para = {h -> (E^(-s))}
para1 = {P ->  1, L ->  1, 
        w ->  1}
para2 = {UnitStep[(L/2)] ->  1, 
        UnitStep[-(L/2)] ->  0, UnitStep[L] -> 0,  
        DiracDelta[L] ->  0, UnitStep[-L] ->  0}

example 7.10 fixed-pin  Load P at x=L/2  p.160

fs = 
      PowerExpand[+ R1 -  M1 s  - P h^( L/2) + R2 h^L /. 
          para]

qs = Expand[fs/s]
qx = InverseLaplaceTransform[qs, s, x]
qxL = qx /. x ->  L /. para2

ms = qs/s;
mx = InverseLaplaceTransform[ms, s, x]
mxL = mx /. x ->  L /. para2

yx = 
      InverseLaplaceTransform[-ms/s^2, s, x]
yxL = (yx /. x ->  L)

ans = Solve[{qxL == 0, mxL == 0, yxL == 0}, {R1, 
              R2, M1}][[1]] /. para2

δx = yx /. ans
% /. x -> L/2 /. para2
N[%]

Simplify[δx - (-11 P x^3/96 + 3 P L x^2/32 + 
            P (x - L/2)^3/6 UnitStep[x - L/2])]

zu11 = 
      Plot[-δx /. para1, {x, 0, 1}, 
        PlotRange -> {{0, 1},  {-0.01, 0}}, 
        PlotStyle ->  Hue[0.9], AspectRatio ->  0.3, 
        GridLines ->  Automatic]

max search

θx = D[δx, x] /. para1;
ans1 = FindRoot[θx == 0, {x, 0.51}]

-δx /. para1 /. ans1

recheck SFD and BMD

qxa = qx /. ans
mxa = mx /. ans

zu12 = 
      Plot[{qxa /. para1, mxa /. para1}, {x, 0, 1}, 
        PlotRange ->  {-0.8, 0.8}, 
        PlotStyle ->  {Hue[0.9], Hue[0.7]}, 
        AspectRatio ->  0.3]

BMD and SFD obtained by operation  D[ Deflection Function ]

mxa = -D[δx /. para1, {x, 
              2}];
qxa = D[mxa, x];
Plot[{qxa /. para1, mxa /. para1}, {x, 0, 1}, 
      PlotRange ->  {-0.8, 0.8}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7]}, 
      AspectRatio ->  0.3]