ATC 78 3 Methodology Concrete Frames_083115

ATC 78-3

Seismic evaluation of older concrete frame buildings for collapse potential

Applied Technology Council Funded by Federal Emergency Management Agency

Applied Technology Council The Applied Technology Council (ATC) is a nonprofit, tax-exempt corporation established in 1973 through the efforts of the Structural Engineers Association of California. ATC’s mission is to develop state-of-the-art, user-friendly engineering resources and applications for use in mitigating the effects of natural and other hazards on the built environment. ATC also identifies and encourages needed research and develops consensus opinions on structural engineering issues in a non-proprietary format. ATC thereby fulfills a unique role in funded information transfer. ATC is guided by a Board of Directors consisting of representatives appointed by the American Society of Civil Engineers, the National Council of Structural Engineers Associations, the Structural Engineers California, the Structural Association of New York, the Western Council Association of Structuralof Engineers Associations, andEngineers four at-large representatives concerned with the practice of structural engineering. Each director serves a three-year term. Project management and administration are carried out by a full-time Executive Director and support staff. Project work is conducted by a wide range of highly qualified consulting professionals, thus incorporating the experience of many individuals from academia, research, and professional practice who would not be available from any single organization. Funding for ATC projects is obtained from government agencies and from the private sector in the form of taxdeductible contributions.

2014-2015 Board of Directors Roberto Leon, President Jim Amundson, Vice President Victoria Arbitrio, Secretary/Treasurer Nancy Gavlin, Past President Leighton Cochran, At Large

Erleen Hatfield Andrew Kennedy Bret Lizundia Robert Paullus, Jr. Don Scott

Michael Engelhardt Kurtis Gurley

Bill Bill Staehlin Warren

ATC Disclaimer While the information presented in this report is believed to be correct, ATC assumes no responsibility for its accuracy or for the opinions expressed herein. The materials presented in this publication should not be used or relied upon for any specific application without competent examination and verification of its accuracy, suitability, and applicability by qualified professionals. Users of information from this publication assume all liability arising from such use.

FEMA Notice Any opinions, findings, conclusions or recommendations expressed in this publication do not necessarily reflect the views of the Federal Emergency Management Agency.

Cover Photo: Reinforced concrete moment frame building, Sherman Oaks, California, ATC-38, Database on the Performance of Structures near Strong-Motion Recordings: 1994 Northridge Earthquake (ATC, 2000).

ATC-78-3 Seismic Evaluation of Older Concrete Frame Buildings for Collapse Potential Prepared by APPLIED TECHNOLOGY COUNCIL 201 Redwood Shores Parkway, Suite 240 Redwood City, California 94065 www.ATCouncil.org Prepared for FEDERAL EMERGENCY MANAGEMENT AGENCY Michael Mahoney, Project Officer Robert D. Hanson, Technical Monitor Washington, D.C. ATC MANAGEMENT AND OVERSIGHT Christopher Rojahn (Project Executive) Anna Olsen (Research Applications Manager) Jon A. Heintz (Program Manager)

PROJECT TECHNICAL COMMITTEE

PROJECT REVIEW PANEL

William T. Holmes (Project Tech. Director) Abbie Liel Michael Mehrain Jack P. Moehle Peter Somers

Craig Comartin (Chair) Michael Cochran Gregory G. Deierlein Ken Elwood Terry Lundeen Robert Pekelnicky John W. Wallace

WORKING GROUP MEMBERS Panagiotis Galanis Cody Harrington Travis Marcilla Siamak Sattar

August 2015

Notice Any opinions, findings, conclusions, or recommendations expressed in this publication do not necessarily reflect the views of the Applied Technology Council (ATC), the Department of Homeland Security (DHS), or the Federal Emergency Management Agency (FEMA). Additionally, neither ATC, DHS, FEMA, nor any of their employees, makes any warranty, expressed or implied, nor assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, product, or process included in this publication. Users of information from this publication assume all liability arising from such use. Cover photograph – Archive photo of a 13-story reinforced concrete frame building located in Sherman Oaks, California, from ATC-38 Database on the Performance of Structures Near Strong-Motion Recordings: 1994 Northridge Earthquake (ATC, 2000).

Preface

Since July 2009, the Applied Technology Council (ATC) has been conducting a series of projects for the Federal Emergency Management Agency (FEMA) focused on the development of procedures for “Identification and Mitigation of Nonductile Concrete Buildings” (the ATC78 Project series). The impetus for this project was based on the recognition that older non-ductile concrete buildings constructed prior to the late-1970s are vulnerable to collapse during strong earthquakes. FEMA’s primary goal for this work has been the development of a low-cost, easily implementable methodology for identifying the subset of concrete buildings in this class that are particularly vulnerable to collapse. The first three projects in this series (ATC-78, ATC-78-1, and ATC-78-2 Projects) were carried out under a “Seismic and Multi-Hazard Technical Guidance Development and Support” contract (HSFEHQ-08-D-0726) awarded by FEMA in 2008. Initial project efforts focused on the concept of “collapse indicators” in certain subclasses of older concrete buildings, with the goal of identifying and prioritizing vulnerable buildings based on building characteristics that could be quantitatively linked to collapse. Although theoretically logical, the development of an evaluation methodology based on collapse-indicator relationships proved difficult and impractical. As a result, the focus changed focus from collapse indicators to story drift demand and capacity relationships. Work in this new focus area, and the completion of the current evaluation methodology, was carried out on the ATC-78-3 Project, funded under a subsequent “Seismic Technical Guidance Development and Support” contract awarded by FEMA in 2012. This report outlines a methodology to evaluate the collapse potential of older concrete frame buildings. In the current methodology, collapse is defined as a loss of gravity support in a story, which is evaluated using a series of simplified calculations of drift demand and drift capacity. Although the resulting calculations are intentionally simplified, the underlying criteria are based on probabilistic concepts and structural reliability theory. This report represents the first version of a complete methodology for the evaluation of reinforced concrete frames and frame components, and is being released for review, comment, and testing on actual buildings. Information from the test program will be used to improve the methodology in future iterations. At

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Preface

iii

present, work is continuing on the expansion of the methodology to address walls and wall components, which are currently excluded from consideration in the calculation procedures. Work in this project series is closely related to, and has been coordinated with, the efforts of many other projects focused on existing concrete buildings. These include the National Science Foundation (NSF) funded Grand Challenge research project Mitigation of Collapse Risks in Older

Reinforced Concrete Buildings, and the National Institute of Standards and Technology (NIST) funded ATC-76-5 Project that developed a Program Plan for the Development of Collapse Assessment and Mitigation Strategies for Existing Reinforced Concrete Buildings (NIST, 2010b). Information contributing to this work has also come from the NIST-funded ATC-95 Project and the resulting report, Review of Past Performance and Further Development of Modeling Techniques for Collapse Assessment of Existing Reinforced Concrete Buildings (NIST, 2014), as well as the NIST-funded project performed by the Building Seismic Safety Council (BSSC) to classify concrete buildings Concrete Model Building Subtypes Recommended for Use in Collecting Inventory Data (NIST, 2010a). ATC is indebted to the leadership of Bill Holmes, Project Technical Director, and to the members of the Project Management Committee, consisting of Abbie Liel, Mike Mehrain, Jack Moehle, and Peter Somers for their efforts in the development of this report and for their oversight and guidance of the Project Working Group, which consisted of Panagiotis Galanis, Cody Harrington, Travis Marcilla, and Siamak Sattar. The Project Review Panel, consisting of Craig Comartin (Chair), Michael Cochrane, Greg Deierlein, Ken Elwood, Terry Lundeen, Robert Pekelnicky, and John Wallace, provided technical review and commentary at key developmental stages of the work. A workshop of invited participants and project team members was convened to receive feedback on the methodology, and input from this group was instrumental in shaping the final product. The names and affiliations of all who contributed to this report are provided in the list of Project Participants. ATC also gratefully acknowledges Michael Mahoney (FEMA Project Officer) and Robert Hansen (FEMA Technical Monitor) for their input and guidance in the preparation if the report, Chris Rojahn and Anna Olsen for ATC project management services, and Amber Houchen for ATC report production services. Jon A. Heintz ATC Executive Director

iv

Preface

ATC-78-3

Table of Contents

Preface ........................................................................................................... iii List of Figures ............................................................................................... ix List of Tables ................................................................................................xv 1.

Introduction ................... .................. ................... .................. ............ 1-1 1.1 Historical Background.............................................................. 1-2 1.2 Seismic Risk of Concrete Buildings versus Unreinforced Masonry Buildings ................................................................... 1-3 1.3 Current Research and Mitigation Projects................................ 1-4 1.3.1 Concrete Coalition....................................................... 1-4 1.3.2 NEES Grand Challenge Project .................................. 1-5 1.3.3 NIST Program Plan and Related Efforts ..................... 1-5 1.3.4 ATC-78 Project Series................................................. 1-6 1.4 ATC-78-3 Evaluation Methodology......................................... 1-7 1.4.1 Scope and Intent .......................................................... 1-7 1.4.2 Comparison with ASCE/SEI 41 Standard ................... 1-9 1.4.3 Policy Implications.................................................... 1-10 1.5 Report Organization and Content ........................................... 1-11

2.

Evaluation Methodology................. .................. .................. ............. 2-1 2.1 Scope and Applicability ........................................................... 2-1

2.2 2.3

2.4

2.5

Overview of the Evaluation Methodology ............................... 2-3 Step 1 – Investigate General Requirements.............................. 2-5 2.3.1 As-Built Information ................................................... 2-5 2.3.2 Site Investigation ......................................................... 2-5 2.3.3 Seismic Hazard ............................................................ 2-6 2.3.4 Material Properties ...................................................... 2-6 2.3.5 Condition of Structural Components........................... 2-7 2.3.6 Column Axial Loads ................................................... 2-7 2.3.7 Component Strength Calculations ............................... 2-9 Step 2 – Determine Relative Story Demand-to-Capacity Ratios ...................................................................................... 2-11 2.4.1 Relative Story Demand-to-Capacity Ratio ................ 2-11 2.4.2 Relative Story Shear Demand ................................... 2-11 2.4.3 Story Shear Capacity ................................................. 2-12 2.4.4 Column Shear Capacity ............................................. 2-12 2.4.5 Column Shear Capacity Controlled by Flexure......... 2-12 Step 3 – Estimate Structural Response ................................... 2-17 2.5.1 Effective Yield Strength ............................................ 2-17 2.5.2 2.5.3 2.5.4

2.6

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Effective Fundamental Period ................................... 2-18 System Strength Ratio ............................................... 2-18 Displacement of an Equivalent Single-Degree-ofFreedom System ........................................................ 2-18 Step 4 – Evaluate Column Components ................................. 2-19

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2.7

2.6.1 Story Drift Demand ................................................... 2-19 2.6.2 Adjusted Column Drift Demand ................................ 2-22 2.6.3 Torsional Amplification Factor ................................. 2-23 2.6.4 Column Drift Factor .................................................. 2-27 2.6.5 Column Drift Demand ............................................... 2-27 2.6.6 Column Drift Capacity .............................................. 2-28 2.6.7 Column Drift Capacity-to-Demand Ratios ................ 2-30 Step 5 – Determine Building Rating ....................................... 2-30 2.7.1 Column Rating ........................................................... 2-31 2.7.2 Story Rating ............................................................... 2-31 2.7.3 Building Rating.......................................................... 2-33 2.7.4 Evaluation of Corner Column and Corner Joint 2.7.5 2.7.6

3.

Conditions.................................................................. 2-33 Evaluation of Discontinuous Column Conditions ..... 2-35 Evaluation of Mezzanine Column Conditions........... 2-35

Development of Procedures for Estimating Drift Demands ......... 3-1 3.1 Overview of the Procedure to Estimate Drift Demands ........... 3-1 3.2 Estimation of the Effective Fundamental Period of the Building .................................................................................... 3-2 3.2.1 Empirical Method ........................................................ 3-2 3.2.2 Eigenvalue Analysis of a Mathematical Model ........... 3-2 3.2.3 Nonlinear Static Procedure .......................................... 3-2 3.2.4 Yield Drift Method ...................................................... 3-2 3.3 Determination of Displacement at the Effective Modal Height ....................................................................................... 3-5 3.3.1 Basic Procedure ........................................................... 3-5 3.3.2 Comparison with Results of Nonlinear Response History Analyses .......................................................... 3-7 3.4 Estimation of Story Drift .......................................................... 3-9

3.4.1

3.5 3.6

3.7 3.8

vi

Development of Coefficient  for Uniform Building Configurations ........................................................... 3-11 3.4.2 Development of Coefficient  for Non-Uniform Building Configurations ............................................ 3-16 Columns with Inadequate Lap Splice Conditions................... 3-18 Torsional Amplification Factor .............................................. 3-23 3.6.1 Overview ................................................................... 3-23 3.6.2 Data Used to Derive the Torsional Amplification Factor, AT,max .............................................................. 3-24 3.6.3 Torsional Amplification Factor Sensitivity to Building Characteristics ........................................................... 3-28 3.6.4 Relationship between Torsional Strength Ratio and Torsional Amplification Factor ................................. 3-28 3.6.5 Calculation of Torsional Strength Ratio .................... 3-30 3.6.6 Relationship between Total Torsional Amplification Factor and Accidental and Real Torsional Amplification Factors ................................................ 3-31 Determination of Critical Components: Column or SlabColumn Connection ................................................................ 3-33 Portion of Drift Taken by the Columns .................................. 3-33 3.8.1 Overview ................................................................... 3-33

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3.8.2

3.9 4.

Data Used to Determine the Portion of Drift Taken by the Columns.......................................................... 3-34 3.8.3 Data Used to Determine the Portion of Drift Taken by the Slab-Column Connections .............................. 3-35 3.8.4 Limitations ................................................................ 3-36 Moment Delivered to Columns from Beams.......................... 3-36

Development of Acceptability and Failure Criteria .................... .. 4-1 4.1 Determination of Column Deformation Capacities .................. 4-1 4.1.1 Overview ..................................................................... 4-1 4.1.2 Development of Column Deformation Capacity Tables .......................................................................... 4-2 4.1.3 Final Deformation Capacity Tables ............................ 4-4 4.1.4 Uncertainty in Drift Capacities.................................... 4-6 4.1.5 Deformation Capacity of Columns with Inadequate Lap Splices .................................................................. 4-7 4.1.6 Overturning Effects on Column Deformation Capacity ....................................................................... 4-8 4.2 Determination of Slab-Column Connection Deformation Capacities ............................................................................... 4-11 4.2.1 Overview ................................................................... 4-11 4.2.2 Development of Slab-Column Connection Deformation Capacity Tables .................................... 4-12 4.3 Method for Determining Column Ratings.............................. 4-14 4.3.1 Overview ................................................................... 4-14 4.3.2 Structural Reliability Methods for Computing the Column Rating .......................................................... 4-14 4.4 Method for Determining Story Ratings .................................. 4-15 4.4.1 Overview ................................................................... 4-15 4.4.2 Probability Theory for Determining Probability of Story Collapse ........................................................... 4-16 4.4.3 Sensitivity of Story Rating to Building and Methodological Assumptions .................................... 4-17 4.5 Studies of Beam-Column Joints ............................................. 4-21 4.5.1 Performance of Connections with Discontinuous Beam Bottom Longitudinal Reinforcement .............. 4-20 4.5.2 Strength of Joints in Beam-Column Connections without Joint Transverse Reinforcement ................... 4-21 4.5.3 Effect of joint eccentricity on joint and column behavior ..................................................................... 4-23 4.5.4 Axial failure of beam-column connections ............... 4-24

Appendix A: Data for Prediction of Drift Demands .................. ............ A-1 Symbols ....................................................................................................... B-1 References .................................................................................................. C-1 Project Participants .................................................................................. D-1

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List of Figures Figure 2-1

Calculation of column shear capacity controlled by flexure in a typical story ............................................................... 2-13

Figure 2-2

Calculation of column shear capacity controlled by flexure in the first story: (a) with a pinned base; and (b) with a fixed base .................................................................................. 2-15

Figure 2-3

Calculation of column shear capacity controlled by flexure in the first story with a frame system extending into the basement .......................................................................... 2-16

Figure 2-4

Calculation of column shear capacity controlled by flexure in the top story of a multistory building ........................... 2-17

Figure 2-5

Example building plan showing locations of the center of mass, center of strength, and frame lines, and illustrating the orthogonal dimensions for earthquake loading in the northsouth direction.................................................................. 2-25

Figure 3-1

Force-displacement curve of a structure, showing the definition of effective stiffness, Ke, for calculation of the effective fundamental period ............................................. 3-3

Figure 3-2

Comparison between empirical data and plots of period relationships based on different values of assumed yield strain .................................................................................. 3-5

Figure 3-3

Ratios of displacement at the effective modal height (nonlinear analysis results versus Equation 3-9) for different values of μstrength (denoted Rindex in the figure) and different column to beam strength ratios .......................................... 3-9

Figure 3-4

Illustration of a building frame, equivalent single-degreeof-freedom oscillator, and two idealized story drift patterns ............................................................................. 3-10

Figure 3-5

Story profiles of coefficient  for the 4-story idealized frame ................................................................................ 3-12

Figure 3-6

Mean values of coefficient  in the first story of the 4-story idealized frame .................................................... 3-12

Figure 3-7


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List of Figures

ix

x

Figure 3-8

Mean values of coefficient  in the first story of the 6-story idealized frame ..................................................... 3-13

Figure 3-9


Figure 3-10

Mean values of coefficient  in the first story of the 8-story idealized frame ..................................................... 3-14

Figure 3-11


Figure 3-12

Mean values of coefficient  in the first story of the 12-story idealized frame ................................................... 3-15

Figure 3-13

Story profiles of coefficient  for the 6-story idealized frame with a critical fourth story ...................................... 3-17

Figure 3-14

Mean values of coefficient  in the first story of the 6-story idealized frame with a critical fourth story .......... 3-18

Figure 3-15

Mean values of coefficient  in the fourth story of the 6-story idealized frame with a critical fourth story .......... 3-18

Figure 3-16

Zero-length plastic hinge rotational behavior for adequate and inadequate lap splice conditions ................................ 3-19

Figure 3-17

Alpha factor story profiles for different variations of the 8-story building with and without inadequate lap-splicing conditions ......................................................................... 3-20

Figure 3-18

Alpha factor story profiles for different variations of the 8-story building with and without inadequate lap-splicing conditions ......................................................................... 3-21

Figure 3-19

Alpha factor story profiles for different variations of the 8-story building with inadequate lap-splicing conditions and the first and fourth stories are critical ........................ 3-22

Figure 3-20

Torsional amplification as a function of Torsional Irregularity Ratio (TIR) for selected 1-story, high gravity load, ordinary moment frames for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse) ............................... 3-25

Figure 3-21

Torsional amplification as a function of Torsional Strength Ratio (TSR) for selected 1-story, high gravity load, ordinary moment frames for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse) ............................................. 3-26

Figure 3-22

Torsional amplification as a function of Torsional Strength Ratio (TSR) for selected 4-story, high gravity load, ordinary moment frames for: (a) linear or near linear response List of Figures

ATC-78-3

(Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse) ............................................. 3-26 Figure 3-23

Torsional amplification as a function of Torsional Strength Ratio (TSR) for selected 1-story, ordinary moment frames with nonlinear or near collapse response ( Sa ≥ 0.90Sa,collapse) and: (a) high gravity load; and (b) low gravity load levels ................................................................................ 3-27

Figure 3-24

Torsional amplification as a function of Torsional Strength Ratio (TSR) for 1-story, symmetric and asymmetric building configurations for: (a) linear or near linear response (Sa < 0.90Sa,collapse) and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse) ............................................. 3-27

Figure 3-25

Relationships between the torsional amplification factor and TSR proposed in Equations 3-17 and 3-18, superimposed on selected nonlinear analysis results, for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse) ............................... 3-29

Figure 3-26

Illustration of parameters for calculation of TSR ............. 3-30

Figure 3-27

Portion of drift taken by the column, Ccol,j (denoted as C in the figure), during nonlinear response from incremental dynamic analysis of selected 6-story buildings................ 3-34

Figure 3-28

Portion of drift taken by the column, Ccol (denoted as C in the figure), assuming a linear relationship with Mc/Mb .......................................................................... 3-35

Figure 3-29

Typical Laboratory Slab-Column Test Setup .................. 3-35

Figure 3-30

Pushover analyses to validate Equation 2-10, illustrating: (a) the pushover setup; (b) pushover results for a model with ΣMc/ΣMb = 1.2 and VpMx/Vn = 0.6; and (c) pushover results for a model with ΣMc/ΣMb = 1.2 and VpMx/Vn = 0.8 ...... ... 3-37

Figure 4-1

Definition of a and b parameters in terms of component force deformation response in ASCE/SEI 41 .................... 4-2

Figure 4-2

Comparison of predictions of drift at axial failure (b or c) by four different methods for condition iii columns .......... 4-5

Figure 4-3

Measured versus predicted plastic rotation at axial failure at different axial load ratios for: (a) ASCE/SEI 41 predictions of b for condition i, ii and iii columns; and (b) predictions of θc for condition iii columns in this methodology ............... 4-6

Figure 4-4

Measured versus predicted plastic rotation at axial failure at different transverse reinforcement ratios for: (a) ASCE/SEI 41 predictions of b for condition i, ii and iii, columns; and (b) predictions of θc for condition iii columns in this methodology ...................................................................... 4-6

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List of Figures

xi

Figure 4-5

Comparison of deformation capacities determined from test results (“empirical”), ASCE/SEI 41-13, and this methodology, for columns with inadequate lap splices ..... 4-8

Figure 4-6

Illustration of simplified method for computing Peq for a 2-dimensional frame ............................................. 4-9

Figure 4-7

Nonlinear analysis results for interior columns in a 6-story building with VpMx/Vn = 0.8 and Mc/Mb = 1.2 ................... 4-10

Figure 4-8

Comparison of simplified relationship for Peq with selected nonlinear analysis results for end columns in a 6-story building with VpMx/Vn = 0.8 and Mc/Mb = 1.2 ................... 4-10

Figure 4-9

Comparison of simplified relationship for Peq with selected nonlinear analysis results for the bottom three stories of a 4-story model ................................................................. 4-11

Figure 4-10

Ratio of measured versus predicted drift at punching shear failure for slab-column connections with different gravity shear ratios ....................................................................... 4-13

Figure 4-11

Comparison of measured versus predicted drift capacities for slab-column connections with different gravity shear ratios ................................................................................. 4-13

Figure 4-12

Assumed model of correlation in column drift capacities for failure of columns i and j, as a function of column separation distance ........................................................... 4-16

Figure 4-13

Relationship between adjusted average column rating,

Figure 4-14

CRavg, and the story rating, SR .......................................... 4-17 Sensitivity of story ratings to assumption about the definition of story collapse ............................................... 4-18

Figure 4-15

Sensitivity of story ratings to assumptions about correlations in column failures ......................................... 4-19

Figure 4-16

Sensitivity of story ratings to assumptions about uncertainty in column drift demand ................................. 4-19

Figure 4-17

Shear strength of unreinforced interior joints................... 4-21

Figure 4-18

Measured and calculated strengths (ASCE/SEI 41) for exterior joints loaded perpendicular to the edge, including corner joints ...................................................................... 4-22

Figure 4-19

Measured and calculated strengths (Equation 4-6) for exterior joints loaded perpendicular to the edge, including

Figure 4-20

xii

corner joints ...................................................................... 4-22 Force transfer at eccentric beam-column connections ..... 4-23

List of Figures

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Figure 4-21

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Maximum observed drift ratios and axial load ratios of corner beam-column connections .................................... 4-24

List of Figures

xiii

List of Tables

Table 2-1

Summary of Steps and Calculations of the Evaluation Methodology ...................................................................... 2-4

Table 2-2

Values for Effective Mass Factor, Cm .............................. 2-18

Table 2-3

Values for Coefficient  .................................................. 2-21

Table 2-4

Column Drift Factor when the Column is the Critical Component ....................................................................... 2-28

Table 2-5

Column Classifications .................................................... 2-29

Table 2-6

Column Plastic Rotation Capacity ................................... 2-29

Table 2-7

Effective Moment of Inertia............................................. 2-30

Table 2-8

Column Drift Capacity Ratio when the Slab-Column Connection is the Critical Component ............................. 2-30

Table 2-9

Column Rating, CR .......................................................... 2-31

Table 2-10

Average Column Rating Adjustment, CRa ...................... 2-32

Table 2-11

Story Rating, SR ............................................................... 2-32

Table 3-1

Parameters of the Idealized Frame Buildings Studied ....... 3-7

Table 3-2

Average Ratios of Displacement at the Effective Modal Height (Nonlinear Analysis Results versus Equation 3-9) for a 6-Story Idealized Frame Building ............................. 3-8

Table 3-3

Mean Values of Coefficient  in the First Story of a 6-story Idealized Frame without Significant Torsional Effects .............................................................................. 3-11

Table 3-4

Mean Values of Coefficient  in the First Story of a 6-story Idealized Frame with a Critical First Story.......... 3-16

Table 3-5

Lap Splice Amplification Factor, s ................................ 3-22

Table 3-6

Building Parameters Used in Pushover Verification Analyses ........................................................................... 3-36

Table 4-1

Number of Columns Used in Developing Predictions of θc, for Each Column Condition .......................................... 4-3

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Table 4-2 Table 4-3

Database Results for Modeling Parameters........................ 4-4 Column Plastic Rotation Capacities at Axial Failure, b = θc, for Condition iii Columns....................................... 4-5

xvi

Table 4-4

Uncertainty in Predictions of Drift Capacity, ln,∆c, as Obtained from Comparison with Experimental Data ......... 4-7

Table 4-5

Probability that Actual Drift Capacity is less than Predicted Drift Capacity for Columns with Inadequate Lap Splices ......................................................................... 4-8

Table 4-6

Slab-Column Drift Capacities .......................................... 4-13

Table A-1

Average Ratios of Displacement at the Effective Modal Height (Nonlinear Analysis Results versus Equation 3-9) for a 4-Story Idealized Frame Building............................. A-1

Table A-2

Average Ratios of Displacement at the Effective Modal Height (Nonlinear Analysis Results versus Equation 3-9) for an 8-Story Idealized Frame Building........................... A-1

Table A-3

Average Ratios of Displacement at the Effective Modal Height (Nonlinear Analysis Results versus Equation 3-9) for a 12-Story Idealized Frame Building........................... A-2

Table A-4

Mean Values of Coefficient  in a 4-story Idealized Frame without Significant Torsional Effects ............................... A-2

Table A-5

Standard Deviation of Coefficient  in a 4-story Idealized Frame without Significant Torsional Effects .................... A-3

Table A-6


Table A-7

Standard Deviation of Coefficient  in a 6-story Idealized Frame without Significant Torsional Effects .................... A-5

Table A-8

Mean Values of Coefficient  in an 8-story Idealized Frame without Significant Torsional Effects ............................... A-6

Table A-9

Standard Deviation of Coefficient  in an 8-story Idealized Frame without Significant Torsional Effects .................... A-7

Table A-10


Table A-11

Standard Deviation of Coefficient  in a 12-story Idealized Frame without Significant Torsional Effects .................. A-10

Table A-12

Mean Values of Coefficient ′ in a 4-story Idealized Frame without Significant Torsional Effects ............................. A-12

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ATC-78-3

Table A-13

Standard Deviation of Coefficient ′ in a 4-story Idealized Frame without Significant Torsional Effects .................. A-13

Table A-14


Table A-15


Table A-16

Mean Values of Coefficient ′ in an 8-story Idealized Frame without Significant Torsional Effects ............................. A-16

Table A-17

Standard Deviation of Coefficient ′ in an 8-story Idealized Frame without Significant Torsional Effects .................. A-17

Table A-18


Table A-19


Table A-20

Mean Values of Coefficient  in a 6-story Idealized Frame with a Critical Fourth Story ............................................ A-22

Table A-21

Standard Deviation of Coefficient  in a 6-story Idealized Frame with a Critical Fourth Story ................................. A-23

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Chapter 1

Introduction As part of its responsibilities under the National Earthquake Hazards Reduction Program (NEHRP), the Federal Emergency Management Agency (FEMA) is charged with supporting mitigation activities necessary to improve technical quality in the field of earthquake engineering and reduce the ever-increasing cost of disasters. One issue that is still of significant concern for existing buildings is the expected seismic performance of older, seismically vulnerable concrete buildings, known as non-ductile concrete buildings. These buildings were constructed prior to the late-1970s, and include archaic construction dating back to the early 1900s. Not all such buildings are hazardous. Problematic issues include inadequate steel reinforcing details, system irregularities, and element discontinuities that result in sudden shear failure and loss of load-carrying ability. Visual inspection of a building cannot identify many of the known seismic deficiencies in this class of buildings. For most buildings, an engineer must perform an analytical evaluation to determine the extent of its vulnerability to failure in strong shaking. In 2009, in recognition of the high collapse potential of some older nonductile buildings during strong earthquake shaking, FEMA began sponsoring what would become a series of projects to develop a reliable, low-cost, easily implemented methodology to identify particularly vulnerable non-ductile concrete buildings (the ATC-78 Project Series). The efforts to develop this methodology were coordinated with, and built on, the results of related projects sponsored by the National Earthquake Hazards Reduction Program (NEHRP), as well as the work of the Structural Engineering Institute (SEI) of the American Society of Civil Engineers (ASCE) to develop the ASCE/SEI 31 and ASCE/SEI 41 standards for the seismic evaluation and retrofit of existing buildings, based on earlier FEMA-sponsored work. The evaluation procedures embodied in this report are currently limited to reinforced concrete frame type structures with rigid diaphragms and a height less than 160 feet. Collapse is defined as the global loss of vertical loadcarrying ability at a story, possibly leading to the collapse of other stories. The story drift demand is calculated probabilistically and combined with a capacity limit, resulting in a probabilistic measure of collapse for each column in each story, and for the collapse of a story as a whole. These

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probabilities can be used to prioritize buildings for mitigation. Since there is no extensive testing or nonlinear analysis required, the engineering time for the evaluation of a non-ductile concrete building is expected to be no greater than that for a Tier 2 evaluation using the ASCE/SEI 41-13 standard, Seismic

Evaluation and Retrofit of Existing Buildings (ASCE, 2013), and significantly less than Tier 3 nonlinear analysis procedures. 1.1

Historical Background

Large earthquakes in the past several decades demonstrated the vulnerability of older concrete buildings. In the 1971 San Fernando, California, earthquake, the Olive View Hospital, a site with newly constructed concrete buildings, sustained major damage including partial collapse (e.g., Lew et al., 1971). In the same event, the Veterans Administration Hospital, a site with two-dozen older concrete buildings, also sustained major damage including several partial or total collapses (e.g., Johnston, 1973) that killed 47 occupants (Murphy, Steinbrugge, and Duke, 1973). In the 1994 Northridge, California, earthquake, several older concrete buildings collapsed. The February 2011 Christchurch, New Zealand, earthquake damaged dozens of concrete buildings beyond repair, and two moderate-sized buildings collapsed, killing 133 people and injuring many more. These experiences, among others, demonstrate that some older concrete buildings are a life safety risk to occupants in earthquakes. During the twentieth century, a wide variety of concrete buildings were built in regions of high seismicity in the United States, and many were constructed prior to the enactment of modern seismic provisions in building codes. Early seismic requirements allowed almost all structural system configurations in concrete buildings that were less than 160 feet tall. For buildings taller than 160 feet, early provisions required the incorporation of a ductile momentresisting frame. At the time, ductile moment frames had to be constructed with steel until the 1967 Uniform Building Code (ICBO, 1967) introduced provisions for ductile concrete moment frames, based on the work of Blume, Newmark, and Corning (1961). Following the 1971 San Fernando earthquake, the 1976 Uniform Building Code introduced requirements for robust detailing of reinforced concrete systems, along with larger seismic design loads than in previous editions of the UBC. Early concrete design codes did not address deformation demands on structural components, such as columns, beams, or walls that were not part of the primary seismic-force-resisting system. In particular, early design codes artificially stipulated that gravity support columns were intended to support gravity loads only, and were not considered part of the seismic-force-

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resisting system. Since 1967, building code provisions for concrete buildings have required increasing ductility capacity and tolerance for story drift. It was not until the late 1990s, after the 1994 Northridge earthquake, that concrete gravity columns were required to be designed to accommodate realistic floor-to-floor drifts. However, the 1976 UBC is often used as the benchmark that distinguishes between concrete buildings that are potentially high seismic risk and concrete buildings that are generally acceptable by current standards. Therefore, concrete buildings built in accordance with codes prior to the 1976 UBC are considered high seismic risk buildings, and recommended for evaluation and potential mitigation.

