STRUCTURAL HEALTH MONITORING WITH THE MODAL STRAIN ENERGY
METHOD DURING SEISMIC LOADING
A Project
Presented to the faculty of the Department of Civil Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Civil Engineering
(Structural Engineering)
by
Kurt Keiichi Plenert Horiuchi
SPRING
2014
STRUCTURAL HEALTH MONITORING WITH THE MODAL STRAIN ENERGY
METHOD DURING SEISMIC LOADING
A Project
by
Kurt Keiichi Plenert Horiuchi
Approved by:
__________________________________, Committee Chair
Dr. Benjamin Fell, P.E.
____________________________
Date
ii
Student: Kurt Keiichi Plenert Horiuchi
I certify that this student has met the requirements for format contained in the University format
manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for
the thesis.
__________________________, Graduate Coordinator ___________________
Dr. Matthew Salveson, P.E.
Date
Department of Civil Engineering
iii
Abstract
of
STRUCTURAL HEALTH MONITORING WITH THE MODAL STRAIN ENERGY
METHOD DURING SEISMIC LOADING
by
Kurt Keiichi Plenert Horiuchi
The purpose of this project is to provide a review of existing literature and research in the field of
Structural Health Monitoring, and exercise the concepts obtained from the review in theoretical
example 4-story steel building. The structural health monitoring system will be designed to
monitor the performance and detect damage in a special steel moment resisting frame and special
steel concentrically braced frame under lateral seismic loading. To aid in the design and
implementation of the structural health monitoring system, pushover analyses were conducted to
assess likely locations of damage. From the pushover analyses, a set of damage cases were
created for each frame that were used in a theoretically confirm the Modal Strain Energy damage
detection algorithm. The modal strain energy method is a vibration-based method of structural
health monitoring that utilizes operational modal analysis and the time domain decomposition
technique. For each damage case, computer modal analysis was used to obtain theoretical mode
shape data of the structure. In each damage case, the modal strain energy method algorithm was
appeared to correctly identify in the story level of the lateral force resisting frame that contained
the damage.
_______________________, Committee Chair
Dr. Benjamin Fell, P.E.
_______________________
Date
iv
TABLE OF CONTENTS
Page
List of Tables…………………………………………………………………………………….viii
List of Figures…………………………………………………………………………………..…ix
Chapter
1. INTRODUCTION ....................................................................................................................... 1
1.1
Motivation .......................................................................................................... 1
1.2
Objectives and Scope.......................................................................................... 3
1.3
Organization and Outline.................................................................................... 4
2. LITERATURE REVIEW AND MODAL STRAIN ENERGY METHODOLOGY................... 6
2.1
Goals and Limitations of Structural Health Monitoring ..................................... 6
2.1.1 The Process of Structural Health Monitoring ......................................... 6
2.1.2 Facts and Limitations of Structural Health Monitoring ........................ 12
2.2
Past and Present Issues in Structural Health Monitoring.................................. 17
2.2.1 Structural Health Monitoring Economy ............................................... 17
2.2.2 Structural Health Monitoring Instrumentation ..................................... 18
2.2.3 Data Processing and Communications ................................................. 19
2.2.4 Research and Development .................................................................. 20
2.3
The Modal Strain Energy Method of Structural Health Monitoring ................ 20
3. ARCHETYPE BUILDING DESCRIPTION AND PUSHOVER ANALYSIS ........................ 27
v
3.1
Project Introduction .......................................................................................... 27
3.1.1 East –West: Special Steel Moment Resisting Frame ............................ 28
3.1.2 North-South: Special Steel Concentrically Braced Frame ................... 29
3.2
Structural Health Monitoring Assumptions ...................................................... 31
3.2.1 Gravity Loads and Lateral Loads ......................................................... 31
3.2.2 Material and Member Assumptions...................................................... 32
3.3
Analysis Layout ................................................................................................ 34
3.4
East-West Frames: Pushover Analysis and Results.......................................... 35
3.4.1 Pushover Analysis ................................................................................ 35
3.4.2 Pushover Analysis Results.................................................................... 35
3.5
North-South Frames: Pushover Analysis and Results ...................................... 37
3.5.1 Pushover Analysis ................................................................................ 37
3.5.2 Pushover Analysis Results.................................................................... 38
4. PROPOSED ARCHETYPE BUILDING INSTRUMENTATION SYSTEM DURING
DYNAMIC LOADING ............................................................................................................ 40
4.1
General Structural Health Monitoring Layout .................................................. 40
4.1.1 Data Collection and Pre-processing...................................................... 40
4.1.2 General Sensor Layout ......................................................................... 41
4.2
E-W: Special Steel Moment Resisting Frame Instrumentation and Analysis .. 41
vi
4.3
N-S: Special Steel Concentrically Braced Frame Instrumentation
and Analysis ..................................................................................................... 46
5. SUMMARY…………… ........................................................................................................... 52
5.1
Summary........................................................................................................... 52
5.2
Conclusions and Recommendations ................................................................. 54
APPENDIX A. AISC SHAPES ..................................................................................................... 56
APPENDIX B. PUSHOVER ANALYSIS RESULTS .................................................................. 58
APPENDIX C. TIME DOMAIN DECOMPOSITION TECHNIQUE .......................................... 63
APPENDIX D. ERRORS ASSOCIATED WITH THE CALCULATION OF
MODAL CURVATURE THROUGH NUMERICAL DIFFERENTIATION .... 68
APPENDIX E. SAP2000 MODAL ANALYSIS RESULTS ........................................................ 71
REFERENCES………………. ..................................................................................................... 97
vii
LIST OF TABLES
Tables
Page
3.1 Reduced Beam Section (RBS) Geometry Values .................................................................... 29
3.2 Dead and Live Loads on the Structure ..................................................................................... 31
3.3 Seismic Design Criteria as per ASCE 7-10 ............................................................................. 32
3.4 Vertical Distribution Ratios ..................................................................................................... 32
4.1 SSMRF Mode Shape Data ....................................................................................................... 43
4.2 SSMRF Mode Period ............................................................................................................... 44
4.3 SSMRF Damage Indicators ..................................................................................................... 45
4.4 SSCBF Mode Shape Data ........................................................................................................ 48
4.5 SSMRF Mode Period ............................................................................................................... 49
4.6 SSCBF Damage Indicators ...................................................................................................... 50
viii
LIST OF FIGURES
Figures
Page
3.1 Plan view of 4-story steel structure for SHM application ........................................................ 27
3.2 Elevation view of E-W Special Steel Moment Resisting......................................................... 28
3.3 Reduced Beam Sections ........................................................................................................... 29
3.4 N-S Special Steel Concentrically Braced Frame Elevation ..................................................... 30
3.5 N-S Concentrically Braced Frame Elevation and Bracing Members ...................................... 30
3.6 A992 Steel Stress vs. Strain ..................................................................................................... 33
3.7: (a) E-W Pushover Collapse Mechanism and (b) Damage States 1-3 considered as
part of the SHM investigation in Chapter 4............................................................................. 36
3.8 E-W Pushover Curve: Base Shear vs. Displacement ............................................................... 36
3.9 (a) N-S Pushover Collapse Mechanism and (b) Damage States 1-3 considered as
part of the SHM investigation in Chapter 4............................................................................. 38
3.10 N-S Pushover Curve: Base Shear vs. Displacement .............................................................. 38
4.1 Instrumentation of the Special Steel Moment Resisting Frame ............................................... 42
4.2 First Three Modes of the Special Steel Moment Resisting Frame ........................................... 43
4.3 Moment Frame Yield Event Diagram...................................................................................... 44
4.4 Instrumentation of the Special Steel Concentrically Braced Frame ........................................ 47
4.5 First Three Modes of the Special Steel Concentrically Braced Frame .................................... 48
4.6 SSCBF Yield Event Diagram .................................................................................................. 49
ix
1
CHAPTER 1. INTRODUCTION
1.1
Motivation
To address the many complexities in structural design of buildings and other structures, and to
simplify the design process of ordinary structures, engineers often rely on provisions from
established organizations such as the American Society of Civil Engineering 7 (2010) for
minimum loads to apply to buildings, and the American Institute of Steel Construction (AISC,
2011) and the American Concrete Institute (ACI, 2012) for nominal strength equations. These
provisions are widely accepted methods for design, providing the designer with loading
information, suitable analysis approaches, and material and member strength capacities. During
design, mean values of material properties are routinely used, as are simplified analysis
techniques to save costly design time. Examples of the latter include soil structure interaction, and
connection behavior – the effects of which are usually ignored. Modern design methodologies,
such as Load Resistance and Factor Design (LRFD) (ASCE, 2010) are useful in resolving the
statistical distribution of building materials and structural loadings, leading to load combination
equations with amplification factors based on the variability of each load. For example, live load
surveys have demonstrated much more scatter about the mean load as compared to dead loads,
and thus, after a structural reliability analysis, are assigned higher load factors (1.6 as compared
to 1.2 for dead loads). In the same reliability analysis, capacity reduction factors are applied to
nominal connection and member strengths according to a target safety index for the component.
The goal of LRFD is to provide an economical design in service load situations, while guarding
against catastrophic failure in overload situations.
While reliable and safe, typical design and the inherent assumptions made therein, leads to a
product, which lacks the level of precision of those in other engineering fields. Additionally, as
2
the building or bridge structure ages, visual inspections are often the only method used to ensure
that society is safe to utilize the structure for its intended purpose. It is not common to precisely
verify, through instrumentation, the actual performance of the structure. Beyond the design
calculations and analysis models, which may be very accurate in some instances, it is difficult to
know the actual load paths and their magnitudes of the built structure. In addition, assessing any
damage or ageing of a structure is left up to annual building inspection. During inspections, when
serious issues are found, the structure may be taken out of service until a solution can be designed
and constructed. At this point, the opportunity may have passed to implement a cheaper and
easier solution to the initially smaller problem.
Structural Health Monitoring (SHM) is a technique that can provide information as to the load
demands on existing structures, while also assessing the overall integrity of the structural system.
There are various methods that are currently used to accomplish this goal, but they all rely on
instruments such as strain gauges and accelerometers, combined with data acquisition systems
and, in some cases, wireless communication technology. With all of these components, a SHM
system creates a neural system of sensors and central data processing to monitor the real-time
performance of the structure and assess potential damage. Damageability can be detected through
a number of post-processing routines using ambient vibration measurements from recorded
accelerations, displacements, or material strains. By applying basic engineering principles all the
way through to sophisticated structural verification, one can find the structural stiffness of a story,
member, connection, or even a critical zone of a member. A permanent reduction in stiffness
implies that the monitored element has undergone damage.
While the SHM system allows engineers to monitor load demands on critical members, as well as
damageability states, the system also provides a feedback loop to help improve future modeling
3
capabilities. The purpose of this project is to describe the basic principles of SHM and provide a
sample instrumentation plan for a 4-story building that uses a moment resisting frame and
concentrically braced frame to resist lateral loads. A seismic loading case is considered for this
sample building, in which the SHM system is designed for using a method of vibration-based
SHM that evaluates structural health through operational and ambient vibrations.
1.2
Objectives and Scope
The first objective of this document is to provide a thorough review of available SHM literature.
This review is to include information from the initial development of SHM through to
contemporary research being conducted. The purpose of the literature review is to compile a set
of information that the reader can use to develop a clear understanding of the background and
requirements of the discipline. The intent is to leave the reader with stronger grasp of how to
design and implement a SHM system, by referencing common methods, challenges, and potential
solutions.
The second objective of this document is to derive the theory behind a specific SHM method
referred to as the Modal Strain Energy Method as it was first stated by Stubbs and Kim in their
research document titled “Field Verification of Non-destructive Damage Localization and
Severity Estimation Algorithm” (Stubbs 1996). This method was developed for dynamic bridge
monitoring, and is now widely implemented for many other structures. Since the initial
development of this theory, others have since expanded it for use in more specific cases such as
bending plates and detailed damage severity estimations.
The final objective of this document is to provide the reader with a detailed example of a partial
design of a SHM system for a 4-story steel building using special moment resisting and special
4
concentrically braced lateral force resisting systems. The design will be based on modal analyses
and pushover analyses to determine important locations to monitor on the building. The layout of
the story specific SHM sensor system will then be created from this information. The pushover
analysis will also provide a collapse event timetable to create realistic damage cases to use the
MSEM to theoretically detect damage after each event. Recommendations will be made as to
further SHM system design, as well as revisit key design concepts mentioned in the literature
review.
1.3
Organization and Outline
Chapter 2 shall be composed of the SHM literature review. The review is to be guided by the
requirements of the SHM system, theoretical implications, and instrumentation obstacles
presented during the general design of the SHM system. Basic guidelines to SHM system design
will follow from this. These guidelines are intended to be extensions of decisions that the
designer will need to make before, and during design process to arrive at the ideal SHM system
for the structure. Lastly, this chapter will lay out the theory behind the Modal Strain Energy
method of SHM. This theoretical development will involve equation derivation for stiffness
calculation and statistical layout of data processing based on the theory originally developed by
N. Stubbs and J.-T. Kim, and using numerical approximations mentioned in Appendix C and D.
Chapter 3 will begin with a description of the design of a SHM system for a four-story steel
building. Details for the lateral force resisting system will be described for both lateral axis of the
building. A list of SHM system assumptions and their reasoning will be defined. The SHM
system will be designed to monitor for seismic loadings as well as system stiffness reduction,
therefore these criteria will guide the reasoning behind the assumptions and the design analysis.
Finally, pushover analyses will be performed and their results summarized for the structure. A
5
brief narrative will be provided to guide the reader through the analysis and how the results will
aid in the SHM system design.
In chapter 4, the information from chapter 3 will be used to instrument the four-story steel
building. Sensor type, and placement will be shown, along with the reasoning behind the layout.
For each lateral force resisting system, the results of the pushover analysis will be used to create
three damage cases that the MSEM algorithm will be used to detect damage in. Damage detection
results will be summarized in a table for each lateral force resisting system.
Chapter 5 will provide a summary of the report as well as recommendations for future
investigation of the SHM system. This chapter will also aim to complete the document with key
observations and conclusions drawn from the four-story steel building example with respect to
the information provided in the literature review.
6
CHAPTER 2. LITERATURE REVIEW AND MODAL STRAIN ENERGY
METHODOLOGY
The purpose of this literature review is to familiarize the reader with past research and
advancements in the field of SHM, and to develop a procedure for the design of the health
monitoring system of a 4-story steel building presented in subsequent chapters of this report. The
review includes literature from the earliest uses of SHM in civil structures, to contemporary
research currently being used in the field. This will provide a basis for the design and
implementation of a SHM system, and the framework of the modal strain-energy method for
SHM.
2.1
Goals and Limitations of Structural Health Monitoring
To monitor the health of a structure, the SHM system gathers data from the structure to infer
loading conditions. The measured data from the SHM system is used, along with engineering,
physics, and mathematics principles to evaluate the status of the structure in terms of damage and
its location. Because SHM infers structural behavior from actual in-field conditions, it becomes
an invaluable asset for, not only damage detection, but design confirmation.
2.1.1
The Process of Structural Health Monitoring
A general process for SHM and system design was described in a book titled “Fibre Optic
Methods for Structural Health Monitoring” (Glisic 2007). The process outlined key phases of
SHM system design and related tasks, and is as follows.
i.
Establishment of the goal(s) of health monitoring
ii.
Identifying and selecting representative parameters to be monitored
iii.
Selecting appropriate monitoring systems
7
iv.
Designing the sensor network
v.
Establishing the monitoring schedule
vi.
Developing the data exploitation system
vii.
Developing the cost of the monitoring system
For the purposes of this document, these concepts will be used and frequently revisited. The
following subsections outline the requirements and deliberation of each of the above mentioned
phases.
2.1.1.1 Establishment of the Goal(s) of Health Monitoring
Clearly defining the goals of the SHM system is important because there are several types of
monitoring systems that each accomplishes a separate set of goals. Defining the goals involve
careful outlining of objectives for the SHM system. A few helpful questions that can help target
the goals are as follows.
a. What is the level of monitoring required (simple low level monitoring or high level
detailed monitoring)?
b. How extensive does the monitoring need to be (comprehensive monitoring or partial
monitoring)?
The intent of defining SHM goals is to outline a scope and avoid designing a system that falls
short of its expectations or includes unnecessary and expensive appurtenances. As mentioned
previously, the goal of SHM for the context of this report will be to detect seismic damage to a
steel moment resisting frame and braced frame.
8
2.1.1.2 Identifying and Selecting Representative Parameters to be Monitored
Some examples of parameters that can be monitored are material strain at a key location on the
structure, or acceleration or deflections at a location in the building. The choice of parameters to
be monitored by the SHM system is related to the list of SHM goals and scope defined in the first
phase of SHM system design. It can be a complicated and tedious process to develop the list of
parameters, so the list of goals and scope can help abate the task. In general, a detailed structural
analysis is important in defining parameters to monitor. Analyses, such as a pushover analysis,
help identify areas of the structure that are at risk of damage. For example, if it is determined that
excessive building drift is a potential problem for the structure, instrumentation at each floor level
to monitor story drift in the building is appropriate; story drift being the parameter being
monitored. This process of choosing parameters gives the engineer a more clear direction to move
in as they begin to develop the finer details of the SHM system.
The parameters that will be monitored as part of this report are the accelerations at each level of a
steel moment frame and braced frame lateral resisting components of a building. In Chapter 3, the
report qualitatively discusses a method to use strain gauges to more precisely determine the
damaged area of specific members. Chapter 4 will then show how the acceleration data at each
level can be used to detect damage in a specific story level. This will be shown performed for
both frame systems.
