The invention provides an attachment member for anchoring a graft system at a desired location in a vessel of a patient's vascular system. The attachment member comprises a cranial zone having a first radial strength, a caudal zone having a second radial strength, and an intermediate zone having multiple joining longitudinal struts located between and connecting the cranial and caudal zones. The intermediate zone has a third radial strength which is less than the first and second radial strengths, and the cranial and caudal zones are formed from a self-expanding material capable of expanding from a first delivery configuration to a second deployed configuration. The invention also provides an endovascular graft system, a method for deploying an aortic attachment member, and a method for repairing an aneurysm in an aorta.