Bending and Shear Stresses in Beams

Introduction to Beam Deformation

• Load on Beam: Causes bending, generating internal stresses.
• Types of Internal Stresses:
  • Shear Force: Resultant of vertical shear stresses, acting parallel to the cross-section.
  • Bending Moment: Resultant of normal stresses (bending stresses), acting perpendicular to the cross-section.

Understanding Bending Stresses

• Pure Bending:
  • Occurs when shear force is zero, resulting in a constant bending moment.
  • Example: Beams loaded by two moments.
• Beam Deflection:
  • Top fibres compress (shorten), bottom fibres stretch (elongate).
  • Neutral Surface: Fibres remain unchanged in length, passes through the centroid.
  • Neutral Axis: Refers to the neutral surface in two dimensions.

Calculating Bending Strains and Stresses

• Bending Strain:
  • Derived from geometric deformation.
  • Strain is positive for fibres below the neutral axis (tension).
• Hooke's Law: For elastic region, uniaxial stress.
• Bending Stress:
  • Related to radius of curvature R and bending moment M.
  • Integral of internal forces equals bending moment.

Flexure Formula

• Links bending moment M and bending stress.
• Area Moment of Inertia I:
  • Resistance to bending, depends on cross-section shape.
• Section Modulus S:
  • I/Y-max, dependent on cross-section geometry.
  • Larger I results in lower stresses.

Bending Stress Distribution

• I-beams: High I, low stresses, stresses max at flanges.
• T-sections: Neutral axis shifted, different stress distribution.

Shear Forces and Stresses

• General Bending:
  • Shear force doesn't significantly affect bending stresses.
  • Shear force V results in shear stresses.
• Shear Stresses Tau:
  • Vertical shear stresses have complementary horizontal shear stresses.
  • Example: Planks of wood under load.
• Shear Stress Calculation:
  • Average shear stress: V divided by cross-sectional area.
  • Shear stress varies parabolically, max at neutral axis.

Shear Stress Equation

• Considers equilibrium of small elements in the beam.
• Factors in Equation:
  • V: Shear force on the cross-section.
  • B: Width of cross-section.
  • I: Area moment of inertia.
  • Q: First moment of area above the point.

Shear Stress in Different Cross-Sections

• Rectangular Section: Max shear stress 1.5 times average.
• Circular Section: Similar equation, different constant.
• Thin-Walled Sections: Shear stress in web, horizontal stresses in flanges.

Conclusion

• Summary of bending and shear stress concepts in beams.
• Encouragement to support further educational content.