Prediction of Effect of Geometrical
Parameters in Trough Shape Folded
Plate Roof Using ANN Modeling
Bhagwan Girish Shanbhag and Y. R. Suresh
Abstract Finite element method is a numerical technique used to obtain approximate solutions to the problems with boundary values. It is simply a technique used
in solving problems which have partial differential equations and boundary conditions. This method gives approximate results at each and every discrete number of
points over the domain. A consistent model is to be developed for easier, faster and
less expensive structural development. In this regard, artificial neural network can
have high possibilities as these networks are universal approximators that can carry
out any uninterrupted mapping and can provide general mechanisms for building
models from data whose input–output relationship is highly nonlinear. In this paper,
the behavior of trough shape folded plate roof is studied in terms of displacement
and stresses for different boundary conditions using the software SAP-2000 (v-20)
by varying geometrical parameters (thickness, bay width and height of FPR) and to
extract the information on the importance of the input parameter on the prediction
of output results using artificial neural network model.
Keywords Artificial neural network · Finite element analysis · Folded plate roof ·
Garson algorithm
1 Introduction
Artificial neural networks are computational systems inspired from the biological
brain in their structure, data processing restoring method and learning ability. Specifically, a neural network is defined as a largely parallel disturbed processor with a
natural propensity to store experimental knowledge and makes it available for future
use in two ways—(a) The network acquires knowledge from a learning process and
B. G. Shanbhag (B) · Y. R. Suresh
Department of Civil Engineering, NMAM Institute of Technology, Nitte, Karkala, Udupi,
Karnataka 574110, India
e-mail: bhagwan.shanbhag13@gmail.com
Y. R. Suresh
e-mail: suresh.yr@jyothyit.ac.in
© Springer Nature Singapore Pte Ltd. 2021
M. C. Narasimhan et al. (eds.), Trends in Civil Engineering and Challenges
for Sustainability, Lecture Notes in Civil Engineering 99,
https://doi.org/10.1007/978-981-15-6828-2_17
221
222
B. G. Shanbhag and Y. R. Suresh
(b) the strengths of inter neuron connection called as synaptic weights are used to
store the knowledge.
Folded plate structures are best defined as flat plate or slab assemblies, inclined
in various directions and attached along their longitudinal edges. The structural
system is thus able to carry loads along mutual edges without the need for additional
supporting beams. Such structures have been described by other names, including
hipped plates, prismatic shells and prismatic pitched slabs. Typically, modern folded
plates structures are made of reinforced concrete cast in situ or precast or steel plate.
Various types of folded plate roofs include V type, trapezoidal type, trough type,
north light type, etc. Folded plate roofs have been used for considerable advantage
due to their rigidity and strength in situations where large areas need to be covered
free from internal columns and other obstructions. For aircraft hangars, with spans of
up to 160 ft. on both side of a centrally anchor wall, a folded plate cantilever system
is used very successfully.
1.1 Related Works
Chauhan [1] developed computer programs in MATLAB based on Simpson’s method
to analyze different types of folded plate roofs like V, trough type and North light type
of folded plate roofs in order to avoid conventional methods which are cumbersome
and time consuming [1]. Desai et al. [2] used the concept of folding, as seen in
nature and origami, to increase load bearing capacity; folded plates are used as
roofing structure for long span. Glass, timber and R.C.C. are the material used for
folded plate structure [2]. Chacko et al. [3] made parametric study on transverse
and longitudinal moment of trough type of folded plate roofs using ANSYS. The
results of the study gave insight to the range of magnitude of various parameters to be
considered for the optimum performance of plate [3]. Elkady and Hasan [4] studied
the impact of various factors like stiffness of end diaphragms, intermediate beam
stiffness, folded plate rise and folded plate thickness on static and dynamic behavior
of quadratic folded plate (QFP) slabs by performing linear static analysis based on
finite element modeling [4]. Lakshmy and Bhavikatti [5] studied the optimization
of simply supported symmetrical trough-type folded plate roofs using improved
move-limit method of sequential linear programming and sequential unconstrained
minimization technique [5].
1.2 Methodology
In the present study, analysis is done for trough shape folded roof (Ref. Fig. 1) square
in plan by changing geometry and boundary conditions. Roofs are supported on all
four ends with fixed and hinge boundary conditions. The variation of displacement
and percentage reduction in displacement is studied (Table 1).
