Cylindrical shells subjected to non-uniform wind pressure display different buckling behaviours from those of cylinders under uniform external pressure. At different aspect ratios, quite varied and complex buckling patterns occur: the results of linear and nonlinear buckling analysis can also be quite different. By contrast, cylinders under uniform external pressure always experience circumferential buckling and are little affected by changes in geometry, except in very short cylinders or changed boundary conditions. This paper presents a wide-ranging study of anchored stocky and intermediate length cylindrical shells of uniform thickness under wind pressure. Its aim is to produce useful information for the design of silos and anchored tanks against buckling under wind. The finite element analyses indicate that both linear and nonlinear analyses predict the circumferential compression buckling mode in stocky cylinders. For intermediate cylinders, pre-buckling ovalization of the cross-section has an important influence on the buckling strength. Empirical expressions are developed to relate the linear and nonlinear critical stagnation pressures under wind to the classical critical value for uniform external pressure. The effects of yielding and imperfection sensitivity are also briefly explored.