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A simplified approach for predicting temperatures in insulated RC members exposed to standard fire
Fire Safety Journal 92 (2017) 80–90
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Fire Safety Journal j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fi r e s a f
A simplified approach for predicting temperatures in insulated RC members exposed to standard fire V.K.R. Kodur *, B. Yu, R. Solhmirzaei Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Reinforced concrete Fire resistance Insulated concrete member Cross sectional temperatures
Fire resistance of concrete structural members can be enhanced through the application of external fire insulation on the surfaces of concrete member. For evaluating fire resistance of such insulated RC members, temperatures in concrete and steel reinforcement are to be known. This paper develops a simplified approach for predicting crosssectional temperatures in an insulated RC structural member exposed to standard fire. The approach is derived by replacing the insulation layer into an equivalent concrete thickness layer and then undertaking statistical regression analysis on temperature data of modified concrete section. The effect of critical parameters, including geometry of concrete member and insulation, thermal properties of concrete and fire insulation, and duration of fire exposure is accounted for in temperature equations. The validity of the approach is established by comparing predictions from the proposed equation with data generated from fire tests and finite element analysis. These comparisons show the proposed equation gives reasonable prediction of temperatures, within a range of ±10%, in insulated concrete members. The applicability of the proposed approach in design situations is illustrated though a numerical example. The simplicity of the proposed method makes it attractive for use in design situations and for incorporation in design manuals.
1. Introduction Provision of appropriate fire resistance is a major design requirement for structural members in buildings. Reinforced concrete (RC) members generally possess good fire resistance and hence no external fire insulation is required. This is mainly attributed to the fact that concrete is a non-combustible material and has a low thermal conductivity. However, in special cases, concrete structural member may not achieve sufficient fire resistance as required in building codes. For example, an RC beam strengthened with fiber-reinforced polymer (FRP) laminates at the beam soffit usually has relative lower fire resistance as compared to conventional steel reinforced concrete beam, due to faster degradation of strength and stiffness properties of FRP at high temperatures [1–3]. Also, in some concrete members at critical locations (such as transfer beams), a high fire resistance is usually required to ensure safety of structures. Therefore, in these situations, external fire insulation is usually applied on the surface of concrete member to enhance fire resistance of such concrete members. Fire resistance analysis on an RC member involves two stages of analysis, i.e. heat transfer analysis followed by strength analysis. When a RC member is subjected to a fire, temperatures in concrete, steel and FRP
* Corresponding author. E-mail address: [email protected] (V.K.R. Kodur). http://dx.doi.org/10.1016/j.firesaf.2017.05.018 Received 16 October 2015; Received in revised form 23 May 2017; Accepted 25 May 2017 0379-7112/© 2017 Elsevier Ltd. All rights reserved.
reinforcement increase with time, resulting in loss of strength and stiffness in the member. This temperature rise is a function of fire characteristics, thermal properties of concrete (and insulation), and crosssectional dimensions. An approach to evaluate cross-sectional temperatures is through fire tests on RC members. Alternatively, design charts specified in codes and standards, such as ACI 216.1 [4] and Eurocode 2 [5] which are applicable to specific types of concrete, can be used to evaluate cross-sectional temperatures in RC members. In addition, there are few simplified approaches for evaluating cross-sectional temperatures, as a function of depth and fire exposure time, in uninsulated RC members [6–8]. Also, detailed thermal analysis on RC members, based on finite difference and finite element methods, can be carried out to obtain cross sectional temperatures. Externally bonded FRP are used in strengthening of RC members and in such cases, external fire insulation is to be provided to achieve required fire resistance in FRP strengthened RC members. Currently fire resistance of such FRP strengthened concrete members in different facilities is established through fire resistance tests [9]. A review of literature indicates that limited studies have been conducted on developing simplified approaches for evaluating temperature profile in insulated RC members. These studies mainly focused on advanced numerical modeling
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Fire Safety Journal 92 (2017) 80–90
Fig. 1. Area divisions in concrete members for temperature calculation.
techniques [1–3,10]. Through numerical models, cross sectional temperatures at critical locations, such as steel rebar, FRP, and concrete center, are generated at any given fire exposure time. However, these approaches are complex, time consuming, and require high level of expertise in numerical analysis. Thus, such detailed analysis may not be feasible in design situations. Therefore, there is a need for simple and reliable approaches for evaluating cross-sectional temperatures in an insulated RC member exposed to fire. Wickstrom and Hadziselimovic [11] have shown that thermal protection capacity of an insulation layer on a concrete member can be expressed by modifying the insulation thickness with an equivalent concrete layer. Such equivalent concrete thickness yields the same temperature distribution in RC member, as the corresponding insulation layer when the structure is exposed to fire. Based on numerical analyses, Wickstrom and Hadziselimovic derived a simple relation between thermal resistance of insulation layer and equivalent concrete layer. However, the proposed relation was not validated against any fire test data for accuracy of temperature predictions and also on its validity over wide range of protection thicknesses. To overcome this gap, empirical equations for predicting temperatures in insulated RC members exposed to standard fire is developed in this paper. The proposed equations are derived utilizing an approach originally developed for predicting temperature in a fire exposed uninsulated RC member [6].
where, t is the fire exposure time in hours, z and y are horizontal and vertical distance from a point in concrete section to the fire exposure surface in meters. The proposed equations are applicable for evaluating cross-sectional temperature when an RC member is subjected to standard fire exposure. atn in above equations represent fire temperature under standard fire exposure [6,12]. For ISO 834 fire, a ¼ 935 and n ¼ 0.168, and for ASTM E119 fire, a ¼ 910 and n ¼ 0.148. c1 and c2 are the coefficients to account for effect of concrete type (properties) on heat transfer where c1 is equal to 1.0, 1.01, 1.12 and 1.12 for NSC-CA, HSCCA, NSC-SA and HSC-SA, respectively; c2 are 1.0, 1.06, 1.12 and 1.20 for NSC-CA, HSC-CA, NSC-SA and HSC-SA, respectively. NSC, HSC, SA, and CA refer to Normal Strength Concrete, High Strength Concrete, and Siliceous Aggregate concrete, and Carbonate Aggregate concrete, respectively. These equations (Eqs. (1)–(3)) can be conveniently applied for predicting cross-sectional temperatures over wide ranging cross sectional dimensions of concrete members subjected to standard fire exposure. One of the major limitations of this approach is that it is only applicable for uninsulated RC members and it is not valid when RC members are protected with fire insulation. This approach does not take into account moisture content of concrete, which might result in slightly conservative predictions. In addition, the proposed equations may not precisely predict cross-sectional temperature when one side of the member is small as compared to the other side, since heat transfer, takes place from multiple fire-exposed sides. The above equations are validated against data generated from a large number of fire tests and finite element analyses [6]. Therefore, these equations are reliable to be used as basis for obtaining equations for cross-sectional temperatures in insulated RC members exposed to fire.
2. Formulae for predicting temperatures in an uninsulated RC member A review of literature indicates that there are a few simplified expressions for evaluating the cross-sectional temperature in an uninsulated RC member [7,8]. More recently, Kodur et al. [6] proposed a set of simplified equations for predicting cross sectional temperatures within an RC member exposed to a standard fire (such as ASTM E119 and ISO 834), and these equations overcome some of the drawbacks of previous empirical equations and provide better temperature predictions. As shown in Fig. 1, for a given point (coordinate is (y,z)) in an RC member, temperature at a given fire exposure time can be obtained as:
For 1 D heat transfer : Tc ¼ c1 ⋅ηz ⋅ðatn Þ In which ηz is given by ηz ¼ 0:155 ln
pffiffi 0:348 z 0:371 z1:5 t
3. Formulae for predicting temperatures in an insulated RC member A simplified formula for predicting cross sectional temperatures in a fire exposed insulated RC member is derived in this section. 3.1. General approach
(1)
Unlike a conventional RC member, temperature profiles of an insulated RC member are influenced by properties of both concrete and fire insulation. Also, thermal properties of concrete and fire insulation can vary significantly. Thus, predicting temperature in an insulated concrete member becomes a more complicated problem. In an RC member (with or without fire insulation) exposed to fire, thermal properties of concrete and insulation primarily influence crosssectional temperature, while thermal properties of steel rebar and FRP
(2)
For 2 D heat transfer : Tc ¼ c2 1:481 ηz ⋅ηy þ 0:985 ηz þ ηy n þ 0:017 ðat Þ (3) 81
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(laminate, strip or rod added to the surface or placed internally) does not significantly affect temperature rise in the member due to their small cross sectional areas as compared to concrete sectional area [13]. Thus, if the insulation layer on concrete member is converted to equivalent concrete layer, temperature profile in an insulated concrete section can be evaluated using the same temperature equation as that for uninsulated RC sections (Eqs. (1)–(3)). For this purpose a derivation of an approximate expression for transforming insulation into an equivalent concrete thickness is required.
Fig. 2 shows a typical RC beam protected by a U-shaped fire insulation, and the thickness of insulation on sides and bottom of the beam are zi and yi respectively. If that fire insulation layer is replaced by an equivalent concrete layer (with the thickness of zec and yec on sides and bottom) which has the same thermal resistance (inertia) as insulation layer, the temperature profiles remain the same within beam cross section after this transformation. Based on basic heat transfer principles, the following equations can be expressed for heat transfer within the insulation or its equivalent concrete layer.
