A Reliability-Based Assessment and Design Calibration of the Modified Strut-and-Tie Model for Steel Fiber Reinforced Concrete Deep Beams
Received: 28 September 2025 | Revised: 19 October 2025, 27 October 2025, and 30 October 2025 | Accepted: 1 November 2025 | Online: 15 December 2025
Corresponding author: Bac An Hoang
Abstract
A reliability analysis is performed on a modified Strut-and-Tie Model (STM), which accurately predicts the shear capacity of Steel Fiber Reinforced Concrete (SFRC) deep beams. However, inherent uncertainties are not considered when calculating the model's performance. This study implements a reliability analysis using Monte Carlo Simulation (MCS) on a dataset of 22 large-scale experimental samples, including samples with a simple supported and a two-span continuous configuration. In addition, a limit state function was established based on the modified STM model. Specifically, the analysis quantified the inherent safety of the uncalibrated model, revealing a reliability index (β) between 0.5 and 2.0. A reliability-based calibration was performed to address the issue of its composition not conforming with modern codes. For the Load and Resistance Factor Design (LRFD) format, a resistance factor (φ) of 0.50 was calculated, ensuring that different beam configurations achieve a consistent target reliability level.
Keywords:
steel fiber reinforced concrete deep beams, strut-and-tie model, structural reliability, reliability-based calibration, resistance factor, LfrdDownloads
References
N. Zhang and K.-H. Tan, "Direct Strut-and-Tie Model for Single Span and Continuous Deep Beams," Engineering Structures, vol. 29, no. 11, pp. 2987–3001, Nov. 2007. DOI: https://doi.org/10.1016/j.engstruct.2007.02.004
J. Schlaich and K. Schäfer, "Zur Druck-Querzug-Festigkeit des Stahlbetons.," Beton- und Stahlbetonbau, vol. 78, no. 3, pp. 73–78, 1983. DOI: https://doi.org/10.1002/best.198300120
Building Code Requirements for Structural Concrete and Commentary, ACI 318-19, American Concrete Institute, Farmington Hills, MI, USA, 2019.
Design of Concrete Structures. Part 1.1: General Rules and Rules for Buildings, EN 1992-1-1, European Committee for Standardization, Brussels, Belgium, 2004.
L. T. Hussein and R. M. Abbas, "A Semi-Empirical Equation based on the Strut-and-Tie Model for the Shear Strength Prediction of Deep Beams with Multiple Large Web Openings," Engineering, Technology & Applied Science Research, vol. 12, no. 2, pp. 8289–8295, Apr. 2022. DOI: https://doi.org/10.48084/etasr.4743
A. Starčev-Ćurčin et al., "Experimental Testing of Reinforced Concrete Deep Beams Designed by Strut-And-Tie Method," Applied Sciences, vol. 10, no. 18, Sept. 2020. DOI: https://doi.org/10.3390/app10186217
A. Ali, Z. Soomro, S. Iqbal, N. Bhatti, and A. F. Abro, "Comparison of Mechanical Properties of Lightweight and Normal Weight Concretes Reinforced with Steel Fibers," Engineering, Technology & Applied Science Research, vol. 8, no. 2, pp. 2741–2744, Apr. 2018. DOI: https://doi.org/10.48084/etasr.1874
K.-F. Weng, J.-X. Zhu, B.-T. Huang, J.-G. Dai, and J.-F. Chen, "Shear strengthening of reinforced concrete beams using FRP-UHPC composite layer," Composite Structures, Dec. 2025, Art. no. 119931. DOI: https://doi.org/10.1016/j.compstruct.2025.119931
A. M. Yousef, A. M. Tahwia, M. S. Al-Enezi, A. M. Yousef, A. M. Tahwia, and M. S. Al-Enezi, "Experimental and Numerical Study of UHPFRC Continuous Deep Beams with Openings," Buildings, vol. 13, no. 7, July 2023. DOI: https://doi.org/10.3390/buildings13071723
M. Deng, F. Ma, X. Wang, and H. Lü, "Investigation on the shear behavior of steel reinforced NC/HDC continuous deep beam," Structures, vol. 23, pp. 20–25, Feb. 2020. DOI: https://doi.org/10.1016/j.istruc.2019.10.002
N. T. Q. Như, T. N. L. Tuyen, N. M. Long, H. B. An, P. T. Le, and T. V. Phuc, "Phân tích thực nghiệm ứng xử cắt của dầm cao bê tông sợi thép liên tục và đơn giản.," Tạp chí Khoa học Công nghệ Xây dựng (TCKHCNXD) - ĐHXDHN, vol. 19, no. 2V, pp. 28–41, May 2025. DOI: https://doi.org/10.31814/stce.huce2025-19(2V)-03
Research on the Bearing Capacity and Deformation of Transfer Beam Structures for High-Rise Buildings Using High-Strength Concrete and High-Strength Concrete Reinforced With Dispersed Steel Fibers, RD 34-22, Ministry of Science and Technology, Ho Minh City, Vietnam, 2024.
