I have a bookshelf similar to this in attachment. I wonder where is the optimal place at the back of bookshelf to mount a metal cross to gain the best stabilization of the whole construction. At first I was thinking about back-top. On the picture is back-bottom. Also what about back-middle?

You don't want it to tip forward, back top shelf (in the middle or two evenly spaced). This presumes there's a solid connection between the shelf and the bars on the sides. Shelving doesn't go any other way but forward unless you tackle someone into it at an angle.

If you're worried about the weight on the shelving and want to reduce it with wall anchors you're shelving is too light for what you want to put on it.

Where the largest weight of books will be. Note that this doesn't necessarily mean where the heaviest books are*, as hard covers will inevitably weigh more than pages.

*but usually it does.

You need a solid base. The bottom is the most stressed part as it supports the whole added mass. I would suggest to strengthen the back-bottom position.

I think I misunderstood, you want to stabilize it rather than mount it to the wall. In that case I'd say: get a heavier bookshelf if you possibly can.

Thank you for all the answers so far. Actually I am not so worried about the weight of books. That bookshelf was/is already tested in that matter. But, so far, it was positioned directly tight between two others bookshelves and all was fine. Now, I would like it to be "standalone" so the most thing I am worried about is to not make it collapsing to left or right side. And that is why I need now this metal cross at back.

I did some graphical experiment (see attachment) and before that I thought that cross on top or bottom will be most stressed, but it seems now that it doesn't depend on its position? It looks like cross at middle and cross at top will have the same stress when bookshelf's sides go right or left side?

On the origional image there appears to be a cross support.

If it were me id rip that off and weld a more substantual version in its place and the same at the top of the shelf unit

Think of it in terms of the sides of the bookcase being like levers, amplifying the force of any movement with the ultimate force the brace experiences being proportional to the distance between the brace and where the initial force is being applied. Now, unless you live in an area very prone to earthquakes the bottom of the shelf is probably going to be standing still, with most of the force being applied towards the top of the bookshelf. Thus you should place the brace near the top so that when stopping any left/right movement it has to experience the least amount of force (and thus least amount of mechanical stress).

As I said I have similar bookshelf - it is not identical. I have "a set" of bookshelf + (unfortunately) only one metal cross "to mount it somewhere". That was my first idea - but I googled and most of similar bookshelves on pictures has this cross on bottom (like in attachment) and some of them in the middle - that was weird to me and that's why I asked this question. I did not know if people mount this cross at bottom/middle because that is (somehow?) better OR because its easier to do that (you need to stand on something to reach the top)?

It could just be a matter of aesthetics- that the cross brace isn't as noticeable at the bottom or center as it is at the top.

I think that wherever you put the cross it will meet your resistance requirements. Just put it in the middle if unsure.

But just for debating sake, I still think that at the bottom is better than up. Think at it as if you would have two shelves that you have to put one over the other. One is reinforced an the other is not. Wouldn't you put the reinforced one below as support for the other?
A good base is always important as it minimizes lateral movement transmitted upwards. This is important if quakes are your primary concern.

You have to think what forces are likely to act upon your shelf:
- earthquake: lateral force applied from the base
- someone pushing the shelf: lateral force applied from the upper part
- heavy load on the shelf: force applied directly downwards
What is your biggest concern regarding these three options?

I think you will make a very good compromise if you put it at the middle as you will have shorter levers from both directions (up and down). And the best would be to anchor the shelf on a wall too, if this is an option.