3.2. Converting fire insulation layer to equivalent concrete layer
ki 2 ∂Ti Q ∇ Ti ¼ ðρcÞi ∂t ðρcÞi
Temperature rise in a structural member results from a fire source occurring through radiation, convection, and conduction. Heat transfer from fire source to the surface of a structural member is through convection and radiation. The heat flux to the surface of a structural member through radiation and convection is expressed as boundary conditions in thermal analysis. Heat transfer within a structural member is through conduction which is expressed as governing equation by Fourier heat transfer equation. Conduction occurs due to temperature difference between two locations. The equation describing this heat transfer is expressed as Eq. (4) [14].
q ¼ k∇T
kc 2 ∂Tec Q ∇ Tec ¼ ðρcÞc ðρcÞc ∂t
∂T ¼ ∇q þ Q ∂t
(4)
(5)
in which ρc is heat capacity of the section, t is time, ∇ is spatial gradient operator, and Q is internal heat generation rate per unit volume. Substituting Eq. (4) into Eq. (5) results in Eq. (6) [14].
k 2 ∂T Q ∇T¼ ρc ∂t ρc
(11)
(6)
The heat flux on the fire exposed boundary due to convection and radiation can be given by following equation [14,15].
qb ¼ ðhcon þ hrad Þ T Tf
Substituting left side of Eqs. (9) and (10) into Eq. (11) results in Eq. (12).
(7)
kc ∂2 Tec ki ∂2 Ti ≈ ðρcÞc ∂z2ec ðρcÞi ∂z2i
where hcon and hrad are convective and radiative heat transfer coefficients, and are defined as:
hrad ¼ 4σε T 2 þ Tf2 T þ Tf
(10)
where ki and ðρcÞi are thermal conductivity and heat capacity of insulation respectively, and kc and ðρcÞc are thermal conductivity and heat capacity of concrete respectively; Q is internal heat generation rate per unit volume; Ti and Tec are the temperatures in insulation and equivalent concrete respectively; and ∇ is the second order derivative of temperature (Ti or Tec ) with respect to the distance to fire exposed surface (z or y); and t is the time. Eqs. (9) and (10) can be applied for calculating temperature at any location within the insulation layer (equivalent concrete layer). At the same fire exposure time, temperature profiles within insulation layer and equivalent concrete layer, which have same thermal resistance, are similar. Moreover, Q which is internal heat generation rate per unit volume is zero, since there is no internal heat source (i.e. no chemical reaction in concrete). Therefore, the following assumption can be made.
where q is the vector of heat flux per unit area, k is thermal conductivity tensor, and T is temperature. The general heat transfer differential equation of a structural member can be expressed as Eq. (5) [14].
ρc
(9)
(12)
where z is the distance from fire exposed surface. Due to smaller thickness of fire insulation or its equivalent concrete layer, 1-D heat transfer dominates temperature distribution within the insulation or concrete layer. Thus the temperature equation corresponding to 1D heat transfer (Eq. (1)) can be applied to derive temperature in the concrete section ð Tec Þ in Eq. (11) and can be expressed as:
(8)
where Tf is fire temperature, σ is Stefan-Boltzman constant, and ε is emissivity factor. The boundary condition of unexposed surface is similar to that of exposed surface with replacing the fire temperature with the ambient temperature.
Fig. 2. Illustration of the equivalent concrete depth method. 82
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Table 1 Characteristics of insulated RC beams used in the regression analysis. Case No.
Section
Fire insulation (type and properties)
Point (location)
Fire exposure time
1
200 300 mm 250 400 mm 300 500 mm
“AESTUVER” k ¼ 0.185 W/m-K (ρc) ¼ 650kJ/m3-K
Corner steel rebar Middle steel rebar NSM FRP strip 1/4 length of FRP laminate Middle of FRP laminate Corner steel rebar Middle steel rebar NSM FRP strip 1/4 length of FRP laminate Middle of FRP laminate
0.5 1.5 2.5 3.5
2
200 300 mm 250 400 mm 300 500 mm
“Tyfo VG” k ¼ 0.116 W/m-K (ρc) ¼ 413kJ/m3-K
pffiffiffiffiffi t Tec ¼ 0:155 ln 1:5 0:348 zec 0:371 :ðatn Þ zec
t ≈ 0:155 ln 1:5 0:371 :ðat n Þ zec
0.5 1.5 2.5 3.5
h, 1 h, 2 h, 3 h, 4
h, 1 h, 2 h, 3 h, 4
where λ and η are the coefficients to be determined. Notice that Eq. (15) describes an approximate relation between insulation thickness ðzi Þ and the equivalent concrete thickness ðzec Þ. Since thermal properties (ki and ðρcÞi , kc and ðρcÞc ) of insulation and concrete vary with fire exposure time (or fire temperature), λ is assumed to be a function of time ðtÞ in hours, accounting for the influence of fire exposure time (or fire temperature) to thermal properties. Finally, the relation between insulation and concrete thickness are expressed as:
h, h, h, h
ffi zec p ¼ αt zi
h, h, h, h
(13)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffi β kc ðρcÞi α z ¼ zc þ zec ¼ zc þ zi t ðρcÞc ki 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffi β kc ðρcÞi α y ¼ yc þ yec ¼ yc þ yi t ðρcÞc ki 0
(14)
where a1 and a2 are the coefficients describing temperature distribution within the insulation layer. Importing Eqs. (13) and (14) into Eqs. (11) and (12), the following relation can be obtained for deriving the ratio of the thickness of insulation layer ðzi Þ to that of concrete layer ðzec Þ.
zec ¼ zi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc ðρcÞi η kc ðρcÞi η 0:155 ≈λ a1 ðρcÞc ki ðρcÞc ki
(16)
where α and β are the coefficients to be determined. Once the values of α and β are obtained, the insulation thickness on a concrete member can be converted to an equivalent thickness of concrete in that member. As an illustration, Fig. 2 shows a point (Point A) located in an insulated RC beam. The distances from Point A to side and bottom concrete surface of beam are zc and yc respectively, and insulation thickness on side and bottom surface of beam are zi and yi respectively. Applying the above equivalent concrete depth method, the equivalent depth from Point A ðz0 or y 0 Þ to fire exposed surface can be expressed as:
Temperature distribution within insulation ðTi Þ follows similar trend to that within concrete (Eq. (13)). Therefore, in the insulation layer
t Ti ≈ a1 ln 1:5 a2 :ðat n Þ z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc ðρcÞi β ðρcÞc ki
(17)
(18)
3.3. Formulae for temperatures in insulated concrete members The equivalent distances, z0 and y 0 , expressed in terms of concrete depth (or width), can be substituted into temperature equations (Eqs. (1)–(3)) in an RC member, to obtain temperature rise in an insulated RC member, as shown below.
(15)
For 1 D heat transfer : Tc ¼ c1 ⋅ηz ⋅ðat n Þ
Fig. 3. FRP strengthened RC beams used in FEA for regression and validation (Units: mm). 83
(19)
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Fig. 4. Comparison of predicted temperatures from the proposed equations (Eqs. (17)–(22)) with those from FEA (Beam 200 300 mm).
where; ηz ¼ 0:155 ln
t ðz0 Þ1:5
pffiffiffi 0:348 z0 0:371
Fig. 6. Comparison of predicted temperatures from the proposed equations (Eqs. (17)–(22)) with those from FEA (Beam 300 500 mm).
element program validated in previous research [6]. The characteristics of these RC beams were varied over a range of parameters as shown in Table 1 and Fig. 3. The width of beam section was varied from 200 to 300 mm, while the depth was varied from 300 to 500 mm. Two types of fire insulation, Aestuver and Tyfo VG (Vermiculite Gypsum) insulations, are assumed to be applied on the beams, and the properties of these fire insulation are tabulated in Table 1. The variation of thermal properties (thermal conductivity, heat capacity, and density) of fire insulation with temperature follows previously reported values [16]. Five different points within the beam section were selected for monitoring temperatures in each beam, and these included typical locations of steel and FRP reinforcement, as shown in Fig. 3. These points represent critical points from fire resistance consideration in a simply supported FRP strengthened RC beam, where steel and rebars are close to fire surface as opposed to compression zone where is farther from fire surface. Due to this reasoning, the temperatures in concrete remain below 500 C for most of the fire duration and FRP properties become more critical as it is sensitive to elevated temperatures due to low tolerance of the polymer matrix to high temperatures. In addition to rapid strength and stiffness degradation of FRP, the bond between FRP and concrete, which is critical to maintain FRP's effectiveness, gets severely reduced at temperatures above glass transition temperature. Once FRP loses its tensile strength and bond at higher temperatures, the rebar temperatures become an important indicator of fire resistance of simply supported FRP-strengthened RC beams [3,17]. Thus FRP and tensile steel rebars are critical points in determining fire resistance of simply supported FRP-strengthened RC beams. In the finite element analysis, each beam was subjected to ASTM E119 fire exposure from three sides for 4 h, and temperature data at each time interval of 0.5 h was output for regression analysis. In total, 240 temperature data points (3 (beams) 2 (insulation types) 5 (points) 8 (time intervals)) with corresponding time and location were generated for regression analysis on temperature equations. A regression analysis was performed using “Solver” function in Excel (2010) to develop an expression for converting insulation to an equivalent concrete layer (Eq. (16)). The “solver” function is able to calculate the optimum coefficients to match the original data with a given format of formula, and applied “constraint” criteria. Since the regression analysis can hardly match all the data points closely, it is necessary to fit the data points in the critical range with the smallest discrepancy, and that have reasonable match in other regions. From the fire resistance point of view, it is well established that the strength of reinforcing steel, FRP, and concrete are not significantly influenced up to 150 C, and there is little interest in temperature data in concrete members beyond 800 C. Therefore, the regression results have to be highly reliable or slightly conservative in temperature-sensitive zone of 150–800 C. In contrast,
(20)
For 2 D heat transferTc ¼ c2 1:481 ηz ⋅ηy þ 0:985 ηz þ ηy þ 0:017 ðatn Þ (21) where t is the fire exposure time in hours, z0 is the distance from the point in concrete section to fire exposed surface using the equivalent concrete depth method (Eqs. (17) and (18)). ηz and ηy are the heat transfer factors resulting from z0 and y 0 side fire exposure, ηy is calculated in the same manner as that of ηz in Eq. (20). The definitions of other symbols (terms) follows the same definition presented in Section 2. Eqs. (17)–(21) provide a simplified expression for evaluating temperatures within an insulated RC beam. The only unknowns to be determined in Eqs. (17) and (18) are α and β. These two unknowns are obtained through nonlinear regression analysis on a large set of database generated using finite element analysis (FEA). 4. Regression analysis A regression analysis was carried out to obtain correlation coefficients α and β, in the above developed empirical equations for predicting temperatures in an insulated RC beam. To generate temperature data for regression analysis, three representative RC beams protected with two types of fire insulation were analyzed using finite
Fig. 5. Comparison of predicted temperatures from the proposed equations (Eqs. (17)–(22)) with those from FEA (Beam 250 400 mm). 84
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Fig. 9. Validation of the proposed approach by comparing predicted and measured temperatures (Palmieri et al. [10]).