2.1.1.3 Selecting Appropriate Monitoring Systems
The selection of the appropriate monitoring system type can be a challenging decision if a scope
is not clearly defined. In general, this decision is guided by the expected loadings and behavior of
the structure. For example, structures that are more predisposed to dynamic lateral loadings, such
9
as earthquakes and wind loads, require a different type of monitoring system then a structure that
is expected to experience larger amounts of creep, (material deformation caused from long term
stress) due to large gravity loads. Listed below, are the three types of SHM systems that currently
exist, and what they entail.
a. Static Monitoring: Monitoring of static parameters such as strains, deflections, and
curvature. These systems are economical and good for long term monitoring damage
from static loadings, such as creep.
b. Dynamic Monitoring: Monitoring dynamic parameters such as acceleration and
dynamic strains. These systems are good for monitoring dynamic loads caused by
earthquakes, wind loads, traffic, heavy machinery, etc.
c. System Identification and Modal Analysis: Monitoring parameters such as system
modal frequencies. These systems integrate elements of dynamic monitoring and
modal response to identify damage. This type of system often uses vibration in the
system to measure the structure’s response.
2.1.1.4 Designing the Sensor Network
In modern SHM systems, computers are used to collect, process, and summarize data given by
the sensors on the structure. The sensor network in the SHM system refers to the interconnected
neural network of sensors and their location on the structure. These sensors report their
measurement data to a central computer for data processing.
Minimizing the complexity of the sensor network is important for efficiency and economy. Aside
from complicated instrumentation requirements, complex systems can sometimes lead to data
10
processing problems and difficulties during instrumentation. This is discussed in further detail
later in this chapter, but is a factor in the design of the sensor network.
A few questions that can help guide the design of the sensor network are as follows.
a. What are the optimum locations for data readings on the structural system (such as areas
on the structure that are more prone to damage)?
b. Is there symmetry or other geometrical advantage in the structure that can allow the SHM
system to function with fewer sensors or by some other less complicated method?
c. Is there enough redundancy in the system to provide supportive data for calculations of
damage location?
2.1.1.5 Establishing the Monitoring Schedule
SHM systems also differ in how they collect data. Some systems collect data on a continuous
basis. Other systems collect data at certain time intervals, such as on a daily basis. It is even
possible to program the system to stay dormant until the structure experiences activity in its
sensors above a certain range, effectively reading data during a particular event.
Continuous systems are ideal for monitoring structures that are expected to change loading often,
such as a bridge with traffic loads, or a tall building that experiences large wind loads. Time
interval systems are optimal for monitoring the structural performance during the life of the
structure. This is good for evaluating structural creep as well as serviceability damage. Time
interval systems are not, however, well equipped for monitoring disastrous events such as
earthquakes. These systems would likely only record data at a single time instance during the
event, rather than at many time instances in the case of a continuous monitoring system. Dormant
systems are particularly useful for monitoring unpredictable events while still being efficient with
11
its data processing and storage. These systems do not require as much backup storage space for
data records since comprehensive data records for the entire structure are not created as often.
However, because data records are not as extensive for this monitoring schedule type, it may not
be an ideal system for special cases where structure loadings are constantly changing.
Regardless of the monitoring schedule type, the SHM system goals, defined earlier, need to
support this decision. Several helpful questions to consider during the decision-making process
are as follows. What are the goals of the SHM system and how do they relate to the monitoring
schedule? What are the chances of an earthquake, heavy wind storm, or other unpredictable
event, in which continuous monitoring would be necessary? Are there any hazards that the system
would not be prepared for if the structure was not continuously monitored? If unpredictable
events and other hazards are not an issue, is there another reason that the structure requires
continuous monitoring? What is the SHM financial budget, and how will it be affected by the
choice of monitoring schedule?
2.1.1.6 Developing the Data Exploitation System
Data exploitation involves extraction and processing of sensor data. The first step in this process
requires the sensors in the sensor network to obtain data by measuring a quality of the structure.
Once the sensors have obtained data from the structure, they must send the data to the central
computer through a communication network. The central computer then processes the data to a
useable state and reports the findings. During the data processing step extensive data
manipulation, using material properties, geometry, and time history data is necessary. The type of
data manipulation depends on the type of SHM system as defined by the previous phases. In the
case of the seismic force resisting 4-story steel building example that will be discussed in chapter
3 and 4, data manipulation involves calculations as per the modal strain-energy method, as
12
described in section 2.3. It is also important to note that data processing can be strenuous in
regards to computer processing, and can thus present a few issues that are discussed later in
section 2.2.
2.1.1.7 Developing the Cost of the Monitoring System
The last phase of SHM system development is evaluating the system in terms of its financial cost.
Costs of design, instrumentation, operation, and maintenance are all factors in this evaluation. In
the cases where the SHM system is deemed too expensive by the owner, it is important to prepare
evidence to support the current system design, as well as prepare a list of elements of the SHM
system that can be reduced or removed and the consequences of doing so. To help identify
elements of the SHM system that can be reduced or removed, the engineer should reconsider the
scope and goals of the SHM system in terms of its sensor network, monitoring schedule, etc.
2.1.2
Facts and Limitations of Structural Health Monitoring
Like other disciplines, SHM has physical restrictions that limit it. In a paper titled “The
Fundamental Axioms of Structural Health Monitoring” (Worden 2007), the authors attempt to
describe the foundation of SHM in terms of its limitations and fundamental scientific truths.
The following is a list of the most important fundamental truths of SHM followed by a brief
description of each item.
i.
All materials have flaws
ii.
Two system states must be compared to assess damage
iii.
There are various depths of damage detection
iv.
Sensors cannot measure damage
v.
SHM sensing systems are defined by how damage is initiated and evolved
13
vi.
There is a tradeoff between damage sensitivity and noise
2.1.2.1 All Materials Have Flaws
Regardless of how a material is formed, flaws are always present because of a variety of
contributing factors. For example, some flaws result from the arrangement of material particles at
the molecular level, in which the presence of electrical signals, heat, or large/small dissimilar
molecules alter the arrangement of molecules in a material lattice, and thus change the behavior
of the material. In the field of material science and engineering, engineers and scientists use
material flaws to gain specific properties in the material. In the case of carbon steel, a small
amount of carbon is alloyed with iron to change the lattice arrangement of the iron atoms to give
carbon steel the properties that it is known for.
Unfortunately, there are no practical methods that allow a material to be formed without any
flaws. This results in uniquely flawed materials, in which determining its actual material
properties, such as elastic strength and yield strength, are difficult to do without extensive
material testing. Consequently, engineers must rely on material classifications that regulate the
formation of a material in order to assume material properties that have been empirically
approximated through extensive material testing.
2.1.2.2 Two System States Must Be Compared To Assess Damage
As described earlier, SHM systems determine the health of a structure from a system of sensors
that measure qualities in the structure. The SHM system does not measure damage directly, but
rather, it compares the structure’s current state to its undamaged state in order to assess damage in
the structure. There are a variety of different methods that take advantage of sensor readings on
the newly built structure to use for its undamaged state, however, in the case of the modal strain
14
energy method of SHM, which shall be the focus of the later chapters of this document, the
undamaged state can either be determined from sensor readings of the structure before damage
has occurred, or a theoretical state determined from elastic properties of the structure’s materials.
2.1.2.3 There are Various Depths of Damage Detection
In general, there are four levels of damage detection in a structure; each requiring a different level
of human supervision. The first level is the detection of damage in the structure. This can be done
solely by means of computer calculation; i.e. minimal human supervision is necessary for this
level of detection. The second level is pinpointing the location of damage. Again, this can usually
be done solely by means of computer calculation, requiring minimal human supervision. The
third level is detecting the type of damage (cracks, yield, breaks). At this level, it becomes
difficult for the SHM system to correctly identify damage types. With special instrumentation, it
is possible for the SHM system to classify the type of damage, however, the only way to confirm
this classification is through physical human inspection. Finally, the last level of damage
detection is ranking the severity of damage. With special instrumentation, a SHM system can give
a preliminary severity ranking to damage, however this must be confirmed with physical human
inspection as many factors can contribute to error in this preliminary severity ranking, such as
noise or uncommon damage types.
In the case of the MSEM, discussed later in this chapter, the level of damage detection will be
focused on existence and location of damage.
2.1.2.4 Sensors Cannot Measure Damage
As previous discussions in this document have alluded to, sensors cannot measure damage
directly. Sensors, instead, measure a quality such as strain or displacement, and through data
15
processing are able to infer the existence or non-existence of damage in the structure. For
instance, standard displacement transducers measure a one-dimensional relative displacement
between two points. However, they are commonly used to estimate the propagation of cracks
(increased damage) by measuring the displacement between opposite sides of the crack, i.e. how
much the crack has opened up. A change of displacement between opposite sides of the crack
results in a change in the propagation of the crack and, thus, the possibility of new damage in the
structure. The sensor only aids in the detection and/or evaluation of damage by providing data to
the SHM system to process.
2.1.2.5 SHM Sensing Systems are Defined by how Damage is Initiated and Evolved
Just as there are different avenues in which damage can occur in a structure, there are different
types of SHM sensing systems that are appropriate for various situations. For instance, damage
from seismic loadings tend to be better evaluated using VBSHM, where as damage from creep is
usually better evaluated using a static monitoring system. Furthermore, impact type loads to a
structure usually cause localized damage requiring more localized monitoring, while cyclic wind
loads can cause system fatigue, which require system wide monitoring. Each SHM system
operates uniquely, thus, it is important for the engineer to understand the type of damage and the
time scales on which the system will operate. The following is a list of properties that should be
considered for the SHM sensing system.
(i.)
Types of data to be acquired.
(ii.)
Sensor types, number, and locations.
(iii.)
Bandwidth, sensitivity, and dynamic range.
(iv.)
Data acquisition/telemetry/storage system.
(v.)
Power requirements.
16
(vi.)
Sampling intervals (continuous monitoring versus monitoring only after extreme
events or at periodic intervals).
(vii.)
Processor/memory requirements.
(viii.)
Excitation source (active sensing).
(Worden 2007)
The following, is a list of factors that should be considered during the selection of SHM
hardware:
(i.)
The length-scales on which damage is to be detected.
(ii.)
The time-scale on which damage evolves.
(iii.)
How will varying and/or adverse operational and environmental conditions affect the
sensing system.
(iv.)
Cost
(Worden 2007)
This list pinpoints key functions of the hardware to consider during selection, and, ultimately,
aids in the selection of ideal hardware in the SHM system.
2.1.2.6 There is a Tradeoff between Damage Sensitivity and Noise
Sensitive equipment is more receptive to small deviations in measurement. However, with this
gain in sensitivity, a susceptibility to data noise is also gained. Since hyper-sensitive equipment
can detect slight measurement changes, it is also more prone to detecting slight measurement
changes due to foreign environmental conditions. Thus, there is a fundamental tradeoff between
monitoring system sensitivity and noise in the data readings. For example, an exposed
17
hypersensitive accelerometer is prone to measuring vibrations of the instrument from wind
blowing past it rather than only measuring vibrations from the structure. This creates small
fluctuations in data readings that, if not handled correctly can lead to compounding error during
data processing.
2.2
Past and Present Issues in Structural Health Monitoring
SHM in civil structures is still a relatively young discipline compared to other engineering fields.
Although SHM systems are currently being implemented for a wide variety of purposes, it still
has much foreseeable growth in the future. Despite this, the field of SHM has overcome many
obstacles in the past, which is one of the reasons it has become such a promising field. While
some of the obstacles have grown obsolete due to technological advancements, others have been
overcome due to refinement in common SHM practices. Unfortunately, some obstacles still exist
and are contemporary research topics in SHM. This chapter seeks to present some of the major
challenges that SHM is faced with and how it has or has not overcome them.
2.2.1
Structural Health Monitoring Economy
SHM systems are expensive due to the large amount of design work, instrumentation, operation,
and maintenance necessary to implement a SHM system. Despite the many benefits of SHM,
owners are often reluctant to agree to the additional costs to implement a SHM system when,
from their perspective, the structure is fully stable without it. Cost-benefit studies are conducted
to provide evidence to the owner that the benefits of a SHM system outweigh the cost of SHM
system.
Generally, in today’s market, sensors and other instrumentation are more widely available at a
relatively inexpensive price compared to the past. The growth of the SHM field has created a
18
higher demand in this market, and has allowed suppliers to respond with lower prices. As the
field continues to develop, it is likely that more standardized SHM practices will be developed,
which will allow SHM systems to be more economically feasible.
However, other than relying on economic supply and demand to solely guide a cost-benefit
assessment, to further alleviate the financial objections of an owner to the implementation of a
SHM system, the cost-benefit assessment should consider how the following benefits relate to the
system.
a. Structural safety protection.
b. Decreased response time for inspection and repair after a traumatic event.
c. Data guided building inspections to improve the efficiency of them.
d. Possibility for cheaper repairs (if caught before failure).
e. Historical data for future research and retrofit design.
2.2.2
Structural Health Monitoring Instrumentation
Instrumentation for SHM can be challenging because it requires that every sensor be accessible
by personnel (for inspection, repair, or replacement) and computer (for data processing) via wires
or radio signal. This can create complications during instrumentation because the locations of
sensors and wires are restricted by the placement of necessary structural elements and other
furnishings. To avoid this, it is important to coordinate with the architect and other engineers
(MEP, etc.) to carefully instrument the structure and minimize these types of conflicts. Wireless
sensors have become popular recently because they avoid complicated wiring and allow their data
transmittal to be centralized in small communication hubs that report to the central computer. The
communication hubs can be placed in easily accessible areas which helps reduce the problem of
19
instrument access and wiring concerns. Currently, wireless sensors have become ideal because
wireless communication networks, capable of transmitting large amounts of data, can be installed
at a relatively low cost.
Historically, over instrumentation has been known to cause problems during data processing.
Computer processors have a finite amount of data that can be processed at one time. Exceeding
the amount of data that the processor is capable of handling can slow the system down and
potentially freeze it due to overload. Since each additional sensor provides more data for the
system to process, over instrumentation is a common cause of processor overload. In today’s
market, however, processing power is more advanced than it was in the past. Data overload is less
of a concern because the processor capacity is higher, and therefore the overload threshold is
more distant.
2.2.3
Data Processing and Communications
SHM actively relies on a system that communicates between sensor and central computer for data
processing and management. In systems that utilize radio signals to send data to the central
computer, an appropriately sized network connection is necessary. Twenty years ago, dial-up
connections were popular, but would not be sufficient for the size of data transfer required by a
SHM system, according to Brownjohn (2006). Although dial-up connections are obsolete in
America, some of the more advanced communications networks can also be considered too slow
for SHM. In a paper entitled “Structural Health Monitoring on Civil Infrastructure” (Brownjohn
2006), the author estimates an average SHM system requiring at least 700 kbps/3G data
connectivity for data compression and pre-processing in the SHM system. However, this
connectivity rate demand is not uncommon and can easily be met by communication networks in
today’s market.
20
2.2.4
Research and Development
In a paper entitled “Structural Health Monitoring on Civil Infrastructure” (Brownjohn 2006), the
author stresses the necessity of continued research. Although researchers in the field have
scientifically tested all concepts referenced by this document, many unverified theories exist.
Additionally, many tested theories still require more verification to be fully recognized as an
acceptable method of SHM. Brownjohn (2006) notes that SHM often loses grant funding
opportunities to fields that have more immediate results and conclusions. In fact, he explains that
much of the early research and testing in SHM was financed by owners rather than grant money
from the scientific community. Currently, grant funded research is more common, but still a
contributing factor to the growth speed of the field.
SHM also requires the contribution of many disciplines: structural engineers, mechanical
engineers, electrical engineers, computer engineers, and communications engineers. Brownjohn
(2006) explains that, in the past, a lack of collaboration has existed within the research realm of
SHM. Unfortunately, this issue is still apparent and the only way to relieve it is to create
awareness of it.
2.3
The Modal Strain Energy Method of Structural Health Monitoring
The Modal Strain Energy Method (MSEM), also known as the Modal Stiffness technique, was
first proposed in the paper entitled “Field Verification of Nondestructive Damage Localization
Severity” (Stubbs 1996). This report presented the theory behind the MSEM and then attempted
to prove its effectiveness with a full scale example.
The MSEM is a SHM method that falls under the umbrella of Vibration-Based SHM (VBSHM).
These methods are rooted in detecting structural damage by using vibrations to extract
21
information from the entire damaged structure to compare it to previous structural data from
when the structure was still undamaged.
The behavior of a structure is dependent on the behavior of its individual elements; therefore
changes in those elements, such as damage, create a global change. This allows the possibility to
sense local damage from a global perspective. In the case of the MSEM the mode shapes of the
structure provide the global perspective and means to sense member stiffness loss, caused by
damage, at a local level.
For VBSHM, vibrations in the structure are necessary for damage detection. In the past, many
methods of evoking vibrations have been used. Some older techniques involved sending constant
vibrations into the structure by the use of special vibration inducing equipment. This method has
since phased out and been replaced with Operational Modal Analysis (OMA). OMA takes
advantage of the ambient and operational vibrations, such as traffic loads, for the damage
detection algorithm. OMA is ideal because it does not require expensive vibration inducing
equipment, and only requires that noise in the system be minimal. The minimization of noise has
been a popular research topic in OMA SHM in the past; however it is not part of the discussion of
this document. Through previous research, OMA has been confirmed to work conjunctively with
the MSEM, and thus is optimal for its implementation.
As mentioned earlier in this chapter, SHM depends on a comparison of two different states. The
basic process of the MSEM is to calculate the modal stiffness of each finite element member
(member) of an undamaged structure and compare it to the modal stiffness of the same member
on the potentially damaged structure (referred to from here on as the damaged structure). These
ratios are then used for damage detection by statistically pinpointing member ratios that indicate a
22
large decrease in stiffness. The result of this analysis allows the SHM system to locate where
damage has occurred in the structure.
The following is a theoretical development of the MSEM described above, as it was first
proposed by Stubbs and Kim (1996). For a more generalized description of the theory in matrix
form, see the “Vibration-Based Structural Health Monitoring of Highway Bridges” (CalTrans
2008).
For the undamaged structure, Equation 2.1 can be used to calculate the ith modal stiffness of the
structure.
𝐿
𝐾𝑖 = ∫0 𝑘(𝑥)[𝜙𝑖′′ (𝑥)]2 𝑑𝑥
Eqn. 2.1
Where L represents the length of the beam, x represents the location along the length of the beam,
𝑘(𝑥) represents the bending stiffness of the beam, and 𝜙𝑖′′ (𝑥) represents the second derivative of
the ith mode shape of the beam, 𝜙𝑖 (𝑥).
