Studia Geotechnica et Mechanica, Vol. 38, No. 2, 2016
DOI: 10.1515/sgem-2016-0017
APPLICATION OF ARTIFICIAL NEURAL NETWORKS
TO PREDICT THE DEFLECTIONS
OF REINFORCED CONCRETE BEAMS
MATEUSZ KACZMAREK, AGNIESZKA SZYMAŃSKA
Wrocław University of Science and Technology, Wrocław, Poland,
e-mail: mateusz.kaczmarek@pwr.edu.pl, a.szymanska@pwr.edu.pl
Abstract: Nonlinear structural mechanics should be taken into account in the practical design of reinforced concrete structures.
Cracking is one of the major sources of nonlinearity. Description of deflection of reinforced concrete elements is a computational
problem, mainly because of the difficulties in modelling the nonlinear stress-strain relationship of concrete and steel. In design prac-
tise, in accordance with technical rules (e.g., Eurocode 2), a simplified approach for reinforced concrete is used, but the results of
simplified calculations differ from the results of experimental studies.
Artificial neural network is a versatile modelling tool capable of making predictions of values that are difficult to obtain in nu-
merical analysis. This paper describes the creation and operation of a neural network for making predictions of deflections of rein-
forced concrete beams at different load levels. In order to obtain a database of results, that is necessary for training and testing the
neural network, a research on measurement of deflections in reinforced concrete beams was conducted by the authors in the Certified
Research Laboratory of the Building Engineering Institute at Wrocław University of Science and Technology. The use of artificial
neural networks is an innovation and an alternative to traditional methods of solving the problem of calculating the deflections of
reinforced concrete elements. The results show the effectiveness of using artificial neural network for predicting the deflection of
reinforced concrete beams, compared with the results of calculations conducted in accordance with Eurocode 2. The neural network
model presented in this paper can acquire new data and be used for further analysis, with availability of more research results.
Key words: reinforced concrete beams, research, deflection, artificial neural network
1. INTRODUCTION
From a structural analysis and design point of
view, reinforced concrete is a very complex composite
material. It is a combination of two materials (con-
crete and steel) with entirely different mechanical
properties. Due to the nonlinear stress-strain relation-
ship of concrete and steel reinforced concrete cannot
be modelled properly by linear elastic behaviour.
Moreover, due to the cracking of concrete, even the
sectional and therefore the structural properties de-
pend on the nature and magnitude of the applied
loads. Cracking of concrete is a significant phenome-
non, as the maximum bending moment is usually sev-
eral times greater than bending moment that causes
cracking. Cracks in structural elements cause a change
of moment of inertia and therefore the stiffness degra-
dation of the element. The cracked reinforced concrete
element shows cracks at certain distances from each
other, and their number is finite.
As the load increases the initial distribution of
stiffness of the element changes and the number of
cracks varies nondeterministically. The distribution of
strains and stresses in concrete and steel, along the
axis of the element, is irregular.
Accurate determination of deflection of reinforced
concrete elements is a computational problem, mainly
because of the difficulties in modelling of the nonlin-
ear stress-strain relationship of concrete and steel.
There are some scientific publications, where hetero-
geneous stiffness of the reinforced concrete element is
described by using continuum function (linear or non-
linear) [1], as well as using constant value of stiffness
in each section of the element [2]. The inelastic char-
acteristics of the structure can be taken into account in
the form of spot-localized defects, described by the
application of distribution calculus [3].
In design practice, when the deflection of a rein-
forced concrete beam is calculated, according to the
standards [4], [5], two states are analysed: cracked
state and uncracked state. The flexural stiffness of
M. KACZMAREK, A. SZYMAŃSKA38
a reinforced concrete beam changes from the un-
cracked state to the cracked state. The equation for the
effective moment of inertia is used to calculate a mo-
ment of inertia somewhere between the uncracked
moment of inertia and the cracked moment of inertia
depending on the applied load. The resulting effective
moment of inertia can be used in the elastic deflection
equations to approximate the actual deflections.
However, this is a simplified approach, which does
not allow the deflection of an RC element to be pre-
cisely calculated. According to experimental research
on bending reinforced concrete beams [6], that was
conducted by Kubicki, the difference in mean values
of deflections calculated according to [5] and experi-
mentally obtained was 21%, with a coefficient of
variation v = 22.6%.
