A Simplified Method for Determining Fire Resistance of RC Columns Using Fire-Resistance-Column-Curves Approach

Abstract

Eurocode-2 (EC2) empirical equation for fire resistance of RC columns is very sensitive to the value of column capacity at normal temperature conditions $N_{Rd}$. Techniques to determine $N_{Rd}$ accurately for eccentric slender columns are difficult and computationally demanding; thus, adopting simplifications leads to unsatisfactory results in many column cases. Another shortcoming of EC2 equation is that it does not include an explicit term regarding the effect of load eccentricity on fire resistance. In this paper, a simplified method, as an attempt to overcome EC2 method defects, is developed to determine the fire resistance of RC columns using fire-resistance-column-curves. A rational numerical model is used to analyze various series of RC columns with different geometric, material, and loading properties at elevated temperatures. The results of the numerical study are utilized to construct different fire-resistance-column-curves from which simplified design equations are developed to predict the fire resistance of fixed- and pinned-end RC columns. The validity of the method is established with the aid of experimental data and it was found that in most cases, there is good agreement between assessed and test columns. It was also found that the proposed equations provide sufficiently safe predictions when appropriate material safety factors are adopted. The applicability of the proposed method to fire resistance design of RC columns is illustrated through numerical examples.

1 Introduction

The provision for sufficient fire resistance for reinforced concrete columns is a fundamental subject in engineering design and is required by most building codes. This resistance to fire is determined primarily with the aid of performance of an isolated column subjected to a standard fire test. Although this fire test may not be representative of an actual fire, it is generally considered as being essential to provide a basis for comparison between different designs and to satisfy the need for reproducibility in test data. The considerable cost in conducting a standard fire test of even a simple column, however, led the researchers to develop thermo-mechanical models to predict analytically the behavior of reinforced concrete columns subjected to fire. Although these models are powerful tools for predicting accurately fire resistance of RC columns, they depend on computer programs, which in many cases are complex, required effort, and can not be implemented in design codes. In practice, designers require simpler methods, which are sufficiently accurate, to be applicable to the large variety of column problems.

When compared to other methods, simple calculation methods serve well in this subject. These simplified methods are based on theoretical and empirical equations, and consider the effects of various parameters on fire resistance. These equations are generally developed using data utilized from analytical and experimental analyses whereby the limited data are used to define empirical constants employed in the equations. These methods furnish a simple design basis not only for column problems in the available data range, but also for extrapolating, within reasonable limits, to situations that are not covered by the database.

Dotreppe et al. [1] proposed a theoretically equation-based method to calculate the ultimate capacity of RC columns at high temperatures. Their formula was developed later by Franssen and Dotreppe [2] to take into consideration the influences of smaller slenderness ratios $\lambda \,$ and circular shape of the column on fire resistance. The basic equations of this method are expressed as follows:

$$N_{u} (t)\, = \,\gamma (t).\eta (\lambda )\,.\,P(t)$$

(1.a)

$$\eta (\lambda )\, = \,\frac{\chi (\lambda )}{{\phi (\lambda )}}$$

(1.b)

where $N_{u} (t)$ is the ultimate axial capacity at a fire duration t, $\gamma (t)$ and $\chi (\lambda )\,$ are spalling and buckling coefficients, and $\phi (\lambda )\,$ is a nonlinear amplification factor that accounts for load eccentricity. The fire resistance in this method can be obtained by performing an iterative process.

Regarding empirically equation-based methods, Franssen [3] suggested a simple equation, which has been later incorporated in Eurocode (EC2-1-2-2004) [4]. This method considers the fire resistance of columns $R_{f}$ in terms of fire ratings as

$$R_{f} \, = \,\,120.\left( {\,\frac{{R_{\eta fi} \, + \,R_{a} \, + \,R_{L} \, + \,R_{b} \, + \,R_{n} }}{120}} \right)^{1.8}$$

(2)

in which $R_{\eta fi}$, $R_{a}$, $R_{\,L}$, $R_{b}$, and $R_{n}$ are the fire ratings values, which account for load ratio, concrete cover thickness, column length, cross-section size, and longitudinal reinforcement, respectively.

Another simple method, in the form of an empirical equation, was developed by Kodur and Raut [5] to evaluate fire resistance of RC columns. According to this method, the fire resistance $R_{f}$ is expected as

$$R_{f} \, = \,C_{t} \left[ {8 \times k_{cp} \times k_{ec} \times (30\, - \,(S_{R} \, + \,5) \times (L_{R} \, - \,0.2))} \right]^{0.94}$$

(3)

where $C_{t}$ is a constant that accounts for aggregate type,$k_{cp}$ is a parameter that depends on the cover thickness and the percentage steel, $k_{ec}$ is a factor that accounts for load eccentricity, load ratio $L_{R}$, and slenderness ratio $S_{R}$.

