 Solution

Use the same first variation on the half-plane, with no restrictions on the edge traces of $\eta$. Write $M=w_{xx}+\nu w_{yy}$. Two $x$ integrations of the $M{\eta }_{xx}$ term produce the boundary term $-M{\eta }_x+M_x\eta$ at $x=0$. The mixed term, after one integration in $x$ and one along the edge in $y$, contributes $2(1-\nu )w_{xyy}\eta$. All other boundary terms vanish at infinity. HenceThe bulk equation makes its area term zero. Stationarity therefore gives exactlyThe two edge traces can be prescribed independently: use $\eta (x,y)=\chi (x)[f(y)+xg(y)]$ with $\chi =1$ near zero and compactly supported smooth $f,g$. The fundamental lemma on the edge then givesThey are zero bending moment and zero effective transverse shear at a free straight edge. The second includes the edge derivative of the twisting moment; omitting the mixed contribution would give the wrong coefficient.