A SIMPLIFIED METHODOLOGY FOR SEISMIC DESIGN AND ASSESSMENT OF

NONSTRUCTURAL ELEMENTS

by

Awanish Kumar

B.Tech., Indian Institute of Technology Kanpur, 2011

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in

THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES

(Civil Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

July 2014

© Awanish Kumar, 2014

ii

Abstract

Seismic safety of the nonstructural elements has drawn considerable attention of design and

research community over past three decades. It is widely recognized that damage to these

elements account for a significant portion of total economic loss after an earthquake. Different

building codes put guidelines for assessment of these elements, however, need of a simplified

and rational method for their seismic design is still felt by practicing engineers. The existing

analysis procedures of floor response spectrum method and modal synthesis method are very

detailed and cumbersome to be efficiently used in a design office. Expressions recommended by

building codes predict a response which, in many cases, is significantly different from observed

behaviour of the components. This thesis proposes a simplified and time-efficient methodology

for seismic assessment of these elements which will enable engineers to take rational and

consistent design decisions. This methodology is based on the concept of floor response

spectrum where the acceleration demand corresponding to a component can be read from

obtained floor spectrum. The procedure makes use of a simplified continuous model proposed by

an earlier researcher to denote structures. An important feature of this methodology is the way of

inputting seismic excitation to structures. The seismic excitation is input in form of ‘Design

Ground Response Spectrum’ provided in building codes rather than commonly expected way of

using groundmotion time-histories. Some results for floor acceleration demands for two sites in

Canada are also presented. The methodology is extended to include base-isolated structures also.

An independent procedure is proposed for assessment of components placed in the irregular

structures. It is based on scaling of the ‘Reference Floor Response Spectrum’. The methodology

presented in this thesis can be developed into an ‘Analyzer’ package to be used by practicing

engineers for components’ design and assessment for all places in Canada. A useful guiding line

for nonstructural element’s assessment to engineer is to decide whether to retrofit, relocate or

replace it. This methodology based on the floor spectrum concept should enable designer in

taking this decision rationally.

iii

Preface

This research was conducted as collaboration between Awanish Kumar and Professor Carlos

Ventura. Contribution of thesis author to research project were (i) design of the research program

(ii) analysis of results and (iii) preparation of this manuscript. The research topic ‘Seismic

Design and Assessment of Nonstructural Elements’ was suggested to the author by Professor

Ventura. Author reviewed existing literature and, subsequently, the idea of a ‘Simplified

Continuous Model’ to denote structures was suggested by Professor Ventura which is presented

in Chapter 3. Author extended formulation for the regular structures to base-isolated structures

presented in Chapter 4. Fruitful discussions led to material of Chapter 5 which gives a qualitative

idea for assessment of nonstructural elements placed at different floor levels of an irregular

structure. In the course of implementation of research program Professor Ventura gave timely

feedback and provided with constructive criticism which was very helpful during preparation of

this manuscript.

iv

Table of Contents

Abstract ......................................................................................................................................................... ii

Preface ......................................................................................................................................................... iii

Table of Contents ......................................................................................................................................... iv

List of Tables ................................................................................................................................................ vi

List of Figures .............................................................................................................................................. vii

1. Introduction .......................................................................................................................................... 1

1.1. Nonstructural Elements ................................................................................................................ 1

1.1.1. Importance of nonstructural elements ................................................................................. 3

1.1.2. Characteristics of nonstructural elements ............................................................................ 5

1.2. Analysis and Assessment Methods ............................................................................................... 6

1.2.1. Floor response spectrum method ......................................................................................... 7

1.3. Steps in Seismic Design ................................................................................................................. 8

1.4. Existing NBCC Method .................................................................................................................. 8

1.5. Objective and Structure of Thesis ............................................................................................... 11

2. Literature Review ................................................................................................................................ 13

2.1. Design Code Procedures ............................................................................................................. 13

2.1.1. ASCE/SEI 7-10 ...................................................................................................................... 13

2.1.2. Eurocode ............................................................................................................................. 14

2.2. Existing Literature ....................................................................................................................... 15

2.2.1. State-of-the-art review by Soong and Chen ....................................................................... 15

2.2.2. Nonlinear behaviour of structure and components ........................................................... 16

2.2.3. Testing on piping systems ................................................................................................... 17

3. Regular Structures ............................................................................................................................... 19

3.1. Floor Acceleration Demands ....................................................................................................... 20

3.1.1. Simplified continuous model of structure .......................................................................... 20

3.2. Generation of Floor Spectra ........................................................................................................ 24

3.2.1. Methodology ....................................................................................................................... 24

3.3. Nonlinearity of Structure and Components ................................................................................ 27

v

3.4. Results for Sites in Canada .......................................................................................................... 28

4. Base-isolated Structures ..................................................................................................................... 36

4.1. Formulation for Base-isolated Structures ................................................................................... 37

4.2. Two Example Problems ............................................................................................................... 40

4.2.1. Discussion ............................................................................................................................ 44

5. Irregular Structures ............................................................................................................................. 46

5.1. Irregularity in Plan ....................................................................................................................... 47

5.1.1. Decision parameter B .......................................................................................................... 48

5.2. Methodology for Irregular Structures ......................................................................................... 49

5.3. Results ......................................................................................................................................... 54

5.3.1. First approach ..................................................................................................................... 54

5.4. Limitations of Proposed Method ................................................................................................ 62

6. Conclusion and Future Work .............................................................................................................. 63

References .................................................................................................................................................. 66

vi

List of Tables

Table 1.1 - Elements of structure, nonstructural components and equipments ........................................... 10

Table 4.1 -Stiffness Ratio SR for base-isolated structures .......................................................................... 40

Table 4.2 - Dynamic properties of 4 story Steel MRF structure; α = 17..................................................... 40

Table 4.3 - Dynamic properties of 4 story concrete shear wall structure; α = 1 ......................................... 41

Table 5.1 - Periods of vibration of 4 story structures in two approaches .................................................... 54

Table 5.2 - Steel sections in four story Steel MRF irregular structures ...................................................... 55

Table 5.3 - Periods of vibration of four story steel MRF structures ........................................................... 55

Table 4.4 -Peak factors for scaling floor spectrum ..................................................................................... 60

Table 4.5 - Scaling factors* to get peak acceleration demands for nonstructural elements........................ 61

vii

List of Figures

Figure 1.1 –Typical structural members and nonstructural elements in a conventional structure ................ 2

Figure 1.2 - Pi chart for costs of damage to the structural and nonstructural elements ................................ 4

Figure 1.3 -Relative costs of investment for three common construction types ........................................... 5

Figure 1.4 - Sliding and overturning of unrestrained contents during lateral movement of floor ................ 5

Figure 1.5 - Generation of Floor Response Spectrum for a groundmotion................................................... 8

Figure 1.6 -Linear variation of design lateral force for nonstructural element design ................................ 11

Figure 3.1 - Schematic diagram of simplified continuous model ............................................................... 21

Figure 3.2 - Acceleration Floor Spectra for third floor of four story structures for four different structure

types; Structure damping: 0.05, Component damping: 0.02 ....................................................................... 29

Figure 3.3 - Acceleration Floor Spectra for third story of 4-story concrete moment frame structure for two

different component damping values .......................................................................................................... 30

Figure 3.4 - Displacement floor spectra for 3rd

floor of 4-story steel MRF structure for two damping

values; Structure damping 0.02 ................................................................................................................... 30

Figure 3.5 - Acceleration Floor Spectra for different floors of a 4-story concrete MRF structure;

Component damping: 0.02 .......................................................................................................................... 31

Figure 3.6 - Displacement floor spectra for three floors of 4-story Steel MRF structure; Structure damping

0.05, component damping 0.02. .................................................................................................................. 32

Figure 3.7 - Displacement and acceleration floor spectra for 3rd

floor of 4-story steel MRF structure;

Structure damping 0.05, component damping 0.02. ................................................................................... 33

Figure 3.8 -Ground Response Spectra for Vancouver and Montreal .......................................................... 34

Figure 3.9 - Acceleration Floor Spectra for 3rd

story of 4-story Concrete MRF structures located in

Vancouver and Montreal; Component damping: 0.02 ................................................................................ 35

Figure 4.1 - Models for a base-isolated structure ........................................................................................ 37

Figure 4.2 - Schematic diagram of base-isolated structure ......................................................................... 38

Figure 4.3 - Displacement floor spectra for 3rd

story of Regular and Base-isolated Shear wall structure;

Structure damping 0.02, Component damping 0.02; .................................................................................. 42

Figure 4.4 - Displacement floor spectra for 3rd

story of Regular and Base-isolated 4 story Steel MRF

structure; Structure damping 0.02, Component damping 0.02; .................................................................. 43

