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Nonlinear Dynamic Analysis of Modular Steel Buildings in Two and Three Dimensions
by
Amirahmad Fathieh
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
Department of Civil Engineering University of Toronto
© Copyright by Amirahmad Fathieh 2013
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Nonlinear Dynamic Analysis of Modular Steel Buildings in Two and Three Dimensions
Amirahmad Fathieh
Master of Applied Science
Department of Civil Engineering
University of Toronto
2013
Abstract
Modular construction is a relatively new technique where prefabricated units are
assembled on-site to produce a complete building. Due to detailing requirements for the
assembly of the modules, these systems are prone to undesirable failure mechanisms
during large earthquakes. Specifically, for multi-story Modular Steel Buildings (MSBs),
inelasticity concentration in vertical connections can be an area of concern. Diaphragm
interaction, relative displacements between modules and the forces in the horizontal
connections need to be investigated. In this study, two 4-story MSBs with two different
structural configurations were chosen to be analyzed. In the first model which was
introduced in a study by Annan et al. (2009 a), some of the unrealistic detailing
assumptions were challenged. To have a more accurate assessment of the structural
capacity, in the second model, a more realistic MSB model was proposed. Using
OpenSees, Incremental Dynamic Analyses (IDA) have been performed and conclusions
were made.
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Acknowledgments
My sincere appreciation goes to my supervisor, Dr. Oya Mercan, for her continuous
support, encouragement, and guidance in all stages of this thesis. During my research, she
supported my attendance at conferences, engaged me in new ideas, and provided me her
valuable insights on structural dynamics.
I would like to express my deep gratitude to Mr. Brent Roberts, whose expertise,
understanding, and comments, added considerably to my research experience. I
appreciate his vast knowledge and skill in modular construction area which have assisted
me at all levels of this study.
I also would like to thank Dr. Oh-Sung Kwon for his help and guidance as well as his
effort in reviewing my work and his attendance as the second reader of my thesis
committee.
In addition, the financial support for this study from NSERC Discovery (Grant 371627-
2009) and the start-up funds from the University of Toronto is gratefully acknowledged.
Finally, I wish to thank my parents and grandparents for their unconditional support they
provided me through my entire life. Without them I would never reach this stage of my
life. To them I dedicate this thesis.
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Table of content
Chapter 1 .............................................................................................................. 1
1. Introduction ....................................................................................................... 1
1.1 Modular Construction: Factors to be Considered, Potential, and Limitations
.......................................................................................................................... 1
1.2 Current State of Modular Construction ........................................................ 3
1.3 Seismic Design of Steel Buildings ............................................................... 5
1.4 Steel Structural Systems ............................................................................. 6
1.4.1 Moment Resisting Frames ..................................................................... 8
1.4.2 Braced Frames ...................................................................................... 8
1.5 Overview of This Study ................................................................................ 9
Chapter 2 ............................................................................................................ 12
2. Incremental Dynamic Analysis (IDA)............................................................... 12
2.1 Introduction ................................................................................................ 12
2.2 Nonlinear Dynamic Analysis ...................................................................... 13
2.3 Step-by-Step Integration Algorithms .......................................................... 15
2.4 Definitions .................................................................................................. 18
2.4.1 Scale Factor (SF) ................................................................................ 18
2.4.2 Intensity Measure (IM) ......................................................................... 20
2.4.3 Damage Measure (DM) ....................................................................... 21
2.5 General Properties of Single-Record IDAs ................................................ 21
2.6 Defining the Limit-State on a Single IDA Curve ......................................... 26
2.6.1 The DM-Based Rule ............................................................................ 27
2.6.2 The IM-Based Rule ............................................................................. 28
2.7 Implementation of the IDA ......................................................................... 31
2.7.1 Selecting the Ground Motions ............................................................. 31
2.7.2 Steps to Perform IDA........................................................................... 34
2.7.3 Defining the Capacity for Single IDA Curves ....................................... 37
2.7.4 Multi-Record IDAs ............................................................................... 42
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2.7.5 Summary of the Outputs...................................................................... 42
Chapter 3 ............................................................................................................ 45
3. Two-Dimensional Nonlinear IDA Analysis of MSB-Braced Frame .................. 45
3.1 Building Configuration................................................................................ 45
3.2 Site Specification ....................................................................................... 48
3.3 Analytical Model......................................................................................... 50
3.4 Results of the Incremental Dynamic Analysis ............................................ 56
3.4.1 IDA Curves .......................................................................................... 56
3.4.2 Summary of the IDA Curves ................................................................ 59
3.5 Inter-Story Drift and Inelastic Distribution along the Height of the Structure
........................................................................................................................ 61
Chapter 4 ............................................................................................................ 65
4. Modified Two and Three Dimensional MSB Structure Analysis ...................... 65
4.1 Common MSB Types and Range of their Application ................................ 65
4.2 Considerations Required in the Design of MSBs ....................................... 66
4.3 Modified 2D MSB Structure ....................................................................... 68
4.3.1 Beams and Columns ........................................................................... 71
4.3.2 Braces ................................................................................................. 72
4.3.3 IDA Results Obtained from the 2D Modified MSB ............................... 76
4.3.4 Selection of the Proper Intensity Measure ........................................... 78
4.3.5 Summary of the IDA Results ............................................................... 79
4.3.6 Inter-Story Drifts .................................................................................. 80
4.4 Modified 3D MSB Structure ....................................................................... 81
4.4.1 Hysteretic and Stiffness Properties ..................................................... 85
4.4.2 Bi-directional Horizontal Shaking ......................................................... 88
4.4.3 IDA Analysis of the 3D MSB Structure ................................................ 88
4.4.4 Summarizing the IDA Results .............................................................. 89
4.5 Comparison of the Two-Dimensional and Three-Dimensional Modified MSB
Structures ........................................................................................................ 90
4.5.1 Effects of Non-SFRS Frames on MSBs Responses ............................ 90
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4.5.2 Inter-Story Drifts in X and Z Directions ................................................ 92
4.5.3 Horizontal Connections and Diaphragm Action ................................... 95
4.5.4 Relative Motions of the Modular Units ............................................... 101
Chapter 5 .......................................................................................................... 103
5. Summary and Conclusions ........................................................................... 103
5.1 Summary ................................................................................................. 103
5.2 Conclusions ............................................................................................. 104
5.3 Future Studies ......................................................................................... 108
Chapter 6 .......................................................................................................... 110
6. References ................................................................................................... 110
Appendix A ....................................................................................................... 117
A.1 OpenSees Code (Modeling and Analysis) ............................................... 117
A.1.1 Modified 2D MSB .............................................................................. 117
A.1.2 Modified 3D MSB .............................................................................. 149
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List of Tables
Table 2.1 Earth quake ground motion records selected from PEER Strong
Ground Motion Database. ................................................................................... 33
Table 2.2 Sequence of runs for a ground motion (hunt & fill tracing algorithm). . 35
Table 3.1 Design spectral acceleration values of S(T) ........................................ 49
Table 3.2 Member sections from the seismic design. ......................................... 52
Table 3.3 Design and analytical periods ............................................................. 54
Table 3.4 Summarized capacities for each limit-state. ........................................ 61
Table 4.1 Member sections from the seismic design. ......................................... 71
Table 4.2 Design and analytical periods. ............................................................ 76
Table 4.3 summarizes the 16%, 50%, and 84% fractile values in terms of DM and
IM for IO, CP, and GI limit-states, for the modified 4-stories MSN-braced frame.
............................................................................................................................ 80
Table 4.4 Summarized capacities for each limit-state for the 3D modified MSB in
Z direction. .......................................................................................................... 90
Table 4.5 Comparison of Collapse Capacities obtained from 2D and 3D analysis
(Z direction). ........................................................................................................ 92
Table 4.6 Maximum Inter-story Drift Demand of the modified 4-Story MSB at the
design intensity level (Z direction). ...................................................................... 92
Table 4.7 Maximum values of connection elements axial, shear, and moment
force in global coordinates ................................................................................ 101
Table 4.8 Maximum values of connection elements nodal displacement and
rotations ............................................................................................................ 101
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List of Figures
Figure 2.1 Ground motion scaling in time domain (scaling on amplitude). .......... 20
Figure 2.2 IDA curves of a 4-story MSB frame subjected to different ground
motion records. ................................................................................................... 23
Figure 2.3 Single IDA curve for a 4-story MSB frame.. ....................................... 24
Figure 2.4 Single IDA curve for a 4-story MSB frame that shows Structural
Resurrection after the structure is pushed to the global instability. ..................... 25
Figure 2.5 The DM-based rule for a T1 = 0.6 sec, 4-story MSB. ......................... 28
Figure 2.6 The IM-based rule for a T1 = 0.6 sec, 4-story MSB using the 20%
slope criterion...................................................................................................... 30
Figure 2.7 IO, CP, and GI limit-states on a single IDA curve .............................. 39
Figure 2.8 Different segments of softening and hardening in a single IDA curve 41
Figure 2.9 IDA curves obtained from the selected ground motions.. .................. 43
Figure 3.1 4-story MSB braced frame. ................................................................ 46
Figure 3.2 Details of a typical MSB structure.3.2 Analysis Characteristics ......... 48
Figure 3.3 Design spectrum of Vancouver with Site Class C .............................. 50
Figure 3.4 Vertical connection model of MSB-braced frame. .............................. 53
Figure 3.5 IDA curves of ‘‘first mode’’ spectral acceleration. ............................... 57
Figure 3.6 IDA curves of Peak Ground Acceleration........................................... 57
Figure 3.7 Summary of IDA curves of the 4-story MSB frame into16th, 50th, and
84th fractiles........................................................................................................ 60
Figure 3.8 Height-wise distribution of peak inter-story drift for the 4-story MSB.. 63
Figure 4.1 Modified 4-story MSB braced frame .................................................. 69
Figure 4.2 Connection detail between the modules ............................................ 71
Figure 4.3 Brace finite element model. ............................................................... 74
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Figure 4.4 Force versus displacement relationship for a sample brace element. 74
Figure 4.5 IDA curves of ‘‘first mode’’ spectral acceleration. ............................... 77
Figure 4.6 IDA curves of Peak ground acceleration ............................................ 77
Figure 4.7 Summary of IDA curves of the modified 4-story MSB frame into16th,
50th, and 84th fractiles ........................................................................................ 79
Figure 4.8 Height-wise distribution of peak inter-story drift ratio for the modified 4-
story MSB ........................................................................................................... 81
Figure 4.9 3D MSB structure. ............................................................................. 83
Figure 4.10 Force transfer between modules. .................................................... 85
Figure 4.11 Different types of materials and elements that can be used in finite
element modeling................................................................................................ 87
Figure 4.12 IDA curves of Peak Ground Acceleration......................................... 89
Figure 4.13 Summary of IDA curves of the three-dimensional 4-story MSB
structure (Z direction). ......................................................................................... 89
Figure 4.14 Height-wise distribution of peak inter-story drift ratio for the 3D 4-
story MSB (Z direction). ...................................................................................... 93
Figure 4.15 Height-wise distribution of peak inter-story drift ratio for the 3D 4-
story MSB (X direction). ...................................................................................... 94
Figure 4.16 Roof displacement of the 2D versus 3D model (Z direction). ........... 94
Figure 4.17 Internal forces in the connections (i-j) direction. ............................... 96
Figure 4.18 Internal forces in the connections (m-n) direction. ........................... 97
Figure 4.19 Internal lateral moment in the floor connections oriented in (i-j) and
(m-n) directions. .................................................................................................. 98
Figure 4.20 End nodes displacements of connections.. ...................................... 99
Figure 4.21 Connection (i-j) rotations. ............................................................... 100
Figure 4.22 Module #7 rotations under the Loma Prieta ground motion. .......... 102
x
List of Appendices
Appendix A ....................................................................................................... 117
A.1 OpenSees Code (Modeling and Analysis) ............................................... 117
A.1.1 Modified 2D MSB .............................................................................. 117
A.1.2 Modified 3D MSB .............................................................................. 149
1
Chapter 1
1. Introduction
1.1 Modular Construction: Factors to be Considered, Potential, and
Limitations
There is a growing interest in developing new design and construction approaches that
are more efficient, safer, environmentally friendly, less labor intensive, and can lead to
buildings that are of higher standards and can be constructed in a compressed schedule
(Jeng, B. DiGiovanni D., and Wan A., 2011; Smith, R. E., 2010). Shortage of skilled
workers, low productivity and increasingly stringent client requirements are the
incentives behind developing innovative approaches (N. Lu, R. Liska, 2008).
Construction industry has been dominated with conventional construction practices
which are less efficient and economical. To meet the needs of client and communities of
future it is required to rethink the design and construction processes and develop
appropriate responsive strategies.
Due to the similarities between the manufacturing industry and construction industry, and
the fact that manufacturing industry has dealt with similar challenges by adopting offsite
fabrication approaches (Smith, R. E., 2010), in many cases, construction industry has
relied on technology transfer from the manufacturing industry (Forbes, L. H., Ahmed, S.
M, 2011). Modular construction is one of the examples as a technology that has been
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transferred from the manufacturing industry. Generally a manufacturing process, carried
out at a specialized facility, where various building materials and components are
assembled to produce a sub-assembly of the final structure/product, is referred to as
modular construction (CIRIA, 1999). The term offsite fabrication is also used when the
prefabrication and pre-assembly are integrated at a remote facility (Gibb, A. G. F., 1999).
Literally the term “module” is defined as “a product resulting from a series of remote
assembly operations”. It usually is the largest transportable unit or component produced
at a facility. Lower waste material, less damaged components and better overall quality
resulted from a more controlled environment, higher safety standards, lower cost, faster
completion and investment return can be listed as the advantages of modular construction
(CIDB, 2003).
The modern construction industry has only recently started to take significant advantage
of this approach. Because of the construction of new towns, suburbs, and large scale
public housing developments in the 1950s, 1960s and the early 1970s, the demand was at
a peak in Eastern and Western Europe (Warszawski, A., 1999). In the early 1970s, the
construction firms in the United States also explored several modular building systems.
Based on the level of the demand modular construction has had a fluctuating growth over
time; however recent technological advances have dramatically increased the scope of
modularization. As a ‘‘new’’ trend, the reemergence of modular construction can be tied
to the rise of Building Information Modeling (BIM) and green projects, as reported in the
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Smart-Market report of 2011 published by the McGraw-Hill Construction (SmartMarket
Report, 2011).
After all, when selecting the type of construction process, deciding between the use of
modular and traditional methods of construction is not easy. There are quite a few
variables involved to be evaluated. These variables can affect the decision making
process in different projects. Some of the parameters to be considered are the complexity
in project organization and planning, monitoring, coordination, transportation, and
possibility of reduced flexibility (Haas, C. T., Song, J., 2004), site conditions, skilled
workers availability, local codes, project schedule and design complexity (Azhar, S.,
Lukkad, M., Ahmad, I., 2012). It should also be noted that considering the
abovementioned parameters, the overall cost of a project may or may not render modular
construction as the preferred method. This decision needs to be made based on the
experience of a senior project manager and a modularization expert. Since, modular
construction is a complex combination of system and techniques, there is always a risk
that a wrong decision may result in poor implementation or even in project failure
(Koskela, L., Ballard, G, 2003).
1.2 Current State of Modular Construction
The modular method of construction is a fast evolving time-saving technique and it is an
alternative to traditional on-site construction. A modular building comprises multiple
prefabricated units called “modules”. Modular components are constructed indoors on an
assembly line then delivered to their intended site and to form a residential or commercial
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building. Each unit is often fully equipped with residential facilities such as plumbing,
flooring, lightening, etc. at the factory. Although the modular concept is similar to
temporary and mobile buildings, it is completely different in terms of structural design
and quality requirements. A modular building is a collection of units joined together to
form a self-supported and load bearing structure which must conform to all local building
codes for its intended use. The applications of modular construction include apartments,
schools, hotels, hospitals, offices, military accommodation, and any other building where
cellular and repetitive units are required.
Improved accuracy and quality, fast on-site installation, and lower final cost of
construction are the main motivations for owners to turn to modular constructions. In
modular construction, the potential in saving time and costs results from simultaneous
module construction and site preparation which leads to reducing the overall completion
schedule by as much as 50% and consequently enabling a faster return of investment. In
door construction will reduce vulnerability to weather condition that may slow or stop
on-site building progress and considerably reduce if not eliminate damaged building
materials. Since units are repeated, there will be reduced waste and site disturbance
resulting in a more environmentally friendly construction process.
Modular construction is widely used for residential buildings for up to eight stories using
different Seismic Force Resistance Systems (SFRS). However, there is always a pressure
to go higher and extend this form of construction to 25 stories or more. Some research
5
has been done on low-rise modular steel building behavior under a series of ground
motions with concentrically braced frame or use of double skinned steel plate; and also
there are some case studies on a couple of mid to relatively high-rise modular steel
buildings in the UK without considering the dynamic response under earthquake loading
(Lawson R. M., Richards J., 2010). Taking into account the manufacturing tolerances and
assembly errors, in taller modular building second-order effect may occur due to sway
and other eccentricities that are often neglected in the design of low-rise buildings.
Although, modular steel building systems differ significantly from traditional on-site
buildings in terms of behavior, detailing requirements and method of construction there is
no specification in CAN/CSA S16-01 and the National Building Code of Canada
(NBCC).
1.3 Seismic Design of Steel Buildings
The seismic design and analysis of low to mid-rise modular steel buildings is a new area
that has not yet been investigated thoroughly. Most modular buildings in the US and
Canada today are not designed for lateral forces due to earthquake except for those in few
areas where seismic analysis is mandatory in the building codes. However, infrequent but
destructive earthquakes in areas with low to moderate seismicity may cause considerable
damage to structures when they have not been designed to have sufficient resistance
against earthquakes. In most of the areas in Canada seismic resistant design is regarded as
uneconomical or too complex for low rise buildings. This may cause a threat to safety of
the occupants and may lead to substantial damage when a moderate earthquake happens.
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Even for regular steel buildings, experience from past earthquakes in the US and in Japan
have shown that many of them experienced moderate to severe damage to the structural
and nonstructural components (Naman S. K., Goodno B. J., 1986). A combination of
material and connection failures can be cited as the major cause of the observed damage
(Degenkolb, H. J., Fratessa, P. F., 1973). Another cause that may contribute is the poor
construction practice on-site and the lack of inspection due to the limited budget typically
available for low to mid-rise structures.
Similar problems may pose a threat to the design and construction of modular steel
buildings in North America especially in Canada. Although due to the considerations
given to the transportation, assembly and fabrication of the modular units, the detailing
and configuration of the elements are noticeably different from the regular steel
buildings, a dedicated study to assess and evaluate the seismic behavior and responses of
these types of structures is lacking. In current practice, when designing modular steel
buildings, static wind pressures are found to govern the designs and are assumed to be a
suitable replacement for earthquake induced inertial forces in many areas. Similarly,
there are a few case studies conducted on some relatively tall modular steel buildings in
the United Kingdom which produced some guidelines for the design and construction
(Lawson R. M., Richards J., 2010).
1.4 Steel Structural Systems
Currently seismic resistant steel buildings are designed in a way that the structural
members are allowed to deform adequately into the inelastic region under large
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earthquakes. Therefore, the members are expected to have enough capacity of yielding
and plastic deformation in way that the strength degradation under seismic load is not
significant. This way, the seismic energy is expected to be dissipated in the form of
hysteretic energy. The ability of structural steel as a base material to dissipate large
amounts of seismic energy through inelastic deformations makes it ideal for structures
undergoing seismic excitations. Steel hardens under cyclic loading; it gains strength as
the number of cycles and the deformation amplitude increase, resulting in large hysteresis
loops.
When a steel element is subjected to loading in tension, the presence of localized
imperfections can result in stress concentrations and associated high strains which may
cause cracks leading to fracture in the material. The imperfections may be pre-existent in
the base material, or may be caused during the erection and fabrication process on-site or
off-site (imperfections such as components damaged or deformed during the
manufacturing and transportation or incomplete and partial connections during
assembly). Undesirable behavior may also be observed when steel elements are loaded in
compression. Local or lateral torsional buckling may occur, which results in gradual
decrease in the strength and stiffness of the element. When the material is subjected to
cyclic loading, crack propagation occurs and on reaching a critical size the crack can
manifest as fractures, which are associated with a sudden deterioration in strength
(Akshay G, Helmut K., 1999).
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1.4.1 Moment Resisting Frames
Traditionally, steel Moment-Resisting Frames (MRFs) are structural systems that are
commonly used in seismic regions. In an MRF, as the inelastic deformation of beams
results in the dissipation of energy, substantive damage to these gravity load carrying
members may be induced. However, in some cases MRFs did not meet anticipated
structural behavior and significant economic losses occurred under ground motions even
less than the design earthquake. Also, excessive lateral deformations of un-braced frames
can result intolerable damage at non-structural elements even under moderate
earthquakes. The damage increases the repair and long-term costs (Mutlu S., Ozgur B.,
2011).
Based on previous experience and experimental research, it is observed that beam-to-
column welded connections can develop catastrophic failures due to their brittle response
(Mahin S. A. 1998; Mahin S. A. et al. 2002). Moreover, keeping the lateral drifts within
the code-mandated limits results in designing larger sections especially for the column
members. Therefore, from an architectural point of view and considering other
limitations in modular buildings such as those imposed by the transportation of the
modules the use of this type of structural systems is not recommended.
1.4.2 Braced Frames
Diagonal steel braced frame or concentrically X-braced frame is another commonly-used
structural system. In this system braces provide lateral stability of the structure and
minimize the lateral drifts. The strong point of this system is that the whole braced frame
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works as a truss and the frame members resist initial axial loads with little or no bending
in the members until the compression braces buckle. The parameters controlling the
behavior of a brace are the effective slenderness, compactness of the cross-section and
end connection details. In braced frames material saving could be achieved as the frame
members are subjected to less bending effect due to the presence of the braces (Gwozdoz
M., Machowski A., 1997). Energy dissipation in X-braced steel frames almost entirely
relies on the cyclic behavior of diagonal braces, which may exhibit significant stiffness
and strength degradation (Maison, B. F., Popov, E. P., 1980; Gugerli, H., Gooel, S. C.,
1982). Deteriorating hysteresis loops due to buckling effects causing a loss of lateral
stiffness and strength of the frame (Khatib I., F., et al. 1988). Thus, the response of the
Concentrically Braced Frame (CBF) is highly sensitive to the compactness and relative
axial strengths of the braces in compression and tension (Redwood, R. G., Channagiri, V.
S., 1991; Georgescu, D., et al. 1992) and the stiffness and strength of the beam into which
the braces frame (Roeder, C. W 1989; Murat D., Anshu M., 2007). This complex
hysteretic behavior exhibits unsymmetrical properties in tension and compression, and
typically showing substantial deterioration (pinching) when loaded monotonically in
compression or cyclically into the inelastic range. This is the reason that makes the
inelastic characteristics such as strength degradation and energy dissipation highly
effective in structural vulnerability under seismic loading (Sabelli, R. et al., 2003).
1.5 Overview of This Study
In this study, two 4-story modular steel buildings with two different structural
configurations were chosen to carry out a detailed seismic evaluation. These structures
10
have been analyzed both in two and three dimensions under a series of ground motions.
To confirm the adequate global capacity of these types of structures and have an overall
conclusion of their safety in comparison with traditional steel buildings and according to
the building code, a series of Incremental Dynamic Analysis (IDA) have been done for
both the two and three-dimensional structures.
In the 2D model, the effects of different floor and ceiling beam configurations have been
verified and the ways that their connection types change the structural response have
been discussed. Inter-story drifts, global collapse capacity, residual deformation of the
structure under different ground motions are some of the dynamic responses evaluated. In
modular construction, the column connections require special attention, which was also
investigated in the numerical studies.
Through the analysis of the 3D model, the effects of considering different horizontal and
vertical connections are assessed. The configuration and specification of the horizontal
connections between the modules in the corner posts are discussed and their contribution
in the overall structural response has been evaluated.
The diaphragm action in the modular steel buildings and the interaction between the
modules are also investigated in this study. Discussions regarding whether it is realistic to
consider each floor of a typical modular building as a whole rigid slab or whether it is
more acceptable to model the diaphragms separately in each unit and then anchor those
through connection elements provided. The locations of the horizontal connections, and
the elements and material used in the model (whether to use rigid element, elastic
11
element or inelastic element) have been verified. Additionally, the axial and shear forces
in the connections that occur due to the relative displacements and rotations between the
modules are also captured and comparisons have been made.
12
Chapter 2
2. Incremental Dynamic Analysis (IDA)
2.1 Introduction
Performance Based Earthquake Engineering (PBEE) aims to make sure that the designed
building satisfies specified performance criteria. Performance criteria include life safety,
post-earthquake functionality and limiting probable repair costs to a specified percentage
of building replacement cost. Performance objectives depend on the functionality of the
structure. For example a stricter performance objective is required for hospitals to remain
operational after a relatively large ground motion; whereas the requirements are less
demanding for less critical facilities.
Evaluation of the performance of a structure requires a method that monitors the structure
behavior from linear elastic region to yielding stage and until it collapses. For MDOF
structures the dynamic interaction of the higher modes can make it hard to predict the
post yield behavior. Incremental Dynamic Analysis (IDA) is a widely used approach to
evaluate the performance of structures. In this method, a set of ground motion records are
chosen, each record is scaled into multiple intensity levels to cover the whole range of
structural response from elastic behavior all the way through yielding and then to
dynamic instability (i.e. collapse or any other limit state targeted).
The IDA curves consist of a set of scaled ground motion records known as Intensity
Measure (IM) and a series of the structural response known as Demand Parameter (DP).
Each DP versus IM produces a single point on the IDA plot. As a result, an IDA curve is
13
generated from a series of IMs and DPs in a way that the curve is produced from different
intensity values of a specific ground motion and their corresponding demand parameters.
To account for the variability of the ground motions and to perform statistical evaluation
of the structure behavior, a sufficiently large number of ground motion records should be
considered. By carrying out the IDA on a given structure with each of the ground motion
records producing a single curve and then by summarizing al the curves considering the
limit-states targeted, valuable information about the seismic behavior of the structure can
be obtained (Vamvatsikos D., Cornell. C. A., 2005).
2.2 Nonlinear Dynamic Analysis
Building systems with large energy dissipation capacity are likely to undergo
significantly greater inelastic deformations than systems with relatively limited energy
dissipation capacity. The behavior of materials in the inelastic range is highly complex.
To predict the seismic performance of structures more accurately -especially in the
inelastic range of response- it is important to simulate the abovementioned factors close
enough to reality in the numerical and experimental studies. In the dynamic response of a
given structures another important factor is the characteristics of the applied earthquake
ground motion. Ground motion selection and scaling is an important component of any
seismic risk assessment study that involves time-history analysis. This is also a parameter
with a very limited guidance provided in the building codes, which results in subjective
choices in design (O’Donnell A.P. et al. 2011). The adoption of performance-based
considerations as pre-requisite in the seismic design and evaluation of building structures,
the use of nonlinear response history analysis (RHA) has gained major importance. This
14
analysis method requires an appropriate suite of ground motion records that are
compatible with the site-specific hazard levels considered.
In a linear RHA, the forces in the structural components are computed, and the
performance is assessed using strength demand/capacity (D/C) ratios. In a nonlinear RHA
though, the performance is assessed using both deformation and strength D/C ratios G.
Powell (2006). The latter type of analysis requires properties such as stiffness
degradation in cyclic loading, post yield behavior, and yield strength in addition to initial
stiffness used in linear analysis. As such, for the components that are supposed to have
ductile behavior, inelastic behavior is captured and the performance of a member is
assessed based on the deformation demand/capacity. Similarly for the members that are
expected to have brittle manner there is no inelasticity and their performance is evaluated
using strength demand/capacity. To get useful information about structures with ductile
members, and with the purpose of retrofitting the vulnerable parts of the system, it is
more reasonable to use nonlinear analysis. In nonlinear analysis a step-by-step integration
is employed which can be computationally expensive. In this study, modeling and
numerical analyses have been done using Open System for Earthquake Engineering
Simulation (OpenSees). In the numerical simulation platform both material and
geometric nonlinearities can be considered.
The change of stiffness and damping matrices can be accommodated from one step to the
next but they are kept constant within each time step. However, it is not really necessary
to consider geometric nonlinearity (large displacement) effects. P-Δ effects usually need
15
to be considered in tall buildings, where equilibrium should be considered in the
deformed shape of the structure and not the initial un-deformed shape. The consideration
of p-δ effects accounts for the equilibrium in the deformed shape of the member itself and
allows the deformation in the member length. This local deformation (δ) is negligible
except for very slender members. Therefore, for seismic analysis it is rarely, if ever,
necessary to consider true large displacement effects, which are significant only at
impractically large drifts. Considering these effects can lead to substantial and
unnecessary increase in computer time. OpenSees includes material nonlinearity and adds
the geometric stiffness matrix to element stiffness matrix to account for P-Δ effects which
is considered as effect of displacement of the element or structure as a whole; however it
does not account for p-δ effect which is considered as effects of deflections within the
length of member. However, if the p-δ effect is likely to be significant in any of the
members a simple approach is to add extra nodes along the length of the elements.
Dividing the elements into segments converts p-δ effect to a P-Δ effect in the overall
structure. The only members that are predicted to have large deformation in a steel braced
frame would be the braces; hence similar approach have been used in modeling the brace
element and will be explained later in the following chapters.
2.3 Step-by-Step Integration Algorithms
Generally, structural dynamic analysis is done using two methods; direct time integration
and modal superposition. However, the former is the only option for nonlinear analysis.
As the most accurate method of structural analysis, nonlinear dynamic analysis solves
differential equation of dynamic equilibrium of motion which is shown in Equation 1:
16
R (u, t) u (t) M (t) r(t) (1)
where,
u, u , and are displacement, velocity, and acceleration, respectively.
M and C are the mass and damping matrices.
R is the restoring force and r(t) is the applied force vector
By having u0 and u 0 as initial displacement and velocity vectors, the initial conditions are:
u (0) = u0 and u ( ) u 0.
To solve the above second order differential equation of nonlinear systems, a numerical
procedure can be used incrementally. Direct time integration method is applicable
without modification to the equation of motion of both single-degree of freedom (SDOF)
and multi-degree of freedom (MDOF) systems. There are many methods (integrators)
used for the direct integration of equation of motion (e.g. Central Difference Method and
Newmark Method) which are based on finite difference method and are classified as
explicit and implicit methods.
In the explicit methods, calculation at the current time step (t Δt) depends only on the
information from previous time steps. For example the central difference method is an
explicit integration method. On the other hand in the implicit method the unknown values
at the end of time step (t Δt) are calculated based on the both previous time step t and
current time step (t Δt) information; Newmark method is an example of an implicit
time integration method. It has been shown that the implicit methods are more accurate
than the explicit ones (Dokainish, M. A., Subbaraj, K, 1989). However, due to the
approximation in the formulation and calculation of these methods compared to exact
17
solution (for linear systems exact solution can be obtained), it is expected to have some
errors in the numerical simulation results. In addition to the method selection, the
accuracy of the results usually depends on the time step size and the frequency content of
the loading (Paultre P., 2011). For most of the explicit methods, if the required time step
Δt is smaller than a critical time step Δtcr, the method is said to be conditionally stable
and when there is no time step limitation the method is considered to be unconditionally
stable.
