Cyclic worst principal stress and strain results

When you load the results of cyclic symmetric analyses, the software calculates the cyclic worst principal stress and strain (for both, scalar and angle). The worst principal stress and strain results have a single scalar component and are evaluated for the primary sector only. Therefore, if you display other sectors in the post view, the software displays the same results for every sector. In addition, the post-processor only shows the absolute value of stress and strain.

Cyclic worst principal stress calculation

The following explains how the software calculates cyclic worst principal stress and strain.

• For harmonic index 0 and N/2 (where N is an even number of sectors), the software obtains a unique eigenvalue for every mode. For harmonic index 0, the response of all structures is identical. Therefore, all the structural responses such as stresses, strains, and displacements are identical between all sectors. For cyclic worst principal stress and strain, the software computes the principal stress and strain at each of the nodes, which corresponds to the cyclic worst principal stress and strain value. The software reports the cyclic worst principal stress angle as 0. For harmonic index N/2 (for even N), the response of neighboring structures are equal in magnitude and opposite in direction. In terms of absolute values, all the sectors exhibit the same results. Similar to the harmonic index 0, for the cyclic worst principal stress and strain value, at each node, the software reports the value of principal stress and strain that corresponds to each node. The software reports the cyclic worst principal stress angle as 0.

• For harmonics index 1 to N/2-1, every eigenvalue repeats. For each repeated eigenvalue, the primary sector has two orthogonal components (cosine and sine) for the corresponding eigenvector. A combination of these two orthogonal components determines the eigenvector for the remaining sectors. For a given harmonic k, the response of the structure is repeated every π/k radians over the entire structure, that is, for harmonic k, 2k repetitions of the response occur over the entire structure. The software clocks and normalizes the eigenvector results so for one of the repeated pairs (the odd number mode), the maximum displacement response occurs in the primary sector. Although, the maximum displacement occurs in the primary sector for the odd number mode, the peak stress and strain response can occur in a sector other than the primary sector. Therefore, to obtain the peak stress and strain response over the primary sector, as well as to reduce the impact of mesh dependency, the software computes the worst principal stress and strain using the following algorithm:The software computes the stresses and strains that correspond to the sine and cosine components of the primary sector eigenvector. Let σa correspond to the stress and strain due to the cosine component and let σb correspond to the stress and strain due to the sine component of the eigenvector. The spatial variation in the structures response varies over π/k radians. Therefore, when computing cyclic worst principal stress and strain, the software subdivides the angle into M increments (user-defined parameter that defaults to 100). Each subdivision is then α = π/(Mk), and at the ith subdivision, the angle is θ = iα = iπ/(Mk). At each subdivision, the software computes a stress and strain tensor corresponding to the combined effect of the sine and cosine component as: σ=σacos(kθ) + σbsin(kθ)The software computes principal stress and strain at each nodal location that corresponds to tensor σ. It compares the absolute value of the maximum principal stress and strain with the previous maximum value. For each node, it stores the maximum value of the worst principal stress and strain and the angle (θ) at which it occurs.

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