final year dissertation
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Development, Calibration and Verification of Finite Element ModelsTRANSCRIPT
Development, Calibration and Verification of Finite Element Models of Laboratory Structures
This thesis is presented in part submission as a requirement of the Bachelor of Engineering (Honours) in Civil Engineering
in the Department of Civil, Construction & Civil Mineral Engineering, Athlone Institute of Technology.
Luke Molloy
2012
Project Supervisor Project Coordinator Head of Department Dr. Paul Archbold Dr. Paul Archbold Mr. Fergal Sweeney
Declaration I declare that the dissertation submitted by me, in whole or in part, has not been submitted to any other university, institute or college as an exercise for a degree or any other qualification. I further declare that, except where reference is given in the text, it is entirely my own work. Signed:______________________ Date:____________
Acknowledgements
I would like to express my sincerest appreciation to my supervisor Dr. Paul
Archbold for his guidance and instruction throughout the production of this
dissertation. Also I would like to thank Athlone Institute of Technology for the
use of the Software and Heavy Structures Laboratories which allowed me to
conduct this project with the highest standard of equipment. I would like to
thank everyone who was involved in this project for their assistance and support
throughout above all my wife for her continuing support.
Abstract
This dissertation sets out to develop and verify finite element models created in
ANSYS of two laboratory structures. The structures modelled were a T shaped
aluminium beam and a scaled 3 hinge arch bridge. The structures were tested in
the Athlone Institute of Technology Heavy Structures Laboratory. The T shaped
beam was tested for strain associated with incremental loading and the bridge
was tested for horizontal reaction forces associated with a transient load across
the bridge deck.
Both structures were tested physically and their geometries replicated in 3D
finite element modelling code ANSYS. The models were subjected to the same
loading and support conditions as the experimental setup to verify the accuracy
of the finite element models.
Model updating parameters implemented were the changing of Young’s modulus
for the beam model based on an experimentally obtained value and structure
geometry for the bridge model. Both models were verified with the experimental
data which showed a successful result from the updating process.
The updated beam model showed an average increase in the model maximum
tensile strain went from 87% to 102% of the experimental value and an average
increase in the model maximum compressive strain went from 88% to 100% of
the experimental value while the 3 hinge arch bridge updated model results
showed the model exceeding the horizontal reaction force by only 4% of the
experimental result.
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Table of Contents
Table of Contents ......................................................................................................i
List of Figures .......................................................................................................... iv
List of Tables ............................................................................................................ ix
1. Introduction .................................................................................................... 1
2 Literature Review ............................................................................................ 3
2.1 Introduction .................................................................................................... 3
2.2 Background to finite element analysis ............................................................ 4
2.3 Fundamental structural concepts ................................................................... 6
2.3.1 Material stiffness..................................................................................... 7
2.3.2 Principle of superposition ..................................................................... 10
2.3.3 Virtual work ........................................................................................... 11
2.4 Finite element direct stiffness method for framed structures ..................... 12
2.5 Matrix representation ................................................................................... 14
2.6 Continuous medium finite element method ................................................ 15
2.7 Model analysis ............................................................................................... 17
2.8 Development of finite element models ........................................................ 18
2.9 Model updating ............................................................................................. 22
2.10 Model validity and verification ................................................................. 30
2.11 Model calibration ...................................................................................... 34
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2.12 FE analysis studies ..................................................................................... 35
2.13 Element type selection .............................................................................. 43
2.14 Unit gravity load check .............................................................................. 45
2.15 Conclusion ................................................................................................. 45
3 Methodology ................................................................................................. 47
3.1 Introduction .................................................................................................. 47
3.2 Experimental data: T shaped aluminium beam ............................................ 48
3.2.1 Experimental strain data ....................................................................... 49
3.2.2 Experimentally derived Young’s modulus ............................................. 51
3.2.3 Mathematically calculated deflection data .......................................... 55
3.2.4 Macaulay’s method ............................................................................... 55
3.2.5 Finite element stiffness method ........................................................... 58
3.3 Experimental data: 3 hinge arch bridge transient load ................................ 62
3.3.1 Experimental procedure ....................................................................... 63
3.3.2 Mathematically calculated data ............................................................ 64
3.4 Finite element modelling .............................................................................. 65
3.4.1 ANSYS self-learning ............................................................................... 65
3.4.2 Modelling methodology ........................................................................ 66
3.4.3 T shaped aluminium beam .................................................................... 70
3.4.4 Transient load on 3 hinge arch bridge .................................................. 79
4 Results ........................................................................................................... 95
4.1 T shaped aluminium beam results ................................................................ 95
4.1.1 Experimentally determined strain values ............................................. 95
4.1.2 Experimentally derived Young’s modulus ............................................. 99
iii
4.1.3 Theoretical strain values ..................................................................... 100
4.1.4 Compressive strain results (ANSYS) .................................................... 101
4.1.5 Tensile strain results (ANSYS).............................................................. 104
4.1.6 Experimental strain investigation ....................................................... 107
4.1.7 Elastic beam bending deflection results ............................................. 110
4.1.8 Deflection results from finite element stiffness method ................... 113
4.1.9 Deflection results from Macaulay’s method ...................................... 118
4.1.10 Finite element model summery ...................................................... 120
4.1.11 Material data ................................................................................... 120
4.2 Three hinge arch bridge results .................................................................. 121
4.2.1 Experimental reaction results ............................................................. 121
4.2.2 ANSYS finite element model results ................................................... 122
4.2.3 Mesh results details ............................................................................ 126
4.2.4 Material data ....................................................................................... 127
4.2.5 Model results ...................................................................................... 128
5 Discussion .................................................................................................... 131
5.1 T shaped aluminium beam .......................................................................... 131
5.1.1 Finite element model .......................................................................... 133
5.1.2 Model updating ................................................................................... 138
5.2 Three hinge arch bridge .............................................................................. 140
5.2.1 Finite element model .......................................................................... 140
6 Conclusion and recommendations ............................................................. 146
7 Bibliography ................................................................................................ 148
8 Appendices .................................................................................................. 152
iv
List of Figures
Figure 2.1 – Explanation of Young’s modulus (Hyperphysics 2012) ........................ 8
Figure 2.2 – Three-member example truss (Colorado 2012) ................................ 12
Figure 2.3 – Disconnection step (Colorado 2012) ................................................. 13
Figure 2.4 – Generic truss member (Colorado 2012) ............................................ 14
Figure 2.5 – Triangular mesh applied to a bracket (VKI 2012) .............................. 16
Figure 2.6 – Cross section of corrugated webs (Chan et al. 2002) ........................ 19
Figure 2.7 – FE mesh representation of beams (Chan et al. 2002) ...................... 20
Figure 2.8 – Finite element model (Chan et al. 2002) ........................................... 20
Figure 2.9 – Experimental setup (Chan et al. 2002) .............................................. 21
Figure 2.10 – Elevation of the experimental modal (Zapico-Valle et al. 2010) ..... 26
Figure 2.11 – Truss geometry (Esfandiari et al. 2010) ........................................... 27
Figure 2.12 – Degrees of freedom of model (Esfandiari et al. 2010) .................... 27
Figure 2.13 – Steel framed scaled benchmark structure (Wu and Li 2006) .......... 28
Figure 2.14 – Elevation of the Svinesund Bridge (Schlune et al. 2009) ................. 29
Figure 2.15 – Finite element model of the bridge (Schlune et al. 2009) ............... 29
Figure 2.16 – Test setup arrangement (Han et al. 2008) ....................................... 32
Figure 2.17 – Test results from lateral load v displacement (Han et al. 2008) ...... 33
Figure 2.18 – Specimen geometry (McCarthy et al. 2005) .................................... 36
Figure 2.19 – Finite element model (McCarthy et al. 2005) .................................. 36
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Figure 2.20 – Modified gripping boundary conditions (McCarthy et al. 2005) ..... 37
Figure 2.21 – Test rig setup (McCarthy et al. 2005)............................................... 38
Figure 2.22 – ISO tank container (Fahy and Tiernan 2001) ................................... 39
Figure 2.23 – Geometry created with ANSYS (Fahy and Tiernan 2001) ................ 40
Figure 2.24 – Meshed model (Fahy and Tiernan 2001) ......................................... 41
Figure 2.25 – Single span slab deck (O’Brien and Keogh 1998) ............................. 42
Figure 2.26 – Up-stand FE model (O’Brien and Keogh 1998) ................................ 42
Figure 2.27 – Overlapping valid ranges of element types (Akin 2012) .................. 44
Figure 3.1 – Beam test rig setup ............................................................................ 49
Figure 3.2 - Cross section dimensions (mm) and strain gauge locations .............. 49
Figure 3.3 - Experimental test rig setup schematic ............................................... 50
Figure 3.4 – Stress/strain curve (Beal 2000) .......................................................... 52
Figure 3.5 - Second moment of area reference data ............................................ 54
Figure 3.6 – Position of x for moment expression ................................................. 57
Figure 3.7 – Experimental setup ............................................................................ 59
Figure 3.8 – Beam discretised into 4 elements and 5 nodes ................................. 59
Figure 3.9 – Numbered system variables .............................................................. 60
Figure 3.10 – Beam element stiffness matrix (Djafour et al. 2010) ....................... 60
Figure 3.11 – Three hinged arch bridge with transient loading ............................ 62
Figure 3.12 – Schematic view of bridge sowing main dimensions (mm) .............. 62
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Figure 3.13 – Free body diagram of arch bridge .................................................... 63
Figure 3.14 – Cylinder dimensions (mm) ............................................................... 63
Figure 3.15 – Force transducer located at horizontal support .............................. 64
Figure 3.16 – ANSYS analysis systems toolbox ...................................................... 67
Figure 3.17 – Static structural stand alone system ................................................ 68
Figure 3.18 – Von Mises stress criterion (Bolognese 2012) .................................. 69
Figure 3.19 – Von Mises yield envelope (Bolognese 2012) ................................... 69
Figure 3.20 – ANSYS workbench project schematic screenshot ........................... 70
Figure 3.21 – ANSYS design modeller screenshot ................................................. 72
Figure 3.22 – Extruded beam ................................................................................. 73
Figure 3.23 – Meshed beam .................................................................................. 74
Figure 3.24 – Solid 186 element ............................................................................ 74
Figure 3.25 – Solid 185 element ............................................................................ 75
Figure 3.26 – View of a quadratic hexahedron Solid 185 element in beam ......... 76
Figure 3.27 – Extruded solid structure................................................................... 81
Figure 3.28 – Solid structure after meshing .......................................................... 82
Figure 3.29 – Remote displacement support 1 ..................................................... 84
Figure 3.30 – Remote displacement support 2 ..................................................... 84
Figure 3.31 – Body parts 1 and 2 ........................................................................... 86
Figure 3.32 – Body parts 2 and 3 ........................................................................... 87
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Figure 3.33 – Experimental setup .......................................................................... 87
Figure 3.34 – Revolute joint connection No's 1 & 2 .............................................. 88
Figure 3.35 – MPC-184 revolute joint (ANSYS) ...................................................... 89
Figure 3.36 – 3-D model after meshing ................................................................. 90
Figure 3.37 – First load stage applied to bridge deck ............................................ 93
Figure 4.1 – Graphical representation for test No. 1 ............................................. 95
Figure 4.2 - Graphical representation of experimental load versus strain for test
No. 2 ........................................................................................................... 96
Figure 4.3 - Graphical representation of experimental load versus strain for test
No. 3 ........................................................................................................... 97
Figure 4.4 - Graphical representation of experimental load versus strain for
average values ............................................................................................ 98
Figure 4.5 – ANSYS compressive strain graphic, updated model ........................102
Figure 4.6 – Maximum compressive strain values ...............................................103
Figure 4.7 – ANSYS tensile strain graphic, post-updated ....................................105
Figure 4.8 – Maximum tensile strain values ........................................................105
Figure 4.9 - % Variation between experimental strain gauge readings and
theoretical values .....................................................................................108
Figure 4.10 – Modified maximum tensile strain values (Gauge 1) ......................109
Figure 4.11 – ANSYS maximum deflection graphic ..............................................111
Figure 4.12 – Maximum deflection values per load step ....................................112
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Figure 4.13 –Beam discretised into 4 elements and 5 nodes ..............................113
Figure 4.14 – Numbered system variables, degrees of freedom ........................117
Figure 4.15 – Deflection results using FE method maximum displacement .......118
Figure 4.16 – Mathematically computed deflections comparison ......................119
Figure 4.17 – First approach half arch .................................................................122
Figure 4.18 – Load located at 0.2m from centre .................................................123
Figure 4.19 – Resulting reaction for load at 0.2 meters ......................................124
Figure 4.20 – Von-Mises stress for load at 0.2m from centre .............................124
Figure 4.21 – Full bridge model ...........................................................................125
Figure 4.22 – Meshed connection at centre of bridge ........................................126
Figure 4.23 – Load positioned at 800mm from left end ......................................129
Figure 4.24 – Equivalent von-Mises stress for load location ...............................130
Figure 4.25 – Left and right hand support reactions graphic ..............................130
Figure 5.1 – Load versus strain experimental results ..........................................132
Figure 5.2 – Initial loading arrangement..............................................................134
Figure 5.3 – Resulting maximum compressive strain ..........................................137
Figure 5.4 – Resulting maximum deflection ........................................................137
Figure 5.5 – Applied moment ..............................................................................138
Figure 5.6 – Remote displacement supports 1 & 2 .............................................140
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List of Tables
Table 3.1 - Second moment of area calculations .................................................. 55
Table 3.2 – Sample material data for aluminium .................................................. 71
Table 3.3 – Table of applied moments .................................................................. 78
Table 3.4 – Structural steel material properties .................................................... 80
Table 3.5 – Tabular data for supports ................................................................... 83
Table 3.6 – Centre hinge revolute joint data ......................................................... 91
Table 3.7 – Remote displacement support details ................................................ 92
Table 4.1 – Experimental strain values for test No.1 ............................................ 95
Table 4.2 – Experimental strain values for test No.2 ............................................ 96
Table 4.3 – Experimental strain values for test No. 3 ............................................ 97
Table 4.4 – Experimental strain values averaged .................................................. 98
Table 4.5 – Experimentally derived Young’s modulus ........................................... 99
Table 4.6 – Theoretical strain values for Young’s modulus of 64GPa .................100
Table 4.7 – ANSYS produced minimum elastic strain ..........................................101
Table 4.8 – Compressive strain comparison ........................................................103
Table 4.9 – ANSYS maximum principal elastic strain ...........................................104
Table 4.10 – Tensile strain comparison ...............................................................106
Table 4.11 – Modified tensile strain comparison ................................................110
Table 4.12 – ANSYS total deformation results .....................................................110
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Table 4.13 – Deflection comparison ....................................................................112
Table 4.14 – Element stiffness matrices ..............................................................114
Table 4.15 – Combined structure stiffness matrix ..............................................115
Table 4.16 – Structure stiffness matrix multiplied by EI ......................................115
Table 4.17 – Computed 10 x 10 inverse matrix ...................................................116
Table 4.18 – Constants for use in FE method (example values) ..........................116
Table 4.19 – Results from FE method maximum displacement ..........................117
Table 4.20 – Required table of values .................................................................118
Table 4.21 – Results from double integration method .......................................119
Table 4.22 – ANSYS element summery ................................................................120
Table 4.23 – Pre-updating isotropic elasticity .....................................................121
Table 4.24 – Updated isotropic elasticity ............................................................121
Table 4.25 – Horizontal reaction force at right hinge (N) ....................................121
Table 4.26 – Horizontal and vertical model results .............................................123
Table 4.27 – Bodies summery ..............................................................................127
Table 4.28 – Element type summery ...................................................................127
Table 4.29 – Material constants ..........................................................................128
Table 4.30 – Isotropic elasticity ...........................................................................128
Table 4.31 – Reaction force results from FE model .............................................128
Table 5.1 – Initial load and support values ..........................................................135
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Table 5.2 – Point loads 1 & 2 stepped load values ..............................................136
Table 5.3 – Applied moment at left end ..............................................................138
Table 5.4 – Mathematically calculated force reactions .......................................141
Table 5.5 –Maximum reactions at left hand support ..........................................142
Table 5.6 – Minimum reactions at left hand support ..........................................143
Table 5.7 – Maximum reactions at right hand support .......................................143
Table 5.8 – Minimum reactions at right hand support........................................144
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1. Introduction
This dissertation aims to develop, calibrate and verify finite element (FE) models
of certain types of structures present in the Heavy Structures Laboratory within
Athlone Institute of Technology. The accuracy in representing the real structure
geometry, supports and loading systems applied in the finite element model are
essential in achieving accurate results in a structural design process. The
structural analysis finite element code used was ANSYS which was available to
use at the institute.
Model verification procedures are important especially when designing large
scale structural elements that depend on the integrity of the component to keep
the structure from collapse. Developing the finite element model in whichever
code is a process that needs to be carried out by an analyst having a good
understanding of structural mechanics concepts in order to competently design a
structure or structural element. Background checks are essential and the use of
test structures can verify the FE model output results. In the absence of a test
structure, mathematical procedures can indicate whether or not the model
results are of the correct magnitude or sense. This dissertation looks at both the
test structure and some mathematical procedures for validation and subsequent
updating of FE models.
Chapter 2 looks at some literature on uses of the finite element method (FEM) as
a structural design tool and the background to its origin. Some basic structural
concepts are explained in the lead up to a simple example of a basic finite
element procedure. The uses of model updating are looked at and the validation,
verification and calibration procedures that are often employed when utilising
finite element analysis software.
Chapter 3 outlines the methodology used which is divided into 2 sections. The
first deals with the experimental setup of both structures while the second
describes the methodology employed for the development of each finite
element model in ANSYS.
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Chapter 4 is the results chapter which produces the results from the
experimental setup, mathematical calculations and results from the finite
element modelling.
Chapter 5 is a discussion chapter where the results from each experimental
setup are compared to the original finite element model and the updated
models. Results are compared in graphic form and a comparison in percentage
form is produced as to how the updated model compared to the pre-updated
and experimental setup.
Finally the dissertation conclusions are stated and some recommendations for
further study are produced in Chapter 6.
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2 Literature Review
2.1 Introduction
Structural analysis is a procedure performed by engineers in order to evaluate
maximum loading that can be applied to a structure for example, static and
dynamic loading on a bridge, lateral wind loading on a wall or bridge truss
reactions to loading. Many methods of analysing structures have been developed
over the years and slow tedious hand calculations can sometimes be large and
complex. In recent years a new method, the finite element Method (FEM), has
been developed and speeds up the process of structural analysis coupled with
the use of increasingly powerful computers which is accelerating the process
even further.
A review of literature from published books and scientific journals is presented in
this chapter of the possible variations in producing finite element models and the
results of forces obtained from finite element analysis software packages, the
verification of these results and the methods of model updating that are used to
calibrate the original models to act more like the real situation.
Commencing with a brief history as to how the method came about and then an
explanation of the basic structural concepts adopted by the method is reviewed.
As many text books refer to the finite element method in the same manner they
do have some different methods of explaining it.
Once the fundamental theory behind the method is dealt with, topics covered
will contain reviewed literature material on previously conducted experiments
using the finite element method. There are many studies and journal articles
published that deal with all of the above topics related to finite element analysis
and the ones that relate to this project will be focused on in order to understand
the task at hand.
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2.2 Background to finite element analysis
The finite element method is widely used in structural analysis along with a wide
use in a range of physical problems including heat transfer, seepage, flow of
fluids and electromagnetic problems. Within this method, a continuum is
idealised as an assemblage of finite elements with specific nodes where an
infinite number of degrees of freedom (displacements, moments, or forces) of
the continuum are replaced by specified known or unknowns at the node
location (Ghali 2009).
The use of the finite element method to solve engineering problems can be
traced back to the early 1900’s when A.A. Griffith (1893–1963) introduced
fracture mechanics while working on stress concentrations around elliptical holes
in glass. By using strain energy equations and the interaction between internal
elements within the structure of the glass he was able to compute the quantity
of energy released for a specific crack depth (Roylance 2012). Also during that
time some investigators were approximating and modelling elastic continua
using discrete equivalent elastic bars.
