...

Wroblewski2012-BeamSolidElements.pdf

by user

on
Category: Documents
7

views

Report

Comments

Transcript

Wroblewski2012-BeamSolidElements.pdf
ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY
Vol. 36, No. 4, 2012
MODELING AND ANALYSIS OF FREE VIBRATION
OF STEEL-CONCRETE COMPOSITE BEAMS
BY FINITE ELEMENT METHOD
Tomasz Wróblewski, Agnieszka Pełka-Sawenko,
Małgorzata Abramowicz, Stefan Berczyński
Summary
Most technological machines generate vibrations which are transferred to either support systems or
foundations. To ensure an object’s safe operation, it is necessary to have adequate knowledge about
dynamic properties of both machines and a supporting construction. A steel-concrete composite floor is
an example of a supporting construction. It consists of steel beams connected with a reinforced
concrete slab in a way that enables mating of both elements. This paper presents a discreet model of a
steel-concrete composite beam that takes into account flexibility of the connection. An analysis of the
beam’s natural vibrations was conducted and the results were compared with those of experimental
studies. Tests were performed on two sets of beams. In the first group of beams B1 a connection that
consisted of steel stud connectors was used whereas perforated steel slats was used in the second
groups of beams B2. The present paper is a report from the analysis that was conducted on the beams
from group B2. The beam model was developed on Abaqus platform using deformable finite element
method. Matlab system was used for the analysis and its environment was applied to control the model
development and to identify the model’s selected parameters. The beam model was made in two
versions that differ in the approach to modelling connection. The developed model, after parameter
identification, yields highly consistent results with those of experimental tests.
Keywords: Composite structures, identification, Abaqus, Matlab, SPRING2
Modelowanie i analiza drgań własnych stalowo-betonowych belek zespolonych
metodą elementów skończonych
Streszczenie
Większość urządzeń technicznych wytwarza drgania, które przekazywane są na konstrukcje wsporcze
lub fundamenty. W celu zapewnienia bezpiecznej eksploatacji obiektu niezbędne jest uwzględnienie
właściwości dynamicznych zarówno maszyn, jak i konstrukcji wsporczej. Konstrukcją wsporczą jest na
przykład stalowo-betonowy strop zespolony. Składa się z belek stalowych połączonych z płytą
Ŝelbetową w sposób umoŜliwiający współpracę obydwu tych elementów. W artykule przedstawiono
model obliczeniowy stalowo-betonowej belki zespolonej uwzględniający podatność zespolenia.
Prowadzono analizę jej drgań własnych i jej wyniki porównano z wynikami wykonanych badań
doświadczalnych dla dwóch grup belek. W pierwszej grupie B1 stosowano zespolenie stalowymi
sworzniami zespalającymi. W drugiej natomiast B2 stosowano zespolenie stalowymi listwami
Address: Prof. Stefan BERCZYŃSKI, Tomasz WRÓBLEWSKI, Ph.D. Eng., Agnieszka PEŁKASAWENKO, M.Sc. Eng., Małgorzata ABRAMOWICZ, M.Sc. Eng., West Pomeranian
University of Technology of Szczecin, Faculty of Civil Engineering and Architecture,
Piastów 50, 70-311 Szczecin, e-mails: [email protected], [email protected],
[email protected], [email protected]
86
T. Wróblewski, A. Pełka-Sawenko, M. Abramowicz, S. Berczyński
perforowanymi. W pracy analizie poddano belki z grupy B2. Model belki opracowano w systemie
Abaqus wykorzystującym metodę odkształcalnych elementów skończonych. Do analizy stosowano
oprogramowanie Matlab sterowania procesem przygotowania modelu oraz identyfikacją wybranych
jego parametrów. Model belki opracowano w dwóch wersjach róŜniących się sposobem modelowania
zespolenia. UŜyto go takŜe do identyfikacji parametrów. Uzyskano wyniki symulacji częstotliwości
drgań o duŜej zgodności z wynikami badań doświadczalnych.
