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This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
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Structural analysis of reinforced concrete chimneys subjected to uncontrolled fireAshkan Vaziri a,∗, Amin Ajdari a, Hosam Ali b, Artemis Agelaridou Twohig b
a Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, United Statesb FM Global, 1151 Boston-Providence Turnpike, Norwood, MA 02062, United States
a r t i c l e i n f o
Article history:Received 2 December 2010Received in revised form12 May 2011Accepted 9 June 2011Available online 20 July 2011
Keywords:Concrete chimneyUncontrolled fireHeat transferFinite elementStructural model
a b s t r a c t
We studied the behavior and residual structural capacity of reinforced concrete chimneys subjected to anuncontrolled fire. We used a combination of a heat transfer finite element model – to obtain the temporaldistributions of temperature during the fire event – and the structural model of concrete chimney designprovided by the American Concrete Institute (ACI 307–08). This approach allows estimating the reductionin the vertical (axial) strength and moment strength of the chimney both during a fire and post-fire,and gives a direct estimate of the reduction in the safety factors of the concrete chimney. Using thismethod, we examined the impact of various design parameters on the residual structural capacity of aconcrete chimney subjected to an internal fire. An iterative finite element method was also presentedas an alternative to the ACI 307 calculations. Moreover, finite element calculations were used to studythe role of thermal stresses on the axial strength of the chimney during fire. Our study provides insightinto possible failure mechanisms of concrete chimneys damaged due to fire and could suggest possibleapproaches for minimizing the risk of chimney failure due to an uncontrolled fire.
Concrete chimneys are used in power plants for venting hotflue gases or smoke to the outside atmosphere. In recent years, theheight of power plant concrete chimneys has increased to enhancethe draw of air for combustion and to disperse pollutants over agreater area to reduce pollutant concentrations. While the numberof reported chimney collapses due to an internal fire is very small,the consequences of chimney damage could be costly in terms ofeconomic and human loss. The popularity of fiberglass reinforcedplastic (FRP) liners – which are combustible materials – has madethe risk of a fire in tall chimneys even more relevant in recentyears. The FRP liners are used in reinforced concrete chimneys toprotect the chimney shell from the effect of hot flue gases. Someof the possible sources of ignition in chimneys with FRP linerscan be hot work inside the chimney during FRP installation ormaintenance, ignition of stored flammable materials at the baseof the chimney, ignition inside the flue gas desulfurization systemor other equipment upstream of the chimney.
A critical effect of an uncontrolled fire in a chimney is thereduction in concrete strength [1–6], which leads to a decrease inthe chimney load carrying capacity and service life. At an elevatedtemperature, the concrete experiences a variety of chemical andphysical changes. For example, large volume changes resulting
from non-uniform thermal expansion of aggregates and shrinkageof the cement paste, results in cracking and spalling. Spallingwhich is usually explosive and critical for structural integrity, isinduced bymechanical and thermal stresses and pore pressure [7].Spalling generally occurs at high temperatures, even though it isalso observed at temperatures as low as 200 °C [8,9]. For spallingto occur, there needs to be a minimum moisture content aswell as a temperature gradient of approximately 5–8 K/mm [6].Temperature gradients induced by heating or uncontrolled firedepend not only on the heating source temperature but also on theheating rate.
In Fig. 1, we have re-plotted the Eurocode 2 [10] data (curve 1)for a concrete with siliceous aggregates which is commonly usedin tall chimneys. These data suggest that the concrete compressivestrength at 600 °C reduces ∼50% from its strength at roomtemperature. This reduction in strength is qualitatively similar tothe behavior of structural steel, where high temperature couldlead to significant weakening and in many cases catastrophicfailure of the structure [11–13]. However, steel strength almostfully recovers as the steel cools down after the fire, so thereis little concern about the behavior of steel structures post-fire.Concrete strength further decreases as it cools down from ahigh temperature, as the incompatibility between the thermaldeformations of the concrete constituents leads to propagations ofexisting microcracks and formation of new ones [4]. The amountof reduction in the strength of the concrete is a complex functionof the cooling method and rate. Sakr and Hakim [4] measuredthe role of cooling methods on the residual compressive strength
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Fig. 1. Compressive strength of concrete at elevated temperatures (from [6]) andafter cooling (from [4]). The cooling of the concrete specimens leads to furtherreduction in the compressive strength of the concrete. Here, the results are shownfor both air cooling and water cooling for samples that are kept at elevatedtemperatures for 2 and 3 h prior to cooling. Samples are then tested at roomtemperature.
of concrete kept at an elevated temperature for different heatingdurations. They considered three different cooling methods: aircooling, water cooling and foam cooling. In Fig. 1, we have showna selected set of results from their study, which show that coolingby water leads to lowest residual compressive strength of theconcrete tested at room temperature. In this study, we used thevalues given by Eurocode 2 [10] to estimate the concrete strengthat elevated temperature, which is applicable for heating ratesbetween 2 and 50 K/min.
