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}]}]

The question remains how to get these inner boundary conditions directly without friendly support by classical beam theory (thanks @Bill Watts) ?

I tried, still without success, to find the NeumannValue of this problem.

Any ideas from FEM experts?

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 condition 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}]}]

The question remains how to get these inner boundary condition directly without friendly support by classical beam theory (thanks @Bill Watts) ?

I tried, still without success, to find the NeumannValue of this problem.

Any ideas from FEM experts?

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}]}]

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}]}]

The question remains how to get these inner boundary conditions directly without friendly support by classical beam theory (thanks @Bill Watts) ?

I tried, still without success, to find the NeumannValue of this problem.

Any ideas from FEM experts?