Evaluation on thermal behavior of concrete-filled steel tubular columns based on modified finite difference method

Abstract

This article presents a modified finite difference method specific for predicting temperatures in concrete-filled steel tubular composite columns. The modifications were based on existing researches on heat transfer analyses of composite columns by finite difference method. The temperature predictions by the modified method were implemented via MATLAB. Validity was established by comparing the predicted temperatures with test results and those obtained from finite element analyses via ABAQUS. The comparisons showed reasonable predictions by the modified finite difference method. Sensitivity studies were also carried out on the effects of emissivity of fire, thermal contact resistance, thermal property model, and cavity ratio based on the modified method. Through the implementation with MATLAB, it is attractive to calculate fire resistance of the concrete-filled steel tubular columns by combining the modified method with the simple calculation models according to Eurocode 4.

Keywords

concrete-filled steel tubular column heat transfer analysis modified finite difference method sensitivity study simple calculation models thermal resistance

Introduction

For evaluating fire resistances of concrete, steel, steel–concrete composite members in accordance with Eurocodes, temperature profiles in these members are required as they give significant influences on such material properties such as strength, elastic modulus, and thermal expansions. For concrete structures, the temperature profiles for slabs, beams, and columns are given in design charts in Eurocode 2 so that they can be directly used for fire-resistant design (EN 1992-1-2:2004, 2004). For steel structures, Eurocode 3 provides a simplified formula to calculate thermal response of both unprotected and protected members (EN 1993-1-2:2005, 2005). The simplified formula is based on section factors of the steel members and assumption of uniform temperature distributions in both steel sections and fire protection materials. Integration in time domain is required and can be implemented in Excel spreadsheets. However, for composite structures, the calculation method for their temperature profiles is not provided by Eurocode 4, although advanced calculation models are briefly introduced (EN 1994-1-2:2005, 2005). The said simplified formula for steel members is not applicable for the composite members since there is temperature gradient existing in the concrete section. It would lead to wrong temperature predictions if uniform temperature distribution is assumed. As a rule of thumb, the concrete section should be discretized to cater for the temperature gradient. As a result, both spatial discretization and time integration are required for calculating the temperature profiles of composite members.

Temperature prediction for concrete-filled steel tubular (CFST) columns has been introduced by Lie and Chabot (1990a) and Kodur and Lie (1997a). In the research of Lie and Chabot (1990a), the temperature profiles were calculated based on one-dimensional (1D) finite difference method (FDM) for unprotected circular CFST columns. The method of Kodur and Lie (1997a) was based on two-dimensional (2D) FDM and for unprotected square CFST columns. The cross section was discretized in triangular network where the elements were square inside the steel and concrete sections and triangular at column surface and at interface of concrete and steel sections. Hence, the temperature calculation was quite complicated in the triangular network due to the combination of triangular and square elements. There might also be incompatibility arising at the interface of the triangular and square elements. For further calculating the fire resistance of CFST columns based on fiber element method or simple calculation models in Eurocode 4, the triangular network needed to be transformed into a square network. In addition, heat convection was ignored in both calculation models of Lie and Chabot (1990a) and Kodur and Lie (1997a). This would affect predictions in low temperature range which is governed by the heat convection. In this study, the FDM was used for predicting temperatures of the circular and square composite columns with the following improvements proposed to the existing methods by Lie and Chabot (1990a) and Kodur and Lie (1997a):

Heat convection was taken into account which is related to the type of fire exposure;

Thermal contact resistance at the interface of steel and concrete was considered;

Given the nonlinear relationship with temperature, the thermal conductivity across two adjacent elements was recommended to be averaged in terms of temperature, rather than directly averaging the individual conductivities;

Square network rather than triangular network was directly used for the square composite columns, which eliminated the above-mentioned transformation to determine the fire resistance;

The proposed model is applicable for heat transfer analysis of concrete-filled double-skin steel tubular (CFDST) columns.

In addition, the level of improvement was discussed by comparisons with the existing model. Sensitivity studies on some key parameters influencing the temperature development have been carried out. The temperature profiles of the CFST and CFDST columns were compared. The accuracy and stability of the proposed model were also discussed. Overall, the validity of the proposed model has been established through comparisons with test results in the available literature and predictions through finite element method (FEM) via ABAQUS.

Basics of heat transfer and FDM

There are three ways of heat transfer, including conduction, convection, and radiation. For temperature determination, partial differential equation of heat transfer can be used and established in accordance with Fourier’s law and the law of conservation of energy (Incropera et al., 2007)

ρ c \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z})

(1)

where T is the temperature and t is the time. λ, ρ, and c are temperature-dependent thermal conductivity, density, and specific heat, respectively. Integration techniques are generally required to solve equation (1). For structures with simple geometries and boundary conditions, analytical solutions can be obtained. However, it is difficult to obtain the analytical solutions for the most cases. Where it is the case, approximate solutions are necessary, and herein FDM is adopted. There are two ways of introducing finite difference algorithm. The former consists of replacing the space and time derivatives in the partial differential equation by finite difference quotients. However, engineers usually prefer to formulate finite difference procedure by writing energy conservation equations for each element (Harmathy, 1993), assuming that for a short time period, quasi-steady-state conditions prevail.

