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A simplified methodology for seismic design and assessment of nonstructural elements Kumar, Awanish 2014

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  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) StructuralElectrical &MechanicalExteriorFinishes5      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 jthand 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 uthfloor.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 102468101214Component period Tc (s)Sa (g)Floor Spectra for 3rd floor of different structure types  Shear WallDual SystemConcrete MRFSteel MRF29  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 102468101214161820Component period Tc (s)Sa (g)Floor Spectra for two different component damping values  2% damping0.5% damping30  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 1020406080100120140Period Tc (s)Spectral displacement (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 102468101214Component period Tc (s)Sa (g)Floor spectra for 3 floors of 4-story structure  2nd floor3rd floor4th floor32   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 101020304050607080Period Tc (s)SD (cm)Displacement spectra for three floors of 4-story Steel MRF structure  2nd floor1st floor3rd floor33  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 101020304050607080Period 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 1.8 200.10.20.30.40.50.60.70.80.91T (sec)Sa (g)Response Spectra for two cities in East and West Canada  VancouverMontreal35   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 102468101214Sa (g)Component period Tc (s)Floor Spectra for two cities in East and West Canada  VancouverMontreal36     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 2020406080100120140Tc (s)Spectral Displacement (cm)Displacement floor spectra for 3rd story of regular and base-isolated shear wall structure  RegularBase-isolated43   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 2050100150200250Tc (s)SD (cm)Displacement floor spectra for regular and Base-isolated structure  Base-isolatedRegular44    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 2012345678Tc (s)Sa (g)Acceleration floor spectra for 3rd story of regular and base-isolated shear wall structure  RegularBase-isolated45  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 Central zone End zone 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 460*235 W 460*128 W 460*128 W 460*235 W 460*128 W 460*113 W 460*128 W 460*213 W 460*113 W 460*113 Stiffer Frame FlexibleFrame 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 105101520253035Tc (s)Sa (g)Floor spectra for 3rd floor of Steel MRF structure (B=1.4)  Stiff edgeFlexible edge57   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 10510152025303540Tc (s)Sa (g)Floor Spectra for 3rd floor of Steel MRF structure (B=1.6)  Stiff edgeFlexible edge58   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 105101520253035Tc (s)Sa (g)Floor spectra for third story of Steel MRF structure  Miranda's modelStiff edgeFlexible edge59   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 105101520253035404550Tc (s)Sa (g)Floor spectra for uppermost floors of 4 and 8 story steel frame structures  4 story8 story60  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 eR 5 m CR CM eR 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  References  [1]  R. Villaverde, "Seismic Response of Nonstructural Elements," in Fundamental Concepts of Earthquake Engineering, 2009, pp. pp. 649-688. [2]  "FEMA E-74, Reducing the Risks of Nonstructural Earthquake Damage," 2011. [3]  A. Filiatrault, "Seismic Design of Nonstructural Elements," New York, 2010. [4]  "Chapter 4," in National Building Code of Canada 2005 Volume 1, 2005, pp. 1-35. [5]  "Seismic Design Criteria, Chapter 11," in ASCE/SEI 7-10, Minimum Design Loads for Buildings and Other Structures, 2010, p. 65. [6]  E. Miranda, "A Comprehensive Study of Floor Acceleration Demands in Multi-story Buildings," in ATC & SEI 2009 Conference on Improving Seismic Performance of Existing Buildings, 2009.  [7]  Y. Chen and S. T, "State-of-the-Art review: Seismic Response of Secondary Structures," Engineering Structures, vol. 10, no. October, pp. 218-229, 1988.  [8]  S. R. Chaudhuri and R. Villaverde, "Effect of Building Nonlinearity on Seismic Response of Nonstructural Components: A parametric study," Journal of Structural Engineering, vol. 134, no. April, pp. 661-670, 2008.  [9]  A. Zaghi, E. Maragakis, A. Itani and E. Goodwin, "Experimental and Analytical Studies of Hospital Piping Assemblies Subjected to Seismic Loading," Earthquake Spectra, vol. 28, no. 1, pp. 367-384, 2012.  [10]  ASCE, "Seismic Design Requirements of Nonstructural Components," in Minimum Design Loads for Buildings and Other Structures, Virginia, ASCE, 2010, pp. 111-126. [11]  T. T. Soong and C. Y, "Seismic behaviour of nonstructural elements: State-of-the-art report," Engineering Structures, vol. 10, no. October, pp. 218-229, 1988.  [12]  E. Miranda and S. D. Akkar, "Generalized Interstory Drift Spectrum," Journal of Structural Engineering, vol. 132, no. 6, pp. 840-852, 2006.  [13]  M. P. Singh, "Generation of Seismic Floor Spectra," Journal of Engineering Mechanics Division, vol. 101, pp. 593-607, 1976.  67  [14]  E. Miranda, "A Comprehensive Study of Floor Acceleration Demands in Multi-Story Buildings," in ATC & SEI 2009 Conference on Improving the Seismic Performance of Existing Buildings and Other Structures, 2009.  [15]  E. Miranda and S. Taghavi, "Approximate Floor Acceleration Demands in Multistory Buildings. I: Formulation," Journal of Structural Engineering, vol. 131, no. 2, pp. 203-211, 2005.  [16]  J. M. Biggs, "Seismic Response Spectra for Equipment Design in Nuclear Power Plants," in First International Conference on Structural Mechanics in Reactor Technology, Berlin, 1971.  [17]  R. Villaverde, "Chapter 9," in Fundamental Concepts of Earthquake Engineering, 2009.  [18]  Maplesoft, Maple 18, Waterloo Maple Inc., 2014.  [19]  A. K. Chopra, "Earthquake Dynamics of Base-isolated Buildings," in Dynamics of Structures, Pearson Education, 2007, pp. 777-802. [20]  F. Pina, C. E. Ventura, G. Taylor and W. D. Finn, "Selection of groundmotions for Low-rise School Buildings in South-western British Columbia, Canada," Lisbon, 2012.  [21]  ASCE, "ASCE/SEI 7-10: Minimum Design Loads for Buildings and Other Structures," 2010. [22]  F. Naeim and M. Mehrain, ""Exact" Three-Dimensional Linear and Nonlinear Seismic Analysis of Structures with Two-Dimensional Models," Earthquake Spectra, vol. 19, no. 4, pp. pp. 897-912, 2003.  [23]  R. Villaverde, "Seismis Design of Secondary Structures: State-of-the-Art," Journal of Structural Engineering, vol. 123, no. August, pp. 1011-1019, 1997.        

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