The artificial neural network (ANN) can be an al-
ternative tool to accurately estimate the deflection of
reinforced concrete beams. ANN as a modelling tool
is suitable for producing prediction systems (such as
the prediction of deflections) based on a set of data
available from real world observations and experi-
ments. ANN uses the independent parameters as input
and predicts the dependent parameters as output. This
requires such training of the network that the resultant
errors are minimised.
This paper presents the application of artificial
neural network in predicting the deflection of rein-
forced concrete beams, as an effective tool for the
analysis of issues in the field of reinforced concrete
structures.
2. ARTIFICIAL NEURAL NETWORKS
AS A TOOL FOR PREDICTION
Scientists have always endeavored to develop
mechanism inspired by a human brain which is capa-
ble of machine learning as well as pattern recogni-
tion. As an effect of this work artificial neural net-
works were created. The history of this discipline
begins with the development of the first artificial
neuron by McCulloch and Pitts in 1943 [7]. Today
artificial neural networks are widely used in classifi-
cation, robotics, data processing and sequence recog-
nition. Main applications are interpolation, approxi-
mation, prediction and grouping. In literature, there
are plenty of examples how to use neural networks
for solving engineering problems such as interpreta-
tion of the results of nondestructive testing [8], plan-
ning of the construction processes [9] or geotechni-
cal problems [10].
Neural networks can be an alternative for predic-
tion of the behaviour of structural elements. Some of
the applications of neural networks in literature, in
the field of structural engineering, include prediction
of various structural quantities [11], [12]. Papers
[13], [14] present the application of ANNs to predict
bending moment in continuous composite beams.
There are some papers that present the application of
artificial neural networks to predict the deflection
of structural elements. Neural networks have been
used for prediction of deflection in steel-concrete
composite bridges incorporating flexibility of shear
connectors, shear lag effect and cracking in concrete
slabs [15]. Paper [16] presents the application of ANN
to predict deep beam deflection using experimental
data from eight high-strength-self-compacting-concrete
(HSSCC) deep beams. These studies reveal the
strength of neural networks in predicting the solutions
of different structural engineering problems.
Neural network can consist of many neurons
grouped in different count of layers. Layer count is
determined by the complexity of the problem to solve
[17]. Every artificial neuron receives one or more
inputs and sums them to produce an output. Usually
the sums of each node are weighted, and the sum is
passed through a function known as an activation
function. Neural networks are trained using different
algorithms such as variable metric methods or back
propagation. During the training of different inputs,
the weight values are changed dynamically until their
values are balanced, so each input will lead to the
desired output. Different measures are used to evalu-
ate the efficacy of the neural network. The most
popular one is MSE – mean squared error (1) or
RMSE – root mean squared error (2). The error is
calculated simultaneously for training and testing data
in the course of the training process.
,)(
1
)(MSE
1
2
P
i
ii
yz
P
P (1)
,)(
1
)(RMSE
1
2
P
i
ii
yz
P
P (2)
where
y
i
– predicted values of output, i = (1, ..., P),
z
i
– actual (measured) values of output,
P – count of elements in database.
Mainly used neural network architecture is MLP
– Multi-Layered Perceptron. The
foundation of this
networks training
is the back propagation algorithm.
MLP networks can estimate many complex mappings.
Network structure is shown in Fig. 1.
Application of artificial neural networks to predict the deflections of reinforced concrete beams 39
The input of the network is vector x. It is multi-
plied by the weight matrix of the hidden layer
IW. In
the next step bias
b
h
is added to the result vector. Af-
ter that hyperbolic tangent activation function is used.
It can be described using the following equation
)tanh(
hh
bxIWy
.(3)
Vector y
h
(3) is the output of the hidden layer. It is
multiplied by the weight matrix of the output layer
LW. Analogously bias b
0
is added. The only differ-
ence is the activation function – in that case a linear
function is used. Final output of the network is given
by the equation
0h
byLW y .(4)
3. RESEARCH
3.1. PURPOSE, SCOPE
AND RESEARCH PROGRAM
Experimental research on bending reinforced con-
crete beams was conducted in order to provide a data-
base used for training and testing artificial neural net-
work. The analysis of three beams on a lab scale had
been planned beforehand.
Experimental investigations carried out on RC
beams were preceded by an experimental determina-
tion of material properties, which included:
determining the average value of Young’s modulus
of reinforcing steel,
determining the average value of Young’s modulus
of concrete.