Fig. 7. Validation of the proposed approach by comparing predicted and measured temperatures (Blontrock et al. [1]).
the regression results in 20–150 C could be set as a secondary target since any variation in this temperature range does not significantly influence the strength of concrete, steel and FRP reinforcement. Then a regression analysis was carried out so as to achieve minimum error between predicted temperature (using Eqs. (19)–(21)) and temperature obtained from FEA. Based on the regression analysis results, α and β are determined to be 4.5 and 1.75 respectively, and thus the relation between insulation thickness and its equivalent concrete thickness can be obtained by substituting for α and β in Eq. (16) as:
zec ¼ zi
pffi t
4:5
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc ðρcÞi ðρcÞc ki
1:75
(22)
The temperature predictions using equivalent concrete method (Eqs. (17)–(22)) are compared with temperature data generated from FEA. These comparisons are plotted in Figs. 4–6 for three RC beams insulated with different fire insulation types (each beam was protected with two types of fire insulation). In these figures, a point below “-10% margin” line indicate that the predicted temperature is to be higher than that obtained in FEA by more than 10%. If a point lies above “þ10% margin” line, the predicted temperature is lower than that obtained in FEA by more than 10%. It can be seen that for three insulated RC beams, most data points lie within ±10% margin zone. Therefore, the proposed equivalent concrete method is capable of predicting temperatures in insulated RC beams exposed to standard fire to a good degree. It is noted that there are a few points outside of “±10% margin” line, indicating that the predicted temperature error is larger than 10%. However, most of these points are below “-10% margin” line, and hence they represent
Fig. 10. Validation of the proposed approach by comparing predicted and measured temperatures (Yu and Kodur [19]).
conservative predictions of steel and/or FRP rebar temperatures. 5. Validation of temperature equations using test results The validity of the proposed approach is established by comparing predicted temperatures using Eqs. (17)–(22) with the measured temperatures from fire tests on FRP strengthened RC beams (protected with
Fig. 8. Validation of the proposed approach by comparing predicted and measured temperatures (Williams [18]).
Fig. 11. Validation of the proposed approach by comparing predicted and measured temperatures (Cree et al. [20]). 85
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Fig. 12. Validation of the proposed approach by comparing predicted and measured temperatures (Kodur et al. [21]).
fire insulation) reported in the literature [1,10,18,19]. The selected FRP strengthened RC beams are cast with concrete made of siliceous and carbonate aggregates with compressive strength of 40 MPa–58 MPa. The validation focuses on the accuracy of temperature prediction in FRP and steel reinforcement, since these temperatures have critical influence on the fire response of FRP strengthened RC beams. Figs. 7–10 show a comparison of predicted temperatures at the locations of steel rebars and FRP in concrete beams from the proposed approach with those recorded in fire tests on FRP-strengthened RC beams. It can be seen that the predicted temperatures at steel and FRP rebar locations are mostly in good agreement with the measured values in insulated FRP strengthened RC beams. The predicted temperatures at the level of FRP are slightly lower than the measured ones, especially in the initial stage of fire exposure. This is mainly due to quick rise in fire temperatures in early stage of fire exposure, leading to significant temperature increase in FRP at beam soffit. Also, if fire insulation develops cracks during fire exposure (as the case of fire tests on MSU beam II), the measured FRP temperatures can suddenly jump to a high level. Since the proposed temperature equations do not account for insulation cracking, predicted temperatures using proposed equations (Eqs. (17)–(22)) are slightly lower than measured temperatures in the fire tests. Also predicted temperatures in an insulated RC beam from the proposed equation is compared against measured temperatures in fire test, as well as with predictions from method proposed by Wickstrom and Hadziselimovic [11]. This comparison (plotted in Fig. 10) shows that predicted temperatures from the proposed equations are closer to test
Fig. 14. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 200 350 mm with Aestver insulation).
results as compared to Wickstrom and Hadziselimovic's equation. Since Wickstrom and Hadziselimovic's equations results in higher equivalent concrete layer. Therefore, higher thermal inertia is predicted and this leads to lower temperatures predictions as compared to test results. It should be noted that Wickstrom and Hadziselimovic's equation is not validated against fire test data and does not account for variation of thermal properties of insulation with temperature. While in the proposed equation, the equivalent concrete layer varies with time of fire exposure to account for degradation of thermal properties of insulation. To further validate the approach on other types of concrete structural members, two FRP strengthened RC columns (protected with fire insulation) were selected [20,21]. The selected columns are made of carbonate aggregate concrete with compressive strength of 28 MPa and 52 MPa. The comparison of predicted temperatures utilizing proposed equations to those obtained by fire tests is shown in Figs. 11 and 12. The trends of predicted temperatures compare well with measured data. However there is a marginal error (±10%) that lies within the error range of a simplified analysis. This difference might be due to dislocation of thermocouples and cracking in insulation during experiments. A further examination of trends in Figs. 7–12 indicate that there is relatively larger discrepancy between predicted and measured temperatures in 20–100 C range. This is because this temperatures range is not primary objective in the regression analysis (or the proposed equations). As indicated in Section 4, the regression analysis cannot match with data closely in the entire range. Thus the regression analysis was performed to fit the data points in the temperature range of 150–800 C. Since temperature variation in 20–150 C range does not significantly influence the strength in steel and FRP, the accuracy of temperature predictions in this range is set as a secondary target in the regression analysis. The slight variation in temperatures at this 20–150 C range does not significantly
Fig. 13. Validation of the proposed approach by comparing predicted temperatures with FEA results (Kodur and Ahmed [22]). 86
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Fig. 15. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 200 350 mm with VG insulation). Fig. 16. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 350 500 mm with Aestver insulation).
affect further strength analysis of FRP strengthened RC beams, since there is no significant strength loss in FRP and steel up to 150 C. Overall predicted temperatures are in reasonable agreement with measured data in most of fire duration, and this demonstrates the validity of the equivalent concrete approach for predicting temperatures in an insulated RC beam.
relatively larger discrepancy with FEA results, as shown in Fig. 17. This is mainly due to the fact that 20–150 C range is not primary target in the regression analysis as explained earlier. However, the discrepancy between predicted temperatures and FEA results is mostly within in 10%. Thus these equations for predicting temperature in an insulated RC members are applicable in a design situation. Overall the comparison of predicted temperatures with data from FEA results indicates the proposed equations are capable of evaluating temperatures in insulated RC beams.
