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Fire-resistance study of restrained steel columns with partial damage to fire protection
ARTICLE IN PRESS Fire Safety Journal 44 (2009) 1088–1094
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Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Fire-resistance study of restrained steel columns with partial damage to fire protection Wei-Yong Wang a,, Guo-Qiang Li b,c a
College of Civil Engineering, Chongqing University, Chongqing 400045, PR China College of Civil Engineering, Tongji University, Shanghai 200092, PR China c State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, PR China b
a r t i c l e in f o
a b s t r a c t
Article history: Received 2 September 2008 Received in revised form 28 July 2009 Accepted 4 August 2009 Available online 9 September 2009
In order to analyze the behavior of steel columns in fire with partial damage of fire protection, an analytical model is presented based on the differential equation of equilibrium, which may be used to predict the ultimate load bearing capacity of steel columns fixed at two ends and to predict the critical temperature of axially restrained steel columns. The imperfection of initial flexure of steel columns is taken into account in the model. The yielding of the edge fiber at the mid-span of a column subjected to elevated temperature is taken as the failure criteria for the fire resistance of the column. A numerical application is carried out to demonstrate the effect of the damage of fire protection on the ultimate load bearing capacity and axial force increase of axially restrained steel column in fire. Comparing with FEM, the model proposed in the paper has been validated and good agreement has been found. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Steel column Fire protection Fire resistance Load bearing capacity Critical temperature
1. Introduction Unprotected steel structures do not have desirable fire resistance because the temperature of unprotected steel elements increase rapidly in fire due to the high thermal conductivity of steel. Thermal insulation materials are widely used to protect steel structures against fire by slowing down heat transfer from the fire to the steel elements, and hence reducing the rate of temperature increase in the steel. However, spray-on insulation materials are often rather fragile, and they may be easily damaged by mechanical actions, such as impact, resulting in a possible fire resistance reduction for the steel elements originally protected. So it is important to investigate the fire resistance of steel members with partial fire protection damage. To the best of our knowledge, there are some published works related to the steel columns or beams with partial damage to fire protection. Ryder et al. [1] carried out a three-dimensional finite element analyses to investigate the effect of fire protection loss on the fire resistance of steel columns. Yu et al [2] carried out a numerical study to investigate the fire resistance reduction of protected steel beams caused by partial loss of spray-on fire protection. The results of a heat transfer analysis of steel columns with partial loss of fire protection using the finite element method were also presented by Milke et al [3]. Fontana and Knobloch [4][5] studied the behavior of steel columns subjected to fire using a three-dimensional finite
Corresponding author.
E-mail address: [email protected] (W.-Y. Wang). 0379-7112/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2009.08.002
element heat transfer and structural model, taking into account geometrical nonlinearities, local temperature distributions, thermal strains, and temperature dependent material properties. Pessiki [6] performed an analysis to examine the behavior of steel H columns in fire with damaged spray-applied fire resistive material subjected to concentric axial compression. A simple model to predict the ultimate load bearing capacity of hinged steel columns with partial fire protection damage is presented in our previous work [7]. A column in a structure is often axially restrained by the adjacent members like beams, slabs, and supports in the steel structures. At ambient temperature, the axial restraint can improve the ultimate load bearing capacity of the column [8], but in fire environment, the additional axial force of the column will be increased, which will cause an earlier yield or buckling of the column. In real structures, the loading conditions of steel columns subjected to the action of a localized fire will change with time, if their axial elongation is restrained. There are also some papers which deal with the restraint of the column in fire. A study by Rodrigues et al [9] showed that neglecting the effect of thermal axial restraint may result in overestimation of the fire resistance of columns. Valente and Neves [10] used a computer program based on the Finite Element Method to analyze the influence of elastic axial and rotational restraints on the critical temperature of columns. Huang and Tan [11] proposed a Rankine approach, which incorporates both the axial restraint and creep strain for the critical temperature prediction of an axially restrained steel column. Wang and Davies [12] carried out some tests to evaluate how bending moments in restrained columns would change and how these changes might affect the column failure temperatures.
ARTICLE IN PRESS W.-Y. Wang, G.-Q. Li / Fire Safety Journal 44 (2009) 1088–1094
x x
x
ld
P
E1I
x
-E2Iy
l-2ld
a0 y2
ld
y
y
y
P
y
M
x x
x P
E2I
''
ld
x
ð1Þ
P
ð0rxrld Þ
E2 Iðy2 y0 Þ00 þ Py2 M ¼ 0
ðld rxrl=2Þ
8 y ð0Þ ¼ 0; y10 ð0Þ ¼ 0 > < 1 y1 ðld Þ ¼ y2 ðld Þ; y10 ðld Þ ¼ y20 ðld Þ > : y 0 ðl=2Þ ¼ 0
E2I
ð3Þ
2
y2 y1
ð2Þ
where E1 is the elastic modulus of steel at temperature T1; E2 is the elastic modulus of steel at temperature T2; T1 is the temperature in the portion of the column with damaged fire protection; T2 is the temperature in the portion of the column with integrated fire protection; I is inertia moment of column; and ld is the length of the damaged fire protection at one end of the column. As the displacement and rotation at the two ends of the column are fixed and the rotation of the cross-section of the column is continuous through the entire length, the following boundary conditions can be adopted as
a0
y
ld
E1 Iðy1 y0 Þ00 þ Py1 M ¼ 0
E1I
l-2ld
y
where l is the length of the column and a0 is the initial flexure at the mid-span of the column. If the lateral displacement of the column at the location without fire protection and with fire protection is represented by y1 and y2, respectively, according to the differential equation for equilibrium, the following equation can be obtained (
y1 E1I
-E1Iy
1 2p x y0 ¼ a0 1 cos l 2
P
y
E2I
2. Ultimate load bearing capacity of steel columns fixed at two ends The fire protection sprayed on steel columns may be damaged, either at the ends or at the center portion of the steel column. If the damage at the two ends, it is assumed that the damage length of fire protection at the two ends of the column is the same. In this paper, it is further assumed that the temperature over both the portions of the column with fire protection and without fire protection due to damage is uniform. Fig. 1 (a) shows the mechanical model of a steel column with fixed boundary conditions and damaged fire protection at its two ends. As we know, every column in the real building has some initial imperfection. The ultimate load bearing capacity for columns with initial imperfection is less than that of perfect columns. The initial flexure of the column may be expressed by
''
It can be seen from the above literature review that most of the works have been carried out using FEM, and fire protection damage has an obvious effect on the fire resistance of steel members. For a restrained steel column, the axial or rotational restraint has an influence on the critical temperature. In this paper, an analytical model to predict the ultimate load bearing capacity of rigidly supported steel columns with partial damage in fire protection is presented. The difference between this paper and the previous FEM modeling is that a simple model is presented and the model can be employed to study the influence of fire protection missing either at the ends or at the central portion of the column. In addition, a numerical application to investigate the ultimate carrying capacity of the steel column and the variety of the axial forces that occur in the column when temperature is raised at two different degrees of axial restraint stiffness is performed.