1.2

Seismic Risk of Concrete Buildings versus Unreinforced Masonry Buildings

Since the 1971 San Fernando earthquake, the life safety risk of non-ductile concrete buildings has been compared to that of unreinforced masonry (URM) buildings. Both classes of buildings are known to have a significant risk of collapse in strong ground shaking. However, the risk for non-ductile concrete buildings is particularly acute when typical occupancies are considered. Non-ductile concrete buildings tend to be larger than URM buildings. As a result, they have higher occupancies, and the collapse of a single non-ductile concrete building could result in as many casualties as the collapse of several URM buildings. The 2011 Christchurch earthquake, in particular, demonstrated this issue. Significant damage to hundreds of URM buildings, including some collapses, resulted in about 40 casualties. The collapse of two moderately sized concrete buildings resulted in nearly 140 casualties. Although there may be fewer collapses of concrete buildings than URM buildings in historical earthquakes, the consequences of concrete building collapses can be more severe. There are also differences in existing strategies for mitigation of non-ductile concrete buildings and URM buildings. Programs to mitigate the seismic risk posed by URM buildings have been implemented in regions of high seismicity in the United States, but no such program has been implemented for older concrete buildings. The absence of mitigation programs for nonductile concrete buildings may be explained, in part, by the following: 

Particularly dangerous concrete buildings can be difficult to identify, whereas URM buildings are easy to identify and inventory.

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

Most older concrete buildings have survived past earthquakes (albeit with some damage and few collapses), but most URM buildings sustain damage, and many collapse.



Owners of URM buildings do not normally form politically powerful advocacy groups, whereas owners of concrete buildings have influential advocates.



Specific retrofit criteria and procedures are available and effective for URM buildings, at moderate construction cost; however, some practicing engineers consider the evaluation and retrofit criteria available for concrete buildings (for example, the ASCE/SEI 41-13 standard) to be conservative and expensive.

1.3

Current Research and Mitigation Projects

Although the potential risk of older concrete buildings has been recognized by the structural engineering since the mid-1970s, only in the last decade have efforts increased to develop practical methods for mitigation. Examples of these efforts include: the formation of the Concrete Coalition; a Network for Earthquake Engineering Simulation (NEES) Grand Challenge research project funded by the National Science Foundation; the development of the National Institute of Standards and Technology (NIST) Program Plan for the Development of Collapse Assessment and Mitigation Strategies for Existing Reinforced Concrete Buildings (the “NIST Program Plan”); and support from the Federal Emergency Management Agency (FEMA) for the ATC-78 Project Series. The current focus of these efforts is to target mitigation programs toward “killer buildings,” that is, the buildings that can be identified as exceptionally vulnerable to strong ground motion. Since evaluation methods and retrofit techniques for concrete buildings are complex and expensive to perform, a methodology for accurately identifying the relatively small proportion of “killer” non-ductile concrete buildings, which can be implemented at low cost, has the potential to enable the beneficial use of limited resources for mitigation of buildings that represent the greatest risk. 1.3.1

Concrete Coalition

The Concrete Coalition is a network of individuals, institutions, and government agencies with a shared interest in assessing the risk associated with non-ductile concrete buildings and developing strategies to mitigate that risk. The Concrete Coalition consists of the Earthquake Engineering Research Institute (EERI), the Pacific Earthquake Engineering Research Center (PEER), the Applied Technology Council (ATC), and their partners,

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including the Structural Engineers Association of California (SEAOC), the American Concrete Institute (ACI), the Building Owners and Managers Association (BOMA) of Greater Los Angeles, and the U.S. Geological Survey (USGS). With funding from the Federal Emergency Management Agency (FEMA) and the California Office of Emergency Services (Cal OES), the Concrete Coalition has been helping California assess the size and scope of the potential risk by providing an educated estimate of the existing non-ductile concrete building inventory. The Concrete Coalition also encourages other similar inventory collection efforts in other regions, particularly those of high seismicity. The Concrete Coalition was an early advocate of developing methods to identify potential “killer buildings.” They did so under the premise that a methodology to easily identify hazardous concrete buildings would enable a systematic mitigation of risk at an increasing pace. The Coalition also recognized that not all older concrete buildings are collapse hazards, that retrofit of these buildings is involved and expensive, and that both the effort and funds to mitigate the risk must be accurately targeted at the most hazardous buildings. 1.3.2

NEES Grand Challenge Project

The NEES Grand Challenge for the Mitigation of Collapse Risk of Older

Concrete Buildings (NEES, 2010), is a project that identifies just how widespread the collapse hazard is in our existing reinforced concrete building stock and develops engineering and policy tools to identify and reduce the risk of these hazardous buildings. In addition to conducting laboratory testing of certain aspects of the performance of concrete building systems considered understudied (e.g., beam column joints), an extensive collection of inventory of older concrete buildings was undertaken in Los Angeles. The purpose of that survey was to identify the number of older concrete buildings, their occupancy, and structural type within the greater Los Angeles area. This information will be used in seismic loss studies to estimate the total risk and benefits of mitigation of these building types. 1.3.3

NIST Program Plan and Related Efforts

The National Institute of Standards and Technology GCR 10-917-7 Report,

Program Plan for the Development of Collapse Assessment and Mitigation Strategies for Existing Reinforced Concrete Buildings (NIST, 2010b), defines a broad program over eight or more years to systematically investigate the vulnerability of non-ductile concrete buildings as well as the

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means to mitigate their risk. The NIST Program Plan is consistent with the aims and philosophy of the Concrete Coalition, builds on the findings of the NEES Grand Challenge project, and defines activities in collaboration with the FEMA-funded efforts described in this report. The NIST Program Plan also suggested a methodology to determine the relative importance of various conditions known to be seismic deficiencies and called them “collapse indicators. Concurrent with the development of the NIST Program Plan, NIST also funded a project conducted by the Building Seismic Safety Council (BSSC) of the National Institute of Building Sciences (NIBS) to identify subclasses of non-ductile concrete buildings with common and quantifiable, characteristic deficiencies. The findings of those efforts are documented in the NIST GCR 10-917-6 Report, Concrete Model Building Subtypes

Recommended for Use in Collecting Inventory Data (NIST, 2010a). The BSSC project provided valuable insight into vulnerabilities of non-ductile concrete buildings, and served as an impetus, in part, for the funding of the ATC-78 series of projects. The initial NIST-funded effort under the Program Plan, which is documented in the NIST GCR 14-917-28 report, Review of Past

Performance and Further Development of Modeling Techniques for Collapse Assessment of Existing Reinforced Concrete Buildings (NIST, 2014), focused on the identification of critical deficiencies for older reinforced concrete buildings (based on building collapse case studies); the identification of collapse mitigation strategies; and the development of guidance on ground motions and component models for collapse simulation. The NIST GCR 14-917-28 report indicates that current simplified evaluation procedures fail to capture the collapse vulnerability of buildings because inadequate attention is paid to the vulnerability of the gravity load-carrying system. 1.3.4

ATC-78 Project Series

The overall goal of the FEMA-funded ATC-78 Project Series, “Identification and Mitigation of Non-Ductile Concrete Buildings,” is to develop an affordable evaluation methodology that: (1) will identify the most seismically hazardous non-ductile concrete buildings; and (2) is less expensive to apply than the full analysis procedures of ASCE/SEI 41-13. To date, four task order projects have been completed in this series, and a fifth is now underway (ATC-78-4 Project). The first two projects (ATC-78 and ATC-78-1) studied the collapse indicator concept identified in the NIST Program Plan, with the purpose of calculating

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a probability of collapse based on the severity of a collapse indicator. Near the end of the ATC-78-1 Project, it was determined that evaluation of all possible combinations and interactions of collapse indicators, for many possible structural configurations and heights, was impractical. Although the collapse indicator concept provided insight into important aspects of predicting global collapse, use of collapse indicators was found to be unsuitable for the purposes of the evaluation methodology envisioned in the NIST Program Plan and by FEMA. These findings were initially documented in the ATC-78 report, Identification and Mitigation of

Seismically Hazardous Older Concrete Buildings: Interim Methodology Evaluation (ATC, 2011), which was an interim document, and then in the ATC-78-1 report, Evaluation of the Methodology to Select and Prioritize Collapse Indicators in Older Concrete Buildings (ATC, 2012), which supersedes the ATC-78 report, and is available on the ATC website (www.ATCouncil.org). Under the ATC-78-2 project in early 2013, the focus of the work changed from using collapse indicators as a parameter for evaluation to considering the effect of story drifts on the collapse potential of concrete columns. Laboratory test data are more abundant than examples of collapse in past earthquakes, and test data can be used to determine the deformation and drift capacities of concrete components. The use of story drift instead of collapse indicators is expected result in a more practical and robust evaluation methodology for non-ductile concrete buildings. Project findings and initial efforts at defining the evaluation methodology were documented in the ATC-78-2 report, Seismic Evaluation for Collapse Potential of Older

Concrete Frame Buildings (ATC, 2013), another interim report, which has now been superseded by this ATC-78-3 report. 1.4

ATC-78-3 Evaluation Methodology

The ATC-78-3 Project continued the work of the ATC-78 Project Series, focusing on the development of an evaluation methodology for non-ductile concrete buildings with frame systems resisting seismic forces. 1.4.1

Scope and Intent

This report presents an evaluation methodology intended to determine the relative risk of collapse of older concrete buildings in strong earthquake shaking. The purpose is to enable identification of the relatively few buildings in this class of buildings that are expected to have a high risk of collapse, without the need for extensive testing or nonlinear analysis. It is expected that evaluation of entire inventories of older concrete buildings

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would thus be affordable and feasible, and the buildings posing the greatest risk of catastrophic life loss could be identified and mitigated. This report also describes the development of the evaluation methodology, including extensive nonlinear analysis used to predict global collapse based on the procedures provided in the FEMA P-695 Report, Quantification of

Seismic Performance Factors (FEMA, 2009), and the use of an existing database of laboratory test data (developed with funding from the National Science Foundation and other organizations) for concrete column behavior. Significant features of the evaluation methodology are: 

The definition of collapse as a global loss of gravity support at a story, possibly leading to the collapse of other stories;



Singular intent of determining the relative risk of global collapse among an inventory of older concrete buildings. Other performance measures, typically based on component performance (e.g., damage levels or downtime) are not considered;



The engineering procedures incorporated in the evaluation methodology are expected to be equal to or less than a Tier 2 evaluation using ASCE/SEI 41-13, and significantly less complicated than the full nonlinear analysis procedures of that document;



The probabilistic characterization of demand (as story drift) and the capacity limit (as column collapse), allowing calculation of the probability of collapse for each individual column at each story;



The estimation of the probability of collapse of a given story by considering the collapse probability of individual columns; and



The use of the probability of collapse as a relative measure for ranking the seismic risk of individual older concrete buildings.

For the purposes of this methodology, all concrete buildings designed to codes not equivalent to the 1976 Uniform Building Code, or not known to meet other locally accepted evaluation or retrofit standards, are considered high seismic risk buildings. This evaluation methodology is intended to identify the subset of these buildings that are considered to be significant collapse hazards, termed exceptionally high seismic risk buildings. The criteria used for this definition are discussed in Sections 1.4.3 and 1.4.4. At present, evaluation procedures in the methodology are limited to reinforced concrete frame type structures with rigid diaphragms and a height less than 160 feet. Procedures for evaluating the collapse potential of reinforced concrete wall systems are currently under development.

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The procedures to test for global collapse require only a drift demand to implement, and are thought to be useable with other analysis procedures. Other estimates of drift demand that are more extensive (and presumably more accurate) than the procedures recommended in this report are possible. Similarly, other, more detailed methods of estimating the effective building period could be used. Voluntary use of the methodology with customized input parameters might be useful to private institutions and individual building owners looking to understand the likely behavior of their buildings and to guide prudent risk-reduction decisions in voluntary situations. However, for policy-level decision making and mandatory programs, it is recommended that the complete methodology described herein be used for all buildings to ensure consistency in the development of relative rankings across an entire building inventory. 1.4.2

Comparison with ASCE/SEI 41 Standard

The protection of life safety has been used as a seismic performance objective for decades, but has not been clearly defined in terms of the level of unacceptable injury and the probability of such an injury. The ASCE/SEI 31 and ASCE/SEI 41 standards for seismic evaluation and retrofit, were srcinally developed with “life safety” as a conceptual basis, although in addition to the notion of “life safety,” ASCE/SEI 41 includes a specific performance level based on prevention of structural collapse. In these documents, the acceptable life safety risk for existing buildings was permitted to be somewhat higher than new buildings due to the cost and disruption of retrofit. In the 2013 update of the standards, the evaluation (ASCE/SEI 31) and retrofit (ASCE/SEI 41) documents were combined into a single document (ASCE/SEI 41-13), and made more internally consistent. The criteria used to define “collapse prevention” in ASCE/SEI 41 are based on the performance of individual building components (e.g., beams, columns, and slabs) rather than global collapse. Often, the failure of one or more components to meet the criteria in ASCE/SEI 41 is not expected to result in building collapse. Some structural engineers experienced with evaluation and retrofit of older concrete buildings estimate that only a small percentage of concrete buildings that fail the “life safety” criterion in ASCE/SEI 41 will actually collapse, even in very rare shaking. Recent studies of building performance, such as those embodied in the FEMA P-695 Methodology, consider global collapse more explicitly and describe performance probabilistically. These studies recognize that most serious injuries and deaths in earthquakes are caused by global structural collapse.

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The intent of this evaluation methodology is specific. It focuses on structural failures of vertical load-carrying elements judged to lead to global collapse, and therefore does not directly address other seismic deficiencies that have been observed to cause structural damage (but not collapse) in past earthquakes. An evaluation using this methodology will result in the approximation of the probability of collapse of a building subject to a demand ground motion. This result is not directly related to “collapse prevention” as defined in ASCE/SEI 41, because it is an approximation of global structural collapse. The probability of global structural collapse is an approximation because of uncertainties related to the determination of ground motion, the methods used to calculate story drifts, the prediction of the probability of failure of individual columns, and the procedure used to estimate story collapse based on column collapse. However, the results of individual building evaluations in a group of buildings can provide a useful indication of the relative potential for collapse across an inventory of buildings. A threshold probability of collapse, above which the risk of collapse should be considered unacceptable by policymakers, and therefore considered exceptionally high seismic risk, has not yet been determined. This evaluation methodology is not intended to replace ASCE/SEI 41-13, although results of trial evaluations using this methodology may be useful to calibrate evaluation and retrofit based on ASCE/SEI 41-13 criteria with respect to structural collapse. However, such a calibration would require the establishment of appropriate probabilistic acceptance criteria to enable use of, and comparison with, the probabilistic results of this methodology. 1.4.3

Policy Implications

As described above, concrete buildings designed to codes not equivalent to the 1976 Uniform Building Code , or not known to meet other locally accepted evaluation or retrofit standards, are considered to be highly vulnerable to earthquake shaking and are herein classified as high seismic risk buildings. The methodology developed in this report is intended to enable relatively inexpensive identification of older concrete buildings with a high probability of collapse, termed exceptionally high seismic risk buildings. For any level of ground motion chosen, this methodology is intended to rank buildings according to their approximate probability of collapse. The probability of collapse for any building is highly dependent on the severity of ground motion at the site and the probability of occurrence for the chosen ground motion, which can include smaller but more frequent earthquakes (72-year return period), or larger but less frequent earthquakes (500-year, 1000-year or 2500-year return periods).

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A more intense basis of ground motion will raise the probability of collapse for all buildings in an area, but in general, the relative ranking will be the same. This may not be true in areas with a steeply sloped hazard curve, such as the New Madrid seismic region in and around Memphis, Tennessee. This area has a relatively low threat from frequent earthquakes, but the ground motion intensity for very rare earthquakes is very high. In such areas, consideration of the probability of collapse over the full range of possible earthquake shaking intensities might be a better measure for ranking buildings; however, this would require analysis of each building for multiple levels of earthquake intensity and a much greater level of engineering effort. Based on observed performance of U.S. style buildings in past earthquakes, it is expected that a relatively low percentage of buildings will be placed in the exceptionally high seismic risk category. However, there are inadequate data to place a number on this percentage or to calibrate this methodology to set an unacceptable probability of collapse. Following evaluation by this methodology, more extensive evaluation using the nonlinear analysis methods of ASCE/SEI 41 could result in reclassification of a building from the exceptionally high seismic risk category to the high seismic risk category. It is anticipated that significant use of this evaluation methodology will include use by political jurisdictions in setting priorities for mandatory seismic risk reduction within their communities, considering collapse as the overriding loss to be avoided. Establishment of an acceptable level of risk is likely to require more input than can be obtained through comparisons with risks implied by traditional performance expectations for new or existing buildings. Further testing of this methodology on real buildings with known collapse characteristics (i.e., from analysis, laboratory testing, or observed performance) will help set such a limit in the future.

1.5

Report Organization and Content

This report outlines a methodology to evaluate the collapse potential of older concrete frame buildings. In this methodology, collapse is defined as a loss of gravity support in a story, which is evaluated using a series of simplified calculations of drift demand and drift capacity. Although the resulting calculations are intentionally simplified, the underlying criteria are based on probabilistic concepts and structural reliability theory. Chapter 2 outlines the evaluation methodology, describing the steps and calculations that are used to identify buildings with an unacceptably high probability of collapse. These include procedures for collecting the necessary building information, estimating structural response, defining

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system and story drift demands, calculating drift capacities, and developing the resulting column, story, and building ratings. Chapter 3 explains the development of procedures to for estimating story drift demands and drift capacities in concrete frame buildings, and presents the results of background studies and analyses used to validate the procedures. Chapter 4 describes the development of acceptability and failure criteria, and summarizes the use of available experimental data to quantify the resulting deformation capacities used in the evaluation methodology. Appendix A provides a record of additional background analytical results that were used in the development of procedures for estimating story drift demands and drift capacities in Chapters 3. A list of symbols used throughout this report, along with a list of references, are provided at the end of this report.

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Chapter 2

Evaluation Methodology Older concrete buildings not in compliance with the strength and detailing requirements of the 1976 or later editions of the Uniform Building Code (ICBO, 1976), and not otherwise determined to have acceptable seismic performance, could be susceptible to significant structural damage in an earthquake and should be considered high seismic risk buildings. This chapter describes the steps and calculations comprising an evaluation methodology that can be used to identify buildings in this group with an unacceptably high probability of collapse, designated to be exceptionally high seismic risk buildings. Buildings categorized as exceptionally high seismic risk using this evaluation methodology should be prioritized for more detailed seismic evaluation or seismic retrofit.

2.1

Scope and Applicability

This evaluation methodology is intended for use in determining the relative risk of collapse within an inventory of buildings, prioritizing buildings for further evaluation, and guiding prudent risk-reduction policy decisions. It is not intended for use in developing or implementing seismic retrofits to improve the seismic performance, or reduce the risk of collapse, in an individual building. At present, the methodology is applicable to reinforced concrete frame structures, 160 feet or less in height, without structural load-bearing walls or shear walls that significantly influence structural response. Reinforced concrete frames that can be evaluated using this methodology include:



frame lines (or bays) that are designed to resist gravity loads, including beam-column systems, slab-column systems, or joist-column systems;



frame lines (or bays) that are designed to resist gravity plus lateral loads, regardless of the level of ductile detailing;



frame lines with partial-height concrete or masonry infill walls that can potentially create short column effects (note: frame lines with full-height



infill walls are excluded from the methodology at this time); or any combination of the above.

Additional requirements for use of the methodology include:

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

Floor and roof diaphragms that are capable of providing a load path to vertically oriented framed bays that resist lateral loads.



Columns that have a cross-sectional aspect ratio no greater than 2.5:1. Up to 10% of the columns may exceed this requirement if it is judged that overall building response will not be adversely affected.



Beam-column joints with limited eccentricity, in which the centerline of the beam is located within the width of the column and at least some of the beam longitudinal reinforcement passes within the column core (as defined by the boundaries of the column longitudinal reinforcement). Up to 10% of the beam-column joints may exceed this requirement if it is judged that overall building response will not be adversely affected.

The evaluation methodology can also be applied to post-tensioned concrete structures with reinforced concrete columns. In such cases, procedures for determining column capacity as a function of the beam or slab capacity, and any other calculations related to the capacity of beams and slabs, must appropriately consider the strength of post-tensioned concrete components, as determined in ACI 318-11, Building Code Requirements for Structural Concrete and Commentary (ACI, 2011). The evaluation methodology is not applicable to precast concrete frame structures because the calculation procedures do not address potential load path and connection deficiencies that are likely to be present in such systems. Because the evaluation methodology is focused on structural collapse, certain aspects considered in conventional seismic evaluation procedures, which identify other sources of potential seismic damage, are not evaluated in this methodology. These include:



Nonstructural components and systems (note: full-height nonstructural concrete or masonry infill walls that influence the response of the structural system will exclude a building from evaluation using this methodology).



Precast concrete or stone cladding with brittle or drift-intolerant connections (note: estimated story drifts can be used to evaluate potential damage to the panels or their connections; however, potential falling hazards due to cladding are not considered in the ratings of this methodology).



Pounding between adjacent buildings (note: buttressing effects from shorter adjacent buildings with little or no separation can be examined using this methodology by designating the story immediately above the shorter building as a “critical story”).

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

Geologic site hazards associated with earthquake-induced ground failure or tsunami effects.

Older concrete buildings that do not qualify for evaluation using this methodology should be considered to be exceptionally high seismic risk

buildings, and prioritized for detailed seismic evaluation by other means. 2.2

Overview of the Evaluation Methodology

This evaluation methodology attempts to approximate the probability of global structural collapse of older concrete buildings subjected to earthquake shaking at a specified seismic hazard level. Implementation of the methodology does not require a structural analysis model, instead relying on linear and hand (spreadsheet) calculations. These features are intended to reduce the time and effort required to evaluate a building and to ensure, through prescriptive criteria, that results for one building can be easily compared to results for another building, enabling consistent ranking of a building inventory in terms of relative collapse risk. By evaluating the likelihood of global collapse, rather than individual component failures, the methodology attempts to avoid some of the conservatisms inherent in other seismic evaluation procedures, such as ASCE/SEI 41-13, Seismic Evaluation and Retrofit of Existing Buildings (ASCE, 2013). The methodology requires estimation of drift demands and drift capacities of key components in the structure (e.g., columns, beam-column joints, or slabcolumn joints). Drift demands and drift capacities are used to compute column ratings, which are then used to determine story ratings, and finally, an overall building rating. These ratings represent the likelihood of collapse for the column, story, or building, and vary between 0.0 and 1.0, with 0.0 representing a low likelihood of collapse and 1.0 representing a high likelihood of collapse. Drift demands depend on building strength, effective period, and the expected vertical distribution of drift among stories. Drift capacities are based on experimental data quantifying the failure of different types of reinforced concrete components with varying levels of ductile detailing. The methodology is intended to require less effort to implement than other more detailed seismic evaluation procedures (e.g., ASCE/SEI 41), and consists of the following steps:

 

Step 1 – Investigate General Requirements Step 2 – Determine Relative Story Demand-to-Capacity Ratios



Step 3 – Estimate Structural Response

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

Step 4 – Evaluate Column Components



Step 5 – Determine Building Rating

A summary of the steps and calculations comprising the evaluation methodology is provided in Table 2-1.

Table 2-1

Summary of Steps and Calculations of the Evaluation Methodology

Step Step 1

Step 2

Step 3

Step 4

Calculations Investigate General Requirements 2.3.1 2.3.2

As-Built Information Site Investigation

2.3.5 2.3.6

Condition of Structural Components Column Axial Loads

2.3.3

Seismic Hazard

2.3.7

Component Strength Calculations

2.3.4

Material Properties

Determine Relative Story Demand-to-Capacity Ratios 2.4.1

Relative Story Demand-to-Capacity Ratio

2.4.5.1 Column Shear Capacity in a Typical Story

2.4.2

Relative Story Shear Demand

2.4.5.2 Shear-Controlled Beams

2.4.3

Story Shear Capacity

2.4.5.3 Slab-Column Frames

2.4.4

Column Shear Capacity

2.4.5.4 Column Shear Capacity in the First Story

2.4.5

Column Shear Capacity Controlled by Flexure

2.4.5.5 Column Shear Capacity in the Top Story

Estimate Structural Response 2.5.1

Effective Yield Strength

2.5.2

Effective Fundamental Period

2.5.3

System Strength Ratio

2.5.4

Displacement of Equivalent Single-Degree-ofFreedom System

2.6.4 2.6.5 2.6.6

Column Drift Factor Column Drift Demand Column Drift Capacity

2.6.6

Column Drift Capacity in Slab-Column Frames

2.6.7

Column Drift Capacity-to-Demand Ratios

2.7.5

Evaluation of Discontinuous Column Conditions

2.7.6

Evaluation of Mezzanine Column Conditions

Evaluate Column Components 2.6.1

Story Drift Demand

2.6.1.1 Critical Stories 2.6.2

Adjusted Column Drift Demand

2.6.2.1 Critical Components 2.6.3

Torsional Amplification Factor

2.6.3.1 Torsional Strength Ratio 2.6.3.2 Maximum Torsional Amplification 2.6.3.3 Accidental Torsional Amplification 2.6.3.4 Real Torsional Amplification Step 5

2-4

Determine Building Rating 2.7.1

Column Rating

2.7.2

Story Rating

2.7.3

Building Rating

2.7.4

Evaluation of Corner Column and Corner Joint Conditions


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The methodology also includes special procedures to address torsional irregularities, lap splice deficiencies, slab-column frames, corner conditions, discontinuous columns, and mezzanines. Certain conditions can result in the assignment of a building rating without further evaluation. For example, buildings that are essentially elastic (Section 2.5.3.1) are assigned a low (good) rating without the need to perform additional calculations. Similarly, very weak buildings (Section 2.5.1) and buildings with excessive torsion (Section 2.6.3.1) are assigned high (poor) ratings without further evaluation. Additional information on the background and derivation of the calculation procedures comprising the methodology can be found in Chapters 3 and 4.

2.3

Step 1 – Investigate General Requirements

In general, the evaluation methodology requires specific knowledge of the as-built configuration and condition of in-place materials and components. 2.3.1

As-Built Information

Documentation of the as-built configuration of a building is necessary for implementation of the evaluation methodology. Required as-built information includes: building size and configuration; structural component size, reinforcement, and detailing; material properties; and site and foundation information. This information is best obtained from complete structural design drawings or as-built drawings. Other potential sources of information include construction specifications, geotechnical reports, structural calculations, and shop drawings. In the case of older concrete buildings, sources of as-built information are often not available. Because it is potentially misleading to make assumptions regarding reinforcement details and member proportioning, it is recommended that any concrete building for which detailed structural drawings are not available should be considered an exceptionally high seismic risk building by default, unless structural details can be confirmed by other means (e.g., destructive and non-destructive site investigations). 2.3.2

Site Investigation

A site investigation consists of visual observation of the condition of the building to verify that as-built information is representative of the existing conditions. The following should be confirmed as part of a site investigation: (1) building configuration, including the presence (or not) of structural additions, alterations, or modifications; (2) layout, proportioning, and condition of structural components; (3) site characteristics; and (4) foundation conditions.

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2-5

It is intended that site investigations be performed using non-destructive means. Concealed conditions should be exposed where feasible and practical (e.g., above suspended ceilings). Destructive removal of finishes and in-situ testing of materials are not required as part of this methodology, although information from destructive investigations could be used to enhance available as-built information. 2.3.3

Seismic Hazard

Seismic hazard due to ground shaking is defined as an acceleration response spectrum based on either a probabilistic or deterministic assessment of hazard presented in United States Geological Survey (USGS) seismic maps. The acceleration response spectrum should be determined in accordance with ASCE/SEI 7-10, Minimum Design Loads for Buildings and Other Structures (ASCE, 2010) using USGS mapped spectral response acceleration parameters (SS, S1) adjusted for site class at the latitude and longitude of the building. Site class (A through F) should be determined in accordance with ASCE/SEI 7-10. Site class as documented in a geotechnical report is acceptable for use. If insufficient information is available to classify the site, default Site Class D shall be used. The recommended seismic hazard level for evaluation is the ASCE 41-13 Basic Safety Earthquake BSE-2E, which corresponds to a 5% probability of exceedance in a 50-year period. The USGS National Seismic Hazard Mapping Project website provides spectral response acceleration parameters (SS, S1) at this and other hazard levels (USGS, 2008). Values should be modified for site class in accordance with ASCE/SEI 7-10, but the resulting parameters need not be greater than those for the risk-targeted maximum considered earthquake (MCER) ground motion (also modified for the site class). 2.3.4

Material Properties

Properties of cast-in-place concrete and reinforcing steel materials shall be taken from design drawings or other as-built information. Physical testing of in-situ concrete or reinforcing steel is not required as part of this methodology. If material properties are not identified in the design drawings or documented in other as-built information, default material properties may be used in accordance with ASCE/SEI 41-13 Section 10.2.2.5. Material properties used in component strength calculations shall be expected material properties, determined as follows:

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

Expected compressive strength of concrete, f'ce, taken as the specified or nominal compressive strength, f'c, multiplied by 1.5.



Expected yield strength of reinforcement, fye, taken as the specified or nominal yield strength, fy, multiplied by 1.25.