6. Validation of temperature equations using FEA results The proposed equations are further validated through comparing temperature predictions in fire exposed FRP strengthened RC beams with those obtained from finite element analysis. To demonstrate the applicability of the equations in a wide range of situations, the selected concrete beams for validation are different from those used in regression analysis, as shown in Table 1 and Fig. 3. The cross-sections of concrete beams used for validation are 200 350 mm and 350 500 mm, and the beams were assumed to be insulated with two types of fire insulation, namely Aestuver and Tyfo VG. In each selected beam, temperatures in corner and middle steel rebars, NSM FRP strip, and center and average temperature of external FRP laminates are evaluated for comparison. Moreover, data reported from finite element analysis carried out by Kodur and Ahmed [22] is selected. In this beam, temperatures in corner and inner rebars are evaluated as illustrated in Fig. 13. A comparison of predicted temperatures from proposed equations (Eqs. (17)–(22)) with those from FEA is plotted in Figs. 13–17. It can be seen that predicted temperatures reasonably match with those obtained from FEA. In a few cases, mainly in the cases of FRP laminates or strips, the predicted temperatures are conservative (higher) as compared to FEA results. This can be attributed to the fact that proposed equations do not account for the variation of thermal properties of insulation at high temperatures. Also, temperature prediction in 20–150 C range has
7. Design example To demonstrate the applicability of the proposed simplified approach in design situations, a numerical example is presented for a typical FRP strengthened concrete beam exposed to ASTM E119 standard fire from three sides. The cross section of beam is 250 mm 450 mm and the top surface is protected from fire by the concrete floor slab. The beam has four 19 mm diameter reinforcing bars at four corners and the clear cover to the reinforcing bars is 38 mm. The stirrups are of 8 mm diameter. One layer of CFRP laminate (250 1 mm) is applied at the beam soffit as external strengthening reinforcement. The beam is insulated with Ushaped Tyfo VG fire insulation of 25 mm thickness. The configuration of this insulated concrete beam is shown in Fig. 18. The temperature rise of this fire exposed beam at 60 min is evaluated using the above developed approach. Using the proposed 1-D and 2-D heat transfer equations, the temperatures in steel rebar and external FRP at critical sections can be calculated at any given fire exposure time. Since the beam is protected by 87
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TC ¼ 1:0 ð 1:481 ð0:08 0:08Þ þ 0:985 ð0:08 þ 0:08Þ þ 0:017Þ 910 10:148 ¼ 150 C For middle rebar,
ηz ¼
0:155 ln
ηy ¼
0:155 ln
pffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:138 0:371 1:5 0:138
¼ 0:04
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0915 0:371 0:09151:5
¼ 0:08
TC ¼ 1:0 ð 1:481 ð 0:04 0:08Þ þ 0:985 ð 0:04 þ 0:08Þ þ 0:017Þ 910 10:148 ¼ 56 C Temperature at the level of FRP can be evaluated using the same equations as
ηz ¼
0:155 ln
ηy ¼
0:155 ln
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0985 0:371 ¼ 0:06 0:09851:5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0365 0:371 0:03651:5
¼ 0:33
TC ¼ 1:0 ð 1:481 ð0:06 0:33Þ þ 0:985 ð0:06 þ 0:33Þ þ 0:017Þ 910 10:148 ¼ 338 C These temperatures can be utilized to evaluate moment capacity of this FRP strengthened RC beam at 60 min and this moment capacity can be used to check future state of the beam. By using spreadsheet the temperatures at any other time can also be calculated. The predicted time-temperature curve of rebars and external FRP are plotted in Fig. 19. To further look at the validity of the proposed equations, this beam was further analyzed using ANSYS and temperature predictions are compared to temperatures predicted by the proposed approach. SOLID70 element was used for discretizing the member in finite element analysis. SOLID 70 has a 3-D thermal conduction capability. The element has eight nodes with a single degree of freedom defined as temperature at each node. The element is applicable to carry on a 3-D, steady-state or transient thermal analysis. LINK33 which is a 3D uniaxial 2-Node conduction bar was used to model the reinforcing steel bar [23]. For the analysis, thermal properties (thermal conductivity, specific heat, and density) of concrete, steel, and FRP as given by Lie [24], Eurocode [5], and Griffis et al. [25] were utilized. For insulation, temperature dependent thermal properties as suggested by Bisby [16] were incorporated in the analysis. It can be seen that predicted temperatures utilizing FEA are in good agreement with simplified approach predictions. But the predicted temperatures in final stage are lower than FEA prediction. This discrepancy can be attributed to the fact that temperature-dependent thermal properties of insulation were considered in FEA, unlike in the proposed approach where room temperature-thermal properties were used.
Fig. 17. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 350 500 mm with VG insulation).
fire insulation, the thickness of insulation layer ðzi Þ needs to be converted to equivalent thickness of concrete layer ðzec Þ. The thermal properties of Tyfo VG insulation (Bisby 2003) and concrete (Lie 1992) are taken to be: kVG ¼ 0.116 W/m-K, (ρc)VG ¼ 413 kJ/m3-K; kc ¼ 1.355 W/m-K, (ρc)c ¼ 2566 kJ/m3-K Applying Eq. (22), the thickness of equivalent concrete layer of 25 mm fire insulation after 1 h fire exposure is evaluated to be:
pffi zec ¼ zi t 4:5
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi kc ðρcÞi 4:5 ¼ 25 1 ðρcÞc ki
1:75
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:355 0:413 ¼ 36 mm 2:566 0:116
1:75
The distance from the center of corner rebar to two fire exposure sides is (36 þ 38þ8 þ 19/2 ¼ 92) mm, so y' ¼ z' ¼ 91.5 mm. The temperatures in rebars at 1 h fire-exposure time can be evaluated by applying Eqs. (19)–(21): For corner rebar,
ηz ¼ηy ¼
0:155 ln
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0915 0:371 ¼ 0:08 0:09151:5
Fig. 18. Layout and cross section of external FRP strengthened RC beam (Beam 1). 88
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2.
3.
Fig. 19. Prediction of temperatures in steel rebar and external FRP and comparison to FEA results.
4.
8. Limitations of applicability 5. Although the proposed equations for predicting temperatures in RC members are applicable over a large range of parameters, the above developed equations have few limitations. This is mainly due to the fact that the coefficients used in the equations are derived for certain set of parameters. Accordingly,
approach is proposed to evaluate cross sectional temperatures in such fire exposed insulated reinforced concrete members. Fire insulation layer attached to an RC member can be transformed into an equivalent concrete thickness layer with consideration of thermal properties and thickness of fire insulation [11]. An equation is proposed to transform external fire insulation into equivalent concrete thickness utilizing theoretical derivation and regression analysis and verified against fire test data. This equation accounts for degradation of thermal properties of insulation with time (temperature) and this provides more reliable temperature predictions as compared to the available equation in literature. A simplified approach is proposed to evaluate cross-sectional temperatures in an insulated RC member. The validity of the approach is partly established by comparing predicted temperatures from proposed equation with a large set of temperature data from fire tests and finite element analysis. The proposed equations give predictions of cross sectional temperatures within a margin of ±10% in insulated reinforced concrete members exposed to standard fire. The applicability of the simplified approach for evaluating crosssectional temperatures of insulated RC beam exposed to fire is illustrated through a design example.
Acknowledgement This material is based upon the work supported by the National Science Foundation (United States) under Grant number CMMI-0855820 to Michigan State University. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.
1) The proposed equations are applicable for evaluating cross-sectional temperatures in RC members exposed to standard fire only. These equations are not applicable for design fires, which have a cooling phase after the initial growth phase. 2) The predictions using proposed equations might have a relatively larger error (more than ±10%) when the temperatures are in 20–150 C range. This is due to the fact that the accuracy of temperature range of 150–800 C is the primary target of the regression analysis. 3) The proposed equations do not take into account the effect of cracking in fire insulation on cross-sectional temperatures of RC member. 4) The proposed equations were obtained based on specific types of fire insulation. For RC members protected by other types of insulation, predicted temperatures may be slightly different from temperature profile in actual situation. 5) The proposed equations yields cross sectional temperatures within a marginal error of ±10% when the cross sectional dimensions (width or depth) of the member are higher than 200 mm. 6) The predicted temperatures using the proposed approach have some level of approximation (error to the extent of ±10%) and may not be accurate as in the case of detailed heat transfer analysis. This is mainly due to the fact that the proposed equations represent simplification to the complex heat transfer problem. 7) The proposed approach is a first step to gauge temperature distribution in insulated reinforced concrete members. Additional validation of the equations to cover wider range of variables to improve the accuracy of the results is needed. Temperature measurements at various depths of concrete in an insulated concrete member and also temperature data at the interface of FRP and concrete surface as well as insulation surface are needed. This type of temperature data is required for members made of different types of concrete.
References [1] H. Blontrock, L. Taerwe, P. Vandevelde, Fire tests on concrete beams strengthened with fibre composite laminates, in: Vienna (Austria), International PhD Symposium in Civil Engineering, 2000. [2] B. Williams, V. Kodur, M.F. Green, L. Bisby, Fire endurance of fiber-reinforced polymer strengthened concrete T-Beams, ACI Struct. J. 105 (2008) 60–67. [3] A. Ahmed, V. Kodur, The experimental behavior of FRP-strengthened RC beams subjected to design fire exposure, Eng. Struct. 33 (2011) 2201–2211. [4] ACI216, Code Requirements for Determining Fire Resistance of Concrete and Masonary Construction Assemblies, American Concrete Institute, Detroit, 2007. [5] Eurocode2, EN 1992-1-2: Design of Concrete Structures, Part 1-2: General Rules Structural Fire Design, European Committee for Standardization, Brussels, Belgium, 2004. [6] V.K.R. Kodur, B.L. Yu, M.M.S. Dwaikat, A simplified approach for predicting temperature in reinforced concrete members exposed to standard fire, Fire Saf. J. 56 (2013) 39–51. [7] U. Wickstrom, A Very Simple Method for Estimating Temperatures in Fire Exposed Concrete Structures, Elsevie Applied Science Publishers, 1986, pp. 186–194. [8] K. Hertz, Simple Temperature Calculations of Fire Exposed Concrete Constructions, Institute of Building Design,Technical University of Denmark, 1981. [9] V. Kodur, L.A. Bisby, M.F. Green, FRP retrofitted concrete under fire conditions, Concr. Int. (2006) 37. [10] A. Palmieri, S. Matthys, L. Taerwe, Experimental investigation on fire endurance of insulated concrete beams strengthened with near surface mounted FRP bar reinforcement, Compos. Part B 43 (2012) 885–895. [11] U. Wickstr€ om, E. Hadziselimovic, Equivalent concrete layer thickness of a fire protection insulation layer, Fire Saf. J. 26 (1996) 295–302. [12] M.M.S. Dwaikat, V.K.R. Kodur, A simplified approach for predicting temperatures in fire exposed steel members, Fire Saf. J. 55 (2013) 87–96. [13] V. Kodur, B. Yu, Rational approach for evaluating fire resistance of FRPstrengthened concrete beams, J. Compos. Constr. 20 (2016) 04016041. [14] J.A. Purkiss, Fire Safety Engineering Design of Structures, Butterworth-Heinemann, Linacre House, Jordan Hill, Oxford 0X2 8DP, UK, 2007. [15] M.B.M. Dwaikat, Flexural Response of Reinforced Concrete Beams Exposed to Fire (Ph.D. thesis), Michigan State University,Michigan, USA, 2009. [16] L.A. Bisby, Fire Behaviour of Fibre-reinforced Polymer (FRP) Reinforced or Confined Concrete (Ph.D. thesis), Queen's University, Kingston, Canada, 2003. [17] B. Williams, L. Bisby, V. Kodur, M. Green, E. Chowdhury, Fire insulation schemes for FRP-strengthened concrete slabs, Compos. Part A 37 (2006) 1151–1160. [18] B.K. Williams, Fire Performance of FRP-strengthened Reinforced Concrete Flexural Members (Ph.D. thesis), Queen's University, Kingston, Canada, 2004. [19] B. Yu, V.K.R. Kodur, Fire behavior of concrete T-beams strengthened with nearsurface mounted FRP reinforcement, Eng. Struct. 80 (2014) 350–361.