Figure 4.5 - Acceleration floor spectra for third story of regular and base-isolated 4 story shear wall

structure; Structure damping 0.05, component damping 0.02 .................................................................... 44

Figure 5.1 - Typical SFRS configurations causing planar irregularity ....................................................... 47

viii

Figure 5.2 - Lateral displacements at floor level caused by eccentric equivalent static force .................... 48

Figure 5.3 - Proposed methodology for seismic assessment of nonstructural components: ....................... 51

Figure 5.4 - Four story irregular structure analysed for proposed procedure; B=1.4 ................................. 53

Figure 5.5 - Floor spectra for stiff and flexible edges of third floor of Steel MRF structure located in

Vancouver; B=1.4, Structure and Component damping 0.02 ..................................................................... 56

Figure 5.6 - Floor Spectra for stiff and flexible edges of third floor of Steel MRF structure located in

Vancouver; B=1.6, Structure and component damping 0.02 ...................................................................... 57

Figure 5.7 - Comparison of floor spectra for third floor of a four story irregular Steel frame structure with

spectrum from Miranda’s model; Structure damping 0.02, component damping 0.02............................... 58

Figure 5.8 - Floor spectra for stiff edges of uppermost floors of a four and an eight story structures; B =

1.4, Structure damping 0.02, component damping 0.02 ............................................................................. 59

Figure 5.9 - Eccentricities in two analysed irregular structures .................................................................. 60

1

CHAPTER 1

INTRODUCTION

1. Introduction

1.1. NonstructuralElements

A Civil Engineering structure primarily consists of the two systems. First system comprises the

load bearing elements, e.g. beams, columns and roof diaphragms. These elements are designed to

resist gravity, wind and earthquake induced loads. The second system consists of the components

of structure such as elevator, refrigerators, bookracks and other electrical and mechanical

equipments and these are referred to as the ‘Nonstructural elements’. These components are not

part of the primary load bearing system of a structure but play an important role in its

functionality. Sometimes they are also referred to as ‘Secondary structures’. Figure 1.1 shows

typical structural members and nonstructural elements in a conventional structure.

2

Figure 1.1 –Typical structural members and nonstructural elements in a conventional structure

Nonstructural elements can be classified into three broad categories[1]- (i) Architectural

components (ii) Mechanical and electrical components and (iii) Building contents. For example,

signboards, chandeliers, elevator penthouses, parapets, partition walls, suspended ceilings etc.

come under first category. The second category is constituted by refrigerators, storage tanks,

electric boilers, smokestacks, switchgears, antennas, control panels etc. Furniture, bookracks and

file cabinets etc. come under third category.

Nonstructural elements can also be classified from modeling and analysis point of view in

following three categories: (i) Rigid (ii) Flexible and (iii) Hanging from the above. Rigid

components are short period elements and their seismic behaviour depends on stiffness and

ductility of their connections. They can be modeled as single degree of freedom system. Flexible

components are modeled as MDOF system with varying stiffness and mass properties. In some

cases they may have multiple points of attachments, e.g. piping systems. Components hanging

from above are modeled as simple pendulum. These types of components are, in general, not

3

analysed as they are seldom damaged by earthquakes. However, one possible hazard posed by

them is that they may undergo large oscillations and collide with the supporting structures or

nearby objects.

1.1.1. Importance of nonstructural elements

Over past three decades it has been realised that damage to nonstructural elements is the major

cause of economic losses due to an earthquake. In a survey of structures after 1971 San Fernando

earthquake it was found that damage to contents and the interior and exterior finishes resulted in

97% of the total economic loss while structural damage was limited to only 3%, as reflected in

figure 1.2 below[2].During 1994 Northridge earthquake major hospitals had to be evacuated not

because of structural damage but due to damage to (i) water storage tanks,sprinklers and the

piping systems (ii) the air conditioning units, cladding systems and broken glass windows and

(iii) elevators, suspended ceilings and the light fixtures. Now it is also recognised that damage to

these elements may pose threat to life safety and impair functionality of a structure. Fall of

suspended ceilings and ornaments, overturning of some heavy equipments and bookshelves,

damage to storage tanks and rupture of the piping systems containing toxic gases etc. may cause

serious injury and affect functionality of critical structures during an earthquake.From

investment perspective, when the costs of various structural and nonstructural elements are

compared, it is found that in most commercial constructions the nonstructural elements and other

contents account for a major portion of total investment. Thus, after an earthquake cost of repair

and replacement of the damaged nonstructural elements may far exceed that of retrofit of the

structural elements. Figure 1.3 below shows the relative costs of contents and structural and

nonstructural elements for three major commercial construction types [2].

4

Figure 1.2 -Pi chart for costs of damage to the structural and nonstructural elements

3% 7%

34% 56%

Damage from San Fernando earthquake (1971)

Structural

Electrical &Mechanical

ExteriorFinishes

5

Figure 1.3 -Relative costs of investment for three common construction types

Unrestrained contents, which are simply placed on the floor, can be damaged in two ways. The

short and stocky contents can slide sideways during a groundmotion. Similarly, slender contents

can overturn and fall on the objects kept nearby. Figure 1.4 below schematically explains this.

(a) Stocky content (b) Slender content

Figure 1.4 - Sliding and overturning of unrestrained contents during lateral movement of floor

1.1.2. Characteristics of nonstructural elements

Nonstructural elements have some specific physical and response characteristics which are

significantly different from those of the structures. The nonstructural elements are attached to or

placed at different floors of a structure. So they are not directly subjected to the groundmotion

generated by an earthquake. Rather they are subjected to the acceleration response of that

particular floor which in turn depends on the dynamic characteristics of that structure.

Nonstructural elements are, in general, lightweight and their mass is much smaller compared to

the floor mass. Also these elements are made up of materials which are not designed to resist

seismic forces like that of the structure. Also, damping properties of the nonstructural elements

are much different from that of the structure and in general their damping ratios are much

6

smaller. In addition, some nonstructural elements, e.g. piping systems, may have multiple points

of attachments to the structure so they may be subjected to differential movements during an

earthquake.

Above physical characteristics of the nonstructural elements impart them specific

response characteristics. Response of a nonstructural element depends not only on the

characteristics of groundmotion but also on dynamic properties of its supporting structure. As

response of an element also depends on the response of floor to which it is attached, so identical

elements placed at two different levels of a structure may have significantly different responses.

Depending on period of the component, there may be significant period interaction between

structure and the component. If period of element is close to one of the higher periods of

structure then combined structure-nonstructural system may result in closely spaced frequencies.

1.2. Analysis and Assessment Methods

For adequate seismic design of nonstructural elements one needs to analyse their response under

expected groundmotion for the corresponding site. Over past four decades researchers have

developed several such analysis methods[1]. A significant portion of this research was motivated

by need of functionality of critical equipments in nuclear power plants, e.g., piping systems and

control panels, during an earthquake. Therefore, some of these methods have strong empirical

basis and others are based on rigorous principles of structural dynamics.

Analysis of nonstructural elements can be performed in two ways. Either the structure andthe

element can be analysed separately and element’s response can be evaluated from two results or

a combined structure-nonstructural element system analysis can be performed. Both of these

methods have some advantages and shortcomings. In former, interaction effects between

structure and element are ignored, whereas in later, analysis is resource-consuming owing to

significant differences between physical and dynamic characteristics of structure and the

element. Since massratio of element and structure is very low and their damping values are

significantly different, a modal analysis of combined system does not yield accurate mode shapes

and natural frequencies. On theother hand, a step by step time integration method is inefficient

and shows convergence issues. Most common methods from each of the above categories are

7

Floor Response Spectrum Method and Modal Synthesis Method. The Floor Response Spectrum

Method is discussed in section 1.2.1 below.

1.2.1. Floor response spectrum method

One of the widely used methods for assessment of nonstructural elements is ‘system-in-cascade’

approach or ‘Floor Response Spectrum Method’. Sometimes it is also called ‘in-structure

response spectrum’ approach. In this method, first, input excitation at that point or floor is found

where component is attached to structure. This input excitation is response of that point or floor

of structure to a suitable ground motion time history. Then response spectrum for this input

excitation is found in much the same way as response spectra for structures are generated. To

distinguish it from Ground Response Spectrum provided in design codes it is called Floor

Response Spectrum. As using a single time-history to get floor spectrum is not accepted for

design purposes,generally, multiple time-histories scaled to required seismic hazard are used and

average or envelope of obtained floor spectra is used.Figure 1.5 schematically explains

generation of a floor response spectrum.