In this study Newmark integration algorithm which is the most common method in
structural dynamics has been adopted as the integrator command in the OpenSees. The
following expressions for the velocity and the displacement are obtained with the help of
Taylor formula:
ut Δt = ut + Δtu t + [(0.5 – β) Δt 2 t [β Δt
2 t Δt (2)
u t Δt u t [( - ) Δt t [ Δt t Δt (3)
where, variables β and are numerical parameters of quadrature formula that control both
the stability and energy dissipation characteristics (e.g., the amount of numerical
damping) of the method. These two parameters are weighting factors on the velocity and
displacement increments (Paultre P., 2011). Two commonly used choices of the
parameters are ( /2, β /4) and ( /2, β /6) which are known as Average
Acceleration Method and Constant/Linear Acceleration Method, respectively. The
Average Acceleration Method is an implicit and therefore unconditionally stable method;
18
however, the Linear Acceleration Method is conditionally stable. The practical ranges for
these numerical parameters are as follows:
/2 ≤ ≤ 3/4 , ≤ β ≤ /4
For /2 there is no numerical damping and for > /2 numerical damping is
introduced. In this study, Average Acceleration Method with the values of /2 and β
= 1/4 have been chosen. While using direct time integration of the equation of motion,
the response of the system is divided into a number of discrete intervals of time. In this
way, the response of the structure will be assessed by computing displacement, velocity,
and acceleration at each time step. Nonlinearity is included through the updated
calculation of the stiffness at the beginning of each step which remains constant over that
time step. The resulting calculated response is considered as the initial condition for the
next time step. This is exactly the same as taking a series of consecutive approximate
linear differential equations.
2.4 Definitions
In what follows concepts required to understand the IDA curves and their properties will
be explained.
2.4.1 Scale Factor (SF)
A base record a1 (ti) is a single ground motion that is a vector of time (ti = 0, t1, t2,… tn-1)
and contains the un-scaled acceleration values. (Vamvatsikos D., Cornell C. A., 2002).
Scaling the amplitude of the ground motion throughout all the frequencies can be done by
19
multiplying the base record with a non-negative scalar, λ∈ [ , ∞), called Scale Factor
(SF). While having the phase information unchanged, this scalar value, aλ λ. a1,
uniformly scales the amplitude up (λ> ) or down (λ< ) to account for severe or mild
ground motions. The goal is scaling each of the spectral acceleration to the target
spectrum of a given site obtained from NBCC. SF values for each record can be
computed using different methods, such as spectral matching in frequency domain,
spectral matching by wavelets, ground motion scaling in time domain, etc. Here, ground
motion scaling in time domain has been used. As it can be seen in Figure 2.1 a given
spectral acceleration is scaled by a) matching to the Peak Ground Acceleration (PGA)
and b) by matching at a specific period such as fundamental period.
20
Figure 2.1 Ground motion scaling in time domain (scaling on amplitude) by a) matching to the
Peak Ground Acceleration (PGA), and b) matching at a specific period such as fundamental period.
2.4.2 Intensity Measure (IM)
By having the scaled acceleration we can introduce another non-negative scalar IM ∈ [0,
∞), called Monotonic Scalable Ground Motion Intensity Measure of a scaled
accelerogram, aλ. Intensity Measure (IM) is made up of the function IM = fa1 (λ), that
21
depends on the un-scaled accelerogram, a1, and it is also proportional to the SF.
Examples of scalable IMs are the PGA, Peak Ground Velocity, and the ξ 5% damped
Spectral Acceleration at the structure’s first-mode period (Sa(T1,5%)). (Vamvatsikos D.,
Cornell C. A., 2002)
2.4.3 Damage Measure (DM)
Damage Measure (DM), which represents the structural response, can be one of the
several outputs available from the results of the nonlinear dynamic analysis. These
include peak inter-story drift/drift angle, peak roof drift, peak story ductility, maximum
base shear, and many other various damage indices can be observed and deduced from
the output results. DMs can be used in performance assessment by providing information
about the limit-states or modes of failure as well as other response characteristics. For
instance, to evaluate the non-structural damage, the peak roof/floor accelerations would
be the proper DMs; whereas to evaluate structural damage -global and local story
collapse- maximum peak inter-story drift angle would be a proper choice for DM.
2.5 General Properties of Single-Record IDAs
The IDA curves are very sensitive to both ground motion records and structural model.
When a given structure is subjected to different ground motion records it produces
various dissimilar IDA curves. This behavior can be observed in Figure 2.2 (a, b, c and d)
where the IDA curves are plotted for a 4-story modular steel building (MSB) with 4
different ground motion records. The plots exhibit various responses from gradual
degradation toward rapid instability to oscillating non-monotonic wavy behavior. The
22
graphs are showing the applied ground motion with increasing intensities versus the
maximum roof inter-story drifts of the structure. There are both similarities and
dissimilarities noticed in the plots which will be discussed.
All the curves in Fig 2.2 exhibit a distinct elastic linear region ranging up to about 0.5g.
This is around where the first brace buckles. This behavior is not exclusive to MSBs but
it is observed in any structural model with initially linearly elastic members. The elastic
region ends when the first element in the system exhibits nonlinearity. The slope of the
elastic portion of the curve can also vary to some extent from record to record; however,
it will be the same for SDOF systems as well as MDOF systems if the higher mode
effects are taken into account (Luco N., 2003). IM/DM slope of the elastic segment of the
curve is called the elastic stiffness for a given IDA curve.
It can be seen in Fig 2.2 (a, b, c, and d) that the IDA curves terminate at various levels of
IMs and their shapes are also considerably different. For instance, in plot (a) the IDA
curve rapidly softens after the first buckling happens by moving to large drifts and
eventually encountering instability. Slight hardening behavior is also observed in plot (b).
However, in case (c) local increase in stiffness is larger, exhibiting severe hardening
response at several IM values. Similarly, plot (d) shows a wavy behavior due to
hardening before the termination of the runs. As a result, although the IMs are increased,
the IDA curve is pulled back to relatively lower DMs and making it a non-monotonic
function of the IM. Therefore, for a single structure under different ground motions,
23
collapse can happen in various ranges of IMs and the structure may show relatively lower
drifts in higher IMs.
Figure 2.2 IDA curves of a 4-story MSB frame subjected to different ground motion records.
To estimate the collapse or dynamic instability the final softening portion of the IDA
curve is considered where it almost has the shape a flat line. The flat line forms when the
pace of DM accumulation is much faster than the corresponding IM increments. This
usually is reached before analysis termination due to numerical non-convergence.
Although hardening in IDA curves resulting in smaller DM values for larger IMs seems
to be counterintuitive, this behavior is not a novel phenomenon. With increasing IM the
weaker, non-effective response at beginning of the time-history becomes stronger and can
result in earlier damage or yielding development in the structure. This will change the
24
initial properties of the structure which will in turn affect the subsequent, more intense
response. For instance, considering a multi-story building under a rather strong ground
motion, once seismic forces amplified by a larger IM cause an earlier yielding of a floor,
this floor will later work as a fuse and may lower the DM value of the structure. Also,
yielding in earlier cycle will result in a period elongation and may modify the following
dynamic response of the structure. This can be observed in Figure 2.3 which is obtained
from the 4-story regular MSB.
Figure 2.3 Single IDA curve for a 4-story MSB frame. Lower maximum roof drift in higher intensity is because of the earlier yielding development in the structure.
25
At high IM values, depending on the program used in the analysis (e.g., OpenSees), the
analysis may stop due to convergence problems, or may result in an infinite DM value. In
the latter case the structure experiences softening behavior and the system experiences
instability and global collapse. However, when the same structure is analyzed again, for
the next (higher) value of the IM, it may appear to be stable. This phenomenon that the
structure remains stable after it experienced collapse at previous cycles is known as
“Structural Resurrection” (Vamvatsikos D., Cornell C. A., 2005). In Figure 2.4 the
structural resurrection for the 4-story MSB is illustrated. The cycle(s) where non-
convergence happened are shown as intermediate collapse area which is located between
two stable analysis cycles.
Figure 2.4 Single IDA curve for a 4-story MSB frame that shows Structural Resurrection after the
structure is pushed to the global instability.
26
2.6 Defining the Limit-State on a Single IDA Curve
As seen before IDA curves are not usually smooth. Moreover, the relation between the
DM and IM is not necessarily one-to-one. For every IM value there is a value of DM,
while there may be one or more values of IM for a single DM. This non-monotonic
relation is apparent in the wavy IDA curves due to the hardening behavior of the system.
Therefore, by definition, an IDA curve is non-differentiable because it contains absolute
values of maximum responses of the time history analysis that may suffer from lack of
one to one mapping.
To assess the real capacity of a structure the information obtained from the analyses
needs to be summarized. In Performance Based Earthquake Engineering it is important to
define a limit state or performance level for the structure. This will be achieved by
introducing a limitation or a statement that when satisfied, commands the program that
the limit-state criterion is reached. FEMA 350 (FEMA 2000a) has defined two structural
performance levels as the recommended criteria, the Collapse Prevention (CP) and the
Immediate Occupancy (IO) structural performance levels. In CP it is assumed that
substantial damage to the structure is large enough to push the structure to the edge of a
total collapse. Although the gravity load resisting system must be stable enough to carry
the gravity loads, the structure may not be safe for re-occupancy and fundamental repairs
of the components is needed. However, in IO, damage is limited to some partial,
structurally unimportant components that would not require repair. Immediate post
occupancy should be safe and utility services are available. Limit states can be defined
using two different rules which are explained in the following sections.
27
2.6.1 The DM-Based Rule
According to the DM-based rule a limit-state is exceeded when DM ≥ DMcollapse. The
value of DMcollapse can be obtained theoretically, experimentally, or they may have a
probability distribution. For example, the θmax = 2% limit is defined as Immediate
Occupancy structural performance level for steel moment-resisting frames in the FEMA
guidelines (FEMA, 2000b). However, determination of such limit-states has randomness
and uncertainty incorporated. For instance, FEMA 350 (FEMA 2000a) defines θmax as
local collapse limit-state when the connection rotation exceeds the gravity load carrying
capacity of the connection. Similarly, maximum inter-story drift is assumed as a primary
damage intensity parameter in the assessment of structural performance level of the
moment resisting frames. It also have been used to evaluate the dynamic response of
ductile concentrically braced frame structures globally (Uriz, P., Mahin, S.A., 2004).
Nevertheless, these are defined based on experiments and engineering judgment for each
connection type and is not unique constant value.
A single DMcollapse may provide multiple limit-state points on the IDA curve, as seen in
Figure 2.5. This is because of the wavy pattern of IDA curves which was discussed
above. According to the DM-based rules the IDA curve pattern itself can be handled by a
conservative approach and by considering the lowest value of IM as the corresponding
DM limit state. This means up to the first dynamic instability the rest of higher IM-DM
points that violate the limit-state will be ignored. Since the DM is a damage indicator all
the DM values larger than the DMcollapse is also assumed to be in the limit-state.
28
Using DMcollapse as the limit-state indicator simplifies the implementation especially
before the instability happens. Moreover, DM-based rules can be representative of a
problem modeling. In case of instability a realistic and quite complicated model will
exhibit numerical non-convergence by terminating the analysis process rather than
showing a finite DM (flat-line) output.
Figure 2.5 The DM-based rule for a T1 = 0.6 sec, 4-story MSB. The DM is θmax, and
DMcollapse is set at 4%.
2.6.2 The IM-Based Rule
Another approach is to set collapse IM as a limit-state criterion. In this case, when IM ≥
IMcollapse the limit-state is exceeded. In the IM-based rule, although there will not be
29
multiple collapse regions, it is required to determine collapse IM points for every IDA
curve separately. Defining such points in a consistent pattern has its own difficulties.
However, a conservative approach is to consider the first point before the flat-line as the
capacity where all the IDA curves preceding are known as non-collapse points. An
example of collapse IM approach is the 20% tangent slope method in FEMA (2000a)
where the last point on the curve with a tangent slope equal to 20% of the initial elastic
slope is taken as the capacity point. In most cases the flat-line cannot be obtained
numerically (which would have a zero slope); therefore, it is assumed that the collapse
happens when the rate of increase of DMs is five times that of IMs. However, due to the
wavy pattern of the IDA curve such slopes may be reached several times before the
actual collapse happens, (i.e. the structure seems to head toward collapse but is recovers
only after a few IM levels). These points with collapse slopes should be ignored as the
capacity candidates. Non-smoothness of the actual IDA curves can also be a problem
when using The IM-based method, but still an interpolation can be used to have a
smoother curve. Figure 2.6 shows the multiple capacity points for a 4-story MSB frame.
In practice, to be able to ensure that the collapse limit state is captured, both of the above
conditions are implemented and checked. Therefore, when collapse modes of a structure
are not detected by a single DM (e.g. DM is still less than DMcollapse), the program can
still detect the collapse by checking individual collapse modes using an OR statement
(e.g. checking the slopes at each mode).
30
Figure 2.6 The IM-based rule for a T1 = 0.6 sec, 4-story MSB using the 20% slope criterion.
An important factor in the estimation of the collapse region is the capabilities of the
numerical simulation platform used for the analysis. Factors such as algorithms used to
solve the nonlinear equation, the integration algorithm, tolerances chosen during the
analysis, and round-off errors are all important in the prediction of the model behavior.
This is because numerical instability of the analysis is considered as the dynamic
instability to get an estimation of the collapse DM. However, this approach would not be
reliable if the analysis tool used has some deficiencies.
While analyzing nonlinear structures convergence problems can be encountered. To
ensure accuracy of the numerical solution, when using OpenSees as the simulation
platform, a Solution Algorithm object can be defined. It determines a sequence of steps to
31
be used to solve the nonlinear equations. In the event of non-convergence, several
different algorithms can be queued to attempt to obtain a convergent solution. The
effectiveness of this approach relies on establishing a correct order for these algorithms
with an increased level of complexity. For example, Newton algorithm command is used
to construct a Newton Raphson algorithm which solves the nonlinear residual equation
and is a robust method for solving nonlinear algebraic equations. However, sometimes
due to the roughness of the residual equation convergence is slow or even out of reach;
therefore, by using a more effective command such as the Newton with Line Search
Algorithm the convergence will be obtained. This trend is continued by using more
complicated algorithms such as Modified Newton Algorithm, Krylov-Newton Algorithm,
BFGS Algorithm, and Broyden Algorithm to assure that ultimately the results are as
complete as possible.
2.7 Implementation of the IDA
2.7.1 Selecting the Ground Motions
Application of a proper IDA to estimate the performance of a structure requires several
steps: selecting suitable ground motions, choosing proper intensity and demand measures,
scaling the selected records, defining appropriate limit-states, and post-processing the
results and performing statistical calculations in order to summarize the results. Here, a 4-
story MSB located in Vancouver is designed for moderate ductility based on the
Canadian standards. With the first mode period of T1=0.5 sec the IDA curves of the
32
structural response have been generated as the damage measure (e.g. peak roof drift ratio
θroof or maximum inter-story drift ratio θmax), versus the ground motion intensity level
(e.g. peak ground acceleration, or the 5% damped first mode spectral acceleration).
According to the NBCC (2010), spectral acceleration is a measure of the ground motion
that includes the seismic energy of an earthquake at a specific period. It describes the
seismic hazard at different periods of 0.2, 0.5, 1.0, and 2.0 seconds. For the structure
considered in this study at the selected location and considering its fundamental period,
the ground motion intensity has been evaluated as 0.67g.
Due to the variation in other seismic hazard parameters such as duration, frequency
content, moment magnitude, and effective number of loading cycles, scaling different
ground motions to the same PGA or Sa(T1,5%) may not result in the same level of
response and damage to a given structure. Therefore, the use of only one ground motion
may not provide enough confidence that the structure will have the same response when
subjected to another ground motion record with the same PGA or Sa (T1,5%). However,
previous studies (Shome N, Cornell C. A., 1999) have shown that for mid-rise buildings,
using an acceptable IM such as Sa(T1,5%) the above approach can be considered reliable
to evaluate the seismic demand of the structure, if a suite of 10 to 20 ground motion
records are used to estimate the seismic demand of the building. Additionally, FEMA
(2000a) has recommended the same number of records as being representative of a site
and hazard level to be used to achieve the capacity of the structure with sufficient
confidence.
33
Here, the same group of ground motions selected in a study by (Vamvatsikos D., Cornell
A. C., 2005) has been used to estimate the seismic demand. A set of 20 ground motion
records listed in table 2.1, have been obtained from the PEER strong motion database.
Source (magnitude, rupture mechanism, directivity, focal depth), path (crustal Structure),
and site (surface geology, topography) characteristics are the factors that have influence a
real accelerogram. These ground motions are obtained from a bin of relatively large
magnitudes from 6.5 to 6.9. Epicentral distances are moderate (15 to 32 km), soil is firm,
and there are no marks of directivity.
Table 2.1 Earth quake ground motion records selected from PEER Strong Ground Motion
Database.
No. Event Year Record station ɸ1 M
2 R
3(km) PGA (g)
1 Imperial Valley 1979 Plaster City 45 6.5 31.7 0.042
2 Imperial Valley 1979 Plaster City 135 6.5 31.7 0.057
3 Imperial Valley 1979 Westmoreland Fire Sta. 90 6.5 15.1 0.074
4 Imperial Valley 1979 Westmoreland Fire Sta. 180 6.5 15.1 0.110
5 Imperial Valley 1979 El Centro Array # 13 140 6.5 21.9 0.117
6 Imperial Valley 1979 El Centro Array # 13 230 6.5 21.9 0.130
7 Imperial Valley 1979 Chihuahua 282 6.5 28.7 0.254
8 Imperial Valley 1979 Cucapah 85 6.9 23.6 0.309
9 Loma Prieta 1989 Agnews State Hospital 90 6.9 28.2 0.159
10 Loma Prieta 1989 Coyote Lake Dam 285 6.5 22.3 0.179
11 Loma Prieta 1989 Sunnyvale Colton Ave 270 6.9 28.8 0.207
34
12 Loma Prieta 1989 Sunnyvale Colton Ave 360 6.9 28.8 0.209
13 Loma Prieta 1989 Anderson Dam Downstream 270 6.9 21.4 0.244
14 Loma Prieta 1989 Hollister Diff. Array 165 6.9 25.8 0.269
15 Loma Prieta 1989 Hollister Diff. Array 255 6.9 25.8 0.279
16 Loma Prieta 1989 WAHO 0 6.9 16.9 0.370
17 Loma Prieta 1989 Hollister South & Pine 0 6.9 28.8 0.371
18 Loma Prieta 1989 WAHO 90 6.9 16.9 0.638
19 Superstition Hill 1987 Wildlife Liquefaction Array 90 6.7 24.4 0.180
20 Superstition Hill 1987 Wildlife Liquefaction Array 360 6.7 24.4 0.200
1 Component
2 Moment magnitude
3 Closest distance to fault rupture
2.7.2 Steps to Perform IDA
As repeated analyses are required to construct a single IDA curve, after selecting the
ground motion, an automated procedure needs to be implemented to perform the required
calculations. This includes appropriately scaling the ground motion for each record to
cover the entire range of response from elastic range all the way to the yielding and
instability. To carry out the required tasks while minimizing the number of runs an
advanced algorithm (Hunt & Fill), (Vamvatsikos D., Cornell A. C., 2002) has been
developed. This algorithm has been implemented in software and analysis programs (e.g.
DRAIN-2DX & SeismoStruct) rendering IDA almost effortless without requiring any
human supervision. Analyses are continued at increasing levels of IM until the global
dynamic instability is reached by encountering numerical non-convergence. Additional
35
analyses may be performed around the numerical non-convergence to achieve a more
accurate estimation of the capacity by bracketing the IM values. Therefore, the user needs
to specify the number of dynamic analyses, and the targeted accuracy for the demand and
capacity. The results of the analysis together with a summary of selected post processed
results are available to the user.
In this study OpenSees has been used to perform the IDA, and there are no built in
functions to perform the analyses or post-process the results. As such, the necessary steps
need to be implemented by the user and they are explained in this section. For the scaling
of the ground motion records, an initial, temporary choice of IM is required. Here, Sa(T1,
5%) is used which can be replaced by any other scalable IM (Vamvatsikos D., Cornell A.
C., 2002). The algorithm is set to have an initial step of 0.1g with a step increment of
0.05g and an arbitrary first elastic run of 0.005g. Targeted global collapse capacity is set
to a maximum of 10% roof drift which means that the numerical non-convergence or a
flat-line are expected to be achieved within that drift span. In addition to the global
capacity limit, a maximum number of 50 runs for each record is allowed.
Table2.2 Sequence of runs for a ground motion (Hunt and Fill tracing algorithm).
NO calculations Sa(T1, 5%) (g) θmax
1 0.005 0.061%
2 0.005 + 0.10 0.105 0.32%
3 0.105 + 0.10 + 1*0.05 0.255 0.54%
36
4 0.255 + 0.10 + 2*0.05 0.455 0.92%
5
…
0.455 + 0.10 + 3*0.05
…
0.705
…
1.81%
…
11 2.705 + 0.10 + 9*0.05 3.255 4.89%
12 3.255+ 0.10 + 10*0.05 3.855 infinity
13 3.255 + (3.855-3.255)/4 3.405 12.23%
14 3.405 + (3.855-3.405)/4 3.517 15.76%
15 3.517 + (3.855-3.517)/4 3.601 infinity
16 (3.517 + 3.405)/2 3.461 13.92%
17 (3.405+ 3.255)/2 3.330 9.76%
18
…
(3.255+ 2.705)/2
…
2.98
…
7.54%
…
In IDA, the first runs are in the elastic range of response and as the IM values increase
the structure experiences nonlinear deformations until the first numerical non-
convergence happens or some extreme values of DM, such as θmax are produced. After
the dynamic analysis algorithm fails to converge additional runs, if necessary, can be
performed to bracket the collapse region. Here, the algorithm places each new run closer-
one and two fourth of the way- to the convergence value providing more information in
comparison to the non-converging case. Additional runs can be performed to fill the IDA
gaps for lower IMs. In this way the demand resolution increases and more confidence is
gained that a possible earlier instability due to structural resurrection was not missed.
37
A simpler stepping algorithm can also be used to produce the IDA curves. In this
algorithm constant steps of 0.1g is chosen starting from zero to collapse; therefore, an
even distribution of points along the curve will be obtained. However, because of the
specific properties of ground motions the size of the steps may be too large or too small
for a given record and that may reduce the computational efficiency of this approach.
Also, it should be pointed out that the number of data points on an IDA curve changes
depending on the earthquake record as the magnitude of the IMcollapse value may be
different for each record.
2.7.3 Defining the Capacity for Single IDA Curves
As mentioned before, FEMA (2000a, b) defines different limit-states for the performance
calculation for PBEE. For instance, θmax 2% is the limit-state for steel Moment
Resisting Frames with Reduced Beam Section (RBS) connections for Immediate
Occupancy (IO) performance level, θmax % corresponds to ollapse Prevention
( P) performance level (θmax % or local tangent slope of 2 % of the elastic slope in
the IDA curve, whichever occurs earlier), and reaching infinite values of damage
parameter (flat-line) for Global Dynamic Instability (GI).
To assess the CP value, calculation of the tangent slope, which is the first-order
derivative of the IDA curve, is required. In addition to that, IDA curvature is needed to
detect the earlier but unacceptable collapse candidates from the local hardening segments
of the IDA curve. Therefore, the targeted point will be somewhere in the softening part of
the curve and can be identified with a stiffness slope equal to 20% of the elastic slope.
38
Another limit-state set to be checked is θmax %, the lowest IM value corresponding
to the two limits mentioned above will be the capacity of the structure for that IDA curve.
Similar approach can be used for the IO performance level. While finding the demand
parameter equal to 2% (θmax 2%) the lowest value of IM is taken as the IO limit-state.
For example, in Figure 2.7 Sa(T1, 5%) < 2.1 g does not satisfy the IO performance level
criterion.
According to Hunt & Fill algorithm, in the collapse estimation using the flat-line, the IM
value lies somewhere before the final numerical non-convergence and after the last
convergence. Since in this intensity span the difference between two consecutive IM
values is negligible, there is no need to know the IM value that corresponds to the flat-
line. For example, for the single IDA curve depicted in Figure 2.7, the last non-
converging result obtained from the analysis, is approximately Sa(T , 5%) ≈ 3. g and can
be considered as a conservative value before the flat-line.
39
Figure 2.7 IO, CP, and GI limit-states on a single IDA curve
Selection of suitable IM and DM as well as the number of earthquake records plays an
important role in the accuracy of the results; and there is no strict set of rules for this
selection. In this study, because of the fact that there are no directivity-influenced
records in the selected suite of ground motions and the modeled modular steel buildings
have medium heights, hence first-mode-dominated, the 5% damped first mode spectral
acceleration Sa(T1,5%) is chosen as the IM. It has been shown to be efficient in
minimizing the number of ground motion records required to get the actual capacity of
the structure by limiting the dispersion of the IDA curves. Moreover, the characterization
40
of the responses does not require magnitude or source-to-site distance information
(Shome N, Cornell C. A., 1999).
DM selection can also be application-specific when performing dynamic analysis. For the
non-structural damages, the peak floor acceleration can be used as an appropriate damage
parameter; while for the structural performance limit-states such as global instability the
maximum inter-story drift ratio θmax provides valuable information (FEMA, 2 a).
Here, maximum inter-story drifts and absolute maximum value of story drifts over time
and along the building height have been used to assess the structural performance. In the
post-processing stage a large amount of data produced during IDA runs should be sorted
out and summarized in a meaningful way. Statistical calculations and graphical outputs
can be obtained through appropriate software with built-in post-processing functions.
Here MATLAB was used to post process the results, for statistical calculations, and
generating various plots.
After running the analyses a large number of discrete points populate the IM-DM plane.
Using a spline interpolation we can approximate the final pattern of the curve and avoid
further analyses. The interpolation formulations are also provided in previous studies
where the spline is defined in n cubic polynomial pieces and is parameterized on a single
non-negative parameter (Vamvatsikos D., Cornell C. A., 2005). After interpolation, a
wide range of information can be extracted from a single IDA curve. As an example in
Figure 2.8 considering a single IDA curve the following details can be observed. The first
segment of the curve starting from zero to almost 0.9g is a straight line representing the
41
elastic range of the response. The slope slightly decreases after 0.9g depicting the initial
yielding and softening portion of the response. In this region the tangent slope of the
curve is less than the initial slope located at the elastic range. The first signs of hardening
can be noticed at around 2.0g where the tangent slope is increasing locally and this trend
continues until about 2.5g with almost the same values of DMs (θmax is about 4. %).
The next softening happens after 2.5g where the local slope decreases again and it is
followed by another hardening behavior before the final softening and an infinite θmax
value. The flat-line is produced after a particular point where the increasing rate of DM
values goes far beyond of the IM ones. This is where the numerical non-convergence
happens and the structure experiences global dynamic instability.
Figure 2.8 Different segments of softening and hardening in a single IDA curve.
42
2.7.4 Multi-Record IDAs
Since the IDA is highly dependent on the record chosen, a single-record IDA study
cannot fully capture the behavior of a structure under seismic loading. Therefore, the
structure should be subjected to a sufficient number of records to cover the entire range
of responses. By definition, a multi-record IDA study is resulted from a collection of
single-record IDA studies of the same structural model subjected to different
accelerograms. As in Figure 2.9 such a study produces series of IDA curves, these curves
are plotted on the same graph with a common selection of IMs and the same DM. The
dispersion and dissimilarities observed in the curves (structural responses) makes them
no longer deterministic and it is a random function DM = f (IM) for a single, monotonic
IM. As a result, the use of a probabilistic characterization to summarize the responses
will be inevitable. The summary technique often used in engineering design is based on
the sample median, 16%, and 84% fractiles. Hence, a large number of data can be
compressed to a probabilistic distribution of a DM given the IM, and a method to
estimate the relation between the variables in 2D IDA data is needed (Vamvatsikos D.,
Cornell C. A., 2002).
2.7.5 Summary of the Outputs
As mentioned before, the large amount of data generated from multiple IDA analyses
needs to be summarized and sorted out in a comprehensible way. To represent the
relationships between the variables “scatterplot smoother” methods such as running
43
mean, running median, and smoothing spline (Hastie, T. J., Tibshirani, R. J., 1990) are
available. The simplest one is the running mean with a zero-length window (or cross-
sectional mean), which calculates the average and standard deviation of DMs at each
level of IM. This is a proper method until the point where the first IDA curve reaches
capacity and the DM becomes infinite. This will produce the mean IDA curve with an
infinite value as well. This is a problem in most of the smoothers; however, cross-
sectional median or cross-sectional fractile is more robust and in the case of this study it
works at all IM levels. In this method, sample median (50%), 16% and 84% fractiles are
calculated and they become infinite only when collapse occurs at 50%, 84%, and 16% of
the records respectively (Figure 2.9). In addition, it fits to the assumption of lognormal
distribution of DM given IM, where the natural “central value” is the median and 6%,
84% fractiles correspond to the median times e±dispersion
(dispersion is the standard
deviation of the logarithms of the values), (Jalayer, F., Cornell, C. A., 2002).
Figure 2.9 a) IDA curves obtained from the selected ground motions. b) The median, 16%, and
84% fractiles of the records.
44
Therefore, a previously defined limit-state capacity can be summarized into some central
value and a measure of dispersion such as standard deviation or the difference between
two fractiles. The cross-sectional fractiles have been used in most of the previous studies
to condense the output data and conclude the distribution of DM values for a given IM
value. The technique is accepted to be efficient for the IDA curves considering infinite
DM values produced in higher intensities. In this way, the probability of exceeding any
particular limit-state required in the code can be estimated.
To produce 16%, 50%, and 84% fractile curves, a suite of IM levels are chosen. Since
there are 20 ground motion records in the 2D plot in Figure 2.9 (a), each level (or stripe)
of IM will be cut by up to twenty IDA curves each corresponding to a specific ground
motion. The resulting DM values corresponding to each record may be finite or even
infinite as a record has already reached its flat-line at a lower IM-level. At each level of
the given IM the 16%, 50%, and 84% fractile values of DM can be calculated as
separated points on a plot. Finally, by interpolating them at every level the 16%, 50%,
and 84% percentiles IDA curves are produced (Figure 2.9 (b)).
45
Chapter 3
3. Two-Dimensional Nonlinear IDA Analysis of MSB-
Braced Frame
3.1 Building Configuration
In this section, a 4-story modular steel frame from a study conducted by Annan et al.