Słowa kluczowe: konstrukcje zespolone, identyfikacja, Abaqus, Matlab, SPRING2, MAC
1. Introduction
Steel-concrete composite beams are very often used in public space and
industrial building engineering as elements of floors or in bridge engineering as
main carrying girders. Special attention should be paid in each case to dynamic
properties of a designed construction. In public buildings, floor vibration control
is required due to meet Serviceability Limit States that ensure comfort of users
of a building. In industrial buildings, machines are often placed on floors.
Machines generate vibrations of various frequency which are transferred to
supporting constructions. Precision machines require a stable floor with defined
and known dynamic characteristics. In bridge engineering, dynamic load is
common and particular attention should be paid to small and medium span beam
bridges that are found along high-speed rail tracks where trains can travel with
speeds exceeding 300 km/h.
Over the past decades many studies on steel-concrete composite
constructions have been published. Elements are modelled usually using Euler or
Timoshenko beam theories [1-5]. Biscontin et al. [1] analysed a composite beam
with Euler beam theory considering connection flexibility along the direction
parallel to the beam’s axis. Analysis results were compared with those from
experimental tests. Flexibility of the connection in two directions was
investigated by Dilena and Morassi [2]. They developed a model to analyse
dynamic properties of composite beams without and with local damage of
connections. In [3] Dilena and Morassi analysed the frequency of flexural and
axial vibrations of a free beam with two connection types as well as their modal
damping ratios and vibration modes. The analysis was conducted for both
undamaged and partly damaged beams. Berczynski and Wróblewski et al. [5]
presented results of experimental tests conducted on three composite beams with
different connections between a steel beam and a reinforced concrete slab. On
the basis of the results, parameters of two computational models were identified.
The first, continuous model was based on Timoshenko beam theory and is
discussed in-depth elsewhere [4]. Motion behavior of the model is described by
a system of partial differential equations. It is difficult and time-consuming to
find a solution of such a system of equations even for one-dimensional systems.
The second model was created in convention of Rigid Finite Element Method –
RFE. The method is often used to developed and identified parameters of the
machine tool supporting systems models [6, 7], however, it can also successfully
Modeling and analysis of free vibration ...
87
be used to analyze the steel-concrete composite elements of the civil engineering
structures. Both models took into account two-directional stiffness of a
connection. The results obtained from the two models were highly consistent
with those of experimental tests.
The discrete model, defined by many scientists, provides an alternative for
the continuous model. The most popular method is the Finite Element Method
(FEM). FEM provides an efficient technique for finding numerical solutions.
Extended, commercial systems, such as Abaqus, Nastran and Ansys, based on
FEM can be used to conduct many comprehensive analyses of three-dimensional
models. Many researchers [8-18] analysed composite beams using FEM. Some
used two-dimensional numerical models, e.g. Gattesco [8] and Pi et al. [9],
others applied layer models in which an element is divided into several layers
(Szajna [10], Madaj [11]). However, most scholars prefer three-dimensional
models [10-16]. A numerical model of a concrete slab and a steel beam can be
defined in many different ways. Thevendran et al. [12] proposed a model in
which a slab and a beam were modelled with shell elements. Prakash et al. [13]
and Vasdravellis et al. [14] developed a model with solid elements to model both
a slab and a beam. A composite beam model which combined solid and shell
elements was used by Mirza and Uy [15], Queiroz et al. [16] and Baskar et al.
[17] – solid elements defined the concrete slab and shell elements defined the
steel beam. Luo et al. [18] applied another approach. They modelled a concrete
slab using solid elements and a steel beam was defined with beam elements.
Various approaches to modelling connection were applied. In [12] and [17]
a connection was modelled as beam elements connecting nodes of elements
modelling a concrete slab with independent nodes of steel beam model. A
comprehensive model, in which connection studs were defined using solid
elements, was applied in [15] and [12]. Likewise, in [18] an (adhesive)
connection was defined with solid elements. To account for shearing, non-linear
spring elements were additionally introduced between the nodes of a concrete
slab and the connection. Spring elements were also used by other authors [14]
and [16]. The complexity degree of a model and model selection depend on the
character of planned analysis. Many aspects of composite elements were
discussed earlier, including beam deflection analysis with partial and full
connections [16], curved plain beams [12], behaviour of beams and connections
in high temperatures [15], stress and slide analysis in concrete [12] and many
others. The issues of dynamic analysis of composite beams have not, however,
been frequently discussed.
