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	40000000.0	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 \(\alpha\):

\[ \alpha = |u_y(G)| m \]

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.

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.