Heat transfer analysis by FDM

Spatial discretization

As CFST column is exposed to fire, heat is transferred into column by convection and radiation at fire–column interface. Inside column, the heat is propagated by thermal conductivity. The heat transfer problem of circular column can be solved by 1D finite difference procedure while 2D for square column. The partial differential equations for the 1D and 2D heat transfer problems are, respectively, given in equations (2) and (3). It should be noted that the partial differential equation for 1D heat transfer is formulated in polar coordinate system (Holman, 1986)

Polar coordinate system : \frac{ρ c}{λ} \frac{\partial T}{\partial t} = \frac{\partial^{2} T}{\partial r^{2}} + \frac{1}{r} \frac{\partial T}{\partial r}

(2)

Cartesian coordinate system : \frac{ρ c}{λ} \frac{\partial T}{\partial t} = \frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}

(3)

Derivations of finite difference equations for the 2D heat transfer are given herein. Discretization of 2D cross section is shown in Figure 1. The domain is discretized into many small areas by the gridlines where the intersection point of two gridlines is a node. The distance between two adjacent nodes is spatial step length, denoted by Δx and Δy. Each node is at the center of hatched area which is enclosed by the perpendicular lines from the midway of two nodes, this area is representative of a finite difference element or control volume of node (m, n, i). Symbol i stands for the ith time step. Assuming that the temperatures at nodes (m, n, i), (m + 1, n, i), (m − 1, n, i), (m, n + 1, i), and (m, n − 1, i) are denoted by $T_{m, n}^{i}$ , $T_{m + 1, n}^{i}$ , $T_{m - 1, n}^{i}$ , $T_{m, n + 1}^{i}$ , and $T_{m, n - 1}^{i}$ , respectively, expanding $T_{m, n - 1}^{i}$ and $T_{m, n + 1}^{i}$ by Taylor series at node (m, n, i) in the x-direction gives

T_{m, n - 1}^{i} = T_{m, n}^{i} + Δ x \frac{\partial T_{m, n}^{i}}{\partial x} + \frac{{(Δ x)}^{2}}{2!} \frac{\partial^{2} T_{m, n}^{i}}{\partial x^{2}} + O ({(Δ x)}^{3})

(4)

T_{m, n + 1}^{i} = T_{m, n}^{i} - Δ x \frac{\partial T_{m, n}^{i}}{\partial x} + \frac{{(Δ x)}^{2}}{2!} \frac{\partial^{2} T_{m, n}^{i}}{\partial x^{2}} + O ({(Δ x)}^{3})

(5)

Figure 1.

Discretization of square column based on 2D heat transfer analysis.

Adding above equations, ignoring $O ({(Δ x)}^{3})$ , and rearranging give

\frac{\partial^{2} T_{m, n}^{i}}{\partial x^{2}} = \frac{T_{m, n - 1}^{i} - 2 T_{m, n}^{i} + T_{m, n + 1}^{i}}{{(Δ x)}^{2}}

(6)

Equation (6) gives the finite difference quotients to replace the space derivative in equation (3). The ignored terms $O ({(Δ x)}^{3})$ are called as local truncation errors which result from the approximation in using the truncated (incomplete) Taylor series. This error is only for one time step (ith step). If we are interested in the solution for the range [0, t] and use constant time step $Δ t$ , then the total number of time steps required is $N = t / Δ t$ . Finally, the error will be accumulated, and approximately the order of error is given in equation (7)

E_{N} ~ O ({(Δ x)}^{3}) \times N ~ O ({(Δ x)}^{2})

(7)

The accumulated error is called as global truncation error. In this case, the approximation has consistency of order 2 and is said to be second-order accurate. The approximation in the y-direction can be similarly derived.

Time integration

Term $\partial T / \partial t$ in equation (3) can be expanded based on forward difference by Taylor series and ignoring the local truncation error $O ({(Δ t)}^{2})$ as follows

\frac{\partial T_{m, n}^{i}}{\partial t} = \frac{T_{m, n}^{i + 1} - T_{m, n}^{i}}{Δ t}

(8)

For this approximation, a global truncation error $O (Δ t)$ is ignored, and thus, it is said to be first-order accurate. It is noted that the approximation for time integration is less accurate than that for the spatial discretization. This indicates that the subdivision in time domain should be finer than that in space domain. Substituting equations (6) and (8) into equation (3) with Δx = Δy = ΔL and α = λ/ρc gives equation (9) through which the temperature at next time step (i + 1) can be explicitly obtained. The calculation procedure is stable when equation (10) is satisfied

\begin{matrix} T_{m, n}^{i + 1} = & \frac{α Δ t}{{(Δ L)}^{2}} (T_{m + 1, n}^{i} + T_{m - 1, n}^{i} + T_{m, n + 1}^{i} + T_{m, n - 1}^{i}) \\ + (1 - \frac{4 α Δ t}{{(Δ L)}^{2}}) T_{m, n}^{i} \end{matrix}

(9)

1 - \frac{4 α Δ t}{{(Δ L)}^{2}} \geq 0

(10)

Equation (10) indicates that the stability of the finite difference procedure depends on time step and element size. To improve the stability, the time step should be reduced, and the element size can be increased. However, the accuracy of results is impaired by the increase in element size. Hence, determinations of time step and element size by trial and error may be necessary to ensure the stability, while the accuracy is maximally achieved.

Energy conservation

It is sometimes difficult to formulate the finite difference equations for calculating nodal temperatures at fire–column interface and steel–concrete interface. The boundary conditions at the fire–column interface are governed by the heat convection and thermal radiation as

q = h (T_{f} - T)

(11)

where q is the heat flux and h is the sum of coefficients of heat convection h_e and thermal radiation h_r. The convection coefficient h_e in this study is taken as 25 W/m K for exposure under the standard fires of ISO-834 (EN 1991-1-2:2002, 2002) and ASTM-E119-12a:2012 (2012). This has been validated by Huo et al. (2011), Lu et al. (2011), Espinos et al. (2012), and Wang and Young (2013). The thermal radiation coefficient is calculated as follows

\begin{matrix} h_{r} = & Φ \cdot ε_{m} \cdot ε_{f} \cdot σ [{(T_{f} + 273)}^{2} + {(T + 273)}^{2}] \\ (T_{f} + T + 546) \end{matrix}

(12)

Φ is the configuration factor which is taken as 1.0 by ignoring the position and shadow effects (EN 1991-1-2:2002, 2002). ε_m is the steel surface emissivity where it is taken as 0.7 for unprotected columns (EN 1993-1-2:2005, 2005). ε_f is the emissivity of the fire and generally taken as 1.0 (Han et al., 2013), although lower values have been used to take account of position and shadow effects in the available literature (Chung et al., 2008; Lu et al., 2011). Sensitivity study on the effect of emissivity of fire is provided in section “Sensitivity study.” The Stephan Boltzmann constant (σ) is equal to 5.67×10⁻⁸ W/m² K⁴. T_f is the effective radiation temperature of the fire environment, and T is the surface temperature of the member.