The present study contains results of frequency of natural vibration analysis
conducted on steel-concrete composite beams. Beam models were defined in
FEM environment. The beam was modelled with both shell and solid finite
elements to model the steel part and reinforced concrete section. The connection
was modelled using one-dimensional finite elements. Stiffness of the connection
in the perpendicular and parallel direction to the beam’s axis was considered and
88
T. Wróblewski, A. Pełka-Sawenko, M. Abramowicz, S. Berczyński
discussed. The model was developed on Abaqus platform. Selected parameters
of the model were identified on the basis of experimental tests that had been
conducted earlier. Perforated steel slats was used as the beam’s connection. The
parameter identification process was controlled in Matlab environment. The
developed model can be used in the future for damage detection purposes in
composite beams.
2. Dynamic equation of motion
The dynamic equation of motion can be written as
Mq Cq Kq P
(2.1)
where M is inertia matrix, C is damping matrix, K is the matrix of system
stiffness, P is the vector of outside forces acting on the system, and q is the
vector of generalised coordinates. For free vibration, where damping and the
influence of external load is neglected, Equation (2.1) takes the following form
Mq Kq 0
(2.2)
The frequency of natural vibration for the modelled composite beam is the
solution of the eigenvalue problem of the following matrix:
detK M 0
(2.3)
where ω is the frequency of natural vibration. The vector of natural vibration
mode can be found using Equation (2.4).
K MΦ 0
(2.4)
3. Experimental tests of composite beams
Experimental tests were conducted on six composite beams: three from
series B1 with connection made using headed steel studs and further three beams
from series B2 in which perforated steel slats welded to the top flange of a
structural steel section was used as the connection. Series B2 beams only are
analysed in the present study. The beam and its cross-section are presented in
Fig. 1. The beam, 3200 mm in length, consists of a reinforced concrete slab 600
mm long and 60 mm thick and a structural steel section IPE 160. The slab was
made from C30/37 concrete, the structural steel section and the connection slats
from S235 steel.
Modeling and analysis of free vibration ...
89
Fig. 1. Composite beam from B2 group: longitudinal view and cross-section
Initially, the beams were tested under static load at the level of 40% of
elastic load capacity. The aim of the test was to check whether or not the
elements were properly manufactured and to prepare the elements for real tests
conducted to find dynamic characteristics. The characteristics were defined for a
beam with both free ends – the beam was suspended on two steel frames using
steel cable. The suspension points were assumed to overlap the nodes of basic
mode of flexural vibrations of the beam. Impulse excitation was applied and
acceleration was measured. The tests were conducted according to the procedure
described elsewhere [4]. A grid of measurement points is presented in Fig. 2.
Acceleration measurements were conducted within the grid using triaxial
piezoelectric sensors. Points were excitation was applied are marked in Fig. 2
with X symbols.
z
x
y
Fig. 2. Grid of measurement points
4. FEM model of composite beams
While defining the computational model, it was assumed that the beam has
a static scheme with free ends and that its cross-section is constant at all its
length. The model was defined on Abaqus platform as a spatial system with
independently modelled reinforced concrete slab, steel beam and connection.
90
T. Wróblewski, A. Pełka-Sawenko, M. Abramowicz, S. Berczyński
A discussion of possible approaches to how a numerical model of the beam
could be developed was presented in the introduction. Given the planned
analysis it was decided that the reinforced concrete slab should be modelled with
solid elements of the solid type. C3D8I solid first-order elements were used.
These are eight-node elements whose standard shape function was enhanced
with so-called “bubble functions”. C3D8I elements are not affected by
“hourglassing” which results in unnatural deformations of the finite element
mesh. Two elements were used on the thickness of the slab. On the width of the
slab, a division into 14 elements was applied with a thickening in the central area
where the slab interacts with the structural steel section. The mesh along the
beam’s axis consisted of elements 50 mm in length.
