OP inquired about stiffening a thin wall tube. It was flexing too much. Much of the discussion went into how to strength the tube which is not the answer he was after. The formula for the center displacement of a simple supported beam with a center load:
Y = P*L^^3/(48*E*I)
Y is the displacement
P is the center load
L is the total span
E is the elastic modulus
I is the moment of inertia of the section
Looking at the formula will give you the solution. Reduce the span will do a lot of good. Changing to titanium will increase
E more than 3 folds, but hardly a cheap solution. A tubular section is most efficient in getting a high
I value. Thicker wall and bigger diameter put you in the right direction.
The most popular suggestion here is filling the tube with concrete. This no doubt will get you a stronger member that is capable in flexing more and take more load without bucking the top or compression side of the tube.
What about displacement? It is inversely proportional to
E value of steel is about 10 times of concrete. Let's say the
I value of the tube is roughly the same as the hollow space or the concrete fill. The increased
E*I is therefore 10% after the concrete fill, and reduction in displacement is roughly 10%. It will not be a reduction of from 40 mm to 5 mm OP was looking for.