Molecular mechanics simulations are carried out on crystals of poly(p-phenylene teraphthalamide) (PPTA) as a function of an applied axial compressive stress. The vibrational frequencies of the long wavelength acoustic modes which propagate along the chain axis and are polarized perpendicular to the plane of the hydrogen-bonded sheets are found to become imaginary when the imposed stress exceeds the modulus for shear between hydrogen-bonded sheets. The imaginary frequencies denote an elastic buckling instability. This instability occurs at a compressive stress of 0.3 GPa, in good agreement with the experimental result for the stress which causes material failure in PPTA fibres. It is suggested that previous overestimates of compressive strengths based on elastic buckling models occurred due to the use of the torsion modulus as the relevant shear modulus; however, the torsion modulus is not relevant because it represents an average of shear moduli in different directions, while elastic buckling takes place along the direction of easiest shear.