Comparison between Computed Shearing Forces by AASHTO - wseas

16 Pages·2009·0.94 MB·English

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INFORMATION SCIENCE and APPLICATIONS Maher Qaqish, Emad Akawwi, Eyad Fadda, Maher Qaqish
Comparison between Computed Shearing Forces by AASHTO
Specifications And Finite Element Method of Two Continuous Spans of
Voided Slab Bridge

MAHER QAQISH 1, EMAD AKAWWI 2 AND EYAD FADDA 3∗

1Associate Professor, College of Engineering, Al-Balqa Applied University, Salt, Jordan
2Assistant professor at Al-Balqa Applied University-Salt-Jordan
3Assistant professor at Al-Balqa Applied University-Salt-Jordan

MAHER QAQISH
Al-Balqa Applied University

SALT- JORDAN
[email protected]

abstract :- Voided slab bridge deck composed of two continuous span is 17.3m Long and 16m width with
all over depth of 0.9m. The voided slab is composed of 20cm top and bottom slabs with circular voided
areas of 50cm in diameter and center to center of these circles of 75cm. The locations of AASHTO
loadings are positioned at certain points of deck slab to give maximum positive and negative shearings.
These locations are determined from one dimensional model. The analysis of the bridge deck is carried
out by two approaches. The first is AASHTO specifications where one dimensional approach is adopted
and the second is three dimensional approach where finite element analysis is considered. The maximum
shears obtained by both methods are found to be in good agreement with negligible differences.

Key-words: Bridges, voided slab, AASHTO Loadings and Finite Element Method.

∗
Corresponding author. E-mail: [email protected]

