The work is concerned with asymmetric buckling of clamped nonuniform orthotropic shallow spherical shells under internal pressure. The effect of degree of orthotropy and material inhomogeneity, the curvature of a shell and the radius of the circular hole in the center of a shell on the buckling load is examined. A mesh-like structure in a human eye called Lamina Cribrosa (LC) may be modeled with such shell. The LC is a part of sclera, where the optic nerve fibres pass through and where the layer of sclera becomes thinner and many little pores appear. The buckling of the LC in the non-axisymmetric state in the neighborhood of the edge could cause edamas and folds at the periphery of the LC, atrophy of the optic nerve fibres and the eventual loss of the sight. The non-symmetric part of the solution is sought in terms of multiples of harmonics in angular coordinate. The numerical method is employed to evaluate the lowest value of the load, which leads to the appearance of waves in the circumferential direction. It is shown that if the modulus of elasticity decreases away from the center of the shell, the critical internal pressure for nonsymmetric buckling is significantly lower than for a shell with constant mechanical properties and the number of waves in the circumferential direction increases with the degree of non-uniformity.