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PHYSICAL VALIDATION OF DIRECT ANALYSIS METHOD FOR STABILITY USING FULL SCALE MEMBERS by JONATHAN JAMES EAST B.S., Colorado State University, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering 2014
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ii This thesis for the Mast er of Science degree by Jonathan East Has been approved for the Civil Engineering Program by Frederick Rutz, Chair Kevin Rens Chengyu Li November 16, 2014
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iii East, Jonathan, James (M.S., Civil Engineering) Physical Validation of Di rect Analysis Method for Stability using Full Scale Members Thesis directed by Assistant Professor Frederick Rutz. ABSTRACT The Direct Analysis Method, introduced into the steel design specification AISC 360 in the 2005 edition, is applicable to all types of structures for the determination of global stability. However, structural engineers have been slow to adopt stability determination by the Direct Analysis Method in lieu of the more traditional Effective Length Method. Research at the University of Colorado Denver has been directed at studying buck ling of full scale trusses for validation of the Direct Analysis Method. The re search was conducted in two parts: a second order Direct Analysis using SAP 2000 to predict a crit ical buckling load and stability experiments conducted on full s cale bar joists to determine the actual critical load to validate the Direct Analysis. The full scale members used inherently include imperfec tions, approximated by the use of notional loads in the Direct Analysis Method. The goal of the research was physical validation of stability analysis by the Direct Analysis Method for full scal e truss members with the inclusion of notional loads in the analysis. In re lating to full scale truss members, the overarching goal of the research conducte d was to validate the Direct Analysis Method for use in determining the global st ability of structures. The validity of global stability analysis of complex stru ctures using 3D st ructural analysis computer programs, common in design o ffices, is demonstrated for full scale
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iv trusses represented by bar joists. Validation of the method will benefit the structural engineering community, and is pa rticularly directed at structural design professionals, to demonstrate the us e of the Direct Analysis Method. Testing results were shown to matc h computer modeling results when appropriate initial conditions were applie d to the computer m odels, validating the use of the Direct Analysis Method for stability analysis. The form and content of this abstract are approved. I recommend its publication. Approved: Frederick Rutz
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v DEDICATION I dedicate this work to my friends family, and especially my fiance Lucy, for supporting and encouraging me.
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vi ACKNOWLEDGMENTS I would like to thank everybody who ha s helped me put this research project together, w ithout the help I received I w ould not have been able to complete this project. First, I would like to thank my gr aduate advisor, Dr. Frederick Rutz, for helping me through this process; from developing an appropriate topic, numerous lunch meetings and for keeping me focused. IÂ’d also like to thank Leslie Foster from the Rocky Mountain Steel Construction Association for reaching out to industry prof essionals to help me acquire bar joists for testing, and Gary Gaulke, from Sigm a Metals, for donating and delivering the bar joists that I used. In addition, there were many pe ople at the University of Colorado Denver that were integral to the su ccess of my testing for this project. Among those who helped me were Tom T huis who helped me design and set up my testing apparatus, Dr. Jimmy Kim who gave me access to a load cell and data acquisition system, and all of the Universi ty of Colorado Denver professors and graduate students who made room for my testing.
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vii TABLE OF CONTENTS CHAPTER I. INTRODUCTION ........................................................................................................... 1 Research Objectives ................................................................................................ 3 Outline of Research Project Thesis ......................................................................... 4 II. HISTORICAL THEORY AND BACK GROUND OF COLUMN THEORY .............. 6 Historical Column Theory ...................................................................................... 6 Elastic Buckling. ............................................................................................... 6 Euler Elastic Buckling. ..................................................................................... 7 Inelastic Buckling. .......................................................................................... 11 TangentModulus and Double Modulus Theories. ......................................... 11 Shanley Theory. .............................................................................................. 13 Conditions Effecting Column Buckling ................................................................ 15 Column Effective Length. ............................................................................... 15 End Restraint of Columns. .............................................................................. 15 Braced and Unbraced Frames. ........................................................................ 18 Residual Stresses. ............................................................................................ 19 Geometric Imperfections. ............................................................................... 21 Buckling Stability of Other Members ................................................................... 24 Beams. ............................................................................................................. 24 BeamColumns. .............................................................................................. 26 Plate Buckling. ................................................................................................ 28 Current Design Practices....................................................................................... 30 Indirect Stability Analysis Methods. ............................................................... 31 Direct Analysis Method. ................................................................................. 31
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viii III. STABILITY AND BUCKLING THEORY DEVELOPMENT ................................. 33 Column Buckling .................................................................................................. 33 Effective Length.................................................................................................... 36 Inelastic Buckling Theory ..................................................................................... 40 Application to State of the Practice ...................................................................... 42 IV. CURRENT DESIGN STANDARDS ......................................................................... 43 Stability Analysis and Design ............................................................................... 43 FirstOrder Analysis Method ................................................................................ 43 Effective Length Method ...................................................................................... 44 Direct Analysis ...................................................................................................... 46 V. PHYSICAL TESTING ................................................................................................ 48 Bar Joists ............................................................................................................... 48 Test Apparatus and Setup ..................................................................................... 50 Testing................................................................................................................... 54 Alpha Joist Testing ............................................................................................... 56 Test 1 Â– Alpha Joist......................................................................................... 58 Test 2 Â– Alpha Joist......................................................................................... 62 Tests 3 and 4 Â– Alpha Joist. ............................................................................ 67 Test 5 Â– Alpha Joist......................................................................................... 73 Beast Joist Testing ................................................................................................ 78 Test 6 Â– Beast Joist. ........................................................................................ 78 Test 7 Â– Beast Joist. ........................................................................................ 83 Test 8 Â– Beast Joist. ........................................................................................ 88 Test 9 Â– Beast Joist. ........................................................................................ 94 Comparison of Joists Alpha and Beast ............................................................... 101
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ix Joist Frame Testing ............................................................................................. 102 Test 10 Â– Frame. ........................................................................................... 102 Test 11 Â– Frame. ........................................................................................... 106 VI. DIRECT ANALYSIS MOD ELING USING SAP2000 ........................................... 115 SAP2000 ............................................................................................................. 115 SAP2000 Buckling Analysis......................................................................... 115 SAP2000 Nonlinear Static Analysis. ............................................................ 117 PDelta Analysis. .......................................................................................... 117 PDelta with Large Displacements Analysis. ............................................... 119 AISC Direct Analysis Method ............................................................................ 120 SAP2000 Models ................................................................................................ 121 PDelta Model Verification........................................................................... 121 Bar Joist Models. .......................................................................................... 122 Model 1 Â– Single Point Load on the Top Chord. .......................................... 123 Model 2 Â– Two Point Loads on the Top Chord. ........................................... 126 Model 3 Â– Point Load on the Top Chord with Pinned Ends. ........................ 128 Model 4 Â– Single Point Load on the Bottom Chord. .................................... 131 Model 5 Â– Frame Loaded on the Top Chord. ................................................ 134 Model 6 Â– Frame Loaded on the Bottom Chord. .......................................... 135 VII. CONCLUSIONS AND RECCOMENDATIONS................................................... 138 Conclusions ......................................................................................................... 138 Recommendations for Future Studies ................................................................. 139 REFERENCES ............................................................................................................... 141 APPENDIX ..................................................................................................................... 144 Asbuilt Drawings............................................................................................... 145
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x LIST OF TABLES TABLE V.1 Summary of Testing ...................................................................................... 54 V.2 Initial OutofPlumbness, in Inch es, of Alpha Joist when Unloaded. ........... 57 V.3 Test 1 Â– Alpha Joist Single Point Load Top Chord PinnedRoller Results. 59 V.4 Test 2 Â– Alpha Joist Two Point Lo ad Top Chord PinnedRoller Results. .... 64 V.5 Test 3 Â– Alpha Joist Point Load T op Chord PinnedPinned (1 of 2 Tests) Results. .................................................................................................................. 69 V.6 Test 4 Â– Alpha Joist Single Point Load Top Chord PinnedPinned (2 of 2 Tests) Results. ....................................................................................................... 69 V.7 Test 5 Â– Alpha Joist Point Load Bottom Chord PinnedRoller Results. ....... 74 V.8 Initial OutofPlumbness, in Inch es, of Beast Joist when Unloaded ............ 78 V.9 Test 6 Â– Beast Joist Single Point Load Top Chord PinnedRoller Results. .. 80 V.10 Test 7 Â– Beast Joist Two Point Lo ad Top Chord PinnedRoller Results. ... 86 V.11 Test 8 Â– Beast Joist Point Load Top Chord PinnedPinned Results. .......... 90 V.12 Test 9 Â– Beast Joist Point Load Bottom Chord PinnedRoller Results. ...... 96 V.13 Summary of Tests 1 through 9. ................................................................. 101 V.14 Test 10 Â– Frame Single Point Load Top Chord PinnedRoller Results. ... 104 V.15 Test 11 Â– Frame Single Point Lo ad Bottom Chord PinnedRoller Results. ............................................................................................................................. 1 08 VI.1 SAP2000 AISC Benchmark Model Results (Moments) ........................... 122 VI.2 SAP2000 AISC Benchmark M odel Results (Deflections) ........................ 122 VI.3 Model 1 Buckling Resu lts and Comparison ............................................... 125 VI.4 Model 2 Buckling Re sults and Comparison. ............................................. 128 VI.5 Model 3 Buckling Re sults and Comparison. ............................................. 131 VI.6 Model 4 Buckling Re sults and Comparison. ............................................. 133
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xi VI.7 Model 6 Buckling Re sults and Comparison. ............................................. 137
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xii LIST OF FIGURES FIGURE II.1 Euler Buckling Load Example. ....................................................................... 9 II.2 Approximate Values of Effective Length Factor, K. Used with permission (AISC 2010). ......................................................................................................... 17 II.3 Euler Buckling Load Example. ..................................................................... 18 II.4 Example of OutofStraightness. ................................................................... 22 II.5 Example of OutofPlumbness. ..................................................................... 23 II.6 Southwell Plot. ............................................................................................... 24 II.7 Example of Pand Pmoments. ............................................................... 27 III.1 Free Body Diagram of Euler Column .......................................................... 35 III.2 Deflected Shape and Critical Load for Different Nodes. ............................. 36 III.3 Effective Length Alignment Charts for Braced and Unbraced Frames. Used with permission (AISC 2010). .............................................................................. 39 V.1 Measuring Bar Joist to Determine Type. ....................................................... 49 V.2 25Foot Long 16K2 Bar Joists above the Strong Floor. ................................ 50 V.3 Bar Joist Testing Apparatus. ......................................................................... 51 V.4 Typical Pinned Connection. .......................................................................... 52 V.5 Typical Roller Connection. ........................................................................... 52 V.6 Half Sphere and Load Cell Bearing Plate. .................................................... 53 V.7 Dial Gauges Used to Measure Lateral Deflection. ....................................... 54 V.8 Dial Gauge Sta nds During a Test. ................................................................. 55 V.9 OutofPlumbness Measurement String. ....................................................... 57 V.10 Test 1 Configuration. .................................................................................. 58 V.11 Test 1 Top Chord Deflection. ..................................................................... 60
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xiii V.12 Test 1 Bottom Chord Deflection .................................................................. 60 V.13 Extrapolated Trendlines for Test 1............................................................... 61 V.14 Deflected Shape of Alpha during Te st 1. The deflected shape is shown looking from the point load towa rds the pinned connection. ................................ 61 V.15 Rotation of Alpha Top Chord during Test 1 ................................................ 62 V.16 Test 2 Configuration. .................................................................................. 63 V.17 Test 2 Configuration Showing the Channel and Bearing Plate. ................. 63 V.18 Test 2 Top Chord Deflection. ..................................................................... 65 V.19 Test 2 Bottom Chord Deflection. ................................................................ 65 V.20 Extrapolated Trendlines for Test 2............................................................... 66 V.21 Deflected S Shape of Alpha during Test 2. Looking from the pinned end toward the load. ..................................................................................................... 66 V.22 Deflected S Shape of Alpha during Test 2. Looking from the roller end toward the load. The deflection is visibly less on this side. ................................. 67 V.23 Test 3 and 4 Pinned End Confi guration at Typical Roller Connected Abutment............................................................................................................... 68 V.24 Test 3 Deflected Shape of Alpha Joist Loaded at 2.12 Kips. Minimal deflection was observed dur ing Tests 3 and 4. ..................................................... 68 V.25 Top Chord Deflection during Test 3. .......................................................... 70 V.26 Bottom Chord Deflection during Test 3. .................................................... 70 V.27 Top Chord Deflection during Test 4. .......................................................... 71 V.28 Bottom Chord Deflection during Test 4. .................................................... 71 V.29 Extrapolated Trendlines for Test 3. ............................................................. 72 V.30 Extrapolated Trendlines for Test 4. ............................................................. 72 V.31 Test 5 Configuration. .................................................................................. 73 V.32 Test 5 Top Chord Deflection. ..................................................................... 75 V.33 Test 5 Bottom Chord Deflection. ................................................................ 76
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xiv V.34 Extrapolated Trendlines for Test 5. ............................................................. 76 V.35 Deflected C Shape of Alpha during Te st 5. View looking from the roller end toward the load. .............................................................................................. 77 V.36 Deflected C Shape of Alpha during Test 5. View looking from the pinned end toward the load. .............................................................................................. 77 V.37 Test 6 Configuration. Looking from the roller connection. ........................ 79 V.38 Test 6 Top Chord Deflection. ..................................................................... 81 V.39 Test 6 Bottom Chord Deflection. ................................................................ 81 V.40 Extrapolated Trendlines for Test 6. ............................................................. 82 V.41 Deflected S Shape of Beast during Test 6. Looking from the pinned end toward the roller end. ............................................................................................ 82 V.42 Deflected S shape of Beast during Test 6. Looking from the roller end towards the load. ................................................................................................... 83 V.43 Test 7 Deflected S Shape of Beas t Joist. Looking from the pinned end toward the roller end. ............................................................................................ 84 V.44 Test 7 Deflected S Shape of Beas t Joist. Looking fr om the roller end toward the pinned end. .......................................................................................... 85 V.45 Test 7 Top Chord Deflection. ..................................................................... 87 V.46 Test 7 Bottom Chord Deflection. ................................................................ 87 V.47 Extrapolated Trendlines for Test 7............................................................... 88 V.48 Test 8 Pinned End Configuration at Typical Roller Connected Abutment. 89 V.49 Top Chord Deflecti on During Test 8. ......................................................... 91 V.50 Bottom Chord Deflection During Test 8..................................................... 91 V.51 Extrapolated Trendlines for Test 8. ............................................................. 92 V.52 Deflected S Shape of Beast Joist U nder a Load of 2.67 kips. Looking from the West side of the joist toward the load. ............................................................ 93 V.53 Deflected S Shape of Beast Joist U nder a Load of 2.68 kips. Looking from the West side of the joist toward the load. ............................................................ 94 V.54 Test 9 Configuration. .................................................................................. 95