Groundmotion input

Tc, ζc

Input floor

acceleration T1

T2

Floor Response

Spectrum

8

Figure 1.5 - Generation of Floor Response Spectrum for a groundmotion

Floor Response Spectrum has been found to give accurate results when mass of

nonstructuralelement is very small compared to mass of its supporting structure and natural

frequency of the element is not too close to that of the structure. A large element to structure

mass ratio causes significant interaction between them. This interaction cannot be captured by

method of floor spectrum as it is based on ‘system-in-cascade’ approach. Also, as the damping

ratio of nonstructural element is much smaller compared to that of the structure, damping of

combined structure-nonstructural system is nonclassical in nature. Floor Response Spectrum

method cannot capture this effect also. Despite these shortcomings it is a simple and rational

method for the earthquake design and assessment of nonstructural elements.

1.3. Steps in Seismic Design

Different steps in the seismic design procedure of nonstructural elements can be listed as

follows:[3]

i. Calculate the design force from analysis or the code expression. Design force is obtained

by multiplying mass of component to the expected acceleration during a seismic event.

ii. Multiply this force by an importance factor larger than 1.0 for critical components.

iii. Divide this force by a factor larger than 1.0 to account for the overstrength and ductility

of element and its connection.

iv. Apply this lateral design force at centre of mass of element.

v. Find response forces at other sections of element and in its connection.

vi. Design nonstructural component itself and its connection to withstand design forces and

support reactions.

1.4. Existing NBCC Method

NBCC 2005(Clause 4.1.8.17) recommends a static force procedure for seismic design and

assessment of elements of structures, nonstructural components and equipments[4].

9

Nonstructural element and its connections should be designed to resist a lateral force Vpapplied

through its centre of mass which is equal to

( )

where

Fa = acceleration based site coefficient

Sa(0.2) = spectral response acceleration value at period 0.2 seconds

IE = importance factor for building

Sp= (CpArAx)/Rp (maximum value of Sp shall be taken as 4.0 and minimum value shall be taken

as 0.7) where

Cp= element or component factor

Ar= component force amplification factor

Ax= height factor (1+2hx/hn); hx and hn are height of floor with component and total

height of structure respectively

Rp= element or component force modification factor

Wp= weight of component

Sa(0.2) is design spectral response acceleration value for short period structures at the site. This

acceleration value is approximately two-third of acceleration due to the Maximum Considered

Earthquake (MCE) at that site [5].

( )

Thus 0.3*Sa(0.2) should approximately denote peak ground acceleration (PGA) at corresponding

site.

Factors Cp, Ar and Ax are listed in Table 4.1.8.17 of code. Component factor Cp varies from 1 to

1.5. For components that are flexible or flexibly connected, code recommends to use a Cp value

of 2. In general Cp value is 1 for most components except for the machineries, pipes or tanks

containing toxic or explosive material or materials having a flash point below 380C, for which a

value of 1.5 is used. The component force amplification factor Ar varies from 1 to 2.5. For

cantilever walls, parapets, chimneys and smokestacks Ar value is 2.5, whereas for horizontally

cantilevered floors, beams, balconies, ducts and cable trays it is 1.00. The element force

10

modification factor Rpvaries from 1 to 5. For nearly half of the component categories code

recommends aRpvalue of 2.5. For rigid components with non-ductile materials or connections it

is 1.0. For machinery and tanks that are rigid or rigidly connected it is 1.25 whereas a value of

2.5 is recommended for flexible or flexibly connected machinery. Table below lists these factors

for some components from Table provided in NBCC.

Table 1.1 - Elements of structure, nonstructural components and equipments

Part or Portion of Building Cp Ar Ax

Towers, chimneys, smokestacks and penthouses 1.00 2.50 2.50

Horizontally cantilevered beams, floors and balconies 1.00 1.00 2.50

Machinery, equipments and ducts

that are rigid or rigidly connected

that are flexible or flexibly connected

1.00

1.00

1.00

2.50

1.25

2.50

Ducts containing toxic or explosive materials 1.50 1.00 3.00

Rigid components with ductile material or connections 1.00 1.00 2.50

Rigid components with non-ductile material or

connections

1.00 1.00 1.00

The height factor Ax accounts for the variation of design force along height of the structure.

Based on statistical analyses of sample set of structures code assumes this variation to be linear,

demand at top of structure being three times that at its base.

11

Figure 1.6 -Linear variation of design lateral force for nonstructural element design

1.5. Objective and Structure of Thesis

Researchers and practicing engineers have reported that the above methodology of nonstructural

elements’ assessment recommended by the Design code might be inadequate in some cases[6].

The existing procedure does not account for effect of type of the structure on floor acceleration

demands. It has also been shown that, when the period ofelement is close to any of the periods of

structure, the element’s response is amplified due to period interaction. This effect is also not

reflected in the results obtained using the code expression.Existing analysis procedures of floor

response spectrum method and modal synthesis method give more accurate results but they are

very detailed to be incorporated in design guidelines in their complete form.

Objective of this thesis is to propose a simplified method of nonstructural element’s assessment

which enables engineer to take rational design and retrofit decisions in an efficient way. This

methodology is based on the concept of floor response spectra but here design engineer does not

need to follow detailed procedure of deriving floor spectra himself. Rather this procedure can be

adapted into a programor ‘Analysis package’ to directly get floor spectra upon input of

structure’s and component’s relevant parameters.

Height factor Ax = 1 + (2hx/hn)

12

Chapter 2 presents a brief overview of existing literature and current design code procedures.

The procedures given in ASCE 41, NBCC 2010 and Eurocode 08 are presented and compared.

Parameters used in their expressions are explained and, in some cases, their physical meaning is

inferred with help of suitable examples. A state-of-the-art review conducted around two decades

ago is presented which broadly covers different analysis methods, their shortcomings and

suggested future works. An analytical work on effect of nonlinear behaviour of structure and the

nonstructural element on element’s response is presented thereafter followed by an experimental

study on behaviour of piping assemblies in a hospital building. These three works broadly

explain characteristics of the nonstructural elements and the methods to enquire their seismic

behaviour.

Chapter 3 presents methodology to generate floor spectra for the regular structures. Here the

modelling procedure for structures and method to input seismic excitation are discussed. Some

results of floor acceleration demands for different structure types and two sites in Canada are

shown thereafter.Chapter 4 extends methodology used for the regular structures to base-isolated

structures. Problem formulation for the base-isolated structures remains same except that the

boundary conditions of model used for regular structure aresuitably modified. Two examples on

base-isolated shear wall and steel MRF structures are presented showing characteristics of

displacement and acceleration responses for such structures.

In Chapter 5 a procedure is proposed for the assessment of nonstructural elements in irregular

structures. This is based on scaling of the acceleration demand from the ‘Reference Floor

Spectrum’ along elevation of an irregular structure and plan of a given floor in it. This procedure

is also based on the concept of floor spectrum itself but it is not as general and exact as that for

regular structures.Conclusion of this study is presented in following and the final chapter. This

work can be developed into a package to be used by design engineers for assessment of

nonstructural elements. Some possible directions of future researchare also mentioned in the

chapter which should fill gaps in this work to be suitably adapted for above purpose.

13

CHAPTER 2

LITERATURE REVIEW

2. Literature Review

Researchers have put a lot of efforts in investigating the seismic behaviour of

nonstructuralelements over past three decades. Much of this research has been driven by need of

survivability of critical components of nuclear power plantsduring an earthquake.The procedures

recommended by various design codes, however, do not reflect some useful implications of

above developments.In the literature review presented in this chapter a brief overview of the

procedures given in different design codes is presented first. Following this, a state-of-the-art

reviewconducted around two decades ago by Soong and Chen[7]is overviewed. During an

earthquake a structure may exhibit nonlinear behaviour which affects response of nonstructural

elements in it. A relevant work by Chaudhuri and Villaverdeon effect of inelastic behavior of

structure and elements on response of elements is briefly discussed[8]. Sometimes lab tests are

performed to check accuracy of analysis results. An experimental work by Zaghi et al[9] on

earthquake response of piping assemblies in a hospital structure is reviewed.

2.1. Design Code Procedures

2.1.1. ASCE/SEI 7-10

ASCE 7, Minimum Design Loads for Buildings and Other Structures, recommends seismic

design procedures for nonstructural components in chapter 13[10]. It provides general guidelines

for design requirements of nonstructural components and its anchorage. It also discusses in detail

14

specific requirements for architectural, mechanical and electrical components and piping

systems. Clause 13.3 quantitatively indicates seismic demand on nonstructural elements. This

demand is primarily based on two factors, one accounting for amplification of the forces in

components and other for the possible response reduction due to ductility of the element and its

anchorage. In this clause seismic design force Fp is written as

(

)

(

)

and minimum and maximum values of design force are given by following relation

where

WPis component operating weight.

SDS is spectral acceleration for short period structures.

apis component’s amplification factor which varies from 1 to 2.5.

IPis component’s importance factor that varies from 1 to 1.5.

RPis component response modification factor that varies from 1 to 12.