The finite element method (FEM) is a numerical procedure that can be applied to
obtain solutions to many different problems that engineers in a wide range of
study encounter. These problems may vary from steady, transient, linear or
nonlinear problems in such areas as stress analysis, heat transfer, fluid flow and
electromagnetism. It was not until 1943 that Richard Courant developed the
mathematical method for an early version of finite element analysis. He used
piecewise polynomial interpolation over triangular sub regions in the
investigation of torsion problems by estimating unknown function values from
known values at nearby points. Moaveni (1999) writes about what took place
after Courant by explaining that the next significant step in the utilization of
finite element methods was taken by Boeing in the 1950’s when they, followed
by others, used triangular stress elements to model airplane wings. Yet it was not
until 1960 that Professor Ray W Clough made the term “finite element” popular.
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In a speech by professor Ray W Clough entitled ‘Early History of the Finite
Element Method from the Viewpoint of a Pioneer’ Clough (2004), in which Clough
addresses the Fifth World Congress on Computational Mechanics (WCCM V), he
comprehensively discusses the subject of the FEM and its origin. In his speech
Clough explains how he came upon problems with the analysis of the vibration
properties of a large ‘delta’ wing structure that had been fabricated in the Boeing
workshop. He was working in a Boeing summer faculty job in June of 1952 at the
time. The problem he faced was different in that standard beam theory applied
to typical aircraft wing stress analysis but could not be applied to this ‘delta’ wing
design. It was this problem that prompted Clough to formulate a mathematical
model of the ‘delta’ wing in which he represented it as an assemblage of 1
dimensional beam components. The results were disappointing and it was not
until 1953 that his boss, John Turner, suggested using an assemblage of 2
dimensional plate elements connected at the corners to evaluate the vibration
properties of the ‘delta’ wing. This concept was the essential definition of the
FEM. From there Clough went on to derive stiffness matrices for these 2
dimensional elements with corner connections and indeed extended them to
triangular plates also. With using the assemblage of triangular elements being
the more accurate representation of the ‘delta’ wing structure Clough was able
to get good agreement between the results of mathematical model vibration
analysis and those measured with a physical model in the lab. He discovered as
the mesh of triangular elements was refined the results of the model converged
with those of the physical laboratory test results. The coining of the name ‘Finite
Element Method’ was to come thereafter.
During the 1960’s, investigators began to apply the finite element method to
other areas of engineering, such as heat transfer and seepage flow problems.
Zienkiewicz and Cheung (1967) wrote the first book entirely devoted to the finite
element method. To understand the method we must appreciate some basic
structural concepts.
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2.3 Fundamental structural concepts
In order to understand the principles behind the finite element method we must
take a look at some basic structural concepts, relating specifically to deflections
of structures and some simple stress/strain relationships when external loads are
applied to a structure of a given material and size.
Any external loading causing a force on a structure will have results. Kong (1997)
says that in any structure the application of some general force system consisting
of both independently variable actions and some dependent reactions will
produce both a system of internal actions and a pattern of deformation. All
points in the structure will move (unless prevented by some postulated external
restraint or by some unexpected combination of effects) and calculations must
involve consideration of the three vector systems of actions (applied forces),
internal actions and displacements, which sums up the basis of a large section of
structural analysis by containing within it reference to Newton’s three laws of
motion which are:
1. Every object in a state of uniform motion tends to remain in that state of
motion unless an external force is applied to it
2. The relationship between an object's mass m, its acceleration a, and the
applied force F is F = ma. Acceleration and force are vectors (as indicated by their
symbols being displayed in slant bold font); in this law the direction of the force
vector is the same as the direction of the acceleration vector
3. For every action there is an equal and opposite reaction
With this in mind it is possible to move forward to examining the effect of a force
on a structure of certain material, with elastic properties.
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2.3.1 Material stiffness
Within the method of finite element analysis there lays a fundamental property
of a material which is an integral part of any structural member. This property is
the stiffness of the member. A significant relationship between the stress (
)
and strain (
is stated by Kong (1997) relating to Robert Hooke’s
experimental observations which it is known, for example, that if a rectangular
block is subjected to uniformly distributed normal stresses σx, then the normal
strain εx which occurs as a result of the application of σx is proportional to σx.
Stress and strain are linked by Young’s modulus which is a material property and
varies with each material.
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Each different material will have a different Young’s modulus as they will strain
at different levels of stress as a result of an applied force. Figure 2.1 shows a
graphical explanation of Young’s modulus and some common material values.
Figure 2.1 – Explanation of Young’s modulus (Hyperphysics 2012)
Any material that contains elastic properties and obey Hooke’s law will have the
relationship between stress and strain as explained in section 2.3.1, this will lead
to the approximate behavioural assumption that when the average stress in a
member is σ then:
σ =
where F is an applied force and A is the cross section of an element,
and the average strain:
ε =
where ΔL is the change in length and L is the original length of the
element. Since the relationship over the elastic region governed by Hooke’s law
by the equation:
σ = Eε, then the combination of these equations will give:
F =
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If we compare this equation with the equation for a linear spring which takes the
form:
F = kx where k is a spring constant, x is the extension and rearranging the
previous equation to state that:
K =
K in this equation is known as the stiffness of a member and as can be seen that
there is a relationship between cross-sectional area A, Young’s modulus E and
the member length L.
This will play an important role in the FEM and will form the basis for the matrix
stiffness method which is used widely in finite element analysis (Moaveni 1999).
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2.3.2 Principle of superposition
An important principal in using finite element analysis for the basis of this
project, i.e. linear elastic, 2 dimensional analyses, will be the principle of
superposition.
Caprani (2007) identifies the principle of superposition as for linear elastic
structures, to be the load effects caused by two or more loadings, are equal to
the sum of the load effects caused by each loading separately. He moves on to
identify some limiting conditions namely:
1. Linearly behaving material only
2. Structures that undergo very small deformations only
It is worth noting that for this project, deformations of individual members will
not be added into the calculations nor will they be considered when establishing
the FE model using the software thus the principle of superposition will apply.
The analysis of a structure by the finite element method is an application of the
displacement method. In frames, trusses and grids, the elements will be bars
connected at the nodes; these elements are considered to be one dimensional
(Ghali 2009).
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2.3.3 Virtual work
The process of finding deflections and force reactions by the finite element
method is fine when dealing with bars which consist within the method exact
element matrices which cannot be generated. The displacements within the
elements are expressed in terms of displacements at the nodes. To establish
element matrices in these cases, the use of assumed displacement fields where
the corresponding strains are determined by differentiation and the stresses by
Hooke’s law. Here the use of the principle of virtual work with respect to nodal
displacement gives the desired nodal displacement (Ghali 2009).
When the elements become very small, the nodal displacements approach the
actual displacement field. In practice, the element sizes are finite, not
infinitesimal; hence the name finite element method (Imechanica 2012).
Kong (1997) while describing the principle of virtual work relates the application
to any structure of a generalised force system that can be represented by a
vector force P will produce both a reaction system accompanied by an internal
stress system characterised by a vector σ.
There are several key components of any structural analysis:
Equilibrium equations (conservation)
Kinematics and compatibility requirements
Constitutive relations
Boundary conditions
There are straightforward ways of deriving equilibrium equations and
compatibility requirements, but solving the resulting equations is not easy
(maybe not even possible) for most realistic engineering structures. Hence,
alternative ways of describing the requirements for equilibrium and compatibility
have been devised that are easier to work with. Two of these ways are based on
an imaginary (virtual) disturbance of a body that is in the deformed state (i.e.
loaded). They are:
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1. The principle of virtual work => alternate way to describe equilibrium
requirements
2. The principle of complementary virtual work => alternate way to describe
compatibility requirements
2.4 Finite element direct stiffness method for framed structures
The direct stiffness method is the most common implementation of the finite
element method. Most commercial computer packages utilize the direct stiffness
method (DSM). The following example taken from Colorado (2012) and explains
the stiffness method quite well. To keep calculations to minimum a simple a
three member truss is used as seen in Figure 2.2 below.
Figure 2.2 – Three-member example truss (Colorado 2012)
To make the transition from the frame truss to the mathematical model it will be
necessary to rename the overall body ‘parts’. This will result in calling individual
members as ‘elements’ and the joints being termed ‘nodes’. To keep the
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housekeeping in order it is necessary to set up a co-ordinate system, a global
system as pictured in red in Figure 2.2. This will be the master co-ordinates that
will represent deflections of nodes in either the x or y direction. Within the frame
an element axis co-ordinate system must also be established for local
deflections. The breakup of the frame into individual elements is known as
‘discretization’ and can be seen in Figure 2.3. This is sometimes known as the
breakdown stage or disconnection stage.
Figure 2.3 – Disconnection step (Colorado 2012)
At this stage it is possible to compute the stiffness matrixes for each element
based on its area and modulus of elasticity, both of which are individual material
properties. It can be seen from Figure 2.3 given the supporting conditions that
there can be no displacement at node 1, horizontal displacement at node 2 and
the possibility of both horizontal and vertical displacements at node 3. The pin at
node 1 will have no displacement assuming that the pin is attached to infinitely
stiff grounds.
Figure 2.4 describes the fundamental principle that allows the numerical values
to be assigned to the matrix. As shown in Figure 2.4, a generic plane truss
member has four joint force components and four joint displacement
components (the member degrees of freedom). The member properties are
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length L, elastic modulus E and cross-section area A. In this member the local axis
of the element lies in the same plane as the global axis. This will result in any
local displacements in the x and y local direction will be transferred without any
change to the global directions. The frame shown in Figure 2.3 will have three
node locations and there are 2 degrees of freedom (DOF) at each node so that
leads to a 6x6 matrix.
Figure 2.4 – Generic truss member (Colorado 2012)
2.5 Matrix representation
If the truss member in Figure 2.4 were to be connected firmly to a support at one
end and pulled by a force at the other, with the member having a constant cross
section and a member stiffness k, of
then based on the theory explained
earlier the force applied can be written as F = k x ΔL. Since the member cannot
displace, say at i, then the two remaining displacements at j can be written in
matrix form.
The matrix for this frame will be in the order of:
A 1x6 matrix for the forces
A 6x6 matrix for the structure stiffness and
A 1x6 matrix for the displacements.
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If the stiffness matrix for each individual element (member in this case) can be
computed then the overall structure stiffness will be the summation of each of
the individual element matrixes. Solving for the unknowns will solve the variables
in the system such as deflections and forces.
A solution can be found for the unknowns based on having some known values
at node locations thus leading to a set of simultaneous equations which are
easily solvable. The process becomes more complicated when variations in
material properties are introduced and with the increase in member numbers. If
a third dimension is added then the process is too complex to be undertaken by
hand and a computer is needed to deal with the large number of calculations
involved.
The method outlined above is ideal for trusses and can lead to direct solutions
for deflections and forces, however, to analyse a continuous medium such as a
plate to compute stresses within it, further techniques need to be adapted.
2.6 Continuous medium finite element method
When an ‘I’ beam is subjected to loading from either a point load or uniformly
distributed load its web is immediately under stress. These stresses are
experienced throughout the web of the beam. The centre section of the beam
resembles a steel plate subjected to the applied forces which may be acting in
any direction on the beam. The structure then can be seen as a continuum and
not as an assemblage of discrete elements connected at nodes, making it
impossible at first glance to apply the same techniques as previously explained.
The finite method of idealising a continuum as a connection of triangular plates
can be attributed to R. Clough while he worked with Boeing in the 1950’s
analysing stresses induced in airplane wings. The assembly of triangular plates
proved to have a great advantage in analysing the ‘delta wing’. Clough (2004)
states “Moreover, the derivation of the in-plane stiffness of a triangular plate was
far simpler than that for a rectangular plate, so very soon I shifted the emphasis
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of my work to the study of assemblages of triangular plate ‘elements’, as I called
them”
The steps involved in this method are as follows:
1. Idealization of the structure, where the plate is idealized as a gathering of
a large number of discrete elements connected only at the nodes
2. Specifying the relation between the internal displacements of each
element and the nodal displacements (based on a mathematical displacement
function relating to expected deformation patterns)
3. Using standard matrix stiffness methods, the analysis of the idealized
assemblage of discrete elements is performed
The process for deriving stresses and stain values over the assemblage of finite
triangular elements involves the placing of virtual loadings in order to establish
element stiffness matrixes and is complex. Figure 2.5 below shows a typical
bracket used in an engine block. The triangular elements are visible as a
superimposed mesh onto the geometry of the shape being analysed.
Figure 2.5 – Triangular mesh applied to a bracket (VKI 2012)
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Each individual element acts like a plate and will have similar physical properties
as the material being tested. The elements here are most likely 3D tetrahedrons.
2.7 Model analysis
Structural analysis must depend on the idealization of structural form and the
various aspects of material behaviour, in order that both maybe reduced to a
form which permits handling by the computational methods available. It is
obviously true that the form of computing equipment available influences the
type of problem which may be tackled. The existence of the computer allows us
to tackle more complicated structures than would be possible with conventional
alone. It is less obvious, but equally true, that the extended computing facilities
now available, and the ease of calculation which they imply, allow computations
to be made with several different idealizations to determine which represents
most adequately the behaviour of the structure. It would frequently be helpful,
however, to have a model of the structure available in order that its behaviour
under load could be readily ascertained and there are cases (even in the
structure in which linear behaviour can be assumed), in which the model can
provide the most ready answer to a problem of analysis. These models take two
forms, depending on the method of use and of interpretation of the results, they
are
a) direct methods and
b) indirect methods as referred to by Kong (1997)
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2.8 Development of finite element models
Finite element analysis (FEA) or the finite element method (FEM) is a process for
finding a numerical solution to a field or global problem. These field problems
might consist of finding the distribution of heat in an engine or distribution of
displacements in a concrete slab. For this project FEA will be used for the
calculation of stress distributions within structural elements like beams and
bridge decks along with associated deflections. Mathematically a field problem is
described by differential equations or by an integral expression, of which are
made up of the collection individual elements. Either description may be used to
formulate finite elements. Finite elements formulations, in ready-to-use form,
are contained in general purpose FEA programs. It is possible to use FEA
programs while having little knowledge of the analysis method or the problem to
which it is applied, inviting consequences that may range from embarrassing to
disastrous, as stated by Cook (2002). Cook also explains that to understand the
method one must visualize a structure not as a single entity but as a collection of
individual finite elements. The term finite alludes to the elements having known
measurable physical properties, like length, mass, thermal conductivity and
thickness etc. with measurable quantities of deflection or stress among others,
as opposed to infinite quantities apportioned within calculus. Within each of
these individual elements a field quantity is only allowed to have a simple spatial
variation which might be described in terms by polynomials terms up to x2, xy
and y2. In this simple case x and y might be horizontal or vertical global
deflections of a node within a truss frame. In general the variation in the region
that is affected by a single element is much more complicated to solve and this is
where FEA provides approximate solutions close to the true overall variation.
However, the basic principles underlying the finite element method are simple
when you consider a body in which the distribution of an unknown variable such
as displacement is required. The difficulty arises when a body has many parts to
it. This leads to a multiple of calculations and is time consuming by hand
therefore the use of a computer program to do the calculations is superlative.
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In order to represent a real structure to perform finite element analysis upon a
mathematical model must be developed. The accurate representation of any
structure, by a model which is to be analysed, depends on the type of finite
element model used to represent the structural members and the structural
properties assigned to the elements (Zárate and Caicedo 2008).
A similar model is developed for the project undertaken in this dissertation,
conducted by Chan et al. (2002) who conducts an experiment where beams of
different formations of web design are loaded to failure and the results are
compared to that of a FE analysis model. In this study the authors propose
testing beams with a plane web, vertically and horizontally corrugated webs and
the results obtained from both the experimental and finite element methods are
compared to verify the finite element models created and to see if it could
closely reflect the behaviour of such beams in the real condition. The various
types of webs are pictured in Figure 2.6 below.
Figure 2.6 – Cross section of corrugated webs (Chan et al. 2002)
The sections shown in Figure 2.6 are the geometric shapes of the different web
configurations with the FE model mesh pictured in Figure 2.7 below.
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Figure 2.7 – FE mesh representation of beams (Chan et al. 2002)
The finite element model for type VCR is shown in Figure 2.8 below.
Figure 2.8 – Finite element model (Chan et al. 2002)
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It can be seen that the developed FE model is represented as series of
trigonometrical shapes which are all joined at points called nodes. Note that
there is in an increased number of elements at turning sections which will refine
the resulting data an increase convergence on the true value of stress at these
locations. It is possible to modify the mesh concentration at sensitive areas;
however, this can lead to a demand on computing power as the calculations
become quite large. The accurate development of the model will reflect in the
results obtained.
With the FE model developed the test rig is setup to manually test each
specimen and record actual deflections and stresses. This resulting measured
data is then used to calibrate the FE model. The test rig is pictured below in
Figure 2.9
Figure 2.9 – Experimental setup (Chan et al. 2002)
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Son and Fam (2008) develop a non-linear FE model to study the flexural
behaviour of hollow and concrete filled fibre reinforced polymer tubes. The
model is developed to account for the geometric and non-linearity of the tubes
and the material to predict the flexural behaviour of both the fibre reinforced
polymer and concrete filled tubes. The model is developed using the FE analysis
program ANSYS, which will also be used for this dissertation. The authors explain
that a finer mesh was used around the lower part of the pole where failure was
expected and gradually changed to a coarser mesh further away to the top of the
pole, accomplished by using the automatic meshing capabilities of the computer
program. The development of the model to represent reality as much as possible
is also explained in the simulation of the translation and rotational degrees of
freedom being restrained along the bottom base of the pole, to mimic a fixed
support condition.
2.9 Model updating
FE models of structures are usually created by simplifying the real structure from
engineering drawings and designs. This simplification process my not exactly
represent fully all of the physical aspects of the original structure. For these
situations the finite element model would need to be calibrated by modification
of the inaccuracies and the possible elimination of such discrepancies wherever
possible. The lesser the level of discrepancy then truer model results can be
predicted. This process is termed ‘model updating’. Updated finite element
models still need to be verified by comparing the calculated results of the
updated model with experimental data so as to address the extent of differences
between the final model and the true value of the structure (Chan et al. 2009).
Model updating can extend from relatively simple models that represent a
simple truss or beam structure in a static two-dimensional analysis to vey
complex dynamic three-dimensional structural analysis problems. It is difficult to
find literature on some of the basic updating techniques for simple problems as
much of the material published focuses on results of vibration analysis and
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dynamic responses by structures which are highly mathematical in nature and
maybe beyond the scope of this project. However, creating a model of a simple
structural setup for analysis can entail some parameters which might need to
updated in the model such as Young’s modulus, fabrication errors, connection
stiffness, possible non-linearity of materials and so forth.
Existing FE models can be updated based on test data. The initial model data
used to design a bridge structure may represent the material properties and
structural response for the bridge in the early years of its existence. As time
passes however these properties might deteriorate and structural damage may
be caused to the structure. Maintenance, upgrading, repair, and replacement of
bridges may lead to high costs and considerable disruption of traffic. For
effective bridge management, accurate and reliable information about the safety
and condition of bridges is essential. In current practice, however, existing
bridges are analysed and evaluated by means of highly simplified structural
models. Structural models that are verified, refined, and tuned with respect to
actual measurements can reduce these uncertainties and provide a better basis
for management decisions. Schlune et al. (2009) explain that FE model updating
can be deceptively simple for small amounts of experimental data when a large
amount of uncertain structural parameters exist. This could lead to the risk of
having an undetermined or ill-posed problem, in which there might be several
non-unique solutions. It is for this reason that a large amount of experimental
data is required to effectively update the FE model. It is noted that physical
phenomena might have to be introduced into the model to obtain a more
realistic description of what is going on in reality.
The finite element method and the models involved have become a widely used
tool in structural mechanics and dynamics, reproducing numerically the static or
dynamic reaction of a structure to the real effects of loading systems. The values
used in the FE model are derived to replicate the physical parameters and are
usually taken from previous tests or experiences similar to the current model.
Model updating consists of estimating some parameters of the model on the
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basis of similar dynamic testing on actual corresponding structures (Zapico-Valle
et al. 2010).
In a study carried out by Chan et al. (2009) the sensitivity based parameter
model updating procedure as applied to model updating concurrent multi-scale
model of structural behaviour of civil infrastructure and was successfully
implemented on a steel truss which formed the basis for their study entitled
‘Concurrent multi-scale modelling of civil infrastructures for analyses on
structural deteriorating—Part II: Model updating and verification’. In this study,
dynamic and static load responses were recorded and a multi-scale FE model
created. The method for model updating of a steel truss in the laboratory
subjected to similar loading was applied to the model for predicting stresses at
locations of the Runyang cable-stayed bridge.