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xv V.55 Test 9 Top Chord Deflection. ..................................................................... 97 V.56 Test 9 Top Chord Deflection. ..................................................................... 98 V.57 Extrapolated Trendlines for Test 9. ............................................................. 98 V.58 Deflected C Shape of Beast Joist dur ing Test 9. Looking from the roller end toward the pinned end. ................................................................................... 99 V.59 Deflected C Shape of Alpha during Test 9. The channel appears to be holding the bottom chord in place during loading. ............................................. 100 V.60 Test 10 Configuration. .............................................................................. 103 V.61 Strut Connection of Frame (Typical). ....................................................... 103 V.62 Test 10 Top Chord Deflection. ................................................................. 105 V.63 Test 10 Bottom Chord Deflection. ............................................................ 106 V.64 Deflected Shape of the Joist Frame during Test 10. The frame loaded at 6.76 kips. ............................................................................................................. 106 V.65 Test 11 Configuration. .............................................................................. 107 V.66 Test 11 Top Chord Deflection. ................................................................. 109 V.67 Test 11 Bottom Chord Deflection. ............................................................ 110 V.68 Extrapolated trend lines for Test 11. .......................................................... 110 V.69 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at 5.00 kips. Photo taken at the first obs ervable sign of buckling during Test 11. Beast joist shown. ............................................................................................... 111 V.70 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at 5.88 kips. Beast joist shown looking fr om pinned end toward the roller end.111 V.71 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at 5.88 kips. Alpha joist shown looking fr om the pinned end towards the roller end. ...................................................................................................................... 112 V.72 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at the max load of 6.74 kips. Beast joist shown. ................................................ 112 V.73 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at the final load of 6.65 kips. Alpha joist shown. .............................................. 113
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xvi V.74 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at the final load of 6.65 ki ps. Beast joist shown. ............................................... 113 V.75 Deflected Shape of the Joist Fram e during Test 11. The frame was loaded at the final load of 6.65 kips. Alpha jois t is shown on the right and Beast joist is shown on the left, looking toward the East end. ................................................. 114 VI.1 SAP2000 Joist Model. ............................................................................... 123 VI.2 SAP2000 Model 1 Setup. .......................................................................... 124 VI.3 SAP2000 Model 1 Buckled Condition (SAP2000 Buckling Analysis). .... 125 VI.4 SAP2000 Model 2 Configuration. ............................................................. 127 VI.5 SAP2000 Model 2 Buckled Condition (SAP2000 Buckling Analysis). .... 127 VI.6 SAP2000 Model 3 Configuration. ............................................................. 129 VI.7 SAP2000 Model 3 Buckled Condition Mode 1 (SAP2000 Buckling Analysis). ............................................................................................................ 130 VI.8 SAP2000 Model 3 Failed Condition (SAP2000 Nonlinear Static Analysis). ............................................................................................................................. 1 30 VI.9 SAP2000 Model 4 Configuration. ............................................................. 132 VI.10 SAP2000 Model 4 Buckled Cond ition (SAP2000 Buckling Analysis)... 133 VI.11 SAP2000 Model 5 Configuration. ........................................................... 135 VI.12 SAP2000 Model 5 Buckled Cond ition (SAP2000 Buckling Analysis)... 135 VI.13 SAP2000 Model 6 Configuration. ........................................................... 136 VI.14 SAP2000 Model 6 Buckled Cond ition (SAP2000 Buckling Analysis)... 137
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xvii LIST OF EQUATIONS EQUATION II.1 EulerÂ’s Critical Buckling Load Equation. ....................................................... 9 II.2 LagrangeÂ’s Modification to the Critical Buckling Load Equation. ............... 10 II.3 PinnedPinned Column Critical Buckling Load. ........................................... 11 II.4 Tangent Modulus Theory Critical Buckling Load Equation. ........................ 12 II.5 Effective Length Factor. ................................................................................ 16 III.1 Modified Euler Critical Buckling Load Equation. ....................................... 34 III.2 Critical Buckling Load Equation with Effective Length Factor. ................. 37 III.3 Critical Buckling Stress Equation with Effective Length Factor ................. 37 III.4 Critical Buckling Load for an Inelastic Column (Double Modulus Theory). ............................................................................................................................... 41 IV.1 AISC Notional Load. ................................................................................... 46 VI.1 Generalized Eigenvalue Problem .............................................................. 116
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xviii LIST OF ABBREVIATIONS AISC American Institute of Steel Construction in inch SJI Steel Joist Institute CSI Computers and Structures Incorporated
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1 CHAPTER I INTRODUCTION Buckling stability research in the st ructural and materials engineering community is not new. It started before Euler published his historical findings on the elastic buckling of perfect column s in 1744, and continues on through this research, and other research, today. Even though research on the subject has been conducted for an extended period of time, a better understandi ng of the stability limit state, including recognizing the signs that exceeding the limit state may be in progress, would be extremely valuable to the structural engi neering community. Furthermore, a greater u nderstanding of buckling is important in todayÂ’s engineering profession because the limit st ate of stability is becoming more and more critical in design an d construction as economic factors and design standards guide designs toward smaller member sizes. In the past, this was not necessarily the case due to the large members used in historic structures selected because they previously performed successfully. However, in todayÂ’s environment, economic factors are clearly pus hing stability into the forefront as a critical limit state that needs to be co nsidered and well understood. This research project was developed to experimentally and analytically study stability, and more specifically, buckling, of full scale truss members. Ultimately, the goal is to compare full s cale member experimental results to predicted Direct Analysis results. Many researchers have looked at the stability of steel columns, beams, and beam column s in the past, but very little research has been conducted on buckling stability of full size members that are used in
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2 construction (Singer et al. 1998). The ma jority of design work is based on analysis of the different components and members of a structure using techniques and methods developed from small scale testing and computer aided design and analysis. Through the research that has been completed, analysis and empirical methods and factors have been developed for the purpose of giving designers the ability to estimate when a specific member will buckle. The latest addition to the AISC Steel Construction Manual Specificati ons concerning stability is the Direct Analysis Method, first in troduced in 2005 (AISC 2005). This research was also intended to expand on research conducted on buckling of scaled models of pony truss br idges that took place at the University of Colorado Denver in 2013 (Fernandez, 2013). The research in 2013 investigated the critical load of pony tru ss bridge models with laboratory testing and computer modeling. In addition, th e focus was on the application of the Pdelta effects in the computer model to determine if they were being accurately accounted for. The study found that the models were accurate and recommended the use of the Direct Analysis Method for design (Fernandez, 2013). Expanding on the work completed in 2013, the primary focus of this research project is to experimentally and computationally analyze the buckling stability limit state of truss members; more specifically, full scale steel bar joists. Although the research is bei ng conducted on bar joists, th e research is intended to be applicable to other structural systems. The experimental research presente d in this thesis documents the deflection and shape of a member lead ing up to and at the critical load;
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3 information that is helpful in understandi ng a structures response to loading. A benefit of using full size members for the e xperimental portion of the research is that it captures the effects of fabricati on tolerances, loading eccentricities, and internal stresses that affect the streng th and stability of actual members. The analytical research is compared to the experimental research to verify the accuracy of the analytical method. The experiments are replicated using a second order analysis perf ormed with SAP2000; a finite element analysis computer program that is currently used in the structural e ngineering industry (CSI 2014). The second order analysis is based upon the Dire ct Analysis Method and is used as a comparative check on the experimental research to verify if the Direct Analysis Method accurately repr esents the buckling strength of truss members, and buckling members in general. Research Objectives The objective of the research is to experimentally validate the Direct Analysis Method for design and analysis of steel structural members and document the behavior of full scale me mbers leading up to the buckling limit state. This AISC Direct Analysis Met hod is used regularly by computer models, and will be validated by demonstrating whether or not it accurately predicts experimental results of full scale members. The specific objectives of the thesis research project are: 1. Investigate the buckling limit state of full scale truss members used in construction and building design.
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4 2. Document the response of full scale members from initial loading until critical load. 3. Compare experimental results with predicted results from a computer program using the Dir ect Analysis Method. Outline of Research Project Thesis The research project initially consisted of a literature review to gain a higher level of understanding of stability and related research. Following the literature review, the testing methodology was developed and the experimental analysis was conducted. In conjunction with the experimental analysis, the SAP2000 computer model was developed based on the experimental setups. After the experimental research was c oncluded, the results were analyzed and compared to the results from the SAP2000 Direct Analysis. An iterative process was applied to the SAP2000 models to better represent the experiments to determine if the Direct Analysis Method could accurately represent experimental findings. A more specific presentation of this thesis report and findings is summarized below. Chapter II summarizes historical re search and developments regarding buckling and the stability limit state. Chap ter II first discusses the development of the buckling limit state and some of the research that has led to the current understanding of buckling. In addition, Ch apter II introduces ot her topics that effect stability, including findings on frames, connections, internal stresses and fabrication imperfections.
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5 Chapter III follows with a more t echnical look at the development of buckling and stability theory and design. Chapter III shows the developmental stages of design and theory and how advancements in research and knowledge have evolved from the theory that Eule r published in 1744 into how it is known in current practice. Chapter IV presents current design st andards for analysis and design of members for the stability limit state. Specifically, Chapter IV introduces the current AISC methods, including a more in depth discussion on the Direct Analysis Method in the AISC code. Chapter V discusses the laboratory tes ting and includes figures, tables and summaries of the results. Chapter VI follows by presenting the SAP2000 models of the different tests for comparison. Th e model results and comparisons to the physical testing results are al so included in Chapter VI. Chapter VII presents the conclusions of this thesis and recommendations for future studies.
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6 CHAPTER II HISTORICAL THEORY AND BAC KGROUND OF COLUMN THEORY Historical Column Theory One of the more basic structural compression members are columns. Although columns in practic e rarely would carry onl y axial compression, the basic analysis of these members neglects bending. Also, columns may be safely designed as members purely in axial compression. Before addressing other aspects of column design, a review of general column theories and their development throughout history is prudent. Elastic Buckling. The first documented experiments pertaining to buckling due to compression were performed by Musschenbroek in 1729, reported in MusschenbroekÂ’s Thesis Â“Introduction to the coherence of solid bodiesÂ” (Truesdell, 1960). Learning from hi s experiments, Musschenbroek stated that the wood members he tested in co mpression Â“Â…exert forces of resistance which vary inversely as the square of the length, directly as the thickness of the side that is not bent, and directly as the square of the side that is bent.Â” In addition, Musschenbroek stated that when compression loading increased on the struts, the struts broke in the middle wh ere bending occurred the most. From his results and observations, Musschenbroek s howed that failure due to compression was not similar to tension, which is a f unction of cross sectional area, but is a function of both length and cross sec tional area. The observation that the members broke in the critical bending section was the first noted observation of
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7 buckling, although it was never distinctly called a buckling failure (Truesdell, 1960). Musschenbroek experimented on co mpression members and made critical observations about buckling, but he did not theorize the results. Euler Elastic Buckling. Although studies on elasticity were ongoing prior to EulerÂ’s contribution to the fi eld, our current understanding of buckling theory started in 1744 with Leonhard Eu ler of Switzerland. The idea that the equation of elastica, or elastic theory, c ould be obtained using material properties was initially brought to EulerÂ’s attention in a letter from Daniel Bernoulli. Euler proceeded to verify BernoulliÂ’s idea and expanded on it by deriving an equation for elastic buckling, very similar to the same equation we use today, in his work Â“De curvis elasticis,Â” an appendix to his Â“Methodus invenieni curvas lineasÂ” (Fraser 1991). Euler was initially working with member s that were fully restrained at one end and free at the other, although simila r principles can be applied to pinnedpinned connection columns that have no re sistance to moments, and therefore, have the lowest buckling strength (Sa lmon et al. 2009). Euler developed an equation to predict the failure load fo r a perfectly elastic column under axial compression. The ideal column that Eule r derived the critic al buckling load equation for was assumed to have no fl aws; no initial internal stresses, was completely straight, and was homogeneous. In addition to an ideal column, the theory was also based upon a perfectly con centric axial load. Therefore, with an ideal column and a perfectly concentric load, the only loading is axial compression (Salmon et al. 2009).
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8 The behavior of the ideal column dur ing loading describes the nature of the critical load. During loading of the ideal column, the column remains perfectly straight and compre ssed as long as the load is less than the critical, or Euler, load. If the load is less than the critical load the column will be in stable equilibrium, and if a small lateral load is applied the column will deflect slightly, but once the load is removed, the column w ill straighten to its original position. Once the load is equivalent to the critical load the column enters an unstable equilibrium condition. If the same small la teral load is applied as before, the column will again deflect. However, unlike before, when the lateral load is removed from the column that is still unde r the critical axial lo ad, the column will remain in the deflected position. Therefore, the critical load is defined as the load in which the column will remain in the slightly deflected form after lateral loading, or, the load causing unstable e quilibrium (Timoshenko and Gere 1961). Any increase in the axial load above the cr itical load will force the column past the unstable equilibrium and put the colu mn in a bent, or buckled, condition. Figure II.1, below, shows an undeformed column, followed by the same column under critical load, and then the column in the buckled state (Mahfouz 1999).
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9 Figure II.1 Euler Buckling Load Example. During EulerÂ’s analysis of elastic bu ckling he derived and equation for the critical load (Timoshenko 1953), Equation II.1, below. Equation II.1 EulerÂ’s Critic al Buckling Load Equation. Euler theorized that unless a load is greater than P from Equation II.1, there should be Â“absolutely no fear of bending; on the other hand, if weight P be greater, the column will be unable to resi st bending.Â” Euler then went on to theorize that the load, P, which the column could safely carry, was inversely proportional to the height of the column squared. The constant C in Equation II.1 is what Euler called the c onstant of Â“absolute elasticity,Â” based on the properties of the material. Although Euler did not have access to the material science that is
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10 common in structural engineering curricu lum today, he established the basis for the buckling of columns used largel y still today (Timoshenko 1953). One of EulerÂ’s pupils and later hi s colleague, JosephLouis Lagrange expanded upon EulerÂ’s work on the elastic buckling of columns. He reproduced the critical buckling formula for a column with hinges at both ends, idealized as a pinnedpinned column, and assuming a small deflection caused by an axial compressive load. Lagrange then derived th at it was possible to have an infinite number of buckling curves by adding a f actor, m, which solved the equation if m was an integer. LagrangeÂ’s equati on, Equation II.2, is presented below (Timoshenko 1953). Equation II.2 LagrangeÂ’s Modificati on to the Critical Buckling Load Equation. LagrangeÂ’s contribution to the equation went past the critical buckling of the column; it explored the various modes that can occur within column buckling. Equation II.2 is the basis for how differe nt end conditions and bracing influences buckling. Today, the buckling equation for a pinnedpinned column is slightly modified with advances in material properties. The term C is replaced with EI the modulus of elasticity multiplied by the moment of inertia. Equation II.3 below, is the current equation for the Eu ler critical buckling load for the pinnedpinned column. It should be noted that to determine the critical load of a certain node, the term n2 would be multiplied into Equati on II.3, similar to LagrangeÂ’s m2.
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11 Equation II.3 PinnedPinned Colu mn Critical Buckling Load. Inelastic Buckling. Although Euler developed the equation to determine the critical axial load a column could undergo prior to buck ling, his theory was turning out to be unconservative for de sign; tests performe d on ordinary length columns were buckling prior to being lo aded by the predicte d critical buckling load (Salmon et al. 2009). After EulerÂ’s contributions to buckling theory, French engineer Anatole Henri Ernest Lamarle was the next to make progress in the field in 1845. Lamarle theorized that EulerÂ’s critical load equation, Equation II.1, should only be used below a proportional limit, now known as the slenderness ratio, otherwise, testing should be used to determine the buckling strength (Vable 2002). TangentModulus and Double Modulus Theories. Following LamarleÂ’s theory, French engineer A.G. Considre published the first comprehensive column testing results in 1889. From these results, Considre deduced that the stress on the concave side of the deflected column increased with the tangent modulus of elasticity, Et, while the stress on the convex side decreased with the elastic modulus of elasticity, E Since they were both not controlled by the elastic modulus of elasticity, Considr e showed that EulerÂ’ s critical load was not applicable for inelastic buckling, but that the effective modulus was actually in between the modulus and tangent modulus of elasticity. Aside from theorizing the effective modulus, Considre made no attempt to determine it (Gere and Goodno 2012).
