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Practical methods for critical load determination and stability evaluation of steel structures

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Title:
Practical methods for critical load determination and stability evaluation of steel structures
Creator:
Fernandez, Pedro ( author )
Language:
English
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1 electronic file. : ;

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Load factor design ( lcsh )
Load factor design ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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The critical load of a column, compression member or structure, calculated from a linear elastic analysis of an idealized perfect structure, does not necessary correspond with the load at which instability of a real structure occurs. This calculated critical load does not provide sufficient information to determine when failure, due to instability of the structure as a whole, will occur. To obtain this information it is necessary to consider the initial geometrical imperfections, eccentricities of loading, and the entire nonlinear load deflection behavior of the structure. However, this process in determining the critical load is too cumbersome and time consuming to be used in practical engineering applications. With today's computer programs that allow for the analysis of complex structures in which they incorporate advanced analytical techniques such as step-by-step large deformation analysis, buckling analysis, progressive collapse analysis, etc., it is just a natural progression that many structures are now analyzed with these tools. The goal of this research is to propose a practical methodology for critical load determination and stability evaluation of structures that are difficult or impossible to analyze with conventional hand-calculation methods, (e.g. the compression cord of a truss pedestrian bridge or a wind girt). The proposed methodology relies on a computer software package that has the ability to perform a second-order analysis taking in consideration end-restraints, reduced flexural stiffness (due to residual stresses in steel or cracked sections in concrete) and initial geometrical imperfections. Further, and importantly, a testing scheme was developed to validate the results from the computer in order to verify the methodology as a practical approach.
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Thesis (M.S.)--University of Colorado Denver.
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Includes bibliographic references.
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Department of Civil Engineering
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by Pedro Fernandez.

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PRACTICAL METHODS FOR CRITICAL LOAD DETERMINATION AND STABILITY EVALUATION OF STEEL STRUCTURES By PEDRO FERNANDEZ B.S., Instituto Tecnologico y de Estudios Superiores de Occidente, 1992 A Thesis submitted to the Faculty of the of the Graduate School of the University of Colorado in partial fulfillment Of the requirements for the degree of Master of Science Civil Engineering 2013

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ii This thesis for the Mast er of Science degree by Pedro Fernandez has been approved for the Civil Engineering Program by Fredrick Rutz, Chair Kevin Rens Chengyu Li June 12, 2013

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iii Fernandez, Pedro. (M.S., Civil Engineering) Practical Methods for Critical Load Determ ination and Stability Evaluation of Steel Structures Thesis directed by Assistant Professor Fredrick Rutz ABSTRACT The critical load of a column, compre ssion member or structure, calculated from a linear elastic analysis of an ideali zed perfect structure, does not necessary correspond with the load at which instability of a real structure occurs. This calculated critical load does not provide sufficient information to determine when failure, due to instability of the structure as a whole, will occur. To obtain this information it is necessary to consider the initial geometrical imperfections, eccentricities of loading, and the entire nonlinear load deflection behavior of the structure. However, this process in determ ining the critical load is too cumbersome and time consuming to be used in practical engineer ing applications. With todayÂ’s computer programs that allow for the analysis of complex structures in which they incorporate advanced analytical techniques such as step-bystep large deformation analysis, buckling an alysis, progressive co llapse analysis, etc., it is just a natural progression that many structures are now analyzed with these tools. The goal of this research is to propo se a practical methodology for critical load determination and stability evaluatio n of structures that are difficult or impossible to analyze with conventiona l hand-calculation methods, (e.g. the

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iv compression cord of a truss pedestrian bridge or a wind girt). The proposed methodology relies on a computer software pack age that has the ability to perform a second-order analysis taking in consideration end-restraints reduced flexural stiffness (due to residual stresses in steel or cr acked sections in c oncrete) and initial geometrical imperfections. Further, and im portantly, a testing scheme was developed to validate the results from the computer in order to verify the methodology as a practical approach. The format and content of this ab stract are approved. I recommend its publication. Approved: Fredrick R. Rutz

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v DEDICATION I dedicate this work to Mary Lynne, my wife, my parents, my brother, Coco and Nina ...and in the loving memory of my grandparents

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vi ACKNOWLEDGMENTS I would like to thank first and foremost Dr. Fredrick Rutz for the support and guidance in completion of this thesis. I would al so like to thank Dr. Rens and Dr. Li for participating on my graduate advisory committee. Lastly, I would like to thank Paul Jones, the artist, who built the test models with so much precision and passion, as well as for being my fearless load operator.

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vii TABLE OF CONTENTS CHAPTER I. INTRODUCTION .................................................................................................. 1Research Program Objectives ........................................................................... 4Outline of Research ........................................................................................... 6II. A HISTORICAL APPROACH ON STRUCTURAL STABILITY ...................... 9Euler .................................................................................................................. 9Effective Length Factors ................................................................................. 11Inelastic Buckling Concepts ........................................................................... 16Tangent Modulus ................................................................................ 19Reduced Modulus ............................................................................... 21Shanley Theory ................................................................................... 22Amplification Factors ..................................................................................... 23Braced Frames .................................................................................... 25Unbraced Frames ................................................................................ 28Column Strength Curves ................................................................................. 30Summary of Present State of Knowledge ........................................... 33III. COLUMN THEORY ........................................................................................... 34Mechanism of Buckling .................................................................................. 35Critical Load Theory ....................................................................................... 37Euler Buckling Load ........................................................................... 40Critical Load of Beam-Columns ......................................................... 45

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viii Second Order Effects .......................................................................... 49Inelastic Buckling of Structures .......................................................... 52Factors Controlling Column Strength and Behavior ...................................... 54Material Properties .............................................................................. 55Length ................................................................................................. 56Influence of Support Conditions ......................................................... 57Moment Frames ...................................................................... 57Members with Elastic La teral Restraints ................................ 61Influence of Imperfections .................................................................. 64Material Imperfections ............................................................ 65Geometrical Imperfections ...................................................... 68IV. METHODS AND PROCEDURES FOR ANALYSIS AND DESIGN OF STEEL STRUCTURES ...................................................................................... 71Structural and Stability Analysis .................................................................... 71First-Order Elastic Analysis ................................................................ 73Elastic Buckling Load ......................................................................... 74Second-Order Elastic Analysis ........................................................... 74First-Order Inelastic Analysis ............................................................. 75Second-Order Inelastic Analysis ......................................................... 76Design of Compression Members ................................................................... 76Determining Required Strength .......................................................... 79Determining Available Strength ......................................................... 80Compression Strength ............................................................. 82

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ix Combined Forces .................................................................... 85Current Stability Requirement s (AISC Specifications) .................................. 87The Effective Length Method ............................................................. 89Direct Analysis Method ...................................................................... 90Imperfections .......................................................................... 91Reduced Flexural and Axial Stiffness ..................................... 92Advanced Analysis ............................................................................. 93Pony Truss Bridges ......................................................................................... 96V. STABILITY ANALYSIS USING NONLINEAR MATRIX ANALYSIS WITH COMPUTER SOFTWARE ................................................................................ 99Matrix Structural Analysis .............................................................................. 99Direct Stiffness Method .................................................................... 101Nonlinear Analysis using Matrix Methods ....................................... 102Computer Software Used .............................................................................. 105Pand P....................................................................................... 106Modeling Geometrical Imperfections ............................................... 108Assessment of the Computer Softwa re with Benchmark Problems from Established Theory........................................................................................ 109VI. A PRACTICAL METHOD FOR CRITICAL LOAD DETERMINATION OF STRUCTURES ................................................................................................. 112Proposed methodology: Step -by-step process .............................................. 113VII. COMPUTER SOFTWARE EVALUATION WITH BENCHMARK PROBLEMS ..................................................................................................... 116Benchmark Problem 1................................................................................... 118Benchmark Problem 2................................................................................... 126

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x Discussion of Results from Benchmark Problems........................................ 133VIII. MODEL DEVELOPMENT ............................................................................. 134Full-Size Analytical Study ............................................................................ 135Experimental Study ....................................................................................... 144Theoretical Models ........................................................................... 145Bridge 1 ................................................................................. 145Bridge 2 ................................................................................. 146Loading ................................................................................. 149Geometric Imperfections ...................................................... 150Procedure .............................................................................. 151Test Models ....................................................................................... 152Test Setup.............................................................................. 155Load Testing Procedure ........................................................ 159IX. EXPERIMENTAL RESULTS ......................................................................... 161Bridge 1 Results ............................................................................................ 164Theoretical Results............................................................................ 164Load Test Results .............................................................................. 168Bridge 2 Results ............................................................................................ 177Theoretical Results............................................................................ 177Load Test Results .............................................................................. 181X. CONCLUSIONS ............................................................................................... 187Research Overview ....................................................................................... 187Research Conclusions ................................................................................... 188

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xi General Recommendations ........................................................................... 189REFERENCES ......................................................................................................... 191APPENDIX 1: TEST MODELS PLANS ................................................................. 197

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xii LIST OF FIGURES Figure 1.1 Pony truss bridge (Excelbridge 2012) ..................................................................... 32.1 (a) Cantilever Column, (b) Pinned End Column .................................................. 102.2 Effective Column Lengths. (a) Pinned End, (b) Cantilever and (c) Fixed-end .... 122.3 Effective Length Factors K for centrally loaded columns (Adapted from Chen 1985) .................................................................................................................... 142.4 Effective length of a moment frame with joint transla tion (Adapted from Geschwindner 1994) ............................................................................................ 142.5 Effective length factors (Adapted from AISC 2005) ............................................ 162.6 Column Critical Stress for va rious slanderness ratios ( L/r ) (Adapted Gere 2009) 172.7 Stress-strain diagram for typical structur al steel in tension (Adapted from Gere 2009) .................................................................................................................... 182.8 Definition of tangent modulus for nonlinear material (Adapted from Geschwindner 1994) ............................................................................................ 202.9 Stress distributions for elastic buckling: (a) tangent modulus theory; (b) reduced modulus theory (Adapted from Geschwindner 1994) ......................................... 222.10 Load-deflection behavior for elastic a nd inelastic buckling (Adapted from Gere 2009) .................................................................................................................... 232.11 Axial loaded column with equal a nd opposite end moments (Adapted from Geschwindner 1994) ............................................................................................ 262.12 Amplified moment: exact and approxi mate (Adapted from Geschwindner 1994) ............................................................................................................................. 272.13 Structure second-order effect: sway (Adapted from Geschwindner 1994) ........ 282.14 Column strength variation (A dapted from Ziemian, 2010) ................................ 323.1 Elastic Stability (Adapted from Gere 2009) ......................................................... 36

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xiii 3.2 Column Stability (Adapted from Chen 1985) ....................................................... 383.3 Beam-Column Stability (Ada pted from Chen 1985) ............................................ 393.4 Pin-ended supported column................................................................................. 413.5 Euler Curve ........................................................................................................... 453.6 Beam-Column (Adapted from Chen 1985) ........................................................... 463.7 Elastic analysis of initially deformed column (Adapted from Chen 1985) .......... 493.8 Second-order effects ............................................................................................. 523.9 Residual stresses and propor tional stress (Chen 1985) ......................................... 543.10 Typical stress-strain curves for different steel classifications ............................ 563.11 Theoretical column strength (Gere 2009) ........................................................... 573.12 Typical load-deflection curves for columns with different framing beam connections (Adapted from Ziemian 2010) ......................................................... 583.13 Column strength curves for members with different fram ing beam connection (Adapted from Ziemian 2010) ............................................................................. 593.14 Connection flexibility (Ada pted from AISC 2005) ............................................ 593.15 Pony truss and analogous top chord (Adapted from Ziemian 2010) .................. 633.16 Influence of imperfections on column behavior (Adapted from ASCE Task Committee 1997) ................................................................................................. 653.17 Column curve for structural steels. (a ) elastic-perfectly pl astic stress-strain relationship; (b) column curve (Adapted from Chen 1985) ................................ 663.18 Lehigh residual stress pattern (A dapted from Surovek 2012) ............................ 684.1 Comparison of load-deflection behavior for different analysis methods (Adapted from Geschwindner 1994) ................................................................................... 734.2 Distinction between effective length and notional load approaches (Adapted from Ziemian 2010). ..................................................................................................... 814.3 Standard column curve (Ada pted from W illiams 2011). ...................................... 85

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xiv 4.4 Interaction curve (Adapted from Williams 2011) ................................................. 874.5 Lateral U-Frame (Adapted from AASHTO 2009) ................................................ 975.1 Schematic representation of incrementaliterative solution procedure. (Ziemian 2011) .................................................................................................................. 1055.2 RISA second-order effects proces s (Adapted from RISA-3D 2012) .................. 1075.3 Benchmark Problem 1 (AISC 2005) ................................................................... 1105.4 Benchmark Problem 2 (AISC 2005) ................................................................... 1116.1 Error message from RISA-3D that Pis no longer converging ....................... 1157.1 Benchmark Problem 1 (AISC 2005) ................................................................... 1177.2 Benchmark Problem 2 (AISC 2005) ................................................................... 1177.3 Benchmark problem 1: a) undeflec ted shape b) deflected shape ........................ 1217.4 Maximum moment values as a function of axial force for benchmark problem 1 (classical solution, Equation 7.2) ....................................................................... 1227.5 Maximum deflection values as a function of axial force for benchmark problem 1 (classical solution, Equation 7.3) ....................................................................... 1227.6 Benchmark problem 1 node arraignments a) Case 1, one additional node at mid height of the beam-column b) Case 2, two additional nodes at equal distance c) Case 3, three additional node s at equal distance ................................................ 1237.7 Maximum moment values as a function of axial force for benchmark problem 1 (classical solution, with one, tw o and three additional nodes) .......................... 1247.8 Maximum deflection values as a function of axial force for benchmark problem 1 classical solution, with one, two and three additional nodes) ............................ 1257.9 Benchmark problem 2, a) undeflecte d shape b) deflected shape ........................ 1277.10 Maximum moment values as a function of axial force for benchmark problem 1 (classical solution, Equation 7.5) ....................................................................... 1297.11 Maximum deflection values as a functi on of axial force for benchmark problem 1 (classical solution, Equation 7.6) .................................................................... 129

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xv 7.12 Benchmark problem 2 node arraignments a) Case 1, no additional node along the height of the beam-column b) Case 2, one additional node at mid height of the beam-column c) Case 3, two additi onal nodes at equal distance ...................... 1307.13 Maximum moment values as a function of axial force for benchmark problem 2 (classical solution, with zero, one and two additional nodes) ........................... 1317.14 Maximum deflection values as a functi on of axial force for benchmark problem 2 (solution, with zero, one and two additional nodes) ....................................... 1328.1 Actual 40 foot long pony truss bridge................................................................. 1378.2 Computer model of the 40 foot long bridge ( analytical model ) ......................... 1378.3 Parametric study.................................................................................................. 1398.4 Geometric imperfection ...................................................................................... 1408.5 Deformed shape at critical load. Slender top chord ............................................ 1428.6 Deformed shape at critical lo ad. Stiffer compression chord ............................... 1428.7 Bridge 1 truss geometry ...................................................................................... 1478.8 Bridge 1 computer model of truss ....................................................................... 1478.9 3D Model of Bridge 1 ......................................................................................... 1478.10 Bridge 2 truss geometry .................................................................................... 1488.11 Bridge 2 computer model of truss ..................................................................... 1488.12 3D model of Bridge 2 ....................................................................................... 1488.13 Load test setup diagram .................................................................................... 1498.14 Bridge 1 computer model loading configuration .............................................. 1508.15 Bridge 2 computer model loading configuration ............................................. 1508.16 Modeling of geometric imperfections ............................................................... 1518.17 Bridge 1 Computer Model ................................................................................ 1538.18 Bridge 1 Test model .......................................................................................... 153

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xvi 8.19 Bridge 2 Computer Model ................................................................................ 1548.20 Bridge 2 Test Model ......................................................................................... 1548.21 Load test setup diagram .................................................................................... 1568.22 Load test setup .................................................................................................. 1568.23 Spreader beam and plates used to uni formly distribute the load on the bridge deck .................................................................................................................... 1578.24 Typical end support configuration and abutments ............................................ 1578.25 Load cell............................................................................................................ 1588.26 Data collection setup ......................................................................................... 1598.27 Test setup top view ........................................................................................... 1609.1 Bridge 1 Naming convention for top chords ....................................................... 1629.2 Bridge 2 Naming convention for the top chords ................................................. 1629.3 Top chord initial imperfection, o (a) Top view of compression chord (b) Crosssection view ....................................................................................................... 1639.4 Bridge 1 Theoretical results ................................................................................ 1679.5 Bridge 1 Theoretical and load test results ........................................................... 1719.6 Bridge 1 Deformed shape at failure .................................................................... 1729.7 Bridge 1 Theoretical (fix-ended diagona ls) and load test results ...................... 1749.8 Deformation of end-diagonal prio r to buckling (test model) .............................. 1759.9 Deformation of end-diagonal (comput er model) at critical load ........................ 1759.10 Deformation of end-diagonal at buckling (test model) ..................................... 1769.11 Deformation of end-diagonal (comput er model) at critical load ...................... 1769.12 Bridge 2 Theoretical results .............................................................................. 1809.13 Bridge 2 Theoretical and load test results ......................................................... 184

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xvii 9.14 Deformed shape at 53% of the critical load ...................................................... 1859.15 Deformed shape at 78% of critical load ............................................................ 1859.16 Deformed shape at 91% of the critical load ...................................................... 1869.17 Deformed shape at critical load ........................................................................ 186

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xviii LIST OF TABLES Table 8.1Parametric Study of Analytical Models ............................................................... 1439.1 Measured initial imperfections, o, of top chords ............................................. 1639.2 Bridge 1 Theoretical Results: Top chord ratio / o ........................................... 1669.3 Bridge 1 Theoretical Results: Critical load ......................................................... 1669.4 Bridge 1 Theoretical and Load Test Results: Top chord ratio / o .................. 1709.5 Bridge 1 Theoretical and Load Test Results: Critical load ................................. 1709.6 Bridge 1 Ratio / o, theoretical results with fix-ended diagonals .................... 1739.7 Bridge 1 Critical load, theore tical results with fi x-ended diagonals ................... 1739.8 Bridge 2 Theoretical Results: Top chord ratio / o .......................................... 1799.9 Bridge 2 Theoretical Results: Critical load ......................................................... 1799.10 Bridge 2 Theoretical and Load Test Results: Top chord ratio / o ................ 1839.11 Bridge 2 Theoretical and Load Test Results: Critical load ............................... 183

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1 CHAPTER I 1 INTRODUCTION A research project on stability ca nnot start without recognizing the contribution of Euler to the problem of stability when he published his famous EulerÂ’s equation on the elastic stability of columns back in 1744. And even though the research on the stability of structures can be traced back to 264 years ago since EulerÂ’s contribution to column stability, prac tical solutions are still not available for some types of structures. Stab ility is one of the most cr itical limit states for steel structures during construction and during thei r service life. One of the most difficult challenges in structural stability is determining the critical load under which a structure collapses due to the loss of stability; this is because of the complexity of this phenomenon and the many material propertie s that are influenced by geometric and material imperfections and material nonlinearity. In addition to the challenges mentioned above, the advancement in industr ial processes in hot -rolled members and the use of high strength steel, which provides a competitive design solution to structural weight reduction, has resulted in increase of memb er slenderness, structural flexibility and therefore more vulnerability to instability. The method proposed in this research pr oject for the evaluation of the lateral stability of structures, and to determine th eir critical load at which they become instable is an innovative approach. It enable s the prediction of th e characteristics of stability of structures by capturing the load-deformation history and the deformed

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2 shape of the structure at the critical load. Such critical information is generally not available through the current stability analysis methods. A lthough the method is oriented for analysis of 3D structures, and structures that are di fficult to analyze with conventional hand-calculation methods, it ca n be used for any 2D structure. Lateral stability of steel structures that are difficult to analyze with conventional hand-calculation methods in clude: columns under axial compression load and biaxial bending in 3D frames, stepped columns, multi-story columns of different lengths and sections, the unbraced compression chord of a wind-girt truss, and the top chord of a pony truss which due to vertical clearance requirements prohibit direct latera l bracing. Pony truss structures, while no longer as common for construction of new highway or rail bridge s, are commonly used in applications similar to Figure 1.1 or as walkways and conveyor systems. The analysis of the compression c hord of a pony truss is typically accomplished by treating the chord as a colu mn with elastic supports (Ziemian 2010). This column, with intermediate elastic restraints, will buckle in a number of halfwaves depending on the stiffne ss of the elastic restraints. The deformed shape or buckled shape of the column will fall some where between the extreme limits of a half-wave of length equating the length of the chord an d a number of half-waves equating the number of spans between the end restraints, that is, the number of panels. From the buckled shape, the effec tive length of the comp ression chord can be

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3 used to determine the critical load. Th e method on how to determine the effective length has long been the focus of th e compression chord buckling problem. Figure 1.1 Pony truss bridge This research will focus primarily on applying the proposed methodology for the determination of the critic al load of pony truss bridges. However, it can be used for any other 3D structures, including fram ed structures. All structural frames in reality are geometrically imperfect, that is lateral deflection commences as soon as the loads are applied. There are typically tw o types of geometric imperfections that are taken into consideration for the stability analysis of structures ; out-of-straightness, which is a lateral deflection of the column relative to a straight line between its end

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4 points, and column out-of-plumbness, which is lateral displacement of one end of the column relative to the other. In the absen ce of more accurate information, evaluation of imperfection effects should be based on the permissible fabrication and erection tolerances specified in the appropriate building code. In the U.S., the initial geometric imperf ections are assumed to be equal to the maximum fabrication and erection tolerances permitted by the AISC Code of Standard Practice for Steel Building and Bridge, AISC 303-10 (AISC 2010). For columns and frames, this implies a member out-of-straightness equal to L /1000, where L is the member length brace or frami ng points, and a frame out-of-plumbness equal to H /500, where H is the story height. For the critical load determination and load-deformation history evaluation in this re search project, three imperfections will be considered: L /300, L /450 and L /700. Research Program Objectives The goal of this research project is to propose a pract ical method for stability evaluation and assessment of structures that are difficult or impossible to analyze with conventional hand-calculation methods The proposed methodology relies on a commercially available computer program in which a second order analysis can be readily accomplished by taking into consideration end-restraints, reduced flexural stiffness, and initial geometrical imperf ections or out-of-plumbness. The method can be used for any frame 3D structure or truss. The specific objectives of this research project are:

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5 1. Develop an analytical process that is based on an iterative approach (with the help of a 3D computer program) that would facilitate the determination of the critical load and stability assessment of steel structures that are difficult or impo ssible to analyze with conventional hand-calculation methods. 2. Investigate the correctness of the computer program to be able to perform a rigorous second order analysis, by using benchmark problems from established theory, which are presented in AISC 36005. 3. Conduct an analytical in vestigation, using the proposed approach, to determine the critical load, the effects of initial imperfections, and the characteristics of the failure m ode of full size pony truss models. 4. Develop two scaled pony truss bridge s based on the results from item 3 that can be load tested with available equipment. 5. Apply the proposed methodology and determine, analytically, the load-deformation history and critical load of the two scaled pony truss bridges developed in item 4. 6. Develop a testing scheme in whic h two scaled pony truss bridges are load tested. The testing scheme must include means to measure both of

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6 the top chords out of plane deforma tions and to determine the load at any instant of the load application. 7. Verify the proposed methodology by comparing the results between the computer model results and the results from the testing scheme. Outline of Research In general terms, this research project consisted of a literature review, development of a methodology for critical lo ad determination of structures, and a testing scheme to validate the proposed me thodology. This thesis is organized in the following manner: Chapter II presents a historical back ground on column stability, from EulerÂ’s early work on column stability and elastic critical load to the development of the column strength curves used in todayÂ’s co lumn strength equations. This historical background includes an introduction to the conc ept of equivalent or effective length, a review of different theories that have b een used to predict column strength and an introduction on the derivation of the amplification factors. In Chapter III, the mechanism of buckling is presented along with the derivation of the critical load for an ideal column with pin supports. An introduction to structural stability of beam-c olumns and second order effects ( Pand P) is discussed. In this Chapter, some of the more critical factors that control column

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7 strength and behavior are reviewed. This re view focuses on the effects of geometrical and material imperfections. The determination of member sizes for a structure, in general terms, is a two step process. First, a stru ctural and stability analysis is performed in order to determine the required strength. Second, empi rical equations are used to determine the available strength and then compared to the required strength. Chapter IV presents an overview of the different methods for stab ility analysis, as well as the procedures and requirements for the determination of the available strength in AISC Specifications for Structural Steel Buildings, AISC 360-05 (AISC 2005) and AASHTO LRFD Guide Specifications for the Design of Pedestrian Bridges, (AASHTO 2012). Because the proposed methodology require s the use of a computer program, Chapter V presents an introduction to the methods of analysis in most computer software packages. Also in this Chapter, an introduction to the capabilities of the computer software package (RISA-3D) is presented as well as a discussion of its ability to perform a rigorous second-order analysis. In Chapter VI the proposed methodology is presents as a step by step process with the goal of determining the load-def ormation history and critical load of a structure. This process addresses the sele ction of the computer software package, modeling, the incorporation of geometrica l imperfections in the model and the determination of th e critical load.

