Optimization of steel roof trusses using machine learning-assisted differential evolution

Tóm tắt Optimization of steel roof trusses using machine learning-assisted differential evolution: ... Nt An fγc − 1 ≤ 0 Nc ϕA fγc − 1 ≤ 0 gλ(A) = λ [λ] − 1 ≤ 0 (6) 102 Hieu, N. T., et al. / Journal of Science and Technology in Civil Engineering To handle constraints, a widely used method called the penalty function is applied in this study. The fitness function is employed... without conducting true fit- ness evaluations. Consequently, the computation time can be shortened. The flowchart of the AdaBoost- DE algorithm is presented in Fig. 3. Journal of Science and Technology in Civil Engineering NUCE 2018 ISSN 1859-2996 10 Figure 3. Flowchart of the AdaBoost-DE ... two algorithms DE and AdaBoost-DE achieve the same best weight (1022.9 kg) but the standard deviation (SD) of the DE is smaller than that of the AdaBoost-DE. It indicates that the DE is more stable than the AdaBoost-DE. However, the difference is not much (the SD of the DE is 48.5 kg while th...

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 and Technology in Civil Engine ring NUCE 2018 ISSN 1859-2996 
11 
Figure 4. Industrial factory roof structure 
 The planar trusses used in this project are the Warren-type truss consisting of 
square hollow section (SHS) members with welded joints. Due to the limitation of 
Vietnamese profile types, this project uses profiles according to European standards. 
Each planar truss has 49 members that are categorized into five groups including: (1) 
top chords; (2) bottom chords; (3) support diagonals; (4) diagonals; (5) posts. 
Members of the same group are assigned the same cross-section which is selected 
from a list of 55 SHS profiles (from o50x3 to o300x12.5). 
 The roof structure is subjected to three load cases: the dead load (DL), the live 
load (LL), and the wind load (WL). The dead load includes the self-weight of the 
structure, the weight of purlins and sheeting (g1=15 daN/m2), and the weight of 
equipment (g2=50 daN/m2). The live load is taken as the maintenance load for unused 
sloping roofs p=30 daN/m2. The project is built in region II.B according to TCVN 
2737:1995 with the basic wind pressure W0=95 daN/m2. The loads distributed on the 
roof are converted to the concentrated loads at the nodes of the truss as shown in 
Figure 5. Three load combinations are considered: (LC1) DL+LL; (LC2) DL+WL; 
(LC3) DL+0.9´LL+0.9´WL. 
4.2. Setting 
 The truss structure is optimized according to both the original DE algorithm and 
the AdaBoost-DE algorithm. For a fair comparison, the parameters of the two 
algorithms are set to the same as follows: the scaling factor F=0.8, the crossover rate 
Cr=0.9, the population size NP=50, the number of generations max_iter=100. For the 
AdaBoost-DE, the first stage lasts n_iter1=10 which means that the training dataset 
contains 500 samples. Two algorithms are coded in Python language with some 
+10.000
L
B
B
h1
Figure 4. Industrial factory roof structure
The planar trusses used in this project are the Warren-type truss consisting of square hollow
section (SHS) members with welded joints. Due to the limitation f Vietnamese profile types, this
project uses profiles according to European standards. Each planar truss has 49 members that are
categorized into five groups including: (1) top chords; (2) bottom chords; (3) support diagonals; (4)
diagonals; (5) posts. Members of the same group are assigned the same cross-section which is selected
from a list of 55 SHS profiles (from 50 × 3 to 300 × 12.5).
The roof structure is subj ct d to three loa cases: the de d load (DL), the live load (LL), and the
wind load (WL). The dead load includes the self-weight of the structure, the weight of purlins and
sheeting (g1 = 15 daN/m2), and the weight of equipment (g2 = 50 daN/m2). The live load is taken
as the maintenance l ad for unused sloping oofs p = 30 daN/m2. The project is built in region II.B
according to TCVN 2737:1995 with the basic wind pressure W0 = 95 daN/m2. The loads distributed
on the roof are converted to the concentrated loads at the nodes of the truss as shown in Fig. 5. Three
load combinations are considered: (LC1) DL+LL; (LC2) DL+WL; (LC3) DL+0.9×LL+0.9×WL.