1089
y
y
y
P
M
Fig. 1. Mechanical model of the steel column with fixed ends. (a) Fire protection to damage at the ends; (b) Fire protection to damage at the center.
where m is the damage length ratio of fire protection of column; e is the initial flexure ratio of column; Pcr1, Pcr2, a and b are intermediate variables and P is the axial force of column. The general solution to Eq. (2) is given by rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 8 P P 2a0 E1 Ip2 2p M > > x þ C2 cos x þ 2 xþ cos > y1 ¼ C1 sin < E1 I E1 I P l Pl 4E1 Ip2 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi > P P 2a0 E2 Ip2 2p M > > y2 ¼ C3 sin x þ C4 cos x þ 2 xþ cos : E2 I E2 I P l Pl 4E2 Ip2
ð5Þ For convenience, the following parameters are used: where the parameters C1, C2, C3, C4, and M are determined by employing the boundary condition Eq. (3), and are given by
m ¼ ld =l; e ¼ a0 =l; Pcr1 ¼ p2 ET1 I=l2 ; Pcr2 ¼ p2 ET2 I=l2 ; a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi P=Pcr1 ; b ¼ P=Pcr2
ð4Þ
C1 ¼ 0
ð6Þ
ARTICLE IN PRESS 1090
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2a0 ða2 b Þfb cos ð2mpÞ ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ
C2 ¼
2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ
þ2 sin ð2mpÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg
ð7Þ
½2 sin ð2mpÞ cos ðmapÞ cos ð2mpÞa sin ðmapÞ
C3 ¼
2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ 2
2a0 ðÞa2 b þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg
ð8Þ
½2 sin ð2mpÞ cos ðmapÞ cos ð2mpÞa sin ðmapÞ
C4 ¼
2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ 2a0 ða2 b Þctgð0:5bpÞ þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg
ð9Þ
2
M¼
2a0 Pða2 b Þfb cos ð2mpÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ 2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ
þ2 sin ð2mpÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg 2a0 P þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg a2 4
ð10Þ
Taking yield of the edge fiber at the mid-span of the column as the failure criterion, the ultimate load bearing capacity of the column with partial loss of fire protection should be the smallest one of the solutions to the following three equations: 8 P M > > > A þ W ¼ fyT1 > > > < P jM Pa max1 j þ ¼ fyT1 > A W > > > P jM Pamax2 j > > : þ ¼ fyT2 A W
3. Behavior of axial restrained steel columns 3.1. Axial force of axial restrained steel columns
2
where A is the area of cross-section of the column; W is the section flexure modulus of the column; fyT1 is the yield strength of steel at temperature T1; fyT2 is the yield strength of steel at temperature T2; amax1 ¼ y1(ld), and amax2 ¼ y2(l/2). Given the dimension of the steel column and the length of damage to the fire protection, at a specific temperature distribution, the ultimate load bearing capacity can be obtained with Eq. (11) by finding the smallest solution. If the damage is at the center of the column, the solution is very similar to the one with the damage at the ends. Fig. 1 (b) shows the mechanical model of the steel column with fixed boundary and the damage to fire protection at the middle of the column. In order to obtain the ultimate carrying capacity, replace E1 with E2 and replace E2 with E1 from Eq (2) to Eq. (11), and ld is the length of the undamaged fire protection at one half of the column.
ðaÞ ðbÞ
ð11Þ
ðcÞ
If an axially restrained column, such as a column in a frame, is heated up by fire, additional axial forces will be produced in the column by thermal expansion, as schematically illustrated in Fig. 2. With the elevation of the temperature, the overall thermal expansion of the column may be obtained by
Dlth ¼ aT ðT1 T0 Þ2ld þ aT ðT2 T0 Þðl 2ld Þ
where aT is the thermal expansion coefficient of steel and T0 is the normal temperature. The column will be shortened due to increase in the axial compressive force and decrease in elastic modulus at elevated temperatures. In addition, bending will also shorten the length of the column. As illustrated in Fig. 3(a), the length change of the column at two different axial force and different elastic modulus can be represented as Dl2Dl1 ¼ (P2l/E2A)(P1l/E1A) as illustrated in Fig. 3(b), since cos y ¼ 1(1/2)y2+(1/24)y4+?E1(1/2)y2 and y ¼ dy/dx, so dx(1cos y) ¼ (1/2)y2 dx. Hence the length change of the column due to the bending can be represented as R R Dl ¼ 0l ð1 cos yÞdx 12 0l ðy0 Þ2 dx.
P20 P20+ΔP
Fk
P20
kr
P20
P20
Δlth Δls
T1 T2
y(P20)
ð12Þ
y(P20+ΔP) T20
T1
Fig. 2. Model of restrained steel column.
