Probabilistic investigation of reinforced concrete structures under fire loading accounting for material and geometric uncertainties

Abstract

The sensitivity of the structural response to variations of material and cross sectional geometrical parameters is investigated for reinforced concrete structures under fire loading. A relatively low cost 2D corotational layered beam finite element is employed in a reduced latin hypercube sampling framework for this purpose. The structural response is assessed in terms of vertical deflection versus time, failure time and cross sectional stress/strain distribution using standard fire curves. The numerical models represent experimentally tested cases in the literature and operate with experimentally derived parameter sets, when possible. A minimum set of three RVs, the bottom concrete cover thickness, steel and concrete yield stregths, out of the initial full set of 12 are identified that drastically reduces the number of RVs and thus the stochastic computation’s cost, while keeping a reasonable envelop for the results. The relationship between structural behavior, material degradation and data extracted from the cross sectional behavior is also successfully established and used to explain why the random set can be reduced to three parameters.

Keywords

reinforced concrete layered beam model fire resistance probabilistic analysis finite elements

Introduction

Computing explicitly the variability of the structural response as a result of the stochastic nature of the problem of reinforced concrete (RC) structures subjected to fire loading is not common practice in everyday engineering, rather focusing on deterministic computations employing safety factors (1992-1-2 (2004) and 216.1 (2007)). During the lifetime of a building, from design through construction to maintenance, there are various sources of uncertainties impacting the fire resistance of a structure that may require a more dedicated stochastic treatment of this engineering problem. In a rigorous approach, the response of RC structures subjected to fire should ideally include the stochastic nature of material properties, construction geometries, loads, etc., as well as their complex interaction (Ellingwood et al., 2007; Thienpont et al., 2021).

Research efforts on the evaluation of the response of RC structures under fire loading are gaining attention in recent years (Almeshal and Bakar (2022); Buttignol and Bittencourt (2021); Jiang et al. (2021); Ni and Gernay (2021a); Pires et al. (2020); Udi et al. (2022)), including employing a probabilistic approach (De Souza et al., 2019; Eamon and Jensen, 2013a, 2013b, Gernay et al., 2019b, Ni and Gernay, 2021b, Roy and Matsagar, 2022, Shi et al., 2013, Thienpont et al., 2021, Van Coile et al., 2013a, 2016). Some studies consider the variation of thermal parameters (e.g. heat conductivity) in the structure (De Souza et al. (2019); Gernay et al. (2019b); Roy and Matsagar (2020); 2022)), while others vary the material and geometrical properties (Gernay et al., 2019b; Molkens et al., 2017; Roy and Matsagar, 2022; Van Coile et al., 2013a, 2013b; Van Coile and Bisby, 2017), defining their reduction ratio as a function of temperature (Gernay et al., 2019b; Molkens et al., 2017; Van Coile et al., 2013b) and applied load (Gernay et al., 2019b; Shi et al., 2013; Van Coile and Bisby, 2017). Some probabilistic approaches (De Souza et al., 2019; Roy and Matsagar, 2020; Van Coile et al., 2011; Van Coile and Bisby, 2017) take into account the uncertainties of different fire scenarios in the analysis of a large range of structural response indicators (e.g. temperature, deflection, failure time, etc.), or when deducing reliability indexes (Eamon and Jensen, 2013b; Molkens et al., 2017; Shi et al., 2013).

Eamon and Jensen (2013b) conducted a reliability analysis focusing on RC columns where load, material and geometrical parameters were taken as random variables (RVs) and the influence of the most significant parameters was investigated. Lange et al. (2014) proposed a probabilistic framework based on the combination of simple analytical techniques, codified methods and random sampling techniques. Heidari et al. (2019) and Guo et al. (2013) conducted parametric studies to determine the resistance of a simply-supported RC slab under the European parametric fire curve, and identified critical uncertain parameters of the design fire curve yielding practical recommendations for fire office design. Some studies, Van Coile and Bisby (2017), Van Coile et al. (2013b), are dedicated to the choice of the used probability distribution function (PDF) and their implications on the global structural response (e.g. bending capacity, deflection). In the majority of studies uncertain parameters are represented by PDFs to evaluate probable responses of the structure from which particular damage states may be defined. Monitoring and detailed analysis of the cross section state (i.e. stress and strain distribution, material degradation) is scarcely done, possibly because of the burden of performing such a detailed analysis in a stochastic approach based on a large number of simulations (Sosso et al. (2020)). Note that such knowledge is particularly useful for the evaluation of the post-fire safety level of a structure, being an outlook of the present work.

The main goals of this work are to contribute to the understanding of the stochastic nature of the behavior of RC structures subjected to fire by performing (i) detailed analyses of the cross sectional behavior resulting from fire loading when employing realistic model parameters fed from the literature, as well as (ii) the identification of a critical minimum set of parameters that could suffice for obtaining reasonable results efficiently and (iii) quantifying their influence on the structural behavior. The methodology here is based on using a layered beam finite element (FE) approach in a correlation reduced Latin Hypercube Sampling based framework (LHS). It involves (i) the sensitivity study of each material and geometrical RV (12 in total) on the structural response separately, (ii) the quantification of the influence of each RV on the global and local structural response, (iii) the classification of the RVs according to their impact on the failure time or deflection, allowing for the identification of the most critical parameters and (iv) a link made in the analyses between the structural response and data extracted from the cross sectional behavior.

Computational formulations and methodology

Multilayered beam formulation

For this study, the low cost thermo-mechanical multilayered beam computational tool developed in Sosso et al. (2021) is used in a stochastic framework in an approach very similar to Sosso et al. (2020) to investigate the influence of material and geometrical model parameters variability on the structural response of RC structures subjected to fire. To ease readability, the main concepts of the computational methodology and the parameters of the constitutive models are recalled first.

