Optimizing a bridge structure

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 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 $y$")

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	-0.001143	0.0	0.0	1000000.0	False	True
1	C	3.0	0.0	-0.000857	-0.00338	0.0	0.0	False	False
2	D	3.0	3.0	0.001429	-0.00338	0.0	0.0	False	False
3	E	6.0	0.0	-0.000571	-0.006188	0.0	0.0	False	False
4	F	6.0	3.0	0.000857	-0.006473	0.0	0.0	False	False
5	G	9.0	0.0	0.0	-0.008139	3000000.0	-1000000.0	True	False
6	H	9.0	3.0	0.0	-0.008139	-3000000.0	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	1000000.0	0.000286	0.000095	20000000.0	False	0.15	3.0	1170.0	1.0	0.0
1	C	D	0.05	7800.0	0.0	0.0	0.0	0.0	False	0.15	3.0	1170.0	0.0	1.0
2	A	D	0.05	7800.0	-1414213.562373	-0.000571	-0.000135	-28284271.247462	False	0.212132	4.242641	1654.629868	0.707107	0.707107
3	C	E	0.05	7800.0	1000000.0	0.000286	0.000095	20000000.0	False	0.15	3.0	1170.0	1.0	0.0
4	D	F	0.05	7800.0	-2000000.0	-0.000571	-0.00019	-40000000.000001	False	0.15	3.0	1170.0	1.0	0.0
5	D	E	0.05	7800.0	1414213.562373	0.000571	0.000135	28284271.247462	False	0.212132	4.242641	1654.629868	0.707107	-0.707107
6	E	F	0.05	7800.0	-1000000.0	-0.000286	-0.000095	-20000000.0	False	0.15	3.0	1170.0	0.0	1.0
7	E	G	0.05	7800.0	2000000.0	0.000571	0.00019	40000000.000001	False	0.15	3.0	1170.0	1.0	0.0
8	F	H	0.05	7800.0	-3000000.0	-0.000857	-0.000286	-60000000.000001	False	0.15	3.0	1170.0	1.0	0.0
9	F	G	0.05	7800.0	1414213.562373	0.000571	0.000135	28284271.247462	False	0.212132	4.242641	1654.629868	0.707107	-0.707107
10	G	H	0.05	7800.0	0.0	0.0	0.0	0.0	False	0.15	3.0	1170.0	0.0	1.0

Dead (or structural) mass#

m0 = model.mass()
m0 * 1.0e-3  # Mass in tons !

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")

	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	1000000.0	0.000286	0.000095	20000000.0	False	0.15	3.0	1170.0	1.0	0.0
1	C	D	0.05	7800.0	0.0	0.0	0.0	0.0	False	0.15	3.0	1170.0	0.0	1.0
2	A	D	0.05	7800.0	-1414213.562373	-0.000571	-0.000135	-28284271.247462	False	0.212132	4.242641	1654.629868	0.707107	0.707107
3	C	E	0.05	7800.0	1000000.0	0.000286	0.000095	20000000.0	False	0.15	3.0	1170.0	1.0	0.0
4	D	F	0.05	7800.0	-2000000.0	-0.000571	-0.00019	-40000000.0	False	0.15	3.0	1170.0	1.0	0.0
5	D	E	0.05	7800.0	1414213.562373	0.000571	0.000135	28284271.247462	False	0.212132	4.242641	1654.629868	0.707107	-0.707107
6	E	F	0.05	7800.0	-1000000.0	-0.000286	-0.000095	-20000000.0	False	0.15	3.0	1170.0	0.0	1.0
7	E	G	0.05	7800.0	2000000.0	0.000571	0.00019	40000000.000001	False	0.15	3.0	1170.0	1.0	0.0
8	F	H	0.05	7800.0	-3000000.0	-0.000857	-0.000286	-60000000.000001	False	0.15	3.0	1170.0	1.0	0.0
9	F	G	0.0333	7800.0	1414213.562373	0.000858	0.000202	42468875.74694	False	0.14128	4.242641	1101.983492	0.707107	-0.707107
10	G	H	0.05	7800.0	0.0	0.0	0.0	0.0	False	0.15	3.0	1170.0	0.0	1.0

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 $α$ :

α = | 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.

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.