## Am

And the confidence, c, calculated according to Eq. (6.28). The three Pre-Northridge model welded steel moment frame (WSMF) buildings described in Appendix B were used to investigate the proposed expression given in Eq. (6.32) for the relationship between exact and approximate building response. Recall from Chapter 4 that the Ml and M2 models of the 3-, 9-, and 20-story buildings were analyzed by the nonlinear time history analysis method using the earthquake ground motion records in the 50 , 10...

## AT[MK at[mKa

Where a is the vector of displacements at each floor level at the target displacement, At. Note that if the static displaced shape is the same as the first mode shape, then Eqs. (3.7) and (3.8) will give the same result. Method 3 Select C0 from the values listed in Table 3.1 given the number of building stories. The C, coefficient represents the ratio between inelastic spectral displacement and elastic spectral displacement. The value of C, is,

## Bse

K + p + any of a. e, I, m or b, f, j, n Enhanced Objectives o Enhanced Objective k alone or p alone s Limited Objectives c, g, d, h Limited Objectives Table 2.7 FEMA 273 general damage descriptions for repair and rehabilitation of buildings (After ATC, 1997) Table 2.7 FEMA 273 general damage descriptions for repair and rehabilitation of buildings (After ATC, 1997)

## Chapter

RELIABILITY-BASED SEISMIC DESIGN 125 5.2 Sources and Types of Uncertainty 131 5.3 Load and Resistance Factor Design 136 5.4 Probabilistic Risk Assessment 139 5 .5 Recent Developments in Reliability-Based Design 144 5.6 Cornell Method - Demand and Capacity Factor Design 151 5.6.2 Relationship Between Seismic Hazard and Building Response 155 5.6.3 Inherent Randomness in Building Response 158 5.6.4 Inherent Randomness in Capacity 164 5.6.5 Application of Demand and Capacity Factor Design 167

## Derivation Of Uncertainty Factors

In the Demand and Capacity Factor Design (DCFD) methodology, the uncertainties associated with inherent randomness and modeling uncertainty are incorporated into the hazard relationships through uncertainty factors. For example, the hazard relationship expressing the probability of exceeding a maximum interstory drift ratio (IDR) demand of S, was derived in Chapter 5, Section 5.4.3, and repeated here as, where e is a random variable representing the uncertainty due to inherent randomness in the...

## Nonlinear Dynamic Analysis Method

The final damage prediction method presented in this study involves the use of the nonlinear time history analysis method and the actual recorded ground motions at the building site. Again, both the Ml and M2 models were used for the predictions and the results used to classify the connections based on their force and deformation demands. The floor displacement and inters to ry drift results from the measured response and the response calculated by a nonlinear time history analysis were...

## EJ Nonlinear Static Analysis Method Without Recorded Ground Motion

In this section, the results of a nonlinear static pushover analysis will be used to predict the locations of damaged connections in the 13-story building without the use of site specific ground motion records. Without a characterization of the earthquake ground motion at the building site, it is not possible to calculate the target roof displacement. Instead, the results of the pushover analysis alone will be used to predict the locations of connection damage. The Ml and M2 nonlinear...

## H

Coefficient are found from the capacity spectrum. The nonlinear response of single degree-of-freedom (SDOF) oscillators were used to calculate uniform seismic hazard spectra relating the yield force coefficient of the oscillator to the period of the oscillator for fixed levels of ductility. The yield force coefficient is equivalent to the spectral acceleration that causes yield of the oscillator and the ductility is the ratio of the maximum displacement demand to the displacement of the...

## Info

NTH Nonlinear Time History Anityiii oimf M2 model CM Target Displacement from Coefficient Method (Actual Itaponsc Spcctnun) CSM Target Displacement from Capacity Speckum Method (Actual Reipoaac Spectram) S CM Target Displacement from Coefficient Method (Smooth Response Spcctnun) SCSM Target Displacement from Capacity Spcctnun Method (Smooth Response Spectrum) ESM Target Displacement from Equivalent System Method NTH Nonlinear Time History Anityiii oimf M2 model CM Target Displacement from...

