Vince,So if we take each piece of 1cm mass and assign an instantaneous centre of rotation to each section, a bending rod will have many instantaneous centres of rotation.
Exactly, because the rod is not a rigid body different sections of the rod will have different instant center of rotation as well as the different angular velocity values that Tobias has pointed out.
And to get the K.E. of the rod you would have to use those different rotation center values (and different omega and mass values along the length of the rod) to calculate the correct K.E. of the moving rod using the rotational dynamics energy value of K.E.=sum(1/2*I*omega.^2) values.