Commonly, fire analysis involves at each time step: (i) a thermal computation of the cross sectional temperature distribution using FE and (ii) a subsequent thermo-mechanical analysis of the structural response. In the proposed multilayered beam approach a computational thermal analysis is never performed, a closed form thermal model proposed by Kodur et al. (2013) is employed instead to approximate the thermal field in the structural members’ cross sections (the interested reader is referred to Kodur et al. (2013) and Sosso et al. (2021) for more details).

Material properties and thermally induced strains depend upon the thermal field in the cross section. The current work is based on the assumption of averaging the temperature along the beam thickness, naturally resulting in a temperature variation only along its height and, specifically, when passing from layer to layer. The equivalent layer temperature is obtained using Simpson’s rule to average the temperature distribution along the width following a procedure summarized in Figure 1.

Figure 1.

Layer temperature calculation procedure.

The thermo-mechanical response is derived through a corotational multilayered Bernoulli beam approach for planar frames (2D), developed in Iribarren et al. (2011), Oliveira et al. (2014), Sosso et al. (2021). In this work the Bernouilli beam theory is employed, as in Kodur and Dwaikat (2008), Prakash and Srivastava (2018a, 2018b), Sosso et al. (2021), a simplification that could be removed by using Timoshenko kinematics (Lin and Zhang (2013)) in the future.

The corotational formulation decouples rigid body rotation from strains and allows for capturing potential catenary effects Oliveira et al. (2014). Constitutive relations are evaluated in the corotating local axes and the multilayered framework consists in the discretization of the beam cross section into a finite number of layers along the height of the cross section, as shown in Figure 2, where $A_{t}^{i}$ (the hatched area), and $A_{s}^{i}$ are the total and the steel area in layer i, respectively. IP_j are the integration points (j = 1–3) of the beam FE. In this approach the generalized stresses (normal force N and bending moment M) are computed at the integration points of the finite element from the generalized strains (i.e. average axial strain $\bar{ε}$ and curvature χ), derived from the nodal displacements.

Figure 2.

Multilayered cross section discretization.

The total axial strain at each layer in a cross section (i.e. at an integration point) $ε_{x, t o t}^{i}$ is computed from the generalized strains based on Bernoulli kinematics:

ε_{x, t o t}^{i} = \bar{ε} - {\bar{y}}^{i} χ

(1)

where

{\bar{y}}^{i}

is the average vertical coordinate of layer i, computed from the sectional center of gravity (Figure 2).

A perfect bonding between the steel reinforcements and concrete is supposed, as in Balaji et al. (2016), Bamonte and Lo Monte (2015), Kodur and Dwaikat (2008), Prakash and Srivastava (2018b). For the layers containing the steel reinforcements the stresses in concrete and steel are computed separately, using 1D constitutive laws fitted to experimental data from the literature.

The constitutive relationships of concrete and steel in this work (Figure 3) were shown to perform well in Berke and Massart (2018), Iribarren et al. (2011), Oliveira et al. (2014), Sosso et al. (2020) for the RC structures and specifically in Sosso et al. (2021) for the stuctures considered here. At each layer 1D strain rate independent elasto-plastic constitutive models are used for both materials, with an ultimate strain limit, ϵ_u, used to detect layer fracture (no stresses are developed in the material once this limit is reached). Concrete is modeled as a material with zero strength in tension with the following softening relationship:

f_{c} = σ_{0}^{c} \cdot \exp (μ^{c} \cdot κ^{c})

(2)

with f_c the compressive yield strength,

σ_{0}^{c}

its initial value, μ^c the softening parameter (≤0) and κ^c the cumulated plastic strain in concrete. Steel is assumed to follow the bilinear hardening model given by:

f_{y} = σ_{0}^{s} (1 + μ^{s} κ^{s})

(3)

with f_y the current yield strength,

σ_{0}^{s}

its the initial value, μ^s the hardening parameter, κ^s the cumulated plastic strain in steel.

Figure 3.

1D elasto-plastic consititutive models of concrete in compression (zero tensile strength) and of steel in traction/compression.

The nominal material parameters used as reference set in the applications treated in this work were identified as best fit to experimental data in Sosso et al. (2021) and are given in Table 1.

Table 1.

Nominal material parameters for the applications.

Application		Concrete			Steel reinforcement\[GPa]
Application		E_c [GPa]	σ_c⁰[MPa]	ɛ_u,c[‰]	E_s[GPA]	σ_s⁰[MPA]	ɛ_u,s[%]
Simply supported beam		27	30	3.5	210	436	20
Reinforced concrete frame	ϕ16	24	24	3.5	190	366	17
Reinforced concrete frame	ϕ19	24	24	3.5	192	383	17

The softening parameter for concrete and the hardening parameter for steel were kept constant in all analyses, at μ_c = 0 (perfect plasticity) and μ_s = 1.52, respectively. Note that the chosen nominal value of the reinforcement ultimate strain (ɛ_u,s) is high, compared to values adopted in the literature, however, the influence of ɛ_u,s is rather limited considering that fracture was never observed as a failure mode in Sosso et al. (2021) even though employing lower ɛ_u,s values.

The layer axial stress $σ_{x, t o t}^{i}$ is obtained considering volume averaging over the layer:

σ_{x, t o t}^{i} = (1 - ρ_{s}^{i}) σ_{x}^{c i} + ρ_{s}^{i} σ_{x}^{s i}

(4)

with

ρ_{s}^{i}

A_{s}^{i}

A_{t}^{i}

the ratio of the cross sectional area of the longitudinal steel reinforcements and the total cross sectional area of the layer i and

σ_{x}^{c i}

σ_{x}^{s i}

are, respectively, the axial stress in concrete and steel. Having

σ_{x, t o t}^{i}

calculated for all layers, the cross section generalized stresses are evaluated as a finite sum of the contributions of each layer (Iribarren et al. (2011)).