Although the authors [1,2,3,4,5] argued that these methods yield safe predictions, these equations still have some drawbacks. Buch and Sharma [6] observed that equations proposed by Refs. [4] and [5] give unsafe results for larger eccentricity values. In addition, fire resistances below 100 min predicted using EC2 equation are mostly unsafe. Moreover, Raut and Kodur equation [5] results in inaccurate predictions in the case of varying reinforcement configurations. In an investigation carried out by Mahmoud [7], similar observations have been indicated. He observed that the abovementioned three methods provide unsatisfactory predictions at certain column cases. In addition, predictions obtained from Ref. [1] are generally conservative. Moreover, equations proposed by Refs. [3] and [4] lead to unsafe results at higher eccentricities and higher load ratios. Furthermore, EC2 equation may provide unsafe predictions at medium and high slenderness ratios. In general, these methods proposed one equation that used at all end conditions in which the influence of end conditions is considered by multiplying the column length by the effective length factor at ambient temperature. However, a study conducted by Mahmoud [8] revealed that the value of effective length factor at high temperature is different from its value at ambient temperature. In addition, effective length factor depends on the slenderness ratio and load eccentricity.

The main contribution of this paper is that it proposes an alternative simple calculation method, which considers appropriate measures as an attempt to overcome the previous simplified methods limitations, to predict fire resistance of RC columns subjected to fire. Whereas previous studies on column fire resistance have focused on developing one empirically- or theoretically-based equation for pin-ended column in which the column length parameter is modified by multiplying by the effective length factor to account for fixed-end conditions; this paper pays attention to consider the effects of end conditions individually and, therefore, presents two different equations for pin-ended and fixed-end columns, which adds to the literature. Another important contribution is that this study is the first to use the concept of fire-resistance-column-curves in which all the influential parameters are related to the slenderness ratio to attain more logical predictions. The equations proposed in this paper are simple and can be safely used for every day design practice. Parametric studies are conducted using a rational numerical model; then, the obtained results are utilized to construct column curves for various parameters; consequently, two different equations are derived using curve-fitting technique. To improve the accuracy of the method, the column curve equation of pinned columns is modified by considering the experimental data. The validity of the method is established with the aid of experimental data obtained from literature. Finally, illustrative examples are presented to explain the applicability of the proposed method.

2 Fire Resistance Column Curves Method

The controversy about the existed simplified methods has always been rooted in the question of the proper representation of fire resistance of all column cases. One of the basic issues of these methods lies in the assumption that the utilized experimental and numerical data in deriving the simplified equations, which mostly based on results of medium length columns, is enough to provide accurate predictions for all column lengths.

In this context, column curves technique furnishes a rational tool to obtain reasonable predictions because these curves are constructed based on a wide range of column lengths. For simplicity, Fire Resistance Column Curves method used in the current study will be denoted FRCC method.

2.1 Concept of Fire Resistance Column Curves

For any RC column of specific geometric, material, and loading properties, and of specific end conditions, a unique fire-resistance-column-curve exists. This curve describes the relationship between fire resistance $R_{f}$ and slenderness ratio $\lambda$; its shape is dependent on all parameters that affect fire resistance. Figure 1 shows two column curves for a RC column with pin-ended and fixed-end conditions. It can be noticed that as slenderness ratio decreases, fire resistance increases and the influence of end condition on fire resistance decreases until a certain value of $\lambda$ is reached at which the effect of end condition diminishes. For RC columns, this value of $\lambda$ was found about 4.3 [8]. This value of $\lambda$ can be treated as the minimum slenderness ratio that affects fire resistance of RC columns and will be denoted by $\lambda^{\min }$. At $\lambda^{\min }$, fire resistance is maximum. It is a characteristic for a specific column since it is independent on the column end conditions. This characteristic fir resistance will be denoted by $\overline{R}_{f}$.

Figure 1

Schematic of two fire resistance column curves for fixed-end and pin-ended column

Experimental-related simplified approaches to determine fire resistance of RC columns have been shown to provide an unsatisfactorily and a much too simple column curves, which are generally described by first-order polynomial curves [7]. However, it would generally not be economical to test a sufficiently large number of columns to construct experimentally fire-resistance-column-curves for a spectrum of possible designs. The theoretically based methods thus provide more practicable means of studying the variation of fire resistance and of constructing the column curves.

2.2 Construction of Column Curves

Numerical analyses are carried out herein to construct fire-resistance-column-curves for different series of RC columns. The influential parameters that are taken into considerations are slenderness ratio, load eccentricity, initial imperfections, load ratio, column size, reinforcement ratio, and concrete cover thickness. Geometric, material, and loading properties of the analyzed column are listed in Table 1, 2, 3, 4, and 5. A two-dimensional heat and mass transfer model [9] was used to predict temperatures inside the columns at different fire exposure times. All columns were analyzed by exposing entire perimeter to ASTM-E119 [10] or ISO 834 [11] standard fire exposure. Concrete thermal conductivities proposed by ASCE manual [12] were used in the analysis. Other temperature dependent properties were considered based on the relationships presented in Eurocode 2 [4]. A rational numerical model [7, 13] was developed by the author and utilized to perform structural analysis. The main feature of the structural model is that it incorporates the nonlinear behavior of RC sections at high temperatures. In addition, the model adopts an iterative technique to obtain the strain distributions on the heated concrete section using Newton–Raphson method. Moreover, the model includes a simple and reliable methodology to determine the lateral deflection of RC columns with different rotational end restraint levels. Furthermore, the model considers material degradations due to elevated temperatures and takes into consideration different time-dependent effects, strain components, and the nonlinear responses of slender columns. Model output includes axial and lateral deformation, bending moment, slope, curvature, and stiffness distributions at different sections through the column length. In structural analysis, the column was divided into 20 segments through its length and each cross-section was further subdivided into 60 × 60 square elements.

