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Are moments relevant to the thicknesses of beams in a structure (ie. the larger the moment the thicker the beam)?

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Yes, moments do determine the thickness of a beam. The relationship is S(x)= M(x)/(Allowable stress)

Where S(x) is the thickness per unit length and M(x) is the moment per unit length. However, this can produce beam thickness values of zero, which would be impossible, so you'll have to constrain this model with allowable sheer force and deflection requirements.

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