The stress in each material is a function of the mechanical strain component obtained from the total strain decomposition (Sosso et al. (2021)):

ε_{x, t o t}^{i} (\bar{ε}, χ) = ε_{x, m e c h}^{i} (σ_{x, t o t}^{i}) + ε_{x, t h}^{i} (T^{i}) + ε_{x, L I T S}^{i} (σ_{x, t o t}^{i}, T^{i})

(5)

where

ε_{x, m e c h}^{i}

ε_{x, t h}^{i}

are, the mechanical (i.e elastic and plastic) and thermal strain, respectively and

ε_{x, L I T S}^{i}

is the Load Induced Thermal Strain (LITS).

σ_{x, t o t}^{i}

and Tⁱ are layer axial stress and the cross section temperature at layer i, respectively. A linear strain measure is used in the formulation, which is appropriate for concrete, considering the relatively low ultimate strain values. It is a more debatable choice for steel at large strains, however, the influence of the variation of the material properties as a result of temperature increase is certainly overwhelming compared to the difference that comes from using a linear, instead of a nonlinear strain measure.

Computational methodology

The displacement-based finite element simulations are using a unilateral thermal-to-mechanical coupling in a dynamic framework. They are driven using physical time increments to which load and temperature discretization is associated, i.e. for a given time increment temperature and/or load variations are prescribed. The primary unknown to determine is the displacement of each node for each physical time. The loading sequence of the simulations in this work is the following: first the mechanical loads are applied to the structure in a single time step of arbitrary length (long enough to not trigger dynamic effects), followed by time steps in which the gas temperature is gradually increased to represent the fire load. The temperature increment generates material parameter variations and thermally induced strain variations that result in a variation of the structural displacements when solving the discrete structural equilibrium problem.

For the evaluation of the cross section temperature distribution, the thickness of the layers that discretize the cross section was taken at most 5 mm. Regarding the material degradation ratios of concrete as a function of temperature, the expressions used in this work are the model of Bahr et al. (2013) for the Young’s modulus, Zhenhai (1993) for the yield strength variation and a trilinear expression fitted to Schneider (1988) for the ultimate strain evolution. Steel elastic modulus and yield strength relative evolution are assumed to follow Xiao and König (2004) and Shi et al. (2004), respectively. Based on a more limited set of experimental data of the ultimate strain of steel at different temperatures, a polynomial fit to data given in Elghazouli et al. (2009) is used. The interested reader is referred to Sosso et al. (2021) where these evolution laws are graphically compared to various sources from the literature.

The probabilistic study is built upon a large set of deterministic computations with random sampling driven by LHS. Two types of analysis are performed: (1) sensitivity analysis, where all RVs except one are set to their nominal (i.e. reference) value, while the individually selected parameter is varied, and (2) simulations where all RVs are convoluted and varied simultaneously. The comparison of these two approaches allows the identification of a minimum set of dominant RVs.

Random sampling method

To investigate stochastically RC members behavior subjected to fire, traditionally Monte Carlo Simulations (MCSs) are adopted (Eamon and Jensen (2013a); Gernay et al. (2019a); Guo et al. (2013); Shi et al. (2013); Thienpont et al. (2021); Van Coile and Bisby (2017); Van Coile et al. (2013b); Wang et al. (2018)). However, this sampling technique can easily become computationally very costly, i.e. beyond what is practically feasible for complex problems with a nonlinear interplay between parameters. Therefore LHS with correlation reduction (Olsson et al. (2003); Olsson and Sandberg (2002)) was used in this work, known to significantly reduce the number of simulations, resulting in more affordable and still accurate results (De Souza et al. (2019); Gernay et al. (2016)). The considered RVs include material properties driving the elasto-plastic behavior and fracture of the constituents, as in Sosso et al. (2020), completed with geometrical parameters, resulting in a total of 12 RVs.

Based on the results of prior attempts aiming at finding the minimum number of samples required for converged results, 500 samples are considered in the cases having up to two RVs. Note that this already resulted in over 10.000 simulations for each of the considered applications. A convergence study on the number of samples is presented later, including cases with more RVs.

Material parameters at ambient temperature

Three material parameters at ambient temperature (i.e. 20°C) were selected as RVs for each constituent, leading to a total of six material RVs, as listed in Table 2. The initial concrete compressive strength

(σ_{0}^{c})

and elastic modulus (E_c) are assumed to follow lognormal distributions with coefficients of variation (CoV), as recommended by Gernay et al. (2019b), Molkens et al. (2017), Sỳkora et al. (2010), Wang et al. (2018) and Van Coile et al. (2013b), respectively. The nominal value of ɛ_u,c is set to 3.5 ‰ corresponding to unconfined concrete with a normal distribution for this parameter (Trezos (1997)). The steel elastic modulus (E_s) is modelled according to Mirza and MacGregor (1979) with a normal distribution whereas the initial yield strength

(σ_{0}^{s})

is assumed to follow Gernay et al. (2019b), Sỳkora et al. (2010), Van Coile et al. (2013b), Wang et al. (2018) and the ultimate strain (ɛ_u,s) follows the Probabilistic Model Code (PMC) suggestions (JCSS (2001)).

Table 2.

Statistical distributions of the chosen material and geometrical RVs.

	RVs names	Symbols	CoV	Distribution	Reference
Concrete at 20°C	Compressive strength	$σ_{0}^{c}$	0.15	Lognormal	Sỳkora et al. (2010)
					Molkens et al. (2017)
					Gernay et al. (2019b)
					Wang et al. (2018)
	Young modulus	E _c	0.15	Lognormal	Van Coile et al. (2013b)
	Ultimate strain	ɛ _u,c	0.143	Normal	Trezos (1997)
Steel at 20°C	Yield strength	$σ_{0}^{s}$	0.07	Lognormal	Sỳkora et al. (2010)
					Gernay et al. (2019b)
					Van Coile et al. (2013b)
					Wang et al. (2018)
	Young modulus	E _s	0.033	Normal	Mirza and MacGregor (1979)
	Ultimate strain	ɛ _u,s	0.09	Normal	JCSS (2001)
Geometrical parameters	Bottom concrete cover	C _b	0.33	Beta [0; 3 C_b,nominal]	Sỳkora et al. (2010)
					Van Coile et al. (2011)
					Van Coile et al. (2013a)
					Van Coile et al. (2013b)
					Van Coile and Bisby (2017)
	Top concrete cover	C _t	0.33	Beta [0; 3 C_t,nominal]	Sỳkora et al. (2010)
					Van Coile et al. (2011)
					Van Coile et al. (2013a)
					Van Coile et al. (2013b)
					Van coile and bisby (2017)
	Cross sectional height	H	0.04	Normal	Eamon and Jensen (2013b)
	Cross sectional width	w	0.04	Normal	Eamon and Jensen (2013b)
	Bottom reinforcement diameter	Db _b	0.02	Normal	Van Coile et al. (2011)
					Van Coile et al. (2013a)
					Van Coile et al. (2013b)
	Top reinforcement diameter	Db _t	0.02	Normal	Van Coile et al. (2011)
					Van Coile et al. (2013a)
					Van Coile et al. (2013b)

