The Euler characteristic is multiplicative in fiber bundles. On the other hand, the signature is not. Atiyah and, independently, Kodaira showed it by giving surface bundles over surfaces with non-zero signatures. Since then, many examples with non-zero signatures have been constructed. The signature of a surface bundle over a surface has some restrictions, for examples, it is dividable by 4 and vanishes if the base genus is 0 or 1. Bryan and Donagi constructed examples over a genus-2 surface with non-zero signatures. The signatures and the genera of their examples are sporadic. In this talk, for any positive integer n, we give a surface bundle of fiber genus g over a surface of genus 2 with signature 4n and a section of self-intersection 0 if g is greater than or equal to 39n. Such example are constructed using mapping class group arguments.