= Solution
The <jump conditions> express a connected <elastic beam> with no concentrated horizontal <force> or applied couple. <Continuity> of <tension>, $[T]=0$, follows from horizontal <force balance>. <Continuity> of the <displacement field> and <slope>, $[w]=[w']=0$, excludes a break or a kink; a kink would also have unbounded bending <elastic energy> in this model. <Continuity> of the <bending moment> in the absence of a point couple gives $[Bw'']=0$, or $[w'']=0$ for constant $B$.
The only allowed discontinuous section resultant is the <transverse shear force>. Integrating vertical <force balance> through an upward <point force> $R_i$ gives $[N]=-R_i$. Since $N=-Bw'''$, this is precisely
$$
\boxed{[Bw''']=R_i.}
$$
The same sign follows by integrating $Bw''''-Tw''+p=R_i\delta(x-x_i)$, where $\delta$ is the <Dirac delta function> and the regular terms have vanishing integrals over a shrinking interval. For nonadhesive <unilateral beam contact>, the required constraint is
$$
\boxed{R_i\geq0.}
$$
A negative <point force> would pull the <elastic beam> towards the obstacle. At an endpoint where $R_i=0$, the third <derivative> is continuous as well.
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