Prediction of Effect of Geometrical Parameters in Trough …
223
Fig. 1 Typical cross section of trough shape FPR; b = bay width; H = height of FPR; a = width
of one fold; = inclination to horizontal
Table 1 Parameters considered for the study
Plan size
20 × 20
Span
20
m
Height
L/15, L/20 and L/25
m
Plate thickness
75–130
mm
Boundary conditions
Fixed and hinged
Live Load
0.8
kN/m2
Compressive strength of concrete [6]
25
N/mm2
Density
25
kN/m3
Young’s modulus
25
GPa
Poison’s ratio
0.2
Mesh size
0.2 × 0.2
m2
m2
1.3 Selection of Geometries
According to IS: 2210-1988 [7], the height of folded plate roof should be L/15.
Hence, heights of L/15, L/20 and L/25 are selected. By using the below formula, the
possible number of bay can be calculated for any height and for any plan dimension.
= tan−1 (H/a) = 30◦ to 60◦
(1)
where = inclination to horizontal; H = L/15 to L/25; a = b/5 (Fig. 1); L = span;
b = bay width.
By trial and error procedure, the number of bays is assumed for a fixed total
width “B” and assumed height “H.” Corresponding value of “a” is calculated from
the obtained bay width “b” and substituted in Eq. (1) for the constant span. Table 2
shows the possible number of bays for various heights. All the number of bay is
selected so that the roof structure is symmetric (Fig. 2).
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B. G. Shanbhag and Y. R. Suresh
Table 2 Geometrical parameter for different height
n
H = 1.33 m
H = 1.00 m
H = 0.8 m
b
b
2
10
33.62
–
–
–
–
3
6.67
45
6.67
36.94
6.67
31.03
4
5
53.06
5
45
5
38.66
5
4
58.98
4
51.34
4
45
6
–
–
3.33
56.17
3.33
50.05
7
–
–
–
–
2.86
54.53
8
–
–
–
–
2.5
57.99
b
H = Height of FPR; n = number of bay; = inclination to horizontal (in degrees)
Fig. 2 Cross section of trough shape folded plate roof with various number of bay; n = number of
bay
1.4 Finite Element Analysis
Using the SAP 2000 (version 20) software, linear static analysis is carried out and 336
models are analyzed for dead load and live load combination [8]. The quadrilateral
element with four node and six degrees of freedom for each node is considered that
has both bending and membrane capabilities.
Prediction of Effect of Geometrical Parameters in Trough …
225
Fig. 3 Artificial neural network modeling procedure in WEKA 3.8
1.5 ANN Modeling
Around 166 data sets from each parameter are considered to develop the respective
ANN model. These data sets are divided for training and validation process. ANN
models are developed using available software WEKA 3.8 by defining input and
output parameters. Multilayer perceptron neural network is used for the optimal
performance of network model (Fig. 3).
1.6 Statistical Measures [9]
Coefficient of determination, COD = R2 , where R is the correlation coefficient given
by
226
B. G. Shanbhag and Y. R. Suresh
(OP O ) ∗ (OP P ) −
(OP O ) ∗ (OP P )
R = N
(OP)2O − (OP)2O ∗ N
(OP)2P − (OP)2P
N
Coefficient of efficiency,
COE = 1 −
{(OP) O − (OP) P }2
,
2
(OP) O − (OP O
Root mean square error,
RMSE =
((OP) O − (OP) P )2
n
where (OP)
O = observed output, (OP) P = predicted output, n = number of observed
output, OP O = mean of the observed output.
1.7 Garson’s Algorithm
Garson’s algorithm [10, 11] determines the relative significance of each input parameter used to model the system outcome. This method uses the weights of connection
obtained from the artificial neural network. The calculation process required for the
Garson’s algorithm with three input neurons can be summarized as follows. Referring to Fig. 4, 1, 2, 3 are the three input neurons; A, B are two hidden neurons, and
Fig. 4 Artificial neural network structure for the illustration of Garson algorithm
Prediction of Effect of Geometrical Parameters in Trough …
227
O is the output neuron. The hidden layer is selected such that the values of COD and
COE are maximum and the value of RMSE is minimum.
• Input-hidden-output neuron connection weights, i.e., referring to Fig. 4, W A1… ,
W B1… , W OA, W OB are tabulated
• Contribution of input neuron to the output through each hidden neuron, i.e., C A1
= W A1 × W OA
• Relative contribution of input neuron given by
rA1 = I CA1 I /(I CA1 I + I CA2 I + I CA3 I ), rB1,rC1, . . .
