Answer: Shade, clay soil, constant rough traffic...I don't think you CAN get a sturdy turf growing in this situation - these conditions are just too tough for grass to adapt. Astroturf may be a better solution!(Only kidding.)
The first problem is the soil. The drainage and structure can be improved if you till in lots of organic matter (compost, aged manure, rotted leaves, etc.). This is a lot of work, though, and you'll have to keep topdressing the lawn with compost to maintain the soil structure and grass health. It won't be easy, with constant traffic slowing you down. Have you considered designing an alternative layout, with raised beds of plants, with timbers or cement block walls? You can plant these with shrubs flowering perennials and annuals, and fill in the areas and walkways between the beds with play sand for the kids and dogs to enjoy. Another alternative is to keep part of the area off limits so you can grow and enjoy some turf in your yard. Before you pursue this type of design, though, check with a local landscaper to see if it will be necessary to install drainage tiles to avoid future problems with pudding moisture during wet spells. I hope this is helpful!
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