Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 407-420, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.407

PARAMETER ESTIMATION OF A DISCRETE MODEL OF

A REINFORCED CONCRETE SLAB

Małgorzata Abramowicz, Stefan Berczyński, Tomasz Wróblewski

West Pomeranian University of Technology, Szczecin, Poland

e-mail: mabramowicz@zut.edu.pl

This paper presents parameter estimation of amathematicalmodel regardingnatural vibra-
tions of a reinforced concrete slab. Parameter estimation is based on experiments conducted
on a real reinforced concrete slab. Estimated parameters include: substitute longitudinal
modulus of elasticity of the reinforced concrete slab, which takes into account longitudinal
reinforcement, effective thickness of the reinforced concrete slab and coefficient of damping.
Using appropriate criteria during, the process of parameter estimation of the reinforcement
concrete slabmodels has a great impact on obtaining precise results. The estimation criteria
are selected in order to achieve consistency of natural vibration frequencies along with the
Frequency Response Function measured during experiments with those calculated with the
mathematical model. The model and all the calculations have been made using MATLAB
programming environment.

Keywords: rigid finite element (RFE) model, reinforced concrete slab, estimation criteria,
modal parameters, vibrations

1. Introduction

Steel-concrete composite beams are the main focus of our previous papers (Berczyński and
Gutowski, 2006; Wróblewski et al., 2013). A composite beam is a connection of two or more
structural elements made of materials with various properties. An example of such a structure
is a steel-concrete composite beam which consists of a steel I-beam and a concrete slab which
rests on it. The present paper focuses on the reinforced concrete slab element.

The main topic of the paper is the modelling of vibration of a concrete slab. A 3D RFE
(RigidFiniteElementMethod)model is presented.Originallydevelopedalgorithmsof parameter
estimation of the reinforced concrete slab model are presented. The estimations were based
on experimental results, including beam natural vibration frequencies and FRF (Frequency
Response Function) determined empirically (Berczyński and Gutowski, 2006).

As there isnocommercially available rigidfinite element-based software, anoriginal computer
program has been developed in MATLAB environment. The created program can be used to
solve theproblemof freevibration and to control parameterswhich canbe introduced todescribe
selected structural elements.

The estimation process is performed in two parallel stages. In the first stage experimental
tests are conducted. In the second stage, the computationalmodel of the concrete slab is defined.
Solving the eigenproblem gives natural frequencies and FRF runs. Then, using optimisation
methods for frequency criteria andFRF runs consistency, parameters of the concrete slabmodel
are chosen to obtain models whose dynamic properties are close to those observed during the
experiment.



408 M. Abramowicz et al.

2. Experimental research of vibrations of a reinforced concrete slab

The experimental stand consisted of two steel bearers at the beam axle spacing of 2m. The be-
arers were bracedwith angle sections. During dynamic tests the beamwas suspended on frames
with four steel wire ropes 3mm in diameter and, in this way, a free beam schemewas implemen-
ted. Bearer deformability and its effect of obtained results were considered to be negligible in
the scheme. Rope deformability was selected so that frequency vibration typical for solid body
in motion was beyond the range of the investigated slab free vibration. A diagram of the test
stand as well as the suspended beam are presented in Fig. 1.

Fig. 1. Test stand: (a) overview, (b) view from the side, (c) head-on view

The aim of the conducted tests was to determine fundamental dynamic characteristics. An
impulse excitation was used. Vibration acceleration was ameasured value whichwas considered
as the response of the system (Ewins, 2000; Wilde, 2008). Acceleration was measured using
triaxial piezoelectric sensors.Thesensorswereattachedwithwaxtocircular steelwashers, 25mm
indiameter, placedon the reinforced concrete slab.Thewasherswerefixedwith amodifiedepoxy
resin.

