[Paper] Simple Method to Predict Fire Resistance of Composite Columns

7/27/2019 [Paper] Simple Method to Predict Fire Resistance of Composite Columns

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Sim ple M e th o d to Predict Fire Resistance ofCom posite ColumnsJUKKA MYLLYMAKI and DJEBAR BAROUDIVTT Building Technology, Fire TechnologyP.O. Box 1803, FIN-02044VTT, FINLAND

ABSTRACTA simplified numerical method to predict the fire resistance of concrete filled tubularcomposite columns is presented. The temperature analysis of the composite column isfornlulated as a one dimensional Galerkin FE-solution of the heat conduction equation. Thesimple design method of the Eurocode 4 has been applied in the calculation of the load-bearingcapacity of the tubular composite columns. Calculated temperatures and load-bearingcapacities are compared to the test results of concentrically loaded circular and square tubularcolum ns filled w ith reinforced and non-reinforced concrete.

KEYWORDS: fire resistance, composite columns, heat conduction, Eurocodes

INTRODUCTIONNumerous research projects and standard fire tests have been conducted on different types ofcomp osite colu mn s in Germany, France, Belgium, England and Canada. Various kinds of designmethods for composite columns have been developed in the literature [I-2,4-111. In theEurocode 4, Design of Composite Steel and Concrete Structures, Part 1.2 [4] calculationmeth ods for the structural fire design are divided into three categories, tabulated d ata designmeth ods, simple and advanced calculation methods.An E xcel-base d FE-code h as been written by the authors for the thermal and structural analysisof composite columns. The thermal analysis part of this code was developed in the Nordtestproject 1381-98 to calculate heat transfer in an axisymmetric duct [14]. The program wasfurther developed for the purposes of the seminar held at Helsinki University of Technology onthe design of steel/concrete composite structures. Structural analysis based on the simple

FIRE SAFETY SCIENCE-PROCEEDINGS OF THE SIXTH INTERN ATION AL SY MP OS IUM Pp 879-890

Copyright International Association for Fire Safety Science


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design method of the Eurocode 4 was adapted in the computer program for non-reinforcedcircular columns [ 1 2 ] . Here the temperature and structural analysis is extended also forreinforced square tubes.The two dimensional temperature field is solved using a one dimensional Galerkin FEMformulation of the heat conduction equation. The dimension reduction of the 2-D heat transferproblem to a I-D problem is achieved assuming that the isothermal contours of thetemperature field are either circular or rectangular. In the axisymmetric case of circularcolumns a circular contour assumption is exact if the temperature field around the column isuniform. In the case of square columns a rectangular contour assumption is often a suitableapproximation for engineering purposes.

CALCULATION OF TEMPERATURESolution of Gen eral Heat Conduction ProblemTh e general two-dimensional partial differential equation of heat conduction is to be solved inorder to get the accurate transient temperature field T ( x , t ) n a given material region. The fieldequation

is the heat conduction equation with r(x,T) as an arbitrary source term. The Fourier heatconduc tion constitutive equation is assumed. Equation (1) is complemented with the appropriateinitial-boundary conditions to get a well-posed problem. Using the standard FE-approach [ 3 ]one so lves the variational form of the problem ( 1 )

by choosing the temperature field approximation T e ( x , t ) N ' ( x ) T e ( t) and the test functionv i ( x )= N i e ( x ) ,where the basis functions N,' are linear.

The semidiscretization of the heat conduction equation (2 ) produces the non-linear initial valueproblemc ( ~ , T ) T ( ~ )f ( t , T )- K ( t , T ) T ( t ) , > 0 ( 3 )T ( o ) = % , t = O ,where T (t) is the global vector of the unknown temperatures of size n x I .

Equation (3) is a set of n x 1- non-linear ordinary differential equations. Notice that the righthand side in the equation ( 2 ) corresponds to the nodal flux vector f ( t , T ) ,which contains theboundary terms and the source terms. Natural boundary conditions are already included in the


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variational form (2). T he essential boundary co nditions are taken into accoun t during thesolution process of the initial value problem. The global matrices and vectors are assembledusing standard FE-assembling techniques. In the axisymmetric case (see Fig. la)( f x ,y , z ; t )= f ( r , z ; t ) , when linear 2-noded elements are used, the following elementcond uctanc e matrix is obtained

r2'Ki = I a ( T ( x ) ) ~ N , ( x ) . ~ N , ( x )a = 277 A1 I a ( ~ ( r ) ) i , , ( r ) N J , , (r ) r d r (4)