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12 Also in 1889, but completely independe nt of Considre, German engineer F. Engesser proposed the Tangent Modulus Theory. Engesser theorized that the critical load of an inelastic beam coul d be estimated in the same way that the critical load could be es timated for an elastic column determined by Euler. Engesser proposed that to determine the crit ical load for an inelastic column the modulus of elasticity needed to be replaced with th e tangent modulus of elasticity (Chen and Atsuta 2008). Therefore, E ngesser modified EulerÂ’s critical load equation to Equation II.4, below. Equation II.4 Tangent Modulus Theory Critical Buckling Load Equation. Equation II.4 was similar to what Considre theorized, but EngesserÂ’s theory directly used the tangent modu lus instead of the effective modulus. Engesser theorized that the column under axial load would remain straight until failure (Salmon et al. 2009), and that th e tangent modulus was the tangent of the stressstrain curve at failure, so the value could be determined with limited material testing (Chen and Atsuta 2008). EngesserÂ’s theory, although closer to experimental results, still did not agree with testing results. In contrast to EulerÂ’s critical buckling load, Engre sserÂ’s critical load resulted in conservative critical load estimates, underestimating the capacity of a column. The assumption that the column remains straight then bends right at failure, assuming that no strain reversal takes place, was considered to be in error (Salmon et al. 2009). In fact, the theory was in error because the assump tion that no strain reversal takes place meant that elastic unloading of the pl astic portion of the column was not
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13 considered. The Tangent Modulus Theory is generally referred to as the Engesser Theory (Gere and Goodno 2012). In 1895 Engresser modified his theory to include the reduced modulus, Er, which included elastic unloading (Doubl e or Reduced Modulus Theory). The reduced modulus was very similar to what Considre initially theorized to be between the elastic modulus and tangent m odulus of elasticity (Chen and Atsuta 2008). The Double Modulus Theory is ge nerally referred to as the ConsidreEngesser Theory (Gere and Goodno 2012). In 1910 Theodore von Krmn derived equations for the Double Modulus of rectangular and ideal Hsection columns. After von KrmnÂ’s derivation many within the field of structural engineer ing debated the differences between the Tangent and Reduced Modulus Theories Theoretically, the Double Modulus theory was more correct than the Tangent Modulus theory, but similar to EulerÂ’s load for elastic buckling experimental re search found that columns tested buckled at loads less than predicted (Johnston 1961). Shanley Theory. The Double Modulus Theory was accepted as the theory for inelastic buc kling during the early 20th century, although it was known to calculate higher strengths then testi ng supported. Further advances in buckling theory beyond the Double Modulus Theory were not completed until 1946 with American aeronautical engin eer Francis R. Shanley. In ShanleyÂ’s journal article Â“The Column ParadoxÂ” (1946), he began to question the assumption made in the Double Modulus Theory (referred to as the Reduced Modulus Theory in ShanleyÂ’s article) that the loaded column will remain straight as the strain
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14 increases from the predicted tangent modul us to the higher strain from the double modulus. Shanely deduced th e following (Shanley 1946): Â“Actually, there is nothing (excep t the columnÂ’s bending stiffness) to prevent the column from bending simultaneously with increasing axial loading. Under such conditions the compressive strain could increase on one edge of the column while remaining constant on the other, or it coul d increase at a different rate on opposite edges. If such action were assumed, the tangent modulus would apply over the entire cro ss section, and the theoretical buckling load would be that predicted by the tangent modulus theory. This creates a paradox, becau se, if all of the strains equal or exceed the tangent modulus value, the average strain will be greater than that predicted by th e tangent modulus theory.Â” Shanely continued, Â“The assump tions involved in the reduced modulus theory also represent a pa radox. The theory predicts that the column will remain straight up to the calculated maximum load, but it also shows that some st rain reversal is needed in order to provide the additional column stiffness required beyond the tangent modulus load. It is impossi ble to have strain reversal in a straight column.Â” Shanley effectively concluded that the critical load for inelastic column buckling is somewhere in between what the Tangent Modulus Theory and the Double Modulus Theory predict, and that it needs to be assumed that axial loading and bending can occur at the same time. Therefore, Shanley disproved the principle of superposition for inelas tic column buckling (Shanley 1946). Shanley continued to expl ore column buckling past his observations in Â“The Column Paradox.Â” Through further experimentation, Shanley determined that the ultimate strength was generally much closer to the Tangent Modulus. Shanley concluded that the Tangent Modulus critical load was the best estimate for buckling in the plastic range (Byskov 2013) because the critical load
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15 determined by the Tangent Modulus Theory is the smallest axial load where bending may begin (Salmon et al. 2009). Conditions Effecting Column Buckling Axial loaded columns will vary significantly in buckling strength based on various factors. The ultimate buckling strength depends on connections of the columns to adjacent members, the length of the column, bracing of the column, and imperfections of the column (Salm on et al. 2009). These factors that influence buckling strength of a column ar e explained in further detail later in the chapter. Column Effective Length. Most of the resear ch discussed above was based on columns with pinned connections at the ends so no mo ment restraint was provided. Using pined connections allowe d the research to take place using the least stable restraints for the columns (Salmon et al. 2009). In practice, many different scenarios will take place, incl uding different connecti ons at each end of the column as well as columns within frame systems of varying degrees of complexity. End Restraint of Columns. The general column buckling theories introduced earlier can be modified based on the end restraints of the column. The typical, worst case scenario, is the pi nnedpinned connection because no moment transfers from the column to the support. Because ideal pinnedpinned connections rarely represent a column in practice significant research has been conducted on the effects of different end restraints and connections (Jones et al. 1982). Jones et al. tested compressi on members with pinned connections,
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16 connections made with double angles, and connections made with topandseat angles. The pinned connections buckled prior to the Euler predicted critical buckling load while the double and topa ndseat angle connections buckled at loads higher than the Euler buc kling load (Jones et al. 1982). The Euler buckling load was develo ped for the ideal pinnedpinned connection, but a derivation of the ideal fixedpinned column shows that the buckling load is increased by 105percent when compared to the Euler pinnedpinned connection. Further e xperimentation showed that the buckling strength of the columns with different end connecti ons other than pinnedpinned and fixedpinned showed that the buckling strength is a multiple of the pinnedpinned elastic buckling load. Therefore, the el astic buckling load of a column can be determined using the Euler buckling load multiplied by a constant based on the end restraints (Craig 2000). To address this finding, the theory of the effective length of a column was developed. The effective length of a column is, for any end condition, the equivalent pinnedpinned column that w ould have the same buckling load (Chen and Lui 2005). The effective length factor, K, is deri ved in Equation II.5 (Chen and Lui 2005), below. Equation II.5 Effective Length Factor. In Equation II.5 Pcr refers to the critical buckling load of the column in question. Although the designer may us e Equation II.5 to develop K through
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17 experimentation, the most common way to de termine K for a certain scenario is to use alignment charts given in th e AISC Steel Construction Manual 14th Edition (2010). In addition to the alignment ch arts, the AISC Steel Construction Manual provides a table of approximate values of K factors for various end conditions, shown in Figure II.2, below. Figure II.2 Approximate Values of Effective Length Factor, K. Used with permission (AISC 2010). In addition to Figure II.2, Figure II .3 expands on the buckled shapes to show the idealized buckled shape as a pi nnedpinned connection to help visualize the theory (Craig 2000).
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18 Figure II.3 Euler Buckling Load Example. The alignment charts and approximations for K are based on the following assumptions: the behavior of the column is purely elastic, all members have a constant cross section, all joints are ri gid, the bending is single curvature when sidesway is inhibited, the bending is reverse curvature when sidesway is uninhibited, and the stiffness of all colu mns (in a system) are equal (AISC 2010). Despite the widespread acceptance of the K factor, the assumptions listed above must be considered. In addition, the K factors may be used with the Tangent Modulus Theory, discussed previously in Chapter II, to determine the buckling strength of inelastic columns with varying end conditi ons (Salmon et al. 2009). It is emphasized in the AISC Steel Constr uction Manual that the K value alignment charts are based on idealized assumptions. Braced and Unbraced Frames. Columns, as part of a frame, will either be braced or unbraced. According to the AISC Steel Construction Manual (2010) a braced frame has Â“lateral stability provided by diagona l bracing, shear walls or
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19 equivalent means.Â” Because the bracing will affect the reactions at the end conditions, bracing needs to be understood as an important component of the system contributing to the effective le ngth of the member. The bracing must adequately prevent buckling of the struct ure, helping maintain the structures lateral stability. A column in this case would have no sidesway movement (Salmon et al. 2009). Since there is no si desway movement the effective length of the column, at a maximum, is equal to the pinnedpinned idealized column. In contrast to the braced frame, the unbraced frame is subjected to sidesway. Therefore, the rotations at either end of the column are equal in magnitude and direction (Chen and Lui 1991). In addition to the deflection caused by the sidesway, the column will also deflect locally based on the effective length of the column, regardless of the sidesway. The theoretically stiffest scenario for an unbraced frame occurs with an infinitely stiff beam; a beam that cannot bend. In that case, the inflection point would be in the middle of the beam, similar to a column with the base fully restrained and the K would be equal to 1.0 (Salmon et al. 2009). Therefore, theoreti cally, the minimum K value is 1.0, which is the K value of a pi nned pinned connection. Residual Stresses. Residual stresses in columns are internal stresses that remain within a member after it has b een fabricated. The stresses may result from: uneven cooling after hot rolling of structural member, cold bending or cambering during fabrication, punching holes and cutting during fabrication, and welding (Salmon et al. 2009).
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20 For hot rolled wide flanged shapes the primary cause of re sidual stresses is due to uneven cooling after hot rolling. Extensive rese arch on residual stresses due to hot rolled shapes has been co nducted, including research at Lehigh University by Goran A. Alpsten in 1968. Alpsten concluded that the principle factors influencing thermal residual stre sses are the shape of the member and cooling conditions. Alpsten also noted that the general distribution of stresses for H shapes were compression in the flange tips and the web and compression where the flange and web join. The compressi on at the flange tips due to residual stresses, the critical region for residual stresses, reached up to 100percent of the assumed yield stress at room temperature (Alpsten 1968). Welded built up shapes also exhibit residual stress effects similar to hot rolled shapes, mostly because they also undergo uneven cooling conditions and are essentially placed under similar c onditions locally (Salmon et al. 2009). Further research conducted at Lehigh Univ ersity in 1972 showed that heat from both welding and flame cutting caused lo calized residual stresses, sometimes much higher than the yield strength of the material, but the effects did not drastically change the residual stresses away from the weld or flame cutting (Bjorhovde et al. 1972). On the other hand, coldformed member s avoid the uneven cooling of hot rolled members, but residual stresses ma y be developed through the cold bending process due to the cold forming. Fr om experiments conducted on coldformed members, it was found that compression re sidual stresses are typically found on the inside surfaces while tensile stresses were found on the outside surfaces. The
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21 stresses caused by coldforming reached up to 70percent of the yield stress of the member and the critical sections were corner regions as expected (Weng and Pekoz 1988). Although hot rolled residual stresse s were greater than coldformed residual stresses, the resear ch discussed above shows that residual stresses for both can reach significant levels and it is appa rent that residual stresses need to be considered (Ziemian 2010). Geometric Imperfections. In addition to residual stresses developed during manufacturing or construction, stru ctural members are also subjected to geometric imperfections causing moments in the member due to axial loads. Imperfections may be in the form of a very minor change in cross section along the member, or a member constructed with a very slight deviation along its axis. Small imperfections can make the actual buckling load substantially lower than the critical load of the pe rfect column (Budiansky 1973). Geometric imperfections such as deviat ions along the axis of a member, or outofstraightness, keep the member from being perfectly parallel to the induced axial load; therefore, the column must re sist a bending moment in addition to the axial load. Figure II.4, below, depicts and column with outofstraightness imperfections.
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22 Figure II.4 Example of OutofStraightness In 1997 the ASCE Task Committee on Effective Length published studies of outofstraightness of cantilever colu mns with various imperfections ranging from initial leaning to unintended camber causing outofstraightness. The results of the studies showed that the stre ngth of the column was reduced to approximately 75 percent of an ideal colu mn (out of straightness equal to 0.1 percent of the length), assuming a sle nderness coefficient of 1 when the imperfection were included in the member. In addition, by varying the slenderness of the column, it was shown th at the magnitude of the effects of outofstraightness were more dependent on the length of the column (ASCE 1997). Another geometric imperfection, outofplumbness, is an imperfection leading to unintended camber of the colu mn (but not outofstraightness), shown in Figure II.5, below.
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23 Figure II.5 Example of OutofPlumbness. The 1997 ASCE task force, discussed above, also considered outofplumbness of a column. Assuming a slende rness coefficient of 1, the initial outofplumbness column (0.2 percent of colu mn length) had a stre ngth equivalent to 82 percent of a perfect column. In a ddition, if outofstraightness and outofplumbness were combined, the strength of the column was reduced to 71 percent of the critical load of a perfect column (ASCE 1997). Prior to the ASCE analysis of outo fstraightness of columns, Richard Southwell, in 1932, proposed a method to re late a real column to the perfect column based on deflection versus load curves. Southwell theorized that the slope of the actual deflection versus th e deflection per unit load curve would predict the critical load; this plot is k nown as the Southwell Plot (Singer et al. 1998). Figure II.6, below, is an example of a Southwell Plot.
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24 Figure II.6 Southwell Plot. The theory presented a way to determin e the critical load of an in place column with nondestructive testing because very small deflections were used to create the plot. SouthwellÂ’s method ha s been extended to other applications through various studies and is still used today as a nondestructive method for approximating the critical load of a column (Singer et al. 1998). Buckling Stability of Other Members Beams. A lot of the focus thus far has been the buckling of the columns. This is because ideally loaded colu mns represent buckling under purely axial loads; although imperfec tions do induce moments that reduce the buckling strength of a column from the ideal Eu ler column. Beams, on the other hand, typically support transverse loading, in addition to the loading caused by the weight of the member itself. Buckling of beams is a similar function to buckling
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25 of columns with the effects of imperfec tions, loading eccentricities, and pdelta effects (discussed late r in the chapter). A beam is a combination of both co mpression and tension elements and both situations and theories must be cons idered. The top flan ge, or plate, under typical loading acts as the compression element and the bottom flange, or plate, acts as the tension element. The web of the member, or side plates for a box section, acts as local late ral bracing for the compression element because the web or plates are parallel to the load stabilizes the compression element along the length of the element (Singer et al. 1998). Typically, design of beams assumes that bending occurs in the plane of symmetry, and the induced moment is also considered the primary design load. Because of those reasons, the design may lead to members that are weak with regard to minor axis bending, thus late ral buckling needs to be a consideration (Singer et al. 1998). In addition, some of the same factors that were discussed earlier, such as loading eccentricities, imperfections, and internal stresses will similarly affect the beam member whic h may lead to deflection and buckling along the minor axis. Lateral buckling of beams wa s realized by the late 19th century after being observed during testing of steel and wrought iron members, but the initial expansive theories and experiments on la teral buckling of be ams were performed simultaneously by Prandtl and Michell in 1899. These were followed up with much more extensive resear ch on the topic into the 20th century (Singer et al. 1998).
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26 Both PrandtlÂ’s and MichellÂ’s resear ch was comprised of experimental research and theoretical analysis of la teral buckling of beams leading to the classical theory of latera l buckling. Their analyses developed the same theory, independently, while generalizing the firs t order effect of the principal bending curvature by deriving an approximate buckling load formula. Additional experiments and research was conducte d by Timenshenko and Gere, as well as Reissner, neither of whom were aware of the previous results. Following their research, Hodges and Peters, in 1975, impr oved on the original lateral torsional buckling equation after finding errors in the previous analyses (Hodges and Peters 1975). In current design practice and research, lateral buckling of members is well documented in terms of stability of structures, although due to various imperfections in members which are not accounted for during design and analysis, experimental data and empirical formul as still account fo r a lot of design decisions. Many of the column buckling methods and theories have been shown to apply to lateral buckling of beams; th is is because both ar e essentially buckling of a member, failing due to the stabil ity limit state (Singer et al. 1998). BeamColumns. Another form of a common compression member is the beamcolumn. Beamcolumns are just as the name implies; a combination of a beam and a column in one member. While a column is primarily subjected to axial loads and beams are primarily s ubjected to bending moments, a beamcolumn is a member subjected to both ma jor bending moments and axial loads. In addition, significant secondary moments are introduced due to deflection of the
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27 member and the axial load acting on the deflected shape must be considered. Two types of secondary moments occur; Pand P, which both need to be considered. The Pmoment is the moment that is developed by the axial force acting through the lateral displacement along the chord of the member while Pis the moment caused by the axial for ce acting through the relative displacement of the ends of the member. Fi gure II.7 shows examples of the Pand Pscenarios (Chen and Lui 1991). It should be noted that both are similar to imperfections during column loading. Figure II.7 Example of Pand Pmoments. Research and experimental developments regarding beamcolumns has been occurring for many decades by stru ctural engineers due to the extensive nature of the use of beamcolumns; infact most elements can be classified as beamcolumns (Singer et al. 1998). Because beamcolumns are merely a combination of the function of a beam a nd column, the theoretical developments discussed above for each independently can be considered applicable to the beamcolumn.
