700 mm2 or equivalently ρp = 0.15% to 1.29%. CFRP tendons have tensile strength of 1840 MPa and elastic modulus of 147 GPa. The initial prestress is 1104 MPa. The tensile reinforcing steel areas over sagging and hogging regions, As1 and As2, are 1600 and 800 mm2, respectively. The compressive reinforcing steel area, As3, is 360 mm2. All the steels have the same yield strength of 450 MPa and
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Fig. 8. Stress increase in external tendons for different concrete strengths and tendon areas. (a) with the applied load; (b) with the deflection.
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Fig. 9. Variation of Δσpu with varying ρp for different concrete strengths. (a) numerical predictions; (a) comparison with ACI 318-14.
4.2. Moment-curvature and load-deflection behavior
is particularly notable at a high ρp level. Fig. 9(a) shows that there is almost a linear relationship between Δσpu (ultimate tendon stress increase) and ρp. As ρp increases the value of Δσpu for NSC decreases rapidly, much quicker than that for HSC. HSC generally leads to a higher Δσpu than NSC, except at a low ρp level. ACI 318-14 [34] recommends an equation for calculating the ultimate unbonded tendon stress:
Fig. 6(a) and (b) show the moment-curvature and load-deflection curves for different concrete strengths and tendon areas, respectively. Prior to the application of live loads (i.e. for the members under selfweight only), there is initial moment consisting of self-weight induced moment and prestress secondary moment. Since external tendons are placed a bit below the concordant profile, the secondary moment produced by external prestressing is positive and the magnitude is rather small. Therefore, the initial moment is a bit lower over the inner support and a bit higher over the midspan than the self-weight induced moment. Because As1 is greater than As2, the sagging yield or ultimate moment is greater than the hogging one. The difference is notable at a low tendon area while it diminishes with increasing tendon area as a result of decreasing contribution of reinforcing steel. As expected, HSC produces higher flexural stiffness, i.e. a smaller curvature or deflection at a given moment or load, when compared to NSC. All the members exhibit favorable ductile behavior, except for the NSC member with a tendon area of 1700 mm2. The latter member does not exhibit an apparent yielding plateau as the tensile reinforcing steel at the midspan just gets to its yield strength at failure. Fig. 7 displays variations of Pu (ultimate load) and δu (ultimate deflection) with varying ρp for different concrete strengths. Fig. 7(a) demonstrates that HSC produces greater ultimate load than NSC. The load difference is slight at low ρp levels and increasingly notable with increasing ρp. At low ρp levels, the ultimate deflection for NSC (fck = 40 MPa) is obviously higher than that for HSC (fck = 60 or 90 MPa), as can be observed in Fig. 7(b). The ultimate deflection decreases as ρp increases. The decrease rate for NSC is greater than that for HSC. As a consequence, at a high ρp level of 1.29%, the ultimate deflection for NSC turns to be obviously lower than that for HSC.
pu
=
pe
+ 70 +
fck 100
p
(8)
where σpe = effective prestress. The value of Δσpu should not be greater than 420 MPa. Eq. (8) is valid for L /dp 35, in which L/dp = span-todepth ratio. Fig. 9(b) shows that for ρp greater than 0.44%, the ACI 318-14 predictions correspond well with the numerical results. Both the effects of the tendon area and concrete strength are satisfactorily reflected in ACI 318-14. However, when ρp increases up to 0.44%, the ACI code significantly overestimates the rate of decrease of the tendon stress and also fails to reflect concrete strength effect. Moreover, at ρp = 0.15%, ACI 318-14 leads to a substantial overestimate of the tendon stress, especially when HSC is used. 4.4. Neutral axis depth (NAD) The initial NAD is highly dependent on the tendon area. At a low tendon area (Ap = 200 mm2), the prestressing effect cannot resist the self-weight effect. In this case, the initial NAD at the critical section is positive, with a large value. At a medium tendon area (Ap = 950 mm2), the initial NAD is negative at the critical section. At a high tendon area (Ap = 1700 mm2), the initial hogging and sagging curvatures are considerable. In this case, the initial NAD at the critical section is also positive but its magnitude is small. Fig. 10(a) an (b) show the NAD versus the applied load and curvature relationships for different concrete strengths and tendon areas, respectively. It is seen that for specific loading levels, the inner support section exhibits smaller NAD than the midspan section. For specific curvature levels, HSC leads to lower NAD than NSC, especially at a higher tendon area.
