Choose the right grass and water it sufficiently. You have to remember that the tree is sucking a tremendous amount of water and doesn't leave much for the grass. We have St. Augustine under our oaks, but the trees range from 50-100 years old and are trimmed up at least 15-20 feet from the ground to allow the sun to get under in the morning and again in the afternoon.
It is also necessary to thin out the limbs inside the drip line of the canopy -- anything that is pointing up or pointing in -- any branches that somehow rub against another. All of the branches should be growing outward. This pruning does two things: 1) keeps the tree vigorous and allows airflow through which helps it defend against strong winds; and 2) permits filtered sunlight to the ground.
Check with your local seed dealer or county extension agent on the shade grasses that thrive in your area.
I have several dirt people friends and they recommend not even trying to grow grass under trees because of the light issue. Grass, regardless of whether it is labeled as “shade tolerant” or not will not grow without sunlight. Your energy time and money might be better spent planting ground covers under the tree like monkey grass or vinca-not the flower vinca but the ground cover vinca.
Just my 2 cents. Good luck.