Uncertainties in steel temperatures during fire

Dilip K. Banerjee n

Materials and Structural Systems Division, Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

a r t i c l e i n f o

Article history:Received 1 November 2012Received in revised form9 May 2013Accepted 4 August 2013Available online 2 September 2013

Keywords:UncertaintyHeat transfer modelMonte Carlo methodFireSteelFire resistance tests

a b s t r a c t

In order to determine the fire resistance of steel members, steel temperatures must be estimated with ahigh confidence. There can be considerable uncertainty in temperatures of both protected andunprotected steels during fire exposure. This is due to uncertainty in the thermal boundary conditionsand thermophysical properties. In this study, uncertainties in both unprotected and protected steeltemperatures are estimated with the use of a Monte Carlo method in conjunction with a “Lumped HeatCapacity” approach for estimating steel temperatures. Computed data are compared with experimentalmeasurements obtained during Cardington fire tests (bare steel) and National Institute of Standards andTechnology (NIST) World Trade Center (WTC) tests (protected). Reasonable agreement was achieved.

Published by Elsevier Ltd.

1. Introduction

Both spatial and temporal variations of temperatures need tobe accounted for when evaluating the fire resistance of steels.The ability to predict with high confidence time-varying tempera-ture profiles in structural members is necessary for a robustperformance-based approach to the fire resistance design ofstructures. Therefore, uncertainties in steel temperatures mustbe accurately estimated. The focus of this study is to demonstrate asimple approach for estimating uncertainty in the predictedthermal response of both unprotected and protected steels duringa fire event.

Temperature profiles in a steel section during fire exposuredepend upon the temperature-dependent thermophysical proper-ties of steel, the thermophysical properties of fireproofing (sprayapplied fire resistive material, SFRM) for protected steel and theconvective and radiative heat transfer parameters associated withfire. However, there can be considerable uncertainty in estimatestypically used for these parameters. For example, although SFRMthickness measurements are reported according to the ASTM E605 standard, individual thickness measurements (as required bythe standard) can vary, while an average measurement is reported.In most cases, the SFRM thickness will be greater than thestipulated value as overspray is normally not penalized. Uncer-tainties in SFRM thickness can result in increased uncertainty insteel temperatures [1]. The variability in SFRM density can also

affect the overall uncertainties in steel temperatures. Density testsare performed following the ASTM E605 standard. Ref. [4] showedair dry density variability in the range of 10–20% for typical floortruss systems. The steel temperatures are influenced because ofthe effect of density variability on the volumetric heat capacity ofSFRM. A sensitivity study can be conducted to determine which ofthese parameters (thermophysical and heat transfer) most sig-nificantly influence the thermal response of the steel. Influentialparameters can be used to quantify the uncertainty in thepredicted temperatures.

Uncertainties can be broadly classified into two basic types:aleatoric (random) and epistemic (systematic). Aleatoric uncer-tainties are due to inherent randomness and cannot be removedby further analysis or testing. For example, fuel load density(MJ/m2) can be classified as inherently random. On the otherhand, epistemic (also known as knowledge-based) uncertaintiescan be reduced by using improved models or algorithms. Estima-tion of both aleatoric and epistemic uncertainties can provide aconfidence interval for time-varying estimates of structural tem-peratures during a fire event.

Uncertainties in measured temperatures of a steel sectionduring fire exposure can be attributed to (a) inherent measure-ment uncertainty associated with measuring devices such asthermocouples, (b) uncertainties associated with thermophysicalproperties of steel due to variability associated with steel compo-sition (e.g., steel web diagonals used in trusses can be sourcedfrom different vendors or from a vendor using various heats forproducing steels), (c) statistical randomness associated with truegas temperatures in fire in the vicinity of a measuring device,(d) uncertainties in heat transfer parameters such as emissivity

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n Tel.: þ1 301 975 3538; fax: þ1 301 869 6275.E-mail addresses: Dilip.Banerjee@nist.gov, banerjee_dilip@yahoo.com

Fire Safety Journal 61 (2013) 65–71

of a steel surface and convective heat transfer coefficients(for example, emissivity of protected steel will vary at a locationif the quality of fireproofing degrades due to unexpected or abruptvariation in gas temperatures). For steel sections, measuredtemperatures can be reported as mean temperatures with theiruncertainty bounds. For example, one can report measured tem-peratures at top flange, bottom flange, and web as mean tempera-tures along with uncertainty bounds for each.