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8 However, it is necessary to evaluate the ability of the software package to perform a rigorous second order analysis. Gi ven in Chapter VII are the results from the benchmark problems used for the evalua tion of RISA-3D. Furthermore, in this chapter, a discussion is give n on the assessment of the Pcapabilities of the program. The development of the testing scheme is given in Chapter VIII and the results are presented in Chapter IX. C onclusions of the current research and recommendations are presented in Chapter X.

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9 CHAPTER II 2 A HISTORICAL APPROACH ON STRUCTURAL STABILITY Euler Any work on structural stability cannot start without a discussion on Leonard Euler (1707-1783), for many, Euler is consid ered the greatest ma thematician of all time (Gere 2009). Euler made gr eat contributions to the fiel d of mathematics, namely, trigonometry, differential and in tegral calculus, infinite series, analytic geometry, differential equations, and many other subjec ts (Gere 2009). However, with regard to applied mechanics, Euler made one of the mo st important contribu tions to structural engineering concerning with elas tic stability of stru ctures, and which is still subject of many studies after almost 270 years: the bu ckling of columns. He was the first to derive the formula for the critical buck ling load of an ideal, slender column. Euler’s work was published in 1744 under the title “Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes” His original work consisted on determining the buckling load of a cantilever column th at was fixed at the bottom and free at the top as shown in Figure 2.1(a) (Timoshenko 1983).

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10 Figure 2.1 (a) Cantilever Column, (b) Pinned End Column In this case, Figure 2.1(a), Euler shows that the equation of the elastic curve can be readily be solved and that the load at which the buckling occurs is given by Equation 2.1 2 24 C P l (2.1) Later on, Euler expanded his work on columns to include other restraint conditions, and determined the buckling lo ad for a column with pinned supports, Figure 2.1(b) as being 2 2 C P l (2.2)

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11 In his work, Euler did not discuss th e physical significance of the constant C which he calls the “absolute elasticity”, merely stating that it depends on the elastic properties of the material (Timoshenko 1983). On e can conclude that this constant is the flexural stiffness of the column, EI Euler was the first to recognize that columns could fail through bending rather than crushing. In his work he states: Therefore, unless the load P to be borne be greater than 224 Clthere will be absolutely no fear of bending; on the other hand, if the weight P be greater, the column will be unable to resist bending. Now when the elasticity of the column and likewise its thickness remain the same, the weight P which it can carry without danger will be inversely proportional to the square of the height of the column; and a column twice as high will be able to bear only one fourth of the load (Timoshenko, 1983). Effective Length Factors The Euler buckling load equation give s the critical load at which a pinsupported column will buckle, Figure 2.2(a) The critical load for columns with various support conditions, other than a pin-pi n support, can be related to this column through the concept of an “effective length”. For example, Figure 2.2(b) a fixed-end column buckles at 4 times the load for a pinned column, and because the Euler buckling load is directly proportional to the inverse square of the length, the effective length of a fixed-end column is one half of that of the same member with pinned ends. At the same time, Figure 2.2(c), a column that is fixe d at the base and free at the top will have an effective length of 2 L and will buckle at a quarter the load of the pinned-end column.

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12 Figure 2.2 Effective Column Lengths. (a ) Pinned End, (b) Cantilever and (c) Fixed-end Because the effective length, Le, for any column is the length of the equivalent pinned-end column, then, the general equation for critical loads, for any column with supports other than pin-pin, can be written as 22 2 2 e E IEI P L Kl (2.3) Where K is the effective length factor and is often included in design formulas for columns. That is, the Euler buckling load formula can be used to obtain the critical load of a column with different end conditions provided the correct effective length of the column is known, illustrated in Figure 2.3.

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13 In simplified terms the concept is merely a method of mathematically reducing the problem of evaluating the critical stress for columns in structures to that of equivalent pinned-end brace columns (Yura 1971). So far columns with ideal boundary conditions have been discussed; however, the effective length concept is equally va lid for any other set of boundary conditions not included in Figure 2.3, except that the determination for the effective length factor is not as simple or straig htforward as in Figure 2.3. For example, in the case of a column that is part of a moment frame, the effective length factor is a function of the stiffness of the beams and columns that frame into the joint. Columns or compression members can be classified based on the type of frame that they form a part. Columns that donÂ’t participate in the lateral stability of the structure are considered to be braced against sway. A braced frame is one for which sidesway or jo int translation is prevented by means of bracing, shear walls or lateral support from adjoining structures. An unbraced frame depends on the stiffness of its members and the rotational rigidity of the joints between the frame members to prevent lateral buckling as s hown in Figure 2.4. Theoretical mathematical analysis may be used to determine effective lengths; however, such procedures are typically too lengthy and unpractical for the average designers. Such methods include trial and error procedures with buckling equations.

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14 Figure 2.3 Effective Length Factors K for centrally loaded columns (Adapted from Chen 1985) Figure 2.4 Effective length of a moment frame with joint translation (Adapted from Geschwindner 1994)

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15 For normal design situations, it is impractic al to carry out a detailed analysis for the determination of buckling strength a nd effective length. Since the introduction of the effective length concep t in the AISC specifications for the first time in 1961 (AISC 1961), several procedures have been proposed for a more direct determination of the buckling capacity and effective lengt h of columns in most civil engineering structures (Shanmugam 1995). The most co mmon method for obtaining the effective length of a column or compression member and to account for the effects of the connected members is to employ the ali gnment charts or nomographs, originally developed by Julian and Lawrence (McCormac 2008). The alignment charts were developed from a slope-deflection analysis of a frame including the effect of column loads (McCormac 2008). Values of K for certain idealized end conditions are given in Figure 2.5. The values of K for columns in frames, depend on the flexural rigidity of adjoining members and the manner in which sidesway is resisted. The values given in Figure 2.5 are based on the assumption that all colu mns in a frame will reach their individual critical loads simultaneously. In certain cases, the effective length factors are not applied correctly given that the underlying assumptions of the method do not apply. The method involves a number of major assumptions which are summarized in section C2.2b of the Commentary of the AISC Specification (AISC 2005). Some of the most significant assumptions are: member s are assumed to behave elastically, all

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16 columns buckle simultaneously, all memb ers are prismatic and the structure is symmetric (AISC 2005). Figure 2.5 Effective length fact ors (Adapted from AISC 2005) Inelastic Buckling Concepts Figure 2.6 shows the theoretical stability behavior of columns as a function of the average compressive strength, P/A versus the slenderness ratio, l/r From this figure and depending on the type of failure, columns can be classified as short columns, intermediate columns, and long columns. Short or stocky columns fail by crushing while very long columns fail by elastic buckling as described by EulerÂ’s

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17 critical load equation. Euler determined that the critical load for a slender column does not necessary correspond to the crushing strength, but a lower values which is a function of the length of the column and the flexural rigi dity as discussed earlier. However, EulerÂ’s critical load for elastic buckling is valid only for relative long columns where the stress in the column is below the proportional limit (Gere 2009). Figure 2.6 Column Critical Stress for various slanderness ratios ( L/r ) (Adapted Gere 2009) Many practical columns are in a range of slenderness where at buckling, portions of the column are no longer linearly elastic, and thus one of the fundamental assumptions of the Euler column theory is violated due to a reduction in the stiffness of the column. This degrada tion of the stiffness may be the result of material

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18 nonlinearity or it may be due to partial yiel ding of the cross sec tion at certain points where compressive stresses pre-exists due to residual stress (Z iemian 2010). This range is considered as the range of the intermediate columns, thus, intermediate columns fail by inelastic buckling. For columns of intermediate length, the stress in the column will reach the proportional lim it before buckling begins. The proportional limit is the limit at which not only the modulus of elasticity or the ra tio of stress-strain is linear but also proportional, this can be seen in Figure 2.7. Various theories have been developed to account for this type of behavior in which columns are considered to be in the inelastic range, some of the theories are presented below. Figure 2.7 Stress-strain diagram for typical structural steel in tension (Adapted from Gere 2009)

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19 Tangent Modulus In the late 19th century, independent research work done by Engesser and Considre recognized the shortcomings of th e Euler theory, in which EulerÂ’s buckling load was based on an el astic material response (Geschwindner 1994). During that time it was already known that various materi als exhibited nonlinear characteristics. Engesser developed a theory in which he assu med that the modulus of elasticity at the instant of buckling was equal to the slope of the material stress-strain curve at the level of the buckling stress. In other words, if the load on a column is increased so that the stress is beyond the proportional lim it of the material, and this increase in load is such that a small increase in stress occurs, the relationship between the increment of stress and the corresponding increment of strain is given by the slope of the stress-strain curve at point A in Figure 2.8. This slope, which is equal to the slope of the tangent line at A, is called the tangent modulus and is designated by ET, (Geschwindner 1994) thus, Td E d (2.4) Engesser rationalized that the fibers in tension (unloading fibers) and the fibers in compression (loading fibers), bot h, would behave according to the tangent modulus as shown in Figure 2.9(a). Ther efore, the buckling load of a column corresponds to the load defined as the ta ngent modulus load. The buckling load can

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20 then be computed from the Euler buckli ng expression by substituting the modulus of elasticity with th e tangent modulus (Geschwindner 1994). 2 2 T T E I P l (2.5) Figure 2.8 Definition of tangent modulus for nonlinear material (Adapted from Geschwindner 1994) Because the tangent modulus, ET, varies with the compressive stress, = P/A the tangent modulus load is obtained by an iterative procedure. From Figure 2.9(a) one can think of the tangent modulus as the average m odulus of elasticity of the complete cross section. Some of the fibers w ill be in the elastic range and others will be in the inelastic range, in other words, some fibers are responding elastically in accordance with the full modulus of elasticity, E while other respond with a modulus of zero (Geschwindner 1994).

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21 Reduced Modulus The reduced modulus theory is also known as the double modulus, Er. When a column bends under loading, in addition to the axial load, bending stresses are added to the compressive stress, P/A due to the axial load. Th ese additional stresses are compression on the concave side and tensi on on the opposite side of the column. By adding the bending stresses to th e axial stresses, it can be seen that on the concave side the stresses in compression are larger than the stresses in tension. Therefore, it can be assumed that on the concave side, the material follows the tangent modulus, Figure 2.9(b) and in the convex side the mate rial follows the unloading line shown in Figure 2.8. One can think as if this column were made of two different materials (Gere 2009). As in the tangent modulus method, the critical load can be found by replacing the modulus of elasticity, in the Euler equation, by the reduced modulus or double modulus. However, the reduced modulus theo ry is difficult to use in practice because Er depends not only on the shape of the cros s section of the column but the stress strain curve of that specific column. This means it needs to be evaluated for every different column sec tion type (Gere 2009).

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22 Figure 2.9 Stress distributions for elastic buckling: (a) tangent modulus theory; (b) reduced modulus theory (Adapted from Geschwindner 1994) Shanley Theory Tests conducted by von Karman and ot hers over many years (Johnston 1981) have demonstrated that actual buckling lo ads tend to more closely follow the loads predicted by the tangent modulus equati on than those from the reduced modulus equation. In 1947, an American aeronautic al-engineering professor F.R. Shanley demonstrated that there would be no stress re versal in the cross section as the column reached the tangent modulus load because the initial deflection at this point was infinitesimal. Shanley proved that the tange nt modulus load was the largest load at which the column will remain straight (Geschwindner 1994). Once the load exceeded the tangent modulus load, there would be a stress reversal a nd elastic unloading would take place and the load carrying capacity is then predicted by the reduced

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23 modulus (Geschwindner 1994). In other words, a perfect inelastic column will begin to deflect laterally when P = PT and P < Pmax < PR, illustrated in Figure 2.10. Figure 2.10 Load-deflection behavior for el astic and inelastic buckling (Adapted from Gere 2009) Amplification Factors Second order effects are the direct result of structural deformations. In order to account for these secondary effects, a nonlinear or second order analysis must be carried out that takes into account equilibrium on the deformed shape of the structure, rather than on the original undeformed shap e as is in the case of the traditional classical methods of analysis. An altern ate approach for this more rigorous and complex method of analysis, is to determ ine the secondary eff ects by amplifying the moments and forces from a first order el astic analysis. The concept of moment

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24 magnification, to account for secondary effects, was introduced in the AISC Specifications in 1961 by including th e following expression (AISC 1961): 1 'm a eC f F (2.6) for the determination of combined stresses in the following equation 1.0 1 'amb a a b efCf F f F F (2.7) The commentary to the AISC Specification of 1961 explains the si gnificance of the new term: The bending stress at any cros s section subject to lateral displacement must be am plified by the factor 1 'm a eC f F (2.8) it recognizes the fact that such displacement, caused by applied moment, generates a secondary mome nt equal to th e product of the resulting eccentricity and the a pplied axial load, which is not reflected in the computed stress, fb (AISC 1961). There are two different magnification factors to account for the secondary effects; one magnification factor is used for columns that are part of a braced frame and a second magnification factor for columns that are part of an unbranced frame. A frame is considered braced if a system, su ch as shear walls or diagonal braces, serves

PAGE 43

25 to resist the lateral loads and to stabilize the frame under gravity loads. In a braced frame the moments in the column due to the gravity loads are determined through a first order elastic analysis, and the additio nal moments resulting from the deformation along the column length will be determ ined through the application of an amplification factor. Unbraced frames depend on the stiffness of the beams and columns for lateral stability under gravity loads and combined gravity and lateral loads. Unlike braced frames, discussed earlier, there is no suppl ementary structure to provide lateral stability. Columns that form part of an unbraced frame are subjected to both axial forces and bending moments and will experi ence lateral transla tion. The latter will cause an additional second order effect to the one discussed earlier. This second order effect is the result from the sway or lateral displacement of the frame. Braced Frames In his paper titled “A Practical Method of Second Order Analysis”, LeMessurier provides the full derivation of the amplification fact ors for both braced frames and unbraced frames (LeMessurieer 1976). A summary of the derivation is presented below. Consider the column in Figure 2.11, the maximum moment at mid height of the column is given by equation 2.9 max M MP (2.9)

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26 Figure 2.11 Axial loaded column with equa l and opposite end moments (Adapted from Geschwindner 1994) The moment magnification or amplif ication factor is given by maxM M P AF M M (2.10) Therefore, 1 1 AF P M P (2.11) For small deformations is sufficiently small that M PM (2.12) And assuming that (Ziemian 2010)

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27 2 228eMEIEI P LL (2.13) The moment magnification factor for a column that is part of a braced frame is given by Equation 2.14 1 1u eAF P P (2.14) Figure 2.12 shows the approximate amplif ication factor, derived above, for a pin-pin simply supported column with oppos ite concentrated moments at its ends. This approximate amplification factor is plotted alongside the exact solution. As it can be seen from Figure 2.12, the approxima te amplification factor is slightly conservative compared to the exact solution because 2 > 8. Figure 2.12 Amplified moment: exact and approximate (Adapted from Geschwindner 1994)

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28 Figure 2.13 Structure second-order effect: sw ay (Adapted from Geschwindner 1994) Unbraced Frames For the case of unbraced frames, and assuming that the lateral stability for the frame in Figure 2.13 is provided by the cant ilever column, the determination of the amplification factor is as follows (Geschwindner 1994): The first order analysis moment and flexural deflections are given by M HL (2.15) 3 13 H L EI (2.16) The displacement 2 is the total displacement including secondary effects, and the total moment including sec ond order effects is given by

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29 max2 M HLP (2.17) An equivalent load may be determined th at will result in the same moment at the bottom of the column as the second order moment. 2 P H L (2.18) It can be seen then that the moments at the bottom of the columns for the second-order and equivalent load cases in Figure 2.13 are the same, thus the second order defection is equal to 3 2 23 PL H LEI (2.19) 333 22 21333 PP LLL H EILEILEI 3 2213 PL LEI Solving for 2 1 2 31 3 PL LEI (2.20) Because the amplification factor, B2 is equal to 2/ 1 and 3 13 H L EI then

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30 2 2 3 1 111 1 1 3B P PL LH LEI 2 11 1 B P HL (2.22) Because the frame is part of a comple te structure, this equation can be modified to include all the loads in a stor y. Therefore the amplification factor for a sway column is given by the following equation (ASIC 2005): 21 1uohB P HL (2.23) Column Strength Curves After obtaining the internals forces, includ ing secondary effect s, of a structure from a structural analysis, the next step is to obtain the column compressive strength from the AISC equations in Chapter H of the AISC Specification. While the effective length factor takes into account global geometrical imperfec tions, that is, column out of plumbness, the column strength curve provides the strength of the column considering local geometrical imperfections and material imperfections. These imperfections are the out of straightne ss and residual stresses respectively. Early versions of column strength date to the 1800Â’s when empirical formulas based on the results of column tests were used. However these formulas were limited

PAGE 49

31 to the material and geometry for which th e tests were performed. Another formula for the strength determination of column s which was very popular in early 1900Â’s, was the Perry-Robertson formula. This form ula considers a column with an initial deformation and the failure of such column will occur when the maximum compression stress at the extreme fiber reach es the yield strength of the material (Ziemian 2010). However, the studies that led to the development of the Perry-Robertson and secant formula, were based only on initial out -of-straightness and eccentrically loaded column models, along with an assumed elastic response; in addition, those expressions did not consider the pres ence of residual stresses. The 1961 AISC Specification presented a new formula for co lumns whose mode of failure was due to inelastic buckling. This new formula wa s based on the recomm endations of the Column Research Council (CRC). The form ula suggested by CRC assumes that the upper limit of elastic buckling fa ilure is defined by an aver age column stress equal to one-half of yield stress. For years the column formulas found in most common specifications were strictly tangent modul us expressions combined with a safety factor (AISC 1961). In the late 1960Â’s and early 1970Â’s, studies provided the basis for the development of a solution method based on an incremental, iterative numerical routine which established the load defl ection curve for the column (Geschwindner 1994). These studies took into account resi dual stress, initial out-of-straightness,

PAGE 50

32 spread of plasticity, and load-deflection curves were determined. The correlation of the computation results with full scale test s was good and in the order of 5 percent of the test values (Bjorhovde 1972). The studies mentioned above, were part of a research program led by Lehigh University for the development of column curves in the U.S. Bjorhovde, in his PH.D dissertation “ Deterministic and probabilistic approaches to the strength of steel columns ” at Lehigh University, examined the char acteristics of column strength of an extensive database for the maximum strengt hs of centrally loaded columns (Figure 2.14). The study encompassed a variety of different shapes, steel grades and manufacturing methods. Bjorhovde observed that there were groupings within the curves and from these, three curves were subdivided. These three column curves are known as the SSRC column strength curves 1, 2 and 3 (Bjorhovde 1972). Figure 2.14 Column strength variation (Adapted from Ziemian 2010)

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33 The commentary to the AISC Specifi cation states that the AISC LRFD column curve represents a reasonable conve rsion of the research data into a single design curve and is essentially the same curve as the SSRC column curve 2P (AISC 2005). The AISC Specification column equations are actually based on the strength of an equivalent pin-ended column of length KL with a mean maximum out-ofstraigtness at is mid length of approximately KL /1500 (Ziemian 2010). Summary of Present State of Knowledge In addition to length, cross-section dime nsions and material properties, the maximum strength of steel columns depends on the residual stress magnitude and distribution, the shape and magnitude of th e initial out-of-straightness and the end restraints. The effects of these three latter variables are discussed in more detail in Chapter III.