The finite difference procedure for predicting temperatures at interfacial nodes can be directly obtained by writing the energy conservation equations. For instance, at node (1, 1, i) in Figure 1, the heat flowing into node (1, 1, i) from adjacent nodes (1, 2, i) and (2, 1, i) should be equal to the energy consumed by the temperature increase at node (1, 1, i). Thus, the nodal temperature at (i + 1)th time step $T_{1, 1}^{i + 1}$ can be solved from equation (13). The finite difference equations for nodal temperatures at the steel–concrete interface can be similarly derived, except thermal contact resistance at the interface should be used to replace the coefficient h

\begin{matrix} λ \frac{T_{2, 1}^{i} - T_{1, 1}^{i}}{Δ y} \frac{Δ x}{2} + λ \frac{T_{1, 2}^{i} - T_{1, 1}^{i}}{Δ x} \frac{Δ y}{2} \\ + h (T_{f}^{i} - T_{1, 1}^{i}) \frac{Δ x + Δ y}{2} = ρ c \frac{Δ x}{2} \frac{Δ y}{2} \frac{T_{1, 1}^{i + 1} - T_{1, 1}^{i}}{Δ t} \end{matrix}

(13)

Temperature calculation for composite columns

Circular columns

Spatial discretization for circular CFST column is shown in Figure 2. The symbols d_c and d_s (d_si or d_so) are the element sizes of concrete and steel tube (inner tube or outer tube), respectively. nc is the number of concrete elements along the radius. Due to the high conductivity, the steel tube is discretized into one annular element. d_t is the time step. Overall, the nodal temperatures can be calculated in the following equations based on the energy conservation. p_c and λ_c are the thermal capacity which is the product of specific heat and density and the thermal conductivity. Subscript c and s, respectively, stand for concrete and steel in this article, unless otherwise stated. For $λ_{c}^{((T_{2}^{i} + T_{1}^{i}) / 2)}$ , $λ_{c}$ is the function of averaged temperature $((T_{2}^{i} + T_{1}^{i}) / 2)$ , which indicates that the thermal conductivity in the path of two adjacent nodes should be considered as the function of the averaged temperature, rather than being directly averaged as shown in equation (14) by Lie and Chabot (1990a). This is due to the fact that the thermal conductivity of concrete is in nonlinear relationship with the temperature

λ_{c} = \frac{λ_{c}^{(T_{1}^{i})} + λ_{c}^{(T_{2}^{i})}}{2}

(14)

\begin{matrix} Node 1 : T_{1}^{i + 1} = T_{1}^{i} \\ + \frac{d_{t}}{p_{c} d_{c}^{2}} λ_{c}^{(\frac{T_{2}^{i} + T_{1}^{i}}{2})} (T_{2}^{i} - T_{1}^{i}) \end{matrix}

(15)

\begin{matrix} Node m (2 \leq m \leq nc) T_{m}^{i + 1} = T_{m}^{i} + \frac{d_{t}}{p_{c} A_{m}} \\ [λ_{c}^{(\frac{T_{m - 1}^{i} + T_{m}^{i}}{2})} \frac{T_{m - 1}^{i} - T_{m}^{i}}{d_{c}} l_{m 1} + λ_{c}^{(\frac{T_{m + 1}^{i} + T_{m}^{i}}{2})} \frac{T_{m + 1}^{i} - T_{m}^{i}}{d_{c}} l_{m 2}] \end{matrix}

(16)

where $l_{m 1} = (m - 1.5) d_{c}$ , $l_{m 2} = (m - 0.5) d_{c}$ , and $A_{m} = (1 / 2) (l_{m 2}^{2} - l_{m 1}^{2})$ .

Figure 2.

Discretization of circular CFST column.

As discussed above, the thermal contact resistance should be considered due to the air gap existing at the steel–concrete interface. The air gap is possibly induced by the shrinkage of concrete and the faster expansion of steel tube subject to higher temperature (Tao and Ghannam, 2013), leading to temperature drop at the interface. Various thermal resistances were proposed in the available literature. Ding and Wang (2008) and Espinos et al. (2010) suggested a constant value of 200 W/m K, whereas a constant value of 100 W/m K was proposed by CIDECT (2004). Perfect bond (thermal resistance equal to infinite) was taken into account by Lie and Chabot (1990a) and Hong and Varma (2009). In addition, the thermal contact resistance was experimentally gauged by Ghojel (2002, 2004) where it is temperature dependent and close to constant value of 100 W/m K. Based on the best fitting, a formula was proposed by Tao and Ghannam (2013) where it is related with the cross-sectional size of column but irrespective of the temperature. Sensitivity study on the thermal contact resistance has been carried out in section “Sensitivity study.” With the introduction of thermal contact resistance h_tr, the nodal temperatures at the steel–concrete interface can be solved from

\begin{matrix} Node (nc + 1) : λ_{c}^{(\frac{T_{nc}^{i} + T_{nc + 1}^{i}}{2})} \frac{T_{nc}^{i} - T_{nc + 1}^{i}}{d_{c}} l_{c} \\ + h_{tr} (T_{nc + 2}^{i} - T_{nc + 1}^{i}) l_{cs} = (p_{c} A_{c}) \frac{T_{nc + 1}^{i + 1} - T_{nc + 1}^{i}}{d_{t}} \end{matrix}