The impulse excitation was performed using amodal hammerKISLER9726A20000 (500g).
LMS SCADAS III analyser connected to the work station fitted with a computer aided system.
Test.Lab package manufactured by LMS was used to record signals. Impact Testing module of
theTest.Labpackagewasused for impulse tests.During each cycle ofmeasurements, acceleration
in ninemeasurement points was recorded. Ten excitation cycles were performed in a predefined
spot of the beam. Signal averaging was conducted automatically according to the algorithm
implemented in Impact Testing module.

The obtained characteristics of frequency response functions were determined as the ratio of
vibration acceleration to the force. The frequency response functionswere used to determine the
so-called modal model usingModal Analysis module of theTest.Lab system (He and Fu, 2001).
A stability analysis method using PolyMAX algorithms was used for parameter estimation of
the modal model. This method is based on the frequency response functions.

3. The analysed reinforced concrete slab

The reinforced concrete slab (dimensions: 60mm thick, 600mm wide and 2200mm long) was
made of C25/30 concrete. The concrete mix was purchased from a local concrete producer. It
was made of cement-based class 42.5 with addition of a BV plasticiser. The ratio was W/C of
0.64 with the consistency of S3. Themaximum size of aggregate was reduced to 8mm owing to
a relatively small size of the investigated elements.



Parameter estimation of a discrete model... 409

Ribbed steel bars, 6mm in diameter made from A-I steel, were used as concrete reinforce-
ment. The longitudinal reinforcement was placed every 75mm while transverse reinforcement
every 150mm. A reinforcing fabric from top and bottom was used.

The impulse excitation was applied to the slab at three points (see Fig. 2a): 2−Z – vertical
impact at the slab axis performedwith themodal hammer, 1−Z – vertical impact at the edge of
slab performedwith themodal hammer, 2+Z – horizontal impact at the face of slab performed
with the modal hammer.

Excitations points in Fig. 2 a are designated with the symbol •. Various excitation points
aimed at producing different vibration forms of the slab are presented in Table 1.

Table 1.Excitation points and directions

Symbol
Excitation

Excited vibration forms
Point Direction

1−Z 1 Z flexural and torsional
2−Z 2 Z flexural
2+X 2 X axial

Measurement pointsweredefinedat the upperarea of the slab, spaced in three rows (9 points
in each row) which gave a total of 27 measurement points (see Fig. 2b).

Fig. 2. Investigated slab: (a) excitation points, (b) measurement point grid

In Fig. 3, some mode shapes of the concrete slab obtained during the dynamic research are
presented.

Dynamic characteristics of the frequency response functionswere obtained and used to iden-
tify the coefficient of damping and other parameters. Peak amplitude values of FRF runs were
determined. The peak amplitude values were corresponding to the next resonance vibrations of
the reinforced concrete slab. Two types of vibrations were analysed: flexural vibrations (impulse
excitation 2−Z) and torsional vibrations (impulse excitation 1−Z). For flexural and torsional
vibrations, the peak amplitude values were determined for five resonances.

The authors developed their own software in MATLAB, which automatically defines the
peak amplitude of FRF runs corresponding to the resonance vibration of the reinforced concrete
slab. The software enabled determination of the values in tabular and graphical forms. For
the analysis of FRF runs, eight representative measurement points were selected. For flexual
vibration (impulse excitation 2−Z) points 2, 5, 23 and 26 were selected, which were located on
the axis of the reinforced concrete slab (see Fig. 2b). For torsional vibration (impulse excitation
1−Z) points 1, 3, 25 and 27 were selected, which were located on the corners of the reinforced
concrete slab (see Fig. 2b). Peak amplitudes of the run of FRF corresponding resonances are
summarised in Table 2.



