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Materials and Structures

, 51:87 | Cite as

A thermo-mechanical model for SFRC beams or slabs at elevated temperatures

  • Peter Heek
  • Jasmin TkoczEmail author
  • Peter Mark
Original Article
  • 418 Downloads

Abstract

The bearing capacity of steel fibre reinforced concrete (SFRC) at elevated temperatures is the subject of significant ongoing research, as the effect of steel fibres on concrete performance at higher temperatures is poorly understood. On one hand, steel fibres increase the average thermal conductivity of the concrete cross section and lead to greater heating within a concrete structural member during fire exposure, and on the other, fibres reduce crack widths and prevent excessive spalling. The former effect negatively impacts SFRC performance at high temperature, whereas the latter effect protects the inner structure from direct fire exposure. Additionally, the decreasing strength of steel at higher temperatures can result in the sudden failure of fibre or traditionally reinforced concrete. Within this contribution, a coupled thermo-mechanical model is developed in order to investigate the influence of steel fibres on the thermal loading of concrete. The effect of fibres on heat transmission within concrete, the length of time for which concrete can sustain thermal loads, and on the bending stiffness of reinforced concrete beams or slabs is investigated. The heat transfer process is modelled using Fourier’s partial differential equation of transient heat conduction. A modified plastic hinge model and moment–curvature relations are used to describe stress-dependent deformations. Thus, two alternative approaches are used to adequately track the localisation of damage for single cracks and for distributed and multiple cracking. Thermo-mechanical coupling is achieved by means of temperature-dependent stress–strain relations. These are derived for SFRC based on experimental data from the literature. Experiments are performed in which concrete slabs reinforced with variable amounts of fibres and rebar are exposed to combined thermal and mechanical loadings. The results of these experiments are used to validate the proposed model. The measured and predicted results agree well and indicate that steel fibres have a positive effect on the fire resistance of structures, assuming additional rebar is provided to prevent crack localisation. Additionally, it is shown that temperature fields within concrete remain almost unaffected by variations in fibre content.

Keywords

Thermo-mechanical modelling Heat transfer Thermal effects Fire resistance duration Moment–curvature relation Plastic hinge model Fibre reinforced concrete Large-scale tests 

List of symbols

a

Thermal diffusivity

A

Cross-sectional area

cnom

Concrete cover

cp

Specific heat capacity

df

Fibre diameter

e, E

Spacing of loading, modulus of elasticity

fc, ft, f1f, f2f

Strength (compressive concrete strength, tensile concrete strength, and post-cracking tensile strengths of SFRC corresponding to predefined strains ε1 and ε2, respectively)

F

Force

g, i, j, k, n, r

Counter variables

I

Moment of inertia

ktf

Ratio of SFRC’s post-cracking tensile strength at elevated temperatures to that at normal temperature

l, lch, lf

Length, characteristic length, fibre length

m, M, \(\bar{M}\)

Mass, bending moment, virtual bending moment

q, Q

Heat flux density, heat flow volume

R

Fire resistance duration in minutes

si, li,k

Element sizes (element length in direction of the heat flow, contact length between two adjacent elements i and k perpendicular to the heat flow)

s

Width of a plastic hinge

SLS

Serviceability limit state

t, T

Time, absolute temperature

ULS

Ultimate limit state

Vf

Fibre volume content

w

Crack width

x, x, y ,z

Vector of a spatial location, spatial coordinates (Cartesian)

αK, αT

Fictive heat-transfer coefficient, thermal expansion coefficient

δ, δth, δσ

Deflections (overall, thermic and stress-dependent)

ε, εc,top, εc,bot, εs1

Strains (overall, concrete strains at the top or bottom of a section, rebar strain)

εf, εm

Emission coefficients (fumes and solid)

θ, θ0, θr

Radiation temperature (overall, initial), rotation angle

ϑ, ϑ0, ϑm, ∆ϑ, ϑID

Temperature (basic, initial, average, difference, equivalent)

κ

Curvature

λ

Thermal conductivity

ρ

Density

σ, σc, σt, σs1

Stress (general, concrete in compression and tension, in longitudinal rebar)

σSB

Stefan–Boltzmann constant

φ, φel, φpl

Rotation angle (overall, elastic and plastic portions)

