In the last years considerable attention has been paid for the orthogonal and non-linear projections of self-similar sets. We consider orthogonal transformation-free self-similar sets in R³. We show that if the dimension of the set is strictly bigger than 1 then the projection of the set under some non-linear functions onto the real line has dimension 1. As an application, we show that the distance set of such self-similar sets has dimension 1. Moreover, the third algebraic product of a self-similar set with itself on the real line has dimension 1 if its dimension is at least 1/3.