Δlc
ARTICLE IN PRESS W.-Y. Wang, G.-Q. Li / Fire Safety Journal 44 (2009) 1088–1094
1091
at an elevated temperature
aT ðT1 20Þ2ld þ aT ðT2 20Þðl 2ld Þ P2
Δl2
Δl1
P1
Δl2-Δl1
¼
PT P20 PT 2ld PT ðl 2ld Þ P20 l þ þ kr ET1 A ET2 A E20 A Z ld Z l=2 þ ½y10 ðPT Þ2 y10 ðP20 Þ2 dx þ ½y20 ðPT Þ2 y20 ðP20 Þ2 dx
ð15Þ
ld
l
0
E2
E1
The axial force of steel columns subjected to fire may increase due to the restraint in the thermal elongation. When the axial force reaches the ultimate load bearing capacity in a fire, the column will fail. The critical temperature of the column with partial fire protection damaged in fire is defined as the temperature in the portion of the column with damaged fire protection at which the axial force increases up to the ultimate load bearing capacity of the column. For a given steel column, the ultimate load bearing capacity of the column can be calculated by Eq. (11). Given a specific restraint stiffness, the axial force of steel columns subjected to fire may be determined by Eq. (15). Therefore, the critical temperature of the columns can be obtained.
Δl
P
3.2. Critical temperature of axial restrained steel columns
l
dy
dx
dx(1-cosθ)
4. Numerical application
x
In order to understand the behavior of a steel column with damaged fire protection at the ends in fire, a column is analyzed using the model proposed above as an example.
Fig. 3. The representation of length change of the column at (a) compression and (b) bending.
According to the formula above, the complete contraction of the column with partial fire protection damage at the two ends may be determined by
Dlc ¼
PT 2ld PT ðl 2ld Þ P20 l þ ET1 A ET2 A E20 A Z ld 2 2 þ ½y10 ðPT Þ y10 ðP20 Þ dx Z
0 l=2
þ
½y20 ðPT Þ2 y20 ðP20 Þ2 dx
ð13Þ
ld
where Pr is the axial force of the column at elevated temperature; P20 is the axial force of the column at the room temperature; and E20 is the elastic modulus of steel at the room temperature. The spring representing the axial restraint will be compressed due to the increase in axial force. The length change of the spring compressed can be given by Hooke’s law
Dls ¼ DP=kr
ð14Þ
where DP ¼ PTP20 and kr is the axial restraint stiffness of the column. The degree of axial restraint depends on the structural framework—for instance, simply supported floors do not restrain axial expansion in a building, but a moment frame may provide some restraint, depending on the framing and connection details. How to confirm the degree of axial restraint is a difficult and complex problem which will be studied in forthcoming papers. As demonstrated in Fig. 2, the following equation can be obtained and employed to determine the axial force in the column
4.1. Geometry and material properties Consider a typical steel column with section of H100 100 6 8. The main characteristic parameters of the column are the area of cross-section, A ¼ 2190 mm2; the inertia moment, I ¼ 383 cm4; the flexure modulus, W ¼ 76.5 cm3; the elastic modulus at normal temperature, E20 ¼ 2.06 105 MPa; the yield strength at normal temperature, fy20 ¼ 235 MPa; the length of column, l ¼ 4 m; and the thickness of fire protection, dc ¼ 30 mm. The modulus and yield strength of the steel for the column at elevated temperatures are determined according to the Chinese Technical Code on Fire Safety of Steel Building Structures [8]. The reduction factor of the elastic modulus and yield strength of the steel are plotted in Fig. 4 and material properties of the steel and the fire protection are listed in Table 1.
1.0 Elastic modulus Yield strength
0.8 Reduction factor
y
0.6
0.4
0.2
0.0 0
200
400 600 Temperature (°C)
800
1000
Fig. 4. Reduction factor of elastic modulus and yield strength of steel.
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Table 1 Material properties of steel and fire protection. Symbol Value
Unit
Steel ls Fire protection lc
45 0.1
W/(m 1C)
Specific heat
Steel Cs Fire protection Cc
600 0.22
J/(kg 1C)
Unit mass
Steel rs Fire protection rc
7850 500
kg/m3
Thermal expansion coefficient Steel
aT
1.4 10
ISO standard temperature
800
Thermal conductivity
Temperature (°C)
Parameters
1000
Temperature of steel column in the portion
600
with damaged FRC
400 200
5
Temperature of steel column in the portion with integrated FRC
m/(m 1C)
0 0
10
20
4.2. Temperatures of the example
ð16Þ
1.0
where t is the duration after start of fire (min). The temperature of the steel column subjected to the fire can be predicted by the following incremental equation [13] as:
0.8
Ts ðt þ DtÞ ¼
½Tg ðtÞ Ts ðtÞDt þ Ts ðtÞ
0.4
where Ts is the temperature of steel; Dt is the increment of time; and B is the general heat transfer coefficient, which can be determined by: If without fire protection
0.2
B ¼ ðac þ ar Þ
1þ
μ = 0.4 μ = 0.5
0.0
ð18Þ
1
lc Fc
cc rc dc Fc 2cs rs V
dc V
200
400 600 Temperature (°C)
800
1000
Fig. 6. Load bearing capacity–temperature curve of the example steel column (the curves are essentially coincident).
and if with non-intumescent fire protection B¼
μ = 0.3
0
F V
60
μ = 0.2
0.6
ð17Þ
cs rs
50
μ = 0.1
ϕT
B
40
Fig. 5. Temperature–time curve of the example steel column.
