We derive a functional relation between the generating functions of connected chord diagrams and 2-connected chord diagrams. This relation enables us to calculate an asymptotic expansion for the number of 2-connected chord diagrams on $n$ chords. The asymptotic information obtained from this expansion refines the last established results and provides a simple alternative for calculating the asymptotic behaviour of certain Green functions in Quenched QED and Yukawa theory in the context of quantum field theory.