DEVELOPMENT OF DEFLECTION MONITORING SYSTEM … · DEVELOPMENT OF DEFLECTION MONITORING SYSTEM FOR...
Transcript of DEVELOPMENT OF DEFLECTION MONITORING SYSTEM … · DEVELOPMENT OF DEFLECTION MONITORING SYSTEM FOR...
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DEVELOPMENT OF DEFLECTION MONITORING
SYSTEM FOR THE KEALAKAHA STREAM BRIDGE
Nathan D. Powelson
Ian N. Robertson
and
Gaur P. Johnson
Research Report UHM/CEE/10-05
December 2010
Prepared in cooperation with the:
State of Hawaii
Department of Transportation Highways Division
and
U.S. Department of Transportation Federal Highway Administration
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ABSTRACT
The Kealakaha Stream Bridge is the first base isolated structure in the state of Hawaii.
As such, it represents an ideal opportunity to monitor the performance of a major bridge structure
equipped with base isolation in a region that experiences numerous earthquakes. Although the
bridge is not located in the most seismic portion of the island, it will experience significant ground
motion during major events elsewhere on the island.
An extensive seismic monitoring system is planned for the bridge structures. This system
will include accelerometers, displacement transducers and rotation sensors. The rotation sensors
provide an opportunity to monitor the deflected shape of the bridge superstructure during ambient
traffic conditions, and to record any change in deflected shape before and after future seismic
events. This report investigates the use of rotation sensors for deflection monitoring of a long-
span bridge structure.
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ACKNOWLEDGEMENTS
This report is based on a Master’s Thesis prepared by Nathan Powelson under the
direction of Ian Robertson and Gaur Johnson. The authors wish to thank Drs. Ronald Riggs and
David Ma for their review of this report.
This project was funded by the Federal Highway Administration and Hawaii Department
of Transportation Research Branch. This support is gratefully acknowledged. The contents of this
report reflect the views of the authors, who are responsible for the facts and accuracy of the data
presented herein. The contents do not necessarily reflect the official views or policies of the State
of Hawaii, Department of Transportation or the Federal Highway Administration. This report does
not constitute a standard, specification or regulation.
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TABLE OF CONTENTS
ABSTRACT ......................................................................................................................... iii
TABLE OF CONTENTS ...................................................................................................... v
LIST OF FIGURES ............................................................................................................. vii
LIST OF TABLES ............................................................................................................... ix
Chapter 1: INTRODUCTION ............................................................................................ 1
1.1 Background .......................................................................................................... 1
1.2 Project Scope ....................................................................................................... 2
1.3 Kealakaha Bridge Description .............................................................................. 2
Chapter 2: LITERATURE REVIEW .................................................................................. 7
2.1 Seismic Instrumentation ....................................................................................... 7
2.2 Deflection ............................................................................................................. 7
2.2.1 Curvature Measurements to Calculate Deflection ........................................... 7
2.2.2 Rotation Measurements to Calculate Deflection .............................................. 8
2.2.3 Using Base Shapes to Calculate Deflection .................................................... 8
2.3 Base Isolation ....................................................................................................... 9
2.4 Inclinometer ........................................................................................................ 11
Chapter 3: SAP2000 FRAME ELEMENT MODEL: STRAIGHT BASE LINE ................ 13
3.1 Development of SAP2000 straight base line model........................................... 13
3.2 Frame Elements ................................................................................................. 14
3.3 Analysis .............................................................................................................. 18
Chapter 4: SAP2000 AREA ELEMENT MODEL: STRAIGHT BASE LINE ................... 21
4.1 Development of SAP2000 straight base line area model .................................. 21
4.2 Area Elements .................................................................................................... 22
4.3 Analysis .............................................................................................................. 24
Chapter 5: SAP2000 AREA ELEMENT MODEL: HORIZONTALLY & VERTICALLY
CURVED BASELINE 25
5.1 Development of SAP2000 area model ............................................................... 25
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5.2 Vertical and Horizontal Geometry ...................................................................... 25
5.3 Comparison with Area Element Model: Straight Base Line (SBL) ..................... 27
Chapter 6: Using Rotation to Calculate Displacement (R2D) ........................................ 29
6.1 Theory ................................................................................................................ 29
6.2 Implementation ................................................................................................... 31
6.2.1 Assumptions ................................................................................................... 31
6.2.2 Model Descretization ...................................................................................... 31
6.2.3 Model Analysis ............................................................................................... 32
6.2.4 Final Model Selection ..................................................................................... 37
Chapter 7: BASE SHAPES ............................................................................................ 39
7.1 Theory ................................................................................................................ 39
7.2 Implementation ................................................................................................... 39
Chapter 8: R2D ANALYSIS ........................................................................................... 49
8.1 Preliminary Locations ......................................................................................... 49
8.2 Number of Inclinometers .................................................................................... 51
8.3 Error Analysis ..................................................................................................... 54
Chapter 9: BASE SHAPES ANALYSIS ......................................................................... 57
9.1 Location of Inclinometers ................................................................................... 57
9.2 Other Deflected Combinations ........................................................................... 60
9.3 Error Analysis ..................................................................................................... 64
Chapter 10: CONCLUSIONS AND RECOMMENDATIONS .......................................... 69
APPENDIX A: R2D GRAPHS ........................................................................................... 71
APPENDIX B: BASE SHAPE GRAPHS ........................................................................... 79
Load Combinations A-I .................................................................................................. 79
Other Load Combinations ............................................................................................. 86
REFERENCES .................................................................................................................. 95
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LIST OF FIGURES
Figure 1.1-1 Location of Kealakaha Bridge ........................................................................ 1
Figure 1.3-1 Normal Deck Section at Drop-In Girder (Looking Down station) (Not to
Scale) [KSF, 2006] ........................................................................................................................... 4
Figure 1.3-2 Normal Deck Section at Pier (Looking Down station) (Not to Scale) [KSF,
2006] ................................................................................................................................................ 5
Figure 1.3-3 Plan and Section along Base Line ................................................................. 6
Figure 2.2-1: North Mentue Bridge, Position of Inclinometers along Bridge [Burdet, 2000]
......................................................................................................................................................... 9
Figure 2.3-1 Typical Friction Bearing Section [Earthquake Protection Systems, 2003] ..... 9
Figure 2.3-2 Modeling the Friction Pendulum Bearings .................................................... 10
Figure 2.4-1 LSPO ±1º ...................................................................................................... 11
Figure 3.1-1 SAP Area Model SBL ................................................................................... 14
Figure 3.2-1 Girder Section used to calculated properties ............................................... 15
Figure 3.2-2 Box Girder Section used to calculate section properties .............................. 15
Figure 3.2-3 Division of elements for SAP2000 ................................................................ 16
Figure 4.1-1 SAP Area model straight base line ............................................................... 22
Figure 5.1-1 SAP Area model straight base line ............................................................... 26
Figure 5.2-1 Profile grade line at centerline of road .......................................................... 26
Figure 5.2-2 Segment Layout of Kealakaha Bridge .......................................................... 27
Figure 6.1-1 Varying Moment of Inertia along the Length of the Bridge ........................... 30
Figure 6.2-1 Location of Inclinometers and Formation of R2D Models 1 – 3 ................... 32
Figure 6.2-2 Plan view of the placement of 9 point load cases ........................................ 33
Figure 6.2-3 HS-20 Truck ................................................................................................. 33
Figure 6.2-4 R2D Model 1 with SAP output ...................................................................... 35
Figure 6.2-5 R2D Model 2 with SAP output ...................................................................... 35
Figure 6.2-6 R2D Model 3 with SAP output ...................................................................... 36
Figure 6.2-7 Load Case C with All models ....................................................................... 36
Figure 6.2-8 Portions of R2D Models used in final deflection curve ................................. 37
Figure 6.2-9 Load Case C with final Delta ........................................................................ 38
Figure 7.2-1 Applied Temperature Gradient along Bridge ................................................ 40
Figure 7.2-2 Location of potential base shapes ................................................................ 41
Figure 7.2-3 HS-20 Truck ................................................................................................. 42
Figure 7.2-4 Base Shape Deflection with SAP output, Load case A ................................ 46
Figure 7.2-5 Base Shape Deflection with SAP output, Load case H ................................ 47
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Figure 8.1-1 Deflection for Load Case H for 6 Possible Inclinometer Locations .............. 50
Figure 8.1-2 Deflections for Load Case A for 6 Possible Inclinometer Locations ............ 51
Figure 8.2-1 Location of Potential Inclinometer Locations ................................................ 52
Figure 8.3-1 Error vs. SAP2000 Deflection ....................................................................... 56
Figure 8.3-2 Error vs SAP2000 Deflection ........................................................................ 56
Figure 9.1-1 Location of Potential Inclinometer Patterns .................................................. 57
Figure 9.1-2 Load Combination F ..................................................................................... 59
Figure 9.1-3 Load Combination I ...................................................................................... 59
Figure 9.2-1 Load Combination 30 ................................................................................... 62
Figure 9.2-2 Load Combination 38 ................................................................................... 62
Figure 9.2-3 Load Combination 44 ................................................................................... 63
Figure 9.2-4 Load Combination 21 ................................................................................... 63
Figure 9.3-1 Error from inclinometer 2 .............................................................................. 65
Figure 9.3-2 Error from inclinometer 2 .............................................................................. 66
Figure 9.3-3 Error envelope for 72 ft spacing, Load Case B ............................................ 66
Figure 9.3-4 Error envelope for 90 ft spacing, Load case B ............................................. 67
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LIST OF TABLES
Table 3-1 Material Properties of Structural Elements ....................................................... 16
Table 3-2 Section Properties ............................................................................................ 17
Table 3-3 Beam Locations and Weights ........................................................................... 17
Table 3-4 Comparing Section Properties .......................................................................... 19
Table 3-5 Deflections at each span .................................................................................. 19
Table 4-1 Section Properties Calculated in Bridge Modeler ............................................. 23
Table 4-2 Dead Load Comparisons .................................................................................. 24
Table 4-3 Deflections at middle of each span ................................................................... 24
Table 5-1 Vertical Geometry ............................................................................................. 27
Table 5-2 Horizontal Geometry ......................................................................................... 27
Table 5-3 Dead Load Comparisons .................................................................................. 28
Table 5-4 Deflections at middle of each span ................................................................... 28
Table 7-1 Potential base shape combinations .................................................................. 41
Table 7-2 Comparison of Different Load Combinations .................................................... 45
Table 8-1 Results .............................................................................................................. 49
Table 8-2 Deflection at Middle of Center Span ................................................................. 53
Table 8-3 Minimum and Maximum Error on Mauka Girder for 18 inclinometers .............. 54
Table 8-4 Maximum and Maximum Error on Mauka Girder for 14 inclinometers ............. 55
Table 9-1 Displacements at Middle of Span 2 .................................................................. 58
Table 9-2 Displacements at the middle of span 2 ............................................................. 61
Table 9-3 Minimum and Maximum Error on Mauka Girder............................................... 64
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Chapter 1: INTRODUCTION
1.1 Background The Kealakaha Stream Bridge is located on the Hawaii Belt Road on the island of Hawaii
(Figure 1-1). The bridge consists of 3 spans composed of cast in place 5 cell box girders over the
piers and precast drop-in girders. The 3 spans are 180 ft., 360 ft., and 180 ft. The Kealakaha
Stream Bridge was designed having an 1800 ft radius and located on a vertical curve. Base
isolation was utilized to compensate for the high seismic zone in which the bridge is located.
