Analytical and Experimental Vibration Analysis of Glass Fibre Reinforced Polymer Composite Beam

Analytical and Experimental Vibration Analysis of Glass Fibre Reinforced Polymer Composite Beam
Master's Degree Thesis
ISRN: BTH-AMT-EX--2007/D-13--SE
Analytical and Experimental
Vibration Analysis of Glass
Fibre Reinforced Polymer
Composite Beam
Oluseun Adediran
Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
Ansel Berghuvud, Ph.D. Mech. Eng.
Analytical and Experimental
Vibration Analysis of Glass
Fibre Reinforced Polymer
Composite Beam
Oluseun Adediran
Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
Thesis submitted in fulfilment of the completion of Master of Science in
Mechanical Engineering by the Department of Mechanical Engineering,
Blekinge Institute of Technology, Karlskrona, Sweden.
The aim of this work is to investigate analytical and experimental
vibration of composite beam. The composite beam is used as composite
footbridge model prototype which is assumed. For the dynamic test,
hammer excitation is used to excite the beam at fixed locations. The modal
parameters are extracted from the time response using a time domain
analysis, i.e. the stochastic subspace identification technique. Finite
element models for different boundary conditions are constructed using
the commercial finite element software package ANSYS for natural
frequencies of the beam to support and verify the dynamic measurements.
The result obtained from analytical solution, dynamic tests and the FEM
are presented and analysed.
Modal Parameters, Frequency, Mode shapes, FE Model, Composite Beam
This work was carried out under the supervision of Dr Ansel Berghuvud,
Blekinge Institute of Technology, Karlskrona, Sweden and Dr. Abdel
Wahab University of Surrey, United Kingdom
I wish to express my gratitude to a lot of people who have supported me
during the past years in one way or the other. To my major professor, Dr.
Abdel Wahab, a special thanks for giving me the opportunity to work with
him and for his patience, tolerance, understanding, and encouragement. He
was more than a professor to me; he was also a friend. It has been a great
pleasure working with him. I wish to thank Dr Ansel Berghuvud.
I am very thankful to my research group members, Prof Andrew Crocombe,
Mr Libardo, Irfan for their patience, and advice. I really appreciate their
time and effort in being part of my research group.
Thanks are also extended to Mr & Mrs Sola Olalekan, Olanrewaju, Sogo,
Opeyemi, Anuoluwapo, Ebunoluwa, Oluwole for their support and help in
many ways. I also want to thank all those that helped me through the
research. The greatest thanks go my mother for her financial support
throughout my degree programme.
A special thanks to my loving mother, for being there for me all the time of
my needs. Also, I thank my friend Babatunde Rasheed.
I am very grateful to my wife, Oluwatoyin, for her encouragement. She
brought more love and strength to my life.
Karlskrona, October 2007
Oluseun. A. Adediran
1 Notations
5 2 Introduction
2.1 Research Context
2.2 Objectives
2.3 Review of previous work
2.3.1 Composite Materials FRP for Bridge Applications
2.3.2 Recent Research in FRP
2.3.3 Characteristics of FRP Structural Beam
2.3.4 Problem with Composite Material
2.3.5 Advantages of Composite Beam
2.3.6 Disadvantages of FRP Composite Beam
2.3.7 Applications of FRP Composite Beam
8 8 9 10 10 12 14 15 15 16 16 3 Basic Concepts and Theory
3.1 Structural dynamics
3.1.1 SDOF model
3.1.2 MDOF model
3.2 Experiment Modal analysis
3.2.1 Introduction
3.2.2 Theoretical Derivation
17 17 17 18 19 19 20 4 Beam Structure
4.1 Introduction
4.2 Finite Element Method Calculations
23 23 25 5 Experimental Work
5.1 Introduction
5.1.1 Measurement Preparations
5.1.2 Identification of Experimental Model
5.1.3 Position of the Accelerometer on the Beam
5.1.4 Point of Excitation
5.2 FRP Composite Beam Test Descriptions
5.3 Measurement Equipment
5.3.1 Data acquisition system
5.3.2 Accelerometer
5.3.3 Impact Hammer
5.3.4 Steel Support
5.3.5 PC with Software
31 31 31 31 32 32 33 35 35 36 36 37 38 3
5.4 Experimental Specifications
5.5 38 Experimental Results and Discussion
5.5.1 Results
40 40 6 Modelling in ANSYS
6.1 Introduction
6.2 Procedure in Modelling ANSYS
6.2.1 Requirement Specification
6.2.2 Idealization Specification
6.2.3 Mesh Generation
6.2.4 Analysis
6.2.5 Post-processing
6.3 ANSYS Graphical Results
6.3.1 Simple-Simple Boundary Condition
6.3.2 Fixed-Fixed Boundary Condition
6.3.3 Fixed-Simple Boundary Condition
6.3.4 Fixed-Free Boundary Condition
47 47 48 48 49 49 50 50 50 50 52 53 55 7 Discussion
7.1 Introduction
7.2 Comparison of Method
7.3 Effect of Boundary Conditions on Natural frequency
57 57 57 60 8 Conclusion
8.1 Future Work
61 61 9 References
62 Appendices
1 Terminology
2 Calculation of FEM Natural frquencies of beam (1 element)
3 Calculation of FEM Natural frquencies of beam (2 element)
List of symbols
C1 , C 2 , C 3 , C 4 Constant-time function in the model
Wave equation for the beam
AR matrix parameters
excitation force at time t
frequency [Hz]
Young Modulus
Elastic force exerted on the mass
Damping force
Inertial force of the mass
H (ω ) Frequency response function
moment of inertia
M, Cp, K
mass, damping and stiffness matrix
z (t ), z& (t ), &z&(t ) Displacement, velocity and acceleration vectors at time t
Damping ratio
Mode shapes
Natural frequency
Constant value
external forces
Density of the beam
Kinetic Energy
Velocity in y-direction
Angular frequency [rad/s]
Density of the composite beam [kg/m3]
Nodal degree of freedom at nodes j
Nodal degree of freedom at nodes i
Bending moment acting at node i
Bending moment acting at node j
External forces
Strain Energy
List of operator
(⋅) T
Generalised System
(⋅) −1
matrix inverse
(⋅) *
complex conjugate
(⋅) i
data or estimate from patch i
Re (⋅)
real part of
Im (⋅)
imaginary part of
matrix transpose
List of abbreviations
Degree of Freedom
Single Degree of Freedom
Multiple Degree of Freedom
Experimental Modal Analysis
Experimental Modal Testing
Finite Element
Not Available
Finite Element Method
Fast Fourier Transform
Frequency Response Function
Peak Picking
Root.Mean Square
Non-Destructive Testing
Fibre reinforced Polymer/Plastic
Glass Fibre reinforced Polymer/Plastic
This chapter contains a general introduction of the research that was carried
out within the frame work of this thesis. The research context is described
in Sec.2.1. The focus of the thesis as well as the main objectives is
discussed in Sec.2.2.
2.1 Research Context
The beam is manufactured from a Glass Fibre Reinforced Plastic (GFRP)
and its box like beam. This beam was actually used as a prototype for
footbridge. The GFRP (Glass Fibre Reinforced Polymer) composite
materials are being utilised in more structures like bridges as the
technology and understand of them improves. GFRP composite are ideal
for structural applications where high strength to weight and stiffness to
weight ratios are needed. As the technology progresses, the cost involved
in manufacturing and designing composite material will reduce, thus
bringing added cost benefits also.
The vibration analyses in footbridge have been a problem for structural
designer for years and have increased recently. The trend in footbridge
design has been towards greater spans and increased flexibility and
lightness. Though, many of our footbridges have natural frequencies with
the potential to suffer excessive vibrations under dynamic load induced by
pedestrians. However, as with any structure, the loading upon it will need
to be considered so that it can perform as intended when it is commissioned
for the usage. This is done by using modal analysis, which allows one to
determine the natural frequencies of the structure, associated mode shapes
and damping. And once natural frequencies are known, thus making
structure suitable for the task designed for.