The jth member contributes the following member stiffness, 𝐶𝑖𝑗 , for the ith mode.
𝐶𝑖𝑗 = 𝑘𝑗 ∫𝑗 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥
Eqn. 2.2
In this equation, 𝑘𝑗 is the stiffness of the jth member.
The fractional modal stiffness of the jth member, Fij, can be calculated with the following equation
𝐹𝑖𝑗 =
𝐶𝑖𝑗
𝐾𝑖
Eqn. 2.3
23
The parameters in Equations 2.1, 2.2, and 2.3 can also be calculated for the damaged structure
where asterisks are used to denote variables pertaining to the undamaged structure.
𝐿
∗
𝐾𝑖 ∗ = ∫0 𝑘 ∗ (𝑥)[𝜙𝑖′′ (𝑥)]2 𝑑𝑥
∗
𝐶𝑖𝑗 ∗ = 𝑘𝑗 ∗ ∫𝑗 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥
𝐹𝑖𝑗 ∗ =
𝐶𝑖𝑗 ∗
𝐾𝑖 ∗
Eqn. 2.4
Eqn. 2.5
Eqn. 2.6
By definition, the sum of all of the ith modal stiffnesses for the structure is equal to unity –
∗
𝑁𝐸
∑𝑁𝐸
𝑗=1 𝐹𝑖𝑗 = ∑𝑗=1 𝐹𝑖𝑗 = 1
Eqn. 2.8
Assuming that the structure is made up of multiple members –
𝐹𝑖𝑗 ≪ 1
Eqn. 2.9
𝐹𝑖𝑗 ∗ ≪ 1
Eqn. 2.10
Thus, Stubbs and Kim proposed the following relationship to compare the undamaged and
damaged states –
1 + 𝐹𝑖𝑗 ≈ 1 + 𝐹𝑖𝑗 ∗
Eqn. 2.11
Stubbs and Kim (1996) were able to show that this was a good assumption for a bridge with 50
elements. In general, they pointed out that with their example, each element contributed roughly
2% of the entire structure’s stiffness. However, based on previous research, it is not apparent
what the exact limit is for Equation 2.11 to be an acceptable assumption.
24
This relationship can be used to find the ratio between the stiffness of the undamaged to the
damaged structure by substituting Equations 2.3 and 2.6 into Equation 2.11 and normalizing –
𝐶𝑖𝑗∗
𝐾𝑖 ∗
𝐶𝑖𝑗
1+
𝐾𝑖
1+
1=
=
(𝐶𝑖𝑗 ∗ +𝐾𝑖 ∗ )𝐾𝑖
Eqn. 2.12
(𝐶𝑖𝑗 +𝐾𝑖 )𝐾𝑖 ∗
Substituting Equations 2.1, 2.2, 2.4, and 2.5 into Equation 2.12 yields the following –
∗
1=
∗
𝐿
𝐿
(𝑘𝑗 ∗ ∫𝑗 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥 +∫0 𝑘 ∗ (𝑥)[𝜙𝑖′′ (𝑥)]2 𝑑𝑥) ∫0 𝑘(𝑥)[𝜙𝑖′′ (𝑥)]2 𝑑𝑥
𝐿
∗
𝐿
(𝑘𝑗 ∫𝑗 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥 +∫0 𝑘(𝑥)[𝜙𝑖′′ (𝑥)]2 𝑑𝑥) ∫0 𝑘 ∗ (𝑥)[𝜙𝑖′′ (𝑥)]2 𝑑𝑥
Eqn. 2.13
Stubbs and Kim then make an assumption that the total stiffness of the structure remains
relatively unchanged, namely that 𝑘 ∗ (𝑥) ≈ 𝑘(𝑥), so that these terms may be canceled. Stubbs and
Kim were able to show that this assumption is valid for damages of at least 30%, while Li et. al
(2006) were able to show that this assumption was valid for locating damage in a building story
level even after completely removing all brace components of one story’s frame, effectively
reducing the story’s stiffness by a large percent. Based on previous research, however, there is not
a clear limit that exists for this assumption to stay valid.
The term 𝑘𝑗 ∗ can be factored from the top equation, and the term 𝑘𝑗 from the bottom equation.
These terms can then be divided out onto the other side of the equation to reveal an important
ratio. This ratio is the ratio of the stiffness of the undamaged structure to the damaged structure.
The following is the resulting equation.
∗
𝐿
∗
𝐿
𝑘𝑗
(∫𝑗 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥+∫0 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥) ∫0 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥
𝑗
∗
𝐿
𝐿
(∫𝑗 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥+∫0 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥) ∫0 [𝜙𝑖′′ (𝑥)]2 𝑑𝑥
𝛽𝑗𝑖 = 𝑘 ∗ =
=
𝑁𝑈𝑀𝑗𝑖
𝐷𝐸𝑁𝑗𝑖
Eqn. 2.14
25
If undamaged,  ij should be equal to one, whereas a positive result would indicate a relative
decrease in modal stiffness of the jth member with respect to mode i. A negative result would
indicate that a relative increase in member modal stiffness has occurred, which can be interpreted
as a shift in structural system stiffness, from a damaged member to the jth member.
By summing these ratios for all modes, a single indicator for the jth member can be found –
𝛽𝑗 =
∑𝑖 𝑁𝑈𝑀𝑗𝑖
∑𝑖 𝐷𝐸𝑁𝑗𝑖
Eqn. 2.15
Using a standard score normalization the normalized classification value can be obtained –
𝑍𝑗 =
(𝛽𝑗 −𝜇𝛽𝑗 )
𝜎𝛽𝑗
Eqn. 2.16
In this equation, 𝜇𝛽𝑗 is the mean value of 𝛽𝑗 ’s measured, and 𝜎𝛽𝑗 is the standard deviation of this.
𝑍𝑗 is called the damage indicator, where a large positive value is a higher indication that damage
has occurred, where negative values, and values closer to zero tend to represent a global shift in
relative stiffness.
As shown in equation 2.14, the MSEM relies solely on the second derivative of the mode shapes.
To obtain these, the SHM sensor system must first obtain the mode shapes from the sensor data.
The process known as Time Domain Decomposition Technique, which is summarized in
Appendix C, is used to extract mode shape data from time variant modal data such as
instantaneous modal acceleration or dynamic modal strain data. To obtain modal data from
measured data, band-pass filters can be used to separate data into their modal contributions using
a frequency range for each mode. Once the modal data has been decomposed using the band-pass
filters, and the mode shapes are extracted using the Time Domain Decomposition Technique, the
26
second derivative of the mode shape/modal curvature, can be obtained through numerical
approximation. See Appendix D for information on numerical methods for estimating modal
curvature.
This method and theoretical framework will form the basis of the design of the SHM system and
the placement of sensors described in Chapters 3 and 4.
27
CHAPTER 3. ARCHETYPE BUILDING DESCRIPTION AND PUSHOVER
ANALYSIS
3.1
Project Introduction
To gain a better understanding of SHM, this chapter and the subsequent chapters will provide an
in depth look at the partial design of a SHM system that monitors seismic activity for a four story
steel building comprised of special steel moment frames and special steel concentrically braced
frames designed to resist lateral loads. The SHM system will be designed to monitor lateral
seismic loading on the building.
The layout of the building is from example 4.3 and 5.3 in the 2nd edition of the AISC Seismic
Design Manual (AISC 2012). Below is a plan view of the building. All floors and roof have the
same geometric layout. The floor and roof levels are also assumed to function as a rigid
diaphragm. This allows the dynamics of the building during a cyclic load to be modeled using a
kinematic condensation approach to decrease the amount of degrees of freedom that are
dynamically contributing to the system.
Figure 3.1 Plan view of 4-story steel structure for SHM application
28
This building is ideal for this example because it integrates two different lateral force-resisting
systems that contrast how different frames can be instrumented with the SHM sensor system.
The SHM sensing system will be largely made up of dynamic strain gauges and accelerometers.
Dynamic strain gauges are sensors that measure the strain, change in length, and its change over
time on the surface of a member at a particular location. Accelerometers measure the
instantaneous acceleration at a particular location. This allows the SHM system to locate damage
as per the MSEM. Details for strain gauge, accelerometer types, and sensor layout for each frame
type can be found in chapter 4.
3.1.1
East –West: Special Steel Moment Resisting Frame
The lateral force resisting system in the east-west direction of the building is comprised of two
identical, symmetric, Special Steel Moment Resisting Frames. The geometry of these frames is
summarized in the following elevation drawing.
Figure 3.2 Elevation view of E-W Special Steel Moment Resisting
29
The following figure and table summarize the reduced beam sections in the frame.
Figure 3.3 Reduced Beam Sections
Table 3.1 Reduced Beam Section (RBS) Geometry Values
D (in.)
bbf (in.)
a (in.)
b (in.)
c (in.)
R (in.)
bbf-RBS (in.)
I0 (in4)
IRBS (in4)
3.1.2
W21x44
20.7
6.5
4
16
1
32.2
5.4
843
833
W24x76
23.9
8.99
5.5
18
2
21.3
6.75
2100
2065
North-South: Special Steel Concentrically Braced Frame
The lateral force resisting system in the north-south direction of the building is comprised of two
identical, symmetric, Special Steel Concentrically Braced Frames. These frames are summarized
in the following elevation drawing.
30
Figure 3.4 N-S Special Steel Concentrically Braced Frame Elevation
The following figure summarizes the tube steel bracing pattern and sections in the frame.
Brace shapes
Story
Designation
4
HSS6.00x0.312
3
HSS6.875x0.500
2
HSS7.500x0.500
1
HSS8.625x0.500
Figure 3.5 N-S Concentrically Braced Frame Elevation and Bracing Members
31
3.2
Structural Health Monitoring Assumptions
Since the MSEM relies on comparison of the potentially damaged structure state to an idealized
undamaged elastic structure state, a few assumptions are necessary to make for the idealized
model.
3.2.1
Gravity Loads and Lateral Loads
Gravity loads include the weight of the building and all of its expected loads. Since in reality,
gravity loads will always be acting on the structure whether it is experiencing seismic activity or
not, a nonlinear representation of the structure (model) will be created to account for geometric
nonlinearity in the structure caused by the gravity loads before any lateral seismic forces are
applied. This representation will ideally give results that more accurately reflect what is
happening in the field. Below is a table that summarizes the assumed loads on the structure for
the ideal gravity load case.
Table 3.2 Dead and Live Loads on the Structure
LOAD
Dfloor
Droof
L0,floor
Lfloor
S
Curtain Wall
VALUE
85 psf
68 psf
80 psf
50 psf (reduced)
20 psf
175 lb/ft
Below is a table that summarizes the Seismic Design Criteria as per ASCE 7-10.
32
Table 3.3 Seismic Design Criteria as per ASCE 7-10
SEISMIC DESIGN
CRITERIA
Risk Category
Seismic Design Category
R
Ω0
Cd
Ie
SDS
ρ
VALUE
FOR
SSMRF
I
D
8
3
5.5
1.0
1.0
1.0
VALUE
FOR
SSCBF
I
D
6
2
5
1.0
1.0
1.0
This criterion is used to find the vertical distribution of seismic force ratios as per the ASCE 7-10.
These ratios will be used to examine the behavior of the structure after member yield has
occurred. The member yield events will be determined through a pushover analysis, which will be
discussed in further detail later in this chapter. Below is a table depicting the vertical distribution
ratios.
Table 3.4 Vertical Distribution Ratios
Level
2
3
4
Roof
3.2.2
C vx
0.115
0.217
0.321
0.347
Material and Member Assumptions
All of the wide flange members are assumed to be made of ASTM A992 Carbon Steel and the
hollow structural section (HSS) braces are assumed to be ASTM A500 Grade B.
33
The members of the building will assume elasto-plastic material yield. For more information on
elasto-plastic material relationships, please refer to the text “Mechanics of Materials” (Gere
2009), or other mechanics of materials text. See below for a representative elasto-plastic stress
versus strain diagram.
Figure 3.6 A992 Steel Stress vs. Strain
All structural steel shapes are assumed to be of ideal dimension matching that in Table 1-1 of the
AISC Steel Construction Manual (AISC 2011). See Appendix A for a full list of structural steel
shape dimensions and properties referenced in this document.
The member connections are assumed to be designed as per the AISC Seismic Design Manual
(AISC 2012) and the AISC Steel Construction Manual (AISC 2011), and therefore stronger than
the structural members. With this assumption, the monitoring scope can be reduced to focus on
instrumentation of the structural members and not locally in the connections. However, with the
MSEM, a damaged connection will provide a loss of calculated stiffness in the structure, and can
thus be detected without physically assigning a sensor to monitor the connection, but physical
verification of damage by a human will be necessary.
34
3.3
Analysis Layout
In the E-W direction, two identical Special Steel Moment Resisting Frames resist the later
seismic forces. Since both identical frames ideally contribute the same rigidity, the assumption
that system torsion effects can be neglected in this direction. In the N-S direction, two identical
Special Steel Concentrically Braced Frames resist the lateral seismic forces. By the same
reasoning as was used the E-W direction, the assumption that system torsion effects can be
neglected in this direction as well.
Since torsion is assumed to not play a role in the behavior of the frames, the frames can be
analyzed as simple 2-dimensional frames rather than a more complicated 3-dimensional frame
representing the entire building. This also allows the identical frames to be instrumented
identically.
For each frame, north-south and east-west, a pushover analysis will be performed to assess
member yield events (damage cases) of the frame. A pushover analysis is a nonlinear analysis
that uses material nonlinearity combined with an increasing lateral load at which structural
members yield. With a pushover analysis, the ultimate capacity of a structure can be tracked as
members yield in sequence until the overall collapse of the structure. The pushover analysis will
be completed using an acceleration based increasing load. The absolute displacement at the roof
level will govern the analysis to structural collapse. The results from this analysis will be used as
a basis for the detail story specific monitoring system, to instrument story levels that are more
susceptible to damage. The results for the pushover analysis will also be used to determine the
most important yield events, to look at further in Chapter 4. Chapter 4 will focus on three yield
events as damage cases to determine the expected system stiffness loss and the expected damage
indication calculation.
35
3.4
East-West Frames: Pushover Analysis and Results
The east-west lateral force resisting system is composed of identical Special Steel Moment
Resisting Frames. The following is a narrative discussing the implementation of the pushover
analysis.
3.4.1
Pushover Analysis
As discussed earlier in this chapter, the pushover analysis in the east-west direction can be
conducted on a 2-dimensional frame. See Figure 3.2 and Figure 3.3 for specifics on the frame’s
geometry. For the analysis, the supports will be assumed as pins (no moment resistance), and the
elements will be frame members (resisting axial, shear, and moment loads).
The pushover analysis was completed for this lateral force resisting system using SAP2000. The
analysis was based on the SAP2000 acceleration-based increasing lateral load. The analysis was
monitored by roof level displacement until structural collapse.
3.4.2
Pushover Analysis Results
Below is a figure of the pushover collapse mechanism, where the circles represent plastic hinge
locations, and the pushover curve depicting base shear versus the monitored displacement at the
roof level.
36
(a)
(b)
Figure 3.7: (a) E-W Pushover Collapse Mechanism and (b) Damage States 1-3 considered as
part of the SHM investigation in Chapter 4.
Figure 3.8 E-W Pushover Curve: Base Shear vs. Displacement
The change of slope in the above graph is the result of member yielding. The resulting maximum
displacement before collapse during the pushover analysis is about 35 inches and maximum
recorded base shear was 461 kips occurring right before collapse.
The pushover analysis showed that most of the initial member yielding occurred in the second
floor level beam members, and later in the first story column members. The upper levels did not
experience member yielding until a collapse mechanism had already formed as shown in Figure
37
3.7. This suggests that the story specific SHM system should be focused more heavily on the first
story members compared to the upper levels since they are likely to be the first members to
experience loss of stiffness due to yielding.
A modal analysis showed that the mode 1, 2, and 3 periods were 1.0, 0.3, and 0.2 seconds
respectively. For more details on the push over analysis please see Appendix B.
3.5
North-South Frames: Pushover Analysis and Results
The north-south lateral force resisting system is composed of identical Special Steel
Concentrically Braced Frames. The following is a narrative describing the implementation of the
pushover analysis.
3.5.1
Pushover Analysis
As discussed earlier in this chapter, the pushover analysis in the north-south direction can be
conducted on a 2-dimensional frame due to the neglect of torsional effects. See Figure 3.4 and
Figure 3.5 N-S Concentrically Braced Frame Elevation and Bracing Members
for specifics on the frame’s geometry. For the analysis, the columns will be modeled as
continuous members that resist axial, shear, and moment loads. The beams will be modeled as
simply supported beams with pin supports. The braces will be modeled as truss members,
supporting only axial loads.
The pushover analysis was completed for this lateral force resisting system using SAP2000. The
analysis was based on the SAP2000 acceleration based increasing lateral load, and was
displacement controlled and monitored through until structural collapse.
38
3.5.2
Pushover Analysis Results
Below is a figure of the pushover collapse mechanism, where circles on braces represent brace
buckling and circles on columns represent plastic hinging, and the pushover curve depicting base
shear versus the monitored displacement at the roof level.
(a)
(b)
Figure 3.9 (a) N-S Pushover Collapse Mechanism and (b) Damage States 1-3 considered as
part of the SHM investigation in Chapter 4.
Figure 3.10 N-S Pushover Curve: Base Shear vs. Displacement
39
The change of slope in the above graph is the result of member yielding. The maximum base
shear recorded was about 690 kips at a displacement of about 20 inches.
The pushover analysis showed that most of the initial member yielding was in the first and second
story brace members. The levels above the second story did not experience member yielding
before a collapse mechanism was formed as per Figure 3.9. This shows that the story specific
SHM system should be focused more heavily on the first and second story braces compared to the
upper levels.
A modal analysis showed that the mode 1, 2, and 3 periods were 1.0, 0.3, and 0.2 seconds
respectively. For more details on the push over analysis please see Appendix B.
