This problem involves analyzing a statically determinate assembly of steel cylinders and calculating support reactions when subjected to a force $P = 260 \, \text{kN}$. Given the gap of $0.15 \, \text{mm}$ between point $C$ and the wall at $D$, we need to determine how the system responds under load.

The assembly is constrained at two points: $A$ and $D$. The primary concepts we will use are the deformation of the steel members and equilibrium conditions.

Given Data:

• Force applied: $P = 260 \, \text{kN}$.
• Distance between supports:
  • $AB = 600 \, \text{mm}$,
  • $BC = 600 \, \text{mm}$,
  • Diameter of section $AB = 50 \, \text{mm}$,
  • Diameter of section $BC = 25 \, \text{mm}$,
  • Initial gap between $C$ and $D = 0.15 \, \text{mm}$.

Material Properties:

• The assembly is made of A-36 steel. The Young’s modulus $E$ for A-36 steel is typically $200 \, \text{GPa}$ (or $200 \times 10^3 \, \text{MPa}$).

Steps to Solve:

1. Compatibility Condition: The deformation of segment $BC$ must close the gap at $D$, and both segments $AB$ and $BC$ will elongate or shorten due to the applied force. Since segment $AB$ is connected at $A$, its deformation will contribute to the force distribution.

2. Force Equilibrium: The sum of forces in the system must equal zero because the system is in static equilibrium. This will give us the relations between reactions at $A$ and $D$.

3. Deformation Equations: The deformation of a segment due to axial force can be calculated using: $\delta = \frac{PL}{AE}$ where:

   • $P$ is the force,
   • $L$ is the length of the segment,
   • $A$ is the cross-sectional area, and
   • $E$ is the Young's modulus of the material.

Let me calculate the reactions at points $A$ and $D$ using these principles.The magnitudes of the support reactions are:

• Reaction at $A$: $235.456 \, \text{kN}$
• Reaction at $D$: $24.544 \, \text{kN}$

These reactions ensure the system is in equilibrium and account for the initial gap between point $C$ and the wall at $D$.

Would you like more details or have any further questions? Here are some related questions for deeper understanding:

1. How does Young's modulus affect the deformations in the system?
2. What happens to the reaction forces if the gap at $D$ increases?
3. How would the system change if different materials were used for segments $AB$ and $BC$?
4. What are the effects of increasing the diameter of segment $BC$?
5. Can this setup be generalized for materials with different mechanical properties?

Tip: Always check material properties like Young's modulus when dealing with deformations, as they play a critical role in determining how much a structure stretches or compresses under load.