Semiempirical Equations to Evaluate Maximum Deflection in RC Beam and Column under Fire
Publication: Journal of Structural Engineering
Volume 150, Issue 10
Abstract
Many design regulations around the globe rely on member deflection as a governing criterion for resistance assessment in fire. The deflection evaluation in fire is generally achieved by conducting expensive experiments or computationally expensive finite-element analyses. This often restricts practicing engineers from using robust performance-based design philosophy for typical structures. Instead, they rely on objective design guidelines, often resulting in inefficient sizes of reinforced concrete (RC) members. Therefore, semiempirical relations are derived in the current study to determine the maximum deflection of the RC beam and RC column in a fire. Three separate end conditions are considered within the beam category: fixed-fixed beam, propped cantilever beam, and simply supported beam. Plausible variables are first identified that could affect the overall deflection of the member, and their proportionality is subsequently determined by performing one-on-one regression analysis. Furthermore, these relations are developed in terms of the most suitable fire intensity measures derived from the literature, which makes them applicable irrespective of the type of fire framework. The credibility of the deflection equations is validated through visual analysis followed by the three popular error indicator parameters, namely, Pearson’s correlation coefficient, relative root-mean squared error (RRMSE), and performance index. Results indicated that all deflection equations accurately predict the RC member behavior under fire.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The research work presented herein is supported by the Government of India’s Ministry of Education under its Scheme for Promotion of Academic and Research Collaboration (SPARC) for the project “Fire Safety in Underground Tunnels” (Project Code: P920). The Ph.D. study of the first author is supported by a fellowship from the UQ-IIT Delhi Research Academy (UQIDRA), a joint Ph.D. program of the Indian Institute of Technology (IIT) Delhi, India, and the University of Queensland (UQ), Australia. However, the opinions expressed in the paper are exclusively those of the authors and not necessarily those of the funding agencies.
References
Anand, N., D. P. Thanaraj, D. Andrushia, É. E. Lublóy, T. Kiran, B. Kanagaraj, and V. Kodur. 2023. “Microstructure investigation, strength assessment, and thermal modelling of concrete exposed to different heating cooling regimes.” J. Therm. Anal. Calorim. 148 (9): 3221–3247. https://doi.org/10.1007/s10973-023-11998-5.
Babanajad, S. K., A. H. Gandomi, and A. H. Alavi. 2017. “New prediction models for concrete ultimate strength under true-triaxial stress states: An evolutionary approach.” Adv. Eng. Software 110 (Jun): 55–68. https://doi.org/10.1016/j.advengsoft.2017.03.011.
Baheti, A., D. Lange, and V. Matsagar. 2021. “Multi-hazard nonlinear analysis of RC structures under earthquake and fire in its current state.” In Proc., 12th Asia–Oceania Symp. on Fire Science and Technology (AOSFST). St. Lucia, QLD, Australia: Univ. of Queensland.
BIS (Bureau of Indian Standards). 2000. Plain and reinforced concrete—Code of practice. IS 456. New Delhi, India: BIS.
CEN (European Committee for Standardization). 2023. Eurocode 2: Design of concrete structures—Part 1-2: General rules—Structural fire design. EN 1992-1-2. Brussels, Belgium: CEN.
CEN (European Committee for Standardization). 2024. Eurocode 1: Actions on structures—Part 1-2: General actions—Actions on structures exposed to fire. EN 1991-1-2. Brussels, Belgium: CEN.
Chaudhary, R. K., T. Roy, and V. Matsagar. 2020. “Member and structural fragility of reinforced concrete structure under fire.” J. Struct. Fire Eng. 11 (4): 409–435. https://doi.org/10.1108/JSFE-02-2019-0015.
Chinthapalli, H. K., and A. Agarwal. 2020. “Effect of confining reinforcement on fire behavior of reinforced concrete columns: Experimental and numerical study.” J. Struct. Eng. 146 (6): 04020084. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002617.
Chinthapalli, H. K., S. Sharma, and A. Agarwal. 2022. “Fire behavior and modeling of short RC columns in pure axial compression: Role of volume, configuration, and spacing of lateral reinforcement.” J. Struct. Eng. 148 (1): 04021245. https://doi.org/10.1061/(ASCE)ST.1943-541X.0003224.
