An analysis of cylindrical sandwich structures with weak orthotropic core subjected to patch loading is presented. A high order theory model combined with a novel formulation technique is used to predict the static response of the structure. The face-sheets are considered as thin shells that follow the first order shear deformation theory, whereas the core is considered as a linear elastic medium. The effects of core elastic and shear stiffness, curvature to length ratio, and stacking sequence and orientation on the core’s stresses and displacements are presented. The case of a moving load is explored and performance charts are generated to design and optimize the structure in response to the patch loads. In particular, considering an orthotropic core with quasi-isotropic lay-up in the face-sheets, limit the variation of the transverse core shear stress and provide an efficient structural design configuration.