40
CHAPTER 4. PROPOSED ARCHETYPE BUILDING INSTRUMENTATION
SYSTEM DURING DYNAMIC LOADING
4.1
General Structural Health Monitoring Layout
4.1.1
Data Collection and Pre-processing
As chapter 2 discussed, the MSEM of SHM will be implemented for the example steel building
described in chapter 3. The MSEM requires modal shapes of the structure determined from
instrument measurement for its damage detection algorithm. For more information on the MSEM
please consult chapter 2.
OMA, as described in chapter 2, is used as the data gathering source for MSEM. From the data
collected (acceleration or dynamic strain data), for use in the MSEM algorithm, mode shapes of
the structure need to be estimated. In the past, this was a challenge because single data points are
difficult to decay into their modal contributions. Presently, with the development of band-pass
filters, a device used to accept data contributions during a certain frequency range and reject data
outside the range, it is possible to focus data collection on multiple frequency ranges to extract
the data’s modal contributions corresponding to the modal frequencies. For the band-pass filter
data extraction to correctly work, a range of frequencies need to be established for each mode
shape that will be used in the MSEM algorithm.
After the modal contribution data is gathered from the sensors using band-pass filter data
extraction, a numerical analysis is necessary to approximate the mode shapes. This analysis is
called Time Domain Decomposition Technique (TDDT) formally described in (Cal Trans 2008).
See Appendix C for more information on the TDDT.
41
The result of the TDDT numerical analysis is an approximation for the mode shapes of a
structure, or element of the structure, calculated over a time range. Modal curvature can be
estimated from the mode shapes as second derivatives of the mode shape curve and then used in
the MSEM algorithm.
4.1.2
General Sensor Layout
For the SHM of the example building, uniaxial accelerometers placed at each floor level, to
measure the motion of the floor slabs as the building experiences lateral excitation, and will be
used to gather mode shape data of the entire frame structure. These sensors will allow damage
detection to be located per story level, and will be the focus of further discussions in this
document. For story level detail monitoring, uniaxial dynamic strain gauges will be installed to
detect damage at the member level.
4.2
E-W: Special Steel Moment Resisting Frame Instrumentation and Analysis
Since Special Steel Moment Resisting Frames are designed for ductility in its beams and use
reduced beam sections to control beam yielding, it is important to monitor the reduced beam
sections. For this case, strain gauges shall be placed at the top flange of each reduced beam
section. Each beam will also have a strain gauge located at the center of the beam on the top
flange. This sensor will allow for monitoring of beam dynamics. This will be sufficient for beams
on the upper floors that are not expected to experience yielding until late in the frame collapse
mechanism. For the lower level beams, a more detailed strain gauge layout is necessary. This is
shown in Figure 4.1.
Strain gauges will also be installed above and below all column connections, as that is where
yielding is expected to occur in the columns. These gauges will be designed to monitor absolute
42
strain levels that exceed the columns elastic capacity and also work as a redundancy in the SHM
system.
For this example, however, the focus is on story level damage detection. As described earlier, this
damage detection is performed using accelerometers at each level to obtain the first three mode
shapes of the structure. Below is a diagram of the instrumentation of the Special Steel Moment
Resisting Frame that resists the lateral loads in the east-west direction.
Figure 4.1 Instrumentation of the Special Steel Moment Resisting Frame
Below is a diagram, generated by SAP2000, depicting the first three modes of the structure
respectively.
43
Figure 4.2 First Three Modes of the Special Steel Moment Resisting Frame
Below is a table with the mode shapes according to accelerometer locations at each floor,
generated by SAP2000’s modal analysis for each of the damage cases (yield events).
Sensor Location/
Story Level
Table 4.1 SSMRF Mode Shape Data
Damage States
Undamaged Case
Damage Case 1
Damage Case 2
Damage Case 3
{q}1
{q}2
{q}3
{q}1
{q}2
{q}3
{q}1
{q}2
{q}3
{q}1
{q}2
{q}3
1
2
3
-0.58
-0.82
-1.02
0.97
0.88
0.00
-1.03
0.16
1.27
-0.59
-0.84
-1.02
0.95
0.88
-0.02
1.04
-0.16
-1.26
0.59
0.86
1.02
0.94
0.87
-0.03
-1.05
0.16
1.25
-0.71
-0.90
-0.99
-1.03
-0.71
0.21
1.05
-0.48
-1.12
4
-1.17
-1.30
-0.77
-1.15
-1.32
0.77
1.13
-1.33
-0.77
-1.04
1.35
0.84
Below is a table summarizing the period of the respective modes for each damage case as per the
SAP2000 modal analysis.
44
Table 4.2 SSMRF Mode Period
Period (s)
Mode
1
2
3
Undamaged
Case
Damage
Case 1
Damage
Case 2
Damage
Case 3
1.0
0.3
0.2
1.1
0.3
0.2
1.2
0.3
0.2
1.8
0.4
0.2
For the following sections, please refer to the following diagram that depicts the order of yield
events from the pushover analysis. These events will be referred to as Damage Case 1, 2, and 3,
respectively.
Figure 4.3 Moment Frame Yield Event Diagram
Using the mode shape data, the second derivative of the mode shape was numerically
approximated and used in equation 2.15 to obtain the 𝛽𝑗 values that represent stiffness loss of
each column and beam member in the system. Using a standard score normalization, shown in
equation 2.16, 𝑍𝑗 values can be obtained which represent the damage indication. Because the
MSEM algorithm does not allow for an accurate member specific damage location within a
45
specific story, the stiffness loss ratios and damage indicators of each member in a story level was
averaged to give the story specific damage ratios and indicators, 𝛽𝑘 and 𝑍𝑘 where k represents the
story level. The following table presents these calculated average stiffness loss ratios and damage
indication values for each damage case.
Sensor
Location/ Story
Level (k)
Table 4.3 SSMRF Damage Indicators
1
2
3
4
Damage States
Undamaged Case
Damage Case 1
Damage Case 2
Damage Case 3
βk
1.000
1.000
1.000
1.000
βk
1.005
1.002
0.998
0.995
βk
1.005
1.001
0.999
0.995
βk
1.055
1.004
0.995
0.952
Zk
0.00
0.00
0.00
0.00
Zk
1.33
0.43
-0.55
-1.21
Zk
1.35
0.26
-0.30
-1.31
Zk
1.42
0.04
-0.17
-1.29
As described in chapter 2, the 𝛽𝑗 ’s represents the undamaged member stiffness divided by the
damaged member stiffness, and thus 𝛽𝑘 represents the average undamaged story stiffness divided
by the average undamaged story stiffness. Therefore, a damaged member of a story would cause a
decrease of story stiffness, and result in a 𝛽𝑘 value greater than 1. To attempt to filter through the
calculated 𝛽𝑗 values, the standard score normalization, 𝑍𝑗 , is used to show the separation in the
calculated 𝛽𝑗 values. As discussed in chapter 2, a relatively large positive 𝑍𝑗 is a good indication
that damage has occurred in that story. The average 𝑍𝑗 for a specific story, 𝑍𝑘 , represents the
damage indicator for the kth story level.
In the case of the SSMRF, all damage cases have damage in the first story. In Table 4.3, a gradual
increase in separation of 𝑍𝑘 values can be seen as damage in the first story develops and the
structure approaches its collapse mechanism. This causes the damage indicator for the first story
46
to become more prominent in successive damage cases. For all damage cases, the damage
detection algorithm correctly detects damage in the first story.
4.3
N-S: Special Steel Concentrically Braced Frame Instrumentation and Analysis
Special Steel Concentrically Braced Frames are designed for ductility in the brace components of
the frame. The ductility is controlled by focusing any member yielding in the brace members
through buckling. For this reason, the braces are important members for the story level detail
monitoring. Therefore, two strain gauges shall be placed at the center length of each brace to
monitor axial strain in the major and minor brace axis. The beams that do not have the concentric
braces converging at its midpoint shall have a strain gauge in the center to monitor its
performance. This can all be summarized in Figure 4.4.
Strain gauges will also be installed above and below all column connections as well as in the
center of the floor to floor length. As were the column strain gauges in the SSMRF, these gauges
will be designed to monitor for absolute strain levels that exceed the columns elastic capacity to
provide a redundancy safe guard for the SHM system.
Below is a diagram of the instrumentation of the Special Steel Moment Resisting Frame that
resists the lateral loads in the east-west direction.
47
Figure 4.4 Instrumentation of the Special Steel Concentrically Braced Frame
For this example, however, the focus is on story level damage detection. As described earlier, this
damage detection is performed using accelerometers at each level to obtain the first three mode
shapes of the structure. Below is a diagram, generated by SAP2000, depicting the first three
modes of the structure respectively for each damage case.
48
Figure 4.5 First Three Modes of the Special Steel Concentrically Braced Frame
Below is a table with the mode shapes according to accelerometer locations at each floor,
generated by SAP2000’s modal analysis for each of the damage cases (yield events).
Sensor Location/
Story Level
Table 4.4 SSCBF Mode Shape Data
Damage States
Undamaged Case
Damage Case 1
Damage Case 2
Damage Case 3
{q}1
{q}2
{q}3
{q}1
{q}2
{q}3
{q}1
{q}2
{q}3
{q}1
{q}2
{q}3
1
2
3
-0.61
-1.10
-1.52
-1.24
-1.37
-0.43
1.73
0.15
-1.83
-1.01
-1.25
-1.43
1.47
0.95
-0.20
-1.45
0.32
1.73
-1.30
-1.33
-1.33
-1.48
-0.75
0.47
-1.36
0.59
1.83
1.18
1.37
1.38
-2.03
0.35
0.65
0.17
1.32
0.22
4
-1.87
1.93
0.94
-1.57
-2.01
-1.10
-1.34
2.14
-1.30
1.38
0.84
-2.15
Below is a table summarizing the period of the respective modes for each damage case as per the
SAP2000 modal analysis.
49
Table 4.5 SSMRF Mode Period
Mode
Undamaged
Case
1
2
3
0.2
0.1
0.0
Period (s)
Damage
Damage
Case 1
Case 2
0.3
0.1
0.1
1.2
0.1
0.1
Damage
Case 3
1.4
0.2
0.1
For the following sections, please refer to the following diagram that depicts the order of yield
events from the pushover analysis.
Figure 4.6 SSCBF Yield Event Diagram
As was done with the SSMRF, using the mode shape data, the second derivative of the mode
shape was numerically approximated and used in equation 2.15 to obtain the 𝛽𝑗 values that
represent stiffness loss of each column and beam member in the system. Using a standard score
normalization, shown in equation 2.16, 𝑍𝑗 values can be obtained which represent the damage
50
indication. Because the MSEM algorithm does not allow for an accurate member specific damage
location within a specific story, the stiffness loss ratios and damage indicators of each member in
a story level was averaged to give the story specific damage ratios and indicators, 𝛽𝑘 and 𝑍𝑘 for
story level k. The following table presents these calculated average stiffness loss ratios and
damage indication values for each damage case.
Sensor
Location/ Floor
Level (k)
Table 4.6 SSCBF Damage Indicators
1
2
3
4
Damage States
Undamaged Case
Damage Case 1
Damage Case 2
Damage Case 3
βk
1.000
1.000
1.000
1.000
Βk
1.023
0.999
1.009
0.974
Βk
1.006
0.995
0.997
1.001
Βk
1.061
1.031
1.005
0.917
Zk
0.00
0.00
0.00
0.00
Zk
1.08
-0.03
0.41
-1.46
Zk
1.34
-1.18
-0.43
0.28
Zk
0.90
0.56
0.07
-1.53
As described in chapter 2, the 𝛽𝑗 ’s represents the undamaged member stiffness divided by the
damaged member stiffness, and thus 𝛽𝑘 represents the average undamaged story stiffness divided
by the average undamaged story stiffness. Therefore, a damaged member of a story would cause a
decrease of story stiffness, and result in a 𝛽𝑘 value greater than 1. To attempt to filter through the
calculated 𝛽𝑗 values, the standard score normalization, 𝑍𝑗 , is used to show the separation in the
calculated 𝛽𝑗 values. As discussed in chapter 2, a relatively large positive 𝑍𝑗 is a good indication
that damage has occurred in that story. The average 𝑍𝑗 for a specific story, 𝑍𝑘 , represents the
damage indicator for the kth story level.
In the case of the SSCBF, damage cases 1 and 2 have damage located in the first story. In Table
4.6, an increase in separation of 𝑍𝑘 values can be seen as damage in the first story develops from
damage case 1 to damage case 2. For these damage cases, the damage detection algorithm
51
correctly detects damage in the first story. However in the third case, damage is now located in
both the first and second stories. The damage indicator, 𝑍𝑘 , seems to indicate that damage has
occurred in both the first and second stories, however, without the clear distinction as was seen in
the earlier cases. This can be attributed to a few conditions. First, 𝑍𝑘 , is based on a standard score
normalization, which can be used to locate large deviations in data, and more importantly,
outliers. In the case where two stories out of a total of four are damaged, it becomes more
difficult to numerically show these deviations. However, examining the 𝛽𝑘 values in damage case
3 reveals that stories 1 and 2 clearly have large decreases in story stiffness suggesting that
damage has occurred, indicated by a 𝛽𝑘 value greater than one. Secondly, to reach the third
damage case, a significant amount of damage is to have occurred in the structure. At this point the
damage detection algorithm may be showing signs of breaking down because of the assumptions
that were made in chapter 2, namely that the total system stiffness of the undamaged and
damaged structure remains relatively unchanged. As mentioned in chapter 2, this assumption has
been proven valid for damages up to 30% (Stubbs 1996), as well as story levels that have
completely lost structural capacity from their brace members (Li 2006), however, damage case 3,
and possibly damage case 2, have experienced damage above 30% to where the original
assumption may come into question. To handle this, in the past, various refinements have been
made to the calculation of 𝛽𝑗 and 𝑍𝑗 . Others have suggested other methods of VBSHM such as
frequency based monitoring rather than time dependent monitoring. For a greater discussion of
this please refer to CalTrans (2008). Despite this downfall, it is clear from the mode shapes (see
Figure 4.5) that a significant change in the structure has occurred supporting the damage
detection results.
52
CHAPTER 5. SUMMARY
5.1
Summary
The purpose of this document was to provide a literature review for the field of SHM and
exercise concepts discovered from the review in a theoretical example.
In chapter 2, basic motivations for implementing a SHM system were discussed. Of these,
structural safety and collapse prevention were among the top benefits for SHM system
implementation. A list of basic fundamental truths of structural engineering with respect to SHM
were established and described. These fundamental truths have been used by researchers in the
field of SHM to develop the established methods that are seen today, and guide much of the
advancement that will occur in the future. A list of guidelines to follow during the decision to
implement a SHM system and the development of it was also established. This list of guidelines
gave important questions and discussions that the engineer and owner should collaborate on
during the system development. Ultimately, before implementing a SHM system, basic goals and
motivations for the SHM system need to be established that guide the rest of the development
process. Details of the SHM system come in the form of recommendations from the engineer but
are usually implemented based on owner preference. In regards to the details of the SHM system,
the cost of many additions, such as intelligent feature extraction or additional instrumentation for
more accurate measurements, result in additional costs that the owner may not desire. For
situations like this, cost-benefit assessments are necessary to establish and depict the necessity of
the SHM system additions in relation to their costs. To guide the development of the SHM system
with minimal complications, a list of issues during implementation and instrumentation was
developed. Historically, most of these were problematic areas for SHM during its more primitive
years. However, as technology advanced and SHM methods became more refined, many of the
53
issues have since been resolved, while others are still problematic but have been shown to be
avoidable to an extent. Finally, the theoretical development of the MSEM of VBSHM was
described. This method presents an algorithm for SHM that was implemented in chapter 4 as an
example.
Chapter 3 presented an example structure that will have a SHM system designed for it. This
chapter discusses important assumptions that must be made in order to efficiently design the
SHM system. Of these assumptions, important ones to note are material properties of the
structural elements, structural boundary conditions, and structure response behavioral
expectations such as rigid diaphragm motion during a lateral loading. The structure was modeled
using SAP2000 and a pushover analysis was performed for both of the lateral force resisting
systems (north-south and east-west on the building). The pushover analysis results were used to
guide the development and theoretical testing of the SHM system in chapter 4.
Chapter 4 narrates how the example in chapter 3 was to be instrumented. The SHM system has a
dual monitoring process that allows the lateral motion of the structural system to be monitored
using accelerometers and the story specific monitoring to be instrumented with dynamic strain
gauges. This accelerometer based component of the system was shown to be capable of locating
damage at the story level. For story specific monitoring, the system of strain gauges are used to
locate damage in specific members of the story, however this component of the SHM system was
not in the scope of this document. The focus of this chapter, however, was the development and
theoretical testing of the structural system specific monitoring component using accelerometers.
For both lateral force resisting systems, SSMRF and SSCBF, the mode shapes were obtained for
the undamaged structure and each of the three damage cases, determined from the pushover
analysis in chapter 3. The mode shapes were used in the MSEM algorithm to locate damage per
54
story level using an average damage indicator for the members of the story. For each of the
damage cases, the monitoring results of the MSEM algorithm were discussed. The discussion
compared and contrasted the results from the calculation to the intuitive expectations. In each
damage case, the algorithm appeared to perform as expected.
5.2
Conclusions and Recommendations
Through the SHM example discussed in this document, the MSEM of VBSHM was shown to be
useful in both the SSMRF and the SSCBF examples. For the SSMRF, the loss of story stiffness in
each damage case seemed to be simple for the algorithm to track. This may be because member
hinging contributed a smaller stiffness decreases than that of brace buckling in the SSCBF. In the
SSCBF, when a brace buckled, the stiffness of the structure changed more drastically. However,
the damage detection algorithm appeared to still correctly identify damage, within some reason,
despite reaching a point where the assumptions of the MSEM algorithm may start to break down.
For the example, the MSEM appeared to be a promising method of SHM. Refinements to the
MSEM have been offered by many researchers in the past, which may be more ideal for a similar
SHM system. It is recommended that these refinements be considered in a separate study.
