response of reinforced concrete columns under fire-induced biaxial bending

610 ACI Structural Journal/September-October 2011

Title no. 108-S58

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

ACI Structural Journal, V. 108, No. 5, September-October 2011.MS No. S-2010-142.R3 received September 14, 2010, and reviewed under Institute

publication policies. Copyright © 2011, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the July-August 2012 ACI Structural Journal if the discussion is received by March 1, 2012.

Response of Reinforced Concrete Columns under

Fire-Induced Biaxial Bending

by Nikhil Raut and Venkatesh Kodur

Reinforced concrete (RC) columns, when exposed to fire, are often

subjected to biaxial bending arising from eccentricity in loading

or one-, two-, and three-sided exposure or due to nonuniform

spalling. The effect of such biaxial bending is often not taken into

consideration in evaluating the fire resistance of RC columns. In

this paper, an approach is presented for modeling the fire response

of RC columns under biaxial bending. This approach accounts

for high-temperature material properties, geometric and material

nonlinearity, fire-induced spalling, and restraint effects and can

be applied under realistic fire and loading scenarios. The validity

of the approach is established by comparing the predictions from

the model with results from full-scale fire resistance tests. The

model is applied to undertake a set of parametric studies and the

results show that the biaxial bending effect arising from one-sided,

two adjacent-sided, and three-sided fire exposure significantly

decreases the fire resistance of RC columns.

Keywords: biaxial bending; computer program; one-, two-, three-, and

four-sided exposure; fire resistance; high-strength concrete; numerical

model; reinforced concrete columns.

INTRODUCTION Reinforced concrete (RC) structural systems are quite

frequently used in high-rise buildings due to the number of advantages they provide over other construction materials. The provision of appropriate fire safety measures for structural members is an important aspect of design because fire represents one of the most severe environmental conditions to which structures may be subjected in their lifetime. RC columns form the main load-bearing component of a structural frame and hence have to satisfy fire resistance requirements specified in building codes. The basis for this requirement can be attributed to the fact that, when other measures for containing the fire fail, structural integrity is the last line of defense.

An RC column is often treated as a two-dimensional (2-D) planar structure and is therefore designed to resist bending moments acting in the plane of the frame. Whereas this idealization is true for peripheral columns, in an actual building framework, columns are frequently subjected to bending moments, acting in two perpendicular directions (biaxial bending) in addition to axial loading. The obvious example is a corner column in a space frame of a building. Also, under fire conditions, a column may be exposed to heat on one, two, three, or four sides. The one-, two-, or three-sided fire exposure causes the development of thermal gradients, which can result in uniaxial (one- or three-sided) or biaxial (two adjacent sides) bending of the column. Also, uneven fire-induced spalling in the concrete cross section and moments induced by thermal expansion of the floor plate during fire can cause bending (uniaxial or biaxial) of the column. Hence, the effect of biaxial bending can play a significant role in determining the fire response of RC columns.

The buildup of such biaxial bending effects in an RC column is illustrated in Fig. 1, where a structural frame in a building is exposed to fire. Figure 1(b) through (d) illustrates the development of thermal gradients in center (four-sided), peripheral (three-sided), and corner column (two adjacent side) fire exposure, respectively, under fire conditions. It can be seen that the occurrence of uniaxial bending (peripheral column) and biaxial bending (corner column) can be a common occurrence in most practical scenarios in buildings. The current fire resistance provisions in codes and standards are mostly based on standard fire tests with all four sides exposed to fire; hence, they may not be fully applicable to one-, two-, or three-sided fire exposure.

The presence of uniaxial bending induces an eccentricity in an originally axially loaded column due to the shift in the neutral axis, resulting from a degradation of the strength and stiffness properties of concrete and reinforcing steel. Thus, the column is subjected to an additional moment along with the axial load; this leads to a reduction in the fire resistance of the column. In the case of corner columns (biaxial bending), the neutral axis shifts and also rotates, thus inducing eccentricity along both the axes. This causes the column to experience additional moments along both the axes. The induced eccentricity in both cases also increases lateral and axial deformations due to P-d effects.

In addition to fire exposure, uneven fire-induced spalling can also produce significant uniaxial or biaxial bending on the column. Such fire-induced spalling may be more prevalent in columns made of high-strength concrete (HSC). Spalling is a phenomenon in which chunks of concrete fall off from the surface of a concrete structure when exposed to high and rapidly rising temperatures. The spalling in concrete is primarily dependent on the rate of temperature rise (fire scenario), permeability (typically related to strength and presence of fines such as silica fume), fire exposure, and other conditions (such as geometry and aggregate type). The development of differential thermal stresses can also significantly influence spalling in concrete.1 Spalling occurs when the pore pressure in a concrete member, due to evaporated moisture, exceeds the tensile strength of concrete. This phenomenon is schematically illustrated in Fig. 1(e), where the pore pressure development and loss of tensile strength is plotted as a function of temperature. Fire-induced spalling can be classified into three stages,

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gfa.com

ACI Structural Journal/September-October 2011 611

Fig. 1—Factors influencing behavior of RC column under fire. (Note: °C = [°F − 32]/1.8.)

namely early spalling, intermediate spalling, and late spalling.2 Early spalling results in the breakup of chunks of concrete and thus may have a detrimental effect on the fire resistance of concrete members. Early spalling results from high thermal gradients developing close to the surface of concrete members. Intermediate spalling results in the form of surface scaling but has a lesser effect on the overall fire resistance of RC members because it generally results in the breakup of thin layers of concrete. Late spalling results from a significant loss of strength and stiffness of concrete due to prolonged fire exposure, leading to the buildup of compressive stresses in the member.

The extent of spalling depends on a number of factors, including thermal gradients, moisture distribution, strength, and porosity of concrete. Depending on these factors, a column may undergo nonuniform spalling, thus causing the applied load to act eccentrically. For example, if the column is concentrically loaded and a piece of concrete disintegrates from one side of the column as a result of spalling, the applied load will then become eccentric due to the shift in the neutral axis. Therefore, one-, two-, or three-sided fire exposure and uneven spalling can produce uniaxial or biaxial bending effects in an RC column and should be accounted for in fire resistance predictions. This paper presents a numerical model for predicting the behavior of RC columns under fire when subjected to biaxial bending. The model is limited to element level analysis (columns) only and system level effects (such as beam-column interaction and floor-column interaction) are not considered. Thus, the presented numerical model can account for biaxial bending occurring due to eccentric loads, different side exposures, and uneven spalling.