Journal of Science and Technology in Civil Engineering NUCE 2018 ISSN 1859-2996 
12 
widely used libraries such as scikit-l arn [28]. The finite element code for fitness 
evaluations is also written in Python based on the direct stiffness method. Each 
algorithm is performed 30 times on a computer with the following configuration: CPU 
Intel Core i5-5257 2.7 GHz, RAM 8.00 Gb. 
Figure 5. Configuration of the planar truss 
4.3. Results 
 The statistical results of 30 optimization times are presented in Table 3. 
Additionally, the convergence histories of two algorithms are displayed in Figure 5. 
Table 3. Comparison of optimal results found by the DE and the AdaBoost-DE 
 DE AdaBoost-DE 
(1) Top chords o150´4 o150´4 
(2) Bottom chords o120´4 o120´4 
(3) Support diagonals o80´3 o80´3 
(4) Diagonals o70´2.5 o70´2.5 
(5) Post o50´3 o50´3 
Best weight (kg) 1022.9 1022.9 
Mean weight (kg) 1060.7 1065.1 
Worst weight (kg) 1213.9 1213.9 
Standard deviation (kg) 48.5 49.6 
Average number of fitness evaluations 5000 3198 
L=12´1.5=24m
h1=1.0m
(1) (1) (1)
(1) (1) (1)
(1) (1)
(2) (2) (2) (2)
(3) (4) (4) (4) (4) (4) (4) (4)
(5)
(5) (5)
(5) (5)
h2=2.2m
3.43 6.86 6.86
6.86 6.86 6.86
6.86 6.86
6.86
3.436.86
6.866.866.86
6.866.866.86
(DL)
1.58 3.15 3.15
3.15 3.15 3.15
3.15 3.15
3.15
1.583.15
3.153.153.15
3.153.153.15
(LL)
2.64
5.27 5.27 5.27
5.27 5.27 5.27
5.27
2.64
(WL)
2.384.764.764.76
4.764.764.76
4.76
2.38
Top chords
Bottom chords
Po
st
Fi r . fi r ti f t lanar truss
106
Hieu, N. T., et al. / Journal of Science and Technology in Civil Engineering
4.2. Setting
The truss structure is optimized according to both the original DE algorithm and the AdaBoost-
DE algorithm. For a fair comparison, the parameters of the two algorithms are set to the same as
follows: the scaling factor F = 0.8, the crossover rate Cr = 0.9, the population size NP = 50, the
number of generations max_iter = 100. For the AdaBoost-DE, the first stage lasts n_iter1 = 10 which
means that the training dataset contains 500 samples. Two algorithms are coded in Python language
with some widely used libraries such as scikit-learn. The finite element code for fitness evaluations is
also written in Python based on the direct stiffness method. Each algorithm is performed 30 times on
a computer with the following configuration: CPU Intel Core i5-5257 2.7 GHz, RAM 8.00 Gb.
4.3. Results
The statistical results of 30 optimization times are presented in Table 3. Additionally, the conver-
gence histories of two algorithms are displayed in Fig. 6.
Table 3. Comparison of optimal results found by the DE and the AdaBoost-DE
DE AdaBoost-DE
(1) Top chords 150×4 150×4
(2) Bottom chords 120×4 120×4
(3) Support diagonals 80×3 80×3
(4) Diagonals 70×2.5 70×2.5
(5) Post 50×3 50×3
Best weight (kg) 1022.9 1022.9
Mean weight (kg) 1060.7 1065.1
Worst weight (kg) 1213.9 1213.9
Standard deviation (kg) 48.5 49.6
Average number of fitness evaluations 5000 3198Journal of Science and Technology in Civil Engineering NUCE 2018 ISSN 1859-2996 
13 
Figure 5. Convergence curves for the optimization of the roof truss 
 Based on the obtained results, some observations can be pointed out as follows. 
First of all, two algorithms DE and AdaBoost-DE achieve the same best weight 
(1022.9 kg) but the standard deviation (SD) of the DE is smaller than that of the 
AdaBoost-DE. It indicates that the DE is more stable than the AdaBoost-DE. 
However, the difference is not much (the SD of the DE is 48.5 kg while the SD of the 
AdaBoost-DE is 49.6 kg). 
