Metal-organic frameworks (MOFs) are promising materials for applications such as separation, catalysis, and gas storage. A key indicator of their structural stability is the shear modulus. By density functional theory calculations in a 106-atom supercell, under the local density approximation, we find c(11)=29.2 GPa and c(12)=13.1 GPa for Zn-based MOF 5. However, we find c(44) of MOF-5 to be exceedingly small, only 1.4 GPa at T=0 K. The binding energy E(ads) of a single hydrogen molecule in MOF-5 is evaluated using the same setup. We find it to be -0.069 to -0.086 eVH(2) near the benzene linker and -0.106 to -0.160 eVH(2) near the Zn(4)O tetrahedra. Substitutions of chlorine and hydroxyl in the benzene linker have negligible effect on the physisorption energies. Pentacoordinated copper (and aluminum) in a framework structure similar to MOF-2 gives E(ads) approximately -0.291 eVH(2) (and -0.230 eVH(2)), and substitution of nitrogen in benzene (pyrazine) further enhances E(ads) near the organic linker to -0.16 eVH(2), according to density functional theory with local density approximation.