Re r iand also the elemen t capacitance m atrixr;

C;: = I p ( T ( x ) ) c ( T ( x) ) N , x ) N j ( x ) dn = 2 n A1 5 p ( T ( r ) ) c ( T ( r )) N , r ) NJ ( r ) r d r ( 5 )Re r,'and eleme nt nodal heat flux vector

The standard basis functions of a 2-noded linear element N , r )= 1- r- q e ) l a' andN 2 ( r )= ( r- ie ) a e are used. The element volume is dQ = 277 A1 rdr , with A1 as thethickness of the element slice in the z-axis-direction. Th e radial length of the eleme nt, numbere , i s a e = r ; - 5 ' .

In the case of a squ are and rectangular tubular column an accurate temperature analysis is twodime nsiona l. Here an approximate one dimensional analysis is presented. It is assumed that theisothermal contours of the temperature field are square (Figure Ib). Using linear eleme nts with2 nodes the follow ing element conductance and capacitance matrices are obtained:

q n , ~and f e i = - A1 [ )Iyn J R q , respectively. The standard basis functions of a linearqn.2

elem ent are used. Th e element volume is d R A1 A ( x ) dx . Th e length of the element e in theradial direction is a' = x; - ,'. Here A ( x )= 8 x is the length of perimeter of the squarecontour at x.


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PerimeterA ( r )= 2m

\ \ I II\\ '\ , ,

- erimeterA(x)=8x

(a ) (b)F I G U R E 1. a) Circular concrete-filled steel tubular column and the axisymm etricdiscretization b) s qua re tubular column and an approximate one-dimensional discretization.

B ounda r y C ond i ti onsIn the special case of a cylindrical composite column the following convection and radiationboundary condition (Neumann) is used at the boundaries of the solution domain;

where T, is the temperature of the furnace hot gases around the colu mn . Sim ilar boundaryconditions are used also in the approximate square solution.

In the equa tions (9) and (10) convection coefficient h2on the fire exposed side was 25 w / ~ ' Kand the resultant em issivity E, = 0,9 as given in [8].

N U M E R IC A L E X A M P LE S O F T H E T H E R M A L A N AL YS ISIn the follow ing, calculated examp les are compared to the results obtained at the tests at NRC ,Canada [ l , 5, 6, 91. The following thermal properties were used; thermal conductivity 1,


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(Wlm K) of normal weight concrete and specific heat of normal-weight concrete c, ( J k g K)given in the Eurocod e 4 (see Fig. 2a);

The moisture content w was considered completing the effective specific heat by a triangularsituated between T,,=100 OC and T,,=200 OC with peak at 100 OC (see Fig. 2a). Th e value ofthe peak was calculated from the equation c,,,,, = 2 w H, l(T,,- T,,,) using value H,=2.257 x 10 Jlkg for the heat of vaporization of water and value w = 10 % for the moisturecontent of concrete. Same moisture content has been used also by Lie and Chabot [8] andWang [ l l ] for the same test results. Specific heat of steel was calculated using the equationsgiven in Eurocod e 4 (Fig. 2b).The furnace temperature was assumed to follow the same ASTM-El19 standard temperature-time curve as in the tests at NRC C see [I];

where Tf s temperature in OC and z is time in hours. This is almost the same standardtemperature-time curve that as given in the standard I S 0 834.

Results of the thermal analysis compared to the test results are shown in Figures 3 - 6. Areasonable agreement between the calculated and tested results can be observed. It must betaken into account that Eurocode material properties were used without any fitting. In thecom putation s by L ie and Chab ot [8] and Wang [11] similar accuracy has been obtained.Thermal properties (Ccncrete) Thermal properties (Steel)

FIGURE 2.The rmal properties used for a) concrete and b) steel


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The problem in thermal analysis of concrete structures is that moisture movement cannot betaken into account properly without solving coupled heat and moisture transfer problem [15].In order to get reasonable results a rather high moisture content had to be used, because of themigration of moisture to the center of the column .