## Info Isv

NTH Nonlinear Time Hiaoiy Analyiix Bang M2 model s cm Target D placement from Coefficient Method (Smooth Fit to Median Rcqionic Spectrum) scsm Target Di iUccmcnt from Capacity Spectrum Method (Smooth Fit to Median Reiponic Spec tram) NTH Nonlinear Time Hiaoiy Analyiix Bang M2 model s cm Target D placement from Coefficient Method (Smooth Fit to Median Rcqionic Spectrum) scsm Target Di iUccmcnt from Capacity Spectrum Method (Smooth Fit to Median Reiponic Spec tram) Table 6.7 Nonlinear static...

## Info Mwv

NTH Nonlinear Time HUloiy Analytis u*ing M2 model cm Target Displacement from Coefficient Metiiod (Actual Response Spectrum) csm Tsrgct Displacement from Capscity Spectrum Method (Actual Response Spectrum) s cm Target Diip la cement from Coefficient Method (Smooth Response Spectrum) s csm Target Displacement from Capacity Spectrum Method (Smooth Reaponie Spectrum) esm Target Diiplaccment from Equivalent Syatem Method NTH Nonlinear Time HUloiy Analytis u*ing M2 model cm Target Displacement from...

## Info Wxb

Logarithmic standard deviation for capacity * 0.2. 1. Logarithmic standard deviation for capacity * 0.2. Table 6.21 Performance objective confidence levels 20-Story Building and Coefficient Method Roof Displacement Ratio Beam Rotation Table 6.21 Performance objective confidence levels 20-Story Building and Coefficient Method Roof Displacement Ratio Beam Rotation

## Multiple Failure Modes

Chapters 5 and 6 describe the development of the Demand and Capacity Factor Design (DCFD) methodology that can be used for the seismic performance evaluation of buildings. The DCFD methodology provides a rational basis for the structural engineering evaluation of seismic building performance because it is capable of incorporating the uncertainties that affect the performance evaluation of buildings into a procedure that can be used to quantify building performance. In the true spirit of...

## Nil

Considering that the bias factor may not be the same for each of the N ground motion records, it is clear that the bias factor is a random variable which represents the uncertainty associated with the use of an approximate analysis method. As noted in Section 5.2, Chapter 5, the analysis method uncertainty is composed of two parts, the systematic error and the random error. If it is assumed that the bias factor random variable has a lognormal probability density function (PDF), then the...

## Nonlinear Time History Analysis Prediction Of

FULL-SCALE BUILDING RESPONSE 265 7.2 13-Story Steel Moment Frame Building 267 7.2.1 Building Description 267 7.2.2 Recorded Building Response and Ground Motion 268 7.2.3 Comparison of Predicted and Recorded Building Response 269 7.3 6-Story Steel Moment Frame Building 273 7.3 .1 Building Description 274 7.3.2 Recorded Building Response and Ground Motion 274 7.3.4 Comparison of Predicted and Recorded Building Response 275

## Ok

Median capacities for FEMA 273 Performance Levels. 2. D C Ratio of factored demand to factory capacity. 3. NG Unacceptable Performance. OK Acceptable Performance. 4. Logarithmic standard deviation for capacity 0.2. Table 5.3 Performance checks and demand-capacity ratios for 9-story building Table 5.3 Performance checks and demand-capacity ratios for 9-story building

## Pc as

Where < I> 1 ( ) is the inverse standard normal cumulative distribution function. Solving Eq. (6.13) for Pfc results in, Substituting Eq. (6.11) into Eq. (6.15) gives, where the confidence correction factor, a CON, is defined as, Equation (6.16) provides an expression for the probability of failure with confidence c. In a state of perfect knowledge, i.e. no modeling uncertainty, aCON is equal to one. Therefore, the probability of failure is equal to the median value given in Eq. (6.11),...

## R

Repeat for all ground motions i l, , 20 Figure 4.18 Flowchart illustrating procedure for calculating bias factors Figure 4.19 Smooth representation of actual response spectrum Figure 4.19 Smooth representation of actual response spectrum Figure 4.20 Median, 16th, and 84th percentile maximum roof displacement bias factors Target displacement method sensitivity Modal load pattern M2 model Figure 4.20 Median, 16th, and 84th percentile maximum roof displacement bias factors Target displacement...

