  The effect of hydrostatic strain and of interstitial hydrogen on the elastic
properties of $\alpha$-iron is investigated using \textit{ab initio}
density-functional theory calculations. We find that the cubic elastic
constants and the polycrystalline elastic moduli to a good approximation
decrease linearly with increasing hydrogen concentration. This net strength
reduction can be partitioned into a strengthening electronic effect which is
overcome by a softening volumetric effect. The calculated hydrogen-dependent
elastic constants are used to determine the polycrystalline elastic moduli and
anisotropic elastic shear moduli. For the key slip planes in $\alpha$-iron,
$[1\bar{1}0]$ and $[11\bar{2}]$, we find a shear modulus reduction of
approximately 1.6% per at.% H.
