Shearing Stress in a Beam and Shear Stress Distribution

Definition

Shearing stress in a beam is the internal force per unit area acting parallel to the cross-section of the beam, caused by the transverse loads (like point loads or distributed loads) that tend to make adjacent layers of the beam slide past one another. Shear stress distribution refers to the variation of this stress intensity from the top surface to the bottom surface of the beam’s cross-section.

Main Content

1. The Concept of Horizontal Shear

When a beam bends, the longitudinal fibers at the top are compressed while those at the bottom are stretched.
Because these adjacent layers try to slide relative to each other, a "horizontal shear stress" must exist to keep the beam acting as a single unit. Due to complementary shear laws, this horizontal shear is equal to the vertical shear stress at any given point.

2. The Shear Formula (Tau-V-Q-I-T)

The magnitude of shear stress ($\tau$) at any level in a beam is given by the formula: $\tau = \frac{VQ}{It}$
Where $V$ is the shear force, $Q$ is the first moment of area, $I$ is the moment of inertia, and $t$ is the thickness of the beam section at that level.

3. Shear Stress Distribution Patterns

The distribution of shear stress is non-uniform across the depth of a beam.
For a rectangular section, the distribution is parabolic, with zero stress at the extreme top and bottom edges and maximum stress occurring at the neutral axis.

Rectangular Beam Cross-Section Shear Distribution:

     |           |  <- Top (Stress = 0)
     |    ____   |
     |   /    \  |
     |  | max  | |  <- Neutral Axis (Max Stress)
     |   \____/  |
     |           |  <- Bottom (Stress = 0)

Working / Process

1. Identify Section Properties

Determine the geometric properties of the beam cross-section, specifically the Moment of Inertia ($I$) about the neutral axis.
Locate the neutral axis, which for symmetrical sections is at the center of the height ($h/2$).

2. Select the Specific Level (y)

Determine the distance ($y$) from the neutral axis to the layer where you want to calculate the stress.
Calculate the First Moment of Area ($Q$) for the portion of the section above (or below) the level $y$. $Q = A' \cdot \bar{y}'$, where $A'$ is the area of the section above $y$ and $\bar{y}'$ is the distance from the neutral axis to the centroid of $A'$.

3. Apply the Shear Formula

Calculate the total shear force ($V$) at the specific cross-section using the beam's shear force diagram.
Plug all values ($V, Q, I, t$) into $\tau = \frac{VQ}{It}$ to obtain the shear stress value. Ensure all units are consistent (e.g., Newtons and millimeters).

Advantages / Applications

Structural Safety: Engineers use shear stress distribution to ensure that beams do not fail due to "web shear," especially in I-beams where the web carries the bulk of the shear.
Material Selection: Helps in choosing materials that have sufficient shear strength, preventing cracking in materials like concrete or wood.
Optimization: Understanding that shear stress is highest at the neutral axis allows for efficient design, where material is concentrated where it is needed most (e.g., flanged sections).

Summary

Shear stress in a beam is the sliding force generated by transverse loads that varies in intensity across the depth of the beam. In most standard sections, this stress follows a parabolic distribution, peaking at the neutral axis and vanishing at the outer edges. Key terms to remember include Neutral Axis, First Moment of Area (Q), Moment of Inertia (I), and the Shear Formula ($\tau = VQ/It$). This topic is essential for structural analysis, ensuring beam integrity against internal sliding forces.