Frequency dependent VBSHM systems have also been shown to be ideal for structures that take
advantage of OMA. It is recommended that this type of SHM system be considered in a separate
study as well.
The installation of additional accelerometers at multiple locations on a floor, and preferably at
story mid-spans, for more accurate mode shape approximations is also recommended. This
however may be challenging since system mode shapes are the desired result of this
instrumentation but there is not a rigid diaphragm located at the story mid-spans, so additional
55
instrumentation (say, at a column mid-span) will be more subject to element specific motion and
noise. To resolve this, it may be more ideal to implement a full member specific SHM system
rather than a story specific SHM system similar to that proposed by Li (2006). However, this also
increases the amount of calculations necessary for damage detection and may run the risk of data
overload for large structures.
Finally, a dormant indication system can be added to the SHM system for the story specific
monitoring. This will allow the story level monitoring to remain dormant if no traumatic events
are occurring or the vertical SHM system is not measuring any damage at that story level. As
discussed in chapter 2, implementing a dormant indication system into the SHM system can help
for data processing and save energy and resources since the system does not require full analysis
at all times. For the addition of this feature, it is recommended that the story level monitoring still
be activated occasionally for historical record as well as measurement redundancy.
56
APPENDIX A. AISC SHAPES
W14X176
W14X132
F
68.0
20.1
23.7
F
44.0
13.0
20.7
F
176
51.8
15.2
F
132
38.8
14.7
ddet
27 1/4
23 7/8
23 3/4
20 5/8
15 1/4
14 5/8
bf
10.1
8.99
8.97
6.50
15.7
14.7
bfdet
10 1/8
9
9
6 1/2
15 5/8
W12X45
W21X44
F
76.0
22.4
23.9
W12X96
W24X68
F
114
33.6
27.3
W14X68
W24X76
T_F
W
A
d
AISC
Manual
Label
W14X82
W27X114
Table 1 Wide Flange Sections
F
82.0
24.0
14.3
F
68.0
20.0
14.0
F
96.0
28.2
12.7
F
45.0
13.1
12.1
14 1/4
14
12 3/4
12
10.1
10.0
12.2
8.05
14 3/4
10 1/8
10
12 1/8
8
tw
0.570
0.440
0.415
0.350
0.830
0.645
0.510
0.415
0.550
0.335
twdet
9/16
7/16
7/16
3/8
13/16
5/8
1/2
7/16
9/16
5/16
twdet/2
5/16
1/4
1/4
3/16
7/16
5/16
1/4
1/4
5/16
3/16
0.575
tf
0.930
0.680
0.585
0.450
1.31
1.03
0.855
0.720
0.900
tfdet
15/16
11/16
9/16
7/16
1 5/16
1
7/8
3/4
7/8
9/16
kdes
1.18
1.09
0.950
1.91
1.63
1 1/2
1 1/8
2 5/8
2 5/16
1 9/16
1 3/8
1 1/16
1 1/16
13/16
1 5/8
1 9/16
1 1/16
1.50
1
13/16
1 1/8
1.08
1 9/16
1.45
1
11/16
1 1/16
1.31
k1
1.53
1
13/16
1 1/8
bf/2tf
5.41
6.61
7.66
7.22
5.97
7.15
5.92
6.97
6.76
7.00
h/tw
42.5
49.0
52.0
53.6
13.7
17.7
22.4
27.5
17.7
29.6
Ix
4080
2100
1830
843
2140
1530
881
722
833
348
Zx
343
200
177
95.4
320
234
139
115
147
64.2
Sx
299
176
154
81.6
281
209
123
103
131
57.7
rx
11.0
9.69
9.55
8.06
6.43
6.28
6.05
6.01
5.44
5.15
kdet
15/16
Iy
159
82.5
70.4
20.7
838
548
148
121
270
50.0
Zy
49.3
28.6
24.5
10.2
163
113
44.8
36.9
67.5
19.0
Sy
31.5
18.4
15.7
6.37
107
74.5
29.3
24.2
44.4
12.4
ry
2.18
1.92
1.87
1.26
4.02
3.76
2.48
2.46
3.09
1.95
J
7.33
2.68
1.87
0.770
26.5
12.3
5.07
3.01
6.85
1.26
Cw
27600
11100
9430
2110
40500
25500
6710
5380
9410
1650
Wno
66.6
52.2
51.8
32.9
54.5
50.2
33.9
33.2
36.0
23.2
Sw1
156
79.8
68.0
24.1
280
190
73.3
59.8
98.8
26.8
Qf
58.4
33.8
28.9
14.0
67.6
49.5
27.6
22.9
30.9
12.8
Qw
170
98.9
87.0
46.8
159
116
68.2
56.0
73.0
31.7
rts
2.65
2.33
2.30
1.60
4.55
4.23
2.85
2.80
3.49
2.23
ho
26.4
23.2
23.1
20.3
13.9
13.7
13.4
13.3
11.8
11.5
PA
82.7
73.0
72.6
59.3
74.8
71.2
56.9
56.2
59.9
46.8
PB
92.8
82.0
81.6
65.8
90.5
85.9
67.0
66.2
72.1
54.9
57
HSS8.625X0.500
HSS7.500X0.500
HSS6.625X0.500
HSS6X0.312
Table 2 HSS Sections
T_F
F
F
F
F
W
43.43
37.42
32.74
18.97
A
11.9
10.3
9.00
5.22
AISC Manual Label
OD
8.63
7.50
6.63
6.00
tnom
0.500
0.500
0.500
0.312
tdes
0.465
0.465
0.465
0.291
D/t
18.5
16.1
14.2
20.6
Ix
100
63.9
42.9
21.3
Zx
31.0
23.0
17.7
9.49
Sx
23.1
17.0
13.0
7.11
rx
2.89
2.49
2.18
2.02
Iy
100
63.9
42.9
21.3
Zy
31.0
23.0
17.7
9.49
Sy
23.1
17.0
13.0
7.11
ry
2.89
2.49
2.18
2.02
J
199
128
85.9
42.6
C
46.2
34.1
25.9
14.2
58
APPENDIX B. PUSHOVER ANALYSIS RESULTS
E-W FRAME: PUSHOVER ANALYSIS
Figure 1 SSMRF Pushover Curve: Displacement vs. Base Shear
Table 1 SSMRF Pushover Analysis SAP2000 Data Tables
TABLE: Pushover Curve - Push
Step Displacement BaseForce
in
Kip
0
2.06E-14
0
1
0.5
19.363
2
1
38.725
3
1.5
58.088
4
2
77.45
5
2.5
96.813
6
3
116.176
7
3.5
135.538
8
4
154.901
9
4.5
174.263
10
5
193.626
11
5.5
212.988
12
6
232.351
13
6.5
251.714
14
7
271.076
15
7.415521
287.167
16
7.937631
303.567
17
8.437631
316.012
18
8.937631
328.456
19
9.535738
342.114
20
10.035738
349.659
21
10.853557
360.613
AtoB
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
159
157
157
157
154
154
150
BtoIO
IOtoLS
LStoCP
CPtoC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
3
3
6
6
10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
CtoD
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
DtoE
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
BeyondE
Total
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
59
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
11.353557
11.569949
12.069949
12.569949
13.069949
13.569949
14.069949
14.569949
15.069949
15.569949
16.069949
16.569949
17.069949
17.569949
18.069949
18.569949
19.069949
19.569949
20.069949
20.569949
21.069949
21.569949
22.069949
22.569949
23.069949
23.569949
24.069949
24.569949
25.069949
25.569949
26.069949
26.569949
27.069949
27.569949
28.069949
28.569949
29.069949
29.569949
30.069949
30.569949
31.069949
31.569949
32.069949
32.569949
33.069949
33.569949
34.069949
34.569949
34.570449
35.070449
35.570449
36.070449
366.692
369.288
371.912
374.536
377.16
379.784
382.407
385.021
387.6
390.18
392.76
395.34
397.92
400.5
403.08
405.66
408.016
410.173
412.33
414.487
416.644
418.801
420.958
423.115
425.271
427.428
429.585
431.742
433.899
435.91
437.92
439.93
441.437
442.943
444.305
445.667
447.028
448.39
449.752
451.096
452.423
453.709
454.943
456.089
457.235
458.353
459.471
460.575
0.158
0.16
0.162
0.165
148
146
146
146
146
146
144
138
138
138
138
138
138
138
138
136
135
135
135
135
135
135
135
135
135
135
135
135
134
134
134
132
132
130
130
130
130
130
128
126
125
124
123
123
119
119
117
117
116
116
116
116
11
13
13
11
11
11
11
15
15
15
15
15
15
15
15
17
18
18
18
18
18
18
18
18
18
18
18
18
19
19
19
21
20
22
22
21
20
19
21
23
24
25
24
24
28
28
30
28
29
29
29
29
1
1
1
3
3
3
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
5
3
3
3
3
4
4
4
5
6
7
7
7
7
7
9
9
9
9
9
11
11
11
11
11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
4
4
4
4
4
4
4
4
4
4
4
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
4
4
4
4
4
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
4
4
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
60
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
36.570449
37.070449
37.570449
38.070449
38.570449
39.070449
39.570449
40.070449
40.570449
41.070449
41.570449
42.070449
42.570449
43.070449
43.570449
44.070449
44.570449
45.070449
45.570449
46.070449
46.570449
47.070449
47.570449
48.070449
48.570449
49.070449
49.570449
50
0.167
0.169
0.171
0.174
0.176
0.178
0.181
0.183
0.185
0.187
0.19
0.192
0.194
0.196
0.199
0.201
0.203
0.205
0.208
0.21
0.212
0.214
0.217
0.219
0.221
0.224
0.226
0.228
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
116
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
160
61
N-S FRAME: PUSHOVER ANALYSIS
Figure 2 SSCBF Pushover Curve: Displacement vs. Base Shear
Table 2 SSCBF Pushover Analysis SAP2000 Data Tables
TABLE:
Pushover
PUSHOVER
Step Displacement
in
0
-9.15E-16
1
0.962822
2
1.389815
3
1.391815
4
1.688707
5
3.67709
6
4.235092
7
4.237092
8
4.476615
9
5.476615
10
6.476615
11
7.476615
12
8.476615
13
9.476615
14
9.740963
15
10.740963
16
11.740963
17
12.740963
18
13.740963
19
14.740963
20
15.740963
21
16.740963
22
17.740963
Curve
-
BaseForce
Kip
0
507.453
630.819
475.494
558.638
600.183
609.721
490.462
503.65
520.745
537.839
554.934
572.028
589.123
593.642
602.973
612.303
621.633
630.963
640.293
649.623
658.953
668.283
AtoB
BtoIO
IOtoLS
LStoCP
CPtoC
40
38
38
38
36
34
34
34
34
34
34
34
34
34
30
30
30
30
30
30
30
30
30
0
2
0
0
2
2
0
0
0
0
0
0
0
0
4
4
4
4
0
0
0
0
0
0
0
1
1
1
3
5
5
5
5
5
5
5
5
5
5
5
5
9
7
7
7
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
CtoD
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
DtoE
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
BeyondE
Total
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
62
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
18.740963
19.740963
20.061787
20.063787
21.063787
22.063787
22.875294
22.877294
23.877294
24.877294
25.877294
26.877294
27.877294
28.877294
29.877294
30.877294
31.877294
32.877294
33.877294
34.877294
35.877294
36.22276
36.22476
37.22476
38.22476
38.769156
38.771156
39.771156
40.771156
41.635568
42.635568
43.635568
44.635568
45.036975
45.038975
45.10352
45.10552
46.10552
47.10552
48.10552
49.10552
50.10552
51.10552
52.10552
53.10552
54.10552
55.10552
56.10552
57.10552
58.10552
59.10552
60.10552
677.613
686.942
689.898
457.447
458.442
459.437
460.244
178.918
179.925
180.931
181.938
182.945
183.952
184.959
185.965
186.972
187.979
188.986
189.993
190.999
192.006
192.354
157.772
168.889
180.007
186.059
161.379
172.497
183.614
193.225
194.231
195.237
196.242
196.646
131.719
131.726
73.581
73.692
73.803
73.914
74.025
74.136
74.247
74.358
74.469
74.58
74.691
74.802
74.913
75.024
75.136
75.247
30
28
28
28
28
28
28
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
0
2
2
2
2
2
2
4
4
4
4
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
7
7
7
7
7
7
7
7
7
7
7
8
9
9
9
5
5
5
5
5
5
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
3
3
2
2
2
2
2
2
2
2
2
6
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
63
APPENDIX C. TIME DOMAIN DECOMPOSITION TECHNIQUE
As stated in section 3.2 of CalTrans “Vibration-Based Structural Health Monitoring of Highway
Bridges” (CalTrans 2008)
“
The vibration response of a linear time-invariant dynamic system can be expressed in terms of its
mode shapes and generalized coordinates as
𝑢(𝑥, 𝑡) = ∑∞
𝑟=1 𝜑𝑟 (𝑥)𝑞𝑟 (𝑡)
Eq. 1
where 𝜑𝑟 (𝑥) is the rth mode shape function and 𝑞𝑟 (𝑡) is the corresponding generalized coordinate
at time instant t. Assuming all modes are well separated, by applying a bandpass filter to the
system responses, it is possible to isolate the individual modal components in the response timehistory (Kim et al. 2002).
𝑢𝑛 (𝑥, 𝑡) = 𝜑𝑛 (𝑥)𝜑𝑛 (𝑥)
Eq. 2
where 𝑢𝑛 (𝑥, 𝑡) is the nth modal contribution to the response, and 𝜑𝑛 (𝑥) and 𝜑𝑛 (𝑥) is the nth
mode shape function and generalized coordinates, respectively.
Consider a system with Nd degrees-of-freedom and assuming the measured
response quantity is an acceleration response sampled at Ns discrete time points, Eq. 1 can be
expressed in discrete time as,
𝑑
𝑇
[𝑈] = ∑𝑁
𝑟=1 𝜑𝑟 𝑞̈ 𝑟
Eq. 3
64
where, [U] is the Nd× Ns response matrix, 𝜑𝑟 is the Nd×1 rth mode shape vector and is the Ns×1
vector containing values of the rth generalized coordinate at each time instant. At the same time,
Eq. 2 can also be expressed in matrix form as
[𝑈𝑛 ] = 𝜑𝑟 𝑞̈ 𝑟𝑇
[𝑈𝑛 ] = [
Eq.4
𝜙1𝑛
𝑢̈ 1𝑛 (1) ⋯ 𝑢̈ 1𝑛 (𝑁𝑠 )
⋮
⋱
⋮
] = [ ⋮ ] [𝑞̈ 1𝑛 (1) … 𝑞̈ 𝑛 (𝑁𝑠 )]
𝜙𝑁 𝑑 𝑛
𝑢̈ 𝑁𝑑𝑛 (1) … 𝑢̈ 𝑁𝑑 𝑛 (𝑁𝑠 )
Eq. 5
The autocorrelation of the nth mode-isolated acceleration time history is thus given by
[𝐸𝑛 ] = [𝑈𝑛 ][𝑈𝑛 ] = 𝜑𝑛 𝑞̈ 𝑛𝑇 𝑞̈ 𝑛 𝜑𝑛𝑇 = 𝜑𝑛 𝑄𝑛 𝜑𝑛𝑇 = 𝑄𝑛 𝜑𝑛 𝜑𝑛𝑇
Eq. 6
where 𝑄𝑛 is a scalar. In expanded matrix form this can be expressed as:
𝜙1𝑛
[𝐸𝑛 ] = 𝑄𝑛 [ ⋮ ] [𝜙1𝑛
𝜙𝑁 𝑑 𝑛
𝜙1𝑛 𝜙1𝑛
𝜙2𝑛 𝜙1𝑛
= 𝑄𝑛 [
⋮
𝜙1𝑛 𝜙2𝑛
⋱
𝜙𝑁𝑑 𝑛 𝜙1𝑛
…
… 𝜙𝑁 𝑑 𝑛 ] =
⋯ 𝜙1𝑛 𝜙𝑁𝑑 𝑛
⋮
]
⋱
⋮
… 𝜙𝑁𝑑 𝑛 𝜙1𝑛
Eq. 7
where [En] is a Nd×Nd symmetric matrix of rank 1.
A close examination of the structure of the [En] matrix reveals that each column of [En] is a
proportional to the modal vector of the nth mode. The Spectral Decomposition Theorem (Lay
2003) states that, the symmetry matrix [En] can be expanded by its eigenvalues and eigenvectors,
65
[𝐸𝑛 ] = 𝑃𝐷𝑃−1 = [𝑢1
⋯ 𝑢𝑁 𝑑 ] [
𝜆1
0
𝑢1𝑇
⋱
][ ⋮ ] =
𝑇
𝜆 𝑁 𝑑 𝑢𝑁
𝑑
0
Eq. 8
𝑇
= 𝜆1 𝑢1 𝑢1𝑇 + ⋯ + 𝜆𝑁𝑑 𝑢𝑁𝑑 𝑢𝑁
𝑑
where 𝜆1 > 𝜆2 > ⋯ > 𝜆𝑁𝑑 are the eigenvalues of matrix [En] and 𝑢1 , 𝑢2 , ⋯, 𝑢𝑁𝑑 are its
eigenvectors. Comparing Eq. 6 with Eq. 8, it becomes clear that if there is no noise in the
measurement response, the spectral decomposition of [En] will generate a single non-zero
eigenvalue 𝜆1 , and the corresponding eigenvector will be proportional to the modal vector 𝜑𝑛 .
Considering the fact that the modal vector can be arbitrarily scaled, the eigenvector 𝑢1 can be
effectively treated as the modal vector.
When noise is present in the measurement, other eigenvalues of the matrix [En] will not be equal
to zero. However, the contribution to system response from the physical mode will usually
dominate the response within the frequency range close to the resonance of that particular mode.
Thus, with appropriate selection of band-pass filter parameters, the largest eigenvalue 𝜆1 always
corresponds to the physical mode and the corresponding eigenvector is same as the modal vector.
The existence of noise does not affect the identification of the modal vector. It is noted that Eq. 8
holds true no matter what kind of motion the system is experiencing, either free vibration or
forced vibration due to some external excitations.
