Determination of bending stresses in simply supported, cantilever, and overhanging beams

Definition

Bending stress is the normal stress induced at any point in a beam due to the internal bending moment acting on the cross-section. It represents the internal resistance of the beam material to the forces that tend to cause it to bend or deflect.

Main Content

1. Beam Classifications

• Simply Supported Beam: Supported at both ends, where the beam is free to rotate but restricted from vertical movement.
• Cantilever Beam: Fixed rigidly at one end and free at the other. It is commonly found in balconies and aircraft wings.
• Overhanging Beam: A beam that extends beyond its supports on one or both sides, creating a combination of simply supported and cantilever behavior.

2. The Theory of Simple Bending

• The fundamental equation for bending stress is derived from the Navier-Bernoulli hypothesis: σ = (M * y) / I
• Where σ is the bending stress, M is the internal bending moment, y is the distance from the neutral axis, and I is the Area Moment of Inertia.

3. Neutral Axis and Stress Distribution

• The neutral axis is the line where the longitudinal fibers of the beam undergo zero stress (no tension or compression).
• Stress varies linearly across the depth of the beam; it is maximum at the extreme fibers (top and bottom) and zero at the neutral axis.

    Bending Stress Profile:
    Top fiber: Max Compression (-)
    Neutral Axis: Zero Stress
    Bottom fiber: Max Tension (+)

    |   /  |
    |  /   |
    | /    |
    |/     |
    ----------------- Neutral Axis
    |\     |
    | \    |
    |  \   |
    |   \  |

Working / Process

1. Calculation of Support Reactions

• Use the equations of static equilibrium: ΣFx = 0, ΣFy = 0, and ΣM = 0.
• For a simply supported beam with a point load at the center, each support takes half the load.

2. Construction of Shear Force and Bending Moment Diagrams (SFD & BMD)

• Identify the maximum bending moment (M_max) along the length of the beam.
• In a cantilever beam, the maximum moment occurs at the fixed support. In a simply supported beam, it typically occurs under the load.

3. Calculation of Section Modulus and Stress

• Determine the Area Moment of Inertia (I) based on the beam's cross-sectional shape (e.g., I = bh³/12 for rectangular sections).
• Calculate the maximum bending stress using the section modulus (Z = I/y_max), leading to the formula σ_max = M_max / Z.

Advantages / Applications

• Structural Safety: Engineers use these calculations to ensure beams do not fail under specific loads (e.g., floors of buildings).
• Material Optimization: Helps in selecting the most economical shape (like I-beams) that provides high resistance to bending with minimum material usage.
• Infrastructure Design: Essential for designing bridges, cantilever trusses, and overhanging roof rafters.

Summary

Bending stress determination is the study of how internal moments deform beam structures. By calculating the maximum bending moment and applying the flexural formula, engineers can predict if a beam will safely support its load.

• Simply Supported Beam: Supported at both ends.
• Cantilever Beam: Fixed at one end, free at the other.
• Overhanging Beam: Extends beyond supports.
• Important terms: Neutral Axis, Area Moment of Inertia (I), Section Modulus (Z), and Bending Moment (M).