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1 Introduction Eight-node shell elements were used to model the
decks, diaphragms, bent caps, and girders. Beam
Bridges can be either concrete, timber and steel.
elements were used to model columns under the
Concrete bridges can be either cast insitu reinforced
bent caps. The analytically predicted distribution
concrete or pre-stressed concrete or a combination
factors were compared with the field test data.
of both. Reinforced concrete is well suited for use in
Finally , the bridge finite element modes were
bridges of all kinds due to its durability, rigidity and
employed to compare load distribution factors for
economy and aesthetic consideration [1].
shear computed using procedures in the AASHTO
The AASHTO specifications [2] specifies that
LRFD and standard specifications.
diaphragms cast within the beam are
Civjan [6] studied the use of integral abutment
recommended at midspan for spans up to
bridges for moderate spans throughout the United
15.0m, at third points for spans is to 23m and at
States. However, design methods and construction
quarter points for spans over 23.0m.
details vary from state to state. Variations between
Trejo [3] identifies the most significant parameters
states are noted in the methods employed to
for evaluating the time to corrosion of reinforced
accommodate deformations in the piles. The
concrete structures.
significance of these differences was evaluated
Suzuki [4] instrumented prestressed concrete (PC)
through a finite-element study. The effects of
box-girder bridge to monitor its surface and body
backfill properties and soil restraint on piles were
temperature. The daily variations of temperature in
evaluated with regard to bridge distortions and
the pavement and upper beam of the bridge were
maximum moment realized in the piles. Results
greater the deeper from the surface. During daytime
show that bridge expansion is predominantly
on a fine day, bridge surface temperatures were
affected by backfill conditions, whereas contraction
higher than air temperatures, and at nighttime on a
is influenced by pile restraint conditions.
fine night, temperatures at the surface were lower
Okui [7] presented a time-dependent finite-element
than air temperatures. It is well known that sensible
analysis of a two I-girder composite bridge with a
heat transfers depend on wind speed and surface
concrete slab. The creep and shrinkage of the
temperature. The wind speed measured at the bridge
concrete slab are considered as sources of time-
and on the road revealed that the wind speed at the
dependent behavior. This analysis ,unlike others,
bridge was higher than at the road due to their
includes the shear – lag effect of the concrete slab
topography. This wind effect during the daytime
on the time – dependent behavior of the two I-girder
causes the surface temperature at the bridge to be
bridges .
lower than that at the road.
Samaan [8] presented the results from a parametric
Thus, the reason why only bridge surfaces may
study on the impact factors for 180 curved
freeze while roads do not can be explained by the
continuous composite multiple-box girder bridges.
fact that the surface temperatures of bridges are
Expressions for the impact factors for tangential
usually lower than those of roads, which can be
flexural stresses, deflection, shear forces and
attributed to their energy budget patterns caused by
reactions are deduced for AASHTO truck loading .
the wind speed differences between bridges and
The finite-element method was utilized to model the
roads.
bridges as three-dimensional structures.
Potisuk [5] presented findings of field tests and
The vehicle axle used in the analysis was simulated
analysis of two conventionally reinforced concrete
as a pair of concentrated forces moving along the
(CRC) deck girder bridges designed in the 1950s.
concrete deck in a circumferential path with a
The bridges are in-service and exhibit diagonal
constant speed. The effects of bridge configurations,
cracks. Stirrup strains in the bridge girders at high
loading positions, and vehicle speed on the impact
shear regions were used to estimate distribution
factors were examined. Bridge configurations
factors for shear Impact factors based on the field
included span length, span-to-radius of curvature
tests are reported. Comparison of field measured
ratio, number of lanes, and number of boxes.
responses with AASHTO factors was performed.
Najm [9] evaluated the behavior and performance of
Three-dimensional elastic finite-element analysis
laminated circular elastomeric bearings and
was employed to model the tested bridges and
compares them to those of square and rectangular
determine distribution factors specifically for shear.
bearings. The experimental investigation studied the
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bearings behavior in compression , compression and typical beam-and-slab bridge decks and the strength
rotation , and compression and shear. Results from enhancement due to compressive membrane action.
this limited study showed that the three bearings This gives a more accurate prediction of the strength
have similar stress-strain behavior in compression of the slab. The method has been developed through
and they are in agreement with the AASHTO LRFD the laboratory testing of concrete specimens and the
guide stress–strain curves. use of nonlinear finite element modeling.
In compression and rotation, the AASHTO LRFD Kirkpatrick [15] presented the results of tests carried
substructure moments are slightly less than the out on a reinforced-concrete beam and slab bridge in
measured values for circular bearings and Northern Ireland that incorporated novel
rectangular bearings rotated about their strong axis reinforcement type and position. The research was
for a compressive stress of 10.3 MPa (1.5 ksi) and aimed at extending previous laboratory tests on 1/3-
slightly higher than those of rectangular bearings scale bridge deck edge panels. The measured crack
rotated about their weak axis. In compression and widths and deflections have been compared with the
shear , the shear stiffness of circular and square current code requirements.
bearings is in agreement with theoretical values Mankar [16] presented the prediction of the load
within the range of design displacement. distribution characteristics of straight composite
Garrett [10] presented a case study that illustrated multi- cell box girder bridges. A finite –element
the effect of live dead loads, geometric nonlinear analytical model based on the commercially
effects, temperature effects and material on load available “ANSYS” software, has been used for the
rating a two-span shallow concrete arch bridge. analyses. The analytical model is substantiated by
Presented are load ratings of the structure's arch ribs published results from tests on composite concrete
using a three-dimensional finite-element model with deck- steel three- cell bridge model.
American Association of State Highway and A parametric study, using the finite- element
Transportation Officials publications. As a result of modeling, was conducted in which composite multi
this study, a refined analysis is recommended for – cell bridge prototypes were analyzed to evaluate
load rating arch bridges. their load distribution factors for shear under IRC
Kayal [11] presented analytical methods to live loading condition. The key parameters
determine the plastic rotation capacity of reinforced considered in this study are: number and stiffness of
cement concrete T-beam bridge girder (or grillage cross- bracing and top- chord system, number of
member) under the combined influence of flexure cells, number of lanes, and loading conditions.
