Optimizing a bridge structure#
Installation of the truss package#
For this session, you will need the Python truss package. The following cell will install it automatically.
# FOR JUPYTER LAB
%matplotlib widget
# FOR JUPYTER NOTEBOOK AND HUB
# % matplotlib notebook
import numpy as np
import matplotlib.pyplot as plt
import sys
import os
import zipfile
import urllib.request
import shutil
from scipy import optimize
url = "https://github.com/lcharleux/truss/archive/master.zip"
file_name = "truss-master.zip"
with urllib.request.urlopen(url) as response, open(file_name, "wb") as out_file:
shutil.copyfileobj(response, out_file)
with zipfile.ZipFile(file_name) as zf:
zf.extractall()
os.remove(file_name)
sys.path.append("truss-master")
try:
import truss
print("Truss is correctly installed")
except:
print("Truss is NOT correctly installed !")
Truss is correctly installed
A short truss tutorial is available here:
Building the bridge structure#
In this session, we will modelled a bridge structure using truss and optimize it using various criteria. The basic structure is introduced below. It is made of steel bars and loaded with one vertical force on \(G\). The bridge is symmetrical so only the left half is modelled.
E = 210.0e9 # Young Modulus [Pa]
rho = 7800.0 # Density [kg/m**3]
A = 5.0e-2 # Cross section [m**2]
sigmay = 400.0e6 # Yield Stress [Pa]
# Model definition
model = truss.core.Model() # Model definition
# NODES
nA = model.add_node((0.0, 0.0), label="A")
nC = model.add_node((3.0, 0.0), label="C")
nD = model.add_node((3.0, 3.0), label="D")
nE = model.add_node((6.0, 0.0), label="E")
nF = model.add_node((6.0, 3.0), label="F")
nG = model.add_node((9.0, 0.0), label="G")
nH = model.add_node((9.0, 3.0), label="H")
# BOUNDARY CONDITIONS
nA.block[1] = True
nG.block[0] = True
nH.block[0] = True
# BARS
AC = model.add_bar(nA, nC, modulus=E, density=rho, section=A, yield_stress=sigmay)
CD = model.add_bar(nC, nD, modulus=E, density=rho, section=A, yield_stress=sigmay)
AD = model.add_bar(nA, nD, modulus=E, density=rho, section=A, yield_stress=sigmay)
CE = model.add_bar(nC, nE, modulus=E, density=rho, section=A, yield_stress=sigmay)
DF = model.add_bar(nD, nF, modulus=E, density=rho, section=A, yield_stress=sigmay)
#DE = model.add_bar(nD, nE, modulus=E, density=rho, section=A, yield_stress=sigmay)
EF = model.add_bar(nE, nF, modulus=E, density=rho, section=A, yield_stress=sigmay)
EG = model.add_bar(nE, nG, modulus=E, density=rho, section=A, yield_stress=sigmay)
FH = model.add_bar(nF, nH, modulus=E, density=rho, section=A, yield_stress=sigmay)
#FG = model.add_bar(nF, nG, modulus=E, density=rho, section=A, yield_stress=sigmay)
GH = model.add_bar(nG, nH, modulus=E, density=rho, section=A, yield_stress=sigmay)
FC = model.add_bar(nF, nC, modulus=E, density=rho, section=A, yield_stress=sigmay)
EH = model.add_bar(nE, nH, modulus=E, density=rho, section=A, yield_stress=sigmay)
# STRUCTURAL LOADING
TotalLoad = -2.0e6
nG.force = np.array([0.0, TotalLoad/4/2.])
nE.force = np.array([0.0, TotalLoad/4.])
nC.force = np.array([0.0, TotalLoad/4.])
nA.force = np.array([0.0, TotalLoad/4/2.])