Table 1 Properties and Results for RC Columns Used in the Parametric Study at Different Initial Imperfection, Eccentricity, Load Ratio, and Concrete Cover Conditions: Series 1–5

Table 2 Properties and Results for RC Columns Used in the Parametric Study at Different Eccentricity, and Loading Ratio Conditions: Series 6, 7, and 8

Table 3 Properties and Results for RC Columns Used in the Parametric Study at Different Initial Imperfection, Eccentricity, and Reinforcement Conditions: Series 9 and 10

Table 4 Properties and Results for RC Columns Used in the Parametric Study at Different Initial Imperfection, Eccentricity, and Size Conditions: Series 11 and 12

Table 5 Properties and Results for RC Columns Used in the Parametric Study at Different Initial Imperfection, Eccentricity, and Reinforcement Conditions: Series 13 and 14

The resulted fire resistances for various column series are listed in Table 1, 2, 3, 4, and 5. For comparison purposes, each column curve is plotted in terms of the nondimensional quantities $\frac{{R_{f} }}{{\overline{R}_{f} }}$ where $\overline{R}_{f}$ is the characteristic fire resistance corresponding to that curve. Figures 2 and 3, respectively, show the band of column curves that has been developed for the fixed-end and pin-ended columns. The width of the band is largest for the intermediate slenderness ratios, and tapers off towards the ends. For low slenderness ratios, the variation of the maximum fire resistance is influenced more by yielding and crushing than any other factor.

Figure 2

Fire-resistance-column-curves band and arithmetic mean for fixed-end column case

Figure 3

Fire-resistance-column-curves band and arithmetic mean for pin-ended column case

It should be noted that the best solution is that one whereby every column could be represented by its own fire resistance curve. However, this would complicate the method in that the practical advantages might be lost. The suitable number of column curves therefore should be such that an optimum of rationale and practicality can be achieved.

It can be observed from Figures 2 and 3 that the number and the density of the curves between the upper and lower limits prevent a meaningful illustration of each separate curve. In addition, the resulted curves in some cases overlaid and in other cases overlap each other and thus, the band width of all columns in each figure is relatively small. Therefore, the curves for each end condition case may be represented by one individual column curve.

The concept of using two individual column curves for pinned- and fixed-end conditions therefore lies in the fact that no one column curve can represent the fire resistance of all columns rationally and adequately. It is to be noted that the use of an appropriate single column curve for each end condition case would not significantly over- or underestimate the fire resistance of many columns. In addition, the difference between the actual and the assessed fire resistance will not be completely eliminated, but rather reduced to an acceptable level. The most important information in Figures 2 and 3 is given by the arithmetic mean curves, which show the gradual shifting of the means; from being located in the vicinity of the upper envelope curve at low $\lambda$-values, to being closer to the median at high $\lambda$-values in the case of fixed-end columns; and from being closer to the upper envelope curve at low $\lambda$-values, to being closer to the lower envelope curve at intermediate and high $\lambda$-values in the case of pin-ended columns.

The bands of column curves shown in Figures 2 and 3 were analyzed statistically throughout the range of slenderness ratios between 4.3 and 94; the results of the statistical computations are show in Figure 4. The mathematical expressions representing the arithmetic mean curves, illustrated in Figure 4, provide a simplified and practical solution that can be easily used. These curves were obtained by originating at the point where ${{R_{f} } \mathord{\left/ {\vphantom {{R_{f} } {\overline{R}_{f} }}} \right. \kern-0pt} {\overline{R}_{f} }} = 1.0$ and $\lambda = \lambda^{\min } = 4.3$. The obtained relationship for fixed-ended columns can be written as follows

$$k_{\lambda }^{F} \, = \,\frac{{R_{f} }}{{\overline{R}_{f} }}\, = \, - 8.43 \times 10^{ - 10} \lambda_{e}^{4} \, + \,7.788 \times 10^{ - 7} \lambda_{e}^{3} \, - \,1.105 \times 10^{ - 4} \lambda_{e}^{2} \, - 1.176 \times 10^{ - 3} \lambda_{e} \, + \,1.0\,$$

(4)

Figure 4

Proposed column curves for fixed-end and pin-ended column cases

For pin-ended column, the relationship can be expressed as

$$k_{\lambda }^{P} \, = \,\frac{{R_{f} }}{{\overline{R}_{f} }}\, = \, - 2.305 \times 10^{ - 8} \lambda_{e}^{4} \, + \,6.008 \times 10^{ - 6} \lambda_{e}^{3} \, - \,4.797 \times 10^{ - 4} \lambda_{e}^{2} \, + 1.671 \times 10^{ - 3} \lambda_{e} \, + \,1\,.0$$

(5)

where $\lambda_{e} = \,\lambda - 4.3$

Equations (4) and (5) may now be used to find the fire resistance $R_{f}$ of a column, provided that the characteristic fire resistance $\overline{R}_{f}$ and the slenderness ratio $\lambda$ are given.