More traditional model updating techniques optimize an objective function; that
is they limit a series of equations to get the best result that fit a mathematical
expression, in order to calculate one single optimal model that replicates the
behaviour of the real structure and represents the physical characteristics of that
structure. Two examples of model updating are reported on by Zárate and
Caicedo (2008) in a paper called ‘Finite Element Model Updating: Multiple
Alternatives’ the first is not of relevance here but the second identifies model
updating alternatives for a finite element model of the Bill Emerson Memorial
Bridge. The proposal in this paper was to create a set of models which hold
similar dynamic characteristics but are physically different and depending on the
final use the analysis can decide on one or more models for further analysis. This
method relies on the experience of the analyst to make an informed decision as
to which model best suits the real setup.
Open to traffic on December 13, 2003, the Bill Emerson Memorial Bridge is a
1206 m long cable-stayed structure. It carries four lanes of vehicular traffic along
Missouri State Highway 34, Missouri State Highway 74 and Illinois Route 146
across the Mississippi River between Cape Girardeau, Missouri, and East Cape
Girardeau, Illinois. The bridge consists of 128 cables, two longitudinal stiffened
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steel girders, and two towers in the cable-stayed spans, and 12 additional piers in
the Illinois approach span. In addition to four pot bearings at two towers, the
superstructure of the cable-stayed span is constrained to the substructure with
16 longitudinal earthquake shock transfer devices at two towers, four tie-down
devices at two ends of the cable-stayed span, and six lateral earthquake
restrainers (UTC 2012).
There are many methods for updating FE models, in essence the physical
variables can be measured at several locations on a structure that is being
modelled and recorded in real time during testing. This data can be transferred
to the model in such a way that discrepancies between the experimental data
and the computed predictions of the model are minimized thus leading to a
more realistic result from the model. Any discrepancies between FEA results and
reference data e.g. test result data, can be due to uncertain available physical
data. Governing physical relations like modelling non-linear behaviour within the
finite element model coupled with inaccurate boundary conditions will lead to
the creation of errors in the FE results. There exists many different types of
model updating techniques and many articles have been published on the
various methods, though most are related to dynamic modelling using the finite
element method. In a study by Zapico-Valle et al. (2010), a new method of finite
element model updating of a small scale bridge was attempted. The model was
created and a corresponding prototype of the experimental model of a multi-
span continuous deck motorway bridge with four identical spans with an
irregular distribution of the bridge piers. The scaled bridge was tested for its
reaction to seismic activity by subjecting it to movement on a shaking table.
Figure 2.10 shows the experimental model on the shaking table.
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Figure 2.10 – Elevation of the experimental modal (Zapico-Valle et al. 2010)
In this study minimisation of an error function in the time domain is carried out
by a novel adaptive sampling algorithm.
A different model updating technique is described by Esfandiari et al. (2010) as
the utilization of the Frequency Response Function (FRF) and measured natural
frequencies as part of a structural damage detection method. Using a non
destructive technique to identify damage to a an existing structure the authors
study the structural model updating using FRF data and measured natural
frequencies of the damaged structure while not enlarging the measured data or
reducing the finite element model. This process involves the excitation of a
structure and recording the response of the structure to the excitation
frequency.
Each part of the structure will have a distinctive natural frequency and frequency
response function, thus a change in the response function of the tested structure
is correlated to change of stiffness, mass and dampening through a change in
measured frequencies of the damaged structure. The effect of excitation
frequencies on finite element modelling has been successfully addressed through
a truss model example which the authors describe in the article. Figure 2.11 and
Figure 2.12 below show the geometry of the truss used and the nodal degrees of
freedom respectively.
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Figure 2.11 – Truss geometry (Esfandiari et al. 2010)
Figure 2.12 – Degrees of freedom of model (Esfandiari et al. 2010)
A two stage finite element model updating method presented by Wu and Li
(2006) in which the authors use the procedure for structural parameter
identification and damage detection of a steel structure, seen here in Figure
2.13, using ambient vibration measurements. The first stage focuses on the
structural parameter identification for the benchmark structure by the finite
element updating approach. The steel structure in question is phase II of the
IASC-ASCE benchmark steel frame structure which is a four-storey; two-bay by
two-bay steel framed scaled structure built by the Earthquake Engineering
Research Laboratory at the University of Colombia.
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Figure 2.13 – Steel framed scaled benchmark structure (Wu and Li 2006)
A methodology for FE model updating proposed by Schlune et al. (2009) where
the choice of measurements, model simplifications, accuracy and reliability of
updated parameters and the analysis of untested load conditions were
examined. The importance of modelling errors, other than model parameters,
was highlighted. The article refers to the complete procedure of modifying a FE
model to better correspond to measured data by methods including manual
model refinement which describes all types of changes which are introduced
manually into the model. In the article a FE model updating through non-linear
optimization is proposed by minimising an objective function. Typical
uncertainties commented on are where elastic modulus is used as a parameter
to model the stiffness changes of a bridge deck in any direction, which is used to
summarise effects, such as the railing system and the asphalt layer, on the
structural performance of the bridge which will lead to uncertainties. Four multi-
response objective functions for FE model updating were proposed and tested in
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the article where modelling the behaviour of the bridge bearings proved it was
not possible to use the same data for static analysis as dynamic analysis for the
updating procedure. The model updating methodology was applied to the
Svinesund Bridge, which connects Norway and Sweden across the Ide Fjord
diagrammatically pictured in Figure 2.14 with the FE model of the bridge pictured
in Figure 2.15 below.
Figure 2.14 – Elevation of the Svinesund Bridge (Schlune et al. 2009)
Figure 2.15 – Finite element model of the bridge (Schlune et al. 2009)
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2.10 Model validity and verification
The development of FE models is a means of predicting future performance of a
structure be it a displacement, moment, stress or reaction to some applied force
or force system. The question could be asked then as to how accurate the model
is performing to the real life action or reaction of the structure. This will depend
on how valid the model is and how well it replicates the given situation.
In a study on The National Cathedral, Washington, DC, by Hinojoso (2010), the
vibration response of arches is experimentally measured to assess the effect of
structural damage. The measurements provide acceleration time series which
are then used to verify and validate predictions of the numerical simulation. They
refer to model validity saying that when a FE analysis model reproduces a match
to a set of physical evidence from tested results, the model is typically
considered validated. However, when there is disagreement between model
predictions and physical evidence, the numerical model can be calibrated.
Korunovic (2011) attempts to validate the results of a FE model of a tyre steady
rolling on a drum relating to cornering and braking behaviour, the use of a
specially developed Computer Aided Design (CAD) package to create geometry
and propagate it to the FE model proved suitable to their study. Two FE models
were used and were performed in FE code ABAQUS. The CAD model contains a
parameterized network of lines and points while following the dimensional
changes of the tyre profile and its structural components, this forms the basis for
the mapped finite element mesh. In this study the authors claim that the results
of the finite element analysis conducted on the model have been directly
compared to experimental results, thus validating the model to a certain degree.
The equipment used and the methods for the experimental determination of
breaking and cornering characteristics of the tire along with experimental
determination of a friction coefficient were shown in the study. The results of
the study show the difference between experimental and numerical results was
smaller after the calibration of the friction coefficient had been included in the
model. This would not have been possible to achieve had they not conducted a
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physical experiment. It is assumed that if tire rolling behaviour is effectively
modelled and verified on one kind of surface for a range of operating
parameters, it may also be used to predict its behaviour in different road
conditions. The use of a drum in this type of tyre performance testing takes up
less space than flat testing systems leading to more efficient testing. The
verification of the FEM used in this experiment proves the model to be effective
and tyre design can be simplified further.
Han et al. (2008) in comparison looks at the behaviour of composite frames
made with square hollow sections (SHS) filled with concrete as column to steel
beams. These types of frames are ideal for construction projects in areas of the
world where there is a high risk of seismic activity. Also known as concrete filled
steel tubular (CFST) columns they possess properties such as high strength and
stiffness, large energy absorption and high ductility. They make reference in the
study as to the complexity in modelling the concrete confinement effect for the
concrete filled tubes leading to limited success in the development of an
accurate model. In order for the authors to establish a valid model that would
replicate the true properties, five components of the frame needed to be
modelled.
The components were;
1. The confined concrete of the square columns
2. The interface and the contact between the concrete and the steel tube
3. The actual steel tubing (hollow steel section)
4. The connection details between the column and the steel
5. The actual steel beams.
Figure 2.16 shows the testing setup for the lateral loading by the MTS actuator.
This is the physical setup and it is necessary to have a valid model to replicate the
true parameters which can then be entered into the FEA software.
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Figure 2.16 – Test setup arrangement (Han et al. 2008)
In addition to the physical setup being correctly modelled, the appropriate mesh
must be applied to the model along with the correct element type, mesh size,
boundary conditions and load applications to provide accurate and reasonable
results which are important in simulating the behaviour of structural frames. In a
paper presented by Han et al. (2008) entitled “Behaviour of steel beam to
concrete-filled SHS column frames: Finite element model and verifications” it was
seen that good agreement was achieved between the experimental curves from
the lateral loading effects on displacement and the numerical curves produced
by the FEM.
Figure 2.17 shows how close the FEM predicted values for displacement were
compared to the actual tested results.
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Figure 2.17 – Test results from lateral load v displacement (Han et al. 2008)
It is also noted that some of the reasons for variations between the model
results and the measured results which are caused at higher load levels, causing
higher axially compressive loads and an increase in the effects of imperfections
caused by unexpected fabrication imperfections in the testing setup, parameters
that might not have been allowed for in the FEM.
The finite element method is an approximation technique and thus will entail
errors. For this reason researchers have designed several pathological tests to
validate any new finite element analysis. The tests should be able to display most
of the parameters which affect finite element accuracy. A representative set of
tests should include patch tests, beam, plate and shell problems. Rao and
Sharinvasa (2012) propose a problem set to help developers of finite element
programs to ascertain the accuracy of particular finite elements in various
applications. This problem set cannot however be used as a bench mark for cost
comparison since the problems are too small for this purpose. Inaccuracies of the
elements are brought in by the presence of spurious mechanisms, locking
(excessive stiffness for particular loadings and or irregular shapes), elementary
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defects like violation of rigid body property and invariance to node numbering
etc. Parameters which affect accuracy are loading, element geometry, problem
geometry, material properties etc. The member being analysed should be
subjected to significant loadings and boundary conditions, for each type of
deformation like: extension, bending, in-plane shear, out-of-plane shear and
twist etc.
2.11 Model calibration
The term model calibration refers to the process of adjusting the finite element
model to better represent field test data. It is the result of the model updating
process and is sometimes referred to as both in some literature. Kangas et al.
(2012) use the process of generating a 3 dimensional finite element model then
calibrating it to field test data. The results of the calibrated model are used to
rate a represented bridge for the University of Cincinnati Infrastructure Institute
(UCII) for condition assessment. The authors choose a bridge in Butler County,
Ohio in America as a case study to illustrate the process. Bridge rating is very
important because a failure to evaluate the health of a bridge correctly may lead
to a catastrophe in the worst case.
There is significant importance in defining model calibration in a larger context
and trying to emphasise its role in relation to model verification and validation.
The terms calibration, validation, and verification are used interchangeably in
some literature, hindering the adequate communication of these principles. For
clarification, the factors to which the accuracy of the FE solutions is dependent
on are listed as:
1. The adequacy of the governing equations involved in the analysis, i.e.,
mathematical definitions for dynamic behaviour of shells or plate elements
2. The precision of numerical solution, i.e., fineness of discretization
3. The accuracy of the physical parameters, i.e., values for material
properties and definitions for boundary conditions, and
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4. The adequacy of the constitutive element models, i.e., assuming linearity
only when the response is predominantly linear (Hinojoso 2010).
Liu (2004) proposes an automatic calibration strategy for 3 dimensional FE
models, going on to say that model calibration starts from a nominal bridge
model and experimental data which is processed from a bridge field test and is
then used for calibration reference. Many of the differences between
experimental and analytic results are due to modelling limitations and
experimental error, thus giving the reason why model calibration is needed to
replicate current bridge structure conditions.
2.12 FE analysis studies
A good example of a combination of the above topics is described in an
experimental study of single lap composite bolted joints by McCarthy et al.
(2005). The paper covers many of the topics previously discussed. In the paper
the authors develop three-dimensional finite element models to study the
effects of bolt-hole clearance on the mechanical behaviour of bolted composite
joints. In this study a single-bolt, single-lap joint type model is constructed in the
non-linear finite element code MSC Marc which is a powerful, general-purpose,
nonlinear finite element analysis solution to precisely simulate the response of
desired products under static, dynamic and multi-physics loading situations. It’s
adaptability in modelling nonlinear material behaviours and transitory
surrounding conditions make it ideal to solve complex design problems. The
specific geometry of the joint is depicted in Figure 2.18.
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Figure 2.18 – Specimen geometry (McCarthy et al. 2005)
The model mesh is displayed in Figure 2.19.
Figure 2.19 – Finite element model (McCarthy et al. 2005)
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Five separate parts were meshed including:
Two laminates
Two washers, one top and one bottom
A combined nut-bolt
It can be seen that there is a high radial mesh density near the hole under the
washer where high strain gradients exist. The increase in mesh density will cause
a convergence on the true strains experienced at this location. The washers are
modelled separately which has the disadvantage of increasing the model size due
to the increase in number of elements. It does provide a more accurate
representation of the real scenario though. The authors explain that improving
the model to replicate the real situation improves the FE results. One of the
examples of better modelling is representation of the clamped section of the
plates illustrated in Figure 2.20
Figure 2.20 – Modified gripping boundary conditions (McCarthy et al. 2005)
It is seen that assuming a fixed nodal system on the surfaces of the plates where
the grips are improves the FE model results and gives a closer result to that of
the experimental data obtained.
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The method for testing the joint stiffness is illustrated below in Figure 2.21.
Figure 2.21 – Test rig setup (McCarthy et al. 2005)
The main aim of the experiment was to study the effects of bolt-hole clearance
on the mechanical behaviour of the joint. In their concluding remarks the authors
explain that a valid model was developed and results verified by experimental
testing in the lab. Efficiencies in the model were found to have improved by
defining the contact bodies as sub-parts of the joint components to see which
bodies would come into contact. The contact tolerance’s and the way in which
they are modelled seem quite important. The results were also compared to
other FE modelling packages such as ABAQUS and STRIPE.
Fahy and Tiernan (2001) attempt to develop a valid FE model of the ISO tank
containers which are used to transport bulk liquids by road, rail and sea and can
contain volumes of 25,000 litres at any one time. The design of these tanks has
arisen by trial and error, due to the lack of a definitive method to analyse the
stiffness of the tank and frame. The main area of concern is where the tank is
attached to the frame as this is difficult to analyse by traditional methods, with
fatigue and vibration analysis being left to the manufacturer which can
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sometimes mean excessively strengthened sections without any analysis. Using
the computer package ANSYS 5.4 the authors model the tank container both
statically and dynamically for road, rail and sea use. During the study the authors
aim to:
Analyse the existing design to determine its safety
Validate the results by conducting static and dynamic tests
Improve the efficiency of the design
The typical ISO tank container is shown in Figure 2.22 and the geometric model
created in ANSYS in Figure 2.23.
Figure 2.22 – ISO tank container (Fahy and Tiernan 2001)
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Figure 2.23 – Geometry created with ANSYS (Fahy and Tiernan 2001)
The size of the tank diameter being 2.285m and length of 6.085m and comparing
this to the thickness of 6mm indicated that plate or shell elements would give
the best results from the FE analysis. Shell63 was chosen for the entire model
and is the simplest shell element having four nodes and six degrees of freedom:
translations in the x, y and z directions and rotations about the nodal x, y and z
directions.
Selection of an appropriate mesh is of paramount importance, starting with an
initial course mesh and refining it in areas of interest with high stresses. The
meshed model can be seen in Figure 2.24.
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Figure 2.24 – Meshed model (Fahy and Tiernan 2001)
The results of the FE model and the verification of the results from testing by
Fahy and Tiernan (2001) proved that a valid model had been developed and
behaves in a similar manner to the actual tank however research is ongoing.
The up-stand FE grillage analogy is used by O’Brien and Keogh (1998) in an effort
to improve modelling of bridge decks with wide transverse edge cantilevers.
Plane grillage analogy is a popular method with bridge designers modelling slabs
in two dimensions, involving the idealisation of the bridge slab as a mesh of
longitudinal and transverse beams located within the same plane. Assuming a
constant neutral axis depth throughout, FE analysis is limited to planar analysis
using plate bending elements similar to the plane grillage method. The slabs in
this study have neutral axes of varying depth so applying the up-stand grillage
analogy improves FE results. Figure 2.25 shows the single slab bridge deck with
cantilevers and Figure 2.26 shows the up-stand grillage FE model of the bridge
deck.
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Figure 2.25 – Single span slab deck (O’Brien and Keogh 1998)
Figure 2.26 – Up-stand FE model (O’Brien and Keogh 1998)
To address the problem of a varying neutral axis elements of infinite flexural
rigidity are placed between the edge cantilever and the base of the deck. Then
depths of the elements were taken to be equal to the depth of the portion of
slab which they represented. The up-stand FEA gave both bending moments and
axial forces in each element ant the total stress value was arrived at by adding
the stress components’ of each of each of these effects. The results compare
well to three dimensional FE analyses.
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2.13 Element type selection
The previous examples of finite element modelling utilize many capabilities of
the method ranging from simple, linear, static analysis to more complex, non-
linear and transient dynamic analysis. In order to correctly model the structure it
is important to resemble it as accurately as possible in the computer program.
This dissertation uses the FE modelling software ANSYS and for that reason the
following section will use terminology associated with this package alone.
ASNSYS element library contains over 150 element types which is very large
considering most structural problems will only ever use a variety of 3 or 4 of
these (Moaveni 1999).
The most up to date version of ANSYS (release 14) utilises the workbench
simulation software which is more user friendly that the classic programme
(ANSYS 2012). ANSYS Workbench in itself is not a product, rather it is a product
development platform and user GUI built for analysis needs with the objective of
providing elegant next generation functionality and intelligent automation to the
engineering community.
The analysis used in this project contains the use of more than one type of
element as the difference in the results will be analysed to see if they will mimic
the true situation.
With today’s advances in computing power there may seem never to be enough
computational resources to solve all the problems that present themselves.
Frequently solid elements are not the best choice for computational efficiency as
a similar result may be obtained from the use of a 2D element and may save
computing power demand. The person analysing the problem should learn when
other element types can be applicable or when they can be utilized to
authenticate a study carried out with a different element type. Solid Works
Simulation offers a small element library that includes bars, trusses, beams,
frames, thin plates and shells, thick plates and shells, and solid elements. There
are also special connector elements called rigid links or multipoint constraints
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however ANSYS element library is far more extensive. Shells and solid elements
are considered to be continuum elements. Plate elements are a special case of
flat shells with no initial curvature. Solid element formulations include the
stresses in all directions and demand more computing power when solving.
Shells are a mathematical simplification of solids of special shape. Thin shells (like
thin beams) do not consider the stress in the direction perpendicular to the shell
surface. Thick shells (like deep beams) do consider the stresses through the
thickness on the shell, in the direction normal to the middle surface, and account
for transverse shear deformations. It is important to choose the correct element
type as to obtain the desired result depending on the analysis required. The
following is a means of choosing a particular type of element.
Let h represent the typical thickness of a component while its typical length is
represented by L. The thickness to length ratio, h/L, gives some guidance as to
when a particular element type is valid for an analysis. When h/L is large then
shear deformation is at its maximum importance and solid elements should be
used. Conversely, when h/L is small then transverse shear deformation is un
important and thin shell elements are the most effective element choice. In the
intermediate range of h/L the thick shell elements will be most effective. The
thick shells are extensions of thin shell elements that contain additional strain
energy terms.
The overlapping h/L ranges for the three continuum element types are
recommended in Figure 2.27.
