To solve the problem of determining the factor of safety $n$ at point $B$, we need to analyze the mechanics and forces acting on the system shown in the diagram.

Steps to Solve:

1. Identify the Forces and Moments:

   • There are forces $P$, $R$, and $F$ applied at specific points along the rods.
   • The geometry indicates distances along the $x$, $y$, and $z$ axes.

2. Evaluate the Stresses at Point $B$:

   • Normal Stress ($\sigma$) due to axial loading: $\sigma = \frac{F}{A}$, where $A$ is the cross-sectional area.
   • Shear Stress ($\tau$) due to torsional or transverse loading: $\tau = \frac{T \cdot r}{J}$, where $T$ is torque, $r$ is the radial distance, and $J$ is the polar moment of inertia.
   • Bending Stress ($\sigma_b$): $\sigma_b = \frac{M \cdot c}{I}$, where $M$ is the bending moment, $c$ is the distance to the neutral axis, and $I$ is the moment of inertia.

3. Superpose Stresses: Combine normal, bending, and shear stresses at point $B$ using the maximum shear stress or von Mises criteria.

4. Factor of Safety Calculation: The factor of safety $n$ is given by: $n = \frac{\text{Yield Strength of Material}}{\text{Maximum Stress at Point } B}$

5. Use Geometry for Moment Arms and Stresses:

   • Determine the lever arms for the forces to calculate moments and torsion.
   • Use the cross-sectional area and material properties for stress computation.

6. Numerical Substitution: If specific numerical values for forces, material properties, and dimensions are provided, substitute them into the derived equations to solve.

To proceed further, I need clarification or additional data:

• Material Properties: Yield strength, ultimate strength, etc.
• Cross-sectional Shape: Circular, square, etc.
• Dimensions: Any missing values for rod radii or areas.
• Force Magnitudes: Exact values of $P$, $R$, and $F$.

Would you like me to continue with general symbolic steps or assist further with specific values? Let me know! 😊

Related Questions:

1. How is the torsional stress calculated at a specific point on a shaft?
2. What is the importance of the factor of safety in mechanical design?
3. Can the von Mises stress criterion be used for brittle materials?
4. How are bending moments distributed in cantilever beams?
5. What are common failure modes in structures under combined loading?

Tip:

Always verify the units of all forces, dimensions, and material properties to avoid errors in calculations!