We formulate generally covariant andGL(n,R)-invariant variational principles for the field of linear co-frames in an-dimensional manifold. The internalGL(n,R)-symmetry is the main difference between presented models and the metric-teleparallel theories of gravitation (including Einstein theory), invariant under the global Lorentz groupO(1,n−1) (local in Einstein theory). There is a structural similarity between our Lagrangians and those used in Born-Infeld electrodynamics. We discuss field equations and Lagrangian constraints. Field equations are nonempty because the existence of certain «vacuumlike» solutions is explicitly shown. These solutions are isomorphic to canonical forms of semi-simple Lie groups (just as in certain field-theoretic models studied by Toller, D’Adda, Nelson and Regge), or to appropriately deformed canonical forms of trivial central extensions of semi-simple Lie groups. The latter applies,e.g., to four-dimensional space-times. The normal-hyperbolic signature is nota priori assumed, however, ifn-4, it turns out to be a property of the most natural solutions; roughly speaking, it is implied by differential equations. On the other hand, if we assume that the signature is to be normal-hyperbolic, thenn=4 is the lowest nontrivial dimension. We present certain arguments in favour of the hypothesis thatGL(n,R) is a promising candidate for the symmetry group of generally covariant fundamental field theories. OurGL(n,R)-invariant Lagrangians for frames provide a first step towards developments in this direction.