model.solve()
xlim, ylim = model.bbox(deformed=False)
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_aspect("equal")
# ax.axis("off")
model.draw(
ax, deformed=False, field="stress", label=True, force_scale=1.0e-6, forces=True, displacement_scale=100
)
plt.xlim(xlim)
plt.ylim(ylim)
plt.grid()
plt.xlabel("Axe $x$")
plt.ylabel("Axe $y$")
Text(0, 0.5, 'Axe $y$')
Read Data#
Detailed results at the nodes#
model.data(at="nodes")
| label | coords | disp | force | block | |||||
|---|---|---|---|---|---|---|---|---|---|
| o | x | y | ux | uy | Fx | Fy | bx | by | |
| 0 | A | 0.0 | 0.0 | -0.001571 | 0.0 | 0.0 | 1250000.0 | False | True |
| 1 | C | 3.0 | 0.0 | -0.001214 | -0.003867 | 0.0 | -500000.0 | False | False |
| 2 | D | 3.0 | 3.0 | 0.000929 | -0.00351 | 0.0 | 0.0 | False | False |
| 3 | E | 6.0 | 0.0 | -0.000643 | -0.006473 | 0.0 | -500000.0 | False | False |
| 4 | F | 6.0 | 3.0 | 0.000571 | -0.006259 | 0.0 | 0.0 | False | False |
| 5 | G | 9.0 | 0.0 | 0.0 | -0.00739 | 2250000.0 | -250000.0 | True | False |
| 6 | H | 9.0 | 3.0 | 0.0 | -0.007318 | -2250000.0 | 0.0 | True | False |
Detailed results on the bars#
model.data(at="bars")
| conn | props | state | geometry | props | direction | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| c1 | c2 | section | density | tension | elongation | strain | stress | failure | volume | length | mass | dx | dy | |
| 0 | A | C | 0.05 | 7800.0 | 1250000.0 | 0.000357 | 0.000119 | 25000000.0 | False | 0.15 | 3.0 | 1170.0 | 1.0 | 0.0 |
| 1 | C | D | 0.05 | 7800.0 | 1250000.0 | 0.000357 | 0.000119 | 25000000.0 | False | 0.15 | 3.0 | 1170.0 | 0.0 | 1.0 |
| 2 | A | D | 0.05 | 7800.0 | -1767766.952966 | -0.000714 | -0.000168 | -35355339.059328 | False | 0.212132 | 4.242641 | 1654.629868 | 0.707107 | 0.707107 |
| 3 | C | E | 0.05 | 7800.0 | 2000000.0 | 0.000571 | 0.00019 | 40000000.0 | False | 0.15 | 3.0 | 1170.0 | 1.0 | 0.0 |
| 4 | D | F | 0.05 | 7800.0 | -1250000.0 | -0.000357 | -0.000119 | -25000000.0 | False | 0.15 | 3.0 | 1170.0 | 1.0 | 0.0 |
| 5 | E | F | 0.05 | 7800.0 | 750000.0 | 0.000214 | 0.000071 | 15000000.0 | False | 0.15 | 3.0 | 1170.0 | 0.0 | 1.0 |
| 6 | E | G | 0.05 | 7800.0 | 2250000.0 | 0.000643 | 0.000214 | 45000000.0 | False | 0.15 | 3.0 | 1170.0 | 1.0 | 0.0 |
| 7 | F | H | 0.05 | 7800.0 | -2000000.0 | -0.000571 | -0.00019 | -40000000.0 | False | 0.15 | 3.0 | 1170.0 | 1.0 | 0.0 |
| 8 | G | H | 0.05 | 7800.0 | 250000.0 | 0.000071 | 0.000024 | 5000000.0 | False | 0.15 | 3.0 | 1170.0 | 0.0 | 1.0 |
| 9 | F | C | 0.05 | 7800.0 | -1060660.17178 | -0.000429 | -0.000101 | -21213203.435597 | False | 0.212132 | 4.242641 | 1654.629868 | -0.707107 | -0.707107 |
| 10 | E | H | 0.05 | 7800.0 | -353553.390593 | -0.000143 | -0.000034 | -7071067.811865 | False | 0.212132 | 4.242641 | 1654.629868 | 0.707107 | 0.707107 |
for bar in model.bars:
mass = bar.mass()
bar.conn[0].force[1] += -mass*9.81/2
bar.conn[1].force[1] += -mass*9.81/2
model.solve()
xlim, ylim = model.bbox(deformed=False)
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_aspect("equal")
# ax.axis("off")
model.draw(
ax, deformed=True, field="stress", label=True, force_scale=1.0e-6, forces=True, displacement_scale=1000000, displacements=True
)
plt.xlim(xlim)
plt.ylim(ylim)
plt.grid()
plt.xlabel("Axe $x$")
plt.ylabel("Axe $y$")
Text(0, 0.5, 'Axe $y$')
Dead (or structural) mass#
m0 = model.mass()
m0 * 1.0e-3 # Mass in tons !
np.float64(14.323889603929565)
Model modification#
Modifing section#
# change section of one bar
FG.section = 0.0333
# solve with updated sections
model.solve()
model.data(at="bars")
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[9], line 2
1 # change section of one bar
----> 2 FG.section = 0.0333
4 # solve with updated sections
5 model.solve()
NameError: name 'FG' is not defined
Changing all section (or other parameters)#
# loop over bars
for bar in model.bars:
# get the normal force of the bar
N = bar.tension
# change the section
bar.section = 0.2
# solve with updated parameters
model.solve()
Questions#
Question 1: Verify that the yield stress is not exceeded anywhere, do you think this structure has an optimimum weight ? You can use the state/failure data available on the whole model.