Two friction pendulum bearings were incorporated at each pier and each abutment.
Figure 1.1-1 Location of Kealakaha Bridge
The existing Kealakaha Stream Bridge is seismically deficient and was scheduled to be
replaced back in 2003. The new bridge was originally designed as a 3 span bridge; 180.5 ft,
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360.9 ft, and 180.5 ft. It incorporated a single cell box girder of varying depth and a horizontal
curve with a radius of 1800 ft. The contractor proposed a value engineering change to
incorporate base isolation and a combined precast cast in place superstructure [Fujiwara, 2010].
1.2 Project Scope The Kealakaha Stream Bridge is located in an ideal location for seismic study, due to the
seismic activity of Hawaii Island. This is an ideal opportunity for the first seismic instrumentation
in Hawai’i to be installed on the first base isolated bridge in Hawaii. Future earthquakes will be
monitored to study the response of the bridge and compared with the analytical model, developed
in SAP 2000. The goal of this thesis is to create an instrumentation plan that will make this
possible, along with monitoring the deflection of the bridge from both temperature and traffic
effects.
Accelerometers will be utilized to monitor the acceleration to help recreate the
earthquake experienced by the bridge in SAP 2000. The original instrumentation plan
incorporated accelerometers and fiber optic strain gages. The use of fiber optic strain gages to
monitor dynamic deflection proved problematic and was discarded in favor of using a taught line
system with linear variable differential transformers (LVDT’s) [Johnson, 2007]. The taught line
system works well for static and traffic deflections but does not give accurate readings for seismic
deflections. In addition, the new value engineered design created clearance and other installation
issues for the LVDT system. Inclinometers will be used to capture the deflection just prior and
immediately after a seismic event. They may be able to capture the deflections during the
shaking itself. The inclinometer system will also be able to capture the deflections of the
superstructure due to traffic and temperature effects.
1.3 Kealakaha Bridge Description The Kealakaha Stream Bridge is a 720 foot long concrete bridge that utilizes both cast in
place and precast concrete sections. The bridge is located on a horizontal curve with a radius of
curvature of 1800 feet; the travel way has a superelevation of 6.2%. The bridge is located at the
end of a vertical curve that transitions into a constant 3.46% grade [Fujiwara, 2010].
Two sections are used in this bridge, a cast in place box girder with varying depth from 9
ft 9 in. to 18 ft 0 in (Figure 1.3-2), and a girder section which incorporated using 6 W95PTG
“super girders” and a 9 inch concrete deck (Figure 1.3-1). The W95PTG girder was developed by
the Washington State Department of Transportation in 1996 and is capable of spanning 200 ft
[Fujiwara 2010]. Abutment 1 is the abutment that is closest to Hilo, Station 206+57, and
abutment 2 is closest to Kona, Station 213+77.
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Due to the curves on the road, the largest precast girder section that could be transported
was 50 ft. The girders are numbered 1 – 6 such that the Mauka girder is 6 and the Makai girder
is 1 (Figure 1.3-3). To connect the precast girders together cast in place beams, beam “B”, were
utilized. They span the width of the girders. Beam “A” was the cast in place pour that connected
all 6 precast girders together at Abutments 1 and 2. Beam “C” was used to connect the precast
girders in span’s 1 and 3 to the cast in place box girder and Beam “D” was used to connect the
girder section to the box girder in the middle span. These beams gave significant increase in
stiffness to the girder section and are spaced every 52.5 ft.
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Figure 1.3-1 Normal Deck Section at Drop-In Girder (Looking Down station) (Not to Scale) [KSF, 2006]
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Figure 1.3-2 Normal Deck Section at Pier (Looking Down station) (Not to Scale) [KSF, 2006]
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Figure 1.3-3 Plan and Section along Base Line
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Chapter 2: LITERATURE REVIEW
2.1 Seismic Instrumentation Todd Stephens created a seismic instrumentation plan for the Kealakaha Stream Bridge
back before the geometry of the bridge changed drastically. The bridge went from a single cell
box girder to a multi cell box girder combined with drop in girders and a base isolation system.
His study used information from the California Strong Motion Instrumentation Program, CSMIP to
determine the instrumentation of the Kealakaha Stream Bridge. The original plan called for 41
seismic accelerometers, 4 relative displacement units and fiber optic strain gages for deflection
measurement. This will record the full motion of the structure, including free field motion, pile cap
translation and rotation, deck and abutment accelerations and joint movements; to be analyzed in
the future [Stephens, 1996].
2.2 Deflection In some cases it is very difficult to use conventional methods to obtain bridge deflection
measurements, be it using LVDT’s, photoelectric, or other means. Photoelectric measurements
are very expensive and are easily affected by natural conditions, rain and fog. Methods which
have no need for any static reference required on site is especially helpful for bridges that are
built over water, deep valleys and other locations to be measured easily [Xingmin, 2005].
2.2.1 Curvature Measurements to Calculate Deflection
Applying beam theory, one can determine the deflection of a beam given the curvature—
double integrating the M/EI diagram along the length of the beam. Others have applied this by
measuring the curvature, via multiple strain measurements, through the depth of a beam, at
various locations along its length [Johnson 2006, Aki & Robertson 2005, Fung & Robertson 2003,
Vurpillot 1998, Xingming 2005]. This method relies on accurately assuming the form of a function
which represents the curvature along the length of the beam. This function is then fit to the
measured data to minimize the difference between measured and predicted curvature. This
method was successfully used to determining the deflection of a static beam with known loading
conditions which were consistent with the exact solution.
This was tested on a beam in the laboratory and on a highway bridge. The laboratory
test was on a 6 meter long timber beam with 112X156 mm^2 cross section on 3 supports. The
Lutrive bridges in Switzerland were tested with success, the Lutrive bridges are twin bridges,
each one supporting lanes of the Swiss National Highway RN9. These bridges were cast in place
box girders, with a varying height from 2.5 to 8.5 meters, constructed using the cantilever method
[Vurpillot 1998]. Fung [2003] had relative success, going from strain measurements to calculate
deflection. Three different types of beams were tested with this method; a single span steel tube,
a three span steel tube, and a prestressed T beam. There were discrepancies with the
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prestressed T-beam as the curvature to deflection algorithm was sensitive to concrete cracking.
Aki [2005] discovered that this method of was unsuccessful in monitoring torsion. Johnson [2006]
identified some weaknesses of the formulation: including susceptibility to signal noise and non-
conforming loading conditions.
2.2.2 Rotation Measurements to Calculate Deflection
Another method that others have used to determine deflection of long span bridges
directly measures the beam rotation with inclinometers [Xingmin 2005]. An advantage to using
inclinometers is that it is possible to measure both static and dynamic deflections and [Xingmin,
2005].
There have been multiple tests, both in the lab and in the field using inclinometers to
measure deflection. Two lab tests that have been performed, using a steel beam 6 meters long
with a constant cross section, one is simply supported and the other involves 3 supports. This
method has also been tested on the Taolaizhaou Songhua River Bridge, a multiple span bridge
[Xingmin, 2005]. These tests demonstrate that inclinometers are an economical choice for
monitoring bridge deflection. Again this research applies beam theory as the basis for calculating
deflection from rotation.
2.2.3 Using Base Shapes to Calculate Deflection
Oliver Burdet and Jean-Luc Zanella of Ecole Polytechnique Federale de Lausanne, in
Switzerland, have developed a method of determining the deflection of a bridge using
inclinometers. They noticed that reconstructing the deflected shape of a bridge from its rotation is
rather difficult in practice. It is sensitive to measurement errors, and becomes very complex
mathematically when the moment of inertia, of the bridge cross-section, is not constant. Because
of these errors a different approach was developed. The deflected shapes are constructed by a
linear combination of a series of pre-calculated shapes known as base shapes. The idea is that
structures deflect following smooth curves obtained from actual deflection calculations under
canonical load cases. Each deflected shape respects the bridge properties and can account for
discontinuities, sudden changes in moment of inertia, much better than polynomials [Burdet,
2000].
Burdet and Zanella tested this method of deriving the deflection from the inclinometer
measurements and used it to accurately model the deflection on the Mentue Bridges, Figure 2-1,
on the A1 motorway from Lausanne to Bern [Burdet, 2000]. This bridge has a single box girder
cross section with varying depth, therefore having significant varying moment of inertia. This is
similar to the original Kealakaha design. However, the Kealakaha Bridge final design and
construction is even more complex with varying depth multi cell box girders over the pier support
transitioning to multiple I beam configuration with a constant cross-section at midspan and
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abutment spans. Application of this rotation-based deflection system to this more complex bridge
geometry will demonstrate a better approach is applicable to complex bridge types.
Figure 2.2-1: North Mentue Bridge, Position of Inclinometers along Bridge [Burdet,
2000]
2.3 Base Isolation The base isolation bearings used in the Kealakaha Stream Bridge are manufactured by
Earthquake Protection Systems, Inc. KSF Inc., the Kealakaha Stream Bridge designer, used the
single pendulum friction pendulum system, Figure 2-2 [Earthquake Protection Systems, 2003].
Using base isolation allows the period of the structure to be increased thus decreasing the
earthquake induced horizontal loads transferred from the superstructure to the substructure
[Chopra, 2007]. The friction pendulum bearings were installed in an inverted fashion as shown in
figure 2.3-1 to reduce the potential for dirt and debris accumulating on the sliding surface.
Figure 2.3-1 Typical Friction Bearing Section [Earthquake Protection Systems,
2003]
The following diagram shows how to model friction pendulum bearings. The equations
displayed were used to determine the properties of friction pendulum bearings.
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Figure 2.3-2 Modeling the Friction Pendulum Bearings
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2.4 Inclinometer The inclinometers that are to be used in this project are LSOC ±1º, manufactured by
Shelborne Sensors (Figure 2.5-1). This model is a gravity referenced servo sensor. These
inclinometers have suggested applications of bore-hole mapping, dam and rock shifts, seismic
and civil engineering studies, and other applications where precise measurements are required.
According to the manufacturer this sensor has a sensitivity of 0.1 arc second resolution,
28*10-6 degrees. This sensor was tested in the Structures Lab at the University of Hawaii and the
best resolution that was obtained was 500*10-6 degrees, nearly 20 times less accurate than
designed by the manufacturer. This resolution was dominated by the base level electrical noise
of the sensor. In other words, the 0.1 arc second resolution is probably only obtained with
averaging over time—not desirable in a seismic measurement application. It is therefore
important to evaluate the effect of reading accuracy on the deflected shape predicted by the
inclinometer – based rotation deflection system.
