Effect of Rotational Restraint on Inelastic Buckling of Steel Column at Elevated Temperatures

Abstract

When a fire develops within the fire compartment of a building, the columns in the compartment are heated, whereas the beams and columns outside the compartment remain cold. Hence, the relative stiffness ratio between the cold members and heated column increases. Consequently, the end-rotational restraint on the heated column is increased, and the effective buckling length of the heated column becomes shorter compared with the ambient temperature condition. However, studies regarding the effect of increased end rotational restraint on heated columns are rare. Although some recommendations to consider this effect are available in representative design standards, they are suggested under limited or ideal conditions and hence not applicable under general conditions. In this study, a procedure for calculating the effective length factor of non-sway steel columns with general end restraints at elevated temperatures is proposed based on the classical Euler column theory and tangent modulus concept. The proposed procedure is validated based on a test-backed nonlinear finite element analysis. It is shown that the proposed equations can be combined with the existing column buckling curve to predict the column buckling strength with significantly increased accuracy compared with the methods suggested by representative design standards.

Appendix

The following symbols are used hereincx

$E\left(T\right)$ :

Elastic modulus of steel at temperature $T$

${E}_{t}\left(T\right)$ :

Tangent modulus of column at temperature $T$

${E}_{0}$ :

Elastic modulus of steel at ambient temperature

${F}_{cr}\left(T\right)$ :

Critical stress of column at temperature $T$

${F}_{e}\left(T\right)$ :

Euler stress of column at temperature $T$

${F}_{FEM}$ :

Critical stress obtained from finite element analysis

${F}_{p}\left(T\right)$ :

Proportional limit of steel at temperature $T$

${F}_{p0}$ :

Proportional limit of steel at ambient temperature

${F}_{r}$ :

Maximum residual stress in wide-flange section at ambient temperature

${F}_{y}\left(T\right)$ :

Yield stress of steel at temperature $T$

${F}_{y0}$ :

Yield stress of steel at ambient temperature

$I$ :

Moment of inertia

$K$ :

Effective buckling length factor of column

${K}_{Euler}$ :

Effective buckling length factor of Euler column

${K}_{FEM}$ :

Effective buckling length factor obtained from finite element analysis

${K}_{pr\left(AISC\right)}$ :

Effective buckling length factor calculated via proposed equation using tangent modulus reduction factor from AISC column curve at elevated temperatures

${K}_{pr\left(EC3\right)}$ :

Effective buckling length factor calculated via proposed equation using tangent modulus reduction factor from EC3 column curve at elevated temperatures

${K}_{pr\left(FEM\right)}$ :

Effective buckling length factor calculated via proposed equation using numerical results

$L$ :

Length of column

${L}_{c}$ :

Effective length of column

${P}_{cr}(T)$ :

Critical load at temperature $T$

${P}_{e}(T)$ :

Euler load at temperature $T$

$T$ :

Temperature of steel column

$k$ :

Parameter to consider applied load in elastic column

${k}_{E}(T)$ :

Retention factor for elastic modulus of steel at temperature $T$

${k}_{p}(T)$ :

Retention factor for proportional limit of steel at temperature $T$

${k}_{y}(T)$ :

Retention factor for yield stress of steel at temperature $T$

${k}_{\theta }$ :

Rotational stiffness of beams and columns adjacent to column end

${n}_{c}$ :

Parameter to consider column continuity

$r$ :

Radius of gyration of section

$u$ :

Parameter to consider applied load in elastic column

${u}_{t}$ :

Parameter to consider applied load in inelastic column

$\alpha$ :

Imperfection factor for column buckling in fire design situation

${\overline{\lambda }}_{0}$ :

Nondimensional slenderness ratio of column at ambient temperature

${\overline{\lambda }}_{p0}$ :

Nondimensional slenderness ratio of column without effect of rotational restraint at ambient temperature

$\overline{\lambda }\left(T\right)$ :

Nondimensional slenderness ratio of column at temperature $T$

${\overline{\lambda }}_{p}(T)$ :

Nondimensional slenderness ratio of column without effect of rotational restraint at temperature $T$

$\mu$ :

Elastic rotational stiffness ratio of column end at temperature $T$

${\mu }_{0}$ :

Elastic rotational stiffness ratio of column end at ambient temperature

${\mu }_{eq}$ :

Equivalent rotational stiffness ratio at temperature $T$

${\mu }_{t}$ :

Inelastic rotational stiffness ratio of column end at temperature $T$

$\varphi$ :

Parameter to determine reduction factor ${\chi }_{fi}$

${\chi }_{fi}$ :

Reduction factor for column buckling in fire design