With modified mesh

    Needs["NDSolve`FEM`"];
    additionalPoints = {{Ls/2 }}
    beamDomain =ToElementMesh[ImplicitRegion[0 <= x <= Ls, x], "IncludePoints" -> additionalPoints, "MeshOrder" -> 1]

and load 

    q[x_] := Piecewise[{{qs, 0 <= x <= lss}, {qss, lss < x <= Ls}}]

we can confirm result derived by @BillWatts using  FiniteElement, if we include correct boundary conditions `moment[Ls/2]==-(219375/32)`.

    solution = 
     NDSolveValue[{{D[moment[x], {x, 2}] + q[x] == 0, 
        moment[x] == -Es Is*D[w[x], {x, 2}]}, {w[0] == 0, 
        DirichletCondition[w[x] == 0, x == Ls/2], w[Ls] == 0, 
        moment[0] == 0, 
        DirichletCondition[moment[x] == -(219375/32), x == Ls/2], 
        moment[Ls] == 0}}, {w, moment}, Element[x, beamDomain]  , 
      Method -> {"FiniteElement"} ] 

    GraphicsRow[{Plot[solution[[1]][x], {x, 0, Ls}], Plot[solution[[2]][x], {x, 0, Ls}]}]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/Kny1eUUG.png