ACI MATERIALS JOURNAL TECHNICAL PAPER

Compressive Strength Relationships for Concrete underElevated Temperaturesby Adam M. Knaack, Yahya C. Kurama, and David J. Kirkner

This paper focuses on compressive strength relationships for thedesign of concrete structures under elevated temperatures fromfire. The development of a database of previous experimentalresearch on the temperature-dependent properties of unreinforcedconcrete is described. A comprehensive statistical analysis of theconcrete strength data from this database is conducted using themethod of multiple least-squares regression with coded variables.High-strength concrete (HSC) and normal-strength concrete(NSC) with normalweight and lightweight North Americanaggregates are considered in the investigation. The results areused to develop predictive relationships for the concrete strengthloss under fire. Compared with existing strength loss relationships,the proposed relationships are based on a much larger data set, thusincreasing statistical robustness. It is shown that a reasonablestatistical fit is achieved with the available data, especiallyconsidering that the proposed relationships use relatively simpleregression models suitable for design. The most significant parametersaffecting the concrete strength loss with temperature are theconcrete strength at room temperature, aggregate type, andheating test type. Through a critical evaluation of the currentdatabase, recommendations are presented for areas where futureresearch should be directed. Recommendations are also made forpresenting the results from future fire tests so that researchers canmost effectively use this data.

INTRODUCTIONThis work aims to develop predictive relationships

between the maximum exposure temperature and theresulting loss in the compressive strength of concrete for usein structural fIre design. The heterogeneous nature ofconcrete leads to signifIcant variability in its properties,making deterministic prediction of its behavior difficult.This has led to the bulk of the previous research on thetemperature-dependent properties of concrete beingexperimental. As a result, a theoretical development isnot sought in this paper; and the proposed models arebased on a statistical analysis of the existing test datafrom previous research.

Research conducted since the 1950s has resulted in aconsiderable amount of data 1-39 on the behavior andproperties of concrete under elevated temperatures. Acomprehensi ve evaluation of this data is lacking,however. The concrete strength models developed inthis paper aim to fill this gap by using a large data set toincrease statistical robustness.

Temperature-dependent concrete stren§th loss models inthe U.S. are provided by ACI 216.1-074 and ASCE.41 Asshown in Fig. I, the ACI 216.1-0740 models (given asfem/femo versus temperature, T relationships, where fem is theconcrete strength at maximum exposure temperature andfemo is the strength at room temperature) are grouped based

on the aggregate type as I) siliceous; 2) calcareous; and 3)lightweight. The models are further divided based on the testtype as 1) residual; 2) stressed; and 3) unstressed. In the"residual" test, the specimen is first heated to a specifiedtemperature, allowed to cool to room temperature, andthen loaded to failure under uniaxial compression. Thistest type is intended to evaluate the remaining strength ofa concrete structure following a fire. In the "stressed" test,the specimen is heated while subjected to an axial preloadof approximately 0.25 to 0.55femo' The preload representsthe axial load that may be present (for example, in acolumn) prior to the start of a fire. Once the specifiedtemperature is reached, further axial compression isapplied to the specimen until failure. In the "unstressed"test type, the concrete specimen is heated to a specifiedtemperature (with no preload) and then subjected to uniaxialcompression until failure. The unstressed test type acts as abaseline for the residual and stressed test types.

Figure 2(a) shows the strength loss model provided byASCE41 for normal-strength concrete (NSC), as well as astrength loss model for high-strength concrete (HSC) developedby Kodur et al.42 based on the ASCE model format.Different from the ACI 216.1-0740 models, the ASCE41 andKodur et al.42 models do not differentiate between the testtype or the aggregate type. Models to estimate the compressivestrength of concrete as affected by fIre are also availablefrom European sources.43-45 For example, Fig. 2(b) showsthe Eurocode 443 strength loss models for NSC and Fig. 2(c)shows the models proposed by the Comites Euro-Intemational duBeton (CEB).44 Different from the ACI 216.I-Oro guidelines,the Eurocode 443 and CEB44 models do not consider the testtype as a parameter.

A few additional models for the concrete strength loss withtemperature are available in the literature. The models by Hertz,16shown in Fig. 3(a) and (b), are for NSC and consider the effects ofaggregate type and test type. According to Hertz, 16the unstressedtest is more conservative than the stressed test (that is, it results inlarger strength loss); and thus, the stressed test type is not includedin the prediction models. The models by Shi et aI.36and Li andPurkiss24 (Fig. 3(c)) are also for NSC; however, the testtype or aggregate type with which these models should beused has not been reported.

Note that the ASCE,41 Kodur et al.,42 Eurocode 4,43and CEB44 models in Fig. 2, and the Hertz, 16Shi et al.,36and Li and Purkiss24 models in Fig. 3 are based on

ACI Materials Journal. V. 107, No.2, March.ApriI201O.MS No. M-2008-323.R4 received May 31, 2009, and reviewed under Institute publication

policies. Copyright © 2010, American Concrete Institute. All rights reserved, including themalcing of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including authors' closure, if any, will be published in the January.February20 II ACI Materials Journal if the discussion is received by October I. 2010.

Adam M. Knaack is a graduate student at the University of Notre Dame, NotreDame, IN. He received his BS in civil engineering from the Rose·Hulman Institute ofTechnology, Terre Haute, IN, in May 2007. His research interests include the properties ofconcrete under fire temperatures.

ACI member Yahya C. Kurama is an Associate Professor of civil engineering at theUniversity of Notre Dame. He is a member of ACI Committees 335, Composite andHybrid Structures, and 374, Performance·Based Seismic Design of ConcreteBuildings; and Joint ACI-ASCE 'Committee 550, Precast Concrete Structures. Hereceived his PhD from Lehigh University, Bethlehem, PA. His research interestsinclude the behavior and design of concrete structures under extreme loads.

David J. Kirkner is an Associate Professor of civil engineering at the University ofNotre Dame. He received his PhD from Case Western Reserve University, Cleveland.OH. His research interests include finite element methods, structural analysis ofpavement systems. damage mechanics, probabilistic mechanics, and stochasticmodels of material heterogeneity.

