Oefeningen ingeklemde liggers

Oefeningen ingeklemde liggers#

Gegeven is de volgende constructie:

../../_images/clamped.svg

Exercise 4

Teken de vorm van de momentenlijn zonder berekeningen te maken voor een kracht op:

  • \(x= 2.5\)

  • \(x = 7.5\)

import micropip
await micropip.install("ipympl")

import numpy as np
import sympy as sym
import matplotlib.pylab as plt
from ipywidgets import widgets, interact

%matplotlib widget
%config InlineBackend.figure_formats = ['svg']

Solution to Exercise 4

Klik –> Live Code in de rechterbovenhoek van dit scherm om de interactieve widget te tonen die gebruikt kan worden om de oplossing te vinden.

Als de grafiek niet reageert op de schuifregelaar, klik dan één keer op de figuur en probeer het opnieuw.

EI, x = sym.symbols('EI, x',real=True)

Av, Am, Bv, phi_S = sym.symbols('Av, Am, Bv, phi_S',real=True)
a = sym.symbols('a',real=True,positive=True)
C1, C2, C3, C4 = sym.symbols('C1, C2, C3, C4')

L1 = 5
L2 = 5
EI = 5000

q = -Av * sym.SingularityFunction(x,0,-1) - Am * sym.SingularityFunction(x,0,-2) - Bv * sym.SingularityFunction(x,L1+L2,-1) + 1 * sym.SingularityFunction(x,a,-1)
V = -sym.integrate(q,x)+C1
M = sym.integrate(V,x)+C2
kappa = M / EI
phi = sym.integrate(kappa,x)+C3 + phi_S * sym.SingularityFunction(x,L1,0)
w = sym.integrate(phi,x)+C4
eq1 = sym.Eq(w.subs(x,0),0)
eq2 = sym.Eq(phi.subs(x,0),0)
eq3 = sym.Eq(w.subs(x,L1+L2),0)
eq4 = sym.Eq(M.subs(x,-1),0)
eq5 = sym.Eq(M.subs(x,L1),0)
eq6 = sym.Eq(V.subs(x,-1),0)
eq7 = sym.Eq(V.subs(x,L1+L2+1),0)
eq8 = sym.Eq(M.subs(x,L1+L2),0)
sol = sym.solve([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8],[C1,C2,C3,C4,Av,Bv,Am,phi_S])
V_sol = V.subs(sol)
V_subs = V_sol
V_numpy = sym.lambdify([a,x],V_sol.rewrite(sym.Piecewise))
M_sol = M.subs(sol)
M_subs = M_sol
M_numpy = sym.lambdify([a,x],M_sol.rewrite(sym.Piecewise))

fig, axs = plt.subplots(2, 1, figsize=(7, 6))
def func(a):
    x = 0
    x = np.append(x,np.linspace(0.0001,a-0.0001,2))
    x = np.append(x,np.linspace(a+0.0001,10,2))
    x = np.append(x,10)
    axs[0].clear()  # Clear the existing plot
    axs[0].plot([0,10],[0,0],color='black',linewidth=2)
    axs[0].grid()
    axs[0].axis('equal')
    axs[0].plot(x,M_numpy(a,x)/2,color='blue')
    #axs[0].annotate('%.2f kNm' % M_numpy(5,5,b,5000,a),xy = [a,M_numpy(5,5,b,5000,a)+0.5])
    #axs[0].annotate(text='', xy=(b,0), xytext=(b,-1), arrowprops=dict(arrowstyle='simple'))
    axs[0].invert_yaxis()
    axs[0].set_ylim(1,-3)
    title0 = 'Momentenlijn'
    axs[0].set_title(title0)

    axs[1].clear()  # Clear the existing plot
    axs[1].plot([0,10],[0,0],color='black',linewidth=2)
    axs[1].grid()
    axs[1].axis('equal')
    axs[1].plot(x,V_numpy(a,x)*2,color='blue')
    #axs[0].annotate('%.2f kNm' % M_numpy(5,5,b,5000,a),xy = [a,M_numpy(5,5,b,5000,a)+0.5])
    #axs[0].annotate(text='', xy=(b,0), xytext=(b,-1), arrowprops=dict(arrowstyle='simple'))
    axs[1].invert_yaxis()
    axs[1].set_ylim(2,-2)
    title0 = 'Dwarskrachtenlijn'
    axs[1].set_title(title0)

    axs[0].set_yticks([])
    axs[0].set_xticks([])
    axs[0].spines['right'].set_color('none')
    axs[0].spines['top'].set_color('none')
    axs[0].spines['bottom'].set_position('zero')
    axs[0].spines['left'].set_position('zero')

    axs[1].set_yticks([])
    axs[1].set_xticks([])
    axs[1].spines['right'].set_color('none')
    axs[1].spines['top'].set_color('none')
    axs[1].spines['bottom'].set_position('zero')
    axs[1].spines['left'].set_position('zero')

