The frame ABC consists of two members AB and BC that are rigidly connected at joint B, as shown in part a of the figure. The frame has pin supports at A and C. A concentrated load P acts at joint B,...

The frame ABC consists of two members AB and BC that are rigidly connected at joint B, as shown in part a of the figure. The frame has pin supports at A and C. A concentrated load P acts at joint B, thereby placing member AB in direct compression. To assist in determining the buckling load for member AB, represent it as a pinned-end column,
as shown in part b of the figure. At the top of the column, a rotational spring of stiffness β_R
represents the restraining action of the horizontal beam BC on the column (note that the horizontal beam provides resistance to rotation of joint B when the column buckles). Also, consider only bending effects in the analysis (i.e., disregard the effects of axial
deformations).
(a) By solving the differential equation of the deflection curve, derive the buckling equation for this column:
[β_RL/EI](kL cot kL-1) - k²L²=0
in which L is the length of the column and EI is its flexural rigidity.
(b) For the particular case when member BC is identical to member AB, the rotational stiffness β_R
equals 3EI/L (see Case 7, Table H-2,
Appendix H). For this special case, determine the critical load P_cr.