Lecture Notes on Bending and Shear Stresses in Beams

Introduction to Beam Deformation

• Applying a load to a beam causes it to deform by bending.
• Deformation generates 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.
• Important to understand these stresses for beam design and analysis.

Bending Stresses

Pure Bending

• A section is in pure bending when the shear force is zero, resulting in a constant bending moment.
• Deflection and Neutral Surface:
  • Fibers at the top compress; fibers at the bottom extend.
  • Neutral Surface: Region where fibers remain the same length, passing through the centroid (neutral axis in 2D).

Calculating Bending Strains

• Strains in the beam can be calculated using the geometry of deformation.
• For pure bending, fibers bend into a circular arc.
• Bending strain equation is derived as a function of distance (y) from the neutral axis.

Calculating Bending Stresses

• Hooke's Law: Used to derive bending stress equation as a function of curvature radius (r).
• Flexure Formula:
  • Relates bending stress with bending moment (m) and area moment of inertia (I).
  • Bending stress increases linearly with increased bending moment and distance from neutral axis.

Section Modulus

• Ratio of area moment of inertia (I) to maximum distance from neutral axis (Ymax).
• Dependent on cross-section geometry.

Shear Stresses

Shear Force and Shear Stresses

• Shear Force: Resultant of vertical shear stresses; denoted by τ.
• Horizontal Shear Stresses:
  • Develop between horizontal layers to maintain equilibrium.
  • Visualized as a tendency for planks to slide, resisted by glue.
  • Explains longitudinal splitting of wooden beams.

Calculating Shear Stresses

• Average shear stress = Shear force (V) / Cross-sectional area.
• Shear stress distribution isn't uniform; zero at top and bottom surfaces.

Shear Stress Equation

• Equation based on equilibrium of stresses in a small beam element.
• Key Variables:
  • Width (b), Area moment of inertia (I), Shear force (V), First moment of area (Q).
  • Q varies with distance from the neutral axis.

Shear Stress Distribution

• Varies parabolically over beam height; maximum at neutral axis.
• Opposite distribution compared to bending stress.

Practical Applications

• Rectangular Sections: Maximum shear stress = 1.5 * average shear stress.
• Circular Sections: Similar shear stress equation with different constant.
• Thin-Walled Sections: Shear stress distributed mainly in the web.

Conclusion

• Understanding bending and shear stresses is critical for effective beam design and analysis.
• Flexure formula is valid for general cases, including presence of shear force.
• Shear stress calculation assumptions limit its application to certain geometries.

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For further understanding, additional resources and videos on related topics are available.