Determining Individual Stresses in an Unsymmetrically-Loaded Circularly-Perforated Tensile Plate by TSA

摘要

This paper extends a previous method for stress analyzing symmetrical shapes by TSA to unsymmetrical geometries. In the previous situation, mechanical and geometric symmetry about the x- and y-axes, combined with analytically (rather than discretely/point-wise) satisfying the traction-free conditions on the edge of a hole, significantly helped to reduce the otherwise large number of coefficients in the Airy stress function. This consequently reduced both the number of equations which must be solved in the least-squares process and the amount of measured input data needed to evaluate these fewer coefficients. For more complicated situations, such as the present finite plate where symmetry does not prevail, the stress function is appreciably more complicated. The present lack of symmetry dramatically increases the number of Airy coefficients needed, and any ability to reduce the number of these coefficients such as by satisfying the traction-free conditions on the edge of the hole analytically now becomes extremely significant. For this unsymmetrical case, satisfying the traction-free conditions on the edge of the hole discretely necessitates at least 55 coefficients. However, utilizing only 25 coefficients provides reliable individual stresses in this unsymmetrical situation if one imposes the traction-free conditions at the hole analytically. Particular attention is paid to determining a realistic minimal number of the real Airy coefficients, as well as reducing the amount of measured input temperature data.