Understanding Beam Deflection
**Beam deflection** is the degree to which a structural element is displaced under a load. It is a critical consideration in structural design to ensure the structure remains within acceptable limits for functionality and aesthetics, even if the stresses are safe.
The Deflection Formula
For a **simply supported beam** (pinned at one end, roller at the other) carrying a single **concentrated load ($P$) at the mid-span**, the maximum deflection ($\delta_{max}$) occurs at the center and is calculated using the following formula:
$$\mathbf{\delta_{\text{max}} = \frac{P L^3}{48 E I}}$$
Key Parameters
P, L (Load & Length)
Deflection is **linearly proportional** to the Load ($P$), but **cubically proportional** to the Beam Length ($L^3$). This means doubling the length increases deflection by $2^3 = 8$ times.
E, I (Material & Shape)
The deflection is **inversely proportional** to the product of the **Modulus of Elasticity** ($E$, material stiffness) and the **Moment of Inertia** ($I$, cross-sectional shape). Increasing either $E$ or $I$ decreases the deflection.
Unit Consistency Warning:
Ensure all inputs are in **consistent SI units** (Newtons, Meters, Pascals, Meters$^4$) to get the final deflection in **Meters**. The calculator converts the final result to millimeters (mm) for practicality ($\text{Meters} \times 1000 = \text{mm}$).