## R FMM t[mKM

Again, considering only the first mode of the building, the roof displacement and spectral displacement are related by the equation, where Ar is a value of roof displacement from the capacity curve and < j> , r is the value of the fundamental mode shape at the roof level. After transforming the capacity curve to spectral coordinates using Eqs. (3.13) and (3.16), the resulting values of spectral acceleration and spectral displacement comprise what is known as the capacity spectrum. An...

## T

Median capacities for FEMA 273 Performance Levels. 2. D C Ratio of factored demand to factory capacity. 3. NG Unacceptable Performance. OK Acceptable Performance. 4. Logarithmic standard deviation for capacity 0.2. Table S.4 Performance checks and demand-capacity ratios for 20-story building

## Uj

Figure 6.1 Relationship between spectral acceleration and maximum interstory drift ratio calculated using the Coefficient Method (Modal M2) for the 3-story building Maximum Interstory Drift Ratio (red) Figure 6.1 Relationship between spectral acceleration and maximum interstory drift ratio calculated using the Coefficient Method (Modal M2) for the 3-story building Figure 6.2 Relationship between spectral acceleration and maximum interstory drift ratio calculated using the Coefficient Method...

## ISL

In addition, random uncertainty includes the statistical uncertainty resulting from the use of a limited number of samples to determine the central value and dispersion in a building performance quantity. For example, the mean value of the yield strength, 47.3 ksi (386 MPa), is termed the sample mean because it is based on the information from twenty observations and therefore is only an estimate of the true mean value. If all twenty samples resulted in a value of the yield strength exactly...

## Rai

Figure 3.2 General shape of the Modal and Uniform lateral load patterns Figure 3.3 Bilinear representation of the capacity curve Figure 3.3 Bilinear representation of the capacity curve Spectral Displacement (Sd) Figure 3.4 Demand and capacity spectrum curves Figure 3.5 Effective damping calculation Figure 3.5 Effective damping calculation Figure 3.6 Composite response spectrum Figure 3.6 Composite response spectrum

## Conclusions And Recommendations For Future Research

The research in this dissertation focuses on quantifying the uncertainty in the nonlinear static analysis method to predict building demands. Such research is necessary in order to provide accurate evaluations of building performance. The nonlinear static analysis method was investigated using three different methods for calculating the target roof displacement (1) Coefficient Method, (2) Capacity Spectrum Method, and (3) Equivalent System Method. These methods were used to predict seismic...

## D

1.2 D + F + T) + I.6(L + H) + 0.5 Lr or S or R) 1 2D +1.6 Lr or S or R) + (O.SL or 0.SW) 1.2D + . JW + 0.5L + 0.5(Lr or S or R) .2D + .0E + 0.5L + 0.2S 0.9D + ( . JW or 7.0E) where D is dead load, L is live load, E is earthquake load, W is wind load, F is load due to fluids, T is self-straining force, S is snow load, R is rain load, H is load due to soil pressure, and Lr is roof live load. Each of these load combinations should be investigated in association with the resistance provided by the...

## Lsm

Where wx and w_ are the seismic weights of floor levels x and , respectively, and n is the total number of floors. If all of the floors are of equal seismic weight, then the force applied to each floor will be the same and C simplifies to, The Modal load pattern is a set of lateral forces applied to the building in proportion to the elastic fundamental mode shape and the seismic floor weights. The Modal load pattern reflects the distribution of lateral inertia forces expected to act on the...

## List Of Figures

Figure 2.1 SEAOC Vision 2000 methodology for performance-based seismic Figure 3.1 Qualitative illustration of the capacity curve 57 Figure 3.2 General shape of the Modal and Uniform lateral load patterns 57 Figure 3.3 Bilinear representation of the capacity curve 58 Figure 3.4 Demand and capacity spectmm curves 59 Figure 3.5 Effective damping calculation 59 Figure 3.6 Composite response spectrum 60 Figure 4.1 Median, 84th percentile, and 16th percentile 2 damped response spectra for 50 in 50...

## List Of Tables

Table 2.1 SEAOC Vision 2000 recommended performance objectives (After Table 2.2 SEAOC Vision 2000 general damage descriptions for new buildings Table 2.3 Description of SEAOC Vision 2000 Performance-Based Seismic Design steps (After OES, 1995) 24 Table 2.4 ATC-40 performance objectives (After ATC, 1996) 25 Table 2.5 ATC-40 general damage descriptions for repair and rehabilitation of Table 2.6 FEMA 273 performance objectives (After ATC, 1997) 27 Table 2.7 FEMA 273 general damage descriptions for...

