Gas foil bearings (GFBs) rely on their underlying elastic foundation to support radial loads at high rotor speeds. The bump foil strip compliance determines the maximum load capacity with large rotor excursions, well in excess of the bearing nominal clearance. GFBs must be designed properly to permit their reliable (predictable) usage in aircraft engines and other heavy load applications because an elastic foundation that is too soft results in a limited load capacity. Configurations in practical use include a second bump-strip layer or stop-pins underneath the original bumps which become active above a certain load threshold. In these last configurations, the overall stiffness of the support structure has piecewise load versus deflection characteristics. Presently, a simple physical model for GFBs with piecewise linear elastic supports follows. The analysis couples the Reynolds equation for the thin film flow of an ideal gas to the elastic supports motion. An exact flow advection model is adopted to solve the partial differential equations for the zeroth- and first- order pressure fields, thus rendering the GFB load capacity and frequency dependent rotordynamic force coefficients. Predictions show that heavily loaded GFBs comprising two bump layers in series and including stop pins prevent too large bump deflections which may induce permanent plastic deformations. The structural damping or loss factor in a GFB with a two bump strip layer enhances the bearing direct damping force coefficients.