The ISO standard temperature–time curve is employed to represent the temperature of the fire around the steel column, which is expressed by Tg ¼ 20 þ 345 logð8t þ 1Þ
30 Time(min)
ð19Þ
where ac is convective heat transfer coefficient, W/(m2 1C); ar is radiative heat transfer coefficient, W/(m2 1C); F is the area per unit length of column, m2/m; V is the volume per unit length of column, m3/m; dc is the thickness of fire protection, m; and Fc is the interior area of fire protection per unit length column, m2/m. With intumescent fire protection, the fire protection will expanded at high temperature and the thermal properties will change. So long as dc and cc in Eq. (19) are replaced with the thickness and thermal conductivity of the fire protection after expansion, the temperature may be obtained. However, the degree of expansion of the intumescent fire protection is influenced greatly by environment and the temperature elevation should be determined experimentally. The increase in the temperature of the example subjected to the ISO fire is calculated and illustrated in Fig. 5.
where Pdcr is the ultimate load bearing capacity of the steel columns with partial fire protection loss and fy20 is the yield strength of steel at normal temperature. When e ¼ 1%, the curve of the load bearing capacity coefficient with temperature is plotted in Fig. 6, with m defined in Eq. (4). As can be seen from Fig. 6, the load bearing capacity is reduced with increase in temperature. However, the length of damaged fire protection has very little influence on the load bearing capacity. 4.4. Axial force of the column The example column axially restrained with partial fire protection loss is also studied to look into the variation of the axial force in the column with the elevation of temperature. The relationship between the temperature of the column at the part with damaged fire protection and the axial force in the column with various restraint stiffness and damage length ratio is shown in Fig. 7, in which z is the axial restraint ratio of the column and is defined as
4.3. Ultimate load bearing capacity of the column
z ¼ kr =ðE20 A=lÞ
The load bearing capacity coefficient of a steel column with damaged fire protection is defined as
As is shown in Fig. 7, the axial force in the column with partial damage to the fire protection and axial restraint firstly increases when the temperature rises, then the axial force decreases to the initial level with the rise in temperature. This may be because the
jT ¼ Pdcr =ðfy20 AÞ
ð20Þ
ð21Þ
ARTICLE IN PRESS W.-Y. Wang, G.-Q. Li / Fire Safety Journal 44 (2009) 1088–1094
1.30
PT/P20
ζ = 0.10, μ = 0.4
6
1000
T1 E1
ζ = 0.10, μ = 0.2
100
8
ζ = 0.05, μ = 0.4
1.20
100
P
ζ = 0.05, μ = 0.2
1.25
1093
8
1.15
Model 1 P = 500kN T1 = 600°C T2 = 20°C E1 = 103000MPa
T2 E2
1.05
4
2000
1.10
1.00 100
200 300 400 Temperature (°C)
500
Model 2 P = 500kN T1 = 20∼820°C T2 = 20°C E1 = 206000∼17490MPa
600
Fig. 7. Axial force in the example steel column.
T1
1000
0
E1
Fig. 8. Sketch of the FEM models.
Proposed model (P = 200KN)
4000
Length of column (mm)
elastic modulus and the yield strength of the steel decrease with the elevation of temperature and the flexure of the column is large enough to release the thermal expansion of the column and then to release the axial force. In addition, at the same damage length of fire protection, for example, m ¼ 0.2, the axial restraint stiffness can significantly increase the degree of the elevation of axial force. Similarly, at the same stiffness of axial restraint, for example, x ¼ 0.10, the length of damage to fire protection can also significantly increase the degree of the elevation of axial force. When the damage to the fire protection is at the center of the column, the procedures to obtain the ultimate load carrying capacity and the axial force in the column are quite similar to the situation in which the fire protection damage is at the ends of the column.
E2 = 206000MPa
FEM (P = 200KN)
3000
Proposed model (P = 500KN) FEM (P = 500KN)
2000
1000
5. FEM validation
0 0
1
2 3 4 5 Deflection (mm)
6
7
Fig. 9. Deflection comparison of the steel column between ANSYS and calculated (the curves are essentially coincident).
12
Displacement (mm)
In order to verify the analytical model proposed in the paper, two FEM models are presented by employing ANSYS software. The element type used in the analysis is BEAM3, which is a uniaxial element with tension, compression, and bending capabilities. The element has three degrees of freedom at each node: translations in the nodal x and y directions and rotation about the nodal z-axis. The length of element is 50 mm. the material of steel at elevated temperature is adopted as recommend in Chinese Technical Code on Fire Safety of Steel Building Structures [13]. One model (Model 1) is performed to verify the flexure shape of steel column, which is the most important parameter in defining the ultimate load capacity of the steel column with partial fire protection damage. In this model, the column section is H100 100 6 8; l ¼ 4 m; m ¼ 0.25; e ¼ 0.1%; P ¼ 500 KN. The temperature of the column with damaged fire protection is 600 1C. The sketch of the model is shown in Fig. 8. The results obtained by ANSYS and the proposed approach in this paper are plotted in Fig. 9. It was found that there is a good agreement between the FEM and the model of this paper. Another FEM model (Model 2) is performed to validate the axial displacement of the steel columns with rise in temperature. The column section is H100 100 6 8. l ¼ 4 m, P ¼ 500 KN, m ¼ 0.25. The thermal expansion coefficient is 1.4 105. The sketch of this model is also shown in Fig. 8. The comparisons of the development of the axial displacement obtained by the proposed model in the paper and FEM method are shown in Fig. 10. It can be seen that the axial displacement predictions by the method proposed in this paper agree well with that obtained by the FEM.
8 4 Results of ANSYS Results of Calculated
0 -4 -8 0
150
300
450 600 Temperature (°C)
750
900
Fig. 10. Axial displacement comparison of the steel column between ANSYS and calculate.
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6. Concluding remarks
References
Using the experimental approach to predict the fire resistance of a steel column is both expensive and time consuming. As an alternative, a theoretical analysis that can be performed simply is a convenient tool for engineers to quickly assess the column fire resistance. In this paper, an analytical model is presented for determining the behavior of restrained steel columns in fire with partially damaged fire protection. The ultimate load bearing capacity and critical temperature of steel columns may be predicted by employing the model proposed. The model can be used to evaluate the fire resistance and predict the critical temperature of steel columns with partial fire protection damage. Through numerical study on a typical steel column with partial fire protection loss, the following conclusions are drawn:
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(1) The load bearing capacity of steel columns with fixed ends exposed to fire is reduced at a given temperature with the damage of fire protection. (2) The length of damaged fire protection has little influence on the load bearing capacity of the steel columns fixed at two ends. (3) For axially restrained steel columns, the length of damaged fire protection has significant influence on the increase of axial force of steel columns in fire.
