# Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Establishment of the Semi-Analytical Approach

#### 2.1. Boltzmann Transformation

#### 2.2. Analytical Solution in a Small Neighborhood

#### 2.3. Approximate Solution in the Whole Domain

- Substitution of the temperature-dependent expressions of the mass density, specific heat capacity, and thermal conductivity, i.e., ${\rho}_{T}={\rho}_{T}\left(T\right)$, ${C}_{T}={C}_{T}\left(T\right)$, and ${K}_{T}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{K}_{T}\left(T\right)$, into the approximate solution of the Boltzmann variable, i.e., Equation (40);
- Determination of two random temperature points within the boundary scenario, i.e., ${T}_{ref}<({T}_{1}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{T}_{2})<{T}_{hot}$, where the approximate solution of the Boltzmann variable (40) accurately satisfies the governing Equation (14), i.e., the last two formulas of Equation (41);
- Calculation of the temperature field at each time instant by recalling the definition of the Boltzmann variable, i.e., Equation (10).

- Substitution of the temperature-dependent expressions of the mass density, specific heat capacity, and thermal conductivity, i.e., ${\rho}_{T}={\rho}_{T}\left(T\right)$, ${C}_{T}={C}_{T}\left(T\right)$, and ${K}_{T}={K}_{T}\left(T\right)$, as well as the time-dependent boundary condition ${T}_{hot}={T}_{hot}\left(t\right)$, into the approximate solution of the x-coordinate, i.e., Equation (44);
- Determination of two random temperature points within the boundary scenario at two random time instants, i.e., ${T}_{ref}<{T}_{1}<{T}_{hot}\left({t}_{1}\right)$ and ${T}_{ref}<{T}_{2}<{T}_{hot}\left({t}_{2}\right)$, where the approximate solution of the x-coordinate (44) accurately satisfies the governing Equation (45), i.e., the last two formulas of Equation (46);
- Calculation of the temperature field at each time instant by recalling the solution of the x-coordinate, i.e., Equation (44).

## 3. Numerical Study for Validation

#### 3.1. Example I: Normal Concrete Plate, Subjected to a Moderate Fire Below 100 °C

#### 3.2. Example II: Self-Consolidating Concrete Plate, Subjected to a Fierce Fire Reaching 400 °C

## 4. Application to a Fire Test

#### 4.1. Experimental Results of a Fire Test of a Subway Station

#### 4.2. Prediction of the Temperature Field and Discussion

## 5. Discussion

#### 5.1. Comparison with the Results Obtained by Linear Heat Conduction

#### 5.2. Discussion on the Nonlinearity Related to the Thermophysical Properties

- S-II: The thermal conductivity was considered to be temperature-dependent, following Equation (51), whereas both the mass density and the specific heat capacity were considered to be constant, with their product equal to the value of Equation (52) by setting $T=20{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$, i.e., ${\rho}_{T}\phantom{\rule{0.166667em}{0ex}}{C}_{T}=2.42\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}\mathrm{J}/({\mathrm{m}}^{3}\xb7{}^{\circ}\mathrm{C})$.
- S-III: All three thermophysical properties were considered to be temperature-independent, reading as the values of Equations (51) and (52) by setting $T=20{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$, i.e., ${K}_{T}=3.03\phantom{\rule{0.166667em}{0ex}}\mathrm{W}/(\mathrm{m}\xb7{}^{\circ}\mathrm{C})$ and ${\rho}_{T}\phantom{\rule{0.166667em}{0ex}}{C}_{T}=2.42\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}\mathrm{J}/({\mathrm{m}}^{3}\xb7{}^{\circ}\mathrm{C})$.

## 6. Conclusions

- The nonlinearity, related to the temperature-dependent mass density, specific heat capacity, and thermal conductivity of concrete, was taken into consideration. The semi-analytical approach was validated by comparison with two independent Finite Element simulations of heated plates, consisting of normal and self-consolidating concrete, respectively. Absolute values of the difference of the semi-analytical solution, with respect to the numerical results, are smaller than $5\%$ in both exemplary studies.
- By referring to the definition of the Boltzmann variable, the semi-analytical solution was further extended to consideration of a time-dependent thermal boundary condition, which is commonly encountered during the service life of concrete structures. The extended solution was compared with the experimental measurements of a fire test of a scaled concrete subway station. Satisfactory agreement was achieved.

