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Welling's winless start to the National League South season continued on Bank Holiday Monday at Park View Road.
They were brushed aside by Chelmsford with a goal in either half and are now seven games without a victory.
While injuries meant that Mark Goldberg was depleted of some of his key players, it was still a largely uninspiring display by Welling.
They were behind as early as the fifth minute when Mitchell Nelson allowed the ball to go over his head and Shamir Mullings rounded keeper Chris Lewington to score.
Lewington then got down well to save from Shaun Jeffers before Mullings somehow fired wide from 12 yards when it seemed easier to score.
Robbie Willmott shot wide of the target with another simple chance seven minutes before half-time.
Welling immediately created their best chance in the first half when Dan Walker crossed for Jamie Slabber, but his volley at the near post went over the bar.
After the break, Lyle Della-Verde looked Welling's best chance of getting back into the game. He crossed for Dan Walker in the 49th minute but the former Aldershot forward totally missed his kick infront of goal.
The game was over on 64 minutes when Welling conceded their sixth penalty in seven games.
Another rash challenge caused little dispute and Jeffers sent Lewington the wrong way from the spot.
Welling responded and former Wings defender Chris Bush nearly sliced into his own net before Nelson headed wide from a corner.
Welling: Lewington, Brand, Brown, Banton, Nelson, Hayles, Joseph-Dubois (Jackson 78mins), Coomes, Slabber (Whitely 46mins), Walker, Della-Verde. Subs not used: Sprague, Johnson, Devyne.
Attendance: 705.