Cross, H. 1932. “Analysis of continuous frames by distributing fixed-end moments.” Trans. Am. Soc. Civ. Eng. 96 (1): 1–10. https://doi.org/10.1061/TACEAT.0004333.
Dai, X., S. Welch, and A. Usmani. 2017. “A critical review of ‘travelling fire.’ scenarios for performance-based structural engineering.” Fire Saf. J. 91 (Mar): 568–578. https://doi.org/10.1016/j.firesaf.2017.04.001.
Frank, I. E., and R. Todeschini. 1994. The data analysis handbook. Amsterdam, Netherlands: Elsevier.
Franssen, J.-M., and T. Gernay. 2017. “Modeling structures in fire with SAFIR: Theoretical background and capabilities.” J. Struct. Fire Eng. 8 (3): 300–323. https://doi.org/10.1108/JSFE-07-2016-0010.
Gandomi, A. H., S. K. Babanajad, A. H. Alavi, and Y. Farnam. 2012. “Novel approach to strength modeling of concrete under triaxial compression.” J. Mater. Civ. Eng. 24 (9): 1132–1143. https://doi.org/10.1061/(ASCE)MT.1943-5533.0000494.
Gandomi, A. H., and D. A. Roke. 2015. “Assessment of artificial neural network and genetic programming as predictive tools.” Adv. Eng. Software 88 (Oct): 63–72. https://doi.org/10.1016/j.advengsoft.2015.05.007.
Gernay, T. 2019. “Fire resistance and burnout resistance of reinforced concrete columns.” Fire Saf. J. 104 (Mar): 67–78. https://doi.org/10.1016/j.firesaf.2019.01.007.
Gernay, T., J.-M. Franssen, F. Robert, R. McNamee, R. Felicetti, P. Bamonte, S. Brunkhorst, S. Mohaine, and J. Zehfuß. 2022. “Experimental investigation of structural failure during the cooling phase of a fire: Concrete columns.” Fire Saf. J. 134 (Jun): 103691. https://doi.org/10.1016/j.firesaf.2022.103691.
Gillie, M., A. S. Usmani, and J. M. Rotter. 2001. “A structural analysis of the first Cardington test.” J. Constr. Steel Res. 57 (6): 581–601. https://doi.org/10.1016/S0143-974X(01)00004-9.
Goel, M. D., and V. A. Matsagar. 2014. “Blast-resistant design of structures.” Pract. Period. Struct. Des. Constr. 19 (2): 04014007. https://doi.org/10.1061/(ASCE)SC.1943-5576.0000188.
Gupta, V. 2021. “Open-plan compartment fire dynamics.” Ph.D. thesis, School of Civil Engineering, Univ. of Queensland.
Gupta, V., J. P. Hidalgo, D. Lange, A. Cowlard, C. Abecassis-Empis, and J. L. Torero. 2021. “A review and analysis of the thermal exposure in large compartment fire experiments.” Int. J. High-Rise Build. 10 (4): 345–364. https://doi.org/10.21022/IJHRB.2021.10.4.345.
Heidari, M., P. Kotsovinos, and G. Rein. 2020. “Flame extension and the near field under the ceiling for travelling fires inside large compartments.” Fire Mater. 44 (3): 423–436. https://doi.org/10.1002/fam.2773.
Jamieson, P., J. Porter, and D. Wilson. 1991. “A test of the computer simulation model ARCWHEAT1 on wheat crops grown in New Zealand.” Field Crops Res. 27 (4): 337–350. https://doi.org/10.1016/0378-4290(91)90040-3.
Jiang, J., L. Jiang, P. Kotsovinos, J. Zhang, A. Usmani, F. McKenna, and G.-Q. Li. 2015. “OpenSees software architecture for the analysis of structures in fire.” J. Comput. Civ. Eng. 29 (1): 04014030. https://doi.org/10.1061/(ASCE)CP.1943-5487.0000305.
Jiang, L. 2016. “Command manual for OpenSees thermal.” Accessed April 16, 2024. https://openseesforfire.github.io/Subpages/commands.html.