- Because of the rather small thermal conductivity, concrete generally exhibits good thermal insulation performance. Therefore, the established semi-analytical approach of the nonlinear heat conduction for a semi-infinite plate can still be widely used in real scenarios, such as the presented examples of the concrete plates and slabs in a subway station, subjected to either moderate or fierce fire loading.
- The thermal insulation property also leads to significant temperature gradients in the vicinity of the heated surface and, therefore, strong nonlinearity of the thermal eigenstrain, as a product of the temperature difference and the thermal expansion coefficient of concrete. This nonlinearity governs the resulting thermal stresses of concrete structures [4,30]. In the case of a fire test of the scaled model of a subway station, tensile cracking can occur at the inaccessible exterior surface, which is a threat to the long-term durability.

- It is recommended to position enough thermocouples close to the fire-exposed surface. With the knowledge of the temperature evolution of the heated surface, the established semi-analytical approach provides access to the evolution of the temperature fields within concrete structures, serving as the basis for the following thermomechanical analysis and damage evaluation.
- It is recommended to carry out careful inspection of the thermally-loaded concrete structures. The strong nonlinearity of the temperature field is very likely to result in significant thermal stresses and potential thermal cracking.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Jeong, J.H.; Zollinger, D.G. Environmental effects on the behavior of jointed plain concrete pavements. J. Transp. Eng.
**2005**, 131, 140–148. [Google Scholar] [CrossRef] - Wang, H.; Yuan, Y.; Mang, H.A.; Ai, Q.; Huang, X.; Pichler, B.L. Thermal stresses in rectangular concrete beams, resulting from constraints at microstructure, cross-section, and supports. Eur. J. Mech.-A/Solids
**2022**, 93, 104495. [Google Scholar] [CrossRef] - Kunz, M.; Sander, J.; Kottmeier, C. Recent trends of thunderstorm and hailstorm frequency and their relation to atmospheric characteristics in southwest Germany. Int. J. Climatol. A J. R. Meteorol. Soc.
**2009**, 29, 2283–2297. [Google Scholar] [CrossRef][Green Version] - Wang, H.; Höller, R.; Aminbaghai, M.; Hellmich, C.; Yuan, Y.; Mang, H.A.; Pichler, B.L. Concrete pavements subjected to hail showers: A semi-analytical thermoelastic multiscale analysis. Eng. Struct.
**2019**, 200, 109677. [Google Scholar] [CrossRef] - Bažant, Z.P.; Kaplan, M.F. Concrete at High Temperatures: Material Properties and Mathematical Models; Longman: Burnt Mill, UK, 1996. [Google Scholar]
- Khoury, G.A. Effect of fire on concrete and concrete structures. Prog. Struct. Eng. Mater.
**2000**, 2, 429–447. [Google Scholar] [CrossRef] - Wang, H.; Binder, E.; Mang, H.; Yuan, Y.; Pichler, B. Multiscale structural analysis inspired by exceptional load cases concerning the immersed tunnel of the Hong Kong-Zhuhai-Macao Bridge. Undergr. Space
**2018**, 3, 252–267. [Google Scholar] [CrossRef] - Wang, H.; Chen, X.; Yang, S.; Caggiano, A.; Ai, Q.; Koenders, E.; Yuan, Y. A dehydration kinetic model of calcium silicate hydrates at high temperature. Struct. Concr. 2022. [CrossRef]
- CEN. Eurocode 2: Design of Concrete Structures-Part 1–2: General Rules–Structural Fire Design; European Committee for Standardization: Brussels, Belgium, 2004. [Google Scholar]
- DeJong, M.J.; Ulm, F.J. The nanogranular behavior of C-S-H at elevated temperatures (up to 700 °C). Cem. Concr. Res.