Figure 2.4-1 LSPO ±1º
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Chapter 3: SAP2000 FRAME ELEMENT MODEL: STRAIGHT BASE LINE
3.1 Development of SAP2000 straight base line model SAP2000 was used to create a simple frame model that did not include the horizontal or
vertical curvature of the bridge. This model was the first iteration of the finite element method. It
is the simplest model using only SAP2000 frame elements and was compared with the results
from the designers, KSF Inc.
This model is a very basic representation of the bridge; it includes the section properties
and mass properties of the bridge. It was analyzed as a 2 - D model and was used to look at the
vertical deflection and dead load of the bridge. This model could not used to determine the
transverse deformations of the bridge.
This model included:
1. Fixed supports between the superstructure and substructure
a. Base Isolation was ignored.
2. Pier Geometry
a. Fixed supports at the base of the pier
3. Girder section
a. Using hand calculated section properties
b. A deck thickness of 9” was used
4. Box section
a. 5 different sections to account for the change in height of the cast in
place box girder.
b. Depth changes from 9’-4” to 18’
c. Bottom slab varies from 8” to 12”
5. Did not include any pre-stressing or post-tensioning
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Figure 3.1-1 SAP Area Model SBL
3.2 Frame Elements The frame elements that were used in this model were calculated based on the section
properties that were in the plans (KSF). The girder section (Figure 3.2-1) was calculated
assuming a constant width throughout the bridge and the box girder section (Figure 3.2-2) was
divided into 5 different sections. The box girder was input as 8 separate non-prismatic element
sections, each of which spans 18.125 feet, and as a single section for the box over the pier
(Figure 3.2-3).
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Figure 3.2-1 Girder Section used to calculated properties
Figure 3.2-2 Box Girder Section used to calculate section properties
These properties were input in SAP2000 as non-prismatic frame sections. They are:
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Figure 3.2-3 Division of elements for SAP2000
These section properties are transformed for the differences in the Young’s modulus
throughout the bridge. The following table shows the different compressive strengths and
Young’s modulus for each member. Young’s Modulus was calculated using equation 3-2; this
formula is from ACI equation 8.5.1.
57000
3-1
Compressive Strength
F’c (28 days) (ksi)
Young’s Modulus
E (ksi)
CIP Deck 6 4415
Precast Deck Planks 6 4415
Precast Girders 8 5098
Box Girder 6 4415
Table 3-1 Material Properties of Structural Elements
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Area
(in^2)(10^6)Ixx
(in^4)(10^6)Iyy
(in^4)(10^6)
Girder Section 0.0138 19.000 374.000
Box 1 0.0162 30.900 434.000
Box 2 0.0171 37.700 966.000
Box 3 0.0194 61.700 523.000
Box 4 0.0216 93.600 1330.000
Box 5 0.0254 173.000 1660.000
Box over Pier 0.114 466.000 248. 000
Pier 0.0223 16.058 357.200
Table 3-2 Section Properties
The way the cross sections were modeled, beams “A”, “B”, “C”, and “D” were not
included. Beam “A” is located over each abutment; it is the end block of the bridge. Beam “B” is
located where the precast girders were cast together, every 52.5 feet. Beam “C” is located the
abutment spans, between the box girder and precast girder sections. Beam “D” is located in the
middle span between the box girder and precast girder sections. To account for the extra weight
these beams added, point loads were added (Table 3-3). The additional stiffness’s were not
accounted for in this model.
Span 1 Span 2 Span 3
Dist. from Abut. 1 (ft)
BEAM DL
(Kip)
Dist. From
Pier 1 (ft) BEAM
DL (Kip)
Dist. From
Pier 2 (ft) BEAM
DL (Kip)
0 A 401 255 D 287 615 C 270
52.5 B 140 307.5 B 140 667.5 B 140
105 C 270 360 B 140 720 A 401
412.5 B 140
465 D 287
Table 3-3 Beam Locations and Weights
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3.3 Analysis There are some differences in the analysis of the 2-D model and KSF’s 2d model. Since
KSF is a designer, they have different tolerances than researchers. An instance of this is the
assumptions that were made when calculating the different section properties of the bridge. The
box girder was divided up into multiple segments but the section properties were linearly
interpolated between each segment instead of parabolically as per the moment of inertia
equation. A comparison of the section properties for the girder section and the box girder section,
where the height was 9’ – 9”, are shown in Table 3-3.
Area
(in^2)(10^6)Ixx
(in^4)(10^6)Iyy
(in^4)(10^6)
Girder Section
(Hand Calc) 0.0138 19 374
Girder Section
(KSF) 0.0145 8.668 437.861
% Diff 5.07 54.38 17.08
Box Section 9’
(Hand Calc) 0.0162 30.9 434
Box Section 9’
(KSF) 0.0236 51.217 815.194
% Diff 45.68 65.75 87.83
Box Section 18’
(Hand Calc) 0.0254 173 1660
Box Section 18’
(KSF) 0.0327 240.185 1185.331
% Diff 28.74 38.84 28.59
Pier (Hand Calc) 0.0223 16.058 357.200
Pier (KSF) 0.0432 29,860 265.628
% Diff 93.72 85.95 25.64
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Table 3-4 Comparing Section Properties
The deflected shape of the bridge from the self-weight gives significant deflections (Table
3-4). These deflections are before the addition of the base isolation and without the prestressing
and post tensioning, so the deflections are very conservative and should be larger than the other
models.
Δ(90 ft) inches
Δ(360 ft) inches
Δ(630 ft) inches
SAP 2000 2-d
Frame Model -0.053 -8.006 0.0613
Table 3-5 Deflections at each span
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Chapter 4: SAP2000 AREA ELEMENT MODEL: STRAIGHT BASE LINE
4.1 Development of SAP2000 straight base line area model SAP2000’s bridge modeler was used to create an area model that did not include the
horizontal or vertical curvature of the bridge. This model was used to determine an algorithm for
determining deflection based on measured rotations and for placing inclinometers in the best
locations. The straight base line area model was compared to the first model with respect to
dead load and mode shapes.
The model includes the section properties and material properties of the bridge. It was
analyzed as a 3d model and was used to look at the vertical deflection and mode shapes of the
bridge.
This model includes:
1. Base Isolation
a. Input as friction isolators
2. Pier Geometry
a. Fixed supports at the base of the pier
3. Abutment Geometry
a. Modeled the abutment as a bent to best capture the deflection and mode
shapes.
b. Fixed supports at bottom of abutment
4. Girder section
a. Used the bridge modeler tool in SAP2000
b. A deck thickness of 9” was used
5. Box section
a. Used the bridge modeler tool in SAP2000
b. Depth varies from 9’-4” to 18’
c. Bottom slab thickness varies from 8” to 12”
6. Included transverse beams, “A”, “B”, “C”, and “D”
7. Includes pre-stressing and post-tensioning
a. Included as steel tendons with axial force applied to them
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Figure 4.1-1 SAP Area model straight base line
4.2 Area Elements The area elements were input in this model by using the bridge modeler. The bridge
modeler is an add-on in the SAP2000 program that allows the properties of the bridge to be input
into a variety of tables which are linked to the bridge model. This is a linked model that can be
easily updated and configured.
The bridge modeler gives options to either update the bridge as an area model or as a
spine model. A spine model is similar to the previous frame model and modeled the bridge using
frames instead of area sections. The area model was chosen to represent the bridge as it has
nodes on the outside fascia of the bridge, allowing the deflections of the Mountainside and
Oceanside girders to be compared readily. Another reason that the spine model was not used
was that when the diaphragms where included in the spine model it did not incorporate the dead
load. This was later modified by emailing SAP2000 which released an updated version,
SAP2000 V14.2.1.
The geometry of the precast girder section and the box girder were input into the bridge
modeler that calculated the section properties. The section properties from SAP2000’s bridge
23
modeler were compared with the hand calculations and the max error was 7%, table 4.2-1. This
was an acceptable value and verified that the bridge modeler was calculating the section
properties correctly.
Section Prop
Girder Box
SAP Hand Calc % Error SAP Hand Calc % Error
Area (in^2) 0.0162 .0153 6.0 0.0168 0.0161 4.1
Ixx (in^4) 23.9000 19.000 6.3 31.300 30.900 1.3
Iyy (in^4) 485.000 374.000 6.0 468.000 434.000 7.3
Table 4-1 Section Properties Calculated in Bridge Modeler
The varying depth of the box girder and the varying thickness of the bottom slab were
accounted for by using parametric variations.
The Area model uses 4 node thin shell elements to model the slab, box girders, and
diaphragms. The precast girders are modeled as frame elements. The abutments and piers are
modeled as bents, using frame elements. The abutments were modeled as bents to give a more
accurate representation of the modal shapes. The diaphragms, beams A, B, C, and D are
modeled using frame elements.
SAP2000 has the ability to model friction pendulum bearings (FPB). The friction
pendulum bearings were modeled using a special link element, a friction isolator. There are two
different types of friction pendulum bearings that were used in the design of the Kealakaha
stream bridge, one design for the abutments and one for the piers. The abutment FPB has a
radius of 88 inches, a displacement capacity of +/- 12.5 inches with coefficients of friction 0.4 and
0.5 for static and dynamic friction respectively. The pier FPB has a radius of 88 inches, a
displacement capacity of 10.5 inches, with coefficients of friction 0.4 and 0.5 for static and
dynamic friction respectively [KSF, 2006]. The FPB at the pier is free to translate in both the
transverse and longitudinal directions. The FPB at the abutment is free to translate in the
longitudinal direction but is fixed in the transverse direction by a shear key. This element was
applied at girders 2 and 5, at the same location as they are located on the plans.
24
4.3 Analysis This model is the model that will be used to determine the rotation to deflection and the
base shape algorithms. Comparing the dead load to the previous model shows a slight increase.
The distribution is much different at the abutments, with 45% difference at abutment 1 and 42%
difference at abutment 2. The
Dead Load
(kip)
DL @
Abut 1
DL@
Abut 2
DL @
Pier 1
DL @
Pier 2
SBL Area
Model 18000 680 700 8325 8300
SBL Frame
Model 17200 1236 1215 7289 7491
% Diff 4.65 44.98 41.67 14.2 10.80
Table 4-2 Dead Load Comparisons
Δ(90 ft) inches
Δ(360 ft) inches
Δ(630 ft) inches
SAP 2000 Thin
Shell Element :
SBL
0.00 -4.83 0.05
SAP 2000 2-d
Frame Model -0.053 -8.006 0.0613
Table 4-3 Deflections at middle of each span
25
Chapter 5: SAP2000 AREA ELEMENT MODEL: HORIZONTALLY & VERTICALLY CURVED BASELINE
5.1 Development of SAP2000 area model SAP2000 was used to create an area model that included the horizontal or vertical
geometry of the bridge. This model used the same methodology as the straight base line area
model and also used the same section properties. The difference is that this model incorporated
the vertical and horizontal geometry.