The principal shortcoming of EC2 equation lies in the fact that techniques to determine the exact value of $N_{Rd}$, the design resistance (capacity) of the column at normal temperature conditions, for eccentric slender columns are very complex and subject to high overheads. A major defect associated with this shortcoming is that EC2 equation does not provide reasonable predictions for all column lengths when simplifications are adopted to calculate $N_{Rd}$. Determination of capacity of slender columns not only exhibit difficulty due to the presence of non-linear stresses resulted from external applied loads, but also due to the presence of initial imperfections. Finding capacity of slender columns considering the combined effect of non-linear stresses, initial imperfections, and second order deflection is rather complicated. The simplified approaches, which is valid only for stocky columns, can not be used here. Instead, other approaches, such as model-column approach or load–deflection approach must be utilized and thus recourse to numerical and iterative methods is inevitable. This shortcoming can be alleviated by modifying EC2 equation by adopting the column curve technique, which essentially means that $N_{Rd}$ is calculated for very short columns and therefore the term including the column length in the equation will be adjusted using Equations (4) and (5).

It has to be noted that the key solution for the proposed FRCC method is the characteristic fire resistance $\overline{R}_{f}$ because it includes almost all the properties of the assessed column. However, there are no experimental fire tests in literature for very small slenderness ratios, particularly at $\lambda^{\min } = 4.3$, regarding fire resistance of RC columns. Therefore, $\overline{R}_{f}$ should be theoretically determined.

Following the recognition that other simplified methods provide unsatisfactory prediction even for short columns [7], the simple characteristic of EC2 equation may form the bases of calculating $\overline{R}_{f}$ since it provides safe predictions for small slenderness ratios.

To establish the applicability of EC2 equation in determining $\overline{R}_{f}$, a comparative study was performed. The analyzed columns were of 305 × 305 mm cross-section and 4.3 slenderness ratio. The concrete and steel were of 40 MPa and 420 MPa compressive and yield strengths, respectively. The investigated parameters were load ratio, cover thickness, steel percentage, and column size. The characteristic fire resistance $\overline{R}_{f}$ was calculated using EC2 equation and the results compared with those obtained using the numerical model as shown in Figures 5, 6, 7 and 8. It is clear that there is good agreement between predictions obtained using Eurocode equation and model results, which indicates that this equation could be safely used to calculate $\overline{R}_{f}$.

Figure 5

Effect of load level on characteristic fire resistance

Figure 6

Effect of concrete cover thickness on characteristic fire resistance

Figure 7

Effect of reinforcement ratio on characteristic fire resistance

Figure 8

Effect of cross-section size on characteristic fire resistance

Another shortcoming of Eurocode equation is that it does not account for load eccentricity even though it is stated that the equation is applicable for eccentricity ratio $e/b$(the ratio of the load eccentricity to the cross-section size) up to 0.40. In fact, EC2 equation implicitly account for load eccentricity by taking its impact on $N_{Rd}$ in the case of slender columns. To determine appropriately the characteristic fire resistance at slenderness ratio $\lambda = 4.3$, EC2 equation should be modified by adding a parameter to account for load eccentricity.

Indeed, the effects of load eccentricity, load ratio, and slenderness ratio on fire resistance are interrelated. Previous studies on intermediate slender columns revealed that fire resistance decreases by about 10% to 24% for every 10% increase in eccentricity ratio $e/b$. In a study performed by Raut and Kodur [14], it was found that at 50% load ratio, fire resistance decreases by about 24% for every 10% increase in eccentricity ratio $e/b$. Mahmoud [13] observed that up to an eccentricity ratio of 0.17, the drop in fire resistance in the case of column bent in double curvature is about 10%, whereas it reaches 20% in the case of column bent in single curvature, for each 10% increase in eccentricity ratio. To establish the influence of load eccentricity on the characteristic fire resistance $\overline{R}_{f}$, an investigation was carried out with the aid of the numerical model. The properties of the analyzed columns and the obtained results are shown in Table 6. To eliminate the effect of column size, the load eccentricity is included in terms of the eccentricity ratio $e/b$. The effect of slenderness ratio is already eliminated because the results in Table 6 are for columns of $\lambda = 4.3$. The variation of characteristic fire resistance with respect to load eccentricity individually (load ratio up to 50%) was obtained and the non-dimensional characteristic fire resistance was plotted against the eccentricity ratio as shown in Figure 9. It can be seen that at this range of load ratio, eccentricity ratio less than 0.05 has no effect on $\overline{R}_{f}$. For eccentricity ratio more than 0.05, however, $\overline{R}_{f}$ decreases by about 14% for each 0.1 increase in eccentricity ratio. From the above, the effect of eccentricity on characteristic fire resistance can be expressed as

$$\begin{gathered} \overline{R}_{f} \, = \,k_{e} .\overline{R}_{f}^{0} \hfill \\ k_{e} = \,1.0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,e\,/b\, \le \,\,0.05 \hfill \\ k_{e} \, = \,1\, - \,1.43\left( {\frac{e}{b}\, - 0.05} \right)\,\,\,\,\,\,\,\,0.05\,\, < \,\,e\,/b\, \le \,\,0.17 \hfill \\ \end{gathered}$$

(6)

where $\overline{R}_{f}^{0}$ is the characteristic fire resistance at zero eccentricity, $k_{e}$ is the load eccentricity coefficient, and $b$ is the dimension of column in direction of e.