(17)

where $l_{c} = (nc - 0.5) d_{c}$ , $l_{cs} = nc \cdot dc$ , and $A_{c} = (1 / 2) [l_{cs}^{2} - l_{c}^{2}]$

\begin{matrix} Node (nc + 2) : h_{tr} (T_{nc + 1}^{i} - T_{nc + 2}^{i}) l_{cs} \\ + λ_{s}^{(\frac{T_{nc + 3}^{i} + T_{nc + 2}^{i}}{2})} \frac{T_{nc + 3}^{i} - T_{nc + 2}^{i}}{d_{s}} l_{s} \\ = (p_{s} A_{s}) \frac{T_{nc + 2}^{i + 1} - T_{nc + 2}^{i}}{d_{t}} \end{matrix}

(18)

where $l_{cs} = nc \cdot dc$ , $l_{s} = nc \cdot d_{c} + 0.5 d_{s}$ , and $A_{s} = (1 / 2) [l_{s}^{2} - l_{cs}^{2}]$

The nodal temperature at the fire–steel interface can be calculated as shown in equation (19), where D is cross-sectional diameter

\begin{matrix} Node (nc + 3) : λ_{s}^{(\frac{T_{nc + 3}^{i} + T_{nc + 2}^{i}}{2})} \frac{T_{nc + 2}^{i} - T_{nc + 3}^{i}}{d_{s}} l_{s 1} \\ + h (T_{f}^{i} - T_{nc + 3}^{i}) l_{s 2} = p_{s} A_{es} \frac{T_{nc + 3}^{i + 1} - T_{nc + 3}^{i}}{d_{t}} \end{matrix}

(19)

where $l_{s 1} = D / 2 - 0.5 d_{s}$ , $l_{s 2} = D / 2$ , $A_{es} = (1 / 2) [l_{s 2}^{2} - l_{s 1}^{2}]$

It is worth noting that the effect of moisture before it is evaporated around 100°C should be considered. The moisture content for normal concrete structure is approximately 1.5%–2.0% in concrete weight (Aulik, 1973). However, the moisture content for the CFST columns could be higher due to less moisture loss inside steel tubes. Different moisture contents were adopted by different researchers. Kodur and Lie (1997b) suggested a value of 4.35%, while 4% was used by Renaud (2007) and Wang (2000). Also, 5% was used by Lu et al. (2009) and Tao and Ghannam (2013), 6% was used by Han et al. (2013), 6.6% was adopted by Chung et al. (2008), and 7% was adopted by Hong and Varma (2009). In addition, Espinos et al. (2010) employed a value of 10% for concrete with calcareous aggregates and 3% for concrete with siliceous aggregates. Generally, the moisture content can be considered by an increase in the thermal capacity of concrete as in EN 1992-1-2:2004 (2004) which is only applicable for a maximum moisture content of 3%. As the moisture content showed in the available literature was generally higher than 3%, the following equation can be used to consider the effect of moisture (Lie and Stringer, 1994)

{\begin{matrix} T < 100 \circ C p_{c} = (1 - ϕ) ρ_{c} c_{c} + ϕ ρ_{w} c_{w} \\ T \geq 100 \circ C p_{c} = ρ_{c} c_{c} \\ ρ_{w} c_{w} = 4.2 \times 10^{6} J / m^{3} \circ C \end{matrix}

(20)

where ρ_cc_c and ρ_wc_w are the thermal capacity of concrete and moisture, respectively, and ϕ is the moisture content (%). The sensitivity study on the moisture content by Tao and Ghannam (2013) showed that the moisture content has negligible influence on the steel temperature but significant influence on the concrete temperature. The higher moisture content generally results in lower concrete temperature. The influence on the concrete temperature at center of cross section is higher than that of outer concrete.

Spatial discretization for circular CFDST column is shown in Figure 3. The nodal temperatures can be calculated similar to those of circular CFST column, except for node 1. Based on energy conservation, the temperature at node 1 is calculated from

T_{1}^{i + 1} = T_{1}^{i} + \frac{d_{t} l_{21}}{d_{si} p_{s} A_{21}} λ_{s}^{(\frac{T_{1}^{i} + T_{2}^{i}}{2})} (T_{2}^{i} - T_{1}^{i})

(21)

where r_i is the inner radius of inner steel tube, $l_{21} = r_{i} + d_{si} / 2$ and $A_{21} = (l_{21}^{2} - {r_{i}}^{2}) / 2$ . It should be noted that the heat transfer occurs between the inner tube and air in the void of inner tube. However, the air in the void is quite limited, and the density and thermal conductivity of air are much smaller than those of steel and concrete. Hence, the heat dissipated into the void was ignored in this study. This has been validated by Lu et al. (2011) and Yang and Han (2005). To formulate the finite difference equations, the boundary condition at the inner surface of inner tube was defined as insulated, indicating no heat flux at this surface.

Figure 3.

Discretization of circular CFDST column.