410 M. Abramowicz et al.

Fig. 3. Flexural and torsional experimental mode shapes for the reinforced concrete slab,
excitation 1−Z: (a) 1flex =50.04Hz, (b) 1tors =113.34Hz, (c) 2tors =232.75Hz

Table 2.Amplitude of frequency response functions for the reinforced concrete slab determined
duringmeasurement, impulse excitation 1−Z, direction Z

Points 1 3 25 27

FRF
FRF f

exp
i FRF f

exp
i FRF f

exp
i FRF f

exp
i

[m·s−2/N] [Hz] [m·s−2/N] [Hz] [m·s−2/N] [Hz] [m·s−2/N] [Hz]
1tors 1.915 113 2.000 113 1.968 113 1.966 113

2tors 3.117 233 3.162 233 3.152 233 3.165 233

3tors 2.63 368 2.599 368 2.676 367 2.598 367

4tors 2.077 526 2.051 526 2.022 526 1.992 526

5tors 2.632 708 0.51 707 2.321 708 0.553 709

In Fig. 4, frequency response functions with marked peak amplitude values (from torsional
vibrations) for point 27 for direction Z, impulsive excitation 1−Z are presented.

Fig. 4. Amplitude of the frequency response functions for the reinforced concrete slab, excitation 1−Z,
point 27, directionZ



Parameter estimation of a discrete model... 411

4. The spatial computational model of the reinforced concrete slab

A discrete computational model of the slab has been developed in the convention of the Rigid
Finite ElementMethod. Themethod consists in dividing a real system into rigid finite elements
which are represented by spring-damping elements (SDEs). While rigid finite elements (RFEs)
are characterised bymasses andmass moments of inertia, spring-damping elements are defined
by stiffness and damping coefficients. The RFE method was developed by Kruszewski et al.
(1999) andWittbrodt et al. (2006).

Modelling of continuous elements in the finite element method starts from primary segmen-
tation. For a slab, the segmentationmust be conducted in two directions, i.e. in the longitudinal
direction along∆L and in transverse direction along∆B. Themodel is divided into segments of
equal or comparable length. The primary segmentation of the present study is shown in Fig. 5a.
Then, at the center of gravity of each element, an SDE is placed which focuses spring and dam-
ping properties of that element. Each SDE is broken down into four smaller SDEs, so that it is
possible to connect the corners of four adjacent finite elements – this is secondary segmentation.
In every set of four SDEs, two of them are parallel to the main axis X and the other two are
parallel to the main axis Y . In the classic approach (Kruszewski et al., 1999) spring properties
of respective elements are reflected by SDEs spaced as shown in Fig. 5b.

Fig. 5. 3D finite element method: (a) primary segmentation, (b) secondary segmentation – classic
positioning of SDEs, (c) secondary segmentation –modified positioning of SDEs

In the classic approach, the elements have 5 degrees of freedom. By placing an SDE at the
corners, it is possible to neglect rotation in the axes perpendicular to the area of a primary
element. The proposed model attempts to define the slab with 6 degrees of freedom (three
translational and three rotational displacements). In order to limit element rotation in the axis



412 M. Abramowicz et al.

perpendicular to the area of the primary element, SDEsmust bemoved from the corners to the
centers of finite elements as shown in Fig. 5c. A similar approach was used in (Adamiec-Wójcik
andWojciech, 2012).

Each rigid finite element has its own independent coordinate system X
(i)
RFE,Y

(i)
RFE,Z

(i)
RFE

which is selected so that it overlaps the principal central axes of inertia of a given RFE. Given
this assumption,mass andmoments of inertia are the only parameters necessary to describe any
RFE. These quantities can be given in form of a diagonal mass matrix

M
(i) = diag

[

m(i),m(i),m(i),J
(i)
X ,J

(i)
Y ,J

(i)
Z

]

(4.1)

The first three terms of the matrix are equal to the mass of the RFE, while the other three

are RFE mass moments of inertia relative to the axes X
(i)
RFE, Y

(i)
RFE, Z

(i)
RFE. The values of the

diagonal element of themass matrixM(i) for modelling of RFEs inside the slab are determined
in the following way

m(i) =hc∆L∆Bρc (4.2)

where hc is thickness of the reinforced concrete slab, ρc – mass density of RFEmaterial

J
(i)
X =

m(i)

12
(∆L2+h2c) J

(i)
Y =

m(i)

12
(∆B2+h2c)

J
(i)
Z =

m(i)

12
(∆L2+∆B2)