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34 CHAPTER III: 3 COLUMN THEORY Since the beginning of human times, columns in structures were built with large cross-sectional dimensions, thus most likely carried loads well below their potential load carrying capacity, and most likely were designed with little or no engineering in the way it is underst ood today (Geschwindner 1994). With the introduction of steel as a c onstruction material and the de velopment of higher strength steels, column stability and structural stability of steel structures has become a primary concern. A column, or compression member, is one of the most critical structural elements of a structure, because it transfer loads from one point of the structure to another, and eventually to the foundati on, through compression or a combination of compression and flexural be nding. Compression members are used as compression chords in trusses as well, and as in the case of pony truss bridges, may be used as pedestrian bridges. In the case of moment or sway frames, the columns and the beams that frame into, and the connections between them, provide lateral stability to those frames. However, for the most part, di scussion of column strength in common literature focuses on an individual colu mn rather than the entire system. This basic column, which is typically us ed for the derivation of stability and column strength formulas, has no imperfectio ns and is supposed to be supported by perfect hinges; in addition, this column, al so known as the perfect column or ideal

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35 column and is referred to as a pinned co lumn, one that rarely exists in actual structures. Nonetheless, the theoretical con cepts of the ideal column are important in understanding column behavior for the study of beam-columns, compression elements and stability of frames. This ideal column is loaded by a vertical force P that is applied through the centroid of the secti on, the column is perfectly straight and follows HookeÂ’s law, that is, it is made of linearly elastic material. Although a compression member or a struct ural column can be seen at as a simple structural member, the interaction between the responses and the characteristics of the material, the cros s section, the method of fabrication, the imperfections and other geometric factors, and end conditions, make the column one of the most complex individual struct ural elements (Geschwindner 1994). Mechanism of Buckling Load carrying members of structures ma y fail in a variety of ways, including flexural bending, crushing, fractur e and shear. Another type of failure is referred as to buckling. For an idealized column, this is called bifurcation buckling. Figure 3.1(a) shows an idealized column consisting of tw o rigid bars and a rotational spring. If a load P is applied to the end of the column, the rigid bars will rotate through small angles and a moment will develop in the sp ring designated as point A. This moment is called the restoring moment. If the load is relative small, the restoring moment will return the column to its initial straight pos ition, once this load is removed. Thus, it is called stable equilibrium. If the axial force is large enough, the displacement at A will

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36 increase and the bars will go through larg e angles until the colu mn collapses. Under these conditions, the column is considered to be unstable and fails by lateral buckling (Gere 2009). Figure 3.1 Elastic Stability (Adapted from Gere 2009) Figure 3.1(b) shows three conditions of equilibrium for the idealized column as a function of the compressive load, P versus the angle of rotation, The two heavy lines, one vertical and one horizonta l, represent the equilibrium conditions. Point B, where the equilibrium diagram bran ches, is called a bifurcation point. That is, up to that point the column or struct ure remains stable and will return to its original position when the load is removed, thus, it is referred as stable equilibrium If the load is removed at the critical load, poi nt B, the column or structure would have

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37 suffered permanent deformation but remains stable, this is called neutral equilibrium If the load exceeds point B, the column or structure becomes unstable and fails due to buckling, this is called unstable equilibrium Critical Load Theory The fundamental requirement for any column strength theory is that it must be based on the basic principles of engineering mechanics and strength of materials, while also taking into account the stress-strain relations hip of the materials and reflecting geometric imperfections of all relevant kinds (Geschwindner 1994). Two basic approaches for the determination of the column critical load can be found within the basic literature for stability an alysis and determination of buckling loads. The two different basic approaches are expl ained below and presented in this chapter. The first or simpler approach, also known as the Euler Load or eigenvalue approach shown in Figure 3.2, attempts to determine th e maximum strength of an ideal column without geometrical or mate rial imperfections in a direct manner without calculating the deflectio n (Chen 1985). This approach will provide the load at which the column will buckle, in other words the load at the bifurcation point described above, without providing a full defo rmation history until instability occurs. This ideal column is assumed to be loaded concentrically and the only deflections that occur at low loads are in the direction of the applied load, that is, the axial load does not produce transverse deflection until the bifurcation or buckling load is reached. This means that up to the critical load, the column remains straight

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38 and at the critical load there exists a bi furcation of equilibrium similar to the one described above, in which the column will r each the point of instability and buckle. Figure 3.2 Column Stability (Adapted from Chen 1985) A second approach for the determination of the buckling load relates to a more realistic column. A real column contains imperfections such as initial out-ofstraightness (Figure 3.3 ), in this case, deflections st art from the initial deflection, o, at the beginning of the application of the axial load and there is no bifurcation or sudden change of deflection as load increases Therefore, the maximum load that an out-of-straightness column can support mu st be calculated using an alternate

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39 approach. A suitable approach that consider s this initial deflection, but it is complex due to its nature, is known as the load-d eflection method. This method is also known as a rigorous second order analysis and will be presented in Chapter IV. Through this approach the solution is found by tracing th e load-deflection beha vior of the beamcolumn through the entire range of loading up to the maximum or peak load, which is also known as the strength of the column (Chen 1985). Figure 3.3 Beam-Column Stability (Adapted from Chen 1985)

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40 Euler Buckling Load As discussed earlier, the perfect-pinned supported co lumn, or namely ideal column, does not exist in real structures; how ever, it provides a simple starting point for the stability analysis of structures that are subjected to instability buckling. For example, the column in Figure 3.4(a) will fail due to instability buckling when the column displaces as shown in Figure 3.4( b). That is, assuming the column span, L is of sufficient length so that failure due to crushing does not take place. When the buckling load is determined for this idea l column, then the following assumptions need to be made (Chen 1985): 1. The column is perfectly straight 2. The ends of the column are simply supported 3. The compressive axial load is static a nd applied at the centr oid of the column 4. The material follows HookÂ’s law and is free of initial or residual stress

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41 Figure 3.4 Pin-ended supported column The critical load is that load for whic h equilibrium for a straight and slightly bent configuration is possible. For relative small deflections it can be seen that the state of equilibrium is given by the following equations (Chen 1985): The external moment is given by Zexternal M Py (3.1) and the internal moment is given by 2 _"ZInternalxxdy M EIEIx dx (3.2)

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42 Where E is YoungÂ’s modulus, I the moment of inertia of the cross section and EI is the flexural rigidity. In order to maintain equilibrium __0ZInternalZExternalMM The differential equation for the deflected column then becomes "0XPyEIy (3.3) Dividing the above equation by EI and re-arranging the terms we obtain the following differential equation "0XP yy EI (3.4) Next, the axial parameter is defined as 2 PP kk EIEI (3.5) And Equation 3.4 takes the form of a linear differential equation with constant coefficients 2"0 yky A general solution is given by

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43 sincos y AkxBkx (3.6) Where A and B are constants of integrat ion. In order to determine A and B it is necessary to apply the boundary cond itions at each end of the column 0 at 0 0 at yx yxL The first boundary condition yields 0010 ABB The second boundary condition yields 0sin0 AkLB Which can be satisfied in two ways, if A = 0 k and P can have any value. This is known as a tr ivial solution. If sin(kL) = 0, then n kLnk L The deformed shape is obtained by sin0sin nx yAkxA L (3.7) Where n can be any integer. The buckling load is

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44 22P kPkEI E I Then 22 2nEI P L (3.8) However, from Figure 3.4, it can be seen that the value of critical load is given by the column bending in single curvature. The critical load is then given by 2 2 cr E I P L (3.9) From the above solution of the ideal column, one can conclude that the column strength is directly proport ional to the modulus of elasticity, E and the moment of inertia, I of the column and inversely pr oportional to the square of its length. Figure 3.5, known as the Euler Curve, ill ustrates the buckling or Euler load as a function of the column length (Gere 2009).

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45 Figure 3.5 Euler Curve Critical Load of Beam-Columns The second approach for the maximum or critical load determination of a beam-column is obtained by a load-deflecti on, also known as the classical differential solution for beam-columns. Beam-columns are members that are subjected to axial forces and bending moments simultaneousl y. A typical load-deflection relation for beam-columns is shown in Figure 3.6. The following discussion will focus on a small deformation analysis (Chen 1985).

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46 Figure 3.6 Beam-Column (Adapted from Chen 1985) Given the beam-column in Figure 3.6, with flexural stiffness EI and loaded simultaneously with an axial load, P, and an arbitrary distributed transverse load q(x) the equations of equilibrium can be derived as follows The transverse force is 0 VdVVqxdx Therefore, 0 dV qx dx (3.10) The moment is

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47 00dMdy MdMMVdxPdyVP dxdx (3.11) Taking the derivative of the moment equation (Equation 3.11) and substituting in Equation 3.10, 22 22 dMdy Pqx dxdx (3.12) From small deformation analysis and assuming that the column is loaded in the elastic range and follows HookÂ’s la w, the defection is related to the curvature by the following expression, 22 22 dyMdy M EIEI dxEIdx (3.13) The differential equation has the form 42 42 dydy E IPqx dxdx (3.14) The axial parameter is 2 PP kk EIEI therefore, 42 2 42qx dydy k dxdxEI (3.15) Equation 3.15 is a forth-order linear diffe rential equation that may be solved rigorously by the use of formal mathem atics (Chen 1985). Figure 3.7 shows the plot of the exact solution for a beam-column w ith two different initial deformations.

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48 Unlike the perfect column, which remains stra ight up to the Euler load, the initially deformed column begins to bend as soon as the load is applied. It can be seen that the deflection in the beam-column increases slowly at a small ratios of the applied load to the Euler load. However, as the applied load increases, the deflection increases more rapidly and it grows exponentia lly with no limit at load ratios near the Euler load. Thus, the carrying capacity of an imperfect column is smaller than the Euler load, regardless of how small the initial imperfection is. The solution of Equation 3.15 is presen ted in Chapter VII along with the solution of a cantilever column. These two different benchmark problems, along with their rigorous differential solutions, will be used to evaluate the computer program employed throughout this research work. The two problems mentioned above are presented in the Appendix 7 of the AI SC Specifications (AISC 2005), (AISC 2010). These benchmark problems are used for the evaluation of any analysis method, including computer programs that would be used as a ri gorous second-or der analysis method for the determination of sec ond-order forces required by the AISC Specifications (AISC 2005), (AISC 2010).

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49 Figure 3.7 Elastic analysis of initially defo rmed column (Adapted from Chen 1985) Second Order Effects So far the critical load of an indivi dual column has been discussed without consideration of the interaction between th e column and other members that are part of a framed structure. This interaction can significantly impact the behavior of a column that is part of a frame structure. For example, the resulting moment from a beam in a moment frame structure, which is transferred to a beam-column from a rigid or moment connection, would cause an initial deformation, and together with the compressive load acting on the column, would cause an additional moment. This additional moment is the result of the eccen tricity between the applied load and the centroid of the column. The additional moment would then cause additional

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50 deflections, these deflections would cau se additional mome nts and so forth. Therefore, second order effects are the direct result of displacements or deformations of the structure. Common elastic methods fo r structural analysis assume that all deformations are small and do not account for these secondary effects. Second order effects may be determined by a complete second order inelastic analysis (also called advanced analysis and is discussed in Chapter IV) that will take into account the actual deformations of the structure and the resulting forces in addition to the sequence of loading and spr ead of plasticity. This approach is more complex than necessary for normal de sign (Geschwindner 1994). Another method which is more common for the determinati on of second order eff ects is known as a Panalysis, which is an iterative elastic analysis to approximate the inelastic conditions. In this method a first order anal ysis, as described above, is carried out. The elastic deformations are determined from the resulting forces of the elastic first order analysis. Secondary moments and defo rmations are determined by applying the loads on the deformed structure. In order to determine the critical load of a structure, this process is repeated, incrementing the load until the analysis shows that the structure has become unstable. A third approach used in common practi ce is to approximate the second order effects by amplifying the first order mo ments by an amplification factor. The derivation of the amplification factors wa s discussed previously in Chapter II

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51 For the critical load assessment and stab ility analysis of a structure, it is important to understand the two different de flection components that will influence the secondary effects in the beam-column. Referring to Figure 3.8, the moment component, P shown in Figure 3.8(a) is the additional moment along the length of the member that results from the eccentricit y between the centroid of the column and the deflected shape of the column. This eccentricity is caused by, either, the firstorder moment, M or by an initial geometrical imperfection (i.e. member out-ofstraightnes). In this case the ends remain in the original position, in other words, there is no joint translation. This effect is known as the member effect or P. The second component is shown in Figure 3.8b. For exampl e, when the column is part of a sway frame (i.e. there is joint translation), the displaced structure will cause an additional moment, M = P( ). This secondary moment is the second order effect on the structure and is known as the structure effect or P. On an unbraced structure, the total second order effect on a column is given by the combined influence of the member effect ( P) and the structure effect (P). However, neglecting the effect of Pin the analysis of sway structures with large gravity loads can result in very erroneous second-order moments and forces. A discussion of neglecting Pand its effects is presented in Chapter VII One thing to keep in mind is that the member effect and the structure effect are not only caused by transverse or lateral forces, but al so by member imperfections

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52 and structure out-of-plumbness. These factors, also known as geometric imperfections, will be discussed later in this chapter. Figure 3.8 Second-order effects Inelastic Buckling of Structures In Chapter II, the inelastic buckling theory was presented where various theories were proposed to overcome some of shortcomings of the EulerÂ’s buckling formula. In the derivation of the Euler fo rmula, it was assumed that the material followed HookeÂ’s law. However, in reality a material has a proportional limit beyond which it does not behave elas tically (Chen 1985). If residual stresses are present in a column cross-section, inelastic buckling can occur when the compressive stress due to applied load plus the local residual co mpressive stress exceed the material

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53 proportional limit, a condition often reached at the tips or corners of wide flange steel columns. Thus, inelastic buckling includes a ny buckling phenomenon during which the proportional limit of the material is exce eded somewhere within the cross section before buckling occurs. The proportional limit is the stress at which the relationship between stress and strain (i.e. modulus of elasticity) is no longer linear and pr oportional. The length at which a column will buckle in this range can be found in the following fashion: From the Euler buckling load Equation, 3.9, the critical stress can be found as follows: Thus 22 cr crcr 22 E IEI P P A KLAKL (3.16) 2 cr 2 E KL r (3.17) Where r is the radius of gyration and is given by, r = I = Moment of inertia in the direction of bending A = Column cross-sectional area The critical length, KLcr, can be found by substituting the critical stress, cr, with the proportional stress, pl, and solving for KL

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54 cr plpl E IE KLr A (3.18) The proportional limit depends on the amount of residual stresses in a column cross-section as shown in Figure 3.9 (resi dual stresses are a type of material imperfection and are discussed la ter in this chapter). For ex ample, for a circular steeltube column of typical material propertie s, this type of buckling would occur at lengths of 28 to 35 times its diameter. Co lumns meeting this criterion will generally buckle inelastically, meaning permanent deformations will occur upon reaching the critical buckling load. Figure 3.9 Residual stresses and proportional stress (Adapted from Chen 1985) Factors Controlling Column Strength and Behavior There are several factors that have an effect on column strength, for example, the characteristics of the material, and the method of fabrication, column cross section, imperfections and end conditions. However, from decades of research and

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55 testing, the most influential factors are di scussed below and include the length of the column, end support conditions, and geom etric and material imperfections (Geschwindner 1994). The most critical fact ors that affect column behavior are summarized below. Material Properties From the stress strain test, Figure 3.10, two very important material properties can be derived for an elastic-plastic material, yield stress, Fy, and modulus of elasticity, E From Figure 3.11, it can be seen that for short columns the strength of the column is directly proportional to the yield strength. However, for long columns the yield stress is irrelevant because the capacity, given in Equation 3.9, is a function of the Euler load and is influenced by the fl exural stiffness. The flexural stiffness is the product of the moment of inertia and the modulus of el asticity. The strength of the material of a long column is not a factor in the determination of the column strength.

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56 Figure 3.10 Typical stress-strain curves fo r different steel classifications Length The length of a column plays a major role on the behavior and strength of the column, as it can be seen in Figure 3.11, the behavior of the column is classified as short, intermediate and long columns. Shor t columns, as descri bed earlier in this chapter, fail by crushing or compression yielding and are controlled by the yield strength. Intermediate columns fail by in elastic buckling and are controlled by inelastic or non linear behavior of the mate rial. This nonlinear behavior is caused by material imperfections and will be discussed below. Long columns on the other hand, fail by elastic buckling and their behavior is governed by the Eu ler load equation.

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57 Figure 3.11 Theoretical column strength (Gere 2009) Influence of Support Conditions Moment Frames Connection flexibility and member inst ability are closely related, and their interaction effects can have a significant in fluence on the overall performance of the frame. The stiffness of the beam-to-column restraining connection plays a major role on the stability behavior of columns. Figures 3.12 and 3.13 illustra te the influence of different beam-column end restraints. As is evident from Figure 3.12, the greater the connection restraint, the stiffer will be the initial response of the column and the greater the maximum load that can be carried as compared to a pin-ended column. In

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58 addition, Figure 3.13 shows column strength curves for members with different end conditions where, the influence of end restra int for columns with different slenderness ratios can be clearly illustrated, and the fact that the influence di minishes for shorter columns. These curves are compared to the Euler curve and the SSRC column curve 2 which is used in the AISC Specifica tions (Ziemmian 2010). Figure 3.14 shows the moment-rotation behavior for different beam-t o-column end restraints or connections. Figure 3.12 Typical load-deflection curves fo r columns with different framing beam connections (Adapted from Ziemian 2010)

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59 Figure 3.13 Column strength curves for memb ers with different framing beam connection (Adapted from Ziemian 2010) Figure 3.14 Connection flexibilit y (Adapted from AISC 2005)

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60 In Chapter II the concept of “effectiv e length” was introduced to account for column supports other than pin-pin. For moment frame structures, the alignment charts which are used for the determination of the effective length factor discussed in Chapter II, take into account the rotationa l restraints by upper and lower assemblages but neglect the inter action of lateral stiffness among the columns in the same story resisting lateral sway buckling of unbraced frames. LeMessurier (1977) presented a method of evaluating the K factor based on the concept of the story-based buckling, which accounts for the lateral restraining e ffect among columns in the same story. Chen and Lui (1991) modified the values of the moment of inertia of the restraining beams while using the alignment chart method in order to incorporate connection flexibility. The modification factors were derived for both the braced and unbraced frames based on the assumption that the beam -to-column connection stiffness at both ends are identical. These modification fact ors were developed in such a way that considered different values of connection st iffness at the ends of the beam (Bjorhovde 1984; Chen et al., 1996; Christopher and Bjorhoved 1999). The m odification factors take into account actual connection stiffness and the influence of beams and effectivelength factors for columns in frames by incorporating an end-restraint relative stiffness distribution factor, Gr given by Equation 3.19 (Zieminan 2010): columns r *EI L G C (3.19)

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61 Where C* is the effective end restraint that is afforded to a column in a beam and column sub assemblage, using connections whose initial stiffness is C the initial slope of the moment-rotation curv e. A significant of drawback of this approach is that the value of C must be known. However, conn ection schemes similar to those provided by Bjorhoved and the approaches of Eurocode 3 are useful (Ziemian 2010). Members with Elastic Lateral Restraints The critical load assessment of structur es with members that are restrained laterally between their ends by intermitte nt elastic lateral supports is not as straightforward as the critical load or buckling load determination of a simply supported column that has been the focus fo r most of the resear ch work and studies on structural stability. Pony trusses, which are widely used as pedestrian structures, have served as the prototype through the years for the development of theory and design procedures that are sometimes utili zed in other similar structures nowadays. Early studies of compression members wi th elastic lateral restrains started with F.S. Jasinsky (1856-1899) in the la te 1800Â’s and with Engesser (1848-1931). JasinkyÂ’s main work on the theory of colu mns deals with problems encountered in bridge engineering. Jasinsky was the firs t to investigate th e stability of the compressed diagonals and to evaluate the strengthening afforded by the diagonals in tension. In his time, several failures of open truss bridge structures (pony truss) occurred in the Western Europe and Russia (Timoshenko 1983). Jasinsky made a theoretical investigation of this probl em and found a rigorous solution for this

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62 complicated case of lateral buckling in whic h he calculated the critical values of compressive forces so that the upper chord and the verticals of an open bridge could be designed rationally. Another engineer that contributed much to the theory of buckling in the same period as Jasisnky was Friedr ich Engesser. He was also in terested in th e problem of buckling of the compressed upper chord of open truss bridges and derived some approximate formulas for calculating the proper lateral flexural rigidity of that chord. However, Egensser which in his analysis assumed an equivalent uniform elastic support, was the first to present a simple rational and approximate formula for the required stiffness ( Creq) of elastic supports equally sp aced between ends of a hingeended column of constant se ction, Figure 3.15 (Ziemian 2010). EngesserÂ’s approach for determining the s tiffness of the elastic restraints and its effects on the pony truss compression chor d was based on the assumption that the connection between the web members and the floor beam was rigid. In this theory, the transverse frame (i.e. the floor beam and the verticals) and diagonal members located at each panel point provide the la teral stiffness of the compression chord. However, this theory may result in a c onservative approach for the actual frame stiffness by underestimating the true behavior of the entire system. In other words, one may argue that investigating the behavi or of the bridge as a three dimensional system may result in a higher stiffness coefficient of these lateral supports.

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63 Other earlier solutions were proposed by methods developed by Hu, Holt, Lutz and Fisher. Hu studied the problem using energy methods for elastically supported chords. He considered non-uniform axial forces, variable chord cross sections and spring stiffness for both simp le and continuous pony truss bridges. Holt included the secondary effects that influen ce the behavior of the pony truss in his studies. Holt presented a method of analysis for determining the critical load of a pony truss top chord (Ziemian 2010). The soluti on presented by Holt is used in common practice for the design of pedestrian bri dges, (AASHTO 2009) a nd is discussed in Chapter IV. Figure 3.15 Pony truss and analogous top chord (Adapted from Ziemian 2010)

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64 Most solutions used for the determinati on of the critical load of a pony truss bridge involve the procedure presented above without providing a di rect approach for the critical load determination of the entire bridge as a whole. Influence of Imperfections It has been stated that the perfect or ideal column do not exist in real structures because actual columns and struct ures have inherent imperfections. These imperfections can be categorized as material imperfections and geometric imperfections which produce nonlinear behavior of a loaded structure. Figure 3.16 illustrates the influence of geometric and material imperfections on the critical load of the column. In addition, Figure 3.16 includes the influence of a column with an initial cu rvature (geometric imperfec tion) only, residual stresses (material imperfection) only and the co mbined effects of both imperfections. In stability design and anal ysis of framed structures it is imperative to account for the effects of imperfections; out-of-plumbness of framing and out ofstraightness of columns are referred to geometric imperfections while residual stresses are referred as to material imperfections. The development of the column strength curves that are used in design specifications was discussed in Chapter II, in which it was mentioned that only two types of imperfections have been considered in the development of such curves: residual stresses and member out of straightness.