This model included:
1. Base Isolation
a. Input as a friction isolator
2. Pier Geometry
a. Fixed supports at the base of the pier
3. Abutment Geometry
a. Modeled the abutment as a bent to best capture the deflection and mode
shapes
4. Girder section
a. Used the bridge modeler tool in SAP2000
b. A deck thickness of 9” was used
5. Box section
a. Used the bridge modeler tool in SAP2000
b. Depth varies from 9’-4” to 18’
c. Bottom slab varies from 8” to 12”
6. Included Diaphragms
7. Includes pre-stressing and post-tensioning
5.2 Vertical and Horizontal Geometry The vertical and horizontal geometry was taken from the contract plans (KSF). The first
365 feet of the bridge is located on a vertical curve and the rest of the bridge has a constant slope
of 3.46% downward toward abutment two. The vertical geometry of the centerline of the road is
defined in Figure 5.2-1. The vertical curve was modified to input into SAP2000, it was split into
multiple straight segments. This is shown in Table 5.2-1, where the grade for each segment was
calculated.
The bridge is built on a curve with a radius of 1800 feet. The precast girders and box
girders were built as straight segments at different angles and the deck was cast such that it has
the radius on the edges. This was modeled in SAP by just using straight sections for the precast
26
girders and box girders but changing the orientation of the direction of each segment, as shown in
Figure 5.1-1. The geometry of the horizontal curve is located in Table 5.2-2.
Figure 5.2-1 SAP Area model straight base line
Figure 5.2-2 Profile grade line at centerline of road
27
Station 1 Elevation Station 2 Elevation Grade (%)
206 + 57 927.44 207 + 92 925.73 -1.63
207 + 92 925.73 209 + 12 922.16 -2.38
209 + 12 922.16 210 + 17 918.90 -3.11
210 + 17 918.90 211+97 912.67 -3.46
Table 5-1 Vertical Geometry
Figure 5.2-3 Segment Layout of Kealakaha Bridge
Point Station deg. Min sec. Azimuth
Honokaa
Abut 206 + 57 104 24 46.3 104.4129
Wash-Box 207 + 62 102 45 27.6 102.7577
Box-Wash 209 + 12 97 55 9.7 97.9194
Wash-Box 211 + 22 91 17 54.8 91.2986
Box-Wash 212 + 72 86 27 36.6 86.4602
Hilo Abut 213 + 77 84 48 17.9 84.8050
Table 5-2 Horizontal Geometry
5.3 Comparison with Area Element Model: Straight Base Line (SBL)
This model was created to see the impacts that the horizontal and vertical geometry have
when compared with the SAP2000 SBL Area model that was used to determine the placement of
the accelerometers and inclinometers. The dead load increased from the straight base line
model, see table 5.3-1. The largest inconsistency between the dead load comparisons was at
Abutment 1 with a 26.6% error. This can be attributed to the geometry differences as abutment 1
28
and pier 1 have much larger portions of the dead load than in the straight base line model. A
possible causes of the increase in loading are the geometry changes from the
Dead Load
(kip)
DL @
Abut 1
DL@
Abut 2
DL @
Pier 1
DL @
Pier 2
Area Model 19450 927 770 9525 8224
SBL Area
Model 18000 680 700 8325 8300
% Error 7.4 26.6 9.1 12.6 0.9
Table 5-3 Dead Load Comparisons
The self weight deflections increased in this model from the SAP2000 SBL Area model.
This is due to the increase in weight of the structure and from change in the geometry of the
bridge.
Δ(90 ft) inches
Δ(360 ft) inches
Δ(630 ft) inches
SAP 2000 Area
Element : 3-D
0.69 -6.28 1.10
SAP 2000 Area
Element : SBL
0.00 -4.83 0.05
Table 5-4 Deflections at middle of each span
29
Chapter 6: Using Rotation to Calculate Displacement (R2D)
6.1 Theory This is the methodology that was used for determining the rotation to deflection
calculations and algorithm. Reviewing beam theory the equation of curvature is:
6-1
Where:
Ψ(x) is the curvature;
M(x) is a function of the moment over the length of the beam;
E is the young’s modulus of the material;
I(x) is a function of the moment of inertia over the length of the beam;
If the moment, M(x) is taken to be:
3 2
6-2
This gives a description of a quadratic bending moment, which is a good representation
of point loads and uniform distributed loads along the span. Given that the Kealakaha Bridge is a
combination of a varying height box girder and precast drop in girders the moment of inertia
varies quite drastically, shown in Figure 1.1-1. The moment of inertia is very difficult to model;
there is a significant increase in the stiffness at 75 feet on either side of both piers. This is where
the section changes between the varying height box girder and the precast drop in girders.
6-3
Assuming small deformations and integrating the curvature gives us the rotation:
3 2
6-4
30
Figure 6.1-1 Varying Moment of Inertia along the Length of the Bridge
Next, the deflection is determined by integrating the rotation (also the second integral of
the curvature equation):
3 2 x
6-5
Instead of doing a direct integration of the rotation the deflection was determined by using
numerical integration, trapezoidal rule.
12
6-6
Where:
δ = deflection
dx is the step size of the integration;
0.0E+00
2.0E+07
4.0E+07
6.0E+07
8.0E+07
1.0E+08
1.2E+08
1.4E+08
1.6E+08
1.8E+08
0 100 200 300 400 500 600 700
Ixx (in^4)
Distance Along Bridge (ft.)
Moment of Inertia along Bridge
Moment of Inertia (Ixx)
31
θ(i) is the rotation at the previous point;
θ(i+1) is the rotation at the point of interest;
The direct integration method is a more accurate method but using the numerical
integration helped reduce the calculation time. The numerical integration method is also a
simpler algorithm to set up for forward matching progress.
6.2 Implementation
6.2.1 Assumptions
To simplify the rotation to deflection equations a few assumptions were made. The
moment of inertia and modulus of elasticity were both taken to be constant instead of varying
along the length of the bridge. This assumption simplified equations 6-4 and 6-5. The deflection
at abutment 1 was taken to be 0; this is where the rotation to deflection algorithm starts and will
be carried through the length of the bridge.
Taking the moment of inertia and modulus of elasticity to be constant reduces the rotation
equation (6-4) to:
6-7
The deflection can then be written as:
4 3 2
6-8
6.2.2 Model Descretization
The R2D method was applied to the Kealakaha bridge model by discretizing the
superstructure into convenient beam segments. These segments are defined such that the four
unknowns in equation (6-7) can be determined from acquired data. In other words, the method
requires four rotations as input to determine the rotation function. The rotation measurements at
the ends of each beam segment provide continuity between adjacent beam segments. In the first
trial, it was assumed that 18 rotation sensors were required to adequately capture the displaced
shape of the super structure. They were assigned to the locations indicated in figure ().
The rotation sensors which are available to calculate the deflection of girder 6 include:
the four sensors at the piers and all the odd numbers sensors along the length of the girder itself.
Starting from sensor location 1, three adjacent beam segments are assigned. The first segment
uses sensors 1, 5, 7 and 2; this segment is called R2D Model 1 Segment 1. The second
32
segment uses sensors 2, 9, 11, and 13; this segment is called R2D Model 1 Segment 2. The
third segment uses sensors 13, 3, 15, and 17; this segment is called R2D Model 1 Segment 3.
Model 1 does not span the whole bridge; it ends at inclinometer 17 or at beam “B” in span
3. To incorporate the whole bridge other models will be needed. Two other models will be used,
Model 2 and Model 3, placed so that each model starts at the next inclinometer along the bridge.
R2D Model 2 Segment 1 uses sensors 5, 7, 2, and 9, R2D Model 2 Segment 2 uses sensors 9,
11, 13, 3, R2D Model 2 Segment 3 uses sensors 3, 15, 17, and 4 R2D Model 3 Segment 1 uses
sensors 7, 2, 9, and 11 and R2D Model 3 Segment 2 uses sensors 11, 13, 3, and 15 (Figure 6.2-
3).
Figure 6.2-1 Location of Inclinometers and Formation of R2D Models 1 – 3
6.2.3 Model Analysis
To test the rotation to deflection theory, the SAP2000 straight base line area model was
used with a variety of loading conditions, A-I, (Figure 6.2-2). This model was used as the
measured values as there aren’t any measurements from the field as of this time. The load
placements are in the middle of each span, at 90 ft, 360 ft, and 630 ft respectively. The model
was loaded with a standard HS-20 truck Figure 6.2-3. The total weight of the HS-20 was applied
as two point loads, 36 Kip each, acting on the two nearest girders (Figure 6.2-3). The loads were
applied to girders 5 and 6 (below the Hilo bound lanes), middle two girders (3 and 4—below the
centerline) and girders 1 and 2 (below the Honokaa bound lanes).
Model 1 Segment 1 Model 1 Segment 2 Model 1 Segment 3
Model 2 Segment 1 Model 2 Segment 2 Model 2 Segment 3
Model 3 Segment 1 Model 3 Segment 2
33
Figure 6.2-2 Plan view of the placement of 9 point load cases
Figure 6.2-3 HS-20 Truck
For the 3 R2D models, for them to be accurately used together a common initial value is
necessary. The first support that is in all 3 models was Pier 1. To use this location as a
boundary condition is not an ideal location as there is some movement at girders 1 and 6 for
loading cases that are not in the center of the bridge. The boundary condition was decided to be
0 inch deflection at the abutment; this is true for all loading conditions. The difficulty with this
location is R2D models 2 and 3 do not include this point.
R2D model 1 was chosen to be the boundary condition for R2D models 2 and 3, δ (R2D
Model 1 @ inclinometer 5) = δ (R2D Model 2 @ inclinometer 5 and δ (R2D Model 1 @
34
inclinometer 7) = δ (R2D Model 3 @ inclinometer 7). This incorporates setting the deflection at
abutment 1 to 0 inches and forces continuity at inclinometer 5 for R2D Model 2 and inclinometer 7
for R2D Model 3 (Figure 6.2-6). The deflections from the R2D models are based off of R2D
Model 1 but quickly diverge as they have different constant values; A, B, C, and D.
The 3 R2D Models have been plotted against the deflections that were determined in
SAP2000 for load case A. R2D Model 1 is a good representation of Span 1, (Figure 6.2-3). R2D
Model 2 doesn’t have as good as a representation of the first span and it does not return to 0
inches of deflection at abutment 2, it is the only model that has a complete deflection for the last
span (Figure 6.2-5). The R2D Model 3 follows the predicted deflection from SAP2000 through
the second span.
6.2-6 shows all 3 R2D models and the calculated deflection from SAP2000 on the same
graph. This illustrates the lack of continuity between the 3 different models and shows how each
model best represents a single span.
35
Figure 6.2-4 R2D Model 1 with SAP output
Figure 6.2-5 R2D Model 2 with SAP output
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0.00 90.00 180.00 270.00 360.00 450.00 540.00 630.00 720.00
Deflection (in.)
Distance Along Bridge (ft.)
Load Case C
SAP
Model 1
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0.00 90.00 180.00 270.00 360.00 450.00 540.00 630.00 720.00
Deflection (in.)
Distance Along Bridge (ft.)
Load Case C
SAP
Model 2
36
Figure 6.2-6 R2D Model 3 with SAP output
Figure 6.2-7 Load Case C with All models
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0.00 90.00 180.00 270.00 360.00 450.00 540.00 630.00 720.00
Deflection (in.)
Distance Along Bridge (ft.)
Load Case C
SAP
Model 3
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0.00 90.00 180.00 270.00 360.00 450.00 540.00 630.00 720.00
Deflection (in.)