**2007**, 37, 1–12. [Google Scholar] [CrossRef] - Fu, Y.; Wong, Y.; Tang, C.; Poon, C.S. Thermal induced stress and associated cracking in cement-based composite at elevated temperatures–Part I: Thermal cracking around single inclusion. Cem. Concr. Compos.
**2004**, 26, 99–111. [Google Scholar] [CrossRef] - Wang, H.; Mang, H.; Yuan, Y.; Pichler, B.L. Multiscale thermoelastic analysis of the thermal expansion coefficient and of microscopic thermal stresses of mature concrete. Materials
**2019**, 12, 2689. [Google Scholar] [CrossRef][Green Version] - Asadi, I.; Shafigh, P.; Hassan, Z.F.B.A.; Mahyuddin, N.B. Thermal conductivity of concrete–A review. J. Build. Eng.
**2018**, 20, 81–93. [Google Scholar] [CrossRef] - Malik, M.; Bhattacharyya, S.; Barai, S.V. Thermal and mechanical properties of concrete and its constituents at elevated temperatures: A review. Constr. Build. Mater.
**2021**, 270, 121398. [Google Scholar] [CrossRef] - Nguyen, T.D.; Pham, D.T.; Vu, M.N. Thermo-mechanically-induced thermal conductivity change and its effect on the behaviour of concrete. Constr. Build. Mater.
**2019**, 198, 98–105. [Google Scholar] [CrossRef] - Shen, L.; Li, W.; Zhou, X.; Feng, J.; Di Luzio, G.; Ren, Q.; Cusatis, G. Multiphysics Lattice Discrete Particle Model for the simulation of concrete thermal spalling. Cem. Concr. Compos.
**2020**, 106, 103457. [Google Scholar] [CrossRef][Green Version] - Gawin, D.; Pesavento, F.; Schrefler, B.A. What physical phenomena can be neglected when modelling concrete at high temperature? A comparative study. Part 2: Comparison between models. Int. J. Solids Struct.
**2011**, 48, 1945–1961. [Google Scholar] [CrossRef] - Kakaç, S.; Yener, Y.; Naveira-Cotta, C.P. Heat Conduction; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Zehfuß, J.; Robert, F.; Spille, J.; Razafinjato, R.N. Evaluation of Eurocode 2 approaches for thermal conductivity of concrete in case of fire. Civ. Eng. Des.
**2020**, 2, 58–71. [Google Scholar] [CrossRef] - Kodur, V.K.; Dwaikat, M.; Dwaikat, M. High-temperature properties of concrete for fire resistance modeling of structures. ACI Mater. J.
**2008**, 105, 517. [Google Scholar] - Kodur, V. Properties of concrete at elevated temperatures. Int. Sch. Res. Not.
**2014**, 2014, 468510. [Google Scholar] [CrossRef][Green Version] - Pimienta, P.; McNamee, R.J.; Mindeguia, J.C. Physical Properties and Behaviour of High-Performance Concrete at High Temperature. State-of-the-Art Report of the RILEM Technical Committee HPB-227; Springer: Cham, Switzerland, 2019. [Google Scholar]
- Wilson, E.L.; Nickell, R.E. Application of the finite element method to heat conduction analysis. Nucl. Eng. Des.
**1966**, 4, 276–286. [Google Scholar] [CrossRef] - Aguirre-Ramirez, G.; Oden, J. Finite element technique applied to heat conduction in solids with temperature dependent thermal conductivity. Int. J. Numer. Methods Eng.
**1973**, 7, 345–355. [Google Scholar] [CrossRef] - Nishizawa, T.; Ozeki, T.; Katoh, K.; Matsui, K. Finite element model analysis of thermal stresses of thick airport concrete pavement slabs. Transp. Res. Rec.
**2009**, 2095, 3–12. [Google Scholar] [CrossRef] - Li, W.; Wu, Y.; Fu, H.; Zhang, J. Long-term continuous in-situ monitoring of tunnel lining surface temperature in cold region and its application. Int. J. Heat Technol.
**2015**, 33, 39–44. [Google Scholar] [CrossRef] - Abid, S.R.; Tayşi, N.; Özakça, M.; Xue, J.; Briseghella, B. Finite element thermo-mechanical analysis of concrete box-girders. In Proceedings of the Structures; Elsevier: Amsterdam, The Netherlands, 2021; Volume 33, pp. 2424–2444. [Google Scholar]
- Louhghalam, A.; Ulm, F.J. Risk of pavement fracture due to eigenstresses at early ages and beyond. J. Eng. Mech.
**2016**, 142, 04016105. [Google Scholar] [CrossRef] - Mukin, D.; Valdaytseva, E.; Turichin, G. Analytical solution of the non-stationary heat conduction problem in thin-walled products during the additive manufacturing process. Materials
**2021**, 14, 4049. [Google Scholar] [CrossRef] [PubMed] - Sorgner, M.; Flores, R.D.; Wang, H.; Hellmich, C.; Pichler, B.L. Hindered thermal warping triggers tensile cracking in the cores of compressed columns of a fire-loaded tunnel segment structure: Efficiency and accuracy of beam theory prediction, compared to FEM. Appl. Eng. Sci.
**2023**, 14, 100128. [Google Scholar] [CrossRef] - Kim, S. A simple direct estimation of temperature-dependent thermal conductivity with Kirchhoff transformation. Int. Commun. Heat Mass Transf.
**2001**, 28, 537–544. [Google Scholar] [CrossRef] - Arslanturk, C. A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Int. Commun. Heat Mass Transf.
**2005**, 32, 831–841. [Google Scholar] [CrossRef] - Joneidi, A.; Ganji, D.; Babaelahi, M. Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity. Int. Commun. Heat Mass Transf.