and torsion. The methods have been validated by Based on the parametric study, load distribution
experiments. The analytical methods are based on factors for shear is deduced. Recommendations to
skew-bending and space truss theories. The tests enhance the torsional resistance to static forces on
have been carried out on 1:6 microconcrete models. this type of bridges are made. A recommendation
The salient conclusions have been enumerated. for optimum spacing of cross bracings has been
Trejo [12] provided a methodology by which design suggested. The results from this practical – design –
engineers can easily identify sensitive variables oriented study would assist the bridge engineer to
affecting the service life of RC structures exposed to design composite cellular bridges more reliably and
potentially corrosive environments. economically.
Guzda [13] developed a methodology incorporating Dwived [17] presented a finite element analysis to
real-time visual data collection from traffic cameras predict the nonlinear temperature distributions and
coupled with structural strain response of girder their response for the design of concrete bridges,
bridges. Average daily truck traffic along with which takes into consideration the various
frequency of multiple presences (same lane as well meteorological, environmental, and other bridge
as adjacent lanes) and lane-wise truck traffic parameters. The proposed model has been validated
distribution were estimated for a steel-girder against the results obtained by an experimental
highway bridge on I-95 in Delaware. These data program on a laboratory model of concrete box-
compared well with predictions from a Poisson girder bridge exposed to solar radiation. The
process based model developed for this study. proposed analytical model has been validated
Hon [14] presented a design method for assessing against the experimentally observed values on
the restraint stiffness that exists for the slab of laboratory models of concrete box girder bridge.
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Good agreement has been observed between the radar to detect common flaws in concrete bridge
computed and experimental thermal profiles. decks is developed and discussed.
Manassa [18] the effect of a skew angle on simple- Results from this study showed the ability of these
span reinforced concrete bridges using the finite- methods to detect defects with varying precision.
element method. The parameters investigated in this Capabilities of the methods were verified and
analytical study were the span length, slab width, comparisons among the methods were made.
and skew angle. The finite-element analysis (FEA) Leem [20] performed Laboratory experiments on
results for skewed bridges were compared to the full-size girder specimens to evaluate possible
reference straight bridges as well as the American deterioration in shear capacity under repeated
Association for State Highway and Transportation loading. Specimen variables included: T and
Officials (AASHTO) Standard Specifications and inverted-T configurations, stirrup spacing, and
LRFD procedures. A total of 96 case study bridges flexural reinforcing details. Test results indicated
were analyzed and subjected to AASHTO HS-20 bond deterioration increased diagonal crack
design trucks positioned close to one edge on each displacements, and analysis methods to predict the
bridge to produce maximum bending in the slab. shear capacity of diagonally cracked reinforced
The AASHTO Standard Specifications procedure concrete girders subjected to high-cycle fatigue
gave similar results to the FEA maximum damage are provided. The AASHTO-LRFD shear
longitudinal bending moment for a skew angle less provides conservatively predicted shear capacity for
than or equal to 20°. As the skew angle increased, the fatigued specimens without stirrup fractures ,
AASHTO Standard Specifications overestimated and shear capacity predictions from computer
the maximum moment by 20% for 30°, 50% for 40°, analysis program response 2000 were very well
and 100% for 50°. correlated with experimental result for fatigued test
The AASHTO LRFD Design Specifications specimens when the input concrete tensile strength
procedure overestimated the FEA maximum was reduced to nearly zero.
longitudinal bending moment. This overestimate Samman [21] studied continuous curved composite
increased with the increase in the skew angle, and multiple-box girder bridges, using the finite-element
decreased when the number of lanes increased; method, to evaluate their natural frequencies and
AASHTO LRFD overestimated the longitudinal mode shapes. Experimental tests are conducted on
bending moment by up to 40% for skew angles less two continuous twin-box girder bridge models of
than 30° and reaching 50% for 50°. The ratio different curvatures to verify and substantiate the
between the three-dimensional FEA longitudinal finite-element model. Empirical expressions are
moments for skewed and straight bridges was deduced from these results to evaluate the
almost one for bridges with skew angle less than fundamental frequency for such bridges.
20°. This ratio decreased to 0.75 for bridges with The parameters considered herein are the span
skew angles between 30 and 40°, and further length, number of lanes, number of boxes, span-to-
decreased to 0.5 as the skew angle of the bridge radius of curvature ratio, span-to-depth ratio, end-
increased to 50°. diaphragm thickness, number of cross bracings, and
This decrease in the longitudinal moment ratio is number of spans.
offset by an increase of up to 75% in the maximum Zhang [22] conducted a theoretical study on the
transverse moment ratio as the skew angle increases determination of the allowable differential
from 0 to 50°. The ratio between the FEA maximum settlement. The maximum transient vibration value
live-load deflection for skewed bridges and straight of the acceleration is recommended as the vibration
bridges decreases in a pattern consistent with that of comfort index. A dynamic response analysis to
the longitudinal moment. This ratio decreased from Man-Vehicle-Road system is carried out by means
one for skew angles less than 10° to 0.6 for skew of the Laplace transform in which the vehicle is
angles between 40 and 50°. modeled as a three-degree-of-freedom system, and
Yehia [19] studied the different nondestructive the bridge approaches with and without approach-
evaluation techniques used in the assessment of relative slopes are simplified as broken lines model
concrete bridge deck conditions. An experimental and step model, respectively. Parametric studies are
investigation to evaluate the ability of infrared carried out to find out what kind of effects some
thermography, impact echo, and ground penetrating parameters such as the driving direction and running
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speed may have on the allowable differential to produce maximum positive or negative moments.
settlement. The type of loading used, whether lane loading or
Finally, the method and computational procedures truck loading and whether the spans are simple or
to determine the allowable differential settlement continuous, shall be the loading which produces the
are presented. maximum stress. Where maximum stresses are
2 Live Loads produced in any member by loading any member of
traffic lanes simultaneously, the following
The live loads of the AASHTO specifications [2]
percentage of the resultant live load stress shall be
consist of standards trucks or of lane loads which
in view of improbable coincident maximum loading:
consist of a uniformly distributed load and a