 Secondly, the AdaBoost-DE carries out fewer fitness evaluations than the DE 
(3198 times for the AdaBoost-DE and 5000 times for the DE). It means using 
AdaBoost helps to reduce approximately 36% fitness evaluations of the DE 
optimization. 
4.4. Comparison of different truss types 
 The proposed method AdaBoost-DE is applied to optimize the weight for 06 
cases with the same design data as described in Section 4.1 but the configurations of 
trusses are different. More specifically, six truss types considered in this section are as 
follows: Case 1: a trapeizoidal Warren truss (Figure 6(a)), Case 2: a parallel chord 
Warren truss (Figure 6(b)), Case 3: a trapeizoidal Pratt truss (Figure 6(c)), Case 4: a 
parallel chord Pratt truss (Figure 6(d)), Case 5: a trapeizoidal Howe truss (Figure 
6(e)), and Case 6: a parallel chord Howe truss (Figure 6(g)). Parameters are set the 
same as Section 4.2. Each case is implemented for 15 independent runs. The best 
designs of six cases are reported in Table 4. Additionally, demand-to-capacity ratios 
for six truss cases are also summarized in this table. 
800
1000
1200
1400
1600
1800
2000
0 1000 2000 3000 4000 5000
W
ei
gh
t (
kg
)
Number of fitness evaluations
DE
AdaBoost-DE
Reducing fitness 
evaluations by 
incorporating AdaBoost 
Figure 6. Convergence curves for the ti i ti of the r of truss
107
Hieu, N. T., et al. / Journal of Science and Technology in Civil Engineering
Based on the obtained results, some observations can be pointed out as follows. First of all, two
algorithms DE and AdaBoost-DE achieve the same best weight (1022.9 kg) but the standard deviation
(SD) of the DE is smaller than that of the AdaBoost-DE. It indicates that the DE is more stable than
the AdaBoost-DE. However, the difference is not much (the SD of the DE is 48.5 kg while the SD of
the AdaBoost-DE is 49.6 kg). Secondly, the AdaBoost-DE carries out fewer fitness evaluations than
the DE (3198 times for the AdaBoost-DE and 5000 times for the DE). It means using AdaBoost helps
to reduce approximately 36% fitness evaluations of the DE optimization.
4.4. Comparison of different truss types
The proposed method AdaBoost-DE is applied to optimize the weight for 06 cases with the
same design data as described in Section 4.1 but the configurations of trusses are different. More
specifically, six truss types considered in this section are as follows: Case 1: a trapeizoidal Warren
truss (Fig. 7(a)), Case 2: a parallel chord Warren truss (Fig. 7(b)), Case 3: a trapeizoidal Pratt truss
(Fig. 7(c)), Case 4: a parallel chord Pratt truss (Fig. 7(d)), Case 5: a trapeizoidal Howe truss (Fig. 7(e)),
and Case 6: a parallel chord Howe truss (Fig. 7(g)). Parameters are set the same as Section 4.2. Each
case is implemented for 15 independent runs. The best designs of six cases are reported in Table 4.
Additionally, demand-to-capacity ratios for six truss cases are also summarized in this table.
Journal of Science and Technology in Civil Engineering NUCE 2018 ISSN 1859-2996 
14 
Figure 6. Configurations of six truss types 
Table 4. Comparison of best designs for six truss types 
 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 
(1) Top chords o150´4 o150´4 o150´4 o150´4 o150´4 o150´4 
(2) Bottom chords o120´4 o150´4 o120´5 o150´5 o120´4 o150´4 
(3) Support diagonals o80´3 o70´2.5 o70´2.5 o50´3 o80´3 o70´2.5 
(4) Diagonals o70´2.5 o50´3 o50´3 o50´3 o70´2.5 o70´2.5 
(5) Post o50´3 o50´3 o50´3 o50´3 o50´3 o50´3 
Weight (kg) 1022.9 1031.7 1118.2 1160.7 1081.8 1095.8 
max(N/Afgc, |N/jAfgc |) 0.933 0.915 0.977 0.890 0.937 0.928 
max(u/[u]) 0.303 0.289 0.374 0.317 0.294 0.268 
max(l/[l]) 0.849 0.677 0.892 0.684 0.849 0.677 
 It is clearly seen that Case 1 has the lowest weight (1022.9 kg), followed by Case 
2 (1031.7 kg), Case 5 (1081.8 kg), Case 6 (1095.8 kg), Case 3 (1118.2 kg), and Case 4 
(1160.7 kg). With the same configuration of web members, the trapeizoidal truss is 
always lighter than the parallel chord truss. In three cases of trapeizoidal shape, the 
weight of the Warren truss (Case 1) is the smallest, followed by the Howe truss (Case 
5), and the Pratt truss (Case 3). For three parallel chord trusses, the order is the same 
when the Warren truss has the smallest weight, followed by the Howe amd the Pratt 
trusses, respectively. Overall, among six common types, the trapeizoidal Warren 
configuration is the most suitable solution for pinned-pinned roof trusses. 
