Improved Predictive Fundamental Period Formula for Reinforced
Concrete Structures through the Use of Machine Learning
Algorithms
Nathan Carstens
1
, George Markou
1
and Nikolaos Bakas
2
1
Department of Civil Engineering, University of Pretoria, South Africa
2
Department of RnD, RDC Informatics, Athens, Greece
Keywords: Machine Learning Algorithms, Fundamental Mode Formulae, Modal Analysis, Soil-structure Interaction,
Finite Element Method, Reinforced Concrete, Hybrid Modelling.
Abstract: With the development of technology and building materials, the world is moving towards creating a better and
safer environment. One of the main challenges for reinforced concrete structures is the capability to withstand
the seismic loads produced by earthquake excitations, through using the fundamental period of the structure.
However, it is well documented that the current design formulae fail to predict the natural frequency of the
considered structures due to their inability to incorporate the soil-structure interaction and other features of the
structures. This research work extends a dataset containing 475 modal analysis results developed through a
previous research work. The extended dataset was then used to develop three predictive fundamental period
formulae using a machine learning algorithm that utilizes a higher-order, nonlinear regression modelling
framework. The predictive formulae were validated with 60 out-of-sample modal analysis results. The numerical
findings concluded that the fundamental period formulae proposed in this study possess superior prediction
ability, compared to all other international proposed formulae, for the under-studied types of buildings.
1 INTRODUCTION
The soil-structure interaction (SSI) phenomenon is a
typical structural and geotechnical engineering issue,
still open regarding its practical applications. Further
investigation is required to develop simplified but
reliable methods to account for such a phenomenon
in routine structural analyses (Ceroni et al., 2012). In
calculating the appropriate seismic loads, the
fundamental period serves as one of the most critical
dynamic characteristics. In the event of a seismic
excitation, the interaction between the superstructure
(building) and substructure (soil) becomes critical as
it commences to alter the distribution of stresses and
strains within the superstructure, which alters the
expected results (Mourlas et al., 2019).
It is well known that computing the fundamental
mode of fixed-base structures through design code
formulae has its challenges (Mourlas et al., 2019).
Furthermore, some shortcomings exist in the stiffness
distribution of the structure due to a lack of adequate
consideration of the effects of shear walls, especially
in the Eurocode 8 design code (Gravett et al., 2019).
These considerations can cause a considerable
amount of over or under designing of reinforced
concrete (RC) structures, which can lead to
inadequate designs liable to seismic conditions. Thus,
it is crucial to establish a design tool that can
successfully predict the dynamic properties of a
variety of different RC structures.
It is usually not in favour of safety to analyse the
response of a fixed-base structure by neglecting the
SSI effect. In some cases, codes provide seismic
design provisions by reducing the base shear of the
fixed-base structures. In others, they suggest
performing advanced analysis to investigate the
overall effect (Mourlas et al., 2020). As a result, there
is a need for more accurate design expressions for RC
structures that can accurately predict their
fundamental period while accounting for SSI effects.
When it comes to the SSI effect, the reaction of a
building to a seismic event is evaluated in conjunction
with the compressibility of its surrounding soil. The
flexibility of the soil can impact its stress distribution
and displacement profiles, which can be distinguished
from standard fixed-base systems (Saadi, 2018,
Markou et al., 2018).
Carstens, N., Markou, G. and Bakas, N.
Improved Predictive Fundamental Period Formula for Reinforced Concrete Structures through the Use of Machine Learning Algorithms.
DOI: 10.5220/0010984500003116
In Proceedings of the 14th International Conference on Agents and Artificial Intelligence (ICAART 2022) - Volume 2, pages 647-652
ISBN: 978-989-758-547-0; ISSN: 2184-433X
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
647
A study conducted by Gravett et al. (2019)
determined that the current design code formula
assumes that all reinforced concrete structures have a
fixed base, resulting in errors of up to 85% for
international codes such as Eurocode 8. Upon further
investigation, it was concluded that a RC structure's
dynamic response could be significantly affected by
SSI and stiffness redistribution when susceptible to
seismic activity (Mourlas et al., 2019). See Table 1
below for a few design code formulae found in the
international literature.
Table 1: International design codes in practice.
Relevant Code Formulae
NEAK (New Greek
Antiseismic Code)
𝑇