Temperature, T ("C)21 446 871J 1.2 . i ACI 216 ~Residual

J ..:~~~;~~~;:~':::::::::L:::::::::T::::::::::::~ ~": :i :_~~£~-~:t--~}~,.;.~ 00 -Siliceous : :

. 70 835Temperature, T ("F)

concrete samples made from non-North Americanconstitutive materials. These models may not be suitable foruse in North America because of differences in thetemperature-dependent properties of the materials usedin the concrete mixture, especially the constituentaggregates. The Phan and Carin030 model is the onlyNorth-American model in Fig. 3 and was developedbased on a large number of tests on HSC and NSC specimens.It was shown that the ACI 216.1-0740 curves in Fig. 1result in unconservative estimates of the strength loss forthe HSC specimens; and thus, the model in Fig. 3(c) wasproposed for high-strength calcareous concrete as a conservativeestimate for any test type.

Temperature, T ("C) Temperature, T ("C)21 446 871 21 446 871J 1.2 i ACI 216 - Stressed J 1.2 !ACI 216 - Unstressed

~ __. .~.__.__.~:~·~~~:~:;S~~::::::~ ~.~_.~_:.~_~.~.-:.I.:,~.:.~.~.:.~.~.~.:.:.~.:.~.t,",~~.:.~_~:.~.~.~:.:.!::, ;:;:: : '.::.:.: .. '. :.:.:.! ----.-------.f------------.f--.------ --~--:>~~\_.bo _ ">--.-~ ..-.---.----.~.--------....~..--.....--+--.-----\ ~ --------..-..l...----..----~----------.~.---->\::.~ :'~~~:f~~~ft--.--.~---..------)--.---------..~:'~~~:f~:~~ft----)-----..-----.~-.--.--'.OJ - Siliceous : : OJ 00 - Siliceous i i~ 0.070 835' 1600 ~ . 70 835

Temperature, T ("F) Temperature, T ("F)

(b)

Temperature, T (0C)446

Temperature, T (0C)446 871

:CEB Model Code 90o 21E 1.2<..:t

----

---_ _-~~-~.~.~.~::~.<~«<,:~:.-.._ ~ -- ::~

Ug":~"g·····r ..-~"':t<,:;_ I---Calcareous --''';--- .. -.. ----; ----, .~

00 -Siliceous : : :§. 68 834 1600 •..•.•

Temperature, T (OF)

---.•.,._._._~------- ---.:-_ ......••.....••. :-----~ ......•• ,_ ~_ .. _ ..-

" .... _-_ ..;.._---------.- -_ ~~.":.~.•.-~ ---- .

, ". --_ .. - .. -- ~.-._- -.r------- - .. -~-~~~~:'"-.•. ---

, .--- Li ht-weight .. -.. ~.. -------- .. -~. - .

0.0 - Sdfceous : :70 835 1600

Temperature, T ("F)

21~ 1.2<..:t

Ternperature, T (0C)446 871

Hertz, Unstressedo 21j 1.2----

E••••u

Temperature, T (0C)446

Temperature, T (0C)446o 21

jl.2

-:.:.:."::.~.~..::.-.------:--------- .._-; .. __ ._---- --

:::.:::::: ..:::::~~~~~.:;~:--::::..~:::::::.-- -',-

.... _--- --- -------------,. ..__ . -~~ ...•.-- •.. --- - ..-, '~

.. ···Shi et a1. : "> ••.~--- Li & Purkiss ---~--.... ----. : ---- •.•...,::----

o 0 - Phan & Carino. 70 ~5

Temperature, T (OF)

Fig. 3-0ther existing models: (a) Hertz, 16 unstressed; (b) Hertz, 16 residual; and (c) Shi et al.,36 Li and Purkiss,24 and Phanand Carino. 30

CURRENT RESEARCH NEEDSCurrent structural design provisions in the U.S. do not

consider fire as a load condition as they do the effects ofother loads (for example, dead, live, and earthquake loads).Performance-based standards, tools, and guidelines(different from the current prescriptive methods) should bedeveloped to enable the design, performance evaluation, andretrofit of structures to resist fire.46 At the most basic level,additional research is needed on the development of temperature-dependent material property models for concrete.

The primary limitation of the ACI 216.1-0740 models inFig. 1 is that these models were developed using a sparse setof data from a single research program (Abrams I), based ona total of 154 data points from 3 x 6 in. (7.5 x 15 cm) cylindricalspecimens, most withfemo < 6 ksi (41.4 MPa), tested using aradiation/electric furnace. A much larger research databaseis available on the properties of North American concreteunder elevated temperatures, and because the strength lossmodels are empirical, a more complete representation of theexisting database is needed. Accordingly, this paperdescribes a statistical analysis on the test results from nineresearch programs on the temperature-dependent compressivestrength of North American concrete. The available resultsare entered into a database and the multiple least-squaresregression method is used to investigate the effects of a largenumber of parameters on the concrete compressive strengthunder fire. Based on this investigation, predictive relationships aredeveloped for the temperature-dependent strength ofconcrete, and the proposed relationships are compared withexisting relationships. In addition, recommendations aremade for further research.

RESEARCH SIGNIFICANCEIn recent years, fire hazard mitigation problems have

become increasingly difficult, in part, due to considerationsof increased fire risk and hazard.46 At the same time, thecurrent fire design provisions in the U.S. date back to theearly 1900s and need a major overhaul. The most fundamentalstep in the rational fire design of structural systems is todevelop basic knowledge on temperature-dependent materialproperties. This paper focuses on this issue.

CIl

.'§ 120o0.. 100

~Cl 80.•..;: 60<I)

E 40::l

;z:: 20

Fig. 4-Room-temperature compressive strength datadistribution.

CONCRETE PROPERTY DATABASEA database of previous experimental research on the properties

of North American unreinforced concrete underelevated temperatures was developed. The primary objectivesof this database are to I) collect, sort, and store the test data; 2)synthesize the existing knowledge on the temperature-dependentproperties of concrete; and 3) make recommendations for futureresearch. In addition, guidelines for presenting future test resultsare developed based on an evaluation of the existing research.

