importance of use of vertical ground motion in seismic

IMPORTANCE OF USE OF VERTICAL GROUND MOTION IN SEISMIC ANALYSIS FOR ISOLATED STRUCTURES

A Thesis submitted to the faculty of San Francisco State University

In partial fulfillment of the requirements for

the Degree

Master of Science

In

Engineering: Structural/Earthquake

by

Faizan Ahmad

San Francisco, California

August 2017

A S

• M G

Copyright by Faizan Ahmad

2017

CERTIFICATION OF APPROVAL

I certify that I have read Importance of Use of Vertical Ground Motion in Seismic Analysis

for Isolated Structures by Faizan Ahmad, and that in my opinion this work meets the criteria

for approving a thesis submitted in partial fulfillment of the requirement for the degree

Master of Science in Structural/Earthquake Engineering at San Francisco State University.

Wong, Assistant Professor

Associate Professor

Importance of Use of Vertical Ground Motion in Seismic Analysis for Isolated Structures

Faizan Ahmad San Francisco, California

2017

Seismic isolation is a mature technology with an excellent record for structural protection. However, the majority of work in this area has focused on purely utilizing horizontal components of ground motions. With the inclusion of vertical excitation, studies show more complex structural behavior and response. This study reviews the influence of vertical motion on isolated structures by investigating moment and shear demands. A 3- story, 2- bay steel frame is isolated using linear elastomeric bearings and modeled in SAP2000. A suite of ground motions is applied to the structure in three different ways. Firstly, the structure is subjected to purely horizontal excitation. Secondly, horizontal and vertical accelerations are applied simultaneously. For last case, structure is initially excited with horizontal acceleration in the isolation layer then excited vertically afterwards. The structure’s maximum allowable drift and moments are recorded at key locations. From these results, trends in the change in response with the vertical component are drawn. The results from these time history analyses are also compared against expected values from ASCE 7 equations for vertical motion. The accuracy of these results is discussed and evaluated for real facilities. With an improved understanding of vertical motion and capacity demands, these results can assist engineers in evaluating the effectiveness of current design guidelines as well as future areas of study to continue moving isolated structures towards performance based designs.

I certify that the Abstract is a correct representation of the content of this thesis.

Date

o S / o I / n

ACKNOWLEDGEMENTS

This report would not have been possible had it not been for my thesis advisor Prof. Jenna Wong. I would like to thank Prof. Wong for her guidance throughout my research work and development of this report. Due to her insightful feedback and supervision, I was able to accomplish my thesis.

I am also indebted to my graduate coordinator Prof. Cheng Chen for his continuous support during my time at San Francisco State University. He has always been approachable and greatly helpful whenever I needed to seek his advice.

I would also like to thank my friends and family who have always supported me and stood by me throughout my years of study.

Lastly, I am thankful to Fulbright program for believing in me and making it possible for me to pursue my dream at San Francisco State university.

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TABLE OF CONTENTS

A B STR A C T................................................................................................................................................................................ iv

1. IN T R O D U C T IO N ....................................................................................................................................................1

2. LITERATURE R E V IE W ...................................................................................................................................... 2

2.1 H istory of Isolation : ..........................................................................................................................................22.1.1 Fundamentals o f Seismic Isolation.................................................................. 3

2.2 T ypes of iso la t o r s ................................................................................................................................................42.2.1 Elastomeric Bearings............................................................................................................42.2.2 Curved Sliders.......................................................................................................................6

2.3 ISOLATION DESIGN PROCEDURES.......................................................................................................................... 72.3.1 Overview o f Equivalent Lateral Force (ELF) procedure....................................................... 72.3.2 ELF procedure step-by-step.................................................................................................. 9

2.4 Vertical g roun d m o t io n s .............................................................................................................................. 102.4.1 Time lag between peak values for vertical and horizontal ground motions...........................12

3. A N ALY TICA L M ODEL D E SIG N .................................................................................................................. 14

3.1 Respo n se Spec tr u m ............................................................................................................................................. 143.2 SAP2000 M odel D ev elo pm en t : ..................................................................................................................... 15

3.2.1 Modal analysis.................................................................................................................... 153.2.2 Pushover analysis..............................................................................•'................................16

3.3 Isolator D e s ig n ................................................................................................................................................... 173.3.1 Design displacement and Checks.........................................................................................19

3.4 Isolated m odel in S A P 2000 ............................................................................................................................203.5 E ffectiv e M odal mass using M odal participation fa c t o r s ..........................................................23

4. ANALYSIS P R O C ED U R E ................................................................................................................................ 24

4.1 G round acceleration h ist o r ie s .................................................................................................................. 254.2 Load cases a nd c o m b in a t io n s ......................................................................................................................29

5. COM PA RA TIV E S T U D Y ..................................................................................................................................35

5.1 P lotting T e c h n iq u e ........................................................................................................................................... 365.2 Earthqu ake 1 -Christ C h u r c h , N ew Z ea l a n d .......................................................................................365.3 Earthqu ake 2 - Lom a Prieta , U S A ...............................................................................................................395.4 Earthqu ake 3 - N o r th r id g e , U S A ...............................................................................................................425.5 Earthqu ake 4 - Im perial V alley , U S A ..................................................................................................... 45

6. CO N C LU SIO N ....................................................................................................................................................... 48

7. R E FE R E N C E S ........................................................................................................................................................49

A P P E N D IC E S ...........................................................................................................................................................................52

LIST OF TABLES

Table 2.1: V/H ratio for seismic events [30].......................................................................... 10Table 3.1: Mass calculation......................................................................... ............................ 14Table 3.2: Comparison of Eigen values with SAP2000 modal analysis for fixed supportstructure........................................................................................................... ............................ 16Table 3.3:Isolator design properties............................................................ ............................ 19Table 3.4: ASCE Elastic Base Shear Requirement............................................................... 19Table 3.5: ASCE Re-centering Req. per ASCE 17.2.4.4......................... ............................ 19Table 3.6: Comparison of Eigen values with SAP2000 modal analysis for Isolatedstructure....................................................................................................................................... 21Table 3.7: Effective modal masses and modal participation factors.... ............................ 23Table 3.8: Modal Participation factors - SAP2000.................................. ............................ 23Table 4.4.1: Ground motion - Imperial Valley, U SA .............................. ............................ 25Table4.4.2: Ground motion - Christchurch, New Zealand..................... ............................ 26Table 4.4.3: Ground motion - Loma Prieta, USA.................................... ............................ 27Table 4.4.4: Ground motion - Northridge, U SA...................................... ............................ 28

LIST OF FIGURES

Figure 2.1: Force-Displacement trade-off for period shift[14]............................................... 4Figure 2.2: Elastomeric Bearings a) Low damping natural rubber bearing b) Lead rubberbearing c) High damping rubber bearing[16]............................................................................5Figure 2.3: Bilinear hysteresis loop for seismic bearing[17]...................................................6Figure 2.4:Single pendulum friction bearing [18].................................................................... 6Figure 2.5: Relationship between Effective damping, Pe and Damping coefficient, Bd.... 8 Figure 2.6:Ground motion time history of El-Centro 1940 a) vertical acceleration b)horizontal acceleration [37]....................................................................................................... 12Figure 3.1 Maximum spectral accelerations............................................................................ 14Figure 3.2: Spectra for (a) Design response and (b) MCER response [25].........................15Figure 3.3: SAP2000 model for analysis..................................................................................16Figure 3.4: a) generalized push-over curve with safety limits b) SAP2000 push-overcurve...............................................................................................................................................17Figure 3.5: High damping rubber isolator[26].........................................................................18Figure 3.6:Isolator properties in SAP2000..............................................................................21Figure 3.7horizontal orthogonal direction properties of isolator........................................22Figure 3.8:Push over Analysis of isolated structure...............................................................22Figure 4.1: Time History- Imperial Valley U SA .................................................................... 26Figure 4.2: Time History- Christchurch, New Zealand......................................................... 27Figure 4.3 Time History- Loma Prieta USA............................................................................28Figure 4.4: Time History- Northridge USA.............................................................................29Figure 4.5: SAP2000 load case for Case -1 ............................................................................. 30Figure 4.6: Load combination for Case-1................................................................................ 30Figure 4.7: SAP2000 load case for Case -2 ............................................................................. 31Figure 4.8: Load combination for Case-2............... 32Figure 4.9: SAP2000 load case for Case -3 .............................................................................33Figure 4.10: Load combination for Case-3.............................................................................. 33Figure 5.1: Time History (Pages Road P.R) - Christ Church, New Zealand...................... 37Figure 5.2:Time History (GLWS) - Christ Church, New Zealand.......................................37Figure 5.3: Percentage difference for moments based on (a) Distance from source and (b)PGA (Christ Church).................................................................................................................. 38Figure 5.4:Percentage difference for shear based on (a) Distance from source and (b)PGA (Christ Church).................................................................................................................. 39Figure 5.5: Time History (LGPC) - Loma Prieta, U SA ........................................................ 40Figure 5.6:Time History (Larkspur Ferry Terminal) - Loma Prieta, U S A ........................ 40Figure 5.7: Percentage difference for moments based on (a) Distance from source and (b)PGA (Loma Prieta)..................................................................................................................... 41

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Figure 5.8:Percentage difference for shear based on (a) Distance from source and (b)PGA (Loma Prieta)........................... 42Figure 5.9: Time History (Jensen Filter Plant) - Northridge, USA..................................... 43Figure 5.10: Percentage difference for moments based on (a) Distance from source and(b) PGA (Northridge).................................................................................................................. 43Figure 5.11: Percentage difference for shear based on (a) Distance from source and (b)PGA (Northridge)....................................................................................................................... 44Figure 5.12:Time History (Brawley Airport) - Imperial valley, U SA ................................ 45Figure 5.13: Time History (Coachella Canal # 4) - Imperial valley, U SA ..........................46Figure 5.14: Percentage difference for moments based on (a) Distance from source and(b) PGA (Imperial Valley)......................................................................................................... 46Figure 5.15Percentage difference for shear based on (a) Distance from source and (b) PGA (Imperial Valley)................................................................................................................47

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1 INTRODUCTION

Engineers use different passive and active earthquake protection devices for energy dissipation during a seismic event. One of the passive approach is seismic isolation that has been proven to be most efficient structural design techniques to limit ground acceleration input during a seismic event. Seismic isolation limits the probability of structural and non-structural damage to the building by reducing the yielding of structural members and relative movement of non- structural components in a building envelope.

Studies [1], [2], and [3] show horizontal accelerations play a significant role in the excitation of a structure under seismic event. Peak vertical accelerations are normally less than those in horizontal planes. But, recent worldwide earthquakes have exhibited peak vertical accelerations exceeding peak horizontal values in near field locations [4]. Generally, peak vertical accelerations decrease with increased distance from the epicenter with higher values usually associated with near-fault recordings [5]. Given this knowledge, it is important to consider how current code provisions address vertical excitation.

Per ASCE 7-05, vertical seismic loads are determined using a fraction of the short period response acceleration values (SDS). Per the current ASCE 7-10 provisions, the design response spectrum in the vertical direction (Sav values) is now used that is dependent on not only Sds but also Cv, a ratio representing horizontal to vertical response [6]. With this update, forces generated in the structure have a modified yet limited correspondence of vertical accelerations. ASCE 7-10 has defined procedures for use of time history analysis for seismic strength designs depending on site locations. Considering the direction and time of application of horizontal and vertical components of strong motion records affect the structural response differently. The application of time histories individually and simultaneously affects the yield behavior and mechanism of members in the isolated superstructure.

The work presented is focused on identifying post-yield structural response variations of an isolated system for near and far field records using vertical ground motions in parallel as well as lagging to horizontal ground motions.

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2 LITERATURE REVIEW

2.1 History of Isolation:

The concept of seismic isolation was first introduced in Tokyo, Japan by John Milne in 1885, as he used ball bearings and dished cast iron plates to isolate the base of a building [7]. In the US, the first application of base isolation was not until 1985 when the Foothill Communities Law and Justice Center in California was seismically isolated using high damping rubber bearings.

Although seismic isolation has existed for over 100 years, there has been slow progression to its inclusion in code standards, especially in the US. Provisions for designing seismically isolated buildings also known as blue book were first published by Structural Engineers Association of California in 1959 and adopted in parts by the Uniform Building Code (UBC) in 1961 [8]. Improved versions were made part of building code by American Society of Civil Engineers (ASCE) first time in ASCE 7-93 standards. These seismic provisions evolved with new research adding to each of future Code revisions from 1995 to 2016 [9]. Chapters 11 through 23 in ASCE 7-16 provide seismic specifications for building components. Chapter 17 deals with seismic design requirements for isolated structures. Even with the inclusion to ASCE codes and well acknowledged performance of this technique, surprisingly, buildings including critical facilities using seismic isolation for retrofits or new construction were very hard to find. By 2013, the United States was said to have about 125 isolated structures compared to those of 6500 building in Japan [10]. The slow movement to wide-use of this technology can be attributed to a number of challenges including: economics, culture and regulations.