• Sum of input neuron contributions, S 1 = r A1 + r B1 + …
• Relative importance of each input parameter is calculated by
R I1 = (S1 /(S1 + S2 + S3 )) × 100
2 Results and Discussion
2.1 Analysis of Folded Plate Roof
Linear static analysis had been carried out for a trough-type folded plate roof of plan
20 m × 20 m. Around 336 models were analyzed under various height, thickness and
bays for both the boundary conditions. The following figures show the displacement
variation with thickness for different height for hinged and fixed boundary conditions.
In the present study, it is observed that:
• For height 1.33 m, for both the boundary conditions, the minimum and maximum
displacements are observed in 5 bay and 2 bay (Ref. Figs. 5 and 6).
• For height 1.00 m, for both the boundary conditions, the minimum and maximum
displacements are observed in 6 bay and 3 bay (Ref. Figs. 7 and 8).
• For height 0.8 m, for both the boundary conditions, the minimum and maximum
displacements are observed in 8 bay and 3 bay (Ref. Figs. 9 and 10).
Tables 3, 4 and 5 refers to the variation of displacement and reduction in percentage
of displacement for different thickness and heights of 1.33, 1.00 and 0.8 m for both
hinged and fixed boundary conditions. Displacement reduces, and hence, increase
in stiffness is observed for all thickness. Reduction in percentage of displacement is
very much helpful in forecasting the economic sections (Table 6).
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B. G. Shanbhag and Y. R. Suresh
Fig. 5 Displacement variation with thickness for H = 1.33 m with hinged boundary condition
Fig. 6 Displacement variation with thickness for H = 1.33 m with fixed boundary condition
Fig. 7 Displacement variation with thickness for H = 1.00 m with hinged boundary condition
Prediction of Effect of Geometrical Parameters in Trough …
Fig. 8 Displacement variation with thickness for H = 1.00 m with fixed boundary condition
Fig. 9 Displacement variation with thickness for H = 0.8 m with hinged boundary condition
Fig. 10 Displacement variation with thickness for H = 0.8 m with fixed boundary condition
229
230
B. G. Shanbhag and Y. R. Suresh
Table 3 Variation of displacement with thickness (75–130 mm) for H = L/15
Number of
Bays
Spacing
between
supports
(m)
Variation of
displacement
(mm) for hinge
condition
% Reduction in Variation of
displacement for displacement
hinge condition (mm) for fixed
condition
% Reduction in
displacement for
fixed condition
2
10
366.17–129.76
64.56
267.61–99.35
62.88
3
6.67
164.39–60.22
63.37
116.38–45.69
60.74
4
5
87.17–36.15
58.53
67.88–29.85
56.03
5
4
58.53–24.87
57.51
46.67–20.92
55.17
Table 4 Variation of displacement with thickness (75–130 mm) for H = L/20
Number of
Bays
Spacing
between
supports (m)
Variation of
displacement
(mm) for hinge
condition
% Reduction
in
displacement
for hinge
condition
Variation of
displacement
(mm) for fixed
condition
% Reduction
in
displacement
for fixed
condition
3
6.67
171.79–65.49
61.88
127.05–51.39
59.55
4
5
96.26–41.49
56.89
78.06–35.13
54.99
5
4
63.81–29.02
54.52
52.83–25.06
52.56
6
3.33
42.26–20.71
50.99
33.24–17.59
47.08
Table 5 Variation of displacement with thickness (75–130 mm) for H = L/25
Number of
Bays
Spacing
between
supports (m)
Variation of
displacement
(mm) for hinge
condition
% Reduction
in
displacement
for hinge
condition
Variation of
displacement
(mm) for fixed
B.C.