Although not required, testing can result in higher material strengths than minimum values specified on the drawings or default values provided in ASCE/SEI 41, particularly in the case of concrete. As a result, physical testing could be used to improve material strength values, and could be considered as part of an enhanced investigation of as-built information. 2.3.5

Condition of Structural Components

This methodology assumes that the building and structure are in generally good condition. Although site investigations do not require destructive investigation or detailed condition assessment, visual observations should include assessment of the general condition of significant structural component in accessible areas. If the overall building or significant structural components are judged to be in poor condition, then the evaluation results should be adjusted to characterize the potentially weakened state of the building. Concrete buildings constructed in the 1940s and earlier are particularly likely to have experienced inadequate quality control on concrete mixing and placement procedures during construction. As a result, the as-built condition of such buildings, and the condition of the concrete columns in particular, should be investigated in greater detail. Guidance on more detailed condition assessment is available in ASCE/SEI 41-13 Section 10.2.3, FEMA 306, Evaluation of Earthquake Damaged Concrete and Masonry Wall Buildings (FEMA, 1998), or ACI 201.2R, Guide to Durable Concrete (ACI, 2008). 2.3.6 2.3.6.1

Column Axial Loads Expected Gravity Loads

For the purpose of this methodology, the determination of expected gravity loads shall be in accordance with this section. Dead loads include the structure self-weight and appropriate superimposed dead loads. Live loads shall be taken as 25% of the unreduced design live loads listed in ASCE/SEI 7-10 Table 4-1. The column axial load due to expected gravity load effects, Pg, is:

Pg = PD + 0.25PL

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(2-1)

2-7

where PD is the axial load due to tributary dead loads, and PL is the axial load due to unreduced tributary live loads. 2.3.6.2

Earthquake Axial Loads

Consideration of column axial loads due to earthquake, Peq, is only required if the spectral acceleration, Sa, at the effective fundamental period of the building, Te, is greater than 0.3 g. The column axial load due to earthquake overturning effects, Peq, at each story x is calculated as:

Vy ( heff  hx ) Peq 

L

(2-2)

where:

Vy

= effective yield strength of a structure, limited by the largest story demand-to-capacity ratio, and reduced to consider P- Δ effects, as defined in Equation 2-15;

heff

= effective height of an equivalent single-degree-of-freedom system (may be taken as 0.7 hn in multistory buildings, and hn in single-story buildings);

hn

= height from the base of the building to the highest level of the

hx

= height from the base of the building to level x (i.e., the bottom

L

= plan dimension between the outermost frame columns in the direction of interest at story x.

seismic force-resisting system (i.e., level n); of story x); and

Axial load, Peq, is taken as positive in compression, and can be assumed to be shared equally by the outermost columns in each frame line parallel to the direction of earthquake loading. Tension loads due to earthquake overturning are not critical and need not be considered. For columns located in a story above the effective height of the equivalent single-degree-offreedom system, or for interior columns in a frame line (i.e., not the outermost columns in the frame line), Peq may be taken as zero. If heff falls within a story, Peq should be computed for the columns within that story. 2.3.6.3

Load Combinations

Where gravity loads are combined with earthquake forces, all load combinations shall be unfactored, with gravity loads determined in accordance with Equation 2-1. Evaluation of components for gravity loads in the absence of earthquake forces is beyond the scope of this methodology.

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2.3.7

Component Strength Calculations

2.3.7.1

Expected Strength

Component strengths used in this evaluation methodology shall be taken as expected strengths. Expected strengths shall be calculated using expected material properties, determined in accordance with Section 2.3.4. Column, beam, slab, and joint component strengths shall be determined using expected material properties and accepted principles of mechanics. Unless specifically indicated otherwise, the procedures of ACI 318-11 may be used to calculate component strengths, except that the strength reduction factor (ϕ) should be taken as unity (i.e., ϕ = 1.0). For structural components constructed using lightweight concrete, strength calculations should be modified in accordance with ACI 318-11 procedures for lightweight concrete. 2.3.7.2

Column Shear Strength

Column shear strength, Vn, is calculated based on ASCE/SEI 4113, as follows:

A f d  6 f ce g P ye v  1  linf / d  s 6 f ce Ag  

Vn  k 

   0.8 Ag     

(2-3)

where:

k

= factor related to displacement ductility demand; can be taken as 1.0 for the purpose of calculating column shear strength in this methodology;

s

= spacing of shear reinforcement;

Av

= area of shear reinforcement within spacing s;

d

= effective depth of the column section;

fye

= expected yield strength of reinforcement;



= 0.75 for lightweight concrete, and 1.0 for normal weight concrete;

f'ce

= expected compressive strength of concrete;

linf

= clear height of an equivalent cantilever column from the face of a joint to the point of inflection (or zero moment); in a typical story, linf may be taken as half of the column clear height, lu; in the first story, refer to Section 2.4.5.4.

Pg

= column axial load (in compression) due to expected gravity load effects (equal to zero if in tension); and

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2-9

Ag

= gross area of concrete column section.

It is permitted to assume d = 0.8hc, where hc is the overall dimension of the column in the direction of shear. The ratio linf /d should not be taken greater than 4, nor less than 2. For columns satisfying the detailing and proportioning requirements of ACI 318-11 Chapter 21, the shear strength equations of ACI 318-11 are permitted. 2.3.7.3

Beam-Column Joint Shear Strength

Beam-column joints satisfying the following conditions may be assumed to have sufficient strength to develop the flexural strength of the beams and columns framing into the joint:



Interior beam-column joints where beams frame into all four faces of the joint.



Any joint with hoop reinforcement within the joint at a spacing not exceeding the lesser of: h/3, where h is the overall dimension of the column in the direction of shear, or 8 inches.

For other joints, shear strength should be calculated in accordance with ASCE/SEI 41-13 Section 10.4.2.3, except that the joint shear strength, Vnj, need not be taken less than:

Vnj  10

hc hb

fA

(2-4)

ce j

where:

hc

= overall dimension of the column in the direction of shear;

hb

= overall depth of the beam;

Aj

= effective cross-sectional area of the beam-column joint.

The ratio hc/hb must be less than or equal to 1.0. The effective area of the joint, Aj, can be taken as hc (bc + b′w)/2, where bc is overall width of the column in the direction perpendicular to joint shear; and b′w is the web width of the beam, excluding portions of the web that extend beyond the width of the column. Joint shear strength calculated in accordance with ASCE/SEI 41-13 is typically less than the value in the exception noted above. Thus, Vnj calculated using Equation 2-4 usually controls. 2.3.7.4

Slab-Column Frame Strength

The flexural strength of a slab, and the shear and moment transfer strength of slab-column connections, shall be calculated in accordance with ASCE/SEI 41-13 Section 10.4.4.3. The flexural strength of the slab, and the flexural

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strength of a joist system, is based on the strength of the column strip, as defined in ASCE/SEI 41-13. The shear and moment transfer strength of slab-column connections must consider the combined action of flexure, shear, and torsion in accordance with ASCE/SEI 41-13.

2.4

Step 2 – Determine Relative Story Demand-toCapacity Ratios

The relative demand-to-capacity ratio for each story is calculated as a ratio between the relative story shear demand and the plastic story shear capacity. Relative story shear demands are calculated using a normalized inverted triangular load pattern as limited by plastic story shear capacities. Plastic story shear capacities are calculated based on controlling cases among column shear strength, column flexural strength, beam or slab flexural strength, and joint shear strength. 2.4.1

Relative Story Demand-to-Capacity Ratio

The relative demand-to-capacity ratio, DCRx, is determined for each story x as the ratio between the relative story shear demand, Vrx, and the plastic story shear capacity, Vpx:

DCR x =

Vrx Vpx

(2-5)

Note that since Vrx is conditioned on Vpx in accordance with Equation 2-6, the relative demand-to-capacity ratio in the first story ( DCR1) will always be equal to 1.0. 2.4.2

Relative Story Shear Demand

The relative story shear demand, Vrx, is calculated at each story x as:

Vrx  Vp

n

1

C vi i x

(2-6)

where Vp1 is the first story plastic shear capacity determined using Equation 2-8, and Cvi is the vertical distribution factor determined as:

Cvx 

wxhx n

 w h

(2-7)

i i

i 1

where: wi, wx = portion of the total effective seismic weight tributary to level i or level x;

hi, hx = height from the base of the building to level i or level x;

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2-11

n

= number of levels in the building.

2.4.3

Story Shear Capacity

The plastic story shear capacity, Vpx, is calculated at each story x as the sum of the individual column plastic capacities, Vpc at story x: nc

V px =

V

(2-8)

pcj

j 1

where nc is the number of columns in the story. 2.4.4

Column Shear Capacity

For each column, the plastic shear capacity, Vpc, is taken as the lesser of the column shear strength, Vn, determined in accordance with Section 2.3.7.2, and the column shear capacity controlled by flexure, VpM, determined in accordance with Section 2.4.5. 2.4.5

Column Shear Capacity Controlled by Flexure

The column shear capacity controlled by flexure, VpM, is calculated based on the expected flexural strengths of the members framing into the joints at the top and bottom of each story. Calculation procedures are provided for the typical story (Section 2.4.5.1), first story (Section 2.4.5.4), and top story (Section 2.4.5.5) of multistory buildings. Additional guidance is provided for structures with shear-controlled beams (Section 2.4.5.2) and slab-column frames (Section 2.4.5.3). 2.4.5.1

Column Shear Capacity in a Typical Story

For columns in a typical story x, the column shear capacity controlled by flexure, VpMx, is:

VpMx 

M cTx  McBx lux

(2-9)

where McTx is the flexural strength at the top of the column, McBx is the flexural strength at the bottom of the column, and lux is the clear height of the column in story x, as shown in Figure 2-1.

McTx is taken as the lesser of the expected flexural strength of the column section, Mn, calculated using expected material properties and ϕ = 1.0, and the flexural strength controlled by the beams or slabs at the top of the column, calculated using Equation 2-10:

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 M bLx( M cTx  

1)

M bRx

( 1) x

hx  h( x1)

 h  M n

(2-10)

where MbL(x+1) is the expected flexural strength of the beam on the left side of the joint, MbR(x+1) is the expected flexural strength of beam on the right side of the joint, hx is the height of story x, and h(x+1) is the height of story x+1, as shown in Figure 2-1. Where the beams are shear-controlled, the expected flexural strength of the beam is limited by the shear strength of the beam in accordance with Section 2.4.5.2. Additionally, the flexural strength at the top of the column shall not be taken greater than the capacity associated with the joint strength, taken as Vnjhb(x+1)/2, where Vnj is the expected shear strength of the joint at the top of the column determined in accordance with Equation 2-4, and hb(x+1) is the overall depth of the beam at level x+1.

McBx is taken as the lesser of the expected flexural strength of the column section, Mn, calculated using expected material properties and ϕ = 1.0, and the flexural strength controlled by the beams or slabs at the bottom of the column, calculated using Equation 2-11: McB(x+1)

Story x+1

MbR(x+1) MbL(x+1)

McTx Story x

VpMx

lux

hx

McBx

MbLx MbRx McT(x-1)

Story x-1

Figure 2-1

Calculation of column shear capacity controlled by flexure in a typical story.

M cBx 

 M bLx  MbRx xh h h x

n M

(2-11)

( x 1)

where MbLx is the expected flexural strength of the beam on the left side of the joint, MbRx is the expected flexural strength of the beam on the right side

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2-13

of the joint, hx is the height of story x, and h(x-1) is the height of story x-1, as shown in Figure 2-1. Where the beams are shear-controlled, the expected flexural strength of the beam is limited by the shear strength of the beam in accordance with Section 2.4.5.2. Additionally, the flexural strength at the top of the column shall not be taken greater than the capacity associated with the joint strength, taken as Vnjhbx/2, where Vnj is the expected shear strength of the joint at the bottom of the column determined in accordance with Equation 2-4, and hbx is the overall depth of the beam at level x. The sense of the beam (or slab) moments, MbL and MbR, applied at a joint must be consistent with the direction of loading and the resulting deformed shape of the frame. In T-beam construction, where slab reinforcement is in tension due to moments at the face of the joint, reinforcement located within an effective flange width determined in accordance with ACI 318-11 Section 8.12 should be assumed to contribute to the flexural strength of the beam. 2.4.5.2

Shear-Controlled Beams

In shear-controlled beams, the expected shear strength, Vb, is less than the shear corresponding to the expected flexural strength at each end of the beam. In such cases, Vb < (Mb1+Mb2)/ln, where Mb1 and Mb2 are the expected flexural strengths at each end of a beam on one side of the column, and ln is the clear span of the beam between the supports. As a result, the flexural strength of shear-controlled beams on either side of a joint, MbL or MbR, is limited to:

M bL  MbR 

Vb ln 2

(2-12)

where Vb is the expected shear strength of the beam on the left or right side of the joint, respectively. 2.4.5.3

Slab-Column Frames

In slab-column frames, the flexural strength at the top and bottom of the column, McTx and McBx, will be limited by the expected flexural strength of the slab, or the shear and moment transfer strength of slab-column connection. 2.4.5.4

Column Shear Capacity in the First Story

In a typical story, equations for column shear capacity controlled by flexure assume point of inflection (i.e., zero moment) located at the mid-height of the columns. In the first story, the point of inflection will vary depending on the level of fixity at the base of the columns. Assuming an effective story height equal to twice the distance from the floor slab to the point of inflection, the

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effective height of the first story will vary along with the location of the point of inflection. Pinned column base.When

the column base is assumed to have zero (or

small) moment capacity as shown in Figure 2-2a, the point of inflection is located at the base of the column. Column moments McB1 = 0, and McT1 is calculated using Equation 2-10 with an effective first story height of 2.0h1. Fixed column base. When

the column base is assumed to be fixed against

rotation (e.g., restrained by basement shear walls or rigid foundation elements) as shown in Figure 2-2b, the point of inflection may be taken as approximately 0.4h1 from the second floor. Column moment McB1 is taken as the lesser of the expected flexural strength of the column section and the maximum moment that can be resisted by the base restraint, and McT1 is calculated using Equation 2-10 with an effective first story height of 0.8h1. Semi-fixed column base.When

the column base is restrained by a frame

system extending into the basement as shown in Figure 2-3, the point of inflection may be taken as approximately at the mid-height of the column, and calculation of McT1 is consistent with the typical story case, using Equation 2-10 and an effective first story height of 1.0h1. Translation (i.e., drift) at the basement level is restrained. As a result, the basement column contributes to rotational restraint along with the beams at level 1, and column moment McB1 is taken as the lesser of the expected McB2

McB2

Story 2 MbR2

MbR2

MbL2

MbL2

McT1

McT1 VpM1

lu1

h1

Story 1

VpM1

lu1

h1

McB1

McB1 = 0 (pin)

Figure 2-2

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(a) (b) Calculation of column shear capacity controlled by flexure in the first story: (a) with a pinned base; and (b) with a fixed base.


2-15

McB2

Story 2

MbR2 MbL2

McT1 Story 1

VpM1

lu1

h1

McB1

MbL1 MbR1 Basement

Figure 2-3

McT (basement)

Calculation of column shear capacity controlled by flexure in the first story with a frame system extending into the basement.

flexural strength of the column section in the first story and the flexural strength controlled by the beam and column components located at the basement level:

M cB1  MbL

1

M bR

1

 McT basement 

(2-13)



where McT (basement) is the expected flexural strength of the top of the basement column, and all other terms are as previously defined. Where the beams are shear-controlled, the expected flexural strength of the beam is limited by the shear strength of the beam in accordance with Section 2.4.5.2. In most cases the expected flexural strength of the column will govern, and McB1 will be taken as Mn. 2.4.5.5

Column Shear Capacity in the Top Story

In the top story of multistory buildings, where columns frame into roof beams as shown in Figure 2-4, McTx is taken as the lesser of the expected flexural strength of the column section and the flexural strength controlled by the beams located at the roof level:

M cTx bLM

 roof 

M  

bR

roof

(2-14)

where MbL (roof) and MbR (roof) are the expected flexural strengths of the roof beams on the left and right sides of the joint, respectively. In the top story of multistory buildings, McBx is calculated using Equation 2-11. Where the

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beams are shear-controlled, the expected flexural strength of the beam is limited by the shear strength of the beam in accordance with Section 2.4.5.2. Roof MbR(roof) MbL(roof)

McTx Story x VpMx

lux

hx

McBx

MbLx MbRx Story x-1

Figure 2-4

2.5

McT(x-1)

Calculation of column shear capacity controlled by flexure in the top story of a multistory building.

Step 3 – Estimate Structural Response

Structural response is estimated based the displacement of an equivalent single-degree-of-freedom (SDOF) system, similar to the procedure in ASCE/SEI 41-13. The displacement of an equivalent SDOF system is defined by the effective fundamental period of the structure and the corresponding spectral ordinate. The effective period is defined by the effective yield strength and an assumed yield displacement. 2.5.1

Effective Yield Strength

The effective yield strength of the structure, Vy, is based on the plastic shear capacity of the first story (Vp1) calculated in accordance with Equation 2-8, adjusted by the ratio of DCR1 and the maximum DCR at any story, and reduced in consideration of P-Δ effects:

Vy 

DCR1 DCR( max)

V p1  0.02W

(2-15)

where W is the total effective seismic weight of the building in units consistent with Vp1. If Vy ≤ 0, the structure is assigned a building rating, BR, of 0.7 without further evaluation, and should be considered an exceptionally high seismic risk building.

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2.5.2

Effective Fundamental Period

The effective fundamental period, Te, is calculated as:

Te  0.078

hnW

(2-16)

Vy

where hn is the total height (in feet) from the base of the building to the highest level of the seismic force-resisting system, W is the total effective seismic weight, and Vy is the effective yield strength in units consistent with

W. 2.5.3

System Strength Ratio

The system strength ratio, strength, for a given direction of earthquake loading is calculated as:  strength 

Sa Vy / W

Cm

(2-17)

where Sa is the spectral acceleration at the effective fundamental period, Te,

Vy is the effective yield strength, and Cm is an effective mass factor determined in accordance with ASCE/SEI 41-13 and taken from Table 2-2. Table 2-2

Values for Effective Mass Factor, Cm

No. of stories

Moment Frame System

1-2 3

≥

2.5.3.1

Pier-Spandrel System

1.0 0.9

1.0 0.8

Essentially Elastic Buildings

A building is considered to be essentially elastic if the following criteria are met:



strength ≤ 0.75 when more than half of the story plastic shear capacity, Vp, in any critical story is controlled by column shear strength, Vn; or



strength ≤ 1.5 in all other cases.

Buildings that are determined to be essentially elastic may be assigned a building rating, BR, of 0.2 without further evaluation. 2.5.4

Displacement of an Equivalent Single-Degree-of-Freedom System

The displacement of an equivalent single-degree-of-freedom (SDOF) system, eff, is calculated as:

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 eff

Te 2 g 4 2

 CC 1 S2 a

(2-18)

where:

C1

= modification factor to relate expected maximum inelastic displacement to displacement calculated for linear elastic response;

C2

= modification factor to represent the effect of pinched hysteresis shape, cyclic stiffness degradation, and strength deterioration on

Sa g

maximum displacement response; = spectral acceleration at the effective fundamental period, Te; and = acceleration of gravity.

Coefficient C1 is determined in accordance with ASCE/SEI 41-13 as:

C1  1 

strength  1

(2-19)

aTe2

where a is a site class factor equal to: 130 for Site Class A or B; 90 for Site Class C; 60 for Site Class D, E, or F; and all other terms are as previously defined. For Te < 0.2 seconds, C1 need not be taken greater than the value at Te = 0.2 seconds. For Te ≥ 1 second, C1 = 1.0. Coefficient C2 is determined in accordance with ASCE/SEI 41-13 as: 2

C2  1  1   strength  1  Te 800  

(2-20)

where all terms are as previously defined. For Te > 0.7 seconds, C2 = 1.0.

2.6

Step 4 – Evaluate Column Components

Columns are evaluated based on story drift demands conditioned on the displacement of an equivalent SDOF system, and adjusted considering approximate increases from torsional effects and potential decreases from rotations at beam-column (or slab-column) joints. Column drift capacities are determined based on the column plastic rotation capacities considering column transverse reinforcement condition, axial load ratio, and shear reinforcement ratio. 2.6.1

Story Drift Demand

Story drift demand, x, at story x is the product of the equivalent SDOF system displacement and a set of factors based on the characteristics of the framing system, determined as follows:

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2-19

x  

x

cx

sx

  eff  heff 

hx 

   

(2-21)

eff

where: x

= coefficient to modify story drifts at story x for building

cx

= coefficient at story x to amplify story drifts in stories that are

sx

= coefficient at story x to amplify story drifts in stories with non-

hx

= height of story x;

eff

= displacement of the equivalent SDOF system;

heff

= effective height of the equivalent SDOF system (may be taken

configuration and strength characteristics; determined to be critical; ductile lap splices in longitudinal column reinforcement;

as 0.7hn in multistory buildings, andhn in single-story buildings); and

hn

= height from the base of the building to the highest level of the seismic force-resisting system.

Story drift demands are calculated in each direction of earthquake loading. Note that the calculated story drift demand in any story must not exceed the displacement of the equivalent SDOF system,δeff. 2.6.1.1

Critical Stories

A critical story is a story with a large relative demand-to-capacity ratio, defined as DCR > 1.2. If the value of DCR at all stories is less than or equal to 1.2, then the first story is designated as a critical story, and all other stories are not critical. Note that a critical story isnot necessarily a weak story. , c, and s. In In critical stories, drift demands are amplified by coefficients

non-critical stories, values of coefficient are reduced, and values of coefficients c and s are set equal to 1.0. 2.6.1.2

Story Ratio of Column Strengths to Beam Strengths

The story ratio of column strengths to beam strengths, ∑Mc/∑Mb, in story x, shall be taken as the ratio of the sum of all column expected flexural strengths to the sum of all beam expected flexural strengths at the top of story x (i.e., the joints at levelx+1):

 M  M  M     M   M    M  

2-20

c

cTx

b

bLx 1


cB x 1

bRx

(2-22)

1

ATC-78-3

Each moment should beevaluated at the face of the joint. Alternatively, moments may be projected to the center of the joint, which may be advantageous in frames with beams or columns with an unusual aspect ratio. 2.6.1.3

Determination of Coefficient

Coefficient  modifies story drifts considering building configuration and strength characteristics.. Values of  depend on the total number of stories in the building, the story ratio of column strengths to beam strengths ∑Mc/∑Mb, and whether or not the story is a critical story. Values for

coefficient  are provided in Table 2-3.

Table 2-3

Values for Coefficient

No. of Stories in the Building

M M

1

(any) 1.0

2

Values of

c b

≤

(1,2,3)

Critical Stories

Other Stories

1.0

(n/a)

2.0

0.5

> 1.4

1.5

1.0 ≤

0.5

2.0

1  0.5

x2 n2

1  0.5

x2 n2

3-6 > 1.4

1.5

Linearly interpolate between the values

≤ 1.0

for 6 and 9 stories Linearly interpolate between the values

7-8 > 1.4 1.0

≥9

≤

for 6 and 9 stories 2.5

> 1.4

1.5 1.5

1.0

Notes: (1) Alternatively, coefficient  may be calculated as 1.2 times mean values linearly interpolated from the appropriate tables in Appendix A. (2) For intermediate values of ∑Mc/∑Mb,  may be calculated using linear interpolation. (3) x is the story under consideration; n is the total number of stories.

2.6.1.4


c

Coefficient c amplifies story drifts in stories that are determined to be critical. If a story is a critical story, thenc is calculated as:

DCR( max ) c

 DCR( avg )

(2-23)

where:

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2-21

DCR(max) = maximum value of DCRx over the height of the building; and DCR(avg) = average value of DCRx over the height of the building. If a story is not a critical story, thenc = 1.0. Note that buttressing effects caused by adjacent buildings with inadequate separation between buildings can be examined by designating the story immediately above the shorter building as a critical story. 2.6.1.5


s

Coefficient s amplifies story drifts in stories with non-ductile lap splices in longitudinal column reinforcement. Non-ductile lap splices are splices that: (1) do not meet the minimum length requirements in ASCE/SEI 41-13; and (2) are not confined over their total length with ties meeting the requirements of ACI 318-11 Section 7.10.5 at a spacing of no more than one third the effective depth of the column. If a story is a critical story, and ∑Mc/∑Mb ≥ 1.2, then s equals 1.3. If ∑Mc/∑Mb < 1.2, or if the story is not a critical story, thens = 1.0. 2.6.2

Adjusted Column Drift Demand

The adjusted column drift demand,D, is determined for each column in a story as a function of the column drift demand, modified considering torsional amplification and a factor that relates story drift to column rotation: D  AT Ccol col

(2-24)

where: T

= torsional amplification factor (Section 2.6.3);

Ccol = column drift factor (Section 2.6.4); and col 2.6.2.1

= the column drift demand (Section 2.6.5). Critical Components

Determination of the column drift factor,Ccol, and the column drift demand, δcol, requires designation of the critical component at each column location.

The critical component is also used to determine the column drift capacity, C, in Section 2.6.6.

The critical component is determined as follows:



If a column frames into a beam-column joint, the critical component at this location is the column.



If a column frames into a slab-column joint, the critical component at this location may bethe column or the slab-column connection. The

2-22


ATC-78-3

critical component at slab-column connections is determined using the requirements for continuity reinforcement in ASCE/SEI 41-13 Table 10-15 as follows: If the area of effectively continuous main bottom bars passing

o

through the column cage in each direction is greater than or equal to 0.5Vg/fye, where Vg is the gravity shear computed using Equation 2-1 acting on the critical section defined in ACI 318-11, then vertical failure in the slab-column connection is not anticipated, and the column is the critical component. Otherwise, the slab-column connection is the critical component.

o

2.6.3


The torsional amplification factor,AT, is calculated as the product of real (inherent) and accidental torsional amplification: AT  ATr ATa

(2-25)

where: r T

= real (inherent) torsional amplification factor; and

a T

= accidental torsional amplification factor.

In a given story, the torsional amplification factor,AT, is calculated for each column, and is calculated independently in each direction of earthquake loading. Torsional amplification factors are afunction of the torsional strength ratio (Section 2.6.3.1), which determines the maximum torsional amplification factor (Section2.6.3.2). The maximum torsional amplification factor is used to compute the accidental torsional amplification factor (Section 2.6.3.3) and the real torsional amplification factor (Section 2.6.3.4) for each column at each story. 2.6.3.1

Torsional Strength Ratio

TSR, in each Torsional amplification depends on the torsional strength ratio, story, which is defined as the sum of the torsional demand and torsional capacity normalized bythe torsional capacity, as follows:

TSRx 

TDx  TCx TCx

(2-26)

where TDx and TCx are determined at each story in accordance with this section. Because torsional irregularities in onestory can result in torsional response in all stories, the largest value of the torsional strength ratio calculated in any one story isused for all stories. If the largest value of the

ATC-78-3


2-23

torsional strength ratio,TSR, exceeds 3.0, the structure is assigned a building rating, BR, of 0.7 without further evaluation, and should be considered an exceptionally high seismic risk building. Torsional demand, TDx, is directional, and must be calculated for each direction of earthquake loading, including both real (inherent) and accidental eccentricities:

TDx V epx (e r 

where: Vpx

er

a

(2-27)

)

= plastic shear capacity of story x; = real (inherent) eccentricity between the center of mass and the center of strength in the direction perpendicular to the direction of earthquake loading in the story under consideration (as shown in Figure 2-5);

ea

= accidental eccentricity defined by an offset in the location of the center of mass equal to 5 percent of the overall plan dimension perpendicular to the direction of earthquake loading (i.e., 0.05L); and

L

= overall plan dimension perpendicular to the direction of earthquake loading.

Note that, for the purpose of computing TDx, the real and accidental torsional moments are assumed to actin the same direction. The center of rotation is taken as the center of strength. The coordinates of thecenter of strength,

 x, y  , are calculated in each direction as the summation of the products of column location and column plastic shear capacity, divided by the summation of the column plastic shear capacities:

x

x V V j

pcj

, y

pcj

y V V j

pcj

(2-28)

pcj

Torsional capacity, TCx, is constant regardless of the direction of loading, and is calculated considering the capacity of all frame lines regardless of orientation: nf

TCx 

R V

fi pfi

(2-29)

i 1

where: Rf

= orthogonal distance between the frame line f and the center of strength (as shown in Figure 2-5);

2-24


ATC-78-3

Vpfi

= plastic capacity of frame line f, taken as the sum of the plastic shear capacities, Vpc, of all columns on frame line f (see Section 2.4.4); and

nf

= number of frame lines in story x (regardless of orientation).

Figure 2-5

2.6.3.2

Example building plan showing locations of the center of mass, center of strength, and frame lines, and illustrating the orthogonal dimensions for earthquake loading in the northsouth direction. Maximum Torsional Amplification

The maximum torsional amplification factor, AT,max, is computed for each story based on the story drift ratio,

δx/hx,

in that story, and is calculated using

the maximum value of the torsional strength ratio, TSR, over the height of the building: for δx/hx ≤ 0.0085, for δx/hx ≥ 0.0115,

AT ,max AT ,max





0.272  1.543TSR TSR

0.317  1.874



0.75

(2-30)



0.65

(2-31)

where δx is the drift and hx is the height at story x. For intermediate values of δx/hx,

the maximum total torsional amplification factor is calculated by linear

interpolation between Equation 2-30 and Equation 2-31.

ATC-78-3


2-25

The maximum torsional amplification factor, AT,max, is decomposed into accidental and real (or inherent) torsional amplification factors in the sections that follow, and the resulting values are used to calculate the torsional amplification factor, AT. 2.6.3.3

Accidental Torsional Amplification

For each story, the maximum torsional amplification associated with accidental torsion is calculated as:

 TSRa

a

AT ,max  max  TSR AT ,max1.0 ,

 

(2-32)

where TSRa is the torsional strength ratio from accidental torsion:

TSRa 

TDa  TCx TCx

(2-33)

TDa is the torsional demand from accidental eccentricity: TDa  V px ea

(2-34)

and Tcx, Vpx, and ea are as previously defined. For the purpose of defining torsional amplification factors, the building is considered to have a “weak/flexible” side and a “strong/stiff” side, determined by the location of the center of strength shown in Figure 2-5. The “weak/flexible” side has the longer dimension between the center of strength and the edge of the building parallel to the direction of earthquake loading (L1 in Figure 2-5), and the “strong/stiff” side has the shorter dimension (L2 in Figure 2-5). Using AaT, max for the story under consideration, the accidental torsional amplification factor for each column in that story is calculated as:



 x    1  L1  

ATa  abs  ATa ,max  1 



(2-35)

where:

x

= distance between the center of strength and the column, measured perpendicular to the direction of earthquake loading, taken as positive if the column is on the “weak/flexible” side of the building and negative if on the “strong/stiff” side; and

L1

= longer dimension between the edge of the building and the center of strength, measured perpendicular to the direction of earthquake loading.

2-26


ATC-78-3

Accidental torsional can act on either side of the calculated center of mass. The equation for the accidental torsional amplification factor ensures that values are greater than or equal to one at all column locations. 2.6.3.4

Real Torsional Amplification

For each story, the maximum torsional amplification associated with real (inherent) torsion is calculated as:

ATr ,max 

AT ,max ATa,max

(2-36)

where AT, max and AaT, max are as previously computed. Using ArT, max for the story under consideration, the real torsional amplification factor for each column in that story is calculated as:

x  1  L1 

ATr   ATr ,max  1 

(2-37)

where all terms are as previously defined. Real (or inherent) torsion acts in the direction of the calculated eccentricity. The sign of x is used to ensure that a drift reduction will occur on the “strong/stiff” side of the building. As a result, the real torsional amplification factor will be less than 1.0 at column locations on the “strong/stiff” side of the building, and greater than 1.0 at column locations on the “weak/flexible” side of the building. 2.6.4

Column Drift Factor

The column drift factor, Ccol, defines the portion of the story drift demand, x, that is taken by an individual column, and is calculated in each direction of earthquake loading. The column drift factor depends on the critical component at the beam-column (or slab-column) joint (as determined in Section 2.6.2.1), and the story ratio of column strengths to beam strengths, ∑Mc/∑Mb, in the story under consideration. Where the column is the critical component, the column drift factor, Ccol, is taken from Table 2-4. Where the slab-column connection is the critical component, the column drift factor, Ccol, is taken as 1.0. 2.6.5

Column Drift Demand

Where the column is the critical component, the column drift demand, δcol, is taken as the story drift demand:

 col  x

ATC-78-3


(2-38)

2-27

Where the slab-column connection is the critical component, the column drift demand, δcol, is taken as half of the story drift demand in the story above the critical component, plus half of the story drift demand in the story below the critical component:  col  0.5 x (

Table 2-4

1)

 0.5 x 

(2-39)

Column Drift Factor when the Column is the Critical Component (1)

Ratio of Column Strengths to Beam Strengths

Column Drift Factor

∑Mc/∑Mb

Ccol

0.6

0.83

≤

0.8

0.77

1

0.71

1.2 1.4

0.66 0.60

≥

Notes: (1) For intermediate values of ∑Mc/∑Mb, the column drift factor, Ccol may be calculated by linear interpolation.

