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


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.


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


Thermal diffusivity


Cross-sectional area


Concrete cover


Specific heat capacity


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)



g, i, j, k, n, r

Counter variables


Moment of inertia


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


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)


Width of a plastic hinge


Serviceability limit state

t, T

Time, absolute temperature


Ultimate limit state


Fibre volume content


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)




Thermal conductivity



σ, σc, σt, σs1

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


Stefan–Boltzmann constant

φ, φel, φpl

Rotation angle (overall, elastic and plastic portions)



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.


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Copyright information

© RILEM 2018

Authors and Affiliations

  1. 1.Institute of Concrete StructuresRuhr-University BochumBochumGermany

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