This is mainly due to the human feeling of vibration while crossing a
footbridge with a frequency close to the first (fundamental) natural
frequency of the bridge, although the vibration caused by the pedestrians
are far from harmful to the bridge. Therefore vibration analysis of such
structure can be considered to be a serviceability issue. Modal parameters
of a structure are frequency, mode shape and damping. Frequency is
directly proportional to structure’s stiffness and inverse of mass.
Nevertheless, modal parameters are functions of physical properties of the
structure. Thus, changes in the physical properties such as, beginning of
local cracks and/or loosening of a connection will cause detectable changes
in the modal properties by reducing the structure’s stiffness.
The design of GFRP (Glass Fibre Reinforced Polymer) bridge deck
established to promote the use of innovative material and lead to use in
footbridge construction to improve the reliability for proper safety and
serviceability. The Aberfeldy cable-stayed was the first GFRP footbridge,
built back in 1992. There are only two GFRP footbridge existing in UK
knows as the Halgavor and Willcot [1&2]. The use of glass or carbon fibre
reinforced polymer was due to its advantages, for they are easily drawn into
having a high strength-to-weight ratio, low maintenance and lightweight.
In this research, Finite element models for different boundary conditions
are constructed using the commercial finite element software package
ANSYS to support and verify the dynamic measurements. Initial a FE 3D
model of FRP (Fibre Reinforced Polymer) composite beam was created
without curvature using beam elements. Furthermore, the FE results of the
straight beam are compared to the exact solution obtained from analytical
solution for understanding of the relationship between the FE results. The
natural frequencies and modes shapes of the composite beam are obtained
after performing modal analysis which the author contribution to this area
of research.
2.2 Objectives
The main objective of this thesis is to study and compare the analytical and
experimental result in vertical forces on the composite Fibre Reinforced
Polymer (FRP) beam. The beam assumed as prototype of bridge in which
pedestrians impart to the model. Special attention is given to the responses
of a structure due to dynamic test (impact excitation) results compare with
theoretical results.
The GFRP (Glass Fibre Reinforced Polymer) composite beam was used to
carry out good experimental research, for comparison of the results, and to
be able to relate to the reality of the footbridge built with composite beam.
2.3 Review of previous work
Composite Materials FRP for Bridge Applications
Many researchers have worked on composite material especially in the
constructions of bridges. The first pedestrian FRP bridge was built by the
Israelis in 1975. Since then, others have been constructed in Asia, Europe,
and North/South America. Many innovative pedestrian bridges have been
constructed throughout the Europe especially United Kingdom using
protruded composite structural shapes which are similar to standard
structural steel shapes. Because of the light- weight materials and ease in
fabrication and installation, many of these pedestrian bridges were able to
be constructed in inaccessible and environmentally restrictive areas without
having to employ heavy equipment [3]. Some of these bridges were flown
to the sites in one piece by helicopters; others were disassembled and
transported by mules and assembled on site. The advancement in this
application has resulted in the production of second generation protruded
shapes of hybrid glass and carbon FRP (Fibre Reinforced Polymer)
composites that will increase the stiffness modulus at very little additional
cost. The recognition of providing high quality fibres at the most effective
regions in a structural member’s cross section is a key innovation to the
effective use of these high performance materials.
With the knowledge gained from working with pedestrian bridges, many
researchers took the next step toward designing FRP composite vehicular
bridges. Many deck systems were developed and tested beginning in the
early 1990's. there are no many FRP bridges around the world, we have
only know of two completed bridges in United Kingdom, Halgavor (firth
2002) and Wilcott which is most recent one. The first world all-composite
vehicular, public bridge was placed in service on 1992 in Aberfeldy. It was
a wet lay-up manufacturing method, a technology transfer from the defence
industry. The FRP (Fibre Reinforced Polymer) deck panels were shopfabricated with composite honeycomb cells sandwiched between two face
It became a common material for construction of deck after the first US in
1996, Kansas. FRP (Fibre Reinforced Polymer) composites have a high
tensile strength; however, in almost all of the demonstration bridge projects
constructed to date, the design has been driven by the stiffness rather than
strength. There is still much room for improvement and advancement of
the composite deck systems in order to capitalize on its material strength.
The key to successful application of the deck superstructure system is to
optimize its geometric cross section and to establish well-defined load
Because most of these deck systems are sealed and enclosed, they are
inaccessible for field inspection. To ensure the composites integrity,
sophisticated non-destructive evaluation/testing (NDE/NDT) devices and
fibre optic sensors have been incorporated into some of the composite deck
systems to monitor the in-service condition of and the presence of moisture
in the bridge deck. With time the effectiveness of the monitoring systems
and the long-term service performance of composites can be ascertained.
The modular panel construction of bridge deck systems enables quick
project delivery. A bridge built of composite materials can be constructed
and put in service in a relatively short time and at a competitive cost. Its
lightweight materials and ease of construction provide tremendous labour
and traffic control cost savings to offset a higher first cost. An FRP deck
could reduce the weight of conventional construction by 70 to 80 percent.
This technology has demonstrated that a bridge structure can be replaced
and put into service in a matter of hours rather than days or months. The
conveniences is replacing composite bridges is comparable, it is due to
innovative technology put to good use.
The FRP composites offer the potential to eliminate the problem of
excessive dead load of long span bridges. Structural components for hybrid
bridge construction such as FRP (Fibre Reinforced Polymer) reinforcing
elements, cable and tendon systems, and laminates have been successfully
demonstrated in highway bridges. The reinforcing elements are fabricated
into 1-D rods, 2-D grids or gratings, and 3-D fabric or cage systems.
The application of composite materials to infrastructure has been limited
due to the lack of industry-recognized design criteria and standards and
standardized test methods [4]
Cable and tendon systems are subject to fatigue, and under sustained loads
will creep. Creep rupture is a major concern for glass fibres. Proper
selection of fibres and adequate design criteria must be established to
ensure the proper use of these products and are essential to the
advancement of FRPs in pre-stressing applications. Laminates are
protruded with unidirectional fibres to form thin and narrow plates
There is much work to be done in developing well-designed anchorages,
connection details, and bonded joints in composites for long-term
durability. Bridge engineers are reluctant to rely solely on epoxy adhesive
bonding technology to connect or join structural components. Electrical
transmission towers out West have been built with connections that were
snapped and locked together without the use of any fasteners. It is a tough
challenge, but when adequate testing and performance data are available,
bridge engineers will change their paradigm.
Since, thousands of concrete bridge piers that were designed with
inadequate ductility, lap splices, and shear capacity have been successfully
retrofitted using FRP (Fibre Reinforced Polymer) composite wrap systems.
In Cali it has been determined that when a wrap system is properly
designed and installed, the ductile capacity can be significantly increased to
allow twice the deformation levels without any reduction in its capacity as
compared to the as-built bridge piers[5].
Several countries have successfully applied FRP (Fibre Reinforced
Polymer) composite materials to the repair of damaged or deteriorated
beams in highway bridges. With this application, the condition of the
deterioration in the concrete behind the composite materials remains
uncertain. A good repair program should include an evaluation of the preexisting condition and structural integrity of the concrete to establish a
baseline reference. After a structural member has been repaired, the inservice condition of the concrete substrate as well as the performance of the
composites should be continuously followed up.
Recent Research in FRP
Research studies and results of the composite technology and its durability
should be published in civil, mechanical and structural engineering
journals. Technical papers should be presented in conferences and
workshops where civil and structural engineers participate. Until recently,
most of the papers have been published in the materials science and testing,
manufacturing and trade journals, which are not read by bridge designers.
Information and knowledge must be openly shared with civil engineers,
bridge designers, and owners. Professional organizations should be
dedicated and/or established to direct the technical advancement of the
composite technology if it is to have a future.
Composite material bridges have become increasingly popular in structural
applications around the world. This is partly due to their excellent
earthquake-resistant properties such as high strength, high ductility, and
large energy absorption capacity. Although the risk of a major earthquake
in United Kingdom is small, this type of structure can offer many other
advantages, for instance the increased speed of construction; positive safety
aspects; the steel tube's function as both formwork and reinforcement for
the concrete core; and possible use of simple standardized connections.
Today's possibility to produce composite with higher compressive strength
allows the design of more slender columns, which leads to greater profits.
To prevent the brittle failure that is normally associated with-high strength
deck and obtain a higher ductility, the stirrup spacing is often decreased.