Pre-multiplying Eq. 4 with the transpose of the identified nth mode shape yields
𝜑𝑛𝑇 [𝑈𝑛 ] = 𝜑𝑛𝑇 𝜑𝑛 𝑞̈ 𝑛𝑇
The response of nth mode in generalized coordinates can then be obtained as
Eq. 9
66
1
𝑞̈ 𝑛𝑇 = 𝜑𝑇 𝜑 𝜑𝑛𝑇 [𝑈𝑛 ]
𝑛 𝑛
Eq. 10
here Eq. 10 represents the response of a single degree-of-freedom system corresponding to the nth
mode. Therefore, the natural frequency and modal damping of the nth mode can be readily
identified using time-domain modal identification techniques such as the Complex Exponential
(CE) method or the Eigensystem Realization Algorithm (ERA).
The described technique is subsequently referred to as the Time Domain Decomposition (TDD)
technique. The general steps of TDD method start with identifying the frequency region where a
certain mode might be located, typically from power spectrum plots of the response signal or
from the Frequency Response Function if input is measured. The second step consists of applying
a band-pass filter to isolate the desired modes while eliminating the contribution from other
modes. In the third step the matrix [En] is formed and the modal vector can be conveniently
extracted using Eq. 8 and Singular Value Decomposition (SVD) algorithm. The last step involves
the construction of SDOF response using Eq. 10 and the identification of natural frequencies and
modal damping. The process is repeated for each mode within the frequency range of interest.
The computationally intensive part of the process, the singular value decomposition, deals with
time domain data only and no Fourier transform is needed. The size of the matrix used for SVD
in the TDD method is Nd × Nd , with Nd equals to the number of measurement sites. If n modes
are to be identified, the SVD process needs to be repeated n times for the Nd × Nd matrix. This
compares favorably with the time domain based ERA technique, where SVD is also used and the
size of the matrix used is sNd × s , with s equaling to the time lag in the Henkel matrix (Juang and
Pappa 1985). For civil engineering applications, s is typically much larger than Ns and nNd . The
67
computation time required by ERA is thus significant longer than TDD when applied to problems
where only a few modes are needed.
“
68
APPENDIX D. ERRORS ASSOCIATED WITH THE CALCULATION OF MODAL
CURVATURE THROUGH NUMERICAL DIFFERENTIATION
As stated in Vibration-Based Structural Health Monitoring of Highway Bridges (CalTrans 2008)
“
For beam-like structures, modal curvature is defined as the second derivative of the
corresponding transverse displacement mode shape 𝜙 , i.e., 𝜅 ≡ 𝜙 ′′ . When an analytical
representation of the mode shape is not available, as is the case of experimentally measured mode
shapes, the calculation of modal curvature has to be performed numerically. If 𝜙(𝑥𝑖 ) is the mode
shape value at a measurement site 𝑥𝑖 , 𝜙(𝑥𝑖+1 ) and 𝜙(𝑥𝑖−1 ) can be expressed in terms of 𝜙(𝑥𝑖 )
using a Taylor series expansion as:
𝜙(𝑥𝑖+1 ) = 𝜙(𝑥𝑖 ) + 𝜙 ′ (𝑥𝑖 )ℎ +
𝜙(𝑥𝑖−1 ) = 𝜙(𝑥𝑖 ) + 𝜙 ′ (𝑥𝑖 )(−ℎ) +
𝜙′′ (𝑥𝑖 ) 2
ℎ
2!
+⋯
Eq. 1
𝜙 ′′ (𝑥𝑖 )
(−ℎ)2 + ⋯
2!
The summation of the two equations in Eq. 1 and reorganize gives
𝜙 ′′ (𝑥𝑖 ) =
𝜙(𝑥𝑖+1 )−2𝜙(𝑥𝑖 )+𝜙(𝑥𝑖−1 )
+
ℎ2
𝑂(ℎ2 ) =
𝜙(𝑥𝑖 +ℎ)−2𝜙(𝑥𝑖 )+𝜙(𝑥𝑖 −ℎ)
+
ℎ2
𝑂(ℎ2 )
Eq. 2
in which, 𝑥𝑖 , 𝑥𝑖−1, 𝑥𝑖+1 are the current, previous, and next measurement sites where displacement
mode shapes are available. 𝜙 ′′ (𝑥𝑖 ) = 𝜅(𝑥𝑖 ) is the modal curvature at data site 𝑥𝑖 , and ℎ is the
spacing between measurement sites. It should be noted that the spacing between measurement
sites must remain constant in order for Eq. 2 to be valid. Eq. 2 is called the second central finite
divided difference, or in short, central difference. It is apparent that Eq. 2 is an approximation due
69
to the truncation error term (ℎ2 ) . The accuracy of Eq. 2 can be further improved following
Chapra and Canale (2001) by including additional terms in the Taylor series expansion, leading to
an expression where the truncation error is of order ℎ4 :
𝜅(𝑥𝑖 ) =
−𝜙(𝑥𝑖+2 )+16𝜙(𝑥𝑖+1 )−30𝜙(𝑥𝑖 )+16𝜙(𝑥𝑖−1 )−𝜙(𝑥𝑖+2 )
12ℎ2
+ 𝑂(ℎ4 )
Eq. 3
Sazonov and Klinkhachorn (2005) demonstrated that the maximum error bound of Eq. 2
considering both truncation error and measurement error in 𝜙(𝑥𝑖 )can be expressed as :
|𝐸[𝜅(𝑥𝑖 )]| ≤
𝜀(|𝜙𝑖+1 |+2|𝜙𝑖 |+|𝜙𝑖−1 |)
ℎ2
+
𝑀4 2
ℎ
12
Eq. 4
where |𝐸[𝜅(𝑥𝑖 )]| is the modal curvature error bound, 𝜀 is the maximum relative random
multiplicative error of mode shape 𝜙, and 𝑀4 is a constant term determined by the maximum
value of the 4th derivative of 𝜙. The first term on the right hand side of Eq. 4 corresponds to the
noise in mode shape data. The second term corresponds to the truncation errors. When the
spacing between measurement sites, ℎ, is relatively large, the second term tends to dominate Eq.
4. With a reduction in ℎ, the first term tends to grow larger and gradually become the dominant
factor in the error.
In most practical cases, modal testing experiments are carried out using accelerometers. The
extracted mode shape sites correspond to the location of accelerometers in a one-to-one fashion.
The number of available sensors thus becomes the main controlling factor for the number of sites
that can be measured. Even with approaches such as multiple set-ups during testing, the number
of measurement sites is often still very limited. Under these conditions, as will be shown later in
this paper, the truncation error term in Eq. 4 will be the dominant factor. In order to reduce the
70
effects of this concern some researchers have proposed the use of sensing equipment with high
spatial resolution such as laser vibrometers (Khan et al. 1999; Pai et al. 2004). However, in modal
testing experiments mode shapes are always prone to be contaminated by noise. With a reduction
of measurement spacing, the first term in Eq. 4 will increase and gradually become the dominate
error factor. Thus it appears that, contrary to common belief, the results of damage detection
method may not be able to benefit from high-spatial resolution measurements if it depends on
modal curvature computed using a numerical differentiation procedure.
“
71
APPENDIX E. SAP2000 MODAL ANALYSIS RESULTS
E-W FRAME MODAL ANALYSIS
Table 1 Undamaged SSMRF SAP2000 Modal Analysis Base Reaction Data
TABLE: Base Reactions
OutputCase CaseType
Text
Text
DEAD
NonStatic
DEAD
NonStatic
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
StepType
Text
Max
Min
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
StepNum
Unitless
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
GlobalFX
Kip
-6.7E-15
-6.7E-15
-9.7E-15
0
-5.2E-11
-1.9E-10
41.333
3.75E-07
-2E-08
92.596
5.17E-08
4.32E-08
161.584
-238.836
6.41E-09
-0.717
7.84E-08
-6.689
1.6E-08
-14.008
5.83E-05
3.07E-05
-46.614
43.901
0.001226
-0.00014
-6.932
-5.6E-06
58.584
-34.887
-7.8E-05
8.41E-05
-0.0002
-197.161
535.18
-435.745
0.000192
-4.196
-0.00045
-0.00033
0.000225
GlobalFY
Kip
0
0
3.95E-09
-1.8E-10
7.858
-3.2E-10
-1.4E-09
51.328
-1.5E-09
-3.4E-08
129.671
-1.5E-08
-8.3E-09
3.91E-09
7.33E-10
2.26E-09
-1.8E-09
-3.1E-09
-2.4E-09
5.72E-09
1.79E-08
3.21E-09
2.85E-09
8.49E-09
8.51E-10
0.000215
6.51E-06
6.43E-06
5.25E-06
1.22E-05
7.47E-06
-1.1E-05
-1.4E-06
-8.8E-07
-1.2E-06
-5.4E-05
2.03E-05
-0.0012
-217.088
-4.6E-05
0.000182
GlobalFZ
Kip
468.294
468.294
2.62E-13
-6.7E-20
1.75E-09
-4.1E-09
2.07E-08
7.32E-06
3.5E-07
3.87E-05
4.85E-14
-3.7E-13
-9.9E-11
7.49E-12
-7892.79
3.39E-09
4489.466
1.49E-08
1900.176
1.22E-06
-596.919
3871.186
0.000416
0.005471
454.196
-8963.96
0.001289
-7076.92
0.000197
0.002577
-600.434
6873.71
4177.515
0.011
-0.00726
-0.016
5423.321
-0.00395
0.006451
-0.017
0.009185
GlobalMX
Kip-in
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
GlobalMY
Kip-in
6.72E-09
6.72E-09
2.83E-10
-2.4E-16
4.74E-06
-1.4E-05
17393.97
0.000799
-0.00046
-25130.6
-1.4E-05
-1.2E-05
-24661.2
2287.257
-2.8E-06
-1977595
-1.3E-05
-2766950
-5.6E-06
-1430002
0.15
-0.077
-825152
2539260
-32.419
-1.346
-2876381
-0.383
2027388
-402398
-3.787
3.928
-11.37
-631685
531188.1
-2080963
-5.954
-896268
14.679
9.224
-0.592
GlobalMZ
Kip-in
0
0
3.85E-07
4.021
-7.5E-07
-3408.09
3.37E-07
-2.1E-05
-21138.2
5.48E-05
2.94E-07
-52322.3
3.96E-07
-1.8E-07
1.17E-09
7.26E-08
1.85E-07
1.45E-07
3.23E-09
4.17E-07
4.45E-07
1.47E-08
3.3E-07
3.57E-07
-5.6E-07
-0.027
-0.024
-0.019
-0.00712
-0.048
-0.00678
0.009988
0.002732
-0.00324
0.003947
0.06
-0.039
-0.553
0.556
133495.6
89690.49
72
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
0.000214
0.000533
-0.00067
9.29E-05
6.84E-05
-0.00254
0.000144
0.001675
-0.00641
0.001216
-395.607
0.003845
-191.328
0.000111
108.393
0.002707
-0.00024
-5.3E-05
6.21E-05
0.000197
0.000425
7.26E-05
0.000451
0.059
-27.484
0.005353
964.592
-0.00087
166.784
-0.044
-0.00041
-123.911
-0.00084
-0.00126
-1.11
1354.791
0.005615
-0.063
-49.937
0.072
-0.00167
-0.117
0.000187
-0.00475
1.205
0.069
-698.822
-719.157
4.54
-0.00145
-0.012
0.004623
-387.332
5.81E-05
1.55E-05
0.000035
5.77E-05
-0.00536
-0.00243
0.00473
-0.015
-0.00014
0.000475
0.000609
0.006011
0.000946
-0.00015
6.86E-05
6.57E-05
0.000382
0.000122
-6.9E-05
0.001336
0.000508
0.000553
0.022
-0.044
-0.00813
0.007587
-0.013
-0.016
-0.06
-8.7E-05
0.000872
14.704
-0.00053
0.003445
-0.00595
-19.799
0.037
0.102
-0.066
-0.0015
0.133
-7.2E-05
-0.00458
-0.231
0.051
0.556
0.406
0.745
-0.0036
-0.00582
0.004951
-0.017
-0.256
0.186
-0.055
-0.042
0.096
0.071
-0.012
0.135
-33707.5
0.031
7064.123
-0.04
-297.346
0.025
-0.092
0.046
0.124
-0.079
0.099
-0.031
-0.054
-0.01
-34541.6
-0.116
68827.03
0.098
-42110.4
0.22
-0.047
-62114.4
0.04
0.032
-0.03
-57408.1
-47.245
-0.263
52254.44
-25.23
2.973
-0.003
-2.247
0.047
-0.08
-4.436
0.085
27507.7
-26737.3
-20.183
0.003306
0.038
-0.00318
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-2.251
-3.779
8.605
-8.189
28.093
9.062
26.192
7.441
-37.305
-65.363
17491937
1.953
3382834
7.52
-20328.6
-2.156
-28.648
-5.655
6.093
19.58
13.495
5.586
29.706
4.059
-5554839
-180.384
-3.3E+07
30.67
4414868
-781.594
25.7
13225622
28.741
30.269
26570.49
-3.2E+07
-107.099
5620.699
9962170
730.15
-1.268
1059.215
-7.709
-1.084
-25208.3
-34.185
14899709
15330348
-92940.4
-0.217
36.307
-19.977
-0.063
0.135
27.375
-0.024
-0.298
0.949
-3.88
-0.876
1.807
0.566
-0.5
-1.688
3.46
-2.71
-0.381
3.042
1.756
389.377
-287.493
-3.533
-5.143
-1.072
-4.231
-17.942
16.234
0.005303
0.032
-0.36
5.262
6351.859
0.001812
-0.0089
-0.0071
7881.575
-5.575
10.59
13.893
13.047
57.525
-54.113
1.198
9.657
1.37
8.684
-31.035
-121.721
78.194
341.395
312.945
19.969
32.802
1464347
73
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
Push
Push
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
NonStatic
NonStatic
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Max
Min
92
93
94
95
96
97
98
99
100
0.051
0.00177
0.067
-0.346
-0.053
0.056
0.069
-0.243
0.023
-6.7E-15
-460.575
4080.62
-0.096
-2537.91
-0.355
-0.019
0.018
0.026
-0.249
0.009475
0
0
-0.946
-0.014
0.001346
-0.029
0.201
-0.079
0.036
-0.068
0.069
468.294
468.294
0
0
0
0
0
0
0
0
0
0
0
-2523.36
-52.041
-212.544
-821.767
124.535
-22.816
-54.976
-560.927
40.094
6.72E-09
-174488
732.388
942466.7
-72.873
653.371
58.493
-56.885
-64.747
486.285
-18.685
0
0
Table 2 Undamaged SSMRF SAP2000 Modal Analysis Joint Displacement Data
TABLE:
Joint
Text
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Joint Displacements
OutputCase
Text
DEAD
DEAD
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
CaseType
Text
NonStatic
NonStatic
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
StepType
Text
Max
Min
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
StepNum
Unitless
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
U1
in
U2
in
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U3
in
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
R1
Radians
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-0.002189
-0.003529
0.004637
0.007681
-2.28E-13
0.009954
0.016179
-1.73E-11
0.013861
0.022064
-1.94E-12
9.04E-13
1.42E-13
3.86E-13
-4.93E-13
-7.21E-13
-4.84E-13
7.95E-13
3.12E-12
5.99E-13
2.98E-13
1.38E-12
6.17E-13
7.49E-09
5.17E-09
3.54E-09
1.88E-10
1.16E-08
-1.69E-10
2.16E-10
-1.39E-10
2E-10
-2.13E-10
-2.89E-09
1.27E-09
1.36E-09
-0.033407
-0.034421
-0.007186
