  We present a unified algebraic Bethe ansatz for open vertex models which are
associated with the non-exceptional
  $A^{(2)}_{2n},A^{(2)}_{2n-1},B^{(1)}_n,C^{(1)}_n,D^{(1)}_{n}$ Lie algebras.
By the method, we solve these models with the trivial K matrix and find that
our results agree with that obtained by analytical
  Bethe ansatz. We also solve the $B^{(1)}_n,C^{(1)}_n,D^{(1)}_{n}$ models with
some non-trivial diagonal K-matrices (one free parameter case) by the algebraic
Bethe ansatz.