2.6.6

Column Drift Capacity

Where the column is the critical component, column drift capacity, c, in a story is calculated based on the column plastic rotation capacity, θc:

C

M c ,avg      2linf  C  3Ec I e     linf  

(2-40)

where:

linf

= clear height of an equivalent cantilever column from the face of a joint to the point of inflection (or zero moment);

θc

= column plastic rotation capacity;

Mc, avg = average expected flexural strength at the top and bottom of the column (i.e., average of McTx and McBx); Ec

= modulus of elasticity of concrete, taken as 57,000

f ce (psi);

and

Ie

= effective moment of inertia of the column, as a fraction of the gross moment of inertia, Ig (from Table 2-7).

Table 2-5 classifies a column as condition i, ii, or iii based on the ratio of flexural to shear strength and detailing of transverse reinforcement. For the purpose of classifying columns as condition i, ii, or iii, the presence of lap splice deficiencies in the longitudinal reinforcement should be ignored.

2-28


ATC-78-3

Table 2-5

Column Classifications (adapted from Li et al., 2014)

Flexural Strength to Shear Strength Ratio VpM/Vn (1)

Transverse Reinforcement Detail Closed Hoops with 90° Hooks

Details with 135° Hooks

Lap-Spliced or Other Reinforcement

(VpM/Vn) < 0.6

i (2) ii

0.6 ≤ (VpM/Vn) ≤ 1.1

ii

ii

iii

(VpM/Vn) > 1.1

iii

iii

iii

ii

Notes: (1) VpM is the shear corresponding to the expected flexural capacity of the column; Vn is the shear strength of the column. (2) If the column Av/bws ≥ 0.002, and s/d ≤ 0.5 within the flexural plastic hinge region, then the column is condition i; otherwise, the column is condition ii; where Av is the area of shear reinforcement, bw is the width, s is the spacing of transverse reinforcement, and d is the column depth.

Table 2-6 defines column plastic rotation capacity, θc, based on column transverse reinforcement condition i, ii, or iii, axial load ratio, P/Agf′ce, and shear reinforcement ratio. The axial load ratio is based on the total column axial load, P = Pg + Peq, where P is positive in compression, and Pg and Peq are determined in accordance with Section 2.3.6.

Table 2-6

Column Plastic Rotation Capacity (1) Axial Load Ratio P/Agf’ce

Condition

i

ii

iii

≤ 0.1 ≥ 0.6

Shear Reinforcement Ratio Av/bws 0.006 ≥ 0.006 ≥

Plastic Rotation Capacity c

0.090 0.030

0.1

≤

= 0.002

0.050

0.6

≥

= 0.002

0.018

≤ 0.1

0.006 ≥

0.082

≥ 0.6

0.006 ≥

0.023

≤ 0.1

0.0005≤

0.025

≥ 0.6

0.0005≤

0.011

≤ 0.1

0.006 ≥

0.075

≥ 0.6

0.006 ≥

0.020

≤ 0.1

0.0005≤

0.016

≥ 0.6

0.0005≤

0.006

Notes: (1) For intermediate values of axial load and shear reinforcement ratio, plastic rotation capacity may be calculated using linear interpolation.

ATC-78-3


2-29

Table 2-7

Effective Moment of Inertia (adapted from ASCE, 2013) (1) Effective Moment of Inertia Ie

Component Columns with compression due to expected gravity loads ≥ 0.5Agf′ce

0.7Ig

Columns with compression due to expected gravity loads ≤ 0.1Agf′ce, or with tension

0.3Ig

Notes: (1) For intermediate values of gravity load, the effective moment of inertia may be calculated using linear interpolation, otherwise, the larger value shall be used.

2.6.6.1

Column Drift Capacity in Slab-Column Frames

In slab-column frames, where the slab-column connection is the critical component, the column drift capacity, c, in a story is taken as the product of the drift capacity ratio from Table 2-8, and the clear height of the column, lu.

Table 2-8

Column Drift Capacity Ratio when the Slab-Column Connection is the Critical Component (1) Gravity Shear Ratio (2) Vg/Vc

Drift Capacity Ratio

≤

0.1

0.045

≥

0.6

0.01

Notes: (1) For intermediate values of gravity load, the drift capacity ratio may be calculated using linear interpolation. (2) The gravity shear ratio is the unfactored gravity shear, Vg, divided by the theoretical punching shear strength, without moment transfer, Vc, determined in accordance with ACI 318-11 Section 11.11.2.1.

2.6.7

Column Drift Capacity-to-Demand Ratios

Column drift capacity-to-demand ratios, CDR, are used to determine column ratings. For each column at each story in each direction, the ratio of column drift capacity to drift demand is calculated as:

CDR  C / D

(2-41)

Note that column drift capacity and column drift demand must share consistent units (i.e., length) before the ratio can be calculated.

2.7

Step 5 – Determine Building Rating

A building rating represents the relative likelihood that a building will lose its ability to support vertical loads under the assumed earthquake loading. Building ratings are determined based on the maximum story rating over the height of the building, and story ratings are a function of the column ratings within a given story.

2-30


ATC-78-3

Column ratings for typical conditions are determined in accordance with Section 2.7.1. Column ratings associated with special column conditions are determined in accordance with Section 2.7.4 (for corner columns), Section 2.75 (for columns supporting discontinuous elements, and Section 2.7.6 (for columns supporting mezzanines). 2.7.1

Column Rating

The column rating, CR, is a number between 0.0 and 1.0, inclusive, representing the relative likelihood that an individual column will lose its ability to support vertical loads under the assumed earthquake loading. Column ratings near 0.0 indicate a low likelihood of failure, and column ratings near 1.0 indicate a high likelihood of failure. The column rating, CR, for each column is obtained from Table 2-9 based on the ratio of column drift capacity to drift demand, C/D. Column ratings are determined for each column in the building, at each story, and in each orthogonal direction. The final column rating for each column is the maximum column rating obtained in either orthogonal direction.

Table 2-9

2.7.2

Column Rating, CR

Drift Capacity to Drift Demand Ratio C/D

Column Rating CR

C/D ≥ 4.0

0.0

2.5 ≤ C/D < 4.0

0.1

1.8 ≤ C/D < 2.5

0.2

1.4 ≤ C/D < 1.8

0.3

1.1 ≤ C/D < 1.4

0.4

0.90 ≤ C/D < 1.1

0.5

0.72 ≤ C/D < 0.90

0.6

0.57 ≤ C/D < 0.72

0.7

0.41 ≤ C/D < 0.57

0.8

0.31 ≤ C/D < 0.41

0.9

C/D < 0.31

0.93

Story Rating

The story rating, SR, is a number between 0.0 and 1.0, inclusive, representing the relative likelihood that an individual story will lose its ability to support vertical loads under the assumed earthquake loading. Story ratings near 0.0

ATC-78-3


2-31

indicate a low likelihood of failure, and story ratings near 1.0 indicate a high likelihood of failure. The story rating, SR, for each story, is based on an adjusted average of the column ratings in the story, CRavg, defined as:

CRavg 

1

nc

nc

 CR

C a R

j

(2-42)

j 1

where CRj is the column rating for each column j, nc is the total number of columns in the story, and CRa is an adjustment based on the range of column ratings in the story. The range of column ratings is equal to the difference between the maximum and minimum values of CR (i.e., CR max – CRmin) in the story. The average column rating adjustment, CRa, is taken from Table 2-10. The resulting story rating, SR, is taken from Table 2-11.

Table 2-10

Average Column Rating Adjustment, CRa (1)

Range of Column Ratings

Average Column Rating Adjustment CRa

CRmax – CR min 0.40

≤

0.80

≥

0 0.15

Notes: (1) For intermediate values of CRmax – CR min, the average column rating adjustment, CRa may be calculated using linear interpolation.

Table 2-11

Story Rating, SR Story Rating SR

Adjusted Average Column Rating CRavg < 0.06CRavg

0.0

0.06 < 0.18 ≤ CRavg

0.1

0.18 < 0.25 ≤ CRavg

0.2

0.25 < 0.31 ≤ CRavg

0.3

0.31 < 0.37 ≤ CRavg

0.4

0.37 < 0.43 ≤ CRavg

0.5

0.43 < 0.49 ≤ CRavg

0.6

0.49 < 0.56 ≤ CRavg

0.7

0.56 < 0.66 ≤ CRavg 0.87 0.66 ≤ CRavg < 0.87

2-32

CRavg ≥


0.8 0.9 1.0

ATC-78-3

2.7.3

Building Rating

The building rating, BR, is a number between 0.0 and 1.0, inclusive, representing the relative likelihood that the building will lose its ability to support vertical loads under the assumed earthquake loading. The building rating, BR, is taken as the maximum story rating, SR, determined for any story over the height of the building. Building ratings can be used to rank the relative risk of collapse among buildings in an inventory of buildings. The portion of the inventory with higher relative ratings should be designated as exceptionally high seismic risk buildings, however, a value defining the threshold for unacceptable risk of collapse is a policy-level decision and has not yet been established. 2.7.4

Evaluation of Corner Column and Corner Joint Conditions

Corner column conditions can result in a change in the column rating. This section shall apply to corner columns where the following two conditions apply:



the joint shear strength is the controlling component at the beam-column joint; and



the story drift ratio exceeds 0.025.

The controlling component at the beam-column joint is determined by the relative strength of the joint and the components framing into the joint. Corner column locations where the joint is the controlling component must be evaluated in accordance with this section. If a corner beam-column joint has sufficient strength to develop the strength of the columns and beams framing into the joint, the joint is not controlling, the column is exempt from the provisions of this section, and the column rating is determined in accordance with Section 2.7.1. Corner column locations with story drift ratios exceeding 0.025 must be evaluated in accordance with this section. Corner columns with story drift ratios less than or equal to 0.025 are exempt from the provisions of this section, and the column rating is determined in accordance with Section 2.7.1. Where required, corner columns are evaluated for axial load, P, due to combined gravity and earthquake loading determined in accordance with this section. The overturning moment at the base of the building, MOT, is determined in each direction of earthquake loading as:

M OT  Vy heff

ATC-78-3


(2-43)

2-33

where Vy is the effective yield strength of the structure, and heff is the height of an equivalent single-degree-of-freedom system (0.7hn in multistory buildings, and hn in single-story buildings). Individual frames providing resistance to MOT in each orthogonal direction shall be assigned a portion of the overturning moment, MOTi, in accordance with principles of structural mechanics. In most cases, the proportion of system overturning resistance assigned to an individual frame can be taken equal to the proportion of the system base shear resistance assigned to that frame. The axial force on corner columns, as required to resist overturning moment plus gravity loads, shall be determined in each direction of earthquake loading, in accordance with principles of mechanics. For buildings with uniform plan and elevation, the axial force due to overturning can be determined as:

POT 

M

OTi

  Mc



L

(2-44)

where:

MOTi = is the overturning resistance provided by an individual frame of which the column is a part;

Mc = summation of the expected flexural strengths of all columns in the frame, calculated at the base of the frame, considering axial force due to gravity loads alone; and

L

= plan dimension between the outermost columns in the frame in the direction under consideration.

Total axial load, P, shall be determined as the sum of axial forces due to gravity loads and overturning moment acting in both orthogonal directions:

P = Pg + POT, x + POT, y

(2-45)

where Pg is determined in accordance with Equation 2-1. Note that the axial earthquake force computed in this section for evaluation of corner columns is different from the axial earthquake force computed in Section 2.3.6.2 for evaluation of typical columns. If the total axial load, P, on a corner column exceeds the value from Equation 2-46:



x 



hx 

0.3 Ag f ce  11 0



(2-46)

where all terms are as previously defined, the column shall be assigned a column rating, CR, of 0.8. Corner columns with total axial load less than or

2-34


ATC-78-3

equal to the value from Equation 2-46 shall have their column rating determined in accordance with Section 2.7.1. 2.7.5

Evaluation of Discontinuous Column Conditions

Columns shall be considered discontinuous at a level where there is no coincidental column in the level below. Discontinuous column conditions can result in a change in the column rating. In the case of discontinuous columns supporting loads from two or more levels, and not supported on cantilever beams or slabs:  The plastic shear capacity of the column, Vpc, for the purposes of determining the story shear capacity shall be calculated in accordance with Section 2.4.5.4 (i.e., for first story columns).



If the plastic capacity of the supporting beam or slab is controlled by shear, the discontinuous column shall be assigned a column rating of 0.8.



If the plastic capacity of the supporting beam or slab is controlled by flexure, no special consideration is necessary.

In the case of discontinuous columns supported on cantilever beams or slabs:



The plastic shear capacity of the column, Vpc, for the purposes of determining the story shear capacity shall be calculated in accordance with Section 2.4.5.4 (i.e., for first story columns).



If the demand-to-capacity ratio of the cantilever element for expected vertical loads is greater than 0.7, the discontinuous column shall be assigned a column rating of 0.8. Expected vertical loads shall be as defined in Section 2.3.6, considering axial loads due to overturning in addition to gravity loads.

2.7.6

Evaluation of Mezzanine Column Conditions

Columns connected to and supporting mezzanines are termed engaged columns. Columns in full stories with mezzanines that are not connected to or supporting the mezzanine are termed non-engaged columns. The presence of mezzanines can result in a change in the column ratings and story ratings for the building. Mezzanine beam-column connections at engaged columns must be evaluated for moment resistance. Mezzanine moment resistance can be considered insignificant if the shear in the column generated by the expected flexural strength of the mezzanine is less than 20% of the shear strength of an equivalent full height column determined in accordance with Section 2.3.7.2.

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2-35



If the mezzanine moment resistance is not significant, the mass of the mezzanine shall be added to the floor above. The plan location of the mezzanine shall be considered in locating the center of mass for the purposes of torsional amplification. No further consideration of the mezzanine is necessary for this condition.



If the mezzanine moment resistance is significant, the mezzanine shall be considered a floor with story xm+ between the mezzanine and the floor above, and story xm- between the mezzanine and the floor below.

The plastic shear strength of story xm+ shall be taken as the sum of column plastic shear capacities of engaged columns between the mezzanine and the floor above, plus the sum of the plastic story capacities of the non-engaged columns calculated over the height xm+ + xm-. The plastic shear strength of story xm- shall be taken as the sum of column shear capacities of engaged columns between the mezzanine and the floor below, plus the sum of the plastic story capacities of the non-engaged columns calculated over the height xm+ + xm-. If story xm+ or story xm- is a critical story, that story shall be assigned a coefficient, , in accordance with Section 2.6.1.3, and the other story shall be assigned  = 1.0. Column ratings shall be calculated for the engaged columns both above and below the mezzanine, and for non-engaged columns over the height xm+ + xm-. The story rating for story xm+ shall consider all the engaged columns above the mezzanine, plus the non-engaged columns extending above and below the mezzanine level. The story rating for story xm- shall consider all the engaged columns below the mezzanine plus the non-engaged columns extending above and below the mezzanine level.

2-36


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Chapter 3

Development of Procedures for Estimating Drift Demands This chapter presents the results of studies used in the development of procedures for estimating story drift demands and drift capacities in concrete buildings. The resulting procedures are based on structural dynamic theories and principles, empirical observations from dynamic analyses of singledegree-of-freedom and multi-degree-of-freedom building prototypes, and engineering judgment.

3.1

Overview of the Procedure to Estimate Drift Demands

In this evaluation methodology, the collapse potential of an existing reinforced concrete building is estimated by comparison of drift demands and drift capacities of critical structural components. The emphasis is on story drift demand, because story drift can most readily be related to collapse potential of critical moment frame buildings and their structural components. The procedure to estimate drift demands is summarized as follows: 1. Estimate the effective fundamental period of the building, Te, along its principal framing directions. 2. Determine the displacement of an equivalent single degree-of-freedom system, eff, at an elevation, heff, corresponding to the effective height of the fundamental vibration mode of the building, using an acceleration response spectrum determined in accordance with ASCE/SEI 7-10, Minimum Design Loads for Buildings and Other Structures (ASCE, 2010) and USGS mapped spectral response acceleration parameters (S S, S1) adjusted for site class at the latitude and longitude of the building (USGS, 2008). 3. Given the drift demand at the effective height and the configuration of the building, estimate the distribution of story drift demands, δx, over the building height. The story drift demands are then compared with drift capacities of critical stories to estimate the collapse risk of the building. The methods used to accomplish each of these steps are described in the sections that follow.

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3: Development of Procedures for Estimating Drift Demands

3-1

3.2

Estimation of the Effective Fundamental Period of the Building

This section presents four methods for estimating the fundamental period of a building. 3.2.1

Empirical Method

Goel and Chopra (1997) reported measured values of building periods, T, for concrete frame buildings as a function of building height. In linear analysis procedures, such as in ASCE/SEI 41-13, Seismic Evaluation and Retrofit of

Existing Buildings (ASCE, 2013), these data have been conservatively modeled (as a shorter period in order to result in a greater base shear) by an empirical equation as a function of the building height (Equation 7-18 in ASCE/SEI 41-13). In this approach, member size, details of reinforcement, and building and component nonlinear characteristics are not considered. 3.2.2

Eigenvalue Analysis of a Mathematical Model

Eigenvalue analysis results in determination of the analytically derived linear-elastic period, T, which considers member stiffness based on gross or cracked section properties and a distribution of mass in the building. A mathematical structural model is developed (usually in commercial software) and eigenvalues and mode shapes are determined. The fundamental mode of vibration corresponds to the fundamental period of the building in each orthogonal direction. In this approach, details of reinforcement and building and component nonlinear characteristics are not considered. 3.2.3

Nonlinear Static Procedure

In nonlinear static (pushover) analysis procedures, such as in ASCE/SEI 41-13, the fundamental period, T, is calculated by an eigenvalue analysis, but the calculated period is increased to an effective period, Te, which considers the progression of yielding that occurs in the pushover curve for the structure. This approach considers element size, details of reinforcement, and nonlinear behavior, however, it requires a nonlinear pushover analysis to implement. As a result, it is not considered a rapid procedure, and has not been incorporated into the evaluation methodology. 3.2.4

Yield Drift Method

In the yield drift method, a value of yield drift is determined for the structure. This value is based on force-displacement cyclic component tests, past experience with pushover analyses of existing concrete frame buildings, and engineering judgment. The yield drift period utilized in this evaluation

3-2


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methodology has been conservatively estimated to result in larger values of story drift. For the yield drift method, the effective fundamental period, Te, is determined using the effective stiffness, Ke, as shown in Figure 3-1.

r a e h s e s a B

t n e m e c a l p s i d

Roof displacement

Figure 3-1

Force-displacement curve of a structure, showing the definition of effective stiffness, Ke, for calculation of the effective fundamental period (FEMA, 2000).

The effective stiffness, Ke is determined from the effective structure yield strength, Vy, and the yield displacement, δy, as follows:

Ke  V y

y

(3-1)

In this evaluation methodology, the effective structural yield strength, Vy, is determined in accordance with Equation 2-15. The value of yield drift has been taken as 0.0075(2/3)hn, where 0.0075 is the assumed yield strain, hn is the height from the base of the building to the highest level of the seismic force-resisting system, and 2/3 convert the building height to the effective height, heff, of an equivalent single-degree-of-freedom system. As a result, the effective fundamental period as a function of building height and strength is derived as follows:

Te  2 W M g

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M Ke

(3-2)

(3-3)


3-3

Vy

Ke 

 2  

 0.0075    hn 3 Te  0.078

(3-4) 12

hnW

(3-5)

Vy

where:

Te M hn

= effective fundamental building period; = building seismic mass; = height of the building from the base to the highest level of the seismic force-resisting system, in feet;

Ke

= effective building stiffness;

W

= building seismic weight, in kips; and

Vy

= effective structural yield strength, in kips, as defined in Equation 2-15.

To compare the results of the yield drift period calculation with the empirical data in Goel and Chopra (1997), the building characteristics for which the periods were measured in the empirical database are needed. If the initial stiffness of a building during the period measurement is assumed to be onethird of the stiffness corresponding to 0.0075(2/3)hn, which corresponds to a yield drift of 0.0025(2/3)hn, then:

Te  at 0.25%   0.045 hnW Vy

(3-6)

The strength of the buildings measured by Goel and Chopra can be estimated from the requirements of the building codes in the 1950s and 1960s. Building codes of this era required working stress base shears of approximately 0.05W/ T . For frames, T was taken as 0.1 n, where n is the number of stories. For simplicity, assuming a story height of 10 feet, T = 0.1hn/10 = 0.01hn. Using a conversion factor of 2.5 for working stress to yield strength, Vy/W = 0.05(2.5)/(0.1 hn ) = 1.25/ hn . Substituting this value of Vy/W into Equation 3-6, yields the following relationship between effective period and building height:

Te  0.041hn0.75

(3-7)

Similarly, when a structure is subjected to large seismic demands, and the yield drift is 0.0075(2/3)hn, the effective period is:

Te  0.058hn0.75

3-4


(3-8)

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Figure 3-2 compares the empirical data with plots of period relationships based on different values of assumed yield strain (Equations 3-7 and 3-8), as well as the empirical equation for period in ASCE/SEI 41.

Figure 3-2

Comparison between empirical data and plots of period relationships based on different values of assumed yield strain.

The plot of Equation 3-8, based on an effective yield strain of 0.0075, can be observed to follow the empirical trend line, but is located above the empirical data, which indicates that the equation is predicting longer than actual periods. Prediction of longer periods is considered appropriate given the nonlinearity implied by Vy and the need to be conservative when estimating displacements for predicting collapse.

3.3

Determination of Displacement at the Effective Modal Height

3.3.1

Basic Procedure

The displacement at the effective modal height is determined using a procedure mirroring the target displacement approach of ASCE/SEI 41-13, calculated as:  eff  C0 C1C2 S a

Te2 4 2

g

(3-9)

where:

Sa

= is the response spectrum acceleration at the effective fundamental period, Te; and

g

= is the acceleration of gravity.

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3-5

C0, C1, and C2 are coefficients used to modify displacement considering several effects, as defined below. In Equation 3-9, the quantity Sa(Te2/4π2)g is the spectral displacement, Sd, at period Te. Coefficient C0 is a modification factor to relate spectral displacement of an equivalent single-degree-of-freedom (SDOF) system to the displacement at the effective modal height of the building multi-degree-of-freedom (MDOF) system. Values of C0 were found to range between 0.9 and 1.1 for typical buildings. Given the small variability of C0, this term was set equal to 1.0, and dropped from the equation used in this evaluation methodology. Coefficient C1 is a modification factor to relate the expected maximum inelastic displacements to displacements calculated for linear elastic response. In ASCE/SEI 41-13, Coefficient C1 is calculated as:

C1  1 

 strength  1

(3-10)

aTe2

where a is a site class factor with values specified in ASCE/SEI 41-13, Te is the effective fundamental period, and μstrength is calculated using the following equation:  strength 

Sa

 Vy    W 

Cm

(3-11)

where:

Sa

= spectral acceleration at the effective fundamental period;

Vy

= yield strength (in this methodology, the effective yield strength in accordance with Equation 2-15);

W

= total effective seismic weight of the building; and

Cm

= effective mass factor.

For Te < 0.2 seconds, C1 need not be greater than the value at Te = 0.2 seconds. For Te ≥ 1 seconds, C1 = 1.0. Coefficient C2 is a modification factor to represent the effect of a pinched hysteresis shape, cyclic stiffness degradation, and strength deterioration on maximum displacement response, calculated as: 2

strength  1  C2  1  1   800  Te 

3-6


(3-12)

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where all terms are as previously defined. For effective fundamental periods greater than 0.7 seconds, C2 =1.0. 3.3.2

Comparison with Results from Nonlinear Response History Analyses

Equation 3-9 is intended to provide an estimate of the maximum displacement at the effective modal height, given a spectral acceleration value that represents the seismic hazard and the modal characteristics of the building. To investigate the accuracy of Equation 3-9, incremental dynamic analyses were carried out on a series of idealized 4-, 6-, 8-, and 12-story frames considered to be representative of existing buildings. The frames included columns controlled by inelastic flexure, or shear and axial failure modes. The frames were designed to have resistance in proportion with demands assuming an inverted triangular lateral load pattern, except that weak stories were introduced into some of the frames for the purpose of this study. These analyses confirmed that cyclic degradation did not appreciably affect the predictive accuracy of Equation 3-9 for buildings with periods exceeding 1.0 second. Furthermore, Equation 3-9 was shown to be more accurate for buildings in which the damage is distributed over the height of the building, and less accurate for buildings with weak stories that lead to concentrated damage. Table 3-1 presents the effective period, Te, and the coefficients C0, C1, and C2 used in Equation 3-9 for the idealized frame buildings that were studied.

Table 3-1

Parameters of the Idealized Frame Buildings Studied Building Height

Parameter (sec) Te

4-Story 1.14

6-Story 1.38

8-Story 1.62

12-Story 1.95

C0

1.09

1.05

1.11

0.94

C1

1.00

1.00

1.00

1.00

C2

1.00

1.00

1.00

1.00

Table 3-2 provides a sample of results presented in the form of average ratios of displacement at the effective modal height taken from nonlinear analysis results versus the displacement calculated from Equation 3-9. Results for the 6-story frame building are shown. Additional results for the 4-, 8-, and 12story frame buildings are provided in Tables A-1 through A-3 of Appendix A.

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3-7

In the tables, ∑Mc/∑Mb is the ratio of the sum of column flexural strengths to the sum of beam flexural strengths at each beam-column joint, and Vpc/Vn is the ratio of column plastic shear capacity (corresponding to development of the column flexural strength at both ends of the column) divided by the column shear strength.

Table 3-2

Average Ratios of Displacement at the Effective Modal Height (Nonlinear Analysis Results versus Equation 3-9) for a 6-Story Idealized Frame Building Average Ratios of Displacement

ΣΜc ΣΜb

/

Vpc/Vn = 0.6

Vpc/Vn = 0.8

Vpc/Vn = 1.0

Vpc/Vn = 1.2

0.6

0.79

0.81

0.90

1.00

0.8

0.79

0.81

0.89

1.00

1.0

0.79

0.81

0.88

0.98

1.2

0.85

0.80

0.81

0.91

1.4

1.03

0.90

0.85

0.83

1.6

1.18

1.03

0.94

0.87

1.8

1.30

1.15

1.02

0.95

Also of interest was the accuracy of displacement estimates, as affected by the lateral strength of the building. This could have been investigated by designing several buildings with different lateral strengths and comparing the resulting analytical and calculated displacements. In this study, however, an alternate approach was used in which idealized frame buildings were subjected to ground motions scaled to progressively increasing intensities, and the displacements obtained from nonlinear analysis for each scaling factor were compared with the displacements calculated using Equation 3-9. This process was repeated for each of the 44 ground motions (22 pairs) in the far-field record set of FEMA P-695, Quantification of Building Seismic Performance Factors (FEMA, 2009). The results are presented as a function of the effective demand-to-capacity ratio, μstrength, defined as:  strength 

S aW Vy

(3-13)

where Vy is the base shear strength calculated from nonlinear static analysis under lateral loads (distributed in an inverted triangular pattern). Figure 3-3 presents a sample of the results. Each plot shows the ratio of displacement at the effective modal height taken from nonlinear analysis results to the displacement calculated from Equation 3-9 (on the vertical axis), for different values of μstrength (designated Rindex in the figure, on the horizontal axis). Each plotted point corresponds to a single ground motion at

3-8


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a particular scaling factor, and the dashed lines represent the mean and plus or minus one standard deviation of the results. In most cases, Equation 3-9 is shown to provide a conservative estimate of the expected maximum displacement at the effective modal height. The accuracy of the displacement estimation varies as the ratio of column to beam strengths changes.

Figure 3-3

3.4

Ratios of displacement at the effective modal height (nonlinear analysis results versus Equation 3-9) for different values of μstrength (denoted Rindex in the figure) and different column to beam strength ratios.

Estimation of Story Drift

Story drift demand, x, for each story x is determined as:  x  x cx

sx

  eff  heff 

hx 

  

(3-14)

where: x

= coefficient to modify story drifts at story x for building configuration and strength characteristics;

cx

= coefficient at story x to amplify story drifts in critical stories (i.e., stories with high variation in strength over height);

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3-9

sx

= coefficient at story x to amplify story drifts in stories with nonductile lap splices in longitudinal column reinforcement;

hx

= height of story x;

eff

= drift of an equivalent SDOF system at the effective modal

heff

= effective modal height of an equivalent SDOF system; and

hn

= height from the base of the building to the highest level of the

height;

seismic force-resisting system. Values of heff were calculated for typical frame or frame-wall buildings, and a value of 0.7hn, where hn is the height from the base of the building to the highest level of the seismic force-resisting system, can be taken as a close approximation of the actual value of heff. Figure 3-4 illustrates several of the terms used to define story drift. In this figure, cx = sx = 1.0 at all stories. For a building with equal story heights, the story drift ratio x/hx in each story is equal to the effective drift ratio eff/heff (see Figure 3-4c), and in this idealized case, x = 1. For a building with a weak story (critical story) drift pattern, in which all of the drift occurs in one story (e.g., the first story in Figure 3-4d), the story drift ratio x/hx of each story is zero except in the critical story. Thus, for the idealized weakstory case: 1 

1 h1  eff heff



heff

(3-15)

h1

for the first story, and x = 0 for all other stories.

Figure 3-4

Illustration of a building frame, equivalent single-degree-offreedom oscillator, and two idealized story drift patterns.

This examples illustrates that values of x can be determined through judgment for building configurations in which the deformed shape can be determined by judgment. For more complicated conditions that are typical of

3-10


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real buildings, values of x must be determined through analysis of the actual building configuration. Results from analyses of several building configurations are presented in the following sections. 3.4.1

Development of Coefficient Configurations

for Uniform Building

The values of coefficient  were explored for a series of uniform planar (two-dimensional) frame configurations. These 4-, 6-, 8-, and 12-story frames had nearly equal story heights in each story, and had design strengths closely matching the demands associated with a load combination of: (a) gravity load; and (b) lateral earthquake force equal to 0.1W in an inverted triangular load pattern. The frames were subjected to the 44 ground motions in the FEMA P-695 far-field record set with gradually increasing scaling factors until collapse occurred. Collapse was defined using two failure modes: (a) sidesway collapse, when the maximum story drift ratio exceeded 10%; and (b) vertical collapse, when more than 50% of the columns in a story failed as a result of shear-induced axial load failure. For each ground motion, values of  were recorded for a scaling factor slightly less than the factor required to achieve collapse. The statistics of  were then determined for the 44 ground motions. Table 3-3 provides a sample of results showing mean values of coefficient  for the first story of a 6-story idealized frame. Figures 3-5 through 3-12 show  coefficient story profiles and plots of 1 versus ∑Mc/∑Mb for different ratios of Vpc/Vn for each of the 4-, 6-, 8-, and 12-story frames. Statistics (mean and standard deviation values) for coefficient  for each frame at each story are provided in Tables A-4 through A-11 of Appendix A.