The database, which can be accessed freely at http://www.nd.edu/-concrete/concrete-fire-database includes atotal of 14 papersl,3,4,6,7,11,14,18,23,29,31,32,38,39 from the1950s to the present, reporting various mechanical, physical,and thermal properties of unreinforced concrete at temperature.The database is divided into independent properties anddependent properties. The independent properties containinformation about each test and are further broken up as 1)mixture properties (for example, water-cement ratio [wlc],aggregate type, and cement content); 2) curing properties(for example, curing duration, humidity, and temperature);3) specimen properties (for example, size and shape); and 4)test properties (for example, furnace type, temperature, andheating rate). The dependent properties include the testresults as I) mechanical properties (for example, compressivestrength, elastic modulus, and tensile strength); 2) thermalproperties (for example, heat diffusivity and thermalconductivity); and 3) physical properties (for example, massloss and spalling).

The primary use of the database is to quickly sort the databased on user-specified criteria. For example, to look at theeffects of the wlc and aggregate type on the concretestrength, a query can be constructed to limit the data to eachspecific type of aggregate (for example, calcareous andsiliceous) and to specific ranges of the wlc. If a specifictemperature range or a specific range of concrete strength isdesired, those limits can easily be applied as well, and thedata can be transferred into a spreadsheet program foranalysis. This database was used as a major tool in thedevelopment of the compressive strength relationshipsdescribed in this paper.

Data rangesA total of 635 data points from nine

papers 1,3,4, 14, 18,23,29,32,39 were used to develop the proposedstrength loss relationships. The other five papers in thedatabase investigate other temperature-dependent propertiesof concrete. Figure 4 shows the distribution of the room-temperature concrete compressive strength, femo' from thenine papers used in this research, which ranges fromapproximately 1.5 to 14.7 ksi (10.3 to 101 MPa). Figure 5 exhibitsfurther distributions for the collected NSC data with femo S 6 ksi(41.4 MPa). Figure 5(a) shows that the unstressed test type is themost common. Figure 5(b) shows that most of the testspecimens were made from calcareous aggregates. This mayhave been because specimens with siliceous aggregatestend to suffer from significant spalling, thus makingstrength measurements difficult (because the effect ofspalling on strength is difficult to quantify). Figures 5(c), (d),and (e) show that the majority of the data is from small,3 x 6 in. (7.6 x 15.2 cm) cylinders tested using radiation/electrictype furnaces. Lastly, Fig. 5(t) shows the number of NSC datapoints collected from each of the reference sources.

The HSC data with femo > 6 ksi (41.1 MPa) has smallerproperty ranges than the NSC data withfemo S 6 ksi (41.1 MPa).

NSC, Test Type NSC, Aggregate Type NSC, Specimen Height

eseres2) 73t

• Residual(36%) DCalcareous (47%) .3.8 in. (9.7 em)(12%) 8 in. (20.3 em) (22%)Stressed(22%) • Lightweight(31%) .4 in. (10.2 em) (8%) 012 in. (30.4em) (8%)

oUnstressed(42%) o Siliceous (22%) 06 in. (15.2 em)(41%) .16 in. (40.6 em) (9"/0)

(a) (b) (c)

SC, Specimen Shape NSC, Furnace Type NSC, Reference Papers

DRadiationlElectric (62%)DConvection/Gas (26%).NOI Available(12%)

o Harmathy & BerndI14(12%) 0 Abramsl (41%)• Saemann & Washal2 (8%) Zoldners39 (26%).Castillo & Durani1 (4%) • Lankard et al.23 (8%)

Fig. 5-Data distributionsforfcmo 5"6 ksi (41.1 MPa): (a) test type; (b) aggregate type;(c) specimen height; (d) specimen shape; (e) furnace type; and (f) reference papers.

• Residual (31%)o Stressed (33%)oUnstressed (36%)

196

.4 in. (10.2 em)(18%)06 in. (15.2 em)(17%)~8 in. (20.3 em) (65%) ~Castilln & Durani' (18%) DAbrams' (6%)

OPhan & Carino" (61%) • Kerr" (11%)DCheng el aI.' (4%)

(C)

Fig 6-Data distributions for fcmo > 6 ksi (41.1 MPa): (a) test type; (b) specimen height;and (c) reference papers.

For example, the HSC data comes only from cylindricalspecimens with calcareous aggregates and tested usingradiation/electric furnaces. Figures 6(a) and (b) show thatthe three test types are almost evenly distributed and 4 x 8 in.(10.2 x 20.3 em) specimens represent the most commonsize used in the HSC tests. Figure 6(c) shows the numberof HSC data points collected from each reference paper.More information about the test specimens (for example,w/c and size) associated with each reference listed inFig. 5(f) and 6(c) can be retrieved from the databaseWeb site.

STATISTICAL ANALYSIS OF CONCRETECOMPRESSNESTRENGTH

Multiple regression analysis was conducted for theconcrete compressive strength loss using the 635 data points,where each data point corresponds to a measured concretestrength,fem' at a measured maximum exposure temperature,T. Note that, in some cases, the average result from a seriesof tests (for example, from three specimens) was used as a"data point" because that was the best information available

from the source document. Comments are provided in thedatabase to identify single point and average test results. Todetermine the form of the strength loss regression model, acorrelation matrix was produced with the independentproperties (that is, mixture properties, curing properties,specimen properties, and test properties) reported for eachtest against the measuredfem'

Using the properties that showed strong correlation withfem (which includefemo' w/c, cement content, slump, curinghumidity, maximum aggregate size, and air dry mass) asvariables in the regression analysis, it was found that someof these properties, although having high correlation with thecompressive strength loss, did not have a large effect on theregression results. For example, the amount of slump in theconcrete mixture showed strong correlation with fem;however, when included as a variable in the regressionmodel, it did not have a significant effect on the overallperformance of the equation. Note that some variables couldnot be included in the regression model because there waseither not enough data or not enough variation in the data toconduct a meaningful statistical analysis.

