 Free straight edge of an isotropic bending plate

For a plate occupying $x>0$, the boundary part of the first variation of its isotropic thin-plate bending energy is, up to an overall nonzero sign,At a free edge the variation and its normal derivative can be prescribed independently. Their coefficients must therefore vanish, giving the two displayed natural boundary conditions: zero bending moment and zero effective transverse shear. The mixed-derivative coefficient includes the derivative along the edge of the twisting moment; keeping only the normal derivative of $w_{xx}+\nu w_{yy}$ would miss that contribution.