Figure 2.27 – Overlapping valid ranges of element types (Akin 2012)
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The thickness of the lines suggests that the regions where a particular element
type is generally considered to be the preferred element of choice. The
overlapping ranges suggest where one type of element calculation can be used
to validate a calculated result obtained with a different element type. Validation
calculations include the different approaches to boundary conditions and loads
required by different element formulations (Akin 2012).
2.14 Unit Gravity Load check
The Unit Gravity Loading validity check verifies that the model will provide
accurate displacements and reactions forces under gravity loading. This is a good
check to perform if a model will be used for quasi-static loads analysis. These are
known as static analysis checks and can be performed simply and quickly to
check a model. The resulting displaced shape should be inspected for its
soundness in terms of whether or not any parts of the structure show any signs
of suspicious displacements or dose the displaced shape look reasonable as
expected under a unit load (NASA 1995).
2.15 Conclusion
The literature reviewed here has described briefly the principles behind the finite
element analysis method of analysing structures. Literature has been reviewed
which covers the main issues that arise when dealing with finite element
analysis. These issues include the initial development of a finite element model,
the calibration and verification of the model in order to have the model produce
results similar to that of measured test data. The main topics are how to
correctly model such variables as: material stiffness, joint stiffness, loading
arrangement and non-linearity among others.
Some examples are given of the dynamic modelling of structures, which will not
form part of the analysis of structures in this dissertation, given its significance it
could not be overlooked. Vibration analysis of structures and the FE model
updating techniques applicable to structural damage detection form a large part
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of journal material. More research on this could be conducted in future but given
the high level of mathematical knowledge required it will not be for some time.
With background research conducted the project problem of developing,
calibrating and verifying static FE models should be a less complicated process.
Models can now be developed to test structures for elastic and plastic bending
and the results compared to laboratory experimental data.
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3 Methodology
3.1 Introduction
Finite element (FE) modelling software (ANSYS) produces results based on the
entered data representing a physical setup for analysis. The results however
need to be proved by some form of hand calculations or experimental test data
in order to ensure the results are within the expected direction or magnitude.
The process of updating FE models can only be accurately conducted by means
of acquiring experimental data from similar physical tests. This data can then be
entered into the FE model to change parameters such as material or section
properties, support conditions, loading arrangements etc.
This chapter has two objectives. The first describes the methods in which the
experimental data was obtained from loading each structure in turn with
mathematical methods included. The second describes the methodology
involved in creating finite element models, in ANSYS simulation software
package, of each structure and how to represent as close as possible the actual
physical scenario. The physical testing took place in the Heavy Structures
Laboratory of Athlone Institute of Technology. The geometry for the FE models
will be created with ANSYS Workbench Design Modeller release 14 based on
measured physical data from each of the test rig setups.
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3.2 Experimental data: T shaped aluminium beam
This structure consists of an aluminium T shaped section 1000mm long
supported at each end giving an effective length of beam of 805mm.
Experimental values of strain were required along the centre of the longitudinal
axis at 9 locations around the face in order to assess how the material reacts to
the applied loading and also used to compute the Young’s Modulus of elasticity
for the material, this value is used later in the ANSYS material properties
(isotopic elasticity) as a model updating parameter. The measured strain values
are used to compare values with those produced in the FE model. The averaged
strain values will be used from the experimental test results to compare with the
FE model to evaluate error percentages. There is also a mathematically
calculated theoretical value for each strain location worked out for the structure
as a further comparison.
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3.2.1 Experimental strain data
The test rig shown in Figure 3.1 was supplied by Hi-Tech Educational Equipment
and consists of a universal frame where the beam was supported as shown.
Figure 3.1 – Beam test rig setup
Load was applied through a turn screw device located under the rig and is
transferred to two locations centred on the beam at C and D identified in the
schematic in Figure 3.3 Strain was recorded for the 9 locations, as indicated in
Figure 3.2, for 15 load cases from 10N to 150N (or closest possible load) in 10N
increments.
Figure 3.2 - Cross Section Dimensions (mm) and Strain Gauge Locations
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Deflection was measured centrally on the beam for each load step by the
deflection gauge located centrally on the top side of the beam. The test was
carried out three times and the average strain values used. Results can be seen
in Chapter 4.
Simple supports were used at A and B which provides vertical reactions indicated
in Figure 3.3. The beam is free to rotate at these locations giving only two
reactions which will make the beam statically determinate. A load cell is attached
to the loading mechanism which measures loading values and sends the
information back to the HDA200 display. Nine strain gauges are located in the
centre of the beam in positions described in Figure 3.2 and these send
information back to the display in units of micro strain (µε). Deflection is
measured via a digital dial gauge placed on the upper surface of the beam
centrally positioned over the load points. A schematic of the test setup is
displayed in Figure 3.3 showing beam dimensions, load locations, support
positions and strain gauge position.
Figure 3.3 - Experimental Test Rig Setup schematic
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3.2.2 Experimentally derived Young’s modulus
Young’s Modulus which applies to materials that obey Hooke’s Law is described
as the ratio of tensile or compressive stress to tensile or compressive strain in a
specimen subject to uni-axial loading as:
E =
(1)
where:
E is Young’s Modulus
σ is tensile/compressive stress
ε is tensile/compressive strain and is unitless
and:
σ =
(2)
where:
F is axial force
A is cross sectional area
and:
ε =
(3)
where:
ΔL is change in length
L is original length
These equations can be manipulated in order to find either value, if the stress is
known, and the related strain measured, then E, the Young’s Modulus, can be
derived for a particular material (Davis and Selvadori 1996). In bending however
the strain is not easily measured physically, as the values are usually quite small,
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so the use of electronic strain gauges are used to quantify it. Equation 1 can be
graphically represented indicating the relationship between stress and strain.
The elastic zone in Figure 3.4 shows that where the angle the line makes with the
strain axis is representative of the modulus of elasticity within the elastic limit
(slope of the line).
Figure 3.4 – Stress/Strain Curve (Beal 2000)
When materials are subjected to a normal or bending stresses, and elastic
behaviour is experienced, the strain developed is recovered immediately the
stress is removed. The limiting value of stress applied in order for this to happen
is noted in Figure 3.4 as the elastic limit. Any further application of stress after
this will result in the strain not fully recovering thus leaving a permanent
deformation of the material. In 1678 Robert Hooke defined his law stating that
the strain developed is directly proportional to the stress producing it. This law
holds for most materials within certain limits (John 1978).
The beam was tested by applying a load causing the beam to bend and recording
the produced strain via electronic strain gauges and subsequently using the
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bending moment equations and calculating the second moment of area for the
shape a value for the Young’s Modulus was obtained.
Bending moment equations
The bending moment equations for a beam can be equated as:
=
(4)
where:
M is the bending moment
I is the second moment of area
σ is the bending stress
y is the distance to the N/A where values are measured
rearranging equation 4 to make bending stress the subject gives:
σ =
(5)
substituting equation 5 into equation 1 will yield a value for E:
E =
(6)
Equation 6 now represents a value for the Young’s Modulus of the material being
tested and since the strain can be recorded as the load increased, all that needs
to be calculated is I, the second moment of area about the neutral axis.
Second moment of area
The second moment of area section property for the bam is calculated with
reference to Figure 3.5.
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Figure 3.5 - Second Moment of Area Reference Data
Second moment of area calculations using the parallel axis theorom are
evaluated by the the equations:
Ix,Total = Ix,1 + Ix,2 (7)
where:
Ix,1 and Ix,2 are the second moment of areas of shapes 1 and 2 respectivle
Values for each Ix are given as:
Ix,1,2 =
+ b.d.ŷ 2 (8)
Section property results with reference to Figure 4 for the beam cross section are
presented in Table 3.1.
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Table 3.1 - Second Moment of Area Calculations
Section Dimensions (mm)
Second Moment of Area (mm4)
b1 25.5 Ix,1= 22285
d1 3.2
b2 3.2 Ix,2= 40484
d2 47.5
y1 16.5 Ix,Total= 62769
y2 8.85
The second moment of area arrived at will be used in Equation 6 and the
recorded strain gauge readings to evaluate Young’s Modulus of elasticity of this
material.
3.2.3 Mathematically calculated deflection data
To check the validity of the deflection values which were obtained in the ANSYS
finite element model, theoretical data for displacements were computed by two
different methods as follows:
1. Double integration (Macaulay’s method)
2. Finite element stiffness method for deflection
3.2.4 Macaulay’s method
Macaulay’s method, also known as the double integration method, is a structural
analysis technique used to analyse deflections of Euler-Bernoulli beams and is
very useful for discontinuous or discrete loading systems.
The Euler-Bernoulli beam bending theory was developed in the mid 1800’s by
Leonard Euler and Daniel Bernoulli to address the problem of finding deflections
in beams subject to loading. Two key assumptions have to be made, the material
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is linear elastic according to Hooke’s law and that the plane sections remain
plane and perpendicular to the neutral axis during bending (Haukaas 2012).
Macaulay’s Method enables the writing of a single equation of the bending
moment for the full length of the beam. When coupled with the Euler-Bernoulli
theory, we can then integrate the expression for bending moment to find the
equation for deflection. This will allow the deflection to be found at any location
on the beam, for the purposes of this experiment only the maximal deflection,
expected at the centre of the beam, will be utilised. The method of finding the
deflection of the beam will give deflected values which can be compared to the
finite element stiffness method and then compared to deflection values obtained
by ANSYS in the modelling stage.
From the Euler-Bernoulli Theory of Bending:
=
(9)
Where:
R is the radius of curvature
For small displacements:
=
(10)
Where:
y is the deflection at the point
x is the distance of the point along the beam
This leads to the fundamental deflection equation:
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=
(11)
This can be rewritten as:
M = EI
(12)
In order to find the deflection, y, at any point on the beam, a bending moment
expression needs to be written from a position on the extreme right end of the
beam in terms of x, the distance from the left end, which takes into account for
all the different loading being applied and the reactions.
Figure 3.6 shows the setup of the beam and the general location for the variable
x.
Figure 3.6 – Position of x for moment expression
The general bending moment equation for the location x is:
Mx = RvA(x) – P1(x – 0.325) – P2(x – 0.48) (13)
Where:
RVA is the vertical reaction at the left end A
P1 and P2 are the applied loads
X is the variable distance from end A
L is the beam length from A to B
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Substituting Equation 13 into Equation 12 and integrating twice to solve for the
deflection in terms of x which introduces two constants of integration, and using
the boundary conditions:
1. Slope:
= 0 at x = L/2 due to symetrical loading
2. Deflection: y = 0 at x = 0 and x = L
This only leaves the second constant of integration and this stays at zero because
of the simplistic nature of the loading arrangement.
The result is an expression for the deflection at any point along the beam by
entering a value for x. This process is simplified by the use of Microsoft Excel to
do the calculations. Values for the constants and the computed deflections for
each of the 15 load stages are presented in the results chapter.
3.2.5 Finite element stiffness method
The finite element method as described in Chapter 2 of this dissertation can be
used in a very simple manner to compute a 1 dimensional deflection value for
the beam being modelled. This will involve descritizing the structure into a series
of finite elements which are connected by nodes. As the beam will have two
support points, two load points and one required deflection location, a 4
element structure with 5 nodes will be required. Figure 3.7 shows the beam in
place in the experimental setup according to how the experimental test was
conducted.
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Figure 3.7 – Experimental setup
The beam was then broken down, or descritized, into 4 elements connected by 5
nodes as indicated in Figure 3.8. Nodes are located at locations of applied
loading, reactions or points of interest for the required measured values. In this
case there is no force, applied or reacted, at node 3. This node is required
however as deflection was measured in the experimental setup at this location
which was where the maximum deflection was expected.
Figure 3.8 – Beam discretised into 4 elements and 5 nodes
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The system variables were assigned as seen in Figure 3.9.
Figure 3.9 – Numbered system variables
Each node has two degrees of freedom, which in turn are represneted as a
vertical force or vertical displacement, a moment force or rotation.
Each of the 4 elements individual stiffness matrixes is computed using methods
of superposition as outlined in Chapter 2 and the summation of fixed end
moments the following member stiffness matrix can be computed for each of the
four members based on the matrix in Figure 3.10.
Figure 3.10 – Beam element stiffness matrix (Djafour et al. 2010)
The results of each member stiffness matrix and the resulting 10 x 10 structure
stiffness matrix are presented in Chapter 4.
There are many known’s associated with the structure, being:
1. Reaction forces at each end: equal to the load due to symmetry
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2. No deflection at the supports
3. Zero moment at supports: simply supported
4. Moments across nodes 2,3 and 4 are equal and opposite: equilibrium
conditions
These known values simplify the solving process significantly thus only leaving
three values of interest to be solved for namely the deflection value at node 3
(max deflection) and both rotation values at nodes 1 and 5 represented in the
global matrix as D3, D6 and D10 respectively.
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3.3 Experimental data: 3 hinge arch bridge transient load
This structure represents a three hinged closed spandrel arch bridge and was
supplied by Hi-Tech Educational Equipment. The bridge set up consists of a
bridge fixed by pin supports at both ends and a third pin at the centre of the
bridge span within the frame as depicted in Figure 3.11.
Figure 3.11 – Three Hinged Arch Bridge with Transient Loading
Dimensions of the experimental setup are shown in the schematic in Figure 3.12.
Figure 3.12 – Schematic View of Bridge Sowing Main Dimensions (mm)
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Hinges are located at A, B and C thus forming the three hinged arch bridge.
The forces involved internally and externally are depicted in Figure 3.13. Force FBX
is where the load measuring cell is located.
Figure 3.13 – Free Body Diagram of Arch Bridge
3.3.1 Experimental procedure
Load was applied by placing a metal cylinder of know mass at 10 equally spaced
locations, identified in Figure 3.12 along the bridge and recording the resulting
horizontal force at end B. Details of the metal cylinder which was the moving
mass can be seen in Figure 3.14.
Figure 3.14 – Cylinder Dimensions (mm)
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Horizontal force was measured through a force transducer located on one of the
supports as seen in Figure 3.15.
Figure 3.15 – Force Transducer located at horizontal support
Three sets of test results were recorded for each of the eleven load positions and
the average value for the horizontal reaction force for each location was used.
3.3.2 Mathematically calculated data
The mathematically determined data was obtained form a simple 3 hinge arch
bridge analysis using the standard three equilibrium equations:
Σ Moments = 0
Σ Horizontal Forces = 0
Σ Vertical Forces = 0
These equations are fine but on analysing the structure and the force reactions it
is visible to see that there are 4 reactions present as a result of the applied
vertical loading, see Figure 3.13. These are:
1. Left hand vertical force – Fay
2. Left hand horizontal force – Fax
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3. Right hand vertical force – Fby
4. Right hand horizontal force – Fbx
Having 4 reactions and only 3 equations is not enough to solve for the resulting
reaction forces. In the 3 hinge arch bridge analysis a further equation is required
and this takes the form of:
Σ Moments about the centre hinge
This method is used to calculate the horizontal and vertical reactions for the
given load of 24.8N at each of the 11 equally spaced locations along the top of
the bridge. Tables of the resulting reactions are presented in the Results Chapter.
3.4 Finite Element Modelling
In order to conduct the finite element model of the given structures a knowledge
of the computer programme ANSYS was required. As this was not part of any
module within the Civil Engineering course a self-educate approach was taken.
This would involve getting to know the basic commands and features of ANSYS in
order to complete the modelling.
3.4.1 ANSYS self-learning
The task of embarking along the self-educate route to have a working
understanding of the ANSYS program seemed daunting initially. This task was
made simpler though by taking a project management approach to the process
with the aim of the project being to gain a fundamental user level as described in
the Fundamental FEA Concepts and Applications publication by ANSYS (2012).
Planning and scoping the project (self-educate) in order to provide clarity on the
overall objectives included defining the project scope and evaluating if it would
be possible to learn enough about the computer programme in order to
complete the dissertation within the given timeframe. This process involved
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searching through the ANSYS help website along with many forums that exist
online. This proved invaluable research as the current version of ANSYS
Workbench was installed in the computer lab within Athlone Institute of
Technology and this version appeared to be more user friendly than the ANSYS
classic version. It was decided at that stage to proceed with the Workbench
element of the computer package and sufficiently educate myself in order to
complete the project objectives.
The initial procedure was to complete some basic tutorials which are easily
accessible either from the ANSYS help section or online tutorials on the internet.
In searching the internet there was many videos on the internet website
Youtube.com which lead to the completion of the modelling process.
3.4.2 Modelling methodology
The finite element models for simple 2 dimensional and 3dimensional problems
are usually generated via the Mechanical ANSYS Parametric Design Language
(APDL) command interface. For complicated assemblies the ANSYS Workbench
product is used as it allows one to define the geometry natively and to set up a
project workflow that allows the entire analysis from model generation to results
processing to occur in a well-defined manner.
Finite Element Analysis is a mathematical representation of a physical system
comprising of an assembly of parts as the model, material properties, and
applicable boundary conditions collectively referred to as pre-processing, the
solution of that mathematical representation known as solving, and the study of
results of that solution known as post-processing. Simple shapes and simple
problems can be, and often are, done by hand. Most real world parts and
assemblies are far too complex to do accurately, let alone quickly, without use of
a computer and appropriate analysis software. The process involved can be
broken down into basic steps including:
1. Creating the geometry
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2. Selection of element type
3. Assigning material properties
4. Defining and generating the mesh
5. Establishment of boundary conditions (supports and load arrangement)
6. Post processing (solving)
7. Analysis of results
To get to the geometry stage an analysis system must be set up first. On entering
the ANSYS Workbench initial screen a list of items is presented in the Toolbox
section. The project has to be built from these initial setting depending on the
type of analysis being conducted. ANSYS Workbench Toolbox is displayed in
Figure 3.16 where the static structural system can be seen amongst several other
analysis type systems should they be required.
Figure 3.16 – ANSYS analysis systems toolbox
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The type of analysis system used for the purpose of this project was the Static
Structural (ANSYS) template. This template is then dragged to the Project
Schematic, seen in Figure 3.20, and creates a standalone system. Other systems
can be added or linked to a current system which will allow the sharing of data
between analysis templates, say for instance the geometry could be the same for
two types of analysis so there is no need to create the geometry twice. Once the
system is in the project schematic there are options available to be completed in
sequential order. The process will not allow the user to continue until the
previous component in the project system is completed correctly. The static
structural system comprises of the following components:
1. Engineering data
2. Geometry
3. Model
4. Setup
5. Solution
6. Results
A static structural analysis system graphic is shown in Figure 3.17. Note all the
cells have a green tick indicating the data has been entered successfully along
with a successful result.
Figure 3.17 – Static structural stand alone system
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The completion of each step is a process in itself and establishing the correct
data to be entered is paramount to a successful result. The methodology
outlined here will be followed through for both of the structures modelled.
Von Mises Yield Criteria
Von Mises postulated in 1913 that a material will yield when the distortional
energy at the point in question reaches a critical value. The distortional energy
written in terms of the 2D principal stresses and the yield stress can be seen in
Figure 3.18.
Figure 3.18 – Von Mises stress criterion (Bolognese 2012)
The associated yield envelope is pictured in Figure 3.19.
Figure 3.19 – Von Mises Yield Envelope (Bolognese 2012)
These are the stresses that are viewed in relation to the principal stresses and
are colour contoured in the ANSYS output as an indication of where the most
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significant concentration of stress is to show where possible failure of a structure
might occur.
3.4.3 T shaped aluminium beam
As discussed in the previous section the structures are to be modelled as 3
dimensional solid structures. The T shaped aluminium beam will be modelled as
a 3 dimensional solid structure. For the purpose of this analysis a static structural
analysis system was used. This type of system determines the displacements,
stresses, strains, and forces in a structure caused by loads that do not induce
significant inertia and damping effects. Steady loading and response conditions
are assumed which means the loads and the structures response are assumed to
vary slowly with respect to time. Figure 3.20 shows three separate stand-alone
static structural systems which were created for this project.
Figure 3.20 – ANSYS workbench project schematic screenshot
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Stage 1: Engineering data
The initial stage in the project system is entering of the engineering data. This
component comprises of material properties of the solid to be modelled. The
material is assumed to be homogenous, linear isotropic. The more relevant
material properties for aluminium alloy set in the default values are presented in
Table 3.2.