Acknowledgments The support of the Natural Science Foundation of China for Innovative Research Groups (Grant No. 50621062), Key Research Project (Grant No. 50738005) and Scientific Research Foundation of Chongqing University for Recruited Scholars for the study reported in this paper is gratefully acknowledged.
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Fire Safety Journal journal homepage: www.elsevier.com/locate/firesaf
Fire-resistance study of restrained steel columns with partial damage to fire protection Wei-Yong Wang a,, Guo-Qiang Li b,c a
College of Civil Engineering, Chongqing University, Chongqing 400045, PR China College of Civil Engineering, Tongji University, Shanghai 200092, PR China c State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, PR China b
a r t i c l e in f o
a b s t r a c t
Article history: Received 2 September 2008 Received in revised form 28 July 2009 Accepted 4 August 2009 Available online 9 September 2009
In order to analyze the behavior of steel columns in fire with partial damage of fire protection, an analytical model is presented based on the differential equation of equilibrium, which may be used to predict the ultimate load bearing capacity of steel columns fixed at two ends and to predict the critical temperature of axially restrained steel columns. The imperfection of initial flexure of steel columns is taken into account in the model. The yielding of the edge fiber at the mid-span of a column subjected to elevated temperature is taken as the failure criteria for the fire resistance of the column. A numerical application is carried out to demonstrate the effect of the damage of fire protection on the ultimate load bearing capacity and axial force increase of axially restrained steel column in fire. Comparing with FEM, the model proposed in the paper has been validated and good agreement has been found. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Steel column Fire protection Fire resistance Load bearing capacity Critical temperature
1. Introduction Unprotected steel structures do not have desirable fire resistance because the temperature of unprotected steel elements increase rapidly in fire due to the high thermal conductivity of steel. Thermal insulation materials are widely used to protect steel structures against fire by slowing down heat transfer from the fire to the steel elements, and hence reducing the rate of temperature increase in the steel. However, spray-on insulation materials are often rather fragile, and they may be easily damaged by mechanical actions, such as impact, resulting in a possible fire resistance reduction for the steel elements originally protected. So it is important to investigate the fire resistance of steel members with partial fire protection damage. To the best of our knowledge, there are some published works related to the steel columns or beams with partial damage to fire protection. Ryder et al. [1] carried out a three-dimensional finite element analyses to investigate the effect of fire protection loss on the fire resistance of steel columns. Yu et al [2] carried out a numerical study to investigate the fire resistance reduction of protected steel beams caused by partial loss of spray-on fire protection. The results of a heat transfer analysis of steel columns with partial loss of fire protection using the finite element method were also presented by Milke et al [3]. Fontana and Knobloch [4][5] studied the behavior of steel columns subjected to fire using a three-dimensional finite
Corresponding author.
E-mail address: [email protected] (W.-Y. Wang). 0379-7112/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2009.08.002
element heat transfer and structural model, taking into account geometrical nonlinearities, local temperature distributions, thermal strains, and temperature dependent material properties. Pessiki [6] performed an analysis to examine the behavior of steel H columns in fire with damaged spray-applied fire resistive material subjected to concentric axial compression. A simple model to predict the ultimate load bearing capacity of hinged steel columns with partial fire protection damage is presented in our previous work [7]. A column in a structure is often axially restrained by the adjacent members like beams, slabs, and supports in the steel structures. At ambient temperature, the axial restraint can improve the ultimate load bearing capacity of the column [8], but in fire environment, the additional axial force of the column will be increased, which will cause an earlier yield or buckling of the column. In real structures, the loading conditions of steel columns subjected to the action of a localized fire will change with time, if their axial elongation is restrained. There are also some papers which deal with the restraint of the column in fire. A study by Rodrigues et al [9] showed that neglecting the effect of thermal axial restraint may result in overestimation of the fire resistance of columns. Valente and Neves [10] used a computer program based on the Finite Element Method to analyze the influence of elastic axial and rotational restraints on the critical temperature of columns. Huang and Tan [11] proposed a Rankine approach, which incorporates both the axial restraint and creep strain for the critical temperature prediction of an axially restrained steel column. Wang and Davies [12] carried out some tests to evaluate how bending moments in restrained columns would change and how these changes might affect the column failure temperatures.
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x x
x
ld
P
E1I
x
-E2Iy
l-2ld
a0 y2
ld
y
y
y
P
y
M
x x
x P
E2I
''
ld
x
ð1Þ
P
ð0rxrld Þ
E2 Iðy2 y0 Þ00 þ Py2 M ¼ 0
ðld rxrl=2Þ
8 y ð0Þ ¼ 0; y10 ð0Þ ¼ 0 > < 1 y1 ðld Þ ¼ y2 ðld Þ; y10 ðld Þ ¼ y20 ðld Þ > : y 0 ðl=2Þ ¼ 0
E2I
ð3Þ
2
y2 y1
ð2Þ
where E1 is the elastic modulus of steel at temperature T1; E2 is the elastic modulus of steel at temperature T2; T1 is the temperature in the portion of the column with damaged fire protection; T2 is the temperature in the portion of the column with integrated fire protection; I is inertia moment of column; and ld is the length of the damaged fire protection at one end of the column. As the displacement and rotation at the two ends of the column are fixed and the rotation of the cross-section of the column is continuous through the entire length, the following boundary conditions can be adopted as
a0
y
ld
E1 Iðy1 y0 Þ00 þ Py1 M ¼ 0
E1I
l-2ld
y
where l is the length of the column and a0 is the initial flexure at the mid-span of the column. If the lateral displacement of the column at the location without fire protection and with fire protection is represented by y1 and y2, respectively, according to the differential equation for equilibrium, the following equation can be obtained (
y1 E1I
-E1Iy
1 2p x y0 ¼ a0 1 cos l 2
P
y
E2I
2. Ultimate load bearing capacity of steel columns fixed at two ends The fire protection sprayed on steel columns may be damaged, either at the ends or at the center portion of the steel column. If the damage at the two ends, it is assumed that the damage length of fire protection at the two ends of the column is the same. In this paper, it is further assumed that the temperature over both the portions of the column with fire protection and without fire protection due to damage is uniform. Fig. 1 (a) shows the mechanical model of a steel column with fixed boundary conditions and damaged fire protection at its two ends. As we know, every column in the real building has some initial imperfection. The ultimate load bearing capacity for columns with initial imperfection is less than that of perfect columns. The initial flexure of the column may be expressed by
''
It can be seen from the above literature review that most of the works have been carried out using FEM, and fire protection damage has an obvious effect on the fire resistance of steel members. For a restrained steel column, the axial or rotational restraint has an influence on the critical temperature. In this paper, an analytical model to predict the ultimate load bearing capacity of rigidly supported steel columns with partial damage in fire protection is presented. The difference between this paper and the previous FEM modeling is that a simple model is presented and the model can be employed to study the influence of fire protection missing either at the ends or at the central portion of the column. In addition, a numerical application to investigate the ultimate carrying capacity of the steel column and the variety of the axial forces that occur in the column when temperature is raised at two different degrees of axial restraint stiffness is performed.