An important factor driving the use of seismic isolation is the additional building cost; this is an increase of approximately 5-15% to the standard framing cost. Most developers as well as consumers look at this cost as a deal breaker, ignoring the performance of this system in case of a seismic event. Buildings with seismic isolation systems have proved to perform extremely well, maintaining functional purposes and alleviating life hazards. Secondly, cultural barriers resist the adoption of seismic isolation techniques on a broader layout. In the United States, the states on the west coast are treated as the ones prone to earthquakes. So, the problem becomes localized. The combined national effort for joining the research and practice of seismic isolation does not seem to grow in this kind of perceptions. Another barrier is complexity of the code and extra requirements for the design of seismically isolated building that does not let the designer and the owner to go through the review process easily. Other seismic strength design techniques have relatively simple procedures and less requirements for the design. Extra regulatory requirements coupled with additional costs for seismic isolation system and lack of social awareness among owners and designers, trend is to opt for a different solution than seismic isolation.

Performance of Ishinomaki Red Cross Hospital in M9.0 Great Tohoku Earthquake provides evidence on its effectiveness. The hospital was seismically isolated as it reduced floor response

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accelerations and prevented the structural damage in the main frame envelope as well as containing the movement of medical equipment [11]. Another example of a base isolated structure performing well in an earthquake is the University of Southern California (USC) hospital building during the 1994 Northridge earthquake. The top floor experienced peak ground accelerations of 0.2lg, that is almost 50 percent less than the peak ground acceleration at the base level, 0.37g. The superstructure stayed elastic during the earthquake and drifts recorded were 30 percent less of the code specifications [12]. Additional examples and more information on case studies for seismic isolation can be found in [12] and [15] for further reading.

2.1.1 Fundamentals of Seismic Isolation

In a typical condition of a fixed support structure, the connection between the foundation and the super-structure is rigid, limiting the translation of building in horizontal planes. A seismically isolated system consists of a layer of flexible devices called base or seismic isolators placed between the foundation and the superstructure [13]. These isolator devices have a maximum displacement dependent upon a design criterion where the system increases the structure’s fundamental period, effectively reducing accelerations and increasing displacement (concentrated in the isolation layer). Most seismic isolators move in the orthogonal horizontal planes while providing relatively high vertical stiffness to support gravity loading.

In general, when ground accelerations are applied to a stiff structure, energy is directly inputted into it causing it to ring similar to a tuning fork. The displacements observed would be minimal but dependent on the structural stiffness of the system. However, floor accelerations will be high with increasing emphasis across the height of the structure. Dissipation of this seismic energy would have to result from the yielding of members throughout the structure. The goal as an engineer is to reduce the effect of ground accelerations on the structure while also limiting deformation and achieving a seismically stable response. This is achieved by increasing the period of the building or adding damping to the structure. Use of base isolation makes the structure more flexible and decreases the input energy into the superstructure. By shifting the period of the system to new higher values this results in an increased displacement but lower accelerations.

Ti ►Period

Figure 2.1: Force-Displacement trade-off for period shift[14]

As the lateral stiffness is reduced the effective period of the super structure is increased. As seen in the equation below, period and stiffness are inversely related.

2.2 Types of isolators

Numerous types of isolation bearings have been developed and used worldwide. All of these bearings can be placed into one of the two categories below:

1. Elastomeric2. Curved sliders

2.2.1 Elastomeric Bearings

Elastomeric bearings are set up using alternative layers of laminated rubber with intermediate layers of steel shims. Depending on the type of elastomeric and steel components used, these bearings can be further divided into three subcategories. These include low damping, high damping and lead plug rubber bearings. In low damping rubber bearings, the natural rubber provides low lateral stiffness allowing a relative displacement for the bearing in the direction of lateral loading. Use of steel shims provides vertical stiffness for the applied gravity loads.

In high damping rubber isolators, carbon black and some other fillers are used in elastomers used to achieve higher level of damping [15]. The shear modulus of high-damping elastomer generally ranges between 0.05 ksi and 0.2 ksi. At large shear strains, the modulus and energy dissipation and shear modulus increase at large shear strains and is beneficial to different levels

2.1

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of input. For a small input the system is stiff, reasonably linear at design level input, and capable to minimize displacements when higher values of forces are exerted than design criteria.

Lead plug rubber bearings are constructed with a lead core in the center and low damping elastomers. Hysteresis energy dissipation is provided by the lead core while the lateral stiffness of elastomers is responsible for seismic displacements in the isolation system.

p

(a) (b) ( 0

Figure 2.2: Elastomeric Bearings a) Low damping natural rubber bearing b) Lead rubber bearing c) High damping rubber bearing [16]

The principle of an isolator is the combined effect of a spring and a damper. A typical isolator requires damping property that provides energy dissipation in isolation system to avoid superstructure yielding, and flexible movement in a horizontal 2D plane enabling the period shift. In addition, bearing is required to come to its original position after going through one or more cycles. Isolators are designed to have an initial lateral stiffness that restrains isolator movement until the lateral force reaches a certain value, Qd. Isolator starts displacing only when the input force is greater than the defined condition for static behavior. Adding a spring element fulfills these two requirements by providing restoring force and initial stiffness.

For an effective isolation system, the following are four basic requirements:

1. Initial Stiffness2. Flexible Movement3. Energy Dissipation4. Re-centering Capability

Different types of isolators provide different mechanisms to meet these requirements. Initial stiffness is usually satisfied with friction between the surfaces or the yield strength of lead members present in form of shims or lead core. For the second requirement, low lateral stiffness of an isolator plays major role. Lead rubber bearings can provide displacements equal to their

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diameter in moving either direction. Use of friction pendulums as flat or sliding bearings enables required displacements.

Inelastic deformation of lead core and steel shims as well as friction present between the rollers in friction pendulums provide energy dissipation that is described as a damping property. In rubber isolators, elastomers also play key role in damping depending on the type of rubber used. High damping rubber bearings mostly depend on material properties of rubber for energy dissipation. The amount of energy dissipated can be calculated via the force-displacement relationship. The area under the curve is equal to the energy dissipated during one cycle. A typical bilinear hysteresis behavior of force displacement loading cycle is displayed in Fig. 2.3.

Force

Re-centering is mostly achieved by curved surfaces in friction pendulums and rubber elasticity in elastomeric isolators.

2.2.2 Curved Sliders

Curved sliders come as single, double or triple pendulum bearings where each bearing provides more displacements than its predecessor. In friction pendulum sliding (FPS) bearings, an articulated slider is placed on concave chrome surface to isolate top surface from bottom part. Fig 1.2 shows illustration of a typical single FPS bearing.

FPS Isolator section

Figure 2.4:Single pendulum friction bearing [18]

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2.3 Isolation design procedures

ASCE 7-10 [19] requires that the design of an isolated system be based on results obtained from the Equivalent Lateral Force (ELF) procedure, response spectrum analysis, or time history analysis. ELF procedure provides preliminary design specifications and usually defines the minimum design requirements when used without any dynamic analysis. ELF assumes the superstructure to be rigid with minimum story level drift as most of the lateral movement is captured by the isolation plane.

On the other hand, dynamic analyses require nonlinear modelling of elements, inclusion of uplift and overturning effects, and accountability of change of isolator properties over the periodic loading. These factors allow dynamic procedures to be more conservation and sometimes under design [20]. In most of the cases time history method is not much useful due to the number of analyses required, complexity of results and need to combine different types of response at each point to consider generalized behavior and need to consider mass eccentricity and accidental torsion at element level.

Table C l3.2-1 inNEHRP commentary specifies lower-bound limits on isolation system design displacements and structural-design forces as a percentage of the values calculated by the ELF method to ensure consistency in the design of isolated structures [21]. Given the capability of ELF procedure to capture maximum response and simplicity, this procedure is widely used for initial design of the isolation system. Dynamic procedures may then be used to verify and refine the design for more conservative results depending on the project scale and importance factor.

2.3.1 Overview of Equivalent Lateral Force (ELF) procedure

ELF is a static design procedure based on force-displacement hysteresis analysis of a structure. The equations used in the equivalent lateral force method are based on 1-second spectral accelerations, Sdi and the assumption of inverse proportional behavior of design response spectrum at long periods. The ELF procedure is a linear analysis so to account for nonlinear isolator behavior equations incorporate amplitude-dependent values of effective stiffness and damping assuming lateral displacement occurs in the isolation plane and the superstructure is rigid [22], Code requires design displacements and forces to be based on deformation characteristics of the isolation system.

Per ASCE 7-10 Section 17.5.3, an isolation system must be designed for a minimum lateral earthquake displacement in both of orthogonal horizontal planes, D d .

i _ gSpiTpD 4n2BD 2.2

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where Bd is a damping factor based on Table 17.5-1. This equation describes the peak (spectral) displacement of a single-degree-of-freedom (SDOF) system. S di in this equation represents 5% damped spectral acceleration response at a site in units of g-s.

Table 17.5-1 Damping Coefficient, BD or BM

Effective Damping, (3D or (3M (percentage of critical)0’* Bd or B m Factor

<2 0.85 1.0

10 1.220 1.530 1.740 1.9

>50 2.0

Figure 2.5: Relationship between Effective damping, fie and Damping coefficient, Bd

The effective period of an isolated structure is dependent on the effective seismic weight and the effective stiffness of the isolation system at the minimum design displacement.

Td = 2n 1 - ^ — 2.3'V j D m i n 9

Equation 2.3 shows the calculation of effective period of an isolated system. Replacing K.Dmin with K.Mmin provides period of structure at minimum displacement. Use of minimum effective stiffness produces larger estimates of effective period to avoid any under-design scenarios.

ASCE 7-10 requires the use of maximum effective stiffness at design displacement in the horizontal direction, kDmax, to determine lateral seismic design forces, Vs, for superstructure and base structure during a seismic event. Isolated superstructure with an increased period now experience reduced accelerations. ASCE 7-10 requires the use of a response modification factor, R, corresponding to the seismic force resisting system applied above the isolation level as a modified factor Ri. Modified Ri is a reduction factor analogous to the R factor that would have been used in case superstructure was not isolated.

Vb = ^Dmax^D 2.4

y fcprnaxPp 2 5s ~ R,

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Equation 2.4 and 2.5 produce the required design forces for the non-isolated base structure and isolated superstructure respectively. In addition, ASCE 7-10 imposes minimum values to be used for total lateral design force or shear force above the isolation system, Vs [23].

Vs must be not less than:

1- Shear force required by for a fixed base structure of the same effective seismic weight, W, and the isolated period, Td.

2- The base shear for the factored design wind load.3- 1.5 times factored lateral seismic force to fully activate the isolation system.

First two limits usually do not govern the seismic design. Mostly 1.5 times factored load is governing limit state to make sure that the isolation system displaces significantly before lateral forces reach the strength of the seismic system.

ELF method is based on a simplified set of equations that determines the isolated response of the superstructure as a rigid body. Detailed procedure for calculating isolated response is discussed in the section 2.3.2.

2.3.2 ELF procedure step-by-step

The procedure begins with an assumed value for the isolator displacement and then determines the effective stiffness, damping and other parameters. This process iterates until the design displacement converges within a tolerance of 3%. This is a tolerance determined at the discretion of the designer.

The following is a summary of the steps involved in the ELF procedure:

1- Assume initial displacement D, and values of Qd and kD using applied seismic weight.

2- Calculate effective stiffness using Eq. 2.6.

K e = - ^ - + kD 2.6Dm ax

3- Calculate effective period using Eq. 2.7.

Te = 2 tc 1 ^ - 2.7

4- Calculate damping provided by the system by simplified method and then estimate value of damping coefficient Bd, from Table 17-5.1 in ASCE 7-10.

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5- Using the spectral accelerations from response spectrum generated from U.S. Seismic Design Maps, U.S. Geological Survey, calculate design displacements using Equation 2 .2 .

6- Reiterate the process until the value of design displacement, Dd converges.

While ELF procedure calculates structural response of super structure using response spectrum variables, time history analysis is required to measure effective design demands for near-field ground motions that was conducted as part of this research in further sections.

2.4 Vertical ground motions

In the majority of earthquakes, seismic records of far-field motions showed the peak vertical ground accelerations were normally less than those in horizontal planes [29]. However, worldwide earthquakes have exhibited peak vertical accelerations exceeding peak horizontal values in near field locations as shown in table 2.1.