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65 Figure 3.16 Influence of imperfections on co lumn behavior (Adapted from ASCE Task Committee 1997) Material Imperfections Residual stresses, which are consider ed the main factors of material nonlinearity, are the direct re sult of the cooling process of hot rolled steel sections. During hot-rolling of steel, the cross section of the steel section is heated to a uniform temperature. Once the rolling is completed, the heat must dissipate. In this process the cooling is not uniform, therefore some of the individual fibers in the steel section cool at different rates, that is, some of the fibers reach room temperature faster than the rest (Geschwindner 1994).

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66 The stress strain relations for structural st eel is of elastic-pe rfectly plastic type shown in Figure 3.17(a) (Chen 1985) Using th is stress strain curve and the Euler buckling load theory described in Chapter II, the column strength curve is shown in Figure 3.17(b). According to this curve, a column will fail as a result of elastic buckling if or will yield if However, a large number of tests have shown that columns of intermedia te slenderness ratios tend to buckle at loads significantly below those given by Figure 3.17(b) (Chen 1985). An extensive investigation at Lehigh University in th e 1950s demonstrated conclusively that residual stresses are responsible for a larg e portion of the disc repancy between the theoretical curve and the tests results. It is noted that residual stresses reduce the critical load of columns with intermediate length (Chen 1985). Figure 3.17 Column curve for structural steels. (a) elastic-perfectly plastic stressstrain relationship; (b) column curve (Adapted from Chen 1985)

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67 For I-section members, the residual stre sses are usually compressive at the flange tips. Compared to sections free of residual stress, the magnitude and distribution of these compressive residual stresses influences the strength of compression members because of the earlier initiation of yielding and subsequent lowering of flexural strength that occurs. The effects of residual stresses are more severe for I-section columns bent about the minor axis the major axis (ASCE Task Committee 1997). This causes a nonlinear behavior of the compression element since the fibers with residual compression will r each the proportional limit before the rest of the fibers. The residual stress pattern employed by Galambos and Ketter (ASCE Task Committee 1997) in their studies of the stre ngth of steel beam-columns is shown in Figure 3.18. Based on numerous measurements of residual stresses made by various researches over the years, analytical m odels have been recommended that follow the residual stress pattern shown in Figure 3.18. In addition, this stress distribution pattern is consistent with measured residual stresses in American wide-flange column-type sections resulting from cooli ng of the section during and after rolling (ASCE Task Committee 1997). Parabolic distributions, varying from 0.35 Fy compression at the flange tips to 0.5 Fy tension at the flange web, have been presented by various researches investig ating the inelastic lateral buckling of beams and beamcolumns (ASCE Task Committee 1997).

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68 However, in order to achieve a better match to experimental distribution, the bi-linear flange distributi ons shown in Figure 3.18 has been employed widely in parametric studies of beam-column and fram e strength in the context of calibration and development of the AISC LRFD inte raction equations (ASCE Task Committee 1997). Figure 3.18 Lehigh residual stress pattern (Adapted from Surovek 2012) Geometrical Imperfections There are two types of geometrical imperfections considered for the assessment of lateral stability of a st ructure: out-of-straightness and out-ofplumbness.

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69 Out-of-straightness refers to the amount of initial bow in a structural steel member. Initial out-of-strai ghtness develops as a result of the cooling conditions for the shape, once it has been rolled to final dimensions. Once the structural shape has passed through the last set of rolls, is then left on the cooling bed to cool in air over a period of time. However, cooling is non-uni form since heat is typically retained longer in the mid length portion of the co lumn. This non-uniform cooling not only causes residual stresses in the cross secti on as mentioned above, but tends to distort the structural shape to the point that en ds up in a curved configuration along its length. Depending on how the shape has been placed on the cooling bed, the curvature may appear in both orthogonal di rections of the cro ss section. Crookedness about the major axis is called camber and cu rvature about the minor axis is designated as sweep (Geschwindner 1994). The maximum out-of-straightness is limited by material standards, for example ASTM A6 in the United States, and is typically ba sed on length of the member. In the case of wide-flange shapes pr oduced in the United States, this limit is approximately L /1000 (AISC 2010). In past specifications, th e effect of initial out-ofstraightness or crookedness was covered through a factor of safety, and used the strength of the straight member as the actual criterion (AISC 1961). As mentioned earlier, in the limit states philosophy of the LRFD Specificat ions all the major paramete rs (residual stresses and an initial out-of-straightne ss) have to be accounted fo r. Therefore, the strength

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70 equations in the AISC Speci fications include the infl uence of initial crookedness (AISC 2005). The inclusion of out-of-plumbness in the design procedure for frames is much more complex. Out-of-plumbness refers to the amount of initial drift or tilt that an actual structure may have. Prior to 2005, the load resistance and factor design (LRFD) AISC specifications did not provide requirements for inclusion of out of plumbness in the design of structures, except through the us e of the effective le ngth method described in chapter C of the AISC Specificatio ns (AISC 1999). Eurocode 3 (ASCE Task Committee 1997) recommends that frame imperf ections be included in the elastic global analysis of the frame. Although the in fluence of the number of columns in a plane and the number of stories is cons idered, only limited guidance is given with respect to the shape and di stribution of imperfections (ASCE Task Committee, 1997). The Australian and Canadian codes incl ude the effect of frame imperfections through the use of an equivalent notional la teral load, a procedure also allowed in Eurocode 3 (ASCE Task Committee, 1997). The 2005 AISC Specifications introduced a new analysis method for the a ssessment of lateral stability of steel structures; this new method of analysis known as the Direct Analysis Method, includes the notional load approach menti oned above. The notional load approach and Direct Analysis Method are di scussed further in Chapter IV.

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71 CHAPTER IV 4 METHODS AND PROCEDURES FOR ANALY SIS AND DESIGN OF STEEL STRUCTURES Structural and Stability Analysis One of the most important aspects during the design pr ocess of structures in civil engineering is structural and stability analysis of such structures. Structural analysis refers to the determination of forces a nd deformations of the structure due to applied loads. Structural design, on the other hand, involves the arrangement and proportioning of structures and their com ponents in such a way that the assembled structure is capable of supporting the designe d loads within the allowable limit states. An analytical model is an idealization of the actual structure. The structural model should relate the actual behavior to material properties, st ructural details, and loading and boundary conditions as accurately as is practicable before proceeding to design the members and connections of a structure (Hib beler 2009). Thus, when designing a structure, a stru ctural analysis must be performed so that the internal forces and stresses are de termined, and an equally important aspect of structural analysis and design, is a lateral stability analys is. This type of analysis requires that all structural members and c onnections of a structure have adequate strength to resist the applied loads w ith equilibrium satisfied on the deformed geometry of the structure (Ziemian 2010). In common design practice it is difficult to satisfy the above statement due to the fact that most practic al methods of analysis are

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72 limited to first order elastic analysis, that is, equilibrium is satisfied only on the original geometry of the structure and the material is assumed to behave elastically without consideration of material and/or geometry nonlinearities. Common methods of analysis for stability include, first-orde r elastic, second-order elastic, first-order elasto-plastic and second-order inelastic, to name some. These types of analyses can be differentiated by whether or not they in clude imperfections such as geometric and material nonlinearity. That is, is equilibrium is satisfied on the deformed or undeformed geometry of the structure? And, ar e residual stresses, sp read of plasticity through the cross section, and yielding considered? During the structural design process, th e serviceability limit state and strength or ultimate limit state need to be considered A serviceability limit state is one in which the structure would become unfit fo r normal service loads, for example, excessive deformations or vibrations, a nd problems with durability. On the other hand, a strength limit state is one in which the stru cture would become unsafe. With regard to the type of structural analysis a linear elastic analysis will provide the engineer with a good representation of the actual response of the structure to service loads, that is, when a serviceability limit st ates is investigated. When a strength limit state is reached, it generally results in an increasing nonlinear elastic or inelastic response of the structure that, sometimes, culminates in stru ctural instability. However, for the most part in common pract ice, the internal load distribution at a strength limit state is ca lculated by a linear method. During the design and sizing of

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73 structural members, the analytical results obtained by the linear method are modified with empirical or curve fitting equations to account for the effects of nonlinearity. Figure 4.1 shows a comparison between se veral analyses approaches and loaddisplacements curves. These analyses are summarized below and range from the most basic first-order linear-elastic analysis to the rigorous second-orde r inelastic analysis. Figure 4.1 Comparison of load-deflection behavior for different analysis methods (Adapted from Geschwindner 2002) First-Order Elastic Analysis This is the most basic and common met hod of analysis in which the material is modeled as linear-elastic This method is also calle d simply elastic analysis. Deformations are assumed to be small and equilibrium is satisfied on the undeformed

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74 configuration of the structure. Because inel astic behavior of the material is ignored, superposition is valid. However, this me thod excludes nonlineari ty, but it generally represents conditions at service loads very well. In addition, the load displacement curve shown in Figure 4.1 is linear. This is the approach used in the development of the common analysis tools of the profe ssion, such as slope-deflection, moment distribution, and the matrix stiffness me thod that is found in most commercial computer software (Geschwindner 2002). Elastic Buckling Load The elastic buckling load (i.e. Euler buckling load) analysis provides the critical buckling load of th e system without a load-deflection history. That is, the results of this analysis do not provide a load-displacement curve but rather the single value of load at which the structure buckles as shown in Figure 4.1. The critical buckling load may be determined thr ough an eigenvalue solution or through the classical solution presented in Chapter III, where equilibrium equations are written with reference to the deformed configurati on. This analysis can provide the critical buckling load of a single column and is th e basis for the effective length factor, K (Geschwindner 2002). Second-Order Elastic Analysis A second-order elastic analysis, also known as a Panalysis, provides the load-displacement history of an imperfect column, typically through an iterative solution thus it is a bit more complex than the first-order elastic analysis. The

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75 equations of equilibrium are written with re ference to the deformed configuration of the structure and the deflections are dete rmined as a function of that deformed configuration. The load-displacement hist ory obtained through this analysis may approach the Euler buckling lo ad as shown in Figure 4.1. Two components of these second-order effects should be included in the analysis. The first effect is known as Por as the member e ffect, this second-order effect is influenced by the member curv ature or member imperfection. The second effect known as Por as the story effect is influenced by the story sidesway. Nonetheless, second order elastic analysis can produce an excellent representation of destabilizing in fluence of the Peffect, but has no provisions for representing inelastic behavior due to spread of plasticity. First-Order Inelastic Analysis In the first order inelastic analysis the equations of equilibrium are written in terms of the geometry of the undeformed structure. Inelastic regions may develop gradually or as abrupt changes in the structure response. When the destabilizing effects of finite displacements are relative insignificant, first-order inelastic analysis can produce an excellent representation of si mple elastic-plastic behavior and failure through hinge formation. However, this an alysis has no provisions for detecting geometric nonlinearity and of principal concer n, their influence on the stability of the system as a whole, (McGuire 2000).

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76 Second-Order Inelastic Analysis The second-order analysis, often referred to as “advanced analysis”, is the most complete analytical approach for st ability assessment of structures in which combines the same principles of second-orde r analysis discussed previously with the inelastic analysis. Although this type of analysis is more complex than any of the other methods of analysis discussed so far, it provides a more complete and accurate picture of the behavior of the structure. In the second order inelastic analysis the equations of equilibrium are written in terms of the geometry of the deformed system. This type of analysis has the ability for accommodating the geometric and material imperfections and elastic factors that influence the response of a structure and is capable of simulating actual behavior. Thus, the inelastic stability limit (i.e. the point at which as system’s continued deformation results in a decrease in load re sisting capacity) can be calculated directly from this type of analysis, that is, this is the true strength of the structural system, (McGuire 2000). Design of Compression Members Most structures behave in a linear el astic manner under service loads. Under ultimate loads the behavior can be quite different, and in most cases, structures loaded close to their limit of resistance woul d most likely exhibit significant nonlinear response. Thus, if a linear elas tic analysis is the highest le vel of analysis available for

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77 design, the design engineer must find anothe r way to account for the effects that the analysis is not able to simulate. That is, the results from an elastic analysis are used with code or material specifications formulas that make allowance for nonlinearity in some empirical or semi-empirical way or supplementary theoretical or experimental studies. (McGuire 2000). It is very important that the approach taken for member design be consistent with the approach chosen for analysis which is a requisite of the AISC Specifications. Chapter B of the 2010 AI SC Specifications requires that: The design of members and connections shall be consistent with the intended behavior of the framing system and the assumptions made in the structural analysis Unless restricted by the applicable building code lateral load resistance and stability may be provided by any combination of members and connections. Currently, three design approaches ar e acceptable for steel structures under U.S. building codes as they incorporat e AISC specifications (AISC, 1999; AISC, 2005; AISC, 2010). The most up-to-date tool for steel design is the load and resistance factor design specification (L RFD). However, the plastic design (PD) approach, now called Design by Inelastic Analysis in the 2010 AISC Specifications, is also permitted and the allowable stress design specification (ASD) may still be used. The LRFD Specification stipulates, in Section C1, that “Second order effects shall be considered in the design of frames”, (AISC 2010). Because th e typical analysis method is first-order, satisfying deforma tion capacity requirements and assuring stability are left to the engineer.

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78 When using the AISC Specifications, stab ility is usually addressed through an estimate of column buckling capacity us ing the effective length approach, and second-order effects may be addressed through a first-order analysis, together with an amplification factor for sec ond-order effects, or through direct use of a second-order analysis (Geschwindner 2002). Another approa ch allowed by the AISC Specifications for structural stability, which is gaini ng more acceptance and popularity and is the basis of this research project is the Direct Analysis Method, whic h will be described in more detail later on in this chapter. However, if the plastic design approach is used, member design and stability are directly obtained from the analysis. Design by Inelastic Analysis is addressed in Appendi x 1 of the AISC Specifications. Section B3 of the 2010 AISC Specifications says: Required Strength The required strength of structural members and connections shall be determined by structural analysis for the appropriate load combinations as stipulated in Section B2. Design by elastic, inelastic or plastic analysis is permitted. Provisions for inelastic and plastic analysis are as stipulated in Appendix 1, Inelastic Analysis and Design. The general requirements of Appendix 1 of the 2010 AISC Specifications require that: Strength limit states detected by an inelastic analysis that incorporates all of the above requirements are not subject to the corresponding provisions of the Sp ecification when a comparable or higher level of reli ability is provided by the analysis. Strength limit states not detected by the inel astic analysis shall be evaluated

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79 using the corresponding pr ovisions of Chapters D, E, F, G, H, I, J and K. Determining Required Strength The ASCE task committee on effective length published a report in 1997 on two different types of approaches for assessing frame stability, the effective length method and the notional load approach (ASCE Task Committee 1997). This document mentions that both of these a pproaches are capable of accounting the effects of imperfections and nonlinearity on the strength of columns and beamcolumns that must be considered in desi gn. However, the effective length approach and the notional load approach differ in the way they approximate these effects. Since 1961, the AISC Specifications has us ed an effective length concept to determine the axial resistan ce term of the beam-column interaction equation. Second order elastic analysis is pe rformed on the structure assu ming perfectly straight and perfectly plumb members. Out-of-straightne ss imperfections and residual stresses are accounted for through the column curve for an equivalent pin-ended member, while out-of-plumbness imperfections are accounted through the effective length of the column, KL where K is the equivalent length factor and L is the actual member length (ASCE Task Committee 1997). An alternative to computing effective length factors to take into account different end restraints in columns of unbr aced frames, is to use the actual column length (i.e. K =1) and add artificial frame imperfections to the analysis of the

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80 structural system through the use of noti onal lateral loads. A second-order elastic analysis is then conducted on the structure. This approach for the assessment of frame stability is termed the notional load approach (ASCE Task Committee 1997). The Australian Standard, AS4100, the Canadian Standard CSA and the European specification Eurocode 3, si nce the early 1990Â’s (ASCE Task Committee 1997), and now the AISC Specification sin ce 2005 (AISC 2005), use notional lateral loads with second-order elastic analysis of the geometrically perfect structures to account for story out-of-plumbness under gr avity loads and then use the actual member length in the beam-column inter action equations to acc ount for member outof-straightness and residual st resses, (ASCE Task Committee 1997). Determining Available Strength Beam-columns need to be designed for the combined effects of flexural moments and compression axial force. The de termination of the compressive strength of a column and the flexural strength of a beam, are typically specified in standard design specifications. In the United States the standard specification ANSI/AISC 360 governs the design of columns (AISC 2005) (AISC 2010). A brief summary is presented here. The AISC Specifications provi de, for general use, a single interaction equation to determine both, in-plane and out of plane member strength. For in-plane instabilities, the AISC intera ction equations either implicitly accounts for geometric imperfections, partial yielding and residual st resses effects by using a column strength curve that is based on effective column lengt hs or explicitly accounts for these effects

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81 by use of the direct analysis method, pr esented below, with column strength calculated on actual member length (Ziemian 2010). Figure 4.2 illustrates the two approaches mentioned above, (the effective length and notional load approach), when us ing the interaction equations specified in the AISC Specifications, w ith the purpose of checking the combined effects of bending and compression. In order to deve lop the interaction equation curve shown in Figure 4.2, two nominal strength points need to be determined first. These two points represent the nominal st rength under pure compression, Pn, and the nominal strength in bending, Mn. Figure 4.2 Distinction between effective leng th and notional load approaches (Adapted from Ziemian 2010).

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82 The nominal compression strength, Pn, represents the nominal axial strength of the column as limited by the effects of late ral buckling either in or out of the plane of the frame. When the effec tive length method is employed, Pn(KL), the nominal axial strength is determined considering the limit of stability of the framing system, or of a local subassembly within the framing system, where all the columns are concentrically loaded. The nominal resistance Pn(KL) is calculated using a single column curve that is a function of the effective buckling length. When the direct analysis method is used, the nominal axial strength, Pn(L), is based on the same curve but in this case is a function of the actual member length. This is the result of the analysis being modified to account for frame stability effects and in turn calculating larger moments and forces associated w ith column out-of-plumbness and reduction of stiffness due to inelastic deformations. The nominal bending strength, Mn, is determined for bending in the plane of the frame considering the potential for elastic or inelastic lateral-torsional buckling out of the plane, (Ziemian 2010). The column strength equations that are presented below are calibrated from results of distri buted plasticity anal yses that include geometric imperfections and residual stresses (Ziemian 2010). Compression Strength For doubly symmetric compact and noncompact compression members, flexural buckling is normally the governi ng limit state (Williams 2011). This capacity is obtained by multiplying the critical stress, Fcr, by the cross sectional area of the

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83 column, Ag. Thus, the nominal capacity of a comp ression member, which is given in chapter E of the AISC Specifications, can be calculated by the following formula (AISC 2005), (AISC 2010): ncrgPFA (4.1) The AISC Specification provides tw o empirical expressions for the determination of the critical compressive stress, Fcr, which is required in order to determine the maximum compressive stre ngth of the column (AISC 2005), (AISC 2010). A historical discussion on the origin of these empirical expressions was presented in Chapter II. For members with slenderness ratios, KL/r of less than: y E 4.71 F (4.2) The critical stress is give n by the following expression: y eF F cryF0.658F (4.3) 2 e 2 E F KL r (4.4) Where KL is the effective length of the column, r is the radius of gyration of the section in the direction being analyzed, E is the modulus of elasticity, Fy is the

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84 yield strength of the material, and Fe is the elastic buckling stress and is given by Equation 4.4. For slenderness ratios of less th an the limit determined in Equation 4.2, the outer fibers of the column cross-section yield before the critical load is reached and the Euler expression is no longer va lid. Therefore, as shown in Figure 4.3, Equation 4.3 provides an empirical expressi on to model the behavior of the actual column in the inelastic range. The behavior of a compression memb er in this range was described in Chapter II. For columns that behave in the elastic range, that is, columns in which the slenderness ratio, KL/r exceeds the limit given in Equation 4.2, Equation 4.5 provides an empirical expressi on to model such behavior. In this case, such column will reach the critical buckling stress before yielding occurs within the cross section. creF0.877F (4.5)

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85 Figure 4.3 Standard column curve (Ada pted from Williams 2011). The effective length factor, K in Equation 4.2, is required in order to account for the effects of different support conditi ons, other than pin-pin as in the basic column, and is illustrated in Chapter II. In a ddition, the effective length factor is used to account for the effects of frame out-ofplumbness. Therefore, if the effects of structure nonlinearity or geometrical im perfections are accounted for through the use of notional loads, as described in the previ ous section, the effectiv e length factor is then equal to 1.0, thus an effective length factor “ K ” is no longer needed. Combined Forces Once the nominal compression and bending strengths of a beam-column element have been determined, the next step is to check the combined effects through

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86 the use of an interaction equation. The AISC Specification (AISC 2005), (AISC 2010) provides the following inte raction equations for design of steel beam-columns: When the ratio of the required axial strength to the available strength is more than 20%, the interaction curve is given by the following equation: ry rx r ccxcyM M P 8 1.0 P9MM (4.6) When the ratio of the required axial strength to the available strength is less than 20%, the interaction curve is given by the following equation: ry rx r ccxcyM M P 1.0 2PMM (4.7) Where Pr is the required axial compressive strength or the axial load effect on the member being considered, Pc is the available or nominal compressive strength, Mrx and Mry are the maximum seco nd-order elastic moment s about the strong and weak axis respectively, within th e unsupported length of the member, and Mcx and Mcy are the corresponding nominal flexur al strengths. The load effects Pr, Mrx and Mry may be determined either directly from second-order elastic analysis or by use of first-order elastic analysis with approxima te amplifiers applied to the first order forces. Equations 4.6 and 4.7 describe a single interaction curve that characterizes the strengths of steel frame members associated with all the possible beam-column limit states.

