ii
COMPARISON BETWEEN GRILLAGE MODEL AND FINITE ELEMENT
MODEL FOR ANALYZING BRIDGE DECK
SAIF LAITH KHALID ALOMAR
A project report submitted in partial fulfilment of the requirement for the award of
the degree of the degree of Master of Engineering (Civil- Structure)
Faculty of Civil Engineering
Universiti Teknologi Malaysia
JUNE 2009
v
ACKNOWLEDGEMENTS
Firstly, I would like to thank Dr. Redzuan for his efforts and time. Without
mentioning the help, which I got from him, I feel this work would be incomplete.
Secondly, it was my pleasure to work with him during my study period where
I gained a wonderful opportunity to learn several things from him, which extend
beyond the technical knowledge that definitely will help me to pursue my career.
Finally, I would like to thank my family for their support, patient;
encouragement and for the love they gave me to complete this project.
vi
ABSTRACT
Since its publication in 1976 up to the present day, Edmund Hambly’s book
“Bridge Deck Behaviors” has remained a valuable reference for bridge engineers.
During this period the processing power and storage capacity of computers has
increased by a factor of over 1000 and analysis software has improved greatly in
sophistication and ease of use. In spite of the increases in computing power, bridge
deck analysis methods have not changed to the same extent, and grillage analysis
remains the standard procedure for most bridges deck. In this study analysis bridge
deck using grillage model are compared with the analysis of the same deck using
finite element model. A bridge deck consists of beam and slab is chosen and
modelled as grillage and finite element. Bending moment, Shear force, Torsion and
Reaction force from both models are compared. Effect of skew deck is also studied.
In general for practical skew bridge deck results from finite element model give
lesser value in terms of displacement, reaction, shear force, torsion, bending moment
compare with the results from grillage model. It can be concluded that analysis of
bridge decks by using finite element method may produce more economical design
than grillage analysis. This is due to the fact that the finite element model resembles
the actual structure more closely than the grillage model.
vii
Abstrak
Semenjak publikasinya dari 1976 ke hari ini, buku Edmund Hambly berjudul
“Bridge Deck Behaviors” merupakan sumber rujukan paling berguna kepada
jurutera-jurutera jambatan. Semasa tempoh ini kuasa pemprosesan dan kapasiti
penyimpanan komputer-komputer telah bertambah dengan satu faktor melebihi 1000
dan
perisian
analisis
telah
banyak
meningkat
dalam
kecanggihan
dan
penggunaannya. Meskipun bertambah dari kuasa pengkomputeran, cara-cara
menganalisis geladak jambatan masih dalam takat yang sama, dan kekisi analisis
kekal sama untuk kebanyakan jambatan-jambatan bertingkat. Dalam analisis kajian
geladak ini, analisis geladak jambatan menggunakan model kekisi dibandingkan
dengan analisis bagi geladak yang sama menggunakan model unsur terhingga.
Geladak jambatan yang mengandungi rasuk dan papak dipilih dan menjadi model
sebagai kekisi dan unsur terhingga. Momen lentur, Daya Ricih, Kilasan dan Daya
Tindakbalas daripada kedua-dua model adalah dibandingkan. Kesan geladak
pencong adalah juga dikaji. Secara keseluruhannya keputusan
analisis daripada
model unsur terhingga memberi nilai yang kurang dalam soal anjakan, tindak balas,
daya ricih, kilasan dan momen lentur berbanding dengan keputusan-keputusan
daripada model kekisi. Di sini dapat disimpulkan bahawa analisis dengan
menggunakan kaedah unsur terhingga mungkin menghasilkan reka bentuk yang lebih
berekonomi daripada kekisi analisis. Ini memandangkan model unsur terhad lebih
menyamai struktur sebenar berbanding model kekisi.
viii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
REPORT STATUS VALIDATION FORM
i
TITLE PAGE
ii
SUPERVISOR DECLARATION
iii
STUDENT DECLARATION
iv
ACKNOWLEDGEMENTS
v
ABSTRACT
vi
ABSTRAK
vii
TABLE OF CONTENTS
viii
INTRODUCTION
1.1 Introduction
1
1.2 Problem Statement
2
1.3 Objectives of Study
2
1.4 Scope and Limitations of the Study
3
1.5 Significant of the Study
3
1.6 Methodology
4
LITERATURE REVIEW
2.1 Introduction
6
2.2 discussion on neutral axis location in bridge deck
7
ix
cantilevers
2.3 Up stand finite element analysis of slab bridges
8
2.4 Bending moment distribution at the main structural
10
elements of skew deck-slab and their implementation
on cost effectiveness
2.5 Effect on support reactions of T-beam skew bridge
13
decks
2.6 Types of bridge deck
3
15
2.6.1 Beam decks
15
2.6.2 Grid deck
16
2.6.3 Slab deck
17
2.6.4 Beam and slab deck
18
2.6.5 Cellular decks
19
2.7 Finite element analysis
21
2.8 Grillage Analysis
23
RESEARCH METHODOLOGY
3.1 Introduction
25
3.2 LUSAS Software
26
3.2.1 LUSAS Software Characteristic
27
3.2.2 Procedure Analysis According to LUSAS
27
Software
3.3 Types of Element
28
3.4 Configuration of Bridge Deck
30
3.5 Materials Description
32
3.6 Loadings Description
33
3.6.1 Dead Load
33
3.6.2 Superimposed Dead Load
34
3.6.3 Live loading
35
3.6.3.1 Vehicle Load – HA Loading
35
3.6.3.2Vehicle Load – HB Loading
36
3.6.4 Loading Combination
3.6.4.1 HA loading for all lanes
38
39
x
3.6.4.2 HB loading for lane1 and HA loading
40
for lane 3&4
3.6.4.3 HB loading for lane 2, HA UDL for lane
40
1, 3 and HA loading for lane 4
3.6.4.4 HB loading on lane 1 only
41
3.6.4.5 HB loading on lane 2 only
41
3.6.4.6 Enveloping the basic live load
41
combinations
3.6.4.7 Smart Load Combination
3.6.5 Vehicle Loading and Load Combinations
4
42
3.7 Grillage Modelling
43
3.8 Finite Element Modelling
44
3.9 Layers and Windows
46
3.10 Viewing Results
46
3.11 Bridge Deck Analysis Result & Discussion
47
RESULT AND ANALYSIS
4.1 Introduction
5
42
48
4.1.1 Displacement Result
49
4.1.2 Reaction Result
52
4.1.3 Shear force Result
57
4.1.4 Torsion Result
60
4.1.5 Moment Result
63
CONCLUSION
5.1 Conclusion
69
LIST OF REFERENCES
71
1
CHAPTER 1
INTRODUCTION
1.1
Introduction
Even though the finite element method has developed to the maturity and
numerous computer software that use the methods are relatively cheap and easily
available. Engineer still prefer to use grillage method for their analysis of bridge
decks. Hambly (1991) listed out reasons why grillage method is a more popular
choice than finite element method. Firstly the finite element method is much more
complicated and expensive than the grillage method .Though the finite element is
thought to be more accurate, in reality does not produce significant different results
as compared with the grillage. According to Hambly (1991), finite element is
cumbersome to use and the choice of element type can be extremely critical and, if
incorrect, the results can be far more inaccurate than those predicted by simpler
models such as grillage or space frame [Hambly 1991].
2
However, perhaps the greatest drawback at present is that while the finite
element technique is developing so rapidly, the job of carrying out finite element
computations is a full time occupation which cannot be carried out at the same time
by the senior engineer responsible for the design. He is unlikely to have time to
understand or verify the appropriateness of the element stiffness’s or to check the
large quantity of computer data. This makes it difficult for him to place his
confidence in the results, especially if the structure is too complicated for him to use
simple physical reasoning to check orders of magnitude [Jenkins, 2004].
