1. Problem Definition:

• Span (L): 3.2 meters (3200 mm)
• Maximum Allowable Bending Stress (σ_max): 50 N/mm²
• Beam Type: Cantilever with circular cross-section

2. Material Selection:

We will initially consider structural steel (e.g., ASTM A36) due to its common usage and availability. However, the final material selection depends on factors like cost, weldability, and desired lifespan. For initial calculations, assume:

• Yield Strength (σ_y): 250 N/mm² (for A36 steel - this is just for reference and not used in the bending stress calculation directly but informs the factor of safety) Engineering ToolBox - ASTM A36 Steel

3. Load Estimation:

This is the most critical and least defined part of the problem. We must estimate the maximum force (F) that the bending press will apply at the end of the cantilever beam. Without this, we cannot complete the design.

Assume for this example: The bending press will exert a maximum force of 10,000 N (1 metric ton) at the end of the beam.

4. Bending Moment Calculation:

For a cantilever beam with a point load (F) at the free end, the maximum bending moment (M) occurs at the fixed end and is given by:

M = F * L

M = 10,000 N * 3200 mm = 32,000,000 N·mm

5. Section Modulus Calculation:

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

σ = M / Z

Therefore, Z = M / σ

Z = 32,000,000 N·mm / 50 N/mm² = 640,000 mm³

6. Diameter Calculation:

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

Z = (π * d³) / 32

Where 'd' is the diameter of the circular cross-section.

Therefore, d³ = (32 * Z) / π

d³ = (32 * 640,000 mm³) / π ≈ 6,517,946 mm³

d = ∛6,517,946 mm³ ≈ 187.1 mm

Therefore, we will choose a diameter of 190 mm as a starting point for our design.

7. Deflection Check:

It's essential to verify that the deflection is within acceptable limits. The maximum deflection (δ) for a cantilever beam with a point load at the free end is:

δ = (F * L³) / (3 * E * I)

Where:

• E is the modulus of elasticity of the material (e.g., for steel, E ≈ 200,000 N/mm²)
• I is the second moment of area (moment of inertia) of the circular cross-section. For a circle, I = (π * d⁴) / 64

Calculate I: I = (π * (190 mm)⁴) / 64 ≈ 63.9 x 10⁶ mm⁴

Calculate Deflection: δ = (10,000 N * (3200 mm)³) / (3 * 200,000 N/mm² * 63.9 x 10⁶ mm⁴) ≈ 8.52 mm

This deflection should be assessed for suitability in the application. If the deflection is too high, consider increasing the diameter further.

8. Factor of Safety:

The design should incorporate an appropriate factor of safety. A common factor of safety for structural steel applications is between 1.5 and 3. Since we are limiting bending stress to 50 N/mm², which is much smaller than the yield strength, the factor of safety is intrinsically high. However, buckling needs to be considered.

9. Buckling Check:

Cantilever beams under bending can be susceptible to lateral-torsional buckling, especially with long spans. This is particularly true for beams with cross-sections that are not very rigid in the lateral direction. A circular cross-section is excellent in this regard, but it should still be checked, particularly if the force estimate turns out to be significantly higher.

A detailed buckling analysis requires more information about the specific application and support conditions. Simplified buckling checks can be performed using Euler's formula with appropriate effective length factors. This is a critical step that requires specialist engineering knowledge, and must be investigated during detailed design.

10. Assembly and Support:

The fixed end of the cantilever beam must be rigidly supported. This usually involves welding or bolting the beam to a robust frame or structure. The support structure must be capable of withstanding the reaction forces and moments from the beam. The design of the support is as important as the beam itself.

• Welding: If welding is used, ensure the weld is strong enough to withstand the bending moment and shear force at the fixed end. Use appropriate welding procedures and qualified welders.
• Bolting: If bolting is used, calculate the required number and size of bolts based on the reaction forces and moments. Use high-strength bolts and ensure proper pre-tensioning.

11. Further Considerations:

• Manufacturing Tolerances: Account for manufacturing tolerances in the diameter and length of the beam.
• Surface Finish: Consider the required surface finish for the application.
• Corrosion Protection: Implement appropriate corrosion protection measures, such as painting or galvanizing, if the beam will be exposed to corrosive environments.
• Dynamic Loading: If the bending press will be subjected to dynamic loads, perform a fatigue analysis to ensure the beam can withstand repeated loading cycles.
• Finite Element Analysis (FEA): For a more detailed analysis, especially concerning buckling and stress concentrations at the support, consider using FEA software.

Summary

• Based on the assumptions above (a 10,000 N load), a 190 mm diameter circular steel beam is a reasonable starting point.
• Buckling must be investigated by a qualified engineer.
• The deflection check is important and must be assessed in the application.
• The fixed end support design is critical and must be robust.