- Original article
- Open Access
Control of Tensile Stress in Prestressed Concrete Members Under Service Loads
- Deuck Hang Lee^{1},
- Sun-Jin Han^{2},
- Hyo-Eun Joo^{2} and
- Kang Su Kim^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s40069-018-0266-3
© The Author(s) 2018
- Received: 22 August 2017
- Accepted: 14 March 2018
- Published: 29 May 2018
Abstract
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 }).
Keywords
- prestressed concrete
- serviceability
- design code
- strand
- stress limit
- effective prestress
1 Introduction
2 Research Significance
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.
3 Net Tensile Stress Limit for PSC Members at Service Loads
According to the ACI318-14 design code, the stress change in prestressed reinforcements at the service loads (Δf_{ ps }) shall be calculated by the cracked section analysis for the PSC members belonging to the Class C category that are cracked in flexure under service load conditions. For the purpose of a proper crack control at the service loads, the value of Δf_{ ps } is limited to 250 MPa (36,000 psi) for the Class C PSC members. As mentioned in the ACI318-14 commentary R24.3.2.2, the maximum stress limit of 250 MPa for the Class C PSC member (Δf_{ ps }) is intended to be similar to the maximum allowable stress of the conventional reinforced concrete (RC) members with the Grade 60 reinforcements (f_{ y } = 420 MPa where f_{ y } is the yield strength of nonprestressed reinforcement), which can be calculated as 2/3 f_{ y } (i.e., 280 MPa). The ACI318-14, however, also allows the use of the Grade 80 reinforcements (f_{ y } = 550 MPa), where the maximum allowable stress is estimated to be about 370 MPa, which is significantly higher than that of the RC members reinforced with the Grade 60 reinforcements (i.e., 280 MPa). On the contrary, the limit value for Δf_{ ps } has been fixed for the Class C partially prestressed concrete members as 250 MPa regardless of the Grades of the nonprestressed reinforcements. This means that Δf_{ ps } is, of course, limited as 250 MPa even for the PSC members reinforced with combinations of 1860 MPa strands and 550 MPa yield strength (Grade 80) rebars.
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 }).
4 Parametric Study for Estimation of Net Tensile Stresses of PSC Members at Service Loads
4.1 Variables for Parametric Study
Summary of parametric study.
Section of type | Details of section^{a} | Tensile strength of strand (f_{ pu }, MPa) | Yield strength of reinforcing bar (f_{ y }, MPa) | Effective prestress ratio (f_{ pe }/f_{ pu }) | Reinforcing index (ω) | PPR (%) |
---|---|---|---|---|---|---|
Rectangular | 12RB16 16RB40 | 1860 (Grade 270) | 420 (Grade 60) 550 (Grade 80) | 0.5 0.55 0.6 0.65 | 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 | 50 67 100 |
Number of rectangular sections: 432 | ||||||
Tee | 12T16 16T40 | 1860 (Grade 270) | 420 (Grade 60) 550 (Grade 80) | 0.5 0.55 0.6 0.65 | 0.0135 0.027 0.0405 0.054 0.0675 0.081 0.0945 0.108 | 50 67 100 |
Number of T sections: 384 | ||||||
Inverted Tee | 28IT20 40IT52 | 1860 (Grade 270) | 420 (Grade 60) 550 (Grade 80) | 0.5 0.55 0.6 0.65 | 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 | 50 67 100 |
Number of inverted T sections: 432 | ||||||
Total number of analyses: 1248 |
4.2 Stress Estimation of Prestressing Strands in PSC Members Under Service Loads
5 Analysis Results of the Parametric Study
5.1 Rectangular Sections
5.2 Flanged Sections: Inverted T (IT) and T Sections
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
5.3 Effects of Tension Stiffening and Section Size
6 Proposed Approaches
6.1 Simple Checking of the Net Tensile Stress
Minimum effective prestress (f_{se,min}) to satisfy the tensile stress limit (Δf_{ps,allow}).
