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A new recursive trapezoid-based algorithm is proposed for calculating the strength- and stiffness-related geometric characteristics of polygonal beam sections. By taking as input the coordinate vertices of the polygon, the approach may effectively be applied to determine the cross-section area and the position of its centroid along with the axial area moments of inertia and product of inertia relative to the cross-sections' centroidal axes. As the method combines the theory of planar moments with the well-known trapezoid numerical method, it provides a straightforward tabular procedure that can be easily implemented using any spreadsheet software. In addition, aiming to determine the principal area moments of inertia, a computation matrix formalism based on the theory of orthogonal diagonalization of symmetric matrices is put forward as an alternative to the classical method of vanishing the first derivative and solving for the principal directions. Some benchmark examples are worked out to prove the relevance and accuracy of the proposed approach for several beam sections ranging from the most common triangular and quadrilateral ones to very complex open and closed polygonal profiles. The solution presented here appears to be virtually foolproof compared to the conventional one, as it does not necessitate the decomposition of complex polygonal cross-section area into elementary surfaces of different shapes, then use the parallel axes theorem, and so forth; rather, it requires only insertion of the coordinates of the vertices of the polygon, turn by turn in the clockwise order, relative to an arbitrary cartesian coordinate system, and no further steps.
Marius Nicolae Baba (Tue,) studied this question.