9. Conclusions Based on the information presented in this paper, the following conclusions can be drawn: 1. RC members strengthened with FRP reinforcement usually require external fire insulation to enhance fire resistance. A simplified 89
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[20] D. Cree, E.U. Chowdhury, M.F. Green, L.A. Bisby, N. Benichou, Performance in fire of FRP-strengthened and insulated reinforced concrete columns, Fire Saf. J. 54 (2012) 86–95. [21] V.K.R. Kodur, L.A. Bisby, M.F. Green, Experimental evaluation of the fire behaviour of insulated fibre-reinforced-polymer-strengthened reinforced concrete columns, Fire Saf. J. 41 (2006) 547–557. [22] V.K.R. Kodur, A. Ahmed, Numerical model for tracing the response of FRPstrengthened RC beams exposed to fire, J. Compos. Constr. 14 (2010) 730–742.
[23] R.A. Hawileh, M. Naser, W. Zaidan, H.A. Rasheed, Modeling of insulated CFRPstrengthened reinforced concrete T-beam exposed to fire, Eng. Struct. 31 (2009) 3072–3079. [24] T.T. Lie, Structural Fire Protection. American Society of Civil Engineers Manuals and Reports on Engineering Practice No. 78. ASCE, New York, NY, 1992. [25] C.A. Griffis, R.A. Masumura, C.I. Chang, Thermal response of graphite epoxy composite subjected to rapid heating, J. Compos. Mater. 15 (1981) 427–442.
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Contents lists available at ScienceDirect
Fire Safety Journal j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fi r e s a f
A simplified approach for predicting temperatures in insulated RC members exposed to standard fire V.K.R. Kodur *, B. Yu, R. Solhmirzaei Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Reinforced concrete Fire resistance Insulated concrete member Cross sectional temperatures
Fire resistance of concrete structural members can be enhanced through the application of external fire insulation on the surfaces of concrete member. For evaluating fire resistance of such insulated RC members, temperatures in concrete and steel reinforcement are to be known. This paper develops a simplified approach for predicting crosssectional temperatures in an insulated RC structural member exposed to standard fire. The approach is derived by replacing the insulation layer into an equivalent concrete thickness layer and then undertaking statistical regression analysis on temperature data of modified concrete section. The effect of critical parameters, including geometry of concrete member and insulation, thermal properties of concrete and fire insulation, and duration of fire exposure is accounted for in temperature equations. The validity of the approach is established by comparing predictions from the proposed equation with data generated from fire tests and finite element analysis. These comparisons show the proposed equation gives reasonable prediction of temperatures, within a range of ±10%, in insulated concrete members. The applicability of the proposed approach in design situations is illustrated though a numerical example. The simplicity of the proposed method makes it attractive for use in design situations and for incorporation in design manuals.
1. Introduction Provision of appropriate fire resistance is a major design requirement for structural members in buildings. Reinforced concrete (RC) members generally possess good fire resistance and hence no external fire insulation is required. This is mainly attributed to the fact that concrete is a non-combustible material and has a low thermal conductivity. However, in special cases, concrete structural member may not achieve sufficient fire resistance as required in building codes. For example, an RC beam strengthened with fiber-reinforced polymer (FRP) laminates at the beam soffit usually has relative lower fire resistance as compared to conventional steel reinforced concrete beam, due to faster degradation of strength and stiffness properties of FRP at high temperatures [1–3]. Also, in some concrete members at critical locations (such as transfer beams), a high fire resistance is usually required to ensure safety of structures. Therefore, in these situations, external fire insulation is usually applied on the surface of concrete member to enhance fire resistance of such concrete members. Fire resistance analysis on an RC member involves two stages of analysis, i.e. heat transfer analysis followed by strength analysis. When a RC member is subjected to a fire, temperatures in concrete, steel and FRP
* Corresponding author. E-mail address: [email protected] (V.K.R. Kodur). http://dx.doi.org/10.1016/j.firesaf.2017.05.018 Received 16 October 2015; Received in revised form 23 May 2017; Accepted 25 May 2017 0379-7112/© 2017 Elsevier Ltd. All rights reserved.
reinforcement increase with time, resulting in loss of strength and stiffness in the member. This temperature rise is a function of fire characteristics, thermal properties of concrete (and insulation), and crosssectional dimensions. An approach to evaluate cross-sectional temperatures is through fire tests on RC members. Alternatively, design charts specified in codes and standards, such as ACI 216.1 [4] and Eurocode 2 [5] which are applicable to specific types of concrete, can be used to evaluate cross-sectional temperatures in RC members. In addition, there are few simplified approaches for evaluating cross-sectional temperatures, as a function of depth and fire exposure time, in uninsulated RC members [6–8]. Also, detailed thermal analysis on RC members, based on finite difference and finite element methods, can be carried out to obtain cross sectional temperatures. Externally bonded FRP are used in strengthening of RC members and in such cases, external fire insulation is to be provided to achieve required fire resistance in FRP strengthened RC members. Currently fire resistance of such FRP strengthened concrete members in different facilities is established through fire resistance tests [9]. A review of literature indicates that limited studies have been conducted on developing simplified approaches for evaluating temperature profile in insulated RC members. These studies mainly focused on advanced numerical modeling
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Fig. 1. Area divisions in concrete members for temperature calculation.
techniques [1–3,10]. Through numerical models, cross sectional temperatures at critical locations, such as steel rebar, FRP, and concrete center, are generated at any given fire exposure time. However, these approaches are complex, time consuming, and require high level of expertise in numerical analysis. Thus, such detailed analysis may not be feasible in design situations. Therefore, there is a need for simple and reliable approaches for evaluating cross-sectional temperatures in an insulated RC member exposed to fire. Wickstrom and Hadziselimovic [11] have shown that thermal protection capacity of an insulation layer on a concrete member can be expressed by modifying the insulation thickness with an equivalent concrete layer. Such equivalent concrete thickness yields the same temperature distribution in RC member, as the corresponding insulation layer when the structure is exposed to fire. Based on numerical analyses, Wickstrom and Hadziselimovic derived a simple relation between thermal resistance of insulation layer and equivalent concrete layer. However, the proposed relation was not validated against any fire test data for accuracy of temperature predictions and also on its validity over wide range of protection thicknesses. To overcome this gap, empirical equations for predicting temperatures in insulated RC members exposed to standard fire is developed in this paper. The proposed equations are derived utilizing an approach originally developed for predicting temperature in a fire exposed uninsulated RC member [6].
where, t is the fire exposure time in hours, z and y are horizontal and vertical distance from a point in concrete section to the fire exposure surface in meters. The proposed equations are applicable for evaluating cross-sectional temperature when an RC member is subjected to standard fire exposure. atn in above equations represent fire temperature under standard fire exposure [6,12]. For ISO 834 fire, a ¼ 935 and n ¼ 0.168, and for ASTM E119 fire, a ¼ 910 and n ¼ 0.148. c1 and c2 are the coefficients to account for effect of concrete type (properties) on heat transfer where c1 is equal to 1.0, 1.01, 1.12 and 1.12 for NSC-CA, HSCCA, NSC-SA and HSC-SA, respectively; c2 are 1.0, 1.06, 1.12 and 1.20 for NSC-CA, HSC-CA, NSC-SA and HSC-SA, respectively. NSC, HSC, SA, and CA refer to Normal Strength Concrete, High Strength Concrete, and Siliceous Aggregate concrete, and Carbonate Aggregate concrete, respectively. These equations (Eqs. (1)–(3)) can be conveniently applied for predicting cross-sectional temperatures over wide ranging cross sectional dimensions of concrete members subjected to standard fire exposure. One of the major limitations of this approach is that it is only applicable for uninsulated RC members and it is not valid when RC members are protected with fire insulation. This approach does not take into account moisture content of concrete, which might result in slightly conservative predictions. In addition, the proposed equations may not precisely predict cross-sectional temperature when one side of the member is small as compared to the other side, since heat transfer, takes place from multiple fire-exposed sides. The above equations are validated against data generated from a large number of fire tests and finite element analyses [6]. Therefore, these equations are reliable to be used as basis for obtaining equations for cross-sectional temperatures in insulated RC members exposed to fire.