Section shapes | PPR 100% (∆f_{ ps } ≤ 250 MPa) | Partially prestressed concrete members (PPR ≥ 50%) | ||
---|---|---|---|---|
f_{ y } = 420 MPa (for ∆f_{ ps } ≤ 250 MPa) | f_{ y } = 550 MPa (for ∆f_{ ps } ≤ 250 MPa) | f_{ y } = 550 MPa (for ∆f_{ ps } ≤ 350 MPa) | ||
R | 0.50 f_{ pu } | 0.50 f_{ pu } | 0.55 f_{ pu } | 0.50 f_{ pu } |
T | 0.50 f_{ pu } | 0.50 f_{ pu } | 0.60 f_{ pu } | 0.50 f_{ pu } |
IT | 0.50 f_{ pu } | 0.50 f_{ pu } | 0.50 f_{ pu } | 0.50 f_{ pu } |
As shown in Fig. 10, in the case of the T-shaped sections, the minimum effective prestress (f_{se,min}) can be determined as 0.5 f_{ pu } for all the full PSC members and the partial PSC members except the partial PSC members with 550 MPa reinforcements. For the partial PSC members with 550 MPa reinforcements, the minimum effective prestress (f_{se,min}) is 0.60f_{ pu } to meet the Δf_{ ps } limit of 250 MPa.
The current ACI318-14 building code allows to use 2/3 f_{ y } for both 420 and 550 MPa reinforcing bars as the steel stress at the service loads when the maximum allowable spacing (s_{max}) of the flexural reinforcements is checked for the proper crack control. Soltani et al. (2013) and Harries et al. (2012) also demonstrated that 2/3 f_{ y } can be taken as the stress in the steel reinforcements at the service loads (f_{ s }) for the high yield strength steels even up to 827 MPa (120,000 psi). Therefore, it is considered that the allowable stress increase of the prestressing steel under the service load (Δf_{ps,allow}) can be increased from 250 to 350 MPa in the partial PSC members reinforced with 550 MPa nonprestressed steel. In that case, the minimum magnitude of the effective prestress (f_{se,min}) can be 0.5 f_{ pu } for all the partial PSC members reinforced with 550 MPa nonprestressed steel.
6.2 Simple Method for Calculating the Net Tensile Stress (Δf _{ ps })
As summarized in Table 3, when the effective prestress (f_{ se }) is greater than 0.50 f_{ pu }, which would be the case in most PSC members, the stress increase of prestressing strands (Δf_{ ps }) in all the full PSC members (PPR = 100%) satisfies the 250 MPa stress limit (Δf_{ps,allow}) specified in the ACI318-14. As aforementioned, in the partial PSC members (PPR ≥ 50%) with the effective prestress (f_{ se }) greater than 0.50 f_{ pu }, the maximum allowable stress of the prestressing strand at the service load (Δf_{ps,allow}) is 250 MPa if 420 MPa nonprestressed steel is used, while it is 350 MPa if 550 MPa nonprestressed steel is used. In order to satisfy the 250 MPa stress limit (Δf_{ps,allow}) in the partial PSC members (PPR ≥ 50%) with 550 MPa nonprestressed steel, however, the effective prestress (f_{ se }) shall be greater than 0.55 f_{ pu }, 0.60 f_{ pu }, and 0.50 f_{ pu } for the rectangular, T-shaped, and IT-shaped sections, respectively. Thus, it is very important to apply a proper magnitude of the effective prestress (f_{ se }) to satisfy the stress limit (Δf_{ps,allow}) for the serviceability design of the PSC members. If the Δf_{ ps } value is, however, smaller than the stress limit (Δf_{ps,allow}), it is not necessary to use the maximum value of Δf_{ ps } (i.e., Δf_{ps,allow}) in the Eq. (11) for calculating the maximum spacing of flexural reinforcements (s_{max}). In that case, the Δf_{ ps } value can be used as f_{ s } in Eq. (11), by which more economical designs can be achieved. As mentioned above, however, the cracked section analysis, which requires complex iterative calculations (Lee and Kim 2011; ACI Committee 318 2014), need to be conducted to estimate Δf_{ ps } of the Class C PSC members. Thus, this study also aimed at proposing a simple method to estimate Δf_{ ps } for the Class C PSC members.
7 Conclusions
- (1)
The nonlinear flexural analysis results of the PSC members showed that the net tensile stress of prestressing strands at the service load (Δf_{ ps }) increases as the yield strength of the nonprestressed reinforcement is greater and as the partial prestressing ratio (PPR) or the effective prestress level (f_{ se }) decreases. It also appeared that the stress change in Δf_{ ps } is more sensitive in the full PSC members compared to in the partial PSC members with respect to the magnitude of the effective prestress (f_{ se }).