The objective of this study is to evaluate the structural responseof reinforced concrete chimneys subjected to uncontrolled fire.Fig. 2 shows the schematic of the scenarios considered here. Duringthe fire, the reinforced concrete shell should be able to resistthermal stresses, its own weight, part of the liners’ weight, andwind loads. Even if the chimney does not collapse during thefire the concrete will be damaged by the heat. After the fire, thedamaged shell should be able to support its weight, the windload, and possibly the inertial loads of an earthquake. Here, westudied the behavior and residual structural capacity of reinforcedconcrete chimneys, with and without an opening, subjected toan uncontrolled fire. We combined a heat transfer finite elementmodel to obtain the temporal distributions of temperature duringthe fire event, with a structural model of concrete chimney designprovided by the American Concrete Institute (ACI 307–08) [14].This approach allows estimating the reduction in the vertical(axial) strength and moment strength of the chimney during afire and post-fire, and gives a direct estimate of the reduction inthe safety factors of the concrete chimney. In our study, we usedcurve 1 in Fig. 1 for estimating the concrete strength at elevatedtemperatures (e.g. during fire). For estimating the residual concretestrength post-fire, we used the data associated with water coolingsince it leads to the greatest reduction in concrete strength (curves3 and 5 in Fig. 1).
The details of the proposed approach are discussed in Section 2.An iterative finite element method is also presented as analternative to the ACI 307 calculations in Section 3. In Section 4, weexamine the impact of various design parameters on the residualstructural capacity of concrete chimneys using, as an example, afire that lasts at most 3 h and causes the average temperatureinside the chimney to be about 870 °C. In Section 5, we discussthe role of thermal stresses on the behavior of reinforced concretechimney during a fire. The key conclusions are drawn in Section 6.
Fig. 2. Scenarios considered in structural analysis.
2. Theoretical approach
To investigate the effect of fire on the residual structural capac-ity of a reinforced concrete chimney, we combined finite elementheat transfer calculations with the ACI structural approach [14].The details of the heat transfermodel developed for calculating thetemperature profile in the chimney shell during the fire event arediscussed in Section 2.1. The elevation of the temperature in thechimney shell results in a reduction of the compressive strengthof the concrete and thus inherently, a reduction in the axial andmoment strengths of the chimney during a fire event and alsopost-fire. The estimated temperature profile from the heat transfermodel was converted to an average reduction in the compressivestrength of concrete and axial strength of the chimney, using anapproximate method (Section 2.2). Subsequently, the reduction inthe vertical strength andmoment strength of the chimney was ob-tained using the structural model provided by the ACI Code as dis-cussed in Section 2.3. In this study, we assumed that the concretestrength is fc = 27.5MPa (4000 psi), the strength of the reinforce-ment steel bars is, fy = 220 MPa (32 000 psi), and the density ofthe concrete is ρ = 2400 kg/m3.
2.1. Heat transfer model
The temperature profile in the chimney shell and its spatial andtemporal variations during the fire event is a complex functionof fire parameters (e.g. temperature of gases inside the chimneyand duration of the fire) and their dynamics, as well as theproperties of the chimney shell. For concrete chimney shells withFRP liners, many fire initiation and propagation scenarios arepossible. Examples include initiation of the fire at a certain heightof the liners followed by vertical propagation both upwards anddownwards at different speeds, or the case that the fire results indetachment of the liners from the chimney and their accumulationat ground level inside the chimney. The latter case can result
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Fig. 3. Heat transfer model of a concrete chimney. (A) Schematic of a one-dimensional heat transfer model used for calculating the temporal variation of the temperatureprofile in a concrete chimney. (B) Temperature profiles in a concrete chimneywith ri = 12.5m (41 ft), t = 0.46m (18 in.) during the fire event and the steady state condition.
in long-lasting fire leading to locally higher temperatures in thechimney shell. Different scenarios can be assumed depending onthe incoming airflow and openings in the chimney shell. If theincoming airflow (from the openings) is adequate during the fire, itis expected that the incoming air ‘pushes’ the flames towards oneside of the chimney causing localized heating of the chimney shell.If the pile of burning FRP rubble causes ‘‘blockage’’ of the airflow, itis conceivable that less air will flow through the chimney and theconcrete shell will be exposed to higher average gas temperatures.That is the scenario considered here, by conservatively assumingthat even for restricted airflow conditions, there is enough oxygenfor well-ventilated burning, and also having the total inner surfaceof the chimney shell subjected to the fire (rather than local heating,which leads to a less pronounced reduction in the chimney residualstrength). Considering that themaximumbending loads due to thewind also occur at the base of the concrete, the case describedabove appears to lead to maximum reduction in the residualstructural capacity of the chimney shell post-fire.