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87 As shown in Figure 4.4, the two interaction expressi ons defined by Equations 4.6 and 4.7, are used to define the lower bound curve for the non-dimensional strengths. The AISC interaction equations were determined in part by curve fitting to strength curves obtained from “exact”, two-dimensional, second-order-inelastic analyses. In these analyses, the spread of plasticity, including the effects of initial residual stresses, is explicitly modeled (ASCE Task Committee 1997). Figure 4.4 Interaction curve (Adapt ed from Williams 2011) Current Stability Requirements (AISC Specifications) The AISC Specifications require that stru ctural stability must be provided for the entire structure and for each of its elements. The Specifications permit any rational method of design for st ability that consid ers all of the effects listed below;

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88 this includes the following methods: the effective length method (AISC 360-10 Specifications Appendix 7), the direct an alysis method (AISC 360-10 Specifications Chapter C) and the first order elastic anal ysis method presented in Appendix 1 of the ASIC Specifications, (AISC 2010). Both elastic methods, the direct anal ysis method and the effective length method, require a second-order analysis of the structure. This may be accomplished using a rigorous, second-order computer an alysis or by an approximate second-order analysis. An Alternate approxi mate approach is presented in AISC Specifications in Appendix 8, this approach is known as the B1-B2 or amplification factors procedure (Williams 2011). The AISC Specification Secti on C1 lists the effects that must be considered in the design of a structure for stability. These are: 1. Flexural, shear, and axial deformations of members 2. All other deformations that contribute to the displacements of the structure 3. P, second-order effects caused by structure displacements 4. P, second-order effects caused by member deformations 5. Geometric imperfections caused by initial out-of-plumbness 6. Reduction in member stiffness due to inelasticity and residual stresses 7. Uncertainty in stiffness and strength The first four of these items are cove red within the method selected for the analysis of the structure. Geometric imperfections are caused by permitted tolerances

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89 in the plumbness of columns, among others. As specified by AISC Code of Standard Practice for Steel Buildings and Bridges (AISC 303) Section 7.13.1.1, the maximum tolerance on out-of-plumbness is L /500 (AISC 2010). Residual stresses cause premature yielding in a member with a co nsequent post-yield lo ss of stiffness and increase in deformations. This effect may be taken into account by reducing the axial and flexural stiffness of members that contri bute to the lateral stab ility of the structure prior to analysis (Williams 2011). The Effective Length Method When the effective length method is us ed, the required member strength must be determined by a second-order analysis, e ither by rigorous analysis using computer software or by the amplification factors in accordance with Appendix 8 of the AISC specifications (AISC 2010). In any case, th e effective length method is not permitted when the ratio of the second-order to th e first-order deflections exceeds 1.5. The AISC Specification allows the ratio of secondorder drift to first-order drift in a story to be taken only as the B2 amplification factor, calculated in accordance with Appendix 8. To account for initial geometrical impe rfections of the st ructure, notional lateral loads are applied at each story in accordance with Section C2.2b of the AISC Specifications, except that the stiffness re duction indicated in Section C2.3 is not required to be applied, thus, the nominal stiffness of all members is used. (AISC 2010). The available strength of the member s is determined using the appropriate

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90 effective length factor K as defined in Appendix 7 of the AISC Specification (AISC 2010). The empirical column curve then accounts for member geometrical imperfections and inelastic softening effects (Williams 2011). Direct Analysis Method The 2005 AISC specifications (AISC 2005), for the first time, permit a more direct assessment of steel framing system s with the inclusion of Appendix 1 (Design by Inelastic Analysis) and A ppendix 7 (Direct Analysis Me thod). In the most recent AISC Specifications (AISC 2010), the direct analysis method has been moved into chapter C as the pref erred method of frame stability as sessment, and the traditionally used effective length approach has been m oved to Appendix 7. By incorporating both, a nominal out-of-plumbness and a nominal stiffness reduction in the structural analysis, the direct design method and desi gn by inelastic analysis provisions focus on obtaining more realistic analysis results for the stability of steel structures. In addition, both methods account, explicitly, for th e effects of imperfections that affect the system and member strengths when ch ecking the strength of steel structures. (Surovek 2012). Some of the major advantag es of the direct analysis method are: 1. It eliminates the need for effective length factors 2. It provides an improved representation of the member frame forces throughout the structure at the ultimate strength limit state. 3. It applies in a logical and consistent fa shion for all types of frames including braced frames, moment frames and combined framing systems.

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91 Another advantage of the direct analysis method is that it can be applicable to all types of structures, including structur es with compression members with elastic supports such as in the case of the comp ression chord of a pony truss bridge or a wind girt truss. When the direct analysis me thod is used, a second-order analysis is required that considers both Pand Peffects. However, in accordance with Section C2.1 of the AISC Specifications (AISC 2010) when Peffects are negligible a Ponly analysis is permitted. This typically occurs when the ratio of second-order drift to first-order drift is less than 1.7 and no more than one-third of the total gravity load on the building is on columns that are part of the moment-resisting frames. Nonetheless, in the direct analysis me thod the effects of inelasticity and frame imperfections are accounted for in the calcu lated forces required for the design of all the structural components. One of the benefits of this approach is that it provides a direct path from elastic analysis and desi gn to advanced analysis and design (Surovek 2012). Imperfections To account for initial imperfections in the members, notional lateral loads are applied at each story in accordance with Section C2.2b of the AISC Specifications (AISC 2010) and are given by iiN0.002Y (4.8)

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92 Where Ni is the notional load applied at level i and Yi is the gravity load applied at level i The notional loads are ad ditive to the applied lateral loads when the sidesway amplification factor is larger than 1.7 when using the reduced elastic stiffness or 1.5 when using the unreduced el astic stiffness. As an alternative to applying the notional loads at each level, the imperfections can be directly modeled in the analysis by including an initial displ acement, at the points of intersection of members, which represents the out-of-plumbness permitted by the Specifications, (Williams 2011). Reduced Flexural and Axial Stiffness To account for residual stresses and inel astic softening effects, the flexural and axial stiffness of members that contribute to the lateral stability of the structure are reduced as specified in Section C2.3 of the AISC Specification (AISC 2010) and are given by b E I0.8EI (4.9) EA0.8EA (4.10) Where b is the stiffness reduction parameter and is given by when bry1.0P0.5P (4.11) when rr bry yyPP 41P0.5P PP (4.12)

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93 Where is an adjustment factor when ASD lo ad combinations are used and is equal to 1.6. When LRFD combinations are used the adjustment factor is equal to 1.0. Pr is the required second-or der axial strength and Py is the member yield strength ( AFy). Because of the allowance for initial ge ometric and material imperfections and inelastic softening effects in the analysis, stability effects are incorporated in the calculated required member stre ngths. Thus, the available strength of members can be determined using as eff ective length factor of K =1.0 for all members, in other words, the K factor is not necessary (Williams 2011). Advanced Analysis In traditional design, the load capacity of the system is assessed on a memberby-member basis, limiting the load carrying cap acity of the system to the strength of the weakest member. Alternatively, the us e of a nonlinear-inelastic analysis to directly determine the strength and stabil ity of a steel framing system as a whole rather than limiting the strength of the stru ctural system at design load levels by the first member failure is the subject of A ppendix 1 of the AISC Specifications. This type of analysis is commonly referred to in the literature as “Advanced Analysis”. Appendix 1 of the 2010 AISC Specifications states that (AISC 2010): Strength limit states detected by an inelastic analysis that incorporates flexural, shear and ax ial member deformations and all other component and connection de formations that contribute to the displacements of the structure, second-order effects (including Pand Peffects), geometric imperfections, stiffness reductions due to inelasticity, including the effect of residual stresses and partial yielding of the cross-s ection, are not subject to the

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94 corresponding provisions of the Sp ecification when a comparable or higher level of reliability is provided in the analysis (AISC 2010). However, Appendix 1 is not prescriptive in how these attributes are to be included in an inelastic analysis. A report prepared by the American Society of Civil Engineers (ASCE) and edited by Andrea Suro vek called “Advanced Analysis in Steel Frame Design: Guidelines for Direct Sec ond-Order Inelastic Analysis”, presents recommendations for the use of second-orde r inelastic analysis in the design and assessment of steel framing systems. One of the objectives of this document is to provide guidelines for how the requirements of Appendix 1 might be met while also providing the necessary background to understand the rationale for the recommendations. In addition, the guideli nes were developed to provide an understanding of the ba seline requirements for directly capturing member and system strength limit states when using a secondorder inelastic analys is (Surovek 2012). The guidelines include analysis and m odeling requirements as well as design considerations (i.e. suggested resistance factors) using the advanced analysis approach in Appendix 1 of the AISC Specifications, such that the behavior and strength of the overall system and the limit states of individual members are checked concurrently (Surovek 2012). Below are summarized these requirements (Surovek 2012): 1. The analysis model must be able to represent the reduction in member stiffness due to spread of plasticity through the member cross-sections and

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95 along the member lengths, and due to in-pla ne stability effect s of axial forces acting on the inelastic member deflect ed geometry. The report recommends two types of analysis approaches for advance analysis: Distributed plasticity analysis and refined pl astic hinge analysis. 2. The analysis and design procedure must account for the attributes that significantly influence member and sy stem strength. Inelastic material behavior must be included, by using one of the analysis approaches recommended above, that accounts for re sidual stresses. The residual stress pattern chosen should be appropriate for the type of cross-section being considered. 3. Effects of geometric imperfections s hould be included, and they include outof-plumbness (i.e. frame nonverticality) and out-of-straightness (i.e. member sweep). The report recommends to mode l out-of-plumbness by modifying the frame geometry. For orthogonal frames imperfections may be modeled through the use of equivalent horiz ontal notional lo ads. The frame nonverticality should be modeled to a de gree that accurately captures the potential out-of-plumbness that can o ccur during construction. Member outof-straightness should be modeled in such a way that it represents the potential deformed shape due to production tolerances. 4. Limit states, (i.e. column strength fle xural buckling, beam yielding, and beamcolumn in-plane flexural buckling) that can be demonstrated being directly

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96 captured within the analysis are not required to be verified with a separate limit state check using the correspondi ng strength resistance equations. 5. Resistance factors must be incorporated in the analysis/design process if advanced analysis methods are to be used in the context of AISC Specifications. Pony Truss Bridges Design of pedestrian bridges is addre ssed in AASHTO “LRFD Guide Specifications for the Design of Pedestrian Bridges” (AASHTO 2009). Section 7.1.2 provides the requirements for stability of the compressi on chord. The Specifications require that the compression chord (top chord) must be considered as a column with elastic supports at panel points, in ot her words, at the location of the vertical members. Thus, lateral support of the top c hord is provided by a transver se U-frame consisting of the floor beams and truss verticals. The stiffn ess of the elastic supports is determined from the bending of this U-frame, show n in Figure 4.5, and it can be found from Equation 4.13.

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97 Figure 4.5 Lateral U-Frame (Adapted from AASHTO 2009) 2 cbE C hb h 3I2I (4.13) Where C is the transverse frame stiffness, E is the modulus of elasticity, h is the height of the vertical, b is the width of the transverse frame, Ic and Ib are the moment of inertias of the vertical and the beam, respectively, of the transverse frame. The general procedure for late ral stability of th e pony truss bridge is as follows: 1. Design loads are obtained from the LRFD load combinations in AASHTO LRFD Bridge Design Specifications 2. A first order elastic analysis of the 2D truss is performed and the axial forces, P on the top chord are determined from such analysis.

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98 3. A factor of 1.33 is applied to the co mpression chord to account for the probability that the maximum compressi on force would occur simultaneously in adjacent truss panels, thus the design compression chord force, Pc, is equal to 1.33 P 4. The transverse frame stiffness, C is determined from Equation 4.13 5. The parameter, CL/Pc, is then determined, where C the transverse frame stiffness, L is the length of the compre ssion chord between panels and Pc is design compression chord force. 6. The effective length factor, K is found from Table 7.1.2-1 of the Specifications (AASHTO 2009), base d on the number of panels, n and the parameter CL/Pc. 7. The capacity of the compression chord is then found from empirical equations given in the AASHTO Bri dge Design Specificati ons, section 6.9.4 (AASHTO 2012) similar to AISC design of compression members.

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99 CHAPTER V 5 STABILITY ANALYSIS USING NONLINEAR MATRIX ANALYSIS WITH COMPUTER SOFTWARE Matrix Structural Analysis Basically, the behavior of all types of structures can be described by means of differential equations. In practice, the writi ng of differential equations for framed structures is rarely necessary because exac t or approximate solutions to the ordinary differential equations for each member, which can be treated as assemblages of one dimension, are well-known. These solutions ca n be set in the form of relationships between the forces and displacements at the end of each member. Proper combinations of these relationships w ith the equation of equilibrium and compatibility at the joints and supports, yi elds a system of algebraic equations that describes the behavior of the structure (McGuire 2000). On the other hand, structures consisting of three dimensional compone nts are more complicated, thus, exact solutions are very difficult to obtain and practical solutions seldom exist for the applicable partial differential equations. In the past years, significant changes have taken place in the way structures are analyzed in common engineering practice. These changes have primarily occurred because of the great developments made in high-speed digital computers and the increasing use of very complex structures Matrix methods of analysis provide a convenient mathematical language for descri bing complex structures and because the

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100 necessary matrix manipulati ons can easily be handled with computers, matrix analysis using computers has almost comp letely replaced the classical methods of analysis in engineering offices (McCormac 2007). Matrix algebra is a mathematical notat ion that simplifies the presentation and solution of simultaneous equations. A key featur e of matrix analysis methods is that it can be use to analyze practic ally all kind of structures, whether they are statically determinate or indeterminate. Additionally, due to the systematic nature of the matrix concepts and techniques, matrix analysis forms the basis of the computer programs used today for the analysis and design of structures. Thus, matrix analysis is used to obtain a concise description of a structur al problem and to create a mathematical model of the structure. The ability of application of matrix methods in struct ural engineering is very important because all linearly elastic, st atically determinat e and indeterminate structures are governed by sy stems of linear equations. A nd therefore, any method of analysis involving linear algebraic equatio ns can be put into matrix notation and matrix operations can be used to obtain th eir solution. Basically there are two general approaches to the matrix analysis of stru ctures: the flexibility matrix method and the stiffness matrix method. The flexibility matrix method is also known as the force or compatibility method. It obtains the solution of a struct ure by determining th e redundant forces. Thus, the number of equations involved is e qual to the degree of indeterminacy of the

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101 structure. The redundant forces may be selected in an arbitrary manner, and their choice is not an automatic procedure. The primary consideration in the selection of the redundant forces is that the resulti ng equations are well conditioned (Williams 2009). The stiffness method is also known as the displacemen t or equilibrium method, in which the displacement of joints (rotations and translations) necessary to describe fully the deformed shape of the stru cture are used in the equations instead of the redundant forces used in the force methods. When the simultaneous equations that result are solved, these displacements are determined and then substituted into the original equations to determine the various internal forces, thus, the solution of a structure is obtained by dete rmining the displacements at its joints. The stiffness matrix method is the most common method utilized in computer programs for the solution of building structures (McCormac 2007). Direct Stiffness Method When a structure is being analyzed with the stiffne ss method, it is subdivided into a series of discrete finite elements and identifyi ng their end points as nodes and the joint displacements (translations and rotations) are treated as unknowns. Equilibrium equations are written for each joint of the structure in terms of: 1. The applied loads 2. The properties of the members framing into the joint 3. The unknown joint displacements

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102 These relationships, for the entire struct ure, are then grou ped together into what is called the structure stiffness matrix, K. This results in a set of linear, algebraic equations that can be solved simultaneously for the joint displacements. These displacements are then used to determin e the internal member forces and support reactions. The general stiffness equation is written symbolically as follows: { P } = [ K ]{ } (5.1) The order of listing the nodal forces in the matrix [ K ] should be the same as the order of listing of the correspon ding displacements of the matrix { }. Thus if the first listed nodal force in { P } is X1 then the first listed displacement in the matrix { } should be u1, and so on. Nonlinear Analysis using Matrix Methods In nonlinear analysis the ai m is to trace the load di splacement history of all material points in a structure as it under goes progressive loading. The only practical way to do this is by approximating their nonlinearity with a piecewise or step-by-step series of linear analyses that employ one or more of the most common equation solution methods to calculate nonlinear stru ctural behavior. Thes e methods are known as the incremental analysis method and the iterative (multi-step) procedures.

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103 Behavior can be traced incrementally and each method can be stated symbolically as a variant of the global stiffness Equation, 5.1, shown below as Equation 5.2. [ Kt]{ d } = { dP } (5.2) Where [Kt] is the tangent stiffness matrix, {d } is a vector of incremental nodal point displacements, and {dP} is a vector of incremental nodal point loads and reactions. The computational problem can be addressed as the conventional one of solving for unknown displacements and back substitution of the results in element stiffness equations to determine element forc es, but doing so in a stepwise fashion in which the total response is determin ed through summation of increments. In second-order elastic an alysis the effects of finite deformations and displacements are accounted for in formulating the equations of equilibrium and Equation 5.2 becomes Equation 5.3: [ Ke + Kg]{ d } = { dP } (5.3) Where [Kg] is the geometric stiffness matrix and represents the change in stiffness that results from the effects mentioned above, and [Ke] is the elastic stiffness matrix. In first order inelastic analysis the e quations of equilibrium are written in terms of the geometry of the undeforme d structure, thus, Equation 5.2 becomes Equation 5.4:

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104 [ Ke + Km]{ d } = { dP } (5.4) Where [Km] is called the plastic reduction matrix and represents the change in stiffness that results from inelastic behavior of the system. In second-order inelastic analysis both geometric and material nonlinearity are accounted for. The equations of equilibrium are written in terms of the geometry of the deformed system and Equation 5.2 becomes Equation 5.5: [ Ke + Kg + Km ]{ d } = { dP } (5.5) In nonlinear analysis the response of the elements is a continuously or intermittently changing function of the applie d loads. It is often impossible to solve the underlying equations in any direct analyti cal way. Thus is necessary to deal with them in some piecewise linear way. The basic objective of the methods men tioned above is the establishment of equilibrium at the end of the load increm ent. They do so by analyzing the imbalance between the applied loads at the end of a li near step and the internal (element-end) forces calculated from the results of that step. Each method is intended to correct, in an iterative fashion, the imbalance between the linear approximation and the actual nonlinear response, or in other words, to reduce it to a tolerable level. In the iterative methods a single repres entative stiffness is not used for each load increment, instead, increments are s ubdivided into a number of steps, each of

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105 which is a cycle in an iterative process aimed at satisfying the requirements of equilibrium to within a specified tolerance. Figure 5.1 Schematic representation of increment al-iterative solution procedure. (Adapted from Ziemian 2011) Computer Software Used Given that the proposed method is targ eted to the practic ing engineer, the computer program selected for this resear ch project is a commercially available structural analysis software and is commonly used in structural engineering firms. The computer program is not an academic or advanced finite element analysis

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106 software. As in a practicing office, the mode ls in this research project were created with simplified assumptions (i.e. pinned end or fixed end members) as commonly performed by practicing engineers, that is without modeling the actual rotational properties of the connections. The computer program used throughout this research project is commercially known as RISA-3D v11 (RISA-3D 2012). Key characteristics of the software as they relate to stability analysis: The software has the ability to perfor m a rigorous second-order analysis (e.i. Panalysis). The software allows the user to select a stiffness reduction in accordance with AISC 360-10 chapter C. Pand PRISA-3D can perform a Panalysis and determine the secondary moments of a column in a frame, for example. Peffects are accounted for whenever the user requests it in the Load Combinations sp readsheet. The actual modeling of these secondary moments is done through the calc ulation of secondary shears (Figure 5.2). These shear forces are applied at the member ends. For a 3D model, this Pcalculation is done for the memberÂ’s local x a nd local z directions. In other words, the secondary effects are calculated simultaneously for in plane and out of plane directions (RISA-3D 2012).

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107 Figure 5.2 RISA second-order effects pro cess (Adapted from RISA-3D 2012) The programÂ’s Psolution sequence is as foll ows (RISA-3D 2012), Figure 5.2: 1. The model is solved with the original applied loads 2. Calculate the secondary shears ( V ) for every member in the model, Equation 5.6. Thus, P PVLV L (5.6) 3. Add these the shears ( V ) to the original loads and re-solve. 4. Compare the displacements for this new solution to those obtained from the previous solution. If they fall within the convergence tolerance the solution has converged. If not, return to step 2 and repeat. When the Pprocess, described above, is diverging dramatically, this process will be stopped before numerical problems develop and an error will be

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108 displayed. When this error is displayed, the Pdisplacements have reached a level where they are more than 1000 times greater than the maximum original displacements. When this happens the mode l is considered to be unstable under the given loads (RISA-3D 2012). RISA's second order analysis method is based entirely on nodal deflections; the effect of Pis not directly accounted for. Thus, addition of nodes at discrete locations along the column, where member displacement effects are at their maximum, will account for the effects of P. There are a number of "benchmark" problems, which are discussed below, given in various publications to determine if a program is capable of prope rly considering this effect. These comparisons can be used as a basis for when the Peffect is significant en ough to consider in the analysis. They can also be used to determine how many intermediate nodes are required to adequately account for the eff ect at a given load level (RISA-3D 2012). Modeling Geometrical Imperfections Geometrical imperfections can be mode led in RISA-3D by either, applying notional loads or directly modeling the im perfection. This can be done by adding an initial displacement to the members at node locations that represents the out-ofplumbness required by code. As it was me ntioned above, the program determines second order effects at joint (node) locations only, thus, it is recommended that the notional loads or modeled initial displ acements to be applied at the joints.

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109 Assessment of the Computer Software with Benchmark Problems from Established Theory Underlying contemporary nonlinear matrix analysis are established classical methods of structural mechanics. These met hods are applied to elementary systems in the following benchmark problems for the purpose to illustrate types of physical behavior and in appraising the matrix me thods used in the computer software. To provide perspective and a basis for comparison with the computer program selected herein, established classical soluti ons of some elementary problems are given in the commentary to Appendix 7 of the AISC Specifications (AISC 2005). The benchmark problems presented in the commentary to Appendix 7 of the 2005 AISC Specifications (AISC 2005) are re commended as a first-level check to determine whether an analysis procedur e meets the requireme nts of a rigorous second-order analysis that is adequate fo r use in the direct analysis method. As discussed earlier, second-order analysis pr ocedures in some software packages may not include the effects of Pon the overall response of the structure. Where Peffects are significant, erro rs arise in approximate methods that do not accurately account for the effect of P moments on amplification of both local ( ) and global ( ) displacements and correspond ing internal moments. Thes e errors can occur with second-order computer analysis programs if additional nodes are not included along the members.