=0.09
𝐻
√
𝐿

𝐻
𝐻+𝜌𝐿
Old Cyprus Code
𝑇

=
𝑁
10
Eurocode
𝑇

=𝐶

𝐻
.
In order to conduct this research report, the
application of finite element modelling, using
advanced modelling software, was utilised to
construct models representing various RC structures.
The finite element method (FEM) is frequently used
in computing engineering, and mathematical models,
the FEM allows for the numerical solution of
differential equations.
For this research project, the constructed FEM
models were analysed using modal analysis to get
results that would help the researcher identify and
understand the dynamic response of the various RC
structures. Eigen-value problems are common in
engineering. The parameter calculates the
fundamental periods of a structural system (Felippa,
2004). The solution method used for this research
project is called the subspace iteration algorithm
(Bathe et al., 1980). The solution is ideally suited for
large-scale structures.
By utilising HYMOD (Markou et al., 2015), one
can decrease the computational demands of the
numerical model, allowing us to perform any type of
analysis of various RC buildings at full-scale. With
Reconan FEA (2020) software, analyses are
performed to capture the complete nonlinear
structural response, either with or without the SSI
effect., while this was software used to perform the
modal analyses. It must be noted here that the modal
algorithm of Reconan FEA was validated through
numerous experimental data.
Based on the procedure described in Taljaard et
al., 2021 and Gravett et al., 2021, developing a dataset
through the use of 3D detailed modeling and then
using machine learning (ML) algorithms to develop
closed form solutions can be a very powerful tool in
developing new fundamental period formulae.
Therefore, the objective of this research work is to
extend the initial dataset developed by Gravett at al.,
2021, and use the extended dataset in developing a
more accurate fundamental period formula.
2 MACHINE LEARNING
This research work used the Julia ML framework.
Similar to Python, this is an open-source, high-level
language for dynamic programming. A mathematical
model in ML is designed to develop generalised
relationships between independent and dependent
variables due to their nonlinear characteristic. As
stated above, the focus of this research work is
developing software generated data that is used to
train ML algorithms to determine the fundamental
period of RC structures.
Table 2, shows the high-order nonlinear
regression algorithm that was used in this research
work to develop the improved formulae. This
algorithm was adopted from Gravet et al., 2021.
Table 2: Higher-Order Nonlinear Regression Algorithm
(Gravett et al., 2021).
Input: XX (matrix of Independent Variables), YY
(Vector of Dependent Variable), nlf (number
of nonlinear features to be kept in the model)
Out
p
ut: Prediction Formulae
1. Create all nonlinear features (anlf)
2. For i from 1 to nl
f
, do:
3. For
j
from 1 to anlf, do:
4. Add
j
th
feature to the model
5. Calculate Prediction Error, MAPE
j
6. END
7. Keep in the model the j
th
feature
which yields the minimum
p
rediction erro
r
8. END
Return: Prediction Formula
3 NUMERICAL CAMPAIGN
3.1 Database Development
In order to construct the extended database that would
consist of various RC structures, various geometrical
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
648
parameters of the initial model were modified. These
parameters include the height of the structures, base
conditions, stiffness distribution throughout the
structure, and the plan area of the structure. Figures 1
and 2 show different RC building models that were
created for the needs of the dataset development.
With all the models created and the parameters for
each model determined, the eigenfrequencies were
determined, and all the data were stored in an Excel
spreadsheet. Table 3 summarises the minimum and
maximum geometrical properties for the models that
were adjusted in this research work.
Figure 1: Different model geometries.
Figure 2: Different model base conditions.
Table 3: Minimum and maximum values of the newly
obtained HYMOD meshes.
Variables Minimum Maximum
Soil Depth (m) 1 60
Soil E (kPa) 65,000 700000
H (m) 3 30
L (m) 3.4 34.4
B (m) 3.4 34.4
%
0 85.29
3.2 Modal Analyses
For each of the numerical models, a modal analysis
was performed to determine the eigenfrequencies of
each model. For each model, only the two
translational modes were used to construct the
dataset. Translation oscillations along the global x-
and y-axis directions.
Figure 3 shows the effect of shear walls on the
computed fundamental period of the under-study RC
structures. It is evident that when shear walls are
added to the RC buildings they significantly lower the
fundamental period of the structure. Figure 4 shows
the relationship of the period of a structure and how
it is affected by the soil depth. It is easy to observe
that the SSI effect reaches a plateau as the depth
increases.
Figure 3: Shear wall effect on fundamental period.
Figure 4: Soil depth effect on the fundamental period.
3.3 Proposed Fundamental Period
Formulae
The ML algorithm is designed to determine the
number of features used within the design formulae.
For this research work, a design formula with 3, 5,
and 20 features were developed and parametrically
investigated. It must be noted here that the total
number of fundamental period results used in the
dataset to train and test was 790. Each fundamental
period formula is constructed through the use of the
following variables:
H the building's height (m)
ρ the percentage shear walls (%)
E
s
the soils' modulus of elasticity (kPa)
L the length of building parallel to the
oscillating direction (m)
B the width of the building perpendicular to the
oscillating direction (m)
D the soil depth (m)
0,1
0,3
0,5
0,7
0,9
1,1
5 152535
Period [s]
Structural Height [m]
No Shear Walls
Added Shear Walls
0,552
0,554
0,556
0,558
0,56
0,562
0 20406080
Period [s]
Soil Depth [m]
Improved Predictive Fundamental Period Formula for Reinforced Concrete Structures through the Use of Machine Learning Algorithms
649
3.3.1 3-Feature Formula
It should be noted that the three feature formulae do
not consider any SSI parameters. However, this
formula still yielded an absolute mean error of 3.43%.
The relationship can be seen in Eq. 1.
𝑇=