2. Formulae for predicting temperatures in an uninsulated RC member A review of literature indicates that there are a few simplified expressions for evaluating the cross-sectional temperature in an uninsulated RC member [7,8]. More recently, Kodur et al. [6] proposed a set of simplified equations for predicting cross sectional temperatures within an RC member exposed to a standard fire (such as ASTM E119 and ISO 834), and these equations overcome some of the drawbacks of previous empirical equations and provide better temperature predictions. As shown in Fig. 1, for a given point (coordinate is (y,z)) in an RC member, temperature at a given fire exposure time can be obtained as:
For 1 D heat transfer : Tc ¼ c1 ⋅ηz ⋅ðatn Þ In which ηz is given by ηz ¼ 0:155 ln
pffiffi 0:348 z 0:371 z1:5 t
3. Formulae for predicting temperatures in an insulated RC member A simplified formula for predicting cross sectional temperatures in a fire exposed insulated RC member is derived in this section. 3.1. General approach
(1)
Unlike a conventional RC member, temperature profiles of an insulated RC member are influenced by properties of both concrete and fire insulation. Also, thermal properties of concrete and fire insulation can vary significantly. Thus, predicting temperature in an insulated concrete member becomes a more complicated problem. In an RC member (with or without fire insulation) exposed to fire, thermal properties of concrete and insulation primarily influence crosssectional temperature, while thermal properties of steel rebar and FRP
(2)
For 2 D heat transfer : Tc ¼ c2 1:481 ηz ⋅ηy þ 0:985 ηz þ ηy n þ 0:017 ðat Þ (3) 81
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(laminate, strip or rod added to the surface or placed internally) does not significantly affect temperature rise in the member due to their small cross sectional areas as compared to concrete sectional area [13]. Thus, if the insulation layer on concrete member is converted to equivalent concrete layer, temperature profile in an insulated concrete section can be evaluated using the same temperature equation as that for uninsulated RC sections (Eqs. (1)–(3)). For this purpose a derivation of an approximate expression for transforming insulation into an equivalent concrete thickness is required.
Fig. 2 shows a typical RC beam protected by a U-shaped fire insulation, and the thickness of insulation on sides and bottom of the beam are zi and yi respectively. If that fire insulation layer is replaced by an equivalent concrete layer (with the thickness of zec and yec on sides and bottom) which has the same thermal resistance (inertia) as insulation layer, the temperature profiles remain the same within beam cross section after this transformation. Based on basic heat transfer principles, the following equations can be expressed for heat transfer within the insulation or its equivalent concrete layer.
3.2. Converting fire insulation layer to equivalent concrete layer
ki 2 ∂Ti Q ∇ Ti ¼ ðρcÞi ∂t ðρcÞi
Temperature rise in a structural member results from a fire source occurring through radiation, convection, and conduction. Heat transfer from fire source to the surface of a structural member is through convection and radiation. The heat flux to the surface of a structural member through radiation and convection is expressed as boundary conditions in thermal analysis. Heat transfer within a structural member is through conduction which is expressed as governing equation by Fourier heat transfer equation. Conduction occurs due to temperature difference between two locations. The equation describing this heat transfer is expressed as Eq. (4) [14].
q ¼ k∇T
kc 2 ∂Tec Q ∇ Tec ¼ ðρcÞc ðρcÞc ∂t
∂T ¼ ∇q þ Q ∂t
(4)
(5)
in which ρc is heat capacity of the section, t is time, ∇ is spatial gradient operator, and Q is internal heat generation rate per unit volume. Substituting Eq. (4) into Eq. (5) results in Eq. (6) [14].
k 2 ∂T Q ∇T¼ ρc ∂t ρc
(11)
(6)
The heat flux on the fire exposed boundary due to convection and radiation can be given by following equation [14,15].
qb ¼ ðhcon þ hrad Þ T Tf
Substituting left side of Eqs. (9) and (10) into Eq. (11) results in Eq. (12).
(7)
kc ∂2 Tec ki ∂2 Ti ≈ ðρcÞc ∂z2ec ðρcÞi ∂z2i
where hcon and hrad are convective and radiative heat transfer coefficients, and are defined as:
hrad ¼ 4σε T 2 þ Tf2 T þ Tf
(10)
where ki and ðρcÞi are thermal conductivity and heat capacity of insulation respectively, and kc and ðρcÞc are thermal conductivity and heat capacity of concrete respectively; Q is internal heat generation rate per unit volume; Ti and Tec are the temperatures in insulation and equivalent concrete respectively; and ∇ is the second order derivative of temperature (Ti or Tec ) with respect to the distance to fire exposed surface (z or y); and t is the time. Eqs. (9) and (10) can be applied for calculating temperature at any location within the insulation layer (equivalent concrete layer). At the same fire exposure time, temperature profiles within insulation layer and equivalent concrete layer, which have same thermal resistance, are similar. Moreover, Q which is internal heat generation rate per unit volume is zero, since there is no internal heat source (i.e. no chemical reaction in concrete). Therefore, the following assumption can be made.
where q is the vector of heat flux per unit area, k is thermal conductivity tensor, and T is temperature. The general heat transfer differential equation of a structural member can be expressed as Eq. (5) [14].
ρc
(9)
(12)
where z is the distance from fire exposed surface. Due to smaller thickness of fire insulation or its equivalent concrete layer, 1-D heat transfer dominates temperature distribution within the insulation or concrete layer. Thus the temperature equation corresponding to 1D heat transfer (Eq. (1)) can be applied to derive temperature in the concrete section ð Tec Þ in Eq. (11) and can be expressed as:
(8)
where Tf is fire temperature, σ is Stefan-Boltzman constant, and ε is emissivity factor. The boundary condition of unexposed surface is similar to that of exposed surface with replacing the fire temperature with the ambient temperature.
Fig. 2. Illustration of the equivalent concrete depth method. 82
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Table 1 Characteristics of insulated RC beams used in the regression analysis. Case No.
Section
Fire insulation (type and properties)
Point (location)
Fire exposure time
1
200 300 mm 250 400 mm 300 500 mm
“AESTUVER” k ¼ 0.185 W/m-K (ρc) ¼ 650kJ/m3-K
Corner steel rebar Middle steel rebar NSM FRP strip 1/4 length of FRP laminate Middle of FRP laminate Corner steel rebar Middle steel rebar NSM FRP strip 1/4 length of FRP laminate Middle of FRP laminate
0.5 1.5 2.5 3.5
2
200 300 mm 250 400 mm 300 500 mm
“Tyfo VG” k ¼ 0.116 W/m-K (ρc) ¼ 413kJ/m3-K
pffiffiffiffiffi t Tec ¼ 0:155 ln 1:5 0:348 zec 0:371 :ðatn Þ zec
t ≈ 0:155 ln 1:5 0:371 :ðat n Þ zec
0.5 1.5 2.5 3.5
h, 1 h, 2 h, 3 h, 4
h, 1 h, 2 h, 3 h, 4
where λ and η are the coefficients to be determined. Notice that Eq. (15) describes an approximate relation between insulation thickness ðzi Þ and the equivalent concrete thickness ðzec Þ. Since thermal properties (ki and ðρcÞi , kc and ðρcÞc ) of insulation and concrete vary with fire exposure time (or fire temperature), λ is assumed to be a function of time ðtÞ in hours, accounting for the influence of fire exposure time (or fire temperature) to thermal properties. Finally, the relation between insulation and concrete thickness are expressed as:
h, h, h, h
ffi zec p ¼ αt zi
h, h, h, h
(13)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffi β kc ðρcÞi α z ¼ zc þ zec ¼ zc þ zi t ðρcÞc ki 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffi β kc ðρcÞi α y ¼ yc þ yec ¼ yc þ yi t ðρcÞc ki 0
(14)
where a1 and a2 are the coefficients describing temperature distribution within the insulation layer. Importing Eqs. (13) and (14) into Eqs. (11) and (12), the following relation can be obtained for deriving the ratio of the thickness of insulation layer ðzi Þ to that of concrete layer ðzec Þ.
zec ¼ zi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc ðρcÞi η kc ðρcÞi η 0:155 ≈λ a1 ðρcÞc ki ðρcÞc ki
(16)
where α and β are the coefficients to be determined. Once the values of α and β are obtained, the insulation thickness on a concrete member can be converted to an equivalent thickness of concrete in that member. As an illustration, Fig. 2 shows a point (Point A) located in an insulated RC beam. The distances from Point A to side and bottom concrete surface of beam are zc and yc respectively, and insulation thickness on side and bottom surface of beam are zi and yi respectively. Applying the above equivalent concrete depth method, the equivalent depth from Point A ðz0 or y 0 Þ to fire exposed surface can be expressed as:
Temperature distribution within insulation ðTi Þ follows similar trend to that within concrete (Eq. (13)). Therefore, in the insulation layer
t Ti ≈ a1 ln 1:5 a2 :ðat n Þ z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc ðρcÞi β ðρcÞc ki
(17)
(18)
3.3. Formulae for temperatures in insulated concrete members The equivalent distances, z0 and y 0 , expressed in terms of concrete depth (or width), can be substituted into temperature equations (Eqs. (1)–(3)) in an RC member, to obtain temperature rise in an insulated RC member, as shown below.