A lot of work is needed in material science development and production.
Design and construction specifications are important protocol for engineers,
and they need to be developed. Bridge owners must know how to inspect,
maintain, and repair FRP (Fibre Reinforced Polymer) composite bridges
and these procedures must be established.
We need to involve more knowledgeable and experienced engineers to help
develop standards, design guidelines, and specifications.
This is an exciting time for civil and structural engineers to be involved
with the FRP (Fibre Reinforced Polymer) composite technology. Although
most engineers are not trained to work with fibre and resin composite
materials, they can quite easily pick up the knowledge. Education and
training should be provided to the critical mass of practicing engineers.
Research studies and results of the composite technology and its durability
should be published in civil and structural engineering journals. Technical
papers should be presented in conferences and workshops where civil and
structural engineers participate. Until recently, most of the papers have
been published in the materials science and testing, manufacturing and
trade journals, which are not read by bridge designers. Information and
knowledge must be openly shared with civil engineers, bridge designers,
and owners. Professional organizations should be dedicated and/or
established to direct the technical advancement of the composite
technology if it is to have a future [6].
There is still more work needed in material science development and
production. Design and construction specifications are important protocol
for engineers, and they need to be developed. Bridges owners must know
how to inspect, maintain, and repair Fibre Reinforced Polymer (FRP)
composite bridges and these procedures must be established.
We need to involve more knowledgeable and experienced engineers to help
develop standards, design guidelines, and specifications when we have an
approved design standard specifications for FRP (Fibre Reinforced
Polymer) composites.
Characteristics of FRP Structural Beam
Nagaraj and Rao [7] have characterized the behaviour of protruded FRP
box beams under static and fatigue or cyclic bending loads. The author
showed that the shear and interfacial slip between adjacent layers had
significant influence on deflection and strain measurements. Davalos and
Qiao [8] conducted a combined analytical and experimental evaluation of
flexural-torsional and lateral-distortional buckling of FRP composite wideflange beams. They also showed that in general buckling and deflections
limits tend to be the governing design criteria for current FRP shapes. The
structural efficiency of protruded FRP (Fibre Reinforced Polymer)
components and systems in terms of joint efficiency, transverse load
distribution, composite action between FRP components, and maximum
deflections and stresses was analyzed by Sotiropoulos, Gangarao, and
Allison [9] by conducting experiments at the coupon level. Structural
performance of individual FRP components was established through threeand four-point bending tests. Barbero, Fu, and Raftoyiannis [10] gave a
theoretical determination of the ultimate bending strength of GFRP beams
produced by protrusion process. Several I-beams and box beams were
tested under bending and the failure modes have been described. The
researchers do attempt to accelerate fatigue damage by testing at loads
much higher than the service load. And there are possibilities for different
damage to occur at different load levels.
FRP (Fibre Reinforced Polymer) beam involves testing the beam in either
three-or-four-point bending using a simply supported geometry, where steel
structural beams are used as supports. Load is applied using hydraulic
actuator to representative wheel patches. The test usually runs in various
span lengths to determine the shear stiffness, since the percentage of shear
deformation increases with decreasing span length.
Bank and Mosallam [11] has characterized creep response of composite
structural element.
They tested a plane portal frame consisting of a girder supported by
columns with fibreglass angles and FRP (Fibre Reinforced Polymer)
threaded rods and nuts used in the connection details.
Problem with Composite Material
There are several inherent difficulties in detecting damage in composite
materials as opposed to traditional engineering such as plastics, FRP (Fibre
Reinforced Polymer). One reason is due to its non-homogeneity and
anisotropy; most metals and plastics are formed by one type of uniformly
isotropic material with very well know properties. Other composite material
on the other hand can have a widely varying set of material properties
based on the chosen fibres, matrix and manufacturing process. This makes
model composite complex and often a mix between materials with widely
differing properties, such as a very good conducting fibre in an insulating
The most difficult is the damage in composite material often occurs below
the surface, which further prevents the implementation of several detection
methods. The importance of damage detection for composite structures is
often accentuated over that of metallic or plastic structures because of their
load bearing requirements.
Advantages of Composite Beam
Basically, one of the primary motivations for utilizing FRP composite in
infrastructure environment is that composite are generally considered to be
more corrosion resistant than metals[12,13]. The superior durability of
composite is attributed to their relative chemical stability and resistance of
fatigue crack growth. An improvement in durability would reduce
maintenance costs and lengthen the service lives of bridges. The high
strength to weight ratio of composite is also attractive and could potentially
reduce labour costs associated with transportation and installation [11].
Other significant advantages include higher damping and energy
absorption, high dielectric strength, and greater suitability for prefabrication
Hence, the advantage of using composite material FRP (Fibre Reinforced
Polymer) instead of traditional concrete is apparent.
Disadvantages of FRP Composite Beam
FRP Composite are generally more expensive than concrete or metallic
structures, the improved durability may also make them more cost-effective
in terms of life-cycle costs [14].
The relatively low stiffness of FRP composite is an obstacle to their usage
in civil engineering applications, as most bridge designs are deflectioncontrol [12, 14]. Serviceability requirement are also difficult to meet with a
composite design. For instance, with fiberglass bridge decks or even
composite superstructures, the large deformations may deteriorate the
concrete overlay and deck-to-support connections prematurely [12, 13].
Applications of FRP Composite Beam
It’s mainly used in bridge structures nowadays. Since environmental
durability is often cited as an advantage for composites, long-term aging
data to support these claim is not conclusive.
Most of experience has been derived from the aerospace industry where the
service live are much shorter than those required in infrastructure [15].
Application of FRP (Fibre Reinforced Polymer) composite in bridges can
be classified in to two members; Primary and Secondary applications.
Primary members include post-stressing and bonded repair applications
while secondary members; guard rails, diaphragms, reinforcement, column
wrapping, stay-in-place forms, and handrails. Also it can be used in
shielding other structural elements from the environment: expansion joint,
bridge bearings, and drainage shielding system [15].
3 Basic Concepts and Theory
3.1 Structural dynamics
Structural dynamics describe the behaviour of a structure due to dynamic
To obtain the responses of the structure a dynamic, the inertia force,
damping force and stiffness force together with the externally applied force:
= f (t )
is the inertial force of the mass .
= mu ,
damping force and is related to the velocity of the structure by
is the
= cu
is the elastic force exerted on the mass and is related to the
displacement of the structure by
= ku , where k is the stiffness, c is the
damping ratio and m is the mass of the dynamic system.
m u + c u + ku = f (t )
Two different dynamic models are presented in the following sections.
SDOF model
Equation of motion
The generalized equation of motion for a SDOF-system is of the form
~ &z& + c~ z& + k~ z = f ( t )
~ , c~, k~ , and ~
where m
f (t ) are defined as the generalized mass, generalized
damping, generalized stiffness and generalized force of the system.
Generalized mass and stiffness can be calculated using the following
~ =
∫ m ( x ) [ψ
( x ) ] dx
k = ∫ EI ( x)[ψ ′′( x)] dx
where m(x) is mass of the structure per unit length, EI(x) is the stiffness of
the structure per unit length and L is the length of the structure [6].
The generalized damping can then be calculated from the expression
~ω )
c~ = ζ (2m
where ω is the natural frequency of the structure.
~ , c~, k~ , and ~
Once the generalized properties m
f (t ) are determined, the
equation of motion (Equation. 3.3) can be solved for z(t) using a numerical
integration method.
MDOF model
All real structures have an infinite number of degrees of freedom (DOF’s).
It is, however, possible to approximate all structures as an assemblage of
finite number of mass less members and a finite number of node
displacements. The mass of the structure is lumped at the nodes and for
linear elastic structures the stiffness properties of the members can be
approximated accurately. Such a model is called a multi-degree-of-freedom
(MDOF) system.
Modal analysis includes the formulation of the eigenvalue problem and a
solution method for solving the eigenvalue problem. Finally, modal
analysis can be used to compute the dynamic response of an MDOF system
to external forces.
Equation of motion
The equation of motion of a MDOF system can now be written on the form:
Mu&& + Cu& + Ku = f (t )
which is a system of N ordinary differential equations that can be solved for
the displacements u due to the applied forces f (t ) . It is now obvious that
Equ. 3.7 is the MDOF equivalent of Equ. 3.3 for a SDOF system [5].