1.2
Problem Statement
Grillage method is a fast and simpler approach compared to the finite element
method, and has been used by engineers to analyses bride deck over a long time on
the other hand the finite element method is thought to be better model for the slab
analysis because of its capability to represent the structure more realistically.
As such there is a need to conduct a though comparison between the two
models to gain better idea on which model may produce more economical design.
1.3
Objectives of Study
The objectives of this study are as follows:
•
To compare the performance between grillage and finite element model
for analysing bridge deck.
•
To conduct analysis of bridge deck using grillage and finite element
model using LUSAS software.
3
1.4
•
To study the effect of deck skew on the analysis result for both models.
•
To propose which model can provide more conservative design.
Scope and Limitations of the Study
In this study, LUSAS software will be used to model and analyse the
bridge deck. Only grillage and finite element using 3D beam and shell elements are
considered. Bending moment, Shear force, Torsion and Reaction force will be
compared.
1.5
Significant of the Study
Grillage method consists of members lying in one plane only while the finite
element method lying in 3D plane. Both of these planar methods of analysis are used
to model a range of bridge forms. Planar methods are among the most popular
methods currently available for the analysis of slab bridges. They can, with
adaptation, be applied to many different types of slab as will be demonstrated.
Further, their basis is well understood and results are considered to be of acceptable
accuracy for most bridges.
However, grillage model and finite element model can also be considerably
more complex and can take much longer to set up. For this reason, planar grillage
and finite element models are at present the method of choice of a great many bridge
designer for most bridge slabs [O'Brien, 1999].
4
The research significance to be obtained from this study will be the results
and analysis of the behaviour of bridge deck. It is necessary to compare between two
models to see which model gives more economical result.
1.6
Methodology
a) The steps adopted in this study are
•
Identify problem and scope of study, obtain realistic bridge deck plan.
•
Literature review of the grillage and finite element model. (Books,
Previous studies, Journal, Case studies)
b) In order to achieve the second objective we have to
•
Choose a realistic a bridge deck section with a different skews.
•
Analyze the bridge deck section properties using each of grillage and
finite element models for each skews in LUSAS software.
•
Application of load cases and vehicles loading.
•
Analysis and result processing.
•
Graphical and report output.
c) Compare between the two models (Grillage and Finite element) by using
the results of first and second objectives.
d) Recommendation & Conclusion.
5
The methodology that will be used for this study is shown in Figure 1.1.
Figure 1.1: methodology
6
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Grillage has been around for some time and the method is practical for use
with and without computer. Although computational power has increased many-fold
since 1960s, and other models method such as finite element is within reach, the
grillage method is still widely used for bridge deck analysis. One of the benefits that
have been quoted for the grillage analysis is that it is inexpensive and easy to use and
comprehend. This benefit has made it more preferable than method of finite-element
analysis. Nowadays the environment is inexpensive, high-powered computers
coupled with elaborate analysis programs and user-friendly graphical interfaces are
available, so the finite element method has begun to replace the grillage method in
many instances, even for more straightforward bridge decks [O'Brien, 1999].
7
2.2
A discussion on neutral axis location in bridge deck cantilevers
Many bridge decks include transverse cantilevers at their edges. If the edge
cantilevers are long and slender the edge cantilevers tend to bend (in the bridge
longitudinal direction) about their own centroids.
The variation in the neutral axis of a bridge with edge cantilevers is
essentially the same phenomenon as the well known concept of shear lag. In the
design of bridge decks, a two-dimensional analysis is often used which takes account
of shear lag through the use of an effective flange width.
A number of three-dimensional modelling techniques have been used which
allow for variations in neutral axis location. In continuous beams the effective flange
width is dependent on the ratio of actual flange width to the length between points of
zero moment and also dependent on the form of applied loading.
A single span simply supported slab bridge deck with edge cantilevers was
analysed by (O'Brien, 1998) and it was found that the neutral axis location was
dependent on the nature of the applied load. This finding verifies a previous study
reported by Green (1975).
It was used two-span bridge deck to demonstrate the degree to which the
neutral axis may vary from its commonly assumed position. A two-span slab bridge
deck with wide edge cantilevers was analysed by (O'Brien, 1998) and it was found
that the neutral axis location varied in the longitudinal direction as well as the
transverse direction.
The implications of this variation in neutral axis location were determined by
comparison of the prediction of longitudinal stresses at the top of the bridge deck
from the NIKE3D analysis with those from a two-dimensional grillage analysis.
8
It was found that the two-dimensional grillage analysis was not capable of
accurately predicting the stresses in the edge cantilever. The greatest discrepancy
was found to exist above the central support towards the edge of the cantilever.
The concept of effective flange width is clearly unsuitable for accurate
structural analysis as there is no simple effective flange width or neutral axis location
that can be used throughout a bridge for all load cases. Hence if greater accuracy is
required than that provided by the two-dimensional grillage analogy, it is necessary
to use an alternative approach. The Up stand Finite Element analogy has been found
by (O'Brien, 1998) to be simple to use and to give excellent results [O'Brien, Keogh,
1998].
2.3
Up stand finite element analysis of slab bridges
The plane grillage analogy is a popular method among bridge designers of
modelling slab bridge decks in two dimensions. Finite element analysis (FEA) is
used extensively by bridge designers, but is most often limited to planar analysis
using plate bending elements, which, like the plane grillage method, assumes a
constant neutral axis depth. It has been reported (O'Brien 1998) that the neutral axis
depth may also vary in the longitudinal direction in some cases, such as close to
concentrated loads or adjacent to intermediate supports in multi span decks.
Variability of neutral axis depth, when significant, results in the incorrect
representation by a planar analysis of the behaviour of the bridge deck. The problem
can be overcome by the use of a three-dimensional model. Such methods do not
require all members or elements representing parts of the deck to be located in the
one plane. Consequently this approach does not require a pre-assumed neutral axis
position, and allows for a rational handling of cases in which the neutral axis depth
varies.
9
The up stand FEA method is simple enough to be used in design offices for
everyday design and is significantly more accurate than plane grillage or plane finite
element analysis, particularly for bridge slabs with wide edge cantilevers.
O'Brien (1998) used Single- and two-span bridge decks with solid and voided
sections are considered for longitudinal bending stresses.
For a single-span bridge deck with wide edge cantilevers, both the plane
grillage and the up stand grillage methods are shown to be inaccurate in predicting
longitudinal bending stresses when compared to a 3D FEA method. Up stand finite
element method gives excellent agreement.
The differences in accuracy are attributed to the inability to model both
variations in the depth of the neutral axis and associated in-plane distortions. Similar
results are reported for longitudinal stresses in a two-span bridge deck.
Up stand FEA and the 3D FEA analyses predicted a similar maximum
transverse stress and a similar distribution across the width of the deck. Both
Predictions of transverse stresses were found to compare reasonably well. However,
a discrepancy was noted in the location of the point of maximum transverse stress.
A voided slab bridge deck with wide edge cantilevers was analysed using the
up stand FEA technique. The presence of the voids complicated the modelling and,
for the software used, required the addition of extra beams in order to maintain both
the correct area and second moment of area of the voided sections.
Isotropic finite elements were used as the void depth was less than 60% of the
slab depth. The up stand finite element model was analysed under the action of selfweight and a 3D FEA was carried out for comparison. The predictions of top
longitudinal stress from the up stand FEA compared very well with those from the
3D FEA [O'Brien, Keogh, 1998].
10
2.4
Bending moment distribution at the main structural elements of skew
...........deck-slab and their implementation on cost effectiveness
Many methods used in analyzing such as grillage and finite element method.