Simplified analytical models are often used for modeling heattransfer in structural members in fire. For example, the “LumpedHeat Capacity Method” is widely used for modeling heat transferin steel members in fire [2]. The lumped heat capacity method isappropriate for steel because of its high thermal conductivity. It isuseful to develop a simple and practical approach for computinguncertainties in structural temperatures when such simplifiedanalytical approaches are used. This paper discusses a simplifiedapproach, e.g., “Lumped Heat Capacity Method”, for modelingtemperatures for both unprotected and protected steel and theuse of the Monte Carlo method for computing uncertainties insteel temperatures. Computed uncertainties are compared withresults of experimental measurements for validation of models forprediction of uncertainties in unprotected and protected steeltemperatures during fire exposure. The following test data wereused for validating computational approach for steel temperaturesand prediction of uncertainties:

1. Cardington Test 1 for unprotected steel [3].2. NIST fire resistance Test 4 for protected steel [4].

In the following section, a brief description is provided firstabout experimental measurements for both unprotected andprotected steels. Then, the “Lumped Heat capacity” approach isdescribed for computation of steel temperatures. Finally, com-puted steel temperatures and uncertainties in steel temperaturesare compared with experimental measurements for both unpro-tected and protected steels.

The validation of the computational approach for determininguncertainties in steel temperatures in fire will allow for reasonableprediction of uncertainties in temperatures when similar steelmembers are exposed to an unknown fire as long as uncertaintiesin key parameters such as gas temperatures in fire are known.

2. Experimental data

2.1. Unprotected steel

Steel temperatures were taken from Cardington Test 1 [3].Cardington Test 1 is a restrained beam test in which a 305�165�40 UB beam (British Universal Beam) was heated with a gasfired furnace over the middle 8 m of its 9 m length. The beam wasinstrumented with a number of thermocouples at the top flange,web, and bottom flange (see Fig. 1). Five sets of thermocoupleswere positioned along the length of the beam for this test.Temperature measurement data collected at the beam web wereused in this study. This is because the heating of the web can beconsidered to be uniform and the influence of the floor slab(positioned above the beam) on the web temperatures waspresumed to be minimal. This is necessary since measurementdata are compared with those computed using a simplifiedapproach that requires assuming no internal temperature gradientacross steel section. This is described later in the text in moredetail.

Fig. 1 shows the positions of the thermocouples at a beam crosssection. Note that thermocouples 51 through 55 represent webtemperatures at this section.

Mean web temperatures and standard deviations were com-puted using the five sets of web temperature data collected alongthe length of the beam. Furnace temperatures (e.g., thermocouples142 through 145 at this section in Fig. 1) were used to yield meanfire temperatures and standard deviations as functions of time.These mean furnace temperatures and standard deviations wereused in computing uncertainties in steel web temperatures asexplained later in the text.

2.2. Protected steel

Protected steel temperature measurement data were takenfrom the NIST fire resistance Test 4 [4]. As part of its investigationinto the World Trade Center (WTC) disaster, NIST conducted fourstandard fire tests of composite floor systems. Two full-scale tests(Test 1 and 2; span 35 ft. (10.7 m)) were conducted at the Under-writers Laboratories (UL) fire testing facility at Toronto, Canadaand the other two (reduced scale; Test 3 and 4; span 17 ft. (5.2 m))were conducted at Northbrook, IL. The UL test furnace was heatedby 80 individual floor mounted burners following the AmericanSociety for Testing and Materials (ASTM) E119 standard time-temperature curve [5], and furnace temperatures were monitoredat 16 locations in the furnace [4]. Time-temperature data werecollected at specific locations along the truss near the top chord, atmid height of the web, and at the bottom chord.