(2009 a, b) has been chosen. At first, using the information from Annan et al. (2009 a) the
same structure has been modeled in OpenSees. This enabled the evaluation of the
response of the structure in comparison to the original study by Annan et al. Realizing
that the original design by Annan et al. (2009a) has not considered some realistic
limitations for modular construction, a new structure with some modifications was
designed. The nonlinear finite element computer program used for modeling and
analyzing the structure in this study is Open System for Earthquake Engineering
Simulation (OpenSees). Incremental Dynamic Analysis has been performed for assessing
the seismic capacity of the building and maximum inter-story drift and global roof drift
were chosen as the targeted demand parameters.
The seismic force resisting system of the 4-story regular braced MSB is shown in Figure
3.1. In terms of size, type, and plan layout this frame in considered as a typical MSB
system. In Annan et al. (2009 a)’s study only the lateral response of a single frame in N-S
direction is considered. Based on the Canadian code, the braced frame is designed
considering moderate ductility. Seismic design forces are in accordance with the National
46
Building Code of Canada (NBCC 2010). As explained by Annan et al. (2009 a), the
building is made up of six modules, labeled M#1 to M#6 connected to each other. The
building is a dormitory with 12 rooms and a corridor crosses the middle section of each
module which can be seen as the un-braced section of the elevation view in Figure 3.1 b.
Each floor level consists of two beams, one as floor beam of the unit from above and the
other as a ceiling beam from the unit underneath. Each of the floor and ceiling framing
has their own separate metal deck with concrete composite floor, two beams and a
number of stringers. The composite floor is assumed to be rigid within each modular unit
and each module is horizontally connected to other units on each side through field-
bolted clip angles. It is also assumed that the rigid floor of the modules and the horizontal
connections between them are sufficiently rigid to transfer the lateral load between the
modular floor units and to the braced frames (Annan et al. 2009 a, b).The only bay
without the ceiling beams is the middle bay (between lines C and D in Figure 3.1 b) to let
the mechanical and electrical facilities run along the building.
Figure 3.1 4-story MSB braced frame a) floor plan, b) elevation of frame 1 or 7.
47
In each story the braces are connected to the floor beam-column connections and ceiling
beam-column connections, and therefore they do not have a single working point. The
connection between the braces and the modular frame is composed of welded gusset
plates at both ends of the braces. In Annan et al. (2009a) `s study the nonlinear finite
element computer program, SeismoStruct (SeismoSoft, 2003) has been used to model the
frame system and the Remennikov steel brace member hysteresis (Remmennikov A.,
Walpole W., 1997) was used to represent all the bracing members. In this study, using
OpenSees, in order to capture the buckling behavior of steel braces, the fiber-based
model developed by (Uriz, P., Mahin, S. A., 2004) is used. Based on this study, each
element is broken into a number of segments and nodes are defined in the middle of each
brace to model large deformations due to the buckling behavior. Moreover, two elements
at each end are introduced as fully rigid elements to simulate the connections behavior.
There is clearance of 15cm between the floor beam of upper module and ceiling beam of
lower unit which can be used for fire protective layer installation. The modules are
connected to each other in vertical direction through the columns. During installation,
since the inner face of the vertical connection is not accessible, only its external portion is
field-welded and this is where independent upper and lower rotation can happen. Figure
3.2 shows the details of vertical (VC) and horizontal (HC) connections for a typical
multi-story MSB.
48
Figure 3.2 Details of a typical MSB structure.
3.2 Site Specification
Mostly, seismic design codes specify seismic hazard in terms of single intensity measure
such as the peak ground acceleration or a spectral ordinate at a given period. In Canada,
design spectra are essentially based on 5% damped site specific Uniform Hazard Spectra
(UHS) obtained for a probability of exceedance of 2% in 50 years (i.e., a return period of
2475 years). Spectral ordinates are specified in building codes at given periods, T, from
which the design spectrum can be built and the seismic hazard is described by spectral
acceleration values at those periods. In the NBCC (2010), the design spectrum is
49
determined from spectral ordinates Sa specified at periods of 0.2, 0.5, 1.0, and 2.0 s and it
is calculated as follows:
The building is located in Vancouver, BC, and the reference ground condition (site class
C) is considered as the site classification for seismic response. Using Table C-2 in the
NBCC 2010, the spectral ordinates are obtained:
Sa (0.2) = 0.94
Sa (0.5) = 0.64
Sa (1.0) = 0.33
Sa (2.0) = 0.17
Acceleration- and velocity-based site coefficients, Fa and Fv are obtained from Tables
4.1.8.4- B and 4.1.8.4- C of the NBCC. Using clause 4.1.8.4- (7) the design spectral
acceleration values of S(T) are determined (Table 3.1) and plotted against the specified
periods (Figure 3.3).
Table 3.1 Design spectral acceleration values of S(T)
Sa(T) Ta
FaSa(0.2) 0.94 0.2
FvSa(0.5) 0.64 0.5
FvSa(1.0) 0.33 1
FvSa(2.0) 0.17 2
0.5*FvSa(2.0) 0.085 4
50
Figure 3.3 Design spectrum of Vancouver with Site Class C
3.3 Analytical Model
Considering earthquake forces and gravity loading, members of the 4-story braced frame
are designed based on traditional strength and stiffness design criteria. The frame section
sizes are then modified according to ductility and capacity design requirements as
necessary. Canadian standard, CAN/CSA-S16.01 (CSA, 2010), was used for the design
purposes. The design load of floor materials is based on a typical floor system where the
weights of the concrete floor, insulation, a steel deck, self-weight of the frame members,
and an all-round metal curtain wall have been considered. Superimposed dead loads of
0.75, 0.32, and 0.7 kN/m2 are also calculated in the program to account for additional
loads on floor, roof, and ceiling respectively. Calculated dead loads are assigned to every
51
node at all the floor and ceiling levels. By assigning the masses to the nodes that the
columns are connected to; each connection takes the mass of 1/2 of each element framing
into it (mass=weight/g). The nodal masses are represented via lumped mass matrices and
used to calculate the eigenvalues during the dynamic analysis. The Rayleigh damping
have been assigned to all elements and nodes. The damping matrix is specified as
stiffness and mass-proportional. To obtain the mode shapes and periods of the structure
genBandArpack solver have been used in the eigenvalue analysis. The design live loads
of 1.9 kN/m2 for the rooms, 4.8 kN/m
2 for the corridors, and a snow load of 1.0 are
assumed in accordance with NBCC (2010) and the seismic loads are for the city of
Vancouver, Canada. CISC Grade 350W steel with a specified yield stress, Fy, of 350 MPa
is assigned to all the structural members. For each element, the least weight required for
strength is selected. Square Hollow Structural Sections (HSS), which are commonly used
in MSB structures, are chosen for all the columns and braces and wide flange sections (W
shape) are assigned to the floor, ceiling and floor beams. Table 3.2 summarizes the frame
sections used for the 4-story MSB. Optimal sections are selected according to
demand/capacity ratio for axial, flexural, and shear forces; and based on factored loads
and factored resistances.
52
Table 3.2 Member sections from the seismic design.
Frame Sections
4-S
toro
ry M
SB
Story # Columns Braces Beams
4 HS 76 x 76 x 5 HS 76 x 76 x 5 W 100 x 19
3 HS 178 x 178 x 5 HS 76 x 76 x 5 W 100 x 19
2 HS 178 x 178 x 5 HS 89 x 89 x 6 W 100 x 19
1 HS 178 x 178 x 6 HS 89 x 89 x 6 W 100 x 19
Similar to the work by Annan et al. (2009 a), the rigid connections for the beam and
columns are introduced in this new design as presented in Figure 3.4. Elastic beam-
column elements are defined as rigid blocks at the end of the beams and columns. The
rigid portions of the connection start at the beam-columns intersection node and continue
up to half of each section's total depth. To capture the independent rotation at the end of
the columns caused by partial field welding of the modules, a short column segment (M1)
is located between the top flange of the lower ceiling beam (J5) and the bottom flange of
the upper unit floor beam (J4). The short column is an inelastic force beam-column
element with the same section properties as the lower column connected to it. The height
of the short column represents the 15 cm vertical clearance allowed between the floor and
ceiling beams. A joint have been introduced at the top end of each short column to
simulate the independent rotation at the connections. Joint are modeled using equalDOF
command to define proper constraints for pinned column connections and pin the element
53
into the common joint with the upper unit column. Therefore, an independent rotation can
be obtained between the modules at the common joint.
Figure 3.4 Vertical connection model of MSB-braced frame.
Based on the NBCC 2010, the fundamental lateral period, Ta, in the direction under
consideration, is determined as 0.025hn (m), for braced frames where hn is the height
above the base. Therefore, here, Ta obtained from the code is 0.34 s. As a measure of
ground motion, the spectral acceleration takes into account the sustained shaking energy
at a specific period. For the selected 4-story modular building and site, the spectral
acceleration, Sa(T), corresponding to the fundamental period of the building is computed
as 0.8g. Table 3.3 shows the periods obtained from the eigenvalue analysis and the
NBCC code:
54
Table 3.3 Design and analytical periods
4-story MSB braced frame periods (sec)
NBCC design 0.34
First mode 0.5
Second mode 0.16
The modal/eigenvalue analysis was conducted for the MSB frame to obtain dynamic
response characteristics of the structure such as frequencies and mode shapes of free
vibration. The behavior of the frame is mostly dominated by the first mode. Shorter
empirical period is estimated based on the NBCC 2010 which would result in a larger
base shear capacity of the structure.
As it was mentioned in section 2.7.1, the response obtained from a single ground motion
may not provide sufficient confidence of the dynamic behavior of the structure. Hence a
suite of ground motion records needs to be considered and in this study the group of
ground motions selected by (Vamvatsikos D., Cornell C. A., 2004), is used. The seismic
inelastic demand of the structure is determined using the Incremental Dynamic Analysis
(IDA) procedure which was described in section 2.7. Similar to the study conducted by
Annan et al. (2009 a, b), here, the spectral acceleration at 5% damping, Sa(T1,5%), is
used as initial Intensity Measure (IM) and a simple stepping algorithm ,described by
(Vamvatsikos D., Cornell C. A., 2002) is employed to scale the ground motion records.
In addition, in order to assess the most appropriate IM parameter for the MSB system,
55
demand distribution with PGA is also provided to evaluate the dispersion of response
parameters.
Inter-story drift can be used as a primary damage parameter to predict the global capacity
of MSB-braced frame; however the validity of using it as a damage measure is
questionable. During a strong ground motion, several local damages can occur in the
structure. Elements such as beams, columns, braces, gusset plates, and vertical
connections of the modules are prone to excessive deformation. Nevertheless, result from
inelastic static (pushover) analyses conducted by Annan et al., (2009 b) and also the
reproduction of his model showed that, when the frame is designed based on capacity
design philosophy, global failure mechanism is restricted only to the failure in brace
members, and beams, columns and brace connections do not show premature inelastic
failure Annan et al., (2009 a, c). Because of the fact that failure of braces is more evident
in maximum inter-story drift, (i.e. high sensitivity of inter-story drift to bucking) it can
represent both the local and global collapse and consequently, can be used as a reliable
Damage Measure (DM).
As a primary DM parameter, maximum inter-story drift is often used in both vulnerability
assessment of moment resisting frames and characterizing global dynamic response of
ductile concentrically braced frame structures (Uriz, P., Mahin, S. A.,2004). In this study
the maximum inter-story drift ratio, θmax, and peak roof drift ratio, θroof, were selected as
global demand parameters to evaluate the structural responses. During the duration of the
ground motion, the inter-story drift ratio is obtained from the ratio between the relative
56
displacement and the specified distance between the corresponding stories (i.e., the story
height); and the peak roof drift ratio is taken as the ratio of the peak roof displacement to
the overall height of the building. Finally, the IDA curves are obtained for each record
and are plotted as DMs against their corresponding IMs.
3.4 Results of the Incremental Dynamic Analysis
3.4.1 IDA Curves
Figures 3.5 - 3.6 show a total number of 80 IDA curve resulted from around 2000
nonlinear time history analyses conducted on the 4-story MSB frame. In each plot, the
DM resulting from a scaled ground motion and a given IM produces a single point of an
IDA curve. By going through the entire range of IMs for each ground motion record a
single IDA curve is generated (for specified IMs versus DMs). The rest of the curves are
produced in a similar manner. In Figure 3.5 the ground motion IM is the ξ 5% damped
Spectral Acceleration of the scaled ground motion at the structure’s first-mode period
Sa(T1, 5%). The corresponding DMs in these plots are the maximum peak inter-story
drift ratio, θmax, (over all stories) and the peak roof drift ratio, θroof, which are expresses as
percentages. Similarly, in Figure 3.6, the DMs are plotted against corresponding PGA of
the scaled ground motion records. The plots are reproductions of Annan’s study.
57
Figure 3.5 IDA curves of ‘‘first mode’’ spectral acceleration, Sa(T , 5%), plotted against a) maximum inter-story drift ratio, θmax, b) peak roof drift ratio, θroof, for the 4-story MSB-braced
frame.
Figure 3.6 IDA curves of Peak Ground Acceleration, PGA, plotted against a) maximum inter-
story drift ratio, θmax, b) peak roof drift ratio, θroof, for the 4-story MSB-braced frame.
58
Comparing the above figures it can be observed that there is a considerable dispersion of
the results in the IDA curves for different ground motion records. However, before the
first signs of nonlinearity, a distinct linear elastic behavior is noticed in all curves. By a
closer examination of the linear elastic region in the IDA curves it is observed that the
Sa(T1, 5%) is wiser choice of intensity measure than the PGA. When PGA in adopted as
a primary IM, the elastic stiffness, which is defined as the ratio of the IM to the DM in
the linear elastic range of response, varies from record to record. On the other hand, the
elastic stiffness is almost the same when using Sa(T1, 5%). This is because first-mode
dominated structures are sensitive to the strength of the frequency content near their first-
mode frequency, which is well characterized by the Sa(T1, 5%) but not by PGA.
A desirable property of an IM candidate is that it has a small dispersion. Smaller
dispersion of DM for a given IM implies that a less number of ground motion records is
needed to estimate the demand of the structure, and as a result fewer nonlinear time
history analyses are necessary. Therefore, based on the above observations, suffice it to
say that the Sa(T1, 5%) is a more consistent intensity measure for the MSB-braced frame
and can be used in the demand and capacity assessment of the selected frame (Annan et
al. 2009a).
In the inelastic range of response of MSBs, increased distortion and complexity in IDA
curves with a wavier behavior is observed. Some display a softening behavior after initial
buckling and accelerate toward large displacements and eventual collapse; however,
59
others display successive segments of softening and hardening before the final instability.
Limited alternative internal force paths due to vulnerable vertical connections in between
the modular units result in a high inelasticity concentration along the building height.
Another reason may be the sensitivity of inter-story drift to brace buckling and possible
significant changes in the dynamic properties of the structure after the brace buckling.
Generally, stiffer frames would experience significant changes in their dynamic
properties after brace buckling. Such frames are expected to produce more complex IDA
curves (than for example a ductile moment resisting frame). In addition, the scaling,
pattern, and record duration also affect the shape of the IDA curves.
3.4.2 Summary of the IDA Curves
As already explained above, to summarize the enormous data produced by IDAs, a
statistical assessment of the demand is required. The data sets obtained under the suite of
ground motions are compressed into probabilistic distribution of a DM given an IM by
defining the 16, 50, and 84% fractile IDA curves (Vamvatsikos D., Cornell C. A., 2002,
2004, 2005; Han, S. W., Chopra, A. K., 2006). This process is explained in detail in
Chapter 2. Figure 3.7 shows the 16, 50, and 84% fractile curves of the DM (demand
parameters, θroof and θmax ) for a given ground motion IM, Sa(T1, 5%) for the 4-story
MSB-braced frame.
As a representative of seismic demand parameter of the frame, the fractile IDA curves
may be used to evaluate the performance of the structure by comparing the calculated
demands with allowable drift demands at any given IM and probability level. For
60
instance, given the design spectral acceleration of the site, the design ground motion
intensity level of Sa(T1, 5%), at 2% in 50 year probability is 0.8g. Accordingly, the
calculated fractiles show that for the design level intensity, 16% of the records would
produce θmax ˂ .25%, 5 % would produce θmax ˂ .45%, and 84% of the records would
yield θmax ˂ .76 %. In the NB (2010) the largest inter-story deflection at any level
which is based on median 2% in 50-year seismic hazard level, shall be limited to 0.01 hs
for post-disaster buildings, 0.02 hs for high importance category buildings, and 0.025 hs
for all other buildings. Hence, it can be inferred that the median ground motions
calculated for the MSB structure provides satisfactory performance in any of the above
building category. Table 3.4 summarizes the 16%, 50%, and 84% fractile values in terms
of DM and IM for Immediate Occupancy (IO), Collapse Prevention (CP), and Global
Instability (GI) limit-states, for the 4-stories MSN-braced frame modeled by Annan et al.
(2009 a).
Figure 3.7 Summary of IDA curves of the 4-story MSB frame into16th, 50th, and 84th fractiles with (a) maximum inter-story drift ratio, (b) Peak roof drift ratio.
61
Table 3.4 Summarized capacities for each limit-state.
Sa(T1, 5%) (g) θmax (%)
IMc16% IM
c50% IM
c84% DM
c16% DM
c50% DM
c84%
IO 1.3 1.6 2.2 2 2 2
CP 1.5 2.8 4.4 10 10 10
GI 1.7 3.2 4.7 ∞ ∞ ∞
The global dynamic capacity varies from record to record. Because of the wavy nature of
most of the IDA curves both of the rules may introduce more than one capacity point. In
such cases, the first capacity point in the IDA curve is recommended for the DM-based
rule and final capacity point is recommended in the IM-based rule (Vamvatsikos D.,
Cornell C. A., 2002). In the study by Annan et al., (2009a) the 20% tangent slope was
adopted as the global capacity of MSB-braced frames under the selected ground motion
records.
3.5 Inter-Story Drift and Inelastic Distribution along the Height of the
Structure
The inter-story drift performance of a multistory building is an important measure of
structural and non-structural damage to the building under various levels of earthquakes
motions. This parameter has become a principal design consideration in performance-
based design (Christopoulos, C., Filiatrault, A., 2006). As mentioned earlier, one of the
distinguishing features of MSB-braced frame from regular steel braced structures is the
existence of the ceiling beams at every level. Ceiling beams also incorporate as horizontal
62
members to brace upper end connections in each level. In the regular MSB developed by
Annan (2009) it is assumed that the vertical connections of different modular units are
not continuous such that the behavior of their configuration may result in independent
rotation of upper and lower module columns at the same joint. This may influence the
inelastic demand especially after brace buckling and cause additional limitation on
redistribution of internal forces in between the modules.
Inter-story drift in the IDA and fractile plots provided above, was obtained from two
consecutive floor beams in modular units and any influence due to ceiling beams between
these floors was ignored. Since the nodal masses were assigned both to the floor and
ceiling nodes, during a ground motion event, the maximum inter-story drift angle may
also change at the ceiling level within the same modular unit. This will alter the lateral
deformation distribution along the building height. Therefore, in Figure 3.8 the inter-story
drift distribution taking into account the ceiling beams incorporation is plotted. In this
figure, the height-wise distribution of peak floor-to-ceiling and ceiling-to-floor drifts of
the 4-story MSB frame for the ground motion recorded at EL Centro Array #13 (Imperial
Valley earthquake) is plotted at three intensity levels (Sa(T1, 5%) = 0.3, 2.0, and 3.0g)
selected by Annan et al. (2009 a).
63
Figure 3.8 - Height-wise distribution of peak inter-story drift ratio for the 4-story MSB at a)
Sa(T1, 5%) = 0.3 g, b) Sa(T1, 5%) = 2.0 g, c) Sa(T1, 5%) = 3.0 g.
When subjected to strong ground motions, multi-story braced frames typically exhibit
large variation in story drift along their height (Perotti, F., Scalassara, P., 1991; Tremblay
R., Robert N., 2001). The reason for that is mostly because of the degradation in brace
compressive resistance that occurs after a number of successive compression load cycles
beyond brace buckling. After the brace buckling, the shear resistance of the story level
diminishes and consequently large story drifts are accompanied with the formation of
story mechanism. The behavior observed in Figure 3.8 is representative of the behavior
under all the other selected ground motions. The intensities are selected such that both
elastic and inelastic responses of the MSB frame are covered. Although the inter-level
drift (drifts between the ceiling level of the lower module and floor level of the upper
one) is larger in inelastic range of response, especially at the maximum inter-story drift
level, it is not significant. Hence, the floor-to-floor inter-story drift can be taken as the
representative of the inter-story demand without considering the ceiling beam level
64
(Annan et al. 2009 a). In the lower ground motion intensities where the structure is in the
elastic range of response, almost no variation in the drift angle is noticed.
65
Chapter 4
4. Modified Two and Three Dimensional MSB Structure
Analysis
4.1 Common MSB Types and Range of their Application
In this section a new modified 2D 4-story MSB structure is introduced. In an attempt to
obtain more accurate and realistic results, in the modified structure some of the detailing
and modeling assumptions of Annan’s model have been improved. IDA and time-history
analyses of the structure are conducted and the results are provided. The structure is then
analyzed in 3D in order to compare the results with those from the 2D model and capture
some of the aspects that are not available in 2D. Summary and conclusions are provided.
Generally, there are two type of modular construction:
1. Load-bearing modules in which loads are transferred through the side walls of the
modules. The determinant element in this type of MSB structures are the side walls
comprising light steel C sections. Depending on the size and the spacing of the C sections
used, the height of this type of construction is typically limited to four to eight stories.
2. Corner-supported modules in which loads are transferred by way of edge beams and
concrete slabs to corner posts (columns). The controlling factor in this type of MSB is the
compressive resistance of the corner columns. Square HSS (Hollow Structural Section) is
mostly used as the corner columns as a result of their high buckling resistance. For the
corner-supported modules, the height of the building is limited only by the size of the
66
HSS that may be used for a given module size. Maximum sensible size of the corner
columns used in the MSBs is 150 x 150 x 12.5 mm, (Lawson R. M., Richards J., 2010).
In this study the second type of modular construction has been considered.
The strategies employed to ensure the adequate stability of modular assemblies are a
function of the height of the building. By increasing the height of the building,
appropriate Seismic Force Resisting Systems (SFRS) should be utilized to provide
sufficient robustness to the structure. The following SFRSs are more common in MSB
structures:
1. Diaphragm interaction of boards, double skinned steel plates or bracing within the
walls of the modules for buildings up to 6 stories.
2. Separated braced structure (supporting structure) located in the elevator and stair area
or external walls using hot-rolled steel members appropriate for six to ten stories.
3. Reinforced concrete or steel-plated core is suitable for buildings with more than ten
stories height.
4.2 Considerations Required in the Design of MSBs
Due to the complexity of the structural interaction within a group of modular units a
detailed model is required to provide more realistic and reliable results. In a MSB
structure, units are tied at their corners so that they act together to transfer lateral loads.
Horizontal forces may be transferred by tension and compression forces in the ties at the
corner of the modules and through the horizontal connections implemented in between
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them. By utilizing the diagram action of the floor and ceiling of each module, these
forces are transferred to the corner connections. Because of potential articulation through
the bolts and connecting plates at the connections, relative displacements and rotations
may occur in between the modules (both horizontally and vertically).
However, in the 2D MSB-braced frame introduced by Annan et al. (2009 a), only one
single modular unit per story level was considered; thus only the effects resulted from the
ceiling beams were included and the interactions between modules (possible rotations or
displacements) could not be captured. Moreover, although in the design section of the
MSBs it was considered that the horizontal connections are designed to be sufficiently
rigid to transfer lateral loads between the modular floor units, practically the existence of
the horizontal connection in the analytical model could not be simulated (there was no
horizontal connection in the model) and as a result the entire floor was assumed to
behave as a rigid plate. The assumption of rigidity of the horizontal connections is also
unrealistic and as it was mentioned, there are possible relative rotations and
displacements in the bolts and connecting plates (within the units). In modular
construction, the composite slab of each module is implemented separately and after the
units are transported to the site they are tied to each other with a specific horizontal
spacing in between them. Thus, there will be several separated rigid diaphragms at each
level of MSB structure.
To conduct a nonlinear analysis, essential characteristics of all elements such as load-
deformation or moment curvature characteristics in the model are required. Therefore, it
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is imperative to have a proper understanding of the behavior of each of the elements. To
achieve the most reliable and realistic results different elements and materials have been
tested both separately and in interaction with other components in the numerical analyses.
In this section, a modified 2D MSB-braced frame is introduced and evaluated with IDA
method. This structure was also modeled and analyzed in 3D. The site specifications and
the suite of ground motion records that have been used are provided in the previous
chapter. The final model considers every possible detail to simulate the behavior of the
structure as accurate as possible. The diaphragm action and the horizontal and vertical
interaction between modules are more realistic in the following 2D and 3D modified
MSB models.
4.3 Modified 2D MSB Structure
Typical modules are generally from 3.3 m to 3.6 m in width and from 6 m to 9 m in
length (internal dimensions) (Lawson, R.M., Ogden, R.G., Bergin, R. 2012). A module
with an area of 20 to 35 m2 is often used for a single-person accommodation. Two
modules are suitable for two-person apartment and three or four modules are generally
suitable for a family (Lifetime Homes 2010). Figure 4.1 shows the elevation and plan
views of the modified MSB-braced frame. Although the number of bays and brace
configurations is different from the model introduced by Annan et al. (2009 a), the same
dead loads and live loads for a dormitory building have been selected. The seismic
loading on each frame was based on the National Building Code of Canada (NBCC,
2010). In Annan`s model there were 6 modules at each level. The dimensions of each
module were 3.6 m wide and 16.5 m long which is much longer than the standard
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dimensions that are typically used and can be problematic when transporting these units
from the plant to the site. Here it is assumed that there are 12 modules at each level
dimensioned 3.5 m by 4 m and a height of 3.5 m.
Figure 4.1 Modified 4-story MSB braced frame a) floor plan, b) elevation of frame 1 or 6.
By providing the structure with inelastic deformation capacity a more efficient and cost-
effective design can be obtained. In capacity design philosophy a proper strength
hierarchy should be adopted to ensure that some specific seismic force resisting system
(SFRS) components are able sustain cyclic inelastic deformations in a ductile and stable
manner while other SFRS elements remain essentially elastic. Diagonal braces in the
MSB frame are considered as the ductile yielding elements that are resisting lateral forces
axially. The design forces for yielding elements are a combination of code specified
gravity, wind, and seismic load effects. These components are deformation controlled
elements, and they are supposed to sustain significant inelastic deformations. After brace
70
yielding or buckling the redistribution of the loads causes alternative load paths. Thus,
the beams carry the horizontal shear forces and the columns work as vertical cantilevers
resisting the overturning moment. Therefore the braces are designed to bear the nominal
tensile strength Ag.Fy (Ag is the gross area of the brace and Fy is the specified yield stress)
and the beams are modeled as beam-columns carrying the design moments obtained from
the code tributary gravity loads as well as the lateral loads corresponding to the forces
induced from ductile mechanism; i.e. the axial compression coming from unequal
capacity of braces in tension and compression. These components are referred to as force
controlled elements and they are design to sustain the maximum force demand in the
elastic range of the response.
Frame member sections obtained from the seismic design of the modified MSB structure
were (Table 4.1) based on the Canadian standard, CAN/CSA-S16-09 (CSA, 2010). Since
each module has its own columns installed off-site (at the remote factory), there may be
more than one column at each axis of the building (one or two for a 2D single-frame
model and one, two or four columns in a 3D model). The column sections comply with
the maximum practical size of the columns stated before and are installed with a
horizontal center to center distance of 0.35 m. The horizontal connections of separately
finished units are achieved by bolting steel plates (Figure 3.2) or shop-welded clip angles
(Figure 4.2) to the floors at the corners of the modules. It is assumed that these
connections are designed so that they remain in the elastic region of response under the
design earthquake. This is achieved by assigning strength of 1.3 times the adjacent beams
at the intersection to the horizontal connections.
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Table 4.1 Member sections from the seismic design.
Frame Sections
Mo
dif
ied
4-S
tory
MS
B
Story # Columns Braces Beams
4 HS 76 x 76 x 5 HS 51 x 51 x 5 W 100 x 19
3 HS 102 x 102 x 6 HS 51 x 51 x 5 W 100 x 19
2 HS 102 x 102 x 6 HS 76 x 76 x 5 W 100 x 19
1 HS 127 x 127 x 5 HS 76 x 76 x 5 W 100 x 19
Figure 4.2 Connection detail between the modules, a) sketch detail; b) actual detail (Modular
design for high-rise buildings, Lawson, Richards, 2010)
4.3.1 Beams and Columns
In the OpenSees model, an inelastic steel beam-column element has been used to
represent all the beams and columns of the MSB frame. Beam-column elements used for
this purpose are actually a ForceBeamColumn element with two nonlinear definable
sections at both ends of the section, and an elastic segment at the middle portion of the
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element. They act as a nonlinear beam-column element with distributed plastic hinges at
both ends. The hinges are defined by assigning rotational springs at both ends of the
element and the length of distributed plastic hinge can be specified based on the length of
the element. While the ForceBeamColumn (nonlinearBeamColumn) element command
considers the spread of plasticity along the entire length of element, the beam with
distributed plastic hinge (beamWithHinge) element localizes the plasticity in specific
hinge regions, therefore the integration points (gauss points) will be limited to that
regions (Scott M. H., G. L. Fenves, 2006). Stiffness modifications should be determined
properly so that the strain hardening of the plastic hinge region in MSB member captures
the actual strain hardening of the frame member (Ibarra L. F., Krawinkler H., 2005).
4.3.2 Braces
Diagonal braces of the MSB frame experience global plastic mechanism and are
subjected to large cyclic deformation during strong earthquake. The mechanism is
achieved through the yielding of a brace in tension and the inelastic buckling of the brace
in compression. The buckling and post-buckling range forms three flexural plastic hinges.
The tension-yielding brace deforms in-elastically through axial inelastic deformation, and
plastic rotations of the flexural plastic hinges occur in compression as the brace buckles.
To model the braces and capture a realistic buckling behavior the fiber-based model
developed by (Uriz, P., Mahin, S. A., 2004) have been used. The uniaxial Giuffre-
Menegotto-Pinto steel material (Steel02) object with isotropic strain hardening has been
used as the material command for the braces. The yield stress of 350 N/mm2, the elastic
modulus of 200 kN/mm2, and the strain-hardening ratio (ratio between post-yield tangent
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and initial elastic tangent), are defined as well as some parameters to control the
transition from elastic to plastic branches. The geometric transformation command used
for the braces is the co-rotational transformation command and it is to be used in large
displacement-small strain problems. To allow the formation of the three above-mentioned
flexural plastic hinges, as it can be seen in Figure 4.3, the brace is divided into different
separated segments. By introducing these segments, behavior of different parts of the
brace can be captured. For example, elastic beam-column elements are adopted at the two
ends of the braces and are modeled to be rigid to simulate the rigid connections and
gusset plates’ behavior. The node introduced in the middle of the brace as well as the
nodes located at end of the rigid section allow the plastic rotation of the flexural plastic
hinges in the compression brace. The model can realistically represent the buckling
behavior of the brace members and captures the inelastic behavior under repetitive axial
tension and compression considering the significant degradation in compressive
resistance of the braces after a few cycles of loading. The force-deformation plot of one
of these elements subjected to a harmonic loading is shown in Figure 4.4.
