Consider a beam with an upwards concentrated force applied to its center. This is equilibrated by a distributed downwards force. There are no displacement boundary conditions. I've tried approaching this with simple beam theory, but I can't get a complete answer (shear, moment, slope, deflection) unless I assume a deflection value somewhere. What is the best way to get the complete answer for this problem? I'm also curious about what happens if the concentrated force is off-center (equilibrium then maintained by applying a moment along with the force). The lack of symmetry complicates things further.