DV-SB-ligger-F

import sympy as sp

w = sp.symbols('w', cls=sp.Function)
F, a, x = sp.symbols('F a x')
L, EI = sp.symbols('L EI')
C1, C2, C3, C4 = sp.symbols('C1 C2 C3 C4')

q = F*sp.DiracDelta(x-a)
DV = sp.Eq(EI*sp.diff(w(x),x,4),q) 
display(DV)

\[\displaystyle EI \frac{d^{4}}{d x^{4}} w{\left(x \right)} = F \delta\left(- a + x\right)\]

w = sp.dsolve(DV, w(x)) 
w = w.rhs 
display(w)

\[\displaystyle C_{1} + x^{3} \left(C_{4} + \frac{F \theta\left(- a + x\right)}{6 EI}\right) + x^{2} \left(C_{3} - \frac{F a \theta\left(- a + x\right)}{2 EI}\right) + x \left(C_{2} + \frac{F a^{2} \theta\left(- a + x\right)}{2 EI}\right) - \frac{F a^{3} \theta\left(- a + x\right)}{6 EI}\]

phi = -sp.diff(w, x)
kappa = sp.diff(phi, x)
M = EI * kappa
V = sp.diff(M, x)

Eq1 = sp.Eq(w.subs(x, 0), 0) 
Eq2 = sp.Eq(w.subs(x, L), 0)
Eq3 = sp.Eq(M.subs(x, 0), 0)
Eq4 = sp.Eq(M.subs(x, L), 0)

sol = sp.solve((Eq1,Eq2,Eq3,Eq4),(C1,C2,C3,C4))
display(sol)

{C1: F*a**3*Heaviside(-a)/(6*EI),
 C2: F*L**4*DiracDelta(L - a, 1)/(36*EI) - F*L**3*a*DiracDelta(L - a, 1)/(12*EI) + F*L**3*DiracDelta(L - a)/(6*EI) + F*L**2*a**2*DiracDelta(L - a, 1)/(12*EI) - F*L**2*a*DiracDelta(L - a)/(3*EI) + F*L*a**3*DiracDelta(a, 1)/(18*EI) - F*L*a**3*DiracDelta(L - a, 1)/(36*EI) + F*L*a**2*DiracDelta(a)/(3*EI) + F*L*a**2*DiracDelta(L - a)/(6*EI) - F*L*a*Heaviside(-a)/(3*EI) + F*L*a*Heaviside(L - a)/(3*EI) - F*a**2*Heaviside(L - a)/(2*EI) - F*a**3*Heaviside(-a)/(6*EI*L) + F*a**3*Heaviside(L - a)/(6*EI*L),
 C3: -F*a**3*DiracDelta(a, 1)/(12*EI) - F*a**2*DiracDelta(a)/(2*EI) + F*a*Heaviside(-a)/(2*EI),
 C4: -F*L**2*DiracDelta(L - a, 1)/(36*EI) + F*L*a*DiracDelta(L - a, 1)/(12*EI) - F*L*DiracDelta(L - a)/(6*EI) - F*a**2*DiracDelta(L - a, 1)/(12*EI) + F*a*DiracDelta(L - a)/(3*EI) - F*Heaviside(L - a)/(6*EI) + F*a**3*DiracDelta(a, 1)/(36*EI*L) + F*a**3*DiracDelta(L - a, 1)/(36*EI*L) + F*a**2*DiracDelta(a)/(6*EI*L) - F*a**2*DiracDelta(L - a)/(6*EI*L) - F*a*Heaviside(-a)/(6*EI*L) + F*a*Heaviside(L - a)/(6*EI*L)}

w_sol = w.subs(sol)
phi_sol = phi.subs(sol)
M_sol = M.subs(sol)
V_sol = V.subs(sol)
display(sp.simplify(phi_sol.subs(x,0)))
display(sp.simplify(phi_sol.subs(x,L)))
display(sp.simplify(w_sol.subs(x,L/2)))