Event Station(Mw) Horl(g) Hor2(g) Ver(g) V/H

Gazli, Uzbeksitan 1976 K.arakyr(6.8) 0.71 0.63 1.34 1.89Imperial valley, USA 1979 El cenro array 6 (6.5) 0.41 0.44 1.66 3.77Nahhani, Canada 1985 Site 1(6.8) 0.98 1.10 2.09 1.90Morgan hill, USA 1984 Gilroy array#7(6.2) 0.11 0.19 0.43 2.25Loma-prieta, USA 1989 LGPC(6.9) 0.56 0.61 0.89 1.47Northridge, USA 1994 Arleta fire station(6.7) 0.34 0.31 0.55 1.61Kobe,Japan 1995 Port Island (6.9) 0.31 0.28 0.56 1.79Chi Chi, Taiwan 1999 TCU 076 (6.3) 0.11 0.12 0.26 2.07

Table 2.1: V/H ratio for seismic events [30]

The seismic response of structures under vertical excitation has been investigated using many numerical and experimental studies. Column members are found to undergo varying axial force when combined horizontal and vertical ground motions are applied. These axial loads then contribute towards larger horizontal displacements and less shear capacity [31], Studies conducted by Elnashai and Papazoglou provide detailed behavior of buildings and bridges when vertical ground motions are applied [32]. These studies have concluded that the vertical forces may cause compressive failure even when the structure is designed with overfreight factors. It was found that vertical excitations produce extensive fluctuations in member axial forces that ultimately reduced shear capacity of columns.

Another study was conducted by Yu using a three-dimensional model to analyze bridge overpass piers considering Northridge earthquake (1994) records [33]. Research indicated that

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the longitudinal moments and axial forces can increase up to 7% and 21% respectively when the ground motion input considered the vertical component with horizontal excitation.

Originally proposed by Newmark et al. (1973), Elnashai and Papazoglou (1996) determined the vertical to horizontal response spectrum ratio (V/H) to be 2/3, independent of period range, to consider the effects of vertical earthquake ground motions in design. However, the V/H ratio is found to be dependent on the frequency content of the ground motion and distance from the epicenter in many analyses of strong motion data. For high frequency ground motion recorded at short epicentral distances, this ratio is larger than those of low frequency content. This study was conducted using ground motion data from Northridge earthquake (1994). Some recorded vertical acceleration at some stations reached upto 1.18g, increasing the V/H peak acceleration ratio to as high as 1.79g [34], During the Chi-Chi earthquake (1999), peak horizontal ground accelerations at station TCU068 were found to be 0.364 g and 0.501 g, while vertical accelerations reached 0.519 g (Wang et al. 2002). In Kobe earthquake (1995), the ratio is calculated to be ranging between 1 to in many cases for peak of the horizontal and vertical ground acceleration [35].

These studies prove the 2/3 rule unconservative in the near-field. Table 2.1 illustrates some of the landmark earthquakes with significant V/H ratio at different stations. Study conducted by Collier and Elnashai (2001) [36] confirm that the V/H ratio exceeds within a 5 km distance from epicenter and gradually decrease at long distances.

Different countries have developed their own design codes to account for seismic strength for both horizontal and vertical earthquake ground motions. In many cases, the code requires detailed static and dynamic analysis for horizontal ground motions; however, for vertical accelerations, additional strength is defined using a simplified method by increasing the applied dead loading. This additional loading ignores the actual effects of ground motion in the vertical direction including peak values, frequency, and soil conditions at a site [37].

Recent studies [32], [38], and [39] on seismic design have involved analyzing vertical response and development of response spectra for vertical ground motion to include vertical excitation effects in design. Research efforts in this direction have developed vertical ground motion spectra, mostly focused on near-field ground motion histories that can be used parallel to horizontal ground motion spectra. This research has opened doors to new niche for seismic strength design.

In the US, ASCE 7-10 requires vertical seismic loads to be determined using a fraction of the short period response acceleration values (SDS). The design response spectrum in the vertical direction (Sav values) is also used while measuring site response factors. Sav is dependent on not only SDS but also Cv, a ratio representing horizontal to vertical response [6].

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2.4.1 Time lag between peak values for vertical and horizontal ground motions

An important characteristic of seismic waves is the arrival time of peak ground accelerations. In some cases, peak values for both directions occur within the same time domain without any time lag. But mostly peak vertical ground accelerations occur before the peak values for horizontal excitation as shown in Figure 2.6 for El Centro earthquake (1940). In this case, the vertical PGA 0.2 lg occurs 1 second before the horizontal PGA in both horizontal directions.

a < -0-2

0 1 2 3 4

0.3

J3§

0 1 2

Figure 2.6:Ground motion time history o f El-Centro 1940 a) vertical acceleration b) horizontalacceleration [37]

Elnashai and Collier (2001) considered 32 records from Imperial Valley (1979) and Morgan Hill (1984) earthquakes to investigate the time interval and the time lags between horizontal and vertical ground motions. For ground motions experienced by structure with in 5 km radius of epicenter, time lag between peak values for vertical and horizontal accelerations was observed to be zero and as the distance from epicenter increases, the time interval also increases although, dependent on the source depth, travel path and soil conditions at the site. In addition, it is important to design structure for vertical excitations separately in addition to horizontal ground motion if peak vertical accelerations are relatively closer to or larger than horizontal peak values and occur significantly before the peak ground accelerations.

The review presented in this section describes the significance of vertical ground motion for strength design of structures in seismic regions. Although in many cases, vertical excitations may be ignored, it is conservative to look into structural behavior when vertical ground motions are applied for many near-field sites. This research outlines structural response of an isolated

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structure in near-field and far-field. Case studies are defined to capture the effect of horizontal and vertical components when applied simultaneously as well as with a time lag between peak values of horizontal and vertical ground motion histories.

14

3 ANALYTICAL MODEL DESIGN

For the purpose of this study a fixed support, two-bay, three-story steel moment frame is defined as a base structure in SAP2000 [24], Story height of 12ft was assumed and bay width was taken as 24ft. For beams a uniform section of W12X106 was used and sections W24xl46, W21X146, and W18X143 are used for the first, second, and third story columns respectively. Acting gravity loads for this model accounted for self-weight of the structure and an additional 3kips/ft of dead load fulfilling the requirement of tributary slab weight and HVAC components installed. The following tables present the gravity load calculations for this structure.

Table 3.1: Mass calculation

Properties 1st Floor 2nd Floor 3rd FloorTributary Length Beams ft 48.000 48.000 48.000Tributary Length Columns ft 36.000 36.000 18.000g in/secA2 386.400 386.400 386.400Dead Load k/ft 3.000 3.000 3.000Self-Weight (Beams)W 12X106 k/ft 0.106 0.106 0.106Self-Weight (Columns)W18X143 k/ft 0.146 0.147 0.143Weight/Floor kips 164.688 164.760 159.324Mass/Floor kip-secA2/in 0.426 0.426 0.412

3.1 Response Spectrum

A site-specific response spectrum was created for a San Francisco location for Latitude 37.793° N, Longitude 122.404° W, on site class D soil with importance factors I, II, and III for seismic design.

Ss = 1.500 g SMS = 1.500 g SDS= 1.000 gS,= 0.600 g SM1= 1.020 g1 SD1= 0.680 g1

Figure 3.1 Maximum spectral accelerations

Figures 3.2 shows the design response spectrum and MCEr response spectrum respectively. These spectra are retrieved from the United States Geological Survey database for our site and are used for structural analysis in SAP2000.

15

V

1,000 V 0.136 1.000

Period, T (sec) Period, T (sec)

Figure 3.2: Spectra for (a) Design response and (b) MCER response [25]

3.2 SAP2000 Model Development:

For this research, SAP2000 was selected as the primary tool for analyzing the isolated and nonisolated structures. The structure was made from A572 Grade 50 steel with the material properties defined to provide sufficient rigidness in the beam elements.

For the first model, fixed supports are assigned and three horizontal degrees of freedom in the XZ plane are defined. Having these conditions set ensures the structure will display unidirectional behavior during analyses as required for this study when response spectra are applied. Response spectrum parameters as discussed in previous section are then assigned to a load case to represent the structure’s behavior for site specific conditions in SAP2000.

3.2.1 Modal analysis

Adequacy checks are performed on the fixed support model to determine and verify the dynamic properties of the system. Eigenvalue analysis was performed along with modal frequencies and periods calculated using MATLAB. These results are later compared with those obtained from SAP2000. For the eigenvalue analysis, the beams were considered to have infinite rigidness such that each story has a singular degree of freedom as shown in Figure 3.1. Using this assumption, the effective stiffness for columns and mass at each story level were calculated.

16

_atiauas—

CD CD CD

Figure 3.3: SAP2000 model for analysis

Modal analysis is then performed to obtain frequencies and periods for the fixed based structure. Table 3.2 below shows the comparison of results between SAP2000 and the hand calculations. They came within a reasonable tolerance of each other. As a result, there was confidence the program was identifying the mass and stiffness correctly.

Table 3.2: Comparison ofEigen values with SAP2000 modal analysis for fixed support structure

Period MATLAB SAP2000 % DifferenceTl(sec) 0.240 0.287 16%T2(sec) 0.093 0.109 14%T3(sec) 0.063 0.073 14%

3.2.2 Pushover analy si s

One more verification study conducted was a pushover analysis to check the elastic behavior and plastic hinge formation under lateral loading. A displacement controlled pushover was conducted per ASCE 41 (formerly FEMA 356). The target displacement is taken as 16 in.

Each frame was assigned hinges on both ends using ASCE 41 specifications for technical requirements for seismic rehabilitation of buildings.For the pushover analysis, a nonlinear load case is defined for dead load of the structure. This is required to account of nonlinear properties of materials defined in during the formation of hinges. This structure is the checked for linear behavior. Formation of hinges is continuously checked for following limit states defined by ASCE 41.

• Immediate Occupancy (10)• Life Safety (LS)• Collapse Prevention (CP)

17

General position of limit states on the pushover curve are shown in Fig. 3.4 (a).

Nonlinear behavior is captured by the pushover analysis confirming the structure’s capability of nonlinear response. But the calculated base shear for this structure being 89 kips is well below the elastic range of structural response which is 480 kips as shown in pushover analysis in Fig. 3.4 (b).

Displacement

M Pushover Curve

(a) (b)

Figure 3.4: a) generalized push-over curve with safety limits b) SAP2000 push-over curve

3.3 Isolator Design

Once the fixed structure was defined and checked for structural performance for elastic conditions, an isolation system was designed at the base level. The ELF procedure was used to design high damping elastomeric bearings similar to the one shown in Fig 3.5, for an isolated period of 2 sec. These isolators are placed above the fixed supports essentially adding one more level to the structure.

18

High damping rubber - / Inner steel plate

v .Cover rubber

X Flange

Figure 3.5: High damping rubber isolator [26]

Base isolation reduces the maximum elastic forces considerably as a result of period shift and energy dissipation. Like other dynamic procedures, static equivalent lateral force procedure increases the structure’s period by reducing the effective stiffness that displaces the super structure under the lateral loading. Eq 2.2 calculates the design lateral displacement of a single- degree-of-freedom, linear-elastic system having a period, T d , and equivalent viscous damping, p. In this equation, the spectral acceleration term, Sd i , represents the spectral acceleration requirements design of a conventional fixed-base structure of period, Td. Damping factor B d, being used instead of equivalent viscous damping, (3 is used to adjust the computed displacement when the equivalent damping coefficient of the isolation system is greater or smaller than 5 percent of critical damping [27]

Manufacturing based properties such as post elastic stiffness, Kd, and characteristic strength, Qd, are approximated during the first iteration. The first approximation is usually based on the total weight of the building. A fraction of the structure’s weight is taken as Kd and Qd for the first pass. These values can be adjusted to satisfy ELF requirements of elastic base shear for an acceptable design and effective stiffness provided by the isolation system.

There are two requirements in ASCE 7-10 guidelines to be satisfied for isolation system design. One requires the system to experience an elastic base shear. This requirement can be described as follows.

This equation represents that that the base shear in isolated system should not exceed 20 percent of the total weight of the buildings for superstructure to stay elastic under the lateral loading. Another requirement to be satisfied using ELF method is lateral restoring force for re-centering of the isolator as per ASCE 17.2.4.4.It is required that the isolated system when displaced to maximum displacement should have lateral force that is 0.025W greater than the lateral force when structure is halfway displaced. Eq. 3.2 shows the relation for this requirement.

KeAd < 0.2W 3.1

19

Fm ax@ D Fm ax@ D / 2 — 40 3.2

3.3.1 Design displacement and Checks

Following the steps for ELF method as discussed in earlier sections, design displacement is calculated and checks are performed. A maximum design displacement of 8 inches is calculated. Table 3.3 shows design and physical properties that will be used in SAP2000 for analysis of isolated structure.

Table 3.3 . Isolator design properties

1 A,)... i

) in OO >1 15.433/>? 12 I k/in 3.787

Weight V 1 kips 0-0284 0 1 1 ^ 1.816I ll llf f f fM I in 5.5 R f l k/in 1112■ ■ ■ ■ - 1.500 1 8 1 in 0.30SIlMlmmBI in Fy(nonlin) W vtSBKi 17

ksi 0.100 I j r 1 - 9.62ESTH ' 1 kip~secA2/in 0.000214 | H U 1 inA2 104.61

The following checks are then performed per ASCE 7-10 requirements for the ELF procedure.

