1. Introduction
The objective of this study is to examine the relationship between battened built-up columns and frames about buckling. Especially, the characteristics of the Bleich's formula are investigated with respect to the upper bound theory and moment distribution. The buckling strengths, buckling modes and bending moment of one bay －n stories frame with rigid beams are calculated by the energy method. The results are compared with those of obtained by the Bleich's formula and correct solutions.
2. Analytical work
The analytical model is shown in figure 3. Buckling strengths are calculated by the three methods. Firstly, the buckling formula presented by Bleich is shown, and analytical method using the energy method is described, where the buckling modes are assumed by the equations (12). The buckling equation is presented as equations (22). Moreover it is shown that the correct buckling strengths can be calculated by the equations (23) and (24).
3. Results and Discussion
As the analytical parameters, the number of stories n and the value of 4Ic/Acb2 are selected. The values of n are 1, 2, 3, 5, 10 and the value of 4Ic/Acb2 ranges from zero to one. Figure 5 and 6 show the correct dimensionless buckling strength p*-4Ic/Acb2 relations and pg*-4Ic/Acb2 relations. Figure 8 shows the ratio of the Bleich's strength to the correct value and the strength obtained by the energy method to the correct value.
Although the buckling strengths by Bleich agree fairly well with correct values, it is shown that the strengths obtained by Bleich equation are neither the upper bound nor the lower bound. Moreover, Figure 12 shows that Bleich's moment distributions correspond to those of obtained by the energy method.
4. Conclusions
The conclusions derived from this study are as follows:
1) Governing parameters of the problem are the number of story n and the value of 4Ic/Acb2.
2) Although the buckling strengths by Bleich agree fairly well with correct values, the strengths by Bleich are neither the upper bound nor the lower bound.
3) Buckling strengths calculated by the energy method estimate the correct strength within 5% error in case of n>2. As is well known, the buckling strength by the energy method gives the upper bound.
4) Bleich's moment distributions correspond to those of obtained by the energy method.