concentrated load. The maximum truck loading is
HS20 and it is designed by the indication HS20-44. 100% One or two lanes
This loading consists of a 3-axle vehicle with a total
weight of 32.5 tons. Axle loads and dimensions of 90% Three lanes
HS20-44 vehicle are shown in Fig.1. The lane
loading as shown in Fig.2 is composed of 75% Four lanes or more
distributed load plus concentrated loads.
The concentrated loads are composed of one load
for bending and the other for shear as shown in
Fig.2 These concentrated loads are located in certain
positions, in addition to the distributed loads to give
the maximum shear stresses.
Live load stresses due to truck loading (or
equivalent lane loading) are increased to allow for
vibration and the sudden application of the load.
The increase is computed by the formula:
15.24
I= <0.3 (1)
L+38
Where:
I = the impact fraction of live load stress.
L = the loaded length in meters.
To encounter the unexpected traffic loadings, these
loadings are multiplied by 1.5. Study of axle weight
in Jordan is illustrated in 56, 57.
Standard truck or lane loading is assumed to occupy
a loaded width of 10ft (3.048m). These loads shall

be placed in 12ft (3.65m) wide design traffic lanes
W = Combined weight on the first two axles
spaced across the entire bridge roadway width in
which is the same as for the corresponding H
number and position required to produce the
truck.
maximum stress in the member under consideration.
V = Variable spacing – 14 feet to 30 feet
The uniform and concentrated load of a lane loading
inclusive. Spacing to be used is that which
shall be considered to be uniformly distributed over
produces maximum stress.
a 10ft (3.048m) width on a line normal to the

centerline of the lane. In computing stresses, each
10ft lane loading or single standard truck shall be
considered as a unit that can occupy position within
its individual traffic lane, so as to a produce
maximum stress. Fractional load lane widths or
fractional trucks shall not be used.
For continuous spans, only one standard H or HS
truck per lane shall be considered and placed so as
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4 Grillage Analysis :
This methods is usually used for analysis of bridges
were based on the consideration of the bridge deck
as an elastic continuum in the form of an
orthogonally anisotropic plate. Using the stiffness
method of structural analysis, it became possible to
analyze the bridge deck structure as an assembly of
elastic structural members connected together at
discrete nodes. There are four distinct techniques
which have been found useful be bridge engineers:
grillage and space frame analysis, folded plate
Fig.1 AASHTO TRUCK LOADING HS 20-44
method, finite element method and finite strip

method.

