Solved Problems

Problem: A W200x46 steel column ($E = 200$ GPa) has a length of 6 m. It is pinned at both ends. Determine the critical buckling load. Properties of W200x46: $I_x = 45.5 \times 10^6$ mm$^4$, $I_y = 15.3 \times 10^6$ mm$^4$.

Problem: A hollow aluminum tube acts as a flagpole fixed at the base and free at the top. The length is 4 m. The outer diameter is 100 mm and the thickness is 5 mm. $E = 70$ GPa. Determine the critical load.

Problem: A solid circular steel rod ($E = 200$ GPa, Yield Stress $\sigma_y = 250$ MPa) is used as a column pinned at both ends. Determine the smallest slenderness ratio ($L/r$) for which Euler's Formula is valid (i.e., buckling occurs before yielding).

Problem: A rectangular wooden column has cross-sectional dimensions of $100 \text{ mm} \times 200 \text{ mm}$ and a length of $3 \text{ m}$. It is pinned at both ends. Determine the critical buckling load ($P_{cr}$) if it is free to buckle in any direction. Assume $E = 12 \text{ GPa}$. Then, determine the critical buckling load if the column is laterally braced at its mid-height against buckling about the weak axis.

Problem: A standard steel pipe ($E = 200$ GPa) is used as a column. It has an outside diameter of 114 mm and an inside diameter of 102 mm. The column is 5 m long, fixed at the base and pinned at the top. Determine the critical buckling load.

Problem: A square timber column ($E = 11$ GPa) has dimensions $150 \text{ mm} \times 150 \text{ mm}$. It is 4 m long and fixed at both ends. Determine the critical buckling load.

Problem: A 7 m long steel column ($E = 200$ GPa) is pinned at both ends. It must support an axial compressive load of 500 kN with a factor of safety of 2.5 against buckling. Determine the minimum required moment of inertia.

Problem: A W250x73 steel column ($E = 200$ GPa, Yield Stress $\sigma_y = 345$ MPa) is 5 m long and pinned at both ends. An eccentric axial load of 800 kN is applied at a distance $e = 50$ mm from the strong axis. Determine the maximum compressive stress in the column. Properties of W250x73: $A = 9280 \text{ mm}^2$, $r_x = 110 \text{ mm}$, $c = 126.5 \text{ mm}$ (half of depth).

Problem: A built-up column is composed of two channels laced together. The overall slenderness ratio of the column acting as a unit is $(KL/r)_o = 60$. The lacing bars connect the channels at intervals of $a = 300$ mm. The minimum radius of gyration of a single channel is $r_i = 15$ mm. Determine the modified effective slenderness ratio $(KL/r)_m$.