3.2 Experiment Modal analysis
The modal parameters of simple structures can be easily established by the
use of Analyzer System. The frequency response function of a structure can
be separated into a set of individual modes. By using an analyzer System
each mode can be identified in terms of frequency, damping and mode
In practice, nearly all vibration problems are related to structural
weaknesses, associated with resonance behaviour (that is natural
frequencies being excited by operational forces). It can be shown that the
complete dynamic behaviour of a structure (in a given frequency range) can
be viewed as a set of individual modes of vibration, each having a
characteristic natural frequency, damping, and mode shape. By using these
so-called modal parameters to model the structure, problems at specific
resonances can be examined and subsequently solved.
The first stage in modelling the dynamic behaviour of a structure to
determine the modal parameters are
The resonance, or modal, frequency
the modal damping
The mode shape
The modal parameters can be determined from a set of frequency response
measurements between a reference point and a number of measurement
points. Such a measurement point, as introduced here, is usually called a
Degree-of-Freedom (DOF). The modal frequencies and damping can be
found from all frequency response measurements on the structure (except
those for which the excitation or response measurement is in a nodal
position, that is, where the displacement is zero). These two parameters are
therefore called “Global Parameters”. However, to accurately model the
associated mode shape, frequency response measurements must be made
over a number of Degrees-of-Freedom, to ensure a sufficiently detailed
covering of the structure under test.
In practice, these types of frequency response measurements are made easy
by using a Dual Channel Signal Analyzer such as the standard four-channel
configuration of Phaser, the Multi-analyzer System. The excitation force
(from either an impact hammer or a vibration exciter provided with a
random or pseudorandom noise signal) is measured by a force transducer,
and the resulting signal is supplied to one of the inputs.
If a vibration exciter is used, a generator module should be installed in the
analyzer. The response is measured by an accelerometer, and the resulting
signal is supplied to another input. Consequently, the frequency response
represents the structure’s accelerance. Since the measured quantity is the
complex ratio of the acceleration to force, in the frequency domain. For
impact hammer excitation, the accelerometer response position is fixed and
used as the reference position. The hammer is moved around and used to
excite the structure at every DOF corresponding to a DOF in the model. For
vibration exciter excitation, the excitation point is fixed and is used as the
reference position, while the response accelerometer is moved around on
the structure.
For structures defined with a large number of DOFs, the Multi-analyzer
Systems can be equipped with up to eight four-channel modules (without
expanding the physical dimensions of the system) to allow for easier and
faster mobility measurements.
Theoretical Derivation
When performing modal analysis, the free vibrations of the structure are of
interest. Free vibration is when no external forces are applied and damping
of the structure is neglected. When damping is neglected the eigenvalues
are real numbers.
The solution for the undamped natural frequencies and mode shapes is
called real eigenvalue analysis or normal modes analysis. The equation of
motion of a free vibration is:
Mu&& + Ku = 0
This equation has a solution in the form of simple harmonic motion:
u = φ n sin ω n t and
u&& = −ω 2nφ n sin ω n t
Substituting these into the equation of motion gives
Kφ n = ω n2 Mω n
which can be re-written as
[K − ω M ]φ
This equation has a nontrivial solution if
det K − ω n2 M = 0
Equation 3.12 is called the system characteristic equation. This equation
has N real roots for ω n2 , which are the natural frequencies of vibration of
the system. They are as many as the degrees of freedom, N . Each natural
frequency ω n has a corresponding eigenvector or mode shape φn, which
fulfils equation 3.11. This is the generalized eigenvalue problem to be
solved in free vibration modal analysis.
After having defined the structural properties; mass, stiffness and damping
ratio and determined the natural frequencies ω n and modes φ n from
solving the eigenvalue problem, the response of the system can be
computed as follows. First, the response of each mode is computed by
solving following equation for q n (t )
M n q&&n + C n q& n + K n q n = f n (t )
Then, the contributions of all the modes can be combined to determine the
total dynamic response of the structure
u (t ) = ∑ φ n q n (t )
n =1
The parameters M n , K n , C n and f n (t ) are defined as follows
mn = φ nT Mφ n , k n = φ nT Kφ n , c n = φ nT Cφ n and f n = φ nT Fφ n , (3.15)
and they depend only on the n th-mode φ n , and not on other modes. Thus,
there are N uncoupled equations like Equ. 3.13, one for each natural mode.
In practice, modal analysis is almost always carried out by implementing
the finite element method (FEM). If the geometry and the material
properties of the structure are known, an FE model of the structure can be
built. The mass, stiffness and damping properties of the structure are
represented by the left hand side of the equation of motion (E.q. 3.7), can
then be established using the FE method. All that now remains, in order to
solve the equation of motion, is to quantify and then to model
mathematically the applied forces F (t ) .
4 Beam Structure
4.1 Introduction
Since beam is continuous structure, it can be modelled using three basics
one dimensional element types are: (a) String element - for vibration in
cables and wires, (b) Bar element- for transverse, (c) Beam element - for
lateral vibration
This formula being used to calculate the natural frequency developed from
equation of motion of the beam. It has stated below
ω n = (β n L )2
Where (β i L ) and table below 4.1 can be used to verify β n L , constant value.
Where c is a constant value, E is Young modulus ( N / m 2 ) , I is the second
moment of area (m 4 ) , ρ is the density (kg / m 3 ) , A is the cross-section
area (m 2 ) , ω n is the natural frequency of vibration (rad/sec), L is the length
of the beam, i represents number of frequencies and β n is determined from
boundary conditions at the two ends of the beam. The beam simply
supported from one end and fixed on the other end. These boundary
conditions with the relevant values of β n L and mode shapes shown in
below table 4.1
Table 4.1: Angular frequencies and mode shapes for a beam in transversal
Values of
( βiL)
( β0L) = 0 (rigid
(β1L) = 4.730
(β2L) = 7.853
(β3L) =10.995
(β4L) =14.137
Mode Shapes
U n = C n (sin β n x + sinh β n x + α n (cos β n x + cosh β n x ))
⎡ sin β n L − sinh β n L ⎤
where ...α n = ⎢
⎣ cosh β n L − cos β n L ⎦
(β1L) =4.730
(β2L) =7.853
(β3L) =10.99
(β4L) =14.13
(β1L) =π
(β2L) = 2π
(β3L) = 3π
(β4L) = 4π
U n = C n (sinh β n x − sin β n x + α n (cos β n x − cosh β n x ))
(β1L) = 3.926
(β2 L) = 7.068
(β3L) = 10.210
(β4 L) = 13.351
U n = C n (sin β n x − sinh β n x + α n (cosh β n x − cos β n x ))
(β1L) = 1.875
(β2 L) = 4.694
(β3 L) = 7.854
(β4 L) = 10.995
U n = C n (sin β n x − sinh β n x − α n (cos β n x − cosh β n x ))
⎡ sinh β n L − sin β n L ⎤
where ...α n = ⎢
⎣ cos β n L − cosh β n L ⎦
U n = Cn sin β n x
⎡ sin β n L − sinh β n L ⎤
where ...α n = ⎢
⎣ cos β n L − cosh β n L ⎦
⎡ sin β n L − sinh β n L ⎤
where ...α n = ⎢
⎣ cos β n L − cosh β n L ⎦
4.2 Finite Element Method Calculations
Consider the beam as uniform beam element as shown in figure 4.1, the
nodes i and j are located at the two beams end. The element is subjected to
a transverse force distribution F, which is function of time and location.
