@phdthesis{oai:tsukuba.repo.nii.ac.jp:00008713,
 author = {Ruangtananurak, Nara},
 month = {},
 note = {When fire takes place in a steel frame, any heated column may buckle at some temperature due to deterioration in both the stiffness and the strength, although the external load to the frame remains unchanged during fire. This column’s buckling is likely a main cause of the instability for steel frames subjected to fire.  Researches on the high temperature buckling of columns span, in fact, for decades.  Culver (1973) was the first to analyze precisely the behavior of the buckling of columns by using finite element analysis. Referring to the experimental studies conducted by Burgess et al.(1992), Franssen has recently made a specific comparison between the test and numerical analysis results (1995,1996).  Although the studies on centrally compressed and simply supported columns provide the basic knowledge on the problem, the behaviors of the heated columns incorporated into the frame are different from those of isolated members. In fact, when fire occurs in a steel frame, the expanded and therefore elongated column is axially restrained by the rest of members, so that thermal compressive force is applied additionally to the heated column (Neves, 1995; Correia Rodrigues et al., 2000). As a result, the buckling temperature of the heated column is lowered due to this thermal stress. On the other hand, since the incorporated column is, at the same time, restrained rotationally at its ends by the neighboring members, the effective length of this column should become shorter than the column height and this raises in turn the buckling temperature of the column.  Another facet of the problem is the overall stability of the frame, when and after the incorporated columns buckle due to fire. Some frames may fall into instability directly after the incorporated column buckle, while the other may remain stable and be able to sustain further increase in member temperature. Once the instability takes place, the former frame cannot maintain static equilibrium any more. This means that, to solve the problem, ordinary load controlled analysis is not applicable in this case, but we need to develop another new numerical method to analyze the unstable frames. For the latter stable frames with heated and buckled columns, on the other hand, stress redistribution capacity of the neighboring members may play an essential role on controlling their ultimate states. Therefore, to clarify the structural performances of the frames with heated and buckled columns, a special emphasis must be placed on their stabilities as well as their ultimate load carrying characteristics when and after several heated columns buckle. The adopted analysis method must be able to trace these natures of the problem accordingly.  In view of the above observations, the main objectives of this study are to be set as follows: 1. To understand and to clarify the mechanics which control the overall stabilities of steel frames with heated and buckled columns.  2. To develop a numerical analysis program that can solve such stability problems of steel frames subjected to fire.  3. To clarify the buckling temperatures of the incorporated columns as well as the frame’s ultimate temperatures.  4. To find the actual ultimate states and the corresponding ultimate temperatures of steel frames subjected to fire.  5. To find the means to prevent the frames subjected to fire from instability.  In order to achieve the above presented objectives, this dissertation is divided into five chapters, which are briefly described below.  Chapter one starts with brief review of previous studies. Fire resistant design of steel structures is reviewed in many relevant aspects, the experimental tests, and numerical analysis. After that, the problem about instability in steel structure that occurred in the collapse of WTC Towers is summarized. This is considered an indispensable background for conceiving the objective of this study, which constitutes the above sections.  In chapter 2, the stabilities and the ultimate behaviors of steel frames after the incorporated heated columns buckle are investigated extensively. If the surrounding members can redistribute the vertical load of the frame which has been carried by the buckled column, it may retain the stability and may not collapse immediately. On the contrary, when the surrounding members are weak, an overall frame will fall into instability directly due to column buckling. This chapter describes and discusses these problems specifically. Furthermore, in this chapter, the development of the numerical analysis method is described specifically, which can simulate both the stable and unstable behaviors of steel frames subjected to fire.  The development is focused on the computational procedure to simulate an unstable and therefore actually a dynamic frame behavior purely by a static means. At the end of this chapter, example numerical solutions solved by the method developed are shown, where complete fire responses and realistic ultimate states due to fire of steel frames are illustrated and discussed extensively.  By using the numerical method whereby the control of the analysis is switched between load and displacement methods, we can examine the behavior of the steel column subjected to fire until it overall collapses. The load controlled analysis is the increase-in-temperature method. While the displacement analysis is the analysis that one degree-of-freedom of the unstable member’s node is gripped, and it is moved in the direction that the unstable state is growing. The displacement analysis is carried out under the temperature that the structure loses its stability.  By using this analysis method to analyze some examples of a steel structure subjected to fire, the following conclusions can be drawn: ･ The main unstable condition of a steel structure subjected to fire is the “snap-through” process. This snap-through easily occurs in the structure when the post-buckling residual force is low and the stiffness and the strength of the restraining members in the structure are low. That means that the structure’s ability to remain stable while the surrounding members restrain the buckling column is intimately connected with the instabilities of the structure itself.  ･ The collapse of the structure is not decided solely by the buckling of the heated column. The ultimate temperature of the structure’s collapse mode varies according to the stress redistribution ability of the surrounding members. If the stress redistribution ability of the surrounding members is low, the structure loses its stability and collapses suddenly after the column buckles. In this case, the ultimate temperature of this kind of structure is lower than the column buckling temperature due to the thermal stress. However, if the stress redistribution ability of the surrounding members is high, the structure still keeps its stability even though the column buckles. The ultimate temperature of this kind of structure is more than the column buckling temperature.  