(4.3)

EverySDEofknumberhas its own independentcoordinate systemwith themainaxesX
(k)
SDE,

Y
(k)
SDE,Z

(k)
SDE. Themain axes of the SDE have such property that forces acting on the SDE in a

direction compatible with these axes result in its translational deformations which occur only in
the direction alongwhich these forces are applied. Themain parameters which describe an SDE
of k number are coefficients defining its spring and damping properties. The spring properties

are described bymeans of twomatrices: a matrix of translational stiffness coefficients K
(k)
T and

amatrix of rotational stiffness coefficients K
(k)
R . Both matrices are diagonal and they are 3×3

in size

K
(k)
T = diag

[

k
(k)
T,X,k

(k)
T,Y ,k

(k)
T,Z

]

K
(k)
R = diag

[

k
(k)
R,X,k

(k)
R,Y ,k

(k)
R,Z

]

(4.4)

The values of translational and rotational stiffness coefficients are determined according to the
following rules:
— for SDEs parallel to the main axis X

k
(k)
T,X−X =

Echc,eff∆B

∆L
k
(k)
T,Y−X =

Gchc,eff∆B

∆Lχ

k
(k)
T,Z−X =

Gchc,eff∆B

∆Lχ
k
(k)
R,X−X =

Gch
3
c,eff∆B

6∆L

k
(k)
R,Y−X =

Ech
3
c,eff∆B

12(1−ν2c)∆L
k
(k)
T,Z−X =

Echc,eff∆B
3

12∆L

(4.5)

— for SDEs parallel to the main axis Y

k
(k)
T,X−Y =

Gchc,eff∆L

∆Bχ
k
(k)
T,Y−Y =

Echc,eff∆L

∆B

k
(k)
T,Z−Y =

Gchc,eff∆L

∆Bχ
k
(k)
R,X−Y =

Ech
3
c,eff∆L

12(1−ν2c)∆B

k
(k)
R,Y−Y =

Gch
3
c,eff∆L

6∆B
k
(k)
T,Z−Y =

Echc,eff∆L
3

12∆B

(4.6)



Parameter estimation of a discrete model... 413

whereEc is the substitute dynamic longitudinal modulus of elasticity of the reinforced concrete
slab (which takes into account the effect of reinforcement used),Gc – substitute dynamic trans-
verse modulus of elasticity of the reinforced concrete slab (which takes into account the effect
of reinforcement used), hc,eff – effective thickness of the reinforced concrete slab, νc – Poisson’s
ratio of concrete, χ – coefficient of cross-section shape (Timoshenko shear coefficient), which
takes into account the nonuniform tangential stress pattern. The method of calculating the co-
efficient was described byBerczyński andWróblewski (2010). TheTimoshenko shear coefficient
of cross-section of a rectangle shape is 1.2.

Dumping properties are described by means of two matrices of damping coefficients: C
(k)
T

andC
(k)
R . Bothmatrices are diagonal and they are 3×3 in size. The relation between equivalent

stiffness k
(k)
i,j and damping coefficients c

(k)
i,j can be given by

c
(k)
i,j =

η

ω
k
(k)
i,j i=T,R j=X,Y,Z (4.7)

where η is the loss ratio, ω – vibration frequency.
Results obtained with the modified RFE method model were compared with the analytical