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28 Plate Buckling. All steel members are composed of different elements, called plates, which are built up to make th e member. In addition to the overall member buckling strength, the local buckli ng strength of these plates needs to be considered when assessing the strength of a member. If local buckling occurs, that section can no l onger carry additional load and th e load must be redistributed to other sections of the member (Salmon et al. 2009). Local buckling of plates needs to be considered because it can lead to a progressive failure of an entire member When analyzing the member for local buckling, the plates can be categorized as stiffened or unstiffened elements. Stiffened elements, such as webs of a Wshape, are supported on two edges in the direction of the load, while unstiffened elements, such as flanges of a Wshape, are not supported on two e dges (Salmon et al. 2009). Plate theory should be recognized as another aspect of compression members that needs to be carefully addr essed in design, because all steel members are essentially composed of various plat es. The first recognized case of plate buckling occurred during construction of a large span suspension bridge for a railway in 1845. Due to concerns ov er buckling of the wrought iron box beam that was to be used, project engineer Robert Stephenson asked William Fairbairn from the University College London to e xperimentally study a scale model of the box beam. The researchers were not e xpecting the results they achieved, commenting that Â“some curious and intere sting phenomena presented themselves Â– many of them are anomalous to our pr econceived notions of the strength of materials, and totally different to a nything yet exhibited in any previous
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29 research.Â” Although the research had st arted with Fairbairn in the mid 1800Â’s, further research was typically theoreti cal without the addition of extensive experimental research (S inger et al. 1998). Through research, both experimental a nd theoretical, ther e is now a much more significant understanding of the limit st ate of local buckling, also known as plate buckling. G. Bryan first derived the buckling load of a simply supported plate in 1891. Bryan derive d that the smallest critical buckling stress could be determined with the following equation (Singer et al. 1998): Where kc is the plate buckling coefficient, h is the thickness of the plate, and b is the width of the plate. The pl ate buckling coefficient is dependent on the length, a, and the width of the plate. The plate buckling coefficient equation is shown below where m and n are buckling nodes: If a short plate is assumed, where th e length is less than the width and assumes that buckling occurs in the first node, the plate buckling equation becomes: Comparing this to the column equation, if L is equivalent to a, the plate buckling equation is similar to a wide co lumn equation. This is different to a
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30 slender column in that the plates provid e restraint to each other where they are built up (Singer et al. 1998). After Bryan, Coan researched ini tially curved plates through large deflections for buckling and post buckling strength in 1951. Coan surmised that although plates buckle similarly to columns, unlike columns, plates can carry significant post buckling stresses. When buckling of a plate o ccurs, the stiffness is decreased, but failure only occurs when the axial stress at the unloaded edges of the plate reach the yield strength of th e material. Components of members such as flanges, webs, angles, and cover pl ates, which when combined make up the steel member (Salamon et al. 2009). Typi cally, these shapes buckle prior to the member in questions itself buckling. Th erefore, local buckling may occur without buckling of the member (Singer et al. 1998). Current Design Practices In current design practice, most of the topics discussed above are factored into design either directly or indirect ly. Column length, cross section shape, material properties, and end restraints as wells as frame connections are all considered in design practice. The current U.S. steel design code the 2010 AISC Steel Construction Manual contains two sections in the Speci fications for designing for stability: the first is Chapter C Â“Design for Stabilit yÂ”, the second is Appendix 7 Â“Alternate Methods of Design for Stability.Â” The methods discussed in the Â“Alternate Methods of Design for StabilityÂ” appendix were moved to the appendix sections
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31 in the 2010 version of the code due to advancements in technology and new methods to design for the stability of structures. Indirect Stability Analysis Methods. Appendix 7 in the 2010 specifications refers to the Effectiv e Length Method, discussed early in the Chapter, and the FirstOrder Analysis Method. The FirstOr der Analysis Method is the only stability design method in th e AISC Steel Construc tion Manual that is a first order method. In the First Orde r Analysis Method the structure being designed is analyzed using nominal ge ometry and elastic stiffness and the required strength is determined using a first order analysis based with the appropriate controlling load scenarios with addition of a lateral notional load at each frame level. This method may only be used if the required compressive strength is less than half of the yield strength in all of the members that contribute to the stiffness of the structure. The eff ective length factor, K, is taken as 1 (Nair 2007). Direct Analysis Method. The Direct Analysis Method was added to the AISC Steel Construction Manual in the Th irteenth Edition, released in 2005, in Appendix 7 of the AISC 360 specifications Appendix 7 was an alternative to Chapter C in the specifications which included the Effective Length Method. The Fourteenth Edition of the Steel Construction Manual, released in 2010, elevated the status of the Direct Analysis Met hod by moving it to Chapter C in lieu of the Effective Length Method, which was moved to Appendix 7 (AISC 2005, AISC 2010). The Direct Analysis Method is the only procedure in the AISC Steel Construction Manual that is applicable to all types of frames (White et al 2006).
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32 The Direct Analysis Method includes a re duced elastic stiffness to account for stiffness reduction in the analysis, ideally leading to more accurate results. In addition, geometric imperfections can be mode led directly or indirectly by using a lateral load based on the vertical load that the structure is subjected to. The intent of the Direct Analysis Method is to provi de a more accurate representation of the structure, including the transfer of forces due to the modified stiffness of the structure. The method directly models the structure, eliminating the need to estimate the effective length of the struct ure with effective length factors (White 2006). Chapter IV expands on the current methods included in the 2010 AISC Steel Construction Ma nual Specifications.
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33 CHAPTER III STABILITY AND BUCKLING THEORY DEVELOPMENT Column Buckling Columns are used in structures of ev ery type of composition; from ancient stone columns common to the Roman Empire to modern steel columns used to construct record breaking skyscrapers. Due to the significant size of columns in the past, buckling was not much of a c oncern, but as member sizes shrink while structures get larger, the consideration for buckling has necessarily increased. The increased understanding of loads and st resses, new design materials, and the desire the keep building co sts down have all led to a decrease in building member size, columns included. Short columns, or very large columns (w ith regard to cross sectional area), can be loaded to their yield stress, alt hough with taller, more slender columns this rarely occurs. Typically, buckling, or s udden bending as a result of instability now occurs prior to the column failing at its yield stress (Salmon et al. 2009). Most column theories are based on the ideal column, one that is impractical in the real world, but still provides the basic theory needed to understand and design or analyze columns. The ideal column is one that is perfectly straight; no outofstraightness or outofplumbness, the cross section of the column is uniform throughout, the column has no residual stresses, is weightless, and is loaded exactly alo ng the centroid of th e column with no bending moment or lateral forces, the co lumn is homogeneous, and is governed by HookeÂ’s Law (Bansal 2010). HookeÂ’s Law st ates that if a ma terial is lightly
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34 stressed it will incur a deformation, but as the stress is removed the material returns to its original shape; in othe r words, the material is elastic. Although a lot of column theory simplifies the member as an ideal, or perfect, column, the real in teraction between the load and the member is quite involved. Everything from the interacti on between the load and the column, the connections between the column and th e system, the columns material, the columns shape, and fabrication all impact the actual strength of the column. Looking at column theory with regard to the stability limit state and the buckling failure mode, we must once again turn back to the Euler column theory for the perfect column, to start the analys is and development of the theory. The Euler column, perfectly elas tic, is represented by the following equation, Equation III.1, modified from its original versi on to incorporate ou r current understanding of material properties. Equation III.1 Modified Euler Cr itical Buckling Load Equation. The assumptions made for the perfect column are: the column ends are perfectly pinned, the column is perfectly straight, the load is applied along the central axis of the column, and the materi al is elastic and act s homogeneously. To derive the Euler buckling load equati on, a free body diagram of the perfect column should be taken through the deflect ed shape, shown in Figure III.1 below.
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35 Figure III.1 Free Body Diagram of Euler Column Analyzing the free body diagram above and taking the moments about the cutting plane, summing the mo ments for equilibrium gives: If the following second order differen tial equation for curvature of a deflected member is used: Replacing the moment equation from the free body diagram into the second order differential equati on, the differential becomes: Now, take the coefficient for the second term in the above equation to be:
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36 The differential equation for the curvature of the column then becomes: Solving the differential equation will yield the results for nontrivial solutions leaves the Euler buckl ing equation, Equation III.1 above. The value for n in Equation III.1 relates to the type of curvature that is occurring. For the pinnedpinned case th at Euler examined, the failure is single curvature. If the column is braced at locations along the length, buckling will occur at higher levels of curvature and thus increase the criti cal strength of the column. Figure III.2, below, shows the deflected shape of columns with various node levels and gives the correspon ding critical strength equation. Figure III.2 Deflected Shape and Cr itical Load for Different Nodes. Effective Length As previously noted, EulerÂ’s buckling theory was based on the pinnedpinned perfect column, although many different boundary conditions are observed in design and practice, all of which contribute to th e buckling strength of the column. For other boundary conditions, if the same perfect column assumptions
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37 are made, the moment at the ends w ill not always be zero, leading to nonhomogeneous differential equations, although solving will lead to equation of a similar form except that a different cons tant is applied (Ges chwindner 2012). To generalize the buckling equation for a ll boundary conditions, Equation III.1 was modified to be Equation III.2: Equation III.2 Critical Buckling Load Equation with Effective Length Factor. Or conversely, Equation III.3 for the critical stress of the column. Equation III.3 Critical Buckling Stre ss Equation with Effective Length Factor Equation III.1 was modified by replacing the variable for length, L, with a variable for the effective length, KL. E quation III.3 is equivalent to Equation III.2 but expresses the critical compressive st ress as opposed to the critical load. The effective length factor, K, as discusse d in Chapter II, is a factor on the length to adapt Equation III.1, the Euler critical load, to columns with different end conditions (Salmon et al. 2009). Fi gure II.4, from the AISC 2010 Steel Construction Manual, shows approximate values of K for various common column end conditions. The effective length of a single colu mn is relatively simple, although it quickly can becomes less trivial when the column is incorporated into the frame of a structure. When a column is part of a frame, the column reacts to the
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38 stiffness of the frame because they are a system of components, rather than individual components. The different members connected to the column in question will all impact the rotation, a nd therefore the reactions at the end restraints, and in turn impact the eff ective length of the column (Geschwindner 2012). If a member is part of a frame that is inhibited from moving laterally, it is considered a braced frame, and the K fact or will range from 0.5 to 1.0, always being less than or equal to 1.0 because values greater than 1.0 imply that the frame is assisting in rotating the column which is not the case if it is inhibited from moving laterally. On the other ha nd, if the ends of the frame are permitted to move laterally, it is an unbraced frame, in which the lowest K value that may be applied is. 1.0. The minimum effective length factor for an unbraced frame is 1.0 because the frame will not, by defini tion of an unbraced frame, assist in reducing the end rotation of th e member (Geschwindner 2012). A significant amount of research ha s been devoted to determining appropriate equivalent length factors for various end restra ints and frame systems. To simplify the design process, various re searchers and design texts and standards have provided charts to assist in easily de termining the effective length factor of a column. For example, charts have been developed for stepped columns, columns with an intermediate axial load, gabled frames, pinbased crane columns, and columns in one story buildings, in additi on to many other scenarios (Salmon et al. 2009). The most common alignment charts are the original charts developed by Julian and Lawrence (Kavanagh 1962). The charts developed by Julian and Lawrence are included as Figure III.3, below.
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39 Figure III.3 Effective Length Alignmen t Charts for Braced and Unbraced Frames. Used with permission (AISC 2010). The Julian and Lawrence alignment ch arts are included in the 2010 AISC Steel Construction Manual Commentary. The alignment charts are all based on ideal conditions, similar to Euler deve loping Equation III.1 based on the perfect column. Perfect conditions and columns neve r exist in the real world. To try and account for the fact that perfect conditi ons and columns do not exist, the AISC Steel Construction Manual provides fact ors to the K values or alternative equations and methods to determine the effective length of a compression member.
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40 Inelastic Buckling Theory EulerÂ’s critical load equation did not ag ree with experiment al results, so it was largely ignored at the time it was pr oposed (Salmon et al. 2009). Considre and Engesser, independently, developed the tangent modulus theory by replacing the modulus of elasticity w ith the tangent modulus of elasticity, as discussed in Chapter II (Salmon et al. 2009). The Tangent Modulus theory, although more accurate than EulerÂ’s theory that it was m odifying, was still not agreeing with test results, was predicting strengths less then what was being observed. To address the issue, Engresser modified the Tangen t Modulus theory to the Double Modulus theory. The Double Modulus theory is based on two different moduli. At the point of unstable equilibrium, where the critical load is being applied, the stress along the neutral axis is the same as it wa s prior to deflection. The side of the neutral axis where strain is increasing, the strain is proportional to the tangent modulus, Et, while the fibers on the other side, the Â“unloading sideÂ”, are still in the elastic range so the strain is pr oportional to the elastic modulus, E. Development of the critical load using the Double M odulus Theory, below, is based on descriptions in Salmon et al. 2009. In the development of the critical buckling load equation for the Double Modulus Theory, f1 depicts the stress based on the tangent modulus in the loading fibers and f2 depicts the stress in the unloadi ng fibers. At the extreme unloaded fiber, applying HookeÂ’s law, the stress is given by the following:
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41 While the loaded fiber stress is given by: The term dz in the equations above is taken as an element along the axis of the column. Since the change in the element dz over the length of the element dz is the slope, the following relationship, is made. Using the slope relationship: Assuming small curvature of the compression member: In the above equation, Er is the reduced modulus. Solving for the bending moment, the equation is reduced into: Rearranging the moment equation base d on a linear stress distribution the critical buckling load for an inelas tic column, based on the double modulus theory, is given by Equation III .4, below (Salmon et al. 2009): Equation III.4 Critical Buckling Load for an Inelastic Column (Double Modulus Theory). Examining Equation III.4 and compari ng it to Equation III.2, it is clear that the equations are of the same form, the only difference being the two different modulus values used in the derivation. Although Shanley proved that
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42 the Double Modulus theory is not represen tative of true column behavior, the difference in the load that causes bifu rcation from the elastic modulus is insignificant for design use (Salmon et al. 2009). Application to State of the Practice Due to the complexity of the equations shown above, and the additional requirements to consider other factors di scussed earlier including imperfections and how the compression element fits in to the structural system, design is typically completed using computer softwa re models that ha ve the capacity to consider many different factors. Development of the buckling theory and methods has been occurring for many years, and as such, second order anal yses have been included in the AISC Steel Construction Manual Specifications to more accurately portray load effects on a structure. Chapter IV discusses the current design standards as they pertain to the stability limit state.
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43 CHAPTER IV CURRENT DESIGN STANDARDS Stability Analysis and Design Chapter C in the 2010 AISC Steel Construction Manual Specification addresses stability analysis and design. In addition to Chapter C, Appendix 7 also includes methods to address stability during analysis a nd design. Both first and second order analysis methods are provi ded by the specifications, although the first order method is an amplified met hod to account for second order effects (AISC 2010). Below is a description of the methods outlined in the 2010 AISC Steel Construction Ma nual Specifications. FirstOrder Analysis Method The most basic of the th ree stability analysis me thods included in the 2010 AISC Steel Construction Manual is the FirstOrder Analysis Method. The FirstOrder Analysis Method is a conservative, simplified approach to the Direct Analysis Method, also incl uded in the specification (W hite et al. 2006). When performing the FirstOrder Analysis Method, the nominal geometry of the structure, and nominal stiffness of the structure, is used for the design and analysis. To account for sec ond order effects in the firs t order analysis, B factors are applied to the nominal moment and fi rst order moments as well as the first order axial force. The B factors are am plifying factors to account for pdelta forces; B1 is used to account for second or der effects caused by displacement between supports and B2 is an amplifier to account for second order effects caused by displacement of end supports (AISC 2005).
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44 The FirstOrder Analysis Method is limited to structures that have a sidesway amplification of second order effects versus first order effects ( 2nd order / 1st order) less than or equal to 1.5 and st ructures in which all compression members have approximately twice the strength than required by the applied loads. When designing with the FirstOrde r Analysis Method an additional lateral load is required ba sed on the initial 2nd order / 1st order is less than or equal to 1.5 assumption. One benefit to the FirstOrde r Analysis is that all members may be assumed to have effective length factors of 1. Effective Length Method The Effective Length Method is a sec ond order analysis method that has been included in the specifications in various forms since the 1963 AISC Specifications, although concepts regardi ng inelasticity were not included until the 1978 Specification (ASCE 1997). The de sign of compression elements using the Effective Length Method are based ar ound determining the effective length of a column, essential, converting any colu mns end conditions to that of a pinnedpinned column for analysis with the us e of a Kfactor, described earlier. In addition to the Kfactor, a notiona l load has been included in the Effective Length Method to induce lateral movement for load combinations that only include gravity loads. The nominal lo ads are addressed further in the Direct Analysis Method section below. Similar to the FirstOrder Analysis Method, the Effective Length Method is restri cted to structure that have a 2nd order / 1st order ratio less than or equal to 1.5.