# Example:
model.data(at="bars").state.failure.values
# ...
array([False, False, False, False, False, False, False, False, False,
False, False], dtype=object)
Question 2: Modify all the cross sections at the same time in order to minimize weight while keeping acceptable stress level.
Question 3: We want to modify the position along the \(\vec y\) axis of the points \(D\), \(F\) and \(H\) in order to minimize the vertical displacement of the node \(G\) times the mass of the structure \(\alpha\):
\[
\alpha = |u_y(G)| m
\]
Where \(u_y(G)\) is the displacement of the node \(G\) along the \(\vec y\) axis and \(m\) the mass of the whole structure.
Do not further modify the sections determined in question 4. Comment the solution.
model.data(at="bars")
| conn | props | state | geometry | props | direction | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| c1 | c2 | section | density | tension | elongation | strain | stress | failure | volume | length | mass | dx | dy | |
| 0 | A | C | 0.2 | 7800.0 | 1376662.547512 | 0.000098 | 0.000033 | 6883312.737561 | False | 0.6 | 3.0 | 4680.0 | 1.0 | 0.0 |
| 1 | C | D | 0.2 | 7800.0 | 1357068.88801 | 0.000097 | 0.000032 | 6785344.440049 | False | 0.6 | 3.0 | 4680.0 | 0.0 | 1.0 |
| 2 | A | D | 0.2 | 7800.0 | -1946894.845503 | -0.000197 | -0.000046 | -9734474.227514 | False | 0.848528 | 4.242641 | 6618.519472 | 0.707107 | 0.707107 |
| 3 | C | E | 0.2 | 7800.0 | 2208398.926019 | 0.000158 | 0.000053 | 11041994.630097 | False | 0.6 | 3.0 | 4680.0 | 1.0 | 0.0 |
| 4 | D | F | 0.2 | 7800.0 | -1376662.547512 | -0.000098 | -0.000033 | -6883312.737561 | False | 0.6 | 3.0 | 4680.0 | 1.0 | 0.0 |
| 5 | E | F | 0.2 | 7800.0 | 806403.869005 | 0.000058 | 0.000019 | 4032019.345024 | False | 0.6 | 3.0 | 4680.0 | 0.0 | 1.0 |
| 6 | E | G | 0.2 | 7800.0 | 2489470.285522 | 0.000178 | 0.000059 | 12447351.427609 | False | 0.6 | 3.0 | 4680.0 | 1.0 | 0.0 |
| 7 | F | H | 0.2 | 7800.0 | -2208398.926019 | -0.000158 | -0.000053 | -11041994.630097 | False | 0.6 | 3.0 | 4680.0 | 1.0 | 0.0 |
| 8 | G | H | 0.2 | 7800.0 | 261477.7 | 0.000019 | 0.000006 | 1307388.5 | False | 0.6 | 3.0 | 4680.0 | 0.0 | 1.0 |
| 9 | F | C | 0.2 | 7800.0 | -1176252.866804 | -0.000119 | -0.000028 | -5881264.33402 | False | 0.848528 | 4.242641 | 6618.519472 | -0.707107 | -0.707107 |
| 10 | E | H | 0.2 | 7800.0 | -397494.928603 | -0.00004 | -0.000009 | -1987474.643015 | False | 0.848528 | 4.242641 | 6618.519472 | 0.707107 | 0.707107 |
S_min = 1e-8
def cost(X):
nD.coords[1] = X[0]
nF.coords[1] = X[1]
nH.coords[1] = X[2]
nD.coords[0] = X[3]
nF.coords[0] = X[4]
model.solve()
# section ajustement
for bar in model.bars:
coef = np.abs( bar.stress/sigmay)
new_section = bar.section * coef
if new_section < S_min:
new_section = S_min
bar.section = new_section
model.solve()
m = model.mass()