The floor system used in the test consisted of a lightweightconcrete floor slab supported by steel trusses. Fig. 2 shows a picture

46 47 48 49 50

56 57 58

5554

53

143

145144

142

52

51

xx

x

x

x

xxx xxx

x

xxxx

x

x

Fig. 1. Schematic representation of the location of the thermocouples used tocollect steel temperatures during the Cardington Test 1 [3].

Fig. 2. Floor system of the WTC towers [6].

D.K. Banerjee / Fire Safety Journal 61 (2013) 65–7166

of a typical floor system tested in UL furnace. The temperaturemeasurement locations on the steel truss are shown in Fig. 3.

The main composite trusses, which were used in pairs, had anominal clear span of 17 ft. (5.2 m). The steel trusses werefabricated using double-angles for the top and bottom chords,and round bars for the webs. The web members protruded abovethe top chord in the form of a “knuckle” which was embedded inthe concrete slab to develop composite action. The floor systemalso included bridging trusses perpendicular to the main trusses(Fig. 3). Fireproofing (SFRM) was applied directly on steel trussesto provide passive fire protection. The applied thickness of fire-proofing was 0.5 in. (12.7 mm) for this test. The diameter of theweb members was 0.92 in. (23.4 mm).

Steel temperatures were measured at 5 locations along thelength of the two main trusses. At each location along the maintruss, eight thermocouples were used to record temperatures. Twothermocouples were positioned on the upper chords, four thermo-couples were at the bottom chords, and two were at the mid-height of the webs (Fig. 3).

Twenty temperature measurements at mid height of the webalong the two main trusses (North and South) were used to obtainmean web temperatures and standard deviations. The standarddeviations in web temperatures (obtained as a function of time)were taken as experimentally observed uncertainties in steel webtemperatures. Sixteen furnace thermocouple readings were usedto obtain furnace mean temperatures and uncertainties in furnacetemperatures.

3. Simplified approach for computation of steel temperatures

The “Lumped Mass” or “Lumped Heat Capacity” method is validwhen there is no temperature gradient in a member (e.g., membertemperatures are uniform). This is an idealized case because atemperature gradient must be present in a member for heat toconduct into or out of a body. In general, the smaller the physicalsize of a member, the more realistic is the assumption of a uniformtemperature throughout the member. The method is valid if thefollowing inequality is maintained [7]:

hðV=FÞk

o0:1 ð1Þ

where h is the heat transfer coefficient, k is the thermal con-ductivity, V is the volume, and F is the surface area of the member.

The left hand side of Eq. (1) is also called the dimensionless Biotnumber. Steel members with typically high thermal conductivityare suitable candidates for the use of this method for computingtemperatures. The following section describes how temperaturesare computed for unprotected and protected steels when exposedto fire.

3.1. Lumped heat capacity method for unprotected steel

The following heat balance equation is used to derive a simpleexpression for the change in steel member temperature duringexposure to fire:

Heat entering¼ heat used to raise temperature_q}FΔt ¼ ρscsVΔTs ð2Þ

where _q}is the heat flux at the member surface (W/m2), Δt is thetime increment, ΔTs is the change in steel temperature, ρs isthe steel density, and cs is the gravimetric steel heat capacity.Since the heat transfer to the exposed member occurs by bothconvection and radiation during a fire event, Eq. (2) can berearranged as follows by expanding _q}:

ΔTs ¼FV

1ρscs

hcðTf�TsÞþsεðT4f �T4

s Þh i

Δt ð3Þ

where hc is the convective heat transfer coefficient, s is the Stefan–Boltzmann constant, ε is the steel effective emissivity, Tf is thegas temperature in fire environment, and Ts is the bare steeltemperature.

Spreadsheets are often used for calculating steel temperaturesfor a certain fire exposure. Gamble proposed such a method [8]with a time increment (Δt) of 5 min. EC3 [9] suggested amaximum time step of 30 s and a minimum value of sectionfactor (F/V) of 10 m�1. Kay et al. [10] reported a very goodprediction of steel temperatures in standard fire resistance testsusing this Lumped Heat Capacity method.