Table 3.4: ASCE Elastic Base Shear Requirement

Elastic Base Shear Req.Fmax/W= 0.19234 <= 0.20

GOOD

Table 3.5: ASCE Re-centering Req. per ASCE 17.2.4.4

Kd>=W/(20D) !>=W/40Kd 1.816 ^ Fmax 89.05r \ r > W/40 11.575W/(20D) 0.986 I FmaxD/2 44.53

GOOD Fmax-FmaxD/2 44.53 GOOD

20

ASCE 7-10 imposes a elastic base shear requirement (Table 3.4) that requires isolator shear force to be less than 20% of total weight of structure. Elastic base shear is the maximum demand force for the structure to remain elastic during an earthquake. This requirement ensures that the superstructure does not yield during a lateral loading and the isolator provides maximum energy dissipation [40]. The isolation system is also required to produce a restoring force to re-center the bearing such that the lateral force at the total design displacement is at least 0.025W greater than the lateral force at 50 percent of the total design displacement as shown in table 3.5.

3.4 Isolated model in SAP2000

The isolators are modeled in SAP2000 using link elements placed at the base level of the structure. Placement of the isolation plane can be at any level depending upon the lateral force requirements, location of sensitive equipment in the building or accessibility issues. For this research, ground level is best suited where isolators will be in direct connection to the ground level and will have access to un-filtered incoming ground accelerations.

The link properties for the rubber isolator are defined for all three translational degrees of freedom. Rotation along the three axes is restricted. Orthogonal horizontal directions are modeled for nonlinear properties. These properties are required when applied lateral forces are large enough to start no linear deformation in links and other structural elements. Nonlinear properties ensure that the response of bearing increases to an ultimate point before softening to a residual strength. Vertical direction is not fixed to avoid the rigidity in vertical direction as having a rigid vertical component in the isolator will not allow use of vertical ground motions. Rather they are defined to be stiff enough to support the system but not extremely flexible either.

Modal analysis is run on the isolated model to compare results against hand calculations. Comparisons show that the values are reasonably close for both the SAP2000 analysis and the hand calculation. Notably, the fundamental model of the isolated system is approximately ten times longer than the fixed based system. Also, this period reflects the desired design period of 2 seconds.

21

Table 3.6: Comparison o f Eigen values with SAP2000 modal analysis for Isolated structure

Period Matlab SAP2000 DifferenceTl(sec) 2.154 2.093889 -3%T2(sec) 0.123 0.141237 13%T3(sec) 0.070 0.078951 11%T4(sec) 0.002 0.0025687 11%

Figures 3.6 and 3.7 show the isolator property inputs for the SAP2000 link elements.

M Link/Support Property Data

Orris/Support Type Rubber feo&tor

Property Name jUNi

Property Notes.

Tote} Masa a»d Weight

U l t i |4.0<30£-0: R&tatema! Jn*rt*a 1

Ratstscnai Inertia 2

Rotational Inert* 3

Link/Support Directional Properties

UN1

factor* fw Lime, Area and Sold Sporvg*

Property is 0 e fined! for This le«sf§» in a Una Spmg

Property «. Defied for This Area M Area and So&l Springs

D s fa flt o a S Psrejs**tse*

i>ect»n Fsxed Umlmsm

S ui □

0 w D &

0 us O a

a ri 0

0 *2 0

a rs 0

....FteAl ;>'i r Dear Al

Modjfy/Sfeow for U1...

ttOdttyfSftOw for U2...

W<xfif>*Stow for US...

Identification

Property Name

Direction

Type

Nonlinear

U1I Rubber isolator

No

Properties Used For AB Analysis C ases

Effective Stiffness

Effective Damping:

1111.

OK Cancel

Caro#*

Figure 3.6.Isolatorproperties in SAP2000

Push over analysis is also performed to confirm elastic behavior of super structure for isolated design displacements. Fig 3.8 illustrates the push over curve for isolated structure under consideration. For design displacement of 8 in, the structure behaves elastically and base shear produced is 89 kips that is less than the base shear required for non-isolated structure, 480 kips.

| Rubber isolator

X Link/Support Directional Properties

Identification

Property Name |

Drection j ̂

TypeNonlinear ]^e*

Properties Used For Linear Analysis Cases

Effective Stiffness

Effective Damping

Shear Deformation Location

Distance from End-J

Properties Used For NonSnear Analysis Cases

Stiffness

Yield Strength

Post Yield Stiffness Ratio

17.

X M lin /̂Support Oirectional Properties

Identificatton

Property Name

Direction

Type

Nonlinear Yes

Properties Used For Linear Analysis Cases

Effective Stiffness

Effective Damping

Shear Deformation Location

Distance from End-J

13.79

10 31

0.1

Properties Used For Non linear Analysis Cases

Stiffness I"1* 2

ElZIoT“

Yield Strength

Post Yield Stiffness Ratio

] [ Cancel Cancel

Figure 3.7: Horizontal orthogonal direction properties o f isolator

m Pushover Curve

File

Static Nonlinear Case

100/

90.'

80

70/

60/

50/

40,

30/

20/

10 .'

Plot Type

Resultant Base Shear vs Monitored Displacement

Displacement

\ \

/f

/..../.'....

1. 2. 3. 4. S. 6. 7. 8.

Mouse Pointer Location Hcriz | Vert

I 0K I

Units

| Kip, in, F

Current Plot Parameters

Add New Parameters...

Add Copy of Parameters...

Modify/Show Parameters...

Figure 3.8: Push over Analysis o f isolated structure

23

3.5 Effective Modal mass using Modal participation factors

Base excitation readily accelerates the modes with relatively high effective masses than the modes with low effective masses. Effective modal masses and modal participation factors were calculated to verify the isolated response of the structure in fundamental mode, Mode 1. Following results are obtained using MATLAB for mass identity matrix.

Table 3.7: Effective modal masses and modal participation factors

Mode Modal participation factors

Effective Modal mass (k g )

PercentageParticipation

1 -1.1249 1.2653 992 0.0044 1.6000e-05 13 -0.0011 1.2100e-06 04 0.0001 1.0000e-08 0

SAP2000 verification also provided comparable results as shown in Table 3.8.

Table 3.8: Modal Participation factors - SAP2000

Case Mode No. Period (sec) Modal participation in UX

MODAL 1 2.120388 0.9987MODAL 2 0.318984 0.0013MODAL 3 0.160383 0MODAL 3 0.116796 0

Results indicate that the structure has maximum participation in its fundamental mode, Mode 1, as an isolated structure. Isolated period of 2.1 sec, elastic behavior in push over analysis and modal participation ensure that the isolated model is properly designed. This model is then used as test subject for this research and multiple ground motions are applied to examine seismic strength and structural design behavior for different test cases as defined in section 4.2.

24

4 ANALYSIS PROCEDURE

A fixed support structure can be assumed to have nearly infinite vertical stiffness at the ground level. On the other hand, an isolation system has a specific vertical stiffness, Kv, dependent on the structure’s dead load. Acknowledging the difference between the vertical stiffness of a fixed and isolated structure, it is understood that the load effect of the vertical ground motion is inherently unique and different for each case. Three loading scenarios based on ASCE 7-10 guidelines are developed with the vertical motion component independent of orthogonal horizontal motion.

Section 12.4.2.3 of ASCE 7-10 provides guidelines for seismic strength of the structural system. Code requires a building system to be designed for both lateral and gravity loads to provide adequate strength and compatibility to the required demands. It also deals with seismiceffects for design ground motions with modified load combinations to be used in seismicconditions. Based on the response spectrum of a site-specific location, seismic load effects are defined. ASCE 7-10 provides the following load combinations for strength design [28]:

(1.2 + 0.2 Sds)D + pQE + L + 0.2 S (4.1)

(0.9 - 0.2Sds)D + pQE + 1.6H (4.2)

The term 0.2SD5D accounts for load effects produced by vertical excitations. As a result, both load combinations above account for both the horizontal and vertical forces. For a seismically isolated system, ASCE 7-10 Section 17.2.4.6 requires Sds to be replaced by Sms. For fixed support structures, code requires the direction of applicable ground motions to be in the direction creating the most critical load effect. Depending on the site class, use of orthogonal horizontal ground motions is permitted for time history analysis. Individual ground motions can be applied for site class B. For site class C, it is possible to run orthogonal ground motions simultaneously. Whereas for site class D through F, in addition to requirements for site class C, there is an additional 20 percent axial load for any column or wall that is part of two or more intersecting seismic force resisting systems.

For this case study, three load cases are introduced using the maximum of the two code specified load combinations (Eq. 4.1, 4.2). The cases are as follows:

• Case-1: The two horizontal ground motions are applied individually with resultsfocused on the direction producing the maximum load. No vertical motion is considered.

• Case-2: The horizontal motion generating the most critical load is applied simultaneously with the vertical ground motion. This case includes any moments developed by axial forces from the vertical motion.

• Case-3: This situation assumes a scenario where there is a lag between the stronghorizontal ground motion and the strong vertical motion input. At the end of the strong horizontal motion, the isolation system is assumed to reach its maximum displacement and there is an amount of energy and momentum in the structure (as the system has not

25

come to a state of rest). It is at this point, the structure can be seen as pre-excited when the strong vertical motion begins. This case develops the scenario of a strong motion record with horizontal high frequency and peak spectral accelerations occurring a bit earlier than the vertical ground motion. For this load case, the term 0.2Sd5D is neglected to achieve conservative results.

4.1 Ground acceleration histories

Twenty sets of ground motions were used for this case study. Time histories for five different stations from four distinctive seismic events around the world were downloaded from PEER ground motion database. The selection criteria for these strong motion records is based on a variety of station distances (both near- and far-field) from the epicenter with a distributed range of (O-lOOKm) and peak ground accelerations (> O.lg). Other factors including Richter magnitude (> 6M), Vs30 (>100 m/s), and arias intensities were also considered. The four seismic events considered include: Imperial Valley (1979), Christchurch (2011), Loma Prieta (1989), and Northridge (1994).

Table 4.4.1: Ground motion - Imperial Valley, USA

Station Mag Mechanism

Rjb(km)

Rrup(km)

PGA(g)

Vs30(m/s)

Lowestuseablefreq(Hz)

Bonds Comer 6.53 Strike Slip 0.44 2.66 0.5987 223.03 0.125Brawley Airport 6.53 Strike Slip 8.54 10.42 0.1626 208.71 0.05375Calexico Fire Station

6.53 Strike Slip 10.45 10.450.277

231.23 0.05

Delta 6.53 Strike Slip 22.03 22.03 0.2357 242.05 0.0875Coachella Canal#4

6.53 Strike Slip 49.1 50.10.116

336.49 0.1375

26

Station • Bonds C orner

i i i r .................. i— • t *--------- Horizontal Acc..... ------------------------------------------------------------------------------------------------------------------------------------------------------— — — -— — * — — --------------— ----------------------------— ---------------

______ 1 S 1 1 1 1

--------- Vertical Acc ^

0 5 10 15 20 25 30 35

T(sec)

Station • Braw ley A irport

--------- Horizontal Acc--------- Vertical Acc *

0 5 10 15 20 25 30 35

T(sec)

Station • C alexico Fire Station

f i l l 1 1--------- Horizontal Acc--------- Vertical Acc

.................................................. I ................................. . .............. 1..................................................1...................................................1.................................................. J................ ..................................1........................ - J

0 5 10 15 20 25 30 35

T(sec)

Station - Delta

I I I ) i f--------- Horizontal Acc--------- Vertical Acc

I 1 1 1........- ----- --------------------- ---- - J ------------------------------------- 1---------------------- -0 5 10 15 20 25 30 35

T(sec)

Station - C oachella Canal #4

[ I I r I i--------- Horizontal Acc--------- Vertical Acc

I I l - ........... 1 - ............................J_____________ - ..........— L _4 2 L™ — ---- — -J— ------------------- ------------- — — ------ —- — i.--------------------— —±—.--_ — ------- — _ -------------------------------------------— —0 5 10 15 20 25 30 35

T(sec)

Figure 4.1: Time History- Imperial Valley USA

Table4.4.2: Ground motion - Christchurch, New Zealand

Station Mag Mechanism Rjb(km) Rrup(km) PGA(g)

Vs30(m/s) Lowestuseablefreq(Hz)

Pages Road P.R

6.2 Reverse Obi 1.92 1.980.573

206 0.1

ChristchurchResthaven

6.2 Reverse Obi 5.11 5.130.3714

141 0.1

Kaiapoi North chool

6.2 Reverse Obi 17.86 17.870.1894

255 0.1

SWNC 6.2 Reverse Obi 25.45 25.45 0.1915 295.74 0.0375GLWS 6.2 Reverse Obi 105.25 105.25 0.01 511.16 0.05

27

Figure 4.2: Time History- Christchurch, New Zealand

Table 4.4.3: Ground motion - Loma Prieta, USA

Station Mag

Mechanism

Rjb(km)

Rrup(km)

PGA(g)

Vs30(m/s)

Lowestuseablefreq(Hz

LGPC 6.93

ReverseObi

0 3.88 0.5631

594.83 0.125

BRAN 6.93

ReverseObi

3.85 10.72 0.456 476.54 0.125

Saratoga - Aloha Ave

6.93

ReverseObi

7.58 8.5 0.5145

380.89 0.125

Hollister-south&Pine

6.93

ReverseObi

27.67 27.93 0.3699

282.14 0.0875

Larkspur Ferry Term.

