تجزیه و تحلیل حساسیت نیروهای حیاتی از خرپاهای با کمانش جانبی

The present research is devoted to the study of out-of-plane buckling of trusses with elastic side bracing. In this paper, a sensitivity analysis of critical buckling loads of a truss due to bracing stiffness is carried out. A method based on the sensitivity analysis for the determination of the threshold bracing stiffness condition for full bracing of a truss is proposed. The influence lines of the unit change of the bracing stiffness on the buckling load, for different initial bracing stiffness, are investigated. The approximations of an exact relation between the buckling load and bracing stiffness are found. The buckling length related to the side-support distance as a function of bracing stiffness is also determined. It is shown that the buckling length of truss chords with elastic side supports is larger than that assumed in design codes.

Steel trusses have a much greater strength and stiffness in their plane than out of their plane, and therefore should be braced against lateral deflection and twisting. The problem of bracing requirements necessary to provide lateral stability of compressed members is present in the Polish design codes [1], and in the Eurocode [2]. The simplified design code requirements allow one to assume that the buckling length of truss chords is equal to the distance between braces. In this approach, only the truss top chord is considered. The effect of the lower chord, verticals and diagonals on the truss stability is neglected. Verticals and diagonals are considered only as vertical supports to the upper truss chord, side bracing of the truss chords is considered as a rigid side support, and normal forces in the truss chords are assumed to be constant along their length. Under the above conditions, the buckling length of compressed truss chords is usually lower than it is described as being in the codes. Such a result was obtained by Biegus and Wojczyszyn [3]. However, the bracing is usually elastic, and other truss elements such as compressed diagonals or verticals may buckle locally, and this may lower the critical loading of the truss. Many solutions of restrained column and beam buckling are presented in the literature (see, for example Trahair [4]). Most code requirements concerning bracing are based on the principles formulated by Winter [5], who introduced a simple model with fictitious hinges at the braced joints. His research was focused on an estimation of the safe lower limit of the necessary rigidity of the bracing, such that the braced element would attain maximal critical force and that buckling occurs between braces. In research conducted by Yura [6], the bracing requirement was extended to cases where the bracing stiffness is less than the full bracing condition. Bracing requirements for frame structures were investigated by Tong and Ji [7], where a threshold bracing condition for full bracing of plane frames was derived and a critical buckling force for weakly braced frames was determined. Studies conducted by Girgin, Özmen and Orakdogen [8] showed that code formulae for determining the buckling lengths of frame columns may yield erroneous results, especially for irregular frames. Similar problems of determining the bracing requirements of trusses are present in only a few studies. The stability of trusses with elastic bracing was investigated in experimental research by Kołodziej and Jankowska-Sandberg [9] and verified by numerical analysis [10] by Iwicki. The results of the author’s numerical studies [11] of two roof trusses with horizontal and sloping elastic bracing have shown that the buckling length of truss compressed chords is greater than the side-support spacing. The spatial stability of trusses designed according to the Polish code [1] is provided even when the buckling length of truss chords is greater than the side-support spacing. The stability of a truss with both linear and rotational bracing was analyzed by Iwicki [12]. For a truss examined with both linear and rotational springs, the limit normal force in the chords was between 20% and 70% greater than in the case without rotational springs. The present research is focused on the determination of the full bracing condition for a truss with elastic bracing. The basic problem under consideration is devoted to investigating the required bracing stiffness that ensures that the out of truss plane buckling occurs between braces, or is prevented, so the buckling occurs in the plane of the truss. The full bracing condition may also be defined as the bracing stiffness that causes the maximal buckling load of the truss, or when an increase in bracing stiffness does not result in a further increase in the buckling load. In the study, the sensitivity analysis method developed by Haug, Choi, and Komkov [13] is used. This method enables one to obtain the influence lines of the buckling load variations due to unit changes in the bracing stiffness. The approximations of an exact relation between the buckling load and bracing stiffness are found. For different stiffnesses of elastic side supports, the critical load and coefficient of buckling length of the truss chord are calculated. The results are compared to established solutions presented by Trahair [4] and Winter [5]. The sensitivity of truss stability was also analyzed in [11] by Iwicki, but that analysis was confined to sensitivities related to the non-linear limit load of imperfect trusses.

The effect of bracing stiffness on the critical buckling load of trusses was investigated. The results of the numerical analysis and sensitivity analysis allow one to draw some conclusions regarding the effect of bracing stiffness on the critical buckling load. • The critical buckling load of a truss depends on the stiffness and spacing of braces. • The sensitivity analysis allows one to obtain the influence lines of the critical buckling load variation due to the location of a new unit stiffness brace. The sensitivity influence lines may be helpful in the design of bracing. • The sensitivity influence lines of the truss top chord normal force variation, corresponding to the critical buckling load, due to bracing stiffness variation depend on the initial bracing stiffness. • The threshold bracing stiffness of the truss top chord can be calculated by means of the sensitivity analysis. The higher-order critical load, calculated for a truss without bracing, which is insensitive to the change in bracing stiffness, is the maximum of the first critical load that may be reached due to the increase in bracing stiffness. • For the examined truss with fewer than five braces, the threshold condition for full bracing corresponds to the out-of-plane buckling of the truss between braces. At a certain number of braces, local buckling in the plane of the truss may occur. In such a case, further increase in bracing stiffness or number of braces is not necessary, because this does not improve the stability of the structure. • The main difference in the stability analysis between the 3D truss model and established solutions of the truss chord models consists, in effect, of the local buckling of other truss elements, which is neglected in the models of the truss chords. • Comparison between the stability analysis conducted and established solutions allows one to conclude that the threshold bracing stiffness and the truss top chord normal force, corresponding to the critical buckling load, of the braced 3D truss is greater than for the similar column model with the same number of braces. • The stability analysis of the classical Winter model allows one to obtain lower critical forces for the same bracing stiffness than the 3D truss model for a low magnitude of bracing stiffness. • In the examined example the buckling length of the truss chord is greater than the distance between braces. The coefficient of effective buckling length for truss chords is greater than that predicted in current codes, and this effect is neglected in simplified code formulae. • The sensitivity analysis allowing for the calculation of threshold bracing stiffness and the maximal critical load in the truss chord may be carried out by means of standard commercial structural analysis programs and commercial spreadsheet programs.