Notes

Acknowledgements

The financial support provided by NV Bekaert SA for the presented experiments is gratefully acknowledged. Additionally, the authors would like to thank Prof. Catherina Thiele as well as Daniele Casucci, M.Sc. from TU Kaiserslautern, Germany for careful execution of all tests. The authors very much appreciate the support of Dr. M. A. Ahrens in elaborating the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    RILEM TC 162-TDF (2003) Test and design methods for steel fibre reinforced concrete—background and experiences. In: Schnütgen B, Vandevalle L (eds) Proceedings of 31 RILEM TC-162-TDF workshop, Bochum, GermanyGoogle Scholar
  2. 2.
    DAfStb (German Committee for Structural Concrete) (2015) DAfStb H614-2015 commentary on the DAfStb guideline “steel fibre reinforced concrete”, BerlinGoogle Scholar
  3. 3.
    Model Code 2010 (2013) fib model code 2010. Fédération Internationale du BétonGoogle Scholar
  4. 4.
    Gödde L, Strack M, Mark P (2010) Bauteile aus Stahlfaserbeton und stahlfaserverstärktem Stahlbeton—Hilfsmittel für Bemessung und Verformungsabschätzung nach DAfStb-Richtlinie Stahlfaserbeton. Beton- und Stahlbetonbau 105(2):78–91CrossRefGoogle Scholar
  5. 5.
    Heek P, Mark P (2016) Load-bearing capacities of SFRC elements accounting for tension stiffening with modified moment–curvature relations. In: ACI-SP 310 + fib bulletin no. 79: fibre reinforced concrete—from design to structural applications, pp 301–310Google Scholar
  6. 6.
    Dehn F, Herrmann A (2014) Steel fibre-reinforced concrete (SFRC) in fire—normative and pre-normative requirements and code-type regulations. In: Proceedings of 1st joint ACI-fib international work. Fibre reinforced concrete—from design to structural applications, Canada, pp 2–8Google Scholar
  7. 7.
    DIN EN 1992-1-2 (2010) Design of concrete structures—part 1–2: general rules—structural fire design, BerlinGoogle Scholar
  8. 8.
    Khaliq W, Kodur V (2011) Thermal and mechanical properties of fibre reinforced high performance self-consolidating concrete at elevated temperatures. Cem Concr Res 41:1112–1122CrossRefGoogle Scholar
  9. 9.
    Aslani F, Samali B (2014) Constitutive relationship for steel fibre reinforced concrete at elevated temperatures. Fire Technol 50:1249–1268CrossRefGoogle Scholar
  10. 10.
    Chen GM, He YH, Yang H, Chen JF, Guo YC (2014) Compressive behaviour of steel fibre reinforced recycled aggregate concrete after exposure to elevated temperatures. Constr Build Mater 71:1–15CrossRefGoogle Scholar
  11. 11.
    Chen B, Liu J (2004) Residual strength of hybrid-fibre-reinforced high-strength concrete after exposure to high temperatures. Cem Concr Res 34(6):1065–1069CrossRefGoogle Scholar
  12. 12.
    Colombo M, di Prisco M, Felicetti R (2010) Mechanical properties of steel fibre reinforced concrete exposed at high temperatures. Mater Struct 43:475–491CrossRefGoogle Scholar
  13. 13.
    Faiyadh FI, Al-Ausi MA (1989) Effect of elevated temperature in splitting tensile strength of fibre concrete. Int J Cem Compos Lightweight Concr 11(3):175–178CrossRefGoogle Scholar
  14. 14.
    Kim J, Lee G-P, Moon DY (2015) Evaluation of mechanical properties of steel-fibre-reinforced concrete exposed to high temperatures by double-punch test. Constr Build Mater 79:182–191CrossRefGoogle Scholar
  15. 15.
    Peng GF, Yang WW, Zhao J, Liu YF, Bian SH, Zhao LH (2006) Explosive spalling and residual mechanical properties of fibre-toughened high-performance concrete subjected to high temperatures. Cem Concr Res 36:723–727CrossRefGoogle Scholar
  16. 16.
    Suhaendi SL, Horiguchi T (2006) Effect of short fibres on residual permeability and mechanical properties of hybrid fibre reinforced high strength concrete after heat exposure. Cem Concr Res 36:1672–1678CrossRefGoogle Scholar
  17. 17.
    Diederichs U (1999) Hochtemperatur- und Brandverhalten von hochfestem Stahlfaserbeton. In: Teutsch M (ed) Betonbau – Forschung, Entwicklung und Anwendung, vol 142. Schriftenreihe des Instituts für Baustoffe, Massivbau und Brandschutz, Braunschweig, pp 67–76Google Scholar
  18. 18.
    Hertel C, Orgass M, Dehn F (2002) Brandverhalten von faserfreiem und faserverstärktem Beton. In: König G, Holschemacher K, Dehn F (eds) Faserbeton – Innovationen im Bauwesen. Bauwerk Verlag, pp 63–76Google Scholar
  19. 19.
    DIN EN 1991-1-2 (2010) Actions on structures—part 1–2: general actions—actions in structures exposed to fire, BerlinGoogle Scholar
  20. 20.
    Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Oxford University Press, OxfordzbMATHGoogle Scholar
  21. 21.
    Mannsfeld T (2011) Tragverhalten von Stahlbetonflächentragwerken unter Berücksichtigung der temperaturbedingten Nichtlinearitäten im Brandfall. Dissertation, TU WuppertalGoogle Scholar
  22. 22.
    Lichte U (2004) Klimatische Temperatureinwirkungen und Kombinationsregeln bei Brückenbauwerken. Ph.D. thesis, München, GermanyGoogle Scholar
  23. 23.
    Sanio D, Mark P, Ahrens MA (2017) Temperaturfeldberechnung für Brücken: Umsetzung mit Tabellenkalkulationen. Beton- und Stahlbetonbau 112(2):85–95CrossRefGoogle Scholar
  24. 24.
    Mark P (2003) Optimierungsmethoden zur Biegebemessung von Stahlbetonquerschnitten. Beton- und Stahlbetonbau 98(9):511–519CrossRefGoogle Scholar
  25. 25.
    Mark P, Strack M (2004) Bending design of arbitrarily shaped steel fibre reinforced concrete sections using optimization methods. In: di Prisco M et al. (eds) 6th RILEM symposium fibre reinforced concrete (BEFIB 2004), Italy, pp 965–974Google Scholar
  26. 26.
    Mark P (2004) Fundamentgestaltung und Sohlspannungsberechnung mit Optimierungsmethoden und Tabellenkalkulation. Bautechnik 81(1):38–43MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tkocz J, Heek P, Mark P (2015) SFRC slabs exposed to fire—experiments, temperature flow and design. ALITinform 6(41):36–53Google Scholar
  28. 28.
     Hosser D (2013) Leitfaden Ingenieurmethoden des Brandschutzes. vfdb TB 04-01/3Google Scholar
  29. 29.
    Fouad N (1998) Numerical simulation of the environmental thermal loading of structures. Fraunhofer-IRB, MunichGoogle Scholar
  30. 30.
    Bhatti A (2002) Practical optimization methods. Springer, New YorkGoogle Scholar
  31. 31.
    König G, Tue NV (2003) Grundlagen des Stahlbetonbaus – Einführung in die Bemessung nach DIN 1045-1. Teubner-Verlag, LeipzigGoogle Scholar
  32. 32.
    Hillerborg A (1980) Analysis of fracture by means of the fictitious crack model, particularly for fibre reinforced concrete. Int J Cem Compos 2(4):177–184Google Scholar
  33. 33.
    Strack M (2008) Modelling of the crack opening controlled load bearing behaviour of steel fibre reinforced concrete under tension and bending. In: Gettu R (eds) Proceedings of 7th RILEM international symposium fibre reinforced concrete—design and applications, BEFIB, India, pp 323–332Google Scholar
  34. 34.
    Bazant ZP, Oh BH (1983) Crack band theory for fracture of concrete. Mater Struct 16(93):155–197Google Scholar
  35. 35.
    Heek P, Ahrens MA, Mark P (2017) Incremental-iterative model for time-variant analysis of SFRC subjected to flexural fatigue. Mater Struct 50(1):62.  https://doi.org/10.1617/s11527-016-0928-z CrossRefGoogle Scholar
  36. 36.
    Heek P, Tkocz J, Thiele C, Vitt G, Mark P (2015) Fasern unter Feuer - Bemessungshilfen für stahlfaserverstärkte Stahlbetondeckenplatten im Brandfall. Beton- und Stahlbetonbau 110(10):656–671CrossRefGoogle Scholar
  37. 37.
    DIN EN 1993-1-2 (2010) Design of steel structures—part 1–2: general rules—structural fire design, BerlinGoogle Scholar
  38. 38.
    Cook DJ, Uher C (1974) The thermal conductivity of fibre-reinforced concrete. Cem Concr Res 4:497–509CrossRefGoogle Scholar
  39. 39.
    Nagy B, Nehme SG, Szagri D (2015) Thermal properties and modeling of fiber reinforced concretes. Energy Procedia 78:2742–2747CrossRefGoogle Scholar
  40. 40.
    DBV (Deutscher Beton- und Bautechnik Verein e.V.) (2001) DBV-2001: Merkblatt StahlfaserbetonGoogle Scholar
  41. 41.
    Balazs G, Lubloy E (2012) Reinforced concrete structures in and after fire. Concr Struct 13:72–80Google Scholar
  42. 42.
    Holschemacher K, Weiße D (2004) Bond of reinforcement in fibre reinforced concrete. In: Proceedings of 6th RILEM symposium on fibre reinforced concrete, BEFIB, Italy, pp 349–358Google Scholar
  43. 43.
    Gödde L, Mark P (2015) Numerical simulation of the structural behaviour of SFRC slabs with or without rebar and prestressing. Mater Struct 48(6):1689–1701CrossRefGoogle Scholar
  44. 44.
    DIN EN 1365-2 (2012) Fire resistance tests for loadbearing elements—part 2: floors and roofs, BerlinGoogle Scholar
  45. 45.
    DIN EN 1992-1-1 (2011) Design of concrete structures—part 1-1: general rules and rules for buildings, BerlinGoogle Scholar

Copyright information

© RILEM 2018

Authors and Affiliations

  1. 1.Institute of Concrete StructuresRuhr-University BochumBochumGermany

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