Free Beam Bending: Find Complete Answer & More

[spovolny]

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.

 

[Simon Bridge]

i.e. the beam is balanced on a fulcrum ... how would you normally do this if, say, the beam were clamped at one end?

 

[spovolny]

i.e. the beam is balanced on a fulcrum ... how would you normally do this if, say, the beam were clamped at one end?

I appreciate the quick reply. Clamped at one end means a cantilevered beam, so moment/shear are zero at the free end and slope/deflection are zero at the clamped end. I see how my problem is like a beam on a fulcrum, but I don't want to necessarily say that the center deflection is zero.

I did impose zero shear/moment at the free ends. I also imposed zero slope at the center, but this breaks down if the concentrated force isn't centered.

 

[Simon Bridge]

Just stick to one problem at a time - treat the simpler case where the concentrated force is centered and then modify the approach to allow for uncentered force.
This is a statics problem - all forces and moments balance, the main trouble is that the beam bends and you want to know how much by right?
What's wrong with modelling as two half-length beams clamped at one end? You will need to account for the reactions due to the other side.
The main thing is to look at how you would treat that situation.