Summary: In this paper, we define cup product on relative bounded cohomology, and study its basic properties. Then, by extending it to a more generalized formula, we prove that all cup products of bounded cohomology classes of an amalgamated free product \(G_1 \ast_A G_2\) are zero for every positive degree, assuming that free factors \(G_i\) are amenable and amalgamated subgroup \(A\) is normal in both of them. As its consequences, we show that all cup products of bounded cohomology classes of the groups \(\mathbb{Z} \ast \mathbb{Z}\) and \(\mathbb{Z}_n \ast_{\mathbb{Z}_d}\mathbb{Z}_m\), where \(d\) is the greatest common divisor of \(n\) and \(m\), are zero for every positive degree.