Table 3.2 – Sample material data for aluminium
Temperature C Young's Modulus Pa Poisson's Ratio
Bulk Modulus Pa
Shear Modulus Pa
22 7.1e+010 0.33 6.9608e+010 2.6692e+010
Compressive Yield Strength Pa
2.8e+008
Tensile Yield Strength Pa
2.8e+008
Tensile Ultimate Strength Pa
3.1e+008
These material properties are set as default and are assumed by the programme
until such time as experimental data is obtained from testing and entered into
the relevant cells. These values are assigned to the individual elements and
effect their interaction with each other and the overall behaviour of the solid
structure as load is applied. Identifying differences in these properties by
physical testing or theoretical assumptions will form part of the updating process
as discussed earlier.
Stage 2: Geometry
Prior to initialising the geometry stage a detailed look at the problem is needed.
This is done to fully understand the structure being analysed to arrive at the
most economical yet thorough way the model can represent the actual structure.
Advantages can be taken of symmetry if possible about the structure which will
reduce the overall time and computer power required to solve the problem.
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The geometry creation is started by opening the Design Modeller by double
clicking in the system cell named geometry. After selecting the desired drawing
units to construct the geometry, the cross section is drawn in the x-y plane as
shown in Figure 3.21.
Figure 3.21 – ANSYS design modeller screenshot
Dimensions are assigned to create a cross section which represents the true
shape of the beam being tested. It is important to ensure that the model is being
created in a 3 dimensional analysis mode. Once the outline of the cross section is
created it is possible to extrude the shape in the Z axis to form a 3 dimensional
solid body. The computer now understands this to be a solid body made of the
pre assigned material. Figure 3.22 shows the extruded body. Note the scale bar
on the bottom of the figure which is there to compare relative size so as not to
be out of scale by some multiple factor. The global coordinates are also displayed
in the corner and clicking on an axis here will rotate the body to a desired view
for inspection.
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Figure 3.22 – Extruded beam
This completes the geometry stage and the model is now ready to setup with the
required loading arrangement and support conditions which will correctly
represent the real scenario. This is done by launching the mechanical application
by clicking on the Model cell in the project schematic.
Stage 3: Mechanical (model)
Launching the mechanical application takes the newly generated solid body and
places it in an environment where the body can be assigned a mesh of elements.
The tree outline in the mechanical window displays the current body part as
created in the geometry application. This is highlighted and the mesh control is
activated by clicking ‘generate mesh’. ANSYS Workbench is ideal for the
fundamental user as the programme chooses the size and type of element mesh
best suited to the body being analysis. In this case 451 quadratic hexahedron
(solid 186) elements are selected by the programme. The meshed solid body can
be seen in Figure 3.23.
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Figure 3.23 – Meshed beam
The quantity of elements can be specified by the programmer as the mesh tool
allows an element size to be selected. For this application a size of 20mm per
element was selected. Depending on the result required and the accuracy more
elements can be specified by reducing the element size, however, this increases
solving time. An example of a solid 186 element is outlined in Figure 3.24.
Figure 3.24 – Solid 186 element
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This type of element is described as a higher order 3 dimensional 20 node solid
element that exhibits quadratic displacement behaviour and is defined by 20
nodes having 3 degrees of freedom per node: translations in the nodal x, y and z
directions. This element is probably a little too advanced for the required
analysis but as the programme has chosen this element it was decided to run
with it though the solve time will increase. Midside nodes (nodes
A,B,Y,Z,V,X,R,T,Q,S,U,W in Figure 3.24 could be dropped for this type of analysis
thus giving linear solution at element edges. This will become a model updating
parameter in itself as lowering the order of the element by allowing the midside
nodes to be dropped will change the element from a Solid 186 element to a Solid
185 element as shown in Figure 3.25.
Figure 3.25 – Solid 185 element
This element is described in the ANSYS element library as being defined by eight
nodes having three degrees of freedom at each node: translation in the nodal x,
y and z directions. This will be an adequate element type to use and should lower
the solve time while still producing the data required. A view of this element can
be seen in Figure 3.26 from the ANSYS finite element modeller.
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Figure 3.26 – View of a quadratic hexahedron Solid 185 element in beam
Stage 4: Setup
Once the body has a mesh allocated to it the loads and support conditions can be
assigned. This is done in the setup stage by clicking on the static structural item
in the tree menu. This activates several options in the main menu ribbon close to
the top of the screen. There is no preference as to whether the load is placed
first or the support.
The selection of the type of support and boundary conditions will be determined
by the actual scenario in which the model is to exist in real life. The model in this
case was represented by a remote displacement support on each end face of the
beam. This option, with reference to the global coordinates, constrains the
movement of the beam at the supports to:
No movement in the X direction
No movement in the Y direction
Free movement in the Z direction (axially)
With regard to rotations:
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No rotation in the Y plane
No rotation in the Z plane
Free rotation in the X plane
This behaviour allows the supports to act as simple supports at the ends which
are representative of the experimental situation.
With the support conditions setup the loading is next to be applied. As the
resulting values obtained from mathematical analysis, as described in section
3.2.2, use moment values in bending stress equations it was thought adequate to
apply moments to each end of the beam to represent a point load as indicated in
the experimental setup. The bending moment values will be applied in a series of
steps. This is known as time history or time dependent tabular loading.
Essentially it means that loads can be placed on the body at a designated time
step. This will allow the 15 load increments to be applied to the body over a 16
second interval, the first being zero. Once the moment load is assigned to each
end face, the amount of load steps to be applied must be specified in the analysis
settings in the tree menu. This is set to 16 steps which is representative of the
experimental setup loading from 0 to 150N or nearest values for increments of
10N. The true values recorded on the day of the experiment were converted into
moments and entered into the tabular data for each load increment. Table 3.3
below shows the values of applied moment to each axis as calculated from
applied loads which were used in the experimental setup.
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Table 3.3 – Table of applied moments
Moments applied LHS (Nm) Moments applied RHS (Nm)
Time(s) x y z Time(s) x y z
0 0 0 0 0 0 0 0
1 -2.04 0 0 1 2.04 0 0
2 -3.5 0 0 2 3.5 0 0
3 -4.9 0 0 3 4.9 0 0
4 -6.7 0 0 4 6.7 0 0
5 -8.4 0 0 5 8.4 0 0
6 -9.7 0 0 6 9.7 0 0
7 -11.5 0 0 7 11.5 0 0
8 -13.2 0 0 8 13.2 0 0
9 -14.6 0 0 9 14.6 0 0
10 -16.3 0 0 10 16.3 0 0
11 -18 0 0 11 18.04 0 0
12 -19.8 0 0 12 19.8 0 0
13 -21.3 0 0 13 21.3 0 0
14 -22.7 0 0 14 22.7 0 0
15 -24.6 0 0 15 24.6 0 0
16 0 0 0 16 0 0 0
Stage 5: Solution
This stage of the analysis allows for the input of solution parameters in which
results will be solved for. The solutions needed for this analysis contain:
Total deformation
Minimum principal elastic strain (compressive strain)
Maximum principal elastic strain (tensile strain)
As these were the only parameters required, where the expected maximum of
the tensile strain was at the bottom most fibres and the maximum compressive
strain was at the top most fibres of the beam, no other parameters were entered
into the solver solution process.
Stage 6: Results
The final stage in the static structural analysis system is to solve the analysis. This
is done by clicking the solve icon and allowing the programme to solve
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in its own time. During this time the computer generates the mathematical
model and solves the thousands of differential equations relating to each node
of each element in the mesh in order to arrive at a solution for the required
parameters. The results are available in graphic format and also tabular data for
each value at a minimum and maximum value for each time step which
represents the moment increments. The results are presented in the results
chapter.
The results from this analysis were compared to the experimental results, and
where possible, changes were made to the FE model in an updating procedure to
calibrate the model to the experimental setup. The results of these are also
presented in the results section.
3.4.4 Transient load on 3 hinge arch bridge
The 3 hinge arch bridge was modelled in ANSYS following the same procedures
as with the T shaped beam described in the previous section. The use of a static
structural analysis system was used again. Each stage is described here assuming
the same procedure but with some differences which were made to the process
to make it specific to the arch bridge model.
The method of analysing this structure with the load changing position along the
top from one end to the other in stages of 100mm was done by firstly setting up
the first 3 stages in the static structural system. It is possible to link the setup of
one analysis system to that of many others. This allowed the use of the first
setup arrangement to be linked to each stage of the load position which was 6 in
this case.
Stage 1: Engineering data
There was no way of assessing the material properties of the steel which makes
up the bridge in the experiment, however, an assumption was made to give it
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the same properties as structural steel. It was decided that the load being
applied was on such a small scale compared to the structural capacity of the
material to resist any real deformations that the use of the preset values were
sufficient.
During the experimental data collection process the self weight of the bridge was
not considered when analysing the horizontal reactions due to the applied
vertical loading, this was done by setting the reading on the force readout to
zero prior to applying the load thus neglecting any horizontal loading from the
self weight.
The data used was that which was set as default in the ANSYS material library
and was as follows in Table 3.4.
Table 3.4 – Structural steel material properties
Temperature C Young's Modulus Pa
Poisson's Ratio
Bulk Modulus Pa
Shear Modulus Pa
2.e+011 0.3 1.6667e+011 7.6923e+010
Compressive Yield Strength Pa
2.5e+008
Tensile Yield Strength Pa
2.5e+008
Tensile Ultimate Strength Pa
4.6e+008
The engineering data presented here was applicable to two design attempts. For
the rest of this section there are two designs attempts presented which were
subsequently analysis for their accuracy in replicating the experimental setup
described in section 3.3
Stage 2: Geometry
The creation of this structure in geometrical terms is dependent on the
perception and understanding as to how best replicate the existing model in
ANSYS. As the aim of this dissertation is to create finite element models, which
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behave as close as possible to the experimental setup data of each tested
structure, different approaches to the modelling of the 3 hinge arch bridge were
attempted. The problems and the subsequent solutions are dealt with in the
results and discussion chapter. Here each of the modelling attempt methods is
described in detail.
Attempt 1
This attempt was created as a 3 dimensional model of only half of the bridge. The
model was constructed in the Design Modeller function of the Model static
structural cell where 3 sketches were generated. The first sketch was one of the
side plates of the bridge and this was then extruded to a depth of 6mm. The
extruded sketch was then copied via the tools dropdown menu and copied to a
distance equal to the experimental bridge. This allowed the correct gap between
the plates to be established. Then the pin, highlighted in green in Figure 3.27 was
drawn and extruded to the same value as the plates were offset. The final sketch
drawn was the 6 connecting rods along the top and these were subsequently
extruded to the same amount. The result is shown in Figure 3.27.
Figure 3.27 – Extruded solid structure
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Stage 3: Mechanical (model)
This stage involves as before the application of the mesh to the newly created
bodies. ANSYS Workbench makes an attempt at creating a mesh which is most
suited to the type of analysis and geometric shape being tested. The choice of
element selected was the Linear Tetrahedron element type. The model mesh
was then generated to take the form pictured in Figure 3.28.
Figure 3.28 – Solid structure after meshing
The size of elements could be controlled by the user and a size of 30mm was
deemed to be substantial as stresses or strains were not of concern in this
model, only reactions at the hinge of the structure were needed. Supporting
conditions and loadings were applied to the model in the setup stage of the
analysis.
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Stage 4: Setup
On initial observations one half of the arch bridge seemed to be sufficient so
modelling the part would have to bear resemblance to the actual experimental
setup in as much as possible. The following support conditions shown in Table
3.5 were applied to the model.
Table 3.5 – Tabular data for supports
X Coordinate 1.1556e-032 m 1.e-002 m 0.5025 m
Y Coordinate 0.2123 m 0.225 m 1.e-002 m
Z Coordinate 3.e-003 m 4.5e-002 m 4.65e-002 m
Location Defined
Definition
Type Remote Displacement
Remote Force Remote Displacement
X Component 0. m (ramped) 0. N (ramped) 0. m (ramped)
Y Component Free -12.4 N (ramped) 0. m (ramped)
Z Component 0. m (ramped) 0. N (ramped) 0. m (ramped)
Rotation X 0. ° (ramped)
Rotation Y 0. ° (ramped)
Rotation Z Free
Suppressed No
Behaviour Deformable Rigid
Define By Components
Rotation X 0. ° (ramped)
Rotation Y 0. ° (ramped)
Rotation Z Free
Remote displacements and the conditions as outlined in Table 3.5 were applied
to the faces of the solid as shown in Figure 3.29 and Figure 3.30.
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Figure 3.29 – Remote displacement support 1
The second support was applied to the hinge face as seen in Figure 3.30.
Figure 3.30 – Remote displacement support 2
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The next stage was to apply the loading. As described at the outset of this
subsection there needed to be 6 stages of loading to account for the variable
load location on the structure. Each system had the same model, mesh, support
conditions and material data. Loading was the only variable in 6 stages from
distances of 0mm (centre of the bridge) to 500mm (over support) in each
separate system.
Stage 5: Solution
The solution requirements of this model were only to assess the support reaction
due to the varying load position. It was thought proper to allow the solution to
contain the equivalent (von-Mises) stress analysis to see if they look like the
predicted distribution. The support reaction was allocated to the hinge location
at the lower right hand side of the model where the remote displacement
support was located.
Stage 6: Results
The results from the model are obtained by solving the model in order to
produce a post-processed result. The required results are decided on in the setup
stage. The results required for this model for each 15 load steps were:
Remote displacement support 1 horizontal and vertical reactions
Remote displacement support 2 horizontal and vertical reaction
Von-Misses stress distribution (for information only)
Results for each load step are displayed and discussed in the results chapter.
Issues in relation to this method are discussed in detail later. Problems with
vertical reaction values prompted a rethink of the modelling setup. A second
attempt was made to represent the true experimental setup as close as possible.
The second attempt follows.
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Attempt two
Stage 1: Engineering data
The material properties in stage one of the analysis setup, engineering data,
remains as structural steel for the second attempt with the same isotropic
elasticity properties. The critical difference is the model geometry which is
attempted to recreate the experimental setup more accurately.
Stage two: Geometry
This second attempt posed a significant challenge as it involved having two
bodies connected by joints. The issues relating to the first attempt needed to be
rectified and a more true representation of the experimental structure was
needed. This was achieved by creating 4 separate body parts shown in Figure
3.31 and Figure 3.32 below highlighted in green.
Figure 3.31 – Body parts 1 and 2
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Figure 3.32 – Body parts 2 and 3
As can be seen by the 3 dimensional solid created in ANSYS Design Modeller,
there is a close representation to the experimental setup displayed in Figure
3.33.
Figure 3.33 – Experimental setup
Stage 3: Mechanical (model)
The solid model consists of 4 body parts which are connected at the centre by
two bodies which need to act as a joint or hinge. This need allows the structure
to behave exactly like the experimental setup of the 3 hinge arch bridge. The
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setting up of the joint types is done in the model stage of the analysis. In order to
replicate the 3 hinge arch structure the interface between the two solid pins at
the centre of the bridge and both sides of the bridge need to be modelled with a
revolute, solid to multiple joint MPC184 element connection between the bodies
at the centre of the bridge on both sides with freedom of rotation about the “Z”
axis as pictured in Figure 3.34 below.
Figure 3.34 – Revolute joint connection No's 1 & 2
MPC184 Revolute Joint Element Description:
The MPC184 revolute joint is a two-node element that has only one primary
degree of freedom, the relative rotation about the revolute (or hinge) axis. This
element imposes kinematic constraints such that the nodes forming the element
have the same displacements. Additionally, only a relative rotation is allowed
about the revolute axis, while the rotations about the other two directions are
fixed which is visible from Figure 3.35 below.
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Figure 3.35 – MPC-184 revolute joint (ANSYS)
This condition allows the connection at this location to only allow rotation about
the Z axis while still maintaining a transmission of shear across the connection.
There should be zero moment experienced at this location also which is
representative of the experimental setup.
The created model consisting of 4 body parts was meshed using the ANSYS
automatic meshing capability which is program controlled to a large extent. The
mesh size was set at 20mm. The meshing of the body parts took the form
depicted in Figure 3.36 below.
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Figure 3.36 – 3-D model after meshing
Once these conditions were completed the next stage in the analysis was to set
boundary conditions and loading in the setup stage.
Stage 4: Setup
To adequately represent the model as the true structure it was important to
mimic the real conditions experienced by the model. The provision of revolute
joints in the setup stage allows for one of the hinges of the three required for
this type of structure.
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Table 3.6 outlines the properties of the joints at the bridge centre.
Table 3.6 – Centre hinge revolute joint data
Object Name Revolute - Solid To Multiple Revolute - Solid To Multiple
State Fully Defined
Definition
Connection Type Body-Body
Type Revolute
Torsional Stiffness 0. N·m/°
Torsional Damping 0. N·m·s/°
Suppressed No
Reference
Scoping Method Geometry Selection
Scope 1 Face
Body Solid
Coordinate System Reference Coordinate System
Behaviour Rigid
Pinball Region All
Mobile
Scoping Method Geometry Selection
Scope 8 Edges
Body Multiple
Initial Position Unchanged
Behavior Rigid
Pinball Region All
Stops
RZ Min Type None
RZ Max Type None
In order to provide the two remaining hinges the use of remote displacements is
required at the two pins at the lower sides of the model with constraints listed in
Table 3.7.
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Table 3.7 – Remote displacement support details
Object Name Remote Displacement
Remote Displacement 2
Force
State Fully Defined
Scope
Scoping Method Geometry Selection
Geometry 1 Face
Coordinate System
Global Coordinate System
X Coordinate 0.5025 m -0.5025 m
Y Coordinate 1.5e-002 m
Z Coordinate 4.8e-002 m
Location Defined
Definition
Type Remote Displacement Force
X Component 0. m (ramped) 0. N (ramped)
Y Component 0. m (ramped) -24.8 N (ramped)
Z Component 0. m (ramped) 0. N (ramped)
Rotation X 0. ° (ramped)
Rotation Y 0. ° (ramped)
Rotation Z Free
Suppressed No
Behaviour Deformable
Rotation X 0. ° (ramped)
Rotation Y 0. ° (ramped)
Rotation Z Free
Define By Components
Coordinate System
Global Coordinate System
Advanced
Pinball Region All
The remote displacement boundary conditions outlined in Table 3.7 are
representative of a hinge support. Neither support allows translations in X, Y or Z
directions. The only allowable movement by the hinge supports is the free
rotation about the Z plane. This allows the body attached to each of these
supports to rotate freely which is representative of the experimental setup.
The application of these boundary conditions ensures reactions in the horizontal
and vertical directions at these locations but no transfer of moment, reflective
conditions of the true scenario.
Loading in the model will be in accordance with the physical experiment and will
be a unit mass of 2536 grams which equates to approximately 24.8N placed at
100mm intervals along the top surface of the ridge deck. This is replicated by
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allowing the load to be placed on each of the connecting pins which for part of
the solid structure. The first load stage is depicted in Figure 3.37.
Figure 3.37 – First load stage applied to bridge deck
The surface of the connecting pin is selected as the application surface then by
entering the load direction using the components selection and entering 0 for Z
and X directions and a value of -24.8N in the Y direction.
As there are 11 load positions to be solved for it is possible to use the same
model and meshing for all stages by linking a separate static structural analysis
system for each load. This keeps all the supporting boundary conditions the same
and there is no need to set these up for each load step. The load application is
repeated for the remaining load steps from 0mm to 1000mm in 100mm
intervals, which is the extent of the bridge deck.
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Stage 5: Solution
Solution results for this structure only require support reactions in the horizontal
and vertical sense as there is no moment reaction capability at any of the 3
hinges. The required information for this structure was horizontal and vertical
reactions at each lower end of the bridge at the remote displacement supports.
As a monitoring exercise the distribution of stresses was selected as a solution
parameter to allow a general comparison with expected locations of maximum
and minimum von-Mises (failure criteria) stresses which are an indication of the
possible failure of the material due to a combination of stresses in the x, y and z
directions.
Stage 6: Results
Solving the model produces the desired results after the computer calculates and
solves the mathematical equations. Based on the data required in the solution
stage the following results were processed:
Remote displacement support 1 horizontal and vertical reactions
Remote displacement support 2 horizontal and vertical reaction
Von-Mises stress distribution (for information only)
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4 Results
4.1 T shaped aluminium beam results
4.1.1 Experimentally determined strain values
Experimental strain values at the locations on the beam for test number 1 values
are in Table 4.1 with the graphical representation shown in Figure 4.1.