Concrete temp erature r=0 mm Concrete temperature r=26 mmtk1 68.3 mm 114.78 mm D=168.3 mm k 4. 78 mm1000 1000800 800

g 800 g 600- -c 400 , 00

200 2000 00 20 40 60 0 20 40 60

Time (rnin) Time (min)

Concrete temperature r=62 mmtk 16 8 .3 mm 1 ~4 .7 8 m Steel temperatureD=168.3 mm k4.78 mm

0 20 40 60 0 20 40 60Time (rnin) Time (rnin)

c) d)FIGURE 3. Calculated and measured temperatures in the concrete and steel (column diameter168.3 mm, wall thickness 4.78 mm), (upper line is the standard furnace temperature, thicksolid lines are calculated values and dotted lines with b oxes are measured values).

CALCULATION OF THE BUCKLING RESISTANCEResistance of Composite Column Applying Eurocode 4 Simple M ethodAccording to Chapter 4.3.6.1 of the Eurocode 4 Part 1.2 [4] the resistance of a compositecolum n in conc entric axial compression is obtained from the equation


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where x is the reduction coefficient and depends on the non-dimensional slenderness ratio-2, =JW.Concrete temperature r=0 mm Concrete temperature h 21 .3 mmD=141 mm h6 .5 mm D=141 mm t.6.5 mm

1000 1000800 800600

D S 600-I- 400 I- 400

200 2000 00 20 40 60 0 20 40 60

Time (min) Time (min)

Concrete temperature r=42.6 mmD=141 mm t.6.5 mm Steel temperatureD=141 mm k 6. 5 mm

0 20 40 60 0 20 40 60Time (min) Time (min)

c) d)FIGURE 4. Calcula ted and measured temperatures in the concrete and steel, colum n diame ter141.3 m m, wall thickness 6.5 m m. (upper solid line is the standard furnace tem perature, thicksolid lines are calculated values and dotted lines with boxes are measured values). Test datafrom Lie and Caron [ 5 ] .

Th e Euler buckling load in the fire situation is Nfi, , ,= x2 EIfi,ef/.1; ,where I , is the bucklinglength of the column in the fire situation. Nfi,,,,,, is the design value of plastic resistance toaxial compression in the fire situation. The non-dimensional slenderness ratio is calculatedusing equation

where a = 0.4 9 for buckling curv e c and a= .21 for buckling curve a.


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According to Eurocode 4 [4] the cross section of a composite column may be divided intovarious parts conce rning the steel profile, concrete and the reinforcing bars. In order to be ab leto calculate the plastic resistance to axial compression Nfi , , ,,, nd effective flexural stiffnessEIfi,eff he cross-sectional area of the column is subdivided into a number of circular or squarering elemen ts (Fig. l a and lb ) that coincide to the FE grid of the temperature analysis.

Concrete temperature r=O mm Concrete temperature rz32.5 mmD=273 mm 1.6.5 mm D=273 mm t-6.5 mm1000 1000800 800g 600- g 600

c 40 0 -c 400200 200

0 00 25 50 75 100 125 150 0 25 50 75 100 125 150Time (min) Time (min)

Concrete temperature r=62.5 m mD-273 mrn tz6.5 m m

1000800g 600-

+ 400200

00 25 50 75 100 125 150

Time (min)

Steel temperatureD=273 mrn k 6 .5 rnm

0 25 50 75 100 125 150Time (min)

FIGURE 5. Calculated and measured temperatures at various depths in the concrete and steel(upp er thin so lid line is the standard furnace tem perature, thick solid lines are calculated va luesand dotted lines are measured values). Test data from circular columns C48 and C49 of size273 m m x 6.3 mm [I] .The plastic resistance to axial compression and effective flexural stiffness of the cross-sectionof the circu lar colu mn a re calculated using following numerical integration form ulas


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where the first term of equations is summation taken over all ring elements, either concrete orsteel. The second term is summation taken over all reinforcement bars located at y , = r, sinp.Temperature T e= (jr;'+ T;") 2 is calculated at the center of the squ are ring elem ent.

Concrete temperature ~ 1 0 . 0 mColumn 254 x 254 x 6.35 (SQ18)-urnace ---[L I I Reinforcement temperatureColumn 254 x 254 x 6.35 (SQ18)0 20 40 60 80 100 0 20 40 60 80

Time (min) Time (min)

Concrete temperature x=60.6 mmColumn 254 x 254 x 6.35 (SQ18) Steel temperatureColumn 254 x 254 x 6.35 (S018)

0 20 40 60 80 100 0 20 40 60 80Time (min) Time (min)

c) d)FIGURE 6 . Calculated and measured temperatures at various depths in the concrete and steel(upper thin so lid line is the standard furnace temperature, thick solid lines are calculated valuesand dotted lines are measured values). Test data of square column 254 mm x 254 mm x 6.3mm [ I ] .The plastic resistance to axial compression and effective flexural stiffness of the cross sectionof the squ are colum n are calculated using following approximate integration formulas


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Note that there is no reinforcement in the thermal analysis. The temperature of eachreinforcement bar 1; is here assumed to be the same as the concrete temperature at that spatiallocation where the reinforcement bar is situated.