(15)
For 1 D heat transfer : Tc ¼ c1 ⋅ηz ⋅ðat n Þ
Fig. 3. FRP strengthened RC beams used in FEA for regression and validation (Units: mm). 83
(19)
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Fig. 4. Comparison of predicted temperatures from the proposed equations (Eqs. (17)–(22)) with those from FEA (Beam 200 300 mm).
where; ηz ¼ 0:155 ln
t ðz0 Þ1:5
pffiffiffi 0:348 z0 0:371
Fig. 6. Comparison of predicted temperatures from the proposed equations (Eqs. (17)–(22)) with those from FEA (Beam 300 500 mm).
element program validated in previous research [6]. The characteristics of these RC beams were varied over a range of parameters as shown in Table 1 and Fig. 3. The width of beam section was varied from 200 to 300 mm, while the depth was varied from 300 to 500 mm. Two types of fire insulation, Aestuver and Tyfo VG (Vermiculite Gypsum) insulations, are assumed to be applied on the beams, and the properties of these fire insulation are tabulated in Table 1. The variation of thermal properties (thermal conductivity, heat capacity, and density) of fire insulation with temperature follows previously reported values [16]. Five different points within the beam section were selected for monitoring temperatures in each beam, and these included typical locations of steel and FRP reinforcement, as shown in Fig. 3. These points represent critical points from fire resistance consideration in a simply supported FRP strengthened RC beam, where steel and rebars are close to fire surface as opposed to compression zone where is farther from fire surface. Due to this reasoning, the temperatures in concrete remain below 500 C for most of the fire duration and FRP properties become more critical as it is sensitive to elevated temperatures due to low tolerance of the polymer matrix to high temperatures. In addition to rapid strength and stiffness degradation of FRP, the bond between FRP and concrete, which is critical to maintain FRP's effectiveness, gets severely reduced at temperatures above glass transition temperature. Once FRP loses its tensile strength and bond at higher temperatures, the rebar temperatures become an important indicator of fire resistance of simply supported FRP-strengthened RC beams [3,17]. Thus FRP and tensile steel rebars are critical points in determining fire resistance of simply supported FRP-strengthened RC beams. In the finite element analysis, each beam was subjected to ASTM E119 fire exposure from three sides for 4 h, and temperature data at each time interval of 0.5 h was output for regression analysis. In total, 240 temperature data points (3 (beams) 2 (insulation types) 5 (points) 8 (time intervals)) with corresponding time and location were generated for regression analysis on temperature equations. A regression analysis was performed using “Solver” function in Excel (2010) to develop an expression for converting insulation to an equivalent concrete layer (Eq. (16)). The “solver” function is able to calculate the optimum coefficients to match the original data with a given format of formula, and applied “constraint” criteria. Since the regression analysis can hardly match all the data points closely, it is necessary to fit the data points in the critical range with the smallest discrepancy, and that have reasonable match in other regions. From the fire resistance point of view, it is well established that the strength of reinforcing steel, FRP, and concrete are not significantly influenced up to 150 C, and there is little interest in temperature data in concrete members beyond 800 C. Therefore, the regression results have to be highly reliable or slightly conservative in temperature-sensitive zone of 150–800 C. In contrast,
(20)
For 2 D heat transferTc ¼ c2 1:481 ηz ⋅ηy þ 0:985 ηz þ ηy þ 0:017 ðatn Þ (21) where t is the fire exposure time in hours, z0 is the distance from the point in concrete section to fire exposed surface using the equivalent concrete depth method (Eqs. (17) and (18)). ηz and ηy are the heat transfer factors resulting from z0 and y 0 side fire exposure, ηy is calculated in the same manner as that of ηz in Eq. (20). The definitions of other symbols (terms) follows the same definition presented in Section 2. Eqs. (17)–(21) provide a simplified expression for evaluating temperatures within an insulated RC beam. The only unknowns to be determined in Eqs. (17) and (18) are α and β. These two unknowns are obtained through nonlinear regression analysis on a large set of database generated using finite element analysis (FEA). 4. Regression analysis A regression analysis was carried out to obtain correlation coefficients α and β, in the above developed empirical equations for predicting temperatures in an insulated RC beam. To generate temperature data for regression analysis, three representative RC beams protected with two types of fire insulation were analyzed using finite
Fig. 5. Comparison of predicted temperatures from the proposed equations (Eqs. (17)–(22)) with those from FEA (Beam 250 400 mm). 84
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Fig. 9. Validation of the proposed approach by comparing predicted and measured temperatures (Palmieri et al. [10]).
Fig. 7. Validation of the proposed approach by comparing predicted and measured temperatures (Blontrock et al. [1]).
the regression results in 20–150 C could be set as a secondary target since any variation in this temperature range does not significantly influence the strength of concrete, steel and FRP reinforcement. Then a regression analysis was carried out so as to achieve minimum error between predicted temperature (using Eqs. (19)–(21)) and temperature obtained from FEA. Based on the regression analysis results, α and β are determined to be 4.5 and 1.75 respectively, and thus the relation between insulation thickness and its equivalent concrete thickness can be obtained by substituting for α and β in Eq. (16) as:
zec ¼ zi
pffi t
4:5
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kc ðρcÞi ðρcÞc ki
1:75
(22)
The temperature predictions using equivalent concrete method (Eqs. (17)–(22)) are compared with temperature data generated from FEA. These comparisons are plotted in Figs. 4–6 for three RC beams insulated with different fire insulation types (each beam was protected with two types of fire insulation). In these figures, a point below “-10% margin” line indicate that the predicted temperature is to be higher than that obtained in FEA by more than 10%. If a point lies above “þ10% margin” line, the predicted temperature is lower than that obtained in FEA by more than 10%. It can be seen that for three insulated RC beams, most data points lie within ±10% margin zone. Therefore, the proposed equivalent concrete method is capable of predicting temperatures in insulated RC beams exposed to standard fire to a good degree. It is noted that there are a few points outside of “±10% margin” line, indicating that the predicted temperature error is larger than 10%. However, most of these points are below “-10% margin” line, and hence they represent
Fig. 10. Validation of the proposed approach by comparing predicted and measured temperatures (Yu and Kodur [19]).
conservative predictions of steel and/or FRP rebar temperatures. 5. Validation of temperature equations using test results The validity of the proposed approach is established by comparing predicted temperatures using Eqs. (17)–(22) with the measured temperatures from fire tests on FRP strengthened RC beams (protected with
Fig. 8. Validation of the proposed approach by comparing predicted and measured temperatures (Williams [18]).
Fig. 11. Validation of the proposed approach by comparing predicted and measured temperatures (Cree et al. [20]). 85
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Fig. 12. Validation of the proposed approach by comparing predicted and measured temperatures (Kodur et al. [21]).
fire insulation) reported in the literature [1,10,18,19]. The selected FRP strengthened RC beams are cast with concrete made of siliceous and carbonate aggregates with compressive strength of 40 MPa–58 MPa. The validation focuses on the accuracy of temperature prediction in FRP and steel reinforcement, since these temperatures have critical influence on the fire response of FRP strengthened RC beams. Figs. 7–10 show a comparison of predicted temperatures at the locations of steel rebars and FRP in concrete beams from the proposed approach with those recorded in fire tests on FRP-strengthened RC beams. It can be seen that the predicted temperatures at steel and FRP rebar locations are mostly in good agreement with the measured values in insulated FRP strengthened RC beams. The predicted temperatures at the level of FRP are slightly lower than the measured ones, especially in the initial stage of fire exposure. This is mainly due to quick rise in fire temperatures in early stage of fire exposure, leading to significant temperature increase in FRP at beam soffit. Also, if fire insulation develops cracks during fire exposure (as the case of fire tests on MSU beam II), the measured FRP temperatures can suddenly jump to a high level. Since the proposed temperature equations do not account for insulation cracking, predicted temperatures using proposed equations (Eqs. (17)–(22)) are slightly lower than measured temperatures in the fire tests. Also predicted temperatures in an insulated RC beam from the proposed equation is compared against measured temperatures in fire test, as well as with predictions from method proposed by Wickstrom and Hadziselimovic [11]. This comparison (plotted in Fig. 10) shows that predicted temperatures from the proposed equations are closer to test
Fig. 14. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 200 350 mm with Aestver insulation).
results as compared to Wickstrom and Hadziselimovic's equation. Since Wickstrom and Hadziselimovic's equations results in higher equivalent concrete layer. Therefore, higher thermal inertia is predicted and this leads to lower temperatures predictions as compared to test results. It should be noted that Wickstrom and Hadziselimovic's equation is not validated against fire test data and does not account for variation of thermal properties of insulation with temperature. While in the proposed equation, the equivalent concrete layer varies with time of fire exposure to account for degradation of thermal properties of insulation. To further validate the approach on other types of concrete structural members, two FRP strengthened RC columns (protected with fire insulation) were selected [20,21]. The selected columns are made of carbonate aggregate concrete with compressive strength of 28 MPa and 52 MPa. The comparison of predicted temperatures utilizing proposed equations to those obtained by fire tests is shown in Figs. 11 and 12. The trends of predicted temperatures compare well with measured data. However there is a marginal error (±10%) that lies within the error range of a simplified analysis. This difference might be due to dislocation of thermocouples and cracking in insulation during experiments. A further examination of trends in Figs. 7–12 indicate that there is relatively larger discrepancy between predicted and measured temperatures in 20–100 C range. This is because this temperatures range is not primary objective in the regression analysis (or the proposed equations). As indicated in Section 4, the regression analysis cannot match with data closely in the entire range. Thus the regression analysis was performed to fit the data points in the temperature range of 150–800 C. Since temperature variation in 20–150 C range does not significantly influence the strength in steel and FRP, the accuracy of temperature predictions in this range is set as a secondary target in the regression analysis. The slight variation in temperatures at this 20–150 C range does not significantly
Fig. 13. Validation of the proposed approach by comparing predicted temperatures with FEA results (Kodur and Ahmed [22]). 86
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Fig. 15. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 200 350 mm with VG insulation). Fig. 16. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 350 500 mm with Aestver insulation).
affect further strength analysis of FRP strengthened RC beams, since there is no significant strength loss in FRP and steel up to 150 C. Overall predicted temperatures are in reasonable agreement with measured data in most of fire duration, and this demonstrates the validity of the equivalent concrete approach for predicting temperatures in an insulated RC beam.
relatively larger discrepancy with FEA results, as shown in Fig. 17. This is mainly due to the fact that 20–150 C range is not primary target in the regression analysis as explained earlier. However, the discrepancy between predicted temperatures and FEA results is mostly within in 10%. Thus these equations for predicting temperature in an insulated RC members are applicable in a design situation. Overall the comparison of predicted temperatures with data from FEA results indicates the proposed equations are capable of evaluating temperatures in insulated RC beams.
