ARTICLE IN PRESS

Fire Safety Journal 44 (2009) 212–218

Contents lists available at ScienceDirect

Fire Safety Journal

0379-71

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/firesaf

Behavior of axially-and-rotationally restrained concrete columns with‘+’-shaped cross section and subjected to fire

Bo Wu a,�, Yu-ye Xu b

a State Key Laboratory of Subtropical Building Science, Department of Civil Engineering, South China University of Technology, Guangdong Province,

Guangzhou 510640, People’s Republic of Chinab College of Civil Engineering, Huaqiao Universtiy, Quanzhou, Fujian Province 362021, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 4 January 2007

Received in revised form

12 June 2008

Accepted 17 June 2008Available online 27 August 2008

Keywords:

Reinforced concrete

Column

‘+’-shaped cross section

Axial restraint ratio

Rotational restraint ratio

Fire

12/$ - see front matter & 2008 Elsevier Ltd. A

016/j.firesaf.2008.07.003

esponding author.

ail addresses: bowu@scut.edu.cn (B. Wu), yuy

a b s t r a c t

Most of the previous investigations studied the structural behavior of concrete columns with ‘+’-shaped

cross section at room temperature, but the behavior of such columns during a fire and the effect of

boundary conditions on the mechanical properties of the heated columns have seldom been examined.

In this paper, the influence of axial-and-rotational restraint on the behavior of heated concrete columns

with ‘+’-shaped cross section is investigated. A self-developed finite element program RCSSCF is applied

in this study. The development of column internal forces as well as deflections at the mid-height of

concrete columns are discussed. Simulation results show that: (1) axial restraint can induce significant

additional axial forces in concrete columns with ‘+’-shaped cross section and subjected to fire, and the

additional axial forces in strongly restrained concrete columns during a fire can reach approximately

65–70% of the axial forces in columns at room temperature; (2) the fire resistance of concrete columns

without rotational restraint decreases significantly with an increase of load eccentricity ratio, but that

with rotational restraint is influenced lightly by load eccentricity ratio; and (3) for columns with

different non-zero rotational restraint ratios, the internal axial forces, and the internal moments and

deflections at the mid-height of these columns appear to follow common trends.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Fig. 1 shows a typical concrete column with ‘+’-shaped crosssection. Concrete columns with ‘+’-shaped cross section aredefined as columns with a ratio of sectional height to sectionalthickness being less than 4.0 [1]. In general, the sectionalthickness is 200–300 mm, and the sectional height is between500 and 1200 mm.

In the past two decades, the construction industry in China hasshown increasing interest in the use of concrete columns with‘+’-shaped cross section. Comparing with concrete columns withrectangular cross section, the columns with ‘+’-shaped crosssection, of which the sectional thickness is equal to the thicknessof adjacent walls, do not extend beyond the walls, enlarging theuseful floor areas. Generally, concrete columns with ‘+’-shapedcross section are used in residential buildings.

Comparing with concrete columns with rectangular crosssection, the columns with ‘+’-shaped cross section have thinnersectional thickness and larger surface area. Hence, the tempera-tures in columns with ‘+’-shaped cross section during a fire rise

ll rights reserved.

exu@hqu.edu.cn (Y.-y. Xu).

more quickly than those in columns with rectangular crosssection, and the fire resistance of columns with ‘+’-shaped crosssection will be less than that of columns with rectangularcross section. However, the behavior of concrete columns with‘+’-shaped cross section in fire situation has seldom beeninvestigated.

In the past two decades, the development of numericaltechniques and an enhanced knowledge of the thermal andmechanical properties of concretes at elevated temperatures havemade it possible to determine the performance of concretecolumns in fire by calculation [2–4]. Compared to experimentaltests that have been used to determine the performance ofconcrete columns subjected to fire, calculation is far less costlyand time-consuming. Although the behavior of concrete columnsin fire has been investigated by experiment/calculation for someyears now [5–8], very little research has been done on the effect ofaxial and rotational restraint on the behavior of concrete columnsin fire situations [9]. When a column in a frame is exposed to fire,the strong interaction between the heated column and itsadjoining cool structure should be treated carefully. Because thecool adjoining structure restrains the thermal expansion as well asthe end rotations of the heated column, both ends of the heatedcolumn are axially and rotationally restrained. Consequently,additional axial load and moment are generated in the heated

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B. Wu, Y.-y. Xu / Fire Safety Journal 44 (2009) 212–218 213

column. In other words, the internal column force does notremain constant, but varies with temperature and the degree ofrestraint [10].