Table 3-3

ΣMc/ΣMb

Mean Values of Coefficient in the First Story of a 6-story Idealized Frame without Significant Torsional Effects Mean Values of Coefficient Vpc/Vn = 0.6

Vpc/Vn = 0.8

Vpc/Vn = 1.0

Vpc/Vn = 1.2

0.8

2.81

2.19

1.91

1.67

1.2

2.31

1.66

1.68

1.56

1.8

1.06

0.97

0.97

1.01

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3-11

Vp/Vn=0.60

Vp/Vn=0.80

4

4

3

3 y r o t S

y r o t S

2

2

1

1 0.00

0.50

1.00

1.50

2.00

0.00

2.50

0.50

1.00

AlphaFactor

1.50

2.00

2.50

AlphaFactor

(a)

(b)

Vp/Vn=1.00

Vp/Vn=1.20

4 4

3 3 y r o t S

y r o t S

2

2

1

1 0.00

0.50

1.00

1.50

2.00

2.50

0.00

0.50

1.00

AlphaFactor

1.50

2.00

2.50

AlphaFactor

(c)

(d)

Story profiles of coefficient  for the 4-story idealized frame.

Figure 3-5

2.50

Vp/Vn=0.60 Vp/Vn=0.80 Vp/Vn=1.00

2.00

Vp/Vn=1.20 1.50

a1 1.00

0.50

0.00 0.6

0.8

1

1.2

1.4

1.6

1.8

ΣΜc/ΣΜb

Figure 3-6

3-12

Mean values of coefficient  in the first story of the 4-story idealized frame.


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Figure 3-7

Story profiles of coefficient  for the 6-story idealized frame.

Figure 3-8


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3-13

Vp/Vn=0.60

Vp/Vn=0.80

8

ΣΜc/ΣΜ b=0.60 ΣΜc/ΣΜ b=0.80

7


6


y r o t S

5

ΣΜc/ΣΜ b=1.80

4 3 2 1 0.00

0.50

1.50

2.00

2.50

(b)

Vp/Vn=1.00

8

y r o t S

1.00

Alpha Factor

(a)

Vp/Vn=1.20

8

7

7

6

6

5

y r o t S

4

5 4

3

3

2

2 1

1 0.00

0.50

1.00

1.50

2.00

0.00

2.50

0.50

1.00

1.50

2.00

2.50

Alpha Factor

Alpha Factor

(c)

(d)

Story profiles of coefficient  for the 8-story idealized frame.

Figure 3-9 3.50

Vp/Vn=0.60 Vp/Vn=0.80 Vp/Vn=1.00 Vp/Vn=1.20 Vp/Vn=0.60 Vp/Vn=0.80 Vp/Vn=1.00 Vp/Vn=1.20

3.00

2.50

2.00

a1 1.50

1.00

0.50

0.00 0.6

0.8

1

1.2

1.4

1.6

1.8

ΣΜc/ΣΜb

Figure 3-10

3-14



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Vp/Vn=0.60

y r o t S

Vp/Vn=0.80

ΣΜc/ ΣΜb=0.60

12

12

11

11

10

10

9

9

8

8

ΣΜc/ ΣΜb=1.60

7

ΣΜc/ ΣΜb=1.80

7

y r o t S

6

ΣΜc/ ΣΜb=0.80 ΣΜc/ ΣΜb=1.00 ΣΜc/ ΣΜb=1.20 ΣΜc/ ΣΜb=1.40

6

5

5

4

4

3

3

2

2 1

1 0.00

0.50

1.00

1.50

2.00

0.00

2.50

0.50

1.00

1.50

(a)

2.50

(b)

Vp/Vn=1.00

Vp/Vn=1.20

12

12

11

11

10

10

9

9

8 y r o t S

2.00

Alpha Factor

Alpha Factor

8

7

y r o t S

6

7 6

5

5

4

4

3

3 2

2

1

1 0

0.5

1

1.5

2

0.00

2.5

0.50

1.50

2.00

2.50

(d)

(c)

Figure 3-11

1.00

Alpha Factor

Alpha Factor

Story profiles of coefficient  for the 12-story idealized frame.

3.00

Vp/Vn=0.60 Vp/Vn=0.80

2.50

Vp/Vn=1.00 Vp/Vn=1.20

2.00

a1

1.50

1.00

0.50

0.00 0.6

0.8

1

1.2

1.4

1.6

1.8

ΣΜc/ΣΜb

Figure 3-12

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3-15

In taller frames, some ground motions cause failures to occur in the first story while other different ground motions cause failure to occur in upper stories. Failures in the upper stories are likely the result of apparent highermode effects. Because failure can occur in one of several stories, the mean drift ratio for any one story can be dominated by collapse realizations in which collapse occurs in other stories. Thus, although coefficient  represents the mean value across all collapse realizations, it does not provide a good indication of the story drift for realizations in which collapse occurs in that particular story. To better understand this relationship, values of ′ were recalculated for each story considering only those cases in which the story under consideration was the story that collapsed. The statistics (mean and standard deviation values) of ′ calculated for each frame at each story are provided in Tables A-12 through A-19 of Appendix A. This evaluation methodology uses recommended values of  for buildings that have been categorized on the basis of the number of stories and the value of Mc/Mb. Because a single “averaged” value of  is used (rather than the best estimate value of  for the specific building under consideration), an additional bias enters into the results, and the standard deviation values for an individual building increase. 3.4.2

Development of Coefficient Configurations

for Non-Uniform Building

In the case of buildings with a characteristic weak story, story drifts are expected to concentrate in the weak story (identified as the critical story). To investigate this behavior, the 6-story idealized frame was modified to include a 50% increase in the strength of all the stories except the critical story. Two cases were explored. In the first case, the critical story was located in the first story, and in the second case, the critical story was located in the fourth story. For the frame with a critical first story, most of the inelastic deformation was concentrated in the first story for all combinations of variables (as shown in Table 3-4).

Table 3-4

ΣMc/ΣMb

3-16

Mean Values of Coefficient in the First Story of a 6-story Idealized Frame with a Critical First Story Mean Values of Coefficient Vpc/Vn = 0.6

Vpc/Vn = 0.8

Vpc/Vn = 1.0

Vpc/Vn = 1.2

0.8

2.56

2.04

1.91

1.93

1.2

2.16

1.53

1.55

1.69

1.8

1.75

1.61

1.46

1.34


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For the frame with a critical fourth story, most of the inelastic deformation was concentrated in the fourth story, however, for frames with ΣMc/ΣMb < 1.0 (weak columns), significant deformations also developed in the first story. Results are shown in Figures 3-13 through 3-15. The statistics (mean and standard deviation values) of  calculated for the 6-story idealized frame with a critical fourth story are provided in Table A-20 and Table A-21 of Appendix A.

Vp/Vn=0.60

Vp/Vn=0.80

(a)

(b)

Vp/Vn=1.20

Vp/Vn=1.00

(c)

Figure 3-13

ATC-78-3

(d)

Story profiles of coefficient  for the 6-story idealized frame with a critical fourth story.


3-17

∑Mc/∑Mb

Figure 3-14

Mean values of coefficient  in the first story of the 6-story idealized frame with a critical fourth story.

∑Mc/∑Mb

Figure 3-15

3.5

Mean values of coefficient  in the fourth story of the 6-story idealized frame with a critical fourth story.

Columns with Inadequate Lap Splice Conditions

In the case of buildings with inadequate lap-splices for column longitudinal reinforcement, including insufficient column confinement as defined by ACI 318-11, Building Code Requirements for Structural Concrete and

Commentary (ACI, 2011), bar slip and eventual failure of the lap splices are expected to occur. Associated damage is likely to result in rapid loss of moment resistance at the location of the lap splice.

3-18


ATC-78-3

To investigate this behavior, a strength-degrading plastic hinge was introduced into the building models at assumed lap splice locations. Inadequate lap splice conditions were modeled in accordance with experimental results provided by Melek and Wallace (2004), with the backbone rotational behavior shown in Figure 3-16.

Figure 3-16

Zero-length plastic hinge rotational behavior for adequate and inadequate lap splice conditions.

Nonlinear dynamic analyses were conducted on frame models with flexurecritical columns (Vpc/Vn=0.6) including lap splices. Frames with different ΣΜc/ΣΜb values, ranging from 0.8 to 1.6, were investigated. The resulting  factor profiles (with and without inadequate lap splices) are shown in Figures 3-17 through 3-19 for the case of lap splices introduced at the bottom of the first and fourth stories of the 8-story idealized frame. Results demonstrated that, for buildings with a tendency to form weak stories (i.e., ΣΜc/ΣΜb ≤ 1.20), drifts were most impacted by the presence of the weak story, and inadequate lap splices had minimal effect. On the other hand, for buildings with a tendency to form a more uniform drift profile (i.e., ΣΜc/ΣΜb > 1.20) inadequate lap splices magnified drifts in the stories where the lap splices occurred. The main observations are:



Very little drift amplification occurs when Vpc/Vn > 1.0. In this case, only limited yielding of the column longitudinal reinforcement occurs, and the lap splices seldom fail.



When Vpc/Vn ≤ 1.0, relatively little amplification occurs for for ΣMc/ΣMb ≤ 1.2. In this case, a weak story is already formed because of weakness in the columns, and further reduction in strength due to lap splice failure does not significantly exacerbate the story failure.



When Vpc/Vn ≤ 1.0, greater amplification occurs for ΣMc/ΣMb > 1.2. In this case, the presence of inadequate lap splices results in reduced

ATC-78-3


3-19

moment strength at one end of the column (if the column yields), which tends to produce a weak-story condition. However, because the columns are stronger than the beams, the extent of column yielding is limited, and the amplification factor is somewhat limited.

Figure 3-17

3-20

Story profiles of coefficient  (with and without inadequate lap splices) for the 8-story idealized frame, for Vpc/Vn=0.6 and Mc/Mb≤1.20.


ATC-78-3

Figure 3-18

ATC-78-3

Story profiles of coefficient  (with and without inadequate lap splices) for the 8-story idealized frame, for Vpc/Vn=0.6 and Mc/Mb>1.20.


3-21

Figure 3-19

Story profiles of coefficient  (with and without inadequate lap splices) for the 8-story idealized frame, for Vpc/Vn=0.6, Mc/Mb>1.20, and the first and fourth stories critical. Thus, in calculating the story drift in Equation 3-14, the coefficient  in stories where inadequate lap splices occur, is multiplied by the s factor corresponding to amplification associated with inadequate lap-splices in column longitudinal reinforcement. Table 3-5 presents values for sx based on the analytical study.

Table 3-5

Lap Splice Amplification Factor,

ΣMc/ΣMb ≤

1.20

≤

>1.20

3-22

s

Vpc/Vn 1.0

1.10

>1.0 1.00

1.30

1.00


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3.6


3.6.1

Overview

In buildings with significant plan irregularities, or limited torsional resistance, the torsional moment is expected to increase story drift demands, which, in turn, increase the drift demands on individual columns. The torsional amplification factor, AT, is a modification factor applied to the column drift demand, δcol. The torsional amplification factor used in this evaluation methodology is derived from nonlinear dynamic analyses of more than 200 threedimensional building models analyzed by DeBock et al. (2014). These building models included reinforced concrete moment frames that varied in terms of:



ductility, including both special and ordinary moment frames designed in accordance with ASCE/SEI 7-10, Minimum Design Loads for Buildings and Other Structures (ASCE, 2010);



height, ranging from 1 to 10 stories;



gravity-load design levels; and



torsional asymmetry and flexibility, including lateral systems with rectangular and I-shaped configurations in plan.

DeBock et al. (2014) showed that both torsional flexibility, defined by a lack of torsional rigidity in the lateral system, and torsional asymmetry, defined by irregular plan layouts, impact story drifts and collapse capacities quantified through nonlinear dynamic analyses. Their study was conducted in terms of the Torsional Irregularity Ratio (TIR), which is defined as the ratio of the maximum story drift, computed including accidental torsion at one end of the structure, to the average of the story drifts at the two ends of the structure, where both drifts are computed in the same direction of interest. An illustration can be found in Figure 12.8-1 of ASCE/SEI 7-10. DeBock et al. (2014) showed that TIR combines torsional flexibility and plan asymmetry into a single measure of a building’s torsional characteristics, and is an effective predictor of torsional responses. A building with inherent torsional flexibility has a larger TIR because the building will rotate under lateral loads acting at the center of mass. In addition, the moment due to accidental torsion will have an effect on the computed drifts, such that even symmetric buildings will have a TIR greater than 1.0. To alleviate the burden of requiring the calculation of stiffness in every column to obtain torsionally induced drifts, a new measure, the Torsional

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3-23

Strength Ratio (TSR), was introduced. This measure is similar to TIR in that it quantifies a building’s torsional characteristics and follows the same trends as TIR. TSR is a ratio of the torsional demand, TDx, on story x, added to the torsional capacity, TCx, of story x, and normalized by TCx. The torsional demand, TDx, on story x includes both the demand due to accidental torsion plus the demand due to real (inherent) torsion caused by a difference in the locations of the center of mass and the center of strength. TSR is then separated into accidental and real (inherent) contributions to torsion. The accidental portion quantifies the amplification of drifts due to accidental torsion moments, which could occur in either direction. The real (inherent) portion quantifies the amplification of drifts due to irregularities. Inherent torsion is accounted for by the eccentricity term er, which represents the actual distance between the center of mass and center of strength of the building. Accidental torsion is accounted for by the accidental eccentricity term, ea, which assumes an accidental eccentricity equal to 5% of the maximum plan dimension (perpendicular to the direction of loading) between the center of mass and the center of strength. An additional adjustment in the evaluation methodology is the replacement of the center of rigidity with the center of strength. Center of strength is used in lieu of center of rigidity to alleviate the additional calculations necessary to compute a relative center of rigidity in addition to the center of strength. If lateral strength is proportional to lateral stiffness, then these two locations are the same. Additionally, the center of nonlinear strength is considered a more accurate measure of post elastic torsional response than the elastic center of rigidity. 3.6.2

Data Used to Derive the Torsional Amplification Factor, AT,max

A subset of the data provided by DeBock et al. (2014) has been repurposed to develop the maximum torsional amplification factor, AT,max, as a function of TSR. Specifically, the torsional amplification factor derived here is based on 1-story to 4-story ordinary moment frames with varying height, gravity loads, and torsional irregularity. Ordinary moment frames were chosen because they are similar in terms of design and ductility to older non-ductile reinforced concrete moment-resisting frames which are the subject of this methodology. In total, analyses of 5 buildings, each of which was assessed with varying torsional flexibility, were considered. Four of these were symmetric with rectangular or I-shaped frame layouts, and one had an offset (asymmetric) layout.

3-24


ATC-78-3

Each building was analyzed under incremental dynamic analysis with a suite of 22 pairs of ground motions. A library of data was generated quantifying response as a function of ground-motion intensity for each building. Symmetric buildings were analyzed with a 5% offset in the modeled center of mass to account for accidental torsion. Asymmetric buildings were analyzed with no offset in the modeled center of mass. Complete details of the modeling process are available in DeBock et al. (2014). In the context of this evaluation methodology, the primary metric of interest is the ratio between the maximum torsionally induced lateral displacement at an edge of the building, in the direction of earthquake loading, to the displacement at the center of strength, in the direction of earthquake loading. This parameter is denoted AT,max. Example results are provided in Figures 3-20 through 3-24 as a function of TIR or TSR, for 1-story and 4-story ordinary moment frames, with high and low gravity load levels, symmetric and asymmetric plan configurations, and linear and nonlinear levels of response. In the figures, results are shown for different ground motion intensity levels on the same plots. 4

4 Response Data Regression Curve +/- Logarithmic Standard Deviation

3.5

3.5

3

3

x a 2.5 ,m T

x a 2.5 ,m T

A

A

2

2

1.5

1.5

1

1 1

Figure 3-20

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1.5

2 TIR a)

2.5

1

1.5

2 TIR b)

2.5

Torsional amplification as a function of Torsional Irregularity Ratio (TIR) for selected 1-story, high gravity load, ordinary moment frames for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse).


3-25

4


3.5

3.5

3

3

x a 2.5 ,m T

x a 2.5 ,m T

A

A

2

2

1.5

1.5

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1

1.1

Figure 3-21

1.3

1.4

Torsional amplification as a function of Torsional Strength Ratio (TSR) for selected 1-story, high gravity load, ordinary moment frames for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse). 4


3.5

3.5

3

3

x a 2.5 ,m T

x a 2.5 ,m T

A

A

2

2

1.5

1.5

1

1 1

1.1

1.2

TSR a)

Figure 3-22

3-26

1.2

TSR b)

TSR a)

1.3

1.4

1.05

1.1

1.15

1.2

1.25

1.3

TSR b)

Torsional amplification as a function of Torsional Strength Ratio (TSR) for selected 4-story, high gravity load, ordinary moment frames for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse).


ATC-78-3

4

4

3.5

3.5

3

3

x a 2.5 ,m T

x a 2.5 ,m T

A

A

2

2

1.5

1.5

1

1 1

1.1

1.2

1.3

1

1.4

1.1

Figure 3-23

1.2

1.3

1.4

TSR b)

TSR a)

Torsional amplification as a function of Torsional Strength Ratio (TSR) for selected 1-story, ordinary moment frames with nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse) and: (a) high gravity load; and (b) low gravity load levels.

4

4 Symmetric Response Data Asymmetric Response Data Regression Curve

3.5

3.5

+/- Logarithmic Standard Deviation

3

3

x a 2.5 ,m T

x 2.5 a m , T

A

A

2

2

1.5

1.5

1

1 1

1.5

2

2.5 TSR a)

Figure 3-24

3

1

1.5

2

2.5

3

TSR b)

Torsional amplification as a function of Torsional Strength Ratio (TSR)(a)forlinear 1-story, symmetric and asymmetric building configurations for: or near linear response (Sa < 0.90S a,collapse) and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse).

ATC-78-3


3-27

Deaggregation of the data revealed that AT,max is not very sensitive to the intensity of ground shaking during elastic response. There is a large variation in AT,max, however, between ground motion intensities that produce elastic structural response and those that produce inelastic structural response. The variability that appears in the plots is largely due to record-torecord variability in response under two-dimensional excitation. 3.6.3

Torsional Amplification Factor Sensitivity to Building Characteristics

Data extracted from the DeBock et al. (2014) study showed that:  Buildings of different heights had similar torsional amplification factors as a function of TIR or TSR (comparison of Figures 3-21 and 3-22).



Buildings with different gravity load levels had similar torsional amplification factors as a function of TIR or TSR (as shown in Figure 3-23).



Buildings governed by torsional asymmetry and buildings governed by torsional flexibility have similar torsional amplification factors as a function of TIR or TSR (as shown in Figure 3-24).

3.6.4

Relationship between Torsional Strength Ratio and Torsional Amplification Factor

The DeBock et al. (2014) data for 1-story to 4-story ordinary moment frames, for high and low gravity load levels, and for buildings with varying plan configurations, are grouped together in Figure 3-25. Figure 3-25a shows the trend between TSR and AT,max for the linear or near linear response, where AT,max represents the maximum torsional amplification. Figure 3-25b shows the trend between TSR and AT,max for the highly nonlinear or near collapse response. In each case, regression analysis is used to derive a mathematical relationship between TSR and AT,max. The torsional amplification factor at the edge of the building, AT,max, depends on the story drift demand, x, and the TSR at story x. If δx ≤ 0.0085 (linear or near-linear range), TSRx

AT , max  0.272 1.543

 0.75

(3-17)

AT , max  0.317 1.874 TSRx  0.65

(3-18)

If δx ≥ 0.0115 (non-linear or near collapse),

3-28


ATC-78-3

Equations 3-17 and 3-18 provide the expected value of AT,max. If 0.0085 ≤ x ≤ 0.015, the torsional amplification factor should be computed by linear interpolation between the values obtained from Equations 3-17 and 3-18.

Figure 3-25

Relationships between the torsional amplification factor and TSR proposed in Equations 3-17 and 3-18, superimposed on selected nonlinear analysis results, for: (a) linear or near linear response (Sa < 0.90Sa,collapse); and (b) nonlinear or near collapse response (Sa ≥ 0.90Sa,collapse).

There are limitations in the methods used to develop Equations 3-17 and 3-18. First, the buildings analyzed by DeBock et al. (2014) have the same properties over the height of the building (i.e., the TSR for each story is the same). It is assumed that the same relationships apply to buildings with different properties at each story, and that these relationships can be used to derive AT,max for each story in a similar manner. Second, the data used to obtain AT,max are based on the maximum ratio of the edge displacement to the displacement at the center of strength, obtained from dynamic analyses in the two orthogonal directions simultaneously. If applied separately in each of the two orthogonal directions, as described herein, the amplification is overestimated (or conservative) in one of the directions. Third, asymmetries in strength (e.g., frames that yield sooner than other frames creating nonlinear torsional response) can have an important influence on the torsional response of the building. This effect was not explicitly analyzed by DeBock et al. (2014), however, the models are nonlinear and included some variation in strength between the frame lines, so the effects of strength

ATC-78-3


3-29

nonlinearities have essentially been included in the derived amplification factors. 3.6.5

Calculation of Torsional Strength Ratio

The torsional strength ratio, TSRx, is the ratio of the torsional demand, TDx, on story x added to the torsional resistance capacity, TCx, of story x and normalized by TCx:

TSRx 

TDx  TCx TCx

(3-19)

Figure 3-26 shows many of the quantities used in the following definitions. Recall that the center of strength is employed in this methodology as a proxy for the center of rigidity in order to simplify calculations associated with determining the center of rigidity. The center of strength is analogous to a center of rigidity. It is computed by summing the product of the frame locations (measured perpendicular from the frame to the center of strength) and their lateral strengths divided by the summation of the frame strengths.

Figure 3-26

Illustration of parameters for calculation of TSR.

Torsional capacity, TCx, is computed as the summation of the plastic capacity of each frame line, Vpf, in story x (regardless of orientation in plan) multiplied by the orthogonal distance from each frame line to the center of strength, Rf: As a result, TCx is constant regardless of the direction of the earthquake loading.

3-30


ATC-78-3

Torsional demand, TDx, is associated with the direction of earthquake loading under consideration, and must be computed for each orthogonal direction. Both accidental and inherent torsion are included in the calculation of the torsional demand:

TDx  V px eT  Vee px 

r



a



(3-20)

where:

Vpx

= plastic shear capacity of story x;

eT

= total eccentricity at story x;

er

= real (inherent) eccentricity between the center of mass and the center of strength in the direction perpendicular to the direction of earthquake loading;

ea

= accidental eccentricity defined by an offset in the location of the center of mass equal to 5 percent of the overall plan dimension perpendicular to the direction of earthquake loading

3.6.6

Relationship between Total Torsional Amplification Factor and Accidental and Real Torsional Amplification Factors

The maximum total torsional amplification factor, AT,max, is disassembled into an accidental torsional amplification, AaT,max, and a real (or “inherent”) torsional amplification, ArT,max, which are assumed to be proportional to the total torsional amplification. This is acomplished by calculating the portion of the total torsional demand that is due to accidental torsion, and the remainder is assumed to be due to real (inherent) torsion. The parameter TSRa represents the torsional strength ratio resulting from accidental torsion alone based on an assumed eccentricity equal to 5% of the overall plan dimension perpendicular to the direction of earthquake loading. TSRa is used to compute the fraction of torsional drift amplification due to accidental torsion, AaT,max, as:

 TSRa  AT , max ,1.0  . TSR  

ATa,max  max 

(3-21)

Once the maximum accidental torsion amplification has been computed, the maximum real torsion amplification is the remainder of the total amplification and computed as:

AT ,max r . AT ,max  Aa T , max

ATC-78-3

(3-22)


3-31

The maximum accidental torsional amplification and real amplification are used to predict the drifts at the edge of the building furthest away from the center of strength (termed the “weak” or “flexible” side), assuming a rigid diaphragm and center of rotation equal to the center of strength.

AaT,max and ArT,max are converted to AaT and ArT at each column location using Equations 3-23 and 3-24. In these equations, the variable x measures the distance between the center of strength and the column location and is taken as positive if the column is on the “weak” side of the building or negative if on the “strong” side. The sign of x is used to ensure that a drift reduction in the real torsional amplification factor will occur on the stiff side of the building. The real torsional amplification factor, ArT is calculated as:

 x .  L1 

ATr   ATr ,max  1 

(3-23)

Thus, if the story of interest contains an inherent irregularity, the real component of the torsional amplification will be less than one for columns on the “strong” or “stiff” side of the building but greater than one on the “flexible” or “weak” side. The accidental torsional moment, however, could act on either side of the center of mass. In this case, the accidental torsional amplification factor, AaT, is restricted to values greater than one to ensure a torsional reduction does not occur from an accidental torsional moment:



 x    1 .  L1  

ATa  abs  ATa,max  1 



(3-24)

The total torsional amplification at the location of any column is then the product of

a T

and

r T

: T

 ATr ATa .

(3-25)

The equations for torsional amplification assume that the floor diaphragm is rigid, which is generally valid for concrete buildings with concrete floor systems. The largest values of ATa, max and ATr , max from any story should be used at every story. This requirement is necessary because irregularities at one story can generate torsional responses in other stories, even if the other stories are more symmetric.

3-32


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3.7

Determination of Critical Components: Column or Slab-Column Connection

For many buildings, the columns will be the critical components governing the collapse mechanism of the structure. However, in some buildings, other components may be critical. The criteria for a slab-column connection to be deemed the critical element are based on the presence, or absence, of continuous bottom reinforcement. The definition of “Continuity Reinforcement” in ASCE/SEI 41-13, Seismic Evaluation and Retrofit of Existing Buildings (ASCE, 2013) has been adopted as a threshold to identify the presence of continuous bottom reinforcement extending from the slab through the column. It is anticipated that a slab-column connection will experience vertical failure, shortly after punching failure, if it lacks continuous bottom reinforcement. As a result, slab-column connections found to have insufficient continuous bottom reinforcement are deemed to be the critical component in the slab-column system. If a slab-column connection exceeds the minimum requirements for continuous bottom reinforcement, it may experience punching failure, but the continuous bottom reinforcement is expected to prevent the slab from experiencing vertical failure. In this case, the column is deemed to be the critical component.

3.8

Portion of Drift Taken by the Columns

3.8.1

Overview

During seismic excitation, the story drift demand is resisted by the flexural strength of the beams and columns. During elastic (or near elastic) response, the drift demand is distributed between beams and columns in proportion to their relative stiffnesses. During nonlinear response, drift tends to concentrate in the components of the building that have yielded (or experienced some other physical damage that leads to a significant reduction in component stiffness and degradation of response). In this methodology, the fraction of the story drift that is taken by each column is denoted by Ccol. The values of Ccol were obtained from dynamic analysis of models of six-story reinforced concrete buildings, subjected to a suite of 44 ground motions. The flexural and shear properties of the columns in the buildings were varied between models to obtain a broad data set capable of capturing the values of Ccol for buildings with varying characteristics.

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3-33

3.8.2

Data Used to Determine the Portion of Drift Taken by the Columns

If the building is responding in the nonlinear range, the portion of the drift taken by the column, Ccol, depends on the relative strength of the columns and the beams framing into a joint. This is because the drift tends to concentrate in the weaker element. To extract values of Ccol from the nonlinear models, the portion of drift taken by the columns was read from lumped-plasticity, nonlinear springs which were present at beam and column ends in the nonlinear models. The maximum ratio of column drift to total story drift was taken from the analysis in which the ground motion was scaled to a spectral acceleration value corresponding to incipient collapse. Results were then averaged over all of the ground motions to obtain the values shown in Figure 3-27. In the figure, each point represents the results obtained from analysis of a 6-story building with a uniform ratio of column to beam strength at every joint, and a uniform ratio of column shear capacity to flexural capacity in every column. The portion of drift taken by the column during nonlinear response depends on both the ratio of column to beam strength, ΣMc/ΣMb, and the ratio of column shear capacity controlled by flexure to column shear capacity controlled by shear, VpMx/Vnx.

Figure 3-27

Portion of drift taken by the column, Ccol,j (denoted as C in the figure), during nonlinear response from incremental dynamic analysis of selected 6-story buildings.

To simplify the calculation of Ccol in the nonlinear range, the value of Ccol was taken to be linear with respect to relative strength of the columns and the beams framing into a joint. This relationship is shown in Figure 3-28. Note that for ΣMc/ΣMb greater than 1.4, Ccol should be taken as 0.6. Limited analyses were conducted for larger Mc/Mb ratios.

3-34


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Figure 3-28

3.8.3

Portion of drift taken by the column, Ccol (denoted as C in the figure), assuming a linear relationship with Mc/Mb.

Data Used to Determine the Portion of Drift Taken by the Slab-Column Connections

If the slab-column connection is found to be the critical component, Ccol will be taken as 1.0. This is because the drift capacities for slab-column connections are based on the entire slab-column subassembly, considering the deformation of the slab, connection, and column. Therefore, there is no need to separate the drift among these different components. If the slabcolumn connection is found to be the critical component, the drift demand placed on the slab is equal to the sum of one-half of the drift demand in the story above the critical slab and one-half of the drift demand in the story below the critical slab. This approach is consistent with the laboratory setups, shown in Figure 3-29, used in the derivation of the slab-column drift capacities, which typically include half of the columns above and below the connection.

Figure 3-29

Typical Laboratory Slab-Column Test Setup (from Gogus and Wallace 2013). ,

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3-35

3.8.4

Limitations

It is assumed that the story drift and deformations are shared by the column and beam or slab elements. Future studies should consider possible deformations in other load-resisting elements, such as joints. In addition, the value of Ccol depends on the amount of softening experienced in a column. The significance of softening in the columns should be investigated in future studies.

3.9

Moment Delivered to Columns from Beams

The flexural capacity at the top of a column is taken as the lesser of the expected flexural strength of the column section (calculated using expected material properties), and the flexural strength controlled by the beams or slabs. The equations used to calculate the flexural strength delivered to the column by the beams or slab were derived from the portal method of approximate structural analysis. These rules are used in computing the effective yield strength of the building. In Equation 2-10, the sum of the beam or slab moments is normalized by hx/(hx+hx+1), where hx+1 is the height of the story above the beam-column connection of interest, and hx is the height of the story below. This relationship implies that force is distributed proportionally to column height. Equation 2-10 is not attempting to distribute force proportionally to the stiffness of the connected columns. To verify this result, nonlinear pushover analyses were conducted. Four 6-story models were analyzed. Details of these models are provided in Table 3-6.

Table 3-6

Building Parameters Used in Pushover Verification Analyses

First Story Height (ft)

Upper Story Height (ft)

ΣMc/ΣMb

VpMx/Vn

13

11

1.2

0.6

20

11

1.2

0.6

13

11

1.2

0.8

20

11

1.2

0.8

The pushover analyses were displacement controlled, with the displacement applied at the roof of the building. Results of the analyses are shown in Figure 3-30. Figure 3-30b and Figure 3-30c show that the models with taller first-story heights develop larger moments at the top of the column during the analysis. This same trend is captured in Equation 2-10. Similar results are expected for column moments at the bottom of a column, and for differences in story height in the upper stories of a building.

3-36


ATC-78-3

(a)

(b)

(c)

Figure 3-30

ATC-78-3

Pushover analyses to validate Equation 2-10, illustrating: (a) the pushover setup; (b) pushover results for a model with ΣMc/ΣMb = 1.2 and VpMx/Vn = 0.6; and (c) pushover results for a model with ΣMc/ΣMb = 1.2 and VpMx/Vn = 0.8.


3-37

Chapter 4

Development of Acceptability and Failure Criteria This chapter presents the results of studies used in the development of acceptability and failure criteria used in the evaluation methodology, which are based on a review of recent available experimental data and structural reliability theory.