**2009**, 36, 757–762. [Google Scholar] [CrossRef] - Rajabi, A.; Ganji, D.; Taherian, H. Application of homotopy perturbation method in nonlinear heat conduction and convection equations. Phys. Lett. A
**2007**, 360, 570–573. [Google Scholar] [CrossRef] - Sajid, M.; Hayat, T. Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. Nonlinear Anal. Real World Appl.
**2008**, 9, 2296–2301. [Google Scholar] [CrossRef] - Wazwaz, A.M. The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations. Appl. Math. Comput.
**2005**, 169, 321–338. [Google Scholar] [CrossRef] - Barna, I.F.; Mátyás, L. General self-similar solutions of diffusion equation and related constructions. arXiv
**2021**, arXiv:2104.09128. [Google Scholar] - Chen, X.; He, D.; Yang, G.; Yuan, Y.; Dai, Y. Approximate analytical solution for Richards’ equation with finite constant water head Dirichlet boundary conditions. Comput. Appl. Math.
**2021**, 40, 1–31. [Google Scholar] [CrossRef] - Larsson, H. On the use of Boltzmann’s transformation to solve diffusion problems. Calphad
**2021**, 73, 102261. [Google Scholar] [CrossRef] - Lu, L.; Qiu, J.; Yuan, Y.; Tao, J.; Yu, H.; Wang, H.; Mang, H. Large-scale test as the basis of investigating the fire-resistance of underground RC substructures. Eng. Struct.
**2019**, 178, 12–23. [Google Scholar] [CrossRef] - Díaz, R.; Wang, H.; Mang, H.; Yuan, Y.; Pichler, B. Numerical analysis of a moderate fire inside a segment of a subway station. Appl. Sci.
**2018**, 8, 2116. [Google Scholar] [CrossRef][Green Version] - Boltzmann, L. Zur Integration der Diffusionsgleichung bei variabeln Diffusionscoefficienten. Ann. Der Phys.
**1894**, 289, 959–964. [Google Scholar] [CrossRef][Green Version] - Strang, G.; Herman, E. Calculus Volume 1; OpenStax: Houston, TX, USA, 2016; Volume 3. [Google Scholar]
- Farouki, R.T. The Bernstein polynomial basis: A centennial retrospective. Comput. Aided Geom. Des.
**2012**, 29, 379–419. [Google Scholar] [CrossRef] - COMSOL AB. COMSOL Multiphysics V. 5.6; COMSOL AB: Stockholm, Sweden, 2020; Available online: www.comsol.com (accessed on 19 February 2023).
- Maplesoft. Maple 2019.0; Maplesoft: Waterloo, ON, Canada, 2019; Available online: www.maplesoft.com (accessed on 19 February 2023).
- Khaliq, W.; Kodur, V. Thermal and mechanical properties of fiber reinforced high performance self-consolidating concrete at elevated temperatures. Cem. Concr. Res.
**2011**, 41, 1112–1122. [Google Scholar] [CrossRef] - Zhao, J.; Zheng, J.j.; Peng, G.f.; van Breugel, K. A meso-level investigation into the explosive spalling mechanism of high-performance concrete under fire exposure. Cem. Concr. Res.
**2014**, 65, 64–75. [Google Scholar] [CrossRef] - Ulm, F.J.; Coussy, O.; Bažant, Z.P. The “Chunnel” fire. I: Chemoplastic softening in rapidly heated concrete. J. Eng. Mech.
**1999**, 125, 272–282. [Google Scholar] [CrossRef] - GB-T 228-2002; Metallic Materials–Tensile Testing at Ambient Temperature. General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China: Beijing, China, 2002.
- GB 50010-2010; Code for Design of Concrete Structures. Ministry of Housing and Urban-Rural Development of the People’s Republic of China: Beijing, China, 2010.
- Lie, T.; Erwin, R. Method to calculate the fire resistance of reinforced concrete columns with rectangular cross section. ACI Struct. J.
**1993**, 90, 52–60. [Google Scholar] - Kodur, V.; Dwaikat, M. A numerical model for predicting the fire resistance of reinforced concrete beams. Cem. Concr. Compos.
**2008**, 30, 431–443. [Google Scholar] [CrossRef] - Sorgner, M.; Flores, R.D.; Pichler, B.; Wang, H. Engineering mechanics analysis of a moderate fire inside a segment of a subway station. In Computational Modelling of Concrete and Concrete Structures; CRC Press: Boca Raton, FL, USA, 2022; pp. 555–563. [Google Scholar]
- Hahn, D.W.; Özisik, M.N. Heat Conduction; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
- Osama, M.E. Solution of Problems in Heat Transfer: Transient Conduction or Unsteady Conduction; LAP LAMBERT Academic Publishing: Saarbrücken, Germany, 2017. [Google Scholar]
- Ulm, F.J.; Acker, P.; Lévy, M. The “Chunnel” fire. II: Analysis of concrete damage. J. Eng. Mech.
**1999**, 125, 283–289. [Google Scholar] [CrossRef]