The steel beam made from structural steel section IPE 160 was modelled
with shell elements (S8R). Three shells were defined in the model. The first shell
defines web of beam while the other two the top and bottom flanges. The FEM
model of the composite beam as well as its cross-section are presented in Fig. 3.
a)
b)
y
x
z
x
z
y
Fig. 3. FEM model of the analysed composite beam: a) a view of the beam, b) cross-section
The connection between the reinforced concrete slab and the steel beam
was modelled in two different ways (Fig. 4) so effectively two independent
models were constructed. In both cases one-dimensional elements were used. In
a model denoted as MB, beam elements defined in Abaqus library as B31 were
used whereas in a model denoted as MS, SPRING2 elements were applied. In
both cases elements that modelled the connection connected shell elements S8R
that modelled the upper flange of the structural steel section with the lower
nodes of C3D81 elements which modelled the reinforced concrete slab. The
elements that modelled the connection were evenly distributed along the whole
area of the upper flange of the steel beam, i.e. 5 elements along the width, every
50 mm, along the beam’s axis at every node of the mesh. This distribution of the
elements ensured a continuous connection between the upper flange and the
lower edge of the slab which is consistent with the character of the connection
used in B2 beams. The connection stiffness was determined independently for
two directions. The stiffness along the horizontal direction (axis z) was denoted
Modeling and analysis of free vibration ...
91
as Kh – in this case the connection is sheared. The stiffness in the vertical
direction (axis y) was denoted as Kv. In MB model the stiffness of the connection
was changed by changing parameters that defined B31 beam elements, i.e.
a change of the cross-sectional area Aconn reflected a change in stiffness Kv
whereas a change of moment of inertia Jy,conn reflected a change in stiffness Kh.
In MS model two independent groups of spring elements SPRING2 were used.
The first group was responsible for stiffness Kh (along axis z direction) while the
second group was responsible for the interaction between steel and concrete
along the vertical direction (axis y) – stiffness Kv. The stiffness of the connection
in axis x direction (horizontal, perpendicular to the beam’s axis) was neglected
owing to the scope of analysed modes of the beam’s vibrations.
a)
b)
Fig. 4. FEM model of connection for composite beam: a) connection in MB model – beam
elements; b) connection in MS model – spring elements
5. Parameter identification of the models
Most parameters for the developed models were obtained from the literature
or from an inventory of three B2 beams. The inventory results were averaged so
that to produce a cumulative model of B2 beams (in fact two models that
differed from each other in the approach to modelling of the connection). As no
precise enough data were available for three parameters, they were identified.
These included substitute Young’s modulus of the reinforced concrete slab Ec
which accounts for the influence of the slab’s longitudinal reinforcement, the
stiffness of connection Kh and Kv . In MB model the stiffness of connection was
indirectly identified by determination of cross-sectional area Aconn and moment
of inertia Jy,conn for B31 elements.
The best fit of computational and experimental dynamic characteristics was
assumed to be the consistency criterion. Consequently, index S can be minimised
and can be given by
92
T. Wróblewski, A. Pełka-Sawenko, M. Abramowicz, S. Berczyński
_ !
#
$%&
"_ "_
#
"_
' _%() !
#
" _%() " $%&
_%()
#
" _%()
'
(5.1)
where S is the sum of squares of relative deviations of computational and
experimental frequencies of the first five modes of flexural vibrations
("_ and frequency of fundamental mode of axial vibration (" _%() .
Additionally, weight index wi_flex was introduced which had the following values:
0.5 for the first frequency of flexural vibration, 0.2 for the second frequency and
0.1 for every successive frequency. The adopted value of index w1_long was 0.1.
Parameter identification was conducted in Matlab environment. To enable
parameter optimisation, the communication between Matlab environment and
Abaqus platform had to be ensured. A script prepared in Python environment
was used to this end. The script was used to create the beam’s model, send it
over to Abaqus for calculations to be conducted, and send received results to
Matlab, where optimisation procedures were used to decide what possible
changes had to be made to the values of sought parameters. A new set of sought
parameters was sent to the script to update the beam’s model and make its new
calculations. This fully automated process was repeated until the minimum of
index S was reached. Table 1 presents a comparison of results of experimental
tests with those obtained using MB and MS models with identified parameters.
