Timoshenko proposed a value of the factor k (kx or ky). He assumed that tsy at the neutral axis NA is the same as in the simplified and in the theoretical models. In the case of the rectangular section:

at y=0 (NA) :

and

we obtain k=2/3 =.666

Theoretical model:

Mindlin without shear correction

Reissner assumed the equivalence between the internal energy associated to the exact distributed shear t and the deformation energy of the simplified beam model. This approach gives a value of k called the Reissner coefficient and in the case of the rectangular section the value of k is 5/6=.833