22.2:
Distribution of Stresses in a Narrow Rectangular Beam
An elemental section of a beam is considered to analyze the distribution of the stresses. The magnitude of the shearing force exerted on the horizontal face of the element is obtained from the horizontal shear expression.
The average shearing stress on that face is determined by dividing horizontal shear by the area of the face.
Shearing stresses on the transverse and horizontal planes of the beam section are equal, representing the average stress along the line on the upper part of the beam. Shear stresses are zero at the upper and lower faces as no forces are being exerted.
In a narrow rectangular beam, shearing stress along the width of the beam section varies less than 0.8% of the average shearing stress as the width is less than a quarter of its depth.
So, the average shearing stress equation is used to determine shearing stress at any point across its cross-section.
In the transverse section of the rectangular beam, the shearing stresses are distributed parabolically, with zero stress at the top and bottom.
22.2:
Distribution of Stresses in a Narrow Rectangular Beam
In studying beam stress distribution, examining an elemental section is essential. To determine the average shearing stress on this face, the calculated shear is divided by the surface area. Importantly, shearing stresses on the beam's transverse and horizontal planes mirror each other, indicating a consistent stress distribution along the upper region of the beam. Notably, shearing stresses are absent at the beam's upper and lower surfaces due to the absence of applied forces in these areas.
For beams with a narrow rectangular cross-section, the variation in shearing stress across the width is minimal, less than 0.8% of the average stress. This minor variation is attributed to the beam's width being significantly smaller than its depth, making the average shearing stress equation a reliable method for assessing stress across any point of the beam's cross-section. In the rectangular beam's transverse section, shearing stresses are distributed parabolically. This distribution features zero stress at the beam's top and bottom edges, with stress increasing toward the center. This parabolic stress profile is essential for understanding how beams manage stress, ensuring their structural integrity and the capacity to bear loads without compromising stability.