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45 One of the perceived drawbacks to the effective length method is the K factor and the rigorous analysis that is sometimes required to accurately portray the K factor. In addition, studies have demonstrated that members with small axial stresses will have large K values based on methods to determine the K value, but the actual struct ural system may not justif y a large K value (White 2006). Examples of these scenarios that require engineering judgment in addition to the calculations include columns in th e higher levels of ta ll building, columns with weak connections, and beams in porta l frames. These judgment cases lead to more complications in the design proce ss than if the Direct Analysis Method, which is the same for any stru cture, is pursued (White 2006). The Effective Length Method is not as sensitive to accuracy of the second order analysis as the Dir ect Analysis method. One of the reasons that the effective length method is not as sensitive is because it is based on the undeformed, nominal geometry and nominal member properties and stiffnesses. Initial imperfections, including outo fplumbness, outofstraightness, construction tolerances, and various ot her imperfections are not directly considered in the analysis. The inpla ne axial strength of the member accounts for the imperfections by use of the e ffective length method braced on the bracing system and connections (AISC 2010). Alt hough, as noted in the commentary to the specifications, because th e magnitude of the increase in internal forces due to imperfections is not included in the Eff ective Length Method, the Direct Analysis Method is preferred for structures whic h are sensitive to stability issues, especially those comprised of slender elements.
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46 Direct Analysis The Direct Analysis Method is a sec ondorder analysis that calculates a reduced stiffness to accurately portray inelastic buckling properties of a compression member. The method was in troduced into the AISC Construction Manual to provide a general design met hod, notably because of the removal of solving for the effective length factor which is always equal to 1; intended to keep the Direct Analysis Method less prone to error than the Effec tive Length Method. The Direct Analysis considers real world second order effects on the structure, Pand P, to more accurately portray the res ponse of a structure to the design loading (AISC 2010). Direct Analysis starts with a mo re accurate model than previously required. The model is required to include initial imperfections, either directly in the model, or represented by Notional Loads. According the AISC Steel Construction Manual, Notional Loads are applied laterally to the structure to account for geometrical imperfections. Th e notional load is based on the gravity loads applied to the structure. The notiona l load prescribed by AISC is shown in Equation IV.1, below (AISC 2010), where N is the notional load, Y is the gravity load, and i represent the specific level being considered: Equation IV.1 AISC Notional Load. The notional load coefficient in Equation IV.1, above, is based on the maximum allowed outofplumbness allowe d by the AISC Specification of 1/500. In addition, a reduced axial a nd or flexural stiffness is to be considered, equal to
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47 approximately 80percent of the unreduced stiffness. The commentary for the AISC Specifications states that the redu ced stiffness is required for the following reasons: first, for frames where elastic stability governs the limit state, the reduction of elastic stiffne ss builds in a margin of safety into the design. Secondly, when a structure includes la rger columns the stiffness reduction accounts for the loss of stiffness that occu rs during high axial lo ads. The loss of stiffness is due to initial outofstraig htness of a compression member. As the axial load nears the critical load of th e member, additional deflection caused the addition load increased asymptotically where very little load increase causes significant deflection of the member. Since all members will have some outofstraightness, the redu ced stiffness is to account for increased deflections at high axial loads (Yura 2011). The steps for the Direct Analysis Method are as follows. First, the structure is modeled with all applicable load combinations, similar to any design process. After modeling of the structur e, imperfections or notional loads to represent imperfections are applied. The second order an alysis is then performed, including Pdelta effects, using the re duced stiffness described above. The members shall be designed with the fo rces and loads obtained by the direct analysis with no modifications require d for effective lengths (Newton and Ericksen, 2010). Chapter VI, following Ch apter V on the physical testing results, describes the use of the Direct Analysis Method with SAP2000, a finite element structural modeling progra m commercially available.
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48 CHAPTER V PHYSICAL TESTING Physical testing of bar joists took place in the University of Colorado Denver Civil Engineering laborator y from August 1, 2014 through August 5, 2014. The Appendix includes asbuilt drawi ngs of one of the bar joists, the University of Colorado Denver Â“strong floor Â” where the tests were conducted, and the test configurations. Bar Joists Bar joists are structural truss members that are typically used in structures to support roof loads becaus e they are economical, light weight, single span truss members (Moran and Jensen 2007). Because of the design intent of joists, they are typically fully braced, or have rela tively short unbraced lengths, along the top chord, effectively elimina ting the buckling limit state of the member. Although they are typically not expos ed to buckling, bar joists were selected for the buckling experiments for the slender qualities of the compression chord (top chord) and the prefabricate d truss configuration. The two bar joists tested were donated by Sigma Metals in Denver, Colorado. For testing and documentation, the joists were named Alpha and Beast. An asbuilt drawing of the bar joists is included in the Appendix. The bar joists were nominally 25feet long and 16inches deep. The joists consisted of a top chord of a 1 1/2inch by 1 1/2inch by 1/ 8inch double angle, a bottom chord of a 1 1/4inch by 1 1/4inch by 1/8inch double angle, and a 9/16inch diameter rod making the truss. No markings on the jo ist could be found to determine the joist
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49 classification, but by using the size (see Figure V.1, below) and weight it was determined that the joists were 16K2 joists weighing approximately 5.5 pounds per foot of length (Vulcraft 2007). Ba sed on the Steel Joist Institute 2005 Specifications, the Â“Kseries joists are open web, parallel chord, load carrying members suitable for the direct support of floors and roof decks in buildings, utilizing hotrolled or co ldformed steelÂ…Â” The bar joists are shown in Figure V.2, below. Figure V.1 Measuring Bar Joist to Determine Type.
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50 Figure V.2 25Foot Long 16K2 Bar Joists above the Strong Floor. Test Apparatus and Setup The bar joist testing apparatus wa s set up on the strong floor in the University of Colorado Denver Civil Engineering Lab located in the North Classroom building. The strong floor is capable of being used as a bearing surface for high compression loads as well as a surface for tension loads to be applied against with the addition of 1 3/4inch diameter steel rods. The rods can be threaded into the strong floor on 3foot centers in each direction. For more details on the layout of the strong floor an asbuilt drawing of the floor is included in the Appendix. The testing apparatus itself was set up in the middle of the strong floor so the point load applied to the bar joists coul d be applied to the center of the joists. The apparatus consisted of six strong floor rods, two approximately 6 feet long and four approximately 18 inches long. The 6 feet long rods were threaded
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51 directly into the floor to transfer the tens ile forces into the st rong floor. Nuts were then threaded onto the rod to support a beam spanning from the rods. Due to the thread pattern of the rods, a second beam had to be connected to the first beam, using the shorter rods, to have a beam low enough to load against to ensure stability of the apparatus and safety duri ng testing. Figure V.3, below, shows the testing apparatus. Figure V.3 Bar Joist Testing Apparatus. In addition to the testing apparatus, two 20 inch deep wide flange beams were used as abutments for the joists. The bar joists were idealized as either pinnedroller or pinnedpinned connectio ns. To make the pinned connection, holes were drilled into the bar joist bear ing flanges as well as the I beams and 3/8 inch diameter bolts were used to secure the joists to the abutments. The roller connection was emulated by using Cclamps on the sides of the bearing flange to restrain lateral movement but allow the jo ist to move along the abutment. Figures V.4 and V.5, below, show the pinned and roller connections, respectively.
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52 Figure V.4 Typical Pinned Connection. Figure V.5 Typical Roller Connection. A 20 kN load cell, placed on a steel b earing plate, was used along with a DATAQ data acquisition system to colle ct loading data. Loads were applied
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53 using an Enerpac hydraulic jack, which lo aded against the beam from the test apparatus to apply load to the load cell and, through the load cell, the joist. A half sphere was used to apply the load from th e jack to the beam to keep the jack stable on the joist as load was increased. The load was transmitted to the joist as a point load on the top chord, a point load on the bottom chord, or two equal point loads on the top chord, depending on whic h test was being conducted. The half sphere and load cell bearing plate are shown in Figure V.6, below. Figure V.6 Half Sphere and Load Cell Bearing Plate.
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54 Testing Eleven different buckling tests were completed on the two bar joists. Table V.1, below, is a summary of the tests that were conducted. Table V.1 Summary of Testing Load Conditions End Conditions Point Load Top Chord Point Load Bottom Chord 2 Point Loads Top Chord Pinned / Roller A/B/F A/B/F A/B Pinned / Pinned A (2)/B Note: A represents Alpha joist tests, B represents Beast joist tests, and F represents a frame test. Due to the number of tests being con ducted, the joists were unloaded prior to yielding or fully buckling. A 20 kN lo ad cell was used to record the applied load. Six dial gauges were used to meas ure lateral displacement of the top and bottom chord, one pair at the midpoint and both third points. Vertical deflection was measured using a ruler with a leveli ng square. Measurements were taken at each loading increment. Figures V.7 and V.8 show the dial gauge assemblies. Figure V.7 Dial Gauges Used to Measure Lateral Deflection.
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55 Figure V.8 Dial Gauge Stands During a Test. To account for twisting and the fact th at the dial gauges were not exactly at the top and bottom chords, dial gauge he ights and lateral deflections were used to determine the angle of twisting at the dial gauge. The angle of twist and the gauge heights were then used based on equilateral triangles to determine the lateral deflection at the top and bottom chor ds. Plates were clamped to the joists to provide a flat surface for the dial gauge s to measure against during the testing. The plates and the dial gauges were leveled vertically and horizontally, respectively, prior to every test to mi nimize errors. In addition, weights were placed on the base of the dial gauge stands to keep the stands from moving during the testing. At various time s during testing the dial gauge s had to be reset if they
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56 reached their measurement capacity, alt hough resets were only performed when required to reduce errors that might be caused by repositioning the dial gauges. The dial gauge measurement points along the joists were named points A, B, and C. A was located at approximately the 1/3 point of the joist, near the East end of the joist. During the pinnedroller tests this was closer to the roller connection. Dial gauge location B was in th e middle, or as close to the middle as allowable due to various testing setups, as possible. Dial gauge C was located at the 1/3 point of the joist nearest to the We st side of the joist. This was also nearest to the fixed end connection a butment. A drawing of the general measurement point layout is included in the Appendix. There were 5 vertical displacement measurement locations, numbered 1 through 5, equally spaced along the joist. Lo cation 1 was nearest the East side of the joist (typically the roller connected end), and Location 5 was nearest the West side of the joist (pinned connected e nd). Measurement points 2 through 4 were equally spaced between 1 and 5 in sequen tial order. A drawing of the general measurement point layout is included in the Appendix. Alpha Joist Testing Tests 1 through 5 were all conducted on Alpha Joist. Comparing the two bar joists, Alpha and Beast, Alpha a ppeared to have higher initial outofplumbness than Beast when unloaded. Th e initial outofplumbness of Alpha in the unloaded position at lateral measuring locations A through C is shown in Table V.2, below. The outofplumbness wa s measured using a string tied to the joist, shown in Figure V.9, below. Once the bar joist was loaded, it was
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57 positioned so that the initial load would stra ighten the joist for stability and safety during testing. Table V.2 Initial OutofPlumbness in Inches, of Alpha Joist when Unloaded. A B C Top Chord 0.0625 0.1250 0.0000 Bottom Chord 0.3125 0.2500 0.0000 Note: Negative values imply outofplu mbness deflection towards the center of the testing apparatus. Figure V.9 OutofPlumbness Measurement String.
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58 Test 1 Â– Alpha Joist. The first test was conducted on Alpha Joist with a point load applied above the middle web section on the top chord, similar to a typical three point test. Th is configuration placed the load approximately in the middle of the joint on the top chord and en sured that local buck ling or yielding of the top chord would not occur. The We st end of Alpha joist was set up as a pinned end connection, bolted to the abut ment. The opposite end, the East end, was set up as a roller end connection, rest rained laterally by C clamps but allowed to move along its main axis. Figure V.10 shows the test setup. A summary of the results is included in Table V.3. Figure V.10 Test 1 Configuration.
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59 Table V.3 Test 1 Â– Alpha Joist Single Point Load Top Chord PinnedRoller Results.
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60 The load was removed from Test 1 at 2.26 kips. The lateral deflection data was used to extrapolate a curve to determine an approximate buckling load. Figures V.11 and V.12 show the load versus deflection curves from the testing for the top and bottom chords, respectively, and Figure V.13 is the extrapolated curve based on the lateral deflection of the top chord at dial gauge C. Figure V.11 Test 1 Top Chord Deflection. Figure V.12 Test 1 Bottom Chord Deflection
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61 Figure V.13 Extrapolated Trendlines for Test 1. Based on observations from Test 1 a nd the trendlines shown in Figure V.13, the buckling load is estimated to be approximately 1.60 kips, although it should be noted that the curve is not as defined as later tests. Figure V.14 shows Alpha joist deflected during Test 1. Figure V.14 Deflected Shape of Alpha during Test 1. The deflected shape is shown looking from the point load towards the pinned connection.
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62 The deflected shape of the top chord appeared to have a node where the load was applied due to the jack and load cell, making an S shape about the point load, although the deflection on the roller end of the joist was less pronounced. The bottom chord deflected in a C shape, as opposed to the S shape that the top chord made. Figure V.15, below, shows th e rotation of the top chord, near the pinned end, during Test 1. Figure V.15 Rotation of Alpha Top Chord during Test 1 Test 2 Â– Alpha Joist. The second test was conducted on Alpha Joist with two point loads applied above web tru ss member on the top chord, distributed using a 5foot long channel shape and two bearing plat es. This configuration placed the load on the web and top chord c onnection to the right and left of the center web, symmetrical about the centerline of th e joist. The West end of Alpha joist was set up as a pinned end connec tion, bolted to the abutment, while the opposite end was set up as a roller end connection, restrain ed laterally by C clamps but allowed to move along its main axis. Figures V.16 and V.17 show the test setup. A summary of the re sults are included in Table V.4.
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63 Figure V.16 Test 2 Configuration. Figure V.17 Test 2 Configuration Showi ng the Channel and Bearing Plate.
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64 Table V.4 Test 2 Â– Alpha Joist Two Point Load Top Chord PinnedRoller Results.
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65 The load was removed from Test 2 at 2.89 Kips. The lateral deflection data was used to extrapolate a curve to determine an approximate buckling load. Figures V.18 and V.19 show the load versus deflection curves from the testing for the top and bottom chords, respectively, and Figure V.20 is the extrapolated curve based on the lateral deflection of the top chord at dial gauge C. Figure V.18 Test 2 Top Chord Deflection. Figure V.19 Test 2 Bottom Chord Deflection.
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66 Figure V.20 Extrapolated Trendlines for Test 2 Using the trendlines, the buckling load for Test 2 was estimated to be approximately 2.58 kips. Figures V.21 and V.22 shows Alpha joist deflected during Test 2. Figure V.21 Deflected S Shape of Alpha during Test 2. Looking from the pinned end toward the load.
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67 Figure V.22 Deflected S Shape of Alpha during Test 2. Looking from the roller end toward the lo ad. The deflection is vi sibly less on this side. Similar to Test 1, the deflected shape of the top chord appeared to have a node at the center of the joist, making an S shape about the channel. The bottom chord deflected in a C shape, as opposed to the S shape that the top chord made. The deflected shape appeared to be less on the roller side of the joist than on the pinned side, presumably because the roller connection could deflect along the axis of the joist. Tests 3 and 4 Â– Alpha Joist. The third and fourth tests were conducted on Alpha Joist with one point load centered on the joist; similar to Test 1, but the joist was pinned on both ends. The fourth test was conducted because buckling occurred suddenly during thir d test, causing the load cell and hydraulic jack to fall off of the joist. It was assumed that the load cell and jack slipped prior to buckling, although the load cell and jack sl ipped again during the fourth test. Both ends of Alpha joist were pinne d during Tests 3 and 4, although the pinned
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68 end on the East side of the joist, ty pically the roller end, was pinned using C clamps instead of bolts. Figure V.23 show s the modified pinned end connection. Summaries of the results for both tests are included in Tables V.5 and V.6. During each test, when loaded at approximately 1.8 kips a loud pop was audible during loading and deflection readings s howed relatively large deflections, but minimal deflections were visible in the jo ist. Figure V.24, below, shows the joist loaded at 2.12 kips during Test 3. Figure V.23 Test 3 and 4 Pinned End Configuration at Typical Roller Connected Abutment. Figure V.24 Test 3 Deflected Shape of Alpha Joist Loaded at 2.12 Kips. Minimal deflection was obser ved during Tests 3 and 4.