Table 4.1 – Experimental strain values for test No.1
Applied Load (N) Channel Number Deflection
Stage Desired Actual 1 2 3 4 5 6 7 8 9 Centre(mm)
1 0 0 0 0 0 0 0 0 0 0 0 0
2 10 16 14 2 -4 -6 -16 -4 -3 -5 6 0.04
3 20 22 25 8 -5 -12 -24 -9 -7 -9 13 0.06
4 30 29 37 13 -8 -16 -29 -16 -13 -10 15 0.09
5 40 42 54 21 -13 -24 -42 -26 -26 -14 28 0.15
6 50 53 65 31 -14 -35 -51 -34 -26 -18 35 0.18
7 60 60 73 32 -19 -37 -58 -36 -32 -19 40 0.21
8 70 69.6 95 38 -19 -40 -66 -42 -40 -24 46 0.24
9 80 83 103 47 -26 -50 -79 -55 -50 -30 56 0.29
10 90 90 110 54 -28 -57 -85 -61 -51 -30 62 0.31
11 100 101 120 62 -33 -62 -93 -73 -63 -35 72 0.35
12 110 110 133 66 -33 -65 -103 -76 -69 -37 77 0.39
13 120 122 146 76 -37 -72 -111 -85 -75 -41 84 0.43
14 130 131 163 82 -39 -79 -118 -95 -84 -44 90 0.46
15 140 140 180 90 -41 -79 -124 -103 -89 -48 99 0.49
16 150 151 192 95 -45 -87 -129 -109 -96 -50 104 0.52
Figure 4.1 – Graphical representation for test No. 1
-200
-100
0
100
200
300
-10 10 30 50 70 90 110 130 150 Mic
rost
rain
(µε)
Load(N)
Test No. 1- µε v Load
1
2
3
4
5
6
7
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The results from test number 2 are displayed in Table 4.2 with the graphical
representation displayed in Figure 4.2.
Table 4.2 – Experimental strain values for test No.2
Applied Load (N) Channel Number Deflection
Stage Desired Actual 1 2 3 4 5 6 7 8 9 Centre(mm)
1 0 0 0 0 0 0 0 0 0 0 0 0
2 10 10.7 14 7 -3 -4 -5 -11 -5 -3 6 0.03
3 20 21.2 25 12 -5 -13 -17 -15 -12 -4 16 0.08
4 30 29.5 37 18 -8 -18 -23 -22 -17 -6 22 0.12
5 40 39 54 27 -11 -27 -31 -26 -26 -10 34 0.16
6 50 51.6 65 33 -13 -25 -33 -32 -31 -12 41 0.21
7 60 59.3 73 37 -15 -27 -40 -38 -36 -11 46 0.24
8 70 70.3 95 44 -18 -35 -45 -44 -39 -17 56 0.28
9 80 82.2 115 52 -21 -40 -56 -52 -50 -24 62 0.32
10 90 89.9 120 59 -24 -46 -64 -61 -51 -27 68 0.35
11 100 101 125 67 -28 -52 -70 -67 -59 -28 75 0.39
12 110 112 133 73 -30 -55 -81 -72 -67 -30 85 0.43
13 120 122 146 81 -34 -66 -86 -83 -73 -34 92 0.47
14 130 133 163 90 -37 -69 -94 -89 -81 -37 99 0.5
15 140 141 180 94 -39 -73 -99 -91 -87 -39 107 0.53
16 150 151 195 96 -40 -80 -107 -97 -93 -44 112 0.56
Figure 4.2 - Graphical representation of experimental load versus strain for test No. 2
-150
-100
-50
0
50
100
150
200
250
-10 10 30 50 70 90 110 130 150 Mic
rost
rain
(µε)
Load(N)
Test No. 2- µε v Load
1
2
3
4
5
6
7
8
9
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The results from test number 3 are displayed in Table 4.3 with the graphical
representation shown in Figure 4.3.
Table 4.3 – Experimental strain values for test No. 3
Applied Load (N) Channel Number Deflection
Stage Desired Actual 1 2 3 4 5 6 7 8 9 Centre(mm)
1 0 0 0 0 0 0 0 0 0 0 0 0
2 10 11 14 9 -1 -7 -5 -9 -6 -2 4 0.05
3 20 21.4 25 12 -7 -14 -14 -17 -18 -10 11 0.09
4 30 32 37 17 -8 -19 -22 -24 -22 -13 19 0.13
5 40 42 54 26 -17 -28 -31 -38 -26 -16 25 0.17
6 50 50 65 28 -15 -35 -38 -43 -31 -17 35 0.2
7 60 59 73 36 -18 -37 -45 -48 -38 -18 38 0.23
8 70 73 90 45 -22 -44 -56 -60 -48 -26 50 0.29
9 80 79 110 50 -27 -52 -60 -65 -50 -22 54 0.31
10 90 89 120 56 -28 -29 -71 -73 -57 -28 65 0.34
11 100 99 125 64 -33 -63 -76 -80 -64 -33 70 0.38
12 110 111 133 71 -39 -70 -85 -92 -72 -36 78 0.42
13 120 122 146 77 -39 -75 -94 -101 -78 -40 89 0.46
14 130 129 163 84 -41 -79 -101 -107 -82 -41 90 0.48
15 140 138 180 88 -46 -83 -106 -111 -88 -47 97 0.51
16 150 152 199 100 -46 -91 -118 -112 -97 -50 108 0.55
Figure 4.3 - Graphical representation of experimental load versus strain for test No. 3
-150
-100
-50
0
50
100
150
200
250
-10 10 30 50 70 90 110 130 150 Mic
rost
rain
(µε)
Load(N)
Test No 3- µε v Load
1
2
3
4
5
6
7
8
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The resulting average strain values from the 3 tests are presented in Table 4.4
with the corresponding graphical representation shown in Figure 4.4.
Table 4.4 – Experimental strain values averaged
Applied Load (N) Channel Number Deflection
Stage Desired Actual 1 2 3 4 5 6 7 8 9 Centre(mm)
1 0 0 0 0 0 0 0 0 0 0 0 0.00
2 10 13 14 6 -3 -6 -9 -8 -5 -3 5 0.04
3 20 22 25 11 -6 -13 -18 -14 -12 -8 13 0.08
4 30 30 37 16 -8 -18 -25 -21 -17 -10 19 0.11
5 40 41 54 25 -14 -26 -35 -30 -26 -13 29 0.16
6 50 52 65 31 -14 -32 -41 -36 -29 -16 37 0.20
7 60 59 73 35 -17 -34 -48 -41 -35 -16 41 0.23
8 70 71 93 42 -20 -40 -56 -49 -42 -22 51 0.27
9 80 81 100 50 -25 -47 -65 -57 -50 -25 57 0.31
10 90 90 117 56 -27 -44 -73 -65 -53 -28 65 0.33
11 100 100 123 64 -31 -59 -80 -73 -62 -32 72 0.37
12 110 111 133 70 -34 -63 -90 -80 -69 -34 80 0.41
13 120 122 146 78 -37 -71 -97 -90 -75 -38 88 0.45
14 130 131 163 85 -39 -76 -104 -97 -82 -41 93 0.48
15 140 140 178 91 -42 -78 -110 -102 -88 -45 101 0.51
16 150 151 195 97 -44 -86 -118 -106 -95 -48 108 0.54
Figure 4.4 - Graphical representation of experimental load versus strain for average values
-150
-100
-50
0
50
100
150
200
250
-5 15 35 55 75 95 115 135 155 Mic
rost
rain
(µε)
Load(N)
Experimental Averaged - µε v Load
1
2
3
4
5
6
7
8
9
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4.1.2 Experimentally derived Young’s modulus
Based on the averaged experimentally derived values for strain, the
experimental value for the modulus of elasticity was derived using Equation 6
from the Methodology Chapter. The values in Table 4.5 are average values for
each strain gauge location over the 15 load stages:
Table 4.5 – Experimentally derived Young’s modulus
Gauge No. 1 2 3 4 5 6 7 8 9
Dist. To NA(mm) 33.96 16.84 6.96 13.93 17.12 17.12 13.93 6.96 16.84
Young's Modulus
of Elasticity (N/mm
2)
78917 91310 84912 79975 64266 69621 97112 67930 102724
75726 88010 68470 59735 52057 69833 62964 50608 70408
71681 82197 67945 61579 54204 64695 62763 56230 70455
66752 72464 54055 56148 52418 60572 56868 55407 61636
69703 73261 66325 58687 56164 62863 63356 59269 60721
71579 74031 61782 63663 55262 64774 60660 66931 62687
66849 73084 65019 64519 56503 64630 60455 57256 61064
71565 71451 59461 62018 55504 62926 58710 57896 61897
67546 69367 60565 73464 54173 61118 60989 57002 60118
71522 67992 57697 61327 55819 60640 58360 56495 60472
73375 69131 58825 63205 54866 61496 57735 58254 60490
73465 68189 59952 61967 55744 60303 58403 57346 60212
70658 66927 60524 62435 55649 59857 57379 58043 61410
68984 67158 59918 64299 56446 60887 57236 56341 60287
68114 68016 62446 63459 56841 63276 57247 56808 61089
Average: 63959 Max value 102724
Min value 50608
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4.1.3 Theoretical strain values
Based on the experimentally derived Young’s modulus of 64GPa, the following
theoretical strain values were computed for the 15 load stages are presented in
Table 4.6.
Table 4.6 – Theoretical strain values for Young’s modulus of 64GPa
Gauge No. 1 2 3 4 5 6 7 8 9
Dist. To N/A (mm) 33.96 16.84 6.96 13.93 17.12 17.12 13.93 6.96 16.84
Theoretical strain x10
-6 values
determined from experimentally derived Young's
modulus
17 9 -4 -7 -9 -9 -7 -4 9
30 15 -6 -12 -15 -15 -12 -6 15
41 21 -8 -17 -21 -21 -17 -8 21
56 28 -12 -23 -28 -28 -23 -12 28
71 35 -15 -29 -36 -36 -29 -15 35
82 40 -17 -33 -41 -41 -33 -17 40
97 48 -20 -40 -49 -49 -40 -20 48
112 55 -23 -46 -56 -56 -46 -23 55
123 61 -25 -51 -62 -62 -51 -25 61
138 68 -28 -57 -69 -69 -57 -28 68
152 76 -31 -63 -77 -77 -63 -31 76
168 83 -34 -69 -84 -84 -69 -34 83
180 89 -37 -74 -91 -91 -74 -37 89
192 95 -39 -79 -97 -97 -79 -39 95
208 103 -43 --85 -105 -105 -85 -43 103
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4.1.4 Compressive strain results (ANSYS)
The results produced by ANSYS are conveniently output in tabular form. Table
4.7 shows the results for the minimum principal elastic strain values for each of
the load steps which are the measure of the compressive strain experienced by
the beam as a result of the loading.
Table 4.7 – ANSYS produced minimum elastic strain
Time [s] Minimum [m/m] Maximum [m/m]
1. -9.2184e-006 -1.5716e-007
2. -1.5816e-005 -2.6964e-007
3. -2.2142e-005 -3.7749e-007
4. -3.0276e-005 -5.1616e-007
5. -3.7958e-005 -6.4713e-007
6. -4.3833e-005 -7.4728e-007
7. -5.1966e-005 -8.8595e-007
8. -5.9648e-005 -1.0169e-006
9. -6.5975e-005 -1.1248e-006
10. -7.3657e-005 -1.2557e-006
11. -8.152e-005 -1.3898e-006
12. -8.9473e-005 -1.5254e-006
13. -9.6251e-005 -1.6409e-006
14. -1.0258e-004 -1.7488e-006
15. -1.1116e-004 -1.8952e-006
16. 0. 0.
The result from the ANSYS finite element model of the beam under the
maximum load condition at load step 15 is shown in Figure 4.5. This figure shows
the minimum principal elastic strain, the dark blue colour indicating the
minimum value, which is negative and represents the maximum compressive
strain produced in the beam by the applied maximum loading as it is the lower
value. The dark red colour running through the beam is indicative of the largest
value which indicates the maximum of the values of negative strain, which is the
larger number, indicating the location of least strain and is coincident with the
neutral axis, as expected.
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Figure 4.5 – ANSYS compressive strain graphic, updated model
Each individual strain gauge value from the experimental setup was not obtained
from the ANSYS model. The values obtained from the model concerned only the
maximum compressive and tensile strain values. Maximum compressive strain
values are graphed in Figure 4.6 and are taken directly from the output file from
the ANSYS report. The full report can be seen on the CD in Appendix D. The
experimental values were obtained by averaging the values from Gauges 5 and 6
(located on the top face of real beam) and are compared to the pre and post-
updated finite element models.
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Figure 4.6 – Maximum compressive strain values
The results obtained from the finite element modelling are tabulated in
Table 4.8 is a percentage comparison basis between the pre and post-updating
model values and the experimental compressive strain results. It can be seen
that the updating procedure has resulted in the model returning on average
100% of the experimental strain produced by the maximum loading.
Table 4.8 – Compressive strain comparison
Model % of Experimental Max Compressive Strain
Load Step Pre-updating Post-updating
1 100 113
2 89 101
3 88 100
4 84 96
5 89 101
6 89 102
7 90 102
8 88 100
9 86 98
10 87 99
11 87 98
12 86 98
13 86 98
14 88 99
15 89 102
Average 88 100
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Stra
in
Load Step
Pre-Updated
Updated
Experimental Data
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4.1.5 Tensile strain results (ANSYS)
Results for tensile strain obtained from the ANSYS model are produced as
maximum principal elastic strain values. Table 4.9 shows the results tubulised for
each of the load steps.
Table 4.9 – ANSYS maximum principal elastic strain
Time [s] Minimum [m/m] Maximum [m/m]
1. 4.7491e-007 1.6563e-005
2. 8.1479e-007 2.8417e-005
3. 1.1407e-006 3.9784e-005
4. 1.5597e-006 5.4399e-005
5. 1.9555e-006 6.8201e-005
6. 2.2581e-006 7.8756e-005
7. 2.6772e-006 9.3371e-005
8. 3.0729e-006 1.0717e-004
9. 3.3988e-006 1.1854e-004
10. 3.7946e-006 1.3234e-004
11. 4.1997e-006 1.4647e-004
12. 4.6094e-006 1.6076e-004
13. 4.9586e-006 1.7294e-004
14. 5.2845e-006 1.8431e-004
15. 5.7268e-006 1.9973e-004
16. 0. 0.
The graphical results from the ANSYS finite element model for maximum tensile
strain at load step 15 are shown in Figure 4.7 where the red colour running along
the extreme bottom of the beam is the maximum positive strain which indicates
tensile strain. This is where the maximum tensile strain value was expected. The
dark blue running through the beam is the minimum value of tensile strain and is
coincident with the neutral axis which is to be expected.
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Figure 4.7 – ANSYS tensile strain graphic, post-updated
Results for the maximum experimental tensile strain are from the average
maximum value of strain gauge number 1. The values for the finite element
model were transferred from ANSYS to Excel and then compared to the
experimental values. The results are compared graphically in Figure 4.8.
Figure 4.8 – Maximum tensile strain values
0
0.00005
0.0001
0.00015
0.0002
0.00025
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Stra
in
Load Step
Pre-Updated
Updated
Experimental Data
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It can be seen from Figure 4.8 that the experimental maximum tensile strain lies
almost midway between the pre and post-updated values. This was noted as
being inconsistent with expectations. When the values were compared as
percentage values of the experimental results, seen in Table 4.10, the finite
element pre-updated model is actually closer to the experimental data than the
post-updated value. Inaccuracies in the performance of gauge number 1 were
witnessed on the day of the experiment.
Table 4.10 – Tensile strain comparison
Model % of Experimental Max Tensile Strain
Load Step Pre-updating Post-updating
1 107 124
2 102 119
3 97 113
4 91 106
5 95 110
6 97 113
7 90 105
8 97 113
9 92 107
10 97 113
11 99 116
12 99 116
13 96 111
14 93 109
15 92 107
Average 96 113
Following the identification of a possible problem with the experimental strain
values for gauge number 1, the following investigation took place.
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4.1.6 Experimental strain investigation
To highlight possible inaccuracies in the measuring equipment a comparison of
the theoretical values as a percentage of the experimental values are displayed
in Figure 4.9.
The values obtained for theoretical strain are based on the experimentally
derived Young’s modulus which was derived in Table 4.5. Using this value the
theoretical strain values were computed as shown in Table 4.6 which was then
used as the basis for the graph in Figure 4.9.
This graph gives an indication as to how each strain gauge performed during the
experimental testing. It was noted on the day of the test that Gauge number 1
was displaying inconsistent readings which highlighted the need for further
investigation. As the reading from this gauge has high significance in being the
position of maximum tensile strain on the beam, the investigation was
warranted. Gauge number 1 is highlighted in yellow and can be seen to be
exceeding the 100% mark in all load steps. As the values shown in Figure 4.9 are
the relationship of how much larger the theoretical strain values were compared
to the measured values of gauge number 1, the conclusion is that the
performance of strain gauge 1 was below that of the expected values. The result
was an average required increase of 11% in the experimental strain data for
gauge number 1.
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Figure 4.9 - % Variation between experimental strain gauge readings and theoretical values
60 80 100 120 140 160
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
% Variation
Load
Ste
p
9
8
7
6
5
4
3
2
1
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Following the % error calculated in Figure 4.9 a percentage error of 11% was
computed in the experimental data for strain gauge No 1 which lead to an
adjustment of the experimental data for this gauge reading. When the
adjustment was accounted for the following graph was produced seen in Figure
4.10 which shows a closer relationship between the experimental tensile strain
and the updated finite element model result.
Figure 4.10 – Modified maximum tensile strain values (Gauge 1)
Table 4.11 shows the comparison between the pre and post-updating and how
the tensile strain values compare to the experimental result. The post-updating
value is almost 100% of the experimental result which was encouraging.
0
0.00005
0.0001
0.00015
0.0002
0.00025
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Stra
in
Load Step
Pre-updated
Updated
Experimental
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Table 4.11 – Modified tensile strain comparison
Model % of Experimental Max Tensile Strain
Load Step Pre-updating Post-updating
1 96 112
2 92 108
3 87 102
4 82 95
5 85 99
6 88 102
7 81 95
8 87 101
9 83 96
10 87 102
11 89 104
12 89 104
13 86 100
14 84 98
15 83 97
Average 87 102
4.1.7 Elastic Beam Bending Deflection Results
As with the previous strain values, ANSYS produces a table of the recorded
deformation values, Table 4.12 shows the maximum and minimum deflection
values for each load step.
Table 4.12 – ANSYS total deformation results
Time [s] Minimum [m] Maximum [m]
1. 1.9114e-007 4.1156e-005
2. 3.2795e-007 7.061e-005
3. 4.5913e-007 9.8854e-005
4. 6.2779e-007 1.3517e-004
5. 7.8708e-007 1.6946e-004
6. 9.0889e-007 1.9569e-004
7. 1.0775e-006 2.3201e-004
8. 1.2368e-006 2.663e-004
9. 1.368e-006 2.9455e-004
10. 1.5273e-006 3.2884e-004
11. 1.6903e-006 3.6395e-004
12. 1.8553e-006 3.9945e-004
13. 1.9958e-006 4.2971e-004
14. 2.127e-006 4.5796e-004
15. 2.305e-006 4.9629e-004
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The graphical result from ANSYS modeller for the maximum deflection is pictured
in Figure 4.11. As expected the maximum deflection occurs at the centre of the
beam. There are 16 images showing each load step increment of approximately
10N and it is not feasible to reproduce every one of them as they all show the
maximum value of deflection for each load step at the midpoint of the beam.
The video clip showing the transition through the load steps is visible on the CD
in Appendix D.
Figure 4.11 – ANSYS maximum deflection graphic
Figure 4.12 shows the relationship between the experimentally produced
deflections at the centre of the beam with those obtained from the finite
element modelling pre and post-updating stages.
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Figure 4.12 – Maximum deflection values per load step
To compare the results in percentage form Table 4.13 displays the relationship
between the pre and post-updated finite element models to the experimental
measured deflection. While the measured deflection was still greater than the
updated model, an improvement was seen in the post-updated model.