If both the convective and radiative heat transfer described inEq. (3) are combined to result in an effective heat transfercoefficient, Eq. (3) can be integrated to yield the followingequation for steel temperature as a function of time:

Ts�Tf

Ti�Tf¼ exp � hef f F

ρscsV

� �t

� �ð4Þ

where heff is the effective heat transfer coefficient, t is the elapsedtime and Ti is the initial steel temperature. Either Eq. (3) or Eq. (4)can be used to compute steel temperatures.

3.2. Lumped heat capacity method for protected steel

The calculation approach for protected steel is similar to theone discussed above for unprotected steel. It is assumed thatthe temperature at the exposed surface of the insulation on steel isthe same as the gas temperature in fire. It is also assumed that theinternal surface temperature of the insulation equals that ofthe steel member. Heat transfer coefficients are not required inthis approach as the method assumes that there is practically noresistance to heat transfer at the exposed surface. The followingequation is obtained by algebraic manipulation [2]:

ΔTs ¼FV

kidiρscs

ρscsðρscsþðF=VÞdiρici=2Þ

� �ðTf�TsÞΔt ð5Þ

where ci, ρi, and di are the gravimetric heat capacity, density, andthickness of the insulation respectively. Gamble proposed a similarequation and formulated a spreadsheet-based approach for com-puting steel temperatures [8]. When the volumetric heat capacityof the insulation (product of ρi and ci) is low, both ECCS [11] andMalhotra [12] suggest omitting the term in square brackets in Eq. (5).

Bridging Trusses

Locations of Temperature Measurement on a Main Truss

West Main Truss

East Main Truss

N

Approximately 0.8m x 0.4 m of concrete slab was dislodged due to spalling

2.0 m

0.7 m

2.0 m

0.7 m

5.4 m

Fig. 3. Thermocouple locations in the main truss in the UL test [4].

D.K. Banerjee / Fire Safety Journal 61 (2013) 65–71 67

This omission is acceptable if the following inequality holds:

ρscsA=24ρiciAi ð6Þ

where Ai is the cross-sectional area of the insulating material. This istypically true for low thermal capacity insulation.

4. Mean steel temperatures and uncertainties in temperatures

4.1. Unprotected steel

Mean steel temperatures and uncertainties in steel tempera-tures were computed using the Monte Carlo simulation method.Monte Carlo simulation uses a sequence of random numbers tosimulate the statistics of a real experiment. In this method, inputsare randomly generated from probability distributions to simulatethe process of sampling from a real population. It is thereforenecessary to choose distributions for the input parameters thatmost closely match available data, or best represent the state ofcurrent knowledge about them. The output data generated fromthe simulation are represented as probability distributions orconverted to error bars or confidence intervals.

A MATLAB1 script was written for the Monte Carlo simulation.Eq. (3) was used to compute steel temperatures. Note that, if theLumped Heat Capacity Method is used, there are four uncertainparameters in Eq. (3): furnace temperature, steel volumetric heatcapacity, convective heat transfer coefficient, and steel effectiveemissivity. Note that temperature dependent volumetric heatcapacity values for steel were used in the simulation [1], whichincluded effects of metallurgical transformation in steel at around727 1C. Normally steel heat capacity increases with temperaturewhile thermal conductivity decreases with increase in tempera-ture. Steel density is considered to be independent of temperature.Steel thermal conductivity values were not required in the lumpedmass approach. Geometric variability in the steel beam is ignored.In the first simulation, only furnace temperatures and steelvolumetric heat capacity were considered to be uncertain para-meters. This is because of a lack of proper knowledge of thedistribution of the convective heat transfer coefficient and steeleffective emissivity as a function of temperature. The steel effec-tive emissivity depends on many factors including the emissivityof the flames, the compartment walls, and the steel itself. Notethat heat transfer in most fires is dominated by radiative heatexchange especially at elevated temperatures and the influence ofthe convective heat transfer is not significant [2]. Paloposki andLiedquist [18] reported that the standard deviations of emissivitycan be as much as 20% of mean values at elevated temperatures.Therefore, one Monte Carlo simulation was also conducted byassuming the steel effective emissivity values were normallydistributed with its mean value of 0.7 [17] and standard deviationof 20% of the mean value. The fire temperatures were consideredto be normally distributed with means and standard deviationsthat are computed from the Cardington Test 1. The base value ofconvective heat transfer coefficient was taken 25 W/m2/K [16]. Themean and standard deviations of the steel volumetric heatcapacity are taken from [6], where standard deviations areapproximately 2% of the mean values [13]. Again, the steelvolumetric heat capacity is considered to be normally distributed.Normal distributions are typically used for measurement ofquantities in physical and engineering applications.