Studies undertaken by Lie and Lin [11] indicated that restraintagainst thermal expansion of reinforced concrete (RC) columnshas little influence on the fire performance of RC columns. It statesthat axial restraint is probably beneficial for the performance,because the surrounding structure can redistribute some of theload when the column contracts. Ali et al. [8] proposed adeveloped method to calculate the axial restraint forces generatedin axially restrained concrete columns in fire, and a compa-rison between the theoretical and the experimental valuesof restraint forces generated in 54 high and normal strengthconcrete columns previously tested at Ulster University waspresented. It was found that the additional axial compressionforces can speed up the failure of concrete columns and reducetheir fire resistance. Twelve high-strength concrete columnswith different load ratios and restraining levels were tested infire by Benmarce and Guenfoud [12]. It was shown that themaximum value of axial deformation is dependent on the loadthat has been applied during the test. Compared with the verylittle research on restrained concrete columns, a little moreresearch has been done on the effect of axial and rotationalrestraint on the behavior of steel columns in fire. Neves used FE

A-A

Thickness b

Hei

ght b

Thic

knes

s a

Height a

AA

Fig. 1. A typical concrete column with ‘+’-shaped cross section.

Table 1Comparison of calculated fire resistance with that measured in tests [20]

Specimen Shape of cross

section

Cubic compressive

strength (MPa)

Reinforcement

ratio (%)

Z1 L 37.6

1.51

Z2 L 37.6

Z3 L 37.6

Z4 L 37.6

Z5 T 37.6

Z6 T 35.2

Z7 T 35.2

Z8 T 37.6

Z9 + 35.2

Z10 + 37.6

Z11 + 37.6

Z12 Square 35.2

program to study the axial restraint effect of steel columns [13],which was also later investigated by Ali et al. [14] throughexperiments. The action of rotational restraint has been investi-gated numerically by Valente and Neves [15], and experimentallyby Ali and O’Connor [16].

To develop guidelines for the fire resistance design of RCcolumns with ‘+’-shaped cross section, both experimental andtheoretical studies are being carried out at the South ChinaUniversity of Technology in China. In this paper, the fireperformance of axially-and-rotationally restrained concrete col-umns with ‘+’-shaped cross section is investigated by calculation.The developments of column internal forces as well as deflectionsat the mid-height of heated columns are discussed.

2. Computational tool

A self-developed finite element program named RCSSCF [17],modeling using a three-dimensional beam element, is used in thisstudy. The transient thermal fields in the ‘+’-shaped columnssubjected to fire are calculated by the finite element method,combining with the finite difference method [18]. The initialcolumn imperfections, which are introduced to account forvarious secondary effects, are taken into account in the program.In the calculations, it is assumed that: (a) for the case that the loadis applied at the sectional centroid, the direction of the initialgeometrical imperfection is on the y-axis; and (b) for the case thatthe load is not applied at the sectional centroid, the direction ofthe initial geometrical imperfection is opposite to the loadingdirection, which is from the sectional centroid to the loadingpoint. Moreover, the mechanical properties of steel bars andconcrete under high temperature given by Guo and Shi [19] areused. The creep of steel bars is ignored.

The RCSSCF has been verified with fire tests of 12 full-scaleconcrete columns (i.e., four concrete columns with L-shaped crosssection, four concrete columns with T-shaped cross section, threeconcrete columns with ‘+’-shaped cross section, and one concretecolumn with square cross section) with fixed–fixed boundaryconditions [20]. Table 1 shows the comparison of the calculatedfire resistance of these concrete columns with that measured inthe tests. The simulation results are in good agreement with theexperimental results on the whole.