RESEARCH SIGNIFICANCERC columns are often subjected to biaxial bending arising

from eccentricity in loading or one-, two-, or three-sided exposure or due to nonuniform spalling when exposed to fire. The current fire resistance provisions in codes and standards and available computer models do not account for fire-induced biaxial bending effects. The approach proposed in this paper can predict the fire response of RC columns subjected to biaxial bending. The results from numerical studies presented herein can be used to gauge the influence of various factors on the fire response of RC columns under biaxial bending conditions.

STATE-OF-THE-ARTA review of the literature indicates that there have been

a number of experimental and numerical studies on the fire performance of axially loaded RC columns; however, there have been only a few experimental and numerical studies undertaken on eccentrically loaded columns. Limited tests carried out at the National Research Council, Canada, by Lie and Woolerton3 and Kodur et al.4 studied the effect of load eccentricity on fire resistance. Also, there have been very few analytical studies on the fire response of RC columns

incorporating biaxial bending; the notable ones include those from Tan and Yao.5,6

Tan and Yao5,6 used a theoretical formulation for deriving the fire resistance of RC columns under uniaxial and biaxial bending due to one-, two-, three-, or four-sided exposure. The approach comprised of evaluating temperature-induced strength reduction factors for concrete and steel through numerical simulations using SAFIR, a special-purpose finite-element-based computer program for the fire analysis of structures. Then, the balanced failure point of the column section was determined so as to evaluate whether the eccentricity of the applied load is small or large. Finally, the failure load was determined through a trial-and-error approach. This method can be regarded as an extension of the ACI method for ultimate strength prediction at ambient temperature. Because SAFIR does not account for fire-induced spalling, a parameter was introduced to account for spalling. Although this method accounts for eccentricity arising from different fire exposures, it is validated for standard fire exposure only. Also, it predicts the load-carrying capacity of the column but does not generate detailed response parameters of the columns, such as axial deformation and temperature gradients as a function of fire exposure time.

The current provisions for the fire design of concrete structures specified in building codes and national standards provide fire resistance values based on concrete cover thickness and minimum sectional dimensions; however, the issue of bi-eccentric loading and biaxial bending arising from one-, two-, or three-sided exposure or fire-induced spalling is not addressed. Although ACI 216.1-077 provides different minimum column dimensions for columns exposed to fire from two parallel faces, it still doesn’t consider fire-induced bending effects, as the thermal gradients are symmetric about both the axes. Also, with the increasing use of newer concretes (such as HSC), which are more prone to fire-induced spalling, the existing fire resistance provisions may not be fully applicable.

NUMERICAL MODEL FOR PREDICTING FIRE RESPONSE

A numerical model based on a macroscopic finite element approach has been developed for tracing the fire response

Nikhil Raut is a PhD Candidate in the Department of Civil and Environmental

Engineering at Michigan State University, East Lansing, MI. He received his MTech

degree from the Indian Institute of Technology, Bombay, India, in 2005. His research

interests include fire resistance of reinforced concrete structures.

Venkatesh Kodur, FACI, is a University Distinguished Professor and Director of

the SAFE-D Center at Michigan State University. His research interests include the

evaluation of fire resistance of structural systems, the characterization of materials

under high temperature, and nonlinear design and analysis of structural systems.


2 Tk T Q c

t

∂∇ + = r

∂

where k is thermal conductivity; rc is heat capacity; T∞ is temperature; t is time; and Q is heat source defined as the amount of heat produced for a unit volume of the material.

The boundary conditions for the heat transfer analysis can be expressed as

( )y z t

T Tk n n h T T

y z∞

∂ ∂+ = − − ∂ ∂

where ht is the heat transfer coefficient; T∞ is fire or ambient temperature depending on the type of exposure; and ny and nz are components of the vector normal to the boundary in the plane of the cross section.

The finite element analysis is applied to solve Eq. (1) and compute the temperature distribution within the cross section of each segment.

(1)( )MT KT F t+ =

where

;

T TT

eA

T

e eA A

N N N NK k k dA N N ds

x x y y

M cNN dA F NQdA N T ds

Γ

∞Γ

∂ ∂ ∂ ∂= + + a∫ ∫ ∂ ∂ ∂ ∂

= r = + a∫ ∫ ∫

where N is the vector of shape functions; and a is the heat transfer coefficient depending on the boundary.

The resulting temperatures in the segment are used to evaluate fire-induced spalling in concrete through pore pressure calculations. Because the early spalling is more detrimental and results from the thermal gradient causing pore pressure development, it has been used in the development of the spalling sub-model. This hydrothermal analysis-based spalling sub-model uses the principles of mechanics and thermodynamics—including the conservation of mass of liquid water and water vapor—to predict fire-induced pore pressure in the concrete element.13 When the resulting pore pressure exceeds the tensile strength of concrete, spalling is said to occur in the element.

As part of strength analysis, M-k relationships are established as a function of time for all the segments and they are, in turn, used to trace the response of the column under fire conditions (Fig. 3). The effect of restraint and applied axial load is included in the generated M-k relationships through the total axial force computed from the axial deformation of the previous time step. Thus, the material and geometrical nonlinearities and the fire-induced restraint effect are implicitly accounted for in the analysis. Various strain components, including mechanical, thermal, and creep strains for both concrete and reinforcing steel and the transient strain in concrete are accounted for in the analysis based on the constitutive relationships proposed by Harmathy14,15 and Anderberg and Thelandersson16 for creep and transient strain, respectively.

Using the previously generated M-k relationships, the flexural rigidity of the segment at a particular time step is evaluated using the segmental stiffness approach and is used to calculate strength, axial, and lateral deformations of the column through the stiffness approach. As part of this segmental stiffness matrix, loading and displacement vectors are set up for each longitudinal segment. These segmental

of RC columns.8 This model has been extended to include the effect of biaxial bending resulting from the thermal gradients, exposure conditions, loading, or spalling. The macroscopic finite element model (FEM) uses a series of moment-curvature relationships for tracing the response of the column in the entire range of behavior from a linear elastic stage to the collapse stage under any given fire and loading scenario. The model incorporates a hydrothermal spalling sub-model to predict the occurrence of fire-induced spalling in HSC columns.

