Using Machine Learning and Finite Element Modelling to Develop a
Formula to Determine the Deflection of Horizontally Curved Steel
Elvis M. Ababu
, George Markou
and Nikolaos Bakas
Department of Civil Engineering, University of Pretoria, Lynnwood Road, Pretoria, South Africa
Department of RnD, RDC Informatics, Athens, Greece
Keywords: Curved Beams, Machine Learning, Steel, Finite Element Method, Design.
Abstract: The use of curved I-beams has been increasing throughout the years as the steel forming industry continues
to advance. However, there are often design limitations on such structures due to the lack of recommendations
and design code formulae for the estimation of the expected deflection of these structures. This is attributed
to the lack of understanding of the behaviour of curved I-beams that exhibit extreme torsion and bending.
Thus, currently, there are no formulae readily available for practising engineers to use to estimate the
deflection of curved beams. Since the design of light steel structures is often governed by serviceability
considerations, this paper aims to analyse the properties of curved steel I-beams and their impact on deflection
as well as develop an accurate formula that will be able to predict the expected deflection of these beams. By
using a combination of an experimentally validated finite element modelling approach and machine learning.
Numerous formulae are developed and tested for the needs of this research work. The final proposed formula,
which is the first of its kind, was found to have an average error of 4.11% in estimating the midspan deflection
on the test dataset.
A beam is a structural element that has been studied
extensively for numerous years and is used widely,
not only in the field of civil engineering but
mechanical engineering as well. The use of curved
steel I-beams have been increasing steadily
throughout the years due to the aesthetically pleasing
designs that they produce in buildings as well as for
industrial applications. Horizontally curved steel I-
sections can be found in various applications ranging
from girders in modern highway bridges, interchange
facilities as well as in industrial buildings where they
are used as crawl beams. Due to advancements in
steel forming technology, almost any section can be
curved with minimum limitations that relate to the
beam’s section size or length. Numerous studies have
been conducted in efforts to understand the behaviour
of curved beams based on different materials as
discussed in (Al-Hassaini, 1962), (McManus et al.,
1969), (Hsu et al., 1978, Mansur and Rangan, 1981),
(Al-Hashimy and Eng, 2005), (Chavel, 2008) and
(Lee et al., 2017). It is widely known that, unlike
straight beams, horizontally curved beams subject to
gravity loads experience a complex system of
combined effects, namely, shear, flexure and torsion.
This loading effect causes a complex structural
response such as warping due to non-uniform torsion.
The study of curved girders began with Umanskii
(1948) who obtained solutions for curved beams
based on several loading conditions by assuming
initial parameters in the solution procedure (Liew et
al., 1995). Numerous other studies and experiments
have been conducted from the 1960s to the present
day, however, these studies mainly focused on the
ultimate limit states of the beam and not on the
serviceability limit states. Those that did focus on
serviceability limit states design, did not analyse the
effect various parameters have on deflection, whereas
those that did analyse the effect on deflection did not
determine an appropriate formula without using
Castigliano’s second theorem. Currently, most design
codes (SANS, AISC, Eurocode 3, etc.) do not have a
section detailing deflection estimation of curved steel
I-beams. AISC however does state that the usage of
finite element modelling (FEM) is appropriate when
Ababu, E., Markou, G. and Bakas, N.
Using Machine Learning and Finite Element Modelling to Develop a Formula to Determine the Deflection of Horizontally Curved Steel I-beams.
DOI: 10.5220/0010982400003116
In Proceedings of the 14th International Conference on Agents and Artificial Intelligence (ICAART 2022) - Volume 3, pages 958-963
ISBN: 978-989-758-547-0; ISSN: 2184-433X
2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
deflection results are of importance (Dowsell, 2018).
The issue arises when Engineers conduct a linear
static analysis on the structure as a whole and check
deflections in that manner. The issue with that
approach is that curved beams experience large
deflections, which leads to nonlinear strain-
displacement and curvature-slope relations, a
phenomenon that necessitates accounting for
geometric nonlinearities. Another approach would be
to use Castigliano’s second theorem which is
numerically tedious (Dahlberg, 2004). Nevertheless,
researchers have attempted to provide a formula for
deflection of curved steel I-beams based on
Castigliano’s second theorem, however, this formula
is often not readily available to practising engineers
and is only applicable for specific support and loading
This research work attempts to derive an easy to
use formula to determine the midspan deflection of
real cantilever types of curved steel I-beams with
loads applied only at the midspan. A machine
learning algorithm that uses nonlinear regression is
implemented herein to create a closed-form equation
based on midspan deflection results, a dataset
developed through the use of Reconan FEA (2020)
(Mourlas and Markou, 2020). Prior to the
development of the models that were used to train the
machine learning algorithm, the software was
validated through the use of experimental results
found in (Shanmugam et al., 1995). It should be noted
herein that the final curved beam properties must be
used because typically beams are curved through a
cold process known as pyramid roll bending which
alters the material properties of the steel (Dowsell,
2018). After developing different formulae through
training the machine learning algorithm on the
developed dataset, an investigation on the prediction
ability of each formula was performed. The factors
considered for developing the formulae that foresaw
different numbers of features, were: section area (A),
curved length of the beam (L), the radius of the beam
(R), yielding strength (f
), Young’s modulus (E).