A comparison of the computed mean web temperatures(“Lumped Heat Capacity Mean”) and those obtained from theCardington Test 1 is shown in Fig. 4. Note that “Cardington Mean”represents the mean temperatures of all the thermocouples of thebeam web as shown in Fig. 1 and “Cardington Mean_1” representsmean temperatures of all thermocouples at the beam web exceptfor the two thermocouples in the vicinity of the top chord (e.g., 54and 55 in Fig. 1). It can be seen that the Lumped Heat Capacitymethod predicts the heating trend reasonably well, although itover-predicts steel temperatures during the heating cycle. Thetemperatures match very well during the cooling phase.

The uncertainties of steel temperatures are discussed next.Uncertainties in experimental web temperatures are obtained bytaking the standard deviations of the web temperatures inCardington Test 1. Note that the standard uncertainty u(y) of ameasurement y is defined as the estimated standard deviation of y[14]. The computed uncertainties in steel temperatures wereobtained by the Monte Carlo simulations in which 1000 randomsamples of the uncertain parameters were used to obtain adistribution of computed web temperatures based on Eq. (3).The mean and standard deviation of this steel temperaturedistribution can be computed by MATLAB at each time duringthe simulation. The standard deviations obtained from this dis-tribution were taken as the computed uncertainties in webtemperatures as a function of time. Fig. 5 compares the uncertain-ties in temperatures. Note that uncertainties in steel emissivityhave minimal influence on the overall uncertainties in steeltemperatures. Steel emissivity has first order influence on steeltemperatures, while fire temperature has fourth order influenceon steel temperatures as evident from Stefan–Boltzmann'slaw. This is also clear from Eq. (3). It is clear from Fig. 5 that

0

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Unc

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in T

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Computed uncertainty including steel emissivity uncertainty

Fig. 5. Uncertainty in the beam web temperatures obtained from the CardingtonTest 1 and those obtained using the Monte Carlo simulation (Eq. (3)).

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Lumped Heat Capacity Mean

Cardington Mean

Cardington Mean_1

Fig. 4. Plots of mean web temperatures obtained from the Cardington Test 1 andthose computed from the Lumped Heat Capacity method.

1 Certain commercial software or materials are identified to describe aprocedure or concept adequately. Such identification is not intended to implyrecommendation, endorsement, or implication by NIST that the software ormaterials are necessarily the best available for the purpose.

D.K. Banerjee / Fire Safety Journal 61 (2013) 65–7168

uncertainties in steel effective emissivity did not have muchinfluence on uncertainties in steel temperatures. It can be seenthat the general trend obtained from the Monte Carlo approachmatches that of the experimental curve quite well. The maximumuncertainty in temperatures was in the range of 40–50 1C. How-ever, the computed values are somewhat lower than thoseobtained from experimental measurements. This discrepancymay be attributed to: (a) the exclusion of uncertainty in convectiveheat transfer coefficient and (b) the influence of thermal massof the concrete slab on web temperatures, especially those nearthe top chord (thermocouples 54 and 55 in Fig. 1), (c) theapproximate solution and error associated with the Lumped HeatCapacity method. Nevertheless, this initial study shows a promis-ing agreement.

4.2. Protected steel

Monte Carlo methods were also used to compute uncertaintiesin the temperatures of the web members used in the NISTfire resistance test described above. Expressions shown in Eq. (5)were used to compute temperatures and the inequality in(6) holds. The constant density and heat capacity values used fortesting this inequality are: steel density 7856 kg/m3, steel heatcapacity 658 J/kg/K, SFRM density 219 kg/m3, SFRM heat capacity1098.2 J/kg/K. These values correspond to those at an averagetemperature of 500 1C [1]. The following parameters were con-sidered to be uncertain:

a. Fire temperatures.b. Volumetric heat capacity of steel.c. Thickness of fireproofing or SFRM.d. Thermal conductivity of SFRM.