6. Validation of temperature equations using FEA results The proposed equations are further validated through comparing temperature predictions in fire exposed FRP strengthened RC beams with those obtained from finite element analysis. To demonstrate the applicability of the equations in a wide range of situations, the selected concrete beams for validation are different from those used in regression analysis, as shown in Table 1 and Fig. 3. The cross-sections of concrete beams used for validation are 200 350 mm and 350 500 mm, and the beams were assumed to be insulated with two types of fire insulation, namely Aestuver and Tyfo VG. In each selected beam, temperatures in corner and middle steel rebars, NSM FRP strip, and center and average temperature of external FRP laminates are evaluated for comparison. Moreover, data reported from finite element analysis carried out by Kodur and Ahmed [22] is selected. In this beam, temperatures in corner and inner rebars are evaluated as illustrated in Fig. 13. A comparison of predicted temperatures from proposed equations (Eqs. (17)–(22)) with those from FEA is plotted in Figs. 13–17. It can be seen that predicted temperatures reasonably match with those obtained from FEA. In a few cases, mainly in the cases of FRP laminates or strips, the predicted temperatures are conservative (higher) as compared to FEA results. This can be attributed to the fact that proposed equations do not account for the variation of thermal properties of insulation at high temperatures. Also, temperature prediction in 20–150 C range has
7. Design example To demonstrate the applicability of the proposed simplified approach in design situations, a numerical example is presented for a typical FRP strengthened concrete beam exposed to ASTM E119 standard fire from three sides. The cross section of beam is 250 mm 450 mm and the top surface is protected from fire by the concrete floor slab. The beam has four 19 mm diameter reinforcing bars at four corners and the clear cover to the reinforcing bars is 38 mm. The stirrups are of 8 mm diameter. One layer of CFRP laminate (250 1 mm) is applied at the beam soffit as external strengthening reinforcement. The beam is insulated with Ushaped Tyfo VG fire insulation of 25 mm thickness. The configuration of this insulated concrete beam is shown in Fig. 18. The temperature rise of this fire exposed beam at 60 min is evaluated using the above developed approach. Using the proposed 1-D and 2-D heat transfer equations, the temperatures in steel rebar and external FRP at critical sections can be calculated at any given fire exposure time. Since the beam is protected by 87
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TC ¼ 1:0 ð 1:481 ð0:08 0:08Þ þ 0:985 ð0:08 þ 0:08Þ þ 0:017Þ 910 10:148 ¼ 150 C For middle rebar,
ηz ¼
0:155 ln
ηy ¼
0:155 ln
pffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:138 0:371 1:5 0:138
¼ 0:04
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0915 0:371 0:09151:5
¼ 0:08
TC ¼ 1:0 ð 1:481 ð 0:04 0:08Þ þ 0:985 ð 0:04 þ 0:08Þ þ 0:017Þ 910 10:148 ¼ 56 C Temperature at the level of FRP can be evaluated using the same equations as
ηz ¼
0:155 ln
ηy ¼
0:155 ln
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0985 0:371 ¼ 0:06 0:09851:5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0365 0:371 0:03651:5
¼ 0:33
TC ¼ 1:0 ð 1:481 ð0:06 0:33Þ þ 0:985 ð0:06 þ 0:33Þ þ 0:017Þ 910 10:148 ¼ 338 C These temperatures can be utilized to evaluate moment capacity of this FRP strengthened RC beam at 60 min and this moment capacity can be used to check future state of the beam. By using spreadsheet the temperatures at any other time can also be calculated. The predicted time-temperature curve of rebars and external FRP are plotted in Fig. 19. To further look at the validity of the proposed equations, this beam was further analyzed using ANSYS and temperature predictions are compared to temperatures predicted by the proposed approach. SOLID70 element was used for discretizing the member in finite element analysis. SOLID 70 has a 3-D thermal conduction capability. The element has eight nodes with a single degree of freedom defined as temperature at each node. The element is applicable to carry on a 3-D, steady-state or transient thermal analysis. LINK33 which is a 3D uniaxial 2-Node conduction bar was used to model the reinforcing steel bar [23]. For the analysis, thermal properties (thermal conductivity, specific heat, and density) of concrete, steel, and FRP as given by Lie [24], Eurocode [5], and Griffis et al. [25] were utilized. For insulation, temperature dependent thermal properties as suggested by Bisby [16] were incorporated in the analysis. It can be seen that predicted temperatures utilizing FEA are in good agreement with simplified approach predictions. But the predicted temperatures in final stage are lower than FEA prediction. This discrepancy can be attributed to the fact that temperature-dependent thermal properties of insulation were considered in FEA, unlike in the proposed approach where room temperature-thermal properties were used.
Fig. 17. Validation of the proposed approach by comparing predicted temperatures with FEA results (Beam 350 500 mm with VG insulation).
fire insulation, the thickness of insulation layer ðzi Þ needs to be converted to equivalent thickness of concrete layer ðzec Þ. The thermal properties of Tyfo VG insulation (Bisby 2003) and concrete (Lie 1992) are taken to be: kVG ¼ 0.116 W/m-K, (ρc)VG ¼ 413 kJ/m3-K; kc ¼ 1.355 W/m-K, (ρc)c ¼ 2566 kJ/m3-K Applying Eq. (22), the thickness of equivalent concrete layer of 25 mm fire insulation after 1 h fire exposure is evaluated to be:
pffi zec ¼ zi t 4:5
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi kc ðρcÞi 4:5 ¼ 25 1 ðρcÞc ki
1:75
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:355 0:413 ¼ 36 mm 2:566 0:116
1:75
The distance from the center of corner rebar to two fire exposure sides is (36 þ 38þ8 þ 19/2 ¼ 92) mm, so y' ¼ z' ¼ 91.5 mm. The temperatures in rebars at 1 h fire-exposure time can be evaluated by applying Eqs. (19)–(21): For corner rebar,
ηz ¼ηy ¼
0:155 ln
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:348 0:0915 0:371 ¼ 0:08 0:09151:5
Fig. 18. Layout and cross section of external FRP strengthened RC beam (Beam 1). 88
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2.
3.
Fig. 19. Prediction of temperatures in steel rebar and external FRP and comparison to FEA results.
4.
8. Limitations of applicability 5. Although the proposed equations for predicting temperatures in RC members are applicable over a large range of parameters, the above developed equations have few limitations. This is mainly due to the fact that the coefficients used in the equations are derived for certain set of parameters. Accordingly,
approach is proposed to evaluate cross sectional temperatures in such fire exposed insulated reinforced concrete members. Fire insulation layer attached to an RC member can be transformed into an equivalent concrete thickness layer with consideration of thermal properties and thickness of fire insulation [11]. An equation is proposed to transform external fire insulation into equivalent concrete thickness utilizing theoretical derivation and regression analysis and verified against fire test data. This equation accounts for degradation of thermal properties of insulation with time (temperature) and this provides more reliable temperature predictions as compared to the available equation in literature. A simplified approach is proposed to evaluate cross-sectional temperatures in an insulated RC member. The validity of the approach is partly established by comparing predicted temperatures from proposed equation with a large set of temperature data from fire tests and finite element analysis. The proposed equations give predictions of cross sectional temperatures within a margin of ±10% in insulated reinforced concrete members exposed to standard fire. The applicability of the simplified approach for evaluating crosssectional temperatures of insulated RC beam exposed to fire is illustrated through a design example.
Acknowledgement This material is based upon the work supported by the National Science Foundation (United States) under Grant number CMMI-0855820 to Michigan State University. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.
1) The proposed equations are applicable for evaluating cross-sectional temperatures in RC members exposed to standard fire only. These equations are not applicable for design fires, which have a cooling phase after the initial growth phase. 2) The predictions using proposed equations might have a relatively larger error (more than ±10%) when the temperatures are in 20–150 C range. This is due to the fact that the accuracy of temperature range of 150–800 C is the primary target of the regression analysis. 3) The proposed equations do not take into account the effect of cracking in fire insulation on cross-sectional temperatures of RC member. 4) The proposed equations were obtained based on specific types of fire insulation. For RC members protected by other types of insulation, predicted temperatures may be slightly different from temperature profile in actual situation. 5) The proposed equations yields cross sectional temperatures within a marginal error of ±10% when the cross sectional dimensions (width or depth) of the member are higher than 200 mm. 6) The predicted temperatures using the proposed approach have some level of approximation (error to the extent of ±10%) and may not be accurate as in the case of detailed heat transfer analysis. This is mainly due to the fact that the proposed equations represent simplification to the complex heat transfer problem. 7) The proposed approach is a first step to gauge temperature distribution in insulated reinforced concrete members. Additional validation of the equations to cover wider range of variables to improve the accuracy of the results is needed. Temperature measurements at various depths of concrete in an insulated concrete member and also temperature data at the interface of FRP and concrete surface as well as insulation surface are needed. This type of temperature data is required for members made of different types of concrete.
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9. Conclusions Based on the information presented in this paper, the following conclusions can be drawn: 1. RC members strengthened with FRP reinforcement usually require external fire insulation to enhance fire resistance. A simplified 89
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