The first four modes of flexural vibrations for MB model are presented in
Fig. 5.
After identification, a comparison of the first four modes of flexural
vibrations was made using MAC (Modal Assurance Criterion) which can be
given by
*+, -.# / .$%& 0-.$%& / .# 0
.# / .# .$%& / .$%&
(5.2)
where .# is the modal vector for mode of natural vibration obtained in the
experimental tests and .$%& is the modal vector for mode obtained from the
computational model. The computational modal vector .$%& was determined
for selected nodes of the model marked in Fig. 5 (red dots). The nodes were
selected so that they overlapped with points on the measurement grid used in the
experimental tests. Modal vector components in the vertical direction (axis y)
were analysed. A comparison of normalized experimental and computational
(MB model) modal vectors corresponding to second flexural mode of vibrations
is presented in Fig. 6.
Modeling and analysis of free vibration ...
93
Table 1. Identification results – frequencies, relative errors, identified parameters
Model Mode of
vibration 1f – first
flexural
2f – second
flexural
3f – third
flexural
4f – fourth
flexural
5f – fifth
flexural
1a – first
axial
S
Ec , MPa
Aconn, m2
Jy,conn, m4
Kh, MPa
Kv, MPa
MB
MS
Experimental Computational Relative Experimental Computational Relative
frequency
frequency
frequency
frequency
error ∆,
error ∆,
fi,exp, Hz
fi,com, Hz
fi,exp, Hz
fi,com, Hz
%
%
75.70
76.01
-0.4
75.70
75.36
0.5
180.00
178.45
0.9
180.00
179.49
0.3
290.00
288.42
0.5
290.00
290.35
–0.1
393.30
394.28
–0.3
393.30
394.21
–0.2
495.60
497.92
–0.5
495.60
494.07
0.3
588.80
594.48
–1.0
588.80
588.80
0.0
-5
-5
2.864∙10
2.947∙104
5.629∙10-6
1.133∙10-11
–
–
1.345∙10
2.927∙104
–
–
337,3
104,4
a)
b)
c)
d)
Fig. 5. Modes of flexural vibrations of beam in MB model: a) first, b) second, c) third, d) fourth
94
T. Wróblewski, A. Pełka-Sawenko, M. Abramowicz, S. Berczyński
Fig. 6. Comparison of normalized modal vectors, second mode of flexural vibrations
– MB model, vertical direction – axis y
MAC values determined for MB and MS models are presented in Fig. 7.
a)
b)
Fig. 7. Histogram of MAC values – direction y: a) MB model, b) MS model
6. Results
The computational models of steel-concrete composite beams with
connection made of perforated steel slats and modelled with FEM are highly
consistent with the real object. Moreover, highly accurate fit of frequency in
experiment and simulation (the maximum relative error ∆ = 0.96%) was
obtained. The MAC values which compare experimental and computational
Modeling and analysis of free vibration ...
95
modal vectors of flexural vibrations also confirm the models’ high consistency
with experimental results.
While comparing the computational models, it can be noticed that MS
model (spring elements) provides more accurate fit than MB model (beam
elements). The degree of fit for vectors of vibration modes is almost at the same
level. It was easier and faster to model beams in Python and Abaqus
environments using MS model. The MS model was also much faster during one
computation loop for the MS model required significantly less time than the MB
model which significantly reduced time of identification process.
The developed computational models of composite beams can be used in
future research for the purposes of analysis and of damage detection in
composite beam components and connection.
Acknowledgments
This research was carried out with the financial support of the Ministry of Science
and Higher Education in Poland in the years 2010-2012.
References
[1] G. BISCONTIN, A. MORASSI, P. WENDEL: Vibrations of steel-concrete
composite beams. Journal of Vibration and Control, 6(2000), 691-714.