Geometrical parameters of the cross section

The geometrical parameters given in Table 2 describe the variation in cross sectional dimensions. The importance of these parameters during fire exposure is non negligible, among others, because they influence the thermal field in the cross section (Choi and Shin (2011); Ellingwood (1980); Lakhani and Hofmann (2019)).

A total of six geometrical parameters are considered as RVs here: the bottom and top concrete cover thickness (C_b and C_t) is described by beta distribution with lower and upper limits, as recommended in Sỳkora et al. (2010), Van Coile et al. (2011, 2013a), Van Coile and Bisby (2017), Van Coile et al. (2013b); the cross sectional height (H) and width (w) are assumed to follow normal distributions with coefficients of variation (CoV) adapted from Eamon and Jensen (2013b); the bottom and top reinforcement diameters (Db_b and Db_t) follow a normal distribution, as recommended in Van Coile et al. (2011, 2013a,b). A bias factor of 1.01 is assumed for H and w, as adopted in Eamon and Jensen (2013b), Nowak and Szerszen (2003). The mean value of a RV is the product of the associated bias factor and its nominal value. The nominal values of the geometrical parameters in the two structural studies are given in Figures 5 and 13. It is emphasized that the majority of the geometrical parameters are naturally convoluted, changing only one parameter affects other parameters, as illustrated in Figure 4.

Figure 4.

Geometrical parameters of a beam/column cross section.

Structural investigations

This section presents two structural studies selected because of the availability of experimental results and a known set of reference numerical model parameters that fit the experimental response well (based upon Sosso et al. (2021)). The structures are subjected to the standard fire ASTM E119 (2008) for the simply supported beam and ISO 834-1 (1999) for the RC frame. In each example the FE length was chosen approximately equal to the cross sectional height to ensure proper energy dissipation when a plastic hinge forms (Iribarren et al. (2011)).

Simply supported beam

The nominal beam and cross section dimensions, boundary conditions and loads maintained constant during the simulation of the experiment in Lin et al. (1981) are shown in Figure 5.

Figure 5.

Geometry, loading and boundary conditions of the simply supported beam model of the Lin et al. (1981) test.

Results of the sensitivity analysis

Figures 6 and 7 show the mid-span vertical displacement versus time results for the geometrical and material model parameters, respectively. The first obvious finding is that the bottom concrete cover has a significant influence on the failure time (Figure 6(c)), as expected based on the literature (Choi and Shin (2011); Lakhani and Hofmann (2019)). It plays a major role by ensuring the thermal isolation of the rebars, as shown later. Note that Figure 6(c) shows a large dispersion in the failure time, defined here corresponding to ≈ 150 mm mid-span deflection, (t_f, min ≈ 39 min and t_f, max ≈ 108 min) for extreme values of the bottom concrete cover thickness varying from 4.9 to 53.3 mm (i.e. for a more than tenfold increase). It is noteworthy about the lower value that such a thin concrete cover is an undesired scenario and would correspond to an impractical and erroneous initial design. Considering the statistical data in the literature on this variable though yields such low probability scenarios that could be result of manufacturing uncertainties or on site issues (e.g. unwanted vibrations sink the reinforcements below their nominal position).

Figure 6.

Mid-span deflection versus time data for the simply supported beam when varying geometrical parameters: experimental data (black), nominal parameter set (blue), 500 stochastic simulations (grey).

Figure 7.

Mid-span deflection versus time data for the simply supported beam when varying material parameters: experimental data (black), nominal parameter set (blue), 500 stochastic simulations (grey).

The top concrete cover thickness variation results in almost no change in the structural response. Due to the loading, the top layers are compressed developing stress in both concrete and steel, while the bottom layers are in tension, supported only by the bottom rebars because of the zero concrete tensile strength assumption. The top zone is thus more failure resisting because of more active (i.e. stress developing) layers and working at the lowest temperature, which explains the lower influence of C_t on the structural response.

The cross section height shows a non negligible influence on the failure time, partially because its large dispersion and in part due to its implicit influence on the beam’s bending capacity (i.e. increasing H leads to rebars distance increase when keeping the other geometrical parameters constant).

The cross section width showed a minor influence on the structural response (Figure 6(b)) partially due to the assumption of averaging the temperature along the beam width in the FE formulation (the temperature distribution along the section height barely changes with varying width) and due to its relatively low influence on the bending response, especially compared to H.

The reinforcement bar diameters variation changed only slightly the reinforcement volume fraction and was observed to influence the structural response for the critical bottom rebars only (Figure 6(e)). The change in Db_b influences R (Figure 4) and the steel volume fraction in the critical heated bottom zone. Figure 6(e) shows quite a large dispersion in fire duration time to reach a vertical displacement of ≈ 150 mm as a result of the bottom bar diameter variation. This could be attributed to the fact that at this stage the bottom reinforcement plays a significant role in the strength of the simply supported beam and therefore their volume fraction (i.e. Db_b variation) affects the structural response.