Distance Along Bridge (ft.)
Load Case C
SAP
Model 1
Model 2
Model 3
37
6.2.4 Final Model Selection
To reduce the 3 R3D Models into a single deflection there needed to be some
manipulation. As shown in figure 6.2-6 the only place where the R2D Models are continuous is in
the first span, from setting the initial value of R2D Models 2 and 3 equal to R2D Model 1. Another
inconsistency is that R2D Model 2 does not return to 0.
This was remedied by determining the difference at abutment 2 of R2D Model 2 and
rotating the whole deflection curve about its initial point. The model was chosen to be rotated
instead of translated to keep continuity with R2D Model 1 in span 1. R2D Model 3 needed to be
updated for continuity with the other 2 models.
Since each model represents a single span; R2D Model 1 – span 1, R2D Model 2 – span
3 and R2D Model 3 – span 2, Each model will only be used for its span that it represents the most
accurately, R2D Model 1 will be used for span 1, 0 ft to 180 ft. R2D Model 3 for 180 ft to 540 ft
and R2D Model 2 for 540 ft. to 720 ft (Figure 6.2-7). Model 3 was transposed to match R2D
Model 1 at pier 1 and then rotated about pier 1 to match R2D Model 2 at pier 2.
Delta 1 is from R2D Model 1 in span 1, Delta 2 is from R2D Model 3 in span 2, and Delta
3 is the portion of R2D Model 2 in span 3. The Deflection
Figure 6.2-8 Portions of R2D Models used in final deflection curve
Model 1 Model 2 Model 3
38
Figure 6.2-9 Load Case C with final Delta
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case C
SAP
Delta
39
Chapter 7: BASE SHAPES
7.1 Theory Using base shapes to calculate the deflection along the bridge is superposition. The
measured rotations from the inclinometers will be related to the pre-calculated rotations at each
point along the bridge by the following equation:
7-1
Where:
θm is a matrix containing the measured rotations of the bridge
θpre is a matrix containing the pre-calculated rotations resulting from the base shapes
α is a matrix of participation factors
Using this equation it is possible to determine the factors, α, that are required to scale the
pre-calculated base shapes rotations to recreate the function of rotations along the length of the
bridge from the measured rotations.
7-2
It is important to note that, θpre, must be a square matrix in order for the inverse function
to work. Using the factors that modify the pre-calculate rotations to the actual measured
rotations; the same can be done for the pre-calculated deflections.
7-3
7.2 Implementation The base shape methodology of determining deflection is a very efficient method. It
takes into account the stiffness and material changes throughout the bridge. This method does
not take much computing time as it only involves matrix inversion and multiplication. The time
consuming portion is to ensure that the pre-calculated shapes are determined for the loads
located in the ideal positions and that the finite element model is correct.
40
The base shape theory of determining deflection is a fairly simple approach. The first
step is to determine the location of loads used to generate the base shapes. This was done by
setting the locations at every 60 feet along the bridge; these are locations 21-38 as labeled in
Figure 7.2-1. The temperature effects were also included as a potential loading condition. The
temperature loading was a linear varying 1 degree change in temperature on the top slab of the
bridge.
The temperature loading was applied in SAP2000 through the bridge modeler as a
temperature gradient. The gradient was applied over the top 9” of the deck as a linear varying
temperature of 1º C decreasing to 0º C at the bottom of the deck (Figure7.2-1).
Figure 7.2-1 Applied Temperature Gradient along Bridge
Locations 21-29 are located on the two Oceanside girders and locations 30-38 are
located on the two mountainside girders. At each of these locations the same loading conditions
were enforced as for the rotation to deflection methodology, placing a HS-20 truck, Figure 7.2-2,
at each location.
Initially only the first 9 load cases, A-I were considered. SAP2000 and the SBL area
model were used to determine the measured rotations due to each load case. This provided a
good number of comparisons when determining which base shapes were to be used. There were
12 combinations of the base shapes that were tested to see which would give an acceptable
deflection when compared with SAP2000. These combinations are shown in Table 7.2-1. The
combinations are split up into pairs with a +T and –T. The +T means that the temperature load
case was taken into account and –T means that the temperature load case was not taken into
account.
41
The temperature was negated out by assigning a value of 1 in the θpre matrix. This was
done to see if the temperature deflection condition was having a positive or negative impact on
the load cases.
Splitting the base shapes into combinations of 9 worked the best. When too many load
combinations are combined into the θpre then they start working against each other and magnify
the error in the deflected shape.
Figure 7.2-2 Location of potential base shapes
Combo Load Location
1 +T 21 22 28 29 30 31 37 38
2 -T 21 22 28 29 30 31 37 38
3 +T 21 23 27 29 30 32 36 38
4 -T 21 23 27 29 30 32 36 38
5 +T 21 24 26 29 30 33 35 38
6 -T 21 24 26 29 30 33 35 38
7 +T 22 23 27 28 31 32 36 37
8 -T 22 23 27 28 31 32 36 37
9 +T 22 24 26 28 31 33 35 37
10 -T 22 24 26 28 31 33 35 37
11 21 22 23 24 25 26 27 28 29
12 30 31 32 33 34 35 36 37 38
Table 7-1 Potential base shape combinations
42
Figure 7.2-3 HS-20 Truck
To determine the best combination to use for the base shapes, the deflections of all 12
load cases were analyzed. Table 7.2-2 shows the results of each load combination at the middle
of each span, 90 ft, 360 ft, and 530 ft respectively. The table shows the difference between
SAP2000’s deflection and each combination. The SAP2000’s calculated rotations act as the
measured rotations as though they are from the inclinometers. These rotations are used in
determining the participation factor, α. Table 7-2 shows the error between the measured;
SAP2000, deflection and the deflection calculated using the base shape method.
%
| |100% 7-4
43
Combo 1
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 48.7 83.4 91.6 26.0 16.9 12.5 101.5 22.0 107.1
360 ft 0.0 212.3 1174.6 10.8 8.8 4.4 406.2 53.2 401.1
630 ft 37.8 82.8 91.3 26.2 16.6 10.5 101.8 17.4 108.0
Combo 2
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 76.3 108.3 103.3 19.4 7.5 29.1 99.5 120.2 97.7
360 ft 246.5 78.3 84.5 4.6 0.9 4.6 89.5 102.4 88.4
630 ft 91.3 102.0 100.9 2.8 19.2 77.5 99.9 105.9 99.3
Combo 3
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 167.2 57.6 84.7 95.0 52.8 21.2 102.9 79.3 122.2
360 ft 61.3 185.3 338.1 23.7 16.9 14.3 189.6 97.7 289.2
630 ft 173.1 56.7 84.1 77.5 46.9 17.9 103.5 78.1 125.6
Combo 4
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 140.7 90.5 96.0 127.1 228.1 2456.2 100.6 76.2 103.1
360 ft 67.7 154.4 128.0 25.9 30.5 36.5 115.8 97.5 118.4
630 ft 146.6 88.7 95.3 97.5 146.8 343.3 100.8 75.4 104.0
Combo 5
44
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 11.8 49.7 164.3 7.8 12.1 0.7 63.1 24.8 48.9
360 ft 5.8 2.5 0.1 4.7 3.9 18.1 8.7 85.6 5.0
630 ft 10.5 44.4 138.7 5.1 10.7 2.9 58.7 26.2 46.0
Combo 6
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 2.1 3.4 12.1 2.5 22.7 91.5 37.1 421.7 208.6
360 ft 1.5 3.1 6.3 3.0 7.1 12.8 12.4 148.6 17.0
630 ft 2.4 2.7 10.9 0.4 20.1 72.6 33.2 802.3 135.6
Combo 7
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 3.8 15.0 44.5 9.8 10.1 22.1 74.9 120.0 3.5
360 ft 7.1 68.8 151.6 3.1 14.6 13.3 17523.8 203.7 2050.5
630 ft 0.4 4.1 9.1 7.9 10.9 22.8 3.1 96.7 116.2
Combo 8
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 0.9 7.0 15.7 11.6 7.7 19.8 66.9 82.4 9.6
360 ft 44.2 682.8 214.3 97.3 70.1 83.7 161.1 96.8 199.7
630 ft 2.9 1.2 4.6 6.3 13.0 24.7 40.7 131.4 78.8
Combo 9
A B C D E F G H I
45
% % % % % % % % %
90 ft 0.4 1.7 2.1 7.9 11.0 22.1 43.8 88.4 56.0
360 ft 5.4 1.3 1.0 8.6 0.9 10.2 10.1 80.9 3.7
630 ft 0.0 1.6 2.5 7.1 9.7 21.1 39.8 81.8 47.7
Combo 10
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 0.0 0.8 0.2 10.1 9.9 21.7 47.2 89.5 54.2
360 ft 1.9 3.4 7.3 20.9 13.6 16.1 0.8 79.3 5.8
630 ft 0.5 0.4 0.1 10.0 8.4 20.5 44.3 83.1 45.8
Combo 11
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 0.0 2.2 4.4 4.2 24.8 57.4 190.5 2924.7 837.8
360 ft 0.0 7.1 8.3 2.2 4.8 9.7 10.5 191.7 11.2
630 ft 0.0 7.5 14.4 3.1 23.0 53.4 497.6 1011.8 5586.5
Combo 12
A
%
B
%
C
%
D
%
E
%
F
%
G
%
H
%
I
%
90 ft 3.2 2.7 0.0 37.8 21.8 1.4 83.0 8.4 55.6
360 ft 14.3 4.0 0.0 35.6 22.9 0.1 53.6 83.8 30.1
630 ft 3.1 2.0 0.0 33.1 18.4 1.4 77.8 8.0 48.7
Table 7-2 Comparison of Different Load Combinations
The results in Table 7.2-2 show that selection of the base shapes has a big influence on
the error between SAP2000’s calculated deflections and the base shape calculated deflections.
Combination 2 has the worst error with a max error of 17500% at the middle of span 2. The best
46
combinations were combinations 9 and 10, with average errors of 21.0% and 22.1%. These two
combinations are very similar, with the difference being whether or not the temperature was
included. Since these two combinations are very similar, other loading conditions will be looked
at. The new loading conditions will be
Figure 7.2-3 shows the deflection along the mountainside (Mauka) girder determined by
using the base shapes approach with combination 10 as the pre-calculated deflected shapes.
Load combination A is the condition where there is one HS-20 Truck located in the middle of the
bridge on the Oceanside (Makai) girders.
Figure 7.2-4 shows the resulting deflection from the base shapes compared to the
deflection output from SAP2000. As shown on the chart the base shape deflection is very close
to SAP2000’s predicted deflection. The differences at the middle of span1, span 2 and span 3
are: 0.009 inches, 0.010 inches, and -0.002 inches respectively.