Machine learning has grown in popularity in recent
years in numerous scientific fields, including civil
engineering. The power of machine learning
algorithms is that even though complex nonlinear
models seem to present no correlation in small sample
spaces, by considering a large enough sample a
pattern emerges, which can then be exploited to
generate a formula to predict out-of-sample data.
Various machine learning and artificial intelligence
methods are applied for engineering problems such as
Random Forests, Gradient Boosting and Artificial
Neural Networks, amongst others. The issue with
such models is that their results often cannot be
interpreted in practical cases unless integrated into
some software. In this work, a higher-order nonlinear
regression modelling framework was utilized
(Gravett et al., 2021) due to its ability to provide an
explicit closed-form formula. The model is based on
the creation of nonlinear terms based on the
independent variables up to the third degree. The
algorithm automatically selects the nonlinear features
which would correspond to the minimum error. The
algorithm found in Table 1 represents the procedure
that would lead to the development of the desired
Table 1: 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 ke
t in the model
Output: Prediction Formulae
1. Create all nonlinear features
2. For i from 1 to nlf, do:
3. For
from 1 to anl
, do:
4. Add
feature to the model
5. Calculate Prediction Error, MAPE
6. END
7. Keep in the model the j
which yields the minimum
rediction erro
8. END
Return: Prediction Formula
This section presents the procedure adopted to
generate the dataset that will be presented thereafter.
Initially, a mesh sensitivity analysis was performed to
determine the optimum mesh size, where the ability
of Reconan FEA (2020) to calculate the mechanical
response of these types of beams is investigated.
Then, beams with varying geometrical and material
properties were developed and analysed for
constructing the dataset that was used to train the
machine-learning algorithm presented in the previous
section. Finally, the predictive models that were
developed as closed-form formulae, were
parametrically tested on additional data to determine
Using Machine Learning and Finite Element Modelling to Develop a Formula to Determine the Deflection of Horizontally Curved Steel
their ability to predict the deformation at the midspan
of the curved steel I-beams.
3.1 Mesh Sensitivity and Validation
A mesh sensitivity analysis was conducted to
determine a mesh size that derives the optimal results
of an experiment found in Shanmugam et al. (1995).
It was decided to compare 8 noded and 20 noded
hexahedral elements with element sizes of 20, 30, 50,
and 100 mm hexahedral element sizes for this
investigation. The developed models can be seen in
Figure 1. The numerically obtained curves were
compared with the experimental curve found in
Shanmugam et al. (1995).
Figure 2 shows the experimental setup of the
beam tested by Shanmugam et al. (1995) and
investigated numerically herein. As it can be seen, the
curved length of the beam (L) is 5 m, the point of load
application (L
) 3.8 m from the left support and the
radius (R) is 20 m.
Figure 1: Mesh sensitivity analysis models with mesh sizes,
(i) 20mm, (ii) 30mm, (iii) 50mm, (iv) 100mm.
Figure 2: Experimental setup (Shanmugam et al., 1995).
Nonlinear analyses were performed using Reconan
FEA (2020) that foresaw 100 load increments and an
energy convergence tolerance of 10
. It is important
to note that only material nonlinearities were
accounted for in the developed models through
adopting the von Mises yielding criterion. The final
results of the mesh sensitivity analysis can be seen in
Figure 3 and Table 2. It is easy to depict that most
beams managed to come close to the ultimate load as
found in the experimental data, however, the beams
with the coarse mesh failed to accurately model the
ductility of the steel and in general reproduce the
overall mechanical response of the curved beam.
Figure 3: Graphically comparing load-deflection results of
various mesh sizes against experimental data.
Table 2: Numerical comparison of the various mesh sizes
against experimental data.
