Behavior of Steel–Concrete Partially Composite Beams Subjected to Fire—Part 2: Analytical Study

Abstract

This paper presents the development of an analytical model of steel–concrete partially composite beams subjected to fire. The model includes consideration of temperature dependent material properties, temperature dependent interface slip between concrete and steel, non-uniform temperature distributions throughout the cross-section and the effect of different rates of thermal expansion at the concrete–steel interface. Model predictions showed good agreement with the results of fire tests on two composite beams reported in an earlier companion paper as well as with limited experimental data published in literature. An extensive parametric study was undertaken by using the proposed model. Parameters considered in this study included geometric dimensions of the composite beam, material grades of steel and concrete, shear connection ratio, reinforcing steel ratio in the concrete slab, and load level on the beam. The parametric study clearly shows that shear connection ratio and load level significantly influence the fire performance of partially composite beams. The critical temperatures with shear connection ratio of 50%, 75% and 100% are 645°C, 602°C and 548°C, respectively, under load level of 0.6. The critical temperatures under load ratio of 0.5, 0.6 and 0.7 are 468°C, 553°C and 633°C respectively, with a shear connection ratio of 50%.

Appendix 1: Details of Composite Beam Model

1.1 Thermal Curvatures Calculation

Figure 14a shows the distribution of temperatures over the depth of the composite beam used in the model. As would be expected when a fire occurs underneath the beam, the steel beam has higher temperatures than the concrete. Because of the low thermal conductivity of concrete, a nonlinear variation of temperature with depth is expected in the concrete slab. Although steel and concrete have similar coefficients of thermal expansion, the different temperature fields within the two materials joined compositely lead to different rates of expansion and, consequently, to a complex stress state, particularly in the shear connectors.

Figure 14

Assumed distributions of temperature thermal strain through the depth of slab and steel beam

If we consider the nonlinear distribution of temperature through the thickness of the slab, the free thermal strain of the concrete slab through the thickness should also be nonlinear, as is shown in Fig. 14b. The free thermal strain can be expressed as:

$$ \varepsilon_{c,th} \,{ = }\,\alpha_{c} \left( {T_{c} - T_{0} } \right) $$

(15)

where, $T_{0}$ is the room temperature.

However, the actual thermal strain through the thickness of the slab must also be linear according to the plane sections remain plane assumption. The thermal strains $\varepsilon_{c1}$ and $\varepsilon_{c2}$ are assumed to be the actual thermal strain at the top and bottom surface of the concrete slab, respectively. Therefore, the difference between the actual thermal strain and free thermal strain can be calculated as

$$ \varepsilon_{c} = \left( {\varepsilon_{c1} - \varepsilon_{c2} } \right)y/h_{c} + \left( {\varepsilon_{c1} - \varepsilon_{c2} } \right)/2 - \varepsilon_{c,th} $$

(16)

Since the composite beam is assumed to be simply supported with a roller at one end, there is no external force and moment developed at the ends when the temperature rises. Based on equilibrium of axial force and bending moment in the concrete slab, as is shown in Eqs. (17)–(18), the values of $\varepsilon_{c1}$ and $\varepsilon_{c2}$ can be obtained.

$$ \sum {F = \int_{{ - h_{c} /2}}^{{h_{c} /2}} {\sigma_{c} b_{eff} dy} } = 0 $$

(17)

$$ \sum {M = \int_{{ - h_{c} /2}}^{{h_{c} /2}} {\sigma_{c} b_{eff} ydy} } = 0 $$

(18)

Hence, the thermal curvature of the concrete slab can be computed as:

$$ \varphi_{c,th} = \left( {\varepsilon_{c2} - \varepsilon_{c1} } \right)/h_{c} $$

(19)

The procedure for calculating the thermal curvature of the concrete slab is summarized in the flow chart shown in Fig. 15. This is an iterative procedure that is assumed to converge if moment M or axial force N in the slab are smaller than 10⁻⁴.

Figure 15

Flow chart for computing the thermal strain in the concrete slab

As for the steel beam, in order to simplify the calculation procedure, the contribution of the web to the thermal curvature of the entire section is neglected and the thermal curvature (Fig. 14c) can then be expressed as:

$$ \varphi_{s,th} = \left( {\varepsilon_{s2} - \varepsilon_{s1} } \right)/h_{s} = \alpha_{s} \left( {T_{s,bf} - T_{s,tf} } \right)/h_{s} $$

(20)

where, $T_{s,bf}$ is the temperature at the bottom flange of steel beam; $T_{s,tf}$ is the temperature at the top flange of steel beam; $h_{s}$ is the height of steel beam and $h_{c}$ is the thickness of concrete slab.