Here, we developed a simplified finite element model tocalculate the temperature profile through the thickness ofthe concrete chimney shell during fire using ABAQUS/CAE 6.9(SIMULIA, Providence, RI). For the scenario mentioned above,the structure and the fire loading is axisymmetric and thus, thetemperature profiles can be estimated using a one-dimensional(through the thickness) axisymmetric finite element model ofthe chimney shell, Fig. 3A. The model was meshed using 8-node quadratic axisymmetric heat transfer elements, and meshsensitivity analysis was carried out to ensure that the results arenot affected by the mesh size. For the concrete, we assumed thecoefficient of thermal conductivity, Cc = 1.72 W/m K, specificheat, Cp = 900 J/kg K [14]. In our calculation, the coefficients ofheat transmission from the outside surface of the chimney shellto surrounding air, Ko = 68 W/m2 K, and the coefficients of heattransmission from the gas to the inner surface of the chimney shell,Ki = 164.5 W/m2 K, were estimated from data provided in theACI code. The above value of Ko indirectly accounts for the effectof convection and conduction of heat from the outer surface to thesurrounding air, while the Ki coefficient considers conduction andconvection along with the radiation of heat from the fire to theinner surface of the chimney shell. It is assumed for this study thatas a result of the fire, the temperature at the inner surface of thechimney is 870 °C with an outer ambient temperature of 27 °C.Fig. 3B shows a set of temperature profiles in a chimney shell withri = 12.5 m (41 ft) and t = 0.46 m (18 in.) calculated at differentfire durations.
2.2. Simple model for axial strength calculation during and post-fire
The axial or vertical strength of the chimney, Pn, with averageradius r , thickness t and concrete nominal compressive strength,fc , can be estimated from Pn = 2π f crt , by neglecting the effect ofthe steel on the axial strength of the chimney. The vertical strengthof chimney generally varies through its height since the radius andthickness of the chimney shell are not constant in most designs.To estimate the residual structural capacity of the chimney duringfire, first, we calculate the concrete strength through the chimneyshell thickness both during the fire event, as well as after the fire,when the chimney has cooled down (i.e. post-fire condition). It wasassumed that the strength of the concrete at elevated temperaturesdepends only on the value of the temperature (i.e. not the rate oftemperature change) allowing the concrete strength to be obtainedfrom the curves in Fig. 1. We related each temperature profile to aconcrete strength profile through the thickness, both during thefire and after cooling, which was used to calculate the averagestrength of the concrete, f ′
c . For the post-fire residual strength,we calculated the average temperature at each point across thechimney shell by integrating the transient temperature data atthe point over the total fire duration. Then, we used Fig. 1 to findthe corresponding strength after cool-down at each point. Theintegration of concrete strength through its thickness provides anestimate for the reduction in the axial (vertical) strength of thechimney section denoted by γ = f ′
c /fc . Here, we conservativelyconsidered the water cooling data which will cause the maximumdecrease in concrete strength post-fire.
Fig. 4A shows the residual axial strength of the concretechimney with ri = 12.5 m, and t = 0.46 m, and no opening,as a function of fire duration for both during-fire and post-fireconditions. The average reduction in the compressive strength ofthe chimney for the steady state temperature profile that is plottedin Fig. 3B, and by considering water cooling data as explainedabove, gives γ = 0.63. This indicates that the axial strength ofthe chimney in the case of steady state heat transfer through itsthickness is reduced by 37%. Note that the fire duration required forreaching a steady state condition is very long and the strength datais only presented here to show the upper bound in the reductionof concrete axial strength.
2.3. ACI methodology for bending strength calculation
From the structural perspective, the stresses acting on achimney cross section result from bending due to lateral forces(wind and earthquake) as well as normal forces (weight of thereinforced concrete, the liners’ and the live loads). The structural
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Fig. 4. Post-fire behavior of a concrete chimney. (A) Reduction in the vertical (axial) strength of the concrete chimney, denoted by γ , as a function of fire duration duringthe fire event and post-fire. (B) Normalized nominal moment strength of the chimney versus the normalized vertical load for a fire duration of 3 h. The results are shown fornormal conditions (before fire), during an ongoing fire event with the duration of 3 h, and post-fire. (C) Reduction in the nominal moment strength versus the normalizedvertical load for the same cases as B. The concrete chimney has ri = 12.5 m (41 ft), t = 0.46 m (18 in.) and no opening.
