Definition of Bending Stress

Bending stress is the internal stress induced in a material when an external bending moment is applied. It is typically categorized as either tensile stress or compressive stress depending on the location within the cross-section of the member.

The bending stress at any point in the cross-section is given by:

σ = (M · y) / I

• σ: Bending stress (Pa or N/m²)
• M: Bending moment (N·m)
• y: Distance from the neutral axis (m)
• I: Moment of inertia of the cross-section about the neutral axis (m⁴)

This equation assumes:

• The material is homogeneous and isotropic.
• The beam experiences pure bending (no shear force).
• The deflections are small.

Key Concepts in Bending Stresses

1. Neutral Axis (NA): The line within the cross-section where the bending stress is zero.
2. Moment of Inertia (I): A geometric property that quantifies a cross-section's resistance to bending.
3. Maximum Bending Stress: The maximum stress occurs at the farthest distance (y_max) from the neutral axis:

   σ_max = (M · y_max) / I

4. Beam Theory: Derived from Euler-Bernoulli beam theory, which assumes that plane sections remain plane and perpendicular to the beam's axis after bending.

Worked Examples

Worked Example 1: Calculating Bending Stress in a Beam

Problem: A simply supported beam with a span of 6 m carries a uniformly distributed load of 20 kN/m over its entire length. The beam has a rectangular cross-section with a width of 300 mm and a height of 500 mm. Calculate the maximum bending stress in the beam.

Solution:

M_max = (w · L²) / 8

Substitute: w = 20 kN/m = 20 × 10³ N/m, L = 6 m

M_max = (20 × 10³ · 6²) / 8 = 90 kN·m = 90 × 10³ N·m

I = (b · h³) / 12

Substitute: b = 0.3 m, h = 0.5 m

I = (0.3 · 0.5³) / 12 = 3.125 × 10^-3 m⁴

y_max = h / 2 = 0.5 / 2 = 0.25 m

σ_max = (M_max · y_max) / I

Substitute: σ_max = (90 × 10³ · 0.25) / (3.125 × 10^-3) = 7.2 MPa

Answer: The maximum bending stress in the beam is 7.2 MPa.

Example 2: Designing a Beam

Problem: Design a simply supported beam to carry a central point load of 50 kN over a span of 4 m. The beam has a rectangular cross-section with a width-to-height ratio of 1:2. The allowable bending stress is 8 MPa. Determine the dimensions of the beam.

Solution:

M_max = (P · L) / 4

Substitute: P = 50 × 10³ N, L = 4 m

M_max = (50 × 10³ · 4) / 4 = 50 × 10³ N·m

σ_max = (6 · M_max) / (b · h²)

Substitute: σ_max = 8 × 10⁶ Pa, M_max = 50 × 10³ N·m

b · h² = (6 · 50 × 10³) / (8 × 10⁶) = 0.0375 m³

h = 2b, substitute into the volume equation:

b · (2b)² = 0.0375

b = 0.21 m, h = 0.42 m

Answer: The beam should have dimensions b = 0.21 m and h = 0.42 m.

Applications of Bending Stresses in Engineering

• Structural Design: Used in beams, bridges, and buildings to ensure structural integrity.
• Machine Components: Design of shafts, levers, and gears subjected to bending loads.
• Material Testing: Understanding material behavior under bending to establish yield and failure criteria.
• Aerospace and Automotive: Analysis of wings, chassis, and frames for safety and performance.

Further Reading: Beam Bending - An Overview

References

1. Beer, F. P., Johnston, E. R., & DeWolf, J. T. (2011). Mechanics of Materials. McGraw-Hill Education.
2. Gere, J. M., & Goodno, B. J. (2012). Mechanics of Materials. Cengage Learning.
3. Hibbeler, R. C. (2017). Mechanics of Materials. Pearson Education.
4. Timoshenko, S. P., & Gere, J. M. (1961). Theory of Elastic Stability. McGraw-Hill.
5. Megson, T. H. G. (2019). Aircraft Structures for Engineering Students. Butterworth-Heinemann.

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