The grillage analogy method bridge involves a plane
grillage of discrete interconnected beams.
The grillage analogy has become popular because of

the following reasons:
Fig.2 LANE LOADING HS 20-44
− It can be used in cases where the bridges

exhibits complicating features such as a heavy
3 Distribution Of Live Loads:
skew, edge stiffening and deep hunches over
When a concentrated load is placed on a bridge
supports.
deck, the load is distributed over an area larger than
− The representation of a bridge as a grillage is
the actual contact area. Hence, a larger portion of
ideally suited to carrying out the necessary
the deck will assist in resisting the load. This is
calculations associated with analysis and design
described as load distribution and has been taken
on a digital computer.
into account in AASHTO specification [2].
− The grillage representation is conductive to
According to AASHTO specification [2], no
giving the designer an idea bout the structure
longitudinal distribution of the wheel loads shall be
behavior of the bridge and the manner in which
assumed in calculating bending moments in
bridge loading is distributed and eventually
longitudinal beams. The lateral distribution shall be
taken to the supports [23].
determined as follows:
It is a horizontal grid consisting of the main
1. The distribution of wheel loads for each
(longitudinal) and cross (transverse) girders are
interior longitudinal beam of a reinforced
orthogonally intersecting and is subjected to vertical
concrete box grinder bridge designed for
loads only. Each of the longitudinal girders having
two or more traffic lanes should be taken as
flexural stiffness (EI), torsional stiffness (GJ) and
S/2. 134, where S is the spacing of
length (L). The longitudinals girder are spaced a
longitudinal beams in meters.
distance (h) apart and are interconnected by a
2. The distribution of wheel loads for exterior
number of equally spaced transverse beams each of
roadway beams of a reinforced concrete box
which has flexural stiffness (EI ) and torisonal
girder bridge should be taken as We/2. 134. T
stiffness (GJ ) [23].
The width (We) to be used in determining T
In a grillage analysis, the elements of a grid are
the wheel line distribution to the exterior
assumed to be rigidly connected, so that the original
girder is the top slab width as measured
angles between elements connected together at a
from the midpoint between girders to the
node remain unchanged. Both torisonal and bending
outside edge of the slab. In no case shall be
moment continuity then exist at the node point of a
exterior beam have less carrying capacity
grid [24]. Each beam element is assumed to have
than an interior beam.
three degree of freedom at each end. These consist
The common methods used in bridge design is
of two rotations and a transverse displacement as
either grillage or finite element or AASHTO.
shown in Figure.3.

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Where:
X − X Z −Z
C =Cosθ= 1 2 ,S =Sinθ= 1 2 (4)
Fig.3 GRID ELEMENT WITH NODAL DEGREES L L
OF FREEDOM AND NODAL FORCES. Where L is length of the element from node i to
node j.
Using the direct stiffness method, we can obtain the
stiffness matrix for an orthogonal grillage members 5 Idealization of Deck:
from the following equations: As a first step in a grillage analysis, the continuum
{F}=[K]* [Δ] (2) of the deck must be idealized into a series of a
discrete members. The idealization was performed

using the following guidelines:
In matrix form as follows:
f1y : shear force in y-axis at node 1.
m1x : torsion moment about x-axis at node 1.
m1z : bending moment about z-axis at node 1.
f2y : shear force in y-axis at node 2.
m2x : torsion moment about x-axis at node 2.
m2z : bending moment about z-axis at node 2.
d1y : displacement in y-axis at node 1.

Where: φ1x : torsion rotation about z-axis at node 1.

φ1z : bending rotation about z-axis at node 1.