In the macroscopic FEM, an RC structural member is divided into a number of segments (Fig. 2(b)) along its length and the midsection of the segment is assumed to represent the behavior of the whole segment. This midsection is divided into elements, as seen in Fig. 2(d). The fire resistance analysis is carried out by incrementing time steps (refer to Fig. 2(e)). At each time step, the model performs analysis through three main steps, namely:

1. Establishing fire temperature due to fire exposure; 2. Carrying out heat transfer analysis to determine temperature

distribution across each segment. The temperature data are used to evaluate the pore pressure in concrete, which is then used to determine the extent of spalling within the cross section; and

3. Performing the strength analysis through the following three sub-steps: (a) calculating the fire-induced axial restraint force in the RC structural member; (b) generating the M-k relationship (using the axial force computed previously) for each segment; and (c) performing structural analysis of the overall member to compute lateral and axial deformations and internal forces.

The fire temperatures are calculated by assuming that the column is exposed to a fire whose temperature follows that of the standard fire exposure such as ASTM E119-079 or any other design fire scenario.10,11 For design fires, the time-temperature relationship specified in Eurocode 111 for typical fuel load and ventilation factors is built into the model. Also, to simulate hydrocarbon fire scenarios, the time-temperature relationship specified in ASTM E1529-9312 is incorporated into the model.

The thermal analysis is carried out for each segment in the column. The temperature is assumed to be uniform along the length of the segment and thus the calculations are performed for a unit length of each segment. The governing equation for heat transfer analysis is written as

Fig. 2—Discretization of column and flowchart of numerical model.


however, the bending and lateral deformations along the other direction also need to be considered. Thus, the 6 x 6 stiffness matrix in Eq. (4) is replaced by a 10 x 10 stiffness matrix (two additional degrees of freedom per node).

The five different degrees of freedom to be considered in a beam-column segment at each node are shown in Fig. 4. The segmental stiffness matrix [k(10 × 10)] is computed considering axial and bending deformations separately. The sectional properties are assumed to be constant within the segment at a given time step. Thus, elastic analysis can be performed. The segmental matrix is derived by separately solving the axial and flexural components using the force-displacement relations. The flexural components in the y- and z-directions and the three force components in the x-, y-, and z-directions are given by

(3); ; x y y z z z y

P EAu m EI u m EI u= − = − = −′ ′′ ′′

where P is the axial force acting on the column; my and mz are the moments about the y- and z-axis; EA is the axial rigidity of the segment; and EIy and EIz are the flexural rigidity of the segment about the y- and z-axes.

The final stiffness matrix for each segment can be written as

(4)( ) ( ) ( )( ) ( )5 5 5 5

10 105 5 5 5

AA AB

BA BB

k kk

k k

× × × = × ×

In Eq. (6), the stiffness matrixes KAA, KBB, and KAB are expressed as

matrixes (vector) are assembled in the form of a nonlinear global stiffness equation as

(2)( ){ } { } { } { } { }

{ } { }

; ;g geo f g f s

s geo

K K P K P P

P K

+ d = ∴ d = +

= − d

where Kg is the global stiffness matrix (computed from the M-k relationship); Kgeo is the geometric stiffness matrix; d is the nodal displacements; Pf is the equivalent nodal load vector due to applied loading; and Ps is the equivalent nodal load vector due to the P-d effect. At every time step, each segment of the column is checked for failure under thermal and strength failure limit states. In the strength limit state, the failure of an RC column is said to occur when the applied axial load (or moments) exceeds the load- (or moment-) carrying capacity of the column. The column is said to have failed in crushing if the curvature of the column exceeds the maximum curvature in the moment-curvature relationship (that is, corresponding to the ultimate strain in concrete) at that time step, or if the global stiffness matrix (elastic stiffness matrix-geometric stiffness matrix) becomes negative or singular, the column is said to have failed in buckling.

ACCOUNTING FOR BIAXIAL BENDINGThe aforementioned procedure is applicable for fire

resistance analysis under axial loading conditions. The solution to a biaxially loaded column (representing the case of a beam-column) requires due consideration of geometry, boundary conditions, and thermal gradients. The instability of the member arising from the magnification of the primary moments by the axial load acting on the laterally deflecting beam-column must also be considered. The effect of biaxial bending resulting from eccentric loading or temperature gradients from one-, two-, or three-sided exposure or due to uneven spalling on the column faces is incorporated through slight modification of the aforementioned steps. The effect of one-, two-, or three-sided fire exposure is incorporated through proper thermal boundary conditions in Eq. (2); thus, the relevant temperature profile across the column cross section is generated. Furthermore, to account for biaxial bending arising from uneven spalling or loading or one-, two-, or three-sided fire exposure, the 2-D stiffness matrix in Eq. (3) is replaced with a three-dimensional (3-D) matrix. The 2-D stiffness matrix Kg in Eq. (4) accounts for bending and lateral deformation in one plane only. Because (in the case of biaxial bending) the column bends in both directions,

Fig. 3—Variation of strain, stress, and internal forces in typical column cross section exposed to fire.

Fig. 4—Independent deformation of column segment.


Additionally, in the case of the biaxial bending problem, generating the stiffness matrix requires flexural rigidity along both axes. Therefore, the moment-curvature relationships for each segment of the column need to be developed along both axes (x-x and y-y). In the model, the moment-curvature relationship along the weaker axis of each segment is generated in the same way as that for the stronger axis, as described previously. Using these time-dependent M-k relationships, the flexural rigidity EI of each segment along both axes can be computed. The global stiffness matrix is generated by assembling the stiffness matrixes for each segment and the strength analysis is carried out.