Each node has two degree of freedom, translation in the y-direction, u y and
rotation φ . The nodal degree of freedom at nodes I are denoted as U I and
φi and those at node j as U j and φ j . Whilst the external forces acting at
node I are FI and M I (bending moment) and those acting at node j are
FJ and M J . The element has a Young’s modulus E, a uniform crosssection area A, a moment of inertia I and density ρ . It is assumed that the
transverse displacement variation in the x-direction is cubic, i.e.,
u y = C1 + C2 . X + C3 . X 2 + C4 .x 3
Figure 4.1 Beam Element Configurations
Where the constant C1 , C2 , C3 and C4 are in general function of time and can
be determined from the boundary conditions. The boundary conditions in
general forms are:
x = 0 ⇒ u x = ui , φ =
∂u y
= φi
x = L ⇒ uy = u j , φ =
∂u y
= φj
Equation (4.1) should satisfy the conditions in equation (4.2) so that C1 and
C2 can be found as
C1 = U I
C2 = φ I
(− 3U I − 2φI L + 3U J − φJ L )
C4 = 3 (2U I + φI L − 2U J + φJ L )
C3 =
Substituting equation (4.3) into equation (4.1), the below is obtained which
equation (4.4)
⎛ 3x2 2x3 ⎞
⎛ 2x2 x3 ⎞ ⎛ 3x2 2x3 ⎞
⎛ x2 x3 ⎞
+ 2 ⎟⎟φi + ⎜⎜ 2 − 3 ⎟⎟U j + ⎜⎜ − + 2 ⎟⎟φ j
uy = ⎜⎜1 − 2 + 3 ⎟⎟Ui + ⎜⎜ x −
L ⎠
L L⎠ ⎝L
L ⎠
⎝ L
⎝ L L⎠
U y = N iU i + N 'iφi + N jU j + N 'jφ j
Where N i , N i' , N j , N 'j are the shape functions.
3x 2 2 x3
Ni = 1 − 2 + 3
N i' = x −
+ 2
Nj = 2 − 3
N 'j = − + 2
The shape functions are shown in figure. 2 below
Figure 4.2: Shape function of a beam element
The kinetics energy, T of the beam is obtained as:
T =
1 ' 2
ρ Vy dx
2 ∫0
Where ρ ' is the mass per unit length and Vy is the velocity in the ydirection. Substituting by ρ ' = ρA and Vy =
∂u y
, into equation (4.6), gives
⎡ ∂u y ⎤
T = ∫ ρA⎢
⎥ .dx
2 0 ⎣ ∂x ⎦
Substituting equation (4.4) into equation (4.7), yields
⎡⎛ 3 x 2 2 x 3 ⎞
⎛ 3x 2 2 x3 ⎞
⎛ x 2 x3 ⎞ ⎤
2 x 2 x3 ⎞
+ 2 ⎟⎟φi + ⎜⎜ 2 − 3 ⎟⎟U j + ⎜⎜ −
+ 3 ⎟⎟φ j ⎥ dx
T = ∫ ρA⎢⎜⎜1 − 2 + 3 ⎟⎟U j + ⎜⎜ x −
L ⎠
L ⎠
L ⎠
2 0 ⎣⎝
⎝ L
⎝ L L ⎠ ⎦
Ui =
∂U i
∂U j
Uj =
φj = j
Integrating equation (4.9), and writing in matrix form, gives:
{U }T [m].{U }
where the superscript T indicates the transpose. The velocity vector, ⎨U ⎬ ,
⎩ ⎭
is given by:
⎧ . ⎫
⎪U i ⎪
⎪. ⎪
⎧ ⎫ ⎪⎪ i ⎪⎪
⎨U ⎬ = ⎨ . ⎬
⎩ ⎭ ⎪U j ⎪
⎪. ⎪
⎪φ j ⎪
⎪⎩ ⎪⎭
And the mass matrix [m] is given by:
22 L 54 − 13L ⎤
⎡ 156
⎢ 22 L
4 L2 13L − 3L2 ⎥⎥
ρAL ⎢
[m] =
13L 156 − 22 L ⎥
420 ⎢ 54
2 ⎥
⎣− 13L − 3L 22 L 4 L ⎦
Using the Euler-Bernoulli beam theory, the strain energy of the beam
element can be expressed as:
⎛ ∂ 2U Y
W ∫ EI ⎜⎜
2 0 ⎝ ∂x 2
⎟⎟ .dx
Differentiating equation (4.4) with respect to x, substituting into equation
(4.13), carrying the integration and writing in matrix form, yield
W =
{U }T [k ].{U }
Where the displacement vector, {U }, is given by:
⎧U i ⎫
⎪φ ⎪
{∪} = ⎪⎨Ui ⎪⎬
⎪ j⎪
⎪φ j ⎪
⎩ ⎭
And the stiffness matrix [k] is calculated as:
6 L − 12 6 L ⎤
⎡ 12
⎢ 6 L 4 L2 − 6 L 2 L2 ⎥
⎥. EI
[k ] = ⎢
⎢− 12 − 6 L 12 − 6 L ⎥ L3
2 ⎥
⎣ 6L 2L − 6L 4L ⎦
The equation of motion is derived using Netwon’s second law and is
defined as:
([k ] − ω [m]){Um} = 0
The mass and stiffness matrices increases along with increasing element of
the beam and more nodes and more degree of freedom will be involved.
5 Experimental Work
5.1 Introduction
The experimental work was carried at University of Surrey, Dynamics
Research Laboratory. The dynamic test was conducted using an intact FRP
composite beam with different boundary conditions. The main aim of this
experimental work is investigate the modal parameters (frequency, mode
shapes and modal damping) of the FRP composite beam. This test beam
was supplied by STRONGWELL Company and the manual (contain both
mechanical and physical properties) provided by the same company being
used during the course of this research.
Measurement Preparations
This is done to ensure that the measurement will be as satisfactory as it can,
pre-preparation are very important. How well the pre-preparation done will
definitely determine the betterment of the expected data in our experiment.
These are significance checks that have been done:
Identification of experimental model
Excitation method
The marked point on the beam sample
Measuring method
Hammer excitation force transducer
Selection of the excitation signal
Identification of Experimental Model
This is done by identifying the experimental model as prototype of
footbridge (Footbridge Bridge). Since, it has been constructed with FRP
(Fibre Reinforced Polymer) composite beam and set up as below in Figure
Figure 5.1: Measurement FRP Beam
Position of the Accelerometer on the Beam
It is very mandatory to know position at which the accelerometer will be
placed. This will be done in order to know the position where to put the
reference and moveable accelerometer during the experiment measurement.
We will use two different accelerometers and the detail will be explained in
next chapter. We have 9 different point where accelerometer being
positioned during the experiment. The distance between each point is
relatively the same
Figure 5.2: Point of the accelerometer
Point of Excitation
It is determine by dividing the model beam into 9 different points. Some
mark has been placed on the FRP composite to indicate these points. The
point of excitation decided on by using residual knowledge on dynamic
test, therefore the excitation point is considered as point 3.
Figure 5.3: Point of Excitation at point 3
5.2 FRP Composite Beam Test Descriptions
In this research work, fibre reinforced polymer (FRP) beams were used as
the test specimens. The dimensions of fibre reinforced polymer beams with
100 mm length, width 5 mm, thickness 3 mm. the FRP beam has the
following properties cross-section area of 0.564e-3 m 2 , young modulus
17.926e9 N / m 2 , density 1827 Kg / m3 and poison ratio 0.3. Figure 5.4
shows the cross sectional details of the test beams.
Figure 5.4: Details of cross-section of the test beam
The length of the beam is about 0.82 with different 9 point as described in
chapter 5, distance between each point on the beam is equivalent
1.025e −1 m. two different accelerometer were used during the experiment,
one being used as reference while other used moveable. The connection
cable pointed toward y, z–direction to the Signal analyzer of the each
accelerometers that is all response measurement was toward vertical
direction. The impulses are used to excite different point on FRP composite
beam to get response for different boundary conditions as indicated.
For each case the response will plot in order to see the behaviour i.e. modal
analysis behaviour of the FRP composite beam. All the vibration data
acquired are imported to SPICE to process to acquired modal parameters
are frequencies, damping ratio and mode shapes. We have two different
technique in SPICE, Stochastic subspace identification technique and peakpicking technique. These methods are being used to process the acquired
data, the sampling frequency, 3000Hz. These acquisition data were
collected in Time-Domain and its being process with Stochastic Subspace
The modal parameters (eigenfrequncies and mode shapes only) are
extracted first using frequency-domain i.e., peak-picking technique, in
order to give a quick look at the dynamic performance of the beam. Then
the modal parameters of the beam i.e., natural frequencies, damping ratios
and mode shapes are extracted from the measured data using time-domain
technique i.e. the stochastic subspace identification technique, with
sampling frequency 3000 Hz. The stochastic method works directly with
the recorded time signals and it is based on linear calculations therefore it is
considered to be more robust and faster than other methods. The
experimental set-up is illustrated in figure 5.5.