-0.005808
-7.08E-09
R2
Radians
-0.000186
-0.000186
-6.11E-19
0
-1.79E-14
8.95E-14
-0.00421
-8.13E-12
7.82E-14
-0.007556
-4.22E-12
-3.53E-12
-0.009367
0.010361
0.000288
0.000324
-0.000671
0.000571
-0.001232
0.001766
0.001855
-0.001114
0.002133
-0.00082
-0.011027
0.008583
0.004543
-0.002176
-0.018032
0.003736
-0.008056
-0.014633
0.004542
0.0044
-0.010674
0.002033
0.00279
-0.001666
1.71E-08
1.07E-08
-7.31E-09
-1.05E-09
1.04E-08
R3
Radians
0
0
-8.84E-11
-0.001098
4.91E-13
0.002145
9.95E-15
1.27E-12
0.003514
-1.57E-12
-1.13E-13
0.003097
-7.24E-14
3.33E-14
1.9E-16
-1.08E-14
-3.08E-14
-2.66E-14
-3.23E-15
-6.78E-14
-7.93E-14
-6.79E-15
-5.57E-14
-5.72E-14
1.01E-13
2.51E-10
3.6E-10
2.6E-10
5.13E-11
7.55E-10
-2.99E-11
4.23E-11
-1.53E-11
2.5E-11
-2.45E-11
-2.78E-10
6.39E-11
-2.17E-11
-1.61E-09
-0.000151
-0.000098
-6.85E-10
-1.63E-09
74
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
Push
Push
DEAD
DEAD
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
NonStatic
NonStatic
NonStatic
NonStatic
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Max
Min
Max
Min
Mode
Mode
Mode
Mode
Mode
Mode
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
1
2
3
4
5
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3.47E-15
3.47E-15
3.6E-17
0
-4.42E-13
5.05E-12
-0.578066
-1.5E-10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.367809
0.592738
-0.703686
-1.158359
1.18E-11
-1.176716
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-0.011224
-0.011224
-4.57E-17
0
-6.79E-13
2.3E-12
-0.00246
1.27E-10
-4.79E-06
2.24E-09
1.49E-08
-3.16E-09
9.23E-07
5.92E-09
-3.26E-08
-1.22E-08
4.26E-08
-7.01E-09
5.34E-08
2.47E-08
3.39E-08
-7.45E-08
-4.13E-08
-0.000071
0.000038
3.05E-08
9.93E-08
1.81E-08
5.67E-08
1.8E-07
-1.05E-07
-9.8E-08
9.54E-08
-1.53E-07
-2.9E-07
-0.000469
-3.04E-09
7.79E-09
0.000284
-0.000509
9.08E-08
-1.53E-07
-0.000347
2.97E-07
9.1E-07
-4.53E-07
-2.16E-08
1.41E-06
-2.62E-09
-8.52E-08
-2.51E-06
6.82E-07
4.18E-06
1.67E-06
4.64E-06
-1.33E-07
-2.08E-07
-0.075994
0.052061
-0.040449
-0.027409
-6.91E-06
-3.78E-07
3.63E-07
4.35E-07
-5.01E-06
1.31E-07
0
0
0
0
-0.002189
-0.003526
0.003275
0.005293
2.52E-13
0.000998
3.35E-08
1.85E-08
1.97E-08
7.58E-08
5.07E-08
-1.19E-08
7.63E-08
-0.016393
0.016326
0.007839
0.007811
-0.00003
-0.000564
-5.55E-08
-1.52E-08
5.58E-08
-2.46E-08
6.46E-08
3.26E-08
-2.34E-09
4.11E-08
0.000729
0.000644
0.032048
-0.032229
0.006003
-0.004765
-4.61E-07
-0.003711
0.004047
4.36E-08
1.23E-08
-0.033827
-0.033837
-9.81E-08
0.001881
0.001968
-1.28E-06
-1.76E-09
3E-06
2.4E-08
-5.33E-08
-0.000029
-9.89E-09
0.029848
0.000418
-0.000101
-5.29E-08
6.64E-08
-1.99E-08
-2.23E-06
-4.24E-08
-2.55E-07
-7.64E-07
-9.33E-08
1.34E-07
2.24E-07
-3.58E-07
1.86E-07
0.264686
-0.000186
0.000427
0.000427
2.11E-18
0
3.24E-14
-1.06E-13
-0.001676
1.57E-11
-6.94E-09
4.04E-10
4.67E-10
-1.83E-10
1.09E-09
1.08E-10
-2.79E-10
-1.2E-09
8.26E-10
3.61E-11
-6.86E-10
4.56E-10
2.37E-10
-6.55E-10
-2.36E-10
-0.000012
7.76E-07
8.11E-11
4.54E-10
1.33E-10
2.77E-10
1.21E-10
1.7E-10
2.11E-10
-6.19E-11
-4.76E-11
-1.19E-09
6.47E-07
-1.13E-11
-4.46E-11
-4.95E-11
9.67E-07
-9.64E-11
1.77E-10
1.82E-10
-1.18E-09
-1.56E-09
1.71E-09
1.5E-11
-4.1E-10
-2.9E-11
5.37E-12
2.44E-09
-3.24E-11
-8.13E-09
-6.02E-09
-1.15E-08
5.21E-11
6.16E-11
-0.009802
4.82E-09
-0.000128
4.55E-10
2.52E-09
2E-10
-1.22E-10
-2.46E-10
1.75E-09
-2.66E-11
0
0
0
0
-8.84E-11
-0.001098
4.91E-13
0.002145
9.95E-15
1.27E-12
75
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
-3.91E-12
-0.970029
-5.42E-10
-4.54E-10
-1.026678
0.890826
-6.71E-11
-0.004417
-8.21E-10
-0.012125
-1.79E-10
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-7.56E-11
1.16E-08
2.55E-11
-1.32E-10
-1.21E-08
6.64E-11
1.21E-08
7.93E-09
1.53E-08
-1.73E-11
-1.86E-10
-1.6E-06
1.01E-08
-0.000019
1.8E-09
1.71E-08
1.86E-10
-5.31E-10
3.34E-10
1.39E-08
-3.22E-10
0
0
81
Table 3 Undamaged SSMRF SAP2000 Modal Analysis Period and Frequency Data
TABLE:
Modal Periods And
Frequencies
OutputCase StepType StepNum
Text
Text
Unitless
MODAL
Mode
1
MODAL
Mode
2
MODAL
Mode
3
MODAL
Mode
4
MODAL
Mode
5
MODAL
Mode
6
MODAL
Mode
7
MODAL
Mode
8
MODAL
Mode
9
MODAL
Mode
10
MODAL
Mode
11
MODAL
Mode
12
MODAL
Mode
13
MODAL
Mode
14
MODAL
Mode
15
MODAL
Mode
16
MODAL
Mode
17
MODAL
Mode
18
MODAL
Mode
19
MODAL
Mode
20
MODAL
Mode
21
MODAL
Mode
22
MODAL
Mode
23
MODAL
Mode
24
MODAL
Mode
25
MODAL
Mode
26
MODAL
Mode
27
MODAL
Mode
28
MODAL
Mode
29
MODAL
Mode
30
MODAL
Mode
31
MODAL
Mode
32
MODAL
Mode
33
MODAL
Mode
34
MODAL
Mode
35
MODAL
Mode
36
MODAL
Mode
37
MODAL
Mode
38
MODAL
Mode
39
MODAL
Mode
40
Period
Sec
167894.0947
57.572964
1.334178
1.178386
1.007748
0.422934
0.378646
0.32047
0.175792
0.158255
0.157332
0.094909
0.066953
0.065785
0.059861
0.059237
0.049647
0.045278
0.041345
0.040534
0.038538
0.037778
0.036668
0.035751
0.035077
0.034674
0.034497
0.033624
0.031709
0.031436
0.030117
0.029829
0.029735
0.028867
0.026002
0.025417
0.017905
0.017795
0.017412
0.017349
Frequency
Cyc/sec
5.9561E-06
0.017369
0.74953
0.84862
0.99231
2.3644
2.641
3.1204
5.6885
6.3189
6.356
10.536
14.936
15.201
16.705
16.881
20.142
22.086
24.186
24.671
25.948
26.47
27.272
27.972
28.508
28.84
28.988
29.74
31.537
31.81
33.203
33.524
33.63
34.641
38.459
39.344
55.85
56.195
57.432
57.64
CircFreq
rad/sec
0.000037424
0.10913
4.7094
5.332
6.2349
14.856
16.594
19.606
35.742
39.703
39.936
66.202
93.845
95.511
104.96
106.07
126.56
138.77
151.97
155.01
163.04
166.32
171.36
175.75
179.12
181.21
182.14
186.86
198.15
199.87
208.62
210.64
211.31
217.66
241.64
247.2
350.92
353.09
360.86
362.16
Eigenvalue
rad2/sec2
1.4005E-09
0.01191
22.178
28.431
38.874
220.71
275.36
384.4
1277.5
1576.3
1594.9
4382.8
8806.9
9122.3
11017
11251
16016
19257
23094
24028
26581
27662
29363
30888
32085
32837
33174
34918
39264
39948
43523
44369
44650
47375
58391
61110
123140
124670
130220
131160
82
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
0.011355
0.01132
0.01131
0.01131
0.010793
0.01076
0.010751
0.010751
0.008938
0.008935
0.008407
0.008388
0.008052
0.008014
0.007314
0.007314
0.0073
0.0073
0.007296
0.007296
0.007296
0.007296
0.007096
0.007007
0.005777
0.005775
0.00557
0.005526
0.004482
0.004371
0.004348
0.004317
0.004144
0.004143
0.004143
0.004013
0.003843
0.003827
0.003776
0.003776
0.003738
0.003738
0.003738
0.003589
0.003589
0.003564
88.069
88.34
88.418
88.418
92.65
92.935
93.017
93.017
111.88
111.93
118.94
119.22
124.19
124.79
136.72
136.72
136.99
136.99
137.07
137.07
137.07
137.07
140.93
142.71
173.1
173.15
179.52
180.97
223.13
228.79
229.97
231.65
241.31
241.39
241.4
249.19
260.25
261.28
264.86
264.86
267.55
267.55
267.55
278.64
278.64
280.56
553.35
555.06
555.55
555.55
582.13
583.93
584.44
584.44
702.98
703.25
747.34
749.06
780.31
784.06
859.05
859.05
860.75
860.75
861.22
861.22
861.22
861.22
885.49
896.69
1087.6
1087.9
1128
1137.1
1402
1437.5
1444.9
1455.5
1516.2
1516.7
1516.7
1565.7
1635.2
1641.7
1664.2
1664.2
1681.1
1681.1
1681.1
1750.7
1750.7
1762.8
306200
308090
308630
308630
338880
340970
341570
341570
494180
494560
558520
561100
608880
614750
737970
737970
740890
740890
741690
741690
741690
741690
784100
804060
1182900
1183600
1272300
1292900
1965500
2066500
2087800
2118400
2298900
2300300
2300500
2451400
2673800
2695200
2769500
2769500
2826000
2826000
2826000
3065100
3065100
3107600
83
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
87
88
89
90
91
92
93
94
95
96
97
98
99
100
0.003564
0.003553
0.003553
0.003553
0.003432
0.003271
0.00324
0.00311
0.002849
0.002849
0.002849
0.002849
0.002821
0.002821
280.56
281.46
281.47
281.47
291.4
305.69
308.62
321.56
350.96
350.97
350.97
350.97
354.53
354.53
1762.8
1768.5
1768.5
1768.5
1830.9
1920.7
1939.1
2020.4
2205.2
2205.2
2205.2
2205.2
2227.6
2227.6
3107600
3127600
3127700
3127700
3352400
3689100
3760200
4082200
4862700
4862900
4862900
4862900
4962000
4962100
84
N-S FRAME MODAL ANALYSIS
Table 4 Undamaged SSCBF SAP2000 Modal Analysis Base Reaction Data
TABLE: Base Reactions
OutputCase CaseType
Text
Text
DEAD
NonStatic
DEAD
NonStatic
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
StepType
Text
Max
Min
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
StepNum
Unitless
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
GlobalFX
Kip
7.28E-13
7.28E-13
1.1E-14
1.31E-10
1.29E-08
-6.4E-08
-5E-08
-1.5E-07
7.08E-07
-1.7E-05
815.821
1.15E-06
5.81E-06
1609.625
-1.5E-06
-2125.34
1722.716
2.58E-05
0.000131
-0.00017
-3.7E-05
0.00106
0.016
-231.173
-30.514
0.000289
0.001833
-542.801
-4802.02
730.787
0.000376
-0.00114
1.518
0.002856
-0.014
-0.025
-0.204
0.164
123.36
0.022
-0.168
-0.01
GlobalFY
Kip
0
0
0.54
-5.084
8.275
-23.502
4.396
-2.992
33.484
-117.486
1.47E-07
7.08E-07
5.94E-06
-3.1E-08
2.27E-06
9.53E-08
-0.00012
1.74E-05
4.09E-05
-248.356
-5.7E-05
0.000983
0.008996
-0.00444
0.000747
-0.00516
0.001731
0.01
-0.015
0.004027
0.000116
-0.00032
-0.00497
-1.546
0.007586
-0.027
-0.138
0.169
0.105
-203.13
-0.102
35795.66
GlobalFZ
Kip
222.461
222.461
-2.8E-14
-1.9E-10
-1.7E-08
9.11E-08
7.04E-08
6.46E-08
-1.2E-06
3.48E-05
8.54E-08
497.805
-456.92
-2.5E-07
7179.724
-3.4E-05
8.44E-06
-3843.66
-10989.4
-8E-05
13086.31
-15164.2
0.009843
0.001808
0.005735
0.000996
-20631
0.001988
0.000558
-0.00247
-0.00134
2.193
-0.00537
0.00253
-0.00364
-0.033
-0.07
127.077
0.096
-0.091
0.064
0.000867
GlobalMX
Kip-in
0
0
-267.127
965.243
-846.051
2388.317
-322.885
900.081
-3537.83
9216.841
-1.2E-07
-6.3E-07
-6.4E-06
2.45E-07
4.28E-06
3.39E-05
-0.00012
-6.9E-06
-8.3E-05
13908.74
3.59E-05
-0.00062
-0.014
0.000216
-0.00604
0.000657
0.000192
-0.00178
-0.00362
0.000424
0.000207
-1.5E-05
-0.00038
26.114
-0.016
0.008842
0.064
-0.152
-0.144
419.409
-0.011
-216.349
GlobalMY
Kip-in
1.2E-10
1.2E-10
1.75E-12
2.24E-08
2.22E-06
-1.1E-05
-8.5E-06
-2.6E-05
0.000118
-0.00274
136951.1
-6.8E-05
-7.4E-05
267735.4
-9E-05
-347987
270622.6
-0.00822
-0.00719
0.008491
-0.00341
-0.00707
-1.012
-37257.2
-1058.34
-0.472
-0.058
-22161.7
-804484
109448.6
-0.098
0.125
-14.24
-0.28
2.769
0.475
15.472
-18.413
-7783.07
4.977
23.813
1.476
GlobalMZ
Kip-in
0
0
1.66E-12
-5.3E-13
8.21E-09
-8.2E-09
7.24E-08
2.02E-08
8.84E-07
1.35E-05
4.54E-05
6.65E-05
8.76E-06
-5.5E-05
9.6E-05
-0.00426
-0.034
0.003159
0.000989
0.017
0.007432
-0.226
-0.768
-0.277
-0.086
-0.162
-0.363
-1.383
2.847
-0.059
0.001246
-0.021
-0.615
-0.024
-0.755
50.854
-14.07
8.952
-4.995
0.952
7.491
0.21
85
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
-0.119
-21011.2
-0.021
0.267
-111403
0.133
0.121
0.38
-0.715
-0.223
-0.248
-0.234
-7.953
-0.081
-0.073
0.001597
4.506
4.147
623.255
0.165
-2.635
0.105
-98915
-0.033
1.877
-0.85
268724.9
-0.032
0.026
0.007382
-1.414
-0.599
-1.414
1.358
0.181
2.371
1016710
-0.319
-1.52
1.106
-746688
-0.148
-0.245
-2.243
2.169
3.781
41136.47
3.794
5219.252
-56920.3
-1.494
-1.7
-0.029
0.031
-0.137
0.1
-0.072
-141694
5871.9
-52293.4
0.004353
-0.019
0.036
0.056
-0.022
52.994
774.377
-1.356
0.02
-0.012
-0.741
2675.196
-2.041
0.058
0.124
-205670
1.33
0.409
-0.119
-322550
0.541
7.975
321.632
3.344
8.33
-1.669
-0.434
-2.918
-0.205
0.119
0.954
-0.901
1.014
0.319
0.336
41942.4
0.067
0.134
0.035
0.098
-0.045
-0.033
2368.619
0.008158
17092.95
0.042
0.038
99132.53
0.019
-0.069
-0.057
-0.091
0.028
0.633
-0.073
-2.776
0.014
0.007667
0.016
0.022
0.133
-285.667
-0.014
0.082
-0.136
84232.27
0.036
-0.01
0.126
241672.3
-0.778
0.153
0.414
0.237
-3.731
78.845
-3.742
292.923
-25.69
4.221
0.759
-0.444
0.293
0.269
0.633
-12262.2
12.864
-2.909
-1.273
-5.792
-0.841
-5.459
4.573
4.698
2.64
1.522
-0.00732
-0.00276
-0.011
0.014
-0.00445
-923.472
-345.547
2516.389
-0.00971
0.154
-0.037
-0.023
0.036
-151.202
8557.351
-158.726
0.361
0.283
0.637
-1600.42
0.826
-0.053
-0.013
-1931.2
-0.731
-2.273
0.911
1253.412
3.209
-13.112
-10740.8
-5.4
-13.498
2.568
0.684
4.733
0.342
-0.194
-1.562
1.5
-1.667
-0.519
-0.561
-963908
-0.108
-0.151
0.011
-0.143
0.116
0.135
-44242.5
0.085
10.571
2693854
11.107
-35.71
15145618
0.345
-11.836
-42.63
-5.564
35.228
18.545
13.305
507.509
13.522
1.454
-3.681
-8.257
-5.266
-58224.4
-7.043
-9.64
-4.939
13393196
0.053
4.537
-185.925
-3.6E+07
-14.89
-141.24
12.867
-215.586
-92.418
-216.706
96.378
45.116
705.85
-2659.95
-157.511
-41.679
8.164
-3287.74
30.484
21.282
-579.237
272.612
164.512
-3753.07
179.316