**Figure 2.**Numerical simulations of the transient nonlinear heat conduction along the length direction of the plate: (

**a**) the Finite Element mesh, and the quantified temperature field at time instants of (

**b**) $10\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, (

**c**) $60\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, and (

**d**) $180\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$ after the thermal shock.

**Figure 3.**Comparison between the temperature field quantified from the semi-analytical solution and from the Finite Element simulation: (

**a**) as a function of the Boltzmann variable $\varphi =x/\sqrt{t}$, (

**b**) along the length of the plate at the time instants of $10\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, $60\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, and $180\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$ after the sudden temperature increase, considering a temperature-dependent thermal conductivity while a constant mass density and specific heat capacity.

**Figure 4.**Numerical simulations of the transient nonlinear heat conduction along the length direction of the plate: (

**a**) the Finite Element mesh, and the quantified temperature field at time instants of (

**b**) $10\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, (

**c**) $60\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, and (

**d**) $180\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$ after the thermal shock.

**Figure 5.**Comparison between the temperature field quantified from the semi-analytical solution and from the Finite Element simulation: (

**a**) as a function of the Boltzmann variable $\varphi =x/\sqrt{t}$, (

**b**) along the length of the plate at the time instants of $10\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, $60\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, and $180\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$ after the sudden temperature increase, considering a temperature-dependent thermal conductivity, mass density, and specific heat capacity.

**Figure 6.**Geometric dimensions of the scaled model of the station hall of a subway station in the fire test: (

**a**) front view, (

**b**) left view, and (

**c**) 3D view (unit: mm), after [40].

**Figure 7.**The fire test: (

**a**) the mechanical loading and supports (unit: mm) and (

**b**) the testing setup, after [40].

**Figure 8.**Temperature history of the air in the furnace, applied to the scaled model, after [40].

**Figure 9.**Experimental measured temperature evolution along the thickness at the right span of the top slab, after [40].

**Figure 10.**Comparison of the experimentally measured [40] and model predicted temperature evolution during the fire test.

**Figure 11.**Comparison of the temperature fields obtained by Sorgner et al. [54] for linear heat conduction and by the semi-analytical approach for heat conduction.

**Figure 12.**(

**a**) The temperature distribution and (

**b**) the corresponding thermal diffusivity along the length of the plate at the time instant of $180\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$, considering temperature-dependent, partly temperature-dependent, and temperature-independent thermophysical properties of concrete.

Loading | ${\mathit{P}}_{1}$ | ${\mathit{P}}_{2}$ | ${\mathit{P}}_{3}$ |
---|---|---|---|

magnitude [$\mathrm{kN}$] | 192.0 | 151.2 | 120.0 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, H.; Chen, X.; Koenders, E.; Dai, Y.; Huang, X.; Ai, Q.; Yuan, Y. Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach. *Entropy* **2023**, *25*, 583.
https://doi.org/10.3390/e25040583

**AMA Style**

Wang H, Chen X, Koenders E, Dai Y, Huang X, Ai Q, Yuan Y. Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach. *Entropy*. 2023; 25(4):583.
https://doi.org/10.3390/e25040583

**Chicago/Turabian Style**

Wang, Hui, Xi Chen, Eduardus Koenders, Ying Dai, Xingchun Huang, Qing Ai, and Yong Yuan. 2023. "Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach" *Entropy* 25, no. 4: 583.
https://doi.org/10.3390/e25040583