I have a flat shelf I want to insert between what are in effect two fixed peg boards on either side.
In short I need to insert - between | and | to make |-|
As I cant move the sides, I need just the tiniest bit of give to put the shelf in. I dont think I can use a flex joint as it would allow the shelf to bend in both directions.
Anyone have any idea to make the shelf bend in one direction only?
How much you can bend the shelf depends on both the material and thickness, which you did not share. Very few materials can be bent only one way. You also did not describe what you want to accomplish, and for these reasons, it is very difficult to propose a solution.
That is just a bit larger than 1/8" which is pretty thin for a shelf. You did not include the span, or length of the shelf, in your reply. Cut your shelf, lay it on a couple of pencils at the end, and put a 2kg (1lb) weight on it for a week or two. then measure the clearance in the middle and the ends. I think you might be surprised how much it sags if your shelf is 300mm (12") or more.
To answer your original question, the thin MDF can be bent, but how much deflection depends on the length. Experimenting is the only way to get a definite answer to your question.
You can also use a peg or tab under the middle of the shelf to discourage it from flexing downward after inserted - that would keep the more visible pieces clean (one peg / tab in the back of the shelf instead of 4 pegs / dowels more up front.