The image presents a problem related to determining the expression for the deflection at the center of a simply supported beam subjected to a combination of a central point load $P$ and a uniformly distributed load $w$.

Given:

• A simply supported beam with length $L$.
• A point load $P$ applied at the center of the beam.
• A uniformly distributed load $w$ across the entire span of the beam.

Required:

Determine the expression for the deflection at the center of the beam using superposition principles.

General Approach:

1. Calculate the deflection due to the point load $P$: For a simply supported beam with a central point load $P$: $\delta_P = \frac{PL^3}{48EI}$ where:

   • $P$ is the point load,
   • $L$ is the length of the beam,
   • $E$ is the modulus of elasticity,
   • $I$ is the moment of inertia.

2. Calculate the deflection due to the uniformly distributed load $w$: For a simply supported beam with a uniformly distributed load $w$: $\delta_w = \frac{5wL^4}{384EI}$ where:

   • $w$ is the load per unit length.

3. Superposition of deflections: The total deflection at the center is the sum of the deflections due to the point load $P$ and the uniformly distributed load $w$: $\delta_{total} = \delta_P + \delta_w = \frac{PL^3}{48EI} + \frac{5wL^4}{384EI}$

Choices Review:

Now, let's compare this with the given options to see which matches the derived expression.

If you have any questions or need further explanation, feel free to ask!

Here are five related questions to further your understanding:

1. How does the position of the point load affect the deflection in the beam?
2. What would the deflection be if only the distributed load $w$ was applied?
3. How would the deflection change if the beam were fixed at both ends instead of simply supported?
4. How does the moment of inertia $I$ affect the deflection?
5. How can superposition principles be applied to other types of loading conditions?

Tip: Always check the boundary conditions when applying formulas for deflection in beams, as they can significantly change the result.