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110 The benchmark problems are intended to assess the ability of the computer program to perform a rigorous second order analysis and to reveal whether or not Peffects are included in the analysis. Th e benchmark problems used to assess the correctness of the computer program are shown in Figures 5.3 and 5.4. A brief description is presented here; however, fu ll descriptions and solutions will be presented later in Chapter VII Benchmark problem 1, shown in Figur e 5.3, is a simply supported beamcolumn subjected to an axial load and a uni formly distributed transverse load between supports. This problem contains only Peffects because there is no relative translation between the ends of the column. Figure 5.3 Benchmark Problem 1 (AISC 2005)

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111 Benchmark problem 2, shown in Figure 5.4, is a fixed-base cantilevered beam-column loaded simultaneously with an ax ial load and a lateral load at its top. This problem contains both Pand Peffects. Figure 5.4 Benchmark Problem 2 (AISC 2005) AISC points out that, when the corre ctness of a proposed second-order analysis method is evaluated, through th ese two benchmark problems, both, moments and deflections should be checked at various levels of axial load on the member and in all cases the results should agree w ithin 3% and 5%, re spectively (AISC 2010).

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112 CHAPTER VI 6 A PRACTICAL METHOD FOR CRITICAL LOAD DETERMINATION OF STRUCTURES The goal of this research project is to propose a pract ical method for stability evaluation and critical load determination of structures that are difficult to analyze with conventional hand calculation methods, (e.g. the compression chord of a pony truss bridge or a wind girt). The pr oposed methodology relies on a commercially available computer program in which a second order analysis can be readily accomplished by taking into consideration en d-restraints, reduced flexural stiffness and initial geometrical imperfections or out-of-plumbness. The method can also be used for any 3D structure. The proposed procedure incorporates an iterative process to determine the critical load at which the structure become s unstable, by systematically increasing the load until the computer program does not converge on a solution. The two key factors in this procedure are the in troduction of an initial impe rfection combined with the ability of the progr am to perform a Panalysis. The methodology is based on combining the direct design method and u tilizing readily available 3D commercial computer software that allows the user to incorporate the two key factors mentioned above. A step-by step process for the impl ementation of the me thodology is presented below.

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113 Proposed methodology: Step-by-step process Step 1: Software Selection. For the methodology proposed here, it is essential to select a 3D computer program for structur al analysis that can perform a rigorous second-order analysis, Pincluding P. The software package selected for this research project was RISA-3D v 11 (RISA3D 2012). An introduction to the software was presented in Chapter V. The following steps (2-5) were implem ented in Chapters VII through IX. Step 2: PCapability Assessment. Not all analysis computer software packages are capable of performing a rigorous second-order analysis. A crucial pa rt of the software selection process is the need to verify th e softwareÂ’s ability to perform a rigorous second-order analysis. This can be done w ith the analysis of benchmark problems. AISC 360-05 and 360-10 both give two benchm ark problems to determine if the analysis procedure meets the requirements of a rigorous second order analysis. AISC requires that the results from the computer program should fall within 3% of the moment amplification values and 5% of th e deflection amplification values, given by the benchmark solutions (AISC 2010). Step 3: Structure Modeling. Once the computer program has been verified to be performing rigorous second-orde r analyses correctly, the ne xt step is to build a representation of the structure within the program. It is recommended to model the structure in the same manner a similar struct ure would be modeled (i.e. same restraint assumptions and boundary conditions). Howeve r, for members that are subjected to

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114 second-order effects, additional nodes may be needed for the determination of Peffects if the softwareÂ’s second-order an alysis method is based entirely on nodal deformations, as is in the case of RISA-3D. Step 4: Geometrical Imperfections. One of the key elements of the methodology is to account for the destabili zing effects of second-or der forces, moments and deformations due geometrical imperfections. He nce, it is necessary to include initial imperfections in the model to initiate these secondary effects on axially loaded members. The initial imperfections can be modeled directly in to the structure by applying initial displacements to the node s or by modeling the nodes in a displaced configuration. A second alte rnative is to apply notiona l loads to the nodes in the direction of the imperfection. As mentioned in step 3, additional nodes may be needed to account for the effects of Pin some software packages. Step 5: Critical Load Determination The determination of the critical load is the final step and goal; this is accomplished through an iterative process in which the gravity load is systematically increased. One way to go about this process is to increase the load factor in the load combin ations tab. This way, the gravity loads in the structure do not need to be changed, thus saving significant time in the analysis. Throughout the process, if th e structure is stable, the Pfeature of the program will lead to successively sma ller displacements until a convergence is reached. The analysis is repeated with larger loads unt il the analytical mode l no longer converges, shown in Figure 6.1, and that is the load that causes instability (i.e. buckling).

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115 Figure 6.1 Error message from RISA-3D that Pis no longer converging

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116 CHAPTER VII 7 COMPUTER SOFTWARE EVAL UATION WITH BENCHMARK PROBLEMS In order to verify that the Pcapability of the program meets the requirements of a rigorous second-order analys is, the computer program used in this research project was evaluated by analyzing the two problems that were described in Chapter V and by following the proposed methodology illustrated in Chapter VI In addition, the effects of Pon the benchmark problems were evaluated by varying the number of nodes along the b eam-column. According to the software developer, additional nodes along the member provide a more accurate representation of the Peffects on a structure. Therefore, diffe rent intermediate node configurations were evaluated and compared to the benchmark solutions presented in the commentary to the AISC Specification. The intermediate node configurations for benchmark problems 1 and 2 are described below. It is important to note that for problem 1, Figure 7.1; at least one intermediate node needs to be included along the member. That is, because there is no translation of the supports, th is is a case of Ponly; in other words, Pdoes not exist for this case. If no intermediate nodes were added to problem 1, the com puter program will not perform a second-order analysis and th e results will be ba sed on a first-order analysis regardless of the compressive force.

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117 The general procedure consisted on a pplying a constant the lateral load ( W shown in Figure 7.1 and H shown in Figure 7.2), that would simulate the inherent geometrical imperfections in real stru ctures, while increasing the axial load, P from zero to the point where the program did not converge anymore. This procedure was followed for each case. Figure 7.1 Benchmark Problem 1 (AISC 2005) Figure 7.2 Benchmark Problem 2 (AISC 2005)

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118 The results of problem 1 a nd problem 2 are presented below and are plotted as the load ratio, P/PeL, versus the moment amplification factor, Mmax/M0, and the load ratio, P/PeL, versus the deflection amplification factor, Ymax/Y0. Where P is the incremental compressive load and PeL is the Euler buckling load given by Equation 7.1. 2 eL 2 E I P L (7.1) The moment amplification factor and de flection amplification factor are the ratios of any given moment to the initia l moment and any given deflection to the initial deflection respectively. According to the commentary to the AISC Specifications, the results from the comput er program should fall within 3% and 5% of the moment amplification values and deflection amplific ation values, respectively, given by the benchmark solutions (AISC 2010). Benchmark Problem 1 The first problem used to evaluate the ability of the computer software to perform a rigorous second-order elastic anal ysis is a simply supported beam-column subjected to a uniform transverse load between the supports and an applied compressive load at its ends. The beam-column used in the analysis is a W14x48 ISection, 28ft long and pin-supported at its top and bottom. The initial deformation, Y0, was created by applying a horizontal unif orm distributed load along the column

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119 equating to 0.20 kip/ft. This in itial displacement represents an initial imperfection of L /1700. The problem contains only the Peffect because there is no translation of one end of the member relative to the other. Problem 1 is the same as the benchmark problem Case 1 presented in the commentary to the AISC Specifications (AISC 2005) and is shown in Figure 7.3. The classi cal solution is presented by Timoshenko (1961), McGuire (2000) and by Chen (1985) which is summa rized in the commentary to the AISC Specification and is presented below (AISC 2005). The maximum moment of problem 1 is given by 2 22secu1 wL M 8u (7.2) And the maximum deflection is given by 2 4 4122secuu2 5wL Y 384EI5u (7.3) Where the parameter u is given by 2PL u 4EI (7.4) P represents the axial lo ad which is increased fro m 0.0 kip until the program no longer converges. L is the distance between supports ; also known as the length of the beam-column. E is the modulus of elasticity of steel and I is the moment of inertia

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120 of the beam-column in the direction of bending. The a bove moment and deflection rigorous solutions ( Mmax and Ymax), are plotted in Figures 7.4 and 7.5 as functions of the applied load ratio P / PeL. As discussed earlier, RISAÂ’s second or der analysis method is based entirely on nodal deflections, thus, for the evaluation of the computer software, three cases of intermediate nodes were evalua ted and are shown in Figure 7.6. For the first case, one intermediate node at mid hei ght was included. As menti oned above, at least one node needs to be included in this case so that the computer program can calculate secondorder effects due to Peffects. For cases 2 and 3, two and three intermediate nodes, respectively, were added to the beam-c olumn as shown in Figure 7.6. The loadmoment history and the load-deflectio n history of all th ree different node configuration cases are pres ented in Figures 7.7 and 7.8 respectively, along with the rigorous solution from the classical solution.

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121 a) b) Figure 7.3 Benchmark problem 1: a) undeflec ted shape b) deflected shape

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122 Figure 7.4 Maximum moment values as a functi on of axial force for benchmark problem 1 (classical solution, Equation 7.2) Figure 7.5 Maximum deflection values as a function of axial force for benchmark problem 1 (classic al solution, Equation 7.3) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 0.00010.00020.00030.00040.00050.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Mmax/M0 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 0.00010.00020.00030.00040.00050.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Ymax/Y0

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123 a) b) c) Figure 7.6 Benchmark problem 1 node arraignm ents a) Case 1, one additional node at mid height of the beam-column b) Case 2, two additional nodes at equal distance c) Case 3, three additi onal nodes at equal distance

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124 Figure 7.7 Maximum moment values as a functi on of axial force for benchmark problem 1 (classical solution, with one, two and three additional nodes) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 0.0005.00010.00015.00020.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Mmax/M0 Classical Solution One additional node (Case 1) Two additional nodes (Case 2) Three additional nodes (Case 3)

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125 Figure 7.8 Maximum deflection values as a function of axial force for benchmark problem 1 classical solution with one, two and three additional nodes) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 0.0005.00010.00015.00020.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Ymax/y0 Classical Solution One additional node (Case 1) Two additional nodes (Case 2) Three additional nodes (Case 3)

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126 Benchmark Problem 2 The second problem is a cantilever beam-c olumn, fixed at the bottom and free at the top with latera l and vertical loads a pplied at its top. The beam-column used is a W14x48 I-Section with a length of 28ft. The initial deformation, Y0, is created by applying a horizontal point load at the t op of the column equating to 1.0 kip. This initial displacement represents an initial imperfection of L /350. Problem 2 contains both Pand Peffects since the top node is free to translate. Problem 2 is the same as the benchmark problem Case 2 presen ted in the commentary to the AISC Specifications (AISC 2005) and is shown in Figure 7.9. The rigorous classical solution is presented by Timoshenko (1961) and McGuire (2000). The solution is summarized in the commentary to the AISC Specification and presented below (AISC, 2005). The solutions presented be low are plotted in Figures 7.10 and 7.11. The maximum moment of problem 2 is given by tan MHL (7.5) And the maximum deflection is given by 3 33tan HL Y 3EI (7.6) Where the parameter is given by 2PL EI (7.7)

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127 P represents the axial load which was increased from 0.0 kip until the program no longer converged. L is the distance between supports; also known as the length of the beam-column. E is the modulus of elasticity of steel and I is the moment of inertia of the beam-column in the direction of bending. a) b) Figure 7.9 Benchmark problem 2, a) undeflec ted shape b) deflected shape

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128 As discussed above, this problem contains both, Pand P, thus it is necessary to add additional nodes along the length of the member to account for the effects of P. Three different node configurations were evaluated and are shown in Figure 7.12. However, as opposed to probl em 1, the first case consisted of no intermediate nodes at mid height. The resu lts from this case are mainly due to P. For cases 2 and 3, one and two intermediate nodes, respectively, were added to the beam-column as shown in Figure 7.11. The load-moment history of all cases and the load-deflection history are presen ted in Figures 7.13 and 7.14 respectively, along with the rigorous solution from the classi cal solution. It is important to note that the maximum load ratio for problem 2 is equal to 0.25( P/Pel), this is due to the fact that for cantilever columns, the effective unsupported length is tw ice the length of an equivalent pin-ended colu mn, thus the Euler buck ling load is given by: For a pin-ended column, 2 eL 2 E I P L and (7.1) For a cantilever column, 2 eL_cant 2 cantilever E I P L (7.8) Because Lcantilever=2L then 22 eL_canteLeL 22EIEI 11 PP0.25P 44 2LL (7.9)

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129 Figure 7.10 Maximum moment values as a functi on of axial force for benchmark problem 1 (classical solution, Equation 7.5) Figure 7.11 Maximum deflection values as a function of axial force for benchmark problem 1 (classic al solution, Equation 7.6) 0.000 0.050 0.100 0.150 0.200 0.250 0.00010.00020.00030.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Mmax/M0 0.000 0.050 0.100 0.150 0.200 0.250 0.00010.00020.00030.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Ymax/Y0

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130 a) b) c) Figure 7.12 Benchmark problem 2 node arraignm ents a) Case 1, no additional node along the height of the beam-column b) Case 2, one additi onal node at mid height of the beam-column c) Case 3, two additional nodes at equal distance

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131 Figure 7.13 Maximum moment values as a functi on of axial force for benchmark problem 2 (classical solution, with zero, one and two additional nodes) 0.000 0.050 0.100 0.150 0.200 0.250 0.0005.00010.00015.00020.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Mmax/M0 Clasical Solution No additional nodes (Case 1) One additional nodes (Case 2) Two additional nodes (Case 3)

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132 Figure 7.14 Maximum deflection values as a function of axial force for benchmark problem 2 (solution, with zero, one and two additional nodes) 0.000 0.050 0.100 0.150 0.200 0.250 0.0005.00010.00015.00020.000Axial Force Normalized by Euler Buckling Load, P/PeLAmplification Factor, Ymax/Y0 Classical Solution No additional nodes (Case 1) One additional nodes (Case 2) Two additional nodes (Case 3)

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133 Discussion of Results fr om Benchmark Problems From the results presented above it can be seen that th e computer program used in this research project meets th e requirements of a rigorous second-order analysis, as defined by AISC (AISC 2005). Therefore, the computer program, RISA3D, can be used with the proposed an alytical method. However, because the computer program calculates secondary eff ects at node locations, it is necessary to add intermediate nodes between joints, al ong the length of compression members, when effects due to Pare significant. Thus, it is necessary to investigate the influence of Pon the structure on a case by case basis.

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134 CHAPTER VIII 8 MODEL DEVELOPMENT As discussed earlier in Chapter I, the aim of this research pr oject is to propose a practical method for stability analysis of structures that are difficult to analyze with conventional methods. Examples of this t ype include, but are not limited to, the compression chord of a pony truss bridge or the compression chord of a wind girt truss in which the buckling mechanism takes place out of the plane of the truss. Nontheless, this method can also be used for any other 3D st ructure. Given the author’s interest in bridges, it was d ecided, for this study, to work with pony truss bridges that can be load tested to validate the proposed methodology. Currently, the design of pedestrian bri dges including pony truss like structures is based on the requirements of AASH TO “LRFD Guide Specifications for the Design of Pedestrian Bridge s” (AASHTO 2009). As disc ussed in Chapter III, the methodology for the design of the top chord used in this document is based on research that was done in the 1950’s. As illustrated earlier, the computer program used throughout this research project was evaluated using tw o benchmark problems to asse ss its ability to carry out a rigorous second-order analysis. The outcome of this evaluation was presented in Chapter VII with satisfactory results. On ce it was determined that the computer program met the requirements for a rigorous second-order analysis, as required by AISC (AISC 2005), the next step was to develop two physical scaled-down test

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135 models that can be load tested using ava ilable load testing equipment to verify the analytical method. Consequently, an analyt ical study was developed to determine the parameters that would affect the load-dis placement behavior of the top chord and stability of the pony truss bridges. This an alytical study started with a parametric scheme in which computer models, that simulated a full-size pony truss bridge, were developed with the purpose of targeting some specific parameters to understand the stiffening effects that the transverse frames have on the stability and overall behavior of the bridge. These computer models of fullsize bridges were labeled the analytical models The results from the parametric study were used to develop an experimental study of scaled computer models for which the critical load was determined by following the methodology presented in Chapter VI. The ultimate goa l of these scaled computer models was to fabricate and load a set of scaled models to validate the results from the computer models. Howeve r, it was also necessary to determine, approximately, the total load at which the scaled models will fail so that hydraulic jack and load cell capaci ties could be determined. Full-Size Analytical Study The analytical evaluation of pony truss br idge models, using RISA-3D, started with a parametric study, in order to assess th e stiffening effects of the lateral restrains, in which a full size pony truss bridge mode l was set up using the computer program. The model, herein referred as the full size model, is a 40 feet long pony truss bridge

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136 similar to the actual pony tr uss bridge shown in Figure 8.1. The full size computer model was modeled with the following dimens ions: The span was set at 40 ft, and the deck width was set at 12 ft. The transverse frame consisted of hollow steel sections (HSS) 8 in x 4 in x in stringer beams and 4 in x 4 in x in verticals and diagonals. The 6 in concrete deck was modeled compos ite with the frames; refer to Figure 8.2. With the purpose of evaluating the influen ce of the transverse frame stiffness, and flexural stiffness of the top compression co rd, a factorial process was developed such that, systematically, all of the variables described below were considered and the critical load was determined for each case. In total, the parametric study, shown in Figure 8.3 included 21 analyses. This study included varying the comp ression chord size by using seven different sections and changi ng the height of the truss wh ile keeping the length of the bridge constant. The analyses included va rying the top chord cr oss-section form a HSS section of 2 in x 1 in x 1/8 in to a sec tion of 6 in x 6 in x in. At the same time, three different truss heights were investigated, 4 f eet, 6 feet and 8 feet which correspond to an aspect ratio, AR of 0.10, 0.15 and 0.20 respectively. The combination of different aspect ratios, AR and compression chord sizes yielded a variety of different out of plane stiffness parameters, for the bridge truss. In total, 21 different stiffness parameters, were generated and are presented in table 8-1. The stiffness parameter correlates the stiffness of the compression chord and

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137 transverse frame to the deformational behavior (i.e. deformed shape) and critical load of the compression chord. A description of the parametric study is presented below. Figure 8.1 Actual 40 foot long pony truss bridge Figure 8.2 Computer model of the 40 foot long bridge ( analytical model )

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138 The aspect ratio, AR refers to the ratio of the height of the truss, h to the length of the bridge, L and is given by equation 8.1. The stiffness parameter, refers to the transverse rigidity of the frame (elastic restraint), C times the length of the chord, Lchord, divided by the cross sect ional area of the chord, Achord, given by equation 8.2. h AR L (8.1) chord chordL C A (8.2) Where C is the transverse rigidity of el astic restraints for the compression chord (AASHTO 2009) is given by equation 8.3 2 cbE C hb h 3I2I (8.3) And h is the height of the truss and verticals, b is the width of the cross-frame, Ic is the moment of inertia of the vertical and Ib is the moment of in ertia of the beam between the verticals. Once the model was completed and the grav ity loads were applied to the deck surface, the first step was to add a geomet rical imperfection. This imperfection was entered as a notional load that simulated an initial imperfection, of the top chord of L /500, in the out of plane direction of the truss, and is shown in Figure 8.4.

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139 Figure 8.3 Parametric study

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140 The next step was to systematically increase the gravity load, following the methodology presented in Chapter VI, until th e critical load was found. In this case, the gravity load was maintained constant while incrementally changing the load factor until the program di d not converge to a solution. Hence, for each case, the gravity load was increased until the Pfeature of the progr am no longer converged to a solution, in which, at this point the maximum load in the compression chord was recorded as the ratio of such maximum load to the yield lo ad of the chord. Figure 8.4 Geometric imperfection The stiffening effects of the chord size and aspect ratio are evident in Figures 8.5 and 8.6. Figure 8.6 shows the deformed shape of the model at the critical load for the case when the top chord is slender with compared to the vertic als of the truss. In comparison, Figure 8.6 shows the same modelÂ’s deformed shape for the case when

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141 the top chord is much stiffer compared to the verticals of the truss. The results of this parametric study are shown in Table 8.1. These results present a correlation between the stiffness parameter of the top chord and th e critical load of that same top chord. By looking at the results in Table 8.1, it is evident that for pony truss bridges with large stiffness parameters and low aspe ct ratios, in other words, bridges with stiffer cross-frames, the critical load in the top chord is close, and in some cases, larger than the yield strength of the chord. This means that it is likely that the top chord would fail in yielding before instabili ty due to buckling o ccurs. Thus, for the experimental study, which includes the theoretical models and the test models mentioned above, it was decided to select pa rameters that would ensure that the top chord would fail due to elasti c buckling before yielding. An aspect ratio of 0.20 and stiffness parameters between 55 and 98 were selected to be used in the experimental study presented below. Based on the results shown in Table 8-1 and the selected parameters it is expected that the critical load of the top chord is on the order of half the yield strength, t hus, ensuring that the failure of the top chords is due to buckling. Ba sed on the proposed test setup, described below, and for practical reasons it was decide d to test two scaled down truss bridges. The first bridge was determined to be 5 feet long and th e second bridge was determined to be 3.333 feet long. The correspon ding truss heights, fo r an aspect ratio of 0.20, are 12 inches and 8 inches re spectively. A full description of the experimental models is provided below.

