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

example 7.11   beam supproted by 3 pins  with Uniform load  p.161

fs = 
      PowerExpand[+ R1 + R2 h^L + R3 h^(2 L) - w/s /. 
          para]

qs = Expand[fs/s]
qx = InverseLaplaceTransform[qs, s, x]
qx2L = qx /. x -> 2 L /. para2

ms = qs/s;
mx = InverseLaplaceTransform[ms, s, x]
mx2L = mx /. x -> 2 L /. para2

ypx = InverseLaplaceTransform[-ms/s^2, s, x]

yx = c1 x + ypx

ans = Solve[{qx2L == 0, 
              mx2L == 0, (yx /. x -> L) == 
                0, (yx /. x -> 2 L) == 0}, {c1, R1, R2, 
              R3}][[1]] /. para2

δx = yx /. ans

Simplify[δx - (w L^3 x/48 - w L x^3/16 + w x^4/24 - 
            5 w L (x - L)^3/24 UnitStep[x - L])]

Plot[-δx /. para1, {x, 0, 2}, 
      PlotRange -> {{0, 2},  {-0.008, 0}}, 
      PlotStyle ->  Hue[0.9], AspectRatio ->  0.3, 
      GridLines ->  Automatic]

max search

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

-δx /. para1 /. ans1

recheck SFD and BMD

qxa = qx /. ans
% /. x ->   0
mxa = mx /. ans
% /. x ->  L /. para2

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