The Flexure Formula
**Bending stress** ($\sigma_b$) is the normal stress that is induced at a point in a beam due to an external bending moment. It is zero at the **neutral axis** and maximum at the extreme fibers (top and bottom surfaces).
Calculation using Section Modulus
The most direct way to calculate the maximum bending stress ($\sigma_{max}$) is by using the applied moment ($M$) and the beam's section modulus ($S$). The relationship is defined by the Flexure Formula:
$$\mathbf{\sigma_{max} = \frac{M}{S}}$$
Key Parameters Explained
Bending Moment ($\mathbf{M}$)
This is the internal moment required to keep a cut section of the beam in equilibrium. It is calculated from the external forces and is generally the **maximum** moment along the beam's length.
Section Modulus ($\mathbf{S}$)
The section modulus is a geometric property of a cross-section used for beam design. A larger $S$ value indicates a greater resistance to bending stress. It is calculated as $S = I/c$, where $I$ is the moment of inertia and $c$ is the distance to the extreme fiber.
Critical Check (Safety Factor):
In design, the calculated $\mathbf{\sigma_{max}}$ must be less than the material's allowable stress (e.g., Yield Strength, $F_y$) divided by a factor of safety (FS), or less than the Design Stress, $F_d$. $\mathbf{\sigma_{max} \le F_d}$.