Temperature field of concrete-filled steel tubular columns in fire

Rodrigues, Fábio Masini; Moreno, Armando Lopes

doi:10.1590/0370-44672015700172

Fábio Masini Rodrigues

Mestrando, Universidade Estadual de Campinas - UNICAMP, Departamento de Engenharia Civil, Campinas - São Paulo - Brasil. fabiosecfmr@gmail.com

Armando Lopes Moreno Júnior

Professor, Universidade Estadual de Campinas - UNICAMP Departamento de Engenharia Civil, Campinas - São Paulo - Brasil. almoreno@fec.unicamp.br


A_c: cross-sectional area of the concrete core	f_ys,θ: yield strength of reinforcing steel at temperature θ
A_{c,θ, i}: cross-section area of the layer "i" of the concrete core	I_c: second moment of area of the concrete
E_c: elasticity modulus of concrete at the room temperature	I_c,θ,i: second moment of area of the concrete at temperature θ, applied to the "i" layer
E_c,θ: elasticity modulus of concrete at the temperature θ	k_c,θ: reduction factor of the resistance of concrete at temperature θ
E_a: elasticity modulus of the structural steel at the room temperature	k_c,θ,i: reduction factor of the concrete strength at temperature θ, applied to the "i" layer
E_a,θ: elasticity modulus of the structural steel at the temperature θ	k_{Ec,θ, i} : reduction factor of the modulus of elasticity of concrete at temperature θ, applied to the "i" layer
E_s: elasticity modulus of reinforcing steel at the room temperature	_{Nfi, Rd}: plastic resistance in axial compression of the concrete core in fire situation
E_s,θ: elasticity modulus of reinforcing steel at the temperature θ
E_c,sec: secant modulus of elasticity of concrete	ε_cu: corresponding deformation of the concrete to the peak stress
E_{c,sec, θ, i}: secant modulus of elasticity of concrete at the temperature θ applied to the "i" layer
(EI)_fi,c: flexural stiffness of concrete in the fire situation	ε_cu,θ: corresponding deformation of the concrete to the peak stress at temperature θ
f_c: compressive strength of concrete	ε_cu,θ,i: corresponding deformation of the concrete to the peak stress at temperature θ, applied to the "i" layer
f_c,θ: compressive strength of concrete at temperature θ
f_{c,θ, i}: compressive strength of concrete at temperature θ, applied to the "i" layer	θ_c,eq,num: equivalent temperature determined for the concrete from numerical models
f_ya: yield strength of structural steel
f_ya,θ: yield strength of structural steel at temperature θ	θ_a,eq,num: equivalent temperature determined for the steel tube from numerical models
f_ys: yield strength of reinforcing steel

θ_c (ºC)	K_c,θ= f_c,θ/ f_c	K_Ec,θ= E_c,θ/E_c	Ka,_θ= fya,_θ/ fya	KEa,_θ= Ea,_θ/Ea	Ks,_θ= fys,_θ/ fys	KEs,_θ= Es,_θ/Es
20	1.00	1.000	1.000	1.000	1.000	1.000
100	1.00	0.625	1.000	1.000	1.000	1.000
200	0.95	0.432	1.000	0.900	1.000	0.870
300	0.85	0.304	1.000	0.800	1.000	0.720
400	0.75	0.188	1.000	0.700	0.940	0.560
500	0.60	0.100	0.780	0.600	0.670	0.400
600	0.45	0.045	0.470	0.310	0.400	0.240
700	0.30	0.030	0.230	0.130	0.120	0.080
800	0.15	0.015	0.110	0.090	0.110	0.060
900	0.08	0.008	0.060	0.0675	0.080	0.050
1000	0.04	0.004	0.040	0.0450	0.050	0.030
1100	0.01	0.001	0.020	0.0225	0.030	0.020
1200	0.00	0.000	0.000	0.000	0.000	0.000