Siliceous lfemo '" 6 ksi [41.4 MPa)) Calcareous lfemo:!: 6 ksi [41.4 MFa))

Kfm; Residual Stressed Unstressed Residual Stressed Unstressed

KftllO 0.963 0.995 0.953 0.997 1.02 0.981

Kftlll 6.45 x I(t4 8.42 x I ()-5 7.89 x I ()-4 6.51 x I ()-5 --4.00 x I ()-4 3.11 x 1()-4

KftllZ -1.64 x I ()-6 -9.02 x I()-8 -1.65 x I ()-6 -3.13 x 1()-7 1.11 x I ()-6 -5.41 x 1()-7

Kftll3 5.46 x I ()-IO -1.89 x I()-IO 5.28 x I()-IO -6.75 x l()-ll -7.00 x I ()-IO 1.76 x I()-II

T data range, 70 to 1472 70 to 1503 70 to 1600 70 to 1472 70 to 1504 70 to 1600OF (0C) (2J to 800) (21 to 817) (21 to 87J) (21 to 800) (21 to 8J8) (21 to 871)

lemo data range, 3.90 to 5.50 3.90 to 5.50 3.90 to 5.50 3.54 to 6.00 3.90 to 5.60 1.15 to 6.00ksi (MPa) (26.9 to 37.9) (26.9 to 37.9) (26.9 to 37.9) (24.4 to 41.4) (26.9 to 38.6) (7.93 to 41.4)

RZ 0.90 0.92 0.93 0.86 0.61 0.69

Lightweight lfemo '" 6 ksi [41.4 MPa)) Calcareous lfemo > 6 ksi [41.4 MPa))

Kftn; Residual Stressed Unstressed Residual Stressed Unstressed

Kft1l0 1.04 1.07 1.02 1.09 1.15 1.10

Kfml -5.48 x 1()-4 -1.06 x I()-3 -3.54 x 1()-4 -1.37 x I ()-3 -2.45 x I ()-3 -1.51 x 1()-3

Kfmz 3.69 x I ()-7 1.83 x I ()-6 2.74 x I ()-7 1.76 x I()-6 4.52 x 1()-6 2.22 x I ()-6

Kftll3 -2.21 x I()-IO -9.20 x I()-IO -2.03 x 1()-IO -1.05 x I ()-9 -2.41 x 1()-9 -1.08 x I ()-9

T data range 70 to 1600 70 to 1501 70 to 1599 70 to 1112 70 to 1292 70 to 1472OF (0C) (21 to 871) (21 to 816) (21 to 871) (21 to 600) (21 to 700) (21 to 800)

lemo data range, 1.60 to 3.90 3.90 to 3.90 2.72 to 3.90 6.44 to 14.0 7.34 to 14.2 7.34 to 14.2ksi (MPa) (11.0 to 26.9) (26.9 to 26.9) (18.8 to 26.9) (44.4 to 96.5) (50.6 to 97.9) (50.6 to 97.9)

RZ 0.86 0.79 0.74 0.79 0.22 0.36

To determine if the model variables contribute significantinformation to the prediction of strength loss, a Student's Ttest47 was performed. This test compares the value of the Tstatistic for any individual regression variable with the critical Tstatistic and determines if the regression coefficient for thevariable could statistically have a value of zero, suggestingthat the variable does not have a significant contribution tothe model. Generally, only those variables that had a significanteffect on the regression results as determined by the Tstatistic were included in the model. Furthermore, as ameasure of the global adequacy of the model, the analysis ofthe variance F test47 was performed. This test is used to determineif any of the regression coefficients in the model couldstatistically have a value of zero, suggesting that the model is nota useful representation of the data.

It was determined that some variables did not need to beincluded in the strength loss model because their effect wasseen through more global predictors. For example, byincludingfcl1Io (strength at room temperature) as a parameterin the regression, many of the concrete mixture propertiesthat were highly correlated with the strength loss (forexample, w/c and cement content) did not need to beincluded in the model because their effect was experiencedthroughfcmo' As a result of this comprehensive investigation,it was found that the most significant independent propertiesaffecting the concrete strength loss with temperature are 1)fClllo; 2) aggregate type; and 3) heating test type. Note thatthese are the same as the parameters used in the ACI 216.1-0740

models in Fig. 1.It was also decided to divide the data based onfcmo as 1)

NSC withfcl1lo:S 6 ksi (41.4 MPa); and 2) HSC withfcmo > 6 ksi(41.4 MPa). The cutoff strength of 6 ksi (41.4 MPa) was usedbecause a distinct difference was observed in the strengthloss for the higher strength concrete specimens, and the twostrength ranges selected produced the best fit to the data.

The aggregate type was grouped into the same threegeneral categories as in ACI 216.1-0740: 1) siliceous (sand-stone and other materials containing significant amounts ofquartz); 2) calcareous (carbonate, limestone, dolomiticlimestone, and dolomite); and 3) lightweight (expandedshale and expanded slag). The test type was alsogrouped into the same categories from ACI 216.1-0740

as I) residual; 2) stressed; and 3) unstressed.Note that the residual test results in the database include

specimens that were quenched in water for a period of timefollowing heating. By introducing water to a heated specimen,there is inherently a significantly different method of heattransfer occurring than otherwise would be experienced by anonquenched (that is, air-cooled) specimen. From the availabledata, it was seen that the quenched specimens tend to havelarger strength losses in the low- to mid-temperature rangesthan the specimens that were cooled in air. Because of theapparent differences between the quenched andnonquenched specimens, a total of 26 data points fromquenched specimens were excluded from the statistical analysisof the residual test data. It is recommended that future workbe conducted to investigate the compressive strength ofquenched concrete specimens. This is especially importantto assess the remaining strength of a structure following theprimary means of fire suppression using water.

Proposed strength loss modelThe final form of the concrete strength loss model

developed in this research is a cubic relationship in temperaturegiven by

whercoeftwithfcmois b~andcoelorlenmcont

Baidelevfron

where "'!m. is the aggregate and test-type dependent regressioncoefficient; and Tis the temperature in degrees Fahrenheit. Alongwith the regression coefficients, Table I shows the ranges offemo and T imposed by the data on which the regression analysisis based. Extrapolation of the equations outside these femaand T ranges is not recommended. Note also that Kfm'coefficients are not provided for high-strength siliceou~or lightweight aggregate concrete because there was notenough data to develop meaningful relationships for theseconcrete types.