Table 6 Properties and Results for RC Columns Used to Determine the Influence of Load Eccentricity on Fire Resistance

Figure 9

Variation of characteristic fire resistance with respect to eccentricity

To account for the modifications in EC2 equation regarding the influence of slenderness ratio in the characteristic fire resistance, Equation (2) can be rewritten as follows

$$\overline{R}_{f}^{0} \, = \,\,120.\left( {\,\frac{{R_{{\eta fi,\lambda^{\min } }} \, + \,R_{a} \, + \,R_{{L^{\min } }} \, + \,R_{b} \, + \,R_{n} }}{120}} \right)^{1.8}$$

(7)

The parameters $R_{{\eta fi,\lambda^{\min } }}$, $R_{a}$, $R_{{\,L^{\min } }}$, $R_{b}$, and $R_{n}$ can be calculated as follows

$$R_{{\eta fi,\lambda^{\min } }} \, = \,83\left( {1.0 - \mu_{{fi,\lambda^{\min } }} \frac{{\left( {1 + \omega } \right)}}{{\left( {0.85/\alpha_{cc} } \right) + \omega }}} \right)$$

(8)

in which $\mu_{{fi,\lambda^{\min } }} = \frac{{N_{Ed,fi} }}{{N_{{Rd,\lambda^{\min } }} }}$. The term $N_{Ed,\,fi}$ represents the design axial load (or failure load) in the fire situation, and $N_{{Rd,\lambda^{\min } }}$ is the design resistance (or ultimate capacity) of the column at normal temperature condition and minimum slenderness ratio. The term $\omega$ can be determined from $\omega \, = \,{{A_{s} \cdot f_{yd} } \mathord{\left/ {\vphantom {{A_{s} \cdot f_{yd} } {A_{c} \cdot f_{cd} }}} \right. \kern-0pt} {A_{c} \cdot f_{cd} }}$ where $f_{yd}$ and $f_{cd}$ are the design yield and compressive strength for steel and concrete, respectively.

For very short columns acted on by concentric load (slenderness ratio 4.3), the design resistance at zero eccentricity $N_{{Rd,\lambda^{\min } }}^{0}$ can be simply calculated as

$$N_{{Rd,\lambda^{\min } }}^{0} \, = \,A_{s} \frac{{f_{yk} }}{{\gamma_{s} }}\, + \,A_{c} \frac{{0.85f_{ck} }}{{\gamma_{c} }}$$

(9)

The ultimate capacity at zero eccentricity can be obtained by replacing $f_{ck}$ by $f_{cm}$.

When the load is eccentrically applied, the problem would be much complex; thus, a specific calculation is required to obtain $N_{{Rd,\lambda^{\min } }}$. Therefore, an investigation was carried out to find a simple expression regarding the impact of load eccentricity on the design resistance (ultimate capacity) at $\lambda^{\min }$. Parameters that are considered include steel ratio, column size, and concrete compressive strength. Properties of the analyzed column and results are shown in Table 7. The ratio $\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}$ was calculated and plotted against $e/b$ as shown in Figure 10. It can be noticed that up to an eccentricity ratio $e/b$ of 0.33, the obtained relationships are very close to each other. Consequently, the results were curve-fitted and the obtained arithmetic mean can be described by

$$k_{R} \, = \,\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}\,\, = \, - 8.49\left( \frac{e}{b} \right)^{3} \, + \,6.91\left( \frac{e}{b} \right)^{2} \, - \,\,3.049\frac{e}{b}\, + \,1.0$$

(10)

Table 7 Properties and Results for RC Columns Used to Determine the Influence of Load Eccentricity on Ultimate Capacity of Very Short Columns at Zero Temperature

Figure 10

Variation of ultimate capacity with respect to eccentricity ($\lambda$ = 4.3)

Equation (10) furnishes a method of estimating $N_{{Rd,\lambda^{\min } }}$, the capacity of very short columns subjected to an eccentric load, given the capacity at zero eccentricity and the load eccentricity.

The value of $R_{a}$ is calculated as

$$R_{a} \, = \,1.6\left( {a - 30} \right)$$

(11)

where a is the distance between the center point of steel rebar and the nearest exposed surface in mm.

The term $R_{{\,L^{\min } }}$ is obtained by

$$R_{{L^{\min } }} \, = \,9.6\left( {5 - L^{\min } } \right)$$

(12)

where $L^{\min }$ is the shortest effective length of the column that corresponds to $\lambda^{\min } = 4.3$. It can be determined as $L^{\min } = 4.3.r$ where $r$ is the radius of gyration.