Variable properties of the concrete
Model	Thermal expansion coefficient of the concrete	Agraggate type	Thermal expansion of the steel	Concrete moisture	Concrete density
1	according Eurocode 4	siliceous	according Eurocode 4	3%	according Eurocode 4
2	according Eurocode 4	siliceous	according Eurocode 4	10%	according Eurocode 4
3	according Eurocode 4	siliceous	according Eurocode 4	3%	2300 kg/m3
4	according Eurocode 4	siliceous	according Eurocode 4	3%	2400 kg/m3

Temperatures according to parametric analysis
Columns	Model	Temperature in the monitoring points (ºC)		Exposure time in fire (min.)
Columns	Model	Point 1b	Point 6a	Exposure time in fire (min.)
PQ-200-5	1	905.0	138.0	80
	2	887.3	51.5
	3	903.6	135.3
	4	901.7	123.9
PQ-140-5	1	854.0	277.8	50
	2	828.8	77.6
	3	852.1	271.0
	4	849.5	254.9

Temperatures obtained in the cross section (ºC )
Column	Points	model with perfect contact	model with thermal resistance between the steel tube and the concrete core as resistence coefficient			model with thermal expansion
Column	Points	model with perfect contact	100 W/m²K	200 W/m²K	300 W/m²K	model with thermal expansion
PQ200-5	1	691	640	634	633	638	30 (min.)
	2	296	142	96	74	98
	3	99	62	49	41	50
	4	48	36	31	28	33
	5	33	27	24	23	26
	1	880	740	723	719	725	60 (min.)
	2	574	283	171	124	173
	3	309	158	103	81	104
	4	176	95	71	60	72
	5	108	71	58	50	59

PARAMETRIC STUDY
Features	Variables
Tube - external dimension "L" (mm):	100	140	200	260	300
Tube thickness "t" (mm):	5.2	5.2	6.4	6.4	9.5
	6.4	6.4	9.5	9.4	12.7
	9.5	9.5
Fire resistance "R" (min):	30 ; 60 ; 90 ; 120

			R30		R60		R90		R120
Column	L/t	u/A (mm^-1)	θc,eq,num (ºC)	θa,eq,num (ºC)	θc,eq,num (ºC)	θa,eq,num (ºC)	θc,eq,num (ºC)	θa,eq,num (ºC)	θc,eq,num (ºC)	θa,eq,num (ºC)
100-5.2	19.2	0.040	550	710	825	920	945	995	1010	1040
100-6.4	15.6	0.040	560	705	830	920	950	995	1010	1040
100-9.5	10.5	0.040	560	705	805	915	940	995	1005	1040
140-5.2	26.9	0.029	450	700	710	910	860	985	935	1030
140-6.4	21.9	0.029	445	690	720	905	855	985	935	1030
140-9.5	14.7	0.029	465	690	730	905	865	985	955	1030
200-6.4	31.3	0.020	395	670	575	890	710	970	820	1020
200-9.5	21.1	0.020	385	660	560	890	740	970	825	1020
260-6.4	40.6	0.015	325	670	525	880	580	960	685	1010
260-9.5	27.4	0.015	330	660	525	880	615	960	740	1015
300-9.5	31.6	0.013	360	630	515	870	595	955	705	1000
300-12.7	23.6	0.013	345	620	520	860	595	950	750	1000

Equivalent temperatures for the concrete and the steel tube
Column	Exposuretimein fire(min.)	u/a(mm^-1)	Points i,a(from the vertex)		Points i,b(from the edge)		Diference
Column	Exposuretimein fire(min.)	u/a(mm^-1)	Concrete(ºC)	Steeltube(ºC)	Concrete(ºC)	Steeltube(ºC)	Concrete(ºC)	Steeltube(ºC)
PQ-200-6.4	30	0.020	395	670	285	620	27.8%	7.5%
PQ-140-5.2	30	0.029	450	700	390	655	13.3%	6.4%
PQ-200-6.4	60	0.020	575	890	570	870	0.9%	2.2%
PQ-140-5.2	60	0.029	710	910	645	890	9.2%	2.2%

