Static Analysis on Cantilever Beam

Definition

A cantilever beam is a structural element that is fixed (supported) at only one end and free at the other. Static analysis involves calculating the internal forces, bending moments, and deflections of the beam when it is subjected to stationary loads that do not change over time.

Main Content

1. Beam Support and Loading

A cantilever beam is characterized by a fixed support that resists both vertical force and rotational movement (moment).
Loads applied to the beam can be point loads (acting at a specific spot) or distributed loads (spread across a length).

Fixed Support |-------Beam-------| Free End
      ||      |                  |
      ||      |    Load (P)      |
      ||      V                  |

2. Shear Force

Shear force is the internal force acting parallel to the cross-section of the beam.
It represents the tendency of one part of the beam to slide against the other due to the applied load.

3. Bending Moment

Bending moment is the internal torque that causes the beam to curve or "bend."
In a cantilever beam, the maximum bending moment always occurs at the fixed support because it must resist the cumulative effect of all loads along the entire span.

Working / Process

1. Mathematical Modeling

Define the beam length ($L$), the material properties (Young's Modulus $E$), and the cross-sectional geometry (Moment of Inertia $I$).
Identify the type of load applied (e.g., a concentrated load $P$ at the free end).

2. Calculation of Reactions

Sum the vertical forces to zero ($\sum F_y = 0$) to find the vertical reaction at the support.
Sum the moments about the fixed support to zero ($\sum M = 0$) to calculate the fixed-end moment.

3. Deflection Analysis

Use the Euler-Bernoulli beam equation to determine how much the beam bends at the free end ($\delta$).
For a point load $P$ at the tip: $\delta = \frac{PL^3}{3EI}$.

Advantages / Applications

Used in architectural structures like balconies and overhanging roofs to provide an unobstructed view or space below.
Essential in mechanical components such as aircraft wings, crane jibs, and micro-electromechanical systems (MEMS) sensors.
Facilitates simple experimental setups in labs for testing material strength and elasticity.

Summary

Static analysis on a cantilever beam is the process of calculating internal forces and deformations on a member fixed at one end. By determining shear force, bending moment, and deflection, engineers can ensure that structural designs remain safe and within material limits.

Important terms: Fixed Support, Shear Force, Bending Moment, Deflection, Young's Modulus, Moment of Inertia.