    # set background transparent
    fig.patch.set_alpha(0.0)
    axs[0].patch.set_alpha(0.0)
    axs[1].patch.set_alpha(0.0)

    plt.draw()  

    
interact(func, a = widgets.FloatSlider(min=0.1, max=9.9, value=2, step=0.1, description="Locatie kracht x_F = ... (m)",readout_format='.1f',style= {'description_width': '180px'},layout = {'width': '400px'}));

Laten we nu kijken naar een verdeelde belasting:

../../_images/clamped_q.svg

Exercise 5

Teken de vorm van de momenten- en dwarskrachtenlijn zonder berekeningen te maken voor een verdeelde belasting tussen:

  • \(x=0\) en \(x=5\)

  • \(x=2.5\) en \(x=7.5\)

  • \(x=5\) en \(x=10\)

  • \(x=7.5\) en \(x=10\)

Solution to Exercise 5

Klik –> Live Code in de rechterbovenhoek van dit scherm om de interactieve widget te tonen die gebruikt kan worden om de oplossing te vinden.

Als de grafiek niet reageert op de schuifregelaar, klik dan één keer op de figuur en probeer het opnieuw.

EI_new, x_new = sym.symbols('EI_new, x_new', real=True)
Av_new, Am_new, Bv_new, phi_S_new = sym.symbols('Av_new, Am_new, Bv_new, phi_S_new', real=True)
a_new, b_new = sym.symbols('a_new, b_new', real=True, positive=True)
C1_new, C2_new, C3_new, C4_new = sym.symbols('C1_new, C2_new, C3_new, C4_new')

L1_new = 5
L2_new = 5
EI_new = 5000

q_new = -Av_new * sym.SingularityFunction(x_new, 0, -1) - Am_new * sym.SingularityFunction(x_new, 0, -2) - Bv_new * sym.SingularityFunction(x_new, L1_new + L2_new, -1) + sym.SingularityFunction(x_new, a_new, 0) - sym.SingularityFunction(x_new, a_new + b_new, 0)
V_new = -sym.integrate(q_new, x_new) + C1_new
M_new = sym.integrate(V_new, x_new) + C2_new
kappa_new = M_new / EI_new
phi_new = sym.integrate(kappa_new, x_new) + C3_new + phi_S_new * sym.SingularityFunction(x_new, L1_new, 0)
w_new = sym.integrate(phi_new, x_new) + C4_new
eq1_new = sym.Eq(w_new.subs(x_new, 0), 0)
eq2_new = sym.Eq(phi_new.subs(x_new, 0), 0)
eq3_new = sym.Eq(w_new.subs(x_new, L1_new + L2_new), 0)
eq4_new = sym.Eq(M_new.subs(x_new, -1), 0)
eq5_new = sym.Eq(M_new.subs(x_new, L1_new), 0)
eq6_new = sym.Eq(V_new.subs(x_new, -1), 0)
eq7_new = sym.Eq(V_new.subs(x_new, L1_new + L2_new + 1), 0)
eq8_new = sym.Eq(M_new.subs(x_new, L1_new + L2_new), 0)
sol_new = sym.solve([eq1_new, eq2_new, eq3_new, eq4_new, eq5_new, eq6_new, eq7_new, eq8_new], [C1_new, C2_new, C3_new, C4_new, Av_new, Bv_new, Am_new, phi_S_new])
V_sol_new = V_new.subs(sol_new)
V_subs_new = V_sol_new
V_numpy_new = sym.lambdify([a_new, b_new, x_new], V_sol_new.rewrite(sym.Piecewise))
M_sol_new = M_new.subs(sol_new)
M_subs_new = M_sol_new
M_numpy_new = sym.lambdify([a_new, b_new, x_new], M_sol_new.rewrite(sym.Piecewise))

