# Stability of Composite Plates with a Dense System of Ribs in Two Directions

## TL;DR

This paper introduces a fast analytical method for analyzing the stability of composite plates with dense ribs, avoiding resource-heavy simulations.

## Contribution

A novel analytical approach using tolerance averaging simplifies stability analysis of ribbed composite plates with discontinuous coefficients.

## Key findings

- The analytical method produces results matching FEM calculations but with much lower computational cost.
- The transformed equations resemble classical plate equations, enabling straightforward analytical solutions.
- Material and geometrical properties significantly influence critical forces under shear and compression.

## Abstract

What are the main findings?
A simple analytical method of stability analysis of a thin plate with a dense system of ribs is derived and verified.Issues of uniaxial compression and shear in-plane loadings are investigated.All presented results are verified with FEM-based calculations, proving the correctness of the derived solution.

A simple analytical method of stability analysis of a thin plate with a dense system of ribs is derived and verified.

Issues of uniaxial compression and shear in-plane loadings are investigated.

All presented results are verified with FEM-based calculations, proving the correctness of the derived solution.

What are the implications of the main findings?
The derived analytical approach allows swift and reliable stability analysis of composites, contrary to FEM-based investigations, where the modelling of microstructure requires a lot of computational resources and long computation times.The performed investigations of the influence of material and geometrical properties on the shear stability of the plate allows us to predict critical forces without the necessity of performing time-consuming calculations.

The derived analytical approach allows swift and reliable stability analysis of composites, contrary to FEM-based investigations, where the modelling of microstructure requires a lot of computational resources and long computation times.

The performed investigations of the influence of material and geometrical properties on the shear stability of the plate allows us to predict critical forces without the necessity of performing time-consuming calculations.

This paper presents an easy-to-use analytical method for stability analysis of composite plates with dense bidirectional microstructure. The main characteristic feature of such a defined composite is that due to its periodic nature the obtainable governing partial differential equations are characterised by discontinuous, strongly oscillating coefficients. Such cases bring many difficulties during derivation of their solution. In order to simplify calculations, the initial governing equations are transformed with the use of the tolerance averaging technique, so a system of partial differential equations with constant coefficients is obtained. The most important finding of the presented work is that the form of the mentioned equations is similar to the classic equations, which describe the stability issue of the thin homogeneous plate. Consequently, the analytical solution to such issues is easily obtainable. Moreover, when compared to, for example, finite element method (FEM) analysis, it requires substantially less computation resources, which can be perceived as its superior feature. Therefore, the proposed method is convenient for engineering applications. In this paper, a comparative analysis of the results obtained from the proposed analytical models with the results obtained from the FEM has been carried out. The impact of materials and dimensions of microstructure on the values of critical normal and shear forces has also been analysed.