Problem Statement:

Design a cantilever beam with a circular cross-section to support a load, limiting the maximum bending stress.

• Span (L): 3.2 meters
• Maximum Allowable Bending Stress (σ_max): 15 N/mm²

Assumptions:

• The load is applied at the free end of the cantilever beam.
• The material is homogeneous and isotropic.
• The beam is perfectly straight.
• Shear stress is not a primary design factor in this case. We are focusing on bending stress.

Design Steps:

1. Determine the Maximum Bending Moment (M):

   First, we need to assume a load (W) at the free end of the beam. Let's assume a load for now and iterate if the resulting diameter is impractical. Assume W = 100 N.

   For a cantilever beam with a point load at the free end, the maximum bending moment occurs at the fixed end and is calculated as:

   M = W * L

   M = 100 N * 3200 mm = 320,000 N·mm

2. Calculate the Required Section Modulus (Z):

   The bending stress (σ) is related to the bending moment (M) and section modulus (Z) by the following formula:

   σ = M / Z

   Therefore, the required section modulus is:

   Z = M / σ_max

   Z = 320,000 N·mm / 15 N/mm² = 21,333.33 mm³

3. Determine the Diameter (d) of the Circular Cross-Section:

   For a circular cross-section, the section modulus (Z) is given by:

   Z = (π * d³) / 32

   Where 'd' is the diameter of the circle.

   Solving for 'd':

   d³ = (32 * Z) / π

   d³ = (32 * 21,333.33 mm³) / π ≈ 217,146.98 mm³

   d = ∛217,146.98 mm³ ≈ 60.07 mm

4. Select a Suitable Diameter:

   Choose a standard or readily available diameter slightly larger than the calculated value. For example, select d = 65 mm. This provides a small safety margin.

5. Verify the Bending Stress:

   Recalculate the section modulus (Z) with the chosen diameter:

   Z = (π * (65 mm)³) / 32 ≈ 27,150.75 mm³

   Calculate the actual bending stress:

   σ = M / Z = 320,000 N·mm / 27,150.75 mm³ ≈ 11.79 N/mm²

   Since 11.79 N/mm² < 15 N/mm², the design is acceptable for the assumed load of 100N.

Iteration (Important!):

The above calculation was based on an assumed load of 100 N. This is unlikely to be the *actual* load. The *actual* load the beam will experience needs to be determined through a different method.

To complete the design:

1. Determine the actual load (W) that the cantilever beam needs to support based on the application.
2. Repeat steps 1-5 using the *actual* load (W).

Without knowing the real load, it is impossible to complete the beam design. This process shows how to select the correct diameter when that load is known.

Important Considerations:

• Material Selection: The material's yield strength must be significantly higher than the calculated bending stress, including a factor of safety. Common materials include steel, aluminum, or composites. The elastic modulus (Young's Modulus) is also important for deflection calculations (not performed here).
• Deflection: Calculate the maximum deflection of the cantilever beam to ensure it meets the application's requirements. The deflection should be within acceptable limits.
• Shear Stress: While bending stress is the primary concern here, check shear stress, especially near the fixed end. For circular cross-sections, shear stress is often less critical than bending stress, but should still be checked.
• Buckling: If the beam is very long relative to its diameter, consider buckling. This is less likely to be an issue for typical cantilever beam proportions but should be checked for slender beams.
• Manufacturing Tolerances: Account for manufacturing tolerances in the diameter.