The second main finding is that when varying separately material parameters, the reinforcement yield strength results in the largest variability in the structural response, which is logical considering that, as observed in Sosso et al. (2021), the bottom reinforcement strength controls the failure of this beam. The other material parameters did barely influence the displacement-time curves, noting that layer fracture never occured in the simulations (like in Sosso et al. (2021)).

It arises from this sensitivity assessment that the bottom concrete cover thickness (influences the thermal field) is the dominant geometrical parameter, while $σ_{0}^{s}$ is dominant among the material properties (influences the structural strength). This is graphically shown in the tornado diagram of failure time in Figure 8, plotted at a vertical displacement of ≈ 150 mm corresponding to the close to collapse regime in which small time increments already result in a significant increase in the deflection.

Figure 8.

Torando diagram for individual RV: fire duration at ≈ 150 mm mid-span deflection.

Convoluted effect of model parameters

From the results of the sensitivity analysis, the combination of three dominant parameters (C_b, H, $σ_{0}^{s}$ ) are taken and their influence is now compared to all geometrical, material and all RVs variation. The geometrical parameter, Db_b is disregarded, because its effect is expected to be masked by (C_b, H).

When different sets of RVs are considered [C_b; (C_b, H); $σ_{0}^{s}$ ; (C_b, $σ_{0}^{s}$ )], the resulting structural response is depicted in Figure 9. The goal is to investigate the interplay of these parameters and conclude whether results of a reduced set of RVs can be used as an acceptable envelop for the complete RV set.

Figure 9.

Mid-span deflection versus time data for the simply supported beam for RV sets: experimental data (black), nominal parameter set (blue), (a)–(d) 500 and (e)–(f) 1000 stochastic simulations.

First, comparing the convoluted effect of (C_b, H) with C_b only (Figure 9(a)), it emerges that the variation of the bottom concrete cover thickness alone results in an identical envelope as when varying simultaneously C_b and H. Actually, the envelop obtained by varying all geometrical RVs is still well approximated by the envelop of C_b only (Figure 9(b)), hence C_b alone may be sufficient to practically estimate the deflection-time envelop due to the variation of the geometrical parameters.

When all material parameters are varied and compared to varying all geometrical parameters (Figure 9(c)), the geometrical RVs are dominant. Among the material parameters $σ_{0}^{s}$ dominates (Figure 9(d)), as expected based on the findings of the sensitivity analysis. Although, C_b has a more significant influence, $σ_{0}^{s}$ is kept in the analysis because of its different, material based nature.

Finally, combining the two dominant material and geometrical parameters (C_b, $σ_{0}^{s}$ ) the resulting displacement-time envelop approximates well the one obtained when considering all of the model parameters as RVs (Figure 9(f)). This shows that the initial set of 12 RVs can be reduced to two dominant parameters for the given structural example, governing the thermal insulation of the bottom rebars (C_b) and the initial bottom rebar strength, $σ_{0}^{s}$ .

In order to confirm the trustworthiness of the results above, a convergence analysis on the number of samples was carried out. The trends observed remained unaltered for an increasing number of samples. As for a quantitative verification, the fire duration required to reach 130–150 mm deflection was considered, specifically, the parameters of a normal PDF fitted to this discrete data were focused upon. For the case where all model parameters are RVs 1000 samples were required to reach converged results (Figure 10), while 500 samples sufficed for (C_b, $σ_{0}^{s}$ ). As expected, substantial computational gains come from the reduction of the number of RVs to the dominant set.

Figure 10.

Fire duration histograms at 130–150 mm deflection, (a) 500 and (b) 1000 stochastic simulations.

Note that with the reduced parameter set the unfavourable part of the response envelop (i.e. rapid collapse) appears to be well captured, rather approximating the long collapse time extremes (low probability convolution of favorable geometrical and material parameters in real life) worse, leading to conservative results.

Structural behavior and cross sectional analysis

The relationship between the structural behavior and cross sectional response is investigated here for the two most dominant model parameters (C_b, $σ_{0}^{s}$ ). In the following, the best case refers to the simulation with the longest failure time, t_f, max, while the worst case indicates the earliest failure, t_f, min. Elapsed fire times are reported at mid-span deflection of 50 and 150 mm.

When varying C_b separately the worst case corresponds to the thinnest concrete cover, C_b,worst ≈ 4.9 mm, while the best case corresponds to the thickest concrete cover C_b,best ≈ 53.3 mm, as expected. For a fixed deflection of ≈ 50 mm (i.e. a horizontal cut in Figure 6(c)) there is already a large difference in the cross sectional temperature distribution (Figure 11(a)) and $σ_{0}^{s}$ values (Figure 11(b)). The reinforcements in the best model, with thicker C_b, operate at a lower temperature for the same deflection in the critical bottom zone in tension, which is beneficial for maintaining a higher yield strength. In the worst case at a deflection of ≈ 150 mm the bottom reinforcement working in the plastic regime develops a higher average stress at t_f, min ≈ 37 min (Figure 11(d)) than the best case at t_f, max ≈ 105 min (Figure 11(c)). Note that Figures 11(c) and 11(d) show a few bottom reinforcement layers in the elastic regime while others yielded. This purely numerical feature will be explained later.

Figure 11.

Cross sectional data for the mid-span when varying C_b separately for the best case (solid lines) and the worst case (dashed lines).

When varying $σ_{0}^{s}$ separately, the difference between best and worst cases is less both in terms of deflection-time and cross sectional data. Figure 12 shows the stress distribution in the cross section at ≈ 150 mm deflection that corresponds to different fire exposure times: 78 minutes (best case) and 63 minutes (worst case). The best case corresponds to $σ_{0, \max}^{s} \approx 546$ MPa while the worst refers to $σ_{0, \min}^{s} \approx 348$ MPA, as expected. Figure 12(a) shows a purely numerical artefact: certain bottom reinforcement layers yielded while others remained elastic. This is attributed to the layered discretization and to the fact that in these layers the reinforcement’s elastic modulus decreases so rapidly that the stress stays below the less diminished yield strength.

Figure 12.