Figure 7.2-4 Base Shape Deflection with SAP output, Load case A
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination A
SAP
Base Shapes
47
Figure 7.2-5 Base Shape Deflection with SAP output, Load case H
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination H
Base Shapes
SAP
48
49
Chapter 8: R2D ANALYSIS
8.1 Preliminary Locations Initially placing the inclinometers at each beam along the bridge was the layout. Looking
at the preliminary layout (figure 6.2-1), it was observed that placing the inclinometers at each
beam was not a very good option. The problem occurred in the middle span, placing the
inclinometers at the end of the box girder did not give the best results for the middle span of the
bridge. The box girder was too stiff, decreasing the rotation measured, thus causing inaccurate
deflection calculations when using the rotation to deflection algorithm described above.
Five different locations were explored, points 1 – 5. Point 1 was located 8.67 feet from
beam D in the middle span and points 2-5 were located 8.67 feet from the previous point. The
five new locations were located in the middle span at equal spacing, 8.67 feet, between beam D
and the first beam B. Table 8-1 shows the results of each location minus the values from
SAP2000. The results are in inches.
A B C D E F G H I
%
Error
%
Error
%
Error
%
Error
%
Error
%
Error
%
Error
%
Error
%
Error
Point 1 7.45 0.02 4.62 2.80 4.93 23.15 5.61 354.68 5.35
Point 2 0.25 1.72 0.46 0.79 6.45 23.50 1.29 532.79 3.43
Point 3 9.77 2.09 4.72 3.65 3.18 18.75 9.54 677.81 0.59
Point 4 14.39 1.88 6.37 3.97 2.40 17.19 12.16 754.75 1.35
Point 5 16.76 1.08 6.91 3.69 2.24 15.76 12.86 779.31 0.15
Beam D 21.75 2.71 13.44 1.94 6.27 25.17 19.80 55.45 12.56
Table 8-1 Results
From Table 8.1 the best result is from Point 2. It is not the best for all cases but for most
cases. It is best for load cases A, C, D, and F. The main inconsistency in choosing to move the
inclinometer from beam “D” to point 2 is load case H (Figure 8.1-1). This load case has an
extreme deflection discrepancy when only the deflections at midspan are looked. The deflected
shapes along the whole center span are similar, the difference being that the deflected shape
when an inclinometer is located at beam “D”, the deflection curves converge at the middle of the
50
bridge. This gives the best results for the middle span; point 2 is a good medium between the
stiffness of the cast in place box girder and the precast girders.
Figure 8.1-1 Deflection for Load Case H for 6 Possible Inclinometer Locations
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case H
SAP
Delta (D)
Delta (Pt 1)
Delta (Pt 2)
Delta (Pt 3)
Delta (Pt 4)
Delta (Pt 5)
51
Figure 8.1-2 Deflections for Load Case A for 6 Possible Inclinometer Locations
8.2 Number of Inclinometers There were 18 inclinometers in the preliminary placement. Due to limited number of
inputs in the data acquisition station, and the amount of accelerometers that are being used in the
bridge, the option of decreasing the number of inclinometers from 18 to 14 is explored. It is more
economical than purchasing additional data acquisition stations and will allow the later installation
of extra accelerometers in the free field stations. More inclinometers can be added at a future
time.
The initial location of the inclinometers was roughly based on 60 foot spacing, where
there are 2 sets of inclinometers in the first and third spans, 3 inclinometers in the second span,
and one inclinometer at each support. To decrease the number of inclinometers, it was
determined to remove 1 set on inclinometers from the first and third span. Two different options
were evaluated, a 72 foot and a 90 foot spacing. These options are shown in Figure 8.2-1.
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case A
SAP
Delta (D)
Delta (Pt 1)
Delta (Pt 2)
Delta (Pt 3)
Delta (Pt 4)
Delta (Pt 5)
52
Figure 8.2-1 Location of Potential Inclinometer Locations
To compare the three inclinometer location patterns; 60, 72 and 90 foot spacing the R2D
method was employed. Each inclinometer schematic was evaluated for load combinations A – I.
53
Load
Combo
Deflection at Middle of Center Span (360 ft)
SAP 2000
Defl. (in)
60 Foot Spacing 72 Foot Spacing 90 Foot Spacing
Defl. (in) % Error Defl. (in.) % Error Defl. (in.) % Error
A -0.162 -0.157 2.98 -0.103 36.597 -0.141 12.594
B -0.310 -0.314 1.44 -0.310 0.261 -0.314 1.377
C -0.470 -0.476 1.22 -0.524 11.602 -0.491 4.440
D 0.169 0.163 3.48 0.122 27.531 0.141 16.671
E 0.162 0.168 3.79 0.123 23.875 0.156 3.749
F 0.155 0.188 21.13 0.102 34.447 0.195 25.623
G -0.301 -0.313 3.98 -0.402 33.405 -0.350 16.338
H -0.007 0.030 556.78 -0.001 83.131 0.053 906.849
I -0.148 -0.146 1.12 -0.187 26.638 -0.159 7.698
Table 8-2 Deflection at Middle of Center Span
Table 8.2 shows the of SAP2000 displacements compared with the R2D displacements
for the 60 foot, the 72 foot and the 90 foot inclinometer spacing. The table shows the deflection,
at the Mauka girder, calculated at each load combination, A – I, using SAP2000 and the R2D
method. The 60 foot spacing of the inclinometers has the best correlation with SAP2000 with the
exception of load combination H. The average error was 4.89%, when load combination H was
not included and 66.21% when it was included. The 90 foot spacing has the second best error
when load combination H is not included with an error of 11.06% and an error of 110.6% when
combination H is included. The 72 foot inclinometer spacing has significant error for all the load
cases other than H but produces the best results for load combination H. The error was 24.29%
when load combination H was not included and 30.83% when it was included.
Disregarding load combination H the optimal spacing is to set the inclinometers at 60 feet
based on error alone. Considering that freeing up 4 inputs in the data acquisition system by
changing the spacing to 90 feet only gives us a 6.17% increase it is a good tradeoff. If the
deflections are not working out, it is easy to convert the 90 foot spacing to 60 foot spacing at a
later time. The inclinometers in the middle span are in the same place for the 60 and 90 foot
spacing, only requiring the addition of 2 inclinometers in spans 1 and 3 and moving the existing
inclinometers by 30 feet.
54
8.3 Error Analysis The preceding comparisons between SAP2000 and R2D deflections have assumed that
the inclinometer measurements are exact. However, the inclinometers can only measure the
rotations of the bridge to a certain precision. The manufacturer claimed that the sensitivity was ±
0.1 arc seconds, which is 28*10^-6 degrees. Laboratory tests at UH determined the sensitivity of
the inclinometer to be 500*10^-6 degrees.
To be conservative the error used to determine the sensitivity of the inclinometers was
0.0005 degrees, 8.727 E-06 radians. To see how an error in one inclinometer propagates
throughout the deflection equation each inclinometer, individually, was given either a
measurement of 0 or 8.727E-06 radians. This shows the amount of deflection of the bridge
calculated by the R2D method when there wasn’t any external loading applied. The other case
was to set all of the inclinometers to 8.727E-06 radians, a worst case scenario assuming that all
inclinometers were all affected by the max error. Table 8-4 shows the maximum positive and
negative errors that were obtained. Positive and negative errors are the amount of deflection
above and below the un-displaced shape.
Sensor Max Pos. Error (in.)
Max Neg. Error (in.)
1 0 ‐0.002
2 0.0000 ‐0.008
3 0.0054 ‐0.001
4 0.0019 0
5 0 ‐0.007
7 0.0009 ‐0.009
9 0.0004 ‐0.007
11 0.0014 ‐0.004
13 0.004837 ‐0.003
15 0.006786 ‐0.002
17 0.00663 ‐3.7E‐06
ALL 0 ‐0.0190
Table 8-3 Minimum and Maximum Error on Mauka Girder for 18 inclinometers
55
Sensor 72 Foot Spacing 90 Foot Spacing
Max (in) Min (in) Max (in) Min (in)
1 0 ‐0.003 0.0000 ‐0.0035
2 0.0007 ‐0.009 0.0017 ‐0.0076
3 0.0047 ‐0.003 0.0042 ‐0.0037
4 0.0026 0 0.0030 0.0000
5 0 ‐0.012 0.0000 ‐0.0124
7 0.0005 ‐0.008 0.0003 ‐0.0081
9 0.0018 ‐0.003 0.0012 ‐0.0045
11 0.0047 ‐0.003 0.0051 ‐0.0027
13 0.009181 0.000 0.0091 ‐0.0001
All 0 ‐0.0190 0.0000 ‐0.0190
Table 8-4 Maximum and Maximum Error on Mauka Girder for 14 inclinometers
As shown in Tables 8-3, 8-4, 8-5; the error is minimal compared to the order of the
deflections. The maximum error is -0.019 inches, resulting from all inclinometers having error
acting in the same direction. The average area is around 0.01 inches, from subtracting the
minimum error from the max error. This is acceptable when comparing this area with the
deflected shapes, load combinations A – I, from SAP2000. Figures 8.3-1 compares the errors
from 60, 72, and 90 ft inclinometer spacing, the effect is minimal. Figure 8.3-2 shows the error
envelope for the 90 ft inclinometer spacing for load case G.
56
Figure 8.3-1 Error vs. SAP2000 Deflection
Figure 8.3-2 Error vs SAP2000 Deflection
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Error All Sensors
SAP (Load Case A)
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 60 120 180 240 300 360 420 480 540 600 660 720
Deflection (in.)
Distance Along Bridge (ft.)
Error Envelope Load Case G
No Error
Error Added (All Sensors)
Error Subtracted (All Sensors)
57
Chapter 9: BASE SHAPES ANALYSIS
9.1 Location of Inclinometers As mentioned in Chapter 8, the number of inclinometers was originally 18 but due to the
number of accelerometers on the bridge and free-field stations, decreasing the inclinometers to
14 was the ideal scenario. Decreasing the inclinometers to 14 posed a problem with the location
of the inclinometers. Two different inclinometer location patterns were tested, Figure 9.1-1.
Figure 9.1-1 Location of Potential Inclinometer Patterns
These two patterns both were based on a 72 ft spacing pattern and a 90 foot pattern.
The spacing distance was measured from Abutment 1 and 2 to place a pair of inclinometers in
spans 1 and 3. To place the inclinometers in span 3, the spacing distance was measured from
either side of the mid-span. These patterns were determined to have a good representation of
58
the bridge, as span 2 was the most critical span having 3 pairs of inclinometers while spans 1 and
3 only had 1 pair of inclinometers. The inclinometers were located on the girder section of the
span, since there would be more rotation at these points and less influence from sensitivity of the
inclinometers.
Load
Combination
Deflection at Midspan (360 ft)
SAP defl.
(in.)
