Distribution of Stresses in a Narrow Rectangular Beam

Average shearing stress is determined by dividing the horizontal shear force by the surface area of the face. The magnitude of shearing force exerted on the horizontal face is obtained from the horizontal shear expression. This calculation provides the average stress distribution along the beam's cross-section.

Shearing stresses are zero at the upper and lower faces because no external forces are being exerted on these surfaces. The absence of applied forces at these locations results in zero stress, while stress develops in the interior regions where forces are transmitted through the beam.

Shearing stresses on the transverse and horizontal planes of a beam section are equal, representing consistent stress distribution along the upper part of the beam. This equality demonstrates that shear stress is uniform across perpendicular planes, ensuring predictable stress behavior throughout the beam's cross-section.

In a narrow rectangular beam, shearing stress variation along the width is less than 0.8% of the average shearing stress. This minimal variation occurs because the beam's width is less than a quarter of its depth, making the average shearing stress equation reliable for determining stress at any point across the cross-section.

In the transverse section of a rectangular beam, shearing stresses are distributed parabolically, with zero stress at the top and bottom edges and maximum stress toward the center. This parabolic profile is essential for understanding how beams manage stress and maintain structural integrity while bearing loads.

The average shearing stress equation is valid for narrow rectangular beams where width is significantly smaller than depth. Since stress variation across the width remains below 0.8% of average stress in such beams, this simplified equation accurately determines shearing stress at any point across the beam's cross-section.

Examining an elemental section of a beam is essential for analyzing stress distribution. By calculating shear on individual elements and dividing by surface area, engineers determine average shearing stress. This approach reveals how stresses vary across the beam and ensures proper understanding of shearing stresses in a beam problem solving context.