Cross sectional data for the mid-span when varying $σ_{0}^{s}$ separately for the best case and worst case.

As a summary from this first application, the number of RVs can be reduced to two significant variables when distinguishing between material and geometrical parameters: C_b and $σ_{0}^{s}$ and 500 samples suffice for converged results with this set.

Reinforced concrete frame

This section focuses on a RC frame studied experimentally in Raouffard and Nishiyama (2017) and more specifically considers in detail the displacement of the extremities of two cantilever beams, one loaded with a vertical force acting upward (Upward Loaded (UL) beam) and one with a force acting downward (Downward Loaded (DL) beam), as shown in Figure 13. The lower part of the frame, including the two cantilever beams is heated and the relative displacement of the extremities of the UL and DL cantilever beams is monitored as a function of time. The objective is to investigate the effects of the loading type combined with dominant geometrical and material parameters variation on the structural response. Note that this frame wasn’t tested up to failure, unlike the cantilever beam in the previous application. In a precursor investigation all RVs have been varied separately with respect to their nominal values, but for the sake of brevity, only the results of the most dominant model parameters are presented in the following.

Figure 13.

Studied RC frame, applied loads and boundary conditions that mimic the experiment in Raouffard and Nishiyama (2017) (cross section dimensions are mm).

Figures 14 and 15 show the relative displacement measured between the load application points and the beam-column connection as a function of time for the UL and DL beams, respectively. The UL beam is the cantilever subjected to a bending moment of the same sign as the single cantilever beam in the previous application, while the DL beam loading is of opposite sign.

Figure 14.

UL beam relative vertical displacement versus time: experimental data (black), nominal parameter set (blue), the number of simulations are between parantheses.

Figure 15.

DL beam relative vertical displacement versus time: experimental data (black), nominal parameter set (blue), the number of simulations are between parantheses.

The bottom concrete cover confirms its dominant influence for both cases. However, now, $σ_{0}^{s}$ has a more limited influence on the structural response of the UL beam (Figure 14(b)). This can be explained by the thick bottom concrete cover at its nominal value (C_b = 40 mm), which provides substantial thermal protection to the bottom reinforcement, leading to a more moderate decay of the beam strength. This illustrates the nonlinear interplay between RVs, i.e. a thinner bottom concrete cover activated a higher sensitivity of the structural response to $σ_{0}^{s}$ in the single cantilever beam in the previous application. As earlier, it was observed that all model parameters variation (taking 2000 samples here to reach convergence) results in a similar envelop of the structural response as varying only the bottom concrete cover thickness (Figure 14(c)). 500 samples for the single RV cases match rather well the envelops obtained for multiple RVs and 2000 samples for both UL and DL beams.

In the DL beam, as opposed to the UL beam, the initial compressive concrete strength, $σ_{0}^{c}$ , shows a noticeable influence on the structural response (Figure 15(b)). This can be explained by the fact that in the DL beam loading, the heated bottom zone is compressed, generating stress in concrete in this zone. Focusing on the cross section stress distribution in sections C (DL) and D (UL) (defined in Figure 13), for the DL beam at the end of the simulation (i.e. 75 min after fire start) several bottom layers still develop compressive stress in concrete and only a few bottom concrete layers are with close to zero stress, as a consequence of material degradation (Figure 16(c)). It is logical that $σ_{0}^{s}$ has a reduced influence in the case of DL beam considering the loading and that the top reinforcements in tension exhibit the lowest temperature increase (Figure 16(a)). For this loading the upper reinforcement working in tension remains elastic at the end of the simulation due to the lesser degradation of the steel yield strength (i.e. $σ_{0, t o p}^{s, 75 \min} \approx 0.8 σ_{0}^{s}$ ). In contrast, the bottom reinforcement in UL case has undergone a significant deterioration $(σ_{0, b o t}^{s, 75 \min} \approx 0.4 σ_{0}^{s})$ due to the high temperature which resulted in the yielding of a few layers of steel (Figure 16(d)).

Figure 16.

Cross sectional data at cross section C and D (defined in Figure 13) for the worst case scenario: elastic and plastic concrete layers are light and dark grey, elastic and plastic rebars are white and black, respectively.

For the DL beam with all model parameters as RVs and 2000 samples a few extreme cases appear in Figure 15(c), detaching from the C_b envelop in the negative displacement range for long fire duration, corresponding to low probability parameter convolutions that are not prone to compromise the conclusions drawn on the general trends.

For the sake of completeness, the following indicators are given on the computational effort: the frame model has 78 degrees of freedom and 24 finite elements, the average computation time on a Laptop with a 2.3 GHz Intel Core TM i7-3610 QM CPU and 6 GB RAM is 9 min for a single simulation having over 75 time increments.

The study of these two cantilever beams indicates that different RV sets should be incorporated in the analysis as function of the applied loading. A set that appears applicable to a general loading incorporates the bottom concrete cover thickness (C_b), the steel initial yield strength $(σ_{0}^{s})$ and the concrete initial compressive strength $(σ_{0}^{c})$ .

Conclusions and outlook

In this study, the influence of material and geometrical variability on RC members failure time when subjected to fire was investigated using in-house developed corotational layered beam finite elements in a reduced Latin Hypercube Sampling framework. The main conclusions of this work are:

(C1) the bottom concrete cover thickness, C_b, is a dominant RV that should always be incorporated in a stochastic fire analysis, since a thicker C_b delays rebar degradation and influences beneficially the bending capacity of RC beams,

(C2) the rebar yield strength, $σ_{0}^{s}$ , plays an important role when the tensile reinforcement governs the structural failure, i.e. with the fire exposed bottom zone in tension,

(C3) the concrete compressive yield strength, $σ_{0}^{c}$ , plays an important role for negative applied bending moment, i.e. with the fire exposed bottom zone in compression.

The following recommendations can be issued from the findings above:

(R1) a reduced set of three random variables (C_b, $σ_{0}^{s}$ , $σ_{0}^{c}$ ) is recommended to be taken on board in a stochastic study for general loading of RC frames subjected to fire,

(R2) in the design and execution of RC structures, special attention should be devoted to the appropriate concrete cover thickness and the quality of steel (their yield strengths).

