Texas is quite a distance from Illinois, yet your questions are similar to those here as well. My thoughts are too much shade. You have an irrigation system that has been in place many years and have to assume you manage this properly so I rule out too little, too much water for your trees. Up here these rings appear as the tree naturally grows taller and wider, limiting sunlight very near the base of the tree. there is also the heavy roots and crown area that just does not allow much of anything to grow. Western exposures are likely to dry out sooner and so here it could be a lack of moisture due to shrub roots and the sandy soil you mention and not sunlight. Compacted soils inhibit water from entering the soil profile. Sandy soils could become compacted, but not in a pattern as you describe. Tree and shrub roots can compete well against grass for soil moisture and I suspect that is part of the situation on the west side.