4.1

Determination of Column Deformation Capacities

4.1.1

Overview

Table 2-6 presents the median column plastic rotation capacities at which column axial failure occurs for columns with different properties, as empirically predicted from experimental data. The development of these capacities began by leveraging data previously collected as part of an update to ASCE/SEI 41-06, Seismic Rehabilitation of Existing Buildings (ASCE, 2007) by Elwood et al. (2007). A similar approach can be taken in future studies to obtain drift capacities of other components that are outside the scope of the current evaluation methodology. As shown in Figure 4-1, ASCE/SEI 41 defines two parameters to quantify component deformation capacity of concrete columns: (1) a, which is the plastic rotation at significant loss of lateral capacity, taken as corresponding to the point at which the column has lost 20% of its lateral strength; and (2) b, which is the plastic rotation at loss of gravity load-carrying ability. These parameters are defined separately for columns with different possible failure conditions:



Condition i columns are those expected to fail in a flexural mode.



Condition ii columns are those expected to fail in a flexure-shear mode that is yielding first in flexure and then failing in shear.



Condition iii columns are those expected to fail directly in shear.



Condition iv columns have inadequate anchorage and development of longitudinal rebar and fail in a bar slip mode, and are addressed in Section 4.1.5.

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4: Development of Acceptability and Failure Criteria

4-1

The criteria used to separate columns into conditions i, ii and iii are defined in Table 2-5, adopted directly from Li et al., (2014). These condition definitions have been modified from those in ASCE/SEI 41 in an attempt to avoid erroneous classification of condition ii columns as condition iii columns, which would be excessively conservative. Condition iv columns are addressed in Section 4.1.5.

Figure 4-1

Definition of a and b parameters in terms of component force deformation response in ASCE/SEI 41 (from Li et al., 2014).

Values of a and b defined in ASCE/SEI 41 are based on experimental data of single columns tested under cyclic static loads, and depend on the properties of the column. Values are intentionally conservative, meaning that the values provided are underestimates of the true deformation capacities. It is likely that a column will have a drift capacity exceeding the values provided in ASCE/SEI 41 (i.e., the probability is greater than 50% that the true drift capacity will exceed the estimate provided in the table). The degree of conservatism depends on the column condition and parameter of interest, but the values in ASCE/SEI 41 are intended to be more conservative for b parameters for columns with higher axial loads (i.e., on the order of 85% probability of exceedance). The ASCE/SEI 41 committee used results from laboratory tests compiled by Berry et al. (2004) to assess the adequacy of the proposed modeling parameters and check that the target probabilities of failure were achieved, and this conservatism was judged appropriate by the committee, given the considerable scatter in the data. 4.1.2

Development of Column Deformation Capacity Tables

Tables defining column plastic rotation capacities, θc, in this methodology were obtained by adapting the tables provided in ASCE/SEI 41-13, Seismic

4-2


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Evaluation and Retrofit of Existing Buildings (ASCE 2013) through the following set of modifications:



Values of θc were based on b (rather than a) values, since the methodology is aiming to quantify failure of columns, and the axial limit state is judged to occur at, or near, the time of failure.



New data were used to supplement the existing column test database such that the total number of experiments used for this study is greater than the number used in ASCE/SEI 41. These new data came from Henkhaus (2010) and Woods and Matamoros (2010). Henkhaus (2010) tested 8 columns classified as condition iii in the lab, continuing the tests until axial failure occurred. Woods and Matamoros (2010) provided data for three columns, which are condition iii, and were tested until axial failure was reached. The number of columns for which experimental data are available for each condition is summarized in Table 4-1Table 4-.



Predictions of θc are intended to produce a constant 50% probability that the observed value will exceed the predicted value, so that all values are medians. These predictions were investigated to ensure that there was no bias with respect to axial load ratio, transverse reinforcement ratio, or any other parameter. As a result, the newly developed tables provide median values, regardless of the properties of the column.



In ASCE/SEI 41, predictions of b for condition ii columns are a function of the shear stress in the column. Since this methodology does not include a detailed structural analysis, the dependence on shear stress in the columns was eliminated to simplify the calculations for drift capacity.

Table 4-1

Number of Columns Used in Developing Predictions of θc, for Each Column Condition Condition

Number of Columns

i

24

ii

24

iii

22

These modifications were intended to reduce some of the conservatism present in the ASCE/SEI 41 values, and to take advantage of additional test data that had become available since the ASCE/SEI 41 values were determined. The same set of data was used for condition i and ii columns, because limited data exists for condition i columns tested to axial failure.

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4-3

Ghannoum and Matamoros (2014) employed a different approach to achieve some of the same goals (i.e., median predictions of plastic rotation capacities). The equations provided by Ghannoum and Matamoros (2014) to predict b are in reasonable agreement with the values provided herein, although the Ghannoum and Matamoros predictions are more conservative in some cases (shown in Figure 4-2). It may be desirable in future versions of the methodology to provide consistency between the different approaches. Two differences exist between the ASCE/SEI 41 drift capacity tables and the drift capacity tables used in this evaluation methodology: (1) the parameter used to designate the drift at axial failure is changed from b to θc; and (2) the shear stress in columns was eliminated as a predictor of column drift capacity. Values of θc for condition ii columns represent high values of shear stress (v/

f c ≥ 6 in ASCE/SEI 41) and values of θc for condition i columns  c ≤ 3 in ASCE/SEI 41 ).

represent lower-bound value of shear stress (v/ 4.1.3

Final Deformation Capacity Tables

The predicted plastic rotation capacities are reported in Table 2-6. Table 4-2 summarizes the modeling parameters, the test database, and the uncertainties in those predictions. The mean ratio of measured to predicted values is less than 1.0 because of the lognormal distribution of the data. A probability of failure of 0.50 indicates that the predicted values represent median results. More details are provided in Table 4-3 for condition iii columns and parameter b = θc. For condition iii columns, the prediction of θc depends on the axial load ratio, P/Agf’ce, and the transverse reinforcement ratio, Av/bws. Comparing the predictions of b from ASCE/SEI 41 and this methodology shows that, in general, values from this methodology are larger, because median values instead of lower bound values are used. Similar observations can be made for the other conditions. Figure 4-2 compares the values of θc to experimental values and other predictive equations for the least ductile (condition iii) columns.

Table 4-2

Database Results for Modeling Parameters

Modeling Parameter

Number of tests

b for Condition 24 i

4-4

Mean (θc_meas/θc_methodology)

0.86

COV (θc_meas/θc_methodology)

0.70

Probability that Actual Drift Capacity is Less Than Predicted Drift Capacity

0.50

b for Condition ii

24

0.86

0.70

0.50

b for Condition iii

22

0.85

0.69

0.50


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The validity of the values of θc developed as part of this methodology is further illustrated in Figures 4-3 and 4-4. Figure 4-3 shows that the predictions for θc used in this methodology re not biased with respect to axial load in the column. Figure 4-4 investigates predictions for θc with respect to bias associated with column transverse reinforcement ratio.

Table 4-3

Column Plastic Rotation Capacities at Axial Failure, b = θc, for Condition iii Columns

P/Agf’ce ≤

0.1

≥ ≤ ≥

0.6

ASCE/SEI 41 b

Av/bws

Methodology c

0.006

≥

0.060

0.075

0.6

0.006

≥

0.008

0.020

0.1

0.0005 ≤ 0.0005

0.006

0.016

0

≤

0.006

0.120 0.100 e r u li 0.080 a F l ia x 0.060 A t a t f i 0.040 r D

Empirical Elwood (2004) Ghannoum (2013) This methodology ASCE 41 (2013)

0.020 0.000 0

5

10

15

20

Column Number

Figure 4-2

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Comparison of predictions of drift at axial failure (b or c) by four different methods for condition iii columns.


4-5

(a) Figure 4-3

(b)

Measured versus predicted plastic rotation at axial failure at different axial load ratios for: (a) ASCE/SEI 41 predictions of b for condition i, ii and iii columns; and (b) predictions of θc for condition iii columns in this methodology. The horizontal line in (b) is a linear regression showing that there is no bias with respect to axial load ratio.

(a) Figure 4-4

(b)

Measured versus predicted plastic rotation at axial failure at different transverse reinforcement ratios for: (a) ASCE/SEI 41 predictions of b for condition i, ii and iii, columns; and (b) predictions of θc for condition iii columns in this methodology. 4.1.4

Uncertainty in Drift Capacities

Experimental data are also used to estimate the uncertainty in the drift capacities. The values shown in Table 4-4 quantify the uncertainty in the ratio of measured to predicted drift capacities. These values range from 0.62 to 0.63. On the basis of this data, the lognormal standard deviation, ln,c, is taken to be 0.60. This value includes uncertainty in test data and in the empirical model (equation) predicting the test result, and is incorporated in the computation of column ratings (described in Section 4.3).

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Table 4-4

4.1.5

Uncertainty in Predictions of Drift Capacity, from Comparison with Experimental Data

, as Obtained

ln, c

Condition

ln,

b for Condition i

0.63

b for Condition ii

0.63

b for Condition iii

0.62

C

Deformation Capacity of Columns with Inadequate Lap Splices

ASCE/SEI 41-13 provides conservative estimates of the deformation capacity of columns with inadequate lap splices in longitudinal reinforcement. In this methodology, columns with inadequate lap splices are defined by Section 12.17 of ACI 318-11, Building Code Requirements for Structural Concrete and Commentary (ACI, 2011). In ASCE/SEI 41 (2013), these columns are classified as condition iv. Here, the deformation capacities of these columns are determined using Table 2-6, and the presence of the lap splice is essentially ignored in determining the column drift capacity. In order to use this table, columns with inadequate lap splices are classified as condition i, ii, or iii, based on the ratio of column shear capacity controlled by flexure to column shear capacity controlled by shear, VpMx/Vn. The application of Table 2-6 to columns with lap splices uses data from seven experimental tests from Melek and Wallace (2004) and Lynn (1999). Each cyclic static test was performed on a column with inadequate lap splices, and continued until the column failed axially. Instead of developing a new column classification and capacity criteria for columns with inadequate lap splices, this methodology assigns deformation capacities for these columns based on their VpMx/Vn ratio. The deformation capacity of a column with inadequate lap splice is taken as the same as the deformation capacity of a column with adequate lap splice and the same flexural and shear strengths. Figure 4-5 compares experimental test results to the predicted deformation capacities. On average, this methodology underpredicts the deformation capacity, as shown in Table 4-5, but is less conservative than ASCE/SEI 41-13. It may seem surprising that the presence of a lap splices do not reduce deformation capacities. In fact, the limited existing test data suggest that the bond failure of a lap splice may actually increase flexibility and deformation capacity in some cases. Due to the limited and scattered nature of the available data, these columns are treated the same as columns without a lap splice. The “penalty” for poor lap splices is instead considered as an increase in the demands on the columns (see Section 3.5).

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4-7

Figure 4-5

Comparison of deformation capacities determined from test results (“empirical”), ASCE/SEI 41-13, and this methodology, for columns with inadequate lap splices.

Table 4-5

Probability that Actual Drift Capacity is less than Predicted Drift Capacity for Columns with Inadequate Lap Splices

Method

Probability that Actual Drift Capacity is Less Than Predicted Drift Capacity

Methodology – condition depending on VpMx/Vn

0.23

ASCE/SEI 41 (2013) – condition iv

0.002

4.1.6

Overturning Effects on Column Deformation Capacity

Most of the parameters needed to determine drift capacities using the tables in this methodology can be read or calculated from structural drawings of the building. However, the axial loads used in Table 2-6 require further discussion. In this methodology, the axial load is taken as the expected gravity load on the column, plus a contribution from seismic overturning effects. The expected gravity load consists of the unfactored dead load plus 25% of the unreduced live load acting tributary to the columns of interest. Use of unfactored dead load plus 25% of the unreduced live load for expected gravity loads is based on data from Ellingwood et al. (1980). The load due to seismic overturning effects, Peq, is difficult to determine without detailed analyses, and depends on the level of seismic excitation, and the position of the column within the building. Since this methodology does not require a detailed structural analysis, a simple computation is included for determining column axial load due to seismic effects. This process is illustrated in Figure 4-6 for a two-dimensional frame.

4-8


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Figure 4-6

Illustration of simplified method for computing Peq for a twodimensional frame.

The resistance of frame overturning depicted in Figure 4-6 is based on the assumption that the outermost columns resist all of the overturning, and Peq can be taken as zero for all interior frame columns. Neglecting the change in axial forces due to seismic effects for interior columns is reasonable based on review of nonlinear analysis results for a 6-story building (shown in Figure 4-7), in which the maximum increase in column axial load, Peq, of 9 kips is only 5% of the gravity load already acting on the column. The methodology also neglects contributions of Peq for columns near the top of building (located above heff). This simplification is supported by Figure 4-8, which shows the small earthquake induced axial force for columns at the top of the six-story building. For columns resisting overturning (end columns outlined in Figure 4-6), the value of Peq can be estimated by the static moment equation:

Peq



Vy  heff



hx 

L

(4-1)

For columns on either side of a frame, Peq is taken as positive (i.e., in compression), because seismic loads reverse directions and increased axial load in columns tends to reduce the column deformation capacity (which is conservative). Note that Figure 4-6 is depicted at a state when the base shear capacity of the building, Vy, is developed or nearly developed. For this reason, the inclusion of Peq is only required if Sa (at the effective fundamental period of the building) > 0.3g. For smaller ground shaking intensities, the contribution of overturning to column axial forces is assumed to be small.

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4-9

Figure 4-7

Nonlinear analysis results for interior columns in a 6-story building with VpMx/Vn = 0.8 and Mc/Mb = 1.2. For reference, the axial load on the first-story interior columns is 192 kips.

Comparison with nonlinear dynamic analysis results in Figure 4-8 indicates that Peq may be slightly underestimated by Equation 4-1. However, this is judged to be acceptable since it is compensated by the conservative assumption that both columns are in compression. st

1 Story nd 2 Story rd

3 Story th

4 Story th

5 Story th

6 Story ‐ Peq=0 for these columns

Figure 4-8

4-10

Comparison of simplified relationship for Peq with selected nonlinear analysis results for end columns in a 6-story building with VpMx/Vn = 0.8 and Mc/Mb = 1.2. Solid lines show the equation prediction for Peq.


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Figure 4-9 provides a more detailed comparison of the simplified relationship for Peq with nonlinear analysis results for a 4-story building. A number of 4-story buildings were considered, each with uniform properties in terms of

Mc/Mb and VpMx/Vn. In this figure, each point corresponds to the average results over all ground motions at either the first, second or third story of the building at the point of incipient collapse. Results show that the simplified relationship for Peq tends to overestimate the axial load contribution from seismic-induced overturning on columns in the upper stories, but underestimates the axial load contribution from seismic-induced overturning on columns in the lower stories.

Figure 4-9

Comparison of simplified relationship for Peq with selected nonlinear analysis results for the bottom three stories of a 4-story model.

This level of approximation is deemed acceptable. However, more research is needed to determine whether the same relationship for Peq is appropriate for taller buildings. Note also that the vertical component of the ground shaking is not considered in the dynamic analyses used to verify the Peq relationship. 4.2

Determination of Slab-Column Connection Deformation Capacities

4.2.1

Overview

Table 4-6 defines total deformation capacity of slab-column systems, and was obtained from the results of 85 laboratory tests. The deformation capacity of a slab-column connection is defined as the drift at which the

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4-11

strength has degraded to 80% of the peak strength. This drift is representative of punching failure in the slab-column connection. It should be noted that this definition of deformation capacity differs somewhat from that used to define column deformation capacity, which is based on vertical (axial) failure. The reasoning behind this definition of punching shear induced collapse is as follows:



In order to protect the laboratory equipment, very few experimental slabcolumn tests were conducted to vertical failure.

 As this methodology is only concerned with slab-column connections that do not satisfy minimum requirements for continuous bottom reinforcement, vertical failure is anticipated to occur within the next few cycles after punching failure. Drift capacities for slab-column connections in Table 4-6 are presented as a function of the gravity shear ratio. The gravity shear ratio, Vg/Vc, represents the unfactored vertical gravity shear, Vg, divided by the theoretical punching shear strength without moment transfer, Vc, determined in accordance with ACI 318-11 Section 11.11.2.1.

Table 4-6

Slab-Column Drift Capacities

Gravity Shear Ratio, Vg/Vc

4.2.2

Median Drift Capacity,

< 0.1

0.045

> 0.6

0.01



C

Development of Slab-Column Connection Deformation Capacity Tables

Drift capacities presented in Table 4-6 represent median drift values (i.e., corresponding to a 50% probability exceedance). Figure 4-10 compares the prediction in this methodology to the experimental data for 85 tests (Aslani, 2005; Gogus and Wallace, 2014) showing that the proposed relationship captures observed trends in slab-column drift capacity. The prediction used in this methodology has an uncertainty (lognormal standard deviation) of 0.4. It can be shown that the residuals (difference between observed and predicted deformation capacities) are independent, normally distributed, unbiased, and homoscedastic. Note that the deformation capacity predictions in this methodology do not depend on the amount of continuous bottom reinforcement in the slab extending through the column. In this methodology, if a slab has sufficient bottom reinforcement, vertical collapse due to punching failure is assumed to be prevented. In this case the columns are critical. If there is insufficient

4-12


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bottom reinforcement, deformation capacities follow the same trend regardless of the amount of reinforcement.

Figure 4-10

Ratio of measured versus predicted drift at punching shear failure for slabcolumn connections with different gravity shear ratios.

Figure 4-11 compares measured versus predicted deformation capacities and the ACI 318-11 regression representing the maximum design story drift ratio for slab-column connections.

Figure 4-11

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Comparison of measured versus predicted drift capacities for slab-column connections with different gravity shear ratios. Predictions from ACI 31811 are also shown for comparison.


4-13

For the purpose of predicting deformation capacity associated with punching shear failure, the ACI 318-11 equation is conservative and has a probability of exceedance of 88.5%.

4.3

Method for Determining Column Ratings

4.3.1

Overview

A column rating represents the probability that the drift demand on a column, ΔD, exceeds the drift capacity of the column, ΔC. Thus, a column rating reflects the probability of failure for that column for the given ground motion intensity. Higher column ratings indicate that the expected performance of columns will be worse. Drift demands and drift capacities are both assumed to be lognormally distributed. A lognormal distribution is commonly used in earthquake engineering applications. The probability that column drift demand exceeds capacity depends on the mean and standard deviations of the probability distributions for drift demand and capacity. In this methodology, uncertainty in drift demand includes record-to-record variability in drift response, as well as uncertainty in the prediction of roof and critical story drifts. Uncertainty in the drift capacity includes variability in material properties, and the predictive model for drift capacity. A column rating should be viewed as a representation of the probability of column failure for the given level of excitation, not as the true probability of failure. As a representation of the probability of failure, column ratings are comparable between different columns in a building and between different buildings. However, due to the simplified nature of this methodology, these values should not be applied in other probabilistic analyses outside of this methodology. 4.3.2

Structural Reliability Methods for Computing the Column Rating

A column rating is computed using structural reliability methods to determine the probability that a column drift demand exceeds the column drift capacity (Melchers, 1999). In the formulation that follows, it is assumed that the random variables representing drift demand and drift capacity are lognormally distributed and statistically independent (Mori and Ellingwood, 1993). In structural reliability terms, the limit state function for this problem can be written as:

where: 

C

4-14

G  ln  C ln D

(4-2)

= median drift capacity for the column of interest; and


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

= is the median drift demand for the column of interest.

D

“Failure” occurs when G  0. It can be shown that the reliability index, β, for this situation can be obtained from the following equation:

  C     D   2 2  ln,    ln,  ln 

C

(4-3) D

where:  ln,C = uncertainty in the column drift capacity, taken as 0.63; and  ln, D = uncertainty in the column drift demand, taken as 0.60. and C and D are as defined above. The probability of failure, pf, for each column can be computed from the reliability index as follows: p f    

(4-4)

where the operator  indicates the cumulative standard normal probability distribution. As a result, the column rating, CR, is given as:

   C   ln     D  CR j      2  2 ln,  ln,  C

D

     

(4-5)

which is the relationship that was used to develop the values shown in Table 2-8.

4.4

Method for Determining Story Ratings

4.4.1

Overview

The story rating represents the probability of a story failure occurring. Similar to the column rating, the story rating represents the probability of collapse of a story, given the level ground shaking being considered. Higher story ratings indicate higher probabilities of collapse. As a representation of the probability of collapse, story ratings are comparable among stories and between buildings. However, due to the simplified nature of this methodology, these values should not be applied in other probabilistic analyses.

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4-15

4.4.2

Probability Theory for Determining Probability of Story Collapse

A number of different methods were explored to relate column failure ratings to story collapse ratings. The method used in this evaluation methodology was developed utilizing probability theory to relate the probability of failure of each column at story x to the probability of collapse in that story. The following assumptions are made in computing the probability of a story collapse:



Story collapse is assumed to occur if at least 25% of the columns in the story fail.



Column drift demands for all columns in a story are assumed to be perfectly correlated.



Column drift (deformation) capacities are correlated assuming the failure model shown in Figure 4-12. This model captures adjacent column relationships, such that the failures of columns that are closer together are assumed to be more highly correlated (or consequential) than columns that are farther apart. The minimum level of correlation applied to the most widely spaced columns corresponds to a correlation coefficient of 0.20.

Figure 4-12

Assumed model of correlation in column drift capacities for failure of columns i and j, as a function of column separation distance.

These assumptions are used to generate the story rating curve shown in Figure 4-13. This curve was obtained using 10,000 Monte Carlo simulations in which random realizations of the drift demand and drift capacity of each column in the story were generated. These realizations were generated using a multi-variate random number generator such that the assumed correlation is considered. Story collapse was identified in a simulation if at least 25% of

4-16


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the columns were identified as having failed in the simulation. Figure 4-13 is represented in tabular form in Table 2-11. 1 0.9 0.8 0.7 g 0.6 in t a r y 0.5 r to S0.4

0.3 0.2 0.1 0 0

0.1

Figure 4-13

4.4.3

0.2

0.3

0.4 0.5 0.6 0.7 Average column rating

0.8

0.9

1

Relationship between adjusted average column rating, CRavg., and the story rating, SR.

Sensitivity of Story Rating to Building and Methodological Assumptions

A number of additional complications affect the development of Figure 4-13. First, this curve assumes that all stories with the same average column rating will have the same story rating. In fact, sensitivity studies showed that stories with a wide variability in column ratings (e.g., some columns with a rating of 0.9 and some columns with a rating of 0.1), had worse story ratings than other stories with the same average column rating. These observations were used to develop the column rating adjustment, CRa, based on the range of column ratings in a story. The column rating adjustment, CRa, increases the computed CRavg if there is a large difference among column ratings for the columns in a story. Buildings with different plan aspect ratios and dimensions were also considered, showing there is not much variability in the story rating with plan aspect ratio. Similar results were obtained for buildings with different column spacing. In Figures 4-14 and 4-15, the sensitivity of the predicted story ratings to other methodological assumptions is examined. One of the key assumptions made in the story rating computation is the assumption that 25% of columns

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4-17

failing constitutes a story collapse. As Figure 4-14 shows, the story rating is highly dependent on this definition of story collapse. As expected, if fewer columns are required to fail for the story to collapse, the story rating (probability of collapse) increases.

Figure 4-14

Sensitivity of story ratings to assumption about the definition of story collapse.

A second critical assumption made is the relationship used to describe correlations in failures among different columns in the building. Although the correlation model shown in Figure 4-12 is rational, there are limited data to suggest whether or not this model is, in fact, correct. In Figure 4-15, the sensitivity of the story ratings to the correlation assumption for drift capacities is investigated (drift demand is assumed to be perfectly correlated in all of the curves on this plot). In the figure, the story rating is higher when the correlation model assumes less correlation, than when higher levels of correlation are introduced between the columns. Essentially, when there is little correlation, it is highly unlikely that none of the columns will fail, and relatively more likely that at least 25% of the columns will fail. On the other hand, when more correlation is introduced, the likelihood that none of the columns will fail increases, leading to smaller story ratings for the case of perfect correlation. Finally, Figure 4-16 shows the effect of the uncertainty assumed in the drift demand. Higher uncertainty has the impact of flattening the relationship between column rating and story rating. In this evaluation methodology,

4-18


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uncertainties in drift demand and drift capacity were assumed to be constant across different structural systems.

Figure 4-15

Sensitivity of story ratings to assumptions about correlations in column failures.

Figure 4-16

Sensitivity of story ratings to assumptions about uncertainty in column drift demand.

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4-19

4.5

Studies of Beam-Column Joints

The behavior of beam-column connections in older-type construction has been widely studied (Moehle, 2014). Of particular interest are:



Performance of connections with discontinuous beam bottom longitudinal reinforcement



Strength of joints in beam-column connections without joint transverse reinforcement



Effect of joint eccentricity on joint and column behavior



Axial failure of beam-column connections.

Each of these topics is reviewed in the following sections. 4.5.1

Performance of Connections with Discontinuous Beam Bottom Longitudinal Reinforcement

Many older buildings have discontinuous beam bottom reinforcement that projects a short distance into the beam-column joint. A typical extension was 6 inches (the minimum permitted by the then-current ACI building codes). Short embedments of this type are insufficient to develop full strength of the embedded bars. However, the embedded bars are well confined by the surrounding concrete, so they are unlikely to cause a brittle splitting failure. For this reason, it is reasonable to calculate the bar stress capacity of the embedded bar in accordance with the provisions of ASCE/SEI 41, to apply this stress to calculate the moment strength of the beam for loading that puts the embedded bars in tension. This reduced moment strength usually can be sustained through relatively large rotations. Figure 4-17 plots measured strengths for interior beam-column joints without transverse reinforcement, both with and without continuous beam bottom longitudinal reinforcement. Cracking strength is notably affected by axial load, whereas ultimate strength is less affected by axial load. Reduced strength associated with discontinuous beam bottom longitudinal reinforcement is apparent.

4-20


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Figure 4-17

4.5.2

Shear strength of unreinforced interior joints. Test specimens with continuous beam bottom bars are noted. All other test specimens have discontinuous bottom bars embedded in the joint (data from Pessiki et al., 1990; and Beres et al., 1992; after Moehle, 2014.)

Strength of Joints in Beam-Column Connections without Joint Transverse Reinforcement

Figure 4-18 presents data for exterior connections (actually, edge connections load parallel to the edge). Measured joint shear strength is at least 10 f c Aj , with higher strengths noted for connections with continuous longitudinal reinforcement. These joints had aspect ratios hb/hc ~ 1.0. Figure 4-18 compares measured and calculated strengths for exterior connections loaded perpendicular to the edge, including corner connections. Calculated strength is based on the ASCE/SEI 41 expression, Vnj



6

f c A j .

In the tests, joints failed without significant inelastic flexural response in the beams or columns. The comparison suggests that: (a) ASCE/SEI 41 calculated strengths are conservative; and (b) joint strength is a function of joint aspect ratio hb/hc. The data from Figures 4-17 and 4-18 suggest an alternative form of joint shear strength could provide improved correlation between measured and calculated strengths. One suggested shear strength expression is:

Vnj



10 hc hb

f c

(4-6)

where hc/hb ≤ 1.0.

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4-21

Figure 4-18

Measured and calculated strengths (ASCE/SEI 41) for exterior joints loaded perpendicular to the edge, including corner joints. Data are for joints failing in shear without significant inelastic flexural deformation in adjacent beams or columns (after Hassan, 2011).

Figure 4-19 compares measured and calculated results, using the same test data shown in Figure 4-18, and using the revised expression (Equation 4-6) to calculate shear strength. Overall correlation is improved considerably.

Figure 4-19

Measured and calculated strengths (Equation 4-6) for exterior joints loaded perpendicular to the edge, including corner joints. Data are for joints failing in shear without significant inelastic flexural deformation in adjacent beams or columns (data after Hassan, 2011).

4-22


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4.5.3

Effect of joint eccentricity on joint and column behavior

Several researchers have studied behavior of eccentric beam-column joints, in which centerlines of beams and columns framing into the joint are eccentric. LaFave et al. (2005) provide a summary of the principal findings. Force transfer at an eccentric beam-column joint is complicated by the misalignment of the framing members. Figure 4-20a depicts the local force transfer mechanism. The diagonal struts forming within the joint depth are oriented in opposite directions on opposite sides of the joint, suggesting the occurrence of joint torsion, which could reduce joint strength. The localization of force transfer at the intersection between the beam and column results in localization of joint damage (Figure 4-20b), which also could justify reduced joint strength for eccentric connections. Some writers have also identified the possibility of column torsion, and have suggested that this column torsion could reduce column shear strength due to the superposition of shear stresses from shear and torsion. This effect appears to be ameliorated by the presence of a floor slab, which is capable of resisting any torsional moments through in-plane bending of the diaphragm (Figure 420c). Physical evidence of column damage due to induced torsion is limited, and does not appear to justify consideration in this methodology.

Figure 4-20

Force transfer at eccentric beam-column connections.

LaFave et al. (2005) recommend including the effect of joint eccentricity by defining the joint area Aj = hc(bc + bw)/2. None of the joints reviewed by LaFave et al. had beams extending outside the plan of the column. Therefore, it is recommended that bw not exceed the width of the beam web that is contained within the outline of the column dimensions. Available test data do not show significant reduction in deformation capacity of eccentric beam-column joints compared with otherwise equivalent concentric joints.

ATC-78-3


4-23

4.5.4

Axial failure of beam-column connections

Several buildings that collapsed in past earthquakes have shown evidence of severe joint damage. On the other hand, physical evidence of building collapses having actually been triggered by joint failures is less clear. Laboratory studies also have not provided convincing proof that building collapse can be triggered by joint failure. One of the reasons for absence of evidence may be that laboratory tests are not subjecting beam-column connections to sufficiently severe loadings because of limitations in testing equipment capabilities. In one study, Hassan (2011) subjected a series of corner beam-column joints to severe lateral and axial loading histories. The joints eventually showed signs of axial shortening through the joint at late stages of testing, although sudden axial failure was never obtained. Hassan also collected laboratory test data from tests conducted by other researchers, delineating, for final loading stages, those tests that showed signs of axial shortening within the joint and those that did not. Figure 4-21 plots the results. The bilinear relation divides those connections showing axial shortening from those not showing axial shortening. According to Hassan, connections falling below the bilinear relation can be considered safe from axial collapse, while those falling above the relation might be suspected of triggering axial collapse. This same relation is adopted for corner beam-column connections in Chapter 2. At present, there are no known test data providing physical evidence that axial failure can be triggered by failure of edge or interior connections.

Figure 4-21

4-24

Maximum observed drift ratios and axial load ratios of corner beam-column connections. (After Hassan, 2011.)


ATC-78-3

Appendix A

Data for Prediction of Drift Demands This appendix includes Tables A-1 through A-21, which present background analytical results used in the development of procedures for estimating story drift demands and drift capacities used in the evaluation methodology. Use of the information in these tables is described in Chapter 3.