TABLE 1.Circula r column data used in the com putation and calculated resistances (c-curve)compared to the tests results from Lie and Caron [5] and iBMB [13j.Column Section Steel Test Fire Steel yieldcylinder -Ze Calc. Calculatedid. n:o diameter thickness Load resistance stress strength resistan- resistance1as in (mm) (mm) (kN) imin) f,. MPa) fc iMPa) ce Test load[5, 131 , in test at test date N l i . ~ d N f i . ~ d' tr\i

(kN)- . ..--- .I [5] 141.3 6.55 131 57 433 31.0 0.912 126 0.962[ 5] 168.3 4.78 218 56 374 35.4 0.952 227 1.043 [5] 219.1 4.78 492 80 347 31.0 0.779 387 0.794 [5] 219 5.56 384 102 347 32.3 0.792 306 0.795 [5] 219 8.2 525 82 396 31.7 0.686 426 0.816 [5] 273 5.6 574 112 445 28.6 0.661 613 1.077 [5] 273 5.6 525 133 445 29.0 0.675 516 0.988 [5] 273 5.6 1000 70 445 27.0 0.596 823 0.82I0 [5] 324 6.4 1050 93 474 24.0 0.51 1 1073 1.0211 [5] 356 6.4 1050 111 446 23.8 0.480 1247 1.1913 [ 5 ] 406 12.7 1900 71 399 27.6 0.349 2519 1.32S3[13] 219 6.3 380 71 428 40. 1.018 428 1.12 -

ean 0.99

Numerical Exam ples of the Resistance CalculationsThe resistances of composite column in axial compression NIL, , , were obtained using theEurocode 4 method in the way described in this paper. The mechanical properties given inEurocode 4 Cha pter 3.2 for both steel and concrete were applied. The m odulus of elasticity ofconcrete is not explicitly given in the Eurocode 4. Here the tangential modulus deduced fromthe stress-strain model of concrete by derivation with respect to E',, was used;

The measured values of yield strength of the steel and concrete (at the date of the test ifknown) were used. In the NRCC [I, 51 fire resistance tests 3,81 m long colum ns w ere testedboth ends clamped and buckling length of 2 m recommended by Lie and Chabot [8] was usedin the calculations. In NR C test [ 9] the column was hinged at both en ds and buckling length3,8 m w as used. Th e iBMB test was conducted one end clamped and the other end hinged andbuckling length of 2,62 m of the test report [I31 was used. Applied strength values and


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calculated results using c-curve are shown in the Table 1 and 2 and compared to the testresults. The resistance of circular non-reinforced columns can be predicted rather well, themean of the ratios between the calculated and tested result being 0,99. The results with thesquare columns are less satisfactory. Mean of the ratios between the calculated and testedresult is 0,73 which is on the safe side. Especially the results with smaller tubes are very muchon the safe side. One reason for this may be that the temperature calculation methodoverestimates the temperature of the composite column at corners where the reinforcement islocated.TABLE 2. Dim ension s of square columns, test loads, fire resistance times, stress values. Testdata from Chabot and Lie [ l ] and Myllymaki, Lie and Chabot [9].Column Section Stee l Test Fire Steel and reinf. Cylinder Calc. Calculatedid. n:o height thickness load resistance yield stress strength @ capacity mest loadas in (mm) (mm) (W (m id fsl fs MPa) fc (MPa) Nfi,Rd NJr,Rd N,,,,[I . 91 in test at test ( k ~ )

dateSQ-3 [9] 150 5 140 83 4181596 37.8 1.571 84 0.60SQ-12 [I ] 203 6.3 500 150 3501400 47 0.598 264 0.53SQ-13 [I ] 203 6.3 930 105 350/400 47 0.652 510 0.55SQ-18 [I ] 254 6.3 1440 113 3501400 48.1 0.549 1210 0.84SQ-19 [I ] 254 6.3 2200 70 3501400 48.1 0.488 1891 0.86SQ-22 [I ] 305 6.3 3400 39 3501400 47 0.364 4088 1.20