-546.2
3339.625
-32.201
-27.569
9.872
6.579
-2.2E+07
-9.581
-5.087
-18.16
-6.927
-0.721
-80.428
4506.917
-29.892
-24.451
3.87
-23.785
-2.273
5.489
-28.375
-24.328
13.499
11.725
26.254
13.562
-0.028
2.161
57534265
91.705
-43.491
16.094
266.929
269.432
276.573
150.952
254.496
39.629
25.661
-23.983
74.633
-88.503
507.27
130.659
204.787
257.631
222.235
139.806
-12.725
168.213
86.098
79.929
3.602
76.92
-42.959
-10.907
86
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
93
94
95
96
-99.357
-0.166
3.031
-0.949
MODAL
MODAL
MODAL
MODAL
PUSHOVER
PUSHOVER
LinModal
LinModal
LinModal
LinModal
NonStatic
NonStatic
Mode
Mode
Mode
Mode
Max
Min
97
98
99
100
-2.595
-1.363
-6.728
-0.392
7.28E-13
-689.898
0.023
-16677.3
0.098
4044577
-0.132
-0.047
-0.033
-13762.6
0
0
0.38
-8.392
-0.687
1.544
0.019
81519.5
-0.3
6605580
-32.734
214.814
221.379
-263.436
75.35
7.039
4.29E+08
16.829
2.603
-15.488
9.4
35.103
222.461
222.461
0.451
0.854
0.881
-44660.9
0
0
-183.793
2249.078
-923.726
-5971.77
1.2E-10
-104206
4.29E+08
-0.581
1.181
-2.013
0
0
Table 5 Undamaged SSCBF SAP2000 Modal Analysis Joint Displacement Data
TABLE:
Joint
Text
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Joint Displacements
OutputCase CaseType
Text
Text
DEAD
NonStatic
DEAD
NonStatic
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
MODAL
LinModal
StepType
Text
Max
Min
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
StepNum
Unitless
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
U1
in
-5E-16
-5E-16
3.7E-18
4.9E-14
5E-12
-2E-11
-2E-11
-6E-11
2.6E-10
-6E-09
-0.605
-5E-12
-1E-11
-1.2388
1.2E-12
1.72975
-1.4833
-3E-11
4.4E-11
5.5E-12
1.6E-12
-2E-10
-5E-10
-0.1698
0.00558
-1E-10
-3E-10
0.06143
-1.7127
0.21318
1.5E-08
U2
in
0
0
-0.2137
0.61536
-0.3483
0.97661
-0.0712
0.66055
-1.5114
2.42459
1.5E-12
-8E-12
-1E-11
4.1E-12
-6E-12
1.2E-10
3.4E-10
-1E-10
-6E-10
0.13396
-8E-10
8.5E-09
2.4E-08
-9E-09
5.1E-09
-6E-09
4.7E-09
-3E-09
-2E-08
1.9E-08
2.3E-08
U3
in
-0.0189
-0.0189
3.2E-18
2.1E-14
1.8E-12
-1E-11
-8E-12
-9E-12
1.3E-10
-4E-09
-2E-12
-0.0443
0.04062
-3E-16
-0.6258
1.7E-11
-3E-10
0.25205
0.45194
-4E-11
-1.613
1.58658
1.4E-09
-7E-10
-5E-11
1.1E-09
2.0816
-4E-09
1.2E-08
-4E-09
1.9E-08
R1
Radians
0
0
0.00238
-0.0058
0.00162
-0.0045
-0.0005
-0.0069
0.00777
0.00701
4.9E-13
4.3E-12
4.2E-12
5.9E-13
-1E-12
3.9E-11
3.7E-10
-4E-11
2E-11
0.07402
8.5E-12
-1E-10
2.9E-09
3.1E-09
3.3E-09
-2E-09
4.4E-09
1.9E-08
-4E-08
2.7E-09
5.6E-10
R2
Radians
-2E-18
-2E-18
1.6E-19
-7E-16
-1E-13
8.3E-13
1.1E-12
2.6E-12
3.3E-12
-4E-11
-0.0035
2.3E-12
4.6E-13
-0.0043
-6E-12
-7E-05
0.01273
-3E-11
-2E-11
-1E-11
-2E-11
3.8E-10
-4E-10
-0.0024
-0.0041
0.06651
4.7E-10
-0.0687
0.00627
0.01158
3.6E-10
R3
Radians
0
0
-1E-05
0.00073
-0.0041
0.02309
-0.008
-0.0263
0.02655
-0.0277
-1E-12
3E-11
5.9E-11
-5E-12
7.7E-11
-4E-10
-5E-09
9.7E-10
1.8E-09
-0.0007
1.4E-09
-4E-08
-4E-08
5.5E-08
6.3E-08
1.3E-07
-2E-09
-9E-08
1.2E-08
-1E-07
-1E-07
87
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
-9E-09
-6E-07
5.1E-08
-6E-08
-3E-08
3.2E-09
-7E-08
0.00018
4E-08
1.1E-07
5.4E-09
-1E-07
-0.0088
-1E-07
8.8E-09
0.01481
8.9E-08
5.5E-08
-4E-08
3.6E-07
-3E-07
-3E-08
-4E-08
4.5E-06
-2E-08
-1E-08
4.3E-08
-4E-07
-2E-07
-0.0001
-4E-08
6.9E-08
-4E-08
0.00226
7E-08
-2E-07
2E-08
0.00594
8.7E-09
2.6E-06
-4E-09
3.6E-07
1.7E-07
3.1E-07
-1E-07
2.9E-08
2.5E-07
0.15599
1.3E-07
-7E-07
1E-06
0.2839
-1E-08
-5E-08
1.1E-05
-4E-08
-1E-07
-1E-07
7.6E-08
-1E-07
0.00057
-1E-08
0.00335
-5E-09
1E-08
2.4E-08
2.8E-08
-3E-08
0.00456
0.00047
-0.0046
-3E-09
-9E-08
9.2E-09
-1E-09
-5E-09
0.00122
1.01173
-0.0026
1.7E-09
1.7E-08
7.2E-09
-0.0004
5.2E-07
-2E-08
-4E-08
-0.0007
-3E-07
3.9E-07
-2E-07
0.0025
-6E-07
-3E-07
0.0008
-5E-07
-6E-07
1E-06
-1E-07
-1E-06
-1E-07
-5E-07
-3E-07
3.3E-07
-7E-07
-0.0002
4.5E-09
3E-08
-2E-08
2.5E-08
8.3E-08
-0.0057
5.3E-08
1.5E-07
-5E-09
-3E-09
0.00574
2E-08
-1E-07
-0.0019
1.5E-08
-3E-08
1.5E-08
-1E-08
7.9E-08
3.9E-07
1.4E-07
6.6E-05
2E-07
2.9E-07
8.3E-08
1.4E-08
-2E-06
0.00169
2.1E-06
-1E-07
3E-06
0.00068
-2E-07
1.8E-08
-2E-06
0.00103
-5E-07
7.8E-08
-1E-06
7.9E-08
-7E-08
-0.0011
-7E-08
-0.0049
0.00044
-2E-07
-1E-08
-3E-08
1.5E-07
-3E-08
9.1E-08
1.4E-10
-8E-10
0.00014
-1E-09
-7E-08
-1E-07
2E-07
-6E-08
0.00228
-3E-08
-0.0012
-2E-08
-2E-08
-1E-08
3.1E-08
-9E-09
-0.0051
-0.0019
0.01379
-1E-06
1.5E-06
-2E-07
-1E-07
1.8E-07
-0.0008
0.02244
-0.0008
2.3E-06
1.1E-06
-3E-06
-0.0086
-1E-05
4.1E-07
1.1E-06
-0.0103
8.2E-06
-8E-06
3.3E-06
0.00663
2.2E-05
-2E-07
-0.0567
3.7E-07
8.1E-07
-7E-07
-6E-08
-6E-07
-2E-07
-2E-07
-3E-07
3.5E-07
-7E-07
4.2E-10
6.8E-05
-1E-09
1.1E-08
9.9E-09
-3E-08
8.3E-08
0.00206
-6E-08
-2E-08
3E-09
-2E-08
0.00239
1.3E-08
1.2E-08
-0.0006
7E-08
2.1E-08
4.6E-08
-8E-07
-8E-07
-5E-07
-1E-06
-0.0001
-1E-06
-5E-07
-1E-07
2.3E-06
1.8E-06
0.00357
-5E-08
1.7E-06
-3E-08
0.00435
6.5E-08
-1E-06
7.9E-07
0.00145
5.4E-08
1.5E-06
4.9E-08
-1E-07
-7E-08
-1E-07
1.8E-07
-3E-09
-1.1385
1.12585
-8E-09
6.9E-08
-0.8777
0.95878
3.9E-08
2.9E-07
-5E-05
-3E-05
-0.0609
8E-07
-6E-07
5.2E-07
-0.0034
8.2E-08
-0.0203
3.1E-08
-6E-08
-1E-07
-2E-07
1.7E-07
-0.0274
-0.003
0.02854
-0.0088
-5.0513
-1E-06
-1E-06
-5E-08
-0.006
-4.896
0.01246
-8E-07
-5E-07
7.1E-06
0.00176
1.4E-05
2.3E-06
-2E-06
0.00284
-8E-06
1.1E-05
-5E-06
-0.0097
1.9E-06
5.4E-06
-0.0007
8.2E-06
4.3E-06
-5E-06
-2E-06
2.2E-06
-2E-06
-8E-07
2.1E-06
-7E-07
-3E-06
88
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
PUSHOVER
PUSHOVER
DEAD
DEAD
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
NonStatic
NonStatic
NonStatic
NonStatic
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
LinModal
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Max
Min
Max
Min
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
-2E-07
-3E-07
4E-07
-8E-07
-6E-07
0.33311
-8E-07
0.04611
-0.677
2.5E-07
-2E-07
0.00045
8.6E-07
-2E-07
2.1E-07
1.5E-08
1.4E-06
1.1E-06
-1E-06
196.875
-5E-16
-6E-16
-6E-16
-1E-18
-3E-14
-4E-12
1.6E-11
1.6E-11
6.3E-11
-8E-11
9.6E-10
-1.103
3.7E-13
-2E-11
-1.3667
-9E-13
0.14936
1.73819
8.3E-11
2.1E-10
1.8E-10
7.4E-11
-2E-09
-8E-09
1.84813
-0.0512
1.2E-09
-1E-09
-0.7168
-0.1496
0.29058
-4E-08
-5E-07
-2E-07
0.15759
-3E-07
-4E-07
2.2E-08
-4E-07
2.9E-07
3.2E-07
0.00977
-6E-08
-2E-08
-0.0736
3.5E-08
0.00519
-4E-08
-1E-07
-1E-07
-0.9307
0
0
0
0
-0.6792
1.43364
-0.1031
0.77615
0.1824
1.56755
-1.0541
-1.8061
-2E-12
4.2E-12
6.7E-12
-2E-12
1.3E-11
4.8E-12
-4E-10
2.1E-10
-6E-11
-0.4348
5.7E-10
-4E-09
-1E-08
2.5E-08
2.9E-08
9.3E-09
-6E-09
1.2E-09
4.1E-08
-5E-08
5.1E-10
1.02571
-0.001
1.4E-08
-2E-08
-6E-08
-2E-09
-6E-08
5.4E-08
5.9E-08
3.7E-08
3.2E-07
7.6E-08
7.1E-08
-7E-07
1.7E-07
3.2E-07
4.7E-07
2E-06
1.4E-06
-0.0189
-0.0893
-0.0323
-0.0323
-9E-19
-1E-14
-7E-13
4.6E-12
3.2E-12
-9E-12
-8E-11
2.7E-09
2.1E-12
-0.0832
0.07622
1.5E-12
-1.1169
1.2E-10
9.7E-10
0.35638
0.56079
-3E-10
-1.7363
0.52594
-2E-09
-1E-08
-7E-09
-3E-09
-2.2462
-1E-08
4.5E-09
6.3E-10
5E-09
-4E-07
-3E-07
-4.8568
-3E-07
-3E-07
1.3E-08
-3E-07
2.3E-07
2.7E-07
-0.2153
5.3E-07
1.2E-07
0.22447
-8E-07
-0.0144
1.3E-06
4.2E-06
4.5E-06
-0.8313
0
0
0
0
0.0037
-0.0043
-0.0056
0.00541
-0.0021
-0.0023
-0.0147
0.01932
-2E-14
-1E-12
-2E-12
-2E-13
-1E-12
-2E-11
-4E-11
2.8E-11
-2E-10
-0.275
-2E-11
2.9E-10
1.1E-10
9.2E-10
8.9E-10
2E-09
-2E-09
-9E-09
2E-08
-3E-11
2.3E-10
3.1E-08
3.3E-08
-1E-07
0.49561
-0.3769
-0.3138
0.21431
-0.0404
0.46182
-9E-08
0.00092
0.0008
5.6E-08
-3E-06
1.2E-06
2.9E-06
5.6E-06
9.6E-06
5.5E-07
0.56553
-2E-18
-9E-20
-9E-20
-7E-19
1.5E-15
3.4E-13
-3E-12
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2E-07
-5E-08
-5E-07
-0.0444
-5E-07
0.42508
0.0074
2.8E-07
-2E-07
-0.1794
-4E-07
2.4E-07
-1E-07
-1E-08
-4E-07
-7E-08
3E-07
200
-9E-16
4.8E-08
-2E-07
-0.0143
-6E-08
-6E-07
-6E-09
-8E-07
3.4E-07
2.1E-07
0.18937
-3E-06
-7E-07
0.00146
-2E-07
9.9E-08
-2E-07
-2E-07
-2E-07
-1E-07
0
0
-4E-06
-0.0047
3.7E-08
5.1E-09
1.9E-07
-2E-09
2E-07
-1E-07
-1E-07
-5E-08
8.1E-07
1.8E-07
-8E-08
2.5E-07
-3E-07
1.6E-07
-8E-07
5.1E-07
2E-06
-0.0476
-0.1509
6.5E-07
4.3E-07
0.30386
9.5E-07
1.6E-05
-2E-07
0.00002
-8E-06
-5E-06
-3.6405
8.6E-05
1.8E-05
-0.0141
4.2E-06
-2E-05
3.8E-06
4.8E-06
3.7E-06
-7E-06
0
0
9.2E-08
7.5E-08
4.1E-07
0.15363
2.069
-0.4268
3.3113
3.91236
0.06284
6E-07
-0.7014
-0.7391
-4E-07
-7E-08
-2E-08
1.7E-07
1E-06
8.2E-07
-6E-07
0.00188
-2E-18
Table 6 Undamaged SSCBF SAP2000 Modal Analysis Period and Frequency Data
TABLE: Modal Periods And Frequencies
OutputCase StepType StepNum
Period
Text
Text
Unitless
Sec
MODAL
Mode
1 6.708358
MODAL
Mode
2 1.612324
MODAL
Mode
3 0.732888
MODAL
Mode
4 0.599393
MODAL
Mode
5 0.530099
MODAL
Mode
6
0.44134
MODAL
Mode
7 0.335253
MODAL
Mode
8 0.189095
MODAL
Mode
9
0.18496
MODAL
Mode
10 0.149752
MODAL
Mode
11
0.13934
MODAL
Mode
12
0.07406
MODAL
Mode
13 0.057658
MODAL
Mode
14 0.049481
MODAL
Mode
15 0.036975
MODAL
Mode
16 0.028589
MODAL
Mode
17 0.026412
MODAL
Mode
18 0.021828
Frequency
Cyc/sec
0.14907
0.62022
1.3645
1.6684
1.8864
2.2658
2.9828
5.2884
5.4066
6.6777
7.1767
13.503
17.344
20.21
27.045
34.979
37.861
45.812
CircFreq
rad/sec
0.93662
3.897
8.5732
10.483
11.853
14.237
18.742
33.228
33.971
41.957
45.092
84.839
108.97
126.98
169.93
219.78
237.89
287.85
Eigenvalue
rad2/sec2
0.87726
15.186
73.499
109.88
140.49
202.68
351.25
1104.1
1154
1760.4
2033.3
7197.6
11875
16125
28876
48303
56591
82856
5.8E-08
3.6E-07
0.00827
1.5E-07
3.3E-07
3.4E-08
5.6E-07
-3E-07
-2E-07
-0.1063
2.3E-06
4.5E-07
-0.0008
1.4E-07
-2E-07
1.2E-07
1.1E-07
1.1E-07
-3E-09
0
0
95
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
0.020487
0.015039
0.014203
0.012779
0.011662
0.01106
0.01094
0.010925
0.009878
0.009819
0.003572
0.003561
0.00356
0.003552
0.003312
0.003295
0.003198
0.003181
0.00318
0.003168
0.002935
0.002925
0.00292
0.00292
0.00287
0.002857
0.002853
0.002843
0.002813
0.002811
0.002466
0.002459
0.002136
0.00213
0.002129
0.002122
0.002113
0.002112
0.001934
0.001924
0.001924
0.001913
0.001793
0.001787
0.001786
0.001781
48.811
66.496
70.406
78.251
85.745
90.416
91.412
91.537
101.24
101.85
279.92
280.81
280.87
281.52
301.92
303.47
312.72
314.38
314.46
315.66
340.72
341.92
342.45
342.46
348.39
350.03
350.56
351.77
355.52
355.73
405.57
406.74
468.16
469.54
469.62
471.27
473.17
473.43
517.11
519.65
519.8
522.78
557.57
559.73
559.84
561.45
306.69
417.81
442.37
491.66
538.75
568.1
574.36
575.14
636.1
639.93
1758.8
1764.4
1764.7
1768.9
1897
1906.8
1964.8
1975.3
1975.8
1983.4
2140.8
2148.3
2151.7
2151.7
2189
2199.3
2202.6
2210.2
2233.8
2235.1
2548.3
2555.7
2941.5
2950.2
2950.7
2961.1
2973
2974.6
3249.1
3265
3266
3284.7
3503.3
3516.9
3517.6
3527.7
94058
174560
195690
241730
290250
322740
329880
330790
404630
409510
3093400
3113100
3114300
3128900
3598600
3635700
3860600
3901800
3903800
3933800
4583200
4615300
4629800
4629900
4791600
4837000
4851500
4885100
4989900
4995900
6493700
6531400
8652600
8703600
8706500
8767900
8838600
8848500
10557000
10660000
10667000
10790000
12273000
12368000
12373000
12445000
96
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
MODAL
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
0.001774
0.001765
0.001763
0.001749
0.001574
0.001574
0.001535
0.001525
0.001519
0.001468
0.001468
0.001386
0.001384
0.001335
0.001335
0.001273
0.001267
0.001256
0.001253
0.001172
0.001125
0.001103
0.001102
0.001102
0.001093
0.001078
0.001067
0.001009
0.001002
0.000806
0.000757
0.000757
0.000756
0.000716
0.000715
0.000681
563.55
566.53
567.33
571.76
635.17
635.17
651.38
655.88
658.16
681.14
681.19
721.62
722.7
748.98
748.98
785.48
789.24
796.06
798.04
852.95
889.18
906.29
907.04
907.09
914.58
927.24
937.31
991.55
998.48
1240.2
1320.9
1321.6
1322.3
1395.9
1397.7
1468.9
3540.9
3559.6
3564.6
3592.5
3990.9
3990.9
4092.8
4121
4135.3
4279.7
4280
4534.1
4540.8
4706
4706
4935.3
4959
5001.8
5014.3
5359.2
5586.9
5694.4
5699.1
5699.4
5746.5
5826
5889.3
6230.1
6273.6
7792.4
8299.2
8303.8
8308.1
8770.8
8782
9229.3
12538000
12671000
12707000
12906000
15927000
15927000
16751000
16983000
17101000
18316000
18319000
20558000
20619000
22146000
22146000
24357000
24591000
25018000
25143000
28721000
31214000
32426000
32480000
32484000
33022000
33943000
34684000
38814000
39358000
60722000
68876000
68953000
69025000
76927000
77123000
85180000
97
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