The extradosed bridge is a variant of the cabled-stayed bridge and the segmental pre-stressed concrete bridge, combining features of both. It has been extensively used in Japan, and its use is spreading to other places.
A cable-stayed bridge usually has a deck that is sufficiently rigid to spread local loads and to prevent undue flexure and torsion, but the deck is unable to support itself without the cables.
The pre-stressed concrete bridge, by contrast, has no external supports apart from the piers, and the forces in it are controlled by tension in the tendons and compression in the concrete.
The extradosed bridge has a deck which is self-supporting, unlike the cable-stayed bridge, but the extra forces live loads are transferred to cables above the deck. Because these cables take only the live loads, they can be thinner than those of an equivalent cable-stayed bridge, and the can be held up by shorter towers.
This sounds very desirable, but what we desire does not always come to pass. What if the extra weight needed to make the deck self-supporting is much greater than the weight of the live loads? Is the extradosed bridge still feasible and economical?
What about the external cables? It is all very well to say that they will take the live loads, but how do we make them do that? What prevents the deck from taking some or all of the live load? If, for example, the external cables are made of an alloy that is too easy to stretch, they will allow the deck to deflect and take load. The same problem occurs in pre-stressing - we have to make sure that the tendons do not stretch enough to allow too much tension in the concrete.
Since the extradosed bridge has a self-supporting deck, this can be built the external cables, using the well established techniques for segmental bridges. The towers and cables can be added after completion of the deck. Since the external cables are not required to support the weight of the deck, they could (in principle) be installed with nominally zero tension. Compare this with the cable-stayed bridge, where the cables have to be installed with the deck sections, and adjusted to specified tensions, which will change as the construction progresses. Even if the external cables of an extradosed bridge are given some small initial tension, the tensions in the cables will not affect each other as much as in a normal cable-stayed bridge.
Here are some URLs about extradosed bridges.