There are many structural situations that involve stiffened plates subjected to a non-uniform stress condition. One of the most common is the flange of I-shaped flexural members subjected to lateral forces or torsion combined with major axis bending. An explicit description for the effect of a linearly varying stress on the elastic buckling of centerline-stiffened plates is lacking. A solution for the buckling of flat rectangular plates with centerline support conditions subjected to non-uniform in-plane axial compression is presented. The loaded edges are simply supported, the non-loaded edges are free, and the centerline is simply supported with a variable rotational stiffness. The Galerkin method is used to establish an eigenvalue problem and a series solution for plate buckling coefficients is obtained by using combined trigonometric and polynomial functions that satisfy the boundary conditions. It is demonstrated that the formulation approaches the classical solution of a plate with a fixed edge as the variable rotational stiffness is increased. The solution is used to generate simple equations and the applicability and implications for design of I-shaped beams that are subjected to biaxial bending or combined flexure and torsion is demonstrated. It is demonstrated that, unlike the uniform stress state, a solution based upon isolating the two sides of an I-beam flange cannot be used when a stress gradient is present. Finally, the conservative approach of considering the peak edge stress as a uniform applied stress condition is critiqued.