model of chimney shells provided by the ACI code [14] accountsfor the dead loads, thermal stresses and wind and earthquakeloads. To quantify the role of the concrete temperature on themoment strength of a chimney, we used the method provided bythe ACI code (ACI 307–08) for reinforced concrete chimney design.In this method, the nominal moment strength of a cross-section ofa concrete chimney is calculated from Mn = Pur(cos θ + K2/K1),where Pu is the factored vertical load at that cross-section andK1and K2 are:
K1 = 1.7Qλ+ 2εmKeωtQ1 + 2ωtλ1
K2 = 1.7QR + εmKeωtQ2 + 2ωtK
where θ is half of the central angle subtended by the neutralaxis, Q is the stress level correction parameter; K , R, Q1 andQ2 are the parameters for nominal moment strength, εm isthe maximum concrete compressive strain; λ and λ1are thegeometrical parameters for each cross-section of chimney shell;Ke and ωt are non-dimensional parameters which are defined inAppendix. Considering the ACI recommendation, for a chimneywith constant cross-section (average radius r and thickness t), theaxial load of the chimney can be calculated from Pu = 1.4 ∗2πrtH
∗ ρg , where H is the height of the chimney located above
the cross-section of interest (e.g. if the calculations are carried outfor the bottom cross-section of the chimney, H is equal to theheight of the chimney). Thus, the normalized axial load is Pu/rtfc =
[2.8πρg/fc ] ∗ H , where g is gravitational acceleration. However,it should be noted that most chimneys have varying thickness
along their height which should be considered when calculatingthe normalized axial load applied to the chimney cross-section.
For a given chimney with known average radius, r , thickness, t ,concrete compressive strength, f ′
c , the angle β which denotes thehalf-opening angle (if any), and the ratio of vertical reinforcementarea to total area of concrete, ρt , the stress envelope for eachcross-section of the concrete chimney can be plotted in non-dimensional form as Mn/fcrt2 against Pu/fcrt (see Appendix fordetails). Examples of the calculated stress envelopes for a concretechimneywith ri = 12.5mand t = 0.46mand no opening for a fireduration of 3 h are shown in Fig. 4B. Fig. 4C shows the reductionin the nominal moment strength of the concrete chimney versusthe nominal vertical load for the cases considered in Fig. 4B. InFig. 4, the results associated with the curves labeled during fireare calculated by excluding thermal stresses. The role of thermalstresses on the chimney strength is discussed in Section 5. Ingeneral, calculations of the residual structural capacity of chimneyshells should be based on the post-fire condition, as the concreteloses additional strength after cooling.
Another example is provided in Fig. 5A, where we plotted thestress envelopes, in non-dimensional form, for the same chimneyat different times of the fire event. The results are obtainedusing the heat transfer results shown in Fig. 3B. Using thesestress envelopes, we also estimated the reduction in the chimneynominal moment strength denoted by Mn/Mn(Normal condition),as quantified in Fig. 5B. The results are shown for various firedurations, as well as the steady state temperature profile whichdenotes the upper bound of reduction in the chimney nominalmoment strength.
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Fig. 5. (A) Normalized nominalmoment strength of the chimney versus the normalized vertical load in normal conditions and at different fire event durations. (B) Reductionin the nominal moment strength, denoted byMn/Mn(Normal condition), versus the normalized vertical load for a chimney subject to uncontrolled fire. The results are shownfor various fire durations, as well as the steady state temperature profile shown in Fig. 3B.
Fig. 6. Finite element model of the concrete chimney and comparison of the finite element results with the ACI structural model for chimney with no opening.
3. Iterative finite element approach for chimney strengthcalculation
As an alternative to the ACI 307 methodology and to evaluatethe validity of our analytical approach, we developed a finiteelement-based iterative method, a schematic of which is shownin Fig. 6. The method consists of a two-dimensional model ofthe chimney cross-section where the following assumptions aremade (in accordance with the ACI standard code): (i) Planestrain condition across the section, (ii) The tensile strength ofthe concrete is ignored, (iii) The steel reinforcement in bothcompression and tension zones is taken into account, and, (iv) Themaximum compressive strain in the concrete is 0.003. Assumingan angle for the compressive zone, we eliminate the portion ofconcretewhich is assumed to be in tension as shown in Fig. 6. Then,
the axial load corresponding to the assumed compressive zone isapplied and afterwards the bending moment is increased until thestrain reaches the maximum allowable compressive strain (εm =
0.003). The size of the compressive zone is examined (e.g. the stateof stress in the concrete was checked to assure that the portion ofconcrete we are modeling is in compression). Based on the results,the size of the compressive zone size is changed and a new modelis developed for the next iteration. This procedure is continued tillconvergence is achieved. In Fig. 6, the results obtained from theACI code are compared with finite element results for a chimneysubjected to a wide range of axial loads, showing good agreementbetween the two approaches. In the calculations, the simulationsare repeated until we get the convergence with an error of 1° forthe half angle of the compressive zone, θ .