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69 Table V.5 Test 3 Â– Alpha Joist Point Load Top Chord PinnedPinned (1 of 2 Tests) Results. Table V.6 Test 4 Â– Alpha Joist Single Point Load Top Chord PinnedPinned (2 of 2 Tests) Results.
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70 The joist buckled during Test 3 when loaded past 2.33 kips. During Test 4 the joist buckled when loaded past 2.04 ki ps, slightly lower than Test 3. Figures V.25 and V.26 show the load versus deflec tion curves from Test 3 for the top and bottom chords, respectively, while Figures V.27 and V.28 show the load versus deflection curves from Test 4 for th e top and bottom chords, respectively. Figure V.25 Top Chord Deflection during Test 3. Figure V.26 Bottom Chord Deflection during Test 3.
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71 Figure V.27 Top Chord Deflection during Test 4. Figure V.28 Bottom Chord Deflection during Test 4. Unlike Tests 1 and 2, the deflected sh apes during Tests 3 and 4 were not that apparent. During both tests an audible pop was heard at approximately 1.8 kips of loading. Figures VI.25 through VI .28 show the increase in deflection per load correlating to the popping noise by the change in slope. Although the deflection increase was visible in the dial gauges during Tests 3 and 4, the joist did not appear to be significantly deflect ed during either test. Failure during both
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72 tests occurred during loading when the jack would become unstable and fall off of the joist. Figures V.29 and V.30 show the trendlines used to determine the buckling load for Tests 3 and 4, respectively. The es timated critical buckling load for Test 3 was 1.90 kips and the estimated critical buckling load for Test 4 was 1.62 kips. Figure V.29 Extrapolated Trendlines for Test 3. Figure V.30 Extrapolated Trendlines for Test 4.
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73 Test 5 Â– Alpha Joist. Test 5 was conducted on Al pha Joist with a single point load applied to the bottom chord al ong the centerline of th e joist. Test 5 was conducted because it appeared that when the jack applied the load to the top chord of the joist it was also acting as a brace. The West end of Alpha joist was set up as a pinned end connection, bolted to the abutment, while the opposite end was set up as a roller end connection, rest rained laterally by C clamps but allowed to move along its main axis. Figure V.31 shows the test setup. A summary of the results is included in Table V.7. Figure V.31 Test 5 Configuration.
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74 Table V.7 Test 5 Â– Alpha Joist Point Load Bottom Chord PinnedRoller Results.
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75 The load was removed from Test 5 at 1.30 kips, significantly less than Tests 1 through 4, all of which loaded the top chord. Unlike loading the top chord, loading the bottom chord showed si gnificant deflection after each loading. Figures V.32 and V.33 show the load versus deflection curves from the testing for the top and bottom chords, respectively. Th e extrapolated trendlines to estimate the critical buckling load is shown in Figure V.34. The estimated buckling load for Test 5 was 0.93 kips, but this may be low since the deflection doesnÂ’t appear to be reaching an asymptote. Figures V.35 and V.36 show Alpha joist deflected during Test 5. Figure V.32 Test 5 Top Chord Deflection.
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76 Figure V.33 Test 5 Bottom Chord Deflection. Figure V.34 Extrapolated Trendlines for Test 5.
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77 Figure V.35 Deflected C Shape of Alpha during Test 5. View looking from the roller end toward the load. Figure V.36 Deflected C Shape of Alpha during Test 5. View looking from the pinned end toward the load.
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78 Unlike the other Alpha joist tests, whic h all loaded the t op chord directly, the deflected shape of the top chord was a C shape without a node in the middle. The bottom chord also deflected in a C sh ape, but not as pronounced, possibly due to bracing caused by the loading. Without the additional support of the jack on the top chord, buckling appeared to occu r at a significantly smaller load than during Tests 1 through 4. Beast Joist Testing Tests 6 through 9 were all conducted on Beast Joist. The same tests that were conducted on Alpha Joist were conduc ted on Beast Joist. Comparing the two bar joists, Alpha and Beast, Alpha appeared to have higher initial outofplumbness than Beast when unloaded. Th e initial outofplumbness of Beast in the unloaded position at lateral measuring locations A through C is shown in Table V.8, below. The out of plumbness was measured using a string tied to the joist, shown for Alpha Joist in Figure V .9, above. Similar to Alpha, once the bar joist was loaded it was positioned so that th e initial load would straighten the joist for stability during testing. Table V.8 Initial OutofPlumbness, in Inches, of Beast Joist when Unloaded A B C Top Chord 0.0625 0.0625 0.0000 Bottom Chord 0.0625 0.0625 0.0000 Note: Negative values imply out of plumbness deflection towards the center of the testing apparatus. Test 6 Â– Beast Joist. Test 6 was conducted on B east Joist with a point load applied above the centered web sect ion on the top chord, similar to a typical three point test. This was the same setup as Test 1 on Alpha Joist. The West end
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79 of Beast joist was set up as a pinned end connection, bolted to the abutment, while the opposite end was set up as a roller e nd connection, restrained laterally by Cclamps but allowed to move along its ma in axis. Figure V.37 shows the test setup. A summary of the results fo r Test 6 are included in Table V.9. Figure V.37 Test 6 Configuration. Looking from the roller connection.
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80 Table V.9 Test 6 Â– Beast Joist Single Point Load Top Chord PinnedRoller Results.
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81 The load was removed from Test 6 at 2.63 Kips. The lateral deflection data was used to extrapolate a curve to estimate the critical buckling load. Figures V.38 and V.39 show the load versus deflection curves from the testing for the top and bottom chords, respectively, and Figure V.40 is the extrapolated curve based on the lateral deflection of the top chord at dial gauge A. Figure V.38 Test 6 Top Chord Deflection. Figure V.39 Test 6 Bottom Chord Deflection.
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82 Figure V.40 Extrapolated Trendlines for Test 6. The trendlines were used to estima te the buckling load for Test 6 at approximately 2.32 kips. Figure VI.41 shows Beast joist deflected during Test 6. Figure V.41 Deflected S Shape of Beast during Test 6. Looking from the pinned end toward the roller end.
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83 The deflected shape of the top chord a ppeared to have a node at the load, making an S shape about the point load. Th e bottom chord deflect ed in a C shape, as opposed to the S shape that the top chord made. Figure V.42, below, shows another picture of the deflected sh ape of Beast jois t during Test 6. Figure V.42 Deflected S shape of Beast during Test 6. Looking from the roller end towards the load. Test 7 Â– Beast Joist. Test 7 was conducted on Beas t joist with two point loads applied above web truss member on th e top chord, distribut ed using a 5foot
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84 long channel shape and two bearing plates. The West end of Beast joist was set up as a pinned end connection, bolted to the abutment, while the opposite end was set up as a roller end connec tion, restrained laterally by C clamps but allowed to move along its main axis. This was the same configuration as Test 2 on Alpha joist. Figures V.43 and V.44 show the defl ection of Beast joist during the testing. A summary of results for Test 7 is in Table V.10, below. Figure V.43 Test 7 Deflected S Shape of Beast Joist. Looking from the pinned end toward the roller end.
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85 Figure V.44 Test 7 Deflected S Shape of Beast Joist. Looking from the roller end toward the pinned end.
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86 Table V.10 Test 7 Â– Beast Joist Two Point Load Top Chord PinnedRoller Results.
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87 The load was removed from Beast joist during Test 7 at 2.84 Kips. The lateral deflection data was used to extrapolate trendlines to determine an approximate buckling load. Figures V.45 and V.46 show the load versus deflection curves from the testing for th e top and bottom chords, respectively, and Figure V.47 is the extrapolated curve ba sed on the lateral deflection of the top chord at dial gauge A. Figure V.45 Test 7 Top Chord Deflection. Figure V.46 Test 7 Bottom Chord Deflection.
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88 Figure V.47 Extrapolated Trendlines for Test 7 The Table Curve 2D program was used to estimate that the buckling load for Test 7 was approximately 2.53 kips. Similar to Tests 1 through 4 on Alpha joist and Test 6 on Beast joist, the deflect ed shape of the top chord appeared to have a node at the center of the joist, making an S shape about the channel. The bottom chord deflected in an S shape as well, but less pronounced, following the top chord of the truss. Test 8 Â– Beast Joist. Test 8 was conducted on Beas t Joist with one point load centered along the joist and both e nds pinned, similar to Tests 3 and 4 on Alpha Joist. The pinned e nd furthest from the loadi ng bay, typically the roller end, was pinned using C clamps instead of bolts. Figure V.48 below, shows the pinned end connection at the typical rolle r connected abutment. A summary of the results for Test 8 is included in Table V.11.
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89 Figure V.48 Test 8 Pinned End Config uration at Typical Roller Connected Abutment.
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90 Table V.11 Test 8 Â– Beast Joist Point Load Top Chord PinnedPinned Results.
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91 The joist didnÂ’t buckle rapidly like Tests 3 and 4 for Alpha Joist. Beast Joist was unloaded from Test 8 after 2.63 Kips. Figures V.49 and V.50 show the load versus deflection curves from Test 8 for the top and bottom chords, respectively. Figure V.49 Top Chord Deflection During Test 8. Figure V.50 Bottom Chord Deflection During Test 8.
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92 Figure V.51 Extrapolated Trendlines for Test 8. The trendlines from point A were used to estimate that the buckling load for Test 8 at approximately 2.28 kips. Th e extrapolated trendlines are shown in Figure V.51, above. Similar to the other te sts with the top chord loaded, the top chord deflected in an S shape. The bo ttom chord deflected in a C shape. The deflections are graphically shown in the deflection charts above. The loading point appeared to act as a restraint along the top chord, similar to other top chord loading tests. Figures V.52 and V.53 s how the deflected shape of Beast Joist during Test 8.
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93 Figure V.52 Deflected S Shape of Beast Joist Under a Load of 2.67 kips. Looking from the West side of the joist toward the load.
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94 Figure V.53 Deflected S Shape of Beast Joist Under a Load of 2.68 kips. Looking from the West side of the joist toward the load. Test 9 Â– Beast Joist. Test 9 was conducted on Beas t Joist with a single point load applied to the bottom chord al ong the centerline of th e joist. Test 9, similar to Test 5, was conducted because it appeared that when the load was applied to the top chord of the joist the lo ad cell and jack were acting as a brace. The West side of Beast Joist was set up as a pinned end conn ection, bolted to the abutment, while the opposite end was set up as a roller end connection, restrained laterally by Cclamps, but allowed to move along its main axis. Figure V.54 shows the test setup. A summary of th e results are included in Table V.12.
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95 Figure V.54 Test 9 Configuration.
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96 Table V.12 Test 9 Â– Beast Joist Point Load Bottom Chord PinnedRoller Results.
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97 The load was removed from Test 9 at 1.76 kips, significantly less than Tests 6 through 8 which loaded the top c hord of the Beast joist. The lateral deflection data was used to extrapolat e trendline curves to estimate an approximate buckling load. Figures V.55 and V.56 show the load versus deflection curves from the testing for th e top and bottom chords, respectively, and Figure V.57 shows the extrapolated curve based on the lateral deflection of the top chord at dial gauge B. Figure V.55 Test 9 Top Chord Deflection.
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98 Figure V.56 Test 9 Top Chord Deflection. Figure V.57 Extrapolated Trendlines for Test 9. The estimated buckling load for Test 9 was approximately 1.45 kips, significantly more than Test 5 on Alpha Jois t. The reverse of direction in Figure V.56 along Gauge B is assumed to be from initial straightening of the joist during
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99 testing before buckling of the top chord began controlling the deflection of the bottom chord. Figures V.58 and V.59 shows Beast Joist deflected during Test 9. Figure V.58 Deflected C Shape of Beast Joist during Test 9. Looking from the roller end toward the pinned end.
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100 Figure V.59 Deflected C Shape of Alpha during Test 9. The channel appears to be holding the bottom c hord in place during loading. The deflected shape of the top chord was a C shape without a node in the middle, similar to Test 5 on Alpha Jo ist. The bottom chord deflected also deflected in a C shape but not as pr onounced, possibly due to bracing caused by the channel used for loading, shown above in Figure V.59. Without the additional support of the load apparatu s on the top chord, buckling appeared to occur at a significantly smaller load th an during Tests 6 through 8.
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101 Comparison of Joists Alpha and Beast A summary of results from Tests 1 through 9 is included in Table VI.13 below. The summary presents the buck ling load based on ob servations and the table curve extrapolation data due to the nature of the testing. Table V.13 Summary of Tests 1 through 9. Test** Buckling Load (kip)* Percent Difference Alpha Joist Beast Joist 1 / 6 1.60 2.32 37 2 / 7 2.58 2.53 2 3 / 8 1.90 2.28 18 4 / 8 1.62 2.28 34 5 / 9 0.93 1.45 44 *Buckling Load based on observations and Table Curve Extrapolation Data ** Test 8 was used to compare to both Te st 3 and 4 since Test 3 and 4 were the same setup Based on the summary in Table VI.13, it appears that the initial outofplumbness, shown in Tables VI.2 and VI .8, significantly impacted the critical buckling load of joists; Beast Joist had a higher buckling load than Alpha Joist for all but the two point load test, which wa s only 2 percent lower than Alpha. In addition, although the pinnedpinned tests ( 3, 4 and 8) were thought to constrain the joists to force buckling, the buckling loads appear to be similar when loaded with a single point load on the top chord with a pinned roller configuration (Tests 1 and 6). This agrees with measurements of the roller end during the tests that indicated no movement along the roller c onnection. Another observation from the tests loading the top chord it that the loading apparatus appeared to slightly restrain the joists where the load wa s applied, creating an S buckling shape instead of the expected C buckling shape.
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102 After the initial tests on Alpha Joist the lower chord test was planned due to the apparent node where the load wa s applied on the top chord. The lower chord tests, Tests 5 and 9, buckled with a C shape, as anticipated. One concern though is the large variance in the tests between Alpha a nd Beast Joists; it appears that the outofplumbness significa ntly impacted the results. Joist Frame Testing The final two tests, Test 10 and Te st 11, were conducted on a frame built out of the two bar joists, Alpha and Beast. Four struts were used to complete the frame; two struts bolted to the top chords and two more struts bolted to the bottom chords. The struts were located at approxi mately third points along the joists. An asbuilt drawing of the joist frame is included in the Appendix. Due to the bracing straightening the joists, no si gnificant initial outofplumbness was observed. Test 10 Â– Frame. The first test conducted on the frame, Test 10, used a point load applied to the top chord of bot h joists. The load was distributed by the channel used during previous tests. The test configuration placed the load approximately in the middle of each joist on the top chord directly above the web, ensuring that local buckling or yielding of the top chord would not occur due to the web. The West ends of the joists were set up as pi nned end connections, similar to the single joist tests, with bo th joists bearing plates bolted to the abutment. The opposite ends of the joists were set up as a roller connection on the abutment, restrained laterally by Cclamps. Figures V.60 and V.61, below,
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103 show the test setup. A summary of the results are included in Table V.14. Asbuilt drawings of the frame and Test 10 are included in the Appendix. Figure V.60 Test 10 Configuration. Figure V.61 Strut Connection of Frame (Typical).
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104 Table V.14 Test 10 Â– Frame Single Point Load Top Chord PinnedRoller Results.
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105 The load was removed from Test 1 at 6.76 Kips with very little deflection or sign of buckling. The load was removed because the channel was starting to flex and it was apparent that the chan nel would have yielded prior to frame buckling. Figures V.62 and V.63 show the load versus deflection curves from the testing for the top and bottom chords, resp ectively. After Test 10, the apparatus was modified to apply the lo ad the lower chords so a smaller load could be used to induce buckling, based on observations fr om the previous tests. Figure V.64 shows the joist frame loaded at 6.76 kips prior to the load being removed. Figure V.62 Test 10 Top Chord Deflection.