solution, the TM (Theoretical Model), and with the FEM model with flexible and deforma-
ble rigid bodies implemented in the Finite Element Method. The FEM model was developed
in Abaqus environment and it was used for calculations. The slab was modelled with 50× 50
second-order cubic elements with reduced integration for second-order functions (C3D20R). The
solution was developed and reported by Liew et al. (1993), Leissa (1973). Leiss (1973) provi-
ded a solution for thin slabs, whereas Liew et al. (1993) found a solution of eigenmodes for
medium-thick rectangular slabs. The authors presented their findings for various boundary con-
ditions (21 cases) for different side length/width ratios (a/b) and different slab thickness/width
ratios (h/b). Their approach was based on the energy function defined using theMindlin theory
of plates with the Rayleigh-Ritz minimisation procedure, providing a solution to the eigenvalue
problem (Szcześniak, 2000). According to the theory of plates ofmediumthickness, it is typically
assumed that a plate has thickness greater or equal 1/10 of the smaller of the remaining two
dimensions. The reinforced concrete slab used in our tests had the thickness to width ratio of
60/600, i.e. 1/10 precisely. Therefore, the theory ofmedium-thick plates could be applied for the
modelling of both the former and the latter. The results were compared for a free-ends slab (the
same scheme was used in empirical investigations). Liew et al. (1993) conducted analytical cal-
culations for a slab with various length/width ratios for comparison purposes. Here, a 1500mm
long, 600mm wide and 60mm thick slab was used in the tests. Therefore, the length to width
ratio of 2.5 and the thickness to width ratio of 0.1 was obtained. The Poisson ratio µc was 0.3,
Young’s modulusEc was 3.0 ·1010N/m2 and themass density ρc was 2400kg/m3. In an earlier
work (Liew et al., 1993), the results were presented in a dimensionless form

λ=
ωB2

π2

√

ρchc
D

(4.8)

where ω is the angular frequency,B – length of the shorter side of the slab width,D – bending
flexural stiffness of the slab.

D=
Ech

3
c

12(1−ν2c)
(4.9)

The results are presented in Table 3. The calculations for the RFE model were conducted for
the initial mesh of 30×12 elements, the slab dimensions were the same as those given above.
Table 3 presents a comparison of solutions for the three models: the theoretical model (TM)
based on the precise solution, the Rigid Finite Element (RFE) model and the Finite Element
Method (FEM) model.



414 M. Abramowicz et al.

Table 3.A comparison of dimensionless vibration frequencies of the slab for the TM,RFE and
FEMmodels

Reinforced concrete slab 60mm×600mm×1500mm

i Vibration forms of the slab
TM RFE FEM
λi TM λi RFE ∆i TM λi FEM ∆i TM

1 0.3455 0.3605 4.3% 0.3457 0%

2 0.5137 0.5144 0.1% 0.5154 0%

3 0.9486 0.9826 3.6% 0.9502 0%

4 1.0952 1.0939 −0.1% 1.0994 0%

5 1.8109 1.8080 −0.2% 1.8199 0%

6 1.8220 1.8952 4.0% 1.8282 0%

7 2.1919 2.1860 −0.3% 2.1981 0%

8 2.3566 2.3884 1.4% 2.3654 0%

9 2.7173 2.7164 0% 2.7332 0.6%

The terms∆i RFE and∆i FEM defined by (4.10) are used to show the percentage differences
for dimensionless values of slab frequency vibrations obtained for the TM, RFEM and FEM
models.

∆i RFE =
λi RFE −λi TM
λi TM

∆i FEM =
λi FEM −λi TM
λi TM

(4.10)

The first column shows the obtained vibration forms of the slab for the dimensionless values of
the slab frequencies.

A very good convergence of the results was obtained for the RFE and TM models. The
largest difference was 4.3%. The largest discrepancy between the FEM and TM models was
observed for higher eigenmodes (0.6% for 9th eigenmode). However, the results obtained for the
torsional (2, 4, 5 and 9) and flexural vibrations (1, 3 and 6), which are significant for parameter
estimation of composite beams, were quite convergent for both the former and the latter.

During the analysis of various values of the parameter ∆ for various mesh densities in the
X axis, a constant mesh density of 12 was used for the Y axis. In the second case, with various
mesh densities in the Y axis, a constant mesh density of 30 was used for the X axis. It was



Parameter estimation of a discrete model... 415

examined that for other constants for X and Y axes the same results were obtained. It was
found that with an increase in mesh density, the eigenmode frequencies were asymptotically
approaching the solution for the sufficientmesh densitywhich provided a good convergence with
the analytical solution. Nevertheless, the problem ofmesh density selection was also affected by
the fact that larger densities significantly increased calculation time.