2.1.2. Eurocode

Eurocode 8 recommends an expression also based on equivalent static force procedure. Seismic

demand Fa is given by

(

)[

(

)

( (

)

) ]

where

αis design spectral acceleration in terms of g

S is soil factor

γais importance factor of component

qais behaviour factor of element

Tais period of component

15

T1is fundamental period of structure.

Similar to other ASCE and NBCC recommendations Eurocode expression is also based on force

amplification factors and response reduction factors. However, a major difference in its approach

is that it also accounts for ratio of period of element to that of structure. Thus seismic demand

force reflects any period interaction between element and structure.

2.2. Existing Literature

A large amount of literature exists on topic of nonstructural elements spanning more than three

decades. In this chapter three works are discussed which give an overview of different areas in

this topic and analytical and experimental approach to study their behaviour.A state-of-the-art

review conducted around two decades ago is presented which broadly covers different analysis

methods, their shortcomings and suggests possible future work. An analytical study conducted

on effect of nonlinear behaviour of structure and nonstructural element on element’s response is

presented thereafter. Then an experimental study on behaviour of piping assembly in a hospital

building is presented. These three works broadly explain special characteristics of nonstructural

elements and method to enquire their seismic behaviour.

2.2.1. State-of-the-art review by Soong and Chen

This review presents an overview of work done till around two decades ago in area of component

analysis and design[11]. In beginning it mentions characteristic features of nonstructural

elements, such as small mass, low damping values and, in some cases, multiple points of

attachment. It discusses existing analysis methods of floor response spectrum approach and

combined primary-secondary system approach.In floor response spectrum approach concepts of

‘Spectrum peak broadening’ and ‘Combined spectrum and Spectrum envelope’ are also

presented. In combined primary-secondary system analysis effect of mass ratio and frequency

ratio on response results are presented. In following section paper presents advances and

developments in analysis methods. Generation of floor response spectra from ‘Design Ground

Spectrum’ in place of groundmotion time-history is mentioned. An area of study at this time was

‘Response Sensitivity to Uncertainties’. Response sensitivity of nonstructural element to different

uncertainties can be accounted for by a nondimensional parameter K where

16

where factors K1, K2 etc. account for different uncertainties separately. These can be due to

uncertainties in mass, stiffness and damping values, structural modeling and material

nonlinearities. In combined system analysis ‘substructuring’ is suggested as a convenient

alternative in cases where combined structure-nonstructural analysis results in a system with very

large number of degrees of freedom. In substructuring combined system is divided into small

subsystems and compatibility and equilibrium conditions are applied at their common points.

This paper suggested to achieve ‘optimization’ in placement of nonstructural elements as a

possible direction of future study where elements are placed at different floor levels of a

structure in an optimum way considering serviceability and damage aspects.

2.2.2. Nonlinear behaviour of structure and components

In past four decades significant development has been made in understanding behaviour of

nonstructural elements. For most of the cases, however, it has been related to behaviour of linear

components mounted on linear structures. But during a major seismic event structure may go

into nonlinear zone of behaviour affecting response of components. Several researchers have

pointed out that nonlinear behaviour of structure affects behaviour of component, either in form

of amplification or reduction in its response. This particular study was aimed at investigating that

to what degree and in what conditions nonlinear behaviour of structure affects seismic response

of component[8].

In this study eight code-designed steel moment frame structures were used. Their models were

subjected to a series of 25 recorded earthquake groundmotions. Structure and nonstructural

elements were alternatively modeled as linear and nonlinear. Structures comprised of four, eight,

twelve and sixteen story moment frames and SDOF components were sequentially placed on

different floors of structures. This study concluded that, as reported earlier by other researchers,

building nonlinearity significantly affects component’s response. Seismic response of

nonstructural element shows large amplification when its supporting structure undergoes

localised nonlinear behaviour compared to it experiences widespread nonlinearity. This response

is also amplified when component is located on lower floor of structure and its frequency is close

to one of the higher frequencies of structure. When groundmotion is narrow-banded and most of

17

its energy content is close to fundamental period of structure then also component’s response is

likely to be significantly amplified.

Overall, it was concluded from this study that nonlinear behaviour of structure has a favourable

effect on seismic response of component. This implied that component can be designed based on

a linear response analysis or even based on a procedure where result from linear analysis is

modified by a response reduction factor.

2.2.3. Testing on piping systems

Functionality of the hospital buildings is desirable after an earthquake. Damage to nonstructural

elements of a hospital building may affect its functionality and cause major economic losses. In

this study a typical piping assembly of hospital buildings was experimentally and analytically

evaluated. During an earthquake piping assemblies are subjected to differential movements. This

may cause damage to its joints and connections and rupture in pipe leading to leakage. This

study had following objectives[9]:

i. To find dynamic properties, frequencies and mode shapes, of system.

ii. To study effect of threaded and welded joints.

iii. To identify damage states such as crack formation and leak initiation in system.

iv. To calibrate a computational model to get aid in modeling of seismic restraints.

Piping assembly was tested on a biaxial shake table. It was found that welded connections

showed no leakage up to a drift ratio of 4.3% whereas threaded connection showed leakage at

2.2% drift ratio and suffered connection failure at drift ratio of 4.3%. A simplified model of test

set-up was built using SAP 2000 and calibrated using experimental data. Seismic restrainers

were also used in this experiment and their effective stiffness was observed to be 10% of actual

stiffness due to initial slack. Authors also suggested that implications of this testshould be judged

before making design decision as boundary conditions of test assembly might be different from

those of piping systems supported by building floors.

A large amount of literature exists in areaof nonstructural elements’ analysis, however,

only a brief overview of representative literature has been presented in this chapter. This thesis

uses a generalized model for structures to obtain the floor acceleration demands. Also, the input

18

of seismic excitation is in the form of Ground Response Spectrum as opposed to commonly

adopted procedure of time-history analysis. The literature referring to these methodologies is

presented in following chapter.

19

CHAPTER 3

REGULAR STRUCTURES

3. Regular Structures

A structure is characterised as regular or irregular based on the configuration of its Seismic Force

Resisting System (SFRS). If configuration of SFRS results in centre of mass of the structure

coinciding with its centre of resistance, structure is said to be regular in its plan. Such structures

show no or least torsional response during a seismic event. In this chapter the procedure for

generating floor spectrum for a regular structure is presented. An earlier study on floor

acceleration demands is reviewed first and its results are compared with those predicted by

existing code expression. In this thesis a generalized modeling procedure for structures is used

and its formulation is discussed[12]. In general practice a set of groundmotion time-histories are

used to obtain design floor acceleration demands but input of seismic excitation in this work is in

form of the ‘Ground Response Spectrum’. The rational and methodology behind this approach

are presented[13]. Finally results obtained for few sites in Canada are shown.

20

3.1. Floor Acceleration Demands

The researchers and practicing engineers have reported over the years that design code

procedures for assessment of nonstructural elements give inconsistent results for some cases. US

and Canadian Codes recommend a trapezoidal variation of peak floor acceleration varying from

peak ground acceleration (PGA) at base of structure to three times of this at its top. This

variation is independent of type of lateral force resisting system. Floor Acceleration Demands in

multi-story buildings were comprehensively studied by Miranda and Taghavi[14]. Results of this

study indicate that existing code procedure has significant shortcomings. In this section,

methodology of this work is discussed and its results are presented to get an insight into research

need.This study usesa simplified continuous model to represent structures. This modelling

approach is based on lateral stiffness ratio parameterα. With varying values of this parameter

model yields representative structure types. In subsections below, first, this modeling approach is

discussed and results of study are presented thereafter.

3.1.1. Simplified continuous model of structure

A continuous model is based on the parameters and, with such a model, closed form expressions

for resulting response quantities can be derived. Continuous models have been used for structural

analysis for around seven decades now. As listed in work by Miranda and Akkar in one of the

references, Jennings and Newmark (1960) used a continuous shear beam model to estimate

lateral deformations in the structures subjected to earthquake motion. Iwan (1997) used shear

beams for his drift spectrum concept. Montes and Rosenblueth (1968) used flexural beams to

estimate overturning moments along height of chimneys. However, these two extreme modes of

structural behaviour are not suitable for few other structure types which show intermediate

behaviour. Some structures have dual lateral force resisting systems consisting of a combination

of walls and frames. Such structures can be represented by a model shown in figure below.

21

Figure 3.1 - Schematic diagram of simplified continuous model

The continuum model consists of a flexural cantilever beam and a shear cantilever beam

deforming in bending and shear configurations respectively. The two beams are connected by a

large number of axially rigid links to transmit lateral forces[15]. This ensures that two beams

have same lateral deflection along height of the structure. Khan and Sbarounis (1964) were first

to propose a model combining shear and flexural behaviour for structures. They used it to study

interaction between shear walls and frames. Miranda and Reyes (2002) used a model where

lateral stiffness and mass was varied along the height.

