State the expressions for limiting values of neutral axis and moment of resistance for mild steel, Fe415 and Fe500.

The limiting values of the neutral axis and moment of resistance depend on the type of steel used in reinforced concrete design. Here's a summary for mild steel, Fe415, and Fe500:

1. Mild Steel (Fe250)

• Limiting Neutral Axis Depth (xu,lim):$$\frac{{𝑥}_{𝑢},\text{lim}}{𝑑}=0.53$$
• Limiting Moment of Resistance (Mu,lim):$${𝑀}_{𝑢},\text{lim}=0.148{𝑓}_{𝑐𝑘}𝑏{𝑑}^2$$where ${𝑓}_{𝑐𝑘}$ is the characteristic compressive strength of concrete, $𝑏$ is the width of the beam, and $𝑑$ is the effective depth.

2. Fe415 Steel

• Limiting Neutral Axis Depth (xu,lim):$$\frac{{𝑥}_{𝑢},\text{lim}}{𝑑}=0.48$$
• Limiting Moment of Resistance (Mu,lim):$${𝑀}_{𝑢},\text{lim}=0.138{𝑓}_{𝑐𝑘}𝑏{𝑑}^2$$

3. Fe500 Steel

• Limiting Neutral Axis Depth (xu,lim):$$\frac{{𝑥}_{𝑢},\text{lim}}{𝑑}=0.46$$
• Limiting Moment of Resistance (Mu,lim):$${𝑀}_{𝑢},\text{lim}=0.133{𝑓}_{𝑐𝑘}𝑏{𝑑}^2$$

These expressions are derived from the limit state design principles in reinforced concrete, accounting for the yield stress of the steel and the behavior of concrete in compression.