Further research of immediate interest is to incorporate the fire action itself as an uncertain parameter. This could be achieved by a proper thermal computation in the cross section, as in (Barros et al., 2018; Prakash and Srivastava, 2018a). Such an extension would also allow addressing post-fire strength analysis.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Péter Z Berke

References

1992-1-2, E (2004). Design of Concrete Structures. Part 1-2: General Rules-Structural Fire Design Eurocode 2. Brussels, Belgium: European Committee for Standardization.

216.1, A (2007) Code Requirements for Determining Fire Resistance of Concrete and Masonry Construction Assemblies”,ACI 216.1- 07/TMS-0216-07. Farmington,Hills, MI, USA: American Concrete Institute.

834-1, I (1999) Fire Resistance Tests- Elements of Buildings Construction, Part 1: General Requirements. Geneva: International Organization for Standardization.

Almeshal

Abu Bakar

Tayeh

(2022) Behaviour of reinforced concrete walls under fire: a review. Fire Technology 58: 2589–2639.

Bahr

Schaumann

Bollen

, et al. (2013) Young’s modulus and Poisson’s ratio of concrete at high temperatures: experimental investigations. Materials and Design 45: 421–429.

Balaji

Luquman K

Nagarajan

, et al. (2016) Studies on the behavior of reinforced concrete short column subjected to fire. Alexandria Engineering Journal 55(1): 475–486.

Bamonte

Lo Monte

(2015) Reinforced concrete columns exposed to standard fire: comparison among different constitutive models for concrete at high temperature. Fire Safety Journal 71: 310–323.

Barros

Pires

Silveira

, et al. (2018). Advanced inelastic analysis of steel structures at elevated temperatures by SCM/RPHM coupling. Journal of Constructional Steel Research, 145:368–385.

Berke

Massart

(2018) Modelling of stirrup confinement effects in RC layered beam finite elements using a 3D yield criterion and transversal equilibrium constraints. International Journal of Concrete Structures and Materials 12: 49.

10.

Buttignol

TET

Bittencourt

(2021) Simplified design procedures for the structural analysis of reinforced concrete columns in fire. Engineering Structures 246: 113076.

11.

Choi

Shin

(2011) The structural behavior and simplified thermal analysis of normal-strength and high-strength concrete beams under fire. Engineering Structures 33(4): 1123–1132.

12.

de Souza

RCS

Andreini

La Mendola

, et al. (2019) Probabilistic thermo-mechanical finite element analysis for the fire resistance of reinforced concrete structures. Fire Safety Journal 104: 22–33.

13.

E119, A. (2008) Standard Test Methods for Fire Test of Building Construction Andmaterials. PA: American Society for Testing and Materials.

14.

Eamon

Jensen

(2013a) Reliability analysis of RC beams exposed to fire. Journal of Structural Engineering 139(2): 212–220.

15.

Eamon

Jensen

(2013b). Reliability analysis of reinforced concrete columns exposed to fire. Fire safety journal, 62:221–229.

16.

Elghazouli

Cashell

Izzuddin

(2009) Experimental evaluation of the mechanical properties of steel reinforcement at elevated temperature. Fire Safety Journal 44(6): 909–919.

17.

Ellingwood

Galambos

MacGregor

Cornell

(1980). Development of a probability based load criterion for Americal national standard A58: Building code requirements for minimum design loads in buildings and other structures. US Department of Commerce, National Bureau of Standards, US Government Printing Office.

18.

Ellingwood

Smilowitz

Dusenberry

, et al. (2007) Best Practices for Reducing the Potential for Progressive Collapse in Buildings. Gaithersburg, USA: Technical report US Department of Commerce, National Institute of Standards and Technology.

19.

Gernay

Elhami Khorasani

Garlock

(2016) Fire fragility curves for steel buildings in a community context: a methodology. Engineering Structures 113: 259–276.

20.

Gernay

Khorasani

Garlock

(2019a) Fire fragility functions for steel frame buildings: sensitivity analysis and reliability framework. Fire technology 55(4): 1175–1210.

21.

Gernay

Van Coile

Elhami Khorasani

, et al. (2019b) Efficient uncertainty quantification method applied to structural fire engineering computations. Engineering Structures 183: 1–17.

22.

Guo

Shi

Jia

, et al. (2013) Probabilistic evaluation of structural fire resistance. Fire technology 49(3): 793–811.

23.

Heidari

Robert

Lange

, et al. (2019) Probabilistic study of the resistance of a simply-supported reinforced concrete slab according to eurocode parametric fire. Fire Technology 55(4): 1377–1404.

24.

Santafé Iribarren

Berke

Bouillard

, et al. (2011). Investigation of the influence of design and material parameters in the progressive collapse analysis of RC structures. Engineering Structures, 33:2805–2820.

25.

JCSS (2001) Probabilistic Model Code: Joint Committee on Structural Safety.

26.

Jiang

Orabi

Jiang

, et al. (2021) Modelling concrete slabs subjected to fires using nonlinear layered shell elements and concrete damage-plasticity material. Engineering Structures 234: 111977.

27.

Kodur

Dwaikat

(2008) A numerical model for predicting the fire resistance of reinforced concrete beams. Cement and Concrete Composites 30(5): 431–443.

28.

Kodur

Dwaikat

(2013) A simplified approach for predicting temperature in reinforced concrete members exposed to standard fire. Fire Safety Journal 56: 39–51.

29.

Lakhani

Hofmann

(2019) Effect of loss in concrete cover during fire on the predicted temperature distribution. IOP Conference Series: Materials Science and Engineering 615(1): 012085.

30.

Lange

Devaney

Usmani

(2014) An application of the peer performance based earthquake engineering framework to structures in fire. Engineering structures 66: 100–115.

31.