Mathematically, model shown in figure 3.1 above in its general form is represented by following

differential equation

Flexural Beam

Shear Beam

Axially rigid links

22

( )

( )

( )

( )

( )

( )

where

ρis mass per unit length of model

EI is flexural rigidity flexural beam

c is damping coefficient per unit length

H is height of structure

√

is lateral stiffness ratio

GA is shear rigidity of shear beam

u(x,t) is lateral deflection of model at normalised height x and time instant t

ug(t) is lateral displacement of ground surface during an earthquake

The lateral stiffness ratio α controls degree of participation of flexural and shear deformation of

continuous model and thus it controls its lateral deflected shape. A value of 0 for α denotes a

pure flexural beam and a value tending to infinity denotes pure shear beam. Miranda and Reyes

(2002) studied influence of α on roof displacements and interstory drift demands. Based on

lateral deflected shape of the model this study indicated that structures, whose lateral force

resisting system consists of structural walls only, can be approximated by a value of α between 0

and 2. Structures with dual lateral force resisting system can be represented by a value of α

between 1.5 and 6 and a value of α between 6 and 20 should represent a structure whose lateral

force resisting system consists only of moment frames.

Dynamic properties of model

For undamped free vibration of model equation (3.1) reduces to

( )

( )

( )

( )

Making appropriate substitutions from method of modal analysis and with method of separation

of variable following equation for model’s mode shapes is obtained.

23

( )

( )

( ) ( )

General solution of above equation is

( ) ( ) ( ) ( √ ) ( √

) ( )

where constants A1, A2, A3 and A4 depend on boundary conditions of model and eigenvalue

parameter γi is related to circular frequency of vibration by

( ) ( )

For a fixed base structure boundary conditions at base of structure, i.e. at x=0, are written as

( ) ( )

and

( )

( )

The continuous model is free at top implying null shear force and moment. This can be written

by following two equations at nondimensional height x=1.

( )

( )

( )

( )

( )

Expression for mode shape can be obtained with use of above equations and normalising modal

deflected shape at top of structure to some constant. Modal participation factor Γi is obtained

from mode shape expression using following equation.

∫ ( )

∫ ( )

( )

Thus dynamic properties of a structure are obtained as above by employing simplified

continuous model. Now the frequencies and modal participation factors of the structure are

known and, with this information, structure’s response to any given seismic excitationcan be

24

found out. In this work, the way of inputting seismic excitation is in form of ‘Ground Response

Spectrum’. Its methodology is presented in section 3.2 below.

3.2. Generation of Floor Spectra

Once structure’s model and its dynamic characteristics are determined, seismic excitation needs

to be input to obtain the floor acceleration demands. A direct way of doing this is to perform a

time-history analysis using a groundmotion expected at that site. However, a single time-history

should not be taken to represent all expected groundmotions at that site and a set of time-

histories should be used for this purpose. These time-histories should be scaled to match Design

Ground Spectrum of that site and the average or envelope of floor spectra thus generated

represents spectrum to be used for design purposes. However, two time-histories scaled to same

spectrum in the same sense can yield very different floor spectra so using this approach can be

argued against.

An indirect and more rational way to generate floor spectra has been proposed by

researchers[13]. In place of time-histories this method uses the Ground Response Spectrum

provided in the building codes. Though Ground Spectrum is also obtained using probabilistic

analysis but, from design perspective, a lot of faith is put into this Spectrum. If floor spectrum is

obtained directly using the Ground Spectrum, groundmotion time-histories can be used to verify

it. This approach is more justifiable as in engineering practice one analyses to design. First such

method was proposed by Biggs which provided magnification curves which were derived by

observing behavior of linear oscillator to groundmotions[16]. However this method was semi-

empirical in nature. The methodology used in this thesis is based on principles of random

vibrationand is derived using transfer characteristics of a linear oscillator. It is overviewed in

subsection below.

3.2.1. Methodology

In this methodology, first, natural periods, mode shapes and modal participation factors of

structure and frequency of element are obtained. Design response of element is obtained from

following expression.

25

( ) ∑

( )[( )

( ) ( ) ( )]

∑ ∑

( ) ( )[(

) ( ) (

) ( )

(

) ( )] ( )

Where

ωo is frequency of oscillator or nonstructural element

ωj is frequency of structure’s jth

mode

Ru(ωo) is response of nonstructural element located at uth

floor of structure

R(ωj) is response of SDOF system with frequency ωjread from Ground Response Spectrum

φj(u) is jth

modal displacement of floor u

Γj is jth

modal participation factor

Aj, Bj etc. are amplification factors for jth

mode

Ajk, Bjk are amplification factors for jth

and kth

mode.

This expression is derived from equation of motion for a general system. The Spectral Density

Function of the earthquake groundmotion is made use of in input of seismic excitation. The

theory of methodology is briefly presented here.

Acceleration response of a SDOF system kept on uth

floor of a structure is obtained from

following equation

( )

Where is absolute acceleration response of uth

floor.ωn and ζ are natural frequency and

damping ratio of SDOF system respectively. Relative floor acceleration needs to be known to

obtain .

Equation of motion of an MDOF system is written as

[ ] [ ] [ ] [ ] ( )

where is ground acceleration.

For a linear structure xu can be written by combining individual modal responses as follows

26

∑

( )

where is modal displacement of jth

mode.

Thus absolute floor acceleration is

∑

( )

Equation (3.16) can be used to obtain the floor acceleration time history . However, for

development of this method groundmotion time-history was assumed to be a random process.

Groundmotion is a Gaussian zero-mean stationary random process. If input seismic excitation

is Gaussian, the response floor acceleration is also Gaussian. Probabilistically, this response can

be characterised by its mean and autocorrelation function. Expected value of response floor

acceleration is zero and its autocorrelation can be written as

{ ( ) ( )}

∑∑ ( ) ( )[

] ( )

The expected values required in above equation can be obtained in terms of power spectral

density function (PSDF) of groundmotion. From above expressions floor acceleration PSDF is

obtained. Similarly after incorporating standard deviation of floor acceleration and making

appropriate statistical substitutions equation (3.12) can be deduced[13].

Now the modal participation factors of structure and its mode shape information are put into

equation (3.12). The spectral demands for structure and nonstructural element are substituted

from corresponding Ground Spectrum.When the structure’s height and floor level of the

nonstructural element are input, the equation yields corresponding ordinate of the floor response

spectrum.

27

3.3. Nonlinearity of Structure and Components

Methodology presented above can be used when structure and the nonstructural element undergo

elastic deformations. But, in general, Civil Engineering structures are designed such that they

exhibit inelastic behaviour in a major seismic event. Similarly connections of nonstructural

elements are designed to impart system of element and its anchorage some amount of ductility.

Apart from few special cases mentioned in Chapter 2, where response of component is amplified,

following procedure can beused to find seismic demands for nonstructural elements when either

of component or its supporting structure shows nonlinear behaviour.

Seismic demand read from an elastic floor spectrumcan bedecreased by a reduction factor as

follows:

with

where R and Rp account for nonlinear behaviour of structure and element respectively. Each of

the factors R and Rp can be found by approach proposed by Newmark and Hall as written by

following expressions[17]

µ if f < 2Hz

Rµ = √ if 2Hz < f < 8Hz

(√ )if 8Hz < f < 33Hz

Thus nonlinear floor spectra can be generated and used for seismic assessment of nonstructural

elements.

28

3.4. Results for Sites in Canada

The formulations of section 3.1 and 3.2 were written in MATLAB and Ground Response

Spectrum data for two sites in Canada was input to it. Model in figure 3.1 was used to denote

different structure types with variation of its parameter α.This section presents some results on

floor acceleration demands obtained using above methodology. Results are shown for 4-story

structures and for different structure types. Demands are shown in terms of displacement and

acceleration floor spectra. Ground Response Spectrum provided in NBCC 2010 corresponds to

5% damped structures. Response spectra for other damping values are obtained using following

expression

(

)

A brief discussion on results is presented thereafter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

Component period Tc (s)

Sa (

g)

Floor Spectra for 3rd floor of different structure types

Shear Wall

Dual System

Concrete MRF

Steel MRF

29

Figure 3.2 -Acceleration Floor Spectra for third floor of four story structures for four different

structure types; Structure damping: 0.05, Component damping: 0.02

Some characteristic features of the floor spectra are reflected in above figure. All spectra have

two distinct peaks, peak at the larger component period having higher ordinate value. These two

periods are first and second periods of the structure as obtained from eigenvalue analysisand the

peaks result from interaction between structure and the ground excitation. Spectra for 3rd

story of

four different 4-story structures are shown in the figure. Here the peak corresponding to spectra

for steel MRF structure lies at largest component period as, having large flexibility, steel MRF

structure has highest natural period of vibration.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

Component period Tc (s)

Sa (

g)

Floor Spectra for two different component damping values

2% damping

0.5% damping

30

Figure 3.3 -Acceleration Floor Spectra for third story of 4-story concrete moment frame

structure for two different component damping values

Figure 3.4- Displacement floor spectra for 3rd

floor of 4-story steel MRF structure for two

damping values; Structure damping 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

Period Tc (s)

Spectr

al dis

pla

cem

ent

(cm

)

Displacement floor spectra for 3rd floor of Steel MRF structure

Damping 0.5%

Damping 2%

31

Figure 3.5- Acceleration Floor Spectra for different floors of a 4-story concrete MRF structure;

Component damping: 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

Component period Tc (s)

Sa (

g)

Floor spectra for 3 floors of 4-story structure

2nd floor

3rd floor

4th floor

32

Figure 3.6 -Displacement floor spectra for three floors of 4-story Steel MRF structure; Structure

damping 0.05, component damping 0.02.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

Period Tc (s)

SD

(cm

)

Displacement spectra for three floors of 4-story Steel MRF structure

2nd floor

1st floor

3rd floor

33

In above figures demands on nonstructural elements are shown in terms of displacement spectra

and acceleration spectra separately. They are referred to by design engineer depending on

whether sensitivity of given element is more critical to floor acceleration or floor displacement.