# Cs = np.abs(model.data(at='bars').state.stress) / sigmay
# Cs = Cs.values.astype(np.float64)
# faillure = np.exp((Cs-1)*1000)
# faillure = np.sum(faillure) + 1
# print(faillure)
return m * np.abs(nG.displacement[1])
import scipy
x0 = np.array([ 3.0, 3.0, 3.0, 3, 6])
sol = scipy.optimize.minimize(cost, x0)
sol
model.data(at="bars")
| conn | props | state | geometry | props | direction | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| c1 | c2 | section | density | tension | elongation | strain | stress | failure | volume | length | mass | dx | dy | |
| 0 | A | C | 0.001546 | 7800.0 | 618481.630734 | 0.005714 | 0.001905 | 399999999.999999 | False | 0.004639 | 3.0 | 36.181175 | 1.0 | 0.0 |
| 1 | C | D | 0.0017 | 7800.0 | 680022.304184 | 0.005819 | 0.001905 | 400000000.0 | False | 0.005193 | 3.054798 | 40.507954 | -0.635002 | 0.77251 |
| 2 | A | D | 0.003773 | 7800.0 | -1509211.481959 | -0.004928 | -0.001905 | -400000000.0 | False | 0.009761 | 2.587077 | 76.136699 | 0.409804 | 0.912173 |
| 3 | C | E | 0.002626 | 7800.0 | 1050294.723583 | 0.005714 | 0.001905 | 400000000.0 | False | 0.007877 | 3.0 | 61.442241 | 1.0 | 0.0 |
| 4 | D | F | 0.003349 | 7800.0 | -1339747.761105 | -0.008845 | -0.001905 | -400000000.0 | False | 0.015553 | 4.643447 | 121.310421 | 0.783952 | 0.620822 |
| 5 | E | F | 0.001794 | 7800.0 | 717778.010285 | 0.010288 | 0.001905 | 399999999.999999 | False | 0.009692 | 5.401287 | 75.600032 | -0.240603 | 0.970624 |
| 6 | E | G | 0.003286 | 7800.0 | 1314332.370967 | 0.005714 | 0.001905 | 400000000.0 | False | 0.009857 | 3.0 | 76.888444 | 1.0 | 0.0 |
| 7 | F | H | 0.00307 | 7800.0 | -1227905.509902 | -0.008223 | -0.001905 | -399999999.999999 | False | 0.013252 | 4.316832 | 103.362912 | 0.996 | 0.089349 |
| 8 | G | H | 0.000654 | 7800.0 | 261477.7 | 0.010721 | 0.001905 | 400000000.0 | True | 0.003679 | 5.628319 | 28.697758 | 0.0 | 1.0 |
| 9 | F | C | 0.0 | 7800.0 | 8.81166 | 0.010498 | 0.001905 | 400000000.017795 | True | 0.0 | 5.511488 | 0.000947 | -0.308525 | -0.951216 |
| 10 | E | H | 0.000485 | 7800.0 | -194182.353104 | -0.012148 | -0.001905 | -400000000.000001 | True | 0.003096 | 6.377929 | 24.150383 | 0.470372 | 0.882468 |
S_min = 1e-2
def cost(X):
nD.coords[1] = X[0]
nF.coords[1] = X[1]
nH.coords[1] = X[2]
nF.coords[0] = X[3]
model.solve()
# section ajustement
for bar in model.bars:
coef = np.abs( bar.stress/sigmay)
new_section = bar.section * coef
if new_section < S_min:
new_section = S_min
bar.section = new_section
model.solve()
m = model.mass()
# Cs = np.abs(model.data(at='bars').state.stress) / sigmay
# Cs = Cs.values.astype(np.float64)
# faillure = np.exp((Cs-1)*1000)
# faillure = np.sum(faillure) + 1
# print(faillure)
return m * np.abs(nG.displacement[1])
import scipy
nD.coords[0] = 0.0
nD.block[0] = True
x0 = np.array([ 5.0, 1.0, 1.0,5.])
