 Solution

Kernel conditions. The clamped-beam Green function must obey $G(0,\xi )=G_x(0,\xi )=G(1,\xi )=G_x(1,\xi )=0$. Away from $x=\xi$, $G_{xxxx}=0$. At the load point, $G,G_x,G_{xx}$ are continuous andA jump in any lower derivative would create a derivative of the Dirac delta, rather than the single point load required by $D_x^{4}G=\delta (x-\xi )$.

Construction. A particularly simple representation isIt already enforces the left-end conditions and the correct source jump. The right-end conditions give $a+b=-(1-\xi {)}^3/6$ and $2a+3b=-(1-\xi {)}^2/2$, henceFor $x<\xi$ this is already factored; for $x>\xi$, expanding and refactoring givesThe formulas agree at the junction. The forced displacement is obtained by integrating this kernel against $f$; for continuous loads it satisfies the equation and all four end conditions. Uniqueness follows because a cubic with zero value and slope at both ends is identically zero.

Reciprocity. In terms of $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$, the same expression isIt is visibly symmetric under interchange of $x$ and $\xi$. Thus the reciprocity of the clamped-beam Green function givesEquivalently, two integrations by parts show self-adjointness of $D_x^4$ under the clamped conditions, with bilinear form $\int u^{\prime \prime }{v}^{\prime \prime }dx$.