fig_new, axs_new = plt.subplots(2, 1, figsize=(7, 6))
def func_new(a_new, b_new):
    x_new = 0
    x_new = np.append(x_new, np.linspace(0.0001, 10 - 0.0001, 100))
    x_new = np.append(x_new, 10)
    axs_new[0].clear()  # Clear the existing plot
    axs_new[0].plot([0, 10], [0, 0], color='black', linewidth=2)
    axs_new[0].grid()
    axs_new[0].axis('equal')
    axs_new[0].plot(x_new, M_numpy_new(a_new, b_new, x_new)/8, color='blue')
    axs_new[0].invert_yaxis()
    axs_new[0].set_ylim(1, -3.5)
    title0_new = 'Momentenlijn'
    axs_new[0].set_title(title0_new)

    axs_new[1].clear()  # Clear the existing plot
    axs_new[1].plot([0, 10], [0, 0], color='black', linewidth=2)
    axs_new[1].grid()
    axs_new[1].axis('equal')
    axs_new[1].plot(x_new, V_numpy_new(a_new, b_new, x_new)/3, color='blue')
    axs_new[1].invert_yaxis()
    axs_new[1].set_ylim(2.5, -1)
    title1_new = 'Dwarskrachtenlijn'
    axs_new[1].set_title(title1_new)

    axs_new[0].set_yticks([])
    axs_new[0].set_xticks([])
    axs_new[0].spines['right'].set_color('none')
    axs_new[0].spines['top'].set_color('none')
    axs_new[0].spines['bottom'].set_position('zero')
    axs_new[0].spines['left'].set_position('zero')

    axs_new[1].set_yticks([])
    axs_new[1].set_xticks([])
    axs_new[1].spines['right'].set_color('none')
    axs_new[1].spines['top'].set_color('none')
    axs_new[1].spines['bottom'].set_position('zero')
    axs_new[1].spines['left'].set_position('zero')

    fig_new.patch.set_alpha(0.0)
    axs_new[0].patch.set_alpha(0.0)
    axs_new[1].patch.set_alpha(0.0)

    plt.draw()
    
interact(func_new, a_new=widgets.FloatSlider(min=0.1, max=9.9, value=0, step=0.1, description="Locatie start verdeelde belasting x_start = ... (m)", readout_format='.1f', style={'description_width': '280px'}, layout={'width': '500px'}), 
         b_new=widgets.FloatSlider(min=0.1, max=9.9, value=5, step=0.1, description="Lengte verdeelde belasting x_end = x_start + ... (m)", readout_format='.1f', style={'description_width': '280px'}, layout={'width': '500px'}));

Exercise 6

Wat gebeurt er met de vorm van de momenten- en dwarskrachtenlijn wanneer je de resultaten van hierboven combineert? Probeer de volgende vragen te beantwoorden:

  • Waar kun je een knik vinden?

  • Waar kun je een sprong vinden?

  • Waar is het buigend moment altijd 0?

  • Waar is de dwarskracht altijd 0?

  • Waar is de buigend momentenlijn altijd lineair of kwadratisch?

  • Waar is de dwarskrachtenlijn altijd (niet-nul-)constant of lineair?

Klik –> Live Code in de rechterbovenhoek van dit scherm om de interactieve widget te tonen.

Als de grafiek niet reageert op de schuifregelaar, klik dan één keer op de figuur en probeer het opnieuw.

EI_alt, x_alt = sym.symbols('EI_alt, x_alt', real=True)
Av_alt, Am_alt, Bv_alt, phi_S_alt = sym.symbols('Av_alt, Am_alt, Bv_alt, phi_S_alt', real=True)
a_alt, b_alt, c_alt = sym.symbols('a_alt, b_alt, c_alt', real=True, positive=True)
C1_alt, C2_alt, C3_alt, C4_alt = sym.symbols('C1_alt, C2_alt, C3_alt, C4_alt')