- (2)
In the full PSC members (PPR = 100%) with the effective prestress (f_{ se }) greater than 0.50 f_{ pu }, which would be the case in most PSC members, the stress increase (Δf_{ ps }) of prestressing strands satisfied the 250 MPa stress limit (Δf_{ps,allow}) specified in the ACI318-14.
- (3)
For the RC members with 550 MPa reinforcing bars as well as 420 MPa reinforcing bars, the current ACI318-14 code permits to use 2/3 f_{ y } as the steel stress in the calculation of the maximum spacing of the flexural reinforcement (s_{max}) for proper crack control; therefore, it is considered that the allowable tensile stress increase of the prestressing steels under the service loads (Δf_{ps,allow}) can be increased from 250 to 350 MPa in the partial PSC members with 550 MPa reinforcing bars.
- (4)
In the partial PSC members (PPR ≥ 50%) with an effective prestress (f_{ se }) greater than 0.50 f_{ pu }, the maximum Δf_{ ps } satisfies the 250 MPa stress limit when 420 MPa reinforcing bar is used, and it satisfies the 350 MPa stress limit when 550 MPa reinforcing bar is used. To satisfy 250 MPa stress limit (Δf_{ps,allow}) in the partial PSC members (PPR ≥ 50%) with 550 MPa reinforcing bar, however, the effective prestress (f_{ se }) shall be greater than 0.55 f_{ pu }, 0.60 f_{ pu }, and 0.50 f_{ pu } for the rectangular, T-shaped, and IT-shaped sections, respectively.
- (5)
A summary table was proposed, which can be used to easily check whether the net tensile stress (Δf_{ ps }) exceeds the specified stress limit (250 or 350 MPa) under the service loads using only the magnitude of the effective prestress (f_{ se }), requiring no complex cracked section analysis, and it can be thus easily applied in practice.
- (6)
The simplified method proposed in this study for estimating the net tensile stress tensile stress of the prestressing strands (Δf_{ ps }) in the Class C PSC sections under the service loads provided conservative analysis results compared to those estimated through the nonlinear flexural analyses, and it is expected to be a useful alternative method for the serviceability design of the PSC members.
- (7)
Since the proposed design method was developed by utilizing non-linear flexural analyses, it is considered that experimental evidences are required for the confirmation of the proposed methods.
Declarations
Acknowledgements
This work was supported by the 2016 Research Fund of the University of Seoul.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- ACI Committee 318. (2014). Building code requirements for structural concrete (ACI 318-14) and commentary. Farmington Hills, MI: American Concrete Institute.Google Scholar
- American Association of state Highway and Transportation Officials. (2010). AASHTO LEFD bridge design specifications: Customary U.S. units (5th ed.). Washington, D.C: AASHTO.Google Scholar
- Atutis, M., Valivonis, J., & Atutis, E. (2015). Analysis of serviceability limit state of GFRP prestressed concrete beams. Composite Structures, 134(15), 450–459.View ArticleGoogle Scholar
- Bentz, E.C. (2000). Sectional analysis of reinforced concrete members. Ph.D. Dissertation, University of Toronto, Ontario, Canada.Google Scholar
- Collins, M. P., & Mitchell, D. (1991). Prestressed concrete structures. Englewood Cliffs, NJ: Prentice Hall.Google Scholar
- Devalapura, R. K., & Tadros, M. K. (1992). Stress-strain modeling of 270 ksi low-relaxation prestresssing strands. PCI Journal, 37(2), 100–106.View ArticleGoogle Scholar
- Frosch, R. J. (1999). Another look at cracking and crack control in reinforced concrete. ACI Structural Journal, 99(3), 437–442.Google Scholar
- Gagely, P., & Lut, L. (1968). Maximum cracks width in reinforcement concrete flexural member. ACI Special Publication SP-20, 20, 87–117.Google Scholar
- Harries, K. A., Shahrooz, B. M., & Soltani, A. (2012). Flexural crack widths in concrete girders with high-strength reinforcement. Journal of Bridge Engineering, 17(1), 29–57.Google Scholar
- Karayannis, C. G., & Chalioris, C. E. (2013). Design of partially prestressed concrete beams based on the cracking control provisions. Engineering Structures, 48(1), 402–416.View ArticleGoogle Scholar
