import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
    matplotlib.RcParams._get = dict.get

Opgave 2

Gegeven is de volgende constructie:

\[EI\left(x\right) = 4000 \cdot e^{-\cfrac{x \left[\rm{m}\right]}{3}} \, \rm{kNm}^2 \]

Fig. 224  

Vind de maximale waarde van het moment en de verplaatsing op \(x=3\) en \(x=3.3\).

Opgave

Maak de opdracht in WebLab. Heb je nog geen toegang? Schrijf je dan eerst hier in voor dit vak op WebLab.

Je kan je antwoord op de vragen in de volgende opgaves invoeren. Maar om je code te checken klik je op ‘Check’ in WebLab.

Opgave

Wat is het maximale moment in \(\rm{kNm}\)?

67.5

Correct!

Incorrect.

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Opgave

Wat is de verplaatsing op \(x=3\) in \(\rm{mm}\)?

182.607;0.025

Correct!

Incorrect.

Wat is de verplaatsing op \(x=3.3\) in \(\rm{mm}\)?

185.476;0.025

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Uitwerking

import sympy as sym
sym.init_printing(use_latex='mathjax')
import numpy as np
import matplotlib.pyplot as plt
from IPython.display import display as ipy_display, Math

%config InlineBackend.figure_formats = ['svg']

def display(*objs):
    for obj in objs:
        if isinstance(obj, dict):
            lines = [f"{sym.latex(k)} &= {sym.latex(obj[k])}" for k in sorted(obj, key=lambda s: str(s))]
            ipy_display(Math(r"\begin{aligned}" + r"\\ ".join(lines) + r"\end{aligned}"))
        else:
            ipy_display(obj)

def plot(w_list, x_range_list,ylabel):
    fig = plt.figure(facecolor='none')
    for i, w in enumerate(w_list):
        # check if x is the only symbol in the expression
        if len(w.free_symbols) > 1:
            raise ValueError('The expression must be a function of x only.')
        
        w_numpy = sym.lambdify(x, w)
        x_vals = np.linspace(x_range_list[i][0], x_range_list[i][1], 100)
        
        # if the expression is a constant, we need to make sure that it is broadcasted correctly
        if isinstance(w_numpy(x_vals),float) or isinstance(w_numpy(x_vals),int):
            w_numpy = np.vectorize(w_numpy)
        plt.plot(x_vals,w_numpy(x_vals))
        plt.xlabel('$x$')
        plt.ylabel(ylabel)
    ax = plt.gca()
    ax.set_facecolor('none')
    ax.spines['right'].set_color('none')
    ax.spines['top'].set_color('none')
    ax.spines['bottom'].set_position('zero')
    ax.spines['left'].set_position('zero')
    ax.invert_yaxis()

import sympy as sym

sym.var('x')
sym.var('C1, C2, C3, C4');

EI = sym.exp(-x/3)*4000
q1 = 15

V1 = sym.integrate(-q1,x)+C1
M1 = sym.integrate(V1,x)+C2
kappa1 = M1 / EI
phi1 = sym.integrate(kappa1,x)+C3
w1 = sym.integrate(-phi1,x)+C4

Eq1  = sym.Eq(  w1.subs(x, 0), 0)
Eq2  = sym.Eq(  M1.subs(x, 0), 0)
Eq3  = sym.Eq(  w1.subs(x, 6), 0)
Eq4  = sym.Eq(  M1.subs(x, 6), 0)

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

\[\begin{split}\displaystyle \begin{aligned}C_{1} &= 45\\ C_{2} &= 0\\ C_{3} &= - \frac{81}{320} + \frac{81 e^{2}}{1600}\\ C_{4} &= - \frac{243}{160}\end{aligned}\end{split}\]

w1_oplossing = w1.subs(sol)
phi1_oplossing = phi1.subs(sol)
M1_oplossing = M1.subs(sol)

plot([w1_oplossing], [(0, 6)], 'w(x)')
plot([phi1_oplossing], [(0, 6)], 'phi(x)')

M1_3 = M1_oplossing.subs(x,3) # kNm
display(M1_3)
display(M1_3.evalf())

\[\displaystyle \frac{135}{2}\]

\[\displaystyle 67.5\]

w1_3 = w1_oplossing.subs(x,3)*1000 # mm
w1_right_of_3 = w1_oplossing.subs(x,3.3)*1000 #mm

display(w1_3)
display(w1_3.evalf())
display(w1_right_of_3)
display(w1_right_of_3.evalf())

\[\displaystyle - \frac{1215 e^{2}}{8} - \frac{6075}{8} + \frac{6075 e}{8}\]

\[\displaystyle 182.607368460995\]

\[\displaystyle 1419.91056562844 - 167.0625 e^{2}\]

\[\displaystyle 185.476381100843\]