Simplified Calculation of the Inertia Moment of the Cross Section of the Console under Loading
Authors: Deryugin E. E. 
Published: July 2024

Introduction. Published studies on the rigidity of consoles under load focus on the issues of their deformation and destruction. Calculations of the inertia moment, fundamentally important characteristic of the strength of the rod, are described. However, the problem of significant time consumption for such calculations has not been solved. The presented study meets the lack. The objective of the work is to describe a new rapid method for analytical calculation of the shear stress distribution in the section of the console corresponding to the action of an external applied force. For the first time, tangential stresses are considered, and examples of calculating the inertia moment for two non-standard sections of the console are given in this context.

Materials and Methods. To develop a new method, the console was presented as a pack of plates oriented parallel to the vector of external force. The source calculations were based on the scheme of a console beam with a dedicated plate. The deformation of the rod elements was modeled taking into account the effect of a uniform shear stress field in the plate section. To validate the simplified calculation of the inertia moment of the sections, schemes of a square, ellipse, triangle, hexagon, six-pointed star, and a figured cross were used. Analytical and mathematical research methods were applied, specifically, the Huygens–Steiner theorem.

Results. A rapid valid method for calculating the inertia moment of the cross section of the console under loading has been developed. Its difference is the rejection of calculations for each section, taking into account the shape and other features. For any shape of the section, the beam is represented as a bundle of infinitely thin plates, their inertia moments are integrated, and a well-known solution for deflection of a thin plate is used. The method allows us to unambiguously show the distribution of tangential stresses at the end of the console, providing a given deflection, and tangential stresses are used for such solutions for the first time. Their profiles are obtained depending on the direction of the external applied force. Formulas for the inertia moments of complex sections — a six-pointed star and a figured cross — are derived for the first time. Each section is correlated with the stress distribution curve and its maximum value. This data is visualized in the form of diagrams. It is found that the inertia moment and the rigidity of the console do not change when the external applied force is rotated by 30° for a star-shaped section and by 45° for a square and a figured cross. In general, the tangent field depends on the geometry and on the orientation of the section relative to the external applied force.

Discussion and Conclusion. The proposed simplified approach to calculating the inertia moment of the cross sections of the consoles makes it possible to uniquely determine the field of tangential stresses at the end, which provides the appropriate value of the external applied force for the given deflection. Engineers and mechanics can use the results of the presented work in the calculations and modeling of deformation of rod structural elements.