3. Scope of the analysis

This paper studies the fire performance of a concrete columnwith ‘+’-shaped cross section in a non-sway space frame (Fig. 2), in

Axial load ratio Fire resistance (min)

Measured Calculated Error (%)

0.35 175 169 �3.4

0.45 147 140 �4.8

0.55 102 123 20.6

0.55 207 227 9.7

0.35 179 156 �12.8

0.45 148 133 �10.1

0.55 124 125 0.8

0.55 190 233 22.6

0.35 245 189 �24.1

0.45 165 163 �1.2

0.55 116 127 9.5

0.55 169 157 �7.1

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B. Wu, Y.-y. Xu / Fire Safety Journal 44 (2009) 212–218214

which only columns ‘AB’, ‘CD’, ‘EF’, ‘GH’ and beams ‘BD’, ‘BF’, ‘DH’,‘FH’ are exposed to a fire whose temperature follows the ISO834standard temperature-time curve, while the rest of the membersremain cool. For simplicity, all columns in Fig. 2 are assumed tohave the same second moment of area Icz about z–z-axis and thesame second moment of area Icy about y–y-axis, while all beams inthe y-direction in Fig. 2 are of the same second moment of area Ibz

about z–z-axis, and all beams in the z-direction are of the samesecond moment of area Iby about y–y-axis.

An isolated column model with axial and rotational springs(Fig. 3) has been proposed by Huang and Tan [10] to investigatethe structural responses of the heated column ‘AB’. Comparingwith the original isolated column model with one rotationalspring about the z–z-axis proposed by Huang and Tan, tworotational springs about the y–y and z–z axes are considered inthis study. The initial column imperfection, which is introduced toaccount for various secondary effects, is assumed to be a singlesine curve with a mid-height deflection equal to one-thousandthof the column height [21]. In Fig. 3, N is the external axial load

h 3

l3y

D

C

B

A

Fi

z

x

h1

h2

h3

h4

h5

l5zl3z l4zl2zl1z

B D

A C

Fire

Fig. 2. A compartment fire within a space frame: (a) A

H

A

B

x

( )y z

lkN

kBrz (kBry) MBz (MBy)

MAz (MAy)kArz (kAry)

A A

200500

�0

Fig. 3. Isolated column model with one axial sp

applied to column ‘AB’ at the beginning of heating; MBz and MAz

are, respectively, the external moments about z–z-axis at thebeginning of heating; MBy and MAy are, respectively, the external

moments about y–y-axis at the beginning of heating; d0 is thedeflection at the mid-height of column ‘AB’ at the beginning ofheating; and e0 and a are, respectively, the load eccentricity andthe load angle. The load angle is positive counter-clockwise fromthe z-axis. In this study, the external moments MBz(MBy) andMAz(MAy) are arranged such that this column bends in a singlecurvature, which is the most unfavourable condition [10]. Forsimplicity, |MBz| in Fig. 3 is assumed to be the same as |MAz|, while|MBy| is the same as |MAy|. Obviously, at the beginning of heating

we have jNe0j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2

Bz þM2By

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2

Az þM2Ay

q.

The axial and rotational elastic springs in Fig. 3 represent theaxial and rotational restraints contributed by the adjoiningstructure, respectively. All springs are assumed to be able tomaintain their stiffnesses during the whole heating process andare not affected by heat [10]. To quantify the respective restraint

l3z

F

Ere

yG

H

h1

h2

h3

h4

h5

l5yl3y l4yl2yl1y

B F

A E

Fire

compartment, (b) Z-direction and (c) Y-direction.

A-A

S

Loading point of N

oe0y

y

z

e0z

ae0

200

500

ring and two rotational springs (unit: mm).

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B. Wu, Y.-y. Xu / Fire Safety Journal 44 (2009) 212–218 215

levels, the axial restraint ratio bl and the rotational restraint ratiosbArz, bAry, bBrz and bBry are defined as follows:

bl ¼kl

E0Ac=H(1)

bArz ¼kArz

4E0Icz=H; bAry ¼

kAry

4E0Icy=H(2)

bBrz ¼kBrz

4E0Icz=H; bBry ¼

kBry

4E0Icy=H(3)

in which kl is the axial spring stiffness; kArz and kBrz are therotational spring stiffnesses about z–z-axis; kAry and kBry are therotational spring stiffnesses about y–y-axis; H is the height ofcolumn ‘AB’ (i.e., h3 in Fig. 2); E0 is the elastic modulus of concreteat room temperature; Ac is the cross-sectional area of column ‘AB’;E0Ac/H represents the elastic axial stiffness of column ‘AB’ at roomtemperature; and 4E0Icz/H and 4E0Icy/H are the elastic flexuralstiffnesses of column ‘AB’ at room temperature about z–z andy–y axes, respectively. Here kl, kArz, kBrz, kAry and kBry can beapproximately written as follows [16]:

