A cage isnt installed to stiffen up the chassis, its a protection device. Read Chassis Engineering. The book talks about several methods of stiffing the chassis. A roll cage doesnt help as much as a properly designed pan really. Which really surprised me. I always thought a roll cage made it a solid unit. Wrong. So if you think about it, the strength of a cage for the chassis stiffness has to travel a long distance. And the bars just arent positioned correctly for chassis stiffening. Great for a roll over, the impact load is applied to the pan, and with proper side bars they help with side impact.
But for chassis stiffness think of the car as a pan, with the front and rear control arms attached to this pan. You want to keep this pan from twisting. Vertical bars and diagonal bars going up to the roof line dont add much in the way of keeping that lower pan from twisting.
Ideally, IMO, would be a box frame. A lower heavy walled tube frame for the lower and an identical frame above, separated by say 8" and linked together with bracing and sheetmetal fully welded.
It would be as long as the wheel base (longest control arm mounts actually) and as wide as the track (again, only as far out as the C/A mounts). Think of a large rectangular box with the control arms attached to this box. No twist. Very heavy and not practical.
Well thats not practical, it would encroach into the entire passengers compartment, or raise the car to a crazy height for the body to sit on this solid platform..
So there is a compromise. Start from the best situation and whittle it down for practical use. We can do the lower frame section of the box, thats what many full frame cars are. But that alone is still kinda weak.
So think about making the "PAN" stronger. Try to build back up to that simple rectangular box configuration. Add where you can, keep the weight as low as you can but still use some of the ideas of the box. The front and rear are easy, there is room.
So, we dont wanna add the top section of the full box, just wont fit and the weight doubles from the single plane frame thats in place. So look at maybe working on the vertical sides of the box, attaching them to points in the car where the upper section of the box frame would be. So it doesnt have to be high, shouldnt be really, the higher you go the less rigidity you get. Think about that 8" tall box.
The weakest spot?? Well open your door. There it is, look at the floor. All it has is the frame and the roof.
And thats why guys see a cage as an obvious improvement because you have a tube "helping" out with this weak link. But unless you are putting in a full nascar cage then the additional stiffness gained from the roof bar is minimal. Even a welded in diagonal door bar isnt helping much with the pan twist. Helps with frame bounce, fore and aft, but not really at all for flex side to side. Left front in relation to right rear.
So what do we do to bridge this gap in chassis stiffness????? Great question... Holy grail of chassis builders.
There are some engineers that are on top of this. And Im not talking about purpose built cars, race cars. Thats easy, clean slate, build a monocoque structure. Im talking about our cars.
Ok, so there are some engineering guys that are on top of this for us too.. I like the idea of a multi tube design frame. With these you can get a box frame within a small area. So in essence is like the 8" box frame but smaller. Upper and lower tubes connected but much closer. Still get a solid boxed pan, but much "thinner" to work with the cars we use..
I have some ideas

Who knows.... Im not gonna build it, but I think some are... Alright!!!!! I know, I went off on a tangent.... Solly

Umm,,, Cages are for protection, not chassis stiffening. As I look at my sagging frame LOL JR