Jiang, L., M. A. Orabi, J. Jiang, and A. Usmani. 2021. “Modelling concrete slabs subjected to fires using nonlinear layered shell elements and concrete damage-plasticity material.” Eng. Struct. 234 (Jun): 111977. https://doi.org/10.1016/j.engstruct.2021.111977.
Khan, A. A., M. A. Khan, R. V. V. Domada, X. Huang, A. Usmani, S. Bakhtiyari, M. J. Ashtiani, S. Garivani, and A. A. Aghakouchak. 2023. “Fire modelling framework for investigating tall building fire: A case study of the Plasco Building.” Case Stud. Therm. Eng. 45 (Jun): 103018. https://doi.org/10.1016/j.csite.2023.103018.
Khan, A. A., M. A. Khan, K. Leung, X. Huang, M. Luo, and A. Usmani. 2022a. “A review of critical fire event library for buildings and safety framework for smart firefighting.” Int. J. Disaster Risk Reduct. 83 (Dec): 103412. https://doi.org/10.1016/j.ijdrr.2022.103412.
Khan, M. A., A. A. Khan, A. S. Usmani, and X. Huang. 2022b. “Can fire cause the collapse of Plasco Building: A numerical investigation.” Fire Mater. 46 (3): 560–575. https://doi.org/10.1002/fam.3003.
Kodakkal, A., P. Jagtap, and V. Matsagar. 2020. “Stochastic response of primary–secondary coupled systems under uncertain ground excitation using generalized polynomial chaos method.” In Handbook of probabilistic models, 383–435. Amsterdam, Netherlands: Elsevier.
Kodakkal, A., S. K. Saha, K. Sepahvand, V. A. Matsagar, F. Duddeck, and S. Marburg. 2019. “Uncertainties in dynamic response of buildings with non-linear base-isolators.” Eng. Struct. 197 (Dec): 109423. https://doi.org/10.1016/j.engstruct.2019.109423.
Kodur, V., and S. Banerji. 2021. “Modeling the fire-induced spalling in concrete structures incorporating hydro-thermo-mechanical stresses.” Cem. Concr. Compos. 117 (Mar): 103902. https://doi.org/10.1016/j.cemconcomp.2020.103902.
Kodur, V., F.-P. Cheng, T.-C. Wang, and M. Sultan. 2003. “Effect of strength and fiber reinforcement on fire resistance of high-strength concrete columns.” J. Struct. Eng. 129 (2): 253–259. https://doi.org/10.1061/(ASCE)0733-9445(2003)129:2(253).
Kodur, V., S. Venkatachari, P. Bhatt, V. A. Matsagar, and S. B. Singh. 2023. “Fire resistance evaluation of concrete beams and slabs incorporating natural fiber-reinforced polymers.” Polymers 15 (3): 755. https://doi.org/10.3390/polym15030755.
Law, A., and L. Bisby. 2020. “The rise and rise of fire resistance.” Fire Saf. J. 116 (Dec): 103188. https://doi.org/10.1016/j.firesaf.2020.103188.
Ma, Q., R. Guo, Z. Zhao, Z. Lin, and K. He. 2015. “Mechanical properties of concrete at high temperature—A review.” Constr. Build. Mater. 93 (Mar): 371–383. https://doi.org/10.1016/j.conbuildmat.2015.05.131.
Newmark, N. M. 1959. “A method of computation for structural dynamics.” J. Eng. Mech. Div. 85 (3): 67–94. https://doi.org/10.1061/JMCEA3.0000098.
Ni, S., and T. Gernay. 2020. “Predicting residual deformations in a reinforced concrete building structure after a fire event.” Eng. Struct. 202 (Dec): 109853. https://doi.org/10.1016/j.engstruct.2019.109853.
Orabi, M. A., J. Qiu, L. Jiang, and A. Usmani. 2023. “Modelling concrete slabs subjected to localised fire action with OpenSees.” J. Struct. Fire Eng. 14 (1): 128–145. https://doi.org/10.1108/JSFE-05-2021-0033.
Pearson, K. 1909. “Determination of the coefficient of correlation.” Science 30 (757): 23–25. https://doi.org/10.1126/science.30.757.23.
Pillai, S. U., and D. Menon. 2021. Reinforced concrete design. New York: McGraw Hill.
