The Walsgrave Hospital born batsman hit 17 fours in an innings of 104 from 210 balls after Warwicks resumed from their overnight total of 15-2.
Bell was the fourth Warwickshire batsman to lose his wicket just before the tea interval. Earlier, skipper Ian Westwood had been trapped lbw by Smith for 57.
In the final session of the day, Jon Trott who had hit one six and eight fours was lbw to Chawla for 56 but Tim Ambrose was 40 not out and Jim Troughton 28 not out to see Warwicks on 302-5 at stumps.