Equation (1) and Table 1 are proposed as a simple designaid to determine the compressive strength of concrete atelevated temperatures from fire. This equation was obtainedfrom the full multiple regression model given by

fcm/fcmo = ~o + ~ I . T + ~2 . r + ~3 . r + ~4 . L + ~5' (2)

C + ~6 . R + ~7 . S + ~8 . L· T + ~9 . L . r + ~10 . L .

r + ~20 . L . R + ~21 . C . R + ~22 . C· S . T + ~23 . C· S . rwhere Pi is the regression coefficient (i = 0 represents theintercept term); L is the coded variable representing light-weight aggregate type; C is the coded variable representingcalcareous aggregate type; R is the coded variable representingresidual test type; and S is the coded variable representingstressed test type. In this format, equations for the siliceousaggregate type and the unstressed test type are obtained byassigning L = 0; C = 0; R = 0; and S = O.

Because the terms included in the regression model rangeover many orders of magnitude (for example, T, T2, andT3), which can affect the conditioning of the equations to besolved, each independent variable was normalized to have amean of zero and a standard $leviation of one. After this step,the regression coefficients Pi were found by following thestandard procedure for minimizing the square of the errorproduced by the regression equation. This minimizationleads to the following set of so-called "normal" equations,the solution of which yields the regression coefficients

where f3 is the regression coefficient vector (m xl); m is thenumber of regression coefficients; X is the independent variablematrix (n x m); n is the number offClifema data points in thedatabase; and y is the response vector (n xl) containing thefem/fema values. The rows in the X matrix represent thedifferent data points and the columns indicate the differentterms in the model (for example, for Eq. (2), the first columnof X contains a column of ones as an intercept term, thesecond column contains the normalized T data, the thirdcolumn contains the normalized T2 data, the fourth columncontains the normalized T3 data, and the fifth columncontains the lightweight aggregate coded variable L).

Note that by definition, fem/fema = 1 at room temperature(70°F [21°Cl). Because the regression model obtained from

a best-fit solution to the data will almost certainly notsatisfy this condition exactly, it was decided to enforcethis simple constraint by employing the classicalpenalty function approach,48 resulting in the following"constrained" normal equations

where a is the penalty parameter (a = 107 was used to achievethe required constraint); b is a vector of ones (r xl); r is thenumber of room-temperature data points; and C is theconstraint matrix (r x n) containing ones and zeros to selectthe y values corresponding to the room-temperature tests.

The constraint equations can be written as Cy = b, whereeach constrained equation is Yj = 1 and the subscriptj indicates atest conducted at room temperature. Constraining theequations in this manner made only small changes to theunconstrained regression coefficients and resulted in modelsthat identically satisfy fem/fema = 1 at room temperature.

Furthermore, because the model variables were normalized,the regression coefficients need to be modified for theequations to be used with unnormalized variables (forexample, a designer would use temperature and not anormalized value for temperature). This was achieved bycomparing the regression equations with and withoutnormalization, resulting in

Pi = (~j i = l. .. (m-l)I

wher~ Po and Pi are the modified regression coefficients; sX'and Xi are the sample standard deviation and sample mean:respectively, of the original unnormalized i-th variable(for example, un normalized T2 data for i = 2).

Lastly, the modified regression coefficients werecombined using the coded variables in the equation to formthe strength loss model in Eq. (1) and the aggregate and test-typedependent '9mi coefficients in Table 1. The coded variables aredesigned such that they are either one or zero.47 For example,the combined equation for the strength loss of calcareous concretefrom an unstressed test is developed as follows:

1. Apply the coded variable values C = 1, L = 0, R = 0, andS = 0 in Eq. (2) as

~20 . 0 . 0 + ~21 . 1 . 0 + ~22 . 1 . 0 . T + ~23 . 1 . 0 . r

Temperature, T ("C)446 87144.8 650021

NSC - Siliceous, Residual

Temperature, T (0C)446 . 87144.8 650 21

NSC - Siliceous, Stressed

Temperature, T (0C)446 87144.8

NSC - Siliceous, Unstressed

- Proposed ...Data

0.070 835Temperature, T ("F)

- Proposed ~ , ..Data

0.070 835 16000.0

Temperature, T ("F)

- Proposed. . .. ~ .Data

0.070 835Temperature, T (OF)

Temperature, T (0C)446 87144.8 650 21

NSC - Calcareous, Residual

Temperature, T ("C)446 871 650 21

NSC - Calcareous, Stressed 44.8····C·I····· ···c .; .

Temperature, T (0C)446 871

448NSC - Calcareous, Unstresse .

- Proposed ~.Data '

0.070

• 0 .~:: 0 ~

0'0--- ...

----- --------~:§, ;...' .c ..

E••..."

'.-"~'- Proposed • Data835

Temperature, T ("F)

Temperature, T (0C)

650 21 446.. 87144,8 650 21NSC - LIght-weIght, ReSidua. _. - .. _ ....~- _. _. -- _ .._~- - - - - _. - - -~._. -- - -----

Temperature, T (0C)

446 87144,8 650 21... l'J.S(;.:cLigh~:weig~t: Stressed

Temperature, T ("C)446 . 87144,8

NSC - Light-weight, Unstressed

- Proposed , .Data

0.070 835Temperature, T (OF)

- ProposedData

0.070

- - __ ,- - - • _~ __ - _._ - __ •• __ l,. •••• _ •••• _ ••, ,, ,, ,, ,.~_. - -- - - - - - --~ - -.- ---

Temperature, T (0C)446 871110 1600 21

HSC - Calcareous, Residual

Temperature, T (0C)

446 . 871110 1600021HSC - Calcareous,........, : r··· c··Stresseij"

Temperature, T ("C)446 871

110HSC - Calcareous, Unstressed---r---,--· ---. --.-

~ ~~ '"6 8E E

••..." ••..."

~ ~~ .;;;

6 8E

E••..." ••..."

-Proposed ....." ' ~.Data

0.070 835Temperature, T (OF)

- ProposedData

0.070 835Temperature, T (OF)

Fig. 7-Comparisons with measured data: (a) NSC siliceous; (b) NSC calcareous; (c) NSC lightweight; and (d)HSC calcareous.

