An extensive test series on circular cylindrical laminated composite stringer-stiffened panels subjected to axial compression, shear loading introduced by shear and combined axial compression and shear was carried out at the Technion, Israel. The test program was an essential part of an ongoing effort undertaken by the POSICOSS project (improved postbuckling simulation for design of fibre composite stiffened fuselage structures) aiming at design of low cost, low weight airborne structures that was initiated and supported by the Fifth European Initiative Program.The first part of this test series, dealing with panels PSC1-PSC9 (blade-stiffened), has already been summarized and published. The results of the tests with panels BOX1-BOX4 (blade-and J-stiffened) have also been reported and published. These tests dealt with two identical stiffened panels, combined together by two flat nonstiffened aluminum webs, to form a torsion box, thus enabling application of shear tractions, through introduction of torsion, and combined axial compression and shear. The present manuscript aims at describing test results and relevant numerical studies on the buckling and postbuckling behavior of another set of four panels, AXIAL1-AXIAL4, stiffened by J-type stringers. Based on the experimental studies carried out within the framework of the POSICOSS project and reported in the literature and on the present study design guidelines were formulated and presented. Accompanying supporting calculations were presented as well; they were performed with a "fast" calculation tool developed at the Technion, and based on the effective width method modified to handle laminated circular cylindrical stringer-stiffened composite panels.