mean 0.76

CONCLUSIONSA simplified numerical method to predict the fire resistance of concrete filled tubularcom posite column s has been presented. The temperature analysis of the composite column wasformulated as a one dimensional Galerkin FE-solution of the heat conduction equation. In theanalysis both axisymmetric elem ents in the cylindrical case and o ne dimensional elem ents in asquare case were applied. In the former case solution is exact but in the latter case the solutionis an approximation for design purposes. The simple design method of the Eurocode 4 hasbeen applied in the calculation of the load-bearing capacity of the tubular composite columns.Both the temperature analysis and structural analysis were conducted with a code w ritten in aspreadsheet program Microsoft Excel. This made it simple and efficient design tool.Calculated temperatures and capacities were compared to the test results of concentricallyloaded circular an d squar e tubular columns filled with reinforced and non-reinforced concrete.Results of the cylindrical columns were in satisfactory agreement with the test results.Calculated load-bearing capacities of the small square tubes filled with reinforced concretewere too conservative. T his may be due to the approximate nature of the temperature analysisespecially when we consider temperature of reinforcement in the corners of the section.REFERENCES1. Chabot, M and Lie, T.T., "Experimental Studies on the F ire Resistance of Hollow Steel Columnsfilled with Bar-Reinforced Concre te". Internal Report No. 628, NRC Canada, 1992


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2. ECCS-TC3-Fire Safety of Steel Structures, Technical Note N055. "Calculation of the FireResistance of Centrally Loaded Composite Steel-Concrete Columns Exposed to the StandardFire". 1988.3. Eriksson, K. Estep , D. Hansbo P. Johnson C . Computational Differential Equa tions. p. 530,Studentlitteratur, Lund, Sweden. 1996.4. Eurocode 4. D es im of Composite Steel and Concrete Structures. Part 1.2: Structural fire de si m .Final edited version. 1994.5. Lie, T.T . and Caron, S. E. "Fire Resistance of Circular Hollow Steel Columns filled with SiliciousAggregate C oncrete: Test Results". Internal Report No. 570. NRC C anada. 1988.6. Lie, T.T., Chabot, M. and Irwin J. "Fire Resistance of Circular Hollow Steel Sections filled withBar-Reinforced Concrete". Internal Report No. 636, NRC Canada, 1992.7 . Lie, T.T., Irwin J. and Caron S.E. "Factors affecting the Fire Resistance of Circular Hollow SteelColum ns filled with Plain C oncrete". Internal Report No. 612, NRC C anada, 1991.8. Lie, T.T. and Chabot, M . (1990) "A Method to predict the Fire Resistance of Circular Concretefilled Hollow Steel Colum ns". Journal of Fire Protection Engineering 2: 4, 11 1-126,1990.9. Myllymkki, J., Lie, T.T., Chabot, M. "Fire Resistance Tests of Square Hollow Steel Columnsfilled with Reinforced Concrete". Internal Report No. 673, NRC Ca nada, 1994.

10. Oksanen, T. and Ala-Outinen T., "Determination of Fire Resistance of Concrete filled HollowSections. Research Notes 1287, Technical Research Centre of Finland, 1991.11. Wang, Y.C., "Some Considerations in the Design of Unprotected Concrete-Filled Steel TubularColum ns U nder Fire Conditions", J. Construct. Steel Res. 44:3, 203-223. 199712. Myllym&ki, J.,"Fire Engineering Design of C omposite Columns", in Seminar on Steel S tructures:Desim of Steel-Concrete Com~osite tructures. ed. Malaska et al., pp.75-87. Helsinki Universityof Technology, S teel Structures. Report 4. 1998.

13. "Priifung von vier Verbundstiitzen auf Brandverhalten bei einer Brandbeanspmchung gemass EI S 0 834", Untersuchungsbericht Nr. 332813089-NauIMd (26.11.1990). Ausg.9 11989. iBMB , T UBraunschweig.1990.14. Baroudi, D. and Hietanierni, J.,"Physical Interpretation of Temperature Data Measured in the SBIFire Test". Nordtest Project 1381-98. Technical Research Centre of Finland. Research Notes (to bepublished)15. Bazant, Z. and Thonguthai, W., "Pore pressure in heated concrete walls: theoretical prediction",Magazin e of Concrete Research 31: 107.67 -76, 1979.

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[Paper] Simple Method to Predict Fire Resistance of Composite Columns

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