6.93

ReverseObi

94.56 94.64 0.1384

169.72 0.288

28

Station ■ LG PC

Figure 4.3 Time History- Loma Prieta USA

Table 4.4A: Ground motion - Northridge, USA

Station Mag

Mechanism

Rjb(km)

Rrup(km)

PGA(g)

Vs30(m/s)

Lowestuseablefreq(Hz)

Jensen Filter Plant

6.69 Reverse 0 5.43 0.5712

525.79 0.14

SepulvedaV A hospital

6.69 Reverse 0 8.44 0.7525

380.06 0.182

LA - N Faring Rd

6.69 Reverse 12.42 20.81 0.28 255 0.1625

LA - CC North

6.69 Reverse 15.53 23.41 0.2557

277.98 0.14

Arcadia Campus Dr

6.69 Reverse 41.11 41.41 0.091 367.53 0.25

29

Station - Jensen Filter Plant

T(sec)

Figure 4.4: Time History- Northridge USA

4.2 Load cases and combinations

The three load cases discussed earlier are then defined in SAP2000 program to apply ground motion. Nonlinear modal histories are defined for each case at each station. Ground motions were scaled by 4g’ without any additional scale factor added.

Case -1 is developed using only horizontal ground motion application without any vertical excitation. Based on historic data of ground acceleration, number of time steps as well as time step interval are used in each loading case. Fig 4.1 illustrates said load case input in SAP2000.

30

M Load Case Data - Nonlinear Modal History (FNA)

Load Case Name

| RJB_0-44_U1

Initial Conditions

® Zero Initial Conditions - Start from Unstressed State

O Continue from State at End of Modal History

Modify/Show...

Important Note: Loads from this previous- case are included in the current case

Modal Load Case

Use Modes from Case

Loads Applied

Load Type Load Name Function Sca le F acto r

Accel v - RJBJM4JJ v 388.4

in .................... f e M I M a

V

Modify

Q Show Advanced Load Parameters

Time Step Data

Number of Output Time Steps

Output Time Step Size Other Parameters

5000

15.000E-03

Modal Damping

Nonlinear Parameters

Constant at 0.05 Modify/Show...

Modify/Show...

Load Case Type

; Time History vjj Design... J

Analysis Type Solution Type

O Linear (§) Modal

(•) Nonlinear O Direct Integration

History Type

0 Transient

O Periodic

Mass Sourcei Previous (MSSSRC1)

Figure 4.5: SAP2000 load case for Case -1

This case is later on used to develop ASCE 7-10 defined combination for design strength in seismic conditions following Eq. 4.1 as shown in Fig. 4.2. For this study, only dead and seismic loads are considered for developing load combinations. Live and snow loads are ignored for comparative study.

Define Combination of Load Case Results

Load! Case Name Load Case Type Scale Factor

DEAD v Nonlinear Static 1.5

lOEAD 111 1.5 IRJB_0-44_U1 Nonlinear Modal History (FNA) 1.

Figure 4.6: Load combination for Case-1

31

Next load case, Case-2, is defined with a combined input of horizontal and vertical ground motion. For each station, maximum of two horizontal ground accelerations and one vertical ground acceleration in upward direction is taken. A nonlinear modal case is used for this input. A new load combination based on Eq. 4.1 is defined that uses this load case and ignores the0.2SDS (0.2SMS for isolated structures) factor that accounts for additional seismic strength in absence of vertical component of ground motion during analysis of structures. Fig 4.2 illustrates the load case defined for combined effect of horizontal and vertical ground motion in SAP2000.


Load Case Name

I RJB_0-44_U1 +U3 Set Def Name Modify/Show..

Initial Conditions

(•) Zero Initial Conditions - Start from Unstressed State

O Continue from State at End of Modal History

Important Mote: Loads from this previous ca se a re Included in the current case

Modal Load Case

Use Modes from Case ! MODAL

Loads Applied

Load Type Load Name Function Scale Factor

i Accel v | U3 v RJB_Q-44_U v- 386.4

U3 ............J E S S S F S E I S l I — AAccel U1 RJBJM4JJ1 386.4

V

Modify

Delete

□ Show Advanced Load Parameters

Time Step Data

Humber of Output Time Steps

O utput Time Step Size Other Parameters

[5.000E-03

Modal Damping


Constant at 0.05

Default

Modify/Show...

Modify/Show...

Load Case Typef..................................I Time History v j Design... J

Analysis Type Solution Type

O Linear ® Modal

(§) Nonlinear O Direct Integration

History Type

0 Transient

C.) Periodic

Mass source| Previous (MSSSRCT)


32

Define Combination of Load Case Resu lts

Load Case Name Load Case Type Scale Factor

DEAD v N o n lin ear Static 1.2

Id e m Monlinear Static 12 1RJB_0-44U1 Nonlinear Modal History (FNA) 1.RJB_0-44_JJ3 Nonlinear Modal History (FNA) 1.


Case -3 as discussed earlier has two ground motions, maximum of horizontal ground motion and a vertical ground motion. In this load case, horizontal and vertical ground motions are not applied at the same time. Instead, input of vertical ground acceleration is initiated after the application of horizontal load case, situation assumes a scenario where there is a lag between the strong horizontal ground motion and the strong vertical motion input. In SAP2000, this scenario is achieved by setting initial conditions for vertical excitation case. SAP2000 allows us to continue a load case from a previous nonlinear load case. Load case 1, containing only horizontal ground motion, is used as initial condition case before vertical acceleration input.

33

Initial Conditions

O Zero Initial Conditions - Start from Unstressed State

(§) Continue from State at End of Modal History

Important Mote: Loads from this- previous c a se a re included in the current c a se

Modal Load Case

Use Modes from Case

Loads Applied


Load Case Name Notes Load Case Type

|rJB_49-1_U3_U1 "1 i Set Def Name Modify/Show... Time History V i Design...

Load Type Load Name Function Scale Factor

| Accel v |;U3 RJB_49-1_U v 3S6.4

I Ac cel ........... A

1

Add

Modify

Delete

[ [] Show Advanced Load Parameters

Time Step Data

Number of Output Time Steps

O utput Time Step Size Other Parameters

[5.000E-03

Modal Damping


Constant at D.05

Default

Modify/Show...

Modify/Show...

Analysis Type

O Linear

(§) Nonlinear

History Type

(§) Transient

( ') Periodic

Mass Source

Solution Type

(§) Modal

O Direct Integration

j Previous {MSS SRC 1)

OK

Cancel


Define Combination of Load Case Results

L o ad Case N a me Load Case Type Scale Factor

DEAD ^ Nonlinear Static 1.2

Nonlinear Modal History (FNA) 1RJ B_G-44_U3_U 1 Nonlinear Modal History (FNA) 1.


Moments are recorded for each ground motion from each load combination and then compared. Two scenarios are developed based on the comparison of Case-1 against Case-2 and Case-3 respectively.

34

1. First Scenario compares Case-2, that is application of vertical and horizontal time histories in the same input while not considering S d s against Case-1 and it evaluates difference between structural demands for isolated structure for these cases.

2. Second scenario compares Case-3 where the system is pre-excited with the horizontal ground motion at the same station and then vertical acceleration is used to analyze the structure, with Case-1.

Results and discussions presented in next section consider these two scenarios for comparative study.

35

5 COMPARATIVE STUDY

Nonlinear time history analysis of the isolated model showed specific trends in the moment demands for the near and far-field motions. An exterior column on the first level and a beam at the top level were selected for review. Moment demands are observed at member ends for multiple time histories with maximum values plotted against the horizontal distance from the source and peak ground accelerations for comparative study.

Two comparative scenarios were developed based on Case-1 against Case-2 and Case-3 respectively.

Scenario 1 (Case-1 vs Case-2): In this scenario, the structural demands for moments and shear are considered for the first story column. Although there are some exceptions, the moment requirements for the column in Case-2 are consistent with Case-1, not exceeding demands and generally less than Code requirements (Eq 3.1 and Eq 3.2) for the majority of ground motions (regardless of distance from epicenter). On the other hand, for beams, values obtained from Case-2 noticeably differ from those related to Case-1 defined combinations.

For sites within 10 km of epicenter the trend is to have higher demand for moments in tension members when vertical ground motions are applied, represented by Case-2. This is true for most of the cases whereas for ground motions recorded from Imperial Valley and Loma Prieta earthquakes this trend is observed for both near-field and far-field events as a constant behavior as shown in Fig 5.11 and Fig 5.8 respectively.

It is observed that as per code prescribed procedure in Case-1 takes vertical seismic effect into account using 0.2SdsD, results obtained from Case-1 are very close to results of combined input of horizontal and vertical components of a specific ground motion except when it’s very close to near-field. It is interesting to note that peak ground accelerations do not affect much the moment demands for columns compared to top story beams in far-field sites for this scenario in this case study as observed in Fig 5.3 (b), Fig 5.8 (b), and Fig 5.11 (b).

Scenario 2 (Case-1 vs Case-3): This scenario compares Case-3 where the system is pre-excited with the horizontal ground motion at the same station and then vertical acceleration is used to analyze the structure. Moment demands are reasonably high in many cases, exceeding more than 50% on occasions compared to code minimum requirements.

These cases mostly occur where peak ground accelerations are quite high. Similar trends to Case-2 are observed. Higher demands for Case-3 compared to code defined procedures are observed for sites that are in close vicinity to epicenter. Beams display more variant behavior than columns at lower values of distance from source and higher peak ground accelerations regardless of seismic event.

The plots in following case studies illustrate the difference between moments for respective cases defined in above mentioned scenarios as a percentage. Values above horizontal axis

36

represents higher moment demands for Case-2 and Case-3 and under-estimation of values using current code procedure for isolated structures. Percentages with negative values mean that Case-1 is conservative enough.

5.1 Plotting Technique

Percentages in the plotted graphs provide a quantitative measure of difference between the strength demands produced by either Case-2 or Case-3, and Case-1. These plots have two possible patterns.

1- Values above the horizontal axis shows that structure is experiencing more strength demands for either Case-2 or Case-3. This represents that the ASCE 7-10 defined procedures underestimate structural demands.

2- If the percentages are below x-axis, Case-1 has higher demands than other two cases. This justifies that using strength design procedure prescribed by ASCE 7-10 is conservative.

5.2 Earthquake 1- Christ Church, New Zealand

To understand the structure’s response for the cases discussed above, it is important to look at the ground motion inputs used to define these cases. The near field ground motion history for the earthquake in Christ Church, New Zealand had a strong vertical component having high frequency and PGA compared to those of the horizontal ground motion as observed in Fig 5.1. The horizontal ground motion had lower peak values (i.e. 0.6 g) than the vertical ground motion where it was very close to 2 g. In addition, this set of motions parallels well with Scenario-2 (higher vertical acceleration input after the horizontal ground acceleration). The horizontal ground acceleration has a peak value of 0.6g and the seismic wave weakens after 6 seconds dropping down to accelerations from 0.25 g to 0.01 g. In parallel, although the vertical ground motion sees a peak value of 1.9 g earlier in the motion, accelerations between 1 g to 0.5 g can still be seen after 6 seconds (after the horizontal ground motion has died down). This behavior is imitated by Scenario-2 in the structure by applying vertical ground motions after the horizontal motion has already excited the structure.

37

Time History, Horizontal ground motion

5 10 15T(sec)

Time History, Vertical ground motion

10 15T(sec)

Figure 5.1: Time History (Pages Road P.R) - Christ Church, New Zealand

Fig 5.2 shows the time history of horizontal and vertical ground motion inputs at 105 km from the fault rupture. Unlike the near-field record, the far-field motion has vertical ground accelerations similar to the horizontal component in terms of both frequency content and peak values.

-0.02


10 20 30 40 50 60T(sec)


30 40T(sec)

Figure 5.2. Time History (GLWS) - Christ Church, New Zealand

38

The near-field ground motions tend to have higher moment and shear demands for both Case- 2 and Case-3 when compared to Case-1. The difference between strength demands for plotted cases is as high as 55% in the case of beams while for columns the variation is close to 20% for both Scenerio-1 and Scenerio-2. This is because of the reason that Case-1 takes care of vertical excitation force as factored addition to the strength based on dead load of the building (Eq 4.1) while at the near-field stations, i.e. Pages Road P.R and Christchurch Resthaven from Christchurch earthquake (Fig 4.2), the vertical ground motion is strong more than the horizontal component.