SFRM is a general category of materials used for steel fire-proofing that includes an array of proprietary products such asBlazeshield D C/F, Monokote etc. Each of these products hasvarying thermal properties that are engineered for providingpassive fire protection. Here, “SFRM” is used only as a genericidentifier of a fire resistance product type, and consequently,thermal properties may vary significantly among different com-mercial products available in this category. Thermal properties ofSFRM vary with increase in temperature. SFRM densities initiallydecrease with temperature until about 600 1C, beyond which theyincrease rapidly. Thermal conductivity of SFRM gradually increaseswith increase in temperature. SFRM products often undergo anendothermic “dehydration spike” that results in very significantenergy absorption within a relatively narrow temperature range.This is reflected in the heat capacity or enthalpy vs. temperaturecurve. A comprehensive finite element analysis employing thesetemperature dependent thermal properties can provide a betterinsight into the fire resistance performance of a SFRM product.Data used for volumetric heat capacity of steel are same as thoseused for the case of unprotected steel. Mean fire temperatures andstandard deviations were obtained from the measurements col-lected by the 16 furnace thermocouples. The distribution of firetemperatures was considered to be normal. Mean thermal con-ductivity data for the SFRM were taken from [6]. The uncertaintiesin thermal conductivity are about 6% of mean values [1]. Thermalconductivity of SFRM is assumed to be normally distributed. TheSFRM thickness is lognormally distributed [15] with mean valueof 0.514 in. (13.1 mm) [4]. The COV (coefficient of variation) ofSFRM thickness is between 0.17 and 0.27 [1]. Since precise COVvalues have not been reported in this test, both COV values of 0.17and 0.27 were used for obtaining the standard deviations of SFRMthickness. A MATLAB script was written to compute the distribu-tion of steel temperatures as a function of time using Eq. (5).

The output temperatures were normally distributed. The standarddeviations of the computed temperatures were compared withexperimental measurements for uncertainty in steel temperatures.

Fig. 6 shows a comparison of mean temperatures for the beamweb. It can be seen that the agreement is very good until about60 min, beyond which the computed temperatures becomeincreasingly greater than the experimentally recorded tempera-tures. At about 55 min into the test, a very loud noise was heardwhen a large piece of concrete fell to the lower part of the furnaceas a result of spalling [4]. See Fig. 3 for the approximation locationof this spalling. It is presumed that this resulted in significant lossof heat from the furnace. Ceramic fiber insulation was placed overthe opening in the concrete floor in order to protect the hydraulicequipment and allow the test to continue. Web temperaturesrecorded during the full scale tests (Test 1 and 2 with 0.75 in.(19 mm) fireproofing) were much higher than those for thereduced scale test (e.g., Test 3 with 0.75 in (19 mm) fireproofing).Large buildup of SFRM was noticed at the intersections of web andchord members during the post-test observations [4]. Note thatthis buildup is not a time-dependent phenomenon and NIST's2 reduced scale tests showed a large buildup of SFRM at theintersections of web and chord sections (much more than that wasnoticed for the other 2 tests, which were full-scale tests). This isdifferent from the traditional overspray. The buildup was becauseof the difficulty of ensuring a constant thickness at the intersectionof chord and web members because of constraint faced by theoperator while spraying. This buildup of SFRM could also havecontributed in resulting in lower measured web temperatures inreduced scale tests.

Fig. 7 shows a comparison of uncertainties obtained in the testand those obtained from the Monte Carlo simulation using two

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mpe

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Experimental uncertainty Computed Uncertainty, COV of di=0.17 Computed Uncertainty, COV of di=0.27

Fig. 7. Uncertainties in beam web temperatures obtained from the NIST test andthose obtained using the Monte Carlo simulation for protected steel.

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Lumped Heat Capacity Model

Fig. 6. A plot of mean web temperatures obtained from the NIST test and thosecomputed from the Lumped Heat Capacity method for protected steel.