sol = scipy.optimize.minimize(cost, x0)
sol
model.data(at="bars")
| conn | props | state | geometry | props | direction | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| c1 | c2 | section | density | tension | elongation | strain | stress | failure | volume | length | mass | dx | dy | |
| 0 | A | C | 0.01 | 7800.0 | 0.0 | 0.0 | 0.0 | 0.0 | False | 0.03 | 3.0 | 234.0 | 1.0 | 0.0 |
| 1 | C | D | 0.01 | 7800.0 | 827224.079561 | 0.00236 | 0.000394 | 82722407.956072 | False | 0.059918 | 5.99175 | 467.356506 | -0.500688 | 0.865628 |
| 2 | A | D | 0.01 | 7800.0 | -1376662.547512 | -0.0034 | -0.000656 | -137666254.751212 | False | 0.051866 | 5.186624 | 404.556675 | 0.0 | 1.0 |
| 3 | C | E | 0.01 | 7800.0 | 580326.451471 | 0.000829 | 0.000276 | 58032645.147143 | False | 0.03 | 3.0 | 234.0 | 1.0 | 0.0 |
| 4 | D | F | 0.01 | 7800.0 | 958672.701654 | 0.002588 | 0.000457 | 95867270.165372 | False | 0.056695 | 5.669543 | 442.22434 | 0.743592 | -0.668634 |
| 5 | E | F | 0.01 | 7800.0 | 1021307.191071 | 0.001102 | 0.000486 | 102130719.107071 | False | 0.022653 | 2.265274 | 176.691404 | -0.787619 | 0.616162 |
| 6 | E | G | 0.01 | 7800.0 | 1451310.01454 | 0.002073 | 0.000691 | 145131001.453976 | False | 0.03 | 3.0 | 234.0 | 1.0 | 0.0 |
| 7 | F | H | 0.01 | 7800.0 | -312682.822253 | -0.000864 | -0.000149 | -31268282.225258 | False | 0.058053 | 5.805264 | 452.810568 | 0.82411 | 0.56643 |
| 8 | G | H | 0.01 | 7800.0 | 261477.7 | 0.000583 | 0.000125 | 26147770.0 | False | 0.046841 | 4.684054 | 365.356178 | 0.0 | 1.0 |
| 9 | F | C | 0.01 | 7800.0 | -252950.881619 | -0.000223 | -0.00012 | -25295088.161927 | False | 0.018511 | 1.85106 | 144.382716 | -0.656827 | -0.754041 |
| 10 | E | H | 0.01 | 7800.0 | -123452.572654 | -0.000327 | -0.000059 | -12345257.265352 | False | 0.055624 | 5.562406 | 433.867649 | 0.539335 | 0.842091 |
model.solve()
xlim, ylim = model.bbox(deformed=False)
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_aspect("equal")
# ax.axis("off")
model.draw(
ax, deformed=False, field="stress", label=True, force_scale=1.0e-6, forces=True
)
plt.xlim(xlim)
plt.ylim(ylim)
plt.grid()
plt.xlabel("Axe $x$")
plt.ylabel("Axe $y$")
Text(0, 0.5, 'Axe $y$')
fix_points = np.array([[0., 0], [9., 0.], [9., 3.], [9., 6.]])
ex_points = np.atleast_2d(np.arange(1,9.,3))
ex_points = np.append(ex_points, np.zeros_like(ex_points), axis=0).T
fix_points = np.append( fix_points, ex_points, axis=0)
fix_points
nb_fix = len(fix_points)
nb_fix
7
np.random.uniform((1,1), (8,6), (nb_fix, 2))
array([[3.77050573, 2.67787623],
[2.17926083, 3.94447292],
[1.88069357, 4.45342859],
[7.53951052, 5.39740586],
[1.7821885 , 3.97033117],
[4.69445305, 4.11213409],
[7.92185874, 4.77569316]])
import matplotlib
Nb_inerpoints = 20
points = np.append(fix_points, np.random.uniform((1,1), (8,6), (nb_fix, 2)), axis=0)
tri = matplotlib.tri.Triangulation(points[:,0], points[:,1])
plt.figure()
plt.plot(points[:,0], points[:,1], 'o')
plt.triplot(tri)
plt.show()
E = 210.0e9 # Young Modulus [Pa]
rho = 7800.0 # Density [kg/m**3]
A = .7e-2 # Cross section [m**2]
sigmay = 400.0e6 # Yield Stress [Pa]
# Model definition
model = truss.core.Model() # Model definition
# NODES
nodes = []
#nA = model.add_node((0.0, 0.0), label="A")
#nodes.append(nA)
#nG = model.add_node((9.0, 0.0), label="G")
#nodes.append(nG)
for i, p in enumerate(points):
nodes.append(model.add_node((p[0], p[1]), label="n{:03d}".format(i)))
for i, e in enumerate(tri.edges):
model.add_bar(nodes[e[0]], nodes[e[1]], modulus=E, density=rho, section=A, yield_stress=sigmay)
# BOUNDARY CONDITIONS
nodes[0].block[1] = True
nodes[1].block[0] = True
nodes[2].block[0] = True
nodes[3].block[0] = True
# STRUCTURAL LOADING
for i in range(nb_fix):
nodes[i].force = np.array([0.0, -1.0e6])/2.