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Figure 4.3 Brace finite element model.
Figure 4.4 Force versus displacement relationship for a sample brace element.
Brace, beam, and column connections need to be designed carefully to carry forces that
are induced by the yielding of the tension braces and buckling of the compression braces.
This is more important in MSB systems where the redistribution of the forces may not be
75
reliable enough due to the partial welding of the vertical connections at the end of the
columns.
Based on the study conducted by Annan et al. (2009 a) it was concluded that within the
entire range of structural response of the MSB-braced frame, floor-to-floor inter-story
drifts can satisfactorily represent inter-story drift demand without explicitly considering
the effect of drift at the ceiling beam levels. This was also confirmed in Chapter 3
(section 3.5) of this study. Moreover, in practice, a common strategy when connecting the
modules is that instead of connecting the columns at each level, the ceiling beams and
floor beams are directly anchored to one another. This can be easily be arranged by
providing a false (drop) ceiling to let the mechanical and electrical facilities run along the
ceilings. This way, the need of a specific clear spacing can be eliminated, and the floor
and ceiling beams can directly be connected. Still, since the working point of the braces
where their frame connections do not intersect at a single location and may cause high
inelasticity and seismic demand by imposing additional moments at vertical connections.
Based on the above observations and insignificant effects of the independent rotations at
the vertical connection on the structural behavior, in the modified model the pinned
connections between the floor and ceiling beams are removed, but still a 0.15 clear space
is provided. In some cases for corner supported modules a gap between the floor and
ceiling beams are allowed to facilitate the bolting or welding procedures.
To have a better evaluation of the seismic vulnerability of MSB structures, the modified
MSB structure shown in figure 20 has been modeled both in 2D and 3D. Dynamic
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response characteristics of both the structures have been assessed using IDA and
nonlinear time history analyses and conclusions are provided.
4.3.3 IDA Results Obtained from the 2D Modified MSB
In this section, the same suite of ground motions and site characteristics used in Chapter 3
are used. As explained before, according to the NBCC, for braced frames, the
fundamental lateral period, Ta, in the direction under consideration can be determined as
0.025hn (m). Here hn is the height above the base. As a result for the structure designed
here, Ta obtained from the code is 0.35 s. The spectral acceleration, Sa(T), corresponding
to the fundamental period of the building is computed as 0.8g. Table 4.2 compares the
periods obtained from the eigenvalue analysis and the NBCC code.
Table 4.2 Design and analytical periods
4-story MSB braced frame periods (sec)
NBCC design 0.35
First mode 0.6
Second mode 0.17
The maximum inter-story drift ratio, θmax, and peak roof drift ratio, θroof, were selected as
global demand parameters, DM, to evaluate the structural responses. These DMs are
plotted against corresponding Sa(T1, 5%) and PGA of the scaled ground motion records
to obtain the IDA curves (Figure 4.5-4.6).
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Figure 4.5 IDA curves of ‘‘first mode’’ spectral acceleration, Sa(T , 5%), plotted against a) maximum inter-story drift ratio, θmax, b) peak roof drift ratio, θroof, for the modified 4-story
MSB-braced frame.
Figure 4.6 IDA curves of Peak ground acceleration, PGA, plotted against a) maximum inter-story drift ratio, θmax, b) peak roof drift ratio, θroof, for the modified 4-story MSB-braced frame.
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4.3.4 Selection of the Proper Intensity Measure
Similar to the results presented in Chapter 3, when using the Sa(T1, 5%) as the primary
intensity measure, smaller dispersion of the DM for a given IM is observed in the elastic
range of the plots (Figure 4.5). Another factor that needs to be considered is the
dispersion of IM for a given DM. The IDA curves can also be used to study how well
particular IMs predict collapse capacity. In the in-elastic range of response smaller IM
dispersion associated with the flat-lines leads to a better prediction of collapse capacity
(Vamvatsikos D., Cornell C. A., 2005). In this case, the results obtained from both
Annan`s and current models show larger dispersion in IM values when using Sa(T1, 5%)
as the primary intensity measure. This could be traced back to the ground motion scaling
where the spectral accelerations were scaled to the target spectrum by matching at the
PGA and fundamental period (scaling on amplitude). Since, in the most of the selected
records the response acceleration (Sa) values at T1 are closer to the corresponding
acceleration value at the design spectrum than the ones at PGA, the scale factors resulted
from the ground motion scaling obtained from matching the PGAs are larger than the
ones obtained from matching the fundamental periods. Therefore, after multiplying each
record by its corresponding scale factor, the structure undergoes ground motions with
higher amplitudes when the PGA is used as the primary intensity measure and as a result
will collapse in lower IMs. Hence, it can be concluded that for a specific design spectrum
and specific set of ground motions, the selection of proper intensity measure for a
structure depends on the structure`s properties such as the fundamental period. As an
example, for the 4-story MSB use of PGA as the intensity measure leads to a smaller
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range of IMs at the collapse DM; however, in a study by (Vamvatsikos D., Cornell C. A.,
2004) Sa(T1, 5%) was found to be more an appropriate IM for a 9-story moment resisting
frame with the fundamental period of T1 = 2.4s.
4.3.5 Summary of the IDA Results
The 16%, 50%, and 84% fractile values are also plotted to summarize and quantify the
randomness introduced by the records. According to NBCC 2010 and for the modified 4-
story MSB frame at its design level intensity of Sa(T1, 5%) = 0.8g, 50% of the records
would produce θmax ˂ .3 %, which is less than the largest allowable inter-story
deflection (1%); hence, the structure provides satisfactory performance in all the building
categories mentioned in the code. The 16%, 50%, and 84% fractile values are also
calculated and gathered for each limit-state and in terms of DM and IM (Figure 4.7).
Table 4.3 shows the Immediate Occupancy (IO), Collapse Prevention (CP), and Global
Instability (GI) limit-states for the modified 4-stories MSN-braced frame. The obtained
results are compared with the demand and capacities of the 3D model in the next section.
Figure 4.7 Summary of IDA curves of the modified 4-story MSB frame into16th, 50th, and 84th fractiles with (a) maximum inter-story drift ratio, (b) Peak roof drift ratio.
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Table 4.3 summarizes the 16%, 50%, and 84% fractile values in terms of DM and IM for IO, CP,
and GI limit-states, for the modified 4-stories MSN-braced frame.
Sa(T1, 5%) (g) θmax (%)
IMc16% IM
c50% IM
c84% DM
c16% DM
c50% DM
c84%
IO 2.2 2.8 3.2 2 2 2
CP 2.4 2.9 3.7 10 10 10
GI 2.5 3.1 3.9 ∞ ∞ ∞
4.3.6 Inter-Story Drifts
Taking into account both the floor and ceiling beams incorporation, the maximum inter-
story drifts distribution along the height of the structure for the ground motion recorded at
EL Centro Array #13 during the 1979 Imperial Valley earthquake are plotted in Figure
4.8. In this figure, the floor to ceiling and ceiling to floor drifts of the modified MSB
frame are plotted at three different intensity levels to cover the elastic and post elastic
rage of responses. These plots are representative of the behavior observed for the rest of
the ground motions and will be compared with the drift results of the 3D model
introduced in the next section.
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Figure 4.8 Height-wise distribution of peak inter-story drift ratio for the modified 4-story MSB
(a) Sa(T1, 5%) = 0.3 g, (b) Sa(T1,5%) = 2.0 g, (c) Sa(T1, 5%) = 3.0 g.
4.4 Modified 3D MSB Structure
In structural analysis, two-dimensional and three-dimensional models can be used for the
analysis. When P-Δ effects are to be considered in the analysis, two-dimensional models
must include the tributary gravity carrying system of the SFRS elements. The gravity
system can be explicitly modeled or represented by means of leaning P-Δ columns.
However, considering the advanced modeling and analysis tools that are now available, it
is generally preferable to use a three-dimensional model of the entire structure for seismic
analysis, even if independent analysis are performed along each orthogonal direction
(Filiatrault et al. 2013). Analyzing the 3D model of a structure has quite a few
advantages. For instance, it provides a three-dimensional representation of the structure
stiffness (for any analysis), mass (for dynamic analysis), and strength (for nonlinear
analysis) properties. Therefore, the torsional response of the structure is explicitly
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included in the analysis and the distribution of the seismic effects in the various
components of the SFRS is directly obtained from the analysis.
In this study, the three-dimensional (3D) finite-element model of the modified 4-story
MSB structure is developed to take the biaxial interaction of building into consideration.
The gravity frames are also included in the model to evaluate their effects on the building
performance. The building has a 15.0 m by 12.7 m square plane configuration and a story
height of 3.5 m. There are 4 modules (bays) in Z direction and 3 modules in X direction
(Figure 4.9). There are 6 of the bays in the perimeter frames that are designed as braced
frames and the rest are gravity frames. When using the IDA to determine the capacity of
the structures, the imposed excitation on the model could have intensity much larger than
the recorded ground motion. It is reasonable not to limit the development of plasticity in
the columns and beams. Hence, elements with plastic hinges (distributed plastic hinges at
the ends of the elements) have been used in both SFRS and non-SFRS frames.
83
Figure 4.9 MSB structure: a) 3D view of the SFRS vertical elements; b) plan view of four of the modules (diaphragms) that are connected through the horizontal connections located at the center
of the modules.
Unlike regular steel buildings that a uniform slab is implemented at every floor level, in
MSB structures each module has its own separated concrete slab that is implemented at
the remote facility. These slabs are then tied to each other through the horizontal
connections. Hence, instead of having a single diaphragm, separated diaphragms (one for
each module) should be considered at each story floor. Another reason that makes
separated diaphragms a better choice to simulate the diaphragm action is the ability to
capture the forces in the horizontal elements. In a single diaphragm covering the whole
floor, the member forces in any of the horizontal elements connecting two or more nodes
of the diaphragm are computed as null from the analysis, which may not be
representative of the actual structural response of these elements. For instance, cumulated
forces in the horizontal connections connecting units to each other, and collectors
connecting horizontal diaphragms to vertical SFRS elements will be omitted if the end
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nodes of the modules at each level are assigned diaphragm constraints. Axial loads and
axial deformations in the beams of the braced fame will also be ignored if all the nodes at
every level of the braced frame are constrained to the rigid diaphragms simulating ceiling
and floor diaphragm responses. Freeing the nodes in between the modules will result in a
more realistic representation of the braced frame lateral stiffness and, thereby, better
predictions of the building drifts and periods. Moreover, by modeling diaphragms
separately, the actual in-plane flexibility of the floors will be obtained more accurately,
leading to more representative distributions of forces and deformations among each of the
horizontal and vertical components.
In the modified 4-story three-dimensional model, rigid diaphragm response is
conveniently reproduced by assigning a diaphragm constraint to the horizontal degree-of-
freedom of the all nodes at each “individual” modular unit. In other words in every
module the perimeter nodes are slaved to a master node located at the center of the
module. These master nodes represent the global lateral movement of the diaphragms at
that level. The horizontal displacements of the slave nodes in both directions are coupled
with the three horizontal degree-of-freedom of the master node of the diaphragm:
horizontal displacements in two orthogonal plan directions and rotation about the vertical
axis. Therefore, floor level and ceiling level of “each module” has three DOFs,
independently, in the resulting model.
During an event of earthquake lateral forces induced by the ground motion are transferred
to the braces through the horizontal connections located at the corners columns or beam
85
edges. Because of the potential articulation in the bolts and connection plates of the
connections within the modules, these forces (axial, shear, and moments) can cause
undesirable rotations and horizontal relative displacements between the units (Figure
4.10). The detailed 3D model is provided to investigate the internal forces in both
horizontal directions and potential shear and bending actions in the connections.
Figure 4.10 Force transfer between modules a) Bending action b) Shear forces.
4.4.1 Hysteretic and Stiffness Properties
To obtain a realistic assessment of building drifts and periods as well as adequate force
distributions, stiffness of all elements must be properly reproduced in the model. In the
NBCC 2010, the verification of possible contribution of stiff elements that are not part of
the SFRS to the lateral stiffness of the structure is required. This could lead to shorter
86
building periods and higher seismic loads which could even be more significant in
modular buildings due to their inherent non-SFRS stiffness (i.e. rigid beam and column
connections in gravity load carrying system components). The stiffening effects from
finite members and connection sizes must also be accounted for in the analysis of MSBs,
which are accomplished by specifying rigid offsets at the member ends in the 3D model.
When producing a 3D model, the modeling technique mostly depends on the anticipated
inelastic deformations and the level of detail required for the results. For example, as
illustrated in Figure 4.11, inelastic deformation in the brace element can be reproduced
using either (a) simple elastic beam elements with concentrated (lumped) plastic hinges at
the member ends (elasticBeamColumn element with rotational springs at the ends); (b)
fiber discretization of the cross-section with nonlinear material properties specified at
discrete locations along the entire member or specific lengths at the ends of the members
(beamWithHinges) allowing for distributed plasticity to be predicted; or (c) complete
discretization of the member using detailed finite element modeling allowing nonlinear
material response combined with local and global member instability (buckling)
response. Similar modeling refinements can also be used for horizontal connections
between the modules and in the SFRS connections. In this study option (c) is chosen for
the brace elements, and option (a) found to be more appropriate for the horizontal
connections. Cyclic strength degradation behavior of seismic force resisting elements and
vertical connections that are expected to undergo significant inelastic deformations is also
incorporated in nonlinear time history analysis.
87
Figure 4.11 Different types of materials and elements that can be used in finite element modeling.
Mass and damping properties of the structure must be specified when performing time-
history dynamic analysis. In the Canadian code the mass must be equivalent to the
seismic weight that is specified for the calculation of the minimum design earthquake
loads. In the 2D analysis explained in the previous section, only the mass associated to
the horizontal DOFs in the direction studied need to be included. However, in the 3D
model, the in-plane distribution of the mass at every level in both horizontal directions
have been reproduced in the structural model so that in-plane torsional moments due to
inertia loads are adequately predicted by analysis. The seismic masses correspond to a
linear combination of portions of the dead, live and snow loads are applied to the
structure. In the 3D model, masses active in both horizontal directions are specified at
every node of the structure. This way the masses are adequately spatially distributed in
the structure and the resultant inertia forces from in-plane torsional response will be
appropriately accounted for in the analysis. In addition, in this case that the modular
units are separately connected to each other through the horizontal connections,
88
distributed masses are more useful because the in-plane dynamic responses of flexible
floor diaphragms need to be included.
4.4.2 Bi-directional Horizontal Shaking
Based on NBCC 2010, time history analysis must be performed with ground motion time
histories that are compatible with the design spectrum S(T) , i.e. their response spectra
must equal or exceed S(T) throughout a period range that includes the periods of the
modes contributing to the structure response in the direction of analysis.
Recommendations for the choice of target spectrum for three-dimensional analysis which
requires two horizontal components of ground motion as input are provided in (National
Institute of Standards and Technology; Selecting and Scaling Earthquake Ground
Motions for Performing Response-History Analyses). For low-to-medium-rise buildings,
the recommendations assume that element deformation and story drift are due to first
mode response and the building is assumed to have principal horizontal directions X and
Z, uncoupled first mode periods of T1X and T1Z , respectively, and an average first-mode
period of T = 0.5(T1X + T1Z). The average first and second mode periods of the 3D
modified 4-story MSB are 0.7 and 0.54 sec.
4.4.3 IDA Analysis of the 3D MSB Structure
The IDA is extended to 3D analyses, in which all the inherent variability in ground
motion, elements, connections, and material properties have been considered. Figure 4.12
shows the IDA curves for the 3D modified 4-story MSB in Z direction. The fractile
curves are also obtained by computing the 16%, 50%, and 84% fractile values of the DM
89
values, θmax and θroof, for the given IM, Sa(T1, 5%), Figure 4.13. The capacity against
different limit-states in terms of drift ratio is obtained and showed on Table 4.4
Figure 4.12 IDA curves of Peak Ground Acceleration, PGA, plotted against a) maximum inter-
story drift ratio, θmax, b) peak roof drift ratio, θroof, for the three-dimensional 4-story MSB structure in Z direction.
4.4.4 Summarizing the IDA Results
Figure 4.13 Summary of IDA curves of the three-dimensional 4-story MSB structure (Z direction) into16th, 50th, and 84th fractiles with (a) maximum inter-story drift ratio, (b) Peak roof drift
ratio.
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Table 4.4 Summarized capacities for each limit-state for the 3D modified MSB in Z direction.
Sa(T1, 5%) (g) θmax (%)
IMc16% IM
c50% IM
c84% DM
c16% DM
c50% DM
c84%
IO 1.5 2.2 3.2 2 2 2
CP 1.8 2.4 3.4 10 10 10
GI 1.9 2.7 3.6 ∞ ∞ ∞
4.5 Comparison of the Two-Dimensional and Three-Dimensional Modified
MSB Structures
4.5.1 Effects of Non-SFRS Frames on MSBs Responses
There are considerable differences between modular method of construction and
conventional steel building construction. For example, non-SFRS beam to column
connections in conventional braced-frame construction are usually achieved by using
shop or site welded/bolted clip angles or by connecting the webs of the beam and
columns and not the flanges. The use of clip angles or partial welding in conventional
construction results in the transfer of the forces at the ends of the beams through shear
action, while allowing for the partial rotation. Hence, the rotational stiffness of the
connections of gravity frames and as a result their lateral resistances are usually ignored
in structural response analysis of these structures. However, beam to column connections
in MSB structures are achieved in a controlled factory environment by direct welding of
the webs of the beams to the HSS columns.
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Based on experimental results the non-SFRS frames do provide some lateral resistance
when a compression force in the composite floor slab is connected to the beam by shear
studs (Liu, J., Astaneh-Asl, A., 2000). Nevertheless, the results obtained from a model
developed by Yun et al. (2002) show that although the lateral resistance from gravity
frames is considerable, most of the contribution is from the columns deformations
connected to the floors and not from the connections. This is because connections do not
provide enough resistant due to their significant loss of strength in the early stages of the
loading. Therefore, the rotational stiffness of the connections of the non-SFRS frames can
be ignored.
The beam to column connection properties is simulated by the rigid end connections
attached to plastic hinges in order to not to restrict the development of plasticity in the
columns provided in the model, and the entire behavior of non-SFRS frames is taken into
consideration in the 3D finite-element model.
Based on the results obtained from the IDA analysis of both the two-dimensional and
three-dimensional modified MSB models introduced in this chapter, (Figures 4.5, 4.7,
4.12, 4.13, tables 4.3, and 4.4) it can be concluded that the structural collapse capacity of
the 3D model is “lower” than of the 2D model. This is because the 2D model fails to
account for torsional effects and as results overestimates the structural capacity against
structural collapse. Given the design level ground motion intensity of Sa(T1, 5%) =
0.8g,Table 4.5 compares the capacities of the two models for CP limit-state, and Table
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4.6 shows the statistics of the maximum inter-story drift ratio demand at 2% probability
of exceedance in 50 years.
Table 4.5 Comparison of Collapse Capacities obtained from 2D and 3D analysis (Z direction).
Model IMc50%: Median Capacity for CP (g)
2D modified MSB-braced frame 2.9
3D modified MSB-braced structure 2.4
Table 4.6 Maximum Inter-story Drift Demand of the modified 4-Story MSB at the design
intensity level (Z direction).
Model θmax50%: Median Inter-story Drift Ratio (%)
2D modified MSB-braced frame 0.31
3D modified MSB-braced structure 0.50
4.5.2 Inter-Story Drifts in X and Z Directions
In Figures 4.14 and 4.15 the maximum drifts of the modified 3D model are plotted in
both Z and X directions. These drifts are obtained from the same ground motion record
(EL Centro Array #13 during the 1979 Imperial Valley earthquake) and at the same
intensity levels (Sa(T1, 5%) = 0.3, 2.0, and 3.0g) as were for the modified 2D model
(Figure 4.8) and are representative of the trend that is obtained from other ground
motions. It is observed that in the elastic range of response the height wise distribution of
the maximum inter-story drift varies from record to record for the both 2D and 3D MSB
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structures. It is also observed that the distribution of the maximum inter-story drifts along
the height of the structure is not uniform in a way that with increasing the ground motion
intensity level, larger drifts are concentrated at a specific story. The concentration of the
inelasticity is mainly at the first story level of the structure and the trend is maintained as
the intensity level of the ground motions increases. In terms of amplitude, the height-wise
distribution of the drifts varies for different ground motions, though following a similar
pattern in both 2D and 3D structures.
Figure 4.14 Height-wise distribution of peak inter-story drift ratio for the 3D 4-story MSB (Z direction) (a) Sa(T1, 5%) = 0.3 g, (b) Sa(T1,5%) = 2.0 g, (c) Sa(T1, 5%) = 3.0 g.
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Figure 4.15 Height-wise distribution of peak inter-story drift ratio for the 3D 4-story MSB (X direction) (a) Sa(T1, 5%) = 0.3 g, (b) Sa(T1,5%) = 2.0 g, (c) Sa(T1, 5%) = 3.0 g.
Figure 4.16 shows the time-history of roof displacement of the modified 2D versus 3D
model (Z direction) at the design intensity level for the above ground motion record. As it
can be seen both structures exhibit comparable displacements in terms of frequency and
amplitude and at this intensity almost no residual drift (permanent displacement) is
observed at the end of ground motion.
Figure 4.16 Roof displacement of the 2D versus 3D model (Z direction).
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4.5.3 Horizontal Connections and Diaphragm Action
By utilizing the diaphragm action of the floor and ceiling of the modules, lateral forces
induced by ground motion are transferred by tension, compression, and shear forces
through the connections located at the horizontal corners of the modules. As illustrated in
Figures 3.2 and 4.2 bolted steel plates or clip angles are used in the connections.
Apparently, in the modified 2D MSB model the axial forces (tension and compression) in
the horizontal connections can be easily obtained; however, shear forces acting on the
connections (out of plane forces which are perpendicular to the frame) cannot be
captured. Realistically, axial and shear forces act on the connection bolts simultaneously,
and base on the CSA S16-9 when designing bolted connections it is required to consider
the combined effects of both the axial and shear forces. In the 3D model all these forces
in two horizontal directions can be calculated as well as their corresponding deformations
and rotations. Figure 4.17 to 4.22 examine these parameters (in global coordinates) for
the modified three-dimensional model under the ground motion recorded at Hollister
Diff. Array during the 1989 Loma Prieta earthquake taking into account all the
independent movements of the modules. In these figures, time-history of internal forces,
rotations, and comparative connection end displacements of a selected horizontal corner
connections which are located at the first floor ceiling level and floor level of the second
floor are plotted (selected connections are highlighted in the Figure 4.9). Noting that in
the selected inter-section, there are two floor beam and two ceiling beam connection
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elements in each direction and the total connection force is the sum of the two floor and
two ceiling connection forces.
Figure 4.17 - 4.18 shows a) axial and b) shear forces for the floor and ceiling connections
in (i-j) and (m-n) directions, and Figure 4.19 shows the internal lateral moments for the
floor connections in both directions.
Figure 4.17 Internal forces in the connections. a) Axial and b) shear forces in ceiling and floor
connections in the (i-j) direction.
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Figure 4.18 Internal forces in the connections. a) Axial and b) shear forces in ceiling and floor
connections in the (m-n) direction.
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Figure 4.19 Internal lateral moment in the floor connections oriented in (i-j) and (m-n) directions.
Correspondingly, the horizontal displacements and rotations of the connections ends are
recorded in the following figures. Apparently, the difference between the two ends
displacements is the relative displacement of the two sides of the connections. (Figures
4.20 (a, b), and figure 4.21).
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Figure 4.20 End nodes displacements of connections. a) Displacement of end nodes of element (i-
j) in X direction, and b) displacement of end nodes of element (m-n) in Z direction.
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Figure 4.21 Connection (i-j) rotations.
Typically, for mid-rise MSB structures, shear resistance is provided by in plane bracing
within the modules, this is assisted by the module to module horizontal connections. For
the modular structures that the overall stability is provided by the modules themselves
(not by an external structure), it is normally assumed that the ties between the modules
withstand a minimum tying force equivalent to half the loaded weight of the module or a
minimum value of 30 kN (The encyclopedia for UK steel construction information). The
forces in the horizontal connections depend on the number of modules at each level and
by having more units use of braced corridor modules would be required for the system. In
that case, shear forces may be transferred through the continuous corridors rather than the
corner connections. Corridor zone can be used to provide in-plane bracing in long
buildings.
Table 4.7 shows the maximum values of corner connection elements internal forces (axial
and shear forces in X and Z direction) as well as moments (about Y axis) at the first floor
and for the ground motion recorded at Hollister Diff. Array during the 1989 Loma Prieta
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earthquake. Maximum floor connection ends global displacement and rotations are also
provided in Table 4.8.
Table 4.7 Maximum values of connection elements axial, shear, and moment forces in global
coordinates
Element Connection Axial
(KN)
Shear
(Lateral)
(KN)
Shear
(Vertical)
(KN)
Moment
(About Y axis)
(KN.m)
i-j Floor 64.96 9.64 5.34 18.68
Ceiling 55.84 8.66 3.58 19.24
m-n Floor 66.98 12.9 3.0 27.8
Ceiling 18.22 11.16 4.06 25.84
Table 4.8 Maximum values of connection elements nodal displacement and rotations
Element Node ΔX (m) ΔY (m) ΔZ (m) ΘY (Rad)
i 0.0604 0.0003 0.0449 0.0064
j 0.0673 0.005 0.0448 0.0064
n 0.0605 0.0002 0.0269 0.0062
4.5.4 Relative Motions of the Modular Units
As it is mentioned before, since the composite slabs of each of the modular units are
usually implemented separately in the remote factory, MSBs floor integrity and relative
displacements of the modules in an event of severe ground motion can be a matter of
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concern. Therefore, in this study, the interactions within the units at each floor and their
relative horizontal rotation and displacements have been examined. Figure 4.22 shows
the global rotations of the second floor module (module #7) under the Loma Prieta
ground motion record. This is representative of the behavior obsereved in the other
modules and the variation in the amount of rotation in all the other moduls is negligible
and it should be noted that this is the rotation in each unit not the entire structure.
Figure 4.22 Module #7 rotations under the Loma Prieta ground motion.
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Chapter 5
5. Summary and Conclusions
5.1 Summary
In this study, theoretical basis and implementation of Incremental Dynamic Analysis
(IDA) were explained. Some of the properties of the IDA curves and their effectiveness
in revealing structural response behavior such as hardening, softening, and structural
resurrection were discussed. Algorithms were presented to reduce the number of
nonlinear runs for each record. Limit-states were defined on each IDA curve using
different rules and techniques. Finally, a method to summarize the IDA curves into the
16%, 50%, and 84% IDA curves using cross-sectional fractiles was introduced.
This was followed by IDA application on investigating some aspects of the seismic
performance of Modular Steel Buildings. Two 4-story MSBs with two different
structural configurations were chosen to carry out a detailed seismic evaluation. In the
first model which was introduced in a study by Annan et al. (2009 a), some of the
unrealistic structural and detailing assumptions were challenged. To have an assessment
of seismic demand and capacity of MSBs, in the second model, a more realistic 4-story
MSB structure was proposed. Using OpenSees, IDA analysis of the proposed structure
was carried out both in two and three dimensions. The National Building Code of Canada
(NBCC 2010) was used to seismically design the buildings for a site located in
Vancouver. The median capacities in terms of spectral acceleration, Sa(T1, 5%), for all
the structures and for each limit-state were estimated. The results of the modified 2D and
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3D models were compared and the responses of some critical elements in the system
were assessed. The interactions between modules as well as unique diaphragm action of
each floor module were captured. Moreover, the shear and moment forces in the
horizontal connections within the modules and the displacement and rotations caused by
them were obtained in this study.
5.2 Conclusions
As a fast evolving, new method of construction, knowledge on the behavior of MSB
structures is limited at this time and since it is a relatively new technique, there is no
record of MSB performance under past earthquakes (Annan et al. ,2009 a). As a useful
tool to quantify the seismic performance of structures, IDA was used to estimate the
severity of damage a MSB might suffer. An understanding of the distribution of
inelasticity along the height of the MSB structure and the effect of ground motion
intensities on maximum drift demand of the building were developed. The capacities at
the Collapse Prevention level with their corresponding probabilities that this performance
level may not be exceeded were estimated. Other conclusions drawn from this study are
listed as follows:
1. The effectiveness of IDA in providing useful intuition into the seismic behavior of
structure was confirmed.
2. Results from the IDA curves produced from both regular and modified MSB models
showed that the choice of Sa(T1, 5%) as the IM in the elastic rage of response is more
effective than PGA. In that range for a given IM the dispersion of DM is smaller and
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therefore smaller sample of records and nonlinear runs are necessary to estimate the
median DM.
3. It was also observed that for these structures and for the collapse prediction, PGA is
more preferable to Sa(T1, 5%), as the dispersion of IM values associated with the flat-
line is smaller in the former.
4. In comparison with the ductile moment resisting frames, it was found that the IDA
curves obtained from MSB-braced frame structures are more complex. In general, stiffer
braced frames experience significant changes in their dynamic properties after brace
buckling. Due to the sensitivity of inter-story drifts to the brace buckling, such frames are
likely to produce scattered IDA points. Moreover, limited alternative load paths after
brace buckling and concentration of inelasticity in one floor level over the height of the
building may further result in a wavier shape of the IDA curves.
5. In all the selected MSB structures the first-mode response was dominant and there
were less sensitivity to higher modes.
6. Within the entire range of structural response of the model presented by Annan et al.
(2009 a) and modified 2D and 3D MSB structures it was observed that floor-to-floor
inter-story drift can satisfactorily represent inter-story drift demand, hence explicitly
considering the ceiling beam contribution is not required (the 0.15 m clear space was
considered in all the above mentioned models).
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7. In the elastic range of response the distribution of the inter-story drift demand along
the height of the structure varies from record to record but generally the upper-story
levels experience maximum drift demands. Due to the inelastic behavior of braces and
limited redistribution of the internal forces within the story levels, in the inelastic range
of response, concentration of the inelasticity is found to be mainly in the first story level.
Although the distribution of the drifts varies in terms of amplitude from one ground
motion record to the next, it follows a similar pattern for all the records. In the inelastic
range of response maximum inter-story drift is more affected by the presence of the
ceiling beams, especially at the first floor level, but it is still insignificant.
8. Based on the above observations and the fact that in practice when connecting the
modules on top of each other, vertical connections can be achieved by anchoring the
ceiling beams to the floor beams directly, (instead of connecting the columns solely), the
assumption of having pinned vertical connections at each level (independent upper and
lower rotation) that was introduced by Annan et al. (2009 b) was eliminated from the
modified 2D and 3D analytical models.
9. In terms of component size and dimensions that are commonly used in modular
construction and in comparison with previous studies by Annan et al. (2009 a, b), more
realistic column section and module sizes were adopted in this study. This resulted in
more accurate estimation of the dynamic response of a real MSB structure.
