Principal Stresses in Terms of Value and Direction in Spatial and plain cases

Abstract

This research presents an analytical study to determine the principal stresses, both in terms of magnitude and direction, based on the normal and shear stresses acting on the faces of a stress cuboid formed around a point in a stressed body. The study is applied to a structural element in the form of a simply supported beam with a rectangular cross-section.

The normal and shear stresses are calculated along the most critical section of the beam, specifically at the section corresponding to the location of the left support. Using these stresses, the first and second principal stresses, along with their angle of inclination relative to the longitudinal axis of the beam, are determined. Mohr’s circle is then constructed for the stress state at a point in the interior of the upper wall of the cross-section. The analytical and graphical results were found to be in exact agreement.

Furthermore, the maximum shear stress theory was employed to assess the strength of the beam. It was found that the beam has a safety margin of 14.3%, even at the most critical points in the cross-section. This result highlights the adequacy of the beam’s strength, fulfilling the primary goal of the study.