On the other hand, for far field ground motions, these demands stay under the code defined Case-1, that has an additional factored load instead of any actual vertical ground motion input. This brings attention to the fact that in the far-field, the effect of vertical ground motion is quite weak as compared to its horizontal component. As a result, the code used S d s values in Case- 1 actually proves sufficient at far-field for seismic resistance of test structure.

An important observation is made in column moment Case-1 vs Case-3. Unlike other plots, the plot line for column moments and shears obtained from Case-3 are quite higher than Case-2. This can be explained by large amount of energy stored in columns members for Case-3 compared to that of Case-2 as this trend is only observed with high vertical PGA inputs compared to horizontal PGA and when the duration of strong motion record is longer i.e. 30 sec or more.

I 30%

I 20%•5 io%QvP 0 % ox

- 10%

— Col Moment Case-1 Vs Case-2- Col Moment Case-1 Vs Case-3

- Beam Moment Case-1 Vs Case-2 •• Beam Moment Case-1 Vs Case-3

ML rML 100% 120

Distance from source (km)(a)

60%50%40%30%20%10%0%

- 10%

- 20%

-30%

i ......................... :.....• Col Moment Case-1 Vs Case-2• Col Moment Case-1 Vs Case-3

Beam Moment Case-1 Vs Case-2 y• Beam Moment Case-1 Vs Case-3 yjjf

PGA (g)(b)

Figure 5.3: Percentage difference for moments based on (a) Distance from source and (b) PGA (ChristChurch)

39

While analyzing the effect of PGA on the response, moment and shear demands tend to increase with higher values of PGA for Case-2 and Case-3 when compared Case-1. Response of beams to vertical accelerations is more than the increased demands experienced by columns. Higher axial forces caused by higher PGA in tension members produce additional shear in members. Column already subjected to structural weight are designed for maximum axial force. As a result, additional axial forces do not have a great impact on column demands. At low PGA, the demands calculated by Eq 4.1 and Eq 4.2 are higher compared to demands shear and moment demands observed using Case-2 and Case-3.

80%

60%<L>c 40%<D

20%

5 °%- 20%

-40%

80%

60%

| 40%2$ 20%5 o%ox

- 20%

-40%

Peak ground acceleration (g)(b)

Figure 5.4: Percentage difference for shear based on (a) Distance from source and (b) PGA (Christ Church)

5.3 Earthquake 2- Loma Prieta, USA

Another observation was made in this study using Loma Prieta input ground motions. Horizontal and vertical near field ground motions for Loma Prieta have the similar frequency content and peak ground acceleration as shown in Fig 5.5. The results obtained from Case-1, Case-2, and Case-3 do not have much variation with the difference between results within a 10% range. Furthermore, Fig 5.6 shows that although the peak ground acceleration of vertical ground motion is less than that of horizontal ground motion, the frequency of vertical ground

uij...— J

l \

* U ' - - “0 2V ____ 4...|...... ......1li

— coi shear Case-1 Vs Case-2— Col Shear Case-1 Vs Case-3

- • • - Beam Shear Case-1 Vs Case-2Beam Shear Case-1 Vs Case-3

“8D--------- rctrf 120

Distance from source (km) (a)

• Col Shear Case-1 Vs Case-2 ■ Col Shear Case-1 Vs Case-3• Beam Shear Case-1 Vs Case-2• Beam Shear Case-1 Vs Case-3

40

motion is higher. High frequency seismic waves create axial demands in tension members that affect shear strength of the members. In the analysis performed, results shown in Fig 5.7, depict the behavior of higher demands for beams when the original vertical ground motion is applied using either Case-2 or Case-3.



Figure 5.5: Time History (LGPC) - Loma Prieta, USA



Figure 5.6: Time History (Larkspur Ferry Terminal) - Loma Prieta, USA

41

An important observation is made with low PGA ground motion for moment demands in beams. Fig 5.7(b) show higher percentage for Case -2 and Case-3. This reflects a different behavior from other earthquakes and may be an interesting point to look into further research.

70% ,60% I * Col Moment Case-1 Vs Case*2

j— «# — Col Moment Case-1 Vs Case-3 ^ *50% |— • * • Beam Moment Case-1 Vs Case-2 #

Beam Moment Case-1 Vs Case-3 . •*c 40% ! . • ..<•••2 ..."| 30% | . ̂j 2°% ;

0% #• . ■..................."11 ~ ~ ~ ~.**"~ ~ - - -i - *....-10%®'*"*.... 2<3“ 40 60 80 * 100

Distance from source (km)(a)

70%60% 0 - Col Moment Case-1 Vs Case-2rrvo/ i i . Col Moment Case-1 Vs Case-3

u 0 * X . o « • Beam Moment Case-1 Vs Case-2c 4 0% *" * * • . * * » * v B earn Mom ent Case-1 Vs Case-3

I 30% .5 20% \ .

10% - | > 4 - : !o% i.............. • . ___J ................,|0% 0 0.1 •' 0.2 0.3 0.4 0.5 0.6

PGA(g).(b)

Figure 5.7: Percentage difference for moments based on (a) Distance from source and (b) PGA (LomaPrieta)

• sA •

l i i ji

Col Moment Case-1 - - Col Moment Case-1

'— ---------------1

Vs Case-2 Vs Case-3fp. .

s.—■—• • • Beam Moment Case-1 Vs Case-2 wn#inr* Beam Moment Case-1 Vs Case-3

* %S.

*»

0 0 .i • ' " o ' 2 0 .3 0' ;;W ...

o » r ' o .

■ Col Moment Case-1 Vs Case-2\ — Col Moment Case-1 Vs Case-3> • - Beam Moment Case-1 Vs Case-2► Beam Moment Case-1 Vs Case-3

'26 40

.o

60 80 1C

42

40%

30%

20%

50%

* 10%

0%

- 10%

- 20%

[ 1-- — Col Shear Case-1 Vs Case-3

%# m

——• • - Beam Shear tCase-1 Vs Case-2 . Beam Sheart Case-1 Vs Case-3

........... .... “T"..‘..i

................,..............................

— • • W~ • • .

§ _ \ ............... ;

.........______

• * — . . ^

•V • I • 1 1 1 1 0 \ f 20 4 o .........3

“ " 8

X...... ....i


Q£

50%

40%

30%

20%

10%0%

- 10%

- 20%

1 • - Col Shear Case-1 Vs Case-2- Col Shear Case-1 Vs Case-3

<«#••• Beam Shear Case-1 Vs Case-2 • - Beam Shearlpa^e-1 Vs Case-3

W' i0.3 0.4 0.5 0.6 0.7


Figure 5.8: Percentage difference fo r shear based on (a) Distance from source and (b) PGA (Loma Prieta)

5.4 Earthquake 3 - Northridge, USA

Effect of vertical ground motion frequency content is also confirmed by Northridge earthquake ground motion. Fig 5.9 represents near field ground accelerations for Northridge earthquake. It is observed that the peak ground accelerations in vertical ground motion are bit higher in addition to the higher frequency. Horizontal ground motion has lower frequency over the time from 0 to 10 (sec) when compared to vertical ground motion. As a result, in both Case-2 and Case-3, where vertical ground motions are used, moments and shear demands are quite high for near field station as shown in Fig 5.10.

% D

iffer

ence

%

Diff

eren

ce

43


10 15 20Tfsec)


Figure 5.9: Time History (Jensen Filter Plant) - Northridge, USA

— Col Moment Case-1 Vs Case-2— Col Moment Case-1 Vs Case-3* - Beam Moment Case-1 Vs Case-2

Beam Moment Case-1 Vs Case-3


60%50%40%30%20%10%0%

- 10%08 - 20%

'30% PGA (g)(b)

Figure 5.10: Percentage difference for moments based on (a) Distance from source and (b) PGA(Northridge)

PGA (g)

44

Far field ground motion data for Northridge earthquake displays the general trend of moment demands as other earthquakes. As the distance from source increases, the effect of vertical acceleration reduces. This is because of the fact that vertical ground motions are primary seismic waves that can travel through solids and fluids as well. Mode of propagation for primary waves is compression and expansion of the medium-layer in the same direction of seismic wave. Depending on the medium, the peak ground accelerations, velocity and the frequency change. In many cases, decrease in the PGA in frequency is caused by a soft medium. This results in the low strength requirements for vertical ground motions and the code define Case-1 suffice.

40%

30%

g 20% ££ 10%

0s

- 10%


40%

30%<D« 20%

10%

0%

- 10%

- 20%

-....Hi►— Col Shear Case-1 Vs b - Col Shear Case-1 Vs

Case-2Case-3

r ~ — r r r z n — '— 5“ ............ 1

---- «► Beam Shear Case-1 Vs Case-2> * - Beam Shear Case-1 Vs Case-3

A j

A....✓ ' ...................... - ................_ ......... .._ ...i

| 6

j O

i /tt1S', i 12 mgU#«.1*- - : 0.5 0.6 0,7 0......I................- ..~f-................... i-..... .....

iI ----- — — ‘ f 1 i : i J


Figure 5.11: Percentage difference for shear based on (a) Distance from source and (b) PGA (Northridge)

Change in peak ground acceleration affects beam strength demands irrespective of epicentral distance. For both near-field and far-filed motions trend is almost similar for variation PGA. Although PGA variation affects beam strength demands more than columns. As discussed earlier, higher axial forces caused by higher PGA in tension members produce additional shear in members. Column already subjected to structural

45

weight are designed for maximum axial force. As a result, additional axial forces do not have a great impact on column demands.

5.5 Earthquake 4 - Imperial Valley, USA

Imperial valley earthquake has a very different set of ground motion input for near and far field sites.

T(sec)

Figure 5.12:Time History (Brawley Airport) - Imperial valley, USA

46

0.1

-0.1

- 0.2


.■ Horizontal Aoc \

10 2015 T(sec)


25 30

Figure 5.13: Time History (Coachella Canal # 4) - Imperial valley, USA

Moment results obtained for these ground motions at near field stations show that the demands are for both beams and columns are much higher with application of vertical ground motion rather using a factored addition for additional demands (Case-1) as shown in fig 5.14.

60%

40%

§ 20%

Q U“/o

° ’ - 20%

-40%

\ ^ •

; lit! Col Moment Case-1 Vs Case-2 Col Moment Case-1 Vs Case-3 Beam MomentCase-1 Vs Case-2 Beam Moment Case-1 Vs Case-3

- £ . - " '

— --j0 0̂ 61m L__ 1 . . . . _J

Distance from source (km)i

(a)

<DOc<Dufc5ox

60%

40%

20%

0%

- 20%

-40%

| - Col Moment Case-1 Vs Case-3immsA v # M om ent 1 \ / q .-2...................Q̂m0mw090gW * * D C oi l 1 iVIUl 11 Vl 11 y V 1 v o vOOV••••♦ ••* Beam Moment Case-1 Vs Case

Jk

gfrj?*.......\ --------- _________ 4 i

b~-- ------------1• -

0 0,Hii —

T o..4 0 .5 0 ON P

(b)

Figure 5.14: Percentage difference for moments based on (a) Distance from source and (b) PGA (ImperialValley)

47

Difference between the defined cases decrease as we move further from epicenter. Increment in PGA displays the higher demand for Case-2 and Case-3 for beams while for columns the demands are less with application of vertical ground motion than the demands produced under the application of load case that uses SDS to capture effect of vertical ground motion.

80%

60%

| 40% 2 £5'•so'-

20%

0%

- 20%

-40%

80%

60%

40%

20%

0%

- 20%

-40%


Figure 5.15. Percentage difference fo r shear based on (a) Distance from source and (b) PGA (Imperial Valley)

O£

—..# — Col Shear Case-1 Vs Case-2---- Oh - Col Shear Case-1 Vs Case-3

^ *•••#■*«« Beam Shear Case-1 VsCase-2 \ — # • - Beam Shear Case-1 Vs Case-3* A\ Z A s ^ «1

1

* .................. ..................................... ..........1►

0 0 . f t 0 .2 0 .4 0 .5 0A...- „ Jfr*¥.6 0.

w * '* r|1... ... :......I....-.:..:..-.:____1. ....1__ _ ., ..L........1— „ ......... L ...........J

Comparison between the three defined cases provide a detailed insight on structural behavior under the effect of vertical ground motions. Plots based on Peak ground accelerations illustrate the dominant behavior of vertical excitations. High PGA in vertical direction require higher shear and moment capacity in structural members. As the distance from source increases, peak values decrease and the plotted curves confirm the behavior of PGA based plots for low peak ground accelerations. The results obtained using ground motions from four different earthquakes display similar trends for structural behavior


48

6 CONCLUSION

This study provides an overview of the structural response of an isolated system when vertical ground motions are applied under a variety of scenarios. The following conclusions are made based on the findings and discussion:

1. Results from the application of vertical ground acceleration in parallel to horizontal ground motions are sensitive to distance from the fault. Lower demands were observed for CASE-2 and Case-3 for far-field motions compared against those calculated from the ASCE 7-10 seismic load combinations.