% Reduction
in
displacement
for fixed
condition
3
6.67
183.39–72.71
60.35
139.65–58.25
58.29
4
5
107.07–48.28
54.91
88.87–41.44
53.37
5
4
70.83–34.97
50.63
59.97–30.63
48.92
6
3.33
46.78–26.13
44.14
38.27–22.79
40.45
7
2.86
37.78–23.36
38.17
31.61–20.69
34.55
8
2.5
36.34–23.48
35.47
32.38–21.48
33.66
2.2 ANN Modeling of Plate
The output obtained from SAP 2000 analysis is divided into two sets of data for
training and validation in the ratio of 70:30, respectively. Considering height, thickness and number of bays as inputs, the following points were observed in the ANN
modeling for hinged and fixed boundary conditions:
Prediction of Effect of Geometrical Parameters in Trough …
231
Table 6 ANN architecture for fixed and hinged boundary conditions
Parameters
Fixed BC
Hinged BC
σ x (Compressive)
3-2-1
3-8-1
σ x (Tensile)
3-3-1
3-3-1
σ y (Compressive)
3-3-1
3-8-1
σ y (Tensile)
3-6-1
3-7-1
τ xy
3-2-1
3-5-1
3-2-1
3-2-1
σ x = Normal stress along “X” direction (along span); σ y = Normal stress along “Y ” direction
(along width); τ xy = shear stress; = displacement
Fig. 11 ANN structure for the illustration of Garsons algorithm
• The scatter diagrams of the observed versus predicted output values of all the
parameters show that the output can be reasonably well simulated by using the
developed ANN model (Ref. Figs. 11, 12 and 13).
• Based on the results obtained from network modeling, the ANN architecture of
3-3-1 has shown the optimal results for most of the parameters.
• The values of COD and COE are very much closer to 1 for both the boundary
conditions.
• COD nearly equal to 1 in most of the ANN model implies that the dependent
variable can be predicted without much error from the independent variable.
• COE nearly equal to 1 in most of the ANN model shows the high prediction
capability of ANN model.
• The results obtained from Garson’s algorithm calculation indicate that the highest
contribution belongs to the number of bays in both fixed and hinged boundary
conditions, relative importance ranging from 45% to 91% and 47% to 94%, respectively, in fixed and hinged boundary conditions (Ref. Tables 7, 8, 9, 10, 11, 12
and 13).
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B. G. Shanbhag and Y. R. Suresh
Fig. 12 Network performance of displacement for training data
Fig. 13 Network performance of displacement for validation data
ANN architecture of different models with minimum prediction errors parameters
is written in the order of input–hidden nodes–output neurons, i.e., referring to Table 6,
3-number of neuron in the input layer, 3-number of neuron in hidden layers, 1-number
of neuron in the output layer.
Prediction of Effect of Geometrical Parameters in Trough …
233
Table 7 COD and COE of training and validation models for different boundary conditions
Outputs
Fixed boundary condition
Hinged boundary condition
Training
Validation
Training
Validation
COD = COE
COD
COE
COD = COE
COD
COE
σ x (Compressive)
0.992
0.997
0.992
0.959
0.999
0.998
σ x (Tensile)
0.994
0.996
0.992
0.983
0.995
0.991
σ y (Compressive)
0.999
0.994
0.992
0.989
0.953
0.946
σ y (Tensile)
0.989
0.998
0.993
0.986
0.976
0.965
τ xy
0.992
0.997
0.991
0.995
0.985
0.948
0.991
0.994
0.992
0.993
0.997
0.996
σ x = Normal stress along “X” direction (along span); σ y = Normal stress along “Y ” direction
(along width); τ xy = Shear stress; = displacement
COD = Coefficient of determination; COE = Coefficient of efficiency
Table 8 Neuron connection weights
Hidden neuron
A
Height
B
0.245
0.069
Thickness
0.605
0.978
Number of bays
1.421
5.012
Displacement
−0.742
−3.197
Table 9 Contribution of input neuron to the output via hidden neuron
Hidden neuron
A
B
Height
−0.182
−0.221
Thickness
−0.449
−3.127
Number of bays
−1.054
−16.023
Table 10 Relative contribution of input neuron and sum of input neuron contributions
Hidden Neuron
A
B
Height
0.108
0.011
0.119
Thickness
0.266
0.161
0.428
Number of bays
0.626
0.827
1.453
Table 11 Relative importance of input variable
Hidden neuron
Relative importance (%)
Height
5.96
Thickness
21.39
Number of bays
72.65
Sum
234
B. G. Shanbhag and Y. R. Suresh
Table 12 Reative importance of input variables for training and validation data with fixed boundary
condition
Output
σ x (c)
σ x (t)
σ y (c)
σ y (t)
