Problem: A short tied column is $400 \times 400 \text{ mm}$ reinforced with 8-25mm bars ($A_{st} = 3927 \text{ mm}^2$). $f'_c = 28 \text{ MPa}$, $f_y = 420 \text{ MPa}$. Calculate the design axial strength $\phi P_{n,max}$ for concentric loading.

Problem: Determine the balanced axial load $P_b$ and moment $M_b$ for a $400 \times 400 \text{ mm}$ tied column with 4-25mm bars (2 on tension face, 2 on compression face). $d' = 65 \text{ mm}$, $d = 335 \text{ mm}$, $A_{s,tension} = A_{s,comp} = 982 \text{ mm}^2$. $f'_c = 28 \text{ MPa}$, $f_y = 420 \text{ MPa}$.

Problem: A column has a factored axial load $P_u = 2000 \text{ kN}$ and moment $M_u = 150 \text{ kN-m}$. From the interaction diagram (approximated), pure compression $\phi P_{n,max} = 2789$ kN and balanced point $(1533, 344)$. Check if the point $(2000, 150)$ is likely within the safe zone using linear interpolation.

Problem: A column has an unbraced length $L_u = 4.0 \text{ m}$. It is pinned at both ends ($k=1.0$). Cross section $400 \times 400$ mm. Check if slenderness effects must be considered.

Problem: For a $300 \text{ mm} \times 400 \text{ mm}$ rectangular tied column with 4-25mm bars (one at each corner, $A_s = 1964 \text{ mm}^2$), $f'_c = 28 \text{ MPa}$, and $f_y = 420 \text{ MPa}$, calculate the nominal axial load $P_{nb}$ and nominal moment $M_{nb}$ at the balanced strain condition. Assume the effective depth $d = 340 \text{ mm}$ and cover to compression steel $d' = 60 \text{ mm}$.

Problem: Following a severe earthquake, inspectors observed that many columns in a hospital building suffered massive "explosive" failures. The vertical rebar buckled entirely outward in large "birdcage" patterns, and the concrete cores were completely crushed into rubble. The lateral ties holding the vertical bars were widely spaced ($>300 \text{ mm}$) and used 90-degree hooks. Analyze the cause of failure.

Problem: A five-story reinforced concrete building completely collapsed during an earthquake, with the entire first floor "pancaking" down while the upper four floors remained relatively intact as a rigid block resting on the rubble. The ground floor was designed as an open parking garage with few walls, while the upper floors had many stiff masonry partition walls. Explain this behavior.