1.2 Mechanical Analysis of Composite Beam

As explained in Sect. 4.1, the composite beam model assumes all shear force at the steel–concrete interface is resisted by the shear studs. Shear forces exist at the boundary of the steel beam and the slab, and these shear forces produce axial force and flexure in the steel beam and concrete slab separately. Figure 16 shows the shear forces at the boundary of the top flange of the steel beam and the bottom of the concrete slab. Figure 16b shows simplified free body diagrams. As shown in this figure, the axial force and bending moment in the concrete slab or steel beam between shear stud i and shear stud j can be calculated as:

Figure 16

Simplified representation of forces of external and internal forces in a composite beam

$$ N_{i} = \sum\limits_{j = 1}^{i} {V_{j} } $$

(21)

$$ M = M_{c} + M_{s} + N_{i} (h_{c} + h_{s} )/2 $$

(22)

where, $V_{j}$ is the shear force on the shear stud j; $M_{c}$ is the bending moment in the concrete slab and $M_{s}$ is the bending moment in the steel beam.

1.3 Curvature and Strain Calculations

According to the second assumption explained in Sect. 4.1, plane sections remain plane within the concrete slab and within the steel beam, although there is a strain discontinuity at the interface. In addition, the concrete slab and steel beam have similar curvatures. The strain on the section can be represented as in Fig. 17a. The curvature of the slab can be expressed as:

Figure 17

Strain on cross section of composite beam and definition of coordinates for strain and stress

$$ \varphi_{c,T} = \varepsilon_{ct,T} /\left( {\gamma h_{c} } \right){ + }\varphi_{c,th} $$

(23)

where $\varphi_{c,T}$ is the curvature of concrete slab at elevated temperature; $\varepsilon_{ct,T}$ is the concrete strain at the top of the slab; $\gamma$ is the relative height of compressive area of the concrete slab; and $\varphi_{c,th}$ is the thermal curvature of the concrete slab induced` by non-uniform temperature expansion.

Therefore, the relative slip strain on the boundary of the steel beam and the concrete slab after taking the different thermal expansion into consideration can be written as

$$ \Delta \varepsilon = \varepsilon_{st,T} - \varepsilon_{{c{\text{b}},T}} - \varepsilon_{th} $$

(24)

where

$$ \varepsilon_{cb,T} = \varepsilon_{ct,T} - \varepsilon_{ct,T} /\gamma = \left( {1 - \gamma } \right)/\gamma \cdot \varepsilon_{ct,T} $$

(25)

$$ \varepsilon_{st,T} = \varepsilon_{{c{\text{t}},T}} h_{s} /\left( {\gamma h_{c} } \right) + \varepsilon_{sd,T} $$

(26)

$$ \varepsilon_{th} = \alpha_{s} T_{s,tf} - \alpha_{c} T_{c,b} $$

(27)

$\varepsilon_{{s{\text{t}},T}}$ is the steel strain at the top boundary of the steel beam; $\varepsilon_{{c{\text{b}},T}}$ is the concrete strain at the bottom of the slab; $\varepsilon_{th}$ is the difference of thermal expansion strain between steel and concrete; $\varepsilon_{sd,T}$ is the steel strain at the bottom of the steel beam; $T_{s,tf}$ is the temperature of the top flange of the steel beam; and $T_{c,b}$ is the temperature at the bottom of the concrete slab.

Substituting Eqs. (25)–(27) into (24), results in:

$$ \Delta \varepsilon = \varepsilon_{ct,T} (h_{s} + h_{c} )/\left( {\gamma h_{c} } \right) - (\varepsilon_{ct,T} - \varepsilon_{sd,T} ){ - }\left( {\alpha_{s} T_{s,tf} - \alpha_{c} T_{c,b} } \right) $$

(28)

1.4 Mechanical Analysis of Concrete Slab

For the concrete slab, a coordinate system is defined, as shown in Fig. 17b. The strain for one position which is $y^{{\prime }}$ away from the top of slab can be given as

$$ \varepsilon_{{y^{{\prime }} ,T}} = \varepsilon_{ct,T} - \varphi_{c,T} y^{{\prime }} = \varepsilon_{ct,T} - \varepsilon_{ct,T} y^{{\prime }} /\left( {\gamma h_{c} } \right) $$

(29)

Based on the strain–stress relationship of concrete at elevated temperatures given in Eurocode 2 [15], the stress in the concrete $\sigma_{c} \left( {y^{{\prime }} } \right)$ can be obtained. Further, the axial force and bending moment in the concrete slab can be calculated as:

$$ N_{c} = \int_{0}^{{h_{c} }} {\sigma_{c} \left( {y^{{\prime }} } \right)} b_{eff} dy^{{\prime }} $$

(30)

$$ M_{c} = \int_{0}^{{h_{c} }} {\sigma_{c} } (y^{{\prime }} )b_{eff} (h_{c} /2 - y^{{\prime }} )dy^{{\prime }} $$

(31)

where $b_{\text{eff}}$ is the effective width of the concrete slab.