The global stiffness matrix KG for members which d2y : displacement in y-axis at node 2.
do not intersect at right angles can be found from
φ2x : torsion rotation about x-axis at node 2.
the following equation:
φ2z : bending rotation about z-axis at node 2.
[K ] = [λ ]T * [K] * [λ ] (3)
G G G
Where: Table 1: Torsional coefficient (K) for
[K]: element stiffness matrix. rectangular sections

[λG]: 6 by 6 transformation matrix.
The transformation matrix relating local to global
degree of freedom is given by:
a | b 1 1.2 1.5 1.75 2.0 2.25

K 0.141 0.166 0.196 0.213 0.229 0.24
a | b 2.5 3.0 4.0 5.0
K 0.249 0.263 0.281 0.291
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Where:
1. The logical choice of longitudinal grillage Iz : second moment of area of the transverse
beams is to have them coincident with the members.
physical (actual) beams. The centre lines of the Jz : torisonal inertia of the transverse members.
grillage beams are coincident with the center Lx : spacing of the transverse members.
lines of the physical beam, as a general rule, Ec : modulus of elasticity of the deck slab
one grillage beam does not replace more than material.
two physical beams. When dealing with Gc : shear modulus of the deck slab material.
member properties, the members are given the
properties of the girders plus associated It is not usually obvious how to define the
portions of the slab, according to the following positions of the transverse grillage beam except
equations [23]. in the case where diaphragms exist within the
span, in which case the idealized beams should
be in the same positions as the physical beams.
I = I (5) For decks with only top slab in the transverse
X comp
L t3 direction, suggested that transverse grillage
J = J + y (6) beams are substituted at [23].
x g 6 a. The abutments (where in most cases
there will be diaphram).
Where: b. Intermediately such that the ratio of
Ix : second moment of area of the longitudinal spacing of transverse grillage beams to
members. the spacing of longitudinal grillage
I : second moment of area of the composite beams is approximately 1.5:1.
comp
girder and slab c. The total number of transversals is odd
section. and not less than five. Two important
Jx :torsional intertia of the longitudinal points should be emphasized regarding:
members. 1. All calculations are based on gross-
Jg : torsional intertia of the girder section. sections, neglecting reinforcement
Ly : spacing of the longitudinal members. and cracking of the concrete.
t : thickness of the deck slab. 2. Poisson's ratio effects where
neglected because they are limited
2. The torsional intertias for an I or T-sections can to the deck slab.
be calculated using the following equation:
3 6 Finite Element Method:
J = Kab (7)
The finite element method is a well known tool for
Where a is greater than b and K coefficient can
the solution of complicated structural engineering
be obtained from Table 1.
problems, as it is capable of accommodating many
3. The transverse deck slab is consist of a number
complexities in the solution. In this method, the
of transverse strips, and each strip is replaced
actual continuum is replaced by an equivalent
by a beam. In which case their properties are
idealized structure composed of discrete elements,
calculated in the same way as for slab bridge
referred to as finite elements, connected together at
[23]. Thus:
a number of nodes.

The finite element method was first applied to

problems of plane stress, using triangular and
L t3 (8) rectangular element. The method has since been
I = x
z 12 extended and we can now use triangular and

rectangular elements in plate bending, tetrahedron
E L t3
J = c x (9) and hexahedron in three-dimensional stress analysis,
z G 12 and curved elements in singly or doubly curved
c
shell problems. Thus the finite element method may

be seen to be very general in application and it is
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sometimes the only valid form of analysis for 7 Idealization of Structure:
difficult deck problems.
A mathematical model of the structure is formulated
The finite element method is a numerical method
in which it is represented as an assemblage of
with powerful technique for solution of complicated
discrete elements. Each element has its own shape,
structural engineering problems. It most accurately
thickness, material properties and even external load
predicted the bridge behavior under the truck axle
effects.
loading [26].
These elements can be in the form of line elements,
The finite element method involves subdividing the
two dimensional plane stress or plane strain
actual structure into a suitable number of sub-
elements, flexural, plate, and/or shell elements, ax
regions that are called finite elements. These
symmetric elements, general three dimensional solid
elements can be in the form of line elements, two
elements. Figure.4 show some frequently used
dimensional elements and three dimensional
elements [26].
elements to represent the structure.
The intersection between the elements are called
nodal points in one dimensional problems where in
two and three dimensional problems are called
nodal lines 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 [24].
Displacements are the most commonly used nodal
variables, with most general purpose programs
limiting their nodal degree of freedom to just