MODEL VALIDATIONThere are no test data on RC columns under uniaxial or

biaxial bending arising from one-, two-, or three-sided fire exposure. However, limited fire test data are available for RC columns under uniaxial eccentric loading. The validity of the aforementioned macroscopic FEM is established by comparing predictions from the analysis with test data reported by Lie and Woolerton3 for normal-strength concrete (NSC) Column III3 and Kodur et al.4 for HSC Column HS2-8. The geometric and material properties of the tested columns are taken from the literature and summarized in Table 1. The eccentrically loaded columns were analyzed by exposing all four sides to the standard time-temperature curve specified in ASTM E119-07.9 Both columns had load applied eccentrically along one axis. The response of the columns was evaluated using the previously described model and fire resistance was computed based on the strength failure criterion. The columns were discretized, as shown in Fig. 2(b) and (d). The length of the column was discretized into 20 segments of 190 mm (7.5 in.) each. A mesh with an elemental size of 25 mm (1 in.) was used across the cross section for the fire resistance analysis based on the mesh sensitivity study presented in Appendix A.* Material properties from the ASCE manual17 were used in the analysis for NSC columns. Properties presented by Kodur et al.18

are used for HSC columns. The predicted results from the

(7)

k

EI k S

C k LS

EI k C

C k LS

EI k

AA

y y y

y y n

y y y

y y n

z z

[ ] =

− −

−( )− −

3 2

3

2 20 0 0

1

2 2

0SS

C k LS

EI k C

C k LS

EA

L

EI k C

z

z z z

z z z

z z z

z z

2 20

1

2 20

0 0 0 0

01

2

2

− −−

−( )− −

−−

zz

z z z

z z z z z

z z z

y y y

C k LS

EI k S k LC

C k LS

EI k C

( )− −

−( )− −

−( )−

2 20

2 20

1

2 2

2

CC k LS

EI k S k LC

C k LSy y n

y y y y y

y y y−

−( )− −

0 0 02 2

(8)

k

EI k S

C k LS

EI k C

C k LS

EI k

BB

y y y

y y n

y y y

y y n

z z

[ ] =

− −

−( )− −

3 2

3

2 20 0 0

1

2 2

0SS

C k LS

EI k C

C k LS

EA

L

EI k C

z

z z z

z z z

z z z

z z

2 20

1

2 20

0 0 0 0

01

2

2

− −−

−( )− −

−−

zz

z z z

z z z z z

z z z

y y y

C k LS

EI k S k LC

C k LS

EI k C

( )− −

−( )− −

−( )−

2 20

2 20

1

2 2

2

CC k LS

EI k S k LC

C k LSy y n

y y y y y

y y y−

−( )− −

0 0 02 2

(9)

k k

EI k S

C k LS

EI k C

C k LS

BA AB

y y y

y y n

y y y

y y

[ ] = [ ] =

−− −

−−( )

− −

3 2

2 20 0 0

1

2 2nn

z z z

z z z

z z z

z z z

EI k S

C k LS

EI k C

C k LS

EA

L

E

02 2

01

2 20

0 0 0 0

0

3 2

−− −

−( )− −

−

II k C

C k LS

EI k S k LC

C k LS

EI k

z z z

z z z

z z z z z

z z z

y y

2

2

1

2 20

2 20

−( )− −

−( )− −

−11

2 20 0 0

2 2

−( )− −

−( )− −

C

C k LS

EI k S k LC

C k LS

y

y y n

y y y y y

y y y

where Cy = cos kyL; Sy = sin kyL; Cz = cos kzL; and Sz = sin kzL

;y z

y z

P Pk k

EI EI= =

Property Column III33 Column HS2-84

Cross section, mm (in.) 305 x 305 (12 x 12) 406 x 406 (16 x 16)

Length, m (ft) 3.8 (12.5) 3.8 (12.5)

Support conditions Fixed-fixed Pinned-pinned

Reinforcement 4f 25 mm (No. 8) bars 8f 25 mm (No. 8) bars

f ′c, MPa (ksi) 34.8 (5) 127 (18.5)

fy, MPa (ksi) 444 (64.5) 400 (58)

Applied total load, kN (kips) 800 (180) 4981 (1120)

Load eccentricity, mm (in.) 25 (1) 25 (1)

Concrete cover thickness, mm (in.) 48 (2) 48 (2)

Aggregate type Siliceous Carbonate

Failure mode Test Buckling Crushing combined with buckling

Macroscopic FEM Buckling Buckling

Fire resistance, minutes Test 214 118

Macroscopic FEM 208 123

Table 1—Properties and results for RC columns used in validation study

*The Appendix is available at www.concrete.org in PDF format as an addendum to the published paper. It is also available in hard copy from ACI headquarters for a fee equal to the cost of reproduction plus handling at the time of the request.


Figure 5(d) shows the axial deformation comparisons for

HSC Column HS2-1. The deformation trend is similar to

that of the NSC column, with expansion in the early stages

followed by contraction of the column. There is lesser expansion

in the HSC columns due to a faster degradation of strength, thus

leading to high axial compression. Also, the applied load ratio is

higher for the HSC column, causing higher compression in the

column. The prediction from the model is in good agreement

with the measured axial deformation.

The fire resistance of the HSC column is higher than that of the

NSC column, but it should be noted that the HSC column is of a

larger cross-sectional area and is made of carbonate aggregate,

which helped in achieving higher fire resistance. Furthermore,

the “bent” ties (135 degrees) in the columns helped to minimize

the spalling beyond the concrete core.19

The results from the analysis were applied to determine the fire

resistance of the columns based on the strength failure criteria.

These fire resistance values are tabulated in Table 1. The fire

resistance of Column III3 obtained from the macroscopic FEM

based on strength failure criteria was 208 minutes, which is close

to the measured fire resistance in the test (214 minutes). The

predicted failure was through buckling and this is in agreement

with the observed failure pattern. Similarly, for Column HS2-8, the

predicted and measured fire resistance was 123 and 118 minutes,

respectively, based on the strength criteria. The model predicted

Column HSC2-8 to fail through buckling and the test reported a

combined crushing and buckling failure. It should be noted that the

model cannot predict combined failure modes because whichever

failure limit state is reached first is taken as failure. Comparing the

behavior of the predicted axial deformation for Column III3 to

the model predictions for the same column but with a concentric

load, it can be seen that the fire resistance increased from 208

to 360 minutes. Similarly, analyzing Column HSC2-8 under

a concentric load, it is seen that the fire resistance increased

to 346 minutes from 118 minutes under eccentricity. Thus,

eccentricity has a significant influence on fire resistance and

needs to be accounted for in the analysis.

analysis are compared to the measured values from the fire test in Fig. 5 and Table 1.

The predicted temperatures at three different cross-sectional locations (middepth, quarter-depth, and reinforcing bar) for Column III3 are compared with the measured values in Fig. 5(a). It can be seen that the measured temperature at the middepth of concrete is lower than that at the reinforcing bars; this is due to the lower thermal conductivity and higher thermal capacity of concrete, which slows down heat penetration to the inner layers of concrete. It can be seen that the predicted temperatures are generally in good agreement with the measured values. The temperatures follow an expected trend with higher values close to the exposed surface. The reinforcing bar temperature is higher than that at the concrete quarter-depth and middepth, as it is closest to the column surface.