R. E. Melchers and A. T. Beck, Structural Reliability Analysis and Prediction. New York, NY, USA: John Wiley & Sons, 2018. DOI: https://doi.org/10.1002/9781119266105
A. S. Nowak and K. R. Collins, Reliability of Structures, Second Edition. Boca Raton, FL, USA: CRC Press, 2012.
T. H. Nguyen, V. D. Le, X. H. Vu, and D. K. Nguyen, "Reliability-based Design Optimization of Steel-Concrete Composite Beams Using Genetic Algorithm and Monte Carlo Simulation," Engineering, Technology & Applied Science Research, vol. 12, no. 6, pp. 9766–9770, Dec. 2022. DOI: https://doi.org/10.48084/etasr.5366
D. Muendacha, J. Teerawong, and P. Chetchotisak, "A Safety-Based Evaluation of Strut-and-Tie Methods for Shear Design of RC Deep Beams in Accordance With International Concrete Codes," Engineering and Applied Science Research, vol. 47, no. 2, pp. 137–144, June 2020.
Probabilistic Model Code, JCSS, Joint Committee on Structural Safety, Zürich, Switzerland, 2006.
R. G. Tuchscherer and A. Quesada, "Replacement of Deformed Side-Face Steel Reinforcement in Deep Beams With Steel Fibers," Structures, vol. 3, pp. 130–136, Aug. 2015. DOI: https://doi.org/10.1016/j.istruc.2015.03.008
K. Ma et al., "Shear Behavior of Hybrid Fiber Reinforced Concrete Deep Beams," Materials, vol. 11, no. 10, Oct. 2018. DOI: https://doi.org/10.3390/ma11102023
T. D. Dang, D. T. Tran, L. Nguyen-Minh, and A. Y. Nassif, "Shear resistant capacity of steel fibres reinforced concrete deep beams: An experimental investigation and a new prediction model," Structures, vol. 33, pp. 2284–2300, Oct. 2021. DOI: https://doi.org/10.1016/j.istruc.2021.05.091
Development of a probability based load criterion for American National Standard A58, National Institute of Standards and Technology, Gaithersburg, MD, USA, 1980.
Minimum Design Loads and Associated Criteria for Buildings and Other Structures, ASCE/SEI 7-22, American Society of Civil Engineers, Reston, VA, USA, 2022.
N. T. Q. Như, T. N. L. Tuyen, N. M. Long, H. B. An, P. T. Le, and T. V. Phuc, "Ảnh hưởng của cường độ bê tông đến ứng xử cắt của dầm cao bê tông sợi thép," Tạp chí Khoa học Công nghệ Xây dựng (TCKHCNXD) - ĐHXDHN, vol. 19, no. 3V, pp. 1–12, Aug. 2025. DOI: https://doi.org/10.31814/stce.huce2025-19(3V)-01
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Copyright (c) 2025 Bac An Hoang, Quynh Nhu Nguyen-Thi, Van Phuc Tran
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