2. Per the code provision, the use of short period response acceleration values (S d s ) to account for vertical excitation is conservative. However, for near field ground motions, the actual demand tends to be higher providing an under-estimation by code defined procedures.

3. Results obtained in Case-3, have diverse variation in moment demands throughout the range of distances from the fault. The overall trend presents higher capacity requirements for near-field site locations for structural members. In some earthquakes, these demands are consistently higher than those code specified regardless of distance to source or peak ground accelerations.

4. The effect of increasing peak ground accelerations is consistent with higher shear and moments; although it shows less variation for moment demands for columns. For beams, the peak ground accelerations represent high load effects compared to those associated with lower peak ground accelerations in most of the cases.

5. Frequency content is an important factor when looking at structural demands for beams and columns. High frequency content in vertical ground motion results in higher moments in tension members.

6. Lastly, it can be concluded that for an isolated structure, vertical ground motions can have an increased effect compared to Code defined provisions for isolated structures. Vertical excitations in isolated structures produce higher strength demands when applied parallel to horizontal accelerations as well as when the structure has relatively displaced, for many near field and some far field sites.

The observations from this study set the framework for more investigation into how vertical ground motions can be address and more importantly transition from purely code based designs to performance based designs.

49

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Loma Prieta Earthquake. Second international conference on recent advances in Geotechnical earthquake engineering and soil dynamics. St Louis, USA.

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13- Naeim, F., and Kelly, J. M. Design o f Seismic Isolated Structures: From Theory to Practice. John Wiley & Sons: Hoboken, NJ, USA, 1999.

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17-A Review o f Seismic Isolation for Buildings: Historical Development and Research Needs Gordon P. Warn 1, and Keri L. Ryan 2

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19-ASCE 7-10, Minimum design loads for buildings and other structures. American Society o f Civil Engineers

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Earthquake Spectra 16.1 (2000): 227-39. Web.22- Kircher, C.A., Seismically Isolated Structures, Femap-751 Chapter 12.23- ASCE 7-10, Minimum design loads for buildings and other structures. American

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Structures, Berkeley, CA. Computer software.25- U.S. Seismic Design Maps. U.S. Geological Survey. Web. 15 Jan. 201726- High Damping Rubber Bearing. Advanced Structural Analysis and Design. N.p., n.d.

Web. 15 July 2017.27-NEHRP Guidelines and Commentary for the Seismic Rehabilitation o f Buildings.

Earthquake Spectra 16.1 (2000): 227-39. Web.28-ASCE 7-10, Minimum design loads fo r buildings and other structures. American

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motions recorded during the Niigata-ken Chuetsu, Japan Earthquake o f 23 October 2004”, Engineering Geology, Vol. 94, No. 1-2.

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33- Yu, C.P. (1996) “Effect o f vertical earthquake components on bridge responses”, PhD Thesis, University o f Texas at Austin, Texas, USA.

34- Papazoglou, A.J. and Elnashai, A.S. (1996). “Analytical and field evidence o f the damaging effect o f vertical earthquake groud motion ”, Earthquake Engineering and Structural Dynamics, Vol. 25, No. 10, pp. 1109-1137.

35- Yang, J. and Lee, C.M. (2007). “Characteristics o f vertical and horizontal ground motions recorded during the Niigata-ken Chuetsu, Japan Earthquake o f 23 October 2004”, Engineering Geology, Vol. 94, No. 1-2.

36- Collier, C.J. and Elnashai, A.S., (2001). “A Procedure for Combining Vertical and Horizontal Seismic Action Effects”, Journal o f Earthquake Engineering, Vol. 5 (4),

521- 539.

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37- Bozorgnia, Y. and Campbell, K.W. (1995). “The vertical-to-horizontal response spectral ratio and tentative procedures for developing simplified V/H and vertical design spectra ”, Journal o f Earthquake Engineering, Vol. 8, No. 2. Vertical ground motions and its effect on Engineering structures: a state-of-the-art review, Bipin

shrestha38- Elnashai, A.S. and Papazoglou, A.J., (1997). “Procedure and Spectra for Analysis

o f RC Structures Subjected to Strong Vertical Earthquake Loads", Journal o f Earthquake Engineering, Vol. 1 (1), 121-156.

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40- Investigation o f Ductility Reduction Factor in Seismic Rehabilitation o f Existing Reinforced Concrete School Buildings, O. Gorgulu and B. Taskin

---------------

APPENDICES

52

53

Floor Masses

Properties\ . ..... ...

'1st Floor 2nd Floor 3rd Floor

Tributary Length Beams ft 48.000 48.000 48.000Tributary Length Columns ft 36.000 36.000 18.000g in/secA2 386.400 386.400 386.400Dead Load k/ft 3.000 3.000 3.000Self Weight (Beams)W12X106 k/ft 0.106 0.106 0.106Self Weight (Columns)W18X143 k/ft 0.146 0.147 0.143Weight/Floor kips 164.688 164.760 159.324Mass/Floor kip-secA2/in 0.426 0.426 0.412

1st Story 2nd StoryV.: : . / ' W24xl46 — ffi U 11 liiiliill!

» . . . mi W21X147

12 - in4 \ 2 - in413 4580 in4 13 3630 in 4E 29000 ksi E 29000 ksiA 43 A 43.2 sqin

Mass 3rd Story

m l 0.4262 kip-secA2/in W 18x143

m2 0.4263 kip-secA2/in 12 - in4m3 0.4262 kip secA2./in 13 2750 in4

Stiffness Columns (12EI/HA3) E 29000 ksi

K1 1601.32 k/in A 42.1 sqin

K2 1269.17 k/inK3 961.49 k/in Frame

L 24 f tH 12 f t

1 %Stiffness - Beams (RIGID) W12X106 I1 .....................i....L,...........................No. of Bays 3

i— ------~™*“[

Mass Matrix 1

M=m l 0 0 0 m2 0 0 0 m3

-0.4262111800

0-0.42630435

0

00

-0.42621118

Stiffness Matrix

K=kl+k2 .-k2 0 .-k2 k2+k3 .-k3

2870.490934 -1269.16956 0 -1269.16956 2230.661651 -961.492091

0 .-k3 k3 0 -961.492091 961.492091

54

g j f jen Values (Fixed) . .

Eigenvalues Excel/Matlat SAP2000

AA2(1) 685 480\ A2(2) 4554 3334

XA2(3) 9984 7369

Frequencies Time Period Matlab SAP2000 Difference

wl 26.173 fl 4.168 Tl(sec) 0.240 0.287 16%

u>2 67.483 f2 10.746 T2(sec) 0.093 0.109 14%

0)3 99.920 f3 15.911 T3(sec) 0.063 0.073 14%

■

Eigen Values (Isolated)

Eigenvalues Matlab SAP2000

AA2(1) 8.5 9.00

XA2(2) 2600 1979.08

\ A2(3) 8000 6333.59

AA2(4) 7539700 12730.67

Frequencies MataLab Time Period SAP2000 Difference

col 2.915 f l 0.464 Tl(sec) 2.154 2.093889 -3%13%0)2 50.990 f2 8.119 T2(sec) 0.123 0.141237

0)3 89.443 f3 14.242 T3(sec) 0.070 0.078951 11%11%0)4 2745.851 f4 437.237 T4(sec) 0.002 0.002569

55

Bearing Design11 .11111 ip

Displacement and Property RequirmentsFILE

No. of Isolators 3 !W(total)= 463 KIPS ~ ".g= 386.4 i n/sA2 Ay=(l/9)*(Qd/l< FPS

Am ax»»A y LRBFormulas:Keff = (Number of Isolators * ((id/Dmax)-y Kd) Fmax = Qd + Kd*DmaxTeff = 2*pi*(W /(Keff*g))A.5 Peff= 6*(Qd/Fmax)

BL=(?/0.05)A.3D(calc)=(A*g*TiSa=Sdl/Teff

- f fA2)/(4*piA2*BL)if(Teff<TL)

Assume Qd=0.1*W 15.433 kips : i z jKd=W/85 1.816 kips/in

Property Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7 J

Dmax(Assumed) 5 6.51 7.27 7.61 7.75 7.81 7.83kd( Ki ps/i n)= 1.82 1.82 1.82 1.82 1.82 1.82 1.82Qd(Kips)= 15.43 15.43 15.43 15.43 15.43 15.43 15.43

Keff (per isolator)= 4.90 4.19 3.94 3.84 3.81 3.79 3.79Keff= 14.71 12.56 11.82 11.53 11.42 11.38 11.36

Teff= 1.79 1.94 2.00 2.03 2.03 2.04 2.04Fmax= 24.51 27.25 28.63 29.24 29.50 29.61 29.65(3eff= 0.38 0.34 0.32 0.32 0.31 0.31 0.31BL= 1.83 1.78 1.75 1.74 1.74 1.73 1.73Sdl= 0.68 0.68 0.68 0.68 0.68 0.68 0.68

Sa(graph)= 0.379 0.350 0.340 0.336 0.334 0.334 0.333

D(calc)= 6.51 7.27 7.61 7.75 7.81 7.83 7.84

...............1 Base Shear = Keff*D or Qd+Kd*Dmax No lsolators*(Qd+Kd*Dmax)

Base Shear= 89.1 K 29.648397 K 88.94519 |k..

Elastic Base Shear Req.Fmax/W= 0*19234 <= 0.20 - y ’ Hi ''? n 't *

GOOD \ 1 1

Kd>=W/(20KdW/(20D)

1,8160.986

I r

FmaxFmaxd/2

•4,ax@D-Fma

89.0544.53

x@D/2>=W/40W/40 11.575

GOOD Fmax-Fmaxd/2 44.53 GOOD

56

Design properties of Isolator (Elastomeric)

REF:Design ofStructures w ith Sesmic Isolation

Ronald L Mayes Farzad naeim

i A/tr 1.500 ti 030 intr A/tf 5.226 in N tr/t i 18G Shear Moc 0.100 ksi S Dia/4ti 9.62Ke 3.787 kips/in Ec 6.G.SA2 55.52 ksiKH=Kd G.A/tr 1.816 kips/in As (Steel Shirrs nDA2/4 104.61 inA2A KH.tr/G 94.883 inA2 ts 0.125 inA rcDA2/4 Ns 17Ku lOKd 18.157 kips/in Fy l.IQ d 16.977 kipsDia 10.994 in Kv Ec.As/tr 1111.49 kips/inUse Dia 12 in

Weight of IsolatorRubber Shims |

Density rubber 910.00 kg/m3 Density Steel 7800.00 kg/m356.78 ibs/ftA3 486.72 lbs/ftA30.00 kips/inA3 0.00 kips/in A3

Dia of Rubber 11.54 in Dia of Steel 11.54 inHeight of Rubber Ni.ti 5.49 in Height of Steel Ns.ts 2.16 inWeight Rubber 0.02 kips Weight Shims 0.06 kips

kip-secA2/i^Total Weight 0,08254 Kips Total Mass 0.000214

57

MATLAB Scripts:Time history plots%Time history plots for Northridge Earthquake

% Horizontal Acceleration

dt=0.005;A=A11';B=B11';C=C11' ;D=Dll';E=E11';

accghzl=[A;B;C;D;E]; accghzl= accghzl(:);

T1=0.005* [1:5725] ;

% Vertical Acceleration

d t = 0 . 0 0 5 ;

F=A12';G=B12';H=C12 ' ;J=D121;K=E12';

accgvrl=[F;G;H;J;K]; accgvrl= accgvrl(:);

T2=0.005*[l:5725];%______________________


d t = 0 . 0 0 5 ;BA=A13';BB=B13';BC=C13’;BD=D13';

BE=E131;

accghzB=[BA;BB;BC;BD;BE]; accghzB= accghzB(:);

TBl = 0.005*[1:6345] ;


d t = 0 . 0 0 5 ;

BF=A14';BG=B14’;BH=C141;BJ=D14’;BK=E141;

accgvrB=[BF;BG;BH;BJ;BK]; accgvrB= accgvrB(:);

TB2 = 0 .005 *[1:6345];%_____________________________


dt=0.005;CA=A15’;CB=B151;CC=C151;CD=D151;CE=E15’;

accghzC=[CA;CB;CC;CD;CE]; accghzC= accghzC(:);

TC1 = 0 . 0 0 5 * [ 1 : 2 9 9 5 ] ;


d t = 0 . 0 0 5 ;

CF=A16’;CG=B161;CH=C161;

CJ=D16'; CK=E16';

accgvrC=[CF;CG;CH;CJ;CK]; accgvrC= accgvrC(:);