4.3. Increase in tendon stress The stress increase in external tendons for different concrete strengths and tendon areas is shown in Fig. 8. At given load levels, HSC produces smaller stress increase than NSC. On the other hand, at a given deflection level, HSC results in a higher stress increase than NSC, which
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Fig. 10. Development of neutral axis depth for different concrete strengths and tendon areas. (a) with the applied load; (b) with the curvature.
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Fig. 11. Variation of cu with varying ρp for different concrete strengths. (a) midspan; (b) inner support.
Fig. 11 shows the relation between cu (NAD at ultimate) and ρp for different concrete strengths. The cu value increases with increasing ρp, and the rate of increase for NSC is much faster than that for HSC. In addition, at given ρp levels, HSC produces lower cu than NSC. The difference is slight for a low ρp level and increasingly pronounced with increasing ρp.
The actual load-moment curves differ from the elastic curves after cracking (Fig. 14(a)), indicating that the moment redistribution does not occur until cracking (Fig. 14(b)). As the applied load increases, the deviation tends to more and more pronounced (Fig. 14(a)), indicating increasing redistribution of moments (Fig. 14(b)). In addition, the deviation at the inner support appears to be more pronounced (Fig. 14(a)), indicating higher moment redistribution in comparison with the midspan (Fig. 14(b)). Moreover, a higher tendon area leads to a smaller deviation between the actual and elastic values (Fig. 14(a)), indicating lower moment redistribution (Fig. 14(b)). The load-moment relationships for different concrete strengths are very similar (Fig. 14(a)), indicating the impact of concrete grade on moment redistribution is not important (Fig. 14(b)). Fig. 15 shows the relation between βu (ultimate moment redistribution) and ρp for different concrete strengths. The βu value diminishes with increasing ρp. At given ρp levels, NSC and HSC members exhibit almost the same moment redistribution, except at a high ρp level of 1.29%. For ρp = 1.29%, NSC produces substantially lower moment redistribution than HSC due to lacking of ductility of the NSC member with external tendons. Therefore, it may be concluded that the concrete grade has negligible influence on moment redistribution in these members, provided that the members have sufficient plastic rotational capacities. In ACI 318-14 [34], the allowable redistribution of moments is determined by
4.5. Stress and strain in reinforcing steel The load versus stresses in reinforcing steels curves for different concrete strengths and tendon areas are illustrated in Fig. 12. It is seen that concrete cracking affects remarkably the stress evolution in tensile reinforcing steel, but its influence on the compressive reinforcing steel is not apparent. The stress increase in the tensile reinforcing steel is slow at first while it turns to be rapid after cracking. The tensile reinforcing steel at the inner support reaches its yield strength firstly, followed by yielding at the midspan. For HSC members, all the compressive reinforcing steels do not get to their yield strength at ultimate. For NSC members, on the other hand, the compressive reinforcing steel may reach its yield strength at a high tendon area. In general, a high tendon area is more likely to cause the yielding of compressive steels at the critical sections. At given load levels, NSC leads to a higher tensile or compressive steel strain (stress also if not having yielded) than HSC. Fig. 13 demonstrates the relation between εsu (strain in reinforcing steel at ultimate) and ρp for different concrete strengths. As ρp increases, the tensile steel strain decreases while the compressive steel strain increases. The midspan section exhibits larger strain in tensile steel at low ρp levels while smaller one at high ρp levels than the inner support section. In addition, the midspan section generally produces higher strain in compressive steel than the inner support section, except for NSC at a high ρp level of 1.29%. NSC generally leads to a lower strain in tensile steel and a higher strain in compressive steel than HSC
u
= (1000 t )%
(9)
where εt = net strain in extreme tensile steel at ultimate. The value of βu should not be greater than 20%. A comparison between the βu-ρp relationships for NSC and HSC members with external tendons by ACI 318-14 and numerical analysis is illustrated Fig. 16(a). It is seen that the influence of the tendon area and concrete strength is satisfactorily reflected in ACI 318-14. However, ACI 318-14 substantially underestimates the moment redistribution, especially at a low ρp level. Based on an extensive parametric study, Lou et al. [18] proposed a simplified equation to calculate βu in externally CFRP prestressed
4.6. Moment redistribution Fig. 14(a) and (b) show the development of moments (actual M and elastic Me) and moment redistribution ( = 1 M /Me ), respectively.
