If I understand the context, I would say that you should use:

clé de voute

Maybe there’s another idiom to express that though.

Le mot « trame » n’est pas un terme aussi imagé, mais à défaut d’une expression consacrée correspondant à l’image que confère « backbone » il convient encore.

As a native french speaker, here is how I would translate that word in a normal conversation :

These ideas are essentially the backbone of what is taught to engineers all around the world.

[...] la base de ce qui est enseigné [...]

These firms are the backbone of our industrial sector.

[...] le pilier de notre secteur industriel.

Other translations such as ‘la clé de voûte‘ or ‘la pierre angulaire‘ are usually not used that often.

Colonne vertébrale is seldom used figuratively but épine dorsale might and is actually almost always used that way.

It is not the best fit for abstract backbones like in your first example but would more or less work for the second one:

Ces entreprises sont l’épine dorsale de notre secteur industriel.

In that case, pilier already suggested by Ob1lan is more idiomatic.

You might also use socle:

Ces idées constituent le socle de ce qui est enseigné…

Finally, a metaphor similar to backbone is ossature:

Ces entreprises forment l’ossature de notre secteur industriel.