Clamped-beam Green function ($G(x,\xi )=x_{<}^2(1-x_{>}{)}^2[3x_{>}-(1+2x_{>})x_{<}]/6$)

For a unit-length elastic beam with unit bending stiffness, the Green function of $D_x^4$ with $G=G_x=0$ at both endpoints is the displayed kernel. It is piecewise cubic and continuous through its second derivative, while $G_{xxx}(\xi +,\xi )-G_{xxx}(\xi -,\xi )=1$. A constructive form is $G=a{x}^2+b{x}^3+(x-\xi {)}_{+}^3/6$, with $a=\xi (1-\xi {)}^2/2$ and $b=-(1-\xi {)}^2(1+2\xi )/6$. The right endpoint conditions determine these coefficients. Thus $y(x)={\int }_0^{1}G(x,\xi )f(\xi )d\xi$ solves the clamped static beam problem.

The clamped-beam Green function is symmetric. Its explicit formula depends only on $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$. More generally, two integrations by parts give $\int u{v}^{\prime \prime \prime \prime }dx=\int u^{\prime \prime }{v}^{\prime \prime }dx=\int u^{\prime \prime \prime \prime }vdx$ when $u=u^{\prime }=v=v^{\prime }=0$ at both endpoints. Applying this identity to two point-load responses shows that the displacement at one point caused by a unit force at a second equals the displacement at the second caused by a unit force at the first.