kl ¼ ðmþ 1Þ �12E0Iby

l32z

þ12E0Ibz

l32y

!(4)

kArz ¼4E0Icz

h2þ

4E0Ibz

l2yþ

4E0Ibz

l3y(5)

kAry ¼4E0Icy

h2þ

4E0Iby

l2zþ

4E0Iby

l3z(6)

kBrz ¼4E0Icz

h4þ

4E0Ibz

l2y(7)

kBry ¼4E0Icy

h4þ

4E0Iby

l2z(8)

where m is the number of building stories over the heatedcompartment (e.g., m ¼ 2 in Fig. 2); and h2, h4, l2z, l3z, l2y and l3y

are described in Fig. 2. To be conservative, the contributions ofheated beams ‘BF’ and ‘BD’ are neglected in the calculation of kBrz

and kBry [21].For simplicity, only the case that bArz ¼ bAry ¼ bBrz ¼ bBry ¼ br

is considered in this study. In the following calculations, thearrangement of longitudinal reinforcements and the dimensionsof column cross section are given in Fig. 3, respectively; theprismatic compressive strength of concrete fc, the elastic modulusof concrete E0, and the yield strength of main reinforcements are,respectively, taken as 30 MPa, 3.0�104 N/mm2, and 375 MPa atroom temperature; the steel ratio (i.e., ratio of longitudinalreinforcement to cross-sectional area of the column) and theclear concrete cover the longitudinal reinforcements are 1.5% and30 mm, respectively; bl takes on the values 0.0, 0.01, 0.025, 0.05,0.10 and 0.25; br takes on the values 0.0, 0.5, 1.0, 2.0 and 10.0.Moreover, it is assumed that the column ‘AB’ is exposed to a fireon all faces, whose temperature follows the ISO834 standardtemperature-time curve.

4. Effect of axial restraint

Fig. 4 shows the variation of the column internal axial force P

with the time of exposure to fire. The fire resistance of the columnis defined as the time at which the column fails in the standardfire test. In the self-developed FE program, the column isconsidered to fail when the program is not able to convergeanymore. The out-of-balance load vector is used in the program to

check for convergence. The fire resistance of column ‘AB’ and theaxial deformation at this failure point, the expansion of which ispositive, are also given in Fig. 4, respectively. If the fire resistanceof the restrained column is 4300 min, the column is considered tobe safe enough in a fire, and it is meaningless to continue thecalculation. Therefore, the analysis beyond 300 min is interruptedartificially in this paper, and only the axial deformation at 300 minis given. In Fig. 4, n is the axial load ratio of column ‘AB’ at thebeginning of heating, and e ¼ e0/ra is the load eccentricity ratio.Here ra is the radius of gyration of the column cross section. Thefollowing can be seen from Fig. 4:

1.

The fire resistance of column ‘AB’, with a small value of n anda large value of e but without rotational restraint (e.g.,Fig. 4(c–e)), decreases with an increase of axial restraint ratiobl, due to the increase of the column internal axial force P

during heating process. The additional axial forces in stronglyrestrained concrete columns with ‘+’-shaped cross sectionduring a fire can reach approximately 65–70% of the axialforces in columns at room temperature.

2.

The fire resistance of concentrically loaded column ‘AB’, with alarge value of n but without rotational restraint (e.g., Fig. 4(b)),increases with an increase of axial restraint ratio bl. This is dueto the expansion of the ‘+’-shaped column with a large value ofn occurring only at the early stage of the heating process, andthen the column becomes shorter and shorter. In this way, thecolumn internal axial force P decreases with an increasing ofaxial restraint ratio bl at the later stage of the heating process,due to the transfer of external axial load from column ‘AB’to the axial spring, this leading to an increase of the fireresistance of column ‘AB’.

3.

