# Is it possible to have curvature without bending moment?

## Main Question or Discussion Point

The simple question is whether it is possible to have curvature in a beam with no bending moment (similar to how there can be strain without stress)?

The main example I have to discuss what lead me to this question is a beam which has been prestressed concentrically and so is undergoing only axial stresses and no bending stresses. The beam at the start therefore just has an axial load at each end representing the PS forces. The next step is that a significant portion of the beam is removed (damaged) from midspan causing the centroidal axis of that part of the member to shift upwards while the prestressing force remains in the same line of action as originally.

Intuitively I would say that if a member under axial loads had a portion of the section removed, the beam would camber upwards and curve at midspan where the damage is. The curvature for this could be represented in a curvature diagram that is zero everywhere that the beam is intact but has a continuous value within the damaged portion where the flexural rigidity is reduced and the centroidal axis has changed. There hasn't necessarily been any forces applied to cause a moment but there would be curvature as far as I can tell.

My thinking is that the shift in the centroidal axis means that you effectively have to apply a pair of moments either side of the damaged portion Md= P χ δy where delta y is the shift upwards in the centroid and P is the prestressing force. The curvature would then be equal to Md/EId within the damaged part of the beam and elsehwere it would be zero as before.