1089
y
y
y
P
M
Fig. 1. Mechanical model of the steel column with fixed ends. (a) Fire protection to damage at the ends; (b) Fire protection to damage at the center.
where m is the damage length ratio of fire protection of column; e is the initial flexure ratio of column; Pcr1, Pcr2, a and b are intermediate variables and P is the axial force of column. The general solution to Eq. (2) is given by rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 8 P P 2a0 E1 Ip2 2p M > > x þ C2 cos x þ 2 xþ cos > y1 ¼ C1 sin < E1 I E1 I P l Pl 4E1 Ip2 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi > P P 2a0 E2 Ip2 2p M > > y2 ¼ C3 sin x þ C4 cos x þ 2 xþ cos : E2 I E2 I P l Pl 4E2 Ip2
ð5Þ For convenience, the following parameters are used: where the parameters C1, C2, C3, C4, and M are determined by employing the boundary condition Eq. (3), and are given by
m ¼ ld =l; e ¼ a0 =l; Pcr1 ¼ p2 ET1 I=l2 ; Pcr2 ¼ p2 ET2 I=l2 ; a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi P=Pcr1 ; b ¼ P=Pcr2
ð4Þ
C1 ¼ 0
ð6Þ
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2a0 ða2 b Þfb cos ð2mpÞ ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ
C2 ¼
2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ
þ2 sin ð2mpÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg
ð7Þ
½2 sin ð2mpÞ cos ðmapÞ cos ð2mpÞa sin ðmapÞ
C3 ¼
2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ 2
2a0 ðÞa2 b þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg
ð8Þ
½2 sin ð2mpÞ cos ðmapÞ cos ð2mpÞa sin ðmapÞ
C4 ¼
2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ 2a0 ða2 b Þctgð0:5bpÞ þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg
ð9Þ
2
M¼
2a0 Pða2 b Þfb cos ð2mpÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ 2
½ða2 4Þðb 4Þfb cos ðmapÞ½cos ðmbpÞ ctgð0:5bpÞ sin ðmbpÞ
þ2 sin ð2mpÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg 2a0 P þa sin ðmapÞ½sin ðmbpÞ þ ctgð0:5bpÞ cos ðmbpÞg a2 4
ð10Þ
Taking yield of the edge fiber at the mid-span of the column as the failure criterion, the ultimate load bearing capacity of the column with partial loss of fire protection should be the smallest one of the solutions to the following three equations: 8 P M > > > A þ W ¼ fyT1 > > > < P jM Pa max1 j þ ¼ fyT1 > A W > > > P jM Pamax2 j > > : þ ¼ fyT2 A W
3. Behavior of axial restrained steel columns 3.1. Axial force of axial restrained steel columns
2
where A is the area of cross-section of the column; W is the section flexure modulus of the column; fyT1 is the yield strength of steel at temperature T1; fyT2 is the yield strength of steel at temperature T2; amax1 ¼ y1(ld), and amax2 ¼ y2(l/2). Given the dimension of the steel column and the length of damage to the fire protection, at a specific temperature distribution, the ultimate load bearing capacity can be obtained with Eq. (11) by finding the smallest solution. If the damage is at the center of the column, the solution is very similar to the one with the damage at the ends. Fig. 1 (b) shows the mechanical model of the steel column with fixed boundary and the damage to fire protection at the middle of the column. In order to obtain the ultimate carrying capacity, replace E1 with E2 and replace E2 with E1 from Eq (2) to Eq. (11), and ld is the length of the undamaged fire protection at one half of the column.
ðaÞ ðbÞ
ð11Þ
ðcÞ
If an axially restrained column, such as a column in a frame, is heated up by fire, additional axial forces will be produced in the column by thermal expansion, as schematically illustrated in Fig. 2. With the elevation of the temperature, the overall thermal expansion of the column may be obtained by
Dlth ¼ aT ðT1 T0 Þ2ld þ aT ðT2 T0 Þðl 2ld Þ
where aT is the thermal expansion coefficient of steel and T0 is the normal temperature. The column will be shortened due to increase in the axial compressive force and decrease in elastic modulus at elevated temperatures. In addition, bending will also shorten the length of the column. As illustrated in Fig. 3(a), the length change of the column at two different axial force and different elastic modulus can be represented as Dl2Dl1 ¼ (P2l/E2A)(P1l/E1A) as illustrated in Fig. 3(b), since cos y ¼ 1(1/2)y2+(1/24)y4+?E1(1/2)y2 and y ¼ dy/dx, so dx(1cos y) ¼ (1/2)y2 dx. Hence the length change of the column due to the bending can be represented as R R Dl ¼ 0l ð1 cos yÞdx 12 0l ðy0 Þ2 dx.
P20 P20+ΔP
Fk
P20
kr
P20
P20
Δlth Δls
T1 T2
y(P20)
ð12Þ
y(P20+ΔP) T20
T1
Fig. 2. Model of restrained steel column.