Square columns

Discretization of square CFST column is shown in Figure 4. Temperatures at nodes (1, 1) and (3, 3) can be determined based on equations (22) and (23), respectively. Temperatures at other boundary nodes can be similarly determined based on the energy conservation

T_{1, 1}^{i + 1} = T_{1, 1}^{i} + \frac{2 d_{t}}{p_{s} d_{s}^{2}} [λ_{s}^{(\frac{T_{1, 2}^{i} + T_{1, 1}^{i}}{2})} (T_{1, 2}^{i} - T_{1, 1}^{i}) + λ_{s}^{(\frac{T_{2, 1}^{i} + T_{1, 1}^{i}}{2})} (T_{2, 1}^{i} - T_{1, 1}^{i}) + 2 h (T_{f} - T_{1, 1}^{i}) d_{s}]

(22)

\begin{matrix} T_{3, 3}^{i + 1} = T_{3, 3}^{i} + \frac{4 d_{t}}{p_{c} d_{c}^{2}} [h_{tr} (T_{2, 2}^{i} - T_{3, 3}^{i}) d_{c} + λ_{c}^{(\frac{T_{4, 3}^{i} + T_{3, 3}^{i}}{2})} \frac{T_{4, 3}^{i} - T_{3, 3}^{i}}{2} + λ_{c}^{(\frac{T_{3, 4}^{i} + T_{3, 3}^{i}}{2})} \frac{T_{3, 4}^{i} - T_{3, 3}^{i}}{2}] \end{matrix}

(23)

Figure 4.

Discretization of square CFST column.

Discretization of square CFDST column is shown in Figure 5. The nodal temperatures can be calculated similar to those of square CFST column, except for nodes on inner surface of inner tube. At the corner, the temperature of node (m = nco + 5, n = nco + 5) is calculated from equation (24)

\begin{matrix} T_{m, n}^{i + 1} = T_{m, n}^{i} + \frac{2 d_{t}}{p_{s} d_{si} (d_{si} + 2 d_{ci})} {\begin{matrix} \frac{d_{si} + d_{ci}}{d_{si}} [λ_{s}^{\frac{(T_{m, n - 1}^{i} + T_{m, n}^{i})}{2}} (T_{m, n - 1}^{i} - T_{m, n}^{i}) + λ_{s}^{\frac{(T_{m - 1, n}^{i} + T_{m, n}^{i})}{2}} (T_{m - 1, n}^{i} - T_{m, n}^{i})] \\ + \frac{d_{si}}{2 d_{ci}} [λ_{s}^{\frac{(T_{m + 1, n}^{i} + T_{m, n}^{i})}{2}} (T_{m + 1, n}^{i} - T_{m, n}^{i}) + λ_{s}^{\frac{(T_{m, n + 1}^{i} + T_{m, n}^{i})}{2}} (T_{m, n + 1}^{i} - T_{m, n}^{i})] \end{matrix}} \end{matrix}

(24)

Figure 5.

Discretization of square CFDST column.

Validation and discussion

Comparison with predictions from finite element analysis

ABAQUS 6.12 (2012) was used to perform the finite element analysis where both linear and quadratic elements were considered. Back difference algorithm is adopted in ABAQUS 6.12 (2012) due to unconditional stability for nonlinear heat transfer problem. In view of the stability, the FEM is more advanced than the proposed FDM where forward difference algorithm is adopted. In terms of accuracy, they would be comparable provided the discretization in time domain is sufficiently fine. It should be noted that the accuracy for FEM can be improved by decrease in element size in space domain. However, it is not the case for the proposed FDM where the stability criterion requires large element size as indicated in equation (10).

The CFST and CFDST composite sections for the validations are shown in Figure 6. The columns were exposed to standard ISO-834 fire. Carbon steel S355 and normal-strength concrete C40/50 were used. A four-node linear heat transfer element DC2D4 and an eight-node quadratic heat transfer element DC2D8 were adopted in ABAQUS. For better comparison, the time step chosen for the FEM and FDM was the same. Regarding the discretization in space domain, the cross sections were similarly meshed for the FDM and FEM in the radial direction for the circular columns and in the directions normal to the sides for the square columns. The fire exposure time was 2 h. The FDM was implemented via MATLAB. The thermal properties of concrete at high temperatures, including density, thermal conductivity, and specific heat, were referred to EN 1992-1-2:2004 (2004). The lower limit of thermal conductivity was used for normal-strength concrete, whereas the upper limit was used for high-strength concrete with cylinder strength not less than 55 MPa (ACI 363R-10:2010, 2010). The moisture content in concrete was taken as 3.0%. The density of concrete was taken as 2400 kg/m³ at ambient temperature. The thermal properties given in EN 1993-1-2:2005 (2005) were used for the steel tubes. The density of steel was taken as 7850 kg/m³ and was not changed with the elevated temperatures.

Figure 6.

Comparisons between temperatures predicted by FDM and FEM: (a) circular CFST, (b) circular CFDST, (c) square CFST, and (d) square CFDST.

The comparisons are shown in Figure 6 where temperatures at three locations were compared for each column. Temperature at point 1 was taken from outer tube and temperature at point 3 was taken from inner tube for CFDST columns. It can be seen that the results based on linear elements and quadratic elements through finite element analyses were the same. This indicates that the linear elements are adequate for heat transfer analyses on steel–concrete composite columns, while it saves computation time. The temperatures predicted by the FDM showed quite good agreements with those by the FEM, although different difference algorithms are adopted. It is worthwhile to note that the thermal properties were input as formulae from EN 1993-1-2 in the FDM, whereas discrete data were input in the FEM. The data input in the FEM should be refined to improve the accuracy of predictions. In this study, the data of thermal properties were input with an increment of 10°C.

Comparison with test results

Tests by Lie and Chabot (1990b) and Lu et al. (2010, 2013) were used for the verification as shown in Table 1. For most of the columns, the time step was set as 1 s. However, for some square columns, 0.25 s was used due to convergence problem. Nevertheless, the time step was determined by trial and error with both convergence and accuracy ensured. The thermal properties proposed by Lie (1984) were used for the validations, which take the effect of the type of aggregate into account. The comparisons with the thermal properties provided in EN 1992-1-2 are given in section “Sensitivity study.” For tests by Lie and Chabot, the moisture content of concrete was taken as 4.35% as proposed by Kodur and Lie (1997b), whereas it was 5.0% for tests by Lu et al. (2009).

