A support structure for an x-ray window comprising a support frame defining a perimeter and an aperture, a plurality of ribs extending across the aperture of the support frame and carried by the support frame, and openings between the plurality of ribs. A rib taper region can extend from a central portion of the ribs to the support frame. The taper region can include a non-circular, arcuate pair of fillets on opposing sides of the ribs and an increasing of rib width from the central portion to the support frame.