Why not just Lagrange polynomials basis functions
The weak form of the euler-bernoulli beam equations has second order weak derivatives. This means that the finite element space requires continuity in the 1st derivatives across each element boundary. Lagrange elements have continuity in the 0th derivative only (the values match at the boundary, but the 1st derivatives don’t). Hermite interpolants can be generalized to ensure continuity to any prescribed derivative order.
There is a theorem which states that for an nth order weak derivative in the weak form, you need (n-1)st order continuity in the interpolants between each element.