# 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%.

### Keywords

Composite beams Analytical approach Partially composite Fire resistance### List of Symbols

- \( \sigma_{c,T} \)
Stress of concrete at elevated temperature

- \( f_{c,T} \)
Ultimate stress of concrete at elevated temperature

- \( \varepsilon_{c,T} \)
Strain of concrete at elevated temperature

- \( \varepsilon_{c1,T} \)
Strain at the ultimate stress of concrete at elevated temperature

- \( \varepsilon_{cu1,T} \)
Ultimate strain of concrete at elevated temperature

- \( E_{c,T} \)
Elastic modulus of concrete at elevated temperature

- \( E_{c} \)
Elastic modulus of concrete at room temperature

- \( T_{c} \)
Temperature of concrete

- \( \sigma_{s,T} \)
Stress of steel at elevated temperature

- \( \varepsilon_{s} \)
Strain of steel

- \( f_{p,T} \)
Proportional limit of steel at elevated temperature

- \( f_{y,T} \)
Yield stress of steel at elevated temperature

- \( E_{s,T} \)
Elastic modulus of steel at elevated temperature

- \( \varepsilon_{y,T} \)
Strain of steel at elevated temperature

- \( \varepsilon_{p,T} \)
Strain at the proportional limit

- \( \varepsilon_{t,T} \)
Limiting strain for yield stress

- \( \varepsilon_{u,T} \)
Ultimate strain of steel at elevated temperature

- \( V_{T} \)
Shear load on the stud at elevated temperatures

- \( V_{u} \)
Shear resistance of the stud at room temperatures

- \( m \)
Material dependant parameters of shear stud

- \( n \)
Material dependant parameters of shear stud

- \( s \)
Slip of the stud at elevated temperatures

- \( s_{u} \)
Ultimate slip of the stud at elevated temperatures

- \( V_{\hbox{max} } \)
Maximum shear force of stud

- \( s_{\hbox{max} } \)
Maximum slip of stud

- \( V_{\hbox{min} } \)
Minimum shear force of stud

- \( s_{\hbox{min} } \)
Minimum slip of stud

- \( V_{j} \)
Shear force on the shear stud j

- \( M_{c} \)
Interior bending moment of concrete slab

- \( M_{s} \)
Interior bending moment of steel beam

- \( h_{c} \)
Thickness of concrete slab

- \( h_{s} \)
Height of steel beam

- \( \varphi_{s,T} \)
Curvature of steel beam at elevated temperature

- \( \varphi_{c,T} \)
Curvature of concrete slab at elevated temperature

- \( \varepsilon_{ct,T} \)
Concrete strain at the top boundary of slab

- \( \gamma \)
Relative height of compressive area of concrete slab

- \( \varphi_{c,th} \)
Curvature of concrete slab results by non-uniform temperature

- \( \varepsilon_{{s{\text{t}},T}} \)
Steel strain at top boundary of steel beam

- \( \varepsilon_{cb,T} \)
Concrete strain at bottom boundary of slab

- \( \varepsilon_{th} \)
Difference of thermal strain between steel and concrete

- \( \varepsilon_{sd,T} \)
Steel strain at bottom boundary of steel beam

- \( \alpha_{s} \)
Thermal expansion coefficient of steel

- \( \alpha_{c} \)
Thermal expansion coefficient of concrete

- \( T_{s,tf} \)
Temperature of top flange of steel beam

- \( T_{c,b} \)
Temperature of concrete slab at the bottom boundary

- \( A_{s} \)
Sectional area of steel beam

- \( \varphi_{s,th} \)
Thermal curvature of steel beam result by non-uniform temperature

*R*Load ratio of composite beam

*η*Shear connection ratio of composite beam

*α*Enforcing steel ratio in concrete slab

*Q*Grade of steel

*C*Grade of concrete

- \( b_{\text{eff}} \)
Effective width of concrete slab in composite beam

## Notes

### Acknowledgments

This material is based upon the work supported by the Open Research fund of State Key Laboratory for Disaster Reduction in Civil Engineering (Grant number:SLDRCE-MB-05). The assistance of the staff in Ferguson Structural Engineering Laboratory in the University of Texas at Austin, and the support of China Scholarship Council are also greatly acknowledged.

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