Generally, grillage analysis is the most common method used in bridge analysis. In
this method the deck is represented by an equivalent grillage of beams. The finer
grillage mesh, provide more accurate results. It was found that the results obtained
from grillage analysis compared with experiments and more rigorous methods are
accurate enough for design purposes. If the load is concentrated on an area which is
much smaller than the grillage mesh, the concentration of moments and torque
cannot be given by this method and the influence charts described in Puncher can be
used. The orientation of the longitudinal members should be always parallel to the
free edges while the orientation of transverse members can be either parallel to the
supports or orthogonal to the longitudinal beams. According to CCA the orthogonal
mesh is cumbersome in input data but the output moments results Mx, My and Mxy
can be used directly in the Wood- Armer equations as in Hambly to calculate the
steel required in any direction [Kakish, 2007].
The other method used in modelling the bridges is the finite element method.
The finite element method is a well known tool for the solution of complicated
structural engineering problems, as it is capable of accommodating many
complexities in the solution. In this method, the actual continuum is replaced by an
equivalent idealized structure composed of discrete elements, referred to as finite
elements, connected together at a number of nodes.
The finite elements method was first applied to problems of plane stress,
using triangular and rectangular element. The method has since been extended and
we can now use triangular and rectangular elements in plate bending, tetrahedron and
hexahedron in three-dimensional stress analysis, and curved elements in singly or
doubly curved shell problems. Thus the finite element method may be seen to be
very general in application and it is sometimes the only valid analysis for difficult
deck problems. The finite element method is a numerical method with powerful
11
technique for solution of complicated structural engineering problems. It most
accurately predicted the bridge behaviour under the truck axle loading.
The finite element method involves subdividing the actual structure into a
suitable number of sub-regions that are called finite elements. These elements can be
in the form of line elements, two dimensional elements and three-dimensional
elements to represent the structure. The intersections between the elements are called
nodal points in one dimensional problem where in two and three-dimensional
problems are called nodal line and nodal planes respectively.
At the nodes, degrees of freedom (which are usually in the form of the nodal
displacement and or their derivatives, stresses, or combinations of these) are
assigned. Models which use displacements are called displacement models and some
models use stresses defined at the nodal points as unknown. Models based on
stresses are called force or equilibrium models, while those based on combinations of
both displacements and stresses are termed mixed models or hybrid models.
Displacements are the most commonly used nodal variable, with most general
purpose programs limiting their nodal degree of freedom to just displacements. A
number of displacement functions such as polynomials and trigonometric series can
be assumed, especially polynomials because of the ease and simplification they
provide in the finite element formulation.
Finite element needs more time and efforts in modelling than the grillage.
The results obtained from the finite element method depend on the mesh size but by
using optimization of the mesh the results of this method are considered more
accurate than grillage. The finite element method is a well-known tool for the
solution of complicated structural engineering problems, as it is capable of
accommodating many complexities in the solution. In this method, the actual
continuum is replaced by an equivalent idealized structure composed of discrete
elements, referred to as finite elements, connected together at a number of nodes.
12
The finite element method was first applied to problem of plane stress, using
triangular and rectangular elements. The method has since been extended and we can
now use triangular and rectangular elements in plate bending, tetrahedron and
hexahedron in three-dimensional stress analysis, and curved elements in singly or
doubly curved shell problems. Thus the finite element method may be seen to be
very general in application and it is sometimes the only valid analysis for difficult
deck problems.
Tiedman shows the finite element method is a numerical method with
powerful technique for solution of complicated structural engineering problems. It
most accurately predicted the bridge behaviour under the truck axle loading.
Qaqish presents the effect of skew angle on distribution of bending moments
in bridge slabs. Qaqish presents comparison between finite element method and
AASHTO specification for the design of T-beam Bridge. The bridge is analyzed by
using the finite element method.
In conclusion the design of bridge deck slab should be carried out in a
structural computer model where the longitudinal beams subjected to vertical wheel
loadings should be designed on the bending moments and shearing force these
girders are subjected to. While the girders which they are not subjected to these truck
loadings should be designed for the actual loadings they are applied on these girders.
This method will achieve economy especially in places where the deck slab is
extended over big area. The vertical displacements are varied from one longitudinal
girder to the other and even these displacements are related to the actual bending
moments these girders are subjected to. So chambering is also different from one
longitudinal beam to the other which makes the constructions cheaper as some of
these longitudinal beams do not need such chambering due to small vertical
displacement [Kakish, 2007].
13
2.5
Effect on support reactions of T-beam skew bridge decks
In order to cater to high speeds and more safety requirements of the traffic,
modern highways are to be straight as far as possible and this has required the
provision of increasing number of skew bridges.
T-beam Bridge is a common choice among the designers for small and
medium span bridges. For the T-beam bridges with small skew angle, it is frequently
considered safe to ignore the angle of skew and analyze the bridge as a right bridge
with a span equal to the skew span. However, T-beam bridges with large angle of
skew can have a considerable effect on the behaviour of the bridge especially in the
short to medium range of spans.
In this study the behaviour of T-beam skew bridges with respect to support
reactions under standard IRC-70R wheeled loading is presented and the study was
based on the analytical modelling of T-beam bridges by Grillage Analogy method.
Effects of support reactions for different spans have been studied. The analysis
provides the useful information about the variation of support reactions with respect
to change in skew. The negative reactions were observed with increase in the span
and skew angles.
The advent of digital computers, computer-aided methods like Finite
Element, Finite Difference and Finite Strip have been developed and are in use to
analysis intricate forms of skew shape of bridges having usual support conditions and
cross-sections. But these methods are highly numerical and always carry a heavy
cost-penalty. Grillage analogy is one of the most popular computer-aided methods
for analysing bridge decks. The method consists of representing the actual decking
system of the bridge by an equivalent grillage of beams. The dispersed bending and
torsional stiffness of the decking system are assumed, for the purpose of analysis, to
be concentrated in these beams. The stiffness of the beams is chosen so that the
prototype bridge deck and the equivalent grillage of beams are subjected to identical
deformations under loading. The method is applicable to bridge decks with simple as
14
well as complex configurations with almost the same ease and confidence. The
method is easy to comprehend and use. The analysis is relatively inexpensive and has
been proved to be reliably accurate for a wide variety of bridges. The grillage
representation helps in giving a feel of the structural behaviour of the bridge and the
manner in which the loading is distributed and eventually taken to the supports.
The grillage analogy method is suitable in cases where bridge exhibits
complicating features such as heavy skew, edge stiffening and isolated supports. The
method is versatile in nature and the contribution of kerb beams and the effect of
differential sinking of girder ends over yielding bearings (such as neoprene bearing)
can also be taken into account and large variety of bridge decks can be analysed with
sufficient practical accuracy.
The method consists of ‘converting’ the bridge deck structure into a network
of rigidly connected beams at discrete nodes i.e. idealizing the bridge by an
equivalent grillage. The deformations at the two ends of a beam element are related
to the bending and torsional moments through their bending and torsional stiffness.
The method of grillage analysis involves the idealization of the bridge deck
as a plane grillage of discrete inter-connected beams. It is difficult to make precise
general rules for choosing a grillage mesh and much depends upon the nature of the
deck to be analysed, its support conditions, accuracy required, quantum of computing
facility available etc. and only a set of guidelines can be suggested for setting grid
lines. It may be noted that such idealization of the deck is not without pitfalls and the
grid lines adopted in once case may not be efficient in another similar case and the
experience and judgment of the designer will always play a major role.
On the basis of analysis it was found that grillage analogy method, based on
stiffness matrix approach, is a reliably accurate method for a wide range of bridge
decks. The method is versatile, easy for engineers to visualize and prepare the data
for a grillage. It has been found that in skew T-beams bridges, the high positive and
negative reactions develop close to each other. The reaction on the obtuse corner
close to load is very high and increases with increasing skew angles 400. With the
15
increasing in the span the negative reaction increases at smaller angles [Gupta,
Misra, 2007].