An acceleration floor spectrum for third floor of a 4-story structure is compared with

corresponding displacement spectrum in following figure.

Figure 3.7 -Displacement and acceleration floor spectra for 3rd

floor of 4-story steel MRF

structure; Structure damping 0.05, component damping 0.02.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

Period Tc (s)

SD

,SA

(cm

, m

/s2)

Displacement and acceleration spectra for 3rd floor of Steel MRF structure

SD (cm)

SA (m/s2)

34

This methodology can be developed into an ‘Analyzer’ package to be used for different locations

in Canada. Floor spectra for third floor of a 4-story concrete MRF structure located in two cities

in west and east Canada are shown in a figure below. Ground Response Spectra for cities are

shown in following figure.

Figure 3.8 -Ground Response Spectra for Vancouver and Montreal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T (sec)

Sa (

g)

Response Spectra for two cities in East and West Canada

Vancouver

Montreal

35

Figure 3.9 - Acceleration Floor Spectra for 3rd

story of 4-story Concrete MRF structures located

in Vancouver and Montreal; Component damping: 0.02

All acceleration and displacement floor spectra in above results reflect characteristic features of

floor spectra mentioned earlier. Figure 3.3 shows acceleration floor spectra for 2% and 0.5%

damped components. In this case, seismic demand for nonstructural element with 0.5% damping

ratio is more than one and a half times greater than that for the element with 2% damping ratio.

The response of a component may be more sensitive to floor acceleration than floor displacement

or vice-versa. Thus, designer should look at both of the acceleration and displacement floor

spectra and decide the critical response. In figure 3.9, floor spectrum for a four story concrete

MRF structure located in Vancouver is compared with that for same structure located in

Montreal. Seismicity in East Canada is less than thatin West Canada and it is reflected in two

floor spectra.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

Sa (

g)

Component period Tc (s)

Floor Spectra for two cities in East and West Canada

Vancouver

Montreal

36

CHAPTER 4

BASE-ISOLATED STRUCTURES

4. Base-isolated Structures

The base-isolation is an emerging technology to protect structures against the damage from

earthquakes. In this method isolation bearings are inserted at first story level of a structure to

modify its dynamic properties. These isolation bearings have high flexibility such that natural

period of the structure is lengthened. Thus the isolated structure is subjected to reduced seismic

forces. In aftermath of an earthquake functionality of the hospital and school buildings is desired.

Therefore these structures are suitable for application of base-isolation technology. In this

chapter the procedure for seismic assessment of nonstructural elements in a base-isolated

structure is presented. The formulation for assessment of components in the regular structures

presented in Chapter 3 is extended to base-isolated structures. The details of this formulation are

presented, followed by two examples on base-isolated shear wall and moment frame structures.

37

4.1. Formulation for Base-isolated Structures

A base-isolated structure can be physically modelled by a translational spring and an equivalent

viscous damper attached to the base of continuous model of structure in Chapter 3. In an

alternative model a rotational spring may also be attached in addition to translational spring.

These two models are shown in figure below.

(a)Translational spring model (b) Rotational and translational spring model

Figure 4.1 - Models for a base-isolated structure

In this work translational spring model is used to represent base-isolated structures. Material

used in the isolation bearings has high flexibility and large damping. A schematic representation

of base-isolated structures is shown in figure below.Isolation story is modeled as a Timoshenko

beam as its lateral deflection is dominated by shear deformation.

38

Figure 4.2 - Schematic diagram of base-isolated structure

Simplified continuous model presented in an earlier chapter is used to model base-isolated

structures also. Thus governing differential equation for regular structures is applicable here with

modified boundary conditions. Boundary conditions at top of structure, i.e. at normalised height

equal to 1, should still satisfy conditions of a free end. Equilibrium and compatibility conditions

at bottom of structure are modified based on properties of isolator. At base of structure slope of

deflected shape should be zero and shear force should equal lateral resisting force provided by

isolator. These conditions can be written by following equations:

At base, normalised height x=0,

( ) ( )

Isolation story

Structure (EI)s

(EI)b

39

( )

( )

and

( )

( )

( )

( )

( )

Combining equation (5.1) and (5.3) yields

( )

( )

(

) ( )

with

( )

( )

√( )

( )

where

is lateral deflection at base of structure

(EI)s is flexural rigidity of structure

(EI)b is flexural rigidity of isolator

GA is shear rigidity of isolator

αbis lateral stiffness ratio of base

hb is height of isolation bearing

With modified boundary conditions equation (3.3) is solved to get expression for mode shape

and other dynamic properties of base-isolated structures. Technical computing software Maple

was used to solve the differential equation[18]. Two example problems to get floor response

spectra for base-isolated structures are solved and are presented in following section. For

example problems αband hb are chosen to be 20 and 0.6 meters, respectively. Damping ratios for

first four modes of base-isolated structures are taken to be 0.096, 0.056, 0.079 and 0.10

respectively[19].

40

4.2. Two Example Problems

A four story concrete shear wall and a steel moment frame structure analysed earlier in Chapter 3

are chosen to apply the solution for base-isolated structures obtained above. Natural periods of

two structures are chosen to shift to 2 seconds after isolation. Methodology used earlier for

regular structures is used to obtain floor spectra from new dynamic properties of isolated

structure. These floor spectra are compared here with those for regular fixed-base structures.

Values of stiffness ratio SR required to shift periods of structures to desired value and modal

properties of regular and base-isolated structures are compared in Tables below.

Table 4.1 -Stiffness Ratio SR for base-isolated structures

Tregular(s) Tbase-isolated (s) SR = (EI)b/(EI)s

Steel MRF 0.54 2 1/24

Concrete Shear wall 0.32 2 1/3350

Table 4.2 -Dynamic properties of 4 story Steel MRF structure; α = 17

Regular Base-isolated

T (s) Γ TBI (s) ΓBI

1st mode 0.54 1.27 2 1.04

2nd

mode 0.18 0.42 0.54 0.013

3rd

mode 0.10 0.25 0.28 -0.043

4th

mode 0.07 0.18 0.14 0.011

41

Table 4.3 -Dynamic properties of 4 story concrete shear wall structure; α = 1

Regular Base-isolated

T (s) Γ TBI (s) ΓBI

1st mode 0.32 0.66 2 1.03

2nd

mode 0.058 0.43 0.2 -0.047

3rd

mode 0.021 0.25 0.04 0.006

4th

mode 0.011 0.18 0.02 0.002

42

Properties obtained above are used to generate floor response spectra for third story of above

four structures. Displacement and acceleration response spectra for these structures are compared

in following figures.