2. Combine the like temperature terms and modify theregression coefficients as

Note that in standard multiple regression analysis,47 thereare four assumptions made about the distribution of the errorbetween the data and the prediction equation as follows: 1)the mean of the error is zero; 2) the error is normally distributed; 3)the variance of the error is constant for each independent variable;and 4) the error is independent for all values of each independent

3. Determine the final strength loss relationship in Eq. (1)where, for calcareous concrete under the unstressed test, the"fmi coefficients are

Temperature, T ("C)o 21 446 871J 1.2 NSC' - Residual-e .:::::-w.;. __ .~ ··,t-------- -----~--.-.---- ---_.

i ::::::::--:-:-:-:l::'~:~~::~~~.~~:~;~~~~~:::::::::::::::~ .....L;~;~:~-i~;-----l-------)~;~'~«.:-.~ --- Calcareous ----~---.~ 0 0 - Siliceous

. 70 835 1600Temperature, T (OF)

Temperature, T ("C)o 21 446 871J 1.2 : NSC - Stressed-S .~. ~~:--~~~~r-~:::~:~.~.,.·~\.,,~·_···__··_·-C+-oU •••••••• :......... .••

~ , ,----_:~~"~.~~::----~ ------------r----------r-----------: .----.'.~:---en ..... ·_·----r-------"---··:-----··-------:-···--- .. ----

<l) , ,

> ....·Light-weight.~ ---Calcareous ----:------- -----:-------------~ 0.0 - Siliceous ' ,

70 835 1600Temperature, T ("F)

Temperature, T (OC)o 21 446 871J 1.2 NSC - Unstressed

J ~~~~~~~t~----------T---·-··_·-·_·~ :::::::::::::::::::::::::._~:::~~~;~~:~.:.:::_:::::~ : ,,"-

~ -~-~-~-~~~~;~~~;--::::[::::::_::----:--~"<,.~ 0 0 - Siliceous

. 70 835 1600Temperature, T ("F)

important. Table 1 shows the sample multiple coefficientof determination R2 for each proposed model, where R2 = 0implies a lack of fit to the data and R2 = 1 implies a perfectfit. For most cases, the regression model fits the datareasonably well (especially considering the inherentvariability of concrete and considering that the objective ofthis research is to generate relatively simple equations foruse in design), with the exception of the calcareous HSCmodel under the stressed and unstressed test cases. Thesesets of HSC data are dominated by two papers (Castillo andDurani3 and Phan and Carin029) with the specimens testedby Phan and Carin029 having smaller fem/femo ratios (thatis, larger relative strength losses) at a given temperaturethan those tested by Castillo and Durani.3

Whereas the specimen sizes and cement contents weredifferent for Castillo and Durani3 and Phan and Carin029

(refer to the database), any differences in concrete strengthloss based on these two parameters could not be observed. Amajor difference between the two research programs is in theway that the test specimens were prepared. In Castillo andDurani,3 the specimens were dried to room humidityprior to testing. In comparison, Phan and Carin029 testedtheir specimens shortly after they were cured in water forseveral days. This may have resulted in more water in theconcrete and in higher pore pressures, leading to moreextensive explosive spalling and larger relative strengthlosses in the specimens from Phan and Carino.29

As a result of these observations, it is possible that a betterfit may be obtained to the test data if the specimen relativehUmidit~ at test time were included in the regressionmodel.4 This could not be investigated for the current data,however, because enough test results and variability do notexist to include relative humidity as a statistically relevantpredictor variable. Further tests are needed to determine theeffect of specimen humidity on the concrete strength loss.Until this data is collected and the regression models areupdated as needed, the proposed equations for HSC shouldbe used with caution.

Effect of aggregate typeFigure 8 shows the proposed fem/femo versus T curves for

NSC grouped based on the three aggregate types. The effect ofaggregate type on the concrete strength loss is mixed. For theresidual and unstressed test types, it can be seen that lightweightconcrete has the least relative strength loss for, approximately,T> 1100°F (593°C). In general, lightweight concrete tends toperform better than siliceous and calcareous concrete under

variable. These four basic regression assumptions were checkedfor the regression models developed in this research. Byexamining the error plots, it was determined thatAssumptions 1,3, and 4 were met for all cases. To checkAssumption 2, a Kolmogorov-Smimov (K-S) test47 wasconducted. It was found that all nine of the NSC modelspassed the K-S test within a 90% confidence interval. Incontrast, none of the HSC models passed the K-S test,suggesting that the error from the HSC data may not comefrom a normal distribution.

The inability of the HSC models to satisfy the error normalityassumption may have been because of the implicit, hard-to-quantify effects of concrete spalling on the strength data.Furthermore, additional parameters not included in thecurrent models may play a significant role in the strengthloss of HSC with temperature. Given the existing database,however, the lack of sufficient test results on potentiallyimportant parameters (for example, specimen humidity asdescribed in the next section) prohibits these parametersfrom being included as statistically relevant predictorvariables in the regression models. It should be noted that outof the four regression assumptions, Assumption 2 is the leastrestrictive because of the overall robustness of the regressionanalysis with respect to error normality. It has beensuggested that moderate departures from this assumptionhave little impact on the regression results.47 Thus, whilefurther research is certainly needed on the temperature-dependent properties of HSC, the relationships proposed inthis paper are recommended for use with caution untiladditional test data is developed.

RESULTSThe results of the multiple regression analysis form the basis for

12 different relationships (nine for NSC and three for HSC), asgiven by Eg. (1) and Table 1.These relationships are discussed inthe following.

Comparisons with test data and evaluationof data fit

Figure 7 shows the prediction band and the measured femdata for each aggregate and test type. The prediction bandsfor fem were generated by multiplying the fedfemo valuesfrom the regression models with the maximum andminimum femo values for each data range. Whereas most ofthe available data falls within the corresponding predictionband, low and high outliers indicate that other variablesnot currently included in the prediction models may be

Temperature, T (0C)446 871

.................L.~~~.-..~i~.ht~v:ei~bt.

..~..~.~~~'-:~._'~~:~:~'~;~:~:~'~'~'~r-',:: .~ - -"; - -- - - - - -. --r _. - - - - ~ :.::- ~"":_. --

, : : .•...•.....-------------r- ._- .. _.... ~ ,..... -------~---- ---~,-

o 21J 1.2

--E••..•u

~c::~~

l1)

;. ..... Stressed.~ --- Unstressed~ 0 0 - Residual

. 70 835Temperature, T (OF)

Fig. 9-Effect of test type on compressive strength loss ofNSC: (a) siliceous aggregates; (b) calcareous aggregates; and (c) lightweightaggregates.