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Fig. 7. Effect of chimney thickness. (A) Temperature profiles in a concrete chimney for various thicknesses after 3 h of fire. (B) Reduction in the vertical (axial) strength ofthe concrete chimney with different thickness (post-fire) as a function of fire duration. (C) Normalized nominal moment strength of the chimneys with various thicknesses(post-fire) versus the normalized vertical load for a 3 h fire event. The plot for the normal condition is also shown. (D) Reduction in the nominal moment strength versusthe normalized vertical load for post-fire behavior of the chimney with different thicknesses after a 3 h fire event. The inner radius of the concrete chimney considered herewas ri = 12.5 m (41 ft) and the section examined was assumed to have no opening. The temperature at the inner surface of the chimney was 870 °C.
4. Parametric study and discussions
In this section, we examine the impact of various designparameters on the residual structural capacity of concretechimneys using as an example, a fire that lasts 3 h and causesthe average temperature inside the chimney to be about 870 °C.The results are presented in the form of stress envelopes andreduction in the chimney nominal axial and moment strengths.Residual strength maps were constructed to give an estimate ofthe reduction in the safety factors of the chimney as a function ofchimney height, thickness and steel relative density, as well as thenumber and size of the opening in the chimney shell.
Fig. 7 shows the effect of chimney shell thickness, t , on thestress envelopes of a chimney with ri = 12.5 m (41 ft) and noopenings. Fig. 7A displays the temperature profiles in a concretechimney with different thicknesses after 3 h of fire. For chimneyswith smaller thickness, the average temperature would be higher,which results in lower concrete strength, and therefore loweraxial strength and moment strength of the chimney. Fig. 7Bshows the average reduction in the concrete strength, which isequal to the reduction in the vertical strength of the chimney,as a function of fire duration for different shell thicknesses. Forrelatively thick chimney shells, the reduction in axial strength ofthe chimney is sensitive to chimney thickness, as well as to fireduration. These results are extended in Fig. 7C andD, by calculatingthe normalized nominal moment strength of the chimneys with
various thicknesses versus the normalized vertical load for post-fire behavior after a 3 h fire event. The plots for the normalcondition are also shown.
In Fig. 8, we look at the role of steel relative density, ρt , on thestructural capacity of the chimney. We considered four differentrelative densities starting as low as 0.5% up to 2.0%, which isthe normal range considered in designing concrete chimneys. Theresults for chimneys as built and after a 3-h fire are presented inFig. 8A and B, respectively. The reduction in the nominal momentstrength, which suggests a small difference for different relativedensities of steel reinforcement, is plotted in Fig. 8C. In this set ofcalculations, the chimney has no opening, and has an inner radius,ri = 12.5 m (41 ft) and thickness, t = 0.46 m (18 in.).
Fig. 9 presents the results for reinforced concrete chimneyswithone opening. A schematic of the chimney cross-section with oneopening is shown in Fig. 9A. The role of opening angle on thestress envelopes of a concrete chimney with ri = 12.5 m, andt = 0.46 m is quantified in Fig. 9B for both normal conditionsand post-fire conditions (3-h fire event). The reduction in nominalmoment strength for chimneys with one opening angle after 3 hof fire exposure (Fig. 9C) shows that, for a cross-section with wideropening, the averagemoment reduction is higher due to the overallweakness of the cross-section compared to the cross-section withno opening.
The results are extended in Fig. 10 for chimneys with twoopenings, where β denotes the half-opening angle and α denotes
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Fig. 8. Effect of steel reinforcement relative density. (A & B) Normalized nominal moment strength of the chimneys with different relative steel densities versus thenormalized vertical load. The results are shown for normal conditions (A) and after a 3 h fire event (B). (C) Reduction in the nominal moment strength versus the normalizedvertical load for post-fire behavior of the chimneys with different relative steel densities for post-fire behavior after a 3 h fire event. The concrete chimneys have ri = 12.5m(41 ft) and t = 0.46 m (18 in.) and no opening.
the half central angle between the two openings. The stressenvelopes of a chimney with two openings, ri = 12.5 m andt = 0.46 m and total opening angle of 2β = 20°, in normalconditions and following a 3 h fire event, are shown in Fig. 10B.Fig. 10C shows the reduction in the nominal moment strength ofa chimney subjected to a 3-h fire. Our study indicates that for aconstant opening angle, the variation of central angle does notmuch affect the structural performance of the chimney.
In Fig. 11, we show contour plots of the reduction in nominalmoment strength for different design parameters with respectto the height and thickness of the chimney cross-section. In thecalculations, the vertical load applied to the chimney cross-sectionis simply calculated by estimating the weight of the concreteabove the cross-section. The value of the vertical load is then usedto obtain the nominal strength of the concrete chimney usingthe stress envelope curves corresponding to post-fire conditionsafter a 3-h fire event. Fig. 11A shows the results for a chimneycross-section with no opening and 1% and 2% relative density ofreinforced steel. Fig. 11B and C compare the results for a cross-section with no opening, with cases for one opening, β = 20°and two openings, β = 10° and α = 40°. Note that the totalopening angle is kept constant in Fig. 11C. These results can be usedfor estimating the reduction in the moment strength of concretechimneys.