model.solve()
xlim, ylim = model.bbox(deformed=False)
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_aspect("equal")
# ax.axis("off")
model.draw(
ax, deformed=False, field="stress", label=True, force_scale=1.0e-6, forces=True
)
plt.xlim(xlim)
plt.ylim(ylim)
plt.grid()
plt.xlabel("Axe $x$")
plt.ylabel("Axe $y$")
Text(0, 0.5, 'Axe $y$')
S_min = 0.2
def cost(X):
for i,id_nodes in enumerate(range(nb_fix,len(nodes))):
nodes[id_nodes].coords[0] = X[i]
nodes[id_nodes].coords[1] = X[Nb_inerpoints+i]
model.solve()
# section ajustement
for bar in model.bars:
coef = np.abs( bar.stress/sigmay)
new_section = bar.section * coef
if new_section < S_min:
new_section = S_min
bar.section = new_section
model.solve()
m = model.mass()
u_n001 = nodes[1].displacement[1]
return np.abs(u_n001)*m
x0 = np.zeros(2*Nb_inerpoints)
for i,id_nodes in enumerate(range(nb_fix,len(nodes))):
x0[i] = nodes[id_nodes].coords[0]
x0[Nb_inerpoints+i] = nodes[id_nodes].coords[1]
print(x0)
cost(x0)
[3.03432345 5.33453408 7.09277802 7.35468322 5.7404693 2.61316389
3.93569183 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0.
0. 0. 1.62018386 2.73126019 4.58523751 2.57478543
3.93090686 4.68994099 1.98526621 0. 0. 0.
0. 0. 0. 0. 0. 0.
0. 0. 0. 0. ]
194.52550722067315
import scipy
x0 = sol.x
sol = scipy.optimize.minimize(cost, x0, method='Nelder-Mead')
sol
final_simplex: (array([[0.04956336, 3.34199724, 4.4005405 , ..., 6.4417869 , 8.90107638,
1.50357903],
[0.04956745, 3.34199865, 4.4005405 , ..., 6.44178451, 8.90107182,
1.50357894],
[0.04955984, 3.34199592, 4.4005405 , ..., 6.44178921, 8.90108017,
1.5035794 ],
...,
[0.04956497, 3.34199772, 4.40054049, ..., 6.44178613, 8.90107435,
1.50357931],
[0.04955511, 3.34199401, 4.40054048, ..., 6.44179275, 8.90108412,
1.50358094],
[0.04955093, 3.34199257, 4.40054048, ..., 6.44179505, 8.90108912,
1.50358075]]), array([162.75216527, 162.75216586, 162.75217362, 162.75217466,
162.75217692, 162.75217706, 162.75217731, 162.75217923,
162.75218011, 162.75218271, 162.75218322, 162.75218336,
162.7521838 , 162.75218451, 162.75218482, 162.75218489,
162.75218718, 162.75218756, 162.75218807, 162.75218817,
162.75218828, 162.752189 , 162.75218962, 162.75218983,
162.75219092, 162.75219106, 162.7521911 , 162.75219127,
162.75219131, 162.75219137, 162.7521925 , 162.75219253,
162.75219405, 162.75219406, 162.7521942 , 162.75219433,
162.75219452, 162.75219594, 162.75219599, 162.75219621,
162.75219636]))
fun: 162.7521652719785
message: 'Optimization terminated successfully.'
nfev: 6161
nit: 4707
status: 0
success: True
x: array([ 0.04956336, 3.34199724, 4.4005405 , 4.40049925, 2.86240419,
4.40054067, 3.48620022, 2.01424681, 8.84364729, 3.86455223,
4.6669124 , 15.45979173, 1.75341207, 1.22990674, 3.84248989,
8.13264685, 2.09854815, 0.74596853, 2.75158541, 8.51588762,
1.12121473, 2.14013607, 5.24332088, 5.24336889, 4.64020194,
5.24332227, 2.90810852, 2.30709751, 4.61702483, 5.75286242,
7.86618182, 35.22141945, 6.88222101, 2.77085482, 6.91561511,
3.2023324 , -0.80314786, 6.4417869 , 8.90107638, 1.50357903])
The Kernel crashed while executing code in the current cell or a previous cell.
Please review the code in the cell(s) to identify a possible cause of the failure.
Click <a href='https://aka.ms/vscodeJupyterKernelCrash'>here</a> for more info.
