Stability of the shell of the minimum surface on a trapezoidal contour, taking into account geometric nonlinearity under thermal and power loading

Abstract

For the calculation of spatial thin-walled structures, it is necessary to perform verification calculations for two groups of limit states. The first group of limit states includes strength and stability calculations. The second group of limit states includes deflection and crack resistance calculations.

The object of study is a shell of minimal surface on a trapezoidal contour, which is a thin-walled spatial structure. This type of shell is used in the following industries: mechanical engineering, chemical industry, aircraft construction, and drone construction. They have high strength, rigidity and lightness, and are easy to manufacture.

The shells of minimal surfaces, which are studied by traditional methods of mathematical physics, are not sufficiently studied due to the unique thin-walled structures that were derived by optimizing the shape on a given contour. Recently, the creation of advanced structures, high-performance machines and special-purpose objects often leads to the need to develop structural shell elements with a complex geometric structure - which are the shells of minimal surfaces.

The essence of the geometrically nonlinear formulation is to find the actual displacements of the finite element model when studying the loss of global stability of the shell of the minimum surface on a trapezoidal contour, as well as to find the special places where the actual loss of stability occurs on the mid-surface. This methodology allows to calculate the loss of stability of the minimum surface shell with a high enough accuracy, which leads to the optimization of stability in multi-criteria parametric optimization.

Numerical studies were carried out using the finite element method (FEM). After creating the finite element model, a combination of external loads was set in accordance with the building codes of Ukraine. Next, an initial stability calculation was performed, which showed that the first four forms of stability loss have a strain energy coefficient of less than one. The thickness of the shell of the minimum surface on the trapezoidal contour was selected as follows, the minimum strain energy on the first form of loss of stability was a factor of 1.0, while the required thickness was 36 mm.