72 foot Spacing 90 Foot Spacing
Defl. (in.) % Error Defl. (in.) % Error
A ‐0.162 ‐0.165 1.89 ‐0.170 5.42
B ‐0.310 ‐0.299 3.30 ‐0.287 7.34
C ‐0.470 ‐0.438 6.82 ‐0.406 13.57
D 0.169 0.140 17.29 0.149 11.57
E 0.162 0.143 11.94 0.159 1.71
F 0.155 0.134 13.83 0.182 17.19
G ‐0.301 ‐0.299 0.76 ‐0.258 14.42
H ‐0.007 ‐0.032 382.09 0.011 265.04
I ‐0.148 ‐0.157 6.21 ‐0.170 15.43
Table 9-1 Displacements at Middle of Span 2
Table 9-1 compares SAP2000’s deflections with the deflections calculated using base
shape method from the 72 and 90 foot inclinometer spacing. The table shows the deflections,
along the Mauka girder, calculated for each load combination A – I. Using the base shape
method for determining the deflection, the 72 foot inclinometer spacing has a better result for
cases A, B, C, F, G, and I. As with comparing the different inclinometer spacing for the R2D
method load combination H has the most error. 383.09% and 265.04% for the base shape
method with 72 and 90 foot spacing respectively. The error for load combination H is an
anomaly, since the error is greater than 10 times the next largest error value. The average error
for the 72 and 90 foot spacing of inclinometers are; 7.76% and 10.83%.
The differences in deflection for the middle of spans 1 and 3 are shown in appendix B.
59
Figure 9.1-2 Load Combination F
Figure 9.1-3 Load Combination I
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination F (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination I (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
60
The calculated deflections of the middle of span 2 are similar to the deflections from
SAP2000. The deviation of the deflected shapes occurs in spans 1 and 3; this is to be expected
as there are less inclinometers in those spans.
9.2 Other Deflected Combinations The base shapes method of determining deflections from rotations was determined
based on load cases A – I. . To determine how well the base shapes method works, it will be
compared against another set of load combinations. The new set of load combinations was
derived from the initial inclinometer locations that were explored back in Chapter 7. From Figure
7.2-1 there are 18 different locations of potential locations for the base shapes, 8 of those
locations were used and 2 of the 18 positions coincided with load cases A and C. There are 8
extra load combinations that will be tested in conjunction with 6 other load combinations.
The 8 extra loading combinations are number 21, 23, 27, 29, 30, 32, 36, and 38. The 6
new load combinations, 39 – 44 are located between beam “B” and the mid-span of the bridge, a
distance of 52.5 feet. Load combinations 39 – 44 were added to see how sensitive the base
shapes method is to loading near the midspan of the bridge.
Table 9-2 shows the displacements of SAP2000 vs. the 72 ft. inclinometer spacing and
SAP2000 vs. 90 ft. inclinometer spacing. This shows the deflection, at the mountainside girder,
calculated at each load combination, for the rest of the load positions, using SAP2000 and the
base shape method. The maximum difference in deflection is at load combination 21, -0.028
inches for 72 ft spacing and -0.054 inches for the 90 foot spacing. The error in load combination
44 is only 4.95% and 11.55% for the 72 and 90 foot inclinometer spacing while the error in load
combination 21 is 40.16% and 28.20% for 72 and 90 foot inclinometer spacing.
There are 5 critical load combinations for the 90 foot spacing and only two for the 72 foot
spacing, where critical means having an error of greater than 10%. The load cases that cause
the most error for the 72 foot spacing are 21 and 29, having errors of 40.16% and 37.32%. The
load cases most affecting the 90 foot spacing are 21, 29, 30, 38, and 44; having errors of 28.2%,
27.19%, 44.96%, 31.79%, and 11.55%.
Load cases 21 and 29 are symmetrical, both with the truck loading 60 feet from the
nearest abutment on the mauka girder. Load cases 30 and 39 are the same loading cases as 21
and 29 but on the makai girder. These load combinations are symmetrical which explains why
the percent error is close as well,
61
Δ @ 360 ft
SAP 2000 72 ft spacing Error SAP 2000 90 ft spacing Error
(in) (in) % (in) (in) %
21 0.071 0.042 40.16 0.071 0.051 28.20
23 ‐0.094 ‐0.100 6.44 ‐0.094 ‐0.098 3.63
27 ‐0.093 ‐0.099 6.24 ‐0.093 ‐0.096 3.47
29 0.070 0.044 37.32 0.070 0.051 27.19
30 0.066 0.071 6.48 0.066 0.096 44.96
32 ‐0.106 ‐0.104 2.56 ‐0.106 ‐0.100 5.88
36 ‐0.105 ‐0.101 3.72 ‐0.105 ‐0.097 7.07
38 0.066 0.061 6.77 0.066 0.086 31.79
39 ‐0.162 ‐0.163 0.64 ‐0.162 ‐0.163 0.88
40 ‐0.164 ‐0.164 0.19 ‐0.164 ‐0.168 2.33
41 ‐0.163 ‐0.166 1.99 ‐0.163 ‐0.171 4.87
42 ‐0.345 ‐0.346 0.34 ‐0.345 ‐0.336 2.43
43 ‐0.406 ‐0.397 2.25 ‐0.406 ‐0.375 7.63
44 ‐0.451 ‐0.429 4.95 ‐0.451 ‐0.399 11.55
Table 9-2 Displacements at the middle of span 2
The differences in deflection for the middle of spans 1 and 3 are shown in appendix B.
For load combination 44 and 72 foot inclinometer spacing the difference between SAP2000 and
the base shape deflections are 0.002 inches and -0.001 inches for spans 1 and 3. For load
combination 44 with 90 foot inclinometer spacing the difference was the same, 0.008 inches and -
0.007 inches. The 4 worst loading conditions are shown below, load combinations 30, 38, 44,
and 21. The Load combinations 30, 38 and 44 all are conditions where the truck is loaded on the
mountainside girder. As shown in Figures 9.2-1, 9.2-1, and 9.2-4 most of the error occurred in
the first and third span. This is expected as there were less inclinometers located in these spans
but they were not too far off. The largest error was around 0.1 inches and it occurred in both
spans 1 and 3 but from different loading conditions.
62
Figure 9.2-1 Load Combination 30
Figure 9.2-2 Load Combination 38
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 30 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 38 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
63
Figure 9.2-3 Load Combination 44
Figure 9.2-4 Load Combination 21
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 44 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 21 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
64
9.3 Error Analysis The base shape method of calculating deflection relies heavily on the accuracy of the
inclinometers. This is a concern when looking at the error of the inclinometers that are to be used
on the Kealakaha stream bridge. As mentioned before in the literature review, the advertised
precision of the inclinometers is 0.1 arc seconds, 28*10^-6 degrees. After testing these
inclinometers in the lab, the actual resolution was determined to be 500*10^-6 degrees.
To test the sensitivity of the base shape method for calculating the deflection, θm was
modified by adding the error to the measured rotations. The error is 500*10^-6 degrees. The
base shape deflection methodology was then run with the 500*10^-6 degree error added on to
each sensor individually and then on to all the sensors. Setting each sensor equal to the results
from the deflected shapes was compared with the deflected shapes with the error.
The maximum and minimum errors are provided in Table 9-3. The errors were
determined by subtracting the deflections that were calculated with error from the deflections that
were calculated assuming perfect conditions. The percent error was determined comparing the
original deflections and the error deflections calculated using the base shapes method.
Sensor 72 Foot Spacing 90 Foot Spacing
Max (in) Min (in) Max (in) Min (in)
1 0.0057 ‐0.0005 0.0037 ‐0.0011
2 0.0106 ‐0.0016 0.0086 ‐0.0032
3 0.0015 ‐0.0105 0.0031 ‐0.0086
4 0.0004 ‐0.0059 0.0009 ‐0.0040
5 0.0029 ‐0.0015 0.0115 0.0000
7 0.0042 ‐0.0013 0.0047 ‐0.0017
9 0.0040 ‐0.0040 0.0049 ‐0.0049
11 0.0013 ‐0.0042 0.0018 ‐0.0044
13 0.0022 ‐0.0029 0.0000 ‐0.0114
All 0.0000 0.0000 0.0000 0.0000
Table 9-3 Minimum and Maximum Error on Mauka Girder
65
The sensor that is most affected by the error was sensor 2, for the 72 foot spacing,
located at Pier 1, maximum positive error of 0.0106 inches and a maximum negative error of -
0.0016 inches. Although this sensor was has the largest difference in deflection it did not have
the largest percent error, only 4.76%. The error was the same for all of the loading conditions.
This was due to the matrix multiplication that is used to determine deflected shapes (Figure 9.3-
1).
Figure 9.3-1 Error from inclinometer 2
Figure 9.3-2 shows the error from inclinometer 2 for both 72 and 90 foot inclinometer
spacing. This inclinometer is located at Pier 1 and had the highest error. Figures 9.3-3 and 9.3-4
show the error envelope from adding and subtracting the error to load case B for 72 and 90 foot
inclinometer spacing. The error is minimal compared to the deflected shapes.
‐0.15
‐0.10
‐0.05
0.00
0.05
0.10
0.15
0 180 360 540 720Delta (in)
Distance Along Bridge (ft)
Error 2A (in.)
B (in.)
C (in.)
D (in.)
E (in.)
F (in.)
G (in.)
H (in.)
I (in.)
25 (in.)
34 (in.)
39 (in.)
40 (in.)
41 (in.)
42 (in.)
43 (in.)
44 (in.)
24 (in.)
33 (in.)
66
Figure 9.3-2 Error from inclinometer 2
Figure 9.3-3 Error envelope for 72 ft spacing, Load Case B
‐0.015
‐0.010
‐0.005
0.000
0.005
0.010
0.015
0 90 180 270 360 450 540 630 720
Deflection (in)
Distance Along Bridge (ft)
Error 2
Error (Sensor 2)(90 ft)
Error (Sensor 2)(72 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 60 120 180 240 300 360 420 480 540 600 660 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case B (72 ft)
No Error
Error Added (Inclinometer 2)
Error Subtracted (Inclinometer 2)
67
Figure 9.3-4 Error envelope for 90 ft spacing, Load case B
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 60 120 180 240 300 360 420 480 540 600 660 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case B (90 ft)
No Error
Error Added (Inclinometer 2)
Error Subtracted (Inclinometer 2)
68
69
Chapter 10: CONCLUSIONS AND RECOMMENDATIONS
Based on the results of this study a method of using inclinometers to calculate deflection
has been determined. The advantages of using inclinometers to measure deflection; it is possible
to measure static and dynamic deflections, ease of installation, and are unobtrusive.
Two different methods of determining deflection from rotation were discussed in this
study, rotation to deflection (R2D) method and base shape method. These methods each have
their advantages and disadvantages. They were tested against results from a finite element
model of the Kealakaha Stream Bridge, around 20 miles north of Hilo. The Kealakaha Stream
Bridge is a 720 foot, 3 span; bridge constructed of precast girders and cast in place box girders
over the Piers.
The R2D method is based on beam theory, where the rotation is integrated to obtain the
deflection. This method has been around for a long time and has been tested on multiple
bridges. It is a forward marching equation, where it starts at one end of the bridge and calculates
the deflection as progresses along the bridge. The mathematics can get very difficult with this
method when the cross section is not constant.
The base shape method of determining deflection is efficient. This method involves using
a combination of predetermined, base, deflections to obtain the displaced shape. The base
shape method accounts for discontinuities, such as changes in the moment of inertia, without all
the complex mathematics. An accurate finite element model is needed for this method, it is
necessary to perform a load test of the bridge to test the model.
The R2D method and base shape method were tested against 9 loading conditions, A-I.