View Jupyter <a href='command:jupyter.viewOutput'>log</a> for further details.
xlim, ylim = model.bbox(deformed=False)
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_aspect("equal")
# ax.axis("off")
model.draw(
ax, deformed=False, field="stress", label=False, force_scale=1.0e-6, forces=True
)
plt.xlim(xlim)
plt.ylim(ylim)
plt.grid()
plt.xlabel("Axe $x$")
plt.ylabel("Axe $y$")
Text(0, 0.5, 'Axe $y$')
model.data(at="bars")
| conn | props | state | geometry | props | direction | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| c1 | c2 | section | density | tension | elongation | strain | stress | failure | volume | length | mass | dx | dy | |
| 0 | n002 | n001 | 0.2 | 7800.0 | -202777.832959 | -0.000014 | -0.000005 | -1013889.164796 | False | 0.6 | 3.0 | 4680.0 | 0.0 | -1.0 |
| 1 | n003 | n002 | 0.2 | 7800.0 | -196022.736335 | -0.000014 | -0.000005 | -980113.681675 | False | 0.6 | 3.0 | 4680.0 | 0.0 | -1.0 |
| 2 | n004 | n000 | 0.2 | 7800.0 | 1316148.537201 | 0.000031 | 0.000031 | 6580742.686005 | False | 0.2 | 1.0 | 1560.0 | -1.0 | 0.0 |
| 3 | n005 | n004 | 0.2 | 7800.0 | 1300154.671608 | 0.000093 | 0.000031 | 6500773.358039 | False | 0.6 | 3.0 | 4680.0 | -1.0 | 0.0 |
| 4 | n006 | n001 | 0.2 | 7800.0 | 2320273.325305 | 0.00011 | 0.000055 | 11601366.626524 | False | 0.4 | 2.0 | 3120.0 | 1.0 | 0.0 |
| 5 | n006 | n005 | 0.2 | 7800.0 | 1530905.225659 | 0.000109 | 0.000036 | 7654526.128295 | False | 0.6 | 3.0 | 4680.0 | -1.0 | 0.0 |
| 6 | n007 | n006 | 0.2 | 7800.0 | 325887.483111 | 0.000028 | 0.000008 | 1629437.415556 | False | 0.731159 | 3.655793 | 5703.036793 | 0.639684 | -0.768638 |
| 7 | n008 | n000 | 0.2 | 7800.0 | -2058562.794836 | -0.000157 | -0.000049 | -10292813.974178 | False | 0.641702 | 3.208509 | 5005.273674 | -0.341733 | -0.939797 |
| 8 | n008 | n004 | 0.2 | 7800.0 | 500255.738335 | 0.000036 | 0.000012 | 2501278.691673 | False | 0.603378 | 3.016889 | 4706.347111 | -0.031971 | -0.999489 |
| 9 | n008 | n005 | 0.2 | 7800.0 | 373200.354373 | 0.000037 | 0.000009 | 1866001.771864 | False | 0.837207 | 4.186036 | 6530.215835 | 0.693627 | -0.720335 |
| 10 | n009 | n002 | 0.2 | 7800.0 | 342133.361145 | 0.000034 | 0.000008 | 1710666.805723 | False | 0.837502 | 4.18751 | 6532.515742 | 0.790577 | -0.612362 |
| 11 | n009 | n003 | 0.2 | 7800.0 | -1623304.514944 | -0.000129 | -0.000039 | -8116522.574721 | False | 0.66782 | 3.339102 | 5208.998524 | 0.991449 | 0.130492 |
| 12 | n009 | n007 | 0.2 | 7800.0 | -70848.700793 | -0.000005 | -0.000002 | -354243.503967 | False | 0.587977 | 2.939884 | 4586.219515 | -0.349675 | -0.936871 |
| 13 | n010 | n001 | 0.2 | 7800.0 | 1177645.013953 | 0.000062 | 0.000028 | 5888225.069767 | False | 0.443368 | 2.216838 | 3458.267357 | 0.802416 | -0.596765 |
| 14 | n010 | n002 | 0.2 | 7800.0 | -413615.046193 | -0.000024 | -0.00001 | -2068075.230963 | False | 0.488949 | 2.444745 | 3813.801866 | 0.727612 | 0.685989 |
| 15 | n010 | n006 | 0.2 | 7800.0 | -160442.836499 | -0.000005 | -0.000004 | -802214.182494 | False | 0.268259 | 1.341293 | 2092.417684 | -0.164896 | -0.986311 |
| 16 | n010 | n007 | 0.2 | 7800.0 | 543483.580128 | 0.000038 | 0.000013 | 2717417.900642 | False | 0.592065 | 2.960324 | 4618.104954 | -0.864678 | 0.502326 |