Lin

Zhang

(2013) Nonlinear finite element analyses of steel/FRP-reinforced concrete beams in fire conditions. Composite Structures 97: 277–285.

32.

Lin

Gustaferro

Abrams

(1981) Fire Endurance of Continuous Reinforced Concrete Beams. Skokie, Ill: Portland Cement Association.

33.

Mirza

MacGregor

(1979) Variability of mechanical properties of reinforcing bars. Journal of the Structural Division 105: 921–937. ASCE 14590 Proceeding.

34.

Molkens

Van Coile

Gernay

(2017) Assessment of damage and residual load bearing capacity of a concrete slab after fire: applied reliability-based methodology. Engineering Structures 150: 969–985.

35.

Gernay

(2021a) Considerations on computational modeling of concrete structures in fire. Fire Safety Journal 120: 103065.

36.

Gernay

(2021b) A framework for probabilistic fire loss estimation in concrete building structures. Structural Safety 88: 102029.

37.

Nowak

Szerszen

(2003) Calibration of design code for buildings (ACI 318): Part 1-statistical models for resistance. ACI Structural Journal 100(3): 377–382.

38.

Oliveira

CEM

Batelo

EAP

Berke

, et al. (2014) Nonlinear analysis of the progressive collapse of reinforced concrete plane frames using a multilayered beam formulation. Revista IBRACON de Estruturas e Materiais 7: 845–855.

39.

Olsson

AMJ

Sandberg

(2002) Latin hypercube sampling for stochastic finite element analysis. Journal of Engineering Mechanics 128(1): 121–125.

40.

Olsson

Sandberg

Dahlblom

(2003). On Latin hypercube sampling for structural reliability analysis. Structural safety, 25(1):47–68.

41.

Pires

Barros

Silveira

, et al. (2020) An efficient inelastic approach using SCM/RPHM coupling to study reinforced concrete beams, columns and frames under fire conditions. Engineering Structures 219: 110852.

42.

Prakash

Srivastava

(2018a) Fully coupled multi-physics nonlinear analysis of structural space frames subjected to fire using the direct stiffness method. Advances in Structural Engineering 22(6): 1266–1283.

43.

Prakash

Srivastava

(2018b). Nonlinear analysis of reinforced concrete plane frames exposed to fire using direct stiffness method. Advances in Structural Engineering, 21(7):1036–1050.

44.

Raouffard

Nishiyama

(2017) Fire response of exterior reinforced concrete beam-column subassemblages. Fire Safety Journal 91: 498–505.

45.

Roy

Matsagar

(2020) Fire fragility of reinforced concrete panels under transverse out of plane and compressive in-plane loads. Fire Safety Journal 113: 102976.

46.

Roy

Matsagar

(2022) A probabilistic framework for assessment of reinforced concrete wall panel under cascaded post-blast fire scenario. Journal of Building Engineering 45: 103506.

47.

Schneider

(1988) Concrete at high temperatures-a general review. Fire safety journal 13(1): 55–68.

48.

Shi

Tan

, et al. (2004) Influence of concrete cover on fire resistance of reinforced concrete flexural members. Journal of Structural Engineering 130(8): 1225–1232.

49.

Shi

Guo

Jeffers

(2013) Stochastic analysis of structures in fire by Monte Carlo simulation. Journal of Structural Fire Engineering 4: 37–46.

50.

Sosso

Andrade

Vieira Jr

, et al. (2020). Probabilistic modelling of the robustness of reinforced concrete frames accounting for material property variability using a layered beam finite element approach. Engineering Failure Analysis 118: 104789.

51.

Sosso

Paz Gutierrez

Berke

(2021) Computational analysis of reinforced concrete structures subjected to fire using a multilayered finite element formulation. Advances in Structural Engineering 24(15): 3488–3506.

52.

Sỳkora

Holickỳ

Jung

, et al. (2010) Structural Assessment of Industrial Heritage Buildings. Prague: CTU in Prague Klokner Institute, p. 155.

53.

Thienpont

Coile

Caspeele

, et al. (2021). Burnout resistance of concrete slabs: probabilistic assessment and global resistance factor calibration. Fire Safety Journal 119: 103242.

54.

Trezos

(1997) Reliability considerations on the confinement of rc columns for ductility. Soil Dynamics and Earthquake Engineering 16(1): 1–8.

55.

Udi

Almeshal

Megat Johari

, et al. (2022) Efficiency of high performance fiber reinforced cementitious composites as a retrofit material for fire-damaged concrete. Materials Today: Proceedings 61: 477–486.

56.

Van Coile

Bisby

(2017) Initial probabilistic studies into a deflection-based design format for concrete floors exposed to fire. Procedia engineering 210: 488–495.

57.

Van Coile

Annerel

Caspeele

, et al. (2011) Probabilistic analysis of concrete beams during fire. In: Applications of Structural Fire Engineering. ASFE 2011, pp. 127–132.

58.

Van Coile

Annerel

Caspeele

, et al. (2013a) Full-probabilistic analysis of concrete beams during fire. Journal of Structural Fire Engineering 4: 165–174.

59.

Van Coile

Caspeele

Taerwe

(2013b) The mixed lognormal distribution for a more precise assessment of the reliability of concrete slabs exposed to fire. In: Proceedings of the 2013 European Safety and Reliability Conference, Vol 29: 9.

60.

Van Coile

Caspeele

Taerwe

(2016) Post-fire safety of concrete columns, an engineering-oriented reliability-based assessment tool. Applications Of Structural Fire Engineering.

61.

Wang

Van Coile

Caspeele

, et al. (2018) A parametric study on concrete columns exposed to biaxial bending at elevated temperatures using a probabilistic analysis. In: High Tech Concrete: Where Technology and Engineering Meet. Springer, pp. 1662–1670.

62.

Xiao

König

(2004) Study on concrete at high temperature in China-an overview. Fire safety journal 39(1): 89–103.

63.

Zhenhai

(1993) Experimental investigation of strength and deformation of concrete at elevated temperature. Journal of Building Structures 14(1): 8.