2.6
Types of bridge deck
The types of bridge deck are divided into beam, slab, beam-slab and cellular,
to differentiate their individual geometric and behavioural characteristics. Inevitably
many decks fall into more than one category, but they can usually be analysed by
using a judicious combination of the methods applicable to the different types
[Hambly, 1991].
2.6.1
Beam decks
A bridge deck can be considered to behave as a beam when its length exceed
its width by such an amount that when loads cause it to bend and twist along its
length, its cross-sections displace bodily and do not change shape. Many long-span
bridges behave as a beam because the dominant load is concentric so that the
direction of the cross-section under eccentric loads has relatively little influence on
the principle bending stresses [Edmund, 1991].
16
Figure 2.1: Beam deck bending and twisting without change of cross-section shape
(Bridge Deck Behaviour, 1991)
2.6.2
Grid deck
The primary structural member of grid deck is a grid of two or more
longitudinal beams with transverse beams (or diaphragms) supporting the running
slab. Loads are distributed between the main longitudinal beams by the bending and
twisting of the transverse beams. Grid decks are most conveniently analysed with the
conventional computer grillage analysis .The analysis in effect sets out a set of
simultaneous slop-deflection equations for the moment and torsions in the beams at
each joint and then solves the equation for the load case required [Edmund, 1991].
17
Figure 2.2: Load distribution in grid deck by bending and torsion of beam members
(Bridge Deck Behaviour, 1991)
2.6.3
Slab deck
A slab deck behaves like a flat plate which is structurally continuous for the
transfer of moments and torsions in all directions within the plane of the plate. When
a load placed on part of a slab, the slab deflects locally in a 'dish' causing a two
dimensional system of moments and torsions which transfer and share the load to
neighbouring pans of the deck which are less severely loaded.
A slab is 'isotropic' when its stiffness’s are the same in all directions in the
plane of the slab. It is 'orthotropic' when the stiffness’s are different in two directions
at right angle.
The beams of precast concrete or steel have a greater stiffness longitudinally
than the in situ concrete bas transversely; thus the deck is orthotropic.
18
If the depth and width of the voids are less than 60% of the overall structural
depth, their effect on the stiffness is small and the deck behaves effectively as a plate.
Voided slab decks are frequently constructed of concrete cast in situ with permanent
void formers, or of precast pre stressed concrete box beams post-tensioned
transversely to ensure transverse continuity. If the void size exceeds 6O% of the
depth, the deck is generally considered to be of cellular construction with a different
behaviour [Edmund, 1991].
Figure 2.3: Load distribution by bending and torsion of slab in two directions (Bridge
Deck Behaviour, 1991)
2.6.4
Beam and slab deck
A beam and slab deck consists of a number of longitudinal beams connected
across their tops by a thin continuous structural slab. In transfer of the load
longitudinally to the supports, the slab acts in concert with the beams as their top
flanges. At the same time the greater deflection of the most heavily loaded beams
19
bends the slab transversely so that it transfers and shares out the load to the neigh
boring beams. Sometimes this transverse distribution of load is assisted by a number
of transverse diaphragms at points along the span, so that deck behaviour is more
similar to that of a grid deck. Beam and slab construction has the advantage over slab
that it is very much lighter while retaining the necessary longitudinal stiffness.
Consequently it is suitable for a much wider range of spans, and it lends itself to
precast and prefabricated construction. Occasionally, the transverse flexibility can be
advantageous; it can help a deck on skew supports to deflect and twist ‘comfortably’
under load without excessively loading the nearest supports to the load or lifting off
those further away.
Beam and slab decks are most conveniently analyzed with the aid of
conventional computer grillage programs [Edmund, 1991].
Figure 2.4: Contiguous beam and slab deck and slab of contiguous beam and slab
deck (Bridge Deck Behaviour, 1991)
2.6.5
Cellular decks
The cross-section of a Cellular or box deck is made up of a number of thin
slabs and thin or thick webs which totally enclose a number of cells. These
complicated structural forms and increasingly used in preference to beam-and-slab
20
decks for spans in excess of 30m (100ft) because in addition to the low material
content, low weight and high longitudinal bending stiffness they have high tensional
stiffness’s which give them better stability and load distribution characteristics. To
describe the behaviour of cellular decks it is convenient to divide them into multi
cellular slabs and box-girders.
When a load is placed on one pan of such a deck, the high torsional stiffness
and transverse bending stiffness of the deck transfer and share out the load over a
wide area. The distribution is not as effective as that of a slab since the thin top and
bottom slabs flex independently when transferring vertical shear forces between
webs, and the cross-section. Such distortion can be reduced by incorporating
transverse diaphragms at various points along the deck but, as with beam and slab
decks, their use is becoming less popular except at support where it is necessary to
transfer the vertical shear forces between webs and bearings [Edmund, 1991].
9
Figure 2.5: Box girder deck (Bridge Deck Behaviour, 1991)
21
Finite Element Analysis
The availability of sophisticated computers over the last three decades has
enabled engineers to take up challenging tasks and solve intractable problems of
earlier years. Nowadays rapid decrease in hardware cost has enabled every
engineering firm to use a desk top computer or micro processor. Moreover they are
ideal for engineering design because they easily provide an immediate access and do
not have the system jargon associated with large computer system. It is to be
expected that software to be sold or leased and the hardware supplied with software.
The development of structural analysis up to its present position can be characterised
by four different eras as shown in Table 1.1. After the initial phase, where only
principles of gravity and statics were enunciated resulting in ambiguity in applying to
structural problem, Mathematicians took over from around 1400 A. D. and presented
a variety of formulations and solutions. Purely, as exercise in basic science, around
1700A.D. these formulations and solutions found practical significance in
applications to structures with proper approximations and adaptations. New methods
exclusive for structural analysis were evolved like slope deflection, moment
distribution and relaxation.
Later part of this period witnessed the emergence of superfast calculation and
later computers. Thus started the era of computers wherein the developments in
structural analysis and design were and are still complementary to those in
computers. A reorientation to the developments and formulation proposed in the
earlier eras took place mainly to use the advantageous features of computers like
high speed arithmetic, large information storage and limited logic, bringing in matrix
methods of analysis and later finite element and boundary integral element methods.
In recent years, the increasing availability of high speed computers have
caused civil engineers to embrace finite element analysis as a feasible method to
solve complex engineering problems. It is common for personal computers for home
use today are more powerful than supercomputer previous years. Therefore, the
increasing popularity of Finite Element Analysis can be attributed to the
advancement of computer technology.
22
Finite element method is a numerical method with powerful technique for
solution of complicated structural engineering problems. It most accurately predicted
the bridge behaviour under the truck axle loading.
The finite element method involves subdividing the actual structure into a
suitable number of sub-regions that are called finite elements. These elements can be
in the form of line elements, two dimensional elements and three-dimensional
elements to represent the structure. The intersection between the elements is called
nodal points in one dimensional problem where in two and three-dimensional
problems are called nodal line and nodal planes respectively. At the nodes, degrees
of freedom (which are usually in the form of the nodal displacement and/ or their
derivatives, stresses, or combinations of these) are assigned. Models which use
displacements are called displacement models and some models use stresses defined
at the nodal points as unknown. Models based on stresses are called force or
equilibrium models, while those based on combinations of both displacements and
stresses are termed mixed models or hybrid models.
Many shapes of elements are available. Thus this method may seem to be
very general in application. Triangular or quadrilateral plate elements can be adopted
to represent a bridge deck as an elastic continuum. The actual choice will depend on
the geometry of the structure, on the importance of local features such as stress
concentrations and also, upon the convergence properties of the bridge deck
[Nicholas M. Baran, 1988].
23
Table 2.1: development in structural analysis (The Finite Element Method 2003)
2.7
Grillage Analysis
The grillage numerical method is known in the static and dynamic analysis of
plate structures. By making use of a grillage discretization, the flexural and torsional
rigidities are determined to closely approximate a plate. The accuracy, simplicity and
speed of the grillage analog make it the most suitable model for bridge analysis
[Zeng, Kuehn, Sun, Stalford, 2000].