Equivalent temperatures for the concrete and the steel tube (ºC)
Columns	Exposure time in fire(minutes)	Simplified equations		Renaud		Diferences
Columns	Exposure time in fire(minutes)	Concrete	Steel tube	Concrete	Steel tube	Concrete	Steel tube
PQ-200-5	30	384	663	340	705	-12.94%	5.96%
PQ-140-5	30	452	684	400	715	-13.00%	4.34%
PQ-200-5	60	596	886	540	885	-10.37%	-0.11%
PQ-140-5	60	705	902	570	895	-23.68%	-0.78%

Temperatures in the reinforcing bars in the corners in columns of square sections
L (mm)	R (min)	u/a (mm^-1)	Simplified equation (ºC)	Renaud (ºC)	Diferences
140	30	0.029	300	302	0.74%
200	30	0.020	289	288	-0.27%
300	30	0.013	280	287	2.36%
140	60	0.029	596	575	-3.61%
200	60	0.020	562	530	-5.95%
300	60	0.013	535	518	-3.27%

\begin{matrix} N_{f^{i, Rd}} = \sum_{i = 1}^{n} (A_{c, θ, i} . k_{c, θ, i} . f_{c}) = f_{c} \sum_{i = 1}^{n} (A_{c, θ, i} . k_{c, θ, i}) \\ fc \sum_{i = 1}^{n} (A_{c, θ, i} . k_{c, θ, i}) = k_{c, θ} (θ_{c, eq}). fc . Ac \to k_{c, θ} (θ_{c, eq 1}) = \frac{Σ_{i = 1}^{n} (A_{c, θ, i} . k_{c, θ, i})}{A_{c}} \end{matrix}

\begin{matrix} (EI)_{fi, c} = \sum_{i = 1}^{n} (E_{c, s ec, θ, i} . I_{c, θ, i}) = \sum_{i = 1}^{n} (\frac{f_{c, θ, i}}{ε_{cu, θ, i}} . I_{c, θ, i}) = \frac{f_{c}}{ε_{cu}} \sum_{i = 1}^{n} (\frac{k_{c, θ, i}}{\frac{ε_{cu, θ, i}}{ε_{cu}}} \cdot I_{c, θ, i}) = \\ = E_{c, s ec} \sum_{i = 1}^{n} (K_{E, c, θ, i} . I_{c, θ, i}), whit k_{E, c, θ, i} = \frac{k_{c, θ, i} . ε_{cu}}{ε_{cu, θ, i}} \\ {EI}_{fi, c} = E_{c, s ec} \sum_{i = 1}^{n} (k_{Ec, θ, i} . I_{c, θ, i}) = k_{Ec, θ} (θ_{c, eq 2}). E_{c, s ec} . I_{c} \\ \to k_{Ec, θ} (θ_{c, eq 2}) = \frac{Σ_{i = 1}^{n} (k_{Ec, θ, i} . I_{c, θ, i})}{I_{c}} \end{matrix}

\begin{matrix} θ c, eq = [(2.9892 . R^{3} - 605 . R^{2} + 29982 . R - 374314). {(\frac{u}{A})}^{2}] + [(- 0.1723 . R^{3} + 33.45 . R^{2} - \\ 1495 . R + 24399). (\frac{u}{A})] + 2.643 . 10^{- 3} . R^{3} - 0.545838 . R^{2} + 33.21 . R - 317 \end{matrix}

\begin{matrix} θ a, eq = [(- 2.2 . 10^{- 3} . R^{3} + 0.6393 . R^{2} - 67 . R + 4023). (\frac{u}{A})] + 7.17 . 10^{- 4} . R^{3} - 0.2127 . R^{2} + \\ 22.54 . R + 108 \end{matrix}

Brasil

Brasil

Temperature field of concrete-filled steel tubular columns in fire

Abstract