Δlc
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1091
at an elevated temperature
aT ðT1 20Þ2ld þ aT ðT2 20Þðl 2ld Þ P2
Δl2
Δl1
P1
Δl2-Δl1
¼
PT P20 PT 2ld PT ðl 2ld Þ P20 l þ þ kr ET1 A ET2 A E20 A Z ld Z l=2 þ ½y10 ðPT Þ2 y10 ðP20 Þ2 dx þ ½y20 ðPT Þ2 y20 ðP20 Þ2 dx
ð15Þ
ld
l
0
E2
E1
The axial force of steel columns subjected to fire may increase due to the restraint in the thermal elongation. When the axial force reaches the ultimate load bearing capacity in a fire, the column will fail. The critical temperature of the column with partial fire protection damaged in fire is defined as the temperature in the portion of the column with damaged fire protection at which the axial force increases up to the ultimate load bearing capacity of the column. For a given steel column, the ultimate load bearing capacity of the column can be calculated by Eq. (11). Given a specific restraint stiffness, the axial force of steel columns subjected to fire may be determined by Eq. (15). Therefore, the critical temperature of the columns can be obtained.
Δl
P
3.2. Critical temperature of axial restrained steel columns
l
dy
dx
dx(1-cosθ)
4. Numerical application
x
In order to understand the behavior of a steel column with damaged fire protection at the ends in fire, a column is analyzed using the model proposed above as an example.
Fig. 3. The representation of length change of the column at (a) compression and (b) bending.
According to the formula above, the complete contraction of the column with partial fire protection damage at the two ends may be determined by
Dlc ¼
PT 2ld PT ðl 2ld Þ P20 l þ ET1 A ET2 A E20 A Z ld 2 2 þ ½y10 ðPT Þ y10 ðP20 Þ dx Z
0 l=2
þ
½y20 ðPT Þ2 y20 ðP20 Þ2 dx
ð13Þ
ld
where Pr is the axial force of the column at elevated temperature; P20 is the axial force of the column at the room temperature; and E20 is the elastic modulus of steel at the room temperature. The spring representing the axial restraint will be compressed due to the increase in axial force. The length change of the spring compressed can be given by Hooke’s law
Dls ¼ DP=kr
ð14Þ
where DP ¼ PTP20 and kr is the axial restraint stiffness of the column. The degree of axial restraint depends on the structural framework—for instance, simply supported floors do not restrain axial expansion in a building, but a moment frame may provide some restraint, depending on the framing and connection details. How to confirm the degree of axial restraint is a difficult and complex problem which will be studied in forthcoming papers. As demonstrated in Fig. 2, the following equation can be obtained and employed to determine the axial force in the column
4.1. Geometry and material properties Consider a typical steel column with section of H100 100 6 8. The main characteristic parameters of the column are the area of cross-section, A ¼ 2190 mm2; the inertia moment, I ¼ 383 cm4; the flexure modulus, W ¼ 76.5 cm3; the elastic modulus at normal temperature, E20 ¼ 2.06 105 MPa; the yield strength at normal temperature, fy20 ¼ 235 MPa; the length of column, l ¼ 4 m; and the thickness of fire protection, dc ¼ 30 mm. The modulus and yield strength of the steel for the column at elevated temperatures are determined according to the Chinese Technical Code on Fire Safety of Steel Building Structures [8]. The reduction factor of the elastic modulus and yield strength of the steel are plotted in Fig. 4 and material properties of the steel and the fire protection are listed in Table 1.
1.0 Elastic modulus Yield strength
0.8 Reduction factor
y
0.6
0.4
0.2
0.0 0
200
400 600 Temperature (°C)
800
1000
Fig. 4. Reduction factor of elastic modulus and yield strength of steel.
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Table 1 Material properties of steel and fire protection. Symbol Value
Unit
Steel ls Fire protection lc
45 0.1
W/(m 1C)
Specific heat
Steel Cs Fire protection Cc
600 0.22
J/(kg 1C)
Unit mass
Steel rs Fire protection rc
7850 500
kg/m3
Thermal expansion coefficient Steel
aT
1.4 10
ISO standard temperature
800
Thermal conductivity
Temperature (°C)
Parameters
1000
Temperature of steel column in the portion
600
with damaged FRC
400 200
5
Temperature of steel column in the portion with integrated FRC
m/(m 1C)
0 0
10
20
4.2. Temperatures of the example
ð16Þ
1.0
where t is the duration after start of fire (min). The temperature of the steel column subjected to the fire can be predicted by the following incremental equation [13] as:
0.8
Ts ðt þ DtÞ ¼
½Tg ðtÞ Ts ðtÞDt þ Ts ðtÞ
0.4
where Ts is the temperature of steel; Dt is the increment of time; and B is the general heat transfer coefficient, which can be determined by: If without fire protection
0.2
B ¼ ðac þ ar Þ
1þ
μ = 0.4 μ = 0.5
0.0
ð18Þ
1
lc Fc
cc rc dc Fc 2cs rs V
dc V
200
400 600 Temperature (°C)
800
1000
Fig. 6. Load bearing capacity–temperature curve of the example steel column (the curves are essentially coincident).
and if with non-intumescent fire protection B¼
μ = 0.3
0
F V
60
μ = 0.2
0.6
ð17Þ
cs rs
50
μ = 0.1
ϕT
B
40
Fig. 5. Temperature–time curve of the example steel column.