Tern perature, T (0C)o 21 446 871J 1.2 HSC - 'Calcareous-- -------------}--------------:---------- :._----_ .••..5 '. : : :~ ··:~·':···:';'·f·;,:,:,:::::::~·+::~':::::~~·i·············~ -------------f---------- -~-----------~\'~-------- ..

C/:J ••••••••• -- --~ - - - - - ------ --~ - - - - -- -- ----~.\~-~\.- - .-.-

.~ ..... Stressed : :"ro --- Unstressed ·······;·············c··· .... ·,····

"0 - Residual : :0::: 0.070 835 1600

Temperature, T (OF)

Fig. lO-Effect of test type on compressive strength loss ofHSC (calcareous aggregates only).

these high temperatures because in creating the expanded shaleand slag aggregates, the material has already undergonethermal processing. In comparison, for all three testtypes, siliceous concrete experiences the most strengthloss for, approximately, T> 800°F (427°C). This may bebecause of the high amount of quartz in siliceous aggregate,which undergoes a phase transformation at 1063°F (573°C)accompanied by a significant volume increase, resulting inmore strength loss than concrete with other aggregate types.

Effect of heating test typeFigures 9 and 10 show the effect of test type on the temperature-

dependent strength ofNSC and HSC, respectively. For each typeof aggregate, the stressed test results in the smallest strengthloss at higher temperatures (with the exception of theHSC in Fig. 10), whereas the largest loss occurs in theresidual test type. It is interesting to note that for calcareousand lightweight aggregate concrete, the stressed specimensdemonstrate some strength regain at moderate temperatures.For NSC, the trends in the concrete strength loss from theunstressed and residual test types are similar, whereas thestressed test type results in a significantly different behavior.

Because the unstressed test type tends to result in largerstrength losses as compared to the stressed test type, theunstressed test has been used as the basis for designrecommendations in the past.30 For calcareous NSC (Fig. 9(b»,however, the unstressed case can be viewed as beingoverly conservative compared to the stressed test. Also, forcalcareous HSC (Fig. 10), the stressed test results in a rapid

strength loss causing the unstressed test to become unconservativeat high temperatures.

HSC versus NSCFigure 11 compares the strength loss models for HSC (that

is,fcmo> 6 ksi [41.4 MPaJ) and NSC (that is,fcmo S 6 ksi[41.4 MP~) with calcareous aggregates. As shown by thetest data3, ,18,29 and represented by the proposed models,HSC behaves in a significantly different manner under fireas compared with NSC. In general, HSC tends to losemore relative strength with temperature than NSC. Thishas been attributed to the larger density (that is, smallerporosity) of HSC,3 which results in larger pore pressuresand, thus, larger strength losses as water is driventhrough the concrete under increased temperatures.

Furthermore, it can be seen from Fig. 11 that the behavior ofHSC is characterized by a sharp strength reduction,followed by a relatively stable range, and then another sharpstrength loss as temperature is increased. The relatively stablerange has been attributed to the stiffening of the cement gel and anincrease in the cohesive properties of the gel particles.3,4 Incomparison, NSC tends to experience a more gradual strength losswith temperature.

COMPARISIONS WITH ACI 216.1-0740 MODELSFigure 12 com£ares the proposed strength loss models and

the ACI 216.1-07 models from Fig. 1.As stated previously, theproposed curves represent a much larger data set than the ACImodels, thus increasing statistical robustness, and also includeHSC, giving designers the ability to predict strength losses for awider range of materials. In general, the proposed curves havesimilar trends as ACI and confirm the significance of theaggregate type and test type based on a more comprehensiveanalysis (note, for example, that the ASCE41 model in Fig. 2(a)does not include thes~arameters). The calcareous residual casefor the ACI 216.1-07 and proposed models shows a differenceof approximately 30% at T = 750°F (399°C). Similarly, for thelightweight residual case, there is a difference of almost 12% at atemperature of approximately 835°F (446°C). Thus, despitethe apparent similarities, the proposed relationships alsorepresent considerable differences with ACI 216.1-07.40

RECOMMENDATIONS FOR FURTHER RESEARCHBy looking at the current test data in Fig. 4 to 6, there is a

major gap in the existing knowledge for HSC, especially forsiliceous and lightweight aggregate types. Other parameters

Temperature, T (OC)446 o 21J 1.2

o 21J 1.2--E,-0.:,"bOc~r/J.,>

.~ -NSCQ) --- HSC0:: 0.0

70

Temperature, T (0C)446 871

Calcareous, Stressed

Temperature, T (0C)446o 21J 1.2

--E,-0.:,"bOc~r/J .-----, - - - - - ..,> , , ,

.~ -NSC ~- - ---------~----.--- - --;-Q) ---HSC :0:: 0.0

70

' ••••••••••••••••• L _....7----....'v

"_ .. - .... _-,. .._-_ .... -~ ~~".__ ..,..- _ ... -

: ':, ,_.~... - _ .._ .... '-. _.-, ,

835Temperature, T (OF)

835Temperature, T (OF)

835Temperature, T (OF)

where additional data is needed include the specimen sizeand relative humidity, axial preload level, and furnace type.Because most of the existing results are from small specimenstested in electric furnaces, it was not possible to include theseparameters in the regression models (especially after subdividingthe data based onfcmo' aggregate type, and heating test type). Thedifferences between air-cooled and water-cooled specimensfrom residual tests should also be investigated. Lastly, thecurrent residual test data only includes unstressed specimens.Residual tests on stressed specimens should be conducted tobetter simulate the remaining strength of an axially loadedmember after a fire. In a "stressed residual test," thespecimen would first be subjected to an axial preload,mimicking typical service gravity load levels. The specimenwould then be heated to a specified temperature; allowed tocool to room temperature (still under preload); and, finally,further loaded to failure under uniaxial compression.