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142 Figure 8.5 Deformed shape at critical load. Slender top chord Figure 8.6 Deformed shape at critical l oad. Stiffer compression chord

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143 Table 8.1Parametric Study of Analytical Models Aspect Ratio ( AR ) Top Chord Size Stiffness Parameter ( ) Top chord Load Ratio Pcr/Py 0.10 2x1x1/8 1989.83 2.39 2x2x1/8 1440.26 1.74 2x2x1/4 801.20 0.97 3x3x1/4 495.83 0.65 4x4x1/4 359.00 0.63 5x5x1/4 281.35 0.62 6x6x1/4 230.88 0.60 0.15 2x1x1/8 694.25 0.73 2x2x1/8 502.51 0.61 2x2x1/4 279.54 0.52 3x3x1/4 172.99 0.53 4x4x1/4 125.25 0.55 5x5x1/4 98.16 0.56 6x6x1/4 80.55 0.54 0.20 2x1x1/8 390.52 0.54 2x2x1/8 282.66 0.54 2x2x1/4 157.24 0.46 3x3x1/4 97.31 0.51 4x4x1/4 70.46 0.51 5x5x1/4 55.22 0.50 6x6x1/4 45.31 0.49

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144 Experimental Study Once the aspect ratio and stiffness pa rameters were selected from the analytical computer trials, such that buck ling of the compression chord occurs before yielding, the next step was to define the member cross-section properties, for the 5.0 foot and 3.333 foot long pony truss bridges, that would result in the same parameters as those selected earlier. For reference purposes, the pony truss bridges were labeled Bridge 1 and Bridge 2 Bridge 1 refers to the smaller bridge, that is, the 3.333 foot long pony truss bridge, while Bridge 2 refers to the bigger bri dge, that is, the 5.0 foot long pony truss bridge. A description for each bridge will be presented below. Following the selection of the crosssection properties of both pony truss bridges, an experimental st udy was developed and divided into two phases. The first phase consisted on modeling th e pony truss bridges in the computer program in order to determine the critical load and load-d eformation history. As mentioned earlier, these computer models were labeled theoretical models The theoretical models are scaled down pony truss bridges that were m odeled within the computer program to determine the critical load at which the br idge would fail due to instability of the compression chord. The second phase consisted on fabricatin g two scaled pony truss bridges with the same cross-sections and properties as the theoretical models (i.e. physically represent the theoretical models ) in order to validate, thr ough load testing, the results from the theoretical models These testing models were labeled test models

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145 Theoretical Models The full-size analytical study provided the frame work for the development of the scaled models that eventually would be load tested for validation of the proposed methodology. Based on the test setup selected, which is described below, and with the purpose of having a baseline for comparison be tween the results of the load tests and the results of the computer analyses, two computer models were developed such that they would replicate all th e properties and the loading condition at which the test models would be subjected to. The co mputer models, referred as to the theoretical models were modeled in RISA-3D as close as practical possible, with consideration of common engineering practice in a design firm by pract icing engineers, to simulate how the test models would be fabricated and load tested. A full description of each model, loading, inclusion of geometric im perfections and procedure, for the first phase ( theoretical models ), is presented below. Bridge 1 The first model, shown in Figures 8.7, 8.8 and 8.9, consisted of a pony truss bridge of 3.33 feet in length an d 8.0 inches in height. The trusses were set 12 inches apart, in other words, the width of the brid ge was set to 12 inches. For this model, the aspect ratio, AR and stiffness parameter, were determined to be 0.20 and 97 respectively. The pony truss bridge was modele d with four panels spaced at 10 inches apart. The top two chords, as well as the vertical and diagonal elements of the truss, were modeled as square tube sections of 0.5 in x 0.5 in x 1/6 in. The bottom chords,

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146 as well as the transverse beams that form the transverse frames (U-frame), were modeled as square tube sections of 1 in x 1 in x 1/6 in. The floor slab was modeled as a 0.75 inch thick shell element with a modulus of elasticity, E of 1,000 ksi and fully composite with the floor beams to simulate the wood plank that wi ll be used in the test models. The vertical elements were m odeled as pin-ended at the connection with the top chord and fix-ended at the connec tion with the bottom chord. On the other hand, the diagonal elements were modele d as pin connected at both ends. Bridge 2 The second model, shown in Figures 8.10, 8.11 and 8.12, consisted of a pony truss bridge of 5.0 feet in length and 12.0 inch es in height. The width of the deck was set to 12 inches. For this model, the aspect ratio, AR and stiffness parameter, were determined to be 0.20 and 63 respectively. The pony truss bridge was modeled with six panels spaced at 10 inches apart. The top two chords, as well as the vertical and diagonal elements of the truss, were modeled as square tube secti ons of 0.5 in x 0.5 in x 1/6 in. The bottom chords, as well as the transverse beams that form the transverse frames (U-frame), were modeled as square tu be sections of 1 in x 1 in x 1/6 in. The floor slab was modeled as a 0.75 inch thick sh ell element with a m odulus of elasticity, E of 1,000 ksi and fully compos ite with the floor beams to simulate the wood plank that will be used in th e test models. Similar to Bridge 1 the vertical elements were modeled as pin-ended at the top chord a nd fix-ended at the bottom chord, and the diagonal elements were modeled as pin connected at both ends.

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147 Figure 8.7 Bridge 1 truss geometry Figure 8.8 Bridge 1 computer model of truss Figure 8.9 3D Model of Bridge 1

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148 Figure 8.10 Bridge 2 truss geometry Figure 8.11 Bridge 2 computer model of truss Figure 8.12 3D model of Bridge 2

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149 Loading It was anticipated that the test models would be loaded using a hydraulic jack and a reaction frame, shown in Figure 8.13. Furthermore, a spreader beam and two spreader plates would be used to dist ribute the load onto the two middle panels adjacent to the bridge center line. Thus, in order to simulate this loading condition, the theoretical models were loaded with a uniform distributed load on the two panels adjacent to the center line of the bridge shown in Figures 8.14 and 8.15. An initial uniform distributed area load of 60 psf was selected. The gravity load was increased by changing the load factor in the load combinations tab of the computer software, as described in Chapter VI. Figure 8.13 Load test setup diagram

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150 Figure 8.14 Bridge 1 computer model loading configuration Figure 8.15 Bridge 2 computer model loading configuration Geometric Imperfections Three different geometric imperfections were incorporated into the computer model study. These three imperfections were modeled as initial imperfections of the top chord (out of plane) of, L /300, L /450 and L /700. As described in Chapter VI the geometric imperfections can be modeled by adding notional loads to the structure or

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151 by physically modeling the compression chord in a deformed shape. For this study the initial imperfections were modeled in bot h ways, by adding notional loads to the top chord, as shown in Figure 8.16, and by disp lacing the chord nodes to a determined distance that would simulate such imperfection. Procedure The procedure used to determine the cr itical load and the load-deformation history of Bridge 1 and Bridge 2 follows the same procedure presented in Chapter VI. The investigation included Ponly, that is, without stiffness reduction as described in AISC 360-10 (AISC 2010). In addition to determining the loaddisplacement history, the results from the theoretical models confirmed that the critical load of the top chords was less th an the yield strength of the chord and the total gravity load on the bridge was below the capacity of the hydraulic jack and load cell. The results from the theoretical models are presented in Chapter IX. Figure 8.16 Modeling of ge ometric imperfections

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152 Test Models As mentioned earlier, the goal of the test models was validate the results obtained from the theoretical models thus, the test models were built with the same dimensions, member cross-sections and material properties as the theoretical models Figures 8.17 and 8.18 show the comput er model and the actual test model respectively for Bridge 1 Figures 8.19 and 8.20 show th e computer model and the actual test model respectively for Bridge 2 The truss members were connected with welded connections. Plans ar e presented in Appendix 1. Prior to fabrication of the test models a welding procedur e specification, WPS, was developed and qualified, in accordance with AWS D1.1 (AWS 2002), in order to verify the quality of the welds. The connections that were modeled as fixed, within the computer program, were fabri cated as “all-around” welded connections with a weld size of 3/16 in. The weld capacity was checked to ensure that the connection developed the full moment capacity of the member. On the other hand, for connections that were modeled as pinned connections, they were fabricated with gusset plates, so that rotation can occu r without developing significant moment capacity. The gusset plate used was 1/8 in thick and the weld size used was 3/16 in. The length of the welds was checked to verify that it could develop the axial capacity of the members. The gusset plate geometry was developed based on the minimum weld length.

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153 Figure 8.17 Bridge 1 Computer Model Figure 8.18 Bridge 1 Test model

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154 Figure 8.19 Bridge 2 Computer Model Figure 8.20 Bridge 2 Test Model

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155 Test Setup A diagram of the test setup and the actu al setup are shown in Figures 8.21 and 8.22 respectively. The load test setup was de signed such that a hydraulic jack, shown in Figure 8.22, would load the bridge by appl ying a thrust force when the jack pushed against a reaction frame. The reaction frame was built with a header beam (HSS steel tube section 6in x 4 in x 3/8 in) held down at the ends by tw o 1- in dia. tiedown rods attached to the concrete reaction floor. The hydrauli c jack, was supported on a transfer beam consisting on four 1 in x 1 in x 1/6 in steel tube sect ion welded together to a in x 4 in flat plate, shown in Figure 8.23. The load from the hydraulic jack was then transferred from the transfer beam onto two spreader plates. Each spreader plat e consisted of a in dia. solid steel bar welded to a in x 4 in flat plate shown in Figure 8.23. The purpose of the solid steel bar was to maintain the same load distribut ion to the bridge deck at all times, even when the bridge deflected downwards. Attached to each spreader plate was 3/8 in neoprene pad, that distributed the load ont o the deck, thus simulating the uniform distributed load assumed in the theoretical models The bridge end supports were designed and built so that the bridge woul d deflect downwards without any horizontal restraint along the center line of the br idge. This was accomplished by welding a in dia. solid bar at each corner of the br idge, and then bearing on in steel plate attached to each abutment. The abutment s were built with concrete-filled CMU blocks resting on a in neopr ene pad to distribute the load evenly onto the floor.

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156 Figure 8.21 Load test setup diagram Figure 8.22 Load test setup

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157 Figure 8.23 Spreader beam and plates used to uniformly distribute the load on the bridge deck Figure 8.24 Typical end support configuration and abutments The load on the bridge was recorded by using a low-profile load cell and a meter, shown in Figures 8.25 and 8.26. The load cell was set between the hydraulic

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158 jack and the reaction frame which was fitted with a smooth plate to distribute evenly the load across the load cell and prevent pun ching deformation of the tube wall. At each load interval, the load on the bridge was read directly from the meter. The working (allowable) cap acity of the load cell and hydr aulic jack were 25,000 lbs and 20,000 lbs respectively. In order to measure the displacement of the compression chord as it deforms during the load test, three dial indicators were placed along each top chord, for a total of six. The dial indicators placement is s hown in Figure 8.26, where it can be seen that the indicators were placed at each end of the top c hords and at the center. In addition, the dial indicators, with a total di splacement of 2 inches, were placed half way so that the displacement of the compression chord can be measured regardless of which way it displaced. Figure 8.25 Load cell

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159 Figure 8.26 Data collection setup Load Testing Procedure The load testing procedure consiste d on loading the br idge in 1,000 lbs increments, by manually pumping the hydrau lic jack. When the target load was reached, the load test was stopped and the measurements from all instruments were

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160 taken. The recordings included: load reading from the load cell meter, out of plane displacement of the truss at the end of each compression chord and out of plane displacement at the center of each compression chord from the dial indicators. Loading continued until each bridge was not able to maintain the load without excessive deformation of the top chords. Figure 8.27 Test setup top view

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161 CHAPTER IX 9 EXPERIMENTAL RESULTS A discussion of the different experiment s, the load testing setup and loading sequence was presented earlier in Chapter VIII, where it was mentioned that the theoretical modes were analyzed with three different initial imperfections, namely o: L /300, L /450 and L/ 700. In addition, two different approaches for modeling the initial imperfection were used. To facilitate the presentation of results in this chapter, the two approaches were labeled: Notional Load (NL) and Dire ct Modeling (DM); a description of these two approach es was presented in Chapter VI. With the purpose of tracking the load-d isplacement history for both top chords of each bridge, the naming convention shown in Figures 9.1 and 9.2 was adopted. Prior to loading the bridges, the top chords of each bridge were measured for out of plane straightness (i.e. to determine the initial out of plane imperfection), illustrated in Figure 9.3. It was determined that the measured initial imperfection, o, for Chord 1 (C1) and Chord 2 (C2) of Bridge 1 were 0.042 inches or L/ 475 and 0.050 inches or L /400, respectively. The meas ured initial imperfection, o, for Chord 1 (C1) and Chord 2 (C2) of Bridge 2 were 0.090 inches or L /445 and 0.105 inches or L /377, respectively. These results are s hown in Table 9.1. The results for Bridge 1 are presented below followed by the results of Bridge 2 For reference purposes, the critical load of the theoretic al models is defined as th e load immediately obtained at the termination of the conve rgence to a solution of the Pfeature.

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162 Figure 9.1 Bridge 1 Naming convention for top chords Figure 9.2 Bridge 2 Naming convention for the top chords Chord 1 (B1 C1) Chord 2 (B1 C2) Chord 1 (B2 C1) Chord 2 (B2 C2)

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163 Figure 9.3 Top chord initial imperfection, o (a) Top view of compression chord (b) Cross-section view Table 9.1 Measured initial imperfections, o, of top chords Bridge Chord 1 (C1) (in) Chord 2 (C2) (in) Bridge 1 0.042 ( L /475) 0.050 ( L /400) Bridge 2 0.090 ( L /445) 0.106 ( L /377)

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164 Bridge 1 Results Theoretical Results The theoretical models were analyzed in accord ance with the proposed method described in Chapter VI, and the results are compared in Table 9.2 and plotted in Figure 9.4. These results are presented as the ratio of the second-order displacement, to the initial imperfection ( / o), of the top cord, for a corresponding gravity load. The critical loads for the th eoretical models are summarized in Table 9.3. From the results presented in Table 9.2, it can be seen th at there is good correlation between the results from the notio nal load (NL) approach and the direct modeling (DM) approach. For example, at a load of 3,000 lb, and for an initial imperfection of L /300, the top chord amplificati ons were 1.515 and 1.530 for the notional load and the modeled imperfec tion methods respectively, compared to amplifications of 1.738 and 1.714 when an initial imperfection of L /450 is applied to the top chord. And for L /700, the top chord amplifica tions were 2.000 and 2.000 for the notional load (NL) and the direct mode ling (DM) approach respectively. At 6,000 lbs the top chord amplifications were 2.439 (NL) and 2.439 (DM) for L /300, 2.929 (NL) and 2.881 (DM) for L/450 and 3.536 and 3.500 (DM) for L /700. At 8,000 lbs the top chord amplifications were 4.333 (NL) and 4.561 (DM) for L /300, 5.262 (NL) and 5.095 (DM) for L /450 and 6.214 and 6.179 (DM) for L /700.

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165 The maximum difference in amplificati on values between the notional load approach and the modeled imperfection appr oach correspond to 5%, and occur when the applied load reached 8,000 lbs and for a initial imperfection of L /300. For the remaining applied load levels the difference is less than 3%. Similarly, the critical loads correlat e well between approaches. The maximum difference between methods was 1.97% a nd corresponds to an imperfection of L /300. However, for the other two initial imperfection analysis, the difference was less than 1%. Based on these results, the critical load of Bridge 1 is expected to be approximately 9,400 lbs. Figure 9.4 provides a good representation of the load-deformation history of the top chord and the de stabilizing effects of P. For example, at load levels up to and less than 74% of the critical load, th e structure remains reasonably stable, this correlates to a load of approximately 7,000 lbs. However, beyond this point, and as shown in Figure 9.4, it can be seen that significant instability starts to take place. At load levels of about 83% of the critical load, approximately 8,000 lbs, the secondary effects due to Plead to large amplifications of the top chord out-of-plane deflection, demonstrating that significant in stability occurs when the applied load approaches the critical load.

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166 Table 9.2 Bridge 1 Theoretical Results: Top chord ratio / o Load (lbs) NL DM NL DM NL DM o= L /300 o= L/ 450 o= L /700 0 1.0 1.0 1.0 1.0 1.0 1.0 1000 1.152 1.152 1.214 1.214 1.321 1.321 2000 1.318 1.333 1.452 1.452 1.643 1.643 3000 1.515 1.530 1.738 1.714 2.000 2.000 4000 1.742 1.758 2.048 2.024 2.393 2.393 5000 2.030 2.045 2.429 2.405 2.893 2.857 6000 2.439 2.439 2.929 2.881 3.536 3.500 7000 3.061 3.076 3.690 3.643 4.464 4.429 8000 4.333 4.561 5.262 5.095 6.214 6.179 9000 11.258 10.924 12.786 12.524 14.643 14.571 NL=Notional load DM=Direct Modeling Table 9.3 Bridge 1 Theoretical Results: Critical load NL=Notional load DM=Direct Modeling Critical Load (lbs) NL DM NL DM NL DM o= L/ 300 o= L /450 o= L /700 9365 9550 9422 9500 9460 9500

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167 NL = Notional Load DM = Direct Modeling Figure 9.4 Bridge 1 Theoretical results 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.0005.00010.00015.00020.00025.000Applied Load, P (lbs)Amplification Factor of top Chord, / 0 NL L/300 DM L/300 NL L/450 DM L/450 NL L/700 DM L/700

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168 Load Test Results The test model for Bridge 1 was load tested in acco rdance with the procedure presented in Chapter VIII. As mentioned above, the expected critical load for Bridge 1 was calculated to be approximately 9,400 lbs. The results from the test model for both top chords (C1 and C2) ar e presented in Table 9.4. Thes e results are compared to the theoretical results that were obtained by using the notional load (NL) approach only. The results presented in Table 9.4 are also plotted in Figure 9.5. It can be observed from table 9.4 that up to a load level of 6,000 lbs, the ratios / o for both, the theoretical models and the test models, correlate well. It is, however, from that point and for higher load levels that the results do not correl ate well. This is due to the fact that the test model exhibited significant deform ation capacity. At about the critical load of the theoretical models, the / o ratio for the test model was less than 4.0, and it wasnÂ’t until a load level of 16,000 lbs, that the test model showed significant loss of stiffness. The critical lo ads are presented in Table 9.5 where it can be seen that the actual critical load from the test model 18,200 lbs, was almost twice as much as the theoretical model From the deformed shape, shown in Figur e 9.6, it was concluded that the joint connections at the diagonals were considerably stiff, hence, increasing the lateral stiffness of the top chord. Thus, by modeli ng the ends of the diagonals as pin-pin connections the computer model underestimat ed the additional stiffening effects of the connection. Therefore, re-modeling the theoretical model with diagonals with

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169 partially fixed-ends would be more appr opriate. However, for simplicity, it was decided to re-model and re-run the Bridge 1 theoretical model with fix-ended diagonals and using the notional load appr oach only. The first attempt at re-running the theoretical model yielded a critical load of 21,000 lbs, significantly higher that the test model. In addition, a re view of the memberÂ’s forces results from the computer program revealed that the axial load in the end diagonals had exceeded the yield strength of the members. Th erefore, from the deformed shape at failure, Figure 9.6, which shows the end-diagonals buckling first, and the fact that the program calculates second-order effects at nodes only, it was d ecided to add an additional node at the midpoint of each end-diagonal and re -run the model for a second time. This change proved to correlate better with the test model as it is shown by the chord amplification-factor results presen ted in Table 9.6. At the same time, the critical load of the theoretical model turned out to be very cl ose to the critical load of the test model as seen in Table 9.7, this differenc e correlates to only 9%. The results of Table 9.6 are plotted in Figure 9.7 where it can be seen that the theoretical model and the test model correlate well. As with previous results, considerable destabilization starts to occur at loads higher than 80% of th e critical load. Figures 9.8 through 9.11 show the sequence of the failur e mode of one of the end-diagonals, for both, the theoretical model and the t est model where it can be seen that there was significant deformation of the end-diagona l before buckling. In addition, one can observe that the deformed shapes are consistent between both models.

