Periodic soft nanostructures are building blocks for small devices. However, mechanical failure in the form of structure buckling or distortion from their original shape is often reported when the dimension of these soft structures were reduced to below submicron scale. Such a phenomenon seriously limits a reliable impact of these nanostructures to greater applications. Current understandings of buckling of soft 2-D nanostructures are limited. The substrate for these soft nanostructures is usually very compliant. Neighboring nanostructures could interact through the deformation of the substrate. We analyze the collective buckling of a two-dimensional array of nanoscale columns with their lower ends built into an elastic substrate. Buckling of these nanostructures is mathematically described by an eigenvalue problem. Numerical analyses show patterned collapse for these 2-D nanostructures, qualitatively matching reported experimental findings. Our efforts are useful toward the understanding and manufacturing of many two-dimensional nanoscale features.