Lecture Notes on Bending and Shear Stresses in Beams

Introduction

• When a load is applied to a beam, it deforms by bending.
• Bending generates internal stresses:
  • Shear Force: Acts vertically; resultant of vertical shear stresses (parallel to cross-section).
  • Bending Moment: Resultant of normal stresses (bending stresses) acting perpendicular to the cross-section.

Importance of Understanding Stresses

• Critical for design and analysis of beams.

Bending Stresses

Definition

• Pure Bending: Occurs when shear force along the beam section is zero, leading to a constant bending moment.

Deformation of Beams

• Beam can be viewed as small fibers:
  • Top fibers shorten (compression).
  • Bottom fibers lengthen (tension).
• Neutral Surface: Contains fibers that remain the same length (neutral axis in 2D).

Quantifying Bending Stresses

1. Calculate Strains:
   • Fibers deform into a circular arc.
   • Strain = Change in length / Original length.
2. Bending Strain Equation:
   • Derived from fiber deformation geometry.
   • Defined with positive y downwards for tension.
3. Bending Stress Calculation:
   • Uses Hooke's law for uniaxial stress.
   • Bending stress as a function of radius of curvature (r).
   • Needs resultant internal forces to equal the bending moment (m).
   • Integrate to find the area moment of inertia (I).
   • Flexure Formula: Relates bending moment, stress, and section geometry.
   • Bending stress increases with:
     • Increasing bending moment.
     • Distance from neutral axis.
     • Section modulus (S = I / Y_max).
   • Maximum stress at outer fibers.

Bending Stress Distribution

• I-beam example: Stress is zero at neutral axis, maximum at flanges.
• T-section: Neutral axis shifted upwards, altering stress distribution.

Shear Stresses

Presence of Shear Force

• In practice, beams often experience shear forces in addition to bending moments.
• Shear stresses ( [ \tau ]) act vertically and require consideration of horizontal shear stresses for equilibrium.

Visualization of Shear Stresses

• Planks analogy: Glue bond failure due to shear stresses.
• Shear Stress Calculation:
  • Average shear stress = Shear force (V) / Cross-sectional area.
  • Not uniformly distributed; maximum shear stress at neutral axis.

Shear Stress Equation

• Assumes constant shear stress across the width of the cross-section.
• Equation: [ \tau = \frac{V \cdot Q}{I \cdot b} ] where:
  • Q = First moment of area above the location of interest.
• For rectangular cross-sections, shear stress varies parabolically; max shear stress occurs at the neutral axis.
• Relationship: Max shear stress = 1.5 * Average shear stress.

Considerations for Circular Sections

• Similar approach but with constant ratios (4/3 for circular vs 3/2 for rectangular).

Shear in Thin-Walled Sections

• I-beams: Vertical shear mainly carried by the web, flanges resist bending moment; shear stresses distributed through height of the web.

Conclusion

• Comprehensive review of bending and shear stresses in beams.
• Understanding these concepts is vital for effective beam design and analysis.