Which value of modulus "transverse or longitudinal" I have to use in finding the critical buckling load?

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Behrad Koohbor

It depends on the orientation of fibers relative to the load direction. For an on-axis condition, i.e. where fibers are aligned with the direction of loading, longitudinal modulus must be used. See this publication for more details:

Article Fiber nonlinear predictive model for combined bending-compre...