Figure 5(b) shows axial deformations as a function of fire exposure time for NSC Column III3. It can be seen that model predictions are in close agreement with the measured axial deformations throughout the fire exposure time. The axial deformations increase initially and this is primarily due to thermal strain in the concrete and reinforcing bars, which results from increasing temperatures. This is followed by a decrease in the axial deformations (contraction) as a result of the loss of strength and stiffness properties in concrete and steel. The faster rate of increase in deformation prior to the failure of the column is mainly due to significant creep deformations resulting from the decreased stiffness at higher temperatures. Also, the increased P-d effect, resulting from bending of the column, contributes to a faster increase in deformations prior to the failure of the column.

Predictions from the fire resistance analysis of HSC Column HSC2-8 are compared with the test data in Fig. 5. Figure 5(c) shows that the predicted temperatures from the model are in good agreement with the measured temperature. Because the column is of HSC, the model predictions indicate the occurrence of fire-induced spalling at approximately 13 minutes, which is close to the time (10 minutes) observed in the test.4 This occurrence of spalling causes a sudden rise in temperatures (measured and predicted) between 15 and 25 minutes.

Fig. 5—Comparison of predicted and measured temperatures and deformations for RC columns.


with a compressive strength of 100 MPa (14.5 ksi). The columns

were reinforced with steel reinforcing bars with a yield strength

of 400 MPa (58 ksi). The NSC columns were assumed to

have a permeability on the order of 10–17, whereas the HSC

columns had a permeability of 10–19. The columns were

designed as per ACI 318-0520 specifications. The fire resistance

was determined as the time at which the column could not carry

the applied load.

Effect of load eccentricity The effect of eccentricity on fire resistance was studied

by analyzing five RC columns, namely Columns CE1,

CE2, CE3, CE4, and CE5 under ASTM E119-079 standard

fire exposure. All the columns had the same geometric

properties, as shown in Fig. 6(a) and (b). The axial load on

the column was assumed to act at an eccentricity (x- and

y-axes), as given in Table 2. When the load is eccentric,

there is additional moment acting on the column along with the

axial force. The fire resistance of an RC column decreases with

increased eccentricity. This can be attributed to the fact that,

as the eccentricity increases, the moment due to applied loading

also increases, thus subjecting the column to higher bending. This

leads to a faster degradation of column strength and stiffness due

to the increased P-d effect. For example, the fire resistance of

uni-eccentrically loaded Columns CE2 (e = 25 mm [1 in.]) and

CE3 (50 mm [2 in.]) is 185 and 134 minutes, respectively, which

is lower than that of axially loaded Column CE1 of 221 minutes.

When the load is applied eccentrically to both axes, a column

will be subjected to additional moments in both axes. Also,

because the neutral axis not only shifts but also rotates (refer

NUMERICAL STUDIESTo quantify the effect of biaxial bending on the fire response

of RC columns, a set of numerical studies was carried out using the aforementioned developed computer model. For the analysis, 22 RC columns, as listed in Table 2, were selected with varying parameters. The fire scenarios and geometric properties of the RC columns used in the analysis are also shown in Table 2. The variables included load eccentricity, type of fire exposure, fire scenario, cross-sectional sizes, load level, and concrete strength. The different loading conditions used in the case studies are represented in Fig. 6(c) and summarized in Table 2. All the NSC columns were assumed to be made of concrete with a compressive strength of 30 MPa (4.35 ksi), whereas the HSC columns were assumed to be made of concrete

Parameter Column Size, mm (in.)Applied load,

kN (kips)

Load eccentricity

Fire scenario Faces exposedFire resistance,

minutesX, mm (in.) Y, mm (in.)

Load eccentricity

CE1 305 (12) 800 (180) 0 (0) 0 (0) ASTM E119 4 221

CE2 305 (12) 800 (180) 25 (1) 0 (0) ASTM E119 4 185

CE3 305 (12) 800 (180) 50 (2) 0 (0) ASTM E119 4 134

CE4 305 (12) 800 (180) 25 (1) 25 (1) ASTM E119 4 122

CE5 305 (12) 800 (180) 50 (2) 50 (2) ASTM E119 4 88

Type of fire exposure

CE6 305 (12) 800 (180) 0 (0) 0 (0) ASTM E119 3 189

CE7 305 (12) 800 (180) 0 (0) 0 (0) ASTM E119 2 (adjacent) 138

CE8 305 (12) 800 (180) 0 (0) 0 (0) ASTM E119 2 (opposite) 245

CE9 305 (12) 800 (180) 0 (0) 0 (0) ASTM E119 1 198

Column size

CD1 406 (16) 1410 (317) 50 (2) 50 (2) ASTM E119 4 131

CD2 508 (20) 2210 (497) 50 (2) 50 (2) ASTM E119 4 190

CD3 610 (24) 3190 (717) 50 (2) 50 (2) ASTM E119 4 274

Fire severity

CF1 406 (16) 1410 (317) 50 (2) 50 (2) Hydrocarbon 4 98

CF2 406 (16) 1410 (317) 50 (2) 50 (2) External 4 155

CF3 406 (16) 1410 (317) 50 (2) 50 (2) Design fire I 4 No failure

CF4 406 (16) 1410 (317) 50 (2) 50 (2) Design fire II 4 No failure

Load level

CL1 406 (16) 940 (211) 50 (2) 50 (2) ASTM E119 4 211

CL2 406 (16) 1175 (264) 50 (2) 50 (2) ASTM E119 4 177

CL3 406 (16) 1645 (370) 50 (2) 50 (2) ASTM E119 4 102

CL4 406 (16) 1880 (423) 50 (2) 50 (2) ASTM E119 4 71

Concrete strength

HE1 406 (16) 2800 (630) 50 (2) 50 (2) ASTM E119 4 62

HE2 406 (16) 2800 (630) 0 (0) 0 (0) ASTM E119 2 (adjacent) 105

Note: All columns were reinforced with 4f 25 mm (No. 8) bars except Columns CD2 and CD3, which have 4f 32 mm (No. 10) bars.

Table 2—Properties and results for RC columns used in numerical study

Fig. 6—Details of column and loading cases used in numerical studies.


in Fig. 7(a). Figure 7(b) shows the progression of lateral

deformation at midheight with time for the five columns.