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170 Table 9.4 Bridge 1 Theoretical and Load Test Results: Top chord ratio / o Load (lbs) NL NL NL B1-C1 B1-C2 L /300 L /450 L /700 L /475 L /400 0 1.000 1.000 1.000 1.000 1.000 1000 1.152 1.214 1.321 1.238 1.357 2000 1.318 1.452 1.643 1.357 1.607 3000 1.515 1.738 2.000 1.714 1.952 4000 1.742 2.048 2.393 1.952 2.190 5000 2.030 2.429 2.893 2.143 2.500 6000 2.439 2.929 3.536 2.310 2.667 7000 3.061 3.690 4.464 2.571 3.024 8000 4.333 5.262 6.214 2.810 3.381 9000 11.258 12.786 14.643 3.143 3.857 10000 N/A N/A N/A 3.500 4.238 11000 N/A N/A N/A 3.976 4.643 12000 N/A N/A N/A 4.452 4.976 13000 N/A N/A N/A 5.024 5.381 14000 N/A N/A N/A 5.167 6.000 15000 N/A N/A N/A 5.762 6.476 16000 N/A N/A N/A 6.476 6.976 17000 N/A N/A N/A 7.714 7.905 18000 N/A N/A N/A 9.619 10.048 NL=Notional Load Table 9.5 Bridge 1 Theoretical and Load Test Results: Critical load Critical Load (lbs) NL L /300 NL L /450 NL L /700 B1-C1 L /475 B1-C2 L /400 9365 9422 9460 18200 18200

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171 NL = Notional Load DM = Direct Modeling Figure 9.5 Bridge 1 Theoretical (pin-ended diagonal s) and load test results 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0.0005.00010.00015.00020.00025.000Applied Load, P (lbs)Amplification Factor of top Chord, / 0 NL L/300 DM L/300 NL L/450 DM L/450 NL L/700 DM L/700 B1 C1 B1 C2

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172 Figure 9.6 Bridge 1 Deformed shape at failure

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173 Table 9.6 Bridge 1 Ratio / o, theoretical results with fix-ended diagonals Load (lbs) NL L /300 NL L /450 NL L /700 B1-C1 L /475 B1-C2 L /400 0 1.000 1.000 1.000 1.000 1.000 1000 1.106 1.167 1.250 1.238 1.357 2000 1.227 1.333 1.500 1.357 1.607 3000 1.348 1.500 1.714 1.714 1.952 4000 1.470 1.690 2.000 1.952 2.190 5000 1.591 1.881 2.250 2.143 2.500 6000 1.742 2.071 2.536 2.310 2.667 7000 1.879 2.286 2.821 2.571 3.024 8000 2.045 2.500 3.143 2.810 3.381 9000 2.212 2.762 3.500 3.143 3.857 10000 2.409 3.024 3.857 3.500 4.238 11000 2.636 3.310 4.286 3.976 4.643 12000 2.879 3.667 4.750 4.452 4.976 13000 3.182 4.071 5.286 5.024 5.381 14000 3.530 4.548 5.929 5.167 6.000 15000 3.985 5.143 6.750 5.762 6.476 16000 4.576 5.929 7.786 6.476 6.976 17000 5.409 7.024 9.250 7.714 7.905 18000 6.773 8.905 11.536 9.619 10.048 NL=Notional Load Table 9.7 Bridge 1 Critical load, theoretical res ults with fix-ended diagonals Critical Load (lbs) NL L /300 NL L /450 NL L /700 B1-C1 L /475 B1-C2 L /400 19850 19820 19840 18200 18200

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174 NL = Notional Load Figure 9.7 Bridge 1 Theoretical (fix-ended diagon als) and load test results 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0.0005.00010.00015.00020.00025.00030.000Applied Load, P (lbs)Amplification Factor of top Chord, / 0 NL L/300 NL L/450 NL L/700 B1 C1 B1 C2

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175 Figure 9.8 Bridge 1 Deformation of end-diagonal pr ior to buckling (test model) Figure 9.9 Bridge 1 Deformation of end-diagonal (c omputer model) at critical load

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176 Figure 9.10 Bridge 1 Deformation of end-diagonal at buckling (test model) Figure 9.11 Bridge 1 Deformation of end-diagonal (c omputer model) at critical load

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177 Bridge 2 Results Theoretical Results Similar to Bridge 1 the theoretical models were analyzed by following the methodology presented in Chapter VI. The analysis consisted on finding the loaddeformation history and critical load of Bridge 2 with three different initial imperfections. In addition, th e imperfections were mode led by using two different approaches: notional loads and modeling the imperfection directly in to the model. The results from the theoretical model of Bridge 2 are compared in Table 9.8 and plotted in Figure 9.12. These results are pr esented as the ratio of the second-order displacement to the initial imperfection ( / o), of the top chor d, for a corresponding gravity load. The critical loads for the th eoretical models are summarized in Table 9.9. The results between the notional load (N L) approach and the direct modeling (DM) approach correlate well. However, the difference is not as small as Bridge 1 At a load 3,000 lb, and for an initial imperfection of L /300, the top chord amplifications were 1.586 and 1.602 for the notio nal load and the modeled imperfection approaches respectively, compared to amplifica tions of 1.600 and 1.667 when an initial imperfection of L /450 is applied to th e top chord. And for L /700, the top chord amplifications were 1.586 and 1.602 for th e notional load (NL) and the direct modeling (DM) approaches respectively. At 4,000 lbs the top chord amplifications were 2.113 (NL) and 2.053 (DM) for L /300, 2.222 (NL) and 2.178 (DM) for L /450

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178 and 2.456 and 2.368 (DM) for L /700. At 6,000 lbs the top chord amplifications were 5.489 (NL) and 4.887 (DM) for L /300, 5.900 (NL) and 5.244(DM) for L /450 and 6.702 and 6.088 (DM) for L /700. The maximum difference, approximat ely 11%, between both approaches occurred when the in itial imperfection was L /450 and corresponded to gravity loads of 6,000 lbs and 7,000 lbs. For the remaining lo ad levels the difference was less than 8%. The difference, between approachess, can be seen in Figure 9.12, where the spread of results is more evident at about an amplification ratio ( / o) of 10 compared to the results from the Bridge 1 test. However, all of the critical loads were only within 0.50% or less.Th e critical load for an initial imperfection of L /300 was 7,155 lbs for the notional load approach and 7,178 lbs for the direct modeling approach. For L /450 the critical load was 7,100 lbs and 7,140 lbs for the notional load and the modeled imperfection approach resp ectively, and for an initial imperfection of L /700, the critical load was 7,180 lbs for bot h, the notional load and the modeled imperfection approaches. Base d on the results from the theoretical models the critical load for Bridge 2 test model is expected to be close to 7,150 lbs. It can be seen in Figure 9.12 that at load levels up to and less than 70% of the critical load, the structure remains reasonabl y stable. It is, however, beyond this point that the models start to lose significant sti ffness and become instable, to the point that at about 85% of the critical load the amplification of the top chord / o, is more than 5 times. For example, between 1,000 lb s and 4,000 lbs the difference in the

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179 amplification ratio, / o, is approximately 38%. Between an applied gravity load of 4,000 lbs and 5,000 lbs the difference is 49 %, while between load 5,000 lbs and 6,000 lbs the difference is 78%. This difference is more than 300% when the applied load goes from 6,000 lbs to 7,000 lbs, demonstrati ng that significant instability occurs when the applied load appr oaches the critical load. Table 9.8 Bridge 2 Theoretical Results: Top chord ratio / o Load (lbs) NL DM NL DM NL DM o= L /300 o= L /450 o= L /700 0 1.0 1.0 1.0 1.0 1.0 1.0 1000 1.128 1.128 1.144 1.144 1.193 1.193 3000 1.586 1.602 1.600 1.667 1.754 1.789 4000 2.113 2.053 2.222 2.178 2.456 2.368 5000 3.098 2.872 3.311 3.078 3.667 3.456 6000 5.489 4.887 5.900 5.244 6.702 6.088 7000 22.594 21.000 23.722 22.594 25.772 22.950 NL=Notional Load DM=Direct Modeling Table 9.9 Bridge 2 Theoretical Results: Critical load Critical Load (lbs) NL DM NL DM NL DM o= L /300 o= L /450 o= L /700 7155 7178 7100 7140 7180 7180

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180 NL = Notional Load DM = Direct Modeling Figure 9.12 Bridge 2 Theoretical (pin-ended diagonals) results 0 1000 2000 3000 4000 5000 6000 7000 8000 0.0005.00010.00015.00020.00025.00030.00035.000Applied Load, P (lbs)Amplification Factor of top Chord, / 0 NL L/300 DM L/300 NL L/450 DM L/450 NL L/700 DM L/700

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181 Load Test Results Similar to Bridge 1 the test model for Bridge 2 was tested in accordance with the procedure presented in Chapter VIII. The results from the test model for both top chords (C1 and C2) are presented in Table 9.10. These results are compared to the results from the theoretical model s. However, only the results from the notional load (NL) approach are presented in Table 9.10 for comparison. These results are also plotted in Figure 9.13. It can be observed that up to a load level of 6,000 lbs the ratios / o for both, the theoretical models and the test models, correlate very well. From this point the test model exhibited higher deformation capacity, although not as much as the capacity exhibited by Bridge 1 At this load, 6,000 lbs, the amplification factor, / o, was 5.111 for chord 1 (C1) and 5.238 for chord 2 (C2) for the test model compared to 5.489, 5.900 and 6.702 for imperfections L /300, L /450 and L /700 respectively for the theoretical models The critical loads are presented in Table 9.11. As mentioned earlier, the expected critical load of Bridge 2 based on the theoretical results, is approximately 7,150 lbs. The actual load from the load test was 7,902 lbs. The difference between the theoretical results and the actual model is only 10%. Although the test model exhibited higher deformation capacity during th e final moments of the load test than the theoretical model, it might seem that, even though the diagonal connections provided some additional stiffness, they di d not contribute with the same level of fixidity as to Bridge 1

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182 Similar to Bridge 1 the test model of Bridge 2 remains reasonably stable up to loads of approximately 64% of the critical load. Beyond that point the model begins to lose stiffness due to second-order effects. At an applied load of 75% of the critical load, the amplification of the top chord is more than 5 times, while at 89% the amplification is more than 12 times. It can be seen that between 75% and 90% of the critical load, the secondary e ffects of the top chord cause significant destabilization of the structure. Figures 9.13 through 9.17 show a series of side-by-side shot s of the bridgeÂ’s top chord deformed shape between the computer model and the actual model for different load values. The intent is to compare the deformed shape between both models for 53%, 78% and 91% of the critical load respec tively. In addition, Figures 9.18 and 9.19 show the deformed shape of the theoretical model and the test model respectively, at the critical load. From thes e pictures it can be observed that results from the computer program demonstrate good agreement with the load test.

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183 Table 9.10 Bridge 2 Theoretical and Load Test Results: Top chord ratio / o Load (lbs) NL o= L/300 NL o= L/450 NL o= L/700 B2-C1 o= L/445 B2-C2 o= L/377 0 1.0 1.0 1.0 1.0 1.0 1000 1.128 1.144 1.193 1.100 1.143 3000 1.586 1.600 1.754 1.689 1.619 4000 2.113 2.222 2.456 2.378 2.238 5000 3.098 3.311 3.667 3.656 3.524 6000 5.489 5.900 6.702 5.111 5.238 7000 22.594 23.722 25.772 12.278 11.905 7500 N/A N/A N/A 15.033 14.762 7900 N/A N/A N/A 25.556 24.762 NL=Notional Load Table 9.11 Bridge 2 Theoretical and Load Test Results: Critical load NL=Notional Load Critical Load (lbs) NL o= L/300 NL o= L/450 NL o= L/700 B2-C1 o= L/445 B2-C2 o= L/377 7155 7100 7180 7902 7902

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184 NL = Notional Load Figure 9.13 Bridge 2 Theoretical and load test results 0 1000 2000 3000 4000 5000 6000 7000 8000 0.0005.00010.00015.00020.000 25.00030.00035.00040.000Applied Load, P (lbs)Amplification Factor of top Chord, / 0 NL L/300 NL L/450 NL L/700 B2 C1 B2 C2

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185 Figure 9.14 Bridge 2 Deformed shape at 53% of the critical load Figure 9.15 Bridge 2 Deformed shape at 78% of critical load

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186 Figure 9.16 Bridge 2 Deformed shape at 91% of the critical load Figure 9.17 Bridge 2 Deformed shape at critical load

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187 CHAPTER X 10 CONCLUSIONS Research Overview The stability and load-deformation be havior of pony truss bridges were investigated using experimental testing a nd computational modeli ng, with the intent of validate an innovative methodology for critic al load determinat ion. The laboratory experiments were conducted on the pony truss bridges that were lo ad tested without intermediate braces. The computational model was developed using the threedimensional computer program, RISA-3D. A va riety of analytical models were used to simulate both as-built and idealized truss models. The methodology computer model was verified by the laboratory test results. A parametric investigation was devel oped using full-size pony truss bridge models, from hollow steel sections, with the purpose of devel oping expressions for determining the stiffness requirements of software generated scaled pony truss models. Based upon results from the parametr ic investigations, tw o scaled computer models were developed with the same aspect ratio, AR of 0.20 and two different stiffness parameters, 63 for the long bridge and 97 for the short bridge. The scaled pony truss computer models were used for predicting the buckling capacity and loaddeformation behavior of the laborator y experiments by following the proposed methodology.

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188 Research Conclusions Overall, the laboratory experiments ag reed well with the computer model results and validate the results from the proposed methodology. In addition, the laboratory experiments demonstrated that the buckling capacity of the pony truss bridge is largely depended on the cross frame stiffne ss, the connection behavior between members and the number of intermediate panels. Both laboratory experiments demonstrated that significant instability starts to occur at load levels above 70% of the calcu lated critical load. In both cases, the top chords deflected towards the center of the bridge. This is because the load on the bridge caused the cross-frame to deflect downward causing the verticals to rotate inwards. The experimental results of Bridge 1 demonstrated that the connection modeling assumptions of the diagonals play a critical role for the prediction of the critical load. In this case, the diagonals provided more stiffness to the cross frame than assumed. However, after modifying th e connections of the theoretical models from pin-pin to fix-fix end restraints, th e load-deformation behavior of the model closely matched the results from the laboratory experiment. On the other hand, good agreement was found between the computer model and the laboratory experiment of Bridge 2 Although the critical load of the test model was 10% higher than the computer mo del, the load-deformation history of the laboratory experiment followed very closely the one from th e computer model. In this

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189 case, because the diagonals were longer than the diagonals of Bridge 1 the connections to the diagonals did not provide as much stiffening effects to the cross frame (U-frame) as with Bridge 1 However, it is still believed that some level of partial fixity, out of plan e of the truss, was provided by the gusset plate. General Recommendations From the results of this research project, the proposed methodology gives a good indication of the critical load of a particular st ructure and th e destabilizing effects of second-order forces due to P. Below are some recommendations when using the proposed methodology: 1. Verify that computer software pack age selected can perform a rigorous second-order analysis. This can be done with benchmark problems given in published literature, in cluding AISC 360-10 (AISC 2010). 2. Because some computer software pack ages calculate secondary effects at node locations, it might be necessary to analyze the structure with different number of additi onal nodes between supports. 3. If moment-rotational information is av ailable for the connections between members, this can be used to provide a more realistic analysis. Otherwise, a conservative assumption should be made. 4. It is recommended to use 75% of the th eoretical critical load for ultimate design.

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190 5. However, because actual design is based on limit states, a factor of safety needs to be incorporated for the fi nal design. Its suggested to use the Direct Analysis MethodÂ’s third constr aint, which requires that the analysis be based on a reduced stiffness ( EI *=0.80 b( EI )), where the 0.80 b factor reduces the stiffness to account for inelastic softening prior to the members reaching their ultimate strength. For slender compression members, the critical load is most likely below 50% the yield strength, thus b=1.0. In this case 0.80 is roughly equi valent to the margin of safety implied by design of slender columns (AISC 2005).

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191 11 REFERENCES AASHTO (2009). LRFD Guide specifications for the design of pedestrian bridges 2th edition American Association of State Highway and Transportation Officials, Washington, DC. AASHTO (2012). AASHTO LRFD bridge design specifications 4th edition American Association of State Highway and Transportation Officials, Washington, DC. AISC (1961). Specification for the design, fabricatio n and erection of structural steel for buildings American Institute of Steel Construction, New York, NY. AISC (1999). Load and Resistance Factor Design Specific ation for structural steel buildings American Institute of Steel Construction, Inc. Chicago. AISC (2005 ). Specification for structural steel buildings (ANSI/AISC 360-05) American Institute of Steel Construction, Inc. Chicago. AISC (2010) Specification for structural steel buildings (ANSI/AISC 360-10). American Institute of Steel Construction, Inc. Chicago. AISC (2010 ) Code of standard practice for steel buildings and bridges (AISC 303-10) American Institute of Steel Construction, Inc. Chicago. ASCE Task Committee on Effective Length. (1997 ) Effective length and notional load approaches for assessing frame stability: implications for American steel design. American Society of Civil Engineers, New York. AWS (2002). Structural Welding Code-Steel, D1.1. American Welding Society, Miami, FL Bjorhovde, R. (1972). Deterministic and probabilistic approaches to the strength of steel columns Ph.D. dissertation, Lehigh University, Bethlehem, PA

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192 Bjorhovde, R. (1984). “Effect of end restraint on column strength –practical applications.” AISC Engineering Journal, 1st Qtr. 1-13 Chen, W. F., and Han, D. J. (1985). Tubular members in offshore structures Pitman Publishing, Inc., Marshfield, MA. Chen, W. F., and Lui, E. M. (1991). Stability design of steel frames CRC Press, Boca Raton, FL. Chen, W.F., Guo, Y. and Liew, J.Y.R., (1996). Stability Design of Semi-Rigid Frames John Wiley & Son, Inc., New York. Christopher, J.E. and Bjorhovde, R. (1999). “S emi-rigid frame design methods for practicing engineers.” AISC Engineering Journal 36 (1), 12-28. Clark, M. and Bridge, R. Q. (1992). “The inclusion of imperfections in the design of beamcolumns.” Proceedings 1992 Annual Technical Session Bethlehem, PA, April. Structural Stability Research Council, Rolla, MO. Deierlein, G. G., Hajjar, J. F., Yura, J. A., Wh ite, D. W., and Baker, W. F. (2002). “Proposed new provisions for frame stability using second-order analysis.” Proceedings 2002 Annual Technical Session Seattle, April. Structural Stability Research Council, Rolla, MO. Ericksen, J. R. (2011). “A how-to approach to notional loads.” AISC Modern Steel Construction, 51 (1). 44-45. Fernandez, P., and Rutz, F.R. (2013). “A practi cal method for critical load determination and stability evaluation of structures.” Proceedings of the 2013 Structures Congress, ed. B. J. Leshko and J. McHugh, Structural Engineering Institute of the American Society of Civil Engineers, May 1-4, 2013, Pittsburgh, PA, ASCE, Reston VA.

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193 Folse, M. D., and Nowak, P. S. (1995). “Steel rigid frames with leaning columns – 1993 LRFD example.” AISC Engineering Journal, 32 (4). 125-131. Galambos, T. V. ed (1998). Guide to stability design criteria for metal structures, 5th Edition John Wiley & Sons, Inc., Hoboken, NJ. Gere, J. M. and Goodno, B. J. (2009). Mechanics of Materials, 7th Edition Cengage Learning, Toronto, ON. Geschwindner, L. F. (2002). “A practical l ook at frame analysis stability and leaning columns.” AISC Engineering Journal, 39 (4). 167-181 Geschwindner, L. F. (2009). “Design for stab ility using 2005 AISC specification.” seminar notes, presented Sept 18, 2009, Sponsored by AISC, Chicago, IL. Geschwindner, L. F., Disque, R. O. and Bjorhovde, R. (1994). Load and resistance factor design of steel structures Prentice Hall, Inc., Englewood, NJ. Hamberger, R. and Whittaker, A. S. (2004). “Design of steel structures for blast-related progressive collapse resistance.” AISC Modern Steel Construction, 44 (3), 45-51. Heyman, J. (1998). “Structural analysis a histor ical approach.” Cambridge University Press, Cambridge, UK. Hibbeler, R. C. (2009). Mechanics of materials 6th Edition Pearson Prentice Hall, New Jersey. Hibbeler, R. C. (2009). “Structural analysis 7th Edition.” Pearson Prentice Hall, New Jersey. Johnston, B. G. (1981). “Column Buckling Theory: Historic Highlights.” ASCE Journal of Structural Engineering 109 (9). 86-96 Kanchanalai, T., and Lu, L.-W. (1979). “Ana lysis and design of framed columns under minor axis bending.” AISC Engineering Journal, 16 (2). 29-41.

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194 LeMessurier, W. J. (1976). “A practical method of second order analysis part 1 – pin jointed systems.” AISC Engineering Journal, 13 (4). 89-96 LeMessurier, W. J. (1977). “A practical me thod of second order analysis part 2 – rigid frames.” AISC Engineering Journal, 14 (2). 49-67 Maleck, A.E., and White, D.W. (1998). “Effects of imperfections on steel framing systems.” Proceedings 1998 Annual Technical Sessions Atlanta, September. Structural Stability Research Council, Rolla, MO. Maleck, A. E., and White, D. W. (2003). “Direct analysis approach for the assessment of frame stability: verification studies.” Proceedings Annual Technical Sessions Baltimore, April, Structural Stability Research Council, Rolla, MO. 18 McCormac, J. C. (2007). Structural analysis using classical and matrix methods John Wiley & Sons, Inc., Hoboken, NJ. McCormac, J. C. (2008). Structural steel design, 4th Edition Pearson Prentice Hall, Upper Saddle River, NJ. McGuire, W., Gallegher, R. H., and Ziemian, R. D. (2000). Matrix structural analysis, 2nd Edition John Wiley & Sons Inc., Hoboken, NJ. Nair, S. R. (2009). “A model specification for stability design by direct analysis.” AISC Engineering Journal, 46 (1). 29-37 Powell, G. H. (2010). Modeling for structural analysis: behavior and basics Computers and Structures, Inc. Berkeley, CA. RISA-3D (2012). RISA-3D v11 General reference manual RISA Technologies, Foothill Ranch, CA.

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195 Salmon, C. G., and Johnson, J. E. (2008). Steel structures: design and behavior, emphasizing load and resistance design 5th Edition Harper Collins, New York. Shanmugam N. E. and Chen, W. F. (1995). “An assessment of K factor formulas.” AISC Engineering Journal, 1 Qtr. 3-11 Surovek, A. E. (2008). “Effects of nonvertica lity on steel framing systems-implications for design.” AISC Engineering Journal, 45 (1). 73-85. Surovek, A. E. ed (2012). Advanced analysis in steel fram e design: Guidelines for direct second-order inelastic analysis American Society of Civil Engineers, New York. Surovek-Maleck, A. E., and White, D. W. ( 2004a). “Alternative approaches for elastic analysis and design of steel frames. I: overview.” ASCE Journal of Structural Engineering, 130 (8). 1186-1196. Surovek-Maleck, A.E., and White, D.W. (2004b) “Alternative approaches for elastic analysis and design of steel frames. II: verification studies.” ASCE Journal of Structural Engineering, 130 (8). 1197-1205. Timoshenko, S. P. (1983). History of strength of materials Dover publications, Inc., Mineola, NY. Timoshenko, S. P., and Gere, J. M. (1989). Theory of elastic stability Dover publications, Inc., Mineola, NY. White, D. W., and Hajjar, J. F. (1991). “App lication of second-order elastic analysis in LRFD: research to practice.” AISC Engineering Journal, 28 (4). 133-148. White, D. W., Surovek, A. E., and Kin, S-C. (2007a). “Direct analysis and design using amplified first-order analysis. part 1 – combined braced and gravity framing systems.” AISC Engineering Journal, 44 (4). 305-322.

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196 White, D. W., Surovek, A. E., and Chang, C-J. (2007b). “Direct analysis and design using amplified first-order analysis. part 2 – moment frames and general framing systems.” AISC Engineering Journal, 44 (4). 323-340. Williams, A. (2009). Structural analysis: in theory and practice Elsevier, Burlington, MA. Williams, A. (2011 ). Steel structures design ASD/LFRD McGraw Hill, Washington, DC Yura, J. A. (1971). “The effective length of columns in unbraced frames.” AISC Engineering Journal, 8 (2). 37-42. Yura, J. A., Kanchanalai, T., and Chotichanathawenwong, S. (1996). “Verification of steel beam-column design based on the AISC-LRFD method.” Proceeding 5th International Colloquium Stability Metal Structures, Lehigh University, Bethlehem, PA, Structural Stability Research Council, Rolla, MO. 21-30. Ziemianm, R. A. ed (2010). Guide to stability design criteria for metal structures, 6th Edition John Wiley & Sons, Inc., Hoboken, NJ. Ziemianm, R. A., McGuire, W., Seo, D. W. (2008). “On the inelastic strength of beamcolumns under biaxial bending.” Proceedings Annual Technical Sessions Nashville, April, Structural Stability Research Council, Rolla, MO. 18

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197 12 APPENDIX 1: TEST MODELS PLANS