TC2 = 0 . 0 0 5 * [ 1 : 2 9 9 5 ] ;%_____________________________


d t = 0 . 0 0 5 ;DA=A1 7 ' ;D B=B1 7 ' ;DC=C1 7 ’ ;DD=D1 7 ' ;DE=E1 7 ' ;

accghzD=[DA;DB;DC;DD;DE]; accghzD= accghzD(:);

TD1 = 0 .005*[1:2000] ;


dt=0.005;

DF=A18';DG=B18’;DH=C18';DJ=D18';DK=E18';

ac cgvrD=[DF;DG;DH;DJ;DK] ; accgvrD= accgvrD(:);

TD2 = 0.005*[1:2000] ;%_____________________________


dt=0.005;EA=A19';EB=B191;EC=C19 ' ;ED=D19';EE=E19';

accghzE= [EA; EB; EC; ED; EE] ; accghzE= accghzE(:);

TE1 = 0 . 0 0 5 * [ 1 : 3 4 9 5 ] ;


d t = 0 . 0 0 5 ;

EF=A2 0 ' ;EG=B2 0 ' ;EH=C2 0 ' ;EJ=D 2 0 1 ;EK=E2 0 1 ;

accgvrE=[EF;EG;EH;EJ;EK]; accgvrE= accgvrE(:);

TE2 = 0 . 0 0 5 * [ 1 : 3 4 9 5 ] ;%____________________________

%plots

figure;subplot(5,1,1);

plot(TI,accghzl,'r ', T2,accgvrl, ' b ' ) title('Station - Jensen Filter Plant') xlabel(1T(sec)1) ylabel(1Acc(g)') xlim([0 15])legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,2);plot(TBl,accghzB,'r ', TB2,accgvrB,'b ') title('Station - SepulvedaVA hospital') xlabel('T (sec)') ylabel('Acc(g)1) xlim([0 15])legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,3);plot(TCI,accghzC,'r', TC2,accgvrC,'b') title ( ' Station - LA - N Faring R d ' ) xlabel('T(sec)') ylabel('Acc(g)') xlim([0 15])

legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,4);plot(TD2,accghzD,'r ', TD2,accgvrD,'b '), title('Station - LA - CC North') xlabel('T(sec)') ylabel('Acc(g)') xlim([0 15])legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,5);plot(TE1,accghzE,'r ', TE2,accgvrE,'b '), title('Station - Arcadia - Campus Dr') xlabel('T(sec)') ylabel('Acc(g)') xlim([0 15])legend('Horizontal Acc', 'Vertical Acc')

%Time history plots for ImpValley earthquake


dt=0.005;A=Al';B=Bl';C=C1';D=Dl';E=E1';

accghz1=[A ;B ;C ;D ;E]; accghzl= accghzl(:);

Tl = 0 .005* [1:7555] ;


dt=0.005;

F=A2';G=B2';H=C2';J = D 2 ' ;K=E2';

accgvrl=[F ;G ;H ;J ;K ]; accgvrl= accgvrl(:);

62

T2=0.005*[l:7555]; %____________________


dt=0 . 005;BA=A3';BB=B3';BC=C3';BD=D3';BE=E3';

accghzB=[BA;BB;BC;BD;BE]; accghzB= accghzB(:);

T B 1 = 0 . 0 0 5 * [ 1 : 7 5 5 5 ] ;


dt=0.005;

BF=A4';BG=B4';BH=C4';BJ=D41;BK=E4';

accgvrB=[BF;BG;BH;BJ;BK]; accgvrB= accgvrB(:);

T B 2 = 0 . 0 0 5 * [ 1 : 7 5 5 5 ] ;%_____________________________


d t = 0 . 0 0 5 ;CA=A5 ' ;C B = B 5 ' ;C C = C 5 ' ;CD=D5 ' ;C E = E 5 1 ;

accghzC=[CA;CB;CC;CD;CE] ; accghzC= accghzC(:);

TC1=0.005*[1:7555];


d t = 0 . 0 0 5 ;

CF=A61;CG=B6';CH=C6';CJ=D6’;CK=E6';

accgvrC=[CF;CG;CH;CJ;CK]; accgvrC= accgvrC(:);

TC2 = 0.005*[1:7555] ;%.


dt=0.005;DA=A71;DB=B7';DC=C7';DD=D7';DE=E7';

accghzD=[DA;DB;DC;DD;DE]; accghzD= accghzD(:);

TD1=0.005*[1:7940];


d t = 0 . 0 0 5 ;

DF=A8';DG=B81;DH=C8';DJ=D8';DK=E8';

ac cgvrD=[DF;DG;DH;DJ;DK] ;

TD2 = 0 .005*[1:7940] ;%_____________________________


dt=0.005;EA=A9';EB=B9';EC=C9';ED=D9';EE=E9';

accgvrD= accgvrD(:);

accghzE=[EA;EB;EC;ED;EE]; accghzE= accghzE(:);

TEl = 0 . 005 * [ 1 : 5 7 1 0 ] ;


dt=0.005;

EF=A10';EG=B10';EH=C10';EJ=D10';EK=E10';

accgvrE=[EF;EG;EH;EJ;EK]; accgvrE= accgvrE(:);

TE2 = 0 .005 *[1:5710];% ___________

%plots

figure;subplot(5,1,1) ;

plot(Tl,accghzl,'r ', T2,accgvrl,'b ') title('Station - Bonds Corner') xlabel('T(sec)1) ylabel(1Acc(g)’) xlim([0 35])legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,2);plot(TBl,accghzB,'r', TB2,accgvrB,'b')

title('Station - Brawley Airport') xlabel('T(sec)') ylabel('Acc(g)') xlim([0 35])legend('Horizontal Acc', 'Vertical Acc')

subplot (5,1,3) ;plot(TCI,accghzC,'r ', TC2,accgvrC,'b ') title('Station - Calexico Fire Station') xlabel('T (sec)') ylabel('Acc(g)') xlim([0 35])legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,4);plot(TD2,accghzD,'r ', TD2,accgvrD,' b ' ) ,title('Station - Delta')xlabel('T(sec)')ylabel('Acc(g)')xlim([0 35])legend('Horizontal Acc', 'Vertical Acc')

subplot(5,1,5);plot(TEl,accghzE, 'r', TE2,accgvrE, 'b ') , title('Station - Coachella Canal #4') xlabel('T(sec)') ylabel('Acc(g)') xlim([0 35])legend('Horizontal Acc', 'Vertical Acc')

Individual plots script%Time history plots for CCNZ Earthquake


dt=0 .005;A= PEERNGASTRONG1;B=MOTIONDATABAS1;C =ER ECO RD l;D=VarName4 ;E=VarName5;

A =A' ;B = B ' ;C=C'D = D ' ;E=E' ;

accghz=[A;B;C;D;E] ; accghz= accghz(:);

Tl = 0 .005* [1:13605] ;


dt=0.005;F=UPEERNGASTRONG;G=UMOTIONDATABAS;H=UERECORD;J=UVarNamel;K=UVarName2;

F =F' ;G=G' ;H=H 1 J=J' ;K = K ' ;

accgvr=[F ;G ;H ;J ;K ]; accgvr= accgvr(:);

T2 = 0 .005 *[1:13605] ;

%plots

figuresubplot(2,1,1) plot(Tl,accghz,'r 1)title('Time History, Horizontal ground motion')xlabel('T(sec)1)ylabel('Acc(g)')legend(’Horizontal Acc ' )

subplot(2,1,2) plot(T2,accgvr, 'b ')title('Time History, Vertical ground motion') xlabel(1T (sec)') ylabel('Acc(g)') legend('Vertical Acc')

67

Eigen Value Non-Isolated structure% Eigen Value Analysis Non-Isolated Structure

Es = 29000; %KSI

LCol = 12*12; %in IxCol = 4580; %inA4 LCol2 = 12*12; %in IxCol2 = 3630; %inA4 LCol3 = 12*12; %in IxCol3 = 2750; %in/'4

ml = .426 %Mass at 1st level - All levels mass divided to each node m2 = .426; %Mass at 2nd level [kip-secA2)/in m3 = .413; %Mass at 3rd level

%Define Equivalent stiffness at each levelKleq = ((12*Es*IxCol)/LColA3)*3; %Equivalent Stiffness for 1st Floor ColsK2eq = ((12*Es*IxCol2)/LCol2A3)*3; %Equivalent Stiffness for 2nd Floor ColsK3eq = ((12*Es*IxCol3)/LCol3^3)*3; %Equivalent Stiffness for 3rd Floor Cols

%Create Mass Matrix M = [ml,0,0;

0 , m2 , 0 ;0 , 0,m3];

%Create Stiffness Matrix K = [Kleq+K2eq, -K2eq, 0;

-K2eq, K2eq+K3eq, -K3eq;0, -K3eq, K3eq];

[phi w2]=eig(K,M); phi w2CircFreqW = diag(w2).A .5 Periods = (2*pi)./CircFreqW Frequency = 1./Periods

68

Eigen Value Isolated structureEs = 29000; %KSI

LCol = 12*12; %in IxCol = 4580; %in/'4 LCol2 = 12*12; %in IxCol2 = 3630; %in/'4 LCol3 = 12*12; %in IxCol3 = 2750; %in/s4

ml = 0.0002; %Mass of isolators - [kip-secA2 )/inm2 = .426; %Mass at 1st level - [kip-secA2)/inm3 = .42699; %Mass at 2nd level- [kip-sec/s2)/inm4 = .41393; %Mass at 3rd level- [kip-secA2)/in

%Define Equivalent stiffness at each level Kleq = 3.80*3 %Equivalent Stiffness for isolatorsK2eq = ((12*Es*IxCol)/LColA3)*3; %Equivalent Stiffness for 1st Floor ColsK3eq = ((12*Es*IxCol2)/LCol2A3)*3; %Equivalent Stiffness for 2nd Floor ColsK4eq = ((12*Es*IxCol3)/LCol3A3)*3; %Equivalent Stiffness for 3rd Floor Cols

%Create Mass Matrix M = [ml,0,0,0;

0 , m2 , 0 , 0 ;0 , 0,m3, 0;0,0,0,m4];

%Create Stiffness MatrixK = [Kleq+K2eq, -K2eq, 0, 0;

-K2eq, K2eq+K3eq, -K3eq, 0;0, -K3eq, K3eq+K4eq, -K4eq;0, 0, -K4eq, K4eq];

[phi w2]=eig(K,M); phi w2CircFreqW = diag(w2).A .5

Periods = (2*pi)./CircFreqW Frequency = 1./Periods;

69

Modal Mass Participation

% Modal Participation mass clcclear all

Es = 29000; %KSI

LCol = 12*12; %in IxCol = 4580; %in/'4 LCol2 = 12*12; %in IxCol2 = 3630; %in^4 LCol3 = 12*12; %in IxCol3 = 2750; %in"4

ml = 0.0002; %Mass of isolators - [kip-sec/v2)/inm2 = .426; %Mass at 1st level - [kip-secA2)/inm3 = .42699; %Mass at 2nd level- [kip-secA2)/inm4 = .41393; %Mass at 3rd level- [kip-sec/v2)/in

%Define Equivalent stiffness at each level Kleq = 3.80*3 %Equivalent Stiffness for isolatorsK2eq = ((12*Es*IxCol)/LColA3)*3; %Equivalent Stiffness for 1st Floor ColsK3eq = ((12*Es*IxCol2)/LCol2A3)*3 ; %Equivalent Stiffness for 2nd Floor ColsK4eq = ( (12 *Es*IxCol3 )/LCol3/N3 ) *3 ; %Equivalent Stiffness for 3rd Floor Cols

Ktoleq = l/(+ (1/K2eq)+ (1/K3eq)+ (1/K4eq))%Create Mass Matrix M = [ml,0,0,0;

0 , m2 , 0 , 0 ;0 , 0,m3,0;0 , 0 , 0 , m4];

%Create Stiffness Matrix K = [Kleq+K2eq, -K2eq,

-K2eq, K2eq+K3eq,# 0, -K3eq,

0, 0,

[phi w2]=eig(K,M); phi; w2;CircFreqW = diag(w2).A .5;

0, 0;-K3eq, 0;K3eq+K4eq, -K4eq; -K4eq, K4eq];

Periods = (2*pi)./CircFreqW;

Frequency = 1./Periods;

phi_t= transpose(phi);

mhat= phi_t*M*phi;

% r_bar Displacement matrix r_bar= [ 1;1;1;1];

% L_bar coefficient vector (mass units) L__bar= phi_t*M*r_bar;

% T Modal participation vector

Ti = L_bar;

mefl= 1 . 1 2 5 A2 % Modal participation mass mef2= 0 . 0 0 4 ^ 2 mef3= 0.0011^2 mef4= 0.0001^2

mm=ml +m2 +m3 +m4

importance of use of vertical ground motion in seismic

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