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Note here that the stress in x direction is not constant as assumed in the analytical method. If we were to pick a section near the middle of the bar, our analytical result would be nearly accurate. The solution, however, no longer applies when considering the stresses at the wall on the top and bottom, and the stresses near the point load. Obviously, the stresses in the x-direction maximizes at the point load.

Sigma_y:

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The analytical method assumed a long bar, therefore by definition, the stresses in the y direction are assumed to be zero. Since this bar does have width, the stresses in the y direction are symmetrical about the middle axis.

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