CHAPTER IV
RESULT AND ANALYSIS
4.1
Introduction
The main objective of the finite element modelling was to determine the
serviceability of the suspension bridge under severe wind loads. However, the
validity of the model can be a major concern for obtaining a reliable result. Variables
and parameters affecting the results should be taken into consideration in modelling
and analysis of the model. Therefore, it is important to determine the reliability of the
model.
There are fundamental requirements that must be observed in order to verify
reliability of a linear elastic model. The equilibrium of forces, compatibility of
displacements and the law of material behaviour conditions must be satisfied to
generate a reliable model. Equilibrium of forces requires that the internal forces
balance the external force. Compatibility of displacements requires that the deformed
structure comply to the loads applied. The law of material behaviour requires the
deformation of the model in accordance to the material behaviour.
54
The stress distribution or stress contour of the bridge subjected by severe
wind loads and heavy live loads were obtained from the analysis. The critical regions
of stress concentration were also identified. The stress obtained from the analyses
were used to determined the serviceability of the bridge section while the high stress
concentration regions will be used to discover potential fatigue point at the structure
of the bridge section. Both these criteria are major determinant for serviceability of
the bridge.
4.2
Static Check
Static check was done in order to verify the reliability of the results obtained
from the analysis. It is essential to determine the correctness of the bridge section
model constructed. The total reactions forces at restraints should be identical to the
total applied loads of the model in order to satisfy the equilibrium of forces, which
mean that the external loads applied are equal to the reacting internal forces. It means
that the bridge section model is in equilibrium states.
4.2.1
Internal Forces
Internal forces can be obtained at the restraints as reaction forces. Reaction
forces are the constraints forces at the supports or boundary conditions. Reaction
forces can be acquired directly from finite elements software, MSC.NASTRAN. The
reactions forces at the supports obtained from the analyses for both load
combinations can be summarised into Table 4.1 and Table 4.2 below.
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Table 4.1: Reaction Forces at Restraints for Load Combination 1
Reactions, F (N)
Nodes
X
Y
Z
Resultant
1263
-2069.62
1336.85
881622
881326
1265
-2073.38
-1339.62
883907
883911
1266
2071.91
1338.85
880837
880840
1267
2071.08
-1338.2
883931
883935
1268
0
0.95771
0
0.95771
1270
0
1.15488
0
1.15488
1271
1.56275
0
0
1.56275
1272
-0.25949
0
0
0.25949
1274
0
0
1.58658
1.58658
1280
0
0
1.59342
1.59342
1282
0
0
-0.25917
0.25917
1283
-1.56116
0
0
1.56116
Sum
-0.2679
-0.00741
3530000
3530021
It can be observed that the sum of reactions in X and Y directions are
insignificant compared to the reactions forces at the Z directions. It is not vary from
direction of the load applied where only vertical loads applied in Z direction. The
conditions also applied for the load combination 2 where reactions forces are noticed
at Y and Z directions.
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Table 4.2: Reaction Forces at Restraints for Load Combination 2
Nodes
Reactions, F (N)
X
Y
Z
Resultant
1263
-1777.75
-34065.7
887339
887995
1265
-2955.25
-36788.5
905150
905902
1266
1782.91
-34059.1
886848
887503
1267
2950.08
-36782
905165
905917
1268
0
-27.2908
0
27.2908
1270
0
-26.9988
0
26.9988
1271
1.29764
0
0
1.29764
1272
0
0
-0.27526
0.27526
1274
0
0
1.5938
1.5938
1280
0
0
1.63199
1.63199
1282
0
0
-0.24966
0.24966
1283
-1.29397
0
0
1.29397
Sum
-0.00633
-141750
3584505
3587378
4.2.2
External Loads
External loads are the loads applied to the model. The loads applied to the
bridge section model are nodal points loads. There are two loads combinations were
applied to the finite elements model. The details of the loads applied to the bridge
section model can be summarised as follow.
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For load combination 1, all loads were applied in negative Z direction. The
loads applied are:

Total Self weight = 1290 kN

Total Monorail Live Point Load = 560 kN

Total Monorail Live Uniform Distributed Load = 1680 kN

Total Loads in Z direction = 3530 kN
For load combination 2, loads were applied in negative Z direction and
positive Y direction. The loads applied are:
 Negative Z Direction

Total Self weight = 1290 kN

Total Monorail Live Point Load = 560 kN

Total Monorail Live Uniform Distributed Load = 1680 kN

Total Vertical Wind Forces = 54.5 kN

Total Loads in Z direction = 3854.5 kN
 Positive Y Direction

4.2.3
Total Horizontal Wind Forces = 141.75 kN
Equilibrium Checking
The criterion of equilibrium of forces of the model was determined by
comparing the external loads applied with the resisting reaction forces at the
restraints. The results obtained from the linear elastic analysis, as shown in Table 4.1
and Table 4.2, have indicated that the sum of the applied forces and the resisting
forces at the constraints or supports in each direction for both load cases is equal to
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zero. This means that the model analysed is in the equilibrium states and the results
obtained are reliable.
4.3
Stress Distribution
The stress distribution or stress contour of the bridge section subjected to
severe wind loads and heavy live loads were obtained from the analyses. The stresses
from the analysis provides significant results for checking the service stress of the
bridge section under severe loads condition. The serviceability of bridge section was
checked with the limiting stress of the materials used for the structure.
The stress distribution or stress contour indicated the region where the high
stress concentration area are occurred. The high stress regions were significant in
providing an approximation of fatigue failure zone of the bridge section.
Identifications of these regions is essentials in order to carry out periodically or risk
based inspections. This method can be the preliminary analysis to determine the hot
spots for inspections which may reduced the costs and time.
There were mainly two types of stress from the analysis which are major
principal stress and minor principal stress or tensile stress and compressive stress.
These two stresses are essentials in recognising the potential fatigue regions as well
as in determination of serviceability of the steel stiffened suspension bridge section.
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4.3.1
Tensile Stress
Tensile stress distribution were obtained from the analyses for both load
cases. The tensile stress is noticed at the bottom part of the girders as shown in
Figure 4.1. High tensile stress concentration regions was discover at the bottom
flange of middle section of the girders, where there is a connection between the
girders and a internal beam. The high tensile stress regions is shown in Figure 4.2.
For load combination 1, where the bridge subjected merely to vertical loads,
the maximum major principal stress or maximum tensile stress is occurred at the high
stress concentration region as shown in Figure 4.2 with a value of 53.687 N/mm2.
Meanwhile for the load combination 2, where the bridge section model with the
addition of horizontal wind load, the maximum major principal stress or maximum
tensile stress value acquired is 59.418 N/mm2.
The values of tensile stress for both load cases are less than the limiting
yielding stress of the steel, y, 355 N/mm2. The service stresses are considered low
even though after the steel is reduced by a factor of safety for the material, 2.0.
Therefore, the steel bridge section is safe from service loads failure.
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Figure 4.1: Tensile Stress Distribution of the Model under Load Combination 1
Figure 4.2: Bottom View of the Model with Tensile Stress Concentration Region
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4.3.2
Compressive Stress
Compressive stress distribution were obtained from the analyses for both load
cases. The compressive stress is noticed at the upper part of the girders as shown in
Figure 4.3. High compressive stress concentration regions was discover at the upper
flange of middle section of the girders, where there is a connection between the
girders and a internal beam. The high compressive stress regions is shown in Figure
4.4.
For load combination 1, where the bridge subjected merely to vertical loads,
the minimum minor principal stress or maximum compressive stress is occurred at
the high stress concentration region as shown in Figure 4.4 with a value of –53.325
N/mm2. Meanwhile for the load combination 2, where the bridge section model was
subjected to the addition of horizontal wind load, the minimum minor principal stress
or maximum compressive stress value acquired is –-56.083 N/mm2.
The values of compressive stress for both load cases are less than the limiting