107
10. Based on NBCC (2010) drift limits, the predicted drift demands for the median
ground motion records and at the design intensity level were satisfactory for both 2D
models and the 3D model.
11. The median capacities in terms of spectral acceleration, Sa(T1, 5%), for both
modified 2D and 3D models and for the Collapse Prevention (CP) limit state were found
to be 1.66 and 1.37 times the median capacities associated to the NBCC drift limit of
2.0% , respectively.
12. Comparing the modified 2D and 3D model, the structural capacity against incipient
collapse of the 3D model was found to be lower than that of the 2D model. This is
because the 2D model fails to account for the torsional response, hence overestimates the
structural capacity. In a 3D model torsional response of the structure is explicitly
included in the analysis and the model provides a three-dimensional representation of the
structural stiffness, mass, and strength properties and includes the distribution of the
seismic effects in all the elements of SFRS and non-SFRS frames.
13. Use of separated diaphragms for each module and not assigning all the end nodes of
the floor and ceiling beams (horizontal element) to one diaphragm (hence nodes in the
diaphragms are not computed as null) result in a more realistic representation of the
braced frame lateral stiffness and, thereby better predictions of the structure drifts and
periods.
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14. In the 3D model, assuming the horizontal connections at the modules' inter-
connections with a strength of 1.3 times of the adjacent beams at the intersection, at the
design level ground motion intensity, all the connections satisfactorily remained in the
elastic range of response The forces in the horizontal connections depend on the number
of modules at each level and by having more units use of braced corridor modules would
be required for the system. In that case, shear forces may be transferred through the
continuous corridors rather than the corner connections.
15. Considering separated composite slabs for each unit, at the design intensity level and
for the selected number of modules, the relative displacements and rotation of the
modules due to the horizontal forces were insignificant.
5.3 Future Studies
For the future studies, the dynamic behavior of high-rise modular buildings needs to be
studied. For taller structures separate braced core structure using hot-rolled steel members
located around the elevator and stair areas are more suitable. To have high-rise modular
steel buildings a reinforced concrete or steel core can be used to stabilize the structure
against lateral forces. Moreover, there are a few issues related to the modules
interconnections and the way that forces are transferred. To have a firm grasp of the
modular building behavior, many complex factors should be considered. Specifically, for
taller buildings, geometric errors that are embedded in manufacturing and installation
procedure have significant contribution to the behavior of the building during ground
motions. Difficulties in installing the modules on top of each other in a way that one
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module is precisely positioned on another to have minimum out of alignment possible
would be a problem in high-rise structures. Additional forces caused by initial
eccentricities, and cumulative positioning errors can be a matter of concern. Therefore,
for taller buildings, question of compression resistance and overall stability require a
clear understanding of the behavior of the column connections and of the interaction
between the modules.
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Chapter 6
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Appendix A
A.1 OpenSees Code (Modeling and Analysis)
A.1.1 Modified 2D MSB
############################################################# # Amirahmad Fathieh, University Of Toronto # 2D Analysis of Modular steel building ############################################################# wipe; # clear memory of all past model definitions wipeAnalysis; #change to reach convergance variable TolDynamic 1.e-6; # Convergence Test:tolerance #define intensity step (Simple Stepping): 0.1g,0.2g,0.3g,... for { set i 1 } { $i < 70 } { incr i 1 } { set s [expr 1*$i] set gmf [expr $s/10.] puts "gmf=[expr $gmf]" puts ****************** file mkdir x[expr $gmf] ############################################################### model BasicBuilder -ndm 2 -ndf 3; # Define the model builder, ndm=#dimension, ndf=#dofs set dataDir Data; # set up name of data directory source Wsection.tcl; # procedure to define fiber W section source HSSsection.tcl; # procedure to define fiber HSS section source DisplayModel2D.tcl; # procedure for displaying a 2D perspective of model source DisplayPlane.tcl; # procedure for displaying a plane in the model source rotSect2DModIKModel.tcl; # procedure for defining bilinear plastic hinge section #units KN & m # define units # Elements are fully rigid when in thuch with gusset plates # Rigid liks of the braces are defined with rigid elastic elements. Beam and column reigid elemnts are defined using rigid offsets inside of geometric transformation. ############################################################### # Define Building Geometry, Nodes, and Constraints ############################################################### # define GEOMETRY ------------------------------------------------------------- # define structure-geometry paramters set LCol 3.5; # Floor height
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set spc 0.15; # Clear space between the ceiling and floor beams set db 0.15; # beam depth set LCol1 [expr $LCol-($spc + 2*$db)]; # column1 height set LCol2 [expr $spc + 2*$db]; # column2 height set LC1 3.05; set LBeam 3.5; # beam length set Lbr1 [expr (sqrt($LCol1)*($LCol1) + ($LBeam)*($LBeam))]; # brace length set dx 0.35; # columns center to center distance at each intersection set LBeam 3.5; # beam length set NStory 4; # number of stories above ground level -------------- you can change this. set NBay 4; # number of bays ------------------------------you can change this. #### Joint offset set jOffbraceX1 [expr $LBeam*0.1]; # joint offset for all braces in X direction at floor 1 set jOffbraceY1 [expr $LCol*0.1]; # joint offset for all braces in Y direction at floor 1 set jOffbraceX2 [expr $LBeam*0.1]; # joint offset for all braces in X direction at floors 2 to 4 set jOffbraceY2 [expr $LCol1*0.1]; # joint offset for all braces in Y direction at floors 2 to 4 set jOffY1 [expr $LCol*0.5]; # middle nodes set jOffX1 [expr $LBeam*0.5]; # middle nodes set jOffY2 [expr $LCol1*0.5]; # middle nodes set jOffX2 [expr $LBeam*0.5]; # middle nodes set joff [expr $db/2]; # beams and columns rigid links set joffs [expr $LCol2*0.05]; ####################################################### # Define NODAL COORDINATES ####################################################### # ORIGINAL # only base and first (level 1 and level 2) floor ceiling nodes: ### ceiling nodes (3) for {set level 1} {$level <=[expr $NStory]} {incr level 1} { set Y [expr ($level-1)*($LCol)-$LCol2]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set X [expr ($pier-1)*$LBeam]; set nodeID [expr 3000+$level*10+$pier] node $nodeID $X $Y; # actually define node } ## floor nodes (7) for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { set Y [expr ($level-1)*$LCol]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set X [expr ($pier-1)*$LBeam]; set nodeID [expr 7000+$level*10+$pier] node $nodeID $X $Y; # actually define node }
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} ################################################################### #Define EXTERA NODAL COORDINATES FOR COLUMNS AT EACH INTERSECTION ################################################################### # EXTERA (2) ### ceiling nodes (32) for {set level 1} {$level <=[expr $NStory]} {incr level 1} { set Y [expr ($level-1)*($LCol)-$LCol2]; for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} { set X [expr ($pier-1)*$LBeam + $dx]; set nodeID [expr 32000+$level*10+$pier] node $nodeID $X $Y; # actually define node } } ## floor nodes (72) for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { set Y [expr ($level-1)*$LCol]; for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} { set X [expr ($pier-1)*$LBeam + $dx]; set nodeID [expr 72000+$level*10+$pier] node $nodeID $X $Y; # actually define node } } ############################################################ # Define extra nodes for rigid links in the braces: ############################################################ #### calculate locations of beam/column intersections: set X1 0.; set X2 [expr $X1 + $LBeam]; set X3 [expr $X2 + $LBeam]; set X4 [expr $X3 + $LBeam]; set X5 [expr $X4 + $LBeam]; set Y1 -0.45 ; set Y2 [expr $Y1 + $LCol]; set Y3 [expr $Y2 + $LCol2]; set Y4 [expr $Y3 + $LCol1]; set Y5 [expr $Y4 + $LCol2]; set Y6 [expr $Y5 + $LCol1]; set Y7 [expr $Y6 + $LCol2]; set Y8 [expr $Y7 + $LCol]; #### define extra nodes for rigid links in the braces:
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# nodeID convention: "4axb" where a= offset is right(1) or left(2) of the intersection x = level # and b = column axis number # right: # level 1: node 4111 [expr $X1+$jOffbraceX1] [expr $Y1+$jOffbraceY1]; node 4114 [expr $X4+$jOffbraceX1 + $dx] [exp $Y1+$jOffbraceY1]; # level 2 : node 4121 [expr $X1+$jOffbraceX1] [expr $Y2-$jOffbraceY1]; node 4124 [expr $X4+$jOffbraceX1 + $dx] [expr $Y2-$jOffbraceY1]; # level 3: node 4131 [expr $X1+$jOffbraceX2] [expr $Y3+$jOffbraceY2]; node 4134 [expr $X4+$jOffbraceX2 + $dx] [expr $Y3+$jOffbraceY2]; # level 4: node 4141 [expr $X1+$jOffbraceX2] [expr $Y4-$jOffbraceY2]; node 4144 [expr $X4+$jOffbraceX2 + $dx] [expr $Y4-$jOffbraceY2]; # level 5: node 4151 [expr $X1+$jOffbraceX2] [expr $Y5+$jOffbraceY2]; node 4154 [expr $X4+$jOffbraceX2 + $dx] [expr $Y5+$jOffbraceY2]; # level 6: node 4161 [expr $X1+$jOffbraceX2] [expr $Y6-$jOffbraceY2]; node 4164 [expr $X4+$jOffbraceX2 + $dx] [expr $Y6-$jOffbraceY2]; # level 7: node 4171 [expr $X1+$jOffbraceX2] [expr $Y7+$jOffbraceY2]; node 4174 [expr $X4+$jOffbraceX2 + $dx] [expr $Y7+$jOffbraceY2]; # level 8: node 4181 [expr $X1+$jOffbraceX2] [expr $Y8-$jOffbraceY2]; node 4184 [expr $X4+$jOffbraceX2 + $dx] [expr $Y8-$jOffbraceY2]; # left: # level 1: node 4212 [expr $X2-$jOffbraceX1] [expr $Y1+$jOffbraceY1]; node 4215 [expr $X5-$jOffbraceX1] [expr $Y1+$jOffbraceY1]; # level 2: node 4222 [expr $X2-$jOffbraceX1] [expr $Y2-$jOffbraceY1]; node 4225 [expr $X5-$jOffbraceX1] [expr $Y2-$jOffbraceY1];
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# level 3: node 4232 [expr $X2-$jOffbraceX2] [expr $Y3+$jOffbraceY2]; node 4235 [expr $X5-$jOffbraceX2] [expr $Y3+$jOffbraceY2]; # level 4: node 4242 [expr $X2-$jOffbraceX2] [expr $Y4-$jOffbraceY2]; node 4245 [expr $X5-$jOffbraceX2] [expr $Y4-$jOffbraceY2]; # level 5:; node 4252 [expr $X2-$jOffbraceX2] [expr $Y5+$jOffbraceY2]; node 4255 [expr $X5-$jOffbraceX2] [expr $Y5+$jOffbraceY2]; # level 6: node 4262 [expr $X2-$jOffbraceX2] [expr $Y6-$jOffbraceY2]; node 4265 [expr $X5-$jOffbraceX2] [expr $Y6-$jOffbraceY2]; # level 7: node 4272 [expr $X2-$jOffbraceX1] [expr $Y7+$jOffbraceY1]; node 4275 [expr $X5-$jOffbraceX1] [expr $Y7+$jOffbraceY1]; # level 8: node 4282 [expr $X2-$jOffbraceX1] [expr $Y8-$jOffbraceY1]; node 4285 [expr $X5-$jOffbraceX1] [expr $Y8-$jOffbraceY1]; #### Center Nodes for the Braces #### # 46axb is for the center nodes of braces going upward and 49axb is for downwards # right # level 1 : node 46111 [expr $X1+$jOffX1] [expr $Y1+$jOffY1]; node 46114 [expr $X4+$jOffX1 + $dx/2] [expr $Y1+$jOffY1]; # level 2 : node 49121 [expr $X1+$jOffX1] [expr $Y2-$jOffY1]; node 49124 [expr $X4+$jOffX1 + $dx/2] [expr $Y2-$jOffY1]; # level 3: node 46131 [expr $X1+$jOffX2] [expr $Y3+$jOffY2]; node 46134 [expr $X4+$jOffX2 + $dx/2] [expr $Y3+$jOffY2]; # level 4: node 49141 [expr $X1+$jOffX2] [expr $Y4-$jOffY2]; node 49144 [expr $X4+$jOffX2 + $dx/2] [expr $Y4-$jOffY2];
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# level 5: node 46151 [expr $X1+$jOffX2] [expr $Y5+$jOffY2]; node 46154 [expr $X4+$jOffX2 + $dx/2] [expr $Y5+$jOffY2]; # level 6: node 49161 [expr $X1+$jOffX2] [expr $Y6-$jOffY2]; node 49164 [expr $X4+$jOffX2 + $dx/2] [expr $Y6-$jOffY2]; # level 7: node 46171 [expr $X1+$jOffX1] [expr $Y7+$jOffY1]; node 46174 [expr $X4+$jOffX1 + $dx/2] [expr $Y7+$jOffY1]; # level 8: node 49181 [expr $X1+$jOffX1] [expr $Y8-$jOffY1]; node 49184 [expr $X4+$jOffX1 + $dx/2] [expr $Y8-$jOffY1]; ##################################################### # determine support nodes where ground motions are input, for multiple-support excitation set iSupportNode "" set level 1 for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set nodeID [expr $level*10+$pier] lappend iSupportNode $nodeID } # BOUNDARY CONDITIONS #fixY -0.45 1 1 1; # pin all Y=0.0 nodes fix 3011 1 1 1; fix 3012 1 1 1; fix 32012 1 1 1; fix 3013 1 1 1; fix 32013 1 1 1; fix 3014 1 1 1; fix 32014 1 1 1; fix 3015 1 1 1; fix 3016 1 1 1; # calculated MODEL PARAMETERS, particular to this model puts "Number of Stories: $NStory Number of bays: $NBay" # Set up parameters that are particular to the model for displacement control set IDctrlNode [expr (700+$NStory+1)*10+1]; # node where displacement is read for displacement control set IDctrlDOF 1; # degree of freedom of displacement read for displacement control set LBuilding [expr $NStory*$LCol]; # total building height
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##################################################### # Define Materials and Sections ##################################################### # Define ELEMENTS & SECTIONS ------------------------------------------------------------- set ColSecTag1 11; # assign a tag number to the column section tag - first floor set ColSecTag23 12; # assign a tag number to the column section tag - second and third floors set ColSecTag4 13; # assign a tag number to the column section tag - fourth floor set ColMatTagFlex 2; # assign a tag number to the column flexural behavior set ColMatTagAxial 3; # assign a tag number to the column axial behavior set BeamSecTag 21; # assign a tag number to the beam section tag set BeamMatTagFlex 5; # assign a tag number to the beam flexural behavior set BeamMatTagAxial 6; # assign a tag number to the beam axial behavior set BrcSecTag12 31; # assign a tag number to the brace section tag - first & second floors set BrcSecTag34 32; # assign a tag number to the brace section tag - third & fourth floors set BrcMatTagFlex 8; # assign a tag number to the brace flexural behavior set BrcMatTagAxial 9; # assign a tag number to the brace axial behavior # define MATERIAL properties ---------------------------------------- set Fy 350.0e3; # yield strenght(KN/m2) set Mybr 16.1236; # yield moment (KN.m) set Es 2.0e8; # Steel Young's Modulus <==??same thing??==>(initial elastic tangent) KN/m2 set nu 0.3; # poisson ratio ????? set Gs 77000000.0; # Torsional stiffness Modulus KN/m2 set btemp 0.01; # strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent) (assumed ? check) ############# steel02 parameters######## # steel02 parameters set R0 18; # control the transition from elastic to plastic branches -- make more like the rest. set cR1 0.925; # control the transition from elastic to plastic branches set cR2 0.15; # control the transition from elastic to plastic branches set Bs 0.01; # strain-hardening ratio uniaxialMaterial Steel02 $BeamMatTagFlex $Fy $Es $Bs $R0 $cR1 $cR2; uniaxialMaterial Steel02 $BeamMatTagAxial $Fy $Es $Bs $R0 $cR1 $cR2; uniaxialMaterial Steel02 $ColMatTagFlex $Fy $Es $Bs $R0 $cR1 $cR2;
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uniaxialMaterial Steel02 $ColMatTagAxial $Fy $Es $Bs $R0 $cR1 $cR2; uniaxialMaterial Steel02 $BrcMatTagAxial $Fy $Es $Bs $R0 $cR1 $cR2; # ELEMENT properties ------------------------------------------------- # Structural-Steel W-section properties ########### beam sections: W100x19 #### command: WSection secID matID d bf tf tw nfdw nftw nfbf nftf ###set d 106.0e-3; # depth ###set bf 103.0e-3; # flange width ###set tf 8.8e-3; # flange thickness ###set tw 7.1e-3; # web thickness ###set nfdw 16; # number of fibers along dw ###set nftw 2; # number of fibers along tw ###set nfbf 16; # number of fibers along bf ###set nftf 4; # number of fibers along tf ###Wsection $BeamSecTag $BeamMatTagAxial $d $bf $tf $tw $nfdw $nftw $nfbf $nftf ######### define sections for braces: # first & second floors: # command: HSSsection secID matID d t nfdy nfty nfdz nftz # HSS 76x76x6.4 HSSsection $BrcSecTag12 $BrcMatTagAxial 0.076 0.00635 12 2 12 2 # third & fourth floors: # command: HSSsection secID matID d t nfdy nfty nfdz nftz # HSS 51x51x4.8 HSSsection $BrcSecTag34 $BrcMatTagAxial 0.051 0.00478 12 2 12 2 ########################################### # Distributed Plastic Hinges ########################################### ## Distributed Plastic Hinges for beams and first floor columns#### # First floor columns and all the beams HSS127x127x4.8 & W100x19: set Mycol 28.22; # yield moment (KN.m) set Acol 2.06e-3; set Icol 5.12e-6; set Mybeam 31.5; # yield moment at plastic hinge location set Abeam 2.48e-3;
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set Ibeam 4.77e-6; # define plastic hinge lengths set Lp_c1 [expr 0.004*$LCol1]; # length of plastic hinge for floor columns set Lp_c2 [expr 0.004*$LCol2]; # length of plastic hinge for space columns set Lp_b2 [expr 0.004*$LBeam]; # length of plastic hinge for beams (corridor beam length is used) # determine stiffness modifications so that the strain hardening of the plastic hinge region captures the actual frame member's strain hardening # Reference: Ibarra, L. F., and Krawinkler, H. (2005). "Global collapse of frame structures under seismic excitations," Technical Report 152, # The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. set n_c1 [expr $LCol1/$Lp_c1]; # rotational stiffness ratio: (floor column plastic hinge region) / (actual floor column) set n_c2 [expr $LCol2/$Lp_c2]; # rotational stiffness ratio: (space column plastic hinge region) / (actual space column) set n_b2 [expr $LBeam/$Lp_b2]; # rotational stiffness ratio: (beam plastic hinge region) / (actual beam) # calculate rotational stiffness for plastic hinges set Ks_col_1 [expr 6.0*$Es*$Icol/$Lp_c1]; # rotational stiffness of floor column hinges set Ks_col_2 [expr 6.0*$Es*$Icol/$Lp_c2]; # rotational stiffness of space column hinges set Ks_beam_2 [expr 6.0*$Es*$Ibeam/$Lp_b2]; # rotational stiffness of beam hinges set Kmem_col_1 [expr 6.0*$Es*$Icol/$LCol1]; # rotational stiffness of floor columns set Kmem_col_2 [expr 6.0*$Es*$Icol/$LCol2]; # rotational stiffness of space columns set Kmem_beam_2 [expr 6.0*$Es*$Ibeam/$LBeam]; # rotational stiffness of beams ########################################################## # Define Rotational Springs for Plastic Hinges ########################################################## # define rotational spring properties and create spring elements using "rotSect2DModIKModel" procedure # rotSect2DModIKModel creates a section with an elastic axial and bilinear flexural response based on Modified Ibarra Krawinkler Deterioration Model # references provided in rotSect2DModIKModel.tcl # input values for Story 1 column springs set McMy 1.05; # ratio of capping moment to yield moment, Mc / My set LS 1000.0; # basic strength deterioration (a very large # = no cyclic deterioration)
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set LK 1000.0; # unloading stiffness deterioration (a very large # = no cyclic deterioration) set LA 1000.0; # accelerated reloading stiffness deterioration (a very large # = no cyclic deterioration) set LD 1000.0; # post-capping strength deterioration (a very large # = no deterioration) set cS 1.0; # exponent for basic strength deterioration (c = 1.0 for no deterioration) set cK 1.0; # exponent for unloading stiffness deterioration (c = 1.0 for no deterioration) set cA 1.0; # exponent for accelerated reloading stiffness deterioration (c = 1.0 for no deterioration) set cD 1.0; # exponent for post-capping strength deterioration (c = 1.0 for no deterioration) set th_pP 0.025; # plastic rot capacity for pos loading set th_pN 0.025; # plastic rot capacity for neg loading set th_pcP 0.3; # post-capping rot capacity for pos loading set th_pcN 0.3; # post-capping rot capacity for neg loading set ResP 0.4; # residual strength ratio for pos loading set ResN 0.4; # residual strength ratio for neg loading set th_uP 0.4; # ultimate rot capacity for pos loading set th_uN 0.4; # ultimate rot capacity for neg loading set DP 1.0; # rate of cyclic deterioration for pos loading set DN 1.0; # rate of cyclic deterioration for neg loading set a_mem [expr ($Mycol*($McMy-1.0)) / ($Kmem_col_1*$th_pP)]; # strain hardening ratio of member set bddm [expr ($a_mem)/(1.0+$n_c1*(1.0-$a_mem))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: Eqn B.5 is incorrect) # define column plastic hinge sections # command: rotSect2DModIKModel id ndR ndC K asPos asNeg MyPos MyNeg LS LK LA LD cS cK cA cD th_p+ th_p- th_pc+ th_pc- Res+ Res- th_u+ th_u- D+ D- set sec_c1 101; # section ID for floor column section rotSect2DModIKModel $sec_c1 $Es $Acol $Ks_col_1 $bddm $bddm $Mycol [expr -$Mycol] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # recompute strain hardening since space column is not the same height as floor column set a_mem [expr ($Mycol*($McMy-1.0)) / ($Kmem_col_2*$th_pP)]; # strain hardening ratio of member set bddm [expr ($a_mem)/(1.0+$n_c2*(1.0-$a_mem))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5)
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set sec_c2 102; # section ID for space column section rotSect2DModIKModel $sec_c2 $Es $Acol $Ks_col_2 $bddm $bddm $Mycol [expr -$Mycol] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # define beam plastic hinge sections # redefine the rotations since they are not the same set th_pP 0.02; set th_pN 0.02; set th_pcP 0.16; set th_pcN 0.16; set a_mem [expr ($Mybeam*($McMy-1.0)) / ($Kmem_beam_2*$th_pP)]; # strain hardening ratio of member set bddm [expr ($a_mem)/(1.0+$n_b2*(1.0-$a_mem))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5) #beam sections set sec_b2 202; # section ID for beams rotSect2DModIKModel $sec_b2 $Es $Abeam $Ks_beam_2 $bddm $bddm $Mybeam [expr -$Mybeam] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; ########################################################### ### Distributed Plastic Hinges for 2&3rd floor columns#### # 2nd & 3rd floor columns HSS102x102x6.4: set Mycol23 21.7; # yield moment (KN.m) set Acol23 2.11e-3; set Icol23 3.16e-6; # define plastic hinge lengths set Lp_c1 [expr 0.004*$LCol1]; # length of plastic hinge for floor columns set Lp_c2 [expr 0.004*$LCol2]; # length of plastic hinge for space columns # determine stiffness modifications so that the strain hardening of the plastic hinge region captures the actual frame member's strain hardening # Reference: Ibarra, L. F., and Krawinkler, H. (2005). "Global collapse of frame structures under seismic excitations," Technical Report 152, # The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. set n_c1 [expr $LCol1/$Lp_c1]; # rotational stiffness ratio: (floor column plastic hinge region) / (actual floor column)
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set n_c2 [expr $LCol2/$Lp_c2]; # rotational stiffness ratio: (space column plastic hinge region) / (actual space column) # calculate rotational stiffness for plastic hinges set Ks_col23_1 [expr 6.0*$Es*$Icol23/$Lp_c1]; # rotational stiffness of floor column hinges set Ks_col23_2 [expr 6.0*$Es*$Icol23/$Lp_c2]; # rotational stiffness of space column hinges set Kmem_col23_1 [expr 6.0*$Es*$Icol23/$LCol1]; # rotational stiffness of floor columns set Kmem_col23_2 [expr 6.0*$Es*$Icol23/$LCol2]; # rotational stiffness of space columns ####Define Rotational Springs for Plastic Hinges##### set a_mem23 [expr ($Mycol23*($McMy-1.0)) / ($Kmem_col23_1*$th_pP)]; # strain hardening ratio of member set bddm23 [expr ($a_mem23)/(1.0+$n_c1*(1.0-$a_mem23))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: Eqn B.5 is incorrect) # define column plastic hinge sections # command: rotSect2DModIKModel id ndR ndC K asPos asNeg MyPos MyNeg LS LK LA LD cS cK cA cD th_p+ th_p- th_pc+ th_pc- Res+ Res- th_u+ th_u- D+ D- set sec23_c1 23101; # section ID for floor column section rotSect2DModIKModel $sec23_c1 $Es $Acol23 $Ks_col23_1 $bddm23 $bddm23 $Mycol23 [expr -$Mycol23] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # recompute strain hardening since space column is not the same height as floor column set a_mem23 [expr ($Mycol23*($McMy-1.0)) / ($Kmem_col23_2*$th_pP)]; # strain hardening ratio of member set bddm23 [expr ($a_mem23)/(1.0+$n_c2*(1.0-$a_mem23))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5) set sec23_c2 23102; # section ID for space column section rotSect2DModIKModel $sec23_c2 $Es $Acol23 $Ks_col23_2 $bddm23 $bddm23 $Mycol23 [expr -$Mycol23] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN;
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####################################################### ### Distributed Plastic Hinges for 4th floor columns### # 4th floor columns HSS76x76x4.8: set Mycol4 9.2; # yield moment (KN.m) set Acol4 1.19e-3; set Icol4 1.0e-6; # define plastic hinge lengths set Lp_c1 [expr 0.004*$LCol1]; # length of plastic hinge for floor columns set Lp_c2 [expr 0.004*$LCol2]; # length of plastic hinge for space columns # determine stiffness modifications so that the strain hardening of the plastic hinge region captures the actual frame member's strain hardening # Reference: Ibarra, L. F., and Krawinkler, H. (2005). "Global collapse of frame structures under seismic excitations," Technical Report 152, # The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. set n_c1 [expr $LCol1/$Lp_c1]; # rotational stiffness ratio: (floor column plastic hinge region) / (actual floor column) set n_c2 [expr $LCol2/$Lp_c2]; # rotational stiffness ratio: (space column plastic hinge region) / (actual space column) # calculate rotational stiffness for plastic hinges set Ks_col4_1 [expr 6.0*$Es*$Icol4/$Lp_c1]; # rotational stiffness of floor column hinges set Ks_col4_2 [expr 6.0*$Es*$Icol4/$Lp_c2]; # rotational stiffness of space column hinges set Kmem_col4_1 [expr 6.0*$Es*$Icol4/$LCol1]; # rotational stiffness of floor columns set Kmem_col4_2 [expr 6.0*$Es*$Icol4/$LCol2]; # rotational stiffness of space columns ########## Define Rotational Springs for Plastic Hinges ####### set a_mem4 [expr ($Mycol4*($McMy-1.0)) / ($Kmem_col4_1*$th_pP)]; # strain hardening ratio of member
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set bddm4 [expr ($a_mem4)/(1.0+$n_c1*(1.0-$a_mem4))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: Eqn B.5 is incorrect) # define column plastic hinge sections # command: rotSect2DModIKModel id ndR ndC K asPos asNeg MyPos MyNeg LS LK LA LD cS cK cA cD th_p+ th_p- th_pc+ th_pc- Res+ Res- th_u+ th_u- D+ D- set sec4_c1 4101; # section ID for floor column section rotSect2DModIKModel $sec4_c1 $Es $Acol4 $Ks_col4_1 $bddm4 $bddm4 $Mycol4 [expr -$Mycol4] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # recompute strain hardening since space column is not the same height as floor column set a_mem4 [expr ($Mycol4*($McMy-1.0)) / ($Kmem_col4_2*$th_pP)]; # strain hardening ratio of member set bddm4 [expr ($a_mem4)/(1.0+$n_c2*(1.0-$a_mem4))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5) set sec4_c2 4102; # section ID for space column section rotSect2DModIKModel $sec4_c2 $Es $Acol4 $Ks_col4_2 $bddm4 $bddm4 $Mycol4 [expr -$Mycol4] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; ################################################## # Define Geometric Transformation ################################################## # Define geometric transformations of elements: # separate columns and beams and braces, in case of P-Delta analysis for columns set IDColTransf 1; # all columns set IDBeamTransf 2; # all beams set IDBraceTransf 3; # all braces set IDRGlinkTransf 4; # all rigid links set IDColTransfs 5; geomTransf PDelta 1 -jntOffset 0.0 $joff 0.0 -$joff; geomTransf PDelta 2 -jntOffset $joff 0.0 -$joff 0.0; geomTransf PDelta 5 -jntOffset 0.0 $joffs 0.0 -$joffs; geomTransf Corotational 3 geomTransf Linear 4 ##################################################### # Define Elements #####################################################
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# ORIGINAL # Define Beam-Column Elements set np 5; # number of Gauss integration points for nonlinear curvature distribution-- np=2 for linear distribution ok # columns # 1st floor columns set N0col 1000; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 3000+$level*10+$pier] set nodeJ [expr 3000+($level+1)*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $IDColTransf; # columns } } # 1st spacing columns set N0col 2000; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 3000+$level*10+$pier] set nodeJ [expr 7000+$level*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $IDColTransf; # columns } } # 2&3rd floor columns set N0col 3000; # column element numbers set level 0 for {set level 2} {$level <=($NStory-1)} {incr level 1} for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 7000+$level*10+$pier] set nodeJ [expr 3000+($level+1)*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $IDColTransf; # columns
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} } # spacing columns set N0col 2000; # column element numbers set level 0 for {set level 3} {$level <=($NStory)} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 3000+$level*10+$pier] set nodeJ [expr 7000+$level*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $IDColTransf; # columns } # last floor columns set N0col 4000; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 7000+$level*10+$pier] set nodeJ [expr 7000+($level+1)*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec4_c2 $Lp_c2 $sec4_c2 $Lp_c2 $Es $Acol4 $Icol4 $IDColTransf; # columns } } ################################################## # Define Elements (2) ################################################## # Extera # columns # 1st floor columns set N0col 11000; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 32000+$level*10+$pier] set nodeJ [expr 32000+($level+1)*10+$pier]
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element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $IDColTransf; # columns # 1st spacing columns set N0col 21000; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 32000+$level*10+$pier] set nodeJ [expr 72000+$level*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $IDColTransf; # columns } # 2&3rd floor columns set N0col 31000; # column element numbers set level 0 for {set level 2} {$level <=($NStory-1)} {incr level 1} { for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 72000+$level*10+$pier] set nodeJ [expr 32000+($level+1)*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $IDColTransf; # columns } # spacing columns set N0col 21000; # column element numbers set level 0 for {set level 3} {$level <=($NStory)} {incr level 1} { for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} { set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 32000+$level*10+$pier] set nodeJ [expr 72000+$level*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $IDColTransf; # columns } } # last floor columns set N0col 41000; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 2} {$pier <= [expr $NBay]} {incr pier 1} {
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set elemID [expr $N0col + $level*10 +$pier] set nodeI [expr 72000+$level*10+$pier] set nodeJ [expr 72000+($level+1)*10+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec4_c2 $Lp_c2 $sec4_c2 $Lp_c2 $Es $Acol4 $Icol4 $IDColTransf; # columns } } ##################################################### # Beam elements ##################################################### # Bay 1 # ceiling beams set N0beam 5000; # beam element numbers (there is no limitation for the number of stories, but change the node names for more than 9 bays) set M0 0 for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam + $level*10 +$bay] set nodeI [expr $M0 +3000+$level*10+ $bay] set nodeJ [expr $M0 +3000+$level*10+ $bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $IDBeamTransf; # beams } } # Bay 2-4 # ceiling beams set N0beam 5000; # beam element numbers (there is no limitation for the number of stories, but change the node names for more than 9 bays) set M0 0 for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0beam + $level*10 +$bay] set nodeI [expr $M0 +32000+$level*10+ $bay] set nodeJ [expr $M0 +3000+$level*10+ $bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $IDBeamTransf; # beams } } # Bay 1 # floor beams set N0beam 6000; # beam element numbers