The fire rating for cross-section size is

$$R_{b} \, = \,0.09b^{\prime}$$

(13)

in which $b^{\prime}$ is determined by $b^{\prime} = {{4A} \mathord{\left/ {\vphantom {{4A} p}} \right. \kern-0pt} p}$ with A and p are the area and perimeter of cross-section.

Finally, the term $R_{n}$ will be

$$\begin{gathered} R_{n} \, = \,0\,\,\,\,\,\,\,\,\,\,\,\,\,for\,n \ge \,\,4 \hfill \\ R_{\,n} \, = \,12\,\,\,\,\,\,\,\,\,\,\,\,\,for\,n < \,4 \hfill \\ \end{gathered}$$

(14)

The design fire resistance $R_{f}$ can be determined as

$$R_{f} \, = \,k_{\lambda } .k_{e} .\overline{R}_{f}^{0}$$

(15)

3 Evaluation

3.1 Evaluation of the Proposed Method by Comparison with Experimental Data

The evaluation of the proposed method is established by comparing its predictions with experimental data found in literature [3, 15, 16]. This comparison demonstrates good examples of different columns behaviors because it was performed over a broad range of section properties and loading conditions. The geometric, material, loading properties, as well as the results of the tested columns are shown in Table 8, 9, and 10. For all columns, the values of $\lambda$, $\lambda_{e}$, $L^{\min }$, and $k_{e}$ were calculated and then, $\overline{R}_{f}^{0}$ and $\overline{R}_{f}$ were determined using Equations (6) and (7). Consequently, the developed column curves (Equations (4) and (5)) and $\frac{{R_{f,test} }}{{\overline{R}_{f} }} - \lambda$ relationships in the cases of fixed- and pinned-end conditions were plotted as illustrated in Figures 11 and 12, respectively. It shall be mentioned that the test columns have a broader range of eccentricity ratio ($\,0.0\,\, \le \,\,e\,/b\, \le \,\,0.5$) than that used to construct the column curves ($\,e\,/b\, \le \,\,0.17$), which might help to demonstrate the usefulness of FRCC method to assess columns with higher eccentricity.

Table 8 Properties and Results for Tested Pin-Ended RC Columns Used in Validation (Group-1 [3])

Table 9 Properties and Results for Tested Pin-Ended RC Columns Used in Validation (Group-2 [3])

Table 10 Properties and Results for Tested Fixed-End RC Columns Used in Validation [12, 13]

Figure 11

Comparison between the proposed fire-resistance-column-curve FRCC and experimental data for fixed-end columns

Figure 12

Comparison between the proposed fire-resistance-column-curve FRCC and experimental data for pin-ended columns

It can be seen that in both cases, the distribution of the experimental results tends to follow a similar trend to that of the developed column curve. In addition, the developed column curve for fixed-end columns represents approximately a median for the results obtained from experimental data (Figure 11). This finding is clearly emphasized by Figure 13 where the ratio $\frac{{R_{f,FRCC} }}{{R_{f,test} }}$ is plotted against $\lambda$ and the resulted average ratio is about 99%. On the other hand, the values of $\frac{{R_{f,test} }}{{\overline{R}_{f} }}$ in the case of short and medium pin-ended columns tend towards the developed column curves (Figure 12); thus, the curves interferes many test results up to a slenderness ratio of about 60. It can be said that up to a slenderness ratio of about 83, FRCC method still accurately predict some of the test data. For very slender pinned columns, however, ($\lambda$ close to 100) FRCC method is conservative (Figure 12). The impact of higher slenderness ratio and the pinned end conditions on the results is clear from Figure 14 where the average ratio (average-1) is only about 90% for columns with $\,\lambda \,\, \le \,\,100$ and $\,e\,/b\, \le \,\,0.17$. When the results are considered for columns of $\,\lambda \,\, \le \,\,83$ and $\,e\,/b\, \le \,\,0.17$, the average reaches 97%. When all tested pinned columns are considered ($\,\lambda \,\, \le \,\,100$ and $\,e\,/b\, \le \,\,0.5$), the average ratio decreases to 83%.

Figure 13

The ratio of the FRCC method predictions to experimental data for fixed-end columns

Figure 14

The ratio of the FRCC method predictions to experimental data for pin-ended columns

Although the above validation reveals that there is reasonably good agreement between predictions of the proposed method and test data that confirm the method capability for assessment, there are some inconsistencies, however, between the predicted and measured results, especially in the case of very slender pin-ended columns. A possible reason for these inconsistencies could be related to the uncertainties in different factors and material parameters that applied in experiments. Performing an ideal heating or loading process could not be optimal in all test cases. In addition, attaining a perfect idealized end condition may not be an easy job for slender pinned columns. Another possible reason could be the higher values of $e/b$ ratio in some column cases, which may lead to some errors in $k_{e}$ estimated by Equation (6). Another possibility is attributed to the difference in definition of failure between test and analysis. Accordingly, if it is of particular interest to reproduce the results of a fire test accurately using numerical analysis, it is imperative that factors used in the calculation be the same as those in the test. Such aim may not always be attained because of the disability to simulate accurately some parameters and the inherent randomness in others.