The bridge motion is composed of many modes, which cannot be predicted by
either simple bending or torsion theory. Therefore, a torsion beam element is
developed as a grillage member. Such a torsion beam element is subjected to a
transverse force distribution and a torsional moment distribution. The nodes undergo
24
not only planar translational and rotational displacements, but also torsional
displacements. Correspondingly, there are transverse joint forces, bending moments
and torsional moments at each node.
Grillage analysis is the most popular computer aided method for analyzing bridge
decks. It is easy to comprehend and use, and most important, it has been proved to be
reliably accurate for a wide variety of bridge types, It can be used in cases where the
bridges exhibits complicating features such as a heavy skew, edge stiffening and
deep hunches over supports,
The grillage representation is conductive to giving the designer an idea about the
structure behaviour of the bridge and the manner in which bridge loading is
distributed and eventually taken to the supports.
This method is to represent the deck by an equivalent grillage of beams. The
dispersed bending and torsion stiffness’s in every region of the slab are assumed for
purpose of analysis to be concentrated in the nearest equivalent grillage beam. The
slab’s longitudinal and transverse stiffness’s are concentrated in the longitudinal and
transverse beams respectively. Ideally, the beam stiffness’s should be such that when
prototype slab and equivalent grillage are subjected to identical loads, the two
structures will deflect identically [Jaeger, LG, Bakht, 1982].
25
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
This project is concerned essentially with the analysis of the bridge deck. A
bridge deck is structurally continuous in the two dimensions of the plane of the slab
so that an applied load is supported two dimensional distributions of shear forces,
moments, reaction, deflection and torques. Normally, an approximate method is
much used to analyze the slab deck behaviour as rigorous solution of the basic
equations for a real deck is seldom possible. In this project, two different analysis
methods are being considered i.e. grillage model and finite element model. Grillage
analysis is a method in which, the deck is represented and analyzed by a twodimensional grillage of beams. As for finite element analysis, the deck is notionally
subdivided into a large number of small elements for each of which approximate
plate bending equations can be written. In short, grillage analysis is a stiffness
method; meanwhile, finite element is plate method. All these analysis are aided with
26
finite element analysis computer software, LUSAS Modeller version 14.1. The
required outputs from the analysis are:
a) A deformed shape plot showing the largest displacement value and its
location.
b) The envelopes of bending moment, shear force and torsional moment in the
Pre-stressed beams for the design load combinations.
c) Maximum and minimum reaction forces at the support.
3.2
LUSAS Software
LUSAS is finite element analysis software it was first developed in 1970 at
London University as a research tool of finite element technology. Since then,
LUSAS has become a powerful tool for the solution of various types of linear and
nonlinear problems.
LUSAS can analyze and organise complex structure problems and shapes
including 3 dimensional structures and can be used in dynamic structural analyses
with temperature changes LUSAS software can solve problems up to 5000 number
of elements.
LUSAS is an associative feature based Modeller. This means the model
geometry is entered in terms of features which are then sub-divided into finite
elements in order to perform the analysis. Increasing the number of elements usually
increase the accuracy of the analysis but the time for the analysis to be done will also
increase. The features in LUSAS form a hierarchy, whereby Points can be joined
27
together to form Lines and then Lines form a Surface and Surfaces in turn can form a
Volume [Lusas Theory Manual, Version 14.1].
3.2.1
LUSAS Software Characteristic
LUSAS software can analysis and organise complex structure problems and
shapes including 3 dimensional structures. This software also can be used in dynamic
structural analyses with temperature changes. LUSAS software can solve problems
up to 5000 number of elements [LUSAS Version 14.1 Software].
3.2.2
Procedure Analysis According to LUSAS Software
There are 3 steps in the finite element analysis using the LUSAS software,
which are as follows:
a)
Pre-processing phase
Pre-processing involves creating a geometric representation of the structure,
then assigning properties, then outputting the information as a formatted data
file (dat.) suitable for processing by LUSAS.
b)
Finite Element Solver
Sets of linear or nonlinear algebra equations are solved simultaneously to
obtain nodal results, such as displacement values at different nodes or
temperature values at different nodes in heat transfer problems.
28
c)
Result-Processing
In this process, the results can be processed to show the contour of
displacements, stresses, strains, reactions and other important information.
Graphs as well as the deformed shapes of a model can be plotted.
In order to perform a full analysis, the LUSAS finite element system consists
of two parts. The two parts are.
i.
LUSAS Modeller - a fully interactive pre-processing and post-processing
graphical user interface.
ii.
3.3
LUSAS Solver - performs the finite element analysis.
Types of Element
The elements used in this study are expected to provide acceptable good
analytical results compared to the experimental results. Since the analysis carried out
in this research consists of 3D analysis, therefore all elements used were in the group
of 3D as it could produce more detail output due to the complex behaviour of the
structure. Table 3.1, below shows the summary of the element types used in this
research.
Three types of nonlinear analysis may be modelled using LUSAS. They are:
a) Geometric Nonlinearity e.g. large deflection or rotation, large strain, non conservative loading.
b) Boundary Nonlinearity e.g. lift-off supports, general contact, compressional load
transfer, dynamic impact.
c) Material Nonlinearity e.g. plasticity, fracture/cracking, damage, creep, volumetric
crushing, rubber material.
29
Table 3.1: Summary of the element types used in this research
Section
Element type
Description of
element
Three dimensional
HX8M
solid hexahedral
elements
comprising 8 nodes
each with 3 degrees
of freedom
Three dimensional
flat facet thick shell
elements
TTS3 and QTS4
comprising either 3
or 4 nodes each
with 5 degrees of
freedom
Three dimensional
bar elements
BRS2
comprising 2 nodes
each with 3 degrees
of freedom
Non-linear contact
gap joint elements
JNT4
and are used to
model the interface
between the end
plate and the
column flange
Shape
30
3.4
Configuration of Bridge Deck
The bridge deck of this project consisted cast-in-situ slabs on top of prestressed wide T beams. Each beam is supported on a bearing at each end and the
deck has a single span of 30 meter, giving width of deck of 11.9 meter, the deck
skew are varied skewed (0o - 40o). The parapet walls are not part of the structure but
to be considered as dead loads in these models. The beams were supported on single
bearings, and were longitudinally fixed at one end and sliding at the other. All the
working unit is KN (force), m (dimension), s (time) and t (mass).
Typical cross section of the bridge deck and T beams are as shown in Figure
3.1, and Figure 3.2 respectively.
Figure 3.1: Typical Cross Section of the Bridge Deck and T beams
31
Figure 3.2: Cross section of Pre-stressed Beam
Figure 3.3: Longitudinal Section of Grillage Model
Figure 3.4: Breadth Dimension - Actual versus Model
32
Figure 3.5: End diaphragm Cross Section
3.5
Materials Description
The following data are to be assumed for the analysis of the entire bridge
deck:
a) Concrete grade C40 for the slab and end diaphragm beams i.e. E slab = 31
x 106 KN/m2.
b) Concrete grade C50 for the pre-stressed beams i.e. E beam = 34 x 106
KN/m2
c) Poisson ratio = 0.2
d) Concrete density = 2.4 t/m3
e) Surfacing 0.05m thick tarmac with density = 2.0 t/m3
33
3.6
Loadings Description
All loading is proportioned to the members and joints (nodes) before the
moments, shears and torsions are calculated. Many programs have the facility for
applying patch loads and point loads which do not necessarily coincide with joints or
members. The program will distribute these loads to the members before calculating
the moments, shears and torsion effects. The bridge is subjected to self-weight,
superimposed dead load and vehicles loading according to BD 37/01.