[2] M. DILENA, A. MORASSI: A damage analysis of steel-concrete composite beams
via dynamic methods: Part II. Analitical models and damage detection. Journal of
Vibration and Control, 9(2003), 529-565.
[3] M. DILENA, A. MORASSI: Experimental modal analysis of steel concrete
composite beams with partially damaged connection. Journal of Vibration and
Control, 10(2004), 897-913.
[4] S. BERCZYŃSKI, T. WRÓBLEWSKI: Vibration of steel-concrete composite
beams using the Timoshenko beam model. Journal of Vibration and Control,
11(2005), 829-848.
[5] S. BERCZYŃSKI, T. WRÓBLEWSKI: Experimental Verification of Natural
Vibration Models of Steel-concrete Composite beams. Journal of Vibration and
Control, 16(2010), 2057-2081.
[6] S. BERCZYŃSKI, P. GUTOWSKI, YU. KRAVTSOV, M. CHODŹKO:
Estimation of nonlinear models' parameters of machine tool supporting systems
using complete vibration test data. Advances in Manufacturing Science and
Technology, 34(2010)4, 5-22.
[7] S. BERCZYŃSKI, P. GUTOWSKI, YU. KRAVTSOV, M. CHODŹKO:
Estimation of nonlinear models' parameters of machine tool supporting systems
using incomplete vibration test data. Advances in Manufacturing Science and
Technology, 35(2011)3, 5-18.
[8] N. GATTESCO: Analytical modeling of nonlinear behavior of composite beams
with deformable connection. Journal of Constructional Steel Research, 52(1999),
195-218.
96
T. Wróblewski, A. Pełka-Sawenko, M. Abramowicz, S. Berczyński
[9] Y.L. PI, M.A. BRADFORD, B. UY: Second order nonlinear inelastic analysis of
composite steel-concrete members. II: Applications. Journal of Structural
Engineering, 132(2006)5, 762-771.
[10] W.S. SZAJNA: Numeryczny model stalowo-Ŝelbetowego dźwigara zespolonego.
Mat. VI Konf. Naukowej „Konstrukcje Zespolone”, Zielona Góra 2002, 175-182.
[11] A. MADAJ: Sztywność giętna stalowo betonowego przekroju zespolonego
z rozciąganą częścią betonową. Mat. Konf. Naukowo-Technicznej „Mosty
zespolone”, Kraków 1998, 233-242.
[12] V. THEVENDRAN, N.E. SHANMUGAM, S. CHEN, J.Y. RICHARD LIEW:
Experimental study on steel-concrete composite beams curved in Plan. Finite
Elements in Analysis and Design, 32(2000), 125-139.
[13] A. PRAKASH, N. ANANDAVALLI, C.K. MADHESWARAN, J. RAJASANKAR, N. LAKSHMANAN: Three Dimensional FE Model of Stud Connected
Steel-Concrete Composite Girders Subjected to Monotonic Loading. Int. Journal of
Mechanics and Applications, 1(2011)1, 1-11.
[14] G. VASDRAVELLIS, B. UY, E.L. TAN, B. KIRKLAND: The effects of axial
tension on the hogging-moment regions of composite beams. Journal of
Constructional Steel Research, 68(2012)1, 20-33.
[15] O. MIRZA, B. UY: Behaviour of headed stud shear connectors for composite
steel-concrete beams at elevated temperatures. Journal of Constructional Steel
Research, 65(2009), 662-674.
[16] F.D. QUEIROZ, P.C.G.S. VELLASCO, D.A. NETHERCOT: Finite element
modelling of composite beams with full and partial shear connection. Journal of
Constructional Steel Research, 63(2007)4, 505-521.
[17] K. BASKAR, N.E. SHANMUNGAM, V. THEVENDRAN: Finite – Element
analysis of steel – concrete composite plate girder. Journal of Structural
Engineering, 9(2002), 1158-1168.
[18] Y. LUO, A. LI, Z. KANG: Parametric study of bonded steel–concrete composite
beams by using finite element analysis. Engineering Structures, 34(2012), 40-51.
Received in August 2012
Fly UP