5. Effect of thermal stresses
In the calculations provided above, the role of thermal stressesin calculating the axial and moment strengths of chimneys duringfire was not considered. This aspect of the problem is consideredhere. It should be noted that the study includes only the effect ofthermal stresses on the chimney’s axial strength, since it is notvery likely that the chimney will experience high design lateralloads during the internal fire [14]. Thus, calculation of the momentstrength was considered less useful in this case.
The ACI code [14] provides a method to calculate the stressesinduced by the steady state (service) temperature profile. The ACIcode accounts for the effect of the thermal stresses, by reducingthe specified compressive strength of concrete according to f ′′
c =
f ′c − 1.20f ′′
CTV , where f ′′
CTV is the maximum vertical stress inconcrete occurring inside the chimney shell due to temperaturevariation through its thickness. This method cannot be applied forthe thermal stresses induced in the reinforced concrete chimneyduring an internal fire, since the temperature profiles across thethickness of the chimney vary significantly with time.
To calculate thermal stresses, we have carried out a coupledheat transfer–structural analysis using axisymmetric models ofthe concrete chimney shell. In the simulations, the concrete wastaken to have different tensile and compressive responses. Thebehavior of the concrete was taken as elastic–plastic under both
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Fig. 9. Residual structural capacity of concrete chimney with one opening. (A) Schematic view of the cross-section of a chimney with one opening. β denotes the half-opening angle. (B) Normalized nominal moment strength of the chimneys with different opening angles versus the normalized vertical load for the normal condition andafter a 3 h fire event. (C) Reduction in the nominal moment strength versus the normalized vertical load for post-fire behavior of the chimneys with different opening anglesafter a 3 h fire event. The concrete chimneys have ri = 12.5 m (41 ft), t = 0.46 m (18 in.), and ρt= 0.01.
tension and compression, however, with different yield strengths.Under compression, the yield strength was set equal to f ′
c , whileunder tension, the yield strength of 3 MPa was assumed in thenumerical simulations [15]. The steel was taken as an elasticmaterial with Es = 200 GPa. Each simulation required severaliterations similar to the finite element–base iterative methodexplained in Section 3. In each iteration, we created a geometricalmodel of the chimney shell by assuming which portions ofthe concrete shell are under compression and tension. Materialproperties were assigned accordingly to each portion and to thesteel. The finite element calculations gave an estimate of both thetemperature profile during the fire, as well as the induced thermalstresses. We examined the stress profile across the thickness ofthe chimney shell and modified the compression and tensionportions accordingly. This procedure continued until the assumedconfiguration was consistent with the induced stress profile.
We used our method to analyze a chimney with a thicknessof 18 in. (0.46 m) and 0.01 steel reinforcement subjected to aninternal fire with temperature 1000 °C for a duration of 50 min.The profile of the thermal stress induced by the internal fire after50min is shown in Fig. 12, where we have also shown the concretestrength at elevated temperatures simply calculated by relating theconcrete strength to the local temperature at each point along thechimney thickness, Fig. 1. The calculated thermal stress profile wasthen used to estimate the residual axial strength of the chimneyshell by accounting for the reduction in the concrete strengthdue to thermal stresses. In our calculations, we used the method
Table 1Reduction in axial strength of the concrete chimney subjected to an internal firewith temperature 1000 °C. The results are presented for a fire duration of 50 min.
Effect of thermal stressesN (%) Y (%)
Reduction in concrete strengthat elevated temperature
N 100 86.14Y 88.52 83.56
provided in the ACI code to select the thickness of the chimney thatis under compression for the first iteration.
In Table 1, we have summarized the results of our calculationsfor the reduction in the axial strength of the concrete chimneyduring fire for four different cases (i.e. no material degradation–nothermal stresses [N–N], no material degradation with thermalstresses [N–Y], material degradation–no thermal stresses [Y–N],and material degradation with thermal stresses [Y–Y]). It shouldbe mentioned that for the Y–Y case, we assumed the compressivestrength of the concrete to be fc(27.5 MPa), rather than varyingit along the thickness according to f ′
c , to simplify the calculations.However, this assumption did not change the steady state resultscompared to the ACI method and will not change the resultsof other cases considerably, since it would slightly change thelocation of zero stress along the thickness.