The expansion of the concentrically loaded column ‘AB’, with asmall value of n but without rotational restraint, is significantduring the heating process (e.g., the axial deformation at theaforementioned failure point is still positive in Fig. 4(a)). In thisway, the column internal axial force P increases with anincrease of axial restraint ratio bl. For the case of bl ¼ 0.25, theadditional axial force in column ‘AB’ is almost 50% of thecolumn internal axial force at the beginning of heating. Thisleads to a decrease of the fire resistance of column ‘AB’ with anincreasing of axial restraint ratio bl.

4.

For ‘+’-shaped columns with identical axial restraint ratio bl butdifferent column heights, although the axial spring stiffnesscorresponding to a slender column is less than that correspond-ing to a stocky column, due to the less elastic axial stiffness ofthe slender column at room temperature, the thermal expansionof the slender column is larger than that of the stocky column.This leads to a little more additional axial force in the slendercolumn in comparison with that in the stocky column for thesame heating time (e.g., Fig. 4(c) and (d)).

5.

With an increase in the load eccentricity ratio, the fireresistance of the axially restrained ‘+’-shaped columns with alow axial load ratio and without rotational restraint decreases(e.g., Fig. 4(a) and (c)).

6.

For columns with rotational restraint, the time that theadditional column axial force reduces to zero (i.e., the columninternal axial force reduces to a value of that at the beginningof heating) is almost the same for ‘+’-shaped columns withdifferent axial restraint ratios (e.g., Fig. 4(f)).

5. Effect of rotational restraint

For given values of n ¼ 0.2, H ¼ 4.5 m, a ¼ �901 and bl ¼ 0.05,the fire resistance of ‘+’-shaped columns with different loadeccentricity ratios is shown in Table 2. It can be seen from Table 2

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Table 2Fire resistance of ‘+’-shaped columns with different load eccentricity ratios (unit:

min)

Rotational restraint ratio br

0.0 0.5 1.0 2.0 10.0

e ¼ 0.0 120 249 249 249 250

e ¼ 0.5 68 243 245 246 247

e ¼ 1.0 43 239 242 243 244

Fig. 4. Variation of column internal axial force with the time of exposure to fire: (a) n ¼ 0.2, H ¼ 3.0 m, e ¼ 0, a ¼ �901, br ¼ 0. (b) n ¼ 0.5, H ¼ 3.0 m, e ¼ 0, a ¼ �901,

br ¼ 0. (c) n ¼ 0.2, H ¼ 3.0 m, e ¼ 1.0, a ¼ �901, br ¼ 0. (d) n ¼ 0.2, H ¼ 4.5 m, e ¼ 1.0, a ¼ �901, br ¼ 0. (e) n ¼ 0.2, H ¼ 3.0 m, e ¼ 1.0, a ¼ �451, br ¼ 0. (f) n ¼ 0.2,

H ¼ 3.0 m, e ¼ 1.0, a ¼ �901, br ¼ 0.

B. Wu, Y.-y. Xu / Fire Safety Journal 44 (2009) 212–218216

that: (1) the rotational restraint is of great benefit to the fireresistance of column ‘AB’; (2) the fire resistance of column ‘AB’without rotational restraint decreases significantly with anincreasing of load eccentricity ratio; and (3) the effect of loadeccentricity ratio on the fire resistance of column ‘AB’ withrotational restraint is limited.

Let Mmid and Umid be, respectively, the internal moment anddeflection at the mid-height of column ‘AB’ in the direction of O–Sin Fig. 3. Fig. 5 shows the variation of the column internal axialforce P, internal moment Mmid, and deflection Umid with the timeof exposure to fire. In Fig. 5, h is the height of column cross section(i.e., h ¼ 500 mm in Fig. 3). The following can be seen from Fig. 5:

1.

The internal moment at the mid-height of column ‘AB’ withoutrotational restraint increases with an increase of the heatingtime, but the internal moment at the mid-height of column ‘AB’with rotational restraint decreases sharply with an increase ofthe heating time first, leading to a significant improvement ofthe fire resistance of the eccentrically loaded column.

2.

With an increase of the heating time, the internal axial force in‘+’-shaped column without rotational restraint increasesmonotonically, but the internal axial forces in ‘+’-shapedcolumns with rotational restraint increase first and then dropsharply. This is probably because the fire resistance of the

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Fig. 5. Variation of the column internal axial force P, internal moment Mmid, and deflection Umid with the time of exposure to fire: (a) n ¼ 0.2, H ¼ 3.0 m, e ¼ 1.0, a ¼ �901,

bl ¼ 0.01. (b) n ¼ 0.2, H ¼ 3.0 m, e ¼ 1.0, a ¼ �901, bl ¼ 0.05. (c) n ¼ 0.2, H ¼ 4.5 m, e ¼ 1.0, a ¼ �901, bl ¼ 0.05.

