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In current design codes, crack control design criterion for prestressed concrete (PSC) members is stricter than conventional reinforced concrete (RC) members. In particular, it is stipulated that the net tensile stress of prestressing strands should be controlled under 250 MPa in the serviceability design of PSC members belonging to the Class C category section that is expected to be cracked due to flexure under service load conditions as defined in ACI318 code. Thus, the cracked section analysis is essentially required to estimate the tensile stress of the prestressing strands under the service loads, which requires very complex iterative calculations, thereby causing many difficulties in the applications of the Class C PSC members in practice. Thus, this study proposed a simple method to estimate the net tensile stress of the prestressing strands (Δf ps ) under the service load conditions, and also provided a summary table to be used for checking whether the net tensile stress (Δf ps ) exceeds the stress limit (250 or 350 MPa) with respect to the magnitude of effective prestress (f se ).

The current ACI318 building code (ACI Committee 318 2014) has stipulated more conservative provisions for the crack control design of prestressed concrete (PSC) members reinforced with high strength prestressing strands compared to conventional reinforced concrete (RC) members. As shown in Fig. 1, the net tensile stress of prestressing strands in the PSC members with cracked section properties, belonging to the Class C category according to the ACI318 code, is expected to be significantly higher at the service load condition compared to that of the Class U and T categories (i.e., uncracked sections). The PSC members exhibit very different flexural behaviors at the service load level depending on the magnitude of the effective prestress (f se ) and the partial prestressing ratio (PPR) even when they have the same flexural strength (Kim and Lee 2011; Lee and Kim 2011; Lee et al. 2013, 2014; Park et al. 2016, 2017; Park and Cho 2017). In the ACI318-14 code, it is specified that the net tensile stress of the prestressing strands (Δf ps ) shall not exceed 250 MPa for the Class C PSC members to ensure proper crack control at the service loads. In order to estimate the net tensile stress of the prestressing strands (Δf ps ) in the Class C flexural members under the service load conditions, the cracked section analysis should be essentially conducted, which unfortunately requires very complex and time-consuming iterative calculations, as pointed out by Skogman et al. (1988). and Mast et al. (2008). Thus, this study aims to develop a simple method to estimate the net tensile stress of the prestressing strands (Δf ps ) under the service load conditions. On the other hand, based on the flexural analysis results of prestressed concrete members with various sectional properties, a summary table is also proposed, which can be used to easily check whether the net tensile stress (Δf ps ) exceeds the specified stress limit (250 MPa) using only the magnitude of the effective prestress (fse) without calculating the net tensile stress of the prestressing strands (Δf ps ).

In this study, nonlinear flexural analyses were performed on a total of 1248 prestressed concrete members with various sectional types, partial prestressing ratios, reinforcing indices, yield strengths of nonprestressed reinforcements, and effective prestresses, based on which a simple method was proposed to estimate the net tensile stress (Δf ps ) of the prestressing strands at the service loads. In order to examine whether the net tensile stress (Δf ps ) of the prestressing strands exceeds the limitation specified in design codes for serviceability check of the Class C PSC members, the proposed method do not require the cracked section analysis that involves very complex and time-consuming iterative calculations.

On the other hand, the cracked section analysis should be essentially conducted to estimate the net tensile stress of the PSC members with the Class C section properties, which requires quite complex iterative calculations, as described by Mast et al. (2008), and PCI design handbook (Prestressed Concrete Institute 2010) in detail. To overcome such limitations, this study proposed a simple method to estimate the net tensile stress of the prestressing strands in the Class C PSC members at service loads (Δf ps ) without the iterative cracked section analysis so that the maximum spacing of the prestressing strands specified in the ACI318 code for the proper crack control can be easily calculated. In addition, this study also presented a summary table to be used for checking whether the net tensile stress (Δf ps ) exceeds the stress limit (250 or 350 MPa) with respect to the magnitude of effective prestress (f se ).

where E p is the elastic modulus of the prestressing strands, and the coefficients A, B, and C are 0.025, 118, and 10, respectively (Devalapura and Tadros 1992). The compressive force of concrete (C c ) was calculated by dividing the cross section into n layers with 5 mm thickness. For the tensile force of concrete, the tension contribution of concrete after cracking, i.e., the so-called tension-stiffening effect, was reflected in the analyses. Therefore, the equilibrium equations on the cross section at an arbitrary loading stage j can be expressed, as follows:

where b i and t are the width and thickness of the ith concrete layer, respectively, and y i is the distance of the centroid of the ith concrete layer from the extreme top fiber. d p and d s are the distance from extreme compression fiber to centroid of prestressing and nonprestressed reinforcements, respectively. When the strain at the extreme compressive fiber of the cross section (ɛ t ) reaches the ultimate strain of concrete (ɛ cu ), where ɛ cu was adopted to be 0.003 in this study, the flexural moment calculated by substituting the sectional force components satisfying Eq. (6) into Eq. (7) is defined as the flexural strength (M n ) of the cross section, and two-thirds of this flexural strength was defined as the flexural moment at the service loads (M service ) (Gagely and Lutz 1968; Frosch 1999; Atutis et al. 2015). In the following discussions, only the Class C sections were considered, excluding the Class U and T sections. In accordance with Tables 24.5.4.1 and 9.6.2.1 in the ACI318-14, if the concrete compressive stress at the service loads exceeds the allowable stress level or M n is smaller than 1.2 times the cracking moment (1.2M cr ) due to very low tensile reinforcement ratio, those PSC members were also excluded. The cracking moment M cr was estimated, as follows:

Figures 9 and 10 show the analysis results of the ITS series and TS series, which are inverted T and T sections, respectively. (See Fig. 2b, c) The ITS series have almost the same flexural strength (M n ) as the RS series with rectangular sections having the same reinforcing index, but the flexural cracking strengths (M cr ) of the IT sections are quite higher than those of the rectangular sections. For this reason, the IT sections have higher stiffness at the service loads and thus have lower Δf ps values than the RS series. Also, because the IT sections have higher stiffness at the service loads, many cases in the ITS series were classified into the Class T or U sections, rather than the Class C section. As shown in Fig. 9a and b, in all the ITS series reinforced with 420 or 550 MPa reinforcing bars, the stress increase in prestressing steel (Δf ps ) was within the stress limitation of 250 MPa, and the maximum value of Δf ps appeared in the reinforcing index (ω) ranging from 0.2 to 0.3.

As shown in Fig. 10, the stress change (Δf ps ) in the full PSC sections with the PPR 100% was more sensitive by the magnitudes of effective prestress (f se ) compared to those in the partial PSC sections, which was also observed the same in the analysis results of the rectangular sections. At the same reinforcement ratio, the TS series had lower cracking strength (M cr ) than the RS series with the rectangular sections, but their flexural moment at the service loads (M service ) were similar to the RS series, and thus the TS series showed higher magnitudes of Δf ps compared to the RS series. The maximum values of Δf ps were estimated in the reinforcing index (ω) ranging from 0.04 to 0.06, after which it gradually decreased. In addition, it can be seen that the Δf ps values of the partially prestressed TS sections are larger when the yield strengths of nonprestressed steel are greater. In particular, as shown in Fig. 10b, in the case of the partial PSC members reinforced with 550 MPa nonprestressed steel and the PPR 67%, the magnitudes of Δf ps exceeded the maximum stress limit of 250 MPa specified in the ACI318-14 when the effective prestress (f se ) was 0.5 f pu and the reinforcing index (ω) was greater than 0.025. For the partial PSC members reinforced with 550 MPa reinforcements and the PPR 50%, the magnitudes of Δf ps also exceeded the 250 MPa limit when the effective prestress (f se ) was less than 0.55 f pu and the reinforcing index (ω) was more than 0.025

It is expected that the effects of the tension stiffening on the flexural strengths of PSC members are marginal because the post-cracking resistance of concrete clearly decreases as the flexural crack width increases; however, it can still play an important role in the service load behavior (Collins and Mitchell 1991; Sahamitmongkol and Kishi 2011; Patel et al. 2016). Figures 11a and b show the effect of the tension stiffening on the stress behaviors of the prestressing strands at the service loads. The effect of the tension stiffening on Δf ps was greater in the full PSC members than that in the partial PSC members with PPR 50%. Especially, as shown in Fig. 11c, the contribution of the concrete in the cracked tension zone to the flexural strength (M tc /M n ) is larger in the PSC sections with low reinforcing index. Accordingly, if the tension stiffening effect is not taken into account, Δf ps can be overestimated in the PSC sections, which would be more serious in those with low tension reinforcement ratios. Therefore, in this study, the tension stiffening effect was considered in the estimations of Δf ps shown in Figs. 6, 7, 8, 9 and 10. Figure 12a shows a comparison of Δf ps values of the RS series with the section height of 400 mm and the RL series with the height of 1000 mm. As the section size increases, the magnitude of Δf ps slightly increases, but their differences were very small. Figure 12b shows a comparison of Δf ps values between the TS series with the height of 400 mm and TL series with the height of 1000 mm. The differences in Δf ps depending on the section sizes were negligible except the sections with the low reinforcing index under about 0.02. The reason of the large differences in Δf ps between the TS and TL series for the sections with the low reinforcing index is because the neutral axis depths are inevitably small in these members due to the low reinforcing ratio, leading to be cracked in flexure at the service loads. After cracking, high tensile strains are developed in the prestressing strands, and thus high levels of Δf ps are also expected. A similar tendency was also found in the analysis results of the IT sections, as shown in Fig. 12c. 2b1af7f3a8