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set M0 0 for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam + $level*10 +$bay] set nodeI [expr $M0 + 7000+$level*10+ $bay] set nodeJ [expr $M0 + 7000+$level*10+ $bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $IDBeamTransf; # beams } } # Bay 2-4 # floor beams set N0beam 6000; # beam element numbers set M0 0 for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0beam + $level*10 +$bay] set nodeI [expr $M0 + 72000+$level*10+ $bay] set nodeJ [expr $M0 + 7000+$level*10+ $bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $IDBeamTransf; # beams } } ########################################### # Brace elements ########################################### # element tag convention : abcd --> a=4--> brace (type of element) b--> level number c--> axsis number d--> upward=6 downward=9 # command arguments: $eleID $iNode $jNode $numIntgrPts $secTag $transfTag #element located on the left side of the middle node # level 1-2: element forceBeamColumn 4116 4111 46111 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 4146 4114 46114 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 4149 4121 49121 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 4119 4124 49124 $np $BrcSecTag12 $IDBraceTransf; # level 3-4: element forceBeamColumn 4316 4131 46131 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 4346 4134 46134 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 4349 4141 49141 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 4319 4144 49144 $np $BrcSecTag12 $IDBraceTransf;
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# level 5-6: element forceBeamColumn 4516 4151 46151 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 4546 4154 46154 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 4549 4161 49161 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 4519 4164 49164 $np $BrcSecTag34 $IDBraceTransf; # level 7-8: element forceBeamColumn 4716 4171 46171 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 4746 4174 46174 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 4749 4181 49181 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 4719 4184 49184 $np $BrcSecTag34 $IDBraceTransf; #element located on the right side of the middle node (8 in added to the element names) # level 1-2: element forceBeamColumn 41168 46111 4222 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 41468 46114 4225 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 41498 49121 4212 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 41198 49124 4215 $np $BrcSecTag12 $IDBraceTransf; # level 3-4: element forceBeamColumn 43168 46131 4242 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 43468 46134 4245 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 43498 49141 4232 $np $BrcSecTag12 $IDBraceTransf; element forceBeamColumn 43198 49144 4235 $np $BrcSecTag12 $IDBraceTransf; # level 5-6: element forceBeamColumn 45168 46151 4262 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 45468 46154 4265 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 45498 49161 4252 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 45198 49164 4255 $np $BrcSecTag34 $IDBraceTransf; # level 7-8: element forceBeamColumn 47168 46171 4282 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 47468 46174 4285 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 47498 49181 4272 $np $BrcSecTag34 $IDBraceTransf; element forceBeamColumn 47198 49184 4275 $np $BrcSecTag34 $IDBraceTransf; ################################################# # define rigid links: ################################################# set Arigbr12 1.82e-2; # rigid link area for braces at 1st and 2nd floor (m2) set Arigbr34 1.19e-2; # rigid link area for braces at 3rd and 4th floor (m2) set Irigbr12 2.05e-5; # rigid link moments of inertia for braces at 1st and 2nd floor (m4)
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set Irigbr34 1.0e-5; # rigid link moments of inertia for braces at 3rd and 4th floor (m4) ################################################ # brace links: # eleID convention: "4ayxb", 4 = rigid link,a = column(1), beam(2), or brace(3) element, y = rigid offset is right(1) or left(2) the intersection , x = level #, b = column axis number # right: # level 1: element elasticBeamColumn 431116 3011 4111 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 431146 32014 4114 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 431249 3021 4121 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 431219 32024 4124 $Arigbr12 $Es $Irigbr12 4; # level 3: element elasticBeamColumn 431216 7021 4131 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 431246 72024 4134 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 431349 3031 4141 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 431319 32034 4144 $Arigbr12 $Es $Irigbr12 4; # level 5: element elasticBeamColumn 431316 7031 4151 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 431346 72034 4154 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 431449 3041 4161 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 431419 32044 4164 $Arigbr34 $Es $Irigbr34 4; # level 7: element elasticBeamColumn 431416 7041 4171 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 431446 72044 4174 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 431549 7051 4181 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 431519 72054 4184 $Arigbr34 $Es $Irigbr34 4; # left: # level 1: element elasticBeamColumn 432116 4212 3012 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 432146 4215 3015 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 432249 4222 3022 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 432219 4225 3025 $Arigbr12 $Es $Irigbr12 4; # level 3: element elasticBeamColumn 432216 4232 7022 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 432246 4235 7025 $Arigbr12 $Es $Irigbr12 4; element elasticBeamColumn 432349 4242 3032 $Arigbr12 $Es $Irigbr12 4;
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element elasticBeamColumn 432319 4245 3035 $Arigbr12 $Es $Irigbr12 4; # level 5: element elasticBeamColumn 432316 4252 7032 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 432346 4255 7035 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 432449 4262 3042 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 432419 4265 3045 $Arigbr34 $Es $Irigbr34 4; # level 7: element elasticBeamColumn 432416 4272 7042 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 432446 4275 7045 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 432549 4282 7052 $Arigbr34 $Es $Irigbr34 4; element elasticBeamColumn 432519 4285 7055 $Arigbr34 $Es $Irigbr34 4; ########################################## # Connection elements ########################################## set ACon 2.48e-3; set ICon 4.77e-6; ########################################## # ceiling connections: set N0beam 7000; # beam element numbers (there is no limitation for the number of stories, but change the node names for more than 9 bays) set M0 0 for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay } {incr bay 1} { set elemID [expr $N0beam + $level*10 +$bay] set nodeI [expr $M0 +3000+$level*10+ $bay] set nodeJ [expr $M0 +32000+$level*10+ $bay] element elasticBeamColumn $elemID $nodeI $nodeJ $ACon $Es $ICon $IDBeamTransf; # beams } } # floor connections: set N0beam 8000; # beam element numbers set M0 0 for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= $NBay } {incr bay 1} { set elemID [expr $N0beam + $level*10 +$bay] set nodeI [expr $M0 + 7000+$level*10+ $bay] set nodeJ [expr $M0 + 72000+$level*10+ $bay] element elasticBeamColumn $elemID $nodeI $nodeJ $ACon $Es $ICon $IDBeamTransf; # beams
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} } ################################################### # Define weight and masses ################################################### # calculate dead load of frame, assume this to be an internal frame (do LL in a similar manner) # calculate distributed weight along the beam length set DLr 7.1; # dead load distributed along the roof beam (kN/m) set DLf 10.41; # dead load distributed along the floor beam (kN/m) set DLc 1.26; # dead load distributed along the ceiling beam (kN/m) set wbeam 0.191; # W-100x19 section weight per length all (kN/m) set wcolf4 0.101; # HSS 76x76x4.8 section weight per length f4 (kN/m) set wcolf32 0.25; # HSS 178x178x4.8 section weight per length f3 & 2 (kN/m) set wcolf1 0.327; # HSS 178x178x6.4 section weight per length f1 (kN/m) set wbrf43 0.101; # HSS 76x76x4.8 section weight per length f4 & 3 (kN/m) set wbrf21 0.153; # HSS 89x89x6.4 section weight per length f2 & 1 (kN/m) set DLroof [expr $DLr + $wbeam]; # total dead load distributed along the roof beam (kN/m) set DLfloor [expr $DLf + $wbeam]; # total dead load distributed along the floor beam (kN/m) set DLceil [expr $DLc + $wbeam]; # total dead load distributed along the ceiling beam (kN/m) set WeightBeamr [expr $DLroof*$LBeam]; # total roof Beam dead load in rooms (kN) set WeightBeamf [expr $DLfloor*$LBeam]; # total floor Beam dead load in rooms (kN) set WeightBeamc [expr $DLceil*$LBeam]; # total ceiling Beam dead load in rooms (kN) set weightColf4 [expr $wcolf4*$LCol]; # total Column weight f4 (kN) set weightColf32s [expr $wcolf32*$LCol2]; # total Column weight f3 & 2 spaces (kN) set weightColf32 [expr $wcolf32*$LCol1]; # total Column weight f3 & 2 (kN) set weightColf1s [expr $wcolf1*$LCol2]; # total Column weight f1 space (kN) set weightColf1 [expr $wcolf1*$LCol1]; # total Column weight f1 (kN) set weightBrf4 [expr $wbrf43*$Lbr1]; # total brace weight f4 (kN) set weightBrf3 [expr $wbrf43*$Lbr1]; # total brace weight f3 (kN) set weightBrf21 [expr $wbrf21*$Lbr1]; # total brace weight f2 & 1 (kN) # assign masses to the nodes that the columns are connected to: # each connection takes the mass of 1/2 of each element framing into it (mass=weight/$g)
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# The nodal mass is used to calculate the eigenvalues and to perform the dynamic analysis. Only the nodal mass in the horizontal direction will be defined in this demonstration ???? why? set g 9.81 # 2nd floor c mass 3021 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; mass 3022 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; mass 32022 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 3023 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 32023 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 3024 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 32024 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; mass 3025 [expr 3*($weightColf1/2 + $weightColf1s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; # 2nd floor f mass 7021 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2 + $weightBrf21/4)/$g] 0. 0.; mass 7022 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2 + $weightBrf21/4)/$g] 0. 0.; mass 72022 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 7023 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 72023 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 7024 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 72024 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2 + $weightBrf21/4)/$g] 0. 0.; mass 7025 [expr 3*($weightColf32/2 + $weightColf1s/2 + $WeightBeamf/2 + $weightBrf21/4)/$g] 0. 0.; # 3rd floor c mass 3031 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; mass 3032 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; mass 32032 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 3033 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 32033 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 3034 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 32034 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.; mass 3035 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf21/4)/$g] 0. 0.;
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# 3rd floor f mass 7031 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf3/4)/$g] 0. 0.; mass 7032 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf3/4)/$g] 0. 0.; mass 72032 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 7033 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 72033 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 7034 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 72034 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf3/4)/$g] 0. 0.; mass 7035 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf3/4)/$g] 0. 0.; # 4th floor c mass 3041 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf3/4)/$g] 0. 0.; mass 3042 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf3/4)/$g] 0. 0.; mass 32042 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 3043 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 32043 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 3044 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2)/$g] 0. 0.; mass 32044 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf3/4)/$g] 0. 0.; mass 3045 [expr 3*($weightColf32/2 + $weightColf32s/2 + $WeightBeamc/2 + $weightBrf3/4)/$g] 0. 0.; # 4th floor f mass 7041 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf4/4)/$g] 0. 0.; mass 7042 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf4/4)/$g] 0. 0.; mass 72042 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 7043 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 72043 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 7044 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2)/$g] 0. 0.; mass 72044 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf4/4)/$g] 0. 0.; mass 7045 [expr 3*($weightColf4/2 + $weightColf32s/2 + $WeightBeamf/2 + $weightBrf4/4)/$g] 0. 0.; # roof floor mass 7051 [expr 3*($weightColf4/2 + $WeightBeamr/2 + $weightBrf4/4)/$g] 0. 0.;
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mass 7052 [expr 3*($weightColf4/2 + $WeightBeamr/2 + $weightBrf4/4)/$g] 0. 0.; mass 72052 [expr 3*($weightColf4/2 + $WeightBeamr/2)/$g] 0. 0.; mass 7053 [expr 3*($weightColf4/2 + $WeightBeamr/2)/$g] 0. 0.; mass 72053 [expr 3*($weightColf4/2 + $WeightBeamr/2)/$g] 0. 0.; mass 7054 [expr 3*($weightColf4/2 + $WeightBeamr/2)/$g] 0. 0.; mass 72054 [expr 3*($weightColf4/2 + $WeightBeamr/2 + $weightBrf4/4)/$g] 0. 0.; mass 7055 [expr 3*($weightColf4/2 + $WeightBeamr/2 + $weightBrf4/4)/$g] 0. 0.; # calculate total Floor Mass: # Considering 4DOF and calculating the weight of ceilings as part of the floor weight set Weightlevel2 [expr 6*($weightColf32 *8.0/2 + $weightColf1 *8.0/2 + $weightColf1s *8.0 + $WeightBeamf *4.0 + $WeightBeamc *4.0 + $weightBrf21*4.0/2 + $weightBrf21*4.0/2 )]; set Weightlevel3 [expr 6*($weightColf32 *8.0/2 + $weightColf32 *8.0/2 + $weightColf32s *8.0 + $WeightBeamf *4.0 + $WeightBeamc *4.0 + $weightBrf21*4.0/2 + $weightBrf3*4.0/2 )]; set Weightlevel4 [expr 6*($weightColf4 *8.0/2 + $weightColf32 *8.0/2 + $weightColf32s *8.0 + $WeightBeamf *4.0 + $WeightBeamc *4.0 + $weightBrf3*4.0/2 + $weightBrf4*4.0/2 )]; set Weightlevel5 [expr 6*($weightColf4 *8.0/2 + $WeightBeamf *5.0 + $weightBrf4*4.0/2 )]; set WeightTotal [expr $Weightlevel2 + $Weightlevel3 + $Weightlevel4 + $Weightlevel5]; # total frame weight set Masslevel2 [expr $Weightlevel2/$g]; set Masslevel3 [expr $Weightlevel3/$g]; set Masslevel4 [expr $Weightlevel4/$g]; set Masslevel5 [expr $Weightlevel5/$g]; set MassTotal [expr $Masslevel2 + $Masslevel3 + $Masslevel4 + $Masslevel5]; # total frame mass ########################################## # Define RECORDERS ########################################## recorder Node -file x[expr $gmf]/DFreeX[expr $gmf].out -time -node 3011 3021 7021 7022 72022 3031 7031 3041 7041 7051 -dof 1 2 3 disp; recorder Node -file x[expr $gmf]/RfloorsX[expr $gmf].out -time -node 3011 3012 32012 3013 32013 3014 32014 3015 -dof 1 2 3 reaction; #recorder Node -file x[expr $gmf]/VbaseX[expr $gmf].out -time -node 3011 3012 32012 3013 32013 3014 32014 3015 -dof 1 reaction; recorder Drift -file x[expr $gmf]/DrroofNodeX[expr $gmf].out -time -iNode 3011 -jNode 7051 -dof 1 -perpDirn 2; # peak lateral roof drift recorder Drift -file x[expr $gmf]/DrISNode13X[expr $gmf].out -time -iNode 3011 -jNode 7021 -dof 1 -perpDirn 2; recorder Drift -file x[expr $gmf]/DrISNode12X[expr $gmf].out -time -iNode 3011 -jNode 3021 -dof 1 -perpDirn 2; # lateral inter-story drift (just floors not ceilings) recorder Drift -file x[expr $gmf]/DrISNode23X[expr $gmf].out -time -iNode 3021 -jNode 7021 -dof 1 -perpDirn 2;
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recorder Drift -file x[expr $gmf]/DrISNode34X[expr $gmf].out -time -iNode 7021 -jNode 3031 -dof 1 -perpDirn 2; recorder Drift -file x[expr $gmf]/DrISNode45X[expr $gmf].out -time -iNode 3031 -jNode 7031 -dof 1 -perpDirn 2; recorder Drift -file x[expr $gmf]/DrISNode56X[expr $gmf].out -time -iNode 7031 -jNode 3041 -dof 1 -perpDirn 2; recorder Drift -file x[expr $gmf]/DrISNode67X[expr $gmf].out -time -iNode 3041 -jNode 7041 -dof 1 -perpDirn 2; recorder Drift -file x[expr $gmf]/DrISNode78X[expr $gmf].out -time -iNode 7041 -jNode 7051 -dof 1 -perpDirn 2; ... # record story 1 column, beam , brace and some connections forces in global coordinates recorder Element -file x[expr $gmf]/FcolX[expr $gmf].out -ele 1011 11011 force; recorder Element -file x[expr $gmf]/FbeamX[expr $gmf].out -ele 5011 6011 6012 force; recorder Element -file x[expr $gmf]/FconnX[expr $gmf].out -ele 7012 8012 7042 force; recorder Element -file x[expr $gmf]/Fbrace4116X[expr $gmf].out -ele 4116 force; ... ################################################## # Eigenvalue Analysis ################################################## # number of modes set numModes 4 # create data directory file mkdir modes; # record eigenvectors for { set k 1 } { $k <= $numModes } { incr k } { recorder Node -file [format "modes/mode%i.out" $k] -node 7021 7031 7041 7051 -dof 1 2 3 "eigen $k" } # perform eigen analysis #----------------------------- set lambda [eigen $numModes]; # calculate frequencies and periods of the structure #--------------------------------------------------- set omega {} set f {} set T {} set pi 3.141593 foreach lam $lambda { lappend omega [expr sqrt($lam)]
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lappend f [expr sqrt($lam)/(2*$pi)] lappend T [expr (2*$pi)/sqrt($lam)] } puts "periods are $T" # write the output file cosisting of periods #-------------------------------------------- set period "modes/Periods.txt" set Periods [open $period "w"] foreach t $T { puts $Periods " $t" } close $Periods # record the eigenvectors #------------------------ record # create display for mode shapes #--------------------------------- set h 3.4; # floor height # $windowTitle $xLoc $yLoc $xPixels $yPixels # record the eigenvectors #------------------------ record # create display for mode shapes #--------------------------------- #set h 3.4; # floor height # $windowTitle $xLoc $yLoc $xPixels $yPixels #recorder display "Mode Shape 1" 10 10 500 500 -wipe #prp $h $h 1; # projection reference point (prp); defines the center of projection (viewer eye) #vup 0 1 0; # view-up vector (vup) #vpn 0 0 1; # view-plane normal (vpn) #viewWindow -10 10 -10 10; # coordiantes of the window relative to prp #display -1 4 5; # the 1st arg. is the tag for display mode (ex. -1 is for the first mode shape) # the 2nd arg. is magnification factor for nodes, the 3rd arg. is magnif. factor of deformed shape #recorder display "Mode Shape 2" 10 510 500 500 -wipe #prp $h $h 1;
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#vup 0 1 0; #vpn 0 0 1; #viewWindow -10 10 -10 10 #display -2 4 5 #recorder display "Mode Shape 3" 10 10 500 500 -wipe #prp $h $h 1; #vup 0 1 0; #vpn 0 0 1; #viewWindow -10 10 -10 10 #display -3 4 5 #recorder display "Mode Shape 4" 10 510 500 500 -wipe #prp $h $h 1; #vup 0 1 0; #vpn 0 0 1; #viewWindow -10 10 -10 10 #display -4 4 5 ########################################### # IDA analysis ########################################### ##EarthQuake Loading puts "groundmotion start!.Time: [getTime]" #define damping set xDamp 0.05; # damping ratio set MpropSwitch 1.0; set KcurrSwitch 0.0; set KcommSwitch 1.0; set KinitSwitch 0.0; set nEigenI 1; # mode 1 set nEigenJ 4; # mode 3 set lambdaN [eigen [expr $nEigenJ]]; # eigenvalue analysis for nEigenJ modes set lambdaI [lindex $lambdaN [expr $nEigenI-1]]; # eigenvalue mode i set lambdaJ [lindex $lambdaN [expr $nEigenJ-1]]; # eigenvalue mode j set omegaI [expr pow($lambdaI,0.5)]; set omegaJ [expr pow($lambdaJ,0.5)]; set alphaM [expr $MpropSwitch*$xDamp*(2*$omegaI*$omegaJ)/($omegaI+$omegaJ)]; # M-prop. damping; D = alphaM*M set betaKcurr [expr $KcurrSwitch*2.*$xDamp/($omegaI+$omegaJ)]; # current-K; +beatKcurr*KCurrent set betaKcomm [expr $KcommSwitch*2.*$xDamp/($omegaI+$omegaJ)]; # last-committed K; +betaKcomm*KlastCommitt
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set betaKinit [expr $KinitSwitch*2.*$xDamp/($omegaI+$omegaJ)]; # initial-K; +beatKinit*Kini rayleigh $alphaM $betaKcurr $betaKinit $betaKcomm; # RAYLEIGH damping puts "damping ok!" #------------------------------- ### change these parameters for each record: ## TmaxAnalysis , DtAnalysis , SaT1Gm, "Series -dt .. - filepath .." set TmaxAnalysis 39.5 ; # maximum duration of ground-motion analysis -- should be 40*$sec set DtAnalysis 0.01; # time-step Dt for lateral analysis # The time interval between the points found in the record (dt) is 0.005 and number of data points found in the record (nPts) is 7990 # defining the number of analysis steps to be performed and defining the analysis increments. The number of analysis steps is set to 3995 (nPts/2) and the analysis increment is set to: DtAnalysis=0.01 (2*dt). Thus every other second point in the record will be skipped during the analysis. #increment gmf set G 9.81; set SaT1 0.57; # NBCC set SaT1Gm 0.27262; # 5% damped first mode spectral acceleration Sa(T1,5%) set SF [expr $SaT1/$SaT1Gm ]; # Scale Factor set accel "Series -dt 0.005 -filePath NGA165282.txt -factor [expr ($gmf)*$G * $SF]" pattern UniformExcitation [expr $i+200] 1 -accel $accel # display displacement shape of the building #recorder display "Displaced shape" 10 10 500 500 -wipe #prp 200. 50. 1; #vup 0 1 0; #vpn 0 0 1; #display 1 5 25 puts "IDA is running ....." ##Dynamic Analysis Parameters variable constraintsTypeDynamic Transformation; constraints $constraintsTypeDynamic ; variable numbererTypeDynamic RCM numberer $numbererTypeDynamic variable systemTypeDynamic BandGeneral; # try UmfPack for large problems
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system $systemTypeDynamic variable maxNumIterDynamic 400; # Convergence Test: maximum number of iterations that will be performed before "failure to converge" is returned variable printFlagDynamic 0; # Convergence Test: flag used to print information on convergence (optional) # 1: print information on each step; variable testTypeDynamic EnergyIncr; # Convergence-test type test $testTypeDynamic $TolDynamic $maxNumIterDynamic $printFlagDynamic; variable maxNumIterConvergeDynamic 2000; variable printFlagConvergeDynamic 0; variable algorithmTypeDynamic KrylovNewton; algorithm $algorithmTypeDynamic; variable NewmarkGamma 0.5; # Newmark-integrator gamma parameter (also HHT) variable NewmarkBeta 0.26; # Newmark-integrator beta parameter variable integratorTypeDynamic Newmark; integrator $integratorTypeDynamic $NewmarkGamma $NewmarkBeta variable analysisTypeDynamic VariableTransient analysis $analysisTypeDynamic #------------------------------- set Nsteps [expr int($TmaxAnalysis/$DtAnalysis)]; #analyze $Nsteps $DtAnalysis #loadConst -time 0.0 set ok [analyze $Nsteps $DtAnalysis]; # actually perform analysis; returns ok=0 if analysis was successful if {$ok != 0} { ; # analysis was not successful. # -------------------------------------------------------------------------------------------------- # change some analysis parameters to achieve convergence # performance is slower inside this loop # Time-controlled analysis set ok 0; set controlTime [getTime]; while {$controlTime < $TmaxAnalysis && $ok == 0} { set controlTime [getTime] set ok [analyze 1 $DtAnalysis] if {$ok != 0} { puts "Trying Newton with Initial Tangent .." test NormDispIncr $TolDynamic 1000 0 algorithm Newton -initial set ok [analyze 1 $DtAnalysis]
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test $testTypeDynamic $TolDynamic $maxNumIterDynamic 0 algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying Broyden .." algorithm Broyden 8 set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying NewtonWithLineSearch .." algorithm NewtonLineSearch .6 set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying ModifiedNewton .." algorithm ModifiedNewton set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying BFGS.." algorithm BFGS set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } } }; # end if ok !0 # introducing a control parameter to stop the analysis at a certain displacement: set IDctrlNode 7051; # node where disp is read for disp control set IDctrlDOF 1; # degree of freedom read for disp control (1 = x displacement) set Dmax [expr 0.1*$LBuilding]; # maximum displacement of pushover: 10% roof drift # applying displacement control stopping point (while loop can be used as well): set roofdisp [nodeDisp $IDctrlNode $IDctrlDOF]; if {abs($roofdisp) > $Dmax} break puts "roofdisp is abs($roofdisp)" puts "Ground Motion DoneX[expr $gmf]. End Time: [getTime]" puts **************************************** puts ****************************************
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wipe all }
A.1.2 Modified 3D MSB
########################################################## # Amirahmad Fathieh, University Of Toronto # 3D Analysis of Modular Steel building ########################################################## ########################################################## # nonlinearBeamColumn element, inelastic fiber section ########################################################### wipe; # clear memory of all past model definitions wipeAnalysis; #change to reach convergance variable TolDynamic 1.e-6; # Convergence Test:tolerance #define intensity step (simple stepping): 0.1g,0.2g,0.3g,... for { set i 1 } { $i < 70 } { incr i 1 } { set s [expr 1*$i] set gmf [expr $s/10.] puts "gmf=[expr $gmf]" puts ****************** file mkdir x[expr $gmf] source 3D-modelConnectSource.tcl ##################################################### # Define RECORDERS ##################################################### # Define RECORDERS ------------------------------------------------------------- # connection ends relative displacement (drift) recorder Drift -file x[expr $gmf]/ConEndsDriftFloYZ[expr $gmf].out -time -iNode 72020302 -jNode 73020302 -dof 2 -perpDirn 3; # Drift (relative displ b/w connection ends) ratio delta-y and z ele 883020302 recorder Drift -file x[expr $gmf]/ConEndsDriftCeilYZ[expr $gmf].out -time -iNode 32020302 -jNode 33020302 -dof 2 -perpDirn 3; # Drift (relative displ b/w connection ends) ratio delta-y and z ele 873020302
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recorder Drift -file x[expr $gmf]/ConEndsDriftFloYX[expr $gmf].out -time -iNode 72020302 -jNode 7020302 -dof 2 -perpDirn 1; # Drift (relative displ b/w connection ends) ratio delta-y and x ele 86020301 recorder Drift -file x[expr $gmf]/ConEndsDriftCeilYX[expr $gmf].out -time -iNode 32020302 -jNode 3020302 -dof 2 -perpDirn 1; # Drift (relative displ b/w connection ends) ratio delta-y and x ele 85020301 # connection end nodes displacements recorder Node -file x[expr $gmf]/DispConNi[expr $gmf].out -time -node 72020302 -dof 1 2 3 5 disp; # displacements of node i (time, disp. in x at node i, disp. in y at node i,... recorder Node -file x[expr $gmf]/DispConNj[expr $gmf].out -time -node 73020302 -dof 1 2 3 5 disp; # displacements of node j (time, disp. in x at node j, disp. in y at node j,... recorder Node -file x[expr $gmf]/DispConNjj[expr $gmf].out -time -node 7020302 -dof 1 2 3 5 disp; # displacements of node j (time, disp. in x at node j, disp. in y at node j,... # connection end nodes reactions recorder Node -file x[expr $gmf]/VConNi[expr $gmf].out -time -node 72020302 -dof 1 2 3 5 reaction; # support reaction recorder Node -file x[expr $gmf]/VConNj[expr $gmf].out -time -node 73020302 -dof 1 2 3 5 reaction; # support reaction recorder Node -file x[expr $gmf]/VConNjj[expr $gmf].out -time -node 7020302 -dof 1 2 3 5 reaction; # support reaction # connection element forces in local coordinates recorder Element -file x[expr $gmf]/LocalForceFloEZ[expr $gmf].out -time -ele 883020302 localForce; # element forces in local coordinates Z dir recorder Element -file x[expr $gmf]/LocalForceCeilEZ[expr $gmf].out -time -ele 873020302 localForce; # element forces in local coordinates Z dir recorder Element -file x[expr $gmf]/LocalForceFloEX[expr $gmf].out -time -ele 86020301 localForce; # element forces in local coordinates X dir recorder Element -file x[expr $gmf]/LocalForceCeilEX[expr $gmf].out -time -ele 85020301 localForce; # element forces in local coordinates X dir # connection element forces in global coordinates recorder Element -file x[expr $gmf]/FConFloZ[expr $gmf].out -ele 883020302 force; # in Z direction recorder Element -file x[expr $gmf]/FConCeilZ[expr $gmf].out -ele 873020302 force; # in Z direction recorder Element -file x[expr $gmf]/FConFloX[expr $gmf].out -ele 86020301 force; # in X direction recorder Element -file x[expr $gmf]/FConCeilX[expr $gmf].out -ele 85020301 force; # in X direction # reactions recorder Node -file x[expr $gmf]/RRBase[expr $gmf].out -time -node 3010102 3010202 31010202 3010302 31010302 3010402 31010402 3010502 -dof 1 2 3 reaction; # reaction recorder Node -file x[expr $gmf]/RFL1React[expr $gmf].out -time -node 7020102 7020202 71020202 7020302 71020302 7020402 71020402 7020502 -dof 1 2 3 reaction; # reaction
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recorder Node -file x[expr $gmf]/RCL1React[expr $gmf].out -time -node 3020102 3020202 31020202 3020302 31020302 3020402 31020402 3020502 -dof 1 2 3 reaction; # reaction recorder Node -file x[expr $gmf]/RlevelsFirstNodeReact[expr $gmf].out -time -node 3020102 7020102 7030102 7040102 7050102 -dof 1 2 3 reaction; # reaction # diaphragms horizontal displ and rot recorder Node -file x[expr $gmf]/DisM1[expr $gmf].out -time -node 75020101 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM2[expr $gmf].out -time -node 75020201 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM3[expr $gmf].out -time -node 75020301 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM4[expr $gmf].out -time -node 75020401 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM5[expr $gmf].out -time -node 75020102 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM6[expr $gmf].out -time -node 75020202 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM7[expr $gmf].out -time -node 75020302 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM8[expr $gmf].out -time -node 75020402 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM9[expr $gmf].out -time -node 75020103 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM10[expr $gmf].out -time -node 75020203 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM11[expr $gmf].out -time -node 75020303 -dof 1 2 3 5 disp; recorder Node -file x[expr $gmf]/DisM12[expr $gmf].out -time -node 75020403 -dof 1 2 3 5 disp; ... ##################################################### # Eigenvalue Analysis ##################################################### # number of modes set numModes 4 # create data directory file mkdir modes; # record eigenvectors for { set k 1 } { $k <= $numModes } { incr k } { recorder Node -file [format "modes/mode%i.out" $k] -node 7020101 7030101 7040101 7050101 -dof 1 "eigen $k"
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} # perform eigen analysis #----------------------------- set lambda [eigen $numModes]; # calculate frequencies and periods of the structure #--------------------------------------------------- set omega {} set f {} set T {} set pi 3.141593 foreach lam $lambda { lappend omega [expr sqrt($lam)] lappend f [expr sqrt($lam)/(2*$pi)] lappend T [expr (2*$pi)/sqrt($lam)] } puts "periods are $T" # write the output file cosisting of periods #-------------------------------------------- set period "modes/Periods.txt" set Periods [open $period "w"] foreach t $T { puts $Periods " $t" } close $Periods # record the eigenvectors #------------------------ record # create display for mode shapes #--------------------------------- set h 3.4; # floor height # $windowTitle $xLoc $yLoc $xPixels $yPixels # record the eigenvectors #------------------------ record ############################################### # Define DISPLAY