Figure 5.5: The Experimental Set-up
5.3 Measurement Equipment
The equipment was supplied by School of Engineering, University of
Surrey, England.
Data Acquisition system
Impact hammer
Steel support
PC with Software
Data acquisition system
Phaser Analyzer (FFT analyzer), figure 5.6 was used as data acquisition
system and has 4 input channels
Figure 5.6: Phaser (FFT) analyzer
The accelerometer (Shear accelerometer) is of type 356B07 PCB
Piezotronics CUBE ACCELEROMETER, figure 5.7 and measures the
accelerometer of the vibrations. They are ICP tri-axial accelerometer with
the following calibration values for reference accelerometer at y-axis,
sensitivity 95.6 mV/g (9.75 mV / m / s 2 ), output bias 11.1 VDC, discharge
time constant 0.3 seconds, transverse sensitivity 3.4 % while the z-axis
sensitivity 96.9 mV/g (9.88 mV / m / s 2 ), output bias 11.3 VDC, discharge
time constant 0.3 seconds, transverse sensitivity 3.0 %
Figure 5.7: Accelerometer
Impact Hammer
The impact hammer is of type figure 5.8 and used to determine component
or system response to impacts of varying amplitude and duration. The
impact has Sensitivity: (±15%) 1 mV/lbf (0.23 mV/N), Measurement
Range: ±5000 lbf pk (±22000 N pk), Hammer Mass: 0.32 kg, Tip Diameter:
0.63 cm, Hammer Length: 22.7 cm, Head Diameter: 2.5 cm.
Figure 5.9: Impact Hammer
Steel Support
The steel support, see figure 5.10 with 0.25m height, the length is 0.12m
and the breadth 0.09m. These support used at both edges used to impose
boundary conditions on the beam except free-free. The two support sides
have bolts and nuts at the screw thread area.
Figure 5.10: Steel Support
PC with Software
When measuring, the software Dactron (the real-time professional) was
used. SPICE was software that was used for data analyzing and verification
5.4 Experimental Specifications
The specified apparatus is based around a Glass Fibre Reinforced
Composite (FRP) square section describe in previous section. The
dimension of the beam are conveyed in figure 5.11
Figure 5.11: Geometry of GFRP Composite beam
In reality, the beam is 1000mm in length, however once it been clamped,
the length between the ends become 820mm, as depicted in figure 5.12
Figure 5.12: Beam position on fixed-fixed position
The beam will be clamped in the apparatus as figure 5.12. figure 5.13 is a
more detailed view of an end of the apparatus which allows the end
conditions to be changed between simply supported and fixed, thus altering
the boundary condition to which it is subjected to. This is achieved by
adjusting the bolts which can be seen on the top of the apparatus.
Figure 5.13: End Condition Adjustment
The experimental apparatus will allow the beam will be clamped in four
different boundary conditions as shown figure 5.14.
Figure 5.14: Boundary Conditions
By considering a number of boundary conditions, it will be possible to
observe how the beam behaves when excited and how the vertical
displacement is affected. The natural frequencies will be monitored and the
manner in which they differ between boundary condition will need to be
5.5 Experimental Results and Discussion
This section presents the results from experimental investigation, where
dynamic test with different boundary conditions were carried out. However,
only output data from modal testing namely the mode shapes, frequencies
were used to compare the analytical results.
The dynamic test was carried out in laboratory with four different boundary
conditions. The measured modal parameters are served as a reference for
further comparison with analytical solution agreement. Using ANSYS
software, an initial finite element model containing 9 elements is
constructed using beam4 element type. The beam supports are simulated as
three translation and rotational stiffness from each side using damper
element. Only the first three bending modes are considered in the vertical
direction. The sensitivity matrix of the stiffness supports in the FE model is
calculated. In tables 5.1 and 5.2, the measured natural frequencies for the
first three bending modes are given and compared to the finite element
Table 5.1 Eigenfrequencies (Hz) of fixed-fixed boundary conditions
FRP Beam
% Difference
Figure 5.15 First bending moment
Figure 5.16 Second bending moment
Figure 5.17: Third bending moment
Table 5.2 Eigenfrequencies (Hz) of Fixed- free Boundary Conditions
FRP Beam
% Difference
Table 5.3 Eigenfrequencies (Hz) of Fixed-Simple Boundary conditions
FRP Beam
% Difference
Figure.5.18: First bending moment
Figure 5.19: Second bending moment
Figure 5.20: Third bending moment
Table 5.4: Eigenfrequencies (Hz) of Simple-Simple Boundary conditions
FRP Beam
% Difference
Figure 5.21: First bending moment
Figure 5.22: Second bending moment
Figure 5.21: Third bending moment
6 Modelling in ANSYS
6.1 Introduction
The finite element simulation was done by FEA package known as
ANSYS. The FEA software package offerings include time-tested,
industry-leading applications for structural, thermal, mechanical,
computational fluid dynamics, and electromagnetic analyses, as well as
solutions for transient impact analysis. ANSYS software solves for the
combined effects of multiple forces, accurately modelling combined
behaviours resulting from "multiphysics" interactions.
This is used to perform the modelling of the beam and calculation of
natural frequencies with relevant mode shapes.This is used to simulate both
the linear & nonlinear effects of structural models in a static or dynamic
environment. Advanced nonlinear structural analysis includes large strain,
numerous nonlinear material models, nonlinear buckling, post-buckling,
and general contact. Also includes the ANSYS Parametric Design
Language (APDL) for building and controlling user-defined parametric and
customized models.
The purpose of the finite element package was utilised to model the Fibre
reinforced polymer (FRP) beam in 3-D as SHELL93 (8node93). This
package enables the user to investigate the physical and mechanical
behaviour of the beam.
The FE-model parameters extracted from the Strongwell manual [19]
provided with the composite beam specimen. The FE-model constructed
along vertical direction only which made it applicable to the real bridge
model. The load applied from pedestrian used to come in vertical directions
during the walking or movement along the bridge that is why the analysis is
being done toward vertical directions. Though, the specimen has anisotropy
properties but we have only considering the vertical direction that is why
the linear isotropic parameter only used.
6.2 Procedure in Modelling ANSYS
There are major and sub important steps in ANSYS model, pre-processing,
solution stage and post-processing stage.
Figure 6.1: FE-Analysis Steps
Requirement Specification
This step is done in pre-processing in ANSYS. In this work the beam
element model used know was SHELL93 and it was specification at the
pre-processing stage. The SHELL93 element is applicable to this model for
the structural meshing and boundary condition applications.
Table 6.1:Input data for Modelling of the beam
Geometry Definition
Young modulus
Length of the beam
Poisson Ratio
The parameter specified in the table above indicated that only vertical
direction analysis was carried on the beam. This is also applicable to the
modal analysis experiment in the previous section.
Idealization Specification
This is sub-stepping procedure in model context represents a 3D shell
definition. This model is optimized for rapid FEM analysis and is
composed of 2D geometry, beam surface model. It is easy to locate and
calculate the numerical position in shell geometry; beam shell model can be
defined of the 3D definition. The analysis type is defined as modal
Mesh Generation
The generation of a mesh on the idealized geometry is done through
meshed model. The meshing depend on the configuration for the model, the
general rules are carried out by setting a density for the mesh. In this
application, loads and boundary conditions are added in the input file. The
solver input file consists of mesh elements, nodes and load cases. The input
file is generated from the application containing mesh elements, nodes and
boundary conditions are added to the file.
This is a stage where solution was conducted. It was the step to preprocessing and different stages of analysis took place. The load is applied
to edges of beam, this was easier to implement in SHELL model. And the
other entire complex algorithm in FEM solved.
At this stage the results of analysis are obtained numerically and
6.3 ANSYS Graphical Results
These are following dynamic analysis for different boundary conditions.
The result obtained are generated inform of graphical view which show the
modal concept and influence of each boundary conditions for the vertical
frequencies of the FE shell model of the beam.