These load combinations are a combination of HS-20 truck loadings in the center of spans 1, 2,
and 3.
Looking at the three different location schematics for the rotation to deflection method,
there was a clear trend of which spacing was more accurate. Suggested inclinometer
installations in the order of preference;
1) 60 foot spacing with 18 inclinometers
2) 90 foot spacing with 14 inclinometers
3) 72 foot spacing with 14 inclinometers, as an advantage for load cases F (abutment
span deflections and H
70
Considering the base shape methodology of determining the deflection one inclinometer
spacing was more accurate. The 72 foot spacing is more accurate for load cases A – I.
The suggested strategy for calculation is to use the base shape method for real time
“look” at the data. Use the R2D method for detailed analysis.
Advantages of R2D method over Base Shape are that R2D is not dependent on the
accuracy of the SAP2000 model. Load testing needed to calibrate the SAP2000 model to
minimize error in the Base shapes formulation.
71
APPENDIX A: R2D GRAPHS
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case A
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case B
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
72
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case C
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case D
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
73
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case E
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case F
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
74
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case G
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case H
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
75
Δ @ 90 ft
Load
Combination SAP 2000 60 ft spacing Error
(in) (in) %
A 0.085 0.083 3.08
B 0.082 0.080 1.94
C 0.078 0.077 1.82
D -0.090 -0.092 2.58
E -0.107 -0.104 2.95
F -0.136 -0.130 4.88
G -0.011 -0.015 35.01
H -0.051 -0.047 9.35
I -0.026 -0.024 6.16
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Case I
SAP
Delta (60 ft)
Delta (72 ft)
Delta (90 ft)
76
Δ @ 90 ft
Load
Combination SAP 2000 72 ft spacing Error
(in) (in) %
A 0.085 0.081 4.57
B 0.082 0.077 6.30
C 0.078 0.072 7.38
D -0.090 -0.094 4.97
E -0.107 -0.093 13.15
F -0.136 -0.106 22.53
G -0.011 -0.022 88.37
H -0.051 -0.024 52.35
I -0.026 -0.017 34.94
Δ @ 90 ft
Load
Combination SAP 2000 90 ft spacing Error
(in) (in) %
A 0.085 0.076 10.87
B 0.082 0.069 15.38
C 0.078 0.062 20.48
D -0.090 -0.087 3.12
E -0.107 -0.078 27.30
F -0.136 -0.056 58.75
G -0.011 -0.025 114.77
H -0.051 0.020 138.17
I -0.026 -0.009 63.29
77
Δ @ 630 ft
Load
Combination SAP 2000 60 ft spacing Error
(in) (in) %
A 0.083 0.078 5.81
B 0.080 0.083 3.82
C 0.077 0.087 13.58
D -0.089 -0.090 1.21
E -0.107 -0.104 2.47
F -0.135 -0.132 1.96
G -0.013 -0.003 72.76
H -0.052 -0.053 2.90
I -0.027 -0.021 21.30
Δ @ 630 ft
Load
Combination SAP 2000 72 ft spacing Error
(in) (in) %
A 0.083 0.088 6.05
B 0.080 0.068 14.96
C 0.077 0.048 37.61
D -0.089 -0.098 10.00
E -0.107 -0.098 8.42
F -0.135 -0.110 18.32
G -0.013 -0.051 295.81
H -0.052 -0.022 57.75
I -0.027 -0.029 9.75
78
Δ @ 630 ft
Load
Combination SAP 2000 90 ft spacing Error
(in) (in) %
A 0.083 0.077 6.91
B 0.080 0.063 20.88
C 0.077 0.048 37.25
D -0.089 -0.088 1.41
E -0.107 -0.079 25.87
F -0.135 -0.057 57.99
G -0.013 -0.040 213.35
H -0.052 0.021 141.35
I -0.027 -0.016 38.72
79
APPENDIX B: BASE SHAPE GRAPHS
Load Combinations A-I
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination A (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
80
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination B (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination C (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
81
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination D (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination E (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
82
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination F (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination G (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
83
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination H (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination I (Mountainside Girder)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
84
Δ @ 90 ft
Load
Combination SAP 2000 72 ft spacing Error
(in) (in) %
A 0.085 0.085 0.02
B 0.082 0.081 0.77
C 0.078 0.078 0.16
D -0.090 -0.100 11.19
E -0.107 -0.098 9.01
F -0.136 -0.112 17.80
G -0.011 -0.022 89.54
H -0.051 -0.027 47.22
I -0.0255 -0.0165 35.15
Δ @ 630 ft
Load
Combination SAP 2000 72 ft spacing Error
(in) (in) %
A 0.083 0.084 0.48
B 0.080 0.080 0.38
C 0.077 0.076 0.09
D -0.089 -0.099 11.07
E -0.107 -0.098 7.75
F -0.135 -0.112 17.04
G -0.013 -0.023 79.45
H -0.052 -0.028 45.37
I 0.083 0.084 0.48
85
Δ @ 90 ft
Load
Combination SAP 2000 90 ft spacing Error
(in) (in) %
A 0.085 0.086 0.61
B 0.082 0.082 0.28
C 0.078 0.078 0.01
D -0.090 -0.108 20.13
E -0.107 -0.093 13.40
F -0.136 -0.059 56.50
G -0.011 -0.030 162.68
H -0.051 0.027 151.69
I -0.026 0.086 435.73
Δ @ 630 ft
Load
Combination SAP 2000 90 ft spacing Error
(in) (in) %
A 0.083 0.083 0.24
B 0.080 0.080 0.80
C 0.077 0.077 0.26
D -0.089 -0.106 18.32
E -0.107 -0.093 12.42
F -0.135 -0.064 52.71
G -0.013 -0.029 128.53
H -0.052 0.020 139.61
I 0.083 0.083 0.24
86
Other Load Combinations
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 21 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 23 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
87
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 27 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 29 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
88
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 30 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 32 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
89
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 36 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 38 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
90
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 39 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 40 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
91
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 41 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 42 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
92
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 43 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
‐0.6
‐0.4
‐0.2
0.0
0.2
0.4
0 90 180 270 360 450 540 630 720
Deflection (in.)
Distance Along Bridge (ft.)
Load Combination 44 (Mountain Side)
SAP
Base Shapes (72 ft)
Base Shapes (90 ft)
93
Δ @ 90 ft
SAP 2000 72 ft
spacing Error SAP 2000
90 ft
spacing Error
(in) (in) % (in) (in) %
21 -0.067 -0.085 26.46 -0.067 -0.101 50.82
23 0.066 0.067 2.30 0.066 0.066 0.99
27 0.018 0.017 6.75 0.018 0.017 8.22
29 -0.008 -0.014 67.66 -0.008 -0.016 95.94
30 -0.112 -0.035 68.96 -0.112 0.000 100.22
32 0.053 0.052 2.15 0.053 0.051 2.92
36 0.018 0.019 3.82 0.018 0.021 12.56
38 -0.008 -0.007 17.16 -0.008 0.000 105.68
39 0.099 0.098 0.64 0.099 0.098 0.43
40 0.097 0.097 0.13 0.097 0.098 0.70
41 0.092 0.092 0.28 0.092 0.093 0.65
42 0.088 0.088 0.43 0.088 0.086 1.84
43 0.087 0.086 1.64 0.087 0.081 7.16
44 0.084 0.082 2.76 0.084 0.076 9.36
94
Δ @ 630 ft
SAP 2000 72 ft
spacing Error SAP 2000
72 ft
spacing Error
(in) (in) % (in) (in) %
21 -0.008 -0.014 70.14 -0.008 -0.016 92.79
23 0.018 0.016 8.96 0.018 0.017 7.53
27 0.065 0.066 1.41 0.065 0.065 0.82
29 -0.068 -0.085 25.71 -0.068 -0.099 45.39
30 -0.008 -0.005 33.92 -0.008 0.005 159.90
32 0.018 0.020 8.54 0.018 0.021 16.55
36 0.052 0.051 3.35 0.052 0.050 4.66
38 -0.112 -0.041 63.66 -0.112 -0.013 88.21
39 0.054 0.054 0.35 0.054 0.054 0.20
40 0.065 0.065 0.35 0.065 0.065 0.23
41 0.075 0.075 0.42 0.075 0.074 0.16
42 0.052 0.051 1.62 0.052 0.054 4.41
43 0.061 0.061 0.58 0.061 0.067 10.46
44 0.069 0.071 2.01 0.069 0.077 10.74
95
REFERENCES
[Aki & Robertson 2005] Aki, Kainoa D. & Robertson, Ian N. “Deflection Monitoring
Systems in Static and Dynamic Conditions,” Research Report UHM/CEE/05-02, University of
Hawaii, Honolulu, HI, 2005.
[Burdet & Zanella, 2000] Burdet, Oliver & Zanella, Jean-Luc. “Automatic Monitoring of
Bridges using Electronic Inclinometers,” 16th Congress of IABSE, Lucerne, 2000.
[Chopra, 2007] Chopra, Anil K. Dynamics of Structures: Theory and Applications to
Earthquake Engineering. Pearson Prentice Hall, Upper Saddle River, NJ, 2007.
[CSI 2010] Computers and Structures, Inc, SAP2000 V14.2, Berkeley CA, 2010
[Earthquake Protection Systems, 2003] Earthquake Protection Systems, “Technical
Characteristics of Friction Pendulum TM Bearings,” Earthquake Protection Systems, Vallejo,
California, 2003.
[Fujiwara, 2010] Fujiwara, David, Hamada, Harold, Iwamoto, Gary, and Robertson, Ian
N., “Project Kealakaha Stream Bridge Replacement,” ASPIRE magazine, Summer 2010, p 22-
24.
[Fung, 2003] Fung, Stephanie S.Y. & Robertson, Ian N., “Seismic Monitoring of Dynamic
Bridge Deformations Using Strain Measurements.” Research Report UHM/CEE/03-02, University
of Hawaii, Honolulu HI, 2003
[Johnson, 2007] Johnson, Gaur P. & Robertson, Ian N. “Structural Health Monitoring
Systems for Civil and Architectural Structures: LVDT-Taught-Wire Baselines, Crack Monitoring
Devices, & Strain Based Deflection Monitoring Algorithms.” Research Report UHM/CEE/07-02,
University of Hawaii, Honolulu, HI, 2007.
[Stephens, 1996] Stephens, Todd W. & Robertson, Ian N. “Kealakaha Stream Bridge
Replacement Seismic Instrumentation Plan”, Research Report UHM/CEE/ 96-04, University of
Hawaii, Honolulu HI, 1996.
[Vurpillot, 1998] Vurpillot, S., Krueger, G., Benouaich, D., Clement, D., and Inaudi, D.,
“Vertical Deflection of a Pre-Stressed Concrete Bridge Obtained Using Deformation Sensors and
Inclinometer Measurements”, ACI Structural Journal, Vol. 95, No. 5, September-October 1998.
[Xingmin, 2005] Xingmin, Hou & Xueshan, Yang. “Using Inclinometers to Measure Bridge
Deflection,” Journal of Bridge Engineering, Vol. 10. No. 5, September/October 2005. p 564-569.