| 17 | n010 | n009 | 0.2 | 7800.0 | 590360.592056 | 0.000063 | 0.000014 | 2951802.960282 | False | 0.90189 | 4.509452 | 7034.744684 | -0.33967 | 0.940545 |
| 18 | n011 | n007 | 0.2 | 7800.0 | 878381.02416 | 0.000056 | 0.000021 | 4391905.120802 | False | 0.539062 | 2.695312 | 4204.685985 | 0.748695 | -0.662915 |
| 19 | n011 | n008 | 0.2 | 7800.0 | -799551.166597 | -0.000042 | -0.000019 | -3997755.832986 | False | 0.442452 | 2.212262 | 3451.128335 | -0.699296 | -0.714832 |
| 20 | n011 | n009 | 0.2 | 7800.0 | -1021845.030704 | -0.000078 | -0.000024 | -5109225.15352 | False | 0.639188 | 3.195942 | 4985.669494 | 0.953074 | 0.302737 |
| 21 | n012 | n003 | 0.2 | 7800.0 | -1627951.671796 | -0.000137 | -0.000039 | -8139758.358982 | False | 0.707193 | 3.535966 | 5516.107313 | 0.998397 | 0.056604 |
| 22 | n012 | n009 | 0.2 | 7800.0 | 743236.866396 | 0.000006 | 0.000018 | 3716184.331982 | False | 0.064431 | 0.322157 | 502.56417 | 0.682112 | -0.731248 |
| 23 | n013 | n009 | 0.2 | 7800.0 | -649257.439574 | -0.000012 | -0.000015 | -3246287.19787 | False | 0.150902 | 0.754509 | 1177.033357 | 0.996087 | 0.08838 |
| 24 | n013 | n011 | 0.2 | 7800.0 | -925057.648724 | -0.000054 | -0.000022 | -4625288.24362 | False | 0.492985 | 2.464926 | 3845.283943 | -0.930824 | -0.365467 |
| 25 | n013 | n012 | 0.2 | 7800.0 | -1286386.287538 | -0.000019 | -0.000031 | -6431931.437692 | False | 0.122341 | 0.611704 | 954.258483 | 0.869389 | 0.494128 |
| 26 | n014 | n000 | 0.2 | 7800.0 | -1228972.895406 | -0.000155 | -0.000029 | -6144864.477031 | False | 1.060529 | 5.302643 | 8272.123232 | -0.498521 | -0.866877 |
| 27 | n014 | n008 | 0.2 | 7800.0 | -766036.201625 | -0.00004 | -0.000018 | -3830181.008125 | False | 0.442452 | 2.212261 | 3451.127801 | -0.699296 | -0.714832 |
| 28 | n014 | n011 | 0.2 | 7800.0 | -1281520.969749 | -0.0 | -0.000031 | -6407604.848746 | False | 0.0 | 0.0 | 0.000639 | 0.19066 | 0.981656 |
| 29 | n014 | n013 | 0.2 | 7800.0 | -971204.707336 | -0.000057 | -0.000023 | -4856023.536682 | False | 0.492985 | 2.464926 | 3845.284285 | 0.930824 | 0.365467 |
| 30 | n015 | n005 | 0.2 | 7800.0 | 232873.762297 | 0.000011 | 0.000006 | 1164368.811485 | False | 0.408019 | 2.040097 | 3182.551982 | -0.120714 | -0.992687 |
| 31 | n015 | n006 | 0.2 | 7800.0 | 688241.861215 | 0.000056 | 0.000016 | 3441209.306076 | False | 0.683649 | 3.418243 | 5332.459306 | 0.805598 | -0.592462 |
| 32 | n015 | n007 | 0.2 | 7800.0 | -8575.299973 | -0.0 | -0.0 | -42876.499865 | False | 0.177571 | 0.887855 | 1385.054124 | 0.467619 | 0.88393 |
| 33 | n015 | n008 | 0.2 | 7800.0 | 155625.251395 | 0.000012 | 0.000004 | 778126.256974 | False | 0.660356 | 3.301782 | 5150.77979 | -0.953974 | 0.299889 |
| 34 | n015 | n011 | 0.2 | 7800.0 | 706808.787136 | 0.000051 | 0.000017 | 3534043.935682 | False | 0.606032 | 3.030159 | 4727.048701 | -0.528945 | 0.848656 |
Question 4: Same question with displacements also along \(\vec x\) of \(C\), \(D\), \(E\) and \(F\). Is it better ?
Question 5: You can now try to perform topological optimization by removing/merging well chosen beams and nodes. In order to make the structure even more efficient.
Question 6: You are now asked to optimize the cross section along with the position of \(C\), \(D\), \(E\) and \(F\) in order to reach the yield stress in each individual beam.