5. Parameter estimation – algorithm I

The estimated parameters included the substitute longitudinalmodulus of elasticity of the rein-
forced concrete slab Ec which took into account longitudinal reinforcement and effective thick-
ness of the reinforced concrete slab hc,eff . Other parameters used for the identification of the
computational model were taken from the literature or from the design of the analysed slab.

The differential equation of free vibration obtained from the general differential equation
of motion, while neglecting outside interaction acting on the slab and damping effects, can be
presented as

Mq̈+Kq=0 (5.1)

where q is the vector of generalised coordinates.

Methods of solving equation (5.1) to determine free vibration frequencies and corresponding
vibration modes are described in detail in literature (Wilde, 2008).
The quantities Sflex and Stors (5.2) are sums of squares of the relative deviations of the

first n numerical frequencies of flexural and torsional vibrationswith their analogous frequencies
determined during the experiment

Sflex =
n
∑

i=1

(

fnumi flex −f
exp
i flex

f
exp
i flex

)2

Stors =
n
∑

i=1

(

fnumi tors−f
exp
i tors

f
exp
i tors

)2

(5.2)

where n is the number of selected frequencies, f
exp/num
i flex – experimental/numerical frequencies

of flexural vibrations, f
exp/num
i tors – experimental/numerical frequencies of torsional vibrations.

The quantity S is a sum of flexural and torsional vibrations in the process of estimation
in which an appropriate weight functionwflex was attributed to flexural and wtors to torsional
frequencies

S =wflexSflex +wtorsStors (5.3)

An additional criterion imposed was the condition of compatibility of the experimental and
mathematical fundamental frequency of longitudinal vibrations:

f
exp
1 long = f

num
1 long (5.4)

The algorithm assumes a limit (5.4) that has the greatest impact on the identified slabmodulus
of elasticity Ec.

In order to determine model parameters that allow the best mapping of frequencies with
those obtained experimentally, the paramter S index was minimised. To solve the problem,
an optimisation procedure implemented in Optimization Toolbox package, which is a part of
MATLAB, was used.

For adopted criterion (5.3), the following numerical experiments was performed. In the 1st
stageofparameter estimation criterion (5.3)wasminimisedassumingwflex =1.0andwtors =0.0.
This means comparison of only flexural frequencies of natural vibration during the estimation



416 M. Abramowicz et al.

process. The estimation resulted in a good compatibility of frequencies of natural vibration
observed in the experiments and in numerical calculations (seeTable 4 –Analysis I). Differences
in compatibility for flexure frequencies for the reinforced concrete slab were under 1.4% and
for torsional frequencies under 3.1%. The iteration algorithm was tested for convergence by
introducing various starting points to determine parameters of the the model. Each time, after
the iteration was complete, the values of estimated variables were similar.

Table 4. Comparison of natural vibration frequencies measured during dynamic tests with
numerical results algorithm I of the estimation, Analysis I-III

Analysis I II III

FRF
f
exp
i f

num
i ∆ f

exp
i f

num
i ∆ f

exp
i f

num
i ∆

[Hz] [Hz] [%] [Hz] [Hz] [%] [Hz] [Hz] [%]

1tors 113.34 110.2 −2.8 113.34 113.4 0.1 113.34 111.33 −1.8
2tors 232.75 227.4 −2.3 232.75 233.97 0.5 232.75 229.7 1.3
3tors 367.33 358.33 −2.5 367.33 368.61 0.3 367.33 361.94 −1.5
4tors 525.61 509.55 −3.1 525.61 524.03 −0.3 525.61 514.64 −2.1
5tors 707.99 686.87 −3.0 707.99 706.18 −0.3 707.99 693.65 −2.0
1flex 50.00 50.64 1.3 50.00 52.13 4.3 50.00 51.16 2.3