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############################################### DisplayModel3D DeformedShape ; # options: DeformedShape NodeNumbers ModeShape #DisplayModel3D NodeNumbers; #####set xPixels 1200; # height of graphical window in pixels #####set yPixels 800; # height of graphical window in pixels #####set xLoc1 10; # horizontal location of graphical window (0=upper left-most corner) #####set yLoc1 10; # vertical location of graphical window (0=upper left-most corner) #####set dAmp 2; # scaling factor for viewing deformed shape, it depends on the dimensions of the model #####DisplayModel3D NodeNumbers $dAmp $xLoc1 $yLoc1 $xPixels $yPixels puts "Model Built" ###################################################### # DYNAMIC Ground-Motion Analysis ###################################################### # Bidirectional Uniform Earthquake ground motion (uniform acceleration input at all support nodes) set iGMfile "NGA778255-x NGA778255-y" ; # ground-motion filenames, should be different files set iGMdirection "1 3"; # ground-motion directions set iGMfact "$gmf $gmf"; # ground-motion scaling factor set Tol 1.0e-8; # convergence tolerance for test # Define DISPLAY ------------------------------------------------------------- # the deformed shape is defined in the build file recorder plot $dataDir/DFree.out DisplDOF[lindex $iGMdirection 0] 1200 10 400 400 -columns 1 [expr 1+[lindex $iGMdirection 0]] ; # a window to plot the nodal displacements versus time recorder plot $dataDir/DFree.out DisplDOF[lindex $iGMdirection 1] 1200 410 400 400 -columns 1 [expr 1+[lindex $iGMdirection 1]] ; # a window to plot the nodal displacements versus time # ----------- set up analysis parameters #source LibAnalysisDynamicParameters.tcl; # constraintsHandler,DOFnumberer,system-ofequations,convergenceTest,solutionAlgorithm,integrator # Set up Analysis Parameters variable constraintsTypeDynamic Transformation; constraints $constraintsTypeDynamic ; # DOF NUMBERER variable numbererTypeDynamic RCM numberer $numbererTypeDynamic
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# SYSTEM variable systemTypeDynamic BandGeneral; # try UmfPack for large problems system $systemTypeDynamic # TEST variable TolDynamic 1.e-8; # Convergence Test: tolerance variable maxNumIterDynamic 400; # Convergence Test: maximum number of iterations that will be performed before "failure to converge" is returned variable printFlagDynamic 0; # Convergence Test: flag used to print information on convergence (optional) # 1: print information on each step; variable testTypeDynamic EnergyIncr; # Convergence-test type test $testTypeDynamic $TolDynamic $maxNumIterDynamic $printFlagDynamic; # for improved-convergence procedure: variable maxNumIterConvergeDynamic 2000; variable printFlagConvergeDynamic 0; # Solution ALGORITHM variable algorithmTypeDynamic KrylovNewton; algorithm $algorithmTypeDynamic; # Static INTEGRATOR variable NewmarkGamma 0.5; # Newmark-integrator gamma parameter (also HHT) variable NewmarkBeta 0.26; # Newmark-integrator beta parameter variable integratorTypeDynamic Newmark; integrator $integratorTypeDynamic $NewmarkGamma $NewmarkBeta # ANALYSIS variable analysisTypeDynamic VariableTransient analysis $analysisTypeDynamic ################## Define & Apply Damping ############## # RAYLEIGH damping parameters, Where to put M/K-prop damping, switches (http://opensees.berkeley.edu/OpenSees/manuals/usermanual/1099.htm) # D=$alphaM*M + $betaKcurr*Kcurrent + $betaKcomm*KlastCommit + $beatKinit*$Kinitial set xDamp 0.02; # damping ratio set MpropSwitch 1.0; set KcurrSwitch 0.0; set KcommSwitch 1.0; set KinitSwitch 0.0; set nEigenI 1; # mode 1 set nEigenJ 4; # mode 4 set lambdaN [eigen [expr $nEigenJ]]; # eigenvalue analysis for nEigenJ modes set lambdaI [lindex $lambdaN [expr $nEigenI-1]]; # eigenvalue mode i
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set lambdaJ [lindex $lambdaN [expr $nEigenJ-1]]; # eigenvalue mode j set omegaI [expr pow($lambdaI,0.5)]; set omegaJ [expr pow($lambdaJ,0.5)]; set alphaM [expr $MpropSwitch*$xDamp*(2*$omegaI*$omegaJ)/($omegaI+$omegaJ)]; # M-prop. damping; D = alphaM*M set betaKcurr [expr $KcurrSwitch*2.*$xDamp/($omegaI+$omegaJ)]; # current-K; +beatKcurr*KCurrent set betaKcomm [expr $KcommSwitch*2.*$xDamp/($omegaI+$omegaJ)]; # last-committed K; +betaKcomm*KlastCommitt set betaKinit [expr $KinitSwitch*2.*$xDamp/($omegaI+$omegaJ)]; # initial-K; +beatKinit*Kini rayleigh $alphaM $betaKcurr $betaKinit $betaKcomm; # RAYLEIGH damping ########## perform Dynamic Ground-Motion Analysis ####### # set up ground-motion-analysis parameters # the following commands are unique to the Uniform Earthquake excitation ## TmaxAnalysis , DtAnalysis , "Series -dt .. - filepath .." set TmaxAnalysis 39.5 ; # maximum duration of ground-motion analysis -- should be 40*$sec set DtAnalysis 0.01; # time-step Dt for lateral analysis set dt 0.005; # Ground motion time-step # The time interval between the points found in the record (dt) is 0.005 and number of data points found in the record (nPts) is 7990 # defining the number of analysis steps to be performed and defining the analysis increments. The number of analysis steps is set to 3995 (nPts/2) and the analysis increment is set to: DtAnalysis=0.01 (2*dt). Thus every other second point in the record will be skipped during the analysis. #increment gmf set G 9.81; set SaT1 0.51; # NBCC set SaT1Gm 0.64514; set SF [expr $SaT1/$SaT1Gm]; # Scale Factor set IDloadTag 400; # for uniformSupport excitation # Uniform EXCITATION: acceleration input foreach GMdirection $iGMdirection GMfile $iGMfile GMfact $iGMfact { incr IDloadTag; set inFile $GMdir/$GMfile.txt set outFile $GMdir/$GMfile.txt; # set variable holding new filename (PEER files have .at2/dt2 extension)
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set GMfatt [expr $G* $GMfact* $SF]; # data in input file is in g Unifts -- ACCELERATION TH set AccelSeries "Series -dt $dt -filePath $outFile -factor $GMfatt"; # time series information pattern UniformExcitation [expr $i+$IDloadTag] $GMdirection -accel $AccelSeries ; # create Uniform excitation puts "IDA is running ..." set Nsteps [expr int($TmaxAnalysis/$DtAnalysis)]; #analyze $Nsteps $DtAnalysis #loadConst -time 0.0 set ok [analyze $Nsteps $DtAnalysis]; # actually perform analysis; returns ok=0 if analysis was successful if {$ok != 0} { ; # analysis was not successful. # -------------------------------------------------------------------------------------------------- # change some analysis parameters to achieve convergence # performance is slower inside this loop # Time-controlled analysis set ok 0; set controlTime [getTime]; while {$controlTime < $TmaxAnalysis && $ok == 0} { set controlTime [getTime] set ok [analyze 1 $DtAnalysis] if {$ok != 0} { puts "Trying Newton with Initial Tangent .." test NormDispIncr $TolDynamic 1000 0 algorithm Newton -initial set ok [analyze 1 $DtAnalysis] test $testTypeDynamic $TolDynamic $maxNumIterDynamic 0 algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying Broyden .." algorithm Broyden 8 set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying NewtonWithLineSearch .." algorithm NewtonLineSearch .6 set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic }
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if {$ok != 0} { puts "Trying ModifiedNewton .." algorithm ModifiedNewton set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } if {$ok != 0} { puts "Trying BFGS.." algorithm BFGS set ok [analyze 1 $DtAnalysis] algorithm $algorithmTypeDynamic } } }; # end if ok !0 # introducing a control parameter to stop the analysis at a certain displacement: set IDctrlNode 7050101; # node where disp is read for disp control set IDctrlDOF 1; # degree of freedom read for disp control (1 = x displacement) set Dmax [expr 0.1*$LBuilding]; # maximum displacement of pushover: 10% roof drift # applying displacement control stopping point (while loop can be used as well): set roofdisp [nodeDisp $IDctrlNode $IDctrlDOF]; if {abs($roofdisp) > $Dmax} break puts "roofdisp is abs($roofdisp)" puts "Ground Motion DoneX[expr $gmf]. End Time: [getTime]" puts **************************************************************** puts **************************************************************** wipe all Model Source : source 3D-modelConnectSource.tcl ########################################## ## SET UP ############################################## # Units KN & m # define units model BasicBuilder -ndm 3 -ndf 6; # Define the model builder, ndm=#dimension, ndf=#dofs set dataDir Data; # set up name of data directory -- remove
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file mkdir $dataDir; # create data directory set GMdir "GMfile"; # ground-motion file directory # source LibUnits.tcl; # define units source DisplayPlane.tcl; # procedure for displaying a plane in model source DisplayModel3D.tcl; # procedure for displaying 3D perspectives of model source Wsection.tcl; # procedure to define fiber W section source HSSsection.tcl; # procedure to define fiber HSS section source rotSect2DModIKModel.tcl; # procedure for defining bilinear plastic hinge section set Ubig 1.e10; # a really large number set Usmall [expr 1/$Ubig]; # a really small number # Elements are fully rigid when in thuch with gusset plates # Rigid liks of the braces are defined with rigid elastic elements. Beam and column reigid elemnts are defined using rigid offsets inside of geometric transformation. ################################################ # Define Building Geometry ################################################ # define structure-geometry paramters set spc 0.15; # Clear space between the ceiling and floor beams set db 0.15; # beam depth set LCol 3.5; # Floor height set LCol1 [expr $LCol-($spc + 2*$db)]; # column1 height (parallel to Y axis) set LCol2 [expr $spc + 2*$db]; # column2 height (parallel to Y axis) set LBeam 4.0; # beam length (parallel to X axis) set LGird 3.5; # girder length (parallel to Z axis) set dx 0.35; # columns center to center distance at each intersection set dz 0.35; # columns center to center distance at each intersection #set Lbrx [expr (sqrt($LCol1)*($LCol1) + ($LBeam)*($LBeam))]; # brace length #set Lbrz [expr (sqrt($LCol1)*($LCol1) + ($LGird)*($LGird))]; # brace length # ------ frame configuration set NStory 4; # number of stories above ground level set NBay 3; # number of bays in X direction set NBayZ 4; # number of bays in Z direction puts "Number of Stories in Y: $NStory Number of bays in X: $NBay Number of bays in Z: $NBayZ" set NFrame [expr $NBayZ + 1]; # actually deal with frames in Z direction, as this is an easy extension of the 2d model ############################################## # Define NODAL COORDINATES ############################################## # Original nodes (1)
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set Dlevel 10000; # numbering increment for new-level nodes set Dframe 100; # numbering increment for new-frame nodes for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { set Z [expr ($frame-1)*$LGird]; ### ceiling nodes (3) ((((((Base nodes are at Y= -0.5 m ))))))) for {set level 1} {$level <=[expr $NStory]} {incr level 1} { set Y [expr ($level-1)*($LCol)-$LCol2]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set X [expr ($pier-1)*$LBeam]; set nodeID [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z; # actually define node } } ## floor nodes (7) for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { set Y [expr ($level-1)*$LCol]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set X [expr ($pier-1)*$LBeam]; set nodeID [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z;; # actually define node } } ############################################ # Define EXTERA NODAL COORDINATES FOR COLUMNS AT EACH INTERSECTION ############################################ # Extra nodes in z dir (2) set Dlevel 10000; # numbering increment for new-level nodes set Dframe 100; # numbering increment for new-frame nodes for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { if {$frame == 1 || $frame == $NFrame} { # do nothing } else { set Z [expr ($frame-1)*$LGird + $dz]; ### ceiling nodes (3) ((((((Base nodes are at Y= -0.5 m ))))))) for {set level 1} {$level <=[expr $NStory]} {incr level 1} { set Y [expr ($level-1)*($LCol)-$LCol2]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} {
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set X [expr ($pier-1)*$LBeam ]; set nodeID [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z; # actually define node } } ## floor nodes (7) for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { set Y [expr ($level-1)*$LCol]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set X [expr ($pier-1)*$LBeam ]; set nodeID [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z;; # actually define node } } ########################################## ########################################## # Extra nodes in x dir (3) set Dlevel 10000; # numbering increment for new-level nodes set Dframe 100; # numbering increment for new-frame nodes for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { set Z [expr ($frame-1)*$LGird]; ### ceiling nodes (3) ((((((Base nodes are at Y= -0.5 m ))))))) for {set level 1} {$level <=[expr $NStory]} {incr level 1} { set Y [expr ($level-1)*($LCol)-$LCol2]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set X [expr ($pier-1)*$LBeam + $dx ]; set nodeID [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z; # actually define node } } } ## floor nodes (7)
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for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { set Y [expr ($level-1)*$LCol]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set X [expr ($pier-1)*$LBeam + $dx ]; set nodeID [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z;; # actually define node } } ############################################## ############################################## # Extra nodes in z and x dir (4) set Dlevel 10000; # numbering increment for new-level nodes set Dframe 100; # numbering increment for new-frame nodes for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { if {$frame == 1 || $frame == $NFrame} { # do nothing } else { set Z [expr ($frame-1)*$LGird + $dz]; ### ceiling nodes (3) ((((((Base nodes are at Y= -0.5 m ))))))) for {set level 1} {$level <=[expr $NStory]} {incr level 1} { set Y [expr ($level-1)*($LCol)-$LCol2]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set X [expr ($pier-1)*$LBeam + $dx ]; set nodeID [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z; # actually define node } } } ## floor nodes (7) for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} {
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set Y [expr ($level-1)*$LCol]; for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set X [expr ($pier-1)*$LBeam + $dx ]; set nodeID [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $X $Y $Z;; # actually define node } } } } ... ################################################ # Rigid Diaphragm ################################################ # Single Rigid Diaphragm # Note: only floor levels has Diaphragm, check if it is correct in real buildings. ## diaphragm nodes (75) set Dlevel 10000; # numbering increment for new-level nodes set Dframe 100; # numbering increment for new-frame nodes for {set frame 1} {$frame <=[expr $NFrame -1]} {incr frame 1} { set Za [expr ($frame-1)*$LGird + $LGird/2 + $dz/2]; for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { set Y [expr ($level-1)*$LCol]; for {set pier 1} {$pier <= [expr $NBay]} {incr pier 1} { set Xa [expr ($pier-1)*$LBeam + $LBeam/2 + $dx/2]; set nodeID [expr 75000000 + $frame*$Dframe+$level*$Dlevel+$pier] node $nodeID $Xa $Y $Za;; # actually define node } } # Constraints for rigid diaphragm master nodes # level 2 fix 75020101 0 1 0 1 0 1 fix 75020102 0 1 0 1 0 1 fix 75020103 0 1 0 1 0 1 fix 75020201 0 1 0 1 0 1 fix 75020202 0 1 0 1 0 1
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fix 75020203 0 1 0 1 0 1 fix 75020301 0 1 0 1 0 1 fix 75020302 0 1 0 1 0 1 fix 75020303 0 1 0 1 0 1 fix 75020401 0 1 0 1 0 1 fix 75020402 0 1 0 1 0 1 fix 75020403 0 1 0 1 0 1 # level 3 fix 75030101 0 1 0 1 0 1 fix 75030102 0 1 0 1 0 1 fix 75030103 0 1 0 1 0 1 fix 75030201 0 1 0 1 0 1 fix 75030202 0 1 0 1 0 1 fix 75030203 0 1 0 1 0 1 fix 75030301 0 1 0 1 0 1 fix 75030302 0 1 0 1 0 1 fix 75030303 0 1 0 1 0 1 fix 75030401 0 1 0 1 0 1 fix 75030402 0 1 0 1 0 1 fix 75030403 0 1 0 1 0 1 # level 4 fix 75040101 0 1 0 1 0 1 fix 75040102 0 1 0 1 0 1 fix 75040103 0 1 0 1 0 1 fix 75040201 0 1 0 1 0 1 fix 75040202 0 1 0 1 0 1 fix 75040203 0 1 0 1 0 1 fix 75040301 0 1 0 1 0 1 fix 75040302 0 1 0 1 0 1 fix 75040303 0 1 0 1 0 1 fix 75040401 0 1 0 1 0 1 fix 75040402 0 1 0 1 0 1 fix 75040403 0 1 0 1 0 1 # level 5 fix 75050101 0 1 0 1 0 1 fix 75050102 0 1 0 1 0 1 fix 75050103 0 1 0 1 0 1 fix 75050201 0 1 0 1 0 1 fix 75050202 0 1 0 1 0 1 fix 75050203 0 1 0 1 0 1 fix 75050301 0 1 0 1 0 1 fix 75050302 0 1 0 1 0 1 fix 75050303 0 1 0 1 0 1 fix 75050401 0 1 0 1 0 1 fix 75050402 0 1 0 1 0 1
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fix 75050403 0 1 0 1 0 1 # Define Rigid Diaphram (dof 2 is normal to floor) # rigidDiaphragm $perpDirn $masterNodeTag $slaveNodeTag1 $slaveNodeTag2 ... set perpDirn 2; # level 2 rigidDiaphragm $perpDirn 75020101 7020101 7020201 7020102 7020202 ; rigidDiaphragm $perpDirn 75020102 72020102 72020202 7020103 7020203 ; rigidDiaphragm $perpDirn 75020103 72020103 72020203 7020104 7020204 ; rigidDiaphragm $perpDirn 75020201 71020201 7020301 71020202 7020302 ; rigidDiaphragm $perpDirn 75020202 73020202 72020302 71020203 7020303 ; rigidDiaphragm $perpDirn 75020203 73020203 72020303 71020204 7020304 ; rigidDiaphragm $perpDirn 75020301 71020301 7020401 71020302 7020402 ; rigidDiaphragm $perpDirn 75020302 73020302 72020402 71020303 7020403 ; rigidDiaphragm $perpDirn 75020303 73020303 72020403 71020304 7020404 ; rigidDiaphragm $perpDirn 75020401 71020401 7020501 71020402 7020502 ; rigidDiaphragm $perpDirn 75020402 73020402 72020502 71020403 7020503 ; rigidDiaphragm $perpDirn 75020403 73020403 72020503 71020404 7020504 ; # level 3 rigidDiaphragm $perpDirn 75030101 7030101 7030201 7030102 7030202 ; rigidDiaphragm $perpDirn 75030102 72030102 72030202 7030103 7030203 ; rigidDiaphragm $perpDirn 75030103 72030103 72030203 7030104 7030204 ; rigidDiaphragm $perpDirn 75030201 71030201 7030301 71030202 7030302 ; rigidDiaphragm $perpDirn 75030202 73030202 72030302 71030203 7030303 ; rigidDiaphragm $perpDirn 75030203 73030203 72030303 71030204 7030304 ; rigidDiaphragm $perpDirn 75030301 71030301 7030401 71030302 7030402 ; rigidDiaphragm $perpDirn 75030302 73030302 72030402 71030303 7030403 ; rigidDiaphragm $perpDirn 75030303 73030303 72030403 71030304 7030404 ; rigidDiaphragm $perpDirn 75030401 71030401 7030501 71030402 7030502 ; rigidDiaphragm $perpDirn 75030402 73030402 72030502 71030403 7030503 ; rigidDiaphragm $perpDirn 75030403 73030403 72030503 71030404 7030504 ; # level 4 rigidDiaphragm $perpDirn 75040101 7040101 7040201 7040102 7040202 ; rigidDiaphragm $perpDirn 75040102 72040102 72040202 7040103 7040203 ; rigidDiaphragm $perpDirn 75040103 72040103 72040203 7040104 7040204 ;
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rigidDiaphragm $perpDirn 75040201 71040201 7040301 71040202 7040302 ; rigidDiaphragm $perpDirn 75040202 73040202 72040302 71040203 7040303 ; rigidDiaphragm $perpDirn 75040203 73040203 72040303 71040204 7040304 ; rigidDiaphragm $perpDirn 75040301 71040301 7040401 71040302 7040402 ; rigidDiaphragm $perpDirn 75040302 73040302 72040402 71040303 7040403 ; rigidDiaphragm $perpDirn 75040303 73040303 72040403 71040304 7040404 ; rigidDiaphragm $perpDirn 75040401 71040401 7040501 71040402 7040502 ; rigidDiaphragm $perpDirn 75040402 73040402 72040502 71040403 7040503 ; rigidDiaphragm $perpDirn 75040403 73040403 72040503 71040404 7040504 ; # level 4 rigidDiaphragm $perpDirn 75050101 7050101 7050201 7050102 7050202 ; rigidDiaphragm $perpDirn 75050102 72050102 72050202 7050103 7050203 ; rigidDiaphragm $perpDirn 75050103 72050103 72050203 7050104 7050204 ; rigidDiaphragm $perpDirn 75050201 71050201 7050301 71050202 7050302 ; rigidDiaphragm $perpDirn 75050202 73050202 72050302 71050203 7050303 ; rigidDiaphragm $perpDirn 75050203 73050203 72050303 71050204 7050304 ; rigidDiaphragm $perpDirn 75050301 71050301 7050401 71050302 7050402 ; rigidDiaphragm $perpDirn 75050302 73050302 72050402 71050303 7050403 ; rigidDiaphragm $perpDirn 75050303 73050303 72050403 71050304 7050404 ; rigidDiaphragm $perpDirn 75050401 71050401 7050501 71050402 7050502 ; rigidDiaphragm $perpDirn 75050402 73050402 72050502 71050403 7050503 ; rigidDiaphragm $perpDirn 75050403 73050403 72050503 71050404 7050504 ; # determine support nodes where ground motions are input, for multiple-support excitation set iSupportNode "" for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { set level 1 for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set nodeID [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] lappend iSupportNode $nodeID } } # BOUNDARY CONDITIONS fixY -0.45 1 1 1 0 1 0; # pin all Y=-0.45 nodes # calculated MODEL PARAMETERS, particular to this model # Set up parameters that are particular to the model for displacement control
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set IDctrlNode [expr int((($NStory+1)*$Dlevel+$NFrame*$Dframe)+1)]; # node where displacement is read for displacement control set IDctrlDOF 1; # degree of freedom of displacement read for displacement control set LBuilding [expr $NStory*$LCol]; # total building height ################################################# # Define SECTIONS ################################################# set SectionType FiberSection; # options: Elastic FiberSection ########### Define section tags ############ set ColSecTag1 11; set ColSecTag23 12; set ColSecTag4 13; set BeamSecTag 21; set GirdSecTag 31; set BrcSecTag12 41; set BrcSecTag34 42; #set CoreSecTag 54; set ColSecTagFiber1 14; set ColSecTagFiber23 15; set ColSecTagFiber4 16; set BeamSecTagFiber 6; set GirdSecTagFiber 7; set BraceSecTagFiber12 43; set BraceSecTagFiber34 44; #set CoreSecTagFiber 88; set ColMatTag 1; set BeamMatTag 2; set GirdMatTag 3; set BraceMatTag 4; #set CoreMatTag 44; #set CoreFiberMatTag 55; set SecTagTorsion 90; ####################################### if {$SectionType == "Elastic"} { # material properties: set Es 350.0e3; # Steel Young's Modulus (KN/m2) set nu 0.3; # Poisson's ratio set Gs [expr $Es/2./[expr 1+$nu]]; # Torsional stiffness Modulus (KN/m2) ???why it is different from 77000000.0; set J $Ubig; # set large torsional stiffness # column sections: HSS 178x178x8 set AgCol 0.0049; # cross-sectional area set IzCol 0.0000234; # moment of Inertia (Local Coordinates) (m4)
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set IyCol 0.0000234; # moment of Inertia (Local Coordinates) (m4) # Brace sections: HSS 89x89x6 set AgCol 0.00188; # cross-sectional area set IzCol 0.0000021; # moment of Inertia (Local Coordinates)(m4) set IyCol 0.0000021; # moment of Inertia (Local Coordinates)(m4) # beam sections: W200x15 set AgBeam 0.00191; # cross-sectional area set IzBeam 0.0000128; # moment of Inertia (Local Coordinates) (m4) set IyBeam 0.000001; # moment of Inertia (Local Coordinates) (m4) # girder sections: W200x15 set AgGird 0.00191; # cross-sectional area set IzGird 0.0000128; # moment of Inertia (m4) set IyGird 0.000001; # moment of Inertia (m4) section Elastic $ColSecTag $Es $AgCol $IzCol $IyCol $Gs $J section Elastic $BeamSecTag $Es $AgBeam $IzBeam $IyBeam $Gs $J section Elastic $GirdSecTag $Es $AgGird $IzGird $IyGird $Gs $J ####set matIDhard 1; # material numbers for recorder (this stressstrain recorder will be blank, as this is an elastic section) ########################################## } elseif {$SectionType == "FiberSection"} { # define MATERIAL properties puts "FiberSection"; set Fy 350.0e3; set Es 2.0e8; # Steel Young's Modulus set nu 0.3; set Gs [expr $Es/2./[expr 1+$nu]]; # Torsional stiffness Modulus ???why it is different from 77000000.0; ######### steel02 parameters########### # steel02 parameters set R0 18; # control the transition from elastic to plastic branches -- make more like the rest. set cR1 0.925; # control the transition from elastic to plastic branches set cR2 0.15; # control the transition from elastic to plastic branches set Bs 0.01; # strain-hardening ratio niaxialMaterial Steel02 $BraceMatTag $Fy $Es $Bs $R0 $cR1 $cR2; ########### ELEMENT properties ############## # Structural-Steel properties ######### define sections for braces:
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# first & second floors: # command: HSSsection secID matID d t nfdy nfty nfdz nftz # HSS 89x89x6.4 HSSsection $BraceSecTagFiber12 $BraceMatTag 0.089 0.00635 12 2 12 2 # third & fourth floors: # HSS 76x76x4.8 HSSsection $BraceSecTagFiber34 $BraceMatTag 0.076 0.00478 12 2 12 2 # assign torsional Stiffness for 3D Model uniaxialMaterial Elastic $SecTagTorsion $Ubig section Aggregator $BrcSecTag12 $SecTagTorsion T -section $BraceSecTagFiber12 ; # ???should i assign torsional Stiffness to the braces at all? section Aggregator $BrcSecTag34 $SecTagTorsion T -section $BraceSecTagFiber34 ; ############################################## # Distributed Plastic Hinges ################################################ ## Distributed Plastic Hinges for beams and first floor columns## # First floor columns and all the beams HSS127x127x4.8 & W100x19: set Mycol 28.22; # yield moment (KN.m) set Acol 2.06e-3; set Icol 5.12e-6; set Icoly 5.12e-6 ; set Jh 7.81e-6; set Mybeam 31.5; # yield moment at plastic hinge location set Abeam 2.48e-3; set Ibeam 4.77e-6; set Ibeamy 1.61e-6; set Jh 6.36e-8 ; # define plastic hinge lengths set Lp_c1 [expr 0.004*$LCol1]; # length of plastic hinge for floor columns set Lp_c2 [expr 0.004*$LCol2]; # length of plastic hinge for space columns set Lp_b2 [expr 0.004*$LBeam]; # length of plastic hinge for beams (corridor beam length is used) # determine stiffness modifications so that the strain hardening of the plastic hinge region captures the actual frame member's strain hardening # Reference: Ibarra, L. F., and Krawinkler, H. (2005). "Global collapse of frame structures under seismic excitations," Technical Report 152, # The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. set n_c1 [expr $LCol1/$Lp_c1]; # rotational stiffness ratio: (floor column plastic hinge region) / (actual floor column)
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set n_c2 [expr $LCol2/$Lp_c2]; # rotational stiffness ratio: (space column plastic hinge region) / (actual space column) set n_b2 [expr $LBeam/$Lp_b2]; # rotational stiffness ratio: (beam plastic hinge region) / (actual beam) # calculate rotational stiffness for plastic hinges set Ks_col_1 [expr 6.0*$Es*$Icol/$Lp_c1]; # rotational stiffness of floor column hinges set Ks_col_2 [expr 6.0*$Es*$Icol/$Lp_c2]; # rotational stiffness of space column hinges set Ks_beam_2 [expr 6.0*$Es*$Ibeam/$Lp_b2]; # rotational stiffness of beam hinges set Kmem_col_1 [expr 6.0*$Es*$Icol/$LCol1]; # rotational stiffness of floor columns set Kmem_col_2 [expr 6.0*$Es*$Icol/$LCol2]; # rotational stiffness of space columns set Kmem_beam_2 [expr 6.0*$Es*$Ibeam/$LBeam]; # rotational stiffness of beams ####################################################### # Define Rotational Springs for Plastic Hinges ####################################################### # define rotational spring properties and create spring elements using "rotSect2DModIKModel" procedure # rotSect2DModIKModel creates a section with an elastic axial and bilinear flexural response based on Modified Ibarra Krawinkler Deterioration Model # references provided in rotSect2DModIKModel.tcl # input values for Story 1 column springs set McMy 1.05; # ratio of capping moment to yield moment, Mc / My set LS 1000.0; # basic strength deterioration (a very large # = no cyclic deterioration) set LK 1000.0; # unloading stiffness deterioration (a very large # = no cyclic deterioration) set LA 1000.0; # accelerated reloading stiffness deterioration (a very large # = no cyclic deterioration) set LD 1000.0; # post-capping strength deterioration (a very large # = no deterioration) set cS 1.0; # exponent for basic strength deterioration (c = 1.0 for no deterioration) set cK 1.0; # exponent for unloading stiffness deterioration (c = 1.0 for no deterioration) set cA 1.0; # exponent for accelerated reloading stiffness deterioration (c = 1.0 for no deterioration) set cD 1.0; # exponent for post-capping strength deterioration (c = 1.0 for no deterioration) set th_pP 0.025; # plastic rot capacity for pos loading set th_pN 0.025; # plastic rot capacity for neg loading set th_pcP 0.3; # post-capping rot capacity for pos loading
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set th_pcN 0.3; # post-capping rot capacity for neg loading set ResP 0.4; # residual strength ratio for pos loading set ResN 0.4; # residual strength ratio for neg loading set th_uP 0.4; # ultimate rot capacity for pos loading set th_uN 0.4; # ultimate rot capacity for neg loading set DP 1.0; # rate of cyclic deterioration for pos loading set DN 1.0; # rate of cyclic deterioration for neg loading set a_mem [expr ($Mycol*($McMy-1.0)) / ($Kmem_col_1*$th_pP)]; # strain hardening ratio of member set bddm [expr ($a_mem)/(1.0+$n_c1*(1.0-$a_mem))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: Eqn B.5 is incorrect) # define column plastic hinge sections # command: rotSect2DModIKModel id ndR ndC K asPos asNeg MyPos MyNeg LS LK LA LD cS cK cA cD th_p+ th_p- th_pc+ th_pc- Res+ Res- th_u+ th_u- D+ D- set sec_c1 101; # section ID for floor column section rotSect2DModIKModel $sec_c1 $Es $Acol $Ks_col_1 $bddm $bddm $Mycol [expr -$Mycol] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # recompute strain hardening since space column is not the same height as floor column set a_mem [expr ($Mycol*($McMy-1.0)) / ($Kmem_col_2*$th_pP)]; # strain hardening ratio of member set bddm [expr ($a_mem)/(1.0+$n_c2*(1.0-$a_mem))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5) set sec_c2 102; # section ID for space column section rotSect2DModIKModel $sec_c2 $Es $Acol $Ks_col_2 $bddm $bddm $Mycol [expr -$Mycol] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # define beam plastic hinge sections # redefine the rotations since they are not the same set th_pP 0.02; set th_pN 0.02; set th_pcP 0.16; set th_pcN 0.16; set a_mem [expr ($Mybeam*($McMy-1.0)) / ($Kmem_beam_2*$th_pP)]; # strain hardening ratio of member set bddm [expr ($a_mem)/(1.0+$n_b2*(1.0-$a_mem))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5)
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#beam sections set sec_b2 202; # section ID for beams rotSect2DModIKModel $sec_b2 $Es $Abeam $Ks_beam_2 $bddm $bddm $Mybeam [expr -$Mybeam] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; ########################################################### #### Distributed Plastic Hinges for 2&3rd floor columns#### # 2nd & 3rd floor columns HSS102x102x6.4: set Mycol23 21.7; # yield moment (KN.m) set Acol23 2.11e-3; set Icol23 3.16e-6; set Icoly23 3.16e-6; set Jh23 4.82e-6 ; # define plastic hinge lengths set Lp_c1 [expr 0.004*$LCol1]; # length of plastic hinge for floor columns set Lp_c2 [expr 0.004*$LCol2]; # length of plastic hinge for space columns # determine stiffness modifications so that the strain hardening of the plastic hinge region captures the actual frame member's strain hardening # Reference: Ibarra, L. F., and Krawinkler, H. (2005). "Global collapse of frame structures under seismic excitations," Technical Report 152, # The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. set n_c1 [expr $LCol1/$Lp_c1]; # rotational stiffness ratio: (floor column plastic hinge region) / (actual floor column) set n_c2 [expr $LCol2/$Lp_c2]; # rotational stiffness ratio: (space column plastic hinge region) / (actual space column) # calculate rotational stiffness for plastic hinges set Ks_col23_1 [expr 6.0*$Es*$Icol23/$Lp_c1]; # rotational stiffness of floor column hinges set Ks_col23_2 [expr 6.0*$Es*$Icol23/$Lp_c2]; # rotational stiffness of space column hinges set Kmem_col23_1 [expr 6.0*$Es*$Icol23/$LCol1]; # rotational stiffness of floor columns set Kmem_col23_2 [expr 6.0*$Es*$Icol23/$LCol2]; # rotational stiffness of space columns ###Define Rotational Springs for Plastic Hinges#### set a_mem23 [expr ($Mycol23*($McMy-1.0)) / ($Kmem_col23_1*$th_pP)]; # strain hardening ratio of member set bddm23 [expr ($a_mem23)/(1.0+$n_c1*(1.0-$a_mem23))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: Eqn B.5 is incorrect)