To improve the accuracy of the method in the case of pinned columns, the column curve illustrated in Figure 4 is modified to take into consideration the distributions of test data. Therefore, the modified curve is based on both numerical results and experimental data. The proposed equation for the new curve, as shown in Figure 15, takes the form

$$k_{\lambda }^{P} \, = \,\frac{{R_{f} }}{{\overline{R}_{f} }}\, = \, - 3.679 \times 10^{ - 8} \lambda_{e}^{4} \, + \,8.653 \times 10^{ - 6} \lambda_{e}^{3} \, - \,6.042 \times 10^{ - 4} \lambda_{e}^{2} \, - 3.493 \times 10^{ - 3} \lambda_{e} \, + \,1\,.0$$

(16)

Figure 15

Proposed modified column curve for pin-ended column case

It is clear from Figure 16 that the distributions of experimental data when compared to the modified column curve are better than those when compared to the old column curve. For pinned columns with $\,\lambda \,\, \le \,\,100$ and $\,e\,/b\, \le \,\,0.5$, the average ratio of ${{R_{f,FRCC} } \mathord{\left/ {\vphantom {{R_{f,FRCC} } {R_{f,test} }}} \right. \kern-0pt} {R_{f,test} }}$ reaches about 99%, as shown in Figure 17.

Figure 16

Comparison between the proposed modified fire-resistance-column-curve FRCC and experimental data for pin-ended columns

Figure 17

The ratio of the FRCC method predictions to experimental data for pin-ended columns (modified column curve)

It is now fairly well established that relations obtained using Equations (4) and (16) provide a good representation to many of the test data. This indicates that by using appropriate safety factors, the developed column curves can be safely used for design purpose.

It is sufficient to state that the solution of Equations (4) and (16), when considered as a design formulae, is simpler to engineers so long as the design value of the characteristic fire resistance, obtained using Equations (7) to (12), is available.

3.2 Comparison with the Results of EC2 Method

The fire resistance of columns listed in Table 8, 9, and 10 are calculated by Franssen [3] using EC2 method. In his calculation, $N_{Rd}$ values were exactly determined using model-column technique. The results were utilized and the ratios $\frac{{R_{f,EC2} }}{{R_{f,test} }}$ were obtained and depicted against $\lambda$ for fixed and pined columns, respectively, as shown in Figures 18 and 19. Comparing the results in Figure 18 with those in Figure 13 reveals that for fixed columns, EC2 and FRCC methods have approximately similar accuracy, with average ratios of 96% and 99% respectively. In addition, the upper and lower limits of the ratio are 144% and 68% for EC2 method, whereas, they are 141% and 83% for FRCC method. Similarly, for pinned columns, the ratios are comparable with an average ratio of 104% for EC2 and 99% and for FRCC (Figure 17).

Figure 18

The ratio of EC2 method predictions to experimental data for fixed-end columns

Figure 19

The ratio of EC2 method predictions to experimental data for pin-ended columns

It can also be noticed for pinned columns that the upper and lower limits of the ratio for EC2 method are 194% and 59%; whereas, they are 159% and 45% in the case of FRCC method. Results in Figure 20 reveal that for the columns with higher eccentricity ratio ($\,0.25\,\, \le \,\,e\,/b\, \le \,\,0.5$), FRCC method provides safe but conservative predictions in some cases. The ratios are in the range between 41% and 94% with a mean value 69%. On the other hand, EC2 method leads to unsafe results in many columns cases. It gives ratios between 81% and 194% with a mean value 117%. Although FRCC method leads to slight improvement in some cases when compared to EC2 method for columns with $\,e\,/b\,\, < \,\,0.25$, there is still good agreement between predictions of both methods. This is quite good result considering the large variability of tests. For columns with higher eccentricity ratio, further improvement would be achieved if higher $e/b$ values were taken in the calculation of $k_{e}$.

Figure 20

The ratio $\frac{{R_{f,\,predicted} }}{{R_{f,\,test} }}$ as a function of eccentricity ratio for pin-ended columns

It has to be mentioned that these averages are susceptible to changes if other test results are included. Moreover, the accuracy of EC2 method would be questionable if simplified techniques in the determination of $N_{Rd}$ were adopted.

4 Numerical Examples

4.1 Fire-Resistance-Column-Curves for Design

To verify the applicability of FRCC method for design, two column examples are presented. The first example is a fixed-end column CF subjected to a concentrated load; whereas, the second example is a pin-ended column CP acted on by an eccentric load. It is assumed that both columns have a cross-section of 250 mm × 250 mm with 35 MPa concrete characteristic compressive strength. The columns have a concrete cover of 48 mm and a length of 4000 mm. The main reinforcement for both columns is four No 20 mm bars with 420 MPa yield strength. The proposed material safety factors $\lambda_{s}$ and $\lambda_{c}$ are 1.15 and 1.5, respectively. Other geometric, material, and loading properties of the columns, as well as the calculation procedure, are shown in Table 11.

