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

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 \(\overline{R}_{f}\).

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.

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.

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.

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

For pin-ended column, the relationship can be expressed aswhere \(\lambda_{e} = \,\lambda - 4.3\)

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 \(\overline{R}_{f}\).

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 aswhere \(\overline{R}_{f}^{0}\) 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.

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

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

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

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 aswhere 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 bywhere \(L^{\min }\) 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 isin 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

The design fire resistance \(R_{f}\) can be determined as

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.

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%.

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

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.

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.

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).

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 \(k_{e}\).

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.

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.

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%.