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106 Figure V.63 Test 10 Bottom Chord Deflection. Figure V.64 Deflected Shape of the Joist Frame during Test 10. The frame loaded at 6.76 kips. Test 11 Â– Frame. The second test conducted on the frame was with a point load applied to the bottom chord of both joists, distributed by the channel
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107 used during previous tests. In addition, to ensure that the channel would not yield during testing, 2inch by 8inch wood memb ers were placed inside the channel to evenly distribute the load across the channe l. This configuration placed the load approximately in the middle of each joist. One concern with this setup was local yielding, although buckling was observed prio r to yielding of the bottom chord. Similar Test 10, the West ends of the jois ts were set up as pinned end connections with both joists bearing plates bolted to the abutment. The opposite ends of the joists were set up as roller end connecti ons on the abutment, restrained laterally by C clamps. Figure V.65 shows the test setup. A summary of the results are included in Table V.15. Asbuilt drawi ngs of the frame and of Test 11 are included in the Appendix. Figure V.65 Test 11 Configuration.
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108 Table V.15 Test 11 Â– Frame Single Point Load Bottom Chord PinnedRoller Results.
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109 The load was removed from Test 11 at approximately 6.67 Kips because the frame began buckling towards yiel ding. No measurements were taken because the frame never reached equilibri um prior to unloading and it appeared the joists would have sustained permanent deformation if the load was maintained. The max load reached was 6.74 kips before significant deflection reduced it to 6.64 kips. Based on the trend lines generated from the data, shown in Figure V. 68, below, the buckling load for the frame was estimated to be 5.85 kips. Figures V.66 and V.67 show the lo ad versus deflection curves from the testing for the top and bottom chords, respectively. Figures V.69 through V.75 show the joist frame deflected unde r various loads throughout Test 11. Figure V.66 Test 11 Top Chord Deflection.
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110 Figure V.67 Test 11 Bottom Chord Deflection. Figure V.68 Extrapolated trendlines for Test 11.
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111 Figure V.69 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at 5.00 kips. Photo taken at the first observable sign of buckling during Test 11. Beast joist shown. Figure V.70 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at 5.88 kips. Beast joist s hown looking from pinned end toward the roller end.
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112 Figure V.71 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at 5.88 kips. Alpha joist s hown looking from the pinned end towards the roller end. Figure V.72 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at the max load of 6.74 kips. Beast joist shown.
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113 Figure V.73 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at the final load of 6.65 kips. Alpha joist shown. Figure V.74 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at the final load of 6.65 kips. Beast joist shown.
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114 Figure V.75 Deflected Shape of the Joist Frame during Test 11. The frame was loaded at the final load of 6.65 kips Alpha joist is shown on the right and Beast joist is shown on the left looking toward the East end.
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115 CHAPTER VI DIRECT ANALYSIS MOD ELING USING SAP2000 SAP2000 SAP2000 is a 3D Finite Element struct ural analysis computer software that is commercially available (CSI 2014). SAP2000 was developed by Computers and Structures Inc. in Berkele y, California. According to the Version 16.0.2 SAP2000 Analysis Reference Manual, SAP2000 specializes in Â“general structures, including stadiums, towers, industrial plants, offshore structures, piping systems, buildings, dams, soils, machine parts and many others.Â” The extensive list provided by Computers and Structures Inc. demonstrates the wide range of applications of the SAP2000 so ftware. SAP2000 was selected for this research project because it is commercially available and the software is capable of second order, nonlinear Pdelta buck ling analysis, critical for use when comparing buckling analysis based on th e Direct Analysis Method and physical buckling test results. Two different SAP2000 analyses were conducted to determ ine the critical buckling load on the bar joist tests. The first analysis was conducted using SAP2000Â’s built in Buckling Analysis which determines the critical load for a specified number of buckling modes. The second analysis was conducted using SAP2000Â’s standard nonlinear static analys is. Both analyses are described in further detail below. SAP2000 Buckling Analysis. SAP2000 will determine the buckling load of a structure based on linear, first order an alysis or second order Pdelta analysis.
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116 The buckling analysis in SAP2000 solves for the buckling modes based on the solution of the generalized eigenva lue problem, Equation VI.1 below. Equation VI.1 Generali zed Eigenvalue Problem In Equation VI.1 K represents the stiffness matrix, G(r) is the Pdelta effect based on the load, r is the eigenvalue matrix, and is the eigenvector matrix (CSI 2014). Solving the equation, each eigenvalue and eigenvector pair correlates to a buckling mode of the structure. The size of the matrix is determined by the number of modes that are specified by the user. The eigenvalues, are the buckling factors co rresponding to each mode. The CSI Analysis Reference Manual descri bes the eigenvalues as buckling scale factors. To determine the buckling load, the scale factor needs to be applied to the load specified during the buckling load case. For each load case specified in the model, either the stiffness matrix of the unstressed structure or the stiffness matrix at the e nd of the nonlin ear (Pdelta) load case may be used. To include the Pdelta effects from a Pdelta analysis the stiffness matrix from the Pdelta load case must be used to apply the modified stiffness (CSI 2014). The results of the buckling analysis were used to compare to the physical testing, al though the buckling analysis results in a pure buckling failure which does not include vertical defl ections if they donÂ’t impact the specific mode being analyzed. Therefore, the t ypical SAP2000 Nonlinear Static Analysis was considered as well.
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117 SAP2000 Nonlinear Static Analysis. The nonlinear static analysis is recommended to analyze a structure with effects of material and geometric imperfections included. In SAP2000, material nonlinearities, geometric nonlinearities, and staged construction may all be modeled. The geometric and material nonlinearity capabilities were used for this thesis. The geometric nonlinearity options in the SAP2000 analysis are Pdelta effects or Pdelta with large displacement effects (CSI 2014). The Nonlinear Static Analysis applies the load gradually to capture nonlinear effects during the analysis. The N onlinear Static Analysis iterates from an initial load of zero to the specified applied load (CSI 2014). The nonlinear static analysis does not directly solve for the critical buckling load. Therefore, to solve for the buckling load an iterative process was used, slightly increasing the specified loads in the mode l until the model failed to c onverge on results. Failure to converge indicated a failure of the model, and in this case, a buckling failure. PDelta Analysis. Two of the nonlinear analysis methods include Pdelta effects; the Pdelta analysis and the Pde lta with large displacements analysis. When the Pdelta analysis option is select ed the transverse bending stiffness of all Frame elements in the model are modified during loading iterations to reflect the Pdelta effect. In the model, in the ca se of the SAP2000 buckling analysis, the Pdelta analysis is performed pr ior to all other analyses to ensure that the modified stiffness is included in the following analyses, including the buckling analysis (CSI 2014). The order of analyses is onl y critical for the buckling analysis; to include Pdelta effects in the SAP2000 Buckling Analysis the Pdelta analysis
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118 needs to be run first. To perform the Pdelta analysis, the following assumptions are made within the SAP2000 program base d on the Pdelta Analysis section of the SAP2000 Analysis Reference guide: Only the largestress effect of an axial force upon transverse bending and shear deformation is considered. All deflections, strains, and rota tions are assumed to be small. The transverse deflected shape of a Fr ame element is assumed to be cubic in bending and linear in shear betwee n the reduced rigid zone offsets. The Pdelta axial forces are assume d to be constant along the element length. Based on the assumptions above, SAP2000 forms equilibrium equations to determine the stresses and deflections of the structure. The equilibrium equations are developed in SAP2000 in two ways. The first way is based on element stiffness matrices that are combined to form the stiffness matrix for the structure. The second way is through stress versus displacement relationships to determine the internal element stresses for results (CSI 2014). Pdelta in SAP2000 is applied along th e length of each frame element to account for deflection within the element, a nd thus, the Pdelta effect. To do this, the transverse deflected shape is assumed to be cubic for bending and linear for shear effects. The assumed shape, al though not completely accurate, is a good approximation except near the buckling load. In that case, dividing frames into multiple frame elements is generally re quired for more accurate results (CSI 2014).
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119 PDelta with Large Displacements Analysis. The large displacements analysis option is only available with an advanced SAP2000 license. The large displacements addition to the Pdelta analysis considers large displacements in the equilibrium equations. This means that the effects caused by a change in position or orientation of a member, due to the di splacements that have occurred, will be accounted for in the analysis (CSI 2104). The PDelta and Large Displacements analysis was used for this study to in clude large displacement effects near buckling if necessary. Specifically for the large displacement option, the displacements and rotations are accounted for in the analysis of the model. Although the Large Displacements option was used in this thesis, using a Pdelta analysis with frames divided into more than one member during analysis is generally sufficient (CSI 2014). As noted in the SAP2000 Referen ce Manual, when a Pdelta load combination is used, an iterative process is required in the analysis to determine the Pdelta forces in the frame elements. A preliminary analysis is performed on the undeformed structure to estimate the axial forces. Then the equilibrium equations are regenerated and solved agai n with the axial forces considered for stiffness reduction. Additional iterations will be performed until the axial forces and deflections converge and minimal ch anges occur during iterations. Axial stiffness changes, due to significant defl ections, require the use of the PDelta with large displacement eff ects setting (CSI 2014).
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120 AISC Direct Analysis Method The complete requirements for the AI SC Direct Analysis Method are to perform a second order analysis, consider Pdelta effects, flexur al shear and axial deformations, geometric imperfections, and member stiffness reduction due to member imperfections. Incorporating th e AISC Direct Analysis Method into SAP2000 only takes a few more steps wh en developing the model. First the Direct Analysis Method speci fies a stiffness reduction for both axial and flexural stiffness. The axial stiffness is reduced to 0.8EA and the flexural stiffness is reduced to 0.8 bEI, where b is based on a ratio of the required axial compressive strength and the yield strength (AISC 2010). In SAP2000, setting the steel design analysis options to use the Direct An alysis Method, the stiffness reduction modifications will automatically be applie d to the model after the initial steel analysis check is completed. Otherwise, property modifications can be applied to frame elements manually (CSI 2014). To account for geometric imperfections, a nominal load is applied in the Direct Analysis Method if the imperfec tions are not directly modeled into the geometry (AISC 2014). In SAP2000, eith er option can be performed. When modeling the physical tests for this study, notional loads were used to represent the geometric imperfections. If notional loads are used in SAP2000, they can be added manually, as was done in the model, or they can be added automatically by SAP2000 to be a percentage of a specifie d load (CSI 2014). In the model for the testing the notional load was added manual to verify that the loads were placed at
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121 critical locations; if SAP2000 were used to place the notional load, the exact location of the notional load is not readily verified. SAP2000 Models PDelta Model Verification. The first step required when using a model that relies on second order an alysis is to verify that th e second order analysis is accurate. The AISC 2010 specifications commentary provides two benchmark problems to verify if a Pdelta second order analysis is being performed accurately; AISC provides the deflection and moments for various loads for the problem and requires that the computer pr ogram being used determine results that are within 5percent. The first problem is a cantilever column with only an axial load applied. The first problem anal yzes the accuracy of the program in determining only Peffects. The second problem is a similar cantilever beam loaded by an axial load with the addition of a lateral load at the top of the column. Benchmark problem 2 analyzes the progr ams ability to determine both Pdelta effects (AISC 2010). Only the benchmark problem 2 was used in the verification because it considers both Pdelta effects. Tables VI.1 and VI.2, below, show the results of the SAP2000 AISC benchmark problem 2 verification models, based on frame segments used (2 or more recommended in the SAP2000 Analysis Reference Manual), and the specified Pdelta for moments and deflections, respectively.
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122 Table VI.1 SAP2000 AISC Benchm ark Model Results (Moments) Model Axial Force (Kip) 0 100 150 200 AISC Results M b ase (kipin) 336 470 601 856 SAP2000 (1 Segment) M b ase (kipin) 336.61 471.33 603.19 857.76 % Difference 0.18% 0.28% 0.36% 0.21% SAP2000 (2 Segments) M b ase (kipin) 336.49 470.83 602.57 861.13 % Difference 0.15% 0.18% 0.26% 0.60% SAP2000 (3 Segments) M b ase (kipin) 336.48 470.80 601.58 861.27 % Difference 0.14% 0.17% 0.10% 0.62% Table VI.2 SAP2000 AISC Benchmar k Model Results (Deflections) Model Axial Force (Kip) 0 100 150 200 AISC Results ti p (in) 0.907 1.34 1.77 2.60 SAP2000 (1 Segment) ti p (in) 0.909 1.346 1.773 2.600 % Difference 0.24% 0.42% 0.19% 0.00% SAP2000 (2 Segments) ti p (in) 0.909 1.345 1.774 2.619 % Difference 0.19% 0.34% 0.23% 0.72% SAP2000 (3 Segments) ti p (in) 0.909 1.345 1.771 2.620 % Difference 0.18% 0.34% 0.06% 0.75% The SAP2000 models for AISC benc hmark problem 2 were both within the 5percent difference range specified by AISC (AISC 2010). Therefore, based on the benchmark results, SAP2000 is an acceptable program to use for second order Pdelta analysis models. Bar Joist Models. The bar joist models were developed based on the physical testing results to determine the model predicted critical buckling load and compare the results with the physica l tests for validation of the Direct Analysis Method in the 2010 AISC Specifica tions. The bar joist dimensions were imported into SAP2000 using the asbuilt drawing included in the Appendix. To determine the material properties of the bar joists, the Stee l Joist Institute 2005
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123 specifications were used (SJI 2003). The 2005 Steel Joist Institu te specifications state that the design of the chords of th e joists Â“shall be based on a yield strength of 50 ksi.Â” It is also noted in the sp ecifications that, Â“t he design of the web sections for KSeries Joists shall be base d on a yield strength of either 36 ksi or 50 ksi.Â” Therefore, in the SAP2000 model, th e A992 steel with a yield strength of 50 ksi was selected for both the double angl es making up the chords and the web. The information obtained from the Vulcraft manual (Vulcraft 2007) about the physical joist dimensions, discussed in th e testing chapter, above, and the Steel Joist Institute specifications (SJI 2003) were used in the assumptions made to develop the SAP2000 models to compare to the results of the physical testing. Figure VI.1 shows an isometric of a joist modeled in SAP2000. Figure VI.1 SAP2000 Joist Model. Model 1 Â– Single Point Load on the Top Chord. The first model developed was the model for Tests 1 and 6 which included one pinned end and
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124 one roller end for the supports as well as a single point load located in the middle of the top chord of the joist. The en ds were modeled similar to the test configuration; the pinned end was modeled to restrain translati on in the x, y, and z axes, as well as torsional rotation (along local axis 1) The roller configuration was modeled to restrain vert ical translation and lateral translation, but allowed the model to move along its main axis. In a ddition to the end restraints, a transverse restraint was added to the top chord wher e the load was applie d; otherwise, the model would not reach the appropriate buckling mode shown during testing. Figure VI.2 shows Model 1 in the undefo rmed shape with load locations and Figure VI.3 shows the buckled shape. Figure VI.2 SAP2000 Model 1 Setup.
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125 Figure VI.3 SAP2000 Model 1 Buck led Condition (SAP2000 Buckling Analysis). The buckling results from Model 1, bot h from the Buckling analysis and the Nonlinear Static analysis, are include d in Table VI.3, below. In addition to the model results, the results from Tests 1 and 6 are included for comparison as well. Table VI.3 Model 1 Buckling Results and Comparison Model/ Test Critical Buckling Load Percent Difference Test 1 Percent Difference Test 6 Buckling 2.88 80 24 Nonlinear Static 2.82 76 21 Test 1 1.60 NA 44 Test 6 2.32 44 NA Both the SAP2000 model and the laborato ry tests show the S shape for the first buckling mode encount ered, but the both SAP2000 buckling loads are greater than what was observed in Tests 1 and 6. The buckling load for Test 1 was assumed to be low but that shouldnÂ’t account for a percent difference of 80
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126 percent. Comparing to Test 6, the 21 to 24 percent difference seems much more reasonable than 76 to 80 percent. Model 2 Â– Two Point Loads on the Top Chord. Model 2 was developed to represent Tests 2 and 7 which include d one pinned end and one roller end for the supports as well as symmetrical point loads located on the first web connection to the top chord on either side of the centerline of the joist. The ends were modeled similar to the test confi guration; the pinned end was modeled to restrain translation in the x, y, and z axes as well as torsional rotation (along local axis 1) The roller config uration was modeled to restra in vertical translation and lateral translation, bu t allowed the model to move along its main axis. In addition to the end restraints, a transverse rest raint was added to the top chord in the middle of the joist. Although the load was applied to the web connection on either side of the joist, during the testing the joist still appeared to form a node in the middle and not at each point load. Figure VI.4 shows Model 2 in with load locations in the undeformed shape and Fi gure VI.5 shows the buckled shape. It should be noted that the geometry and re straints of Model 2 were identical to those of Model 1, the only differences between the two models is the load assignments.