The 50 mm, 20-noded isoparametric hexahedral
finite elements derived the best results with a 1.5%
0 20406080
Load (kN)
Deflection (mm)
8 noded with 0.1m elements
20 noded with 0.1m elements
8 noded with 0.05m elements
20 noded with 0.05m elements
8 noded with 0.03m elements
20 noded with 0.03m elements
8 noded with 0.02m elements
20 noded with 0.02m elements
Experimental data
8 noded with 0.1m elements 1.376 8.588 00:21.9
20 noded with 0.1m elements 1.108 3.049 01:22.6
8 noded with 0.05m elements 1.813 5.040 01:04.5
20 noded with 0.05m elements 0.985 1.199 05:16.3
8 noded with 0.03m elements 0.950 1.288 06:17.0
20 noded with 0.03m elements 1.425 2.371 15:12.3
8 noded with 0.02m elements 0.985 1.218 16:20.9
20 noded with 0.02m elements 1.662 2.874 03:06:05
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
error in predicting the ultimate failure load, less than
20% error for capturing the max deflection at failure
and 6.33% error in estimating deflection at 50% of the
total load. It is evident that this experiment did not
develop any local or global buckling prior to ultimate
failure, thus the nonlinear detailed model was able to
capture the overall mechanical response with
acceptable accuracy.
3.2 Finite Element Models
Figure 4 shows the 20-noded hexahedral mesh of one
of the beams that were analysed in this paper for
different material properties, while other beams were
developed with different geometric properties. It was
chosen to place a fixed support on the left side of the
beam and a roller support on the right side of the beam
which closely represents the boundary conditions as
experienced by crawl beams in industrial buildings.
The load was placed at the centre span of the beam
and were divided into 100 load increments to
accurately determine the midspan deflection. The
loads were placed on the web of the member to avoid
local failure of the flange.
Numerous beams were created to make sure the
formula encompasses a large variety of parameters.
The parameters considered in this study included the
curved length of the beam (L), the radius of the curve
(R), the Young’s modulus of steel (E), and the section
sizes. It is also noted that specific combinations of
these input parameters are of significance in
analysing curved beams, specifically, the R/L ratio
which is an indication of the amount of curvature a
beam experiences. This shall be considered in the
sensitivity analysis later but will lead to
complications when developing the formula. A total
of 270 beams were initially created, where, 3 I-beam
sections were considered, namely a 305x165x46UB,
533x210x82UB and a 203x133x25UB. These beam
sections represent the extreme ends of I-beam
sections that are commercially available in South
Africa as well as an intermediate beam size. All
beams were analysed until failure, where the ultimate
load was recorded. Various points along the elastic
load-deflection diagrams were considered which lead
to a total number of 1890 unique data points to train
the machine learning algorithm. To ensure that the
points were on the elastic region, the points that were
selected foresaw a maximum of 50% load level of the
maximum load level as computed from the analysis.
A statistical summary of the various properties
can be seen in Table 3. Where A is the section area in
and I
is the second moment of area of the cross-
section about the x-axis in mm
. A large number of
radii were considered as this was hypothesised to be
the parameter to have a significant impact on the
Figure 4: General beam undeformed mesh.
Table 3: Statistical summary of independent variables.
mean 2.02x10
5.02 58.99 205.10
std 1.99 x10
1.64 47.55 8.38
median 9.93 x10
5.00 60.00 205.10
min 2.35 x10
3.00 3.50 194.85
max 4.75 x10
7 140.00 215.36
Figure 5: General beam deformed shape and Von Mises
stress contour.
Figure 5 shows the general deflected shape that the
beam experienced as well as the von Mises stresses
developed throughout the beam at 50% of the
ultimate load. It is easy to observe that torsion
dominates the beam while controlling the type of
failure. This was expected due to the eccentricity of
the load and shows that the mechanical behaviour of
the beam is as expected. The maximum stress
experienced was close to the pin-like support, where
the failure occurred at the bottom flange of the section
due to excessive stresses. This was a mechanical
response that was noted in all understudy curved steel
I-beams. The beams that had a low R/L ratio (beams
with lower curvature) failed close to the pin-like
Using Machine Learning and Finite Element Modelling to Develop a Formula to Determine the Deflection of Horizontally Curved Steel
Figure 6: Correlation between numerically determined deflection and (i) 40 term, (ii) 25 term, (iii) 20 term, (iv) 15 term, (v)
13 term, (vi) 10 term, (vii) 7 term and (viii) 5 term formulae.
support, while failure occurred at the web and not
within the flange. It should, however, be noted that
failure analysis was not the subject of this research
work but rather, the deflection experienced during the
loading within the elastic region, which is what
interests Civil Engineering designers.
Once the models were analysed, a numerical
database was created that contained the parameters of
each beam and the respective midspan deflections
(one row per deflection point). This numerical
database was then used to train the machine learning
algorithm, which was able to provide formulae as
well as other descriptive statistics about the training
and testing process (sensitivity analysis that will be
presented in an extended version of this manuscript).
