Column Failure Modes and Balance Condition
📌 The balance condition (C=Cᵦ) occurs when the concrete reaches its ultimate strain ($\epsilon_c = 0.003$) simultaneously with the steel yielding ($\epsilon_s = \epsilon_y$).
⚖️ If the calculated neutral axis depth ($c$) is less than the balance depth ($c_B$), the failure is tension-controlled (Tension Failure), indicated by the tensile strain area being dominant over the compression area.
💥 If $c$ is greater than $c_B$, the failure is compression-controlled (Compression Failure), meaning the compression zone is wider than the tension zone.

Axial Load and Moment Interaction (P-M Diagram)
🔝 Three primary failure modes govern the behavior of a column subjected to axial load ($P$) and bending moment ($M$) with eccentric loading ($e$):
1. Pure Compression Failure (AB Region): Occurs when eccentricity ($e$) is zero, or the eccentricity is small enough that the entire section remains in compression (or near balance).
2. Balance Failure (Point B): The transition point where concrete crushing and steel yielding happen simultaneously.
3. Tension Failure (BC Region): Occurs when eccentricity is significant, leading to failure dominated by steel yielding in tension due to bending.
📉 The dotted line region on the P-M interaction curve signifies behavior still within the section limits, while the solid line indicates conditions outside the nominal sectional boundaries.

Practical Application Using Interaction Graphs
📏 A practical example involves determining the required steel area ($A_s$) for a $508 \text{ mm} \times 508 \text{ mm}$ column with an axial load of $2380 \text{ kN}$ and eccentricity of $146 \text{ mm}$, using concrete strength $f'_c = 20.7 \text{ MPa}$ and steel yield strength $f_y = 414 \text{ MPa}$.
📐 The method requires first calculating the normalized eccentricity ratio, $g = e/h$, and checking if it falls within the acceptable graph range of $0.6 \le g \le 0.9$.
📊 By plotting the normalized load ($P_u / (f'_c b h)$) and normalized moment ($M_u / (f'_c b h^2)$) onto the interaction graph, the required reinforcement ratio ($\rho = A_s / (b d)$) is found at the intersection point, leading to the required area $A_s$.

Key Points & Insights
➡️ Understanding the relationship between the neutral axis depth ($c$) and the balance depth ($c_B$) is crucial for classifying column failure as Tension ($\mathbf{c < c_B}$) or Compression ($\mathbf{c > c_B}$).
⚙️ The P-M interaction diagram summarizes the performance envelope of a reinforced concrete column under combined axial load and bending for specific material and reinforcement properties.
🔧 Using pre-established interaction graphs simplifies the design process for columns, allowing quick determination of the necessary reinforcement ratio ($\rho$) based on applied normalized loads and moment ratios.

📸 Video summarized with SummaryTube.com on Dec 01, 2025, 04:20 UTC