2flex 138.34 138.94 0.4 138.34 143.00 3.4 138.34 140.36 1.5

3flex 266.76 270.53 1.4 266.76 278.36 4.3 266.76 273.28 2.4

4flex 441.54 443.22 0.4 441.54 455.9 3.3 441.54 447.67 1.4

5flex 649.34 654.85 0.8 649.34 673.31 3.7 649.34 661.34 1.8

1long 933.12 933.12 0.0 933.12 933.12 0.0 933.12 933.12 0.0

Ec [N/m
2] 4.248E+10 4.127E+10 4.205E+10

hc,ff [m] 0.057 0.059 0.058

Stors 3.718E-03 5.500E-05 1.548E-03

wtors 0.0 1.0 1.0

Sflex 4.711E-04 7.262E-03 1.890E-03

wflex 1.0 0.0 1.0

S 4.711E-04 5.500E-05 3.438E-03

The following Section presents results of the 2nd stage of estimation for weight functions
wflex = 0.0 and wtors = 1.0. This means comparison of only torsional frequencies of natural
vibration during the estimation process. The estimation results are presented in Table 4 – Ana-
lysis II. Differences in compatibility for torsional frequencies for the reinforced concrete slab
were under 0.5% and for flexural frequencies under 4.3%.
The following part of the paper presents results of the 3rd stage of estimation for weight

functionswflex =1.0 andwtors =1.0.Thismeans comparison of flexual and torsional frequencies
of natural vibration during the estimation process. The estimation results are presented in
Table 4 – Analysis III. Differences in compatibility for torsional frequencies for the reinforced
concrete slab were under 2.1% and for flexural frequencies under 2.4%.

6. Parameter estimation for the model – algorithm II

While working on the finite element method model of the slab, an assumption was made that
the substitute dynamic longitudinal modulus of elasticity of the reinforced concrete slab Ec,
effective thickness of the reinforced concrete slab hc,eff , and the loss ratio of concrete ηc would
be determined based on identification. The value of the loss ratio is dependent on frequency,
temperature and other factors.



Parameter estimation of a discrete model... 417

The loss ratio of concrete varies in the range (2-6)·10−4 according to (Silva, 2000). High
diversity of concrete types results in a very different range of concrete damping values. Damping
depends on concrete density, amount of cement slurry, load history, intensity of stress, etc.
Parameter estimation was conducted by fitting the frequency response functions calculated

with the finite element method model to the characteristics obtained in experimental research.
A system of differential equations defining the oscillating motion with damping can be given by

Mq̈(t)+Cq̇(t)+Kq(t)= f(t) (6.1)

where q is the vector of generalised displacement, M, C, K – inertia, dumping and stiffness
matrices, f – vector of generalised forces. The vectors q and the system response f are functions
of time t. The above system of differential equations can be solved, depending on the form of
the excitation signal, using either Fourier or Laplace integral transform (Marchelek, 1991). By
using the Laplace transform, it is possible tomove from time domain over to the domain of the
complex frequency s. Given zero initial conditions, while performing the Laplace transform, the
system of equations (6.1) takes the following form

(Ms2+Cs+K)q(s)= f(s) (6.2)

A consequence of using the Laplace transform is algebraisation of the system of equ-
ations (6.1). While solving a system of linear algebraic equations, we assume that the matrix
(Ms2+Cs+K) is not singular, i.e. that there is a matrix inverse to it. As a result, Equation
(6.2) takes the form

q(s)= (Ms2+Cs+K)−1f(s) (6.3)

To find a solution in the frequency form, if the excitation applied to the system is periodic
(solution for a steady state), the Fourier transform can be used. A solution is found directly in
the Laplace solution by substituting it to Equation (6.3) s= jω, where j=

√
−1

q(jω)= (K−ω2M+jωC)−1f(jω) (6.4)

where

A(jω)=K−ω2M+jωC (6.5)

is referred to as the dynamic stiffness matrix, while

W(jω)=A−1(jω)= (K−ω2M+jωC)−1 (6.6)

is the dynamic flexibility matrix.
Dynamic flexibility is a characteristic obtained on the premise that the system input is a

force and its output is a displacement.During experimental research, accelerationwasmeasured.
A characteristic found given the condition that the system input is the force and its output is
the displacement is called inertanceG(jω). Both dynamic flexibility and inertance are frequency
characteristics defined for steadymotion and they are therefore closely interrelated (Uhl, 1997).