Table 11 Properties and Results for RC Columns Used in Illustrative Examples

For the column in first example, the eccentricity is zero and thus, the design resistance $N_{{Rd,\lambda^{\min } }}$ = $N_{{Rd,\lambda^{\min } }}^{0}$ = 1673 kN. The load ratio $\mu_{{fi,\lambda^{\min } }}$ is calculated as 1000/1673 = 0.6. The characteristic fire resistance $\overline{R}_{f}^{0}$ is obtained from Equations (6) as 156 min. The coefficients $k_{e}$ and $k_{\lambda }$ are equal to 1.0 and 0.75, respectively; hence, the design fire resistance $R_{f} \, = \,k_{\lambda } .k_{e} .\overline{R}_{f}^{0}$ = 117 min.

For column CP, $N_{{Rd,\lambda^{\min } }}^{0}$ equals 1673 kN. Because the load is eccentrically applied, $k_{R}$ is 0.8 and consequently $N_{{Rd,\lambda^{\min } }}$ = 0.8 × 1673 = 1332 kN. The value of $\mu_{{fi,\lambda^{\min } }}$ is 1000/1332 = 0.75 and $\overline{R}_{f}^{0}$ equals 128 min. The values of $k_{e}$ and $k_{\lambda }$ are 0.96 and 0.50 ($k_{\lambda }$ is determined using Equation (16)), respectively, and the design fire resistance $R_{f}$ in this case is 61 min. For comparison, $k_{\lambda }$ is determined using Equation (5). Its values in this case is 0.47 and $R_{f}$ is 58 min. This reveals that at this range of $\lambda$, Equations (16) and (5) lead to close predictions.

The most striking result to emerge from these two examples is the impact of load eccentricity on the design fire resistance. It affects $R_{f}$ not only through the reduction factor $k_{e}$, but also through the multiplier $k_{R}$, which reduces the design resistance $N_{{Rd,\lambda^{\min } }}$, increases the load ratio $\mu_{{fi,\lambda^{\min } }}$, and consequently decreases $\overline{R}_{f}^{0}$. These two examples have further strengthened the conviction that end condition has a big influence in fire resistance. The coefficient $k_{\lambda }$ reduced from 0.75 when the column is fixed to 0.50 when the column is pinned, indicating that changing the end condition for this column from fixed to pinned reduced the fire resistance by about 48%.

5 Limitations

The proposed FRCC method presents a practical tool for determining the fire resistance of RC columns exposed to fire. As the method has been developed with respect to numerical studies, test data, and EC2 equation, the applicability of this method is limited to the range of parameters that were considered in these sources. Therefore, the limitations for the method are:

• Columns subjected to ASTM E119 standard fire, ISO 834 standard fire, or any design fire of similar profile as that of standard fire.

• Dimension of the column cross-section b (square or rectangular): up to 600 mm.

• Concrete cover: 25–80 mm.

• Percentage of longitudinal bars: 1%–4%.

• Concrete compressive strength: 24 MPa–53 MPa.

• Eccentricity ratio e/b: 0–0.4.

• Slenderness ratio $\lambda$: 0–100.

• Load ratio: 0.15–0.7.

6 Conclusions

In general, this study has primarily been concerned with developing an accurate and more rational solution of RC column design fire resistance problem. EC2 method does not have an explicit term that account for load eccentricity. In addition, a rigorous exact solution of design capacity at normal temperature conditions, which highly affects EC2 method predictions, is quite a formidable task. The theoretically based FRCC method, using the concept of fire-resistance-column-curves, thereby appears to be the method most viable and practical.

Based on the information presented in this study, the key findings can be concluded as follows:

• The results illustrated that by using an individual column curve for each end condition case, the deviation between the actual and the assessed fire resistance of the columns will be reduced significantly. Influential parameters such as slenderness ratio, load ratio, load eccentricity, cross-section size, concrete cover thickness, and reinforcement ratio imply a realistic basis for the computation. By explicitly taking into consideration the initial out-of-straightness, the solution appears to assume even further closeness to reality.

• Two equations were suggested to describe the proposed column curves. These equations are applicable for slenderness ratio in the range between 4.3 and 100, which cover most RC column lengths in practice.

• The characteristic fire resistance, which includes almost all the parameters that affect fire resistance, implies a realistic basis for the computation. This characteristic fire resistance is determined using EC2 equation at slenderness ratio of 4.3. The equation was modified by adding a parameter that accounts for the effect of load eccentricity.

• An equation was derived to calculate the column capacity at normal temperature conditions for very short eccentric columns. The proposed equation is simple, accurate, and helpful in determining the characteristic fire resistance.

• It was found that the resulted column curves represent a median for many experimental data. Therefore, these fire-resistance-column-curves can be safely used for design purpose when appropriate safety factors are adopted.

• Results revealed that for columns with smaller eccentricity ratio, explicitly taking the effect of load eccentricity in FRCC method leads to only slight improvement when compared to EC2 method. For columns with high eccentricity ratio, FRCC method is safe but conservative. EC2 method, however, is unsafe in many column cases.

• By using FRCC method, the difficulty in determining exact solution of column capacity at normal temperature conditions that encountered in EC2 method has been eliminated.

• The proposed FRCC method can be treated as a modification for EC2 method. The method is practical, simple, and can be conveniently used by engineers for every day design; thus, it can be easily incorporated in design codes.