To calculate the maximum deflection of the steel beam using the principle of superposition, we need to consider the contributions to deflection from both the point load (PL) at mid-span and the uniformly distributed load (UDL) over the beam's length. Here’s the step-by-step approach:

Given Data:

• Point Load (P) = 50 kN (at mid-span)
• Uniformly Distributed Load (UDL) = 30 N/mm = 30,000 N/m
• Length of Beam (L) = 8 m
• Moment of Inertia (I) = $56.8 \times 10^6 \, \text{mm}^4$
• Modulus of Elasticity (E) = 205,000 N/mm²

Deflection Formulas:

For a simply supported beam with:

1. Point Load (P) at Mid-span:
   The maximum deflection ($\delta_{\text{PL}}$) at mid-span due to a point load is given by: $\delta_{\text{PL}} = \frac{P L^3}{48 E I}$

2. Uniformly Distributed Load (UDL): The maximum deflection ($\delta_{\text{UDL}}$) at mid-span due to a UDL is given by: $\delta_{\text{UDL}} = \frac{5 w L^4}{384 E I}$ where $w$ is the UDL in terms of force per unit length.

Step-by-Step Calculation:

1. Convert Units:

   • Convert $P$ into N: $50 \, \text{kN} = 50,000 \, \text{N}$.
   • Convert UDL $w$ to N/m (already given as 30,000 N/m).
   • Convert $L$ into mm for consistency: $L = 8 \, \text{m} = 8000 \, \text{mm}$.

2. Calculate Deflection Due to Point Load: $\delta_{\text{PL}} = \frac{50,000 \times (8000)^3}{48 \times 205,000 \times 56.8 \times 10^6}$

3. Calculate Deflection Due to UDL: $\delta_{\text{UDL}} = \frac{5 \times 30,000 \times (8000)^4}{384 \times 205,000 \times 56.8 \times 10^6}$

4. Total Deflection (Superposition Principle): Since deflections due to different loads can be superimposed, the total maximum deflection $\delta_{\text{total}}$ at mid-span is: $\delta_{\text{total}} = \delta_{\text{PL}} + \delta_{\text{UDL}}$

Would you like me to proceed with the detailed calculations, or is there a specific step you’d like more clarification on?

Here are some related questions to consider:

1. How does the moment of inertia affect the deflection of a beam?
2. What is the principle of superposition, and why is it applicable here?
3. How would the deflection change if the point load were not at mid-span?
4. How does the modulus of elasticity (E) influence the beam’s deflection?
5. What changes would occur if the UDL were applied only on a portion of the beam?

Tip: When dealing with beam deflections, always ensure unit consistency across all parameters to avoid errors.