|G(jω)|=ω2|W(jω)| (6.7)

To find the inertance of a system based on a finite element model, it is necessary to know
the stiffness matrix K, inertia matrix M and damping matrix C. The methods were described
in-depth elsewhere in the literature (Kruszewski et al., 1999; Wróblewski et al., 2013).
Our identification criterion wasminimization of the coefficient JFRF (6.7) which is a double

sum for m-th measurement points, a sum of relative quadratic deviation of the first n-th me-
asurement points for a given FRF amplitude to the same amplitude determined in experimental



418 M. Abramowicz et al.

research. While determining loss ratios, an attempt was made to fit the calculated amplitudes
with those determined experimentally

JFRF =
m
∑

i=1

n
∑

i=1

(

FRFnumi,amp−FRF
exp
i,amp

FRF
exp
i,amp

)2

(6.8)

The above algorithm allowed one to select an analysed point and, as a result, it was possible
to choose vibration forms (flexural and torsional vibration) and vibration modes (1,2, . . . ,n).
Both flexural and torsional vibration formswere taken into account in the estimation procedure.
An optimization procedure implemented in Optimization Toolbox package, which is a part of
MATLAB, was used.

The results of analysis are presented in Table 5. First analysis was conducted for torsional
vibration forpoint1 (vibrationmodes from2to4)and forflexuralvibration forpoint2 (vibration
modes from 2 to 4). In the identification, 5 points from each amplitude of every FRFwere taken
into consideration.

Table 5.Results of estimated parameters during optimisation – algorithm II of the estimation,
Analysis I-III

Analysis I II III

Ec [n/m
2] 4.230E+10 4.300E+10 4.224E+10

hc,eff [m] 0.0570 0.0577 0.0570

ηc [–] 0.0090 0.0112 0.0091

J 9.4703 4.7114 3.7924

Second analysis was conducted for torsional vibration for point 1 (vibration modes from 2
to 4) and for flexural vibration for point 2 (vibrationmode3). In the identification, 5 points from
each amplitude of every FRF were taken into consideration. Third analysis was performed for
torsional vibration for point 1 (vibrationmode2) and for flexural vibration for point 2 (vibration
modes from 2 to 4). In the identification, 5 points from each amplitude of every FRFwere taken
into consideration.

Fig. 6. A comparison of the frequency response functions for the reinforced concrete slab – Analysis III



Parameter estimation of a discrete model... 419

A comparison of FRFs determined using the above estimated parameters with those deter-
mined experimentally is presented in Fig. 6. As the consistency of the FRFs was high, it was
fair to state that the identified parameters were determined correctly.

7. Conclusions

The numerical tests show a very good convergence of the algorithms developed for estimating
the parameters of amathematical model of the reinforced concrete slab. They enable creation of
a model representing the actual reinforced concrete slab and allowed one to determine dynamic
characteristics very similar to those measured during the experiments.

Following the identification, it can be observed that the identified parameters for algorithms
I and II are very similar. Both algorithms provide convergent results of identification, which
indicates the appropriateness of the model and algorithms.

A very good fit of FRFs calculated using themodel with those obtained in the experimental
research is achieved. The originally developed spatial finite element model allows one to take
into account both flexural and torsional vibration forms. The goal was to achieve consistency of
frequencies and FRFs observed during tests with those calculated using the analytical model.

This well-developed 3D model of the reinforced concrete slab has allowed the authors to
develop a model of a steel-concrete composite beam which has been the focus of their research
attempts.

Acknowledgments

The researchwork reported in this paper wasmade possible throught financial support fromMNiSW

Poland in the years 2011-2013, completed within the project NN501192138.

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Manuscript received April 7, 2015; accepted for print August 23, 2016