The lateral deformations of all the columns are zero until

the columns are in the expansion zone. This is as expected,

as an expanding column cannot deform laterally. The

lateral deformations start after the column begins to

contract. These deformations are very small until just

prior to failure and start to increase at a faster rate in the

last few minutes of fire exposure. This can be attributed to

the fact that the moments get magnified with time due to

the P-d effect and thus the lateral deformations increase

exponentially. Also, the creep strains play a higher role

and the concrete has lost a significant amount of strength.

Effect of type of fire exposureTo study the effect of different fire exposure scenarios,

Columns CE6, CE7, CE8, and CE9 were subject to three-,

two-(adjacent), two-(opposite), and one-sided fire exposure,

respectively, as shown in Fig. 6(c) and Table 2. Figure 7(c)

to Fig. 1(c)) due to biaxial bending, the sectional area available farther away from the neutral axis (necessary for higher moment capacity) is much less. This further reduces the fire resistance of the RC column. It can be seen that the fire resistance of bi-eccentrically loaded Column CE4 (122 minutes) is considerably lower than that of eccentrically (uniaxial) loaded Column CE2 (185 minutes), as Column CE4 has moments acting along both axes. Similar to the uniaxially eccentric loading case, fire resistance decreases with increased biaxial eccentricity as the moments acting on the column increase.

Figure 7(a) shows the variation of the axial deformation of Columns CE1 to CE5 as a function of fire exposure time for different loading cases. It can be seen that the columns with increased eccentricity have a shorter expansion zone (the time zone in which the column is expanding [elongating] due to increasing temperature and is seen as an increase in axial deformation) because the eccentricity causes bending of the column and the total expansion of the column is lower, as can be seen

Fig. 7—Effect of various parameters on RC columns under biaxial bending.


dimensions does not significantly increase the flexural strength of the column under biaxial bending.

Effect of fire severityTo illustrate the effects of the fire scenario combined with

biaxial bending, four RC columns (Columns CF1, CF2, CF3, and CF4) were analyzed under standard and design fire scenarios. The column is analyzed under three standard and two design fire exposures (refer to Fig. 7(e)) by subjecting them to a bi-eccentric load (50 mm [2 in.]). The three standard fire scenarios are ASTM E119-079 standard fire, hydrocarbon standard fire, and external standard fire.9,10,21 There is no decay phase in the standard fire curves but, in realistic fires, a decay phase always exists because the fuel or ventilation runs out, leading to compartment burnout. Thus, the remaining two fire scenarios are used to represent more realistic (design) fire exposure.

It can be seen from Table 2 and Fig. 7(f) that the lowest fire resistance of 98 minutes is obtained for Column CF1 exposed to the hydrocarbon fire as compared to 131 minutes for Column CD1 under the ASTM E119-079 standard fire exposure. This is due to the steep increase in temperature for the hydrocarbon fire, as shown in Fig. 7(e). The axial deformation of Column CF1 is also lower than that of Column CD1 in the expansion phase due to a higher loss in strength and stiffness in the column. Also, the fire resistance for the column exposed to the external fire is higher than that for the column under ASTM E119-079 standard fire. This is on the expected line and is due to the fact that the maximum fire temperature attained for external fire exposure is low when compared to that of ASTM E119-079 standard fire. The columns exposed to design fire scenarios (Columns CF3 and CF4) do not attain failure. In spite of the severe fire scenario (Fire I), no failure is attained in Column CF3.

It is evident from previous studies on axially loaded columns8 that the fire scenario has a significant effect on fire resistance. Comparing axially loaded columns8 to Columns CF1 to CF4, it can be seen that the reduction or increase in the fire resistance of a column due to the change in the fire scenario is more pronounced when the column is subjected to bending. For example, under axial loading, the fire resistance of Column C18 reduced by 13% when the fire scenario changed from ASTM E119-079 to hydrocarbon fire (Column C2). Under bi-eccentric loading, the reduction in fire resistance is approximately 25%. This can be attributed to the fact that a change in the time-temperature profile affects the temperature in the outer layer of concrete more; this concrete contributes more toward the flexural strength of the column.

Effect of load levelFour RC columns, namely Columns CL1, CL2, CL3, and

CL4 were analyzed to study the effect of load level on fire resistance under a 30, 40, 60, and 70% load ratio applied at an eccentricity of 50 mm (2 in.) along both axes. All columns were analyzed under ASTM E119-079 standard fire and the results are presented in Table 2. Figure 7(g) shows the axial deformation for these columns as compared to Column CD1, which is loaded to 50% of its capacity. As seen in Table 2, the fire resistance decreases with an increase in the load level. The drop in the fire resistance with an increased load level is higher than that observed from the literature3 (wherein a drop of approximately 10% in fire resistance per 25 mm [1 in.] of load eccentricity was recorded) under an axial loading case only. This can be attributed to the fact that an increase in load levels will cause an increase in moments acting on the column due to the eccentricity of the

shows the thermal gradients developed in Columns CE1 and

CE6 to CE9 after 45 minutes of fire exposure. These columns

were subject to an axial load but the bending occurred due

to nonuniform thermal gradients resulting from uneven fire

exposure. Column CE6 was subjected to uneven heating (three-

sided exposure) and thus thermal gradients developed along the

y-axis (refer to Fig. 7(c)). Thus, the neutral axis of the column

shifted toward the cooler side, inducing eccentricity in the

axially applied load. Thus, this column experienced uniaxial

bending and its fire resistance was lower (189 minutes) than

that of Column CE1 (221 minutes) with four-sided exposure.

Column CE7 is subjected to thermal gradients along both axes

and thus its neutral axis shifted toward the cooler section of the

column. Also, the neutral axis rotated due to the asymmetry in

the thermal gradients and thus the column experienced biaxial

bending. This resulted in much lower fire resistance

(138 minutes) as compared to Columns CE1 and CE6.

Column CE8 exhibited a higher fire resistance of 245 minutes. The

slower rise in temperature in the concrete due to exposure from

two opposite sides only and the symmetrical thermal gradient

(that is, no bending) helped the column attain higher fire

resistance as compared to Column CE1. Column CE9, on the

other hand, was exposed to fire on only one side, so its behavior

was similar to Column CE6 with the three sides exposed to

fire, as they both underwent uniaxial bending. The temperature

rise in Column CE9 was slower, however, as it was exposed on

only one side, leading to a slower loss of strength and stiffness

properties and thus its fire resistance was higher (198 minutes).