3.6.1
Dead Load
The Dead load is made up of self-weight of the structure and any permanent
load fixed thereon, which is defined as acceleration due to gravity. The dead load is
initially assumed and checked after design is completed. The load intensity is:
DL (KN/m) = m (t/m3) * g (m/s2) * A (m2)
Dead load is applied to the main longitudinal members. Some programs will
automatically generate dead load by applying a density to the cross-sectional area of
the member. Care is needed to avoid double accounting for the weight of the deck
slab.
34
3.6.2
Superimposed Dead Load
Superimposed dead load (carriageway surfacing, footpath fill and surfacing
and parapets) are input as uniformly distributed loads along the length of the
longitudinal members. Some programs have the facility of applying patch loads
which can be used for the surfacing providing it is of constant thickness. The load
intensity for the tarmac is as shown: w = (2.0) t/m3 * (9.81) m/s2 * 0.05 m = 0.981
KN/m2
This load is applied through discrete load type where the load intensities at
four corners are specified. Figure 3.6 of loading application is as shown.
Reference point for assigning load
Figure 3.6: Superimposed Dead Load
35
3.6.3
Live loading
Live loading can consist of HA (UDL + KEL) load, HB load, Pedestrian load,
Accidental Wheel load and Wind load. Collision load on parapets is only included if
high containment parapets are required. Horizontal loads such as traction or braking
and skidding are generally not included as the deck is very stiff in resisting these
loads and they will have negligible effect on the results from the grillage analysis.
3.6.3.1 Vehicle Load – HA Loading
HA loading represent normal actual vehicles loads on bridges. In this project,
UK vehicle loading code BD 37/01 is referred. The HA loading consists of UDL
(equivalent uniform dead load) and KEL (knife edge loading).
Table 3.2: Vehicle load − HA Loading
Loaded length
30m
Width of carriageway
11.5m
Number of notional lane (Clause
3.2.9.3.1)
Notional lane width
HA UDL per linear meter of loaded
length
HA KEL (Clause 6.2.2)
4
11.5m /4 = 2.875m
34.4kN/m
(Table 13)
120kN / notional lane
The HA loading is being assigned to each notional lane in the centre
coordinate as shown in Figure 3.7. To obtain maximum shear and reaction force,
separate HA KEL should be applied at location near the support.
36
Notional lane 4
Notional lane 3
Notional lane 2
Notional lane 1
Figure 3.7: Notional Lanes & Center Coordinates
3.6.3.2 Vehicle Load – HB Loading
The HB vehicle consists of 4 axles. One unit of HB loading is taken as 10kN.
6m axle spacing with 45 units of HB load per axle is selected for this project.
Static vehicle, lane, and knife edge loading types are provided for many
regional codes of practice. These currently include: AASHTO LFD & LRFD (USA),
BD21/97 (UK), 21/01 (UK), BD37/88 (UK), 37/01 (UK), BRO94 and BRO2002
Vehicle and Classification loads and BRO Train loading (Sweden), Korean,
Israel, Norway, and HK (Hong Kong), Australia, China, Eurocode vehicle and train
loading, Finland, India, New Zealand and Poland. Additional loading types are being
added all the time.
Table 3.3: Vehicle load − HB Loading
HB loading for one axle of 45 units
45 x 10kN/ unit = 450kN
Point load for each wheel
450/4 = 112.5kN
37
Figure 3.8: Dimension of HB Load
3.5
Notional lane 1
2.25
Notional lane 2
Figure 3.9: HB at lane 1
Notional lane 4
Figure 3.10: HB at lane 2
2.55
Notional lane 1
3.5
Notional lane 2
2.55
Notional lane 3
38
3.6.4
Loading Combination
A key feature of LUSAS Bridge is the Basic, Smart and Code-specific load
combination facilities which allow manual or fully automated assembly of design
load combinations. From these, envelopes, contour and deflected shape plots, and
results graphs can be readily obtained for any load case under consideration.
Basic load combinations allow for manual definition of load cases and load
factors. The Smart Combinations facility, unique to LUSAS Bridge, automatically
generates maximum and minimum load combinations from the applied loadings to
take account of adverse and relieving effects. This enables the number of
combinations and envelopes required to model a bridge to be substantially reduced.
Absolute maximum envelopes are included.
Load Combination Wizards use predefined bridge load cases for country-specific
design codes and help automate the definition of load combinations for bridges.
When used in conjunction with a design code template, combinations of load
combinations are automatically created to give the resultant maximum and minimum
ULS or SLS load cases. The loading combination made including of:
39
3.6.4.1 HA loading for all lanes
Table 3.4: HA loading for all lanes
Adverse factor
Load case name
HA + KEL lane 1
HA + KEL lane 2
HA + KEL lane 3
HA + KEL lane 4
γf3
Lane factor
(Table 14)
1.10
β1 = 0.89
1.10
β1 = 0.89
1.10
β3 = 0.60
1.10
βn = 0.54
γf1
ULS
1.5
SLS
1.2
ULS
1.5
SLS
1.2
ULS
1.5
SLS
1.2
ULS
1.5
SLS
1.2
Factors to be
applied
1.47
1.17
1.47
1.17
0.99
0.79
0.89
0.71
3.6.4.2 HB loading for lane1 and HA loading for lane 3&4
The HA load is not required on lane 2 because the clear space of lane 2 is
2.25m which is less than 2.5m.
Table 3.5: HB loading for lane1 and HA loading for lane 3&4
Adverse
factor
Load case name
γf1
HB lane 1
HA lane 2
ULS
1.3
SLS
1.1
−
−
γf3
Lane factor
(Table 14)
1.10
−
−
−
Factors to be
applied
1.43
1.21
−
40
HA + KEL lane 3
HA + KEL lane 4
ULS
1.3
SLS
1.1
ULS
1.3
SLS
1.1
1.10
β1 = 0.89
1.10
β2 = 0.89
1.27
1.08
0.27
1.08
3.6.4.3 HB loading for lane 2, HA UDL for lane 1, 3 and HA loading for lane 4
Table 3.6: HB loading for lane 2, HA UDL for lane 1, 3 and HA loading for lane 4
Adverse factor
Load case name
γf1
ULS
HB lane 1
HA lane 2
HA lane 3
HA + KEL lane 4
γf3
Lane factor
(Table 14)
1.10
βbL=2.5 = 0.84
1.10
−
1.10
βbL=2.5 = 0.84
1.10
β2 = 0.89
1.3
SLS
1.1
ULS
1.3
SLS
1.1
ULS
1.3
SLS
1.1
ULS
1.3
SLS
1.1
Factors to be
applied
1.20
1.02
1.43
1.21
1.20
1.02
1.27
1.08
41
3.6.4.4 HB loading on lane 1 only
Table 3.7: HB loading on lane 1 only
Adverse factor
Load case name
γf1
ULS
HB lane 1
SLS
γf3
1.3
1.1
1.10
Lane factor
(Table 14)
−
Factors to be
applied
1.43
1.21
3.6.4.5 HB loading on lane 2 only
Table 3.8: HB loading on lane 2 only
Adverse factor
Load case name
γf1
ULS
HB lane 2
SLS
γf3
1.3
1.1
1.10
Lane factor
(Table 14)
−
Factors to be
applied
1.43
1.21
3.6.4.6 Enveloping the basic live load combinations
Enveloping is used to perform results processing on maximum and minimum
values of results of load cases. LUSAS modeller does this by creating two load
cases, one for the maximum values and one for the minimum of the specified
load cases.
42
3.6.4.7 Smart Load Combination
Smart load combinations take account of adverse and relieving effects for the
load cases being considered. The Self-weight, Superimposed Dead Load, and the
Live Load Envelope will all be combined using the Smart Load Combination
facility to give the design combination.
