The structural performance and design of concrete-filled double skin tubular (CFDST) cross-sections with square stainless steel outer tubes are studied herein. A total of 17 four-point bending tests on CFDST cross-sections with varying concrete grades, together with accompanying material tests, were first conducted. The details of the test rig and procedures, as well as the key experimental results are reported. Following the physical testing, a numerical modelling campaign was carried out. A finite element (FE) model was initially validated against the tests, and then adopted to conduct a parametric study to acquire further FE data, covering a broader spectrum of material strengths and cross-section slendernesses. The obtained test and FE results were used to evaluate the applicability of the general design provisions for concrete-filled carbon steel members in the current European and American design codes. Overall, the examined design codes are shown to provide unduly conservative (less so for the higher concrete grades) and rather scattered moment resistance predictions, though some moment resistances predicted using the European code were on the unsafe side. Modifications to the European design treatment in relation to the assumed stress distribution, to take due account of the partial spread of plasticity in the outer tube, and the effective compressive strength of the concrete infill, to reflect the reduced relative effectiveness of using higher concrete grades, are proposed and shown to improve the consistency of the resistance predictions.