Sympy voorbeeld 3

import sympy as sp
x, C1, C2, C3, C4, C5, C6, C7, C8, q, T, EI, L1, L2 = sp.symbols('x C1 C2 C3 C4 C5 C6 C7 C8 q T EI L1 L2')

Definieer de verplaatsingen \(w_1\) en \(w_2\).

\(w_1\) voor \(0 < x < L_1\)

\(w_2\) voor \(L_1 < x < L_1+L_2\)

w1 = C1 * x**3 / 6 + C2 * x**2 / 2 + C3 * x + C4

w2 = q * x**4 / (24 * EI) + C5 * x**3 / 6 + C6 * x**2 / 2 + C7 * x + C8

Druk de verdraaiingen (\(\phi\)), de momenten (\(M\)) en de dwarskrachten (\(V\)) uit in \(w_1\) en \(w_2\).

\(\phi_1\), \(\kappa_1\), \(M_1\), \(V_1\) voor \(0 < x < L_1\)

\(\phi_2\), \(\kappa_2\), \(M_2\), \(V_2\) voor \(L_1 < x < L_1+L_2\)

phi1 = -w1.diff(x)
phi2 = -w2.diff(x)
kappa1 = phi1.diff(x)
kappa2 = phi2.diff(x)
M1 = EI * kappa1
M2 = EI * kappa2
V1 = M1.diff(x)
V2 = M2.diff(x)

Geef de rand- en overgangsvoorwaarden om de onbekenden (\(C_1\) t/m \(C_8\)) op te lossen. (8 onbekenden, dus 8 vergelijkingen nodig).

eq1 = sp.Eq(phi1.subs(x,0),0)
eq2 = sp.Eq(w1.subs(x,0),0)
eq3 = sp.Eq(w1.subs(x,L1),0)
eq4 = sp.Eq(w2.subs(x,L1),0)
eq5 = sp.Eq(phi1.subs(x,L1),phi2.subs(x,L1))
eq6 = sp.Eq(M1.subs(x,L1),M2.subs(x,L1))
eq7 = sp.Eq(V2.subs(x,L1+L2),0)
eq8 = sp.Eq(M2.subs(x,L1+L2),-T)

Laat Maple de 8 onbekenden oplossen met behulp van de 8 hierboven gegeven vergelijkingen.

sol = sp.solve((eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8),(C1,C2,C3,C4,C5,C6,C7,C8))
display(sol)
w1_sol = w1.subs(sol)
phi1_sol = phi1.subs(sol)
M1_sol = M1.subs(sol)
V1_sol = V1.subs(sol)
w2_sol = w2.subs(sol)
phi2_sol = phi2.subs(sol)
M2_sol = M2.subs(sol)
V2_sol = V2.subs(sol)

{C1: (3*L2**2*q + 6*T)/(4*EI*L1),
 C2: (-L2**2*q - 2*T)/(4*EI),
 C3: 0,
 C4: 0,
 C5: (-L1*q - L2*q)/EI,
 C6: (L1**2*q + 2*L1*L2*q + L2**2*q + 2*T)/(2*EI),
 C7: (-4*L1**3*q - 12*L1**2*L2*q - 9*L1*L2**2*q - 18*L1*T)/(24*EI),
 C8: (L1**4*q + 4*L1**3*L2*q + 3*L1**2*L2**2*q + 6*L1**2*T)/(24*EI)}

w1_sub = w1_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
phi1_sub = phi1_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
M1_sub = M1_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
V1_sub = V1_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
w2_sub = w2_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
phi2_sub = phi2_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
M2_sub = M2_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])
V2_sub = V2_sol.subs([(T,50),(L1,4),(L2,2),(q,15),(EI,1000)])

Je kunt checken of het klopt. Zijn de momenten \(M_1\) en \(M_2\) nu hetzelfde op \(x = 3m\)? Ja! Je kunt natuurlijk alle andere rand- en overgangsvoorwaarden ook checken.

display(M1_sub.subs(x,4).evalf())
display(M2_sub.subs(x,4).evalf())

\[\displaystyle -80.0\]

\[\displaystyle -80.0\]

Nu kunnen we de twee velden aan elkaar plakken, zodat je een gehele zakkings-, momenten- en dwarskrachtenlijn van de constructie krijgen.

w = sp.Piecewise((w1_sub,(x>=0) & (x<=4)),(w2_sub,(x>4)&(x<=6)))
phi = sp.Piecewise((phi1_sub,(x>=0) & (x<=4)),(phi2_sub,(x>4)&(x<=6)))
M = sp.Piecewise((M1_sub,(x>=0) & (x<=4)),(M2_sub,(x>4)&(x<=6)))
V = sp.Piecewise((V1_sub,(x>=0) & (x<=4)),(V2_sub,(x>4)&(x<=6)))
display(w)
display(phi)
display(M)
display(V)

\[\begin{split}\displaystyle \begin{cases} \frac{x^{3}}{200} - \frac{x^{2}}{50} & \text{for}\: x \geq 0 \wedge x \leq 4 \\\frac{x^{4}}{1600} - \frac{3 x^{3}}{200} + \frac{4 x^{2}}{25} - \frac{16 x}{25} + \frac{4}{5} & \text{for}\: x \leq 6 \wedge x > 4 \end{cases}\end{split}\]

\[\begin{split}\displaystyle \begin{cases} - \frac{3 x^{2}}{200} + \frac{x}{25} & \text{for}\: x \geq 0 \wedge x \leq 4 \\- \frac{x^{3}}{400} + \frac{9 x^{2}}{200} - \frac{8 x}{25} + \frac{16}{25} & \text{for}\: x \leq 6 \wedge x > 4 \end{cases}\end{split}\]

\[\begin{split}\displaystyle \begin{cases} 40 - 30 x & \text{for}\: x \geq 0 \wedge x \leq 4 \\- \frac{15 x^{2}}{2} + 90 x - 320 & \text{for}\: x \leq 6 \wedge x > 4 \end{cases}\end{split}\]

\[\begin{split}\displaystyle \begin{cases} -30 & \text{for}\: x \geq 0 \wedge x \leq 4 \\90 - 15 x & \text{for}\: x \leq 6 \wedge x > 4 \end{cases}\end{split}\]

Ook kunnen we de lijnen \(w\), \(\phi\), \(V\) en \(M\) plotten in Python

sp.plot(-w,title='$w$');
sp.plot(-phi,title='$\phi$');
sp.plot(-M,title='$M$');
sp.plot(-V,title='$V$');