## Nonlinear Time History Analysis Prediction Of Fullscale Building Response

The research in this dissertation has so far focused on quantifying the uncertainty in the nonlinear static (NS) analysis method to predict building demands. In Chapter 4 and Chapter 6, building response predicted using various forms of the NS analysis method were compared to the demands predicted using the nonlinear dynamic (ND) analysis method. The assumption is these chapters was that the demands calculated using the ND analysis method are exact measures of the building response and that the...

## Analysis Method Uncertainty

The Demand and Capacity Factor Design (DCFD) methodology provides a foundation for reliability-based seismic design and performance evaluation of new and existing buildings. In Chapter 5, the DCFD method was introduced by only taking into consideration the inherent randomness of the building demands and capacities, and neglecting modeling uncertainty. Recall that inherent randomness is uncertainty that is part of nature and is always present, while modeling uncertainty results from the use of...

## Model Steel Frame Buildings

B. 1 Description of P re-North ridge Model Buildings Three welded steel moment frame (WSMF) model buildings, a low-rise (3-story), mid-rise (9-story), and high-rise (20-story) building, were designed as part of the SAC Joint Venture1 research program by the Los Angeles structural engineering firm of Brandow and Johnston (Mercado and Ungermann, 1997). The designs follow the 1994 Uniform Building Code (UBC) (ICBO, 1994) requirements for office buildings located in Los Angeles, California on a...

## Estimating Building Response Using Nonlinear Static Analysis Methods

The design of buildings is fundamentally concerned with ensuring that the components of the building, e.g. lateral force resisting system, can adequately serve their intended function. In the case of seismic design of the lateral force resisting system, the design problem can be reduced simply to the problem of providing adequate force and deformation capacity to resist the seismic demands. However, due to the uncertainties associated with the structural materials, the demand and capacity...

## Organization of Dissertation

The motivations and objectives for the research in this report are described in Chapter 1. A detailed description of the performance-based design methodology is presented in Chapter 2. A review of state-of-the-art code provisions and recommended guidelines for the design of new buildings and rehabilitation of existing building is presented. A review of nonlinear static pushover analysis methods used to quantify the demands on buildings subjected to earthquake ground motions is presented in...

## Performancebased Seismic Design

This chapter provides a description of the Performance-Based Seismic Design methodology and the state-of-the-art guidelines and code provisions for the design of new buildings and the rehabilitation of existing buildings. Section 2.2 describes the Performance-Based Seismic Design methodology. Section 2.3 describes the state-of-the-art guidelines and code provisions for the design of new buildings. Section 2.4 describes the state-of-the-art recommended guidelines for the rehabilitation of...

## Multi-mode Spectral Method Seismic

Where T0 is the characteristic period of the earthquake ground motion response spectrum and R is the ratio of the elastic base shear demand to the yield base shear determined from the bilinear representation of the capacity curve. The value of R is calculated from the following, where W is the total seismic weight of the building. The characteristic period of the earthquake ground motion response spectrum is calculated based on a smooth representation of the response spectrum from the...

## Performance Based Seismic Design

The objective of Performance-Based Seismic Design (PBSD) is to accurately predict, in definable terms, the performance of a building during any intensity of earthquake ground motion that may occur at the building site over the lifetime, or design life, of the building. Definable performance can be accomplished by designing the building to meet a wide range of performance objectives. A single performance objective consists of a level of performance in terms of damage, coupled with a level of...

## Vision2000 Oes1995

Equation (5.61) is used to modify the probability of failure for the uncertainty due to inherent randomness in the collapse capacity. Thus, the expected value of the annual probability of failure considering the uncertainty due to inherent randomness in the capacity and in the maximum IDR demand is, Figures 5.14, 5.15, and 5.16 show the probability of failure calculated using Eq. (5.62) for the 3-, 9-, and 20-story buildings, respectively. These figures show the probability of failure as a...

## S Steel Moment Frame Buildings

Buildings that use a system of steel beam and columns, with connections capable of transferring bending moments, for the primary lateral force resisting system are called steel moment frame buildings. Steel moment frames are those frames that develop their seismic resistance through bending of beams and columns and shearing of the beam-column panel zones (ATC, 1997). Steel moment frames classified as special moment resisting frames (SMRF) by the 1997 AJSC Seismic Provisions for Structural Steel...