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# define column plastic hinge sections # command: rotSect2DModIKModel id ndR ndC K asPos asNeg MyPos MyNeg LS LK LA LD cS cK cA cD th_p+ th_p- th_pc+ th_pc- Res+ Res- th_u+ th_u- D+ D- set sec23_c1 23101; # section ID for floor column section rotSect2DModIKModel $sec23_c1 $Es $Acol23 $Ks_col23_1 $bddm23 $bddm23 $Mycol23 [expr -$Mycol23] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # recompute strain hardening since space column is not the same height as floor column set a_mem23 [expr ($Mycol23*($McMy-1.0)) / ($Kmem_col23_2*$th_pP)]; # strain hardening ratio of member set bddm23 [expr ($a_mem23)/(1.0+$n_c2*(1.0-$a_mem23))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5) set sec23_c2 23102; # section ID for space column section rotSect2DModIKModel $sec23_c2 $Es $Acol23 $Ks_col23_2 $bddm23 $bddm23 $Mycol23 [expr -$Mycol23] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; ####################################################### ### Distributed Plastic Hinges for 4th floor columns### # 4th floor columns HSS76x76x4.8: set Mycol4 9.2; # yield moment (KN.m) set Acol4 1.19e-3; set Icol4 1.0e-6; set Icoly4 1.0e-6 ; set Jh4 1.52e-6 ; # define plastic hinge lengths set Lp_c1 [expr 0.004*$LCol1]; # length of plastic hinge for floor columns set Lp_c2 [expr 0.004*$LCol2]; # length of plastic hinge for space columns # determine stiffness modifications so that the strain hardening of the plastic hinge region captures the actual frame member's strain hardening # Reference: Ibarra, L. F., and Krawinkler, H. (2005). "Global collapse of frame structures under seismic excitations," Technical Report 152, # The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. set n_c1 [expr $LCol1/$Lp_c1]; # rotational stiffness ratio: (floor column plastic hinge region) / (actual floor column) set n_c2 [expr $LCol2/$Lp_c2]; # rotational stiffness ratio: (space column plastic hinge region) / (actual space column)
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# calculate rotational stiffness for plastic hinges set Ks_col4_1 [expr 6.0*$Es*$Icol4/$Lp_c1]; # rotational stiffness of floor column hinges set Ks_col4_2 [expr 6.0*$Es*$Icol4/$Lp_c2]; # rotational stiffness of space column hinges set Kmem_col4_1 [expr 6.0*$Es*$Icol4/$LCol1]; # rotational stiffness of floor columns set Kmem_col4_2 [expr 6.0*$Es*$Icol4/$LCol2]; # rotational stiffness of space columns #### Define Rotational Springs for Plastic Hinges ### set a_mem4 [expr ($Mycol4*($McMy-1.0)) / ($Kmem_col4_1*$th_pP)]; # strain hardening ratio of member set bddm4 [expr ($a_mem4)/(1.0+$n_c1*(1.0-$a_mem4))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: Eqn B.5 is incorrect) # define column plastic hinge sections # command: rotSect2DModIKModel id ndR ndC K asPos asNeg MyPos MyNeg LS LK LA LD cS cK cA cD th_p+ th_p- th_pc+ th_pc- Res+ Res- th_u+ th_u- D+ D- set sec4_c1 4101; # section ID for floor column section rotSect2DModIKModel $sec4_c1 $Es $Acol4 $Ks_col4_1 $bddm4 $bddm4 $Mycol4 [expr -$Mycol4] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; # recompute strain hardening since space column is not the same height as floor column set a_mem4 [expr ($Mycol4*($McMy-1.0)) / ($Kmem_col4_2*$th_pP)]; # strain hardening ratio of member set bddm4 [expr ($a_mem4)/(1.0+$n_c2*(1.0-$a_mem4))]; # modified strain hardening ratio (Ibarra & Krawinkler 2005, note: there is mistake in Eqn B.5) set sec4_c2 4102; # section ID for space column section rotSect2DModIKModel $sec4_c2 $Es $Acol4 $Ks_col4_2 $bddm4 $bddm4 $Mycol4 [expr -$Mycol4] $LS $LK $LA $LD $cS $cK $cA $cD $th_pP $th_pN $th_pcP $th_pcN $ResP $ResN $th_uP $th_uN $DP $DN; ############################################## } else { puts "No section has been defined" return -1 } ############################################## # Define Geometric Transformation
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############################################## # set up geometric transformations of element # separate columns and beams, in case of P-Delta analysis for columns # in 3D model, assign vector vecxz set IDColTransf 1; # all columns set IDColTransfs 6; set IDBeamTransf 2; # all beams (x dir) set IDGirdTransf 3; # all girders (z dir) set IDBrXTransf 4; # all braces in x direction set IDBrZTransf 5; # all braces in z direction set joff [expr $db/2]; set joffs [expr $LCol2*0.05]; geomTransf PDelta $IDColTransfs 0 0 1 -jntOffset 0.0 $joffs 0.0 0.0 -$joffs 0.0; geomTransf PDelta $IDColTransf 0 0 1 -jntOffset 0.0 $joff 0.0 0.0 -$joff 0.0; # only columns can have PDelta effects (gravity effects) geomTransf PDelta $IDBeamTransf 0 0 1 -jntOffset $joff 0.0 0.0 -$joff 0.0 0.0; geomTransf PDelta $IDGirdTransf 1 0 0 -jntOffset 0.0 0.0 $joff 0.0 0.0 -$joff; geomTransf Corotational $IDBrXTransf 1 1 0; geomTransf Corotational $IDBrZTransf 0 1 1; ######################################## # Define Elements ######################################## # (1) Original # Define Column Elements set numIntgrPts 5; # number of Gauss integration points for nonlinear curvature distribution # columns set N0col [expr 1000000-1]; # column element numbers set level 0 for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { # 1st floor columns for {set level 1} {$level <=1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns
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} # 1st spacing columns set N0col [expr 2000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } # 2&3rd floor columns set N0col [expr 3000000-1]; # column element numbers set level 0 for {set level 2} {$level <=$NStory-1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } # spacing columns set N0col [expr 2000000-1]; # column element numbers set level 0 for {set level 3} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} {
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set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } # last floor columns set N0col [expr 4000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec4_c2 $Lp_c2 $sec4_c2 $Lp_c2 $Es $Acol4 $Icol4 $Icoly4 $Gs $Jh $IDColTransf; # columns } } ############################################ ############################################ # (2) extra set N0col [expr 11000000-1]; # column element numbers set level 0 for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { if {$frame == 1 || $frame == $NFrame} { # do nothing } else { # 1st floor columns for {set level 1} {$level <=1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 31000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier]
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element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } # 1st spacing columns set N0col [expr 21000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } # 2&3rd floor columns set N0col [expr 31000000-1]; # column element numbers set level 0 for {set level 2} {$level <=$NStory-1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 31000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } # spacing columns set N0col [expr 21000000-1]; # column element numbers set level 0 for {set level 3} {$level <=$NStory} {incr level 1} {
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for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } # last floor columns set N0col [expr 41000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 71000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec4_c2 $Lp_c2 $sec4_c2 $Lp_c2 $Es $Acol4 $Icol4 $Icoly4 $Gs $Jh $IDColTransf; # columns } } } ######################################## ######################################## # (3) extra set N0col [expr 12000000-1]; # column element numbers set level 0 for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { # 1st floor columns for {set level 1} {$level <=1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier]
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set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 32000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } # 1st spacing columns set N0col [expr 22000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } } # 2&3rd floor columns set N0col [expr 32000000-1]; # column element numbers set level 0 for {set level 2} {$level <=$NStory-1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 32000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier]
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element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } } # 2&3rd spacing columns set N0col [expr 22000000-1]; # column element numbers set level 0 for {set level 3} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } } # last floor columns set N0col [expr 42000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 72000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec4_c2 $Lp_c2 $sec4_c2 $Lp_c2 $Es $Acol4 $Icol4 $Icoly4 $Gs $Jh $IDColTransf; # columns } } }
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} ########################################## ########################################## # (4) extra set N0col [expr 13000000-1]; # column element numbers set level 0 for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { if {$frame == 1 || $frame == $NFrame} { # do nothing } else { # 1st floor columns for {set level 1} {$level <=1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 33000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } } # 1st spacing columns set N0col [expr 23000000-1]; # column element numbers set level 0 for {set level 2} {$level <2} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$pier]
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set nodeJ [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec_c1 $Lp_c1 $sec_c1 $Lp_c1 $Es $Acol $Icol $Icoly $Gs $Jh $IDColTransf; # columns } } } # 2&3rd floor columns set N0col [expr 33000000-1]; # column element numbers set level 0 for {set level 2} {$level <=$NStory-1} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 33000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns } } } # 2&3rd spacing columns set N0col [expr 23000000-1]; # column element numbers set level 0 for {set level 3} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec23_c2 $Lp_c2 $sec23_c2 $Lp_c2 $Es $Acol23 $Icol23 $Icoly23 $Gs $Jh $IDColTransf; # columns }
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} } # last floor columns set N0col [expr 43000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == $NBay + 1} { # do nothing } else { set elemID [expr $N0col +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 73000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] element beamWithHinges $elemID $nodeI $nodeJ $sec4_c2 $Lp_c2 $sec4_c2 $Lp_c2 $Es $Acol4 $Icol4 $Icoly4 $Gs $Jh $IDColTransf; # columns } } } } } ################################################ ########## Define Beam Elements ################ ################################################ ###### beams -- parallel to X-axis######## # (1) # bay1 # ceiling beams set N0beam 5000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams
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} } } # bay2 # ceiling beams set N0beam 5000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay3 # ceiling beams set N0beam 5000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 3} {$bay <= 3} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ########################################## ########################################## # floor beams # bay1
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# floor beams set N0beam 6000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay2 # floor beams set N0beam 6000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay2 # floor beams set N0beam 6000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 3} {$bay <= 3} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay+1]
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element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ########################################## ########## Define Beam Elements ########## ########################################## ###### beams -- parallel to X-axis######## # (4) # bay1 # ceiling beams set N0beam 53000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay2 # ceiling beams set N0beam 53000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay+1]
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element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay3 # ceiling beams set N0beam 53000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 3} {$bay <= 3} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ###################################### ###################################### # floor beams # bay1 # floor beams set N0beam 63000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } }
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} # bay2 # floor beams set N0beam 63000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay3 # floor beams set N0beam 63000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 3} {$bay <= 3} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ############################################ ############################################ ########## girders -- parallel to Z-axis#### # (2) # Frame 1 # ceiling beams set N0gird 7000000; # gird element numbers for {set frame 1} {$frame <= 1} {incr frame 1} {
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for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 2 # ceiling beams set N0gird 7000000; # gird element numbers for {set frame 2} {$frame <= 2} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 3 # ceiling beams set N0gird 7000000; # gird element numbers for {set frame 3} {$frame <= 3} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay]
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element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf;; # Girds } } } # Frame 4 # ceiling beams set N0gird 7000000; # gird element numbers for {set frame 4} {$frame <= 4} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame n ... (copy and past each bay then just change "n" in: for {set frame n} {$frame <= n} {incr frame 1} { ############################################## ############################################## # Frame 1 # floor beams # girders -- parallel to Z-axis set N0gird 8000000; # gird element numbers for {set frame 1} {$frame <= 1} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } }
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# Frame 2 # floor beams # girders -- parallel to Z-axis set N0gird 8000000; # gird element numbers for {set frame 2} {$frame <= 2} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 3 # floor beams # girders -- parallel to Z-axis set N0gird 8000000; # gird element numbers for {set frame 3} {$frame <= 3} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 4 # floor beams # girders -- parallel to Z-axis set N0gird 8000000; # gird element numbers for {set frame 4} {$frame <= 4} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} {
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for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame n ... (copy and past each bay then just change "n" in: for {set frame n} {$frame <= n} {incr frame 1} { ############################################# ############################################# ########## girders -- parallel to Z-axis##### # (4) # Frame 1 # ceiling beams set N0gird 73000000; # gird element numbers for {set frame 1} {$frame <= 1} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 32000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 2 # ceiling beams set N0gird 73000000; # gird element numbers for {set frame 2} {$frame <= 2} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} {
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set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 32000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 3 # ceiling beams set N0gird 73000000; # gird element numbers for {set frame 3} {$frame <= 3} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 32000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 4 # ceiling beams set N0gird 73000000; # gird element numbers for {set frame 4} {$frame <= 4} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 32000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } ###############################################
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############################################### # Frame 1 # floor beams # girders -- parallel to Z-axis set N0gird 83000000; # gird element numbers for {set frame 1} {$frame <= 1} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 72000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 2 # floor beams # girders -- parallel to Z-axis set N0gird 83000000; # gird element numbers for {set frame 2} {$frame <= 2} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 72000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 3 # floor beams # girders -- parallel to Z-axis set N0gird 83000000; # gird element numbers for {set frame 3} {$frame <= 3} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} {
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for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 72000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame 4 # floor beams # girders -- parallel to Z-axis set N0gird 83000000; # gird element numbers for {set frame 4} {$frame <= 4} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 72000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element beamWithHinges $elemID $nodeI $nodeJ $sec_b2 $Lp_b2 $sec_b2 $Lp_b2 $Es $Abeam $Ibeam $Ibeamy $Gs $Jh $IDGirdTransf; # Girds } } } # Frame n ... (copy and past each bay then just change "n" in: for {set frame n} {$frame <= n} {incr frame 1} { ... ################################################ # # Define brace elements: ################################################ # xdir for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { if {$frame == 1 || $frame == 5} { # 1st floor braces # upward set N0br [expr 16000000-1]; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} {
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for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } # downward set N0br [expr 19000000-1]; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 32000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 3000000 + $frame*$Dframe+$level*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } # 2nd floor braces # upward set N0br [expr 26000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } # downward set N0br [expr 29000000-1]; # column element numbers set level 0
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for {set level 2} {$level <=2} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 32000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } # 3rd floor braces # upward set N0br [expr 26000000-1]; # column element numbers set level 0 for {set level 3} {$level <=3} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } # downward set N0br [expr 29000000-1]; # column element numbers set level 0 for {set level 3} {$level <=3} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 32000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns
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} } # last floor braces # upward set N0br [expr 36000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+($level+1)*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } # downward set N0br [expr 39000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set pier 2} {$pier <= 2} {incr pier 1} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 72000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + $frame*$Dframe+$level*$Dlevel+($pier+1)] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } #################################################### # z dir (4) & (2) for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == 4} { # 1st floor braces # upward
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set N0br [expr 46000000-1]; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # downward set N0br [expr 49000000-1]; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # 2nd floor braces # upward set N0br [expr 56000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier]
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set nodeJ [expr 3000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # downward set N0br [expr 59000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # 3rd floor braces # upward set N0br [expr 56000000-1]; # column element numbers set level 0 for {set level 3} {$level <=3} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } # downward set N0br [expr 59000000-1]; # column element numbers set level 0
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for {set level 3} {$level <=3} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 3000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } # last floor braces # upward set N0br [expr 66000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } # downward set N0br [expr 69000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 1 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 7000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns
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} } } } } #################################################### # z dir (2) & (1) for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} { if {$pier == 1 || $pier == 4} { # 1st floor braces # upward set N0br [expr 46000000-1]; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4} { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # downward set N0br [expr 49000000-1]; # column element numbers set level 0 for {set level 1} {$level <=1} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns }
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} } # 2nd floor braces # upward set N0br [expr 56000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # downward set N0br [expr 59000000-1]; # column element numbers set level 0 for {set level 2} {$level <=2} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag12 $IDColTransf; # columns } } } # 3rd floor braces # upward set N0br [expr 56000000-1]; # column element numbers set level 0
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for {set level 3} {$level <=3} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 3000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } # downward set N0br [expr 59000000-1]; # column element numbers set level 0 for {set level 3} {$level <=3} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 31000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } # last floor braces # upward set N0br [expr 66000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+($level+1)*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns
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} } } # downward set N0br [expr 69000000-1]; # column element numbers set level 0 for {set level [expr $NStory]} {$level <=$NStory} {incr level 1} { for {set frame 1} {$frame <=[expr $NFrame-1]} {incr frame 1} { if {$frame == 4 } { set elemID [expr $N0br +$level*$Dlevel + $frame*$Dframe+$pier] set nodeI [expr 71000000 + $frame*$Dframe+($level+1)*$Dlevel+$pier] set nodeJ [expr 7000000 + ($frame+1)*$Dframe+$level*$Dlevel+$pier] element nonlinearBeamColumn $elemID $nodeI $nodeJ $numIntgrPts $BrcSecTag34 $IDColTransf; # columns } } } } … ###################################################### ###################################################### ########## Define horizontal connection Elements ##### ###################################################### # It is assumed that the strength of horizontal connection is 1.3 times its adjacent element set Arig [expr 2.48e-3 * 1.3]; # elastic link area for beams at all floor (m2) set Irigy [expr 4.77e-6 * 1.3]; # elastic link moments of inertia for beams at all floor (m4) set Irigz [expr 1.61e-6 * 1.3]; # elastic link moments of inertia for beams at all floor (m4) set Jc [expr 6.36e-8 * 1.3]; ################################################ ###### connections -- parallel to X-axis######## # (1) # bay1 # ceiling connections set N0beam 85000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} {
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for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay2 # ceiling connections set N0beam 85000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ######################################### ######################################### # floor connections # bay1 # floor connections set N0beam 86000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay]
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set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay2 # floor connections set N0beam 86000000; # beam element numbers for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ###################################################### ###################################################### ########## Define horizontal connections Elements #### ###### connections-- parallel to X-axis######## # (4) # bay1 # ceiling connections set N0beam 853000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay+1]
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element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay2 # ceiling connections set N0beam 853000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 31000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 33000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay n ... (copy and past each bay then just change "n" in: for {set bay n} {$bay <= n} {incr bay 1} { ####################################### ####################################### # floor connections # bay1 # floor connections set N0beam 863000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= 1} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } # bay2
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# floor connections set N0beam 863000000; # beam element numbers for {set frame 2} {$frame <=[expr $NFrame -1]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= 2} {incr bay 1} { set elemID [expr $N0beam +$level*$Dlevel + $frame*$Dframe+ $bay] set nodeI [expr 71000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 73000000 + $frame*$Dframe+$level*$Dlevel+$bay+1] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDBeamTransf; # beams } } } ############################################# ############################################# ########## horizontal connections -- parallel to Z-axis######### # (2) # ceiling connections set N0gird 87000000; # gird element numbers for {set frame 1} {$frame <= [expr $NFrame -2] } {incr frame 1} { for {set level 2} {$level <= [expr $NStory]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 31000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDGirdTransf; # Girds } } } ############################################# ############################################# # floor connections # girders -- parallel to Z-axis set N0gird 88000000; # gird element numbers for {set frame 1} {$frame <= [expr $NFrame -2]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 1} {$bay <= $NBay+1} {incr bay 1} {
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set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 71000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDGirdTransf; # Girds } } } ############################################### ############################################### ########## girders -- parallel to Z-axis####### # (4) # ceiling connections set N0gird 873000000; # gird element numbers for {set frame 1} {$frame <= [expr $NFrame -2]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 32000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 33000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDGirdTransf; # Girds } } } ############################################### ############################################### # floor connections # girders -- parallel to Z-axis set N0gird 883000000; # gird element numbers for {set frame 1} {$frame <= [expr $NFrame -2]} {incr frame 1} { for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { for {set bay 2} {$bay <= $NBay} {incr bay 1} { set elemID [expr $N0gird + $level*$Dlevel +$frame*$Dframe+ $bay] set nodeI [expr 72000000 + $frame*$Dframe+$level*$Dlevel+$bay] set nodeJ [expr 73000000 + ($frame+1)*$Dframe+$level*$Dlevel+$bay] element elasticBeamColumn $elemID $nodeI $nodeJ $Arig $Es $Gs $Jc $Irigy $Irigz $IDGirdTransf; # Girds
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} } } ################################################ ################################################ # Mass Calculations # ################################################ ################################################ ####set DLroof 5.8; #(KN/m) set DLfloor 7.86; #(KN/m) set DLceiling 1.26; #(KN/m) set ColWeight 0.2; # HSS-section weight per length (KN/m) set BeamWeight 0.1; # W-section weight per length (KN/m) #Note: brace weight is assumed to be as part of the external wall weight. set QdlCol $ColWeight; # dead load distributed along columns (KN/m) set QdlBeamf $BeamWeight; # dead load distributed along floor beam (one-way slab) (KN/m) set QdlGirdf [expr $DLfloor + $BeamWeight ]; # dead load distributed along floor girders (KN/m) set QdlBeamc $BeamWeight; # dead load distributed along ceiling beam (one-way slab) (KN/m) set QdlGirdc [expr $DLceiling + $BeamWeight ]; # dead load distributed along ceiling girders (KN/m) # Note: Mode shapes and eigen vectors are function of mass and stiffness of the structure. They do not depend on the external load applied on the structure. # i.e. Mode shapes of the structure in the initial condition is not function of the load applied to it. # The seismic weight to be considered (weight assigned to mass command)is DL + 0.25 SL or it can even be just DL (seemingly for important structures we can assume DL + 0.2 LL) # Here seismic weight = DL set g 9.81; # -------------------------------------------------------------------------------------------------------------------------------- # Define GRAVITY LOADS, weight and masses # calculate dead load of frame, assume this to be an internal frame (do LL in a similar manner) # calculate distributed weight along the beam length set WeightCol1 [expr $QdlCol*$LCol1]; # total Column weight set WeightCol2 [expr $QdlCol*$LCol2]; set WeightBeamf [expr $QdlBeamf*$LBeam]; # total Beam weight set WeightGirdf [expr $QdlGirdf*$LGird]; # total Beam weight set WeightBeamc [expr $QdlBeamc*$LBeam]; set WeightGirdc [expr $QdlGirdc*$LGird]; # assign masses to the nodes that the columns are connected to # each connection takes the mass of 1/2 of each element framing into it (mass=weight/$g)
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#### ceiling levels #### set iFloorWeight1 "" set WeightTotal1 0.0; set sumWiHi1 0.0; # sum of storey weight times height, for lateral-load distribution for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { set FloorWeight1 0.0; if {$frame == 1 || $frame == $NFrame} { set GirdWeightFact 1; # 1x1/2girder on exterior frames } else { set GirdWeightFact 2; # 2x1/2girder on interior frames } for {set level 2} {$level <=[expr $NStory]} {incr level 1} { ; set FloorWeight 0.0 if {$level == [expr $NStory+1]} { set ColWeightFact 1; # one column in top story } else { set ColWeightFact 2; # two columns elsewhere } for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} {; if {$pier == 1 || $pier == [expr $NBay+1]} { set BeamWeightFact 1; # one beam at exterior nodes } else {; set BeamWeightFact 2; # two beams elewhere } set WeightNode1 [expr $ColWeightFact*($WeightCol1 + $WeightCol2)/4 + $BeamWeightFact*$WeightBeamc/2 + $GirdWeightFact*$WeightGirdc/2] set MassNode1 [expr $WeightNode1/$g]; set nodeID [expr 3000000 + $frame*$Dframe+$level*$Dlevel+$pier] mass $nodeID $MassNode1 0. $MassNode1 0. 0. 0.; # define mass set FloorWeight1 [expr $FloorWeight1+$WeightNode1]; } lappend iFloorWeight1 $FloorWeight1 set WeightTotal1 [expr $WeightTotal1+ $FloorWeight1] set sumWiHi1 [expr $sumWiHi1+$FloorWeight1*($level-1)*$LCol]; # sum of storey weight times height, for lateral-load distribution } } set MassTotal1 [expr $WeightTotal1/$g]; # total mass # floor levels set iFloorWeight2 "" set WeightTotal2 0.0; set sumWiHi2 0.0; # sum of storey weight times height, for lateral-load distribution
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for {set frame 1} {$frame <=[expr $NFrame]} {incr frame 1} { set FloorWeight2 0.0; if {$frame == 1 || $frame == $NFrame} { set GirdWeightFact 1; # 1x1/2girder on exterior frames } else { set GirdWeightFact 2; # 2x1/2girder on interior frames } for {set level 2} {$level <=[expr $NStory+1]} {incr level 1} { ; set FloorWeight 0.0 if {$level == [expr $NStory+1]} { set ColWeightFact 1; # one column in top story } else { set ColWeightFact 2; # two columns elsewhere } for {set pier 1} {$pier <= [expr $NBay+1]} {incr pier 1} {; if {$pier == 1 || $pier == [expr $NBay+1]} { set BeamWeightFact 1; # one beam at exterior nodes } else {; set BeamWeightFact 2; # two beams elewhere } set WeightNode2 [expr $ColWeightFact*($WeightCol1 + $WeightCol2)/4 + $BeamWeightFact*$WeightBeamf/2 + $GirdWeightFact*$WeightGirdf/2] set MassNode2 [expr $WeightNode2/$g]; set nodeID [expr 7000000 + $frame*$Dframe+$level*$Dlevel+$pier] mass $nodeID $MassNode2 0. $MassNode2 0. 0. 0.; # define mass set FloorWeight2 [expr $FloorWeight2+$WeightNode2]; } lappend iFloorWeight2 $FloorWeight2 set WeightTotal2 [expr $WeightTotal2+ $FloorWeight2] set sumWiHi2 [expr $sumWiHi2+$FloorWeight2*($level-1)*$LCol]; # sum of storey weight times height, for lateral-load distribution } } set MassTotal2 [expr $WeightTotal2/$g]; # total mass set MassTotal [expr $WeightTotal1/$g + $WeightTotal2/$g] #puts $MassTotal1 #puts $MassTotal2 #puts $MassTotal
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Acknowledgement
The financial support for this study from NSERC Discovery (Grant 371627-2009) and
the start-up funds from the University of Toronto is gratefully acknowledged.