To calculate the pressure exerted by an agitated fluid on the walls of a non-pressurized vessel, consider the effects of extreme accelerations and decelerations, as these can significantly influence wall stress. Minor agitation typically has negligible impact, so designing for the liquid's mass with an appropriate safety factor is often sufficient. If agitation involves vibrations, it's crucial to address potential fatigue and resonance issues. Surge loading can also be managed with proper design considerations. Focus on critical areas, particularly the upper and lower parts of the container, and incorporate wave analysis, hydraulics, and the water hammer effect in your calculations.