## Quantifying Analysis Method Uncertainty Sac Method

The SAC Joint Venture1 research program is currently in the process of finalizing a series of seismic design guidelines (FEMA, 2000) which incorporates the Cornell Demand and Capacity Factor Design (DCFD) methodology for the seismic performance evaluation of steel frame buildings. When an approximate analysis method is used to conduct a seismic performance evaluation of a building, the reliability framework must account for the uncertainty in the analysis method. This section describes the...

## Introduction

As part of any seismic building design or evaluation procedure, the structural engineer must perform an analysis of the building, incorporating the seismic hazard at the building site, to obtain building response quantities. The building response quantities are calculated from a mathematical model of the building subjected to gravity forces and lateral earthquake forces. Building performance is deemed acceptable if these quantities are within the limits of acceptable building response. Building...

## Building Weights and PDelta Effects

The seismic weight and gravity loads of each building floor were calculated using the design loading assumptions described in Section B.l. One-half of the total floor seismic weight was assigned to the analytical model of the perimeter frame since there are two frames resisting seismic forces in each building direction. The gravity load tributary to the frame was applied as point loads on the columns and as uniformly distributed loads on the beams. In addition to the earthquake-induced shear in...

## Info Idt

Figure 7.7 13-story instrumented building East-West 6th Floor displacement time history comparison Figure 7.8 13-story instrumented building North-South Roof displacement time history comparison Figure 7.8 13-story instrumented building North-South Roof displacement time history comparison

## Abstract Of The Dissertation

Reliability-Based Seismic Performance Evaluation of Steel Frame Buildings using Nonlinear Static Analysis Methods Matthew John Skokan Doctor of Philosophy in Civil Engineering University of California, Los Angeles, 2000 Professor Gary C. Hart, Chair Performance-based seismic design enables the structural engineer to quantify the probability that a building design will satisfy specified seismic performance objectives. A probabilistic assessment of seismic building performance requires the...

## Performance Evaluation Level

Quality Assurance During Construction Figure 2.1 SEAOC Vision 2000 methodology for performance-based Table 2.1 SEAOC Vision 2000 recommended performance objectives After OES, 1995 Indicates the earthquake return period. Basic Basic Objective Essential Essential Hazardous Objective Critical Safety Critical Objective Unacceptable Unacceptable Performance Table 2.2 SEAOC Vision 2000 general damage descriptions for new building designs After OES, 1995 Table 2.2 SEAOC Vision 2000 general damage...

## Research Motivation

Starting in the early 1990's, the direction of building design in seismic regions has been towards Performance-Based Seismic Design. The objective of this building design method is to accurately predict, in definable terms, the performance of a building during any intensity of earthquake ground motion that may occur at the building site during the design life of the building. Definable building performance can be accomplished by designing the building to meet a range of performance objectives....

## Reliabilitybased Seismic Design

The primary purpose of performance-based seismic engineering design is to produce a structure that has a definable and predictable performance during future earthquakes. This design concept was described in detail, as it applies to building structures, in Chapter 2. The performance-based seismic engineering philosophy can be applied to both new and existing buildings and includes a performance evaluation stage whereby the reliability, or alternately the probability of failure, of the building...

## Rd

Where the term in brackets in Eq. 5.49 is the conditional probability of exceedance expressed by Eq. 5.44 . Rearranging Eq. 5.49 gives, Cornell now assumes that the random variable eu gt has a lognormal probability distribution. Therefore, with this carefully selected set of mathematical relationships, the integral in Eq. 5.50 can be solved and results in the expression, Appendix A presents the complete derivation of the integral in Eq. S.50 . In Eq. 5.51 , is the median value of the random...

## Earthquake Ground Motion

A total of sixty earthquake ground motion records were developed by Somerville, et al. 1997 as part of the SAC Joint Venture1 research program. Suites of twenty ground motion records were developed for each of three seismic hazard categories having exceedance probabilities of 50 in 50 years, 10 in 50 years, and 2 in 50 years. These exceedance probabilities correspond to 72-yr, 475-yr, and 2475-yr return periods, respectively. The ground motion records for each seismic hazard category are listed...