internal forces in beams
Cut a loaded beam anywhere and the exposed section shows the internal forces that hold that piece in equilibrium. They are the axial force, the shear force, and the bending moment, and sweeping the cut along the beam draws the shear diagram and the bending-moment diagram and decides where the stress is greatest. Opening the inside with a cut is the tool of this lesson.
Drag the cut and what happens inside shows itself. For the piece you keep to stay in equilibrium, the section has to supply exactly the internal shear and moment that cancel the loads on that side, and that is what really happens inside a solid beam. The place you cut is the place you read.
A section can carry three internal actions. An axial force N that pulls or pushes along the axis, a shear force V that cuts across, and a bending moment that bends the beam, and the toggle shows how each one deforms the material. Pulling, cutting, and bending are the three branches.
The shear on a cut is the sum of every transverse force on one side. On a beam with a central load, dragging the cut makes the shear jump like a step from +P2 to -P2 as it passes the load, and that is how the shear diagram is drawn. The sign flips the instant the cut crosses the load.
The bending moment on a cut is the moment that the loads on one side make about the cut. Dragging it, the moment climbs straight up to a peak of P L4 under the load and then comes down, and that is the triangular bending-moment diagram. The middle bends the most.
The two diagrams are tied together. The slope of the moment curve is the shear, so dM/dx = V, and therefore the bending moment is greatest where the shear passes through 0. That is the very spot where the beam is most ready to break, and dragging the marker finds the place. The top sits where the slope vanishes.