The results show that if the effect of thermal stresses isneglected (case [Y–N]), the estimated axial strength of the chimneyat elevated temperatures is ∼0.885 of chimney axial strength innormal conditions. On the other hand, by considering the thermal
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Fig. 10. Residual structural capacity of a concrete chimney with two openings. (A) Schematic view of the cross-section of a chimney with two openings with equal openingangle, 2β . (B) Normalized nominal moment strength of the chimneys with different α versus the normalized vertical load for normal conditions and after a 3 h fire event.Note that the results for α = β = 10°, are very close to the results shown in Fig. 6 for a chimney with one opening and β = 20°. (C) Reduction in the nominal momentstrength versus the normalized vertical load for post-fire behavior of the chimneys with different α after a 3 h fire event. The concrete chimneys have ri = 12.5 m (41 ft),t = 0.46 m (18 in.), and ρt = 0.01.
stresses and neglecting the reduction in the compressive strengthof the concrete at elevated temperatures (case [N–Y]), the axialstrength of the chimney is ∼0.861 of chimney axial strength innormal conditions. Considering the effects of both thermal stressesand reduction in the concrete strength at elevated temperatures(case [Y–Y]) gives a reduction factor of ∼0.836. The residual axialstrength of the chimney shell for post-fire conditions is∼0.885 forthe same fire conditions.
6. Conclusions
Reduction in the axial strength and moment strength of areinforced concrete chimney, due to a fire event, was investigatedfor a range of design and geometrical parameters. The temporalvariation in the concrete chimney during the fire was calculatedusing an axisymmetric heat transfer finite element analysis. Thereduction of the concrete strength due to the fire was calculatedusing experimental data available in the literature, which givesa direct estimate of the reduction in the axial strength of thechimney at different cross sections. The moment strength of thedamaged concrete was calculated using the standard method ofACI. Based on the proposed approach, the reduction in the axialstrength depends only on the thickness and fire duration. On theother hand, the reduction in the moment strength is a complexfunction of chimney design parameters (e.g. thickness, height,steel reinforcement and opening(s) size and location(s)) and fireduration. The ACI code [14] provides a method to calculate the
stresses induced only by the steady state temperature profile andthus, cannot be used directly to estimate the reduction in thechimney strength due to the thermal stresses during the fire.We used finite element simulation to study the role of thermalstresses on the behavior of reinforced concrete chimney duringfire. The results show that considering thermal stresses does notsignificantly affect the estimated reductions in chimney’s axialstrength.
Our study showed that the detrimental effect of uncontrolledfire on the residual strength of the chimney is stronger for tallchimneys with thinner shells. Openings in the chimney shell resultin further reduction in the residual strength of the chimney. Thus,shell openings and their exact geometry should be consideredwhen evaluating the residual strength of concrete chimneys.
Acknowledgments
The authors thank Franco Tamanini and Louis Gritzo of FMGlobal for their valuable comments. This work was supported inpart by FM Global, in part by the US Department of HomelandSecurity and in part by the Department of Mechanical andIndustrial Engineering at Northeastern University. This material isbased upon work supported by the US Department of HomelandSecurity under Award Number 2008-ST-061-ED0001. The viewsand conclusions contained in this document are those of theauthors and should not be interpreted as necessarily representingthe official policies, either expressed or implied, of the USDepartment of Homeland Security.
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A. Vaziri et al. / Engineering Structures 33 (2011) 2888–2898 2897
Fig. 11. Reduction in the nominalmoment strength for post-fire behavior of chimneyswith a range of heights and thicknesses. The results are presented for fire temperature870 °C and duration 3 h and considering the post-fire condition. The numbers of the plots show the corresponding value of Mn/Mn(normal condition). (A) Reduction in thenominal moment strength for two chimneys with 1% and 2% vertical steel reinforcement. Solid and dashed lines are plotted for chimneys with ρt = 0.01 and ρt = 0.02,respectively. (B) Reduction in the nominal moment strength for two chimneys with and without opening. Solid and dashed lines are plotted for chimneys with no openingand with one opening with β = 20°, respectively. (C) Reduction in the nominal moment strength for two chimneys with one opening and two openings. The concretechimneys had ri = 12.5m (41 ft), t = 0.46m (18 in.), and ρt = 0.01. Solid and dashed lines are plotted for chimneys with one opening with β = 20° andwith two openingswith β = 10° for each opening and α = 40°.
Fig. 12. Induced thermal stresses and the calculated compressive strength ofconcrete at elevated temperatures through the chimney shell thickness. Thestresses and strength are normalized by the concrete compressive strength.
Appendix
For a given chimney with known radius, r , shell thickness,t , concrete compressive strength, f ′
c , the angle β which denotesthe half-opening angle (if any), and the ratio of total verticalreinforcement to total area of concrete, ρt :
Pu/rtf ′
c = K1 = 1.7Qλ+ 2εmKeωtQ1 + 2ωtλ1
where
λ = τ − n1β
Q1 =sinψ − sinµ− (ψ − µ) cos θ
1 − cos θλ1 = µ+ ψ − π
µ, τ , ψ are geometrical parameters for a cross-section that arediscussed in [14], and
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2898 A. Vaziri et al. / Engineering Structures 33 (2011) 2888–2898
cos τ = 1 − β1 (1 − cos θ)
cosψ = cos θ −
1 − cos θεm
fyEs
≥ −1.0
cosµ = cos θ +
1 − cos θεm
fyEs
≤ 1.0
where Ke = Es/fy, ωt = ρt fy/f ′c , n1 is the number of openings
entirely in the compression zone (the maximum allowed base onthe ACI code is two openings), θ is one-half the central anglesubtended by the neutral axis, and,
β1 = 0.85 for f ′
c ≤ 4000 psiβ1 = 0.85 − 0.05(f ′
c − 4000)/1000 ≥ 0.65for f ′
c ≥ 4000 psiεm = 0.07 (1 − cos θ) / (1 + cos θ) ≤ 0.003.