D.K. Banerjee / Fire Safety Journal 61 (2013) 65–71 69

different values of standard deviations of SFRM thickness (di)corresponding to COV values of 0.17 and 0.27. The uncertaintiesin fire temperature, steel volumetric heat capacity, and thermalconductivity of SFRM were included in the simulation as discussedin the previous paragraphs. Computed uncertainties in steel webtemperatures obtained for the COV values of 0.17 for di matchreasonably well with those from experimental measurementsuntil about 60 min, beyond which there is an abrupt increase inexperimentally obtained uncertainties in steel web temperatures(Fig. 7). Computed uncertainties for the COV values of 0.27 for diprovide an upper bound. It appears from these plots that the COVvalue for di in the experiment was probably close to 0.17. Concretespalling and SFRM buildup at the intersection of web and chordmembers (as explained in the previous paragraph) appear to havecontributed significantly to the enhanced measured uncertaintybeyond 60 min into the fire test. The maximum uncertainty incomputed results was about 20 1C and 30 1C respectively for theCOV values of 0.17 and 0.27 for the SFRM thickness. The MonteCarlo simulations show another interesting behavior. Uncertaintiesin steel temperatures quickly increase to a maximum value withinabout 20–25 min into the test and then start decreasing afterabout 60 min into the test. This is because the absolute values ofsensitivity coefficients, ∂T=∂Ki (Ki is an uncertain parameter), havemuch higher magnitudes during initial transients and thesecoefficients decrease toward the end of the test as temperaturesin the furnace become more uniform.

5. Summary

A Monte Carlo based approach was used to compute uncer-tainties in temperatures in both unprotected and protected steelmembers during fire resistance tests. The well-known “LumpedHeat Capacity” method was used to obtain equations required forthe computation of steel temperatures. Computed time-dependentmean steel temperatures and uncertainties in steel temperatureswere compared with those obtained from fire resistance tests(Cardington Test 1 data were used for unprotected steel and NISTWorld Trade Center fire resistance Test 4 data were used forprotected steel).

Computed time-dependent mean temperatures and uncertain-ties in steel temperatures agreed reasonably well with thoseobtained from experimental measurements. The maximum uncer-tainties in unprotected steels during fire resistance tests werehigher than those obtained in protected steels. Loss of heat fromthe furnace due to concrete spalling and SFRM buildup at web/chord member intersection for the reduced-scale NIST fire testpossibly resulted in somewhat lower steel temperatures andincreased uncertainties in steel temperatures during the later partof the test.

It is quite clear from this study that it is difficult to predictuncertainties in protected steel temperatures (when steel beamsare used in fire tests as part of composite floor systems). This isbecause of many unknowns such as (a) proper knowledge of thequality and thickness of steel fireproofing at different locations(e.g., it is difficult to ascertain the quality of adherence of fire-proofing on steel as test progresses), (b) changes in thermalproperties of fireproofing as fireproofing degrades possibly dueto rapid and uneven heating, (c) inadequate knowledge of gastemperatures near the vicinity of structural members, and(d) concrete spalling that could possibly change macroscopic heatflow in the furnace and impact the quality of SFRM bonding on thesteel etc. In addition, it is well known that during a standardfurnace test, SFRM is first applied on steel members and then themajority of the design load is applied via load actuators. Thisapproach may induce SFRMmicro-cracking during the onset of the

fire resistance test. The possible existence of such variable SFRMmicro-cracking at the beginning of the test may become amplifiedduring the subsequent elevated temperature furnace exposure asthe SFRM dehydrates, and may lead to increased and varying rateof member heating. This phenomenon could add to the variabilityof SFRM thickness distribution on steel members during fireexposure in standard fire tests.

It is demonstrated that the Monte Carlo method can beeffectively used to compute uncertainties when the computationalapproach is simple as is the case here. Further research is neededto understand how the uncertain parameters considered in thisstudy could influence the overall uncertainties in steel tempera-tures with more advanced models such as FE (finite element)models. Use of probabilistic models such as Monte Carlo methodin conjunction with robust FE model will be needed to accuratelydetermine the uncertainties in steel temperatures in fire especiallyfor protected steels.