Rackauskaite, E., C. Hamel, A. Law, and G. Rein. 2015. “Improved formulation of travelling fires and application to concrete and steel structures.” Structures 3 (Dec): 250–260. https://doi.org/10.1016/j.istruc.2015.06.001.
Rackauskaite, E., P. Kotsovinos, A. Jeffers, and G. Rein. 2019. “Computational analysis of thermal and structural failure criteria of a multi-storey steel frame exposed to fire.” Eng. Struct. 180 (Mar): 524–543. https://doi.org/10.1016/j.engstruct.2018.11.026.
Rajnish, K., A. Kodakkal, D. H. Zelleke, R. E. Meethal, V. A. Matsagar, K.-U. Bletzinger, and R. Wüchner. 2023. “Machine learning driven damper for response control in vehicle–bridge interaction systems.” Proc. Inst. Civ. Eng. Bridge Eng. 1–23. https://doi.org/10.1680/jbren.21.00090.
Roy, T., and V. Matsagar. 2020. “Fire fragility of reinforced concrete panels under transverse out-of-plane and compressive in-plane loads.” Fire Saf. J. 113 (Jun): 102976. https://doi.org/10.1016/j.firesaf.2020.102976.
Smith, G. N. 1986. “Probability and statistics in civil engineering: An introduction.” In Collins professional and technical books, 244. London: Collins.
Song, Y., C. Fu, S. Liang, I. Topilin, and X. Song. 2023. “Residual shear capacity of post-fire RC beams under indirect loading.” Buildings 13 (4): 969. https://doi.org/10.3390/buildings13040969.
Stern-Gottfried, J., and G. Rein. 2012. “Travelling fires for structural design—Part I: Literature review.” Fire Saf. J. 54 (Jun): 74–85. https://doi.org/10.1016/j.firesaf.2012.06.003.
Thompson, M. L. 1978. “Selection of variables in multiple regression: Part I. A review and evaluation.” In International statistical review/Revue Internationale de Statistique, 1–19. Hague, Netherlands: International Statistical Institute.
Timoshenko, S. 1983. History of strength of materials: With a brief account of the history of theory of elasticity and theory of structures. Chelmsford, MA: Courier Corporation.
Usmani, A. S., and N. Cameron. 2004. “Limit capacity of laterally restrained reinforced concrete floor slabs in fire.” Cem. Concr. Compos. 26 (2): 127–140. https://doi.org/10.1016/S0958-9465(03)00090-8.
Usmani, A. S., Y. Chung, and J. L. Torero. 2003. “How did the WTC towers collapse: A new theory.” Fire Saf. J. 38 (6): 501–533. https://doi.org/10.1016/S0379-7112(03)00069-9.
Usmani, A. S., J. M. Rotter, S. Lamont, A. Sanad, and M. Gillie. 2001. “Fundamental principles of structural behaviour under thermal effects.” Fire Saf. J. 36 (8): 721–744. https://doi.org/10.1016/S0379-7112(01)00037-6.
Walkup, S., E. Musselman, and S. Gross. 2019. “Effect of service load levels on long-term deflection multiplier.” ACI Struct. J. 116 (2): 89–100. https://doi.org/10.14359/51711137.
Wilson, W., F. E. Richart, and C. Weiss. 1918. Analysis of statically indeterminate structures by the slope deflection method Bulletin, 108. Urbana, IL: Univ. of Illinois.
Zelleke, D. H., and V. A. Matsagar. 2022. “Multi-hazard stochastic response assessment of base-isolated buildings.” Struct. Infrastruct. Eng. 20 (3): 301–325. https://doi.org/10.1080/15732479.2022.2088812.
Zhang, H. Y., Q. Y. Li, V. Kodur, and H. R. Lv. 2021. “Effect of cracking and residual deformation on behavior of concrete beams with different scales under fire exposure.” Eng. Struct. 245 (Mar): 112886. https://doi.org/10.1016/j.engstruct.2021.112886.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 150 • Issue 10 • October 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Nov 16, 2023
Accepted: May 15, 2024
Published online: Aug 5, 2024
Published in print: Oct 1, 2024
Discussion open until: Jan 5, 2025
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.