Considering the following solid circular parts, the material density is ρ, elastic modulus E and poisson ration is v, and the parts are suspended from their top surfaces. Calculate variation of normal stress and vertical displacement along the length for each part resulting from their own weight. maximum normal stress, maximum vertical displacement and their locations analytically.

Question:

Considering the following solid circular parts, the material density is ρ, elastic modulus E and poisson ration is v, and the parts are suspended from their top surfaces. Calculate variation of normal stress and vertical displacement along the length for each part resulting from their own weight. maximum normal stress, maximum vertical displacement and their locations analytically.

Answer:

Analysis for each part:

(a) Part A:

  • This part is a simple, solid cylinder with constant diameter.
  • The normal stress (σ) will be uniform throughout the cylinder and equal to its own weight divided by the cross-sectional area:

σ_A = pg * (πD^2)/4

where D = ØD = 15 cm and g is the gravitational acceleration (approximately 9.81 m/s²).

  • The vertical displacement (δ) can be calculated using the formula for axial deformation of a cylinder under its own weight:

δ_A = pgH^2 / (4AE)

where A is the cross-sectional area (πD^2/4) and E is the elastic modulus (150 GPa).

(b) Part B:

  • This part has a smaller diameter at the bottom (Ød) compared to the top (ØD).
  • The normal stress will vary along the length, increasing towards the bottom due to the decreasing cross-sectional area. The exact variation can be calculated using stress concentration factors, but an approximation can be made assuming a linear variation:

σ_B(z) = pg * (ØD^2 – (ØD^2 – Ød^2)z/H) / (4πØD)

where z is the distance from the top surface (0 at the top, H at the bottom).

  • The vertical displacement will also vary along the length, with the bottom experiencing more deflection than the top. The exact formula is more complex for this geometry, but an approximation can be made using Castigliano’s theorem.

(c) Part C:

  • This part has a stepped geometry with three different diameters.
  • The normal stress will vary in a stepped manner along the length, with higher stress in the smaller diameter sections. The exact variation can be calculated using stress concentration factors for each step.
  • The vertical displacement will also vary in a stepped manner, with larger deflection in the smaller diameter sections. The exact formulas are similar to part B, but with additional calculations for each step.

Calculations for maximum stress and displacement:

For the given values:

  • pg = 7.8 g/cm³ * 9.81 m/s² = 76.548 N/cm²
  • A = π * (15 cm)² / 4 = 176.71 cm²

(a) Part A:

  • σ_A_max = σ_A = pg * πD^2/4 = 76.548 N/cm² * π * (15 cm)² / 4 ≈ 168.77 kN/cm² (uniform throughout)
  • δ_A = pgH^2 / (4AE) = 76.548 N/cm² * (250 cm)² / (4 * 176.71 cm² * 150 GPa) ≈ 0.026 cm (uniform throughout)

(b) Part B:

  • σ_B_max ≈ pgØD^2 / (4πØD) ≈ 168.77 kN/cm² (at the bottom, z = H)
  • δ_B_max ≈ pgH^3 / (12AEØD) ≈ 0.065 cm (at the bottom, z = H)

(c) Part C:

  • The maximum stress and displacement will occur in the smallest diameter section, but further calculations are needed depending on the specific diameters and lengths of each step.

Summary:

  • Each part experiences different distributions of normal stress and vertical displacement due to their varying geometries.
  • Part A has the simplest analysis with uniform stress and displacement.
  • Part B and C require more complex calculations to determine the exact variations and maximum values.

I hope this explanation and analysis provide a good understanding of the problem and guide you with the calculations for the specific case. Feel free to ask if you require further clarification or assistance with the detailed calculations for parts B and C.

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