The third chapter clarifies specifically the buckling temperatures of steel columns incorporated into frames. The buckling temperature is defined as the one at which a column begins to show an apparent buckling deformation. The incorporated columns are restrained rotationally by the adjacent members when they buckle. Therefore the effective buckling length of the column is shorter than its height. In this chapter, simple closed form formulae are presented to estimate both the effective buckling length and the buckling temperature of an incorporated heated column. Finally, comparisons are made between the numerically solved buckling temperatures of incorporated columns and the corresponding theoretical predictions, and the accuracy and the applicability of the formula are discussed in detail.  From the investigated results, the following conclusions can be drawn: ･ The rotationally restraining effect of the connecting members has an effect on the column’s buckling temperature. However, the thermal expansion of the beam does not have a direct affect on the buckling temperature.  ･ The exterior column is not axially restrained by the adjacent members during the fire. The lower end of the heated column is rotationally restrained by the unheated connecting members, so its boundary condition can be assumed to be fixed end. On the other hand, for the upper end of the heated column’s case, the rotationally restraining effect of the upper-story unheated column can be neglected due to the thermal expansion of the beam. The buckling temperature of the exterior column can be assumed to be the theoretical buckling temperature when the effective length is determined from the proposed equation.  ･ The interior column is strongly rotationally restrained by the adjacent members, so its effective length is equal to 0.5 h . However, it is also axially restrained by the adjacent members. As a result it buckles at a temperature that is lower than the estimated buckling temperature. In this case, when considering the buckling temperature, the thermal effect should be included.  An incorporated column is not only restrained rotationally as discussed in the previous chapter but, if it is heated, it is also restrained axially by the surrounding members. Chapter 4 clarifies the latter problems specifically. The axial restraint of the surrounding members plays two contrary roles on the stability action of the frame. Firstly, it adds axial compressive force to the incorporated column and so increased compressive force lowers the column’s buckling temperature. On the contrary, once the column buckles, the surrounding members turn to work so as to redistribute a part of the axial compressive force carried by the bucked column to other sound columns, which is therefore helps to strengthen the overall stability and to raise the ultimate temperature of the frame. This is called herein stress redistribution effect of the surrounding members. In this chapter, the lowering of the column’s buckling temperature and the raise of the frame’s ultimate temperature, which are both brought by the effects of the surrounding members, are formulated respectively in theoretical closed forms. Finally the numerical analysis, which are developed in Chapter 2, are conducted extensively to obtain the realistic column’s buckling and the frame’s ultimate temperatures of a lot of practical steel frames, which are used to verify the accuracy and the applicability of the above derived formulae. In this chapter, the following conclusions can be drawn: ･ The effects of the axially restraining members have two mechanical roles.  The first role is to restrain an expanded heated column. As a result of this action, the thermal compressive force is added to the heated column. This causes a drop in buckling temperature of heated columns. On the contrary, the second role is to redistribute a part of the compressive force of the buckled column to other sound columns. As a result of this action, the structure can retain its stability after the column buckles. It causes a rise of the ultimate temperature of a frame. However, if the second stress redistribution ability is not enough, it falls into instability immediately after the heated column buckles. The ultimate temperature for this frame is lower than the theoretical buckling temperature, since the buckling is affected by the first thermal effect.  ･ Based on the above observation, it is found that a higher ultimate temperature of a frame is obtained if it has higher stress redistribution capacity. A way to improve the fire-resistance capacity of a frame is to install a suitably proportioned hat truss. A hatted frame is found to have definitely an improved redundancy.  ･ Closed form formulae to predict both the buckling temperature of a heated column and the beam plastified temperature of a frame is presented herein.  Conducting the numerical fire response analysis, the predicted temperatures are found to be in good agreement with the numerical results for various structural and heating conditions. The prediction to estimate the improved ultimate temperature of hatted frames is also proposed herein.  In chapter 5, the conclusion of this thesis is drawn: ･ The “snap-through” phenomenon is the main unstable condition of steel structures subjected to fire. This snap-through easily occurs in the structure when the post-buckling residual force is low and the stiffness and the strength of the restraining members in the structure are low.  ･ The column’s buckling temperature can be estimated by the theoretical buckling temperature when the effective length is determined from the proposed equation. The exterior column’s effective length ratio can be calculated from the proposed equation while the interior column’s effective length ratio is 0.5.  ･ In the case that the stress redistribution ability is not enough to retain the stability of the steel frame, the structure falls into the instability state and failure at the buckling temperature under thermal effect. It means the ultimate temperature of that structure is lower than the theoretical buckling temperature.  However, in the case that the stress redistribution ability is high, a part of the compressive force of the buckled column is redistributed to other sound columns. As a result the structure can retain its stability after the column buckles.  ･ The ultimate temperature can be determined by the higher temperature among the bucking temperature under thermal effect and the beam plastified temperature.  ･ A way to improve the fire-resistance capacity of a steel frame is to increase the redundancy of the axial force to the steel frame., 2003, Includes bibliographical references},
 school = {筑波大学, University of Tsukuba},
 title = {Ultimate temperatures and stabilities of steel frames subjected to fire},
 year = {2004}
}