5. Conclusions 
 This paper uses an effective algorithm called AdaBoost-DE to optimize steel 
roof trusses. The AdaBoost-DE adopts the machine learning classification technique 
to enhance the performance of the DE algorithm. In more detail, the DE is used as a 
search engine while the AdaBoost classifier serves as a coarse filter to remove poor 
trial vectors during the optimization process. The applicability of the AdaBoost-DE 
(a)
(c)
(e)
(b)
(d)
(g)
Figure 7. Configura s of six truss types
Table 4. Comparison of best designs for six truss types
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6
(1) Top chords 150×4 150×4 150×4 150×4 150×4 150×4
(2) Bottom chords 120×4 150×4 120×5 150×5 120×4 150×4
(3) Support diagonals 80×3 70×2.5 70×2.5 50×3 80×3 70×2.5
(4) Diagonals 70×2.5 50×3 50×3 50×3 70×2.5 70×2.5
(5) Post 50×3 50×3 50×3 50×3 50×3 50×3
Weight (kg) 1022.9 1031.7 1118.2 1160.7 1081.8 1095.8
max (N/A fγc, |N/ϕA fγc|) 0.933 0.915 0.977 0.890 0.937 0.928
max(u/[u]) 0.303 0.289 0.374 0.317 0.294 0.268
max(λ/[λ]) 0.849 0.677 0.892 0.684 0.849 0.677
It is clearly seen that Case 1 has the lowest weight (1022.9 kg), followed by Case 2 (1031.7 kg),
Case 5 (1081.8 kg), Case 6 (1095.8 kg), Case 3 (1118.2 kg), and Case 4 (1160.7 kg). With the same
108
Hieu, N. T., et al. / Journal of Science and Technology in Civil Engineering
configuration of web members, the trapeizoidal truss is always lighter than the parallel chord truss.
In three cases of trapeizoidal shape, the weight of the Warren truss (Case 1) is the smallest, followed
by the Howe truss (Case 5), and the Pratt truss (Case 3). For three parallel chord trusses, the order is
the same when the Warren truss has the smallest weight, followed by the Howe amd the Pratt trusses,
respectively. Overall, among six common types, the trapeizoidal Warren configuration is the most
suitable solution for pinned-pinned roof trusses.
5. Conclusions
This paper uses an effective algorithm called AdaBoost-DE to optimize steel roof trusses. The
AdaBoost-DE adopts the machine learning classification technique to enhance the performance of the
DE algorithm. In more detail, the DE is used as a search engine while the AdaBoost classifier serves
as a coarse filter to remove poor trial vectors during the optimization process. The applicability of the
AdaBoost-DE algorithm in the practical design is demonstrated through an example to design a steel
roof truss of an industrial factory. The numerical example shows that the AdaBoost-DE achieves the
same optimal design as the original DE algorithm but it reduces about 36% of fitness evaluations.
Obviously, the proposed AdaBoost-DE method requires additional times to train models. There-
fore, this method is effective when solving problems having computationally expensive fitness eval-
uation, for example, large-scale structures where the training time is very small in comparison with
the time of finite element analysis. In the future, the application of the AdaBoost-DE for optimizing
other structures like double-layer grids, lattice towers can be investigated. Moreover, the study on
integrating other classification techniques into evolutionary algorithms has great potential.
Acknowledgments
The first author was funded by Vingroup Joint Stock Company and supported by the Domes-
tic PhD. Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data
Institute (VINBIGDATA) code VINIF.2020.TS.134.
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