In this paper we study the action of fiber preserving involutions of elliptic surfaces. In the case where there is a distinguished section considered as the zero element of the Mordell-Weil group, we have the so called elliptic involution. It is well known that the quotient of the elliptic surface by the elliptic involution is smooth, and the elliptic surface can be realized as a double cover of a ruled surface, which is the so called Weierstrass model. On the other hand, when the elliptic surface has a non-zero section, the composition of the elliptic involution and translation by this section also becomes an involution. We show that the action of this involution does not have any isolated fixed points and by taking the quotient, we can obtained a second model for realizing the elliptic surface. We analyze and compare the branch loci of the two models. In particular, we consider the case of rational elliptic surfaces and as an application we obtain results concerning the concurrence of bitangent lines of plane quartic curves with at most nodes.