Simple-Simple Boundary Condition
The results shown below are the graphical solution of deformed and undeformed shape for first 2 modes.
Figure 6.2: Mode 1 under simple-simple boundary conditions
Figure 6.3: Mode 2 under simple-simple boundary conditions
Table 6.2: the table shows the mode frequencies in Hz predicted theory and
Percent Error
Fixed-Fixed Boundary Condition
The results shown below are the graphical solution of deformed and undeformed shape for first 2 modes.
Figure 6.4: Mode 1 under fixed-fixed boundary conditions
Figure 6.5: Mode 2 under fixed-fixed boundary conditions
Table 6.3: the table shown the mode frequencies in Hz predicted theory and
Mode Theory
Percent Error
Fixed-Simple Boundary Condition
The results shown below are the graphical solution of deformed and undeformed shape for first 2 modes
Figure 6.6: Mode 1 under fixed-simple boundary conditions
Figure 6.7: Mode 2 under fixed-simple boundary conditions
Table 6.4: the table shown the mode frequencies in Hz predicted theory and
Mode Theory
Percent Error
Fixed-Free Boundary Condition
The results shown below are the graphical solution of deformed shape for
first 2 modes
Figure 6.8: Mode 1 under fixed-free boundary conditions
Figure 6.9: Mode 2 under fixed-free boundary conditions
Table 6.5: the table shown the mode frequencies in Hz predicted theory and
Mode Theory
Percent Error
The ANSYS results also show small relative errors compared with the
analytical solution as shown in table 6.2, 6.3, 6.4 and 6.5.
7 Discussion
7.1 Introduction
With the completion of experimental, analytical and ANSYS section of this
research work, it is now possible to analyse result found in chapter 5 of this
work. This section will focus on the findings of the studys, the comparisons
which can be made between the different methods, and most importantly,
the effect that frequency has on the FRP composite beam.
7.2 Comparison of Method
Four different area of analysis was conducted on the beam, analytical
method, finite element methods (1 and 2 elements), Experimental study,
ANSYS Model.
Different observations were made in this work from previous sections. It is
mandatory to mention the density values used within the calculations and
ANSYS study. This manual provided with the composite beam specimen
was used in the experiment, the density was quoted as having a value
ρ = 1716 − 1937kg / m 3 . there are different density values considered when
determining the natural frequencies using analytical and finite element
methods and ANSYS. The mean average of the maximum and minimum
density values was taken and found to be ρ = 1827kg / m 3 . these three
density values adequately covered the density range provided.
When comparing natural frequencies calculated using different methods
and the three density values, it can be seen that although the natural
frequencies obviously vary, the spread of the results is fairly small (due to
the small density change). It was noticed that the value of
ρ = 1827kg / m 3 best suitable the experimental results and it was decided to
use this density values in all the comparisons in the every part of this work.
Reference to the appendix 3, when comparing analytical method to the
finite element with 1 element, it can be seen that although the values
obtained for the natural frequencies are in a similar, there are some issues
with inaccuracy. This is because finite element method is an approximate
solution which is intended to be use when analytical solution is difficult to
obtain. Significant improvement may be seen when the number of elements
considered is increased from one to two in Appendix 4. by comparing
results and calculating the relative errors between the analytical method and
the finite method at the first natural frequency, table 7.1 below was
Table 7.1: % Error between FEM & Analytical Solution for Mode 1
Finite Element Method-1
Finite Element Method-2
Element (Percent error)
Element (Percent error)
From the table above, it can be seen that the relative errors between the
finite element method and the analytical solution are significant reduced
when the number of elements is increased, there will be a greater number of
nodes in the structure which means that a better idea of the displacement
and deformation of the structure can be found. This will lead to mode
shapes of the system becoming more similar to the analytical mode shape.
In a similar manner, the ANSYS results also show small relative errors with
the analytical solution as found in table 6.2, 6.3, 6.4 and 6.5.
When considering the natural frequency, further observation about the
errors between these methods may be made. The increase in number of
elements reducues the errors. By referring to mode shapes provided from
ANSYS in chapter 6 of this work, it can be seen that a relatively smooth
curves is produce with increase in number of elements.
Some observations were also made in experimental results as indicated in
figure 5.15 – 5.21, the natural frequency may be determined by the peak on
the Frequency-Amplitude graphs. In theory, the maximum displacement of
the beam should be at the point 5 (that is the centre of the beam) and in
most cases this has been achieved. If this is not case then either point 4 or
point 6, just off the centre line of the beam has had the maximum
From the result in table 5.1 – 5.4, there are discrepancy with the first
natural frequency recorded during the experimental study to that of the
analytical solution, finite element and ANSYS. Initially it was thought that
it was partially due to the fact the density value of the beam was different to
that used in the calculation but it was then decided that since the density
does not vary to a great extent compared to the value of the natural
frequency then this couldn’t be the case. A more likely cause of these
discrepancies could be due the way that the boundary conditions are
applied to the beam during the experiment. In the analytical, finite element
and ANSYS methods, all boundary conditions are considered to be ideal,
i.e perfectly for a fixed end and perfectly simply supported for a simply
supported end. In experimental setup, a fixed end is achieved by combining
two simply supported setup’s as shown in figure 7.1.
Figure 7.1: Experimental Fixed End
The issue of increase in stiffness at the boundary to a single simply
supported end, there will still be some movement at the end of the beam
and will therefore not achieve the ideal fixed end condition. It would
therefore be useful to design a method of producing a more effective fixed
end for future experimental work. Also, clamping system on the apparatus
was over-tightened when applying a simply supported condition to the
beam. That means the beam a lot stiffer at the end which will result in the
higher than natural frequencies.
7.3 Effect of Boundary Conditions on Natural
The observation made in table 5.1 to 5.4, the natural frequency is lower
under simply supported condition than fixed conditions (except for
experimental result which was explained in the previous section). This is
due to the below equation:
ωn =
The fixed-fixed condition will generate greater stiffness than simply
supported since the mass remain constant in the situation. The overall
natural frequency is reduced under simply supported due to the reduced
stiffness. But, fixed-fixed boundary condition is best applicable to the real
8 Conclusion
In this section of the report, series of reasonable observations were made
and discussed below.
The dynamic investigation of a fibre reinforced polymer beam was carried
out in this work. The modal analysis was performed to the natural
frequencies and mode shapes. The fundamental vertical frequencies for four
boundary conditions were estimated in experimental and analytical ways.
When comparing the experimental results (frequencies, mode shapes etc)
with analytical results, there are some discrepancies with the error between
the two values. It was likely that this is connected with the manner in which
boundary conditions are applied in the beams. It seems that the fixed
boundary conditions allows for too much movement at the end and that
simply supported end is much stiffer than would be experienced under the
analytical solution. It is advisable to redesign the experimental apparatus to
give better results with different boundary conditions.
The finite element method results compared to the analytical solution, it can
be observed that the accuracy of the finite element method increases as the
number of the element is increased. It has been seen that when comparing
the results of second mode from the finite element method with analytical
solution, that a greater number of elements is required to adequately
reproduce the more complex mode shapes
Finally, there are fair agreements between experimental results and
analytical which was the target of this research work. The reason for the
discrepancies has been discussed in previous section.
8.1 Future Work
The work can be extended by apply to Structural Health Monitoring
(SHM). We can apply this work in the case of damage detection techniques
in NDT. The experimental work can also be improved by recording cross
channel where displacement of the beam is measured as a function of the
applied force.
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Appendix 1, Terminogoly
Several definitions of the terminology critical to this study are contained
within this section.
Dynamic Force - a force that changes with respect to time (not static).
Vibration - Oscillation of a system in alternately opposite directions from
its position of equilibrium, when that equilibrium position has been
disturbed. Two types are free vibration and forced vibration. Forced
vibration takes place when a dynamic force disturbs equilibrium in the
system. Free vibration takes place after the dynamic force becomes static
(or zero).
Amplitude - The offset of equilibrium of the system at a given time, also
known as the magnitude of the wave when plotting displacement, velocity,
or acceleration against time.
Period- The amount of time it takes for one cycle.
Cycle - A complete motion of a system starting at any given point of
magnitude and direction that ends with the same magnitude and direction
(i.e., the motion over a full period).
