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

Q 7.4 (a) Cantilever with load P at x=L  p.174

fs = PowerExpand[ P -  P L s  - P h^ L /. para]

yx = InverseLaplaceTransform[-fs/s^2/s^2, s, x]

δL = (yx /. x ->  L)
δx = Simplify[yx/δL /. x ->  ξ L]

Plot[-δx /. para1, {ξ, 0, 1.5}, 
      PlotRange -> {{0, 1.5},  {-2, 0}}, 
      PlotStyle ->  Hue[0.9], AspectRatio ->  0.3, 
      GridLines ->  Automatic]

Q 7.4 (b)    pin-pin beam  with load P at x=a  p.174

fs = 
      PowerExpand[+ R1 + R2 h^L  - P h^ (a 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 = δ0 + θ0 x + 
        InverseLaplaceTransform[-ms/s^2, s, x]

ans = 
      Simplify[Solve[{qxL == 0, 
                mxL == 0, (yx /. x ->  0) == 
                  0, (yx /. x ->  L) == 0}, {R1, 
                R2, δ0, θ0}][[1]] /. para2]

δx = Simplify[yx/δa /. ans /. para2]

δa = 
      Simplify[(yx /. x ->  a L) /. ans /. 
          para2]
δx = 
      Simplify[yx/δa /. ans /. para2 /. 
          x ->  ξ L]

FullSimplify[(δx - ((b ξ (1 - (ξ^2) - (b^2)) + (((ξ - a))^3) UnitStep[L (-a + ξ)])/(2 (b^2) (a^2)))) /.  b -> 1 - a]  (*for check*)

Plot[-δx /. para1 /. a ->  7/10, {ξ, 0, 1}, 
      PlotRange -> {{0, 1},  {-1.2, 0}}, 
      PlotStyle ->  Hue[0.9], AspectRatio ->  0.3, 
      GridLines ->  Automatic]

δa /. a ->  1/2
Simplify[δx /. a ->  1/2]

Q 7.4 (c) Cantilever with load Moment at x=L  p. 174

fs = PowerExpand[- M s  + M s h^ L /. para]

yx = InverseLaplaceTransform[-fs/s^2/s^2, s, x]

δL = 
      Simplify[(yx /. x ->  L)]
θL = (D[yx, x] /. x ->  L)
δx = 
      Simplify[yx/δL /. para2 /. x ->  ξ L]

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

Q 7.4 (d) Cantilever with uniform load w at x=0-L  p.174

fs = PowerExpand[R1 -  M1 s  - w/s]

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 = δ0 + θ0 x + 
        InverseLaplaceTransform[-ms/s^2, s, x]

ans = Solve[{qxL == 0, 
              mxL == 0, (yx /. x ->  0) == 
                0, (D[yx, x] /. x ->  0) == 0}, {R1, 
              M1, δ0, θ0}][[1]] /. para2

δL = (yx /. x ->  L) /. ans
δx = 
      Simplify[yx/δL /. ans /. x ->  ξ L]

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