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Fig. 12. Development of the stress in reinforcing steel for different concrete strengths and tendon areas. (a) midspan; (b) inner support.
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Fig. 13. Variation of εsu with varying ρp for different concrete strengths. (a) tensile reinforcing steel; (b) compressive reinforcing steel.
• HSC generally produces higher ultimate stress in external tendons
members. The equation is a modification of Eq. (9) by including a parameter λ: (10)
= [ (1000 t )]%
u
where λ is a parameter related to As2/As1, i.e. when As2 /As1 1,
= 0.65
•
(11a)
1.2 ln(As2 / As1 )
•
and when As2 /As1 > 1
= 0.65 + 0.67 ln(As2 /As1 )
2.76ln2 (As2 /As1 )
(11b)
A comparison of the simplified equations (i.e. ACI 318-14 and modified ACI 318-14) against the numerical predictions for βu of the prestressed NSC and HSC members with external CFRP tendons investigated is illustrated in Fig. 16(b). It is seen that by introducing the structure-related factor As2/As1, the modified ACI 318-14 equation, i.e. Eq. (10), demonstrates better correlation than ACI 318-14.
•
5. Conclusions The results of analysis of two-span continuous prestressed NSC and HSC members with external CFRP tendons lead to the following conclusions:
Acknowledgments The work has been supported by the Fundamental Research Funds for the Central Universities under Grant No. 2018ⅣA006 and by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 751921.
• NSC members with a high tendon area may fail to exhibit favorable •
than NSC. In calculating the ultimate tendon stress, ACI 318-14 satisfactorily reflects the impact of concrete grade and tendon area for ρp greater than 0.44%. However, ACI 318-14 is non-conservative at a low ρp level, especially when HSC is used. HSC produces a lower cu value than NSC, and the difference is increasingly significant with increasing ρp. The cu value for the inner support is lower than that for the midspan. HSC leads to higher ultimate strain in tensile reinforcing steel than NSC, except at low ρp levels. The ultimate strain in compressive reinforcing steel for HSC is lower than that for NSC. The inner support section exhibits smaller strain in tensile reinforcing steel that the midspan section at low ρp levels, while the observation is opposite at high ρp levels. The concrete strength has negligible impact on moment redistribution in these members provided that the members have sufficient plastic rotational capacities. The predictive model of Eq. (10) is more accurate than ACI 318-14 when predicting moment redistribution in this structural typology.
ductile behavior, while the counterpart members made of HSC are able to develop sufficient rotation of plastic hinges. HSC produces larger crack widths at the critical sections than NSC except at a low ρp level. HSC leads to higher ultimate load-carrying capacity than NSC, particularly obvious at high ρp levels. HSC produces smaller ultimate deflection at low ρp levels but larger one at high ρp levels than NSC.
Data availability statement The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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Fig. 14. Development of moments and moment redistribution for different concrete strengths and tendon areas; (a) load-moment behavior; (b) load versus moment redistribution.
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Fig. 15. Variation of βu with varying ρp for different concrete strengths. (a) midspan; (b) inner support.
Fig. 16. Comparison of βu values. (a) ACI 318-14 and numerical predictions; (b) simplified equations against numerical analysis.
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