Shear forces and moments

Cantilever Beam with Concentrated Loads

This interactive visualization demonstrates the internal forces (shear force \(V\) and bending moment \(M\)) in a cantilever beam subjected to concentrated forces.

The beam is clamped at the left end (\(x = 0\)) and free at the right end (\(x = L\)).

How to use

  1. Enter a position along the beam (0 to 10)
  2. Enter the force magnitude (positive = downward)
  3. Click “Add force” to apply the load
  4. Observe how the shear force and bending moment diagrams change

Sign conventions

  • Positive shear force \(V > 0\): Acts upward on the left face of a beam section
  • Positive bending moment \(M > 0\): Causes compression on the top fiber

Theory

For a cantilever beam with concentrated force \(F\) at position \(a\), the internal forces are:

Shear force: \[ V(x) = \begin{cases} F & \text{if } x < a \\ 0 & \text{if } x > a \end{cases} \]

Bending moment: \[ M(x) = \begin{cases} F(a - x) & \text{if } x < a \\ 0 & \text{if } x > a \end{cases} \]

The reaction forces at the clamped support (\(x = 0\)) are:

  • Reaction force: \(R = -\sum F_i\)
  • Reaction moment: \(M_R = -\sum F_i \cdot a_i\)