Degrees of freedom of an internal hinge in plane beam?

Question

What are the degrees of freedom of an internal hinge in a plane beam?

I understand that the bending moment=0, does this mean there is no slope either?

Or does it have both degrees of freedom of deflection (V) and slope (θ)?

Answer 1

Let's start talking not about hinges, but supports. Specifically, why do supports generate the forces (including bending moments, if applicable) they do (or don't)?

Think of a simply supported beam under a downward force. Its supports generate upward reaction forces. Why? Because if they didn't, the system would be unbalanced, and we wouldn't have a structure but a mechanism accelerating according to $F = ma$. More specifically, though, the supports create their reactions because the specific nodes they are attached to would move otherwise. For example, you can create structures where the deflection at a given unsupported point is zero. See the example below, where the deflection at midspan is zero and vertical reactions at each support.

If you then put a support at the midspan, that support's reaction will be zero; after all, it's not having to resist that point's movement.

This is a trivial example of a fundamental relationship between reactions and deflections (including rotations): boundary conditions create reactions when they are inhibiting a deflection. If a given boundary condition doesn't create a reaction, it doesn't resist that deflection.

So, as you've stated, hinges have zero bending moment. What does that tell us? That hinges don't resist rotations. They also don't have force reactions (though axial and shear forces aren't necessarily zero at hinges; they transferred through the hinges, not absorbed), so we also know hinges don't resist deflections.

So, to sum up: hinges can have non-zero deflections and non-zero rotations (specifically, the slope on one side of the hinge can be different from the other side).

Answer 2

You seem to be mixing up the conditions on the forces and moments acting across the hinge, and the displacements and slopes there.

There is no bending moment transmitted across the hinge, but there can be equal and opposite shear forces on each side, and the two slopes can be different non-zero values.

The fact that the slopes can be non-zero should be obvious if you think about how a real-life hinge (e.g a door hinge) works!