When generating the formulae, as was seen in the
machine learning section of this manuscript, the beam
features are considered as input parameters. A range of
5 to 40 features were considered when training the
formulae and the various results were compared to
determine the most accurate formula in terms of
predictability. Figure 6 shows the graphs comparing
the generated formulae to the numerically computed
deflection. As can be seen, there is a strong correlation
between the numerically determined deflections and
the deflections predicted by the various formulae.
In order to quantify the accuracy of the proposed
formula on the training and test set, several error
metrics were used, namely, the correlation coefficient
), alpha metric (α), the root mean square error
(RMSE), the mean absolute error (MAE), the mean
absolute percentage error (MAPE), the max absolute
percentage error (MAXPE), and the quotient error
It was seen that as the number of features
decreases, the error increases, a numerical response
that is in line with the finding reported by Gravett et
al. (2021). Some error metrics, such as the r
, barely
change as the number of features changes (the r
was seen to be approximately 0.95 regardless of the
number of features), however, other metrics such as
MAXPE change drastically as the number of features
increase (MAXPE ranged from 1.9607 when 5
features were used to 0.7850 when 40 features were
used). This shows the importance of considering
numerous error metrics in determining the optimum
formula that will show advanced predictive
capabilities. It must be noted here that the dataset was
divided into 85% and 15%, training and testing data,
4.1 Validation
After training and testing the numerous formulae that
were discussed in the previous section, the proposed
formulae were then further validated by creating an
out of sample model and estimating the midspan
deflection. The beam considered was a 305x127x42
UB with a curved length of 5 m and a radius of 20 m.
The Young’s modulus of the beam was also varied
(205.1 GPa, 200 GPa and 210 GPa). Various ultimate
load percentages were also considered in this
validation process, where a total of 15 new data points
were developed for the needs of this investigation. It
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
was found that the function that consisted of 10
features derived the lowest average error, of only
4.11% in estimating the deflection of 15 out-of-
sample data points, while the model with 40 features
resulted in the largest error of 13.05%. This is a
numerical phenomenon that is usually attributed to
overfitting during the training and testing procedure.
The proposed formula used to estimate the
deflections that consisted of 10 terms can be seen in
Equation 1. The various independent variables are E
which is the Youngs modulus in GPa, L is the curved
length of the beam in metres, A is the section area in
, f
is the yielding stress, I
is the second moment
of area about the strong axis in mm
and Q is the
percentage of ultimate loading applied on the beam as
a number (50% = 50). The resulting deflection from
the formula is in mm.
𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 = 1.08128 10
𝑄 𝐿 8.41124
𝑄 𝐿 𝐴 9.81969
𝑄 𝐸 𝑅 + 1.82604
𝑄 𝑅 𝑅 2.71029
𝑄 𝑅 𝐿 + 6.22991
𝑄 𝐿 𝐿 + 2.55963
𝑄 𝐿 𝐼𝑥𝑥 + 2.99150
𝑄 𝑄 𝑓𝑦 + 4.19580
𝑄 𝑓𝑦 𝑓𝑦 1.07754
𝑄 𝐸 𝐿 8.95792
A formula was successfully developed for the
prediction of the deflection of curved steel I-beams.
When comparing the proposed formula with the out-
of-sample data, it was found that the formula
containing 10 features was the most accurate, having
an average error of 4.11%, while the formula with 40
features was the least accurate having an error of
13.05%. The lack of accuracy in the 40 feature
equation was attributed to an over-fitting
phenomenon but can also be attributed to another
phenomenon known as the “interaction effect”, which
can greatly increase the effect of the independent
variables on the dependent variable.
Based on the parametric and sensitivity
investigation, it was concluded that the variables with
the largest impact on deflection are the curved length
and radius of the beams. Due to the page limitations
of this manuscript, the results of the in-depth
sensitivity analysis could not be shared, however,
these will be published at a later stage. Even though
the results of this study are positive seeing as very low
error metrics were observed, the study has to be
expanded in the future by developing additional
models with a larger spectrum in terms of geometries.
Various boundary conditions, as well as different
yield strengths of steel, will also be considered.
Experimental curved steel I-beams will also be tested
to validate the proposed formula developed in this
study. Future research work will foresee the
development of similar formulae on curved concrete
beams. Finally, the long-run objective is to develop
machine learning models that will be able to evaluate
the response of full scale-structures
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Using Machine Learning and Finite Element Modelling to Develop a Formula to Determine the Deflection of Horizontally Curved Steel