τ xy
Displacement
Relative importance (%)
Height
Thickness
Number of bays
Training
23.53
24.30
52.17
Validation
1.40
22.32
76.27
Training
15.99
23.73
60.28
Validation
3.49
31.06
65.45
Training
0.86
18.77
80.37
Validation
1.14
15.19
83.66
Training
32.57
21.78
45.66
Validation
2.63
12.04
85.33
Training
0.86
7.48
91.65
Validation
1.35
13.34
85.31
Training
5.96
21.39
72.65
Validation
1.05
18.57
80.38
Table 13 Relative importance of input variables for training and validation data with hinged
boundary condition
Output
σ x (c)
σ x (t)
σ y (c)
σ y (t)
τ xy
Displacement
Relative importance (%)
Height
Thickness
Number of Bays
Training
5.63
5.15
89.22
Validation
0.67
5.25
94.08
Training
3.87
5.91
90.22
Validation
1.12
13.91
84.97
Training
6.96
40.34
52.71
Validation
0.73
26.23
73.04
Training
4.63
30.89
64.47
Validation
1.12
29.14
69.73
Training
12.80
28.26
58.93
Validation
2.15
50.62
47.23
Training
5.21
22.71
72.09
Validation
1.09
18.86
80.06
Prediction of Effect of Geometrical Parameters in Trough …
235
3 Conclusions
In the present study, the behavior of trough shape folded plate roof is studied in terms
of displacement by changing various geometrical parameters. Analytical results are
then used to develop artificial neural network model to predict the behavior of folded
plate roof and the relative importance of input variable.
From the analysis of folded plate roof using the software SAP 2000 (v-20), the
following conclusions are drawn:
• Displacement decreases as the thickness increases for all the heights and
corresponding possible number of bays.
• Displacement in the FPR with fixed boundary condition is less when compared
to hinged boundary condition due to the restraint capacity at the boundary.
• As the number of bay increases the percentage reduction in displacement decreases
for both fixed and hinged boundary conditions due to increase in the resisting
capacity at the supports.
• Stiffness and rigidity of folded plate roof increases with increase in thickness and
number of bays which leads to reduction in displacement.
• Stiffness and rigidity of folded plate roof reduces with increase in height of folded
plate roof.
• Economic sections may be selected when reduction in percentage of displacement
is found to be the least.
The following points were observed in the ANN modeling for hinged and fixed
boundary conditions:
• By using the developed ANN model, the output can be reasonably well simulated.
• A very minute difference is observed between the simulated FEM analysis values
and the predicted values, which show good agreement.
• Based on the results obtained from network modeling, the ANN architecture of
3-3-1 is suitable for most of the parameters.
• Results obtained from Garson’s algorithm indicate that the highest contribution
belongs to the number of bays in both fixed and hinged boundary conditions when
compared to thickness and height of FPR.
• Similar studies have been carried out for predicting the stresses whose sample
results are provided in the appendix.
Acknowledgements I would like to express my special thanks of gratitude to the second author
and institute for supporting this work.
Appendix
Sample Design Tables 14, 15 and 16.
236
B. G. Shanbhag and Y. R. Suresh
Table 14 Values of stress for height 1.33 m for hinged boundary condition
n
t = 130 mm
σx
σy
τ xy
C
T
C
T
–
2
202.61
163.62
69.34
54.62
41.21
3
50.92
48.13
54.69
44.63
33.47
4
105.03
74.04
39.65
29.03
27.01
5
88.50
59.38
34.13
23.77
24.25
Table 15 Values of stress for height 1.00 m for hinged boundary condition
n
t = 130 mm
σx
σy
τ xy
C
T
C
T
–
3
49.39
43.88
52.10
41.40
36.03
4
101.75
64.19
37.44
26.47
27.72
5
84.04
49.6
33.44
21.30
24.46
6
26.8
25.27
21.84
30.46
22.17
Table 16 Values of stress for height 0.8 m for hinged boundary condition
n
t = 130 mm
σx
σy
τ xy
C
T
C
C
T
3
48.95
40.87
50.55
39.01
39.29
4
101.96
57.65
40.69
24.68
29.1
5
83.06
43.11
37.18
19.62
25.39
6
24.93
22.69
29.52
19.88
22.9
7
20.87
20.00
27.64
16.81
19.69
8
58.98
23.88
33.41
11.92
21.12
n = number of bay; σ x = longitudinal stress in X direction along span (MPa); σ y = transverse
stress in Y direction along width (MPa); τ xy = shear stress (MPa); C = compression; T = tension
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