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127 Figure VI.4 SAP2000 Model 2 Configuration. Figure VI.5 SAP2000 Model 2 Buck led Condition (SAP2000 Buckling Analysis). The buckling results from Model 2, bot h from the Buckling analysis and the Nonlinear Static analysis, are include d in Table VI.4, below. In addition to the model results, the results from Tests 2 and 7 are included for comparison as well.
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128 Table VI.4 Model 2 Buckling Results and Comparison. Model/ Test Critical Buckling Load Percent Difference Test 2 Percent Difference Test 7 Buckling 2.72 5 8 Nonlinear Static 2.65 3 5 Test 2 2.58 NA 2 Test 7 2.53 2 NA One observation made during the mode ling was that the two point load model resulted in a lower buckling load th an the single point load model, opposite to what was observed in the testing. Th e reason for this is unknown except it is believed that the lateral restraint provided by the load in Model 1 also relieved the test of some second order deflection. The percent difference between the test s and the models were all within 8 percent, while the SAP2000 Nonlinear Static analysis estimated the buckling load within 3 percent of Test 2 a nd 5 percent of Test 7. Anot her observation is that the tests themselves buckled within 2 percen t of each other. The larger testing apparatus, spaced along the chord of the joists, may have provided enough support to the joists to keep them straighter during the tests, therefore, with both joists in a more stable (straighter) cond ition the tests would be more likely to be replicated between the two jo ists. In addition to the tests, the idealized model would also more accurately represent the tests. These observations are based on the low percent differences between the tests and the SAP2000 models, as well as when comparing the tests themselves. Model 3 Â– Point Load on the Top Chord with Pinned Ends. Model 3 was developed to represent Tests 3, 4 and 8 which pinned both ends of the joists to the abutments during testing. In add ition, similar to Model 1, a point load was
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129 placed at the centerline of the top chord of the joist. The ends were modeled similar to the test config uration; both pinned ends we re modeled to restrain translation in the x, y, and z axes. Unlike the other models which fixed the ends for torsional rotation (along local axis 1), this was not done for the pinned end model. When both ends were fixed, th e model buckled at unrealistically low loads. In addition to the e nd restraints, a transverse re straint was added to the top chord in the middle of the joist where th e load was applied, similar to Model 1. Figure VI.6 shows Model 3 in the undeformed shape with load locations. Figures VI.7 and VI.8 show the buckled shape for the Buckling and Nonlinear Static Analyses, respectively. Figure VI.6 SAP2000 Model 3 Configuration.
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130 Figure VI.7 SAP2000 Model 3 Buck led Condition Mode 1 (SAP2000 Buckling Analysis). Figure VI.8 SAP2000 Model 3 Failed Co ndition (SAP2000 Nonlinear Static Analysis). The buckled shape of the SAP2000 Nonlinear Static model is a C shape, similar to what was observed during Test 3 and 4. The Buckling analysis buckled shape shows the bottom chord deflecting with little movement of the top chord. The Buckling analysis showed the S sh ape for the second buckling mode under a vertical point load, alth ough the corresponding buckling load was unrealistically high at 7.64 kips. The buckling results from Model 3, both from the Buckling
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131 analysis and the Nonlinear Static analysis are included in Table VI.5, below. In addition to the model results, the results from Tests 3, 4 and 8 are included for comparison as well. Table VI.5 Model 3 Buckling Results and Comparison. Model/ Test Critical Buckling Load Percent Difference Test 3/4 Percent Difference Test 8 Buckling (Mode 1) 2.43 38 7 Nonlinear Static 1.67 5 27 Test 3/4** 1.76 NA 26 Test 8 2.28 26 NA ** Tests 3 and 4 were averaged. Model 3 was the only model that had significantly different results when comparing the Buckling Analysis and the N onlinear Static Analysis. Besides the critical load, the Nonlinear Static Analys is failed shape was a C shape, while the Buckling Analysis showed the bottom chor d deflecting out. Interestingly, the Nonlinear Static analysis, which formed the same C shape as Tests 3 and 4, was within 5 percent of the buckling mode, wh ile the Buckling analysis was within 7 percent of the buckling load for Test 8, although the shape was not replicated. This could represent that a different fa ilure mode was occurring during Tests 3 and 4, which would be captured in the Non linear Static analysis, but would not be captured during the Buckling Analysis. Likewise, the Buckling analysis, which only considers buckling failure modes, pred icted a similar buckling load to Test 8, although the buckling shap e is different. Model 4 Â– Single Point Load on the Bottom Chord. The fourth model developed was the model for Tests 5 and 9, th e last of the single joist tests. The model, similar to Models 1 and 2, was modeled using one pinned end and one
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132 roller end for the supports as well as a single point load located in the middle of the bottom chord of the joist. The pinned end was modeled to restrain translation in the x, y, and z axes, as well as torsional rotati on (along local axis 1). The roller configuration was modeled to restrain vert ical translation and lateral translation, but allowed the model to move along its main axis. Unlike the other models, no lateral restraint was provid ed along the span of the joist. Figure VI.9 shows Model 4 in the undeformed shape with load locations and Figure VI.10 shows the buckled shape. Figure VI.9 SAP2000 Model 4 Configuration.
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133 Figure VI.10 SAP2000 Model 4 Buckled Condition (SAP2000 Buckling Analysis). The buckling results from Model 4, bot h from the Buckling analysis and the Nonlinear Static analysis, are include d in Table VI.6, below. In addition to the model results, the results from Tests 5 and 9 are included for comparison as well. Table VI.6 Model 4 Buckling Results and Comparison. Model/ Test Critical Buckling Load Percent Difference Test 5 Percent Difference Test 9 Buckling 1.48 59 2 Nonlinear Static 1.47 58 1 Test 5 0.93 NA 44 Test 9 1.45 44 NA The Buckling analysis and the Nonlinear Static analysis were within 1 percent difference, again showing th e correlation between the two SAP2000 analyses. In addition, the Buckling anal ysis was 2 percent greater than the buckling load of Test 9 and the Nonlinear Static analysis wa s 1 percent greater than Test 9. Both analyses were approxi mately 60 percent greater than Test 5.
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134 The large initial outofplumbness is believe d to be a factor in the significant difference in buckling loads. Model 5 Â– Frame Loaded on the Top Chord. Model 5 was developed for the first frame test, Test 10, which loaded the frame on the top chord. Although the frame did not buckle during test ing, the model was created to verify that SAP2000 would predict similar results The model included one pinned end and one roller end for the supports, as well as a single point lo ad located in the middle of the top chord of both joists ma king the frame. The ends were modeled similar to the single bar joist model c onfigurations; the pinned end was modeled to restrain translation in the x, y, and z axes, as well as torsional rotation (along local axis 1). The roller connections were modeled to restrain vertical translation and lateral translation, but al lowed the model to move along its main axis. Struts were added to the model in the approximate location that they were included in the tests. The struts were modeled with moment releases so they would not transfer moment since they were only single bolt connections during the testing. In addition to the end restraints and struts a transverse restraint was added to the top chord where the load was applied ba sed on the observations made during the testing. Figure VI.11 shows Model 5 in the undeformed shape and Figure VI.12 shows the buckled shape of Model 5.
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135 Figure VI.11 SAP2000 Model 5 Configuration. Figure VI.12 SAP2000 Model 5 Buckled Condition (SAP2000 Buckling Analysis). The SAP2000 model predicted that th e model would buckle at 9.01 kips. Although a critical buckling lo ad was not estimated duri ng the testing this shows that SAP2000 agrees that the load wa s too low to buckle the frame. The maximum load achieved duri ng testing was 6.79 kips. Model 6 Â– Frame Loaded on the Bottom Chord. Model 6 was developed for the second frame test, Test 11, which loaded the frame on the bottom chord. The model was the same as Model 5, except the load was placed on the bottom chord and the strut repres enting the channel was removed. The model included one pinned end and one ro ller end for the supports. The pinned end was modeled to restrain translation in the x, y, and z axes, as well as torsional
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136 rotation (along local axis 1). The roller connections were modeled to restrain vertical translation and lateral translati on, but allowed the model to move along its main axis. Struts were added to the m odel in the approximate location that they were included in the tests. The struts we re modeled with moment releases so they would not transfer moment since they were connected by only single bolt connections during the tes ting. Figure VI.13 shows Model 6 in the undeformed shape and Figure VI.14 shows th e buckled shape of Model 6. Figure VI.13 SAP2000 Model 6 Configuration.
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137 Figure VI.14 SAP2000 Model 6 Buckled Condition (SAP2000 Buckling Analysis). The buckling results from Model 6, bot h from the Buckling analysis and the Nonlinear Static analysis, are include d in Table VI.7, below. In addition to the model results, the results from Tests 11 are included for comparison. Table VI.7 Model 6 Buckling Results and Comparison. Model/ Test Critical Buckling Load Percent Difference Test 11 Buckling 5.88 <1 Nonlinear Static 6.10 4 Test 11 5.85 NA The Buckling analysis and the Nonlinear Static analysis were within 4 percent difference, showing the correla tion between the two SAP2000 analyses. The Buckling analysis was less than 1 per cent greater than th e buckling load of Test 11 and the Nonlinear Static analys is was 4 percent gr eater than Test 11.
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138 CHAPTER VII CONCLUSIONS AND RECCOMENDATIONS Conclusions The test results and the model results do appear to validate the use of the Direct Analysis Method for predicting the critical buckling load, although not without some caveats. First, a signifi cant difference was seen in most of the single joist testing results when compari ng Alpha and Beast Jo ist, and typically Alpha Joist was lower. This is likely due to the initial outofplumbness which was much greater for Alpha than B east; compounding the second order effects significantly lower the buckling load. The SAP2000 analysis closely matche d the Beast Joist buckling loads for Models 2 and 4 while the frame model, Model 6, was within 5 percent of the buckling load from Test 11. Model 1 ha d a significantly higher buckling load than Tests 1 and 6. It is po ssible that the single point load on the top chord started pushing the top chord when a little move ment at Gauge B occurred, accelerating buckling of the joist. Model 3 predic ted a buckling load (Buckling analysis) similar to Beast Joist, although the predic ted shape was incorrect. In addition, the Nonlinear Static Analysis predicted a much smaller load to failure, which matched Alpha Joists tests, although this may not have been strictly a buckling failure. The discrepancies in Model 3 be tween the buckled shape of the Buckling versus Nonlinear Static analysis provide a sense of uncertainty with the pinnedpinned model. Finally, the frame model co rrelated very closely with the frame test. These results also support the hypot hesis that the outofplumbness of the
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139 joists significantly impact the comparis ons to the model, especially since the model was modeled straight, although a notional load was added to account for imperfections within standard tolerances. Although the testing and m odels appear to validate the Direct Analysis Method, there were also numer ous possible sources of erro r in the test. First, a more accurate deflection measurement that does not rely on verti cally static dial gauges and angles to get the final deflect ion should be used. In addition, longer dial gauges would have been helpful so that they would not need to be reset during the testing. Another possible sour ce of error was the way the load was applied. The jack loading the top of th e chord was maintained as straight as possible, but there was likely some late ral resultant force, leading to a lower buckling load by adding an unintended a nd unmeasurable lateral load. Another observation from the tests was that the loading apparatus was observed to be providing a restraint where the load was applied. This was unexpected, but it certainly makes sense after seeing the test s. If the load was not acting as a restraint, the load would have sli pped off of the joist every test. Recommendations for Future Studies As mentioned in the introduc tory chapters of this thesis, continued studies on stability and buckling are critical to the continued development of structural engineering. The testing completed for this thesis could be m odified to be more accurate in various ways. First, the deflection measurement may be able to be taken in a way to reduce the possible errors discussed above. Additionally, a different testing apparatus, one that provides a more vertical load or less restraint
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140 would reduce error. An example of this could be a cable used to pull down on the top or bottom chord instead of a jack that us es the chord it is loading for stability. Another testing improvement that could be included in a future study is to measure the outofplumbness prior to each te st (when the initial load is applied) and model that in lieu of the notional load. Finally, models on more complex frames or structures would be signifi cant to advancements in buckling theory, especially since the Direct Analysis Method is intende d for complete structures, without restriction.
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141 REFERENCES 2007. Vulcraft Steel Joists and Girders. edited by Vulcraft: Nucor Group. Alpsten, Goran. 1968. Thermal Residual Stresses in HotRolled Steel Members. In Fritz Laboratory Report Bethlehem, Pennsylvani a: Lehigh University. American Institute of Steel Construction. 2005. Steel construction manual 13th ed. 1 vols. Chicago, Ill.: American Institute of Steel Construction. American Institute of Steel Construction. 2011. Steel construction manual 14th ed. 1 vols. Chicago, Ill.: American Institute of Steel Construction. Bansal, R.K. 2010. A Textbook of Strength of Materials : Laxmi Publications. Bjorhovde, Reidar, Jacques Brozzetti, Goran Alpsten, and Lambert Tall. 1971. Residual Stresses in Thick Welded Plates, June 1971 (7226). In Fritz Laboratory Reports Bethlehem, Pennsylvania, USA: Lehigh University. Budiansky, B., and Harvard University. Division of Engineering and Applied Physics. 1973. Theory of Buckling and Postbuckling Behavior of Elastic Structures : Division of Engineering and Applied Physics, Harvard University. Byskov, Esben. 2013. Elementary continuum mec hanics for everyone : with applications to structural mechanics Dordrecht: Springer. Chen, WaiFah, and Toshio Atsuta. 2008. Theory of beamcolumns 2 vols, J Ross Publishing classics Ft. Lauderdale, FL: J. Ross Pub. Chen, WaiFah, and E. M. Lui. 1991. Stability design of steel frames New directions in civil engineering Boca Raton: CRC Press. Chen, WaiFah, and E. M. Lui. 2005. Handbook of structural engineering 2nd ed. 1 vols. Boca Raton, Fla.: CRC Press. Computers and Structures, Inc. 2014. CSI Analysis Reference Manual. In For SAP2000, ETABS, SAFE and CSiBridge Berkeley, California: Computers and Structures, Inc. Craig, Roy R. 2000. Mechanics of materials 2nd ed. New York: Wiley. Fernandez, P. 2013. Practical Methods for Critical Load Determination and Stability Evaluation of Steel Structures : University of Colorado Denver.
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142 Fraser, Craig G. 1991. "Mathematical Tec hnique and Physical Conception in Euler's Investigation of the Elastica." Centaurus 34 (3):211246. doi: 10.1111/j.16000498.1991.tb00695.x. Gere, James M., Goodno, Barry J. 2012. Mechanics of materials 8e Ed. ed. Mason, OH: Cengage Learning. Geschwindner, Louis F. 2012. Unified design of steel structures 2nd ed. Hoboken, NJ: Wiley. Hodges, Dewey, and David Peters. 1975. "On the Lateral Buckling of Uniform Slender Cantilever Beams." International Journal of Solids and Structures 11 (April):12. Institute, Steel Joist. 2003. American National Standard SJIJG1.1. Fox River Grove, IL: Steel Joist Institute. Johnston, Bruce. 1961. Behavior of an Inelastic buckling Model Between the Tangent Modulus and Shanley Loads. The University of Michigan: Industry Program of the College Engineering. Kavanagh, Thomas. 1962. "Effective Length of Framed Columns." Transactions of the American Society of Civil Engineers 127 (2):21. Length, Task Committee on Effective. 1997. Effective Length and Notional Load Approaches for Assessing Frame Stab ility: Implications for American Steel Design Edited by ASCE: American So ciety of Civil Engineers. Mahfouz, S. Y. 1999. "Design optimization of structural steelwork [electronic resource]." University of Bradford. Moran, P.J., and Eric Jens en. 2007. "Joist Causse." Modern Steel Construction 3. Nair, Shankar