The normal stress developed in each rod (AB and CD) due to the applied vertical force P is given by the formula: σ = (4 * P) / (π * D^2).
To determine the normal stress developed in each rod (rods AB and CD) when a vertical force P is applied at the end of the beam, we can use the principles of statics and mechanics of materials. We'll assume that the rods are in tension due to the applied load. Here are the steps to calculate the normal stress:
1. Determine the Force in the Beam (P):
- The vertical force P applied at the end of the beam will create an equal and opposite reaction force at the pinned joint F. This force is also P.
2. Identify the Geometry and Cross-sectional Area of the Rods:
- The rods AB and CD are identical and have a length (L) and diameter (D).
- The cross-sectional area (A) of each rod is given by the formula for the area of a circle: A = π * (D/2)^2 = (π/4) * D^2.
3. Calculate the Axial Load in Each Rod:
- Since the rods are in tension, the axial load in each rod is equal to the applied force P.
- Axial Load (F) in each rod = P.
4. Use Hooke's Law to Calculate Normal Stress:
- Hooke's Law relates stress (σ), modulus of elasticity (E), and strain (ε): σ = E * ε.
- In this case, stress (σ) is the normal stress in the rods, and we want to find it.
- Strain (ε) is the ratio of deformation to the original length of the rod due to the axial load.
- Since the rods are in tension and the deformation is along the length (L), ε = ΔL / L, where ΔL is the change in length due to the applied load.
5. Determine the Deformation (ΔL) for Each Rod:
- To find ΔL, we can use the formula: ΔL = F * L / (A * E), where F is the axial load, L is the length of the rod, A is the cross-sectional area, and E is the modulus of elasticity.
- ΔL for each rod = P * L / ((π/4) * D^2 * E).
6. Calculate Normal Stress (σ) for Each Rod:
- Now that we have ΔL, we can calculate the normal stress using Hooke's Law: σ = E * ε.
- σ for each rod = E * (P * L) / ((π/4) * D^2 * E * L).
7. Simplify the Expression:
- The length (L) cancels out, and you are left with the final expression for the normal stress in each rod:
- σ for each rod = (4 * P) / (π * D^2).