The folding of epithelial cell sheets is a fundamental process that sculpts developing tissues and organs into their proper shapes required for normal physiological functions. In the absence of detailed biochemical regulations, the epithelial sheet folding may simply proceed through buckling due to mechanical compression arising extrinsically from the surroundings or intrinsically within the sheets. Previous studies hypothesized that the formation of an expanding supracellular actomyosin ring within epithelial sheets could result in compression that ultimately leads to epithelial folding during tracheal development in the Drosophila (fruit fly) embryo. However, the exact mechanism by which the formation of epithelial folds is coordinated by the ring expansion remains unclear. Using a vertex-based mechanical model, here I systematically study the dependence of epithelial fold formation on the physical properties of expanding supracellular contractile rings. The simulations show that depending on the contractile strength, epithelial cell sheets can undergo distinct patterns of folding during ring expansion. The formation of folds in particular is robust against fluctuations in the ring properties such as ring numbers and tensions. These findings provide a systematic view to understand how the expansion of supracellular contractile rings in epithelial sheets mediates epithelial folding morphogenesis.