Citation :

Madhusudhan M S, Umesha P K, "Reliability-Based Evaluation of Long-Term Deflection in Reinforced Concrete Flanged Beams: A Unified FOSM-Monte Carlo Simulation Framework with Dual Sensitivity Analysis and Design Implications," International Journal of Civil Engineering, vol. 13, no. 8, pp. 560-578, 2026. Crossref, https://doi.org/10.14445/23488352/IJCE-V13I8P132

Abstract

The potential for long-term deflection to govern the serviceability of flanged beams results from the combined effects of the sustained loading on the beams, as well as factors like cracking, creep, and shrinkage that contribute to the reduction of the margin between calculated and prescribed deflection limits. While deterministically determining the deflection of beams ensures that the deflection specification is satisfied, no information is obtained regarding the probability that the beam will experience deflection that reaches the prescribed limit state. Eight flanged beams of various section depths were evaluated, each with a span length of 5,000 mm. For each beam, the total deflection limit of l/250, or 20 mm as prescribed in the Indian standard code IS 456:2000 was utilized to define the serviceability limit state. Additionally, the limit state can be defined in terms of the individual components of deflection: instantaneous, shrinkage, and creep. Eighteen structural and material variable characteristics were determined for each of the eight beams. For the reliability analysis, eight of the variables were utilized: the sustained bending moment, the span length, the elastic modulus of the concrete, the effective moment of inertia, the shrinkage curvature of the beam, the modulus of elasticity of the concrete that has been creep-modified, the long-term effective moment of inertia, and the moment that is applied to the beam under the sustained load. The reliability of each beam was calculated using the First Order Second Moment (FOSM) method, as well as Monte Carlo simulation utilizing 100,000 samples to determine the reliability of each beam. Furthermore, methods were utilized to determine the local sensitivity of the beams, including analyzing the inclusion of variables, the suppression of individual variables, and the importance of the variance of each variable. Results of the analysis indicate that as the tabulated depth of each beam decreases from 450 mm to 310 mm, the mean total deflection of each beam increases from 10.464 mm to 16.982 mm, or an increase of 62.3%. Additionally, the reliability of each beam calculated via FOSM indicates that as depth decreases, the FOSM reliability index decreases from 4.634 to 1.018, with the corresponding failure probabilities increasing from 1.79×10−6 to 1.54×10−1. Monte Carlo method results indicate that the mean total deflection of each beam increases from 10.547 mm to 17.131 mm, an increase of 62.4%, and that the failure probability increases from 5.00×10−5 to 1.37×10−1, with reliability indices of 3.891 and 1.094, respectively. Comparison of the two methods indicates that the FOSM approximation is non-conservative for the deeper sections, returning a reliability index 19.1% above the simulated value at a depth of 450 mm, and mildly conservative for the shallowest sections, returning a value 6.9% below the simulated one at a depth of 310 mm, the sign of the deviation changing at an effective depth of approximately 340 mm. Additionally, each of the variables related to the sustained load on the beams contributed to approximately 60.3% to 65.8% of the total deflection of each beam. Furthermore, analysis of each of the eight beams indicates that the shallowest depth of 410 mm is the shallowest depth of any of the beams studied at which each method achieved a reliability index of 3.0 or more; thus, the depth-to-reliability relationship is only preliminary and should be applied to beams of the same span length, similar material properties, similar support conditions, and with a sustained load of 40%. A more thorough application of these methods to other beams would require additional experimental validation of the results.

Keywords

First-Order Second-Moment method, IS 456:2000, Long-term deflection, Monte Carlo simulation, Probabilistic design, Reinforced concrete flanged beams, Reliability index.