Table 3.9: Smart Load Combination
Variable factor
Load case
γf1
name
Dead load
Super dead
load
ULS 1.15
SLS
1.0
ULS 1.75
SLS
Live load
envelope
1.0
γf3
1.20
1.20
Load
factor
1.38
1.20
2.1
1.2
Permanent
factor
1.0
1.0
Factors to be
applied
0.38
0.2
1.1
0.2
1.0
1.0
1.0
0
1.0
1.0
1.0
1.0
0
1.0
(max)
Live load
envelope
(min)
3.6.5
Vehicle Loading and Load Combinations
LUSAS Bridge provides static vehicle loading options for many worldwide
bridge design codes. These loadings can be used either on their own or with a
moving load generator.
Moving vehicle/train load generators can be used to automatically generate the
required loadcases for a vehicle as it tracks across a bridge.
43
For special heavy vehicles, an Abnormal Indivisible Load generator is
included which can generate the load pattern for all possible combinations of
vehicles/axle load/axle spacing.
An optional Vehicle Load Optimisation facility allows an optimised load
pattern to be generated in accordance with the chosen code of practice.
3.7
Grillage Modelling
Longitudinal beam is placed at the pre-stressed beam position, i.e. at 1.9 m
c/c, while the transverse beam representing the slab stiffness in the transverse
direction is placed at 2m c/c. The slab can act as a flange to the end diaphragm beam.
The recommended length of the flange is 0.3*beam spacing = 0.57 m. Dummy beam
is used in the model for assigning parapet wall load. 1 meter wide solid diaphragm
beams are provided at each end and no additional transverse beams are provided
within the span. The parapet walls are not part of the structure but to be considered as
dead loads in the model.
Since only one element is used to represent the beam and slab which have
different material stiffness, it is important to calculate the equivalent breadth section
by adopting same material stiffness. In this project, the material stiffness of beam is
selected i.e. the stiffness of slab is factor by a modular ratio, m = Es /Eb.
Equivalent breadth of slab, beq = mb = (31/34) x 1.9 = 1.730m
44
Figure 3.11: Grillage and Finite element model
3.8
Finite Element Modelling
The finite element model of this project is shown in Figure 3.12; Types of
beams to be used in the model are longitudinal beam, end diaphragm beam and
dummy beam. Longitudinal beam is placed at the pre-stressed beam position, i.e. at
1.9 m c/c. Unlike the grillage model, the original pre-stressed I section shall be
adopted. 1 meter wide solid diaphragm beams are provided at each end. The parapet
walls are not part of the structure but to be considered as dead loads in the model.
Dummy beam is used in the model for assigning parapet wall load.
45
Slab thickness = 0.2m &
6 nos. of Longitudinal I Beam
Figure 3.12: Finite Element Model
Figure 3.13: Finite Element Model
As shown in Figure 3.13, the span of a skew bridge measured along an
unsupported edge of the bridge in plan is called “Skew Span”. The directions parallel
and perpendicular to the flow of traffic on the bridge are still called the longitudinal
and transverse directions respectively.
46
3.9
Layers and Windows
Models are formed of layers where the visibility and properties of each layer
can be controlled and accessed via the layer name which is held in the Tree view. As
the model is built up, model features may be grouped together and manipulated to
speed up data preparation or to enable parts of the model to be temporarily hidden.
3.10
Viewing Results
A whole host of facilities and wizards are available to help you evaluate your
results.
Results can be viewed using separate layers for diagram, contour, vector and discrete
value data.
•
Plot bending moments, shears forces, and deflections.
•
Contour ranges and vector/diagram scales can be controlled locally in
each window or set globally to apply to all windows.
•
Load cases are selected on a window basis allowing multiple views of the
model with each window displaying results for different load cases.
•
Results can be displayed in global or local directions, in element
directions, or at any specified orientation.
•
Results can be plotted on deformed or undeformed fleshed or unfleshed
beam sections.
•
Multiple slices may be cut through 3D solid models on arbitrary planes
and made visible or invisible in any window.
47
3.11
Bridge Deck Analysis Result & Discussion
It is always a good practice to carry out approximate checks of the output as
the job proceeds. One simple check is to obtain the total reactions for each load case
to see if they agree with an estimate of the total load applied in each load case.
All the results from these two different models shall be discussed in this
chapter for the beam. The beam results are taken only for both models (grillage,
finite element) after removing (making invisible) all of dummy, transverse, end
diaphragm for grillage model; because the grillage analysis is meant to get the result
for rectangular beam while the rest is not that useful in the design. In finite element
model, removing (making invisible) all of shell split divisions y = 3, y = 2 and end
diaphragm for the finite element model. The division (x,y) is same for all skews for
each of the models, When the surface is divided differently, the result is a little bit
different because of the interpolation. Also the loadings in both models with different
skews are same (value, coordinate). Deflection results are illustrated for better
visualization. Also, the summaries of bending moment, shear force, reaction and
torsion are mentioned and tabulated, while the objective of the contents of this thesis
is to court all the case of the modelling by Lusas software.
Figure 3.14: Simple of bridge deck
48
CHAPTER 4
RESULT AND ANALYSIS
4.1
Introduction
In recent time’s research has been carried out into improving vehicle models,
road surface models and numerical models for the bridge-truck dynamic interaction.
It is hoped to advance the knowledge of the dynamic interaction between bridges and
crossing trucks using a finite element approach, whereby different load cases and
simulation of critical load events can be carried out, using complex models [Rattigan,
Obrien, Gonzalez, 2005].
It is important in the design of highway bridges that adequate consideration is
given to the level of bridge excitation resulting from the dynamic components of a
bridge-truck interaction system [Rattigan, Obrien, Gonzalez, 2005].
The presence of skew in a bridge makes the analysis and design of bridge decks
intricate [Gupta, Misra, 2007]
49
4.1.1
Displacement Result
Graphs of the largest displacement value and its location for each modelling
are plotted below. According to the result obtained, minimum combination is taken
for displacement because there is no upward movement. The result depends on the
geometry and the position of a point load from (HB) loading, because there’s no
interpolation at that point.
The minimum displacement for both models is occurring close to mid span
Figure 4.3. When the skew increases, the load (the position of HB loading varies
with increasing the skew) will move far away from the mid span experiencing a
decrease in displacement value. For instance, in grillage analysis, the result of 0°
skew was (– 0.0717 m) while it was (– 0.0687 m) with 40° skew. Whilst in finite
element analysis the minimum value at 0° skew angle was (– 0.043 m) and will
reduce with increasing the skew till (– 0.0407 m) at 40° skew. Both analysis models
yield to the same reduction value of the displacement while increasing the skew,
Table 4.2.
From 0° onward the maximum displacement for both models (grillage model
and finite element model) decrease approximately linearly till 40° skew.
The reason for that is because the displacement value affected by (supports)
and (loads), since the bridge is simply supported for both models, all the beams are
symmetrical in properties (thickness, width, materials, loading ...) and the value of
loading is the same for each model with different skew values. Consequently,
difference in load position (HB loading) and mesh (decreasing mesh by increasing
the skew) result in a difference in the value of displacement between the beams when
the skew is increasing for both models. For both models the position of maximum
displacement was in the right position of bridge span beam 6 because (HB loading)
lies in the right position of the bridge span Figure 4.3.
50
In the table (4.2) the value of grillage model result gives almost one and half
times the finite element model result, because finite element model is more precise
than grillage model and possesses more discretization (more accurate). Therefore
finite element gives less response than grillage model.