L1_alt = 5
L2_alt = 5
EI_alt = 5000

q_alt = -Av_alt * sym.SingularityFunction(x_alt, 0 , -1) - Am_alt * sym.SingularityFunction(x_alt, 0, -2) - Bv_alt * sym.SingularityFunction(x_alt, L1_alt + L2_alt, -1) + sym.SingularityFunction(x_alt, a_alt, 0)- sym.SingularityFunction(x_alt, a_alt + b_alt, 0) + 10 * sym.SingularityFunction(x_alt, c_alt, -1)
V_alt = -sym.integrate(q_alt, x_alt) + C1_alt
M_alt = sym.integrate(V_alt, x_alt) + C2_alt
kappa_alt = M_alt / EI_alt
phi_alt = sym.integrate(kappa_alt, x_alt) + C3_alt + phi_S_alt * sym.SingularityFunction(x_alt, L1_alt, 0)
w_alt = sym.integrate(phi_alt, x_alt) + C4_alt
eq1_alt = sym.Eq(w_alt.subs(x_alt, 0), 0)
eq2_alt = sym.Eq(phi_alt.subs(x_alt, 0), 0)
eq3_alt = sym.Eq(w_alt.subs(x_alt, L1_alt + L2_alt), 0)
eq4_alt = sym.Eq(M_alt.subs(x_alt, -1), 0)
eq5_alt = sym.Eq(M_alt.subs(x_alt, L1_alt), 0)
eq6_alt = sym.Eq(V_alt.subs(x_alt, -1), 0)
eq7_alt = sym.Eq(V_alt.subs(x_alt, L1_alt + L2_alt + 1), 0)
eq8_alt = sym.Eq(M_alt.subs(x_alt, L1_alt + L2_alt), 0)
sol_alt = sym.solve([eq1_alt, eq2_alt, eq3_alt, eq4_alt, eq5_alt, eq6_alt, eq7_alt, eq8_alt], [C1_alt, C2_alt, C3_alt, C4_alt, Av_alt, Bv_alt, Am_alt, phi_S_alt])
V_sol_alt = V_alt.subs(sol_alt)
V_subs_alt = V_sol_alt
V_numpy_alt = sym.lambdify([a_alt, b_alt, c_alt, x_alt], V_sol_alt.rewrite(sym.Piecewise))
M_sol_alt = M_alt.subs(sol_alt)
M_subs_alt = M_sol_alt
M_numpy_alt = sym.lambdify([a_alt, b_alt, c_alt, x_alt], M_sol_alt.rewrite(sym.Piecewise))

fig_alt, axs_alt = plt.subplots(2, 1, figsize=(7, 6))
def func_alt(a_alt, b_alt, c_alt):
    x_alt = 0
    x_alt = np.append(x_alt, np.linspace(0.0001, 10, 100))
    x_alt = np.append(x_alt, 10)
    axs_alt[0].clear()  # Clear the existing plot
    axs_alt[0].plot([0, 10], [0, 0], color='black', linewidth=2)
    axs_alt[0].grid()
    axs_alt[0].axis('equal')
    axs_alt[0].plot(x_alt, M_numpy_alt(a_alt, b_alt, c_alt, x_alt) / 23, color='blue')
    axs_alt[0].invert_yaxis()
    axs_alt[0].set_ylim(1.75, -4.25)
    title0_alt = 'Momentenlijn'
    axs_alt[0].set_title(title0_alt)

    axs_alt[1].clear()  # Clear the existing plot
    axs_alt[1].plot([0, 10], [0, 0], color='black', linewidth=2)
    axs_alt[1].grid()
    axs_alt[1].axis('equal')
    axs_alt[1].plot(x_alt, V_numpy_alt(a_alt, b_alt, c_alt, x_alt) / 8, color='blue')
    axs_alt[1].invert_yaxis()
    axs_alt[1].set_ylim(2.5, -1.5)
    title1_alt = 'Dwarskrachtenlijn'
    axs_alt[1].set_title(title1_alt)

    axs_alt[0].set_yticks([])
    axs_alt[0].set_xticks([])
    axs_alt[0].spines['right'].set_color('none')
    axs_alt[0].spines['top'].set_color('none')
    axs_alt[0].spines['bottom'].set_position('zero')
    axs_alt[0].spines['left'].set_position('zero')

    axs_alt[1].set_yticks([])
    axs_alt[1].set_xticks([])
    axs_alt[1].spines['right'].set_color('none')
    axs_alt[1].spines['top'].set_color('none')
    axs_alt[1].spines['bottom'].set_position('zero')
    axs_alt[1].spines['left'].set_position('zero')

    fig_alt.patch.set_alpha(0.0)
    axs_alt[0].patch.set_alpha(0.0)
    axs_alt[1].patch.set_alpha(0.0)

    plt.draw()

interact(func_alt, a_alt=widgets.FloatSlider(min=0, max=10, value=0, step=0.1, description="Locatie start verdeelde belasting x_start = ... (m)", readout_format='.1f', style={'description_width': '280px'}, layout={'width': '500px'}), 
         b_alt=widgets.FloatSlider(min=0.1, max=10, value=0, step=0.1, description="Lengte verdeelde belasting x_end = x_start + ... (m)", readout_format='.1f', style={'description_width': '280px'}, layout={'width': '500px'}),
         c_alt=widgets.FloatSlider(min=0, max=10, value=7.5, step=0.1, description="Locatie kracht x_F = ... (m)", readout_format='.1f', style={'description_width': '280px'}, layout={'width': '500px'}));