Looking at the plot I get for the moment at x=30, I don't see how you can solve it the way you are doing it. Try this.

(*Define parameters*)`
Ls = 60 ;(*Length of the beam in inches*)
Es = 10000000; (*Young's modulus in psi*)
Is = 10; (*Moment of inertia in in^4*)
qs = -50;(*Uniform load in pli applied over full length of beam *)
qss = -100; (* seam load over 4 inches*)
lss = 45; (*seam start location from left edge *)

We have three supports

p1 at x = 0

p2 at x = 30

p3 at x = Ls

Being statically indeterminant we can only find two of the three in terms of the other using only forces and torques. Find p2 and p3 in terms of p1.

The torque about x = 0.

eq1 = Integrate[qs*x, {x, 0, lss}] + Integrate[qss*x, {x, lss, Ls}] + p2*(Ls/2) + p3*Ls == 0

The second equation comes from the sum of the forces = 0

eq2 = p1 + p2 + p3 + qs*lss + qss*(Ls - lss) == 0

solp = Solve[{eq1, eq2}, {p2, p3}] // Flatten
(*{p2->1/2 (6375-4 p1),p3->1/2 (2 p1+1125)}*)

Now the load function for all the loads is

p[x_] = p1  DiracDelta[x] + p2  DiracDelta[x - Ls/2] + p3  DiracDelta[x - Ls] + qs + qss/2  UnitStep[x - 3  Ls/4] - qss  UnitStep[x - Ls]

EI = Es  Is

Now instead of using the differential equation for the moment, use the differential equation for the displacement.

ode = v''''[x] == p[x]/EI /. solp

or write as

v4[x_] = p[x]/EI /. solp

Integrate to get v'''[x]

v3[x_] = Integrate[v4[x], x] + c1 // Simplify

Integrate to get v''[x]

v2[x_] = Integrate[v3[x], x] + c2 // Simplify

And v'[x]

v1[x_] = Integrate[v2[x], x] + c3 // Simplify

and the displacement v[x]

v[x_] = Integrate[v1[x], x] + c4

Now we know v2[x] is proportional to the moment which is zero at the endpoints. So solve for some of the c's.

at x = 0

v2[0] == 0
(*c2==0*)

c2 = 0

v2[Ls] == 0 // Expand
(*60 c1==0*)

c1 = 0

The displacements at the supports are zero.

v[0] == 0
(*c4==0*)

We can solve for p1 and c3 by use of the other supports

solpc = Solve[{v[Ls] == 0, v[Ls/2] == 0}, {p1, c3}] // Flatten
(*{p1->33375/64,c3->-(9/40960)}*)

p1 = p1 /. solpc
c3 = c3 /. solpc

The displacement

Plot[v[x], {x, 0, Ls}]

the slope

slope[x_] = v1[x] // Simplify

Plot[slope[x], {x, 0, Ls}]

The moment

m[x_] = EI  v2[x]

Plot[m[x], {x, 0, Ls}]

The shear

shear[x_] = -EI  v3[x]

Plot[shear[x], {x, 0, Ls}, Exclusions -> None]