Table A-1


ΣΜc ΣΜb

/

Vp/Vn = 0.6

Vp/Vn = 0.8

Vp/Vn = 1.0

Vp/Vn = 1.2

0.6

0.88

0.80

0.82

0.91

0.8

0.88

0.80

0.82

0.91

1.0

0.87

0.81

0.81

0.90

1.2

0.92

0.82

0.83

0.85

1.4

0.99

0.85

0.85

0.84

1.6

1.12

0.91

0.86

0.85

1.8

1.17

0.94

0.91

0.87

Table A-2

Average Ratios of Displacement at the Effective Modal Height (Nonlinear Analysis Results versus Equation 3-9) for an 8-Story Idealized Frame Building Average Ratios of Displacement

ΣΜc ΣΜb

/

Vp/Vn = 0.6

Vp/Vn = 0.8

Vp/Vn = 1.0

Vp/Vn = 1.2

0.6

0.84

0.84

0.85

0.94

0.8

0.84

0.83

0.85

0.92

1.0

0.84

0.83

0.84

0.91

1.2

0.87

0.82

0.81

0.83

1.4

1.01

0.90

0.87

0.86

1.6

1.12

0.97

0.93

0.91

1.8

1.23

1.01

0.98

0.96

ATC-78-3

A: Data for Prediction of Drift Demands

A-1

Table A-3


ΣΜc ΣΜb

/

Vp/Vn = 0.6

Vp/Vn = 0.8

Vp/Vn = 1.0

Vp/Vn = 1.2

0.6

0.95

0.96

0.97

0.99

0.8

0.94

0.96

0.97

1.00

1.0

0.94

0.94

0.96

0.96

1.2

NA

0.96

0.93

0.90

1.4

1.25

1.10

1.02

0.96

1.6

1.42

1.18

1.10

1.05

1.8

NA

1.24

1.16

1.12

Table A-4

Mean Values of Coefficient in a 4-story Idealized Frame without Significant Torsional Effects Mean Value of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 4

0.30

0.34

0.34

0.30

0.36

0.42

0.57

3

0.54

0.63

0.56

0.50

0.58

0.71

0.86

2

0.43

0.47

0.51

0.66

0.93

1.14

1.17

1

2.29

2.18

2.21

2.00

1.60

1.30

1.14

Vp/Vn =0.8 4

0.45

0.45

0.46

0.45

0.44

0.50

0.62

3

0.81

0.82

0.78

0.73

0.75

0.80

0.90

2

0.69

0.69

0.73

0.87

1.00

1.09

1.13

1

1.93

1.92

1.88

1.67

1.42

1.26

1.14

Vp/Vn =1.0 4

0.50

0.50

0.50

0.49

0.48

0.54

0.61

3

0.89

0.89

0.85

0.80

0.78

0.85

0.90

2

0.77

0.77

0.79

0.91

1.02

1.09

1.13

1

1.81

1.82

1.76

1.55

1.37

1.24

1.13

Vp/Vn =1.2

A-2

4

0.53

0.52

0.52

0.51

0.54

0.56

0.62

3

0.81

0.81

0.81

0.81

0.85

0.89

0.93

2

0.86

0.86

0.87

0.95

1.05

1.11

1.14

1

1.68

1.67

1.66

1.51

1.31

1.21

1.13


ATC-78-3

Table A-5

Standard Deviation of Coefficient in a 4-story Idealized Frame without Significant Torsional Effects Standard Deviation of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 4

0.11

0.17

0.18

0.17

0.34

0.32

0.42

3

0.26

0.43

0.31

0.22

0.30

0.33

0.32

2

0.08

0.15

0.14

0.16

0.21

0.12

0.08

1

0.20

0.37

0.30

0.30

0.27

0.22

0.21

Vp/Vn =0.8 4

0.10

0.11

0.13

0.19

0.23

0.29

0.37

3

0.33

0.35

0.29

0.22

0.24

0.24

0.24

2

0.08

0.08

0.09

0.10

0.09

0.06

0.06

1

0.25

0.26

0.25

0.21

0.17

0.13

0.14

Vp/Vn =1.0 4

0.12

0.12

0.14

0.26

0.25

0.30

0.37

3

0.36

0.36

0.31

0.27

0.24

0.24

0.24

2

0.08

0.08

0.09

0.09

0.08

0.05

0.04

1

0.23

0.23

0.22

0.24

0.19

0.13

0.13

Vp/Vn =1.2 4

0.14

0.14

0.15

0.23

0.34

0.35

0.36

3

0.17

0.17

0.18

0.23

0.31

0.29

0.26

2

0.09

0.09

0.09

0.09

0.06

0.04

0.04

1

0.24

0.24

0.23

0.21

0.20

0.15

0.14

ATC-78-3


A-3

Table A-6

Mean Values of Coefficient in a 6-story Idealized Frame without Significant Torsional Effects Mean Value of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Vp/Vn =0.6 6

0.30

0.31

0.30

0.24

0.22

0.24

0.31

5

0.54

0.55

0.55

0.44

0.39

0.44

0.55

4

0.66

0.68

0.64

0.55

0.57

0.65

0.79

3

0.52

0.52

0.51

0.60

0.80

0.97

1.09

2

0.45

0.46

0.48

0.80

1.27

1.31

1.22

1

2.84

2.81

2.79

2.31

1.59

1.26

1.06

Vp/Vn =0.8 6

0.44

0.43

0.42

0.34

0.30

0.31

0.32

5

0.74

0.74

0.74

0.66

0.58

0.57

0.60

4

0.90

0.90

0.88

0.85

0.81

0.88

0.92

3

0.73

0.73

0.75

0.87

0.99

1.12

1.18

2

0.72

0.72

0.74

1.06

1.21

1.22

1.21

1

2.20

2.19

2.14

1.66

1.31

1.12

0.97

Vp/Vn =1.0 6

0.47

0.47

0.46

0.42

0.35

0.36

0.35

5

0.81

0.81

0.82

0.79

0.69

0.64

0.63

4

1.01

1.00

0.98

0.96

0.90

0.90

0.97

3

0.80

0.80

0.80

0.89

1.01

1.11

1.18

2

0.82

0.82

0.83

1.00

1.18

1.23

1.22

1

1.92

1.91

1.89

1.68

1.30

1.12

0.97

Vp/Vn =1.2

A-4

6

0.51

0.50

0.51

0.48

0.41

0.38

0.36

5

0.84

0.83

0.85

0.85

0.80

0.71

0.63

4

0.99

1.01

0.98

1.01

1.05

0.95

0.94

3

0.84

0.84

0.84

0.88

1.03

1.10

1.16

2

0.93

0.92

0.92

0.95

1.10

1.21

1.24

1

1.69

1.67

1.70

1.56

1.36

1.16

1.01


ATC-78-3

Table A-7

Standard Deviation of Coefficient in a 6-story Idealized Frame without Significant Torsional Effects Standard Deviation of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Vp/Vn =0.6 6

0.07

0.07

0.07

0.10

0.19

0.17

0.26

5

0.23

0.23

0.23

0.22

0.28

0.25

0.37

4

0.29

0.31

0.22

0.20

0.26

0.31

0.37

3

0.16

0.16

0.14

0.20

0.26

0.25

0.19

2

0.06

0.06

0.06

0.18

0.19

0.13

0.14

1

0.26

0.33

0.23

0.42

0.38

0.29

0.27

Vp/Vn =0.8 6

0.10

0.10

0.10

0.12

0.17

0.22

0.24

5

0.25

0.25

0.28

0.32

0.30

0.35

0.34

4

0.36

0.36

0.29

0.32

0.30

0.34

0.29

3

0.11

0.11

0.14

0.19

0.20

0.18

0.13

2

0.08

0.09

0.09

0.19

0.13

0.14

0.11

1

0.27

0.27

0.29

0.34

0.31

0.26

0.23

Vp/Vn =1.0 6

0.12

0.12

0.11

0.12

0.17

0.23

0.26

5

0.28

0.26

0.29

0.35

0.33

0.34

0.36

4

0.39

0.39

0.30

0.37

0.30

0.30

0.32

3

0.09

0.09

0.09

0.19

0.18

0.17

0.13

2

0.09

0.09

0.10

0.15

0.13

0.14

0.13

1

0.28

0.27

0.26

0.26

0.32

0.28

0.23

Vp/Vn =1.2 6

0.20

0.19

0.18

0.15

0.18

0.18

0.22

5

0.32

0.31

0.36

0.38

0.41

0.35

0.33

4

0.30

0.37

0.29

0.36

0.41

0.31

0.29

3

0.08

0.09

0.08

0.13

0.17

0.17

0.14

2

0.06

0.08

0.05

0.11

0.15

0.14

0.13

1

0.20

0.25

0.21

0.23

0.30

0.28

0.24

ATC-78-3


A-5

Table A-8

Mean Values of Coefficient in an 8-story Idealized Frame without Significant Torsional Effects Mean Value of Coefficient

Prior to Collapse

ΣΜc Σ Μb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 8

0.64

0.69

0.67

0.52

0.43

0.44

0.43

7

1.01

1.07

1.02

0.88

0.71

0.65

0.62

6

1.02

1.09

1.11

1.01

0.85

0.80

0.77

5

1.38

1.42

1.24

0.97

0.86

0.89

0.91

4

0.57

0.60

0.58

0.73

0.79

0.96

1.03

3

0.84

0.89

0.79

0.93

1.22

1.34

1.35

2

0.53

0.56

0.56

1.04

1.41

1.46

1.41

1

2.98

2.91

2.92

2.36

1.74

1.41

1.27

Vp/Vn =0.8 8

0.91

0.83

0.84

0.76

0.55

0.53

0.55

7

1.36

1.34

1.31

1.16

0.92

0.81

0.79

6

1.37

1.38

1.43

1.31

1.09

0.99

0.97

5

1.51

1.49

1.38

1.24

1.07

1.05

1.09

4

0.78

0.77

0.76

0.89

0.96

1.06

1.15

3

1.00

0.98

0.92

1.03

1.23

1.29

1.34

2

0.73

0.72

0.74

1.13

1.35

1.36

1.34

1

2.17

2.14

2.18

1.75

1.46

1.29

1.18

Vp/Vn =1.0 8

0.87

0.87

0.90

0.77

0.61

0.58

0.56

7

1.42

1.40

1.41

1.26

1.04

0.91

0.84

6

1.52

1.50

1.52

1.40

1.22

1.11

1.04

5

1.50

1.49

1.45

1.27

1.17

1.13

1.14

4

0.85

0.84

0.84

0.93

1.01

1.09

1.17

3

1.02

1.01

0.98

1.07

1.19

1.27

1.31

2

0.79

0.78

0.80

1.14

1.30

1.33

1.31

1

1.86

1.89

1.90

1.54

1.34

1.24

1.12

Vp/Vn =1.2

A-6

8

0.92

0.92

0.94

0.84

0.73

0.61

0.59

7

1.52

1.54

1.52

1.41

1.21

0.99

0.91

6

1.61

1.60

1.63

1.54

1.37

1.20

1.12

5

1.49

1.49

1.46

1.35

1.25

1.18

1.18

4

0.93

0.93

0.92

0.97

1.02

1.10

1.18

3

1.16

1.15

1.08

1.06

1.15

1.24

1.30

2

0.89

0.88

0.89

1.11

1.25

1.29

1.29

1

1.49

1.53

1.57

1.36

1.22

1.16

1.09


ATC-78-3

Table A-9

Standard Deviation of Coefficient in an 8-story Idealized Frame without Significant Torsional Effects Standard Deviation of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 8

0.24

0.31

0.27

0.22

0.23

0.25

0.30

7

0.51

0.57

0.54

0.33

0.36

0.35

0.36

6

0.27

0.40

0.32

0.34

0.39

0.36

0.36

5

0.57

0.57

0.44

0.35

0.36

0.33

0.33

4

0.08

0.11

0.08

0.20

0.22

0.25

0.26

3

0.24

0.35

0.21

0.23

0.22

0.17

0.16

2

0.08

0.12

0.10

0.20

0.23

0.24

0.26

1

0.69

0.72

0.61

0.62

0.50

0.36

0.34

Vp/Vn =0.8 8

0.43

0.36

0.37

0.35

0.24

0.28

0.27

7

0.79

0.76

0.70

0.49

0.38

0.37

0.32

6

0.44

0.42

0.45

0.45

0.38

0.38

0.33

5

0.59

0.56

0.44

0.39

0.34

0.30

0.26

4

0.13

0.13

0.13

0.17

0.18

0.15

0.16

3

0.33

0.32

0.28

0.21

0.19

0.17

0.17

2

0.13

0.13

0.15

0.21

0.26

0.25

0.25

1

0.54

0.57

0.58

0.53

0.36

0.31

0.29

Vp/Vn =1.0 8

0.38

0.39

0.43

0.39

0.27

0.29

0.29

7

0.75

0.75

0.73

0.59

0.43

0.37

0.36

6

0.52

0.52

0.54

0.49

0.42

0.39

0.38

5

0.45

0.44

0.45

0.37

0.35

0.32

0.29

4

0.12

0.12

0.13

0.19

0.20

0.16

0.15

3

0.37

0.37

0.27

0.23

0.19

0.17

0.17

2

0.15

0.14

0.16

0.22

0.28

0.26

0.26

1

0.54

0.54

0.52

0.46

0.39

0.32

0.31

Vp/Vn =1.2 8

0.42

0.41

0.45

0.46

0.43

0.30

0.31

7

0.79

0.81

0.78

0.72

0.61

0.41

0.37

6

0.54

0.54

0.52

0.54

0.53

0.41

0.38

5

0.39

0.39

0.37

0.36

0.35

0.33

0.30

4

0.12

0.12

0.12

0.18

0.20

0.17

0.15

3

0.44

0.44

0.35

0.23

0.17

0.18

0.18

2

0.16

0.16

0.17

0.21

0.29

0.28

0.27

1

0.52

0.56

0.50

0.37

0.37

0.33

0.31

ATC-78-3


A-7

Table A-10 Mean Values of Coefficient in a 12-story Idealized Frame without Significant Torsional Effects Mean Value of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 12

0.50

0.51

0.49

NA

0.26

0.23

NA

11

0.50

0.51

0.49

NA

0.26

0.23

NA

10

1.25

1.26

1.17

NA

0.59

0.50

NA

9

0.96

0.98

1.14

NA

0.70

0.60

NA

8

2.00

1.99

1.76

NA

0.77

0.67

NA

7

0.69

0.71

0.83

NA

0.72

0.68

NA

6

1.48

1.48

1.38

NA

0.79

0.79

NA

5

0.69

0.71

0.78

NA

0.79

0.85

NA

4

0.79

0.79

0.73

NA

0.85

0.98

NA

3

0.51

0.53

0.51

NA

0.88

1.03

NA

2

0.54

0.56

0.61

NA

1.54

1.44

NA

1

2.68

2.69

2.55

NA

1.57

1.34

NA

Vp/Vn =0.8

A-8

12

0.59

0.59

0.58

0.44

0.34

0.31

0.30

11

0.59

0.59

0.58

0.44

0.34

0.31

0.30

10

1.50

1.48

1.45

1.11

0.79

0.68

0.63

9

1.22

1.21

1.38

1.21

0.91

0.81

0.74

8

1.88

1.88

1.69

1.31

1.00

0.89

0.85

7

0.87

0.87

0.98

1.07

0.94

0.90

0.90

6

1.62

1.61

1.46

1.09

1.02

1.02

1.04

5

0.86

0.87

0.98

0.95

0.99

1.05

1.09

4

1.03

1.03

0.92

0.87

1.00

1.11

1.16

3

0.64

0.64

0.65

0.81

0.98

1.09

1.14

2

0.67

0.68

0.75

1.41

1.35

1.24

1.18

1

1.88

1.90

1.90

1.54

1.26

1.11

0.99


ATC-78-3

Table A-10 Mean Values of Coefficient in a 12-story Idealized Frame without Significant Torsional Effects (continued) Mean Value of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.6

0.6

0.6

0.6

0.6

0.6

Vp/Vn =1.0 12

0.63

0.64

0.63

0.51

0.43

0.36

0.34

11

0.63

0.64

0.63

0.51

0.43

0.36

0.34

10

1.59

1.61

1.50

1.23

0.98

0.80

0.72

9

1.24

1.24

1.38

1.32

1.08

0.92

0.84

8

1.74

1.74

1.60

1.43

1.17

1.01

0.94

7

0.93

0.95

1.03

1.18

1.07

1.00

0.99

6

1.53

1.52

1.49

1.19

1.11

1.10

1.12

5

0.93

0.94

1.05

1.04

1.06

1.11

1.15

4

1.06

1.06

0.99

0.95

1.06

1.14

1.19

3

0.70

0.71

0.71

0.85

1.04

1.12

1.15

2

0.73

0.74

0.79

1.31

1.28

1.20

1.15

1

1.77

1.76

1.66

1.38

1.18

1.04

0.94

Vp/Vn =1.2 12

0.70

0.70

0.68

0.59

0.68

0.44

0.37

11

0.70

0.70

0.68

0.59

0.68

0.44

0.37

10

1.60

1.59

1.54

1.40

1.54

0.93

0.78

9

1.36

1.36

1.42

1.52

1.42

1.04

0.91

8

1.62

1.61

1.65

1.56

1.65

1.14

1.01

7

1.04

1.05

1.09

1.30

1.09

1.11

1.05

6

1.53

1.50

1.46

1.29

1.46

1.17

1.16

5

1.03

1.04

1.10

1.12

1.10

1.16

1.19

4

1.06

1.07

1.04

1.00

1.04

1.16

1.21

3

0.79

0.79

0.78

0.90

0.78

1.13

1.16

2

0.82

0.83

0.85

1.23

0.85

1.18

1.15

1

1.60

1.60

1.49

1.25

1.10

1.00

0.91

ATC-78-3


A-9

Table A-11 Standard Deviation of Coefficient in a 12-story Idealized Frame without Significant Torsional Effects Standard Deviation of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 12

0.15

0.17

0.16

NA

0.12

0.10

NA

11

0.89

1.00

0.83

NA

0.29

0.23

NA

10

0.60

0.57

0.49

NA

0.33

0.27

NA

9

0.33

0.33

0.49

NA

0.37

0.30

NA

8

1.70

1.69

1.70

NA

0.36

0.29

NA

7

0.20

0.21

0.25

NA

0.26

0.28

NA

6

0.72

0.78

0.65

NA

0.21

0.28

NA

5

0.16

0.18

0.16

NA

0.18

0.25

NA

4

0.26

0.26

0.16

NA

0.16

0.15

NA

3

0.12

0.14

0.11

NA

0.14

0.14

NA

2

0.12

0.14

0.13

NA

0.28

0.29

NA

1

0.75

0.78

0.65

NA

0.37

0.36

NA

Vp/Vn =0.8

A-10

12

0.20

0.20

0.20

0.17

0.13

0.12

0.13

11

1.11

1.01

0.98

0.57

0.37

0.32

0.30

10

0.97

1.05

0.74

0.58

0.37

0.33

0.33

9

0.53

0.48

0.74

0.71

0.40

0.39

0.37

8

1.36

1.10

1.17

0.75

0.41

0.37

0.37

7

0.26

0.23

0.30

0.43

0.33

0.31

0.32

6

0.58

0.53

0.52

0.32

0.25

0.24

0.25

5

0.18

0.16

0.14

0.20

0.19

0.18

0.18

4

0.26

0.22

0.15

0.15

0.15

0.11

0.09

3

0.14

0.13

0.12

0.11

0.10

0.10

0.10

2

0.15

0.14

0.14

0.27

0.27

0.25

0.23

1

0.55

0.47

0.58

0.35

0.31

0.29

0.26


ATC-78-3

Table A-11 Standard Deviation of Coefficient in a 12-story Idealized Frame without Significant Torsional Effects (continued) Standard Deviation of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =1.0 12

0.20

0.47

0.23

0.21

0.19

0.15

0.13

11

1.03

0.14

1.06

0.67

0.49

0.36

0.32

10

1.04

0.13

0.83

0.61

0.53

0.37

0.33

9

0.49

0.22

0.72

0.71

0.59

0.42

0.39

8

1.09

0.16

1.03

0.76

0.63

0.44

0.38

7

0.23

0.53

0.33

0.51

0.46

0.36

0.33

6

0.53

0.23

0.46

0.38

0.31

0.27

0.27

5

0.16

1.10

0.15

0.23

0.22

0.19

0.18

4

0.22

0.48

0.15

0.17

0.13

0.12

0.09

3

0.13

1.05

0.12

0.11

0.11

0.10

0.11

2

0.14

1.01

0.14

0.28

0.24

0.24

0.23

1

0.47

0.20

0.49

0.35

0.29

0.29

0.29

Vp/Vn =1.2 12

0.27

0.27

0.26

0.26

0.26

0.22

0.15

11

1.22

1.19

1.10

0.79

1.10

0.48

0.35

10

1.03

0.97

0.84

0.70

0.84

0.53

0.35

9

0.55

0.50

0.70

0.85

0.70

0.60

0.41

8

0.71

0.69

0.94

0.83

0.94

0.60

0.44

7

0.22

0.23

0.29

0.55

0.29

0.44

0.36

6

0.62

0.61

0.50

0.42

0.50

0.28

0.25

5

0.21

0.21

0.16

0.24

0.16

0.19

0.17

4

0.17

0.18

0.16

0.15

0.16

0.11

0.11

3

0.12

0.12

0.11

0.11

0.11

0.11

0.11

2

0.14

0.13

0.13

0.28

0.13

0.22

0.22

1

0.41

0.40

0.39

0.35

0.39

0.28

0.28

ATC-78-3


A-11

Table A-12 Mean Values of Coefficient in a 4-story Idealized Frame without Significant Torsional Effects ′

Mean Value of Coefficient ’ Prior to Collapse /

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 4

NA

NA

NA

NA

2.09

1.34

1.58

3

NA

1.67

1.66

1.39

1.45

1.34

1.27

2

NA

NA

NA

NA

NA

1.19

1.18

1

2.29

2.29

2.22

2.02

1.64

1.43

1.33

4

NA

NA

NA

NA

NA

NA

1.43

3

1.71

1.73

1.84

1.21

1.32

1.23

1.30

Vp/Vn =0.8

2

NA

NA

NA

NA

NA

1.16

1.17

1

1.97

1.96

1.90

1.68

1.43

1.29

1.24

4

NA

NA

NA

1.31

1.19

1.30

1.51

3

1.70

1.69

1.52

1.33

1.27

1.30

1.27

Vp/Vn =1.0

2

NA

NA

NA

NA

NA

1.12

1.16

1

1.88

1.89

1.82

1.61

1.42

1.29

1.23

4

NA

NA

NA

NA

1.44

1.60

1.51

3

1.53

1.53

1.48

1.35

1.42

1.36

1.38

Vp/Vn =1.2

A-12

2

NA

NA

NA

1.15

NA

1.16

1.17

1

1.69

1.68

1.67

1.55

1.38

1.29

1.25


ATC-78-3

Table A-13 Standard Deviation of Coefficient in a 4-story Idealized Frame without Significant Torsional Effects ′

Standard Deviation of Coefficient ’ Prior to Collapse /

ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 4

NA

NA

NA

NA

NA

0.15

0.02

3

NA

0.11

NA

NA

0.11

0.21

0.18

2

NA

NA

NA

NA

NA

0.10

0.05

1

0.21

0.22

0.29

0.27

0.25

0.13

0.11

4

NA

NA

NA

NA

NA

NA

0.18

3

0.11

0.08

NA

NA

0.11

0.17

0.21

Vp/Vn =0.8

2

NA

NA

NA

NA

NA

0.04

0.06

1

0.21

0.22

0.22

0.20

0.17

0.09

0.06

4

NA

NA

NA

0.16

NA

NA

0.14

3

0.21

0.21

0.15

0.23

0.13

0.13

0.18

Vp/Vn =1.0

2

NA

NA

NA

NA

NA

0.06

0.04

1

0.16

0.16

0.18

0.18

0.14

0.10

0.05

4

NA

NA

NA

NA

NA

NA

0.06

3

NA

NA

NA

0.22

0.22

0.22

0.09

Vp/Vn =1.2

2

NA

NA

NA

NA

NA

0.01

0.04

1

0.24

0.24

0.22

0.17

0.13

0.10

0.08

ATC-78-3


A-13



ΣΜc ΣΜb

Story

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Vp/Vn =0.6 6

NA

NA

NA

NA

NA

NA

NA

5

NA

NA

NA

NA

1.70

NA

1.32

4

NA

1.62

NA

NA

NA

1.60

1.52

3

NA

NA

NA

NA

1.35

1.32

1.26

2

NA

NA

NA

1.14

1.34

1.29

1.28

1

2.84

2.85

2.79

2.34

1.76

1.49

1.41

Vp/Vn =0.8 6

NA

NA

NA

NA

NA

NA

NA

5

NA

NA

NA

1.40

1.27

NA

NA

4

2.23

2.22

1.48

1.47

1.92

1.59

1.37

3

NA

NA

NA

1.24

1.27

1.33

1.27

2

NA

NA

NA

1.13

1.31

1.30

1.27

1

2.24

2.23

2.17

1.74

1.52

1.40

1.39

6

NA

NA

NA

NA

NA

NA

NA

5

NA

NA

NA

1.85

1.37

NA

1.58

4

2.27

2.27

1.86

1.56

1.46

NA

1.46

3

NA

NA

NA

NA

1.25

NA

1.29

2

NA

NA

NA

NA

1.29

NA

1.30

1

1.96

1.95

1.93

1.73

1.49

NA

1.31

Vp/Vn =1.0

Vp/Vn =1.2

A-14

6

NA

NA

NA

NA

NA

NA

5

1.79

NA

1.88

1.97

1.95

1.52

NA

4

2.18

2.36

2.02

1.90

1.63

1.43

1.53

3

NA

NA

NA

1.24

1.18

1.29

1.30

2

NA

NA

NA

NA

1.25

1.30

1.30

1

1.69

1.70

1.71

1.59

1.53

1.43

1.34


NA

ATC-78-3



ΣΜc ΣΜb

Story

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Vp/Vn =0.6 6

NA

NA

NA

NA

NA

NA

5

NA

NA

NA

NA

NA

NA

NA NA

4

NA

NA

NA

NA

NA

NA

0.20

3

NA

NA

NA

NA

0.23

0.16

0.09

2

NA

NA

NA

NA

0.16

0.08

0.08

1

0.26

0.24

0.23

0.39

0.24

0.13

0.07

6

NA

NA

NA

NA

NA

NA

NA

5

NA

NA

NA

NA

NA

NA

NA

4

0.39

0.39

NA

0.25

NA

0.33

0.09

3

NA

NA

NA

0.15

0.07

0.10

0.10

2

NA

NA

NA

NA

0.08

0.08

0.07

1

0.20

0.20

0.24

0.27

0.17

0.12

0.10

6

NA

NA

NA

NA

NA

NA

NA

5

NA

NA

NA

NA

0.14

NA

NA

4

0.33

0.34

0.12

0.29

0.28

NA

0.32

3

NA

NA

NA

NA

0.03

NA

0.08

2

NA

NA

NA

NA

0.05

NA

0.07

1

0.24

0.24

0.23

0.24

0.19

NA

0.05

Vp/Vn =0.8

Vp/Vn =1.0

Vp/Vn =1.2 6

NA

NA

NA

NA

NA

NA

NA

5

0.17

NA

0.28

0.31

NA

NA

NA

4

NA

0.25

NA

0.21

0.22

0.27

0.38

3

NA

NA

NA

NA

0.08

0.04

0.07

2

NA

NA

NA

NA

0.03

0.07

0.08

1

0.20

0.22

0.20

0.22

0.20

0.16

0.10

ATC-78-3


A-15

Table A-16 Mean Values of Coefficient in an 8-story Idealized Frame without Significant Torsional Effects ′

Mean Value of Coefficient ’ Prior to Collapse ΣΜc/ΣΜb

Story 0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 8

NA

NA

NA

NA

NA

NA

NA

7

NA

NA

3.50

NA

1.84

NA

NA

6

NA

NA

NA

1.79

1.61

1.61

1.84

5

2.42

2.46

2.13

1.63

1.50

1.50

1.45

4

NA

NA

NA

NA

NA

NA

1.38

3

NA

2.58

NA

NA

1.48

1.45

1.41

2

NA

NA

NA

NA

1.44

1.54

1.56

1

3.13

3.10

3.07

2.50

2.03

1.77

1.78

8

NA

NA

NA

NA

NA

NA

NA

7

3.22

3.51

4.32

2.28

1.67

1.72

NA

6

2.15

NA

2.35

1.72

1.78

1.64

1.58

5

2.45

2.35

2.04

1.80

1.53

1.46

1.47

4

NA

NA

NA

NA

1.17

1.20

1.36

3

NA

NA

NA

1.40

1.51

1.40

1.43

Vp/Vn =0.8

2

NA

NA

NA

NA

1.43

1.48

1.48

1

2.40

2.37

2.43

1.97

1.71

1.68

1.48

8

NA

NA

NA

NA

NA

NA

NA

7

2.99

2.98

3.07

2.28

1.92

1.61

1.77

6

2.58

2.58

2.24

2.00

1.87

1.90

1.78

5

2.11

2.05

2.13

1.60

1.52

1.51

1.50

4

NA

NA

NA

1.29

NA

1.29

1.33

3

2.90

2.90

2.18

1.34

1.24

1.30

1.39

Vp/Vn =1.0

2

NA

NA

NA

1.30

1.39

1.51

1.50

1

2.12

2.16

2.12

1.75

1.68

1.69

NA

Vp/Vn =1.2

A-16

8

NA

NA

NA

NA

NA

NA

NA

7

3.22

2.86

2.98

2.64

2.10

1.81

1.64

6

2.13

2.16

2.14

2.01

1.89

1.98

1.85

5

1.93

1.96

1.89

1.68

1.60

1.49

1.52

4

NA

NA

NA

NA

NA

1.33

1.32

3

2.10

2.10

2.10

1.57

1.29

1.29

1.41

2

NA

NA

NA

1.29

1.38

1.46

1.51

1

1.96

2.07

1.89

1.56

1.60

1.78

1.64


ATC-78-3

Table A-17 Standard Deviation of Coefficient in an 8-story Idealized Frame without Significant Torsional Effects ′

Standard Deviation of Coefficient ’ Prior to Collapse ΣΜc/ΣΜb Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 8

NA

NA

NA

NA

NA

NA

7

NA

NA

NA

NA

NA

NA

NA NA

6

NA

NA

NA

NA

0.30

0.60

0.26

5

0.16

0.12

0.21

0.12

0.38

0.09

0.20

4

NA

NA

NA

NA

NA

NA

0.11

3

NA

NA

NA

NA

0.08

0.17

0.20

2

NA

NA

NA

NA

0.19

0.19

0.15

1

0.53

0.53

0.45

0.52

0.37

0.24

0.50

8

NA

NA

NA

NA

NA

NA

NA

7

0.86

0.82

NA

0.27

NA

NA

NA

6

0.41

NA

0.27

0.53

0.29

0.35

0.34

5

0.37

0.27

0.24

0.31

0.23

0.14

0.25

Vp/Vn =0.8

4

NA

NA

NA

NA

NA

0.07

0.05

3

NA

NA

NA

NA

0.29

0.20

0.12

2

NA

NA

NA

NA

0.12

0.16

0.19

1

0.39

0.45

0.41

0.39

0.26

0.22

0.25

8

NA

NA

NA

NA

NA

NA

NA

7

0.80

0.79

0.75

0.41

0.18

0.09

NA

6

0.26

0.26

0.21

0.50

0.35

0.35

0.41

5

0.30

0.33

0.26

0.27

0.31

0.23

0.18

4

NA

NA

NA

NA

NA

0.03

0.07

3

NA

NA

NA

NA

0.04

0.12

0.16

Vp/Vn =1.0

2

NA

NA

NA

NA

0.08

0.16

0.16

1

0.33

0.38

0.39

0.29

0.22

0.19

NA

Vp/Vn =1.2 8

NA

NA

NA

NA

NA

NA

NA

7

1.10

1.16

0.98

0.77

0.67

0.14

0.11

6

0.70

0.80

0.51

0.43

0.51

0.28

0.32

5

0.41

0.41

0.32

0.29

0.24

0.23

0.18

4

NA

NA

NA

NA

NA

0.00

0.07

3

0.47

0.47

0.50

0.59

0.07

0.07

0.15

2

NA

NA

NA

0.04

0.14

0.15

0.17

1

0.52

0.49

0.34

0.25

0.16

0.02

NA

ATC-78-3


A-17



ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 12

NA

NA

NA

NA

11

3.53

3.51

3.00

NA

10

2.85

2.85

2.46

NA

9

NA

NA

NA

NA

8

5.07

5.03

4.92

NA

NA

NA

NA

NA

NA

NA

NA

NA

1.91

1.67

NA

1.25

1.10

NA

NA

7

NA

NA

NA

NA

NA

NA

NA

6

2.72

3.06

2.74

NA

1.21

NA

NA

5

NA

NA

NA

NA

1.33

1.62

NA

4

NA

NA

NA

NA

NA

1.14

NA

3

NA

NA

NA

NA

1.26

1.36

NA

2

NA

NA

NA

NA

1.49

1.52

NA

1

3.06

3.07

2.84

NA

1.79

1.64

NA

12

NA

NA

NA

NA

NA

NA

11

4.13

4.23

3.55

NA

NA

NA

NA

10

3.36

3.23

2.50

2.27

1.88

NA

NA

Vp/Vn =0.8

A-18

NA

9

NA

NA

NA

2.88

1.74

1.79

2.24

8

3.94

3.94

3.50

2.27

1.66

1.20

1.83

7

NA

NA

NA

NA

NA

1.56

1.66

6

2.22

2.21

2.20

1.54

1.42

1.34

1.54

5

NA

NA

NA

NA

NA

1.15

1.21

4

NA

NA

NA

NA

1.22

1.38

1.18

3

NA

NA

NA

NA

NA

1.09

1.22

2

NA

NA

NA

1.64

1.43

1.35

1.38

1

2.27

2.28

2.14

1.65

1.63

1.58

NA


ATC-78-3

Table A-18 Mean Values of Coefficient in a 12-story Idealized Frame without Significant Torsional Effects (continued) ′


ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =1.0 12

NA

NA

NA

NA

NA

NA

NA

11

3.46

3.44

3.14

2.05

1.58

NA

NA

10

3.45

3.54

3.03

2.13

1.96

1.72

NA

9

NA

NA

NA

2.77

2.44

1.67

1.72

8

3.08

3.06

3.43

2.04

2.00

1.94

NA

7

NA

NA

NA

NA

1.59

1.46

1.60

6

2.25

2.19

1.88

1.49

1.39

1.46

1.49

5

NA

NA

NA

NA

1.13

NA

1.23

4

NA

NA

NA

NA

1.24

1.24

1.20

3

NA

NA

NA

NA

1.18

1.23

1.24

2

NA

NA

NA

1.44

1.41

1.32

1.39

1

2.07

2.07

1.97

1.60

1.80

1.66

1.46

Vp/Vn =1.2 12

NA

NA

NA

NA

NA

NA

NA

11

3.85

3.42

2.86

1.80

2.86

NA

NA

10

3.61

3.57

3.18

1.90

3.18

1.70

1.75

9

1.67

1.67

NA

3.22

NA

3.03

1.53

8

2.42

2.62

2.63

2.12

2.63

2.33

1.91

7

NA

NA

NA

NA

NA

1.55

1.55

6

2.16

2.44

1.92

1.90

1.92

1.41

1.59

5

NA

NA

NA

1.36

NA

1.18

1.28

4

NA

NA

NA

NA

NA

1.21

1.22

3

NA

NA

NA

NA

NA

1.27

1.27

2

NA

NA

NA

1.42

NA

1.42

1.40

1

1.86

1.84

1.80

1.43

1.80

1.65

1.44

ATC-78-3


A-19



ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =0.6 12

NA

NA

NA

11

0.16

2.26

NA

NA

10

NA

NA

NA

NA

NA

NA

NA

9

NA

NA

NA

NA

0.14

NA

NA

8

2.27

2.28

3.03

NA

NA

NA

NA

7

NA

NA

NA

NA

NA

NA

NA

6

0.57

0.86

0.30

NA

NA

NA

NA

5

NA

NA

NA

NA

NA

0.33

NA

4

NA

NA

NA

NA

NA

0.05

NA

3

NA

NA

NA

NA

NA

0.21

NA

2

NA

NA

NA

NA

0.27

0.23

NA

1

0.44

0.46

0.43

NA

0.29

0.14

NA

NA

NA NA

NA

NA

NA

NA

Vp/Vn =0.8

A-20

12

NA

NA

NA

NA

NA

NA

11

0.11

0.09

0.08

NA

NA

NA

NA

NA

10

0.85

0.91

0.32

0.15

0.01

NA

NA

9

NA

NA

NA

0.85

NA

0.42

NA

8

1.47

1.37

1.31

1.13

0.31

NA

NA

7

NA

NA

NA

NA

NA

0.10

NA

6

0.15

0.15

0.58

0.16

0.21

0.25

0.16

5

NA

NA

NA

NA

NA

0.08

0.09

4

NA

NA

NA

NA

0.20

NA

0.06

3

NA

NA

NA

NA

NA

NA

0.11

2

NA

NA

NA

0.19

0.21

0.20

0.19

1

0.28

0.29

0.40

0.27

0.36

0.08

NA


ATC-78-3

Table A-19 Standard Deviation of Coefficient in a 12-story Idealized Frame without Significant Torsional Effects (continued) ′


ΣΜc ΣΜb

Story

0.6

0.8

1

1.2

1.4

1.6

1.8

Vp/Vn =1.0 12

NA

NA

NA

NA

NA

NA

NA

11

1.11

1.11

0.87

NA

NA

NA

NA

10

1.01

0.93

0.66

0.43

0.12

0.00

NA

9

NA

NA

NA

0.11

0.42

NA

0.39

8

1.21

1.16

1.00

0.94

0.87

0.42

NA

7

NA

NA

NA

NA

0.15

0.28

0.14

6

0.39

0.38

0.12

0.26

0.23

0.21

0.25

5

NA

NA

NA

NA

0.05

NA

NA

4

NA

NA

NA

NA

0.16

0.11

0.08

3

NA

NA

NA

NA

NA

0.13

0.09

2

NA

NA

NA

0.19

0.15

0.22

0.20

1

0.33

0.33

0.32

0.20

NA

NA

NA

Vp/Vn =1.2 12

NA

NA

NA

NA

NA

NA

NA

11

0.70

0.80

0.85

0.37

0.85

NA

NA

10

1.03

1.04

0.66

0.48

0.66

0.38

0.15

9

NA

NA

NA

0.92

NA

NA

NA

8

0.54

0.39

1.04

0.86

1.04

0.99

0.47

7

NA

NA

NA

NA

NA

0.24

0.22

6

0.45

0.90

0.27

0.70

0.27

0.21

0.22

5

NA

NA

NA

NA

NA

0.05

0.13

4

NA

NA

NA

NA

NA

0.07

0.09

3

NA

NA

NA

NA

NA

0.10

0.08

2

NA

NA

NA

0.25

NA

0.17

0.21

1

0.32

0.32

0.29

0.24

0.29

NA

NA

ATC-78-3


A-21

Table A-20 Mean Values of Coefficient Critical Fourth Story

in a 6-story Idealized Frame with a

Mean Value of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.8

1.4

1.8

Vp/Vn =0.6 6

0.27

0.17

0.47

5

0.46

1.03

1.23

4

2.11

1.75

1.81

3

0.64

1.24

1.59

2

0.47

0.78

1.08

1

1.50

0.97

0.82

Vp/Vn =0.8 6

0.35

0.31

0.34

5

0.61

1.09

1.11

4

1.68

1.91

1.74

3

0.81

1.21

1.34

2

0.60

0.79

0.91

1

1.54

0.94

0.81

Vp/Vn =1.0 6

0.37

0.34

0.36

5

0.62

1.14

1.14

4

1.56

1.92

1.76

3

0.83

1.20

1.32

2

0.63

0.80

0.90

1

1.52

0.99

0.86

Vp/Vn =1.2

A-22

6

0.49

0.36

0.37

5

0.78

1.07

1.18

4

1.30

1.78

1.82

3

0.90

1.19

1.26

2

0.87

0.80

0.86

1

1.41

1.01

0.91


ATC-78-3

Table A-21 Standard Deviation of Coefficient with a Critical Fourth Story

in a 6-story Idealized Frame

Standard Deviation of Coefficient

Prior to Collapse

/

ΣΜc ΣΜb

Story

0.8

1.4

1.8

Vp/Vn =0.6 6

0.08

0.06

0.34

5

0.18

0.62

0.43

4

0.96

0.51

0.40

3

0.18

0.29

0.29

2

0.11

0.21

0.36

1

0.63

0.45

0.30

Vp/Vn =0.8 6

0.14

0.10

0.15

5

0.29

0.45

0.41

4

0.67

0.53

0.46

3

0.20

0.15

0.12

2

0.13

0.19

0.24

1

0.57

0.26

0.23

Vp/Vn=1.0 6

0.14

0.12

0.17

5

0.27

0.46

0.43

4

0.64

0.46

0.48

3

0.18

0.13

0.12

2

0.13

0.17

0.25

1

0.56

0.33

0.26

Vp/Vn=1.2 6

0.20

0.15

0.17

5

0.29

0.36

0.43

4

0.51

0.32

0.44

3

0.12

0.11

0.13

2

0.15

0.14

0.23

1

0.29

0.25

0.28

ATC-78-3


A-23

Symbols a

site class factor, or plastic rotation at significant loss of lateral capacity (from ASCE/SEI 41) a

AT torsional amplification factor associated with accidental torsion AaT, max maximum torsional amplification associated with accidental torsion Ag

gross area of concrete section, in2

Aj

effective cross-sectional area of a joint, in 2

ArT

torsional amplification factor associated with real (inherent) torsion

ArT, max

maximum torsional amplification associated with real (inherent) torsion

As

area of longitudinal tension reinforcement, in2

AT, max maximum torsional amplification factor AT

torsional amplification factor

Av

area of shear or transverse reinforcement within spacing s, in2

b

plastic rotation at loss of gravity load-carrying ability (from ASCE/SEI 41)

bc

overall width of a column in the direction perpendicular to shear, inches

BR

building rating

bw

web width of a beam or column, inches

b′w

web width of a beam, excluding portions of the web that extend beyond the width of a column, inches

C0

modification factor to relate spectral displacement to the

C1

modification factor to relate expected maximum inelastic

displacement at the first mode effective height of the building

displacement to displacement calculated for linear elastic response

C2

modification factor to represent the effect of pinched hysteresis shape, cyclic stiffness degradation, and strength deterioration on maximum displacement response

ATC-78-3

Symbols

B-1

Ccol

column drift factor

CDR

column drift capacity-to-demand ratio

Cm

effective mass factor

CRa

column rating adjustment based on the range of column ratings in a story

CR CRavg Cvx d

column rating adjusted average column rating in a story vertical distribution factor effective depth or the distance from extreme compression fiber to centroid of longitudinal tension reinforcement, inches

DCR(avg) average value of DCRx over the height of the building DCR(max) maximum value of DCRx over the height of the building DCRx relative demand-to-capacity ratio for story x ea

accidental eccentricity defined by an offset in the location of the

er

real (inherent) eccentricity between the center of mass and the center

center of mass equal to 5 percent of the overall plan dimension, feet

of strength, feet

eT

total eccentricity, including eccentricity between center of mass and

Ec

center of strength and accidental eccentricity modulus of elasticity of concrete, psi

f′c

specified compressive strength of concrete, psi

f′ce

expected compressive strength of concrete, psi

fy

specified yield strength of reinforcement, psi or ksi

fye

expected yield strength of reinforcement, psi or ksi

g

acceleration of gravity, in/sec2

G

limit state function defining failure of a column

hb

overall depth of a beam, inches

hc

overall dimension of a column in the direction of shear, inches

heff

effective height of an equivalent single-degree-of-freedom system,

hi, hx

height from the base of the building to level i or level x, feet

feet or inches

B-2

Symbols

ATC-78-3

hn

height from the base of the building to the highest level of the seismic force-resisting system, feet or inches

hx

height of story

x

Ie

effective moment of inertia of a concrete section, in 4

Ig

moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, in4

k

factor related to ductility demand for computing column shear strength

Ke

effective stiffness of a building, k/in

linf

height of an equivalent cantilever column, from the face of a joint to the point of inflection (or zero moment), inches

ln

clear span of beam measured face-to-face of supports, inches

lux

clear height of a column in story x, inches or feet

L

overall building plan dimension, or plan dimension between the

L1

longer dimension between the edge of the building and the center of

outermost frame columns in a story, feet

strength

L2

shorter dimension between the edge of the building and the center of

MbLx

flexural strength of a beam on the left side of a joint at level x, kip-

strength

inches

MbRx

flexural strength of a beam on the right side of a joint at level x, kipinches

Mc, avg average expected flexural strength at the top and bottom of a column, kip-inches McBx

flexural strength at the bottom of a column in story x, kip-inches

McTx

flexural strength at the top of a column in story x, kip-inches

Mn

expected flexural strength, calculated using expected material properties and strength reduction factor (ϕ) equal to 1.0, kip-inches

MOT

overturning moment at the base of a building, kip-feet

MOTi

overturning resistance provided by an individual frame in a building, kip-feet

Mc

summation of column expected flexural strengths, kip-inches

ATC-78-3

Symbols

B-3

Mb

summation of beam expected flexural strengths, kip-inches

n

number of levels or number of stories in a building (Eq 2-6/Table 2-3)

nc

number of columns in story x

nf

number of frame lines in story x

pf

probability of column drift demand exceeding column drift capacity

P

total column axial load, kips

PD

axial load due to tributary dead loads, kips

Peq

axial load due to earthquake overturning effects, kips

Pg

axial load due to gravity effects, kips or pounds

PL

axial load due to unreduced tributary live loads, kips

POT

axial load due to earthquake overturning effects on corner columns, kips

Rf

perpendicular distance from frame line f to the center of strength (for

s

spacing of transverse reinforcement, inches

torsion), feet

Sa

spectral acceleration at the effective fundamental period of the building, percentage of g

Sa, collapse spectral acceleration at collapse at the effective fundamental period of the building, percentage of g Sd

spectral displacement at the effective fundamental period of the building

SS

mapped spectral response acceleration parameter at short periods,

S1

mapped spectral response acceleration parameter at a period of one

percentage of g

second, percentage of g

TCx

torsional capacity of story x, kip-feet

TDx

torsional demand at story x, considering both real (inherent) and

TD,a

torsional demand considering accidental eccentricity, kip-feet

Te

effective fundamental period of a building, seconds

TSR

largest value of torsional strength ratio, TSRx, in a building

accidental torsion, kip-feet

B-4

Symbols

ATC-78-3

TSRx

torsional strength ratio at story x

TSRa

torsional strength ratio from accidental torsion at story x

v

shear stress, psi

Vb

expected shear strength of a beam, kips

Vc

punching shear strength of concrete determined in accordance with

Vg

unfactored gravity shear demand, kips

Vn

expected shear strength of a column, calculated using expected

ACI 318-11

material properties and strength reduction factor (ϕ) equal to 1.0, pounds or kips

Vnj

expected joint shear strength, pounds or kips

Vpc

plastic shear capacity of a column, pounds or kips

Vpf

plastic shear capacity of frame line f, pounds or kips

VpMx

column shear capacity controlled by flexure in story x, kips

Vp1

plastic story shear capacity in the first story, pounds or kips

Vpx

plastic story shear capacity at story x, pounds or kips

Vrx

relative story shear demand at story x, pounds or kips

Vy

effective yield strength of a structure, adjusted by the relative demand-to-capacity ratio, and reduced in consideration of P-Δ effects, pounds or kips

W

total effective seismic weight of a building, kips

wi

portion of the total effective seismic weight tributary to level i, kips

wx

portion of the total effective seismic weight tributary to level x, kips

x, y

distance from center of strength to column of interest, feet



coefficient to modify story drifts at a story for building configuration and strength characteristics



ratio of post-yield stiffness to initial effective stiffness before yield

′

coefficient to modify story drifts at a given story for building configuration, recalculated from  considering only the simulations in which the story collapsed

β

reliability index for a given column

ATC-78-3

Symbols

B-5

βc

coefficient to amplify story drifts in stories that are determined to be critical

βs

coefficient to amplify story drifts in stories with non-compliant lap splices in longitudinal column reinforcement

eff

displacement of an equivalent single degree-of-freedom system, inches

col

column drift demand, inches

x

story drift demand at story x, inches

y

yield displacement at the roof of a building, inches

ΔC

drift capacity of a column, inches

ΔD

column drift demand, adjusted for torsional amplification and column drift factor, inches



median drift capacity of a particular column, inches

C



median drift demand on a particular column, inches

D

ϕ

strength reduction factor

λ

modification factor for lightweight concrete

μstrength system strength ratio c

column plastic rotation capacity, radians

ln, C

standard deviation of the drift capacity (in log space)

ln, 

standard deviation of the drift demand (in log space)

D

B-6

Symbols

ATC-78-3

References

ACI, 2008, Guide to Durable Concrete , ACI 201.2R-08, American Concrete Institute, Farmington Hills, Michigan. ACI, 2011, Building Code Requirements for Structural Concrete and

Commentary, ACI 318-11, American Concrete Institute, Farmington Hills, Michigan. ASCE, 2010, Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-10, American Society of Civil Engineers, Structural Engineering Institute, Reston, Virginia. ASCE, 2007, Seismic Rehabilitation of Existing Buildings, ASCE/SEI 41-06, American Society of Civil Engineers, Structural Engineering Institute, Reston, Virginia. ASCE, 2013, Seismic Evaluation and Retrofit of Existing Buildings, ASCE/SEI 41-13, American Society of Civil Engineers, Structural Engineering Institute, Reston, Virginia. Aslani, H., 2005, Probabilistic Earthquake Loss Estimation and Loss

Disaggregation in Buildings, Ph.D. Thesis, Stanford University, Stanford, California. ATC, 2000, Database on the Performance of Structures Near Strong-Motion

Recordings: 1994 Northridge Earthquake, ATC-38, Applied Technology Council, Redwood City, California. ATC, 2011, Identification and Mitigation of Seismically Hazardous Older

Concrete Buildings: Interim Methodology Evaluation, ATC-78, Applied Technology Council, Redwood City, California. ATC, 2012, Evaluation of the Methodology to Select and Prioritize Collapse

Indicators in Older Concrete Buildings, ATC-78-1, Applied Technology Council, Redwood City, California. ATC, 2013, Seismic Evaluation for Collapse Potential of Older Concrete

Frame Buildings, ATC-78-2, Applied Technology Council, Redwood City, California. Beres, A., White, R.N., and Gergely, P., 1992, Seismic Behavior of Reinforced Concrete Frame Structures with Nonductile Details: Part

I - Summary of Experimental Findings of Full-Scale Beam-Column

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Performance Database User’s Manual, Version 1.0, Pacific Earthquake Engineering Research Center, University of California, Berkeley, California. Blume, J.A., Newmark, N.M., and Corning, L.M., 1961, Design of Multi-

story Reinforced Concrete Buildings for Earthquake Motions, Portland Cement Association, Chicago, Illinois. DeBock, D.J., Liel, A.B., Haselton, C.B., Hooper, J.D., and Henige, R.A., 2014, “Importance of seismic design accidental torsion requirements for building collapse capacity,” Earthquake Engineering and

Structural Dynamics, Vol. 43, No. 6, pp. 831-850. Ellingwood, B., Galambos, T.V., MacGregor, J.G., and Cornell, C.A., 1980,

Development of a Probability Based Load Criterion for American National Standard A58, National Bureau of Standards, Special Publication 577, Washington, D.C. Elwood, K.J., 2004, “Modelling failures in existing reinforced concrete columns,” Canadian Journal of Civil Engineering, Vol. 31, No. 5, pp. 846-859. Elwood, K.J., Matamoros, A.B., Wallace, J.W., Lehman, D.E., Heintz, J.A., Mitchell, A.D., Moore, M.A., Valley, M.T., Lowes, L.N., Comartin, C.D., and Moehle, J.P., 2007, “Update to ASCE/SEI 41 Concrete Provisions,” Earthquake Spectra, Vol. 23, No. 3, pp. 493-523. FEMA, 1998, Evaluation of Earthquake Damaged Concrete and Masonry

Wall Buildings, FEMA 306, prepared by the Applied Technology Council for the Federal Emergency Management Agency, Washington, D.C. FEMA, 2000, Prestandard and Commentary for the Seismic Rehabilitation

of Buildings, FEMA 356, prepared by the American Society of Civil Engineers for the Federal Emergency Management Agency, Washington, D.C. FEMA, 2009, Quantification of Building Seismic Performance Factors, FEMA P-695, prepared by the Applied Technology Council for the Federal Emergency Management Agency, Washington, D.C. Ghannoum, W.M., and Matamoros, A.B., 2014, Nonlinear Modeling

Parameters and Acceptance Criteria for Concrete Columns,

C-2

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American Concrete Institute, ACI Special Publication 297, Farmington Hills, Michigan. Goel, R., and Chopra, A.K., 1997, Vibration Properties of Buildings

Determined from Recorded Earthquake Motions, Earthquake Engineering Research Center, University of California, UCB/EERC97/14, Berkeley, California. Gogus, A., and Wallace, J.W., 2013, “Fragility assessment of slab-column connections,” Earthquake Spectra, (in press). Hassan, W.M., 2011, Analytical and Experimental Assessment of Seismic Vulnerability of Beam-Column Joints without Transverse

Reinforcement in Concrete Buildings, Ph.D. Dissertation, University of California, Berkeley, California. Henkhaus, K., 2010, Axial Failure of Vulnerable Reinforced Concrete

Columns Damaged by Shear Reversals , Ph.D. Dissertation, Purdue University, West Lafayette, Indiana. ICBO, 1967, Uniform Building Code, International Council of Building Officials, Whittier, California. ICBO, 1976, Uniform Building Code, International Council of Building Officials, Whittier, California. Johnston, R.G., 1973, “Veterans Administration Hospital,” in San Fernando,

California, Earthquake of February 9, 1971 , Vol. I, Pt. B, N.A. Benfer and J.L. Coffman, Editors, L.M. Murphy, Scientific Coordinator, National Oceanic and Atmospheric Administration, Washington, D.C. LaFave, J.M., Bonacci, J.F., Burak, B., and Shin, M., 2005, “Eccentric beamcolumn connections,” Concrete International, Vol. 27, No. 9, pp. 5862. Lew, H.S., Leyendecker, E.V., and Dikkers, R.D., 1971, Engineering Aspects

of the 1971 San Fernando Earthquake, National Bureau of Standards, Building Science Series 40, Washington, D.C. Li, Y., Elwood K.J., and Hwang S.-J., 2014, Assessment of ASCE/SEI 41

Concrete Column Provisions using Shaking Table Tests, American Concrete Institute, ACI Special Publication 297, Farmington Hills, Michigan. Lynn, A., 1999, Seismic Evaluation of Existing Reinforced Concrete Building

Columns, Ph.D. Thesis, University of California, Berkley, California.

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Melchers, R.E., 1999, Structural Reliability Analysis and Prediction, Second Edition, John Wiley and Sons Ltd., West Sussex, England. Melek, M., and Wallace, J.W., 2004, “Cyclic behavior of columns with short lap splices,” ACI Structural Journal, Vol. 101, No. 6, pp. 802-811. Moehle, J., 2014, Seismic Design of Reinforced Concrete Buildings, McGraw-Hill, New York, New York. Mori, Y., and Ellingwood, B.R., 1993, “Reliability-based service-life assessment of aging concrete structures,” Journal of Structural

Engineering, Vol. 119, No. 5, pp. 1600-1621. Murphy, L. M., Steinbrugge, K. V., and Duke, C. M., 1973, “The San Fernando Earthquake,” in San Fernando, California, Earthquake of February 9, 1971, Vol. I, Part A, L.M. Murphy, Scientific Coordinator, National Oceanic and Atmospheric Administration, Washington, D.C. NEES, 2010, Mitigation of Collapse Risk in Older Concrete Buildings, Grand Challenge Research, Pacific Earthquake Engineering Research Center and the Network for Earthquake Engineering Simulation, http://peer.berkeley.edu/grandchallenge/index.html, last accessed November 5, 2014. NIST, 2010a, Concrete Model Building Subtypes Recommended for Use in

Collecting Inventory Data, NIST GCR 10-917-6, prepared by the Building Seismic Safety Council for the National Institute of Standards and Technology, Gaithersburg, Maryland. NIST, 2010b, Program Plan for the Development of Collapse Assessment

and Mitigation Strategies for Existing Reinforced Concrete Buildings, NIST GCR 10-917-7, prepared by the NEHRP Consultants Joint Venture, a partnership of the Applied Technology Council and the Consortium of Universities for Research in Earthquake Engineering, for the National Institute of Standards and Technology, Gaithersburg, Maryland. NIST, 2014, Review of Past Performance and Further Development of

Modeling Techniques for Collapse Assessment of Existing Reinforced Concrete Buildings, NIST GCR 14-917-28, prepared by the NEHRP Consultants Joint Venture, a partnership of the Applied Technology Council and the Consortium of Universities for Research in Earthquake Engineering, for the National Institute of Standards and Technology, Gaithersburg, Maryland.

C-4

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Pessiki, S.P., Conley, C.H., Gergely, P., and White, R.N., 1990, Seismic

Behavior of Lightly-Reinforced Concrete Column and Beam-Column Joint Details, National Center for Earthquake Engineering Research Technical Report, NCEER-90-0014, Buffalo, New York. USGS, 2008, NSHM Figures, Earthquake Hazards Program, U.S. Geological Survey, http://earthquake.usgs.gov/hazards/products/conterminous/ 2008/maps/, last accessed November 5, 2014. Woods, C., and Matamoros, A., 2010, “Effect of longitudinal reinforcement ratio on the failure mechanism of R/C columns most vulnerable to collapse,” Proceedings, The 9th U.S. National Conference and 10th Canadian Conference on Earthquake Engineering, Toronto, Canada.

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References

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Project Participants ATC Management and Oversight Christopher Rojahn (Project Executive) Applied Technology Council 201 Redwood Shores Parkway, Suite 240

Anna Olsen (Research Applications Manager) Applied Technology Council 201 Redwood Shores Parkway, Suite 240

Redwood City, California 94065

Redwood City, California 94065

Jon A. Heintz (Program Manager) Applied Technology Council 201 Redwood Shores Parkway, Suite 240 Redwood City, California 94065

FEMA Project Officer

FEMA Technical Monitor

Michael Mahoney Federal Emergency Management Agency 500 C Street, SW, Room 416 Washington, DC 20472

Robert D. Hanson Federal Emergency Management Agency 5885 Dunabbey Loop Dublin, Ohio 43017

Project Technical Committee William T. Holmes (Project Technical Director) Rutherford + Chekene 55 Second Street, Suite 600

Jack P. Moehle University of California, Berkeley Dept. of Civil and Environmental Engineering

San Francisco, California 94105

760 Davis Hall Berkeley, California 94720

Abbie Liel University of Colorado, Boulder Dept. of Civil, Environ. and Architectural Engin. ECOT 441, UCB 428 Boulder, Colorado 80309

Peter Somers Magnusson Klemencic Associates 1301 Fifth Avenue, Suite 3200 Seattle, Washington 98101

Michael Mehrain AECOM 915 Wilshire Boulevard, Suite 700 Los Angeles, California 90017

Project Review Panel Craig Comartin (Chair) CDComartin Inc. 535 La Honda Drive

Michael Cochran Weidlinger Associates 4551 Glencoe Avenue, Suite 350

Aptos, California 95003

Marina del Rey, California 90292

ATC-78-3

Project Participants

D-1

Gregory G. Deierlein Stanford University Dept. of Civil and Environmental Engineering 473 Via Ortega, Room 314 Stanford, California 94305 Ken Elwood The University of Auckland Dept. of Civil and Environmental Engineering 3 Grafton Road Auckland, New Zealand

Robert Pekelnicky Degenkolb Engineers 235 Montgomery Street, Suite 500 San Francisco, California 94104 John W. Wallace University of California, Los Angeles Dept. of Civil and Environmental Engineering 5731 Boelter Hall Los Angeles, California 90095

Terry Lundeen Coughlin Porter Lundeen, Inc. 801 Second Avenue, Suite 900 Seattle, Washington 98104

Working Group Members Panagiotis Galanis University of California, Berkeley 760 Davis Hall Berkeley, California 94720

Travis Marcilla University of Colorado, Boulder ECOT 441, UCB 428 Boulder, Colorado 80309

Cody Harrington University of Colorado, Boulder ECOT 441, UCB 428 Boulder, Colorado 80309

Siamak Sattar University of Colorado, Boulder ECOT 441, UCB 428 Boulder, Colorado 80309

Workshop Participants – November 18, 2014, Burlingame, California Russell Berkowitz Gregory G. Deierlein (Invited Participant) (Project Review Panel) Forell/Elsesser Engineers, Inc Stanford University 160 Pine Street, Suite 600 473 Via Ortega, Room 314 San Francisco, California 94111 Stanford, California 94305 Michael Cochran (Project Review Panel) Weidlinger Associates, Inc. 4551 Glencoe Avenue, Suite 350 Marina del Rey, California 90292

Kenneth Elwood (Project Review Panel) The University of Auckland 3 Grafton Road Auckland, New Zealand

Craig D. Comartin (Project Review Panel) CDComartin, Inc. 535 La Honda Drive

Robert Hanson (FEMA Subject Matter Expert) 5885 Dunabbey Loop Dublin, Ohio 43017

Aptos, California 95003

D-2


ATC-78-3

Cody Harrington (Project Working Group) University of Colorado, Boulder ECOT 441, UCB 428 Boulder, Colorado 80309

Michael Mehrain (Project Technical Committee) AECOM 915 Wilshire Boulevard, Suite 700 Los Angeles, California 90017

William T. Holmes (Project Technical Director) Rutherford + Chekene 55 Second Street, Suite 600 San Francisco, California 94105

Jack Moehle (Project Technical Committee) University of California, Berkeley 760 Davis Hall Berkeley, California 94720

Ifa Kashefi (Invited Participant) Los Angeles Department of Building and Safety 201 North Figueroa Street, Suite 1080 Los Angeles, California 90012

Farzad Naeim (Invited Participant) Farzad Naeim, Inc. 32 Mapleton Irvine, California 92620

Colin Kumabe (Invited Participant) Los Angeles Department of Building and Safety 201 North Figueroa Street, Suite 1080 Los Angeles, California 90012

Anna H. Olsen Applied Technology Council 201 Redwood Shores Parkway, Suite 240 Redwood City, California 94065

Abbie B. Liel (Project Technical Committee) University of Colorado at Boulder ECOT 441, UCB 428 Boulder, Colorado 80309 Terry Lundeen (Project Review Panel) Coughlin Porter Lundeen, Inc. 801 Second Avenue, Suite 900 Seattle, Washington 98104 Michael Mahoney Federal Emergency Management Agency 500 C Street, SW, Room 416 Washington, DC 20472 Travis Marcilla (Project Working Group) University of Colorado, Boulder ECOT 441, UCB 428 Boulder, Colorado 80309

Robert Pekelnicky (Project Review Panel) Degenkolb Engineers 235 Montgomery Street, Suite 500 San Francisco, California 94104 Christopher Rojahn Applied Technology Council 201 Redwood Shores Parkway, Suite 240 Redwood City, California 94065 Siamak Sattar (Invited Participant) National Institute of Standards and Technology 100 Bureau Drive, Building 226 Gaithersburg, Maryland 20899 Peter Somers (Project Technical Committee) Magnusson Klemencic Associates 1301 Fifth Avenue, Suite 3200 Seattle, Washington 98101-2699 John W. Review WallacePanel) (Project University of California, Los Angeles 5731 Boelter Hall Los Angeles, California 90095

ATC-78-3


D-3

ATC 78 3 Methodology Concrete Frames_083115

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