Properties of reinforcing steel were determined in
the state of axial tension and PN-EN10002-1:2004
standard [18] was followed. The average value of
Young’s modulus of concrete was determined as a result
of cyclic loading of cylindrical samples. Material
properties were determined one day prior to the re-
search on RC beams.
3.2. RESEARCH ON REINFORCED
CONCRETE BEAMS
Three reinforced concrete beams with rectangular
cross-section (100
200 mm) and the same length
were prepared for the tests. RC beams differed in the
degree of reinforcement. There were two tensile rein-
forcement rods with a diameter of 10 mm (beam
Fig. 1. The structure of MLP neural network used for prediction
Fig. 2. Beam B-001
M. KACZMAREK, A. SZYMAŃSKA40
B-001), 12 mm (beam B-002) and 14 mm (beam
B-003). The beams that were prepared for the tests are
presented in Figs. 2–4.
The study assumed a static diagram of a simply
supported beam. Steel rollers provided freedom of
rotation on the ends of the beam. Research methodol-
Fig. 3. Beam B-002
Fig. 4. Beam B-003
Fig. 5. Test stand. EI – stiffness of supporting beam, EI
B
– stiffness of test beam
Fig. 6. Scheme of the measurement system
Application of artificial neural networks to predict the deflections of reinforced concrete beams 41
ogy assumed a three-point bending scheme. A scheme
of the test stand is presented in Fig. 5.
Deflections were measured by using inductive
sensors, accurate to 0.001 mm. A scheme of the
measurement system is shown in Fig. 6.
Deflections were measured for each load level. At
the next load level the force was increased by 0.4 kN.
Finally, there were measured 293 values of deflection
(74 values of deflection of beam B-001, 96 values of
deflection of beam B-002 and 123 values of deflection
of beam B-003). Table 1 presents the tabulation of
measured values of deflection.
Table 1. Measured values of deflection
Number
of test
Force
[kN]
Bending moment
M [kNm]
Deflection
a [mm]
Beam B-001
1 0.5 0.238 0.057
2 1.0 0.475 0.106
3 1.5 0.713 0.152
... ... ... ...
72 36.0 17.100 12.417
73 36.5 17.338 12.681
74 37.0 17.575 13.035
Beam B-002
75 0.5 0.238 0.038
76 1.0 0.475 0.073
77 1.5 0.713 0.104
... ... ... ...
168 47.0 22,325 12,140
169 47.5 22.563 12.582
170 48.0 22.800 13.184
Beam B-003
171 0.5 0.238 0.022
172 1.0 0.475 0.042
173 1.5 0.713 0.059
... ... ... ...
291 60.5 28.738 12.492
292 61.0 28.975 12.857
293 61.5 29.213 13.529
The graph (Fig. 7) presents a comparison of de-
flections a in the middle of the span versus the
value of the bending moment M for all three RC
beams.
The comparison of deflections a in the middle of
the span versus the value of the bending moment M
demonstrates the difference in the stiffness of three
RC beams tested.
Fig. 7. Graph of deflections of RC beams versus value
of the bending moment M
4. APPLICATION OF ARTIFICIAL
NEURAL NETWORKS
A database made up of the results of experimental
investigations carried out on RC beams (the values of
the deflections in the middle of the span) was used to
train and test the network. Finally, 293 patterns, corre-
sponding to 293 tests performed on beams (sets of
input and output data) were obtained. The database
was randomly divided into two parts: a training set
(197 patterns – 67%) and a testing set (96 patterns
– 33%).
In order to predict the deflection of beams, as a re-
sult of training and testing, a unidirectional Multi-
Layered Perceptron network (MLP) with error back
propagation algorithm was constructed. This type of
network was chosen since it is the most suitable for
solving the problem considered. Input vector x con-
sisted of four elements: surface area of tension rein-
forcement A
s
, the value of Young’s modulus of rein-
forcing steel E
s
, the value of Young’s modulus of
concrete E
c
and the value of bending moment M in the
cross section.
x = {A
s
, E
c
, E
s
, M}. (5)
The deflection of reinforced concrete beam also
depends on the geometry of the element (cross-
sectional dimensions, span) and the static scheme.
In view of the fact that all of the test beams have
the same geometrical dimensions and the same
static scheme, these components are omitted in the
input vector. For the analysis of more diverse
beams vector with a larger number of inputs should
be adopted in the structure of the artificial neural
network.