1.5 Mechanical Analysis of Steel Beam

As for the steel beam, another coordinate system is defined, as shown in Fig. 17c. The axial force and bending moment in the steel beam are denoted as $N_{\text{s}}$, $M_{s}$, respectively. Since it assumed that no external axial force is applied on the composite beam, $N_{\text{s}}$ can be calculated as:

$$ N_{s} = - N_{c} $$

(32)

The axial strain in the steel beam generated by axial force is:

$$ \varepsilon_{s0,T} = - N_{s} /\left( {E_{s,T} A_{s} } \right) $$

(33)

where $A_{s}$ is cross sectional area of steel beam.

The curvature of the steel beam after taking the non-uniform thermal expansion into consideration can be expressed as

$$ \varphi_{s,T}^{{\prime }} = \varphi_{s,T} - \varphi_{s,th} $$

(34)

where $\varphi_{s,T}$ is the mechanical curvature of the steel beam at elevated temperature; and $\varphi_{s,th}$ is the thermal curvature of the steel beam.

The strains at the top and bottom of the steel beam induced by bending moment are denoted as $\varepsilon_{st,T}$ $\varepsilon_{sb,T}$, respectively, and their corresponding expressions are:

$$ \varepsilon_{{s{\text{b}},T}} \,{ = }\, - \varphi_{s,T}^{{\prime }} h_{s} /2\,{\text{and}}\,\varepsilon_{st,T} { = }\varphi_{s,T}^{{\prime }} h_{s} /2 $$

(35)

Therefore, the total strain at the top of the steel beam is the sum of axial strain and bending strain, and can be written as

$$ \varepsilon_{st,T}^{{\prime }} = \varphi_{s,T}^{{\prime }} h_{s} /2 + \varepsilon_{s0,T} $$

(36)

The strain for a fiber that is $y^{{{\prime \prime }}}$ away from the top of the steel beam can be given as:

$$ \varepsilon_{s,T} \, = \,\varepsilon_{st,T}^{{\prime }} - \varphi_{s,T}^{{\prime }} y^{{{\prime \prime }}} \, = \,\varphi_{s,T}^{{\prime }} h_{s} /2\, + \,\varepsilon_{s0,T} - \varphi_{s,T}^{{\prime }} y^{{{\prime \prime }}} $$

(37)

Based on the strain–stress relationship of steel at elevated temperatures given in Ref. [18], the stress in the steel beam $\sigma_{s} \left( {y^{{{\prime \prime }}} } \right)$ can be obtained. Further, the axial force and bending moment in the steel beam can be calculated as:

$$ N_{s} = \int_{0}^{{h_{s} }} {\sigma_{s} } \left( {y^{{{\prime \prime }}} } \right)b_{s} dy^{{{\prime \prime }}} $$

(38)

$$ M_{s} = \int_{0}^{{h_{s} }} {\sigma_{s} } \left( {y^{{{\prime \prime }}} } \right)b_{s} (h_{s} /2 - y^{{{\prime \prime }}} )dy^{{{\prime \prime }}} $$

(39)

where $b_{s}$ is the width of the steel beam.

The procedure for the strain calculation in the concrete slab and steel beam is summarized in the flow chart shown in Fig. 18. Using this procedure, the strain distribution in the composite beam can be obtained by equilibrium between internal and applied external forces on the composite beam.

Figure 18

Flow chart for computing mechanical strain in the concrete slab and steel beam

1.6 Deflection and Slip Calculation

A computer program using the software package MATLAB [23] was developed based on the analytical model presented above. First, thermal strains in the concrete slab were computed according to flow chart in Fig. 15. Next, mechanical strains in the steel beam and concrete slab were computed according to flow chart in Fig. 18, and then the deflection and slip in the composite beam can be computed based on the flow chart shown in Fig. 19. Finally, the defection-temperature curve and slip-temperature curve can be plotted. From the curves, the critical temperature in the bottom flange of the steel beam can be determined.

Figure 19

Flow chart for computing deflection and slip for composite beam