Effect of column sizeThree RC columns, namely Columns CD1, CD2, and CD3,

were analyzed to study the effect of the cross-sectional size

of columns on fire resistance. All the columns were analyzed

under ASTM E119-079 standard fire. Columns CD1, CD2,

and CD3 had cross-sectional sizes of 406, 508, and 610 mm

(16, 20, and 24 in.), respectively. All columns were assumed

to carry an axial load equal to 50% of their design capacity

applied at an eccentricity of 50 mm (2 in.) along both axes. As

seen in Table 2, Column CD1 (406 mm [16 in.]) has a higher

fire resistance of 131 minutes as compared to Column CE5

(305 mm [12 in.]), which has a fire resistance of 88 minutes.

This is as expected, as there is a larger mass of concrete

available to act as a heat sink in Column CD1; thus, the overall

temperature rise within the column is lower than that of

Column CE5. This results in a slower strength loss in concrete

and steel reinforcing bars and thus has a higher load-carrying

capacity. Thus, it is seen that the fire resistance increases with

the increase in cross-sectional size.

It should be noted that the thermal expansion increases

with column size (refer to Fig. 7(d)) due to a larger mass

of concrete being heated up before the column starts losing

a significant amount of strength and contracting. Figure 7(d)

shows the axial deformation for the three columns as

compared to Column CE5. It should be noted that under

biaxial bending, the increase in fire resistance with column

size is lower than the condition when the applied load is

only axial. This can be attributed to the fact that under biaxial

bending, the neutral axis not only moves down but also rotates.

So, the area available farther away from the neutral axis (which

is most effective in providing flexural strength) is less than

under axially loaded columns. Thus, the increase in the column


than axially loaded columns. The fire resistance drops by approximately 30 minutes for each 10% increase in the load ratio when the column undergoes biaxial bending.

5. Increased column size leads to higher fire resistance (40% for every 100 mm [4 in.] increase in size). The increase is smaller when columns are subjected to biaxial bending. Because the area farthest away from the central core is more efficient in resisting bending, the fire resistance increases exponentially with an increase in column size.

6. HSC columns are highly affected by biaxial bending due to the occurrence of spalling. The fire resistance drops by approximately 50% for the HSC columns due to the reduction of area contributing to flexural rigidity.

REFERENCES1. Khoury, G. A., “Concrete Spalling Assessment Methodologies

and Polypropylene Fibre Toxicity Analysis in Tunnel Fires,” Structural

Concrete, V. 9, No. 1, 2009, pp. 11-18.

2. Kodur, V. R., and Dwaikat, M. B., “Fire Induced Spalling in

Concrete—State-of-the-Art and Research Needs,” First International

Workshop on Concrete Spalling Due to Fire Exposure, V. 1, Leipzig,

Germany, 2009, 548 pp.

3. Lie, T. T., and Woolerton, J. L., “Fire Resistance of Reinforced

Concrete Columns—Test Results,” Internal Report No. 569, National

Research Council Canada, Ottawa, ON, Canada, 1988, pp. 1-302.

4. Kodur, V. R.; McGrath, R.; Leroux, P.; and Latour, J. C., “Experimental

Studies for Evaluating the Fire Endurance of High-Strength Concrete

Columns,” Internal Report No. 197, National Research Council Canada,

Ottawa, ON, Canada, 2005, pp. 1-153.

5. Tan, K. H., and Yao, Y., “Fire Resistance of Four-Face Heated

Reinforced Concrete Columns,” Journal of Structural Engineering, ASCE,

V. 129, No. 9, 2003, pp. 1220-1229.

6. Tan, K. H., and Yao, Y., “Fire Resistance of Reinforced Concrete

Columns Subjected to 1-, 2-, and 3-Face Heating,” Journal of Structural

Engineering, ASCE, V. 130, No. 11, 2004, pp. 1820-1828.

7. Joint ACI-TMS Committee 216, “Code Requirements for Determining

Fire Resistance of Concrete and Masonry Construction Assemblies (ACI 216.1-07),”

American Concrete Institute, Farmington Hills, MI, 2007, 28 pp.

8. Kodur, V.; Dwaikat, M.; and Raut, N., “Macroscopic FE Models for

Tracing the Fire Response of Reinforced Concrete Members,” Engineering

Structures Journal, V. 31, No. 10, 2009, pp. 2368-2379.

9. ASTM E119-07, “Standard Test Methods for Fire Tests of Building Construction

and Materials,” ASTM International, West Conshohocken, PA, 2007, 22 pp.

10. SFPE, “Fire Exposures to Structural Elements—Engineering Guide,”

Society of Fire Protection Engineers, Bethesda, MD, 2004, 150 pp.

11. EN 1991-1-2, “Eurocode 1: Actions on Structures. Part 1-2: General

Actions—Actions on Structures Exposed to Fire,” European Committee for

Standardization, Brussels, Belgium, 2002, pp. 1-59.

12. ASTM E1529-93, “Standard Test Method for Determining Effects

of Large Hydrocarbon Pool Fires on Structural Members and Assemblies,”

ASTM International, West Conshohocken, PA, 1993, 15 pp.

13. Dwaikat, M. B., and Kodur, V. K., “Hydrothermal Model for

Predicting Fire Induced Spalling in Concrete Structural Systems,” Fire

Safety Journal, V. 44, No. 3, 2009, pp. 425-434.

14. Harmathy, T. Z., “A Comprehensive Creep Model,” Journal of Basic

Engineering, V. 89, No. 2, 1967, pp. 496-502.

15. Harmathy, T. Z., Fire Safety Design and Concrete, Longman Scientific

and Technical, London, UK, 1993, pp. 1-272.

16. Anderberg, Y., and Thelandersson, S., “Stress and Deformation

Characteristics of Concrete at High Temperatures. Part 2—Experimental

Investigation and Material Behavior Model,” Lund Institute of Technology,

Lund, Sweden, 1976, pp. 1-85.

17. ASCE Manual No. 78, “Structural Fire Protection,” American Society

of Civil Engineers, New York, 1992, pp. 1-241.

18. Kodur, V. K. R.; Wang, T.; and Cheng, F., “Predicting the Fire

Resistance Behavior of High Strength Concrete Columns,” Cement and

Concrete Composites, V. 26, No. 2, 2004, pp. 141-153.