The procedures and results from future tests should bereported in a more consistent manner to facilitate better use of thedata. Table 2 shows the recommended data to be reported. Ingeneral, it is important to describe how each measurement wastaken (for example, ASTM designations). Under specimenproperties, the papers in the current database report the densitiesand masses in different ways (for example, oven versus airdry), which makes an assessment of the data difficult.Information on test properties should describe the chamberswhere the specimens were kept before and after heating (thatis, conditioning and residual chambers), as well as information onthe furnace (for example, heating rate). The load rate used inthe compression testing also needs to be reported, includinghow the specimen deformations were measured. Inpresenting the results, it is important to report not only thedata at elevated temperatures but also at room temperature.Furthermore, a clear definition of each reported property shouldbe provided (for example, definitions for concrete stiffnessand ultimate strain).

Note that these recommendations and the list in Table 2may seem trivial. The variability in the type and presentationof the information in the current literature, however, justifiesthe need for these recommendations. For many of the oldersources used in this research, it was not possible to obtainany information other than the published papers and reports.Thus, it is important that future researchers present their testsin a consistent and complete manner to allow the data fromdifferent projects to be analyzed collectively.

Table 2-Recommended data to be presentedfrom future fire tests

Mixture Curing Specimen Testproperties properties properties properties

Sand/cement Initial curing Specimen Test typeratio temperature shape

Aggregate/ Initial curing Length or Preload levelcement ratio humidity height

w/c Initial curing Cross- Axial displacement (orduration section area load) rate

Unit weight Subsequent curing Volume Conditioning chambertemperature conditions

Slump Subsequent Specimen Heating furnace typecuring humidity mass

Cement type Subsequent Specimen Heating furnacecuring duration density specification

Cement content End Heating furnace sizeconditions

Aggregate type Heating furnacehumidity

Aggregate origin Heating furnacetemperature

Maximum Location(s) of furnace

aggregate size temperaturemeasurements

d50 size Heating rate

Sand type Specimen age whenin furnace

Sand origin Heating furnaceduration

Air-entrainment Fumace durationtype/amount - at equilibrium

-Air content Water quenching

duration

Water reducer Residual chambertype/amount conditions

Retarder type/ Subsequent residualamount chamber conditions

Silica fume type/amount

Fly ash type/amount

-Admixture type/

amount

Fiber type/amount

SUMMARY AND CONCLUSIONSThis paper proposes compressive strength loss relationships for

use in the design of concrete structures under elevatedtemperatures from fire. The proposed relationships are based

"~~-~-'-~~~<:~~~~L<~:-.- ...•,'.: -~ _.

",~ .

o 21J 1.2

--E'-u

Temperature, T (0C)446 871

Siliceous, Residual21J 1.2

"-'-:-""':"--~ ....•......... ""-... .~

" '..": ............... ,

'«-----.30 .••• _

'"

--E'-u.s=bbc~en~ -AC1216 :

.~ · .... Proposed HSC - -.---.; ---Proc<: 0.070

ic~en">.~ -AC1216di --- Proc<: 0.070

o 21J 1.2

--E'-u.s=bbc~en">.~ -AC1216-.; ---Proc<: 0.070

o 21J 1.2

--E'-u

Temperature, T ("C)446 871

Calcareous, Residual

on Hi1990,

4. IStrenEngi/

5.1

faun6.

TemILabc

7.Past198

8EfffPasl

9ofCor

IHigHi!pp.

IR&

JSlPSIr

o 21J 1.2--E'-u.s=bbc~en">.~ -AC1216-.; --- Proc<: 0.070

o 21J 1.2

--E'-u

Temperature, T ("C)446 871

Light-weight, Stressed

o 21J 1.2- ..-------E'-u

Temperature, T ("C)446 87\

Siliceous, Unstressedo 21

JI.2

" -------------~----- ••• ""'L ••••••

"""-5"OIlc~en">.~ -AC1216-.; ---Proc<: 0.070

Temperature, T (0C)446 871

Calcareous,' Unstressedo 2\

JI.2

--E'-u.s=~~en">.~ -AC1216-.; --- Proc<: 0.070

" ,.... '~.....•......... _._ ... _----.•..•."- ~_..

.................

.s=~ - ... ----.-.-~en~ -AC1216

.~ ..... Proposed HSC-.; 00 --- Proposed NSCc<: . 70 835

Temperature, T ("F)

Fig. 12-Comparisons between proposed models and ACI 216.1-0ro models: (a) residual test; (b) stressed test;and (c) unstressed test.

on a comprehensive multiple regression analysis of theexisting experimental data on North American concrete. Ingeneral, a good fit is observed between the test data and themodel predictions. As compared with previous models, theproposed relationships represent a larger data set, thusincreasing statistical robustness for a wider range of materials. Indeveloping the proposed relationships, significant gaps areidentified in the existing knowledge. The regression modelsshould be reevaluated and revised as necessary once futureresearch is conducted to fill these research gaps. Some of theconclusions from the study are:

1. For NSC, siliceous aggregates result in the moststrength loss at medium to high temperatures. For hightemperatures, lightweight concrete performs better than siliceousand calcareous concrete, except under the stressed test type.

2. For NSC, the stressed test results in the smallest strengthloss at higher temperatures, whereas the largest loss occursunder the residual test type. For calcareous and lightweightconcrete, the stressed test specimens demonstrate somestrength regain at moderate temperatures.

3. As compared to NSC, HSC tends to experience largerrelative strength loss with temperature. The behavior ofHSC is characterized by a sharp strength reduction,

followed by a relatively stable range, and then another sharpstrength loss as temperature is increased.

4. In general, the proposed concrete strength loss modelshave similar trends as ACI 216.1-0740 and confirm thesignificance of the aggregate type and heating test type basedon a more comprehensive statistical analysis. Despite theapparent similarities, the proposed relationships also representconsiderable differences with ACI 216.1-07.40

ACKNOWLEDGMENTSThis research was funded by the Portland Cement Association (PCA)

through a PCA Education Foundation Fellowship. This support is gratefullyacknowledged. In addition, the authors would like to thank D. N. Bilow,and T. D. Lin, for providing guidance. The opinions, findings, and conclusionsexpressed in the paper are those of the authors and do not necessarilyreflect the views of the individuals or institutions acknowledged.

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