The nominal moment strength of a cross-section, Mn can beestimated from Mn = PurK3, where K3 = cos θ + K2/K1 andK2 = 1.7QR + εmKeωtQ2 + 2ωtK .
For θ ≤ 5 °
Q =−0.523 + 0.181θ − 0.0154θ2
+
41.3 − 13.2θ + 1.32θ2
(t/r) .
For 5 < θ ≤ 10 °
Q =−0.154 + 0.01773θ − 0.00249θ2
+ (16.42 − 1.980θ
+ 0.0674θ2(t/r) .
For 10 < θ ≤ 17 °
Q = (−0.488 + 0.076θ)+ (9.758 − 0.640θ) (t/r) .
For 17 < θ ≤ 25 °
Q =−1.345 + 0.2108θ − 0.004434θ2
+ (15.83 − 1.676θ
+ 0.03994θ2(t/r) .
For 25 < θ ≤ 35 °
Q = (0.993 − 0.00258θ)+ (−3.27 + 0.0862θ) (t/r) .
For θ > 35 °
Q = 0.89.
Other parameters are calculated from the following equations:
Q2 =
(ψ − µ)1 + 2 cos2 θ
+ (1/2) (4 sin 2θ + sin 2ψ − sin 2µ)
−4 cos θ (sin θ + sinψ − sinµ)
(1 − cos θ)
K = sinψ + sinµ+ (π − ψ − µ) cos θR = sin τ − (τ − n1β) cos θ − (n1/2) [sin (γ + β)− sin (γ − β)]
where γ is the one-half angle between center lines of two openings(For a cross-section with no opening, n1 = γ = β = 0; for a cross-section with one opening in the compression zone, n1 = 1, γ = 0;for two openings in the compression zone, n1 = 2).
References
[1] Arioz O. Effects of elevated temperatures on properties of concrete. Kidlington,ROYAUME-UNI: Elsevier; 2007.
[2] Mohamedbhai GTG. Effect of exposure time and rates of heating and coolingon residual strength of heated concrete. Mag Conc Res 1986;38:8.
[3] Nassif A. Postfire full stress–strain response of fire-damaged concrete. FireMater 2006;30:323–32.
[4] Sakr K, El-Hakim E. Effect of high temperature or fire on heavyweight concreteproperties. Cement Concr Res 2005;35:590–6.
[5] Xiao J, Falkner H. On residual strength of high-performance concrete with andwithout polypropylene fibres at elevated temperatures. Fire Safety J 2006;41:115–21.
[6] Fletcher I, Borg A, Hitchen N, Welch S. Performance of concrete in fire: areview of the state of the art, with a case study of the Windsor tower fire.in:Proceedings of the 4th international workshop in structures in fire. 2006.p. 779–90.
[7] Tenchev R, Purnell P. An application of a damage constitutive model toconcrete at high temperature and prediction of spalling. Int J Solids Struct2005;42:6550–65.
[8] Canisius TDG, Waleed N, Mattews SL. A preliminary study of the CardingtonConcrete Building under a compartment fire. In: Proc structural faults & repairconf. 2003.
[9] Both C, van de Haar P, Tan G, Wolsink G. Evaluation of passive fire protectionmeasures for concrete tunnel linings. In: Proc int conf on tunnel fires andescape from tunnels. 1999.
[10] Eurocode 2: Design of concrete structures—General rules, Structural firedesign (BS EN 1992-1-2:2004). British-Adopted European Standard; 2005.
[11] Ali HM, Senseny PE, Alpert RL. Lateral displacement and collapse of single-story steel frames in uncontrolled fires. Eng Struct 2004;26:593–607.
[12] Ali HM, Senseny PE, Alpert RL. Clearance between single-story steel framesand firewalls. Eng Struct 2005;131:13.
[13] Wang Y. Steel and composite structures: behavior and design for fire safety.New York, NY: Taylor and Francis; 2002.
[14] Design and construction of reinforced concrete chimneys (ACI 307-08).American Concrete Institute.
[15] Hillerborg A, Modéer M, Petersson PE. Analysis of crack formation and crackgrowth in concrete by means of fracture mechanics and finite elements.Cement Concr Res 1976;6:773–81.
