Statics

Internal Forces: Cut a Beam Open to See What It Carries Inside

Cutting exposes internal N, V, M, read from one-side FBD, the shear diagram, the moment diagram, dM/dx = V so M peaks where V=0

Cut a loaded beam at any point and the exposed face reveals the internal forces holding that piece in equilibrium: a normal force, a shear force, and a bending moment. Sweep the cut along the beam and these trace the shear and bending-moment diagrams that decide where it is most stressed.

Drag the cut. To keep the piece you kept in equilibrium, the cut face must supply exactly the internal shear and moment that balance the loads on that side - that is what is happening inside the solid beam.

A cut face can carry three internal actions: a normal force N pulling or pushing along the axis, a shear force V slicing across it, and a bending moment M that bends the beam. Toggle to see how each one deforms the material.

The shear V at a cut is just the net transverse force from everything on one side. Drag the cut across a center-loaded beam and the shear steps from +P2 to -P2 as you pass the load - tracing the shear diagram.

The bending moment M at a cut is the moment of those same one-side loads about the cut. Drag along and M rises straight to its peak P L4 under the load, then falls - the triangular bending-moment diagram.

The two diagrams are linked: the slope of the moment curve is the shear, dM/dx = V. So where the shear passes through zero, the bending moment reaches its maximum - exactly the spot a beam is most likely to fail. Drag the marker to find it.

In PracticeCutting a loaded member exposes the internal forces that keep each side in equilibrium: a normal force N along the axis, a transverse shear V, and a bending moment M. Each one is read straight off a free-body diagram of one side - V is the net transverse load there, M is its moment about the cut - and sweeping the cut along the beam traces the shear and bending-moment diagrams. Those two are tied by dM/dx = V, so the moment peaks where the shear crosses zero, pinpointing the most stressed section. This is the doorway from statics into mechanics of materials, where those internal forces become stresses the material must survive - and it closes the statics arc that began with a single force vector.

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