19. Kodur, V. K. R., “Fire Resistance Design Guidelines for High Strength

Concrete Columns,” Internal Report No. 46116, National Research Council

Canada, Ottawa, ON, Canada, 2003, pp. 1-11.

20. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-05) and Commentary,” American Concrete Institute,

Farmington Hills, MI, 2005, 467 pp.

21. Buchanan, A. H., Structural Design for Fire Safety, John Wiley &

Sons, Inc., Chichester, UK, 2002, 444 pp.

load. This will result in early failure of the column. Also, the rate of deformation increases significantly in the later stages of fire exposure for highly loaded columns; this is mainly due to higher creep strains and the P-d effect in these columns (refer to Fig. 7(g)).

Effect of concrete strength Two RC columns, namely Columns HE1 and HE2 (made of

100 MPa [14.5 ksi] concrete), were analyzed to study the effect of concrete strength on fire resistance under ASTM E119-079 standard fire. Column HE1 was assumed to carry a concentrated load equal to 50% of its design capacity applied at an eccentricity of 50 mm (2 in.) along both axes, whereas Column HE2 carried the same load concentrically but was exposed to fire on two adjacent sides. The HSC columns were assumed to have a permeability of 10–19 as compared to a permeability of 10–17 for the NSC columns. Thus, the moisture in HSC columns cannot escape, leading to pore pressure buildup, leading to spalling. Figure 7(h) shows the progression of axial deformation with time for these columns and NSC Column CD1. Comparing the behavior of Columns HE1 to CD1, it can be seen that the fire resistance drops to 62 minutes from 131 minutes. This can be attributed to spalling occurring in the HSC columns. This not only reduces the cross section of the column, but also considerably reduces the second moment of inertia about the neutral axis (due to the loss of mass farther away from the axis) and thus leads to early failure of the column. Similarly, for Column HE2, the fire resistance is higher than in Column HE1. This is due to the fact that the load is not applied eccentrically and the fire exposure is only on two sides. So, even though the column is subjected to biaxial bending, the loss of strength and stiffness is smaller due to a slower rise in thermal gradients. Also, the amount of spalling and thus the amount of concrete area lost is less. The fire resistance values for the two columns are provided in Table 2.

CONCLUSIONSBased on the numerical study presented in this paper, the

following conclusions can be drawn. These conclusions are valid within the range and set of parameters considered and may not fully apply for columns beyond the tested parameters.

1. RC columns in a structural frame can experience biaxial bending either due to one-, two-, or three-sided fire exposure, uneven spalling, or eccentric loading. Such biaxial bending, which is often not accounted for in the fire resistance evaluation, can have a significant influence on the response of RC columns.

2. The proposed macroscopic FEM is capable of accounting for the biaxial bending of RC columns arising from one-, two, or three-sided exposure; spalling; or eccentric loading. Also, the model accounts for other critical parameters, such as different fire scenarios, high-temperature material properties, various strain components, fire-induced spalling, and restraint effects.

3. Load eccentricity has a significant influence on fire resistance and hence needs to be incorporated through detailed fire resistance analysis. The fire resistance reduces by approximately 20% for every 25 mm (1 in.) increase in load eccentricity arising due to uniaxial bending or P-d effects. This fire resistance further reduces by approximately 30% for biaxial bending situations. Different exposure conditions affect the fire resistance of RC columns, with two adjacent sides exposed being the worst case (least fire resistance) and two opposite sides exposed being the best case (highest fire resistance) scenario.

4. The reduction in fire resistance due to an increase in load level is higher for columns subjected to biaxial bending

26

APPENDIX A 1

The given RC column is idealized to be a set of longitudinal frame elements 2

(segments). For hydro-thermal and mechanical analysis of RC columns, the cross-section of 3

the column is idealized as a mesh of elements as shown in Figure 2. The number of 4

longitudinal segments and the number of elements and the grid size in each direction are to be 5

specified in the input file. The program allows for non-uniform grid size in the cross-6

sectional mesh. The program determines the element size based on the specified number of 7

elements. The program also allows for any rebar configuration. Since finite element analysis 8

is utilized the model is sensitive to the cross-sectional mesh size. This is very important since 9

the spalling sub-model predicts spalling by eliminating elements. Larger the size of the 10

elements (coarse mesh) the spalling the prediction is less accurate and less continuous. In 11

order to quantify the effect and to recommend the appropriate mesh size for the analysis a 12

mesh-sensitivity study was undertaken and the results from the same are shown in Figure 7. 13

Figure A1(a) shows the progression of spalling for various element sizes. It can be 14

seen that there are sudden jumps in spalling prediction when the mesh is coarse. This 15

smoothens out as we decrease the element size making the mesh finer. The predictions from a 16

10 mm element mesh and a 1 mm element mesh are identical. There is a slight difference in 17

the prediction from the 25 mm mesh and 10 mm mesh. The 10 mm mesh predicts spalling to 18

begin earlier than the 255 mm mesh. This is due to the small element size making it capable 19

of killing elements earlier in the analysis. 20

Figure A1(b) shows the rebar temperature throughout the fire exposure time for the 21

different meshes. Again the rise in temperature with a coarse mesh is more sudden due to the 22

loss of am element thus making the inner element (and eventually the rebar) exposed to 23

higher temperatures. But it can be seen that the predictions with mesh 25 mm and finer are 24

27

similar and thus it can be said that an element size of 25 mm or smaller should be used for the 1

analysis. For all further calculations in this paper an element size of 25 mm has been used. 2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 30 60 90 120 150 180 210 240

Time (minutes)

Are

a S

palle

d (

%)

100 mm mesh

50 mm mesh

25 mm mesh

10 mm mesh

1 mm mesh

3

(a) Prediction of spalled area using different mesh sizes 4

32

392

752

1112

1472

1832

2192

0

200

400

600

800

1000

1200

0 30 60 90 120 150 180 210 240

Reb

ar

tem

pera

ture

(°F

)

Reb

ar te

mp

era

ture

(°C

)

Time (minutes)

100 mm

mesh

50 mm mesh

25 mm mesh

5

(b) Prediction of rebar temperature using different mesh sizes 6

Fig. A1– Mesh sensitivity study on fire response of RC columns 7

response of reinforced concrete columns under fire-induced biaxial bending

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