- Kim, K. S., & Lee, D. H. (2011). Flexural behavior of prestressed composite beams with corrugated web: Part II. Experiment and verification. Composite Part B: Engineering, 42(1), 1617–1629.View ArticleGoogle Scholar
- Kim, K. S., Lee, D. H., Choi, S. M., Choi, Y. H., & Jung, S. H. (2011). Flexural behavior of prestressed composite beams with corrugated web: Part I. Development and analysis. Composite Part B: Engineering, 42(6), 1603–1616.View ArticleGoogle Scholar
- Lee, D. H., Hwang, J. H., Kim, K. S., Kim, J. S., Chung, W., & Oh, H. (2014). Simplified strength design method for allowable compressive stressed in pretensioned concrete members at transfer. KSCE Journal of Civil Engineering, 18(7), 2209–2217.View ArticleGoogle Scholar
- Lee, D. H., & Kim, K. S. (2011). Flexural strength of prestressed concrete members with unbonded tendons. Structural Engineering Mechanics, 38(5), 675–696.View ArticleGoogle Scholar
- Lee, J. Y., Lee, D. H., Hwang, J. H., Park, M. K., Kim, Y. H., & Kim, K. S. (2013). Investigation on allowable compressive stresses in pre-tensioned concrete members at transfer. KSCE Journal of Civil Engineering, 17(5), 1083–1098.View ArticleGoogle Scholar
- Marí, A., Bairán, J. M., Cladera, A., & Oller, E. (2016). Shear design and assessment of reinforced and prestressed concrete beams based on a mechanical model. Journal of Structural Engineering, 142(10), 1–17.View ArticleGoogle Scholar
- Mast, R. F., Dawood, M., Rizkalla, S. H., & Zia, P. (2008). Flexural strength design of concrete beams reinforced with high-strength steel bars. ACI Structural Journal, 105(5), 570–577.Google Scholar
- Mattock, A. H. (1979). Flexural strength of prestressed concrete sections by programmable calculator. PCI Journal, 24(1), 32–54.View ArticleGoogle Scholar
- Nawy, E. G. (2010). Prestressed concrete: A fundamental approach (5th ed.). Upper Saddle River, NJ: Pearson Prentice Hall, Pearson Education Inc.Google Scholar
- Park, H., & Cho, J. Y. (2017). Ductility analysis of prestressed concrete members with high-strength strands and code. ACI Structural Journal, 114(34), 407–416.Google Scholar
- Park, H., Jeong, S., Lee, S. C., & Cho, J. Y. (2016). Flexural behavior of post-tensioned prestressed concrete girders with high-strength strands. Engineering Structures, 112(1), 90–99.View ArticleGoogle Scholar
- Park, J. H., Park, H., & Cho, J. Y. (2017). Prediction of stress in bonded strands at flexural. ACI Structural Journal, 114(56), 697–705.Google Scholar
- Patel, K. A., Chaudhary, S., & Nagpal, A. K. (2016). A tension stiffening model for analysis of RC flexural members under service load. Computers and Concrete, 17(1), 29–57.View ArticleGoogle Scholar
- Prestressed Concrete Institute. (2010). PCI design handbook: Precast and prestressed concrete. Chicago, IL: Precast/Prestressed Concrete Institute.Google Scholar
- Rodriguez-Gutierrez, J. A., & Aristizabal-Ochoa, J. D. (2000). Partially and fully prestressed concrete sections under biaxial bending and axial load. ACI Structural Journal, 97(4), 553–563.Google Scholar
- Sahamitmongkol, R., & Kishi, T. (2011). Tension stiffening effect and bonding characteristics of chemically prestressed concrete under tension. Materials and Structures, 44(2), 455–474.View ArticleGoogle Scholar
- Scholz, H. (1990). Ductility, redistribution, and hyperstatic moments in partially prestressed members. ACI Structural Journal, 87(3), 341–349.Google Scholar
- Skogman, B. C., Tadros, M. K., & Grasmick, R. (1988). Flexural strength of prestressed concrete members. PCI Journal, 33(5), 96–123.View ArticleGoogle Scholar
- Soltani, A., Harries, K. A., & Shahrooz, B. M. (2013). Crack opening behavior of concrete reinforced with high strength reinforcing steel. International Journal of Concrete Structures and Materials, 7(4), 253–264.View ArticleGoogle Scholar
- Vecchio, F. J., & Collins, M. P. (1986). The modified compression-field theory for reinforced concrete elements subjected to shear. ACI Journal, 83(2), 219–231.Google Scholar