This research undertakes a comprehensive parametric study of the ASTM D 6641 Combined Loading Compression (CLC) test fixture from Wyoming Test Fixtures (WTF) using numerical analysis to evaluate various test parameters associated with compression testing of unidirectional test specimens. A two-dimensional finite element analysis of the D 6641 CLC fixture was used to identify loading conditions and tabbing configurations that minimized the peak axial, shear, and through-thickness stress concentrations in the test specimen. Combined shear and end loading appears to offer a good compromise between reducing the tab tip stress concentrations inherent in shear loading, and avoiding premature failure by end brooming associated with end loading. Thin, tapered tabs made of compliant materials were found to have the smallest stress concentrations at the tab tip. Specimens with these tab configurations should yield higher measured compressive strengths since premature failure at the tab tip is less likely to occur. Partial debonding of the tab tip was found to increase the stress concentration compared to the fully-bonded case, but to move its location away from the gage section of the specimen. Tab tip debonding should be avoided for compressive strength testing.