displacements. A number of displacement functions Fig.4 ELEMENT MODELS.
such as polynomials and trigonometric series can be
assumed, especially polynomials because of the ease The model of elements depends on the geometry of
and simplification they provide in the finite element the structure, the local features such as stress
formulation. concentration and the convergence properties of the
To develop the element matrix, it is much easier to element. For slab deck, the element model may be
apply a work or energy method. The principle of taken as a triangular or quadrilateral plate element.
virtual work, the principle of minimum potential 8 Generation of Element Stiffness
energy and castigliano's theorem are methods
Matrix :
frequently used for the purpose of derivation of
For an element, a stiffness matrix K is a matrix such
element equation.
that f = kd, where k relates local coordinate (x, y, z)
The finite element method has a number of
nodal displacements d to local forces f of a single
advantages, they include the ability to [25]:
element.
− Model irregularly shaped bodies and
The stiffness matrix k and the local forces matrix f
composed of several different materials.
are evaluated based on the virtual work principle
− Handle general load condition and unlimited
according to the following equation:
numbers and kinds of boundary conditions.

− Include dynamic effects.
K = ∫BTEBdv (10)
− Handle nonlinear behavior existing with

large deformation and non-linear materials. F = ∫NTTds +∫NTXdv + NTP (11)

s v
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element, the following points should be considered:
Where: − The program automatically generates a fifth
node (center node) at the element center.
To obtain the global stiffness matrix for the whole − Element nodes should be specified either
structure, the element stiffness matrix K for each clock wise or counter clock wise.
element can be added together using a method of − The precise orientation of local coordinates
superposition called the direct stiffness method is determined as follows:
whose basis is nodal force equilibrium. Implicit in a. Designate the mid points of the four
the direct stiffness method is the concept of elements edges IJ, JK, KL, LI by M, N,
continuity, or compatibility, which requires that the O, P respectively.
structure remain together and that no tears or b. The vector pointing from p to N is
overlaps occur anywhere in the structure. defined to be the local x-axis.
The global stiffness matrix is singular, that is, its c. The cross product of vectors PN and
determinate vanishes, and there can be an infinite MO defines the local z-axis,
number of possible solutions. To remove this 1. i. e., Z = PN x MO
singularity, it must invoke certain boundary d. The cross product of vectors Z and X
conditions. When introducing the boundary defines the local Y-axis,
conditions, the displacement or stresses usually are i.e. Y = Z x X
called primary unknowns or primary quantities can Figure.5 shows the local coordinates
be found at all nodes of the structure. With these orientation.
unknowns established, the quantities such as strains,
− Element aspect ratio should be on the order
shear forces can be evaluated.
1:1, and preferably less than 4:1.

− The angles between two adjacent element
9 Analysis of Finite Element Using sides should not be much larger than 90 º
STADD III: and never larger than 180°.
As excellent summary of the scope of finite element
using STADD III [27] can be done by taking into
consideration the following three basic phases:

9.1 Theoretical Basis:
The principle of the STADD III [27] plate finite
element is based on hybrid finite element
formulations. The distinguishing features of this
finite element are as follows:
− These elements are available as triangles
and quadrilaterals, with corner nodes only,
with each node having six degree of
freedom.
Fig.5 ELEMENT LOCAL AXIS SYSTEM.
− These elements may be connected to plan or

10 Structural Idealization
B : ∂N, where N: shape function matrix.
The bridge is composed of 2 continuous spans each
E : modulus of elasticity matrix.
17.3m with 20cm top and bottom slabs and 50cm
T : surface force matrix. the height of the web. Plane of the bridge is shown
X : body force matrix. in Fig.6 Transverse section is shown in Fig.7.
Longitudinal section is shown in Fig.8. The width of
P : nodal load matrix.
top slab is 16m and the width of bottom slab is
space frames.
15.0m. One Fixed electrometric bearing pad support

at the middle pier and two guided electrometric
9.2 Geometry Modeling:
bearing pads supports at the abutment are provided.
When the STADD III [27] generating the finite
ISSN: 1790-0832 630 Issue 4, Volume 6, April 2009