B. Wu, Y.-y. Xu / Fire Safety Journal 44 (2009) 212–218 217

column without rotational restraint is much less than that ofthe columns with rotational restraint. At the same time, theadditional axial force in column ‘AB’ without rotationalrestraint is larger than that with rotational restraint.

3.

The increase of the deflection at the mid-height of column ‘AB’with rotational restraint is slower than that without rotationalrestraint.

4.

For concrete columns with different non-zero rotationalrestraint ratios, the internal axial force P, internal momentMmid, and deflection Umid appear to follow common trends,respectively. At the same time, the column internal momentMmid and deflection Umid decrease with an increasing of therotational restraint ratio br.

For given values of n ¼ 0.2, H ¼ 4.5 m, e ¼ 1.0, a ¼ �901,bl ¼ 0.05, and (a) br ¼ 0.0 and (b) br ¼ 1.0, the internal bending

moments along the height of column ‘AB’ corresponding todifferent heating times are shown in Fig. 6, where tR is the failuretime of the column. It can been seen from Fig. 6 that: (a) incomparison with the bending moments in the column at roomtemperature, the bending moments in the heated column withoutrotational restraint increase with the fire exposure time; (b) theadditional moments at the top and bottom ends of the rotationallyrestrained column increase with the fire exposure time in theopposite direction of the initial end moments; and (c) the bendingmoment at the mid-height of the heated column with rotationalrestraint reduces significantly at the early stage of the heatingprocess due to the additional end moments, and increase at thelater stage of the heating process due to a continual increase ofthe lateral deflection at the mid-height of the column. Obviously, theadditional end moments is the reason for the significant increase ofthe fire resistance of ‘+’-shaped columns with rotational restraint.

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0.05 0.06 0.07 0.08 0.09 0.10 -0.06 -0.04 -0.02 -0.00 -0.02 -0.04 -0.060

1

2

3

4

x/m

M /( fcAh )

t =0 mint =20 mintR=42 min

1

2

3

4

t=0 mint=60 mint=120 mint=180 mintR=242 min

x/m

M /( fcAh )

Fig. 6. Internal bending moments along the height of restrained columns: (a) br ¼ 0.0 and (b) br ¼ 1.0.

B. Wu, Y.-y. Xu / Fire Safety Journal 44 (2009) 212–218218

6. Conclusions

Based on the results obtained in this study, the followingconclusions can be drawn:

1.

The additional axial forces in strongly axially restrainedconcrete columns with ‘+’-shaped cross section during a firecan approximately reach 65–70% of the axial forces in columnsat room temperature.

2.

In the case of the concentrically loaded ‘+’-shaped columnwithout rotational restraint, the fire resistance increases withan increase in axial restraint ratio for a large value of n; but itdecreases with an increase in axial restraint ratio for a smallvalue of n.

3.

For ‘+’-shaped columns with identical axial restraint ratio butdifferent column heights, the additional axial force in a slendercolumn is a little larger than that in a stocky column for thesame heating time.

4.

The internal moment at the mid-height of ‘+’-shaped columnwithout rotational restraint increases with an increase ofheating time, but that with rotational restraint decreasessharply with an increasing of heating time first, leading to asignificant improvement of the fire resistance of eccentricallyloaded column.

5.

The increase of the deflection at the mid-height of ‘+’-shapedcolumn with rotational restraint is slower than that withoutrotational restraint.

6.

For ‘+’-shaped columns with different non-zero rotationalrestraint ratios, the internal axial forces, and the internalmoments and deflections at the mid-height of these columnsappear to follow common trends, respectively. At the sametime, the internal moment and deflection at the mid-height of‘+’-shaped column decrease with an increasing of the rota-tional restraint ratio.

Acknowledgements

The research reported was financially supported by theNational Natural Science Foundation of China (50738005,50478078), the Ministry of Education of PR of China (NCET-04-0819) and the Ministry of Science and Technology of PR of China(2006BAJ03A03-12).

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