The ISO standard temperature–time curve is employed to represent the temperature of the fire around the steel column, which is expressed by Tg ¼ 20 þ 345 logð8t þ 1Þ
30 Time(min)
ð19Þ
where ac is convective heat transfer coefficient, W/(m2 1C); ar is radiative heat transfer coefficient, W/(m2 1C); F is the area per unit length of column, m2/m; V is the volume per unit length of column, m3/m; dc is the thickness of fire protection, m; and Fc is the interior area of fire protection per unit length column, m2/m. With intumescent fire protection, the fire protection will expanded at high temperature and the thermal properties will change. So long as dc and cc in Eq. (19) are replaced with the thickness and thermal conductivity of the fire protection after expansion, the temperature may be obtained. However, the degree of expansion of the intumescent fire protection is influenced greatly by environment and the temperature elevation should be determined experimentally. The increase in the temperature of the example subjected to the ISO fire is calculated and illustrated in Fig. 5.
where Pdcr is the ultimate load bearing capacity of the steel columns with partial fire protection loss and fy20 is the yield strength of steel at normal temperature. When e ¼ 1%, the curve of the load bearing capacity coefficient with temperature is plotted in Fig. 6, with m defined in Eq. (4). As can be seen from Fig. 6, the load bearing capacity is reduced with increase in temperature. However, the length of damaged fire protection has very little influence on the load bearing capacity. 4.4. Axial force of the column The example column axially restrained with partial fire protection loss is also studied to look into the variation of the axial force in the column with the elevation of temperature. The relationship between the temperature of the column at the part with damaged fire protection and the axial force in the column with various restraint stiffness and damage length ratio is shown in Fig. 7, in which z is the axial restraint ratio of the column and is defined as
4.3. Ultimate load bearing capacity of the column
z ¼ kr =ðE20 A=lÞ
The load bearing capacity coefficient of a steel column with damaged fire protection is defined as
As is shown in Fig. 7, the axial force in the column with partial damage to the fire protection and axial restraint firstly increases when the temperature rises, then the axial force decreases to the initial level with the rise in temperature. This may be because the
jT ¼ Pdcr =ðfy20 AÞ
ð20Þ
ð21Þ
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1.30
PT/P20
ζ = 0.10, μ = 0.4
6
1000
T1 E1
ζ = 0.10, μ = 0.2
100
8
ζ = 0.05, μ = 0.4
1.20
100
P
ζ = 0.05, μ = 0.2
1.25
1093
8
1.15
Model 1 P = 500kN T1 = 600°C T2 = 20°C E1 = 103000MPa
T2 E2
1.05
4
2000
1.10
1.00 100
200 300 400 Temperature (°C)
500
Model 2 P = 500kN T1 = 20∼820°C T2 = 20°C E1 = 206000∼17490MPa
600
Fig. 7. Axial force in the example steel column.
T1
1000
0
E1
Fig. 8. Sketch of the FEM models.
Proposed model (P = 200KN)
4000
Length of column (mm)
elastic modulus and the yield strength of the steel decrease with the elevation of temperature and the flexure of the column is large enough to release the thermal expansion of the column and then to release the axial force. In addition, at the same damage length of fire protection, for example, m ¼ 0.2, the axial restraint stiffness can significantly increase the degree of the elevation of axial force. Similarly, at the same stiffness of axial restraint, for example, x ¼ 0.10, the length of damage to fire protection can also significantly increase the degree of the elevation of axial force. When the damage to the fire protection is at the center of the column, the procedures to obtain the ultimate load carrying capacity and the axial force in the column are quite similar to the situation in which the fire protection damage is at the ends of the column.
E2 = 206000MPa
FEM (P = 200KN)
3000
Proposed model (P = 500KN) FEM (P = 500KN)
2000
1000
5. FEM validation
0 0
1
2 3 4 5 Deflection (mm)
6
7
Fig. 9. Deflection comparison of the steel column between ANSYS and calculated (the curves are essentially coincident).
12
Displacement (mm)
In order to verify the analytical model proposed in the paper, two FEM models are presented by employing ANSYS software. The element type used in the analysis is BEAM3, which is a uniaxial element with tension, compression, and bending capabilities. The element has three degrees of freedom at each node: translations in the nodal x and y directions and rotation about the nodal z-axis. The length of element is 50 mm. the material of steel at elevated temperature is adopted as recommend in Chinese Technical Code on Fire Safety of Steel Building Structures [13]. One model (Model 1) is performed to verify the flexure shape of steel column, which is the most important parameter in defining the ultimate load capacity of the steel column with partial fire protection damage. In this model, the column section is H100 100 6 8; l ¼ 4 m; m ¼ 0.25; e ¼ 0.1%; P ¼ 500 KN. The temperature of the column with damaged fire protection is 600 1C. The sketch of the model is shown in Fig. 8. The results obtained by ANSYS and the proposed approach in this paper are plotted in Fig. 9. It was found that there is a good agreement between the FEM and the model of this paper. Another FEM model (Model 2) is performed to validate the axial displacement of the steel columns with rise in temperature. The column section is H100 100 6 8. l ¼ 4 m, P ¼ 500 KN, m ¼ 0.25. The thermal expansion coefficient is 1.4 105. The sketch of this model is also shown in Fig. 8. The comparisons of the development of the axial displacement obtained by the proposed model in the paper and FEM method are shown in Fig. 10. It can be seen that the axial displacement predictions by the method proposed in this paper agree well with that obtained by the FEM.
8 4 Results of ANSYS Results of Calculated
0 -4 -8 0
150
300
450 600 Temperature (°C)
750
900
Fig. 10. Axial displacement comparison of the steel column between ANSYS and calculate.
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6. Concluding remarks
References
Using the experimental approach to predict the fire resistance of a steel column is both expensive and time consuming. As an alternative, a theoretical analysis that can be performed simply is a convenient tool for engineers to quickly assess the column fire resistance. In this paper, an analytical model is presented for determining the behavior of restrained steel columns in fire with partially damaged fire protection. The ultimate load bearing capacity and critical temperature of steel columns may be predicted by employing the model proposed. The model can be used to evaluate the fire resistance and predict the critical temperature of steel columns with partial fire protection damage. Through numerical study on a typical steel column with partial fire protection loss, the following conclusions are drawn:
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(1) The load bearing capacity of steel columns with fixed ends exposed to fire is reduced at a given temperature with the damage of fire protection. (2) The length of damaged fire protection has little influence on the load bearing capacity of the steel columns fixed at two ends. (3) For axially restrained steel columns, the length of damaged fire protection has significant influence on the increase of axial force of steel columns in fire.
Acknowledgments The support of the Natural Science Foundation of China for Innovative Research Groups (Grant No. 50621062), Key Research Project (Grant No. 50738005) and Scientific Research Foundation of Chongqing University for Recruited Scholars for the study reported in this paper is gratefully acknowledged.