References

1. L.M. Abdel-Hafez, “Reliability of the Use of Codes of Practices Deflection Equations to Compute the Short Time Deflection,” Journal of Engineering Sciences, vol. 34, no. 1, pp. 13-36, 2006.
   [CrossRef] [Google Scholar] [Publisher Link]
2. Seyed Bahram Beheshti Aval, and Mohsen Ghabdian, “Time-Dependent Reliability Analysis of RC Deep Beams Considering Linear/Nonlinear Creep and Shrinkage using ANFIS Network and MCS,” Advances in Civil Engineering, vol. 2019, no. 1, pp. 1-15, 2019.
   [CrossRef] [Google Scholar] [Publisher Link]
3. H. Charkas, H.A. Rasheed, and M.H. Melhem, “Simplified Load-Deflection Calculation of FRP Strengthened RC Beams Based on a Rigorous Approach,” Proceeding 15^th ASCE Engineering Mechanics Conference, Columbia University, New York, 2002.
   [Google Scholar]
4. Wenjiao Dan et al., “Prediction and Global Sensitivity Analysis of Long-Term Deflections in Reinforced Concrete Flexural Structures using Surrogate Models,” Materials, vol. 16, no. 13, pp. 1-33, 2023.
   [CrossRef] [Google Scholar] [Publisher Link]
5. Lin Feng et al., “Reliability-Based Design Analysis for FRP Reinforced Compression Yield Beams,” Polymers, vol. 14, no. 22, pp. 1-23, 2022.
   [CrossRef] [Google Scholar] [Publisher Link]
6. Afshin Firouzi, Amirmasoud Taki, and Saeed Mohammadzadeh, “Time-Dependent Reliability Analysis of RC Beams Shear and Flexural Strengthened with CFRP Subjected to Harsh Environmental Deteriorations,” Engineering Structures, vol. 196, 2019.
   [CrossRef] [Google Scholar] [Publisher Link]
7. Mohsen Ghabdian et al., “Reliability of Reinforced Concrete Beams in Serviceability Limit State via Microprestress-Solidification Theory, A Structural Health Monitoring Strategy,” Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, vol. 236, no. 5, 2022.
   [CrossRef] [Google Scholar] [Publisher Link]
8. R. Ian Gilbert, and Robert F. Warner, “Tension Stiffening in Reinforced Concrete Slabs,” Journal of Structural Division, ASCE, vol. 104, no. 12, pp. 1885-1900, 1978.
   [CrossRef] [Google Scholar] [Publisher Link]
9. Marin Grubišić, Jelena Ivošević, and Ante Grubišić, “Reliability Analysis of Reinforced Concrete Frame by Finite Element Method with Implicit Limit State Functions,” Buildings, vol. 9, no. 5, pp. 1-22, 2019.
   [CrossRef] [Google Scholar] [Publisher Link]
10. Bingcheng Guo et al., “Advanced Analytical Model and Reliability Analysis for Long-Term Deflection of FRP Reinforced Concrete Beams with Compression Yielding Mechanism,” Structures, vol. 80, pp. 1-11, 2025.
    [CrossRef] [Google Scholar] [Publisher Link]
11. Abraham M. Hasofer, and Niels C. Lind, “Exact and Invariant Second-Moment Code Format,” Journal of the Engineering Mechanics Division, vol. 100, no. 1, pp. 111-121, 1974.
    [CrossRef] [Google Scholar] [Publisher Link]
12. Rami A. Hawileh, Haya H. Mhanna, and Jamal A. Abdalla, “Effect of Flange Geometry on the Shear Capacity of RC T-Beams,” Procedia Structural Integrity, vol. 42, pp. 1198-1205, 2022.
    [CrossRef] [Google Scholar] [Publisher Link]
13. IS 456:2000, Plain and Reinforced Concrete — Code of Practice, 4^th Revision, Bureau of Indian Standards, New Delhi, India, 2000.
    [Google Scholar] [Publisher Link]
14. Joanna Kaliszuk, “Hybrid Monte Carlo Method in the Reliability Analysis of Structures,” Computer Assisted Mechanics and Engineering Sciences, vol. 18, pp. 205-216, 2011.
    [Google Scholar] [Publisher Link]
15. Niels C. Lind, “Consistent Partial Safety Factors,” Journal of Structural Division, ASCE, vol. 97, no. 6, pp. 1651-1669, 1971.
    [CrossRef] [Google Scholar] [Publisher Link]
16. Mohammed S. Al-Ansari, “Reliability and Flexural Behavior of Triangular and T-Reinforced Concrete Beams,” International Journal of Advanced Structural Engineering, vol. 7, pp. 377-386, 2015.
    [CrossRef] [Google Scholar] [Publisher Link]
17. Osama Ali, “Time-Dependent Reliability of FRP Strengthened Reinforced Concrete Beams Under Coupled Corrosion and Changing Loading Effects,” Ph.D Thesis, University of Angers, pp. 1-363, 2014.
    [Google Scholar] [Publisher Link]
18. Ravindra P. Patil, and K. Manjunath, “An Empirical Expression for Reliability Index of Flanged RC Beams in Limit State of Deflection,” International Journal of Performability Engineering, vol. 6, no. 4, pp. 373-380, 2010.
    [CrossRef] [Google Scholar] [Publisher Link]
19. Ravindra P. Patil, and K. Manjunath, “Time Dependent Performance Analysis of Flanged RC Beam in Limit State of Deflection,” International Journal of Performability Engineering, vol. 7, no. 4, pp. 379-386, 2011.
    [CrossRef] [Google Scholar] [Publisher Link]
20. David Pustka et al., “Probabilistic Reliability Analysis of High-Performance Reinforced Concrete Beam using MATLAB Software,” International Journal of Mechanics, vol. 8, pp. 101-111, 2014.
    [Google Scholar] [Publisher Link]
21. Y.K. Wen, “Reliability and Performance-Based Design,” Structural Safety, vol. 23, no. 4, pp. 407-428, 2001.
    [CrossRef] [Google Scholar] [Publisher Link]
22. JCSS Probabilistic Model Code (PMC), Joint Committee on Structural Safety, 2000. [Online]. Available: https://www.jcss-lc.org/jcss-probabilistic-model-code/