A fundamental task in structural stability analysis is to ensure the safety of structures throughout their operational life so as to prevent catastrophic consequences either at ambient or elevated temperatures. This thesis concerns the stability of unbraced steel frames due to abnormal loadings or fire loads, and develops practical methods to evaluate the stability capacity of unbraced steel frames at ambient temperature or in fire. The problem of determining the elastic buckling strengths of unbraced steel frames subjected to variable loadings can be expressed as an optimization problem with stability constraints based on the concept of storey-based buckling. The optimization problem can be solved by the linear programming method, which is considerably simpler and more suitable for engineering practice than the nonlinear programming method. However, it was found that the frame buckling strength obtained from the linear programming method based on Taylor series approximation on column stiffness may be overestimated in some cases. Thus, a secant approximation of the column stiffness was introduced, and a modified linear programming method based on the secant approximation was proposed. Numerical examples show that the linear programming method in light of the secant approximation can yield conservative results and maintain simplicity. In spite of the convenience of the modified linear programming method, numerical examples show that the linear programming method cannot accurately detect the maximum and minimum frame buckling strengths in some cases. Therefore, an alternative method to assess the lateral stiffness of an axially loaded column derived by using two cubic Hermite elements to signify the column is proposed. Unlike the column stiffness obtained from the Euler-Bernoulli beam theory containing transcendental functions, the stiffness in the proposed method includes only polynomials. Thus, the column stiffness within the proposed method enables the minimization and maximization problems to be solved by efficient gradient-based nonlinear programming algorithms, which overcome the inability of linear programming algorithm to detect the minimum frame buckling strength in some cases. The accuracy of the column stiffness associated with the proposed method was compared with that of the Euler-Bernoulli beam theory. Four unbraced steel frames were investigated to demonstrate the efficiency of the proposed method. It is known that the evaluation of the lateral stability of steel frames subjected to elevated temperatures is different from that at ambient temperature due to the degradation of material strength. Thus, the storey-based buckling method at ambient temperature was extended to evaluating the stability of unbraced steel frames subjected to elevated temperature. To simulate a steel column exposed to the elevated temperature, an analytical model was proposed to examine the effects of axial loading, elevated temperature, and thermal boundary restraints on the lateral stiffness of steel columns in unbraced frames. The procedure of evaluating the stability capacity of unbraced steel frames at elevated temperature was then concluded. Numerical examples are presented to demonstrate the evaluation procedure of the proposed method. The column model was then refined to evaluate the lateral stiffness of steel column subjected to non-uniform elevated temperature distributions along the longitudinal direction. The lateral stiffness equation of the column model was derived based on the Euler-Bernoulli beam theory. The procedure to evaluate the stability capacity of unbraced steel frames subjected to non-uniform elevated temperature distributions was then concluded. The numerical examples were investigated with the proposed method for non-uniform elevated temperature distributions. Finally, initial attempts were made to evaluate the stability of unbraced steel frames with fire-protected columns at different fire scenarios. A degradation factor charactering the variation of the Young's Modulus of steel at elevated temperature was introduced. The objective and constraint functions were constructed, and optimal tools were used to determine the buckling strength of an unbraced steel frame at different fire scenarios.

A fundamental task in structural stability analysis is to ensure the safety of structures throughout their operational life so as to prevent catastrophic consequences either at ambient or elevated temperatures. This thesis concerns the stability of unbraced steel frames due to abnormal loadings or fire loads, and develops practical methods to evaluate the stability capacity of unbraced steel frames at ambient temperature or in fire. The problem of determining the elastic buckling strengths of unbraced steel frames subjected to variable loadings can be expressed as an optimization problem with stability constraints based on the concept of storey-based buckling. The optimization problem can be solved by the linear programming method, which is considerably simpler and more suitable for engineering practice than the nonlinear programming method. However, it was found that the frame buckling strength obtained from the linear programming method based on Taylor series approximation on column stiffness may be overestimated in some cases. Thus, a secant approximation of the column stiffness was introduced, and a modified linear programming method based on the secant approximation was proposed. Numerical examples show that the linear programming method in light of the secant approximation can yield conservative results and maintain simplicity. In spite of the convenience of the modified linear programming method, numerical examples show that the linear programming method cannot accurately detect the maximum and minimum frame buckling strengths in some cases. Therefore, an alternative method to assess the lateral stiffness of an axially loaded column derived by using two cubic Hermite elements to signify the column is proposed. Unlike the column stiffness obtained from the Euler-Bernoulli beam theory containing transcendental functions, the stiffness in the proposed method includes only polynomials. Thus, the column stiffness within the proposed method enables the minimization and maximization problems to be solved by efficient gradient-based nonlinear programming algorithms, which overcome the inability of linear programming algorithm to detect the minimum frame buckling strength in some cases. The accuracy of the column stiffness associated with the proposed method was compared with that of the Euler-Bernoulli beam theory. Four unbraced steel frames were investigated to demonstrate the efficiency of the proposed method. It is known that the evaluation of the lateral stability of steel frames subjected to elevated temperatures is different from that at ambient temperature due to the degradation of material strength. Thus, the storey-based buckling method at ambient temperature was extended to evaluating the stability of unbraced steel frames subjected to elevated temperature. To simulate a steel column exposed to the elevated temperature, an analytical model was proposed to examine the effects of axial loading, elevated temperature, and thermal boundary restraints on the lateral stiffness of steel columns in unbraced frames. The procedure of evaluating the stability capacity of unbraced steel frames at elevated temperature was then concluded. Numerical examples are presented to demonstrate the evaluation procedure of the proposed method. The column model was then refined to evaluate the lateral stiffness of steel column subjected to non-uniform elevated temperature distributions along the longitudinal direction. The lateral stiffness equation of the column model was derived based on the Euler-Bernoulli beam theory. The procedure to evaluate the stability capacity of unbraced steel frames subjected to non-uniform elevated temperature distributions was then concluded. The numerical examples were investigated with the proposed method for non-uniform elevated temperature distributions. Finally, initial attempts were made to evaluate the stability of unbraced steel frames with fire-protected columns at different fire scenarios. A degradation factor charactering the variation of the Young's Modulus of steel at elevated temperature was introduced. The objective and constraint functions were constructed, and optimal tools were used to determine the buckling strength of an unbraced steel frame at different fire scenarios.