Data from a number of similar structural fire tests employingdifferent furnaces can be used to properly construct the uncer-tainty bounds and distribution of member temperatures duringelevated temperature exposure. Such information can be used toproperly develop temperature dependent load and resistancefactors that are used by design engineers. Uncertainties in struc-tural temperatures have effects on both the load and resistancefactors. There is ongoing active research in determining the loadand resistance factors at elevated temperatures and the presentpaper is directed toward this end.

Acknowledgments

The writer wishes to gratefully acknowledge helpful exchangeswith E. Simiu concerning the application of the Monte Carloapproach to the problems discussed in this paper.

Thanks are also extended to F. Lombardo for his help inimplementing a few MATLAB functions in the script developedto compute uncertainties.

References

[1] NIST (National Institute of Standards and Technology), Federal building andfire safety investigation of the World Trade Center disaster: final report on thecollapse of the World Trade Center Towers, NIST NCSTAR 1-5G, Gaithersburg,Maryland, 2005.

[2] A.H. Buchanan, Structural Design for Fire Safety, John Wiley & Sons, U.K., 2001.[3] British Steel plc., The behavior of multi-story steel framed buildings in fire,

Swinden Technology Institute, Rotherham, U.K., 1999.[4] NIST (National Institute of Standards and Technology), Federal building and

fire safety investigation of the World Trade Center disaster: final report on thecollapse of the World Trade Center Towers, NIST NCSTAR 1-6B, Gaithersburg,Maryland, 2005.

[5] ASTM, Standard test methods for fire tests of building construction andmaterials, ASTM E119-73, ASTM International, Conshohocken, PA, 1973.

[6] NIST (National Institute of Standards and Technology), Federal building andfire safety investigation of the World Trade Center disaster: final report on thecollapse of the World Trade Center Towers, NIST NCSTAR 1-6, Gaithersburg,Maryland, 2005.

[7] J.P. Holman, Heat Transfer, sixth Ed., McGraw-Hill, New York, 1986.[8] W.L. Gamble, Predicting protected steel member fire endurance using spread-

sheet programs, Fire Technology 25 (1989) 256–273.[9] EC3, Eurocode 3: design of steel structures, ENV 1993-1-2: general rules –

structural fire design, European Committee for Standardization, Brussels,Belgium, 1995.

[10] T.R. Kay, B.R. Kirby, R.R. Preston, Calculation of the heating rate of anunprotected steel member in a standard fire resistance test, Fire Safety Journal26 (1996) 327–350.

[11] ECCS, Design manual on the European recommendations for the fire safety ofsteel structures, European Commission for Constructional Steelwork, Brussels,Belgium, 1985.

[12] H.L. Malhotra, Design of Fire Resisting Structures, Surrey University Press, U.K.,1982.

[13] NIST (National Institute of Standards and Technology), Federal building andfire safety investigation of the World Trade Center disaster: final report on the

D.K. Banerjee / Fire Safety Journal 61 (2013) 65–7170

collapse of the World Trade Center Towers, NIST NCSTAR 1-3E, Gaithersburg,Maryland, 2005.

[14] B.N. Taylor, C.E. Kuyatt, Guidelines for evaluating and expressing the uncer-tainty of NIST measurement results, NIST Technical Note 1297, Gaithersburg,Maryland, 1994.

[15] NIST (National Institute of Standards and Technology), Federal building andfire safety investigation of the World Trade Center disaster: final report on thecollapse of the World Trade Center Towers, NIST NCSTAR 1-6A, Gaithersburg,Maryland, 2005.

[16] ECI, Eurocode 1: Basis of design and design actions on structures, Part 2-2:action on structures exposed to fire, ENV 1991-2-2, European Committee forStandardization, Brussels, Belgium, 1994.

[17] S. Lamont, A.S. Usmani, D.D. Drysdale, Heat transfer analysis of the compositeslab in the Cardington frame fire tests, Fire Safety Journal 36 (2011) 815–839.

[18] T. Paloposki, L. Liedquist, Steel emissivity at high temperatures, VTT ResearchNotes 2299, VTT Technical Research Centre of Finland, Vuorimiehentie,Finland, 2005.

D.K. Banerjee / Fire Safety Journal 61 (2013) 65–71 71