Rigidity Analysis of Braced Grid Models via Best Constants of Discrete Sobolev Inequalities

Abstract

Abstract. This paper discusses the application of discrete Sobolev inequalities to grid bracing problems in structural engineering. We first consider the planar graph corresponding to a braced grid model and derive the Moore-Penrose generalized inverse matrix (Green matrix) of its discrete Laplacian matrix. By defining the underlying suitable Hilbert space, we show that the Green matrix serves as a reproducing kernel, from which we derive a discrete Sobolev inequality via the reproducing property. The best constant of this inequality provides a quantitative measure of the “rigidity” (stiffness) of the bracing model. In this study, we focus on the 1×n and 2×n braced grid model to investigate how bracing patterns and stiffness parameters affect the best constant. Our results mathematically clarify the effectiveness of alternating bracing directions and demonstrate that the optimal bracing pattern can shift depending on the specific values of the stiffness parameters.