\[\displaystyle \frac{F \left(L \left(- L^{4} \delta^{\left( 1 \right)}\left( L - a \right) + 3 L^{3} a \delta^{\left( 1 \right)}\left( L - a \right) - 6 L^{3} \delta\left(L - a\right) - 3 L^{2} a^{2} \delta^{\left( 1 \right)}\left( L - a \right) + 12 L^{2} a \delta\left(L - a\right) - 2 L a^{3} \delta^{\left( 1 \right)}\left( a \right) + L a^{3} \delta^{\left( 1 \right)}\left( L - a \right) - 12 L a^{2} \delta\left(a\right) - 6 L a^{2} \delta\left(L - a\right) + 12 L a \theta\left(- a\right) - 12 L a \theta\left(L - a\right) + 6 a^{3} \delta\left(a\right) - 18 a^{2} \theta\left(- a\right) + 18 a^{2} \theta\left(L - a\right)\right) + 6 a^{3} \left(\theta\left(- a\right) - \theta\left(L - a\right)\right)\right)}{36 EI L}\]

\[\displaystyle \frac{F \left(2 L^{5} \delta^{\left( 1 \right)}\left( L - a \right) - 6 L^{4} a \delta^{\left( 1 \right)}\left( L - a \right) + 6 L^{4} \delta\left(L - a\right) + 6 L^{3} a^{2} \delta^{\left( 1 \right)}\left( L - a \right) - 6 L^{3} a \delta\left(L - a\right) + L^{2} a^{3} \delta^{\left( 1 \right)}\left( a \right) - 2 L^{2} a^{3} \delta^{\left( 1 \right)}\left( L - a \right) + 6 L^{2} a^{2} \delta\left(a\right) - 6 L^{2} a^{2} \delta\left(L - a\right) - 6 L^{2} a \theta\left(- a\right) + 6 L^{2} a \theta\left(L - a\right) + 6 L a^{3} \delta\left(L - a\right) + 6 a^{3} \theta\left(- a\right) - 6 a^{3} \theta\left(L - a\right)\right)}{36 EI L}\]

\[\displaystyle \frac{F \left(L^{5} \delta^{\left( 1 \right)}\left( L - a \right) - 3 L^{4} a \delta^{\left( 1 \right)}\left( L - a \right) + 6 L^{4} \delta\left(L - a\right) + 3 L^{3} a^{2} \delta^{\left( 1 \right)}\left( L - a \right) - 12 L^{3} a \delta\left(L - a\right) + 2 L^{3} \theta\left(\frac{L}{2} - a\right) - 2 L^{3} \theta\left(L - a\right) + L^{2} a^{3} \delta^{\left( 1 \right)}\left( a \right) - L^{2} a^{3} \delta^{\left( 1 \right)}\left( L - a \right) + 6 L^{2} a^{2} \delta\left(a\right) + 6 L^{2} a^{2} \delta\left(L - a\right) - 6 L^{2} a \theta\left(- a\right) - 12 L^{2} a \theta\left(\frac{L}{2} - a\right) + 18 L^{2} a \theta\left(L - a\right) + 24 L a^{2} \theta\left(\frac{L}{2} - a\right) - 24 L a^{2} \theta\left(L - a\right) + 8 a^{3} \theta\left(- a\right) - 16 a^{3} \theta\left(\frac{L}{2} - a\right) + 8 a^{3} \theta\left(L - a\right)\right)}{96 EI}\]

w_subs = w_sol.subs([(EI,1000),(F,5),(a,2),(L,10)])
phi_subs = phi_sol.subs([(EI,1000),(F,5),(a,2),(L,10)])
M_subs = M_sol.subs([(EI,1000),(F,5),(a,2),(L,10)])
V_subs = V_sol.subs([(EI,1000),(F,5),(a,2),(L,10)])

sp.plot(-w_subs,(x,0,10),title='$w$');
sp.plot(-phi_subs,(x,0,10),title='$\phi$');
sp.plot(-M_subs,(x,0,10),title='$M$');
sp.plot(-V_subs,(x,0,10),title='$V$');