Consider two axially loaded columns, namely. $$1$$ and $$2,$$ made of a linear elastic material with Young's modulus $$2 \times {10^5}\,\,MPa,$$ square cross-section with side $$10$$ $$mm$$, and length $$1$$ $$m.$$ For Column $$1,$$ one end is fixed and the other end is free. For column $$2,$$ one end is fixed and the other end is pinned. Based on the Euler's theory, the ratio (up to one decimal place) of the buckling load of Column $$2$$ to the buckling load of column $$1$$ is ___________

If the following equation establishes equilibrium in slightly bent position, the mid-span deflection of a member shown in the figure is
$${{{d^2}y} \over {d{x^2}}} + {P \over {EI}}y = 0$$

If $$'a'$$ is amplitude constant for $$y,$$ then

The sketch shows a column with a pin at the base and rollers at the top. It is subjected to an axial force $$P$$ and a moment $$M$$ at mid-height. The reaction$$(s)$$ at $$R$$ is/are

Cross-section of a column consisting of two steel strips, each of thickness $$t$$ and width $$b$$ is shown in the figure below. The critical loads of the column with perfect bond and without bond between the strips are $$P$$ and $${P_0}$$ respectively. The ratio $$P/{P_0}$$ is