Table 4.13 – Deflection comparison
Model % of Experimental Deflection
Load Step Pre-updating Post-updating
1 93 103
2 83 92
3 79 87
4 76 84
5 78 86
6 78 86
7 77 86
8 78 87
9 80 88
10 79 88
11 79 88
12 79 88
13 81 90
14 81 90
15 82 91
Average 80 89
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
De
fle
ctio
n (
mm
)
Load Step
Updated FE Model
Pre-Updated FE Model
Experimental Data
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As a further check on the performance of the finite element model on the
deflection values, two other methods were implemented to predict the
maximum deflection as described in the Methodology chapter. They were the
finite element direct stiffness method and the double integration or Macaulay’s
method for deflection. The following section is the results produced from those
mathematical calculations.
4.1.8 Deflection results from finite element stiffness method
Following the procedure outlined in the Methodology chapter the beam was
broken down, or discretised into 4 elements and 5 nodes as seen in Figure 4.13.
Figure 4.13 –Beam discretised into 4 elements and 5 nodes
Based on the discretisation of the beam into 4 elements, an individual element
stiffness matrix was computed and all 4 element stiffness matrices are produced
in Table 4.14.
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Table 4.14 – Element stiffness matrices
Element 1
1 6 2 7
k1 350 56.8 -350 56.8 1
L = .325 56.8 12.3 -56.8 6.1 6
-350 -56.8 350 -56.8 2
56.8 6.15 -56.8 12.3 7
Element 2
2 7 3 8
k2 25779 1000 -25779 1000 2
L = 0.0775 1000 51.6 -1000 25.8 7
-25779 -1000 25779 -1000 3
1000 25.8 -1000 51.6 8
Element 3
3 8 4 9
k3 25779 1000 -25779 1000 3
L = 0.0775 1000 51.6 -1000 25.8 8
-25779 -1000 25779 -1000 4
1000 25.8 -1000 51.6 9
Element 4
4 9 5 10
k4 350 56.8 -350 56.8 4
L = .325 56.8 12.3 -56.8 6.1 9
-350 -56.8 350 -56.8 5
56.8 6.15 -56.8 12.3 10
Combining all 4 element matrices into 1 single structure matrix results in a 10 x
10 structure stiffness matrix and is presented in Table 4.15.
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Table 4.15 – Combined structure stiffness matrix
1 2 3 4 5 6 7 8 9 10
1 350 -350 0 0 0 56.8 56.8 0 0 0
2 -350 26129 -25779 0 0 -56.8 943.2 1000 0 0
3 23 -25779 51558 -25779 0 0 -1000 0 1000 0
4 0 0 -25779 26129 -350 0 0 -1000 -943.2 56.8
5 0 0 0 -350 350 0 0 0 -56.8 -56.8
6 56.8 -56.8 0 0 0 12.3 6.15 0 0 0
7 56.8 943.2 -1000 0 0 6.15 63.9 25.8 0 0
8 0 1000 0 -1000 0 0 25.8 103.2 25.8 0
9 0 0 1000 -943.2 -56.8 0 0 25.8 63.9 6.15
10 0 0 0 56.8 -56.8 0 0 0 6.15 12.3
The combined structure stiffness matrix was then multiplied by the Young’s
modulus (E) and the second moment of area (I) and results in the 10 x 10 matrix
produced in Table 4.16.
Table 4.16 – Structure stiffness matrix multiplied by EI
1 2 3 4 5 6 7 8 9 10
1 1405250 -1E+06 0 0 0 228052 228052 0 0 0
2 -1E+06 1E+08 -1E+08 0 0 -228052 3786948 4015000 0 0
3 92345 -1E+08 2.1E+08 -1E+08 0 0 -4E+06 0 4E+06 0
4 0 0 -1E+08 1E+08 -1E+06 0 0 -4E+06 -4E+06 228052
5 0 0 0 -1E+06 1405250 0 0 0 -228052 -228052
6 228052 -228052 0 0 0 49384.5 24692.3 0 0 0
7 228052 3786948 -4E+06 0 0 24692.3 256559 103587 0 0
8 0 4015000 0 -4E+06 0 0 103587 414348 103587 0
9 0 0 4015000 -4E+06 -228052 0 0 103587 256559 24692.3
10 0 0 0 228052 -228052 0 0 0 24692 49384.5
The inverse of the matrix displayed in Table 4.16 was computed using the matrix
function in Excel and the result is produced in Table 4.17 and is required as part
of the solving process.
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Table 4.17 – Computed 10 x 10 inverse matrix
1 2 3 4 5 6 7 8 9 10
1 1.1E-05 1.1E-05 1.1E-05 1.1E-05 1.1E-05 2E-19 7.1E-20 -2E-20 -7E-20 -2E-19
2 5.5E-05 2E-05 1.1E-05 1.4E-06 -4E-05 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001
3 6.6E-05 2.2E-05 1.1E-05 -8E-07 -5E-05 -0.0001 -0.0001 -0.0001 -0.0002 -0.0002
4 7.8E-05 2.4E-05 1.1E-05 -3E-06 -6E-05 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002
5 0.00013 3.4E-05 1.1E-05 -1E-05 -0.0001 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003
6 0.00013 2.8E-05 -3E-18 -3E-05 -0.0002 -0.0003 -0.0004 -0.0004 -0.0004 -0.0004
7 0.00014 2.8E-05 -3E-18 -3E-05 -0.0002 -0.0004 -0.0004 -0.0004 -0.0004 -0.0004
8 0.00015 2.9E-05 -3E-18 -3E-05 -0.0001 -0.0004 -0.0004 -0.0004 -0.0004 -0.0004
9 0.00015 2.9E-05 -3E-18 -3E-05 -0.0001 -0.0004 -0.0004 -0.0004 -0.0004 -0.0004
10 0.00015 2.9E-05 -3E-18 -3E-05 -0.0001 -0.0004 -0.0004 -0.0004 -0.0004 -0.0003
To simplify the method the use of known constants wherever possible is a huge
advantage. In the case of the beam being considered, and given the symmetrical
nature of the beam itself, loading and support conditions, the reaction force at
each end equated to half the total loading for each individual load step. Table
4.18 shows an example of the application of the total load applied in the
experimental setup and entered into the cell which returns the required values
for forces F1, F2, F4 and F5, refer to Figure 4.14 for force numbers. Full details of
the method are available in CD format on in Appendix D.
Table 4.18 – Constants for use in FE method (example values)
Total Point load (N) -151.3
Constants
F1 75.6
F2 -75.6
F3 0
F4 -75.6
F5 75.6
F6 0
F7 0
F8 0
F9 0
F10 0
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Figure 4.14 – Numbered system variables, degrees of freedom
A sample of the results produced from the previously mentioned data entry is
produced in Table 4.19.
The letter D in the table represents displacements, D3 and D10 being rotational
displacements measured in radians and the required maximum deflection value
in meters displayed as D3.
Table 4.19 – Results from FE method maximum displacement
Solution of Displacements -1.95156E-18 D1 -0.000368818 D2 Deflection (m) -0.000387258 D3 Centre of beam deflection
-0.000368818 D4 8.67362E-17 D5 Rotation(Radians) -0.001465468 D6 Rotation of beam at left end
-0.000475381 D7 1.14492E-16 D8 0.000475381 D9 Rotation(Radians) 0.001465468 D10 Rotation of beam at right end
To represent the deflection data graphically a graph is produced in Figure 4.15
where only the maximum load step of 151.3N is applied to the method. The
maximum value of displacement being – 0.387mm as a result.
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Figure 4.15 – Deflection results using FE method maximum displacement
4.1.9 Deflection results from Macaulay’s method
Using the double integration method described in the Methodology chapter the
following Table 4.20 shows the required table of values. Px1 is the position of the
first point load from the left support and Px2 is the position of the second point
load from the left support. L is the length of the beam.
Table 4.20 – Required table of values
EI L (m) Px 1 (m) Px 2 (m)
4015 0.805 0.325 0.48
Table 4.21 shows the resulting maximum deflection for each load step. The
calculations were performed using Excel which can be seen on the spreadsheet
in Appendix D.
-0.001
-0.0008
-0.0006
-0.0004
-0.0002
-4E-18
0.0002
-0.095 0.005 0.105 0.205 0.305 0.405 0.505 0.605 0.705 0.805 D
isp
lace
me
nt
(m)
Beam Length
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Table 4.21 – Results from double integration method
Constant A Load(N) Load each point(N)
Deflection (m)
0 0.00 0.00 0.00000
-0.4901 12.57 6.28 -0.00003
-0.8398 21.53 10.77 -0.00006
-1.1765 30.17 15.08 -0.00008
-1.599 41.00 20.50 -0.00011
-2.0098 51.53 25.77 -0.00013
-2.3179 59.43 29.72 -0.00015
-2.7677 70.97 35.48 -0.00018
-3.1746 81.40 40.70 -0.00021
-3.4957 89.63 44.82 -0.00023
-3.913 100.33 50.17 -0.00026
-4.329 111.00 55.50 -0.00028
-4.758 122.00 61.00 -0.00031
-5.109 131.00 65.50 -0.00034
-5.447 139.67 69.83 -0.00036
-5.8968 151.33 75.60 -0.00039
The comparison between the experimental, Macaulay’s, finite element stiffness
method and the ANSYS updated model for the beam deflections are produced in
graphical form in Figure 4.16.
Figure 4.16 – Mathematically computed deflections comparison
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
De
fle
ctio
n (
mm
)
Load Step
Experimental
Machauly
FE Method
Updated Model
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Both the stiffness method and Macaulay’s method are in agreement with each
other but differ to some extent to the experimental and updated ANSYS model of
the beam. The close relationship between the experimental deflection results
and the updated finite element model results are encouraging. The variation of
the mathematical results is discussed later in the following chapter.
4.1.10 Finite element model summery
Table 4.22 shows the summery description of the element types used by ANSYS
in the meshing section of the setup.
Table 4.22 – ANSYS element summery
Description Quantity
Total Nodes 3420
Total Elements 451
Total Body Elements 451
Total Contact
Elements
0
Total Spot Weld
Elements
0
Element Types 1
Coordinate Systems 0
Materials 1
Generic Element
Type Name
Mechanical APDL
Name
NASTRAN
Name
ABAQUS
Name
STL
Name
Quadratic Hexahedron Solid186 CHEXA C3D20 N/A
4.1.11 Material data
One of the updating parameters of this model was the material data. For this
reason there are two material data sets. The manual changes to the Young’s
modulus formed part of the model updating procedure. Table 4.23 shows the
isotropic elasticity pre-updating while Table 4.24 shows the updated values.
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Table 4.23 – Pre-updating isotropic elasticity
Temperature C
Young's Modulus Pa
Poisson's Ratio
Bulk Modulus Pa
Shear Modulus Pa
22 7.1e+010 0.33 6.9608e+010 2.6692e+010
Table 4.24 – Updated isotropic elasticity
Temperature C
Young's Modulus Pa
Poisson's Ratio
Bulk Modulus Pa
Shear Modulus Pa
6.4e+010 0.33 6.2745e+010 2.406e+010
4.2 Three hinge arch bridge results
4.2.1 Experimental reaction results
The results from experimentally testing the three hinge arch bridge are not many
as only the horizontal reactions from the resulting loading at the 11 locations
were to be recorded. The values recorded by the force transducer of the HDA
200 interface are produced in Table 4.25 and are expressed in Newtons. The
position for each load location is from the left end of the bridge. The values do
not contain the self weight of the bridge as the force readings were set to zero
prior to loading.
Table 4.25 – Horizontal reaction force at right hinge (N)
Load Position (mm) Test 1 Test 2 Test 3 Test 4 Test 5 Average
0 1 1.8 1.6 0 1.1 1.1
100 6.3 6.9 6.8 4.7 6.4 6.22
200 11.4 12.1 11.9 10.2 11.6 11.44
300 16.7 18 17.7 15.5 17.4 17.06
400 21.7 22.8 22.2 20.3 22 21.8
500 26.9 28.5 28 26.2 27.7 27.46
600 23.6 24.9 24.1 22.3 24 23.78
700 18.2 19 18.7 16.6 18.4 18.18
800 12.4 13.1 12.9 10.9 13 12.46
900 6.1 6.8 6.8 4.6 6.4 6.14
1000 0.2 0.9 0.6 1.2 0.5 0.68
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4.2.2 ANSYS finite element model results
First approach
The three hinge arch was modelled by two different approaches as outlined in
the Methodology section. The first is pictured in Figure 4.17 and represents only
half of the bridge. The idea was to only model the setup as half the bridge as it
was assumed that symmetry could be taken advantage of.
Figure 4.17 – First approach half arch
The results of placing the load, (12.4N) which was deemed half of the
experimental load due to only using half of the bridge, are presented in Table
4.26. It was noted the value of the vertical reaction should change as the load is
being moved along the bridge; this prompted a change in methodology.
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Table 4.26 – Horizontal and vertical model results
Load distance (m) 0 0.1 0.2 0.3 0.4 0.5
Horizontal (N) 30.189 24.059 17.93 11.79 5.66 0.15
Vertical (N) 12.4 12.4 12.4 12.4 12.4 12.4
Some of the other results for the bridge were also obtained such as the
equivalent von-Misses stress. As an example of this the load at one section, 0.2
meters from the centre is chosen. The load being applied is depicted in Figure
4.18 with the right end support resulting reaction shown in Figure 4.19.
Figure 4.18 – Load located at 0.2m from centre
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Figure 4.19 – Resulting reaction for load at 0.2 meters
The associated stress is depicted in Figure 4.20.
Figure 4.20 – Von-Mises stress for load at 0.2m from centre
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Using the von-Mises stress failure criterion ANSYS produces a colour contour of
the relevant stresses which are lightly to cause failure. The model showed
agreement in such that the curved arch area was where concentrations of
stresses were mostly felt. This was indicated by the dark red colour visible in
Figure 4.20.
Second approach
Following the errors witnessed in the first attempt, a second attempt was made
to replicate the experimental setup and to have the model bridge perform like a
three hinge arch bridge as close as possible. Reasons for doing so are outlined
later in the discussion chapter. Figure 4.21 depicts the full 3 hinge arch bridge as
modelled in ANSYS Design Modeller. The modelled bridge consist of 4 separate
parts, the two main sections which make up the body of the bridge and two
small hinges at the centre of the bridge which connect the two parts.
Figure 4.21 – Full bridge model
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The types of joints which connect the body parts are outlined in the
Methodology Chapter and their implications are discussed later in the Discussion
Chapter.
This method of modelling the bridge as a whole allows the structure to act like
the experimental setup by the functioning revolute joint at the centre.
4.2.3 Mesh results details
Meshing was performed automatically by ANSYS during the mechanical stage of
the modelling process as outlined in the Methodology section. On closer
examination of the centre of the bridge, as seen in Figure 4.22, there consist
different element types for the hinges.
Figure 4.22 – Meshed connection at centre of bridge
It was possible to view the results of the elements by the use of the Finite
Element Modeller as a system in the project schematic which allows details to be
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viewed of the various types of elements being implemented by ANSYS and if
adjustment needs to be made to the type of element it can be done here. Figure
4.22 above shows the blue elements as linear tetrahedron elements while the
grey pins are linear hexahedron elements. The method of achieving a hinge at
this point is by creating a joint element type for the pins as discussed in the
Methodology Chapter.
Table 4.27 and Table 4.28 show a breakdown of the bodies and element types
respectively. The larger element number is the main body while the smaller
number is the pin connections at the centre of the bridge.
Table 4.27 – Bodies summery
Body Name Nodes Elements
Solid 867 2229
Solid 874 2245
Solid 27 8
Solid 27 8
Table 4.28 – Element type summery
Generic Element Type Name
Mechanical APDL Name
NASTRAN Name
ABAQUS Name
STL Name
Linear Tetrahedron Mesh200 CTETRA C3D4 N/A
Linear Hexahedron Mesh200 CHEXA C3D8 N/A
4.2.4 Material data
The material used in the model was structural steel and was pre-setup in the
ANSYS material library. Table 4.29 and Table 4.30 show the values for the
material constants and isotropic elasticity respectively.
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Table 4.29 – Material constants
Density 7850 kg m^-3
Coefficient of Thermal Expansion 1.2e-005 C^-1
Specific Heat 434 J kg^-1 C^-1
Thermal Conductivity 60.5 W m^-1 C^-1
Resistivity 1.7e-007 ohm m
Table 4.30 – Isotropic elasticity
Temperature oC
Young's Modulus Pa
Poisson's Ratio
Bulk Modulus Pa
Shear Modulus Pa
22 2.e+011 0.3 4.1667e+011 1.9231e+011
4.2.5 Model results
Table 4.31 shows the model results of the reaction forces at the supports for the
applied loading at each position from the left end of the bridge.
Table 4.31 – Reaction force results from FE model
Position (mm) Load (N)
Left Horizontal
Reaction(N)
Left Vertical Reaction(N)
Right Horizontal
Reaction(N)
Right Vertical Reaction (N)
0 24.8 0.14811 24.751 0.14811 4.94E-02
100 24.8 6.4385 22.283 6.4385 2.5171
200 24.8 12.673 19.815 12.673 4.9847
300 24.8 18.677 17.348 18.677 7.452
400 24.8 24.094 14.879 24.094 9.9208
500 24.8 27.598 12.4 27.598 12.4
600 24.8 24.048 9.8698 24.048 14.93
700 24.8 18.662 7.4033 18.662 17.397
800 24.8 12.612 4.9353 12.612 19.865
900 24.8 6.3426 2.4676 6.3426 22.332
1000 24.8 0.022441 -5.48E-05 2.24E-02 24.8
The loading was applied at 11 locations along the bridge. The full report output
from ANSYS can be viewed on the accompanying CD as the report is quite
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detailed. For an example, one load location is presented here and the full results
for that load. The load location chosen is at 800mm from the left end support.
Figure 4.23 shows the load position on the model.
Figure 4.23 – Load positioned at 800mm from left end
The equivalent von-Mises stress is represented in Figure 4.24 where it can be
seen that the largest area of stress is where the load is located which is to be
expected.
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Figure 4.24 – Equivalent von-Mises stress for load location
The resultant force reactions at the supports are depicted in Figure 4.25 showing
the direction of the reaction force. The corresponding values for the horizontal
and vertical components of the resultants are shown in Table 4.31 next to the
800mm distance.
Figure 4.25 – Left and right hand support reactions graphic
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5 Discussion
5.1 T shaped aluminium beam
The aluminium T shaped beam is relatively simple structure however it has many
significant important uses in construction and engineering. The process of
designing beams for use as structural elements is mostly done nowadays by
computers which utilise finite element analysis in their computation of resulting
reactions, stresses and strains which occur due to applied loading. It is not a
difficult process to setup a model of the required structure, nor is it complicated
to apply the desired loading arrangement. It is however very important to know
and understand the results that are being produced by such computer programs
such as ANSYS or similar finite element analysis packages. The question of
understanding the results and knowing where the finite element analysis
program might be returning errors is the subject of this dissertation. The best
way to compare the results from such finite element analysis programs is to
conduct an experiment which is similar to the model, or in this case a model
similar to the experimental setup.
The first structure to be experimentally tested was the T shaped aluminium
beam which was to be examined for the level of strain which was experienced in
the beam at certain locations due to successive incremental loading. The beam
was then modelled using ANSYS finite element analysis software to assess the
performance of the model and compare the strain results with real data
recorded from the experimental setup.
The beam was subjected to loading increments of 10 Newtons (N) from zero up
to approximately 150 N in the experiment. As this loading was not large enough
to cause the material to yield, the results were linear as expected obeying
Hooke’s Law.
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Figure 5.1 shows the graph of the averaged values of strain recorded as the load
was increased. It can be seen that the lines are generally strait indicating the
material linearly elastic. The value for gauge number 1 appeared to be higher
than the other values which prompted an investigation early on in the
experiment as to the functionality of this gauge. This gauge reading was
important as it was the location of maximum tensile strain experienced by the
beam at each individual load increment. The results from Excel and the
supporting graph are discussed in Section 4.1.6 in the Results chapter.
Figure 5.1 – Load versus strain experimental results
The graph in Figure 5.1 shows 3 strain gage values as positive, positive strain
indicating the tensile zone below the neutral axis and this is where gauges 1, 2
and 9 are located. The rest of the gauge values are negative, negative strain
indicating shortening which is where the beam is experiencing compressive
strain. Theoretically all lines on the graph should be straight in the case of a
linear elastic isotropic material but in the case of these results there is some
variation in the lines. This is mainly due to the electronic strain gauges
-150
-100
-50
0
50
100
150
200
250
-5 15 35 55 75 95 115 135 155 Mic
rost
rain
(µε)
Load(N)
Experimental Averaged - µε v Load
1
2
3
4
5
6
7
8
9
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themselves showing variations as the extension or contraction of the various
gauges is diminutive.
