In the paper, we first introduce a new concept of generalized convex structure and obtain its useful properties, and then a pair of new generalized enriched contractions is defined in this convex metric space. Based on these we present some new common fixed point theorems for this pair of generalized enriched contractions. Furthermore, by applying this technique to other mappings, we define several pairs of new contractions including a pair of generalized enriched Kannan contractions, a pair of generalized enriched Ćirić-Reich-Rus contractions and a pair of generalized enriched Chatterjea contractions. And then we obtain their corresponding common fixed point results, which extend and improve many existing results in the literature, such as the results of Kannan, Ćirić, Reich, Rus, Chatterjea, Berinde et.al. Finally, simple numerical experiments are presented to illustrate the impact of the choice of the control parameters a,b on the convergence rate of sequence.