yielding stress of the steel, y, 355 N/mm2. The service compressive stresses are
considered low even though after the steel is reduced by a factor of safety for the
material, 2.0. Therefore, the steel bridge section is safe from service loads failure.
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Figure 4.3: Compressive Stress Distribution of the Model under Load Combination 1
Figure 4.4: Plan View of the Model with Compressive Stress Concentration Region
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4.4
Deflection
Displacement values were obtained from the analyses for both load cases.
The maximum displacement is noticed at the middle span of the bridge section. The
deformation due to subjected loads are shown in Figure 4.5 and Figure 4.7
respectively.
For load combination 1, where the bridge subjected merely to vertical loads,
the maximum displacement in Z direction is –5.28 mm. The deformed shape pf
bridge section model due to vertical loads is shown in Figure 4.6. Meanwhile for
load combination 2, where the bridge section model was subjected to the addition of
horizontal loads, the maximum displacement in Y direction and Z direction is 14.55
mm and –-5.74 mm respectively.
The values of displacement in all directions for both load cases are less than
the limiting deflection, L/800, where L is the span of the girder. Therefore, the steel
bridge section is considered safe for serviceability limit states.
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Figure 4.5: Deformed Shape of the Model under Load Combination 1
Figure 4.6: Deflection in Longitudinal Elevation View of the Model Under Load
Combination 1
65
Figure 4.7: Deformed Shape of the Model under Load Combination 2
Figure 4.8: Deflection in Plan View of the Model Under Load Combination 2
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4.5
Discussions
The results obtained from the linear elastic analysis conducted in the study
can be summarised into Table 4.3 below.
Table 4.3: Maximum Stress and Maximum Displacement for the Study
Maximum Stress, 
Maximum Displacement, 
(N/mm2)
(mm)
Load
Combination
Direction
Direction
Direction
X
Y
Z
-53.325
0.862
0.704
-5.280
-56.084
0.950
14.549
-5.740
Tensile
Compressive
Load Case 1
53.687
Load Case 2
59.418
The maximum stress values obtained from the analysis for both load cases
were found far less than the yielding stress of the structural steel used in the design
of the suspension bridge. For serviceability consideration, this result shows that the
steel stiffened suspension bridge section is within the allowable stress under service
loads.
The maximum deflection or displacement () of the girders, in all directions,
under combination of service loads was found much lesser than the limiting value as
stated by AASHTO. This result indicated that the bridge section is within the
allowable deflection for serviceability limit states consideration.
From the linear elastic analysis conducted, the maximum stress regions were
identified for both load cases. It is obviously indicated by the previously figures that
the stress concentration regions were in the same regions for both load cases. This
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result means that the stress concentration either compressive or tensile occurred at
the middle span of the bridge section.
For long terms considerations, the fluctuating stresses generated by repetitive
wind loads and cyclic monorail live loads causing stress reversals in the steel plate
girders. This phenomenon always leads to reduce material resistance strength, which
is defined as fatigue failure. The fatigue limit is the maximum stress that can be
repeated indefinitely without causing fatigue failure when applying fluctuating loads.
Tests [AISC Mkt, 1986] performed on a large number of different steels indicate that
the fatigue limit of steel is about 50 percent of its tensile strength and fatigue strength
decreases with increasing number of loading cycles.
Although the stresses under combination of load cases are not in the range of
fatigue failure, however, the periodical inspection of the structure should be carried
out throughout the design life of the bridge to avoid the fatigue failure due to time,
fluctuating stresses, stress corrosion and other environmental effects. This is because
the fatigue strength of the steel is reduced with the respect of number of cycles of
loads.
Attention should be paid to areas that were identified as high stress
concentration region and regions with high stress range because these are the regions
highly prone to fatigue failure. High stress region identified through finite element
modelling and analysis reduced the cost for detail monitoring of the behaviour of
structures under cyclic wind and live loads. This is because the monitoring procedure
can be focused at the predicted fatigue failure region and reduced the frequency of
inspection for other parts of the structure.