Table 4.1: Displacement, Dz (m) versus skew
Table 4.2: largest value of displacement, Dz
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Figure 4.1: location of displacement, Dz, Grillage model skew 200
Figure 4.2: location of displacement, Dz, Finite element model skew 200
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Figure 4.3: location of displacement, Dz
4.1.2
Reaction Result
According to the result obtained in Table 4.4, the maximum support reaction
(largest value) in the grillage model at 0° was (2130 KN) at the end of girder support
beam 6. However, at 40° the maximum reaction (largest value) was (3190 KN) in the
same position beam 6. Meanwhile in finite element model, the maximum support
reaction (largest value) at 0° was (1340 KN), (1530 KN) at 40° skew and all in the
same position beam 6, see Figure 4.8. The results for Grillage and finite element
analysis are increasing linearly while incrementing skews.
The minimum support reaction (smallest value) for the grillage model at 0°
was (439kN) at the end of girder support to the right position beam 5. However, at
40° the minimum reaction (smallest value) was (-139 KN) in the same position (beam
6), grillage model decrease continuously. While in finite element model, the location
happened to be in beam 1 and the minimum support reaction (smallest value) at 0°
and 40° was (130 KN), (144 KN) respectively.
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The maximum support reaction (largest value) for grillage model is almost
one and half times the finite element at 0° and twice in 40°, because finite element is
obviously more precise than grillage model.
In both models with different skews, all the beam symmetrical properties
(dimension, materials...), have same value of loading in all cases with only position
difference for (HB loading) because all different skews’ positions are dependent on
beam 6. The result increases constantly while increasing the skew because the
loading becomes closer to the support (right support); we can say increasing the skew
will increase the value of reaction.
In grillage model case, when the skew becomes larger, the maximum reaction
increases but minimum reaction reduces (can become negative at 40°) which means
that the reaction behaviour is an upwards movement (moving up). As a conclusion,
finite element model gives less response than grillage model in reaction force.
Table 4.3: Reaction, Fz (m) versus skew
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Table 4.4: largest & smallest value of reaction, Fz
Figure 4.4: location of largest reaction, Fz, Grillage model skew 200
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Figure 4.5: location of smallest reaction, Fz, Grillage model skew 200
Figure 4.6: location of largest reaction, Fz, Finite element model skew 200
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Figure 4.7: location of smallest reaction, Fz, Finite element model skew 200
Figure 4.8: location of largest reaction, Fz
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Figure 4.9: location of smallest reaction, Fz
4.1.3
Shear force Result
Shear element on the deck has been analyzed to affect the entire model in
skewed analysis, where the piers or the abutment has been recorded with the greatest
effect of the shear loading. We will take the magnitude for shear force in the beam,
after that take the largest between them. Therefore, the condition at the two models
has recorded a desirable effect of the loading.
The grillage has been provided in the grillage wizard in such a way that the
model does not need to take special consideration of providing more elements and
nodes for analysis. Finite element analysis has been divided into better small
partitions for more discretization. Grillage and finite element models with different
skews give same position they give in beam 6 Figure 4.12, for example the result in
grillage model at 0° skew was (1680 KN), and value was increasing till 20° skew,
after that it starts to decrease a little bit at 30° skew, whereupon it increases again,
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this also happens with finite element model. Grillage value is one and a half times
the finite element value. Again finite element model gives a less response than
grillage model.
Table 4.5: Shear Force, Fz (m) versus skew
Table 4.6: largest value of Shear Force, Fz
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Figure 4.10: location of Shear Force, Fz, Grillage model skew 200
Figure 4.11: location of Shear Force, Fz, Finite element model skew 200
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Figure 4.12: location of Shear Force, Fz
4.1.4
Torsion Result
As mentioned before, all the results must be taken on the beam because the
grillage analysis is to get the result for rectangular beam only, the others (end
diaphragm ...) is not that useful for design. Torsion is induced when there is a vertical
loading, when bridges are curved or crooked in plane. The analysis’ result for Finite
element model and grillage model increases gradually when increasing the skew, the
value of grillage model at 0° skew was (401 KN.m) and was continuously increasing
till (561 KN.m) at 40° skew, while in finite element, it was (262 KN.m),(336 KN.m),
at 0°, 40° skews respectively. See Table 4.8. The difference between them is only the
position. In grillage model, it happened in beam 6, while in finite element model, it
happened in beam 5, Figure 4.15.
The difference is because the position of (HB loading) value increases
whenever the skew increases, because the (HB loading) is moving to the right if
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compared with beam 6 (stable). Since torsion depends on the position of loading, the
position of (HB loading) is closer to the support when skew increases. Again grillage
value is one and half times the finite element value.
Table 4.6: largest value of Shear Force, Fz
Table 4.8: largest value of Torsion, Mx (KN.m)
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Figure 4.13: largest value of torsion, Mx (KN.m), Grillage model skew 200
Figure 4.14: largest value of torsion, Mx (KN.m), Finite element model skew 200
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Figure 4.15: largest value of torsion, Mx (KN.m)
4.1.5
Moment Result
Moment depends on the value of loading and the position of loading. Since it
is different in position because of the variation of skews, therefore we get different
value of moment.
Ultimate bending moment =
𝑤𝑤 l2
8
In addition to the longitudinal loading, horizontal loading in bridges can affect the
design of bearings and generate bending moment in substructures and throughout
Frame Bridge [O'Brien, Keogh, Lehane, 1999].
In grillage model (largest value) the 0° skew result was (2220 KN.m). It was
increasing regularly till (2940 KN.m) at 40° skew, It also happens in finite element
model in which the result in 0° skew was (948 KN.m), the value increases to (973
KN.m) at 20° skew. Suddenly, the amount of value decreases to (957 KN.m) at 30°
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skew, the quantity appears to be continuously reducing, the value (941 KN.m)
resulted at 40° skew, see Table 4.10. The position resulting from each of the two
models happened in beam 6. The difference was that in grillage model, the location
was on (right support) position, while in finite element; it was on the left support
Figure 4.20.
Again in both of the models (using smallest value), grillage model behaves
the same with the (largest value) in value but the difference was in position, it was in
the same position for both models and skews in beam 6 Figure 4.21.
Finite element gives the smallest positive and negative because of the
difference in mesh (more discretization), while all the beams are symmetrical in
properties (thickness, width, materials, loading ...) and the value of loading is same
for each model with different skews. Finite element model gives less response than
grillage model.
Table 4.9: moment, My (KN.m) versus skew
65
Table 4.10: largest & smallest value of moment, My (KN.m)
Figure 4.16: largest value of moment, My (KN.m), Grillage model skew 200
66
Figure 4.17: smallest value of moment, My (KN.m), Grillage model skew 200
Figure 4.18: largest value of moment, My (KN.m), Finite element model skew 200
67
Figure 4.19: smallest value of moment, My (KN.m), Finite element model skew 200
Figure 4.20: largest value of moment, My (KN.m)
68
Figure 4.21: smallest value of moment, My (KN.m)
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CHAPTER 5
CONCLUSION
5.1
CONCLUSION
The focus of this modelling is to find the reason of the results’ differences of the
two models (Grillage, Finite Element), while the objective of this thesis is to simulate
the behaviour of bridge structure in terms of (displacement, reaction, shear force,
bending moment, and torsion) by varying the skew angle value. All done by Lusas
software.
In general for practical skew bridge deck result for finite element give lesser
value in terms of displacement, reaction, shear force, torsion, bending moment
compare with grillage model, therefore can be concluded that analysis by using finite
element method made produce more economical design then compare with the
grillage analysis. In grillage model the result got for both the slab and beam, whilst in
finite element (separate) the result only for beam.
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All the characteristic results increase during the increase of skew except the
displacement result (decreasing), the reason for that is because the position of HB
loading becomes near to the support by increasing the skew angle, in addition to the
element (the load distribution) the distribution is transfer throw the element, of
course the mesh is also important, the last reason is the stiffness change due to skew
into the element skew.
In conclusion, Finite Element model is less precise than grillage model and
possesses more discretization (more accurate), so the design base of this response
with the smaller element gives less amount of materials and so on (that is the
economical factor), subsequently finite element model is more economical design
than grillage model.
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