Overview
When a beam is subjected to external loads, it experiences internal shear forces and bending moments. These internal actions induce stresses and strains within the material, which are governed by the geometry of the beam and the nature of loading.
Bending Stress
The axial stresses developed due to bending moments are called bending stresses. These vary across the cross-section of the beam and are maximum at the outermost fibers.
At any section, the bending stress \( \sigma \) is a function of the bending moment \( M \), the distance from the neutral axis \( y \), and the moment of inertia \( I \):
$$ \sigma = \frac{M y}{I} $$Strain Distribution
Due to varying bending stresses, different layers (laminae) of the beam experience different axial strains. This results in a linear strain distribution across the depth of the beam, assuming the material obeys Hooke’s law and plane sections remain plane.
Longitudinal Shear
The differential axial strain between adjacent laminae leads to longitudinal shear stresses. These act parallel to the beam axis and are crucial for maintaining structural integrity between layers.
These shear stresses will be described in detail in the following subsections, including their derivation and distribution across the cross-section.