Table 1.

Details of fire tests for verification.

	D (mm) × t (mm)	f_y (MPa)	f_ck (MPa)
C-02	CHS141.3 × 6.55	350	33.1
C-04	CHS141.3 × 6.55		31
C-06	CHS168.3 × 4.78		32.7
C-08	CHS168.3 × 4.78		35.5
C-11	CHS219.1 × 4.78		31
C-13	CHS219.1 × 4.78		32.3
C-15	CHS219.1 × 8.18		31.9
C-17	CHS219.1 × 8.18		31.7
C-20	CHS273.1 × 5.56		28.6
C-21	CHS273.1 × 5.56		29
C-23	CHS273.1 × 12.7		27.4
C-29	CHS355.6 × 12.7		25.4
C-31	CHS141.3 × 6.55		30.2
C-32	CHS141.3 × 6.55		34.8
C-34	CHS219.1 × 4.78		35.4
C-35	CHS219.1 × 4.78		42.7
C-37	CHS219.1 × 8.18		28.7
C-40	CHS273.1 × 6.35		46.5
C-41	CHS273.1 × 6.35		50.7
C-44	CHS273.1 × 6.35		38.7
C-45	CHS273.1 × 6.35		38.2
C-50	CHS323.9 × 6.35	300	42.4
C-53	CHS355.6 × 6.35		42.4
C-55	CHS355.6 × 12.7		40.7
C-60	CHS406.4 × 12.7		45.1
SQ-01	SHS152.4 × 6.35	350	58.3
SQ-02	SHS152.4 × 6.35		46.5
SQ-07	SHS177.8 × 6.35		57
SQ-17	SHS254 × 6.35		58.3
SQ-20	SHS254 × 6.35		46.5
SQ-24	SHS304.8 × 6.35		58.8
CC2	CHS300 × 5/CHS125 × 5	320	38
CC3	CHS300 × 5/CHS225 × 5
C1-C3-SCC2	CHS406 × 8/CHS165.1 × 3	401/399	63.4
C2-C4-SCC2	CHS219.1 × 5/CHS101.6 × 3.2	426/426
SS1	SHS280 × 5/SHS140 × 5	320	38
S1-S3-SCC2	SHS350 × 8/SHS150 × 5	514/504	63.4
S2-S4-SCC2	SHS200 × 6/SHS89 × 3.5	506/506

In practice, the fire temperature inside the furnace during testing is influenced by factors such as gas supply rate, burner location, air–gas ratio, gas pressure, and turbulence. It is difficult to achieve uniform and smooth temperature distribution inside the furnace. Hence, the temperature measurements in the CFST columns generally showed fluctuations, whereas the measurements for infilled concrete showed greater fluctuations than those for steel tubes as observed in the referred tests. For temperature predicted by the FDM as shown in Figures 7 and 8, the temperatures of outer steel tubes were better predicted compared with those of infilled concrete. It is worthwhile to note that at the early stage of fire exposure, the concrete temperatures during the evaporation of water (around 100°C) were generally not well predicted. This is due to the migration of moisture (Lie and Chabot, 1990a). Although the moisture content was considered in the FDM, it is stationary. The heat flow by the migration of moisture was not captured, resulting in deviations between the predicted and measured temperatures. However, there are good agreements at the later stages of fire exposure which are important from the point of view of determining the fire resistance of the composite columns.

Figure 7.

Comparison between calculated and measured temperatures in Lie’s tests: (a) C-02, (b) C-04, (c) C-06, (d) C-08, (e) C-11, (f) C-13, (g) C-15, (h) C-17, (i) C-20, (j) C-21, (k) C-23, (l) C-29, (m) C-31, (n) C-32, (o) C-34, (p) C-35, (q) C-37, (r) C-40, (s) C-41, (t) C-44, (u) C-45, (v) C-50, (w) C-53, (x) C-55, (y) C-60, (z) SQ-01, (aa) SQ-02, (bb) SQ-07, (cc) SQ-17, (dd) SQ-20, and (ee) SQ-24.

Figure 8.

Comparison between calculated and measured temperatures for CFDST columns: (a) CC2, (b) CC3, (c) SS1, (c) C1-C3-SCC2, (d) C2-C4-SCC2, (e) S1-S3-SCC2, and (f) S2-S4-SCC2.

Comparison with Lie’s model

As mentioned above, the improvements to Lie’s model are the specific considerations for thermal contact resistance at steel–concrete interface, thermal conductivity across two adjacent elements, and the inclusion of heat convection. Through heat transfer analysis on the specimen C-02 by Lie and Chabot (1990b), the comparison between Lie’s model and the modified model is given in Figure 9. It can be seen that the steel temperature from Lie’s model shows large deviations from the test data. This is due to the ignorance of heat convection which is dominant in the early stage of fire exposure. At the later stage, the concrete temperatures deviate more from the test data compared with those from the modified model. This is attributed to the omission of the said thermal contact resistance and the nonlinear behavior of thermal conductivity of concrete with temperature.

Figure 9.

Comparison between Lie’s model and modified model.

Sensitivity study

Overview

Unless otherwise stated, the sensitivity study was based on the heat transfer analysis on the specimen C-02 by Lie and Chabot (1990b). Temperatures at point 1 and point 5 were used to illustrate the discrepancies between the predicted and measured values. The emissivity of fire of 1.0 provided in Eurocode 3 (EN 1993-1-2:2005, 2005), the thermal contact resistance of 100 W/m K proposed by CIDECT (2004), the moisture content of 4.35% recommended by Kodur and Lie (1997b), and the thermal properties of steel and concrete proposed by Lie (1984) were adopted, unless its sensitivity was being studied.

