Optimizing a bridge structure - POSITRON: PythOn for Science In The Reblochon cOuNtry

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 notebook
# 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:

http://truss.readthedocs.io/en/latest/tutorial.html

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)

# STRUCTURAL LOADING
nG.force = np.array([0.0, -1.0e6])


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 $y$")

Read Data¶

Detailed results at the nodes¶

model.data(at="nodes")

Detailed results on the bars¶

model.data(at="bars")

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")

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([np.False_, np.False_, np.False_, np.False_, np.False_, np.False_,
       np.False_, np.False_, np.False_, np.False_, np.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

(1)

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.

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.