0.0310197 ∙ 𝐻

−

0.00011254 ∙ 𝜌 ∙ 𝐻

+

0.0000129093 ∙ 𝐻∙ 𝐵


+ 0.0110165
(1)
Figure 5: 3-Feature formula prediction vs numerically
predicted results.
Figure 6: 5-Feature formula prediction vs numerically
predicted results.
3.3.2 5-Feature Formula
Eq. 2 shows the 5-feature formula as it derived from
the training and testing of the extended dataset. The
absolute mean error for the 5-feature formula was
calculated as 2.70%, which is more accurate
compared to the 3-feature formula. This is attributed
to the inclusion of additional parameters that affect
the final predictions.
𝑇=

0.0296602 ∙ H

−

0.000154717 ∙ ρ ∙ H

+

0.0000210854 ∙ L ∙ B ∙ H

+

0.0000042983 ∙ 𝐻

∙
ρ

−

0.00000785228 ∙ ρ ∙ B ∙ H

+ 0.0229
(2)
3.3.3 20-Feature Formula
Finally, the most accurate formula is presented in Eq.
3. The absolute mean error of the 20-feature formula
was calculated as 1.49%. It is evident that the use of
SSI related parameters in this relationship, makes this
formula the most accurate when used on the training
and testing datasets. Fig. 7 shows the comparison
between the predictions derived from the proposed
formula and the numerical results.
𝑇=