Effect of emissivity of fire

The effects of emissivity of fire taken as 0.8, 0.9, and 1.0, respectively, by EN 1993-1-2:2005 (2005), Chung et al. (2008), and Lu et al. (2011) were compared as shown in Figure 10. It can be seen that the temperatures increased with the increase in the emissivity of fire, due to the rise of heat energy by the thermal radiation. Furthermore, the temperatures based on the different emissivities of fire were almost the same at the early stage of fire exposure. This is because of the dominant effect of heat convection in this stage compared with the thermal radiation. At the later stage, the temperatures showed discrepancies. The discrepancies of the steel temperatures were larger than those of the concrete temperatures. Overall, the discrepancies were minor; the emissivity of fire ranging from 0.8 to 1.0 could be used to well predict the temperature profiles of steel–concrete composite columns under standard ISO-834 fire. However, for conservative fire resistance design, an emissivity of fire equal to 1.0 may be used.

Figure 10.

Sensitivity study on the effect of emissivity of fire.

Effect of thermal contact resistance

Comparisons between the effects of thermal contact resistance taken as 100 W/m K by CIDECT (2004), 200 W/m K by Ding and Wang (2008) and Espinos et al. (2010), and infinite by Lie and Chabot (1990a) and Hong and Varma (2009) are shown in Figure 11. As the thermal contact resistance increased, the steel temperature decreased and the concrete temperature increased. At the early stage of fire exposure, the influence of the contact resistance was more significant on the magnitude of steel temperature but more significant on that of concrete temperature at the later stage. The thermal contact resistance almost had no influence in terms of magnitude for concrete temperature before 15 min and for steel temperature after 35 min. Given the thermal contact resistance doubled from 100 to 200 W/m K, the concrete temperature at the center, in terms of percentage, increased by 12.2% at 25 min and by 8.6% at 50 min. The steel temperature was reduced by 3.0% at 25 min and 0.1% at 50 min. Comparing the temperatures from h_tr = 100 W/m K with those from contact resistance equal to infinite, the maximum decrease in steel temperature was 21.5% at 11 min, and the maximum increase in concrete temperature was 30.7% at 21 min. It could be concluded that as the thermal contact resistance increases, the discrepancies for both steel and concrete temperatures in terms of percentage reach maximum at the early stage of fire exposure. As the fire exposure time goes, the discrepancies decrease, although it is not the case in terms of magnitude (refer to concrete temperature). Overall, the thermal contact resistance at steel–concrete interface should be taken into account for heat transfer analysis on composite columns. Provided the fire protection exists, the contact resistance at protection–steel interface should also be considered, although there is little information reported in the available literature. For present column specimen, both 100 and 200 W/m K yielded reasonable temperature predictions. The validity of 100 W/m K has been further established in section “Comparison with test results” (refer to Figures 7 and 8).

Figure 11.

Effect of thermal contact resistance at steel–concrete interface.

Effect of models of thermal properties of concrete

The thermal properties of concrete provided in EN 1992-1-2:2004 (2004) and Lie (1984) were compared. Lie proposed different models for different types of aggregate, but this is not considered by the model in Eurocode 2. The comparisons are shown in Figure 12. The various models have negligible influence on steel temperature but significant influence on concrete temperature. Lie’s model for siliceous aggregates yielded the highest concrete temperature, followed by Lie’s model for carbonate aggregates. The model in Eurocode 2 resulted in the lowest concrete temperature. This is because the thermal conductivity of concrete proposed by Eurocode 2 is lowest, followed by that by Lie’s models for carbonate aggregates and siliceous aggregates. The thermal capacities (p_c) in the three models are quite similar, but for Lie’s model for concrete with carbonate aggregates, there is an abrupt increase in the range of 600°C–800°C, due to the decomposition of hydration products such as calcium silicate hydrate and calcium hydroxide. The influence of the abrupt increase was not found in the present case study since the concrete temperature was low. It is believed that the concrete with the carbonate aggregates would experience a slow temperature rise when the temperature goes over 600°C, which is similar to the effect of moisture at approximately 100°C.

Figure 12.

Comparisons between different models of thermal properties of concrete.

Effects of cavity ratio and section factor

For CFDST columns, the cavity ratio is defined as the area of internal void over the gross cross-sectional area. The section factor is defined as the ratio between the exposed surface area and the volume of the column. The section factor is used to measure the rate of temperature increase. The higher the section factor, the faster the column section heats up. The cavity ratios and section factors of the CFDST–CFST columns used for the sensitivity study are given in Table 2. Carbon steel S355 and normal-strength concrete C40/50 were used for the columns which were exposed to standard ISO-834 fire. The thermal properties for concrete with siliceous aggregates proposed by Lie (1984) were used in the heat transfer analyses. The moisture content in concrete was 4.35%. Figure 13 shows the steel and the averaged concrete temperatures in terms of different cavity ratios. It is found that the cavity ratio has negligible influence on steel temperature, except for the CFST column where there is no internal tube. The cavity ratio has significant influence on the averaged concrete temperature which increases with the increase in cavity ratio. Figure 14 gives the ratio between the averaged concrete temperature and the section factor. It is well known that the increase in temperature of unprotected steel member is in linear relationship with the section factor as indicated in EN 1993-1-2:2005 (2005). However, it is not the case for the unprotected CFST–CFDST columns. It is found in Figure 14 that the average concrete temperature per section factor decreases with the increase in section factor in the early stage of fire exposure, whereas it increases as the fire exposure time goes. It is worthwhile to note that the discrepancy between the various averaged concrete temperature per section factor decreases, although the discrepancy between the section factors increases. This is reasonable since the thickness of concrete section in the CFDST column becomes smaller as the section factor increases. As a result, the concrete temperature approaches the same as the concrete section is vanishing (the section factor of column approaches that of external tube). Overall, in terms of temperature rise and with the same perimeter, the solid CFST columns show better thermal behavior than the hollow CFDST columns as the concrete temperature is lower.