Frequency- Number of cycles over a given time, usually cycles per second
(also called Hz).
Natural Frequency - a frequency at which the system will vibrate freely
when excited by a sudden force.
Fundamental Natural Frequency - The lowest natural frequency for the
system at which a system will vibrate.
Resonance - a condition where a system is excited at one of its natural
Damping - a property of energy dissipation within the system. More
damping results in a quicker decay of amplitude in free vibration. When
less damping is present, the system retains its energy for a longer amount of
Fast Fourier Transform (FFT) - An algorithm for computing the
fourier transform of a set of discrete data values. The FFT expresses the
data in terms of its component frequencies.
FFT Spectrum - The relative contribution of frequencies in a trace of
amplitude over a time range. This is obtained by performing an FFT of
the data.
Mode shapes - The shape of a system showing
displacements when undergoing vibration.
Node - The point location on a mode shape that undergoes zero relative
Appendix 2, Calculation for FEM for
Natural frequencies of Beam (1 Element)
22 L
⎡ 156
4 L2
ρAL ⎢ 22 L
Mass Matrix: [m] =
420 ⎢ 54
⎣− 13L − 3L
− 13L ⎤
13L − 3L2 ⎥⎥
156 − 22 L ⎥
− 22 L 4 L2 ⎦
6 L − 12 6 L ⎤
⎡ 12
⎢ 6 L 4 L2 − 6 L 2 L2 ⎥
Stiffness Matrix: [k ] = 3 ⎢
L ⎢− 12 − 6 L 12 − 6 L ⎥
− 6 L 4 L2 ⎦
⎣ 6L 2L
Equation of Motion:
⎛ ⎡ 12 6L −12 6L ⎤
13L ⎤ ⎞⎧Umi ⎫
⎡ 156 22L
⎜ ⎢
2 ⎥
13L − 3L2 ⎥⎥ ⎟⎪φmi ⎪
⎜ EI ⎢ 6L 4L − 6L 2L ⎥ 2 ρAL ⎢ 22L 4L
⎜ L3 ⎢−12 − 6L 12 − 6L⎥ − ω 420 ⎢ 54 13L 156 − 22L⎥ ⎟⎨Um ⎬ = 0
⎟⎪ j ⎪
⎜ ⎢
2 ⎥
2 ⎥ ⎟⎪
⎣−13L − 3L − 22L 4L ⎦ ⎠⎩φmj ⎪⎭
⎝ ⎣ 6L 2L − 6L 4L ⎦
Where: λ
ρAL4ω 2
420 EI
(Note: these calculations were completed using a density of
ρ = 1827kg / m 3 . for calculations with different density values, this figure
was substituted).
Not possible for Single Element
⎡4 L2 − 4 L2 λ
⎢ 2
⎣ 2 L + 3L λ
(4 L
2 L2 + 3L2 λ ⎤ ⎧φmi ⎫
4 L2 − 4 L2 λ ⎦ ⎩φm j ⎭
− 4 L2 λ )(4 L2 − 4 L2 λ ) − (2 L2 + 3L2 λ )(2 L2 + 3L2 λ ) = 0
12 L4 − 44 L4 λ + 7 L4 λ2 = 0 ⇒ 3.165λ2 − 19.89λ + 5.43 = 0
19.89 ± 19.89 2 − (4 × 3.165 × 5.43)
2 × 3.165
λ = 0.286
λ =6
420 × 17.926 × 10 9 × 2.085 × 0.286
= 981.31rad / s ∴ f1 =
1827 × 5.64 × 10 × 0.82
f1 = 156.18Hz
ω1 =
420 × 17.926 × 10 9 × 2.085 × 6
= 4496.95rad / s ∴ f 2 =
1827 × 5.64 × 10 × 0.82
f 2 = 715.7 Hz
ω2 =
ω1 =
− 4 L2 λ {φm j } = 0 ⇒ λ = 1
420 × 17.926 × 2.085 × 10 −7 × 1
= 1835.87rad / s ∴ f1 =
= 292.19 Hz
1827 × 5.64 × 10 × 0.82
Appendix 3, Calculations for FEM for
Natural Frequencies of Beam (2 Elements)
ρAL ⎢
Globa Mass Matrix: [m] =
⎢ 0
420 ⎢
⎢ 0
54 −13L
13L − 3L2
− 3L
312 0
0 8L2
54 13L
−13L − 3L2
EI ⎢
[k] = 3 ⎢ 0
Global Stiffness Matrix:
L ⎢
⎢ 0
0 ⎥
13L − 3L2 ⎥
156 22L ⎥
− 22L 4L2 ⎥
0 8L2 − 6L 2L2 ⎥
−12 − 6L 12 − 6L⎥
6L 2L2 − 6L 4L2 ⎥
−12 6L
4L2 − 6L 2L2
− 6L 24 0
(Note: these calculations were completed using a density of
ρ = 1827kg / m 3 . for calculations with different density values, this figure
was submitted)
Equation of motion:
⎛ EI
⎜ 3
0 ⎤ ⎞⎪⎧Um j ⎫⎪
⎡24 0 ⎤
2 ρAL ⎡312
⎢ 0 8 L2 ⎥ − ω 420 ⎢ 0 8L2 ⎥ ⎟⎟⎨ φm ⎬ = 0
⎦ ⎠⎪⎩ j ⎪⎭
⎛ ⎡24 0 ⎤
⎡312 0 ⎤ ⎞⎧⎪Um j ⎫⎪
⎢ 0 8L2 ⎥ ⎟⎟⎨ φm ⎬ = 0
⎜ 0 8L2
⎦ ⎠⎪⎩ j ⎪⎭
Where: λ =
ρAL4ω 2
420 EI
⎡24 − 312λ
⎤ ⎧Um j ⎫
Thus: ⎢
2 ⎥⎨
8 L − 8 L λ ⎦ ⎩ φm j ⎭
To resolve λ ,∴ λ = 0.077 or λ = 1
Substituting parameters to the above formula,
ω1 = 2036rad / s,
∴ f 1 = 318.89 Hz
ω 2 = 7343rad / s
∴ f 2 = 879.24 Hz
Equation of Motion
⎜ EI
⎜ 3
⎡ 4 L2 − 6 L
⎢− 6 L
⎢ 2L
⎣⎢ 0
2 L2
8 L2
2 L2
⎡ 4 L2
0 ⎤
6L ⎥
2 ρAL ⎢ − 13L
420 ⎢− 13L2
2 L2 ⎥
4 L2 ⎦⎥
⎣⎢ 0
13L − 13L
.0 ⎤ ⎞⎟⎧ φmi ⎫
⎥ ⎪
− 13L ⎥ ⎟⎪Um j ⎪
− 3L ⎥ ⎟⎪ φn j ⎪
− 13L − 3L2 4 L2 ⎥⎦ ⎟⎠⎪⎩ φnk ⎪⎭
Thus the natural frequencies are as follows:
ω1 = 887.32rad / s
∴ f 1 = 140.72
ω 2 = 3550.0rad / s
∴ f 2 = 562.86 Hz
Equation of Motion:
⎜ EI
⎜ 3
⎡ 24 0
⎢ 0 8L2
⎢⎣6 L 2 L2
6L ⎤
⎡ 312
2 ρ AL ⎢
2L ⎥ − ω
⎢⎣− 13L − 3L2
4 L2 ⎥⎦
− 13L ⎤ ⎞⎧Um j ⎫
− 3L2 ⎥⎥ ⎟⎨ φm j ⎬ = 0
4 L2 ⎥⎦ ⎟⎠⎪⎩φmk ⎪⎭
Using equation solving program, the first two eigenvalues were found to
The natural frequencies were calculated as follows:
ω1 = 1393rad / s
∴ f 1 = 219.75 Hz
ω 2 = 5234rad / s
∴ 712.25 Hz
From the previous procedure, fixed-free boundary condition can be
calculated where:
f1 = 50.0952,
f 2 = 313.964
Department of Mechanical Engineering, Master’s Degree Programme
Blekinge Institute of Technology, Campus Gräsvik
SE-371 79 Karlskrona, SWEDEN
+46 455-38 55 10
+46 455-38 55 07
[email protected]
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