Based on the experimental recorded strain values and the relationship between
stresses, strain and Young’s modulus as described in Section 4.1.2, an average
value for Young’s modulus was computed. This was found to be approximately
64 Gigapascals (GPa’s). This experimentally derived value is of significant
importance to the model setup as the value is a material property and affects the
bending behaviour of the material subjected to loading.
5.1.1 Finite element model
The initial stage of the modelling was trial and error to some degree. Several
attempts were made at creating a model to replicate the actual setup and some
were with limited success. One issue was the supports at either end; the beam
was tending to be adequately responsive to the loading and experienced lateral
twisting when loading was applied. The problem was solved later with the
application of remote displacement supports which allowed free or fixed
rotation of the beam at either end in any desired plane. This lead to the selection
of the plane normal to the length of the beam to be the only plane of rotation
thus the other two planes could remain fixed in rotation. The remote
displacement supports also allow the free or fixed condition for translation in the
X, Y and Z directions to be selected. Allowing the relevant directions to be fixed
resulted in the supports replicating a simple support condition which was
representative of the experimental setup.
The application of loading in the 3D model had its own difficulties. The initial
method of applying the loads as two point loads pictured in Figure 5.2, which is
representative of the experimental situation, proved to be inadequate.
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Figure 5.2 – Initial loading arrangement
The location of the loads was set by applying a remote force but the method in
which ANSYS applies the load did not produce the desired strain or displacement
results. It seemed irrelevent where the point loads were placed on the beam, via
entering coordinates entered in the remote displacement field, as to the
possition of maximum deformation or strain values. The problem was with the
selection of a reference edge or face in order to apply the load. The face selected
was the top face shown as red in Figure 5.2 and this face becomes the origin
point for the load being applied. This was not representative of the real situation.
A possible future approach would be to create a small raised face at the points of
loading and select this face as the reference for load application. Table 5.1 shows
the allocation of the loadings Points 1 & 2 at their respective Z coordinates and
the geomotry selection required as being 1 face.
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Table 5.1 – Initial load and support values
Object Name Point Load 1
Point Load 2
Remote Displacement
Remote Displacement 2
State Fully Defined
Scope
Scoping Method Geometry Selection
Geometry 1 Face
Coordinate System
Global Coordinate System
X Coordinate 1.275e-002 m
Y Coordinate 5.07e-002 m 4.75e-002 m
Z Coordinate 0.327 m 0.478 m 0.805 m 0. m
Location Defined
Definition
Type Remote Force Remote Displacement
Define By Components
X Component 0. N (ramped) 0. m (ramped)
Y Component Tabular Data 0. m (ramped)
Z Component 0. N (ramped) Free
Suppressed No
Behaviour Deformable
Rotation X Free
Rotation Y 0. ° (ramped)
Rotation Z 0. ° (ramped)
Rotation X Free
Rotation Y 0. ° (ramped)
Rotation Z 0. ° (ramped)
Advanced
Pinball Region All
The tabular data in Table 5.1 for the Y directional compnent of the applied forces
refers to the stepped incremental load application in ANSYS which is how the
loading was applied in the experimental setup. Load steps of 1 second intervals
were adopted and the values entered corresponded to the experimental values.
The values corresponding to the applied loadings recorded in the experimental
setup were entered as negative loading in the Y direction and are displayed in
Table 5.2. The values in the table are half of the load recorded on the HDA 200
from the experiment as the load was applied in two locations in the real and
experimental setup.
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Table 5.2 – Point loads 1 & 2 stepped load values
Steps Time [s] X [N] Y [N] Z [N]
1 0. 0. 0. 0.
1. -6.3
2 2. = 0. -12.25 = 0.
3 3. -15.1
4 4. -20.5
5 5. -25.75
6 6. -29.7
7 7. -35.5
8 8. -40.7
9 9. -44.8
10 10. -50.15
11 11. -55.5
12 12. -61.
13 13. -65.5
14 14. -69.85
15 15. -75.65
16 16. 0.
The resulting strain at each load step is graphed in Figure 5.3 which shows the
largest difference in the model compressive strain, at the maximum total applied
load of approximately 151 N, is 61% less than the same experimental strain
under the same conditions. This was an important observation as a difference of
this magnitude could have significant replications if it was not noticed in a design
process. Under estimation of strain values could lead to the beam being over
strained if put into service under false data from design modelling software.
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Figure 5.3 – Resulting maximum compressive strain
The resulting maximum deflection for each load step is presented in Figure 5.4
where a 51% lower value was experienced by the model beam compared to that
of the experimental setup.
Figure 5.4 – Resulting maximum deflection
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Stra
in x
10
-6
Load Step
Two loads
Experimental
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Max
De
lect
ion
(mm
)
Load Step
Two Loads
Experimental
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5.1.2 Model updating
To update the model a different approach to the loading situation was adopted.
Since the theoretical strain and experimental strain are linked to bending theory
and beam bending equations the application of moment at each beam support
was considered. Some analysis of the loading and support reactions was needed
first, a simple static analysis. As the load applied in the experimental setup was
applied in a symmetrical manner, being an equal distance from each support
which gave an equal reaction at both supports, an equivalent moment could be
applied at each end of the beam to replicate the effects of the combined point
loads. The value for each moment, opposite in sense at either end, was a simple
calculation of the force reaction at that end multiplied by the distance of the
point load location from that end. This was computed in Excel for each of the
applied loads for the 15 load steps and the results for the moment at the left end
are presented in Table 5.3 with the corresponding ANSYS model graphic
presented in Figure 5.5. The similar values, except positive in sense, were applied
to the far end of the beam.
Table 5.3 – Applied moment at left end
Figure 5.5 – Applied moment
Steps Time [s] X [N·m] Y [N·m] Z [N·m]
1 0. 0. 0. 0.
1. -2.04
2 2. -3.5 = 0. = 0.
3 3. -4.9
4 4. -6.7
5 5. -8.4
6 6. -9.7
7 7. -11.5
8 8. -13.2
9 9. -14.6
10 10. -16.3
11 11. -18.04
12 12. -19.8
13 13. -21.3
14 14. -22.7
15 15. -24.6
16 16. 0.
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The resulting strain and deflection values from the application of the moments as
the representative applied load moved closer to the experimental strain values.
The initial values from this approach were still not close enough to the
experimental results so another model updating technique was implemented. It
was noticed that the isotropic elastic properties, Young’s modulus value, of the
ANSYS material library value for aluminium was set at 71GPa’s which was
resulting in a stiffer material being used by the model elements. The results from
the experimentally derived Young’s modulus as seen in section 4.1.2 returned a
value of 64GPa’s. This new value was used as a model updating parameter and
proved to bring a closer correlation between the maximum tensile and
compressive strain values of the model and the experimental results. The change
in the maximum compressive strain experienced by the model beam went from
88% of the experimental compressive strain in the pre-updated model to 100% in
the updated model. The change in the maximum tensile strain went from 87% to
102% of the experimental tensile strain values recorded. This proved a successful
model updating procedure based on these results.
Deflection values at the centre of the beam in the model also correlated well
with the experimental results and were almost identical in the updated model.
The deflection results from the mathematical methods, double integration and
finite element stiffness, showed good correlation with each other but were on
average for each load step 77% of the updated model deflection and were only
69% of the experimental deflection values. This could indicate a possible over
estimation of the model deflection results but this can be ruled out by the
correlation between the experimental results for deflection and the updated
model values being almost equal. The remaining observation then is to assume
the finite element stiffness method and Macaulay’s method for beam deflections
under estimate the true deflection due to the assumptions made in section 3.2.3
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5.2 Three hinge arch bridge
The three hinge arch bridge is a significant structural form which is more
economical in terms of performance in wide spans than beams. The horizontal
reactions produced at the supports reduce the bending moment experienced in
the bridge.
The experimental setup for the bridge recorded horizontal reactions at one end
only as a known mass was positioned at 100mm intervals from one end to the far
end. These results were compared to mathematically calculated values for
verification before they were used to verify the ANSYS finite element model of
the bridge.
5.2.1 Finite element model
The first approach to the modelling of the bridge is explained in section 3.4.4 and
outlines the initial model as half of the experimental bridge. The attempt was
hoped to make use of the symmetrical nature of the real bridge geometry and by
utilising remote displacements shown in Figure 5.6.
Figure 5.6 – Remote displacement supports 1 & 2
The remote displacement in the graphic on the left in Figure 5.6 was modelled as
having zero translation in the X and Z directions with free translation in the Y
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direction and free rotation in the Z plane which was aimed at replicating a pin
type connection at this location like in the experimental setup. A similar remote
displacement supporting condition was placed at the right hand hinge but free
rotation in the Z plane was allowed here thus providing horizontal and vertical
reactions like the experimental setup but allowing rotational hinge like
behaviour.
The results output for the ANSYS finite element model of the bridge did not
compare well with the expected results based on the experimental setup. The
vertical reaction remained a constant value even though the load was moved in
100mm increments from one end to the other; this should not be the case as
shown in the mathematically calculated values based on the principles of the 3
hinge arch bridge displayed in Table 5.4 which were calculated in Excel.
Table 5.4 – Mathematically calculated force reactions
Position (mm)
Load (N)
Left Horizontal
Reaction(N)
Left Vertical Reaction(N)
Right Horizontal
Reaction(N)
Right Vertical Reaction (N)
0 24.8 0.00 24.8 0.00 0.00
100 24.8 5.51 22.32 5.51 2.48
200 24.8 11.02 19.84 11.02 4.96
300 24.8 16.53 17.36 16.53 7.44
400 24.8 22.04 14.88 22.04 9.92
500 24.8 27.56 12.4 27.56 12.4
600 24.8 22.04 9.92 22.04 14.88
700 24.8 16.53 7.44 16.53 17.36
800 24.8 11.02 4.96 11.02 19.84
900 24.8 5.51 2.48 5.51 22.32
1000 24.8 0.00 0.00 0.00 24.8
It is clear to see from the results in Table 5.4 that both end support reactions
(vertical and horizontal) of the bridge change reciprocally with the varying
position of the mass. This was the data which prompted a new and more
detailed approach to the modelling of the bridge as a full bridge.
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The full bridge was modelled as described in section 3.4.4 attempt two. The
model represented the function of a three hinge arch bridge as shown by the
results in Table 4.31 in the Results Chapter.
To assess the performance of the finite element analysis a percentage
comparison between the experimental, mathematical and model reaction results
was performed. The following tables make use of Excel to analyse the data from
each result origin. Maximum and minimum values of both the horizontal and
vertical reaction were searched for and the resulting maximum or minimum
value was returned in a cell with the corresponding origin of that result placed in
the cell next to it. A sample of the formula used was:
=IF(Z22=H29,"Experimental",(IF(Z22=Z7,"Mathematical",(IF(Z22=AJ7,"Model")))))
The following Table 5.5, Table 5.6, Table 5.7 and Table 5.8 are the results from
the analysis and compare the performance of the ANSYS model support reaction
results to those of the experimental and mathematically calculated results.
Table 5.5 –Maximum reactions at left hand support
Load Stage (m)
Max value of LH Hz (Newtons)
% Exp
% Model
Max value of LH V (Newtons)
% Math
% Model
0 1.10 Exp 100.00 742.69 24.8 Math 100 100.2
0.1 6.44 Model 103.51 100.00 22.32 Math 100 100.2
0.2 12.67 Model 110.78 100.00 19.84 Math 100 100.1
0.3 18.68 Model 109.48 100.00 17.36 Math 100 100.1
0.4 24.09 Model 110.52 100.00 14.88 Math 100 100.0
0.5 27.60 Model 100.50 100.00 12.4 Math 100 100.0
0.6 24.05 Model 101.13 100.00 9.92 Math 100 100.5
0.7 18.66 Model 102.65 100.00 7.44 Math 100 100.5
0.8 12.61 Model 101.22 100.00 4.96 Math 100 100.5
0.9 6.34 Model 103.30 100.00 2.48 Math 100 100.5
1 0.68 Exp 100.00 3030.17 0 Math 100 0.0
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Table 5.6 – Minimum reactions at left hand support
Load Stage (m)
Min Value of LH Hz (Newtons)
% Exp
% Model
Min value of LH V (Newtons)
% Math
% Model
0 0.00 Math 0.00 0.00 24.751 Model 99.8 100
0.1 5.51 Math 88.60 85.60 22.283 Model 99.8 100
0.2 11.02 Math 96.35 86.97 19.815 Model 99.9 100
0.3 16.53 Math 96.91 88.52 17.348 Model 99.9 100
0.4 21.80 Exp 100.00 90.48 14.879 Model 100.0 100
0.5 27.46 Exp 100.00 99.50 12.4 Math 100.0 100
0.6 22.04 Math 92.70 91.67 9.8698 Model 99.5 100
0.7 16.53 Math 90.94 88.59 7.4033 Model 99.5 100
0.8 11.02 Math 88.46 87.39 4.9353 Model 99.5 100
0.9 5.51 Math 89.76 86.89 2.4676 Model 99.5 100
1 0.00 Math 0.00 0.00 -5.5E-05 Model 0.0 100
Table 5.7 – Maximum reactions at right hand support
Load Stage(m)
Max Value of RH Hz (Newtons)
% Exp
% Model
Max Value of RH V (Newtons)
% Math
% Model
0 1.10 Exp 100.00 742.69 4.94E-02 Model 100.0 100
0.1 6.44 Model 103.51 100.00 2.52E+00 Model 101.5 100
0.2 12.67 Model 110.78 100.00 4.98E+00 Model 100.5 100
0.3 18.68 Model 109.48 100.00 7.45E+00 Model 100.2 100
0.4 24.09 Model 110.52 100.00 9.92E+00 Model 100.0 100
0.5 27.60 Model 100.50 100.00 1.24E+01 Math 100.0 100
0.6 24.05 Model 101.13 100.00 1.49E+01 Model 100.3 100
0.7 18.66 Model 102.65 100.00 1.74E+01 Model 100.2 100
0.8 12.61 Model 101.22 100.00 1.99E+01 Model 100.1 100
0.9 6.34 Model 103.30 100.00 2.23E+01 Model 100.1 100
1 0.68 Exp 100.00 3030.17 2.48E+01 Math 100.0 100
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Table 5.8 – Minimum reactions at right hand support
Load Stage
Min Value of RH Hz (Newtons)
% Exp
% Model
Min Value of RH V (Newtons)
% Math
% Model
0 0.00 Math 0.00 0.00 0.00E+00 Math 100 0.0
0.1 5.51 Math 88.60 85.60 2.48E+00 Math 100 98.5
0.2 11.02 Math 96.35 86.97 4.96E+00 Math 100 99.5
0.3 16.53 Math 96.91 88.52 7.44E+00 Math 100 99.8
0.4 21.80 Exp 100.00 90.48 9.92E+00 Math 100 100.0
0.5 27.46 Exp 100.00 99.50 1.24E+01 Math 100 100.0
0.6 22.04 Math 92.70 91.67 1.49E+01 Math 100 99.7
0.7 16.53 Math 90.94 88.59 1.74E+01 Math 100 99.8
0.8 11.02 Math 88.46 87.39 1.98E+01 Math 100 99.9
0.9 5.51 Math 89.76 86.89 2.23E+01 Math 100 99.9
1 0.00 Math 0.00 0.00 2.48E+01 Math 100 100.0
The tables were created using Microsoft Excel with the following abbreviations
used:
Math: mathematical results
Exp: experimentally recorded results
Model: ANSYS finite element modelled results
Three result origins are represented in the results tables above which show the
model results for the maximum left hand horizontal reaction being returned for
almost all loading positions. When compared to the experimental result for the
same reaction the model exceeds the experimental result by approximately 4%.
The left hand vertical reaction maximum value was returned by the
mathematically obtained value but only exceeded the model result by 0.3% on
average for each load location. This result shows the successful outcome from
the modelling process. There were no experimental vertical reaction results for
the bridge.
The minimum reaction values show good correlation between the results with a
slight variation the horizontal results which show the mathematically calculated
results being the minimum value for almost all of the load stages. The model
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value was never a minimum value for this reaction. The minimum reaction values
of the left hand vertical reactions show good correlation between the
mathematical and model results.
In a similar fashion the reactions for the right hand supports show good
correlation between model, mathematical and experimental results. It can be
deduced from this that the modelling procedure implemented in the second
attempt is representative of the experimental setup and could be used as a
design tool with confidence.
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6 Conclusion and recommendations
This dissertation shows methods of verifying the results output from finite
element analysis software for structural analysis. The finite element analysis
code ANSYS was used.
Two laboratory structures were tested and a 3D model of each was created in
ANSYS and the loads applied in a similar manner to the experimental setup. The
result of each model output was then compared to the experimental results for
verification. Simple model updating was performed to calibrate each model to
replicate the physical experimental setup such as material properties and actual
model representation.
The first structure to be experimentally tested was a T shaped aluminium section
of beam 805mm long. This structure was tested for the strain effects at the
centre of the beam while being subjected to 10 Newton (N) incremental loads
from 0N up to 150N. The strain was tested using electronic strain gauges at 9
locations around the periphery of the beam face. The resulting strain from each
location was then used to assess the Young’s modulus of the material. A 3D finite
element model was created in ANSYS and the load applied, using a similar
moment value to replicate the point loading, and maximum compression and
tensile strain values and maximum deflection were recorded. It was observed
that the initial results were of less magnitude than the experimental result. This
was due to the ANSYS isotropic material properties for aluminium as set by the
program for Young’s modulus was 71GPa’s. Having obtained a value of 64GPa’s
for the experimental beam this became a model updating parameter. The
updated model showed an average increase in the model maximum tensile strain
from 87% to 102% of the experimental value and an average increase in the
model maximum compressive strain from 88% to 100% of the experimental
value thus showing a successful model updating procedure.
The second structure to be experimentally tested was the three hinge arch
bridge. The bridge was tested for horizontal support reactions at one end while a
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147
mass of 2.536kg (24.8N) was moved to 11 locations along the bridge deck. Self
weight of the bridge was ignored in the test. The bridge was then modelled with
ANSYS to compare its performance with the experimental results. The main issue
with this structure was the method of modelling in ANSYS which lead to two
separate attempts at modelling. The first was attempt was modelled as half the
bridge with an attempt at replicating a hinge at the centre by the use of remote
displacements and setting up the appropriate boundary conditions. This proved
to be unsuccessful as the vertical reactions in the model were unresponsive to
the changing load position. The second attempt updated the model itself by
creating a modified geometry of the entire 3 hinge arch bridge. The main feature
of the model was the application of revolute joints (MPC 184, ANSYS library
element) at the centre connection of the bridge to allow the joint to function as a
hinge. This proved a successful model updating procedure as the updated results
showed the model exceeding the horizontal reaction force by only 4% of the
experimental result.
The validity of finite element models is essential in trusting the output results.
Proving the results is not always made possible by replicating a physical test
similar to the model. For this reason more study needs to conducted as to the
actual performance of individual element types by other means. The effects of
static loading were only tested here but more emphasis on the dynamic effects
on structures like bridges subjected to moving vehicles or seismic activity need to
be conducted and a means of calibrating these effects without conducting
physical experiments on the structure.
The analyst must convey apt knowledge of the procedures involved in producing
finite element models In spite of the great power of FEA, the disadvantages of
computer solutions must be kept in mind when using this method as they do not
necessarily reveal how the output values are influenced by important problem
variables such as materials properties and geometrical features, and errors in
input data can produce wildly incorrect results that may be overlooked revealing
drastic consequences.
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148
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8 Appendices
Appendix A – Dissertation Mind Map.................................................................A-1
Appendix B – ANSYS Full Report: Beam....................................Separate Document
Appendix C – ANSYS Full Report: 3 Hinge Arch Bridge ............Separate Document
Appendix D – CD Containing Electronic Support Information.....Inside Front Cover
Molloy 2012
A-1
A. Appendix A – Dissertation Mind Map
Appendix A – Dissertation Mind Map
Figure A.1 – Dissertation Mind Map