0.0292939 ∙ H

−

0.000150825 ∙ ρ ∙ H

+

0.00000582242 ∙ H ∙ B


+

0.00000330369 ∙ ρ ∙
𝐻


+

0.000215881 ∙ H ∙ L

− 1.89375x10

∙
𝐸


∙D+

0.00000323855 ∙ L ∙ H ∙D

−

0.00000646154 ∙ ρ ∙ B ∙ H

−

0.0000000000925478 ∙ 𝐻∙ 𝐸

∙𝐷

−

0.0000000000406192 ∙ 𝜌∙ 𝐸

∙𝐷

+

0.000000194394 ∙ 𝐷∙ 𝜌


+

0.0037148 ∙ 𝐵

+

0.000000358861 ∙ 𝜌 ∙𝐻∙ 𝐷

+

0.0000000000662381 ∙ 𝐸

∙𝐷


−

0.000000278639 ∙ 𝐷


−

0.000000000113737 ∙
𝐿∙𝐸

∙𝐷

−

0.0000016727 ∙ 𝐵


+

0.0000309934 ∙ 𝐿∙ 𝐷

−

0.00178654 ∙ 𝐿

+

0.000000645744 ∙ 𝐿


+ 0.00239996
(3)
Figure 7: 20-Feature formula prediction vs numerically
predicted results.
4 VALIDATIONS OF RESULTS
To further test the ability of the proposed formulae to
predict the fundamental period of RC structures, a
dataset was developed for validation purposes. For
this reason, 60 out-of-sample building models were
constructed and used to further validate the ability of
the proposed formulae in predicting the fundamental
period of RC buildings with and without SSI effects.
From Figures 8 - 10 it is evident that comparing
the proposed formulae manage to predict the out-of-
sample data with high accuracy. The most significant
improvement was seen with the 3-feature proposed
0
0,2
0,4
0,6
0,8
1
1,2
00,511,5
Formula Predicted Values [s]
Numerically Predicted Values [s]
0
0,2
0,4
0,6
0,8
1
1,2
0 0,5 1 1,5
Formula Predicted Values [s]
Numerically Predicted Values [s]
0
0,2
0,4
0,6
0,8
1
1,2
0 0,5 1 1,5
Formula Predicted Values
[s]
Numerically Predicted Values [s]
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
650
formula that improved with 4.19% from the results
obtained by Gravett et al., (2021). The 20-feature
formula also had a significant improvement of 0.22%.
Figure 8: Correlation of 3-feature formula on validation
dataset.
Figure 9: Correlation of 5-feature formula on validation
dataset.
5 CONCLUSIONS AND FUTURE
WORK
790 fundamental period results were used to train an
ML algorithm and develop accurate design formulae
to calculate the fundamental period of RC structures.
The three proposed formulae were then tested with
out-of-sample data comprising 60 new RC models
constructed in a manner that foresaw the use of new
parameters compared to the models used to train and
test the formulae. This served as the validation phase
in which the design formulae showcased a high
degree of correlation, effectively proving their
accuracy and extendibility.
According to the numerical investigation, the
most accurate proposed formula on train, test and
validation data was the 20-feature, which was found
to be improved compared to the one proposed by
Gravett et al., 2021.
Figure 10: Correlation of 20-feature formula on validation
dataset
Finally, this research work will foresee a further
dataset extension and also take into account the infill
walls of RC buildings. The asymmetry of buildings
should also be investigated in the near future and how
that affects the fundamental period of RC structures
when the SSI effect is accounted for.
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y = 0,9873x + 0,041
R² = 0,9789
0
0,2
0,4
0,6
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1
1,2
00,511,5
Formula Predicted Period [s]
Numerically Predicted Period [s]
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0
0,2
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1
1,2
0 0,5 1 1,5
Formula Predicted Period [s]
Numerically Predicted Period [s]
y = 1,0159x + 0,0267
R² = 0,9888
0
0,2
0,4
0,6
0,8
1
1,2
00,511,5
Formula Predicted Period [s]
Numerically Predicted Period [s]
Improved Predictive Fundamental Period Formula for Reinforced Concrete Structures through the Use of Machine Learning Algorithms
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