Table 2.

Columns for sensitivity study on the effects of cavity ratio and section factor.

Column	Size of external tube (mm)	Size of internal tube (mm)	Cavity ratio ( $η_{o}$ )	Section factor (Hp/V, m⁻¹)
C1	CHS 508 × 10	CHS 323.9 × 6.3	37.55%	12.61
C2	CHS 508 × 10	CHS 219.1 × 6.3	16.52%	9.43
C3	CHS 508 × 10	CHS 114.3 × 6.3	4.01%	8.20
C4	CHS 508 × 10	–	0	7.87

Figure 13.

Steel and averaged concrete temperatures at various cavity ratios.

Figure 14.

Ratio of the averaged concrete temperature over section factor.

Combination with simple calculation models in EC4 for fire resistance design

The design resistance of a CFST column under axial compression in fire situation is calculated from equation (25) according to the simple calculation model in EN 1994-1-2:2005 (2005)

N_{fi, Rd} = χ (\sum_{j} \frac{A_{a, θ} f_{ay, θ}}{γ_{M, fi, a}} + \sum_{k} \frac{A_{c, θ} f_{c, θ}}{γ_{M, fi, c}})

(25)

To determine the design resistance, the cross section of the column should be discretized where A_a,θ and A_c,θ are the areas of steel and concrete elements at temperature θ. The χ is the reduction factor which depends on the buckling curve c. The elemental temperatures can be obtained through the modified FDM. Thus, the same mesh network can be used in both the heat transfer analysis and the fire resistance calculation, avoiding the transformation in the model of Kodur and Lie (1997a). Herein, the buckling resistance and the buckling reduction factors for all columns with a diameter of 219.1 mm and fixed–fixed ends by Lie and Chabot (1990b) were calculated based on the combination of the modified FDM and the simple calculation method. For the calculations, the buckling length of column under standard fire test was determined according to Xiong and Yan (2015). The mechanical properties of concrete at elevated temperatures were referred to EN 1992-1-2:2004 (2004), whereas their counterparts in EN 1993-1-2:2005 (2005) were used for steel.

Figure 15 shows that the buckling resistance was rapidly reduced at the early stage of fire exposure and then decreased smoothly at the later stage. The fire resistance time can be determined when the buckling resistance approaches the test load. Regarding the buckling reduction factor, it decreased first and then slightly increased as the fire exposure continued. The reduction factor was less than 1.0 until the column failed, indicating a global buckling failure. The failure mode can be directly obtained herein through the buckling reduction factor, rather than by observing the failed column specimen. It is worthwhile to note that it is quite tedious to obtain the continuously developed buckling resistance and buckling reduction factor by the FEM via commercial software, such as ABAQUS. since each analysis only contributes one data point on the curves. Figure 15 also indicates that the simple calculation method in EC4 generally gives conservative predictions on the fire resistance of composite columns, which is safe for fire-resistant design.

Figure 15.

Fire resistance of CFST columns according to simple calculation method: (a) C-11, (b) C-13, (c) C-17, (d) C-34, (e) C-35, and (f) C-37.

Conclusion

A modified finite difference model for predicting temperatures of steel–concrete composite columns has been introduced in this article. The modifications focused on discretization of space and time domains, inclusion of heat convection, determination of thermal conductivity between adjacent elements, consideration of thermal contact resistance at steel–concrete interface, and calculation for double-skin composite columns. The level of improvement of the modifications has been gauged by predicting column temperatures in real fire tests. It is found that the modified model resulted in better predictions, compared with the existing model.

The modified finite difference model was implemented through MATLAB, and the predicted temperatures were almost the same with those predicted by finite element analyses via ABAQUS. Hence, the modified model, characterizing first-order accuracy and explicit forward difference algorithm in time domain, and second-order accuracy in space domain, is comparably effective with the FEM. The validity of the modified model has also been established by comparing the predictions with test temperatures from the available literature. In total, 38 column tests were used for the validation.

Regarding the temperature development in composite columns, sensitivity studies on the effects of emissivity of fire, thermal contact resistance, models of thermal properties of concrete, cavity ratio, and section factor were carried out based on the validated model. It is found that the steel and concrete temperatures slightly increase as the emissivity of fire increases from 0.8 to 1.0, the influence is minor, and an emissivity of fire of 1.0 is recommended for conservative fire resistance design. The increase in thermal contact resistance at steel–concrete interface gives a decrease in steel temperature and an increase in concrete temperature, and a thermal contact resistance of 100 W/m K would be reasonable to predict temperatures of composite columns. The three models for thermal properties of concrete have negligible influence on the steel temperature but significant influence on the concrete temperature. Lie’s model for siliceous aggregates gives the highest concrete temperature, followed by Lie’s model for carbonate aggregates. The model in Eurocode 2 yields the lowest concrete temperature due to the lowest thermal conductivity of concrete. The cavity ratio has negligible influence on steel temperature but significant influence on the averaged concrete temperature. The increase in the averaged concrete temperature of the unprotected CFST-CFDST columns is not in linear relationship with the section factor, which is different from the unprotected steel columns. The averaged concrete temperature per section factor decreases with the increase in section factor in the early stage of fire exposure but increases at the later stage.

Finite element analysis has been widely used for determining the fire resistance of composite columns. However, it does not exactly follow Eurocode simplified design model. For the modified finite difference model with implementation via MATLAB, it is convenient to evaluate the said fire resistance through combination with the simple calculation models in Eurocode 4. The continuous development of buckling resistance and buckling reduction factor can be directly obtained, which is tedious by finite element analysis.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Singapore A*STAR for research project “Steel-concrete composite systems employing ultra-high strength steel and concrete for sustainable high-rise construction” under SERC grant no. 092 142 0045.

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