Figure 4.3 - Displacement floor spectra for 3rd

story of Regular and Base-isolated Shear wall

structure; Structure damping 0.02, Component damping 0.02;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140

Tc (s)

Spectr

al D

ispla

cem

ent

(cm

)

Displacement floor spectra for 3rd story of regular and base-isolated shear wall structure

Regular

Base-isolated

43

Figure 4.4 - Displacement floor spectra for 3rd

story of Regular and Base-isolated 4 story Steel

MRF structure; Structure damping 0.02, Component damping 0.02;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

250

Tc (s)

SD

(cm

)

Displacement floor spectra for regular and Base-isolated structure

Base-isolated

Regular

44

Figure 4.5 - Acceleration floor spectra for third story of regular and base-isolated 4 story shear

wall structure; Structure damping 0.05, component damping 0.02

4.2.1. Discussion

The dynamic properties of base-isolated structures can be obtained by using simplified

continuous model with changed boundary conditions. Above two examples indicate some

distinct differences between properties of regular and the base-isolated structures. The first mode

dominates overall response of base-isolated structures as evident from very large value of modal

participation factors for this mode compared to those for subsequent modes. For regular

structures also contribution of first mode is significant but modal participation factors decrease in

a gradual manner. Floor acceleration demands in the base-isolatedstructures are very low which

is attributed to their large flexibility. For example, for a regular concrete shear wall structure the

peak floor acceleration (PFA) is 8g, whereas after base-isolation it decreases to a very low value

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

Tc (s)

Sa (

g)

Acceleration floor spectra for 3rd story of regular and base-isolated shear wall structure

Regular

Base-isolated

45

of approximately 0.5g. Thus acceleration sensitive rigid nonstructural elements pose relatively

lesser hazard in base-isolated structures. On the other hand, displacement response of floors for

such structures is comparable to that for regular structures. Base-isolated structures, being

flexible, undergo large displacements resulting in large displacements for long period

components andwith increasing period a component’s displacement tends to Peak Floor

Displacement. A useful inference from presented examples might be about stiffness ratio

(EI)b/(EI)s. The Stiffness Ratio required to shift period of structures to 2 seconds is very large for

steel MRF structure compared to that for concrete shear wall structure. For concrete shear wall

this ratio is only about 1/3400 compared to a ratio of 1/25 for steel MRF structure. This may be

attributed to high Modulus of Elasticity of Steel for steel MRF structure and large cross-sectional

area of walls for concrete shear wall structure.

46

CHAPTER 5

IRREGULAR STRUCTURES

5. Irregular Structures

The irregular structures have a configuration of their Seismic Force Resisting System (SFRS)

such that they undergo significant torsional movement during an earthquake groundmotion. In

such structures either SFRSs are unsymmetrically located or they have significantly different

lateral stiffness. From earthquake safety and design perspective only regular structures should be

desirable but architectural and other considerations cause some structures to be irregular.

A structure can have either vertical irregularity or irregularity in its plan. Unlike regular

structures, nonstructural elements located at different places of same story of an irregular

structure undergo different accelerations. A component placed farther from Centre of Resistance

of story undergoes largest acceleration. It is difficult to model irregular structures in a

generalized way as it was done for the regular structures in an earlier chapter. In this chapter a

methodology is proposed to find design earthquake demand for nonstructural elements in an

irregular structure. This methodology also is based on the concept of floor response spectrum. A

‘Reference Floor Response Spectrum’ is proposed for a particular structure type and its ordinates

are linearly scaled along plan and height of the structure to obtain seismic demand for element.

47

5.1. Irregularity in Plan

Planar irregularity in a structure can result from various factors. Core wall in a regular square-

shaped structure can be unsymmetrically located. Two opposite edges of a similar structure can

have braced frame and shear wall, respectively, causing eccentricity of lateral stiffness. Some

typical configurations resulting in structural irregularity are shown in figure below.

Figure 5.1 - Typical SFRS configurations causing planar irregularity

(a) Asymmetric core wall (b) Nonorthogonal SFRS

(c) Irregular plan (d) Different SFRS types

48

Earlier the regular structures were mathematically modelled using Miranda’s method. But, given

the variety of irregular structures, it is difficult to model them in a generalised way. It is difficult

to classify irregular structures based on a single parameter like lateral stiffness ratio.However, a

common parameter simultaneously applicable to different types of irregular structures is needed

to suggest a general assessment methodology. For this purpose torsional sensitivity index ‘B’

defined in NBCC is used in methodology proposed in this thesis.

5.1.1. Decision parameter B

Torsional Sensitivity Index B, as defined in NBCC, is

where

( )

( )

δ1, δ2, δ3, δ4 are lateral displacements at extreme points of structure in direction of earthquake

when equivalent static force is applied at distance 0.1D from centre of mass of floor.

Figure 5.2 - Lateral displacements at floor level caused by eccentric equivalent static force

For methodology proposed in this thesis index B is used as single parameter to address structures

with different irregularities in plan.

49

5.2. Methodology for Irregular Structures

In the proposed methodology a ‘Reference Floor Response Spectrum’shall be provided for a

particular structure type. The reference spectrum is obtained using Miranda’s simplified

continuous model used in an earlier chapter. This spectrum shall be scaled using scaling factors

from provided Design Table to find acceleration demand for any component located at a

particular place within the structure. These scaling factors shall be determined by analyses of a

set of structures representing a particular degree of torsional sensitivity. The acceleration demand

from reference spectrum shall be scaled along plan of given floor and along height of the

structure. This concept is schematically illustrated in figure below.

50

(a)

(b)

Miranda’s

model

Structure type

(Concrete moment frame)

Component frequency fc (Hz)

Sa

(g)

Reference Floor Response Spectrum

Regular structure

Symmetric

response

CM

CR

Largest acceleration

Smallest acceleration

Irregular structure

51

(c)

Figure 5.3 -Proposed methodology for seismic assessment of nonstructural components:

(a) Obtaining base floor response spectrum using Miranda’s approach (b) Unsymmetrical

acceleration demands in a torsionally sensitive structure (c) Zones of similar acceleration

demands and spectra in central and end zones

Acceleration demand shall be scaled by linear interpolation using following formula:

( )

( )

where for a component with period Tc

aBis acceleration demand from Reference Spectrum in central zone

aLis acceleration demand obtained from end zone

azis acceleration demand for zone located at xzdistance from centre of resistance

bis dimension of structure perpendicular to groundmotion direction

ris distance of centre of resistance from edge of structure with higher lateral stiffness

l

Cen

tral

zone

End z

one

r

b-r

x

Component frequency fc (Hz)

S a (

g)

Reference

spectrum in

central zone

Spectrum in

end zone

52

The procedure discussed above is for obtaining acceleration demands at any particular floor

level. It can beextended to obtain seismic demands along height of that structure. Variation

of acceleration demand along height also depends on type of the structure. The ‘Reference

Response Spectrum’ provided for structure type shall beappropriately scaled to find required

acceleration demand.Following figure shows a four story steel moment frame structure

which was one of the structures analysed for this work. It has a square plan and moment

frames on back and front faces have different flexibilities. Two frames are connected by

girders at floor levels. The beams and columns of two frames are chosen to impart structure a

certain degree of irregularity in plan. The structure shown in figure has a B value of 1.4.

Similarly other four and eight story structures are modeled with higher B values denoting

increased torsional sensitivity.

53

Figure 5.4 -Four story irregular structure analysed for proposed procedure; B=1.4

W 460*213

y

x

z

W 4

60

*23

5

W 4

60

*12

8

W 4

60

*12

8

W 4

60

*23

5

W 4

60

*12

8

W 460*113 W

46

0*1

28

W 460*213

W 460*113

W 460*113

Stiffer

Frame

Flexible

Frame

5 m

5 m

54

5.3. Results

Initial results of this proposed methodology are presented in this section. There may be two

approaches to this methodology. In first approach natural period of structure matches code

recommended value of 0.54 sec for period of four story steel MRF structure. However deflection

in this mode is pure translational and is orthogonal to direction of earthquake groundmotion. In

alternate approach period of lateral torsional mode is matched to code recommended value of

0.54 sec. Period values in these two approaches are provided in table below.

Table 5.1 - Periods of vibration of 4 story structures in two approaches

Period (sec)

Mode First approach

(B=1.4)

Alternate approach

(B=1.4)

Vibration

1st 0.54 0.80 Lateral

2nd

0.32 0.54 Lateral torsional

Models are subjected to a suite of 10 groundmotions. These groundmotions were used in another

project earlier and were selected and scaled to match seismicity of Vancouver [20].

Results provided in this section outline qualitative idea behind proposed methodology with some

initial quantitative results. This work deals with only steel MRF structures with two different B

values of 1.4 and 1.6.

5.3.1. First approach

In this approach natural period of the structure is matched to code recommended period value for

a 4 story steel MRF structure. Steel W sections used in modeling two structures and dynamic

properties of models are listed in two tables below.

55

Table 5.2 - Steel sections in four story Steel MRF irregular structures

B=1.4 B=1.6

Stiff edge Flexible edge Stiff edge Flexible edge

Column W460*235 W460*128 W460*384 W460*113

Beam W460*213 W460*113 W460*213 W460*82

Table 5.3 - Periods of vibration of four story steel MRF structures

Mode B=1.4 B=1.6 Vibration

1st 0.54 0.53 Lateral

2nd

0.32 0.34 Lateral torsional

3rd 0.23 0.22 Lateral torsional

56

Figures below compare floor spectra for stiff and flexible edges of structures. In one of the plots

these results are compared with corresponding floor spectrum for a regular structure obtained

using Miranda’s model.

Figure 5.5 - Floor spectra for stiff and flexible edges of third floor of Steel MRF structure

located in Vancouver; B=1.4, Structure and Component damping 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

Tc (s)

Sa (

g)

Floor spectra for 3rd floor of Steel MRF structure (B=1.4)

Stiff edge

Flexible edge

57

Figure 5.6 - Floor Spectra for stiff and flexible edges of third floor of Steel MRF structure

located in Vancouver; B=1.6, Structure and component damping 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

Tc (s)

Sa (

g)

Floor Spectra for 3rd floor of Steel MRF structure (B=1.6)

Stiff edge

Flexible edge

58

Figure 5.7 - Comparison of floor spectra for third floor of a four story irregular Steel frame

structure with spectrum from Miranda’s model; Structure damping 0.02, component damping

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

Tc (s)

Sa (

g)

Floor spectra for third story of Steel MRF structure

Miranda's model

Stiff edge

Flexible edge

59

Figure below compares the floor spectra for uppermost floors of four and eight story

structures. Here the eight story structure was modeled such that its natural period

matched with code recommended value of 0.92 seconds for period of eight story steel

MRF structure. Steel W sections of lower four stories were different from those of upper

four stories. Lower four stories and upper four stories had B values of 1.4 each.

Figure 5.8 - Floor spectra for stiff edges of uppermost floors of a four and an eight story

structures; B = 1.4, Structure damping 0.02, component damping 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

Tc (s)

Sa (

g)

Floor spectra for uppermost floors of 4 and 8 story steel frame structures

4 story

8 story

60

Following figure shows eccentricities in the two analysed irregular structures. Eccentricity values

for structures with B value of 1.4 and 1.6 were found to be 0.84 meter and 1.44 meters

respectively. Floor spectra were generated for points A, CR, CM and B. Peak acceleration

demands for nonstructural elements were compared. These values are shown in following table.

Values in table indicate that variation of acceleration demand from stiff edge to flexible edge is

almost twice in structure with B value of 1.6 compared to structure with B value of 1.4.

Figure 5.9 - Eccentricities in two analysed irregular structures

Table 5.4 -Peak factors for scaling floor spectrum

Peak factors

Location B = 1.4 B = 1.6

A 0.64 0.61

CR* 1 1

CM* 1.19 1.6

B 1.72 2.8

*- CR = Centre of resistance, CM = Centre of mass

A

B

CM

CR e

R

5 m

CR

CM

e

R

A

B 5 m

5 m

B = 1.4; e = 0.84 m B = 1.6; e = 1.44 m

Stiff edge Stiff edge

Flexible edge Flexible edge

61

Based on methodology proposed here a design table can be prepared to be used by engineers in

seismic assessment and retrofit of nonstructural elements. Such a table is shown below. It shows

scaling factors for three structure types Concrete Shear wall and Concrete and Steel Moment

Resisting Frames.

Table 5.5 - Scaling factors* to get peak acceleration demands for nonstructural elements

Scaling Factors

B SFSFRS SFfloor SFplan

1.4 SFRS Factor Floor level Factor Plan zone Factor

Steel MRF 1 2 0.72 CM+2e 0.64

Shear wall 1.03 4 1.3 CR 1

Concrete MRF 1.08 6 1.94 CM 1.19

8 2.6 CM-1e 1.44

10 3.17 CM-2e 1.72

1.6 SFRS Factor Floor level Factor Plan zone Factor

Steel MRF 1 2 0.69 CM+2e 0.61

Shear wall 1.04 4 1.24 CR 1

Concrete MRF 1.10 6 1.92 CM 1.6

8 2.56 CM-1e 2.17

10 3.21 CM-2e 2.8

1.8 SFRS Factor Floor level Factor Plan zone Factor

Steel MRF 1 2 0.74 CM+2e 0.56

Shear wall 1.06 4 1.38 CR 1

Concrete MRF 1.11 6 1.97 CM 2.04

8 2.63 CM-1e 2.72

10 3.14 CM-2e 3.34

*- Factors for 2nd and 4th floor levels for Steel MRF only are accurate in above table. Other factors of the table are not exact and

have been provided only to give a tentative idea of proposed procedure. They should be verified and substituted after

analysis.

62

The ‘Reference Floor Response Spectrum’ obtained using Miranda’s model is for third floor of

any structure type. In theabove table scaling factors are provided for second to tenth floor levels.

Similarly these factors are also provided along plan of floor. Acceleration demand at a given

location in the structure is obtained by multiplying Reference Spectrum ordinate with appropriate

scaling factor. This factor is obtained by interpolation of scaling factors provided in above table.

This can be written by following equation:

( )

Where

Acomponentis acceleration demand for a component at a given location

ARSis acceleration read from Reference Spectrum

SFs, SFfand SFp are scaling factors SFSFRS, SFfloor and SFplanrespectively provided in above table.

Thus seismic demand for a nonstructural element at a given location in a structure can be found.

5.4. Limitations of Proposed Method

Above proposed procedure can be used for all structures with planar irregularities. But it is not

appropriate for structures with other irregularity types. A structure, for example, may have

Concrete shear wall as the lateral force resisting system for lower some floors and concrete

moment frames for remaining floors above. This gives structure irregularity along elevation. This

procedure cannot be used for such structures as scaling factors in its design table are obtained for

structures with planar irregularities only.

63

CHAPTER 6

CONCLUSION AND FUTURE WORK

6. Conclusion and Future Work

This study indicates that existing design and assessment procedures for nonstructural elements

recommended by building codes might be inadequate in some cases. Some shortcomings in

existing codes’ methodologies are reported by researchers and practicing engineers for around

two decades now. More recently few researchers comprehensively studied floor acceleration

demands indicatingsignificant differences between observed response of nonstructural elements

and that predicted by code procedures. The literature review conducted for this work also

indicated that suggested improvements in assessment procedures are yet to be incorporated by

existing design guidelines.The existing procedures have two major shortcomings that they do not

account for component’s period and type of structure, though it is well known that interaction

between periods of nonstructural element and structure results in significantly increased demand

for element.

This study was aimed at providing a simplified method to practicing engineers which

enables them to take nonstructural element design and assessment decisions in a time efficient,

more accurate and consistent way. This simplified method is based on rational concept of floor

response spectrum. By looking at a set of floor spectra for a given structure engineer can decide

whether to retrofit, remove or relocate a given component within that structure. These floor

spectra can be readily generated by input of structure’s parameters and its location into

‘Analyzer’ package. An important aspect of this procedure is that seismic excitation to structure

is in form of ‘Design Ground Response Spectrum’ provided in building codes. Input of

64

structure’s location feeds corresponding Ground Response Spectrum into the package. With this

procedure, floor spectra for a regular structure at any given location in Canada can be obtained.

Above methodology can be extended to the base-isolated structures also. It is proposed to

address these structures in NBCC 2014.A base-isolated structure can also be represented by

simplified continuous model used earlier for regular fixed-base structure. However, boundary

conditions of model need to be suitably modified for this case. Two example problems on such

structures are presented in this work. If similar more problems are worked-out, useful design

inferences can be drawn about required ratios of flexural stiffness of isolation bearings to that of

the structure. A specialized tool suitable to get symbolic solution of the system of differential

equations is needed for this problem. In this work Technical Computation SoftwareMaple is used

for this purpose. Once dynamic properties of the model are obtained, displacement and

acceleration floor spectra can be obtained in the same way as forfixed-base structures.

An attempt is made to propose a methodology for assessment of nonstructural elements

in irregular structures also. A ‘Reference Floor Response Spectrum’ for a particular structure

type is provided. A design table is provided which lists scaling factors for irregular structures

with increasing degree of torsional sensitivity. The acceleration demand for a nonstructural

element in a given structure is obtained by interpolation along plan and elevation of structure

using listed scaling factors.

This study puts forth a wide scope of future research. Work on regular structures in

Chapter 3 has a direct industrial application. Its methodology can be developed into an

‘Analyzer’ package to be used for assessment of regular structures at any site in Canada. The

Ground Response Spectra of different cities in Canada would be included in this package

database. The designer needs to only input few identifying parameters of structure and its

location to get corresponding floor spectra. This will enable designer to take time-efficient

assessment decisions in a rational way. The procedure proposed for irregular structures,

however, is not very exact. It makes an attemptto provide an indirect approach to address this

problem. The direct approach of modeling irregular structure can be resource consuming and

cumbersome. If merit found, design table in Chapter 4 should be refined and provided scaling

factors should be validated.In chapter on the base-isolated structures, developing an approximate

way to find modal damping ratios can be an area of future research. The methodology used in

65

this thesis is based on simplified continuous model of structure and, in the procedure, mass and

damping matrices need not be formed. In the problem of base-isolated structure designer knows

damping ratios of isolation bearings and structure separately, however, it is difficult to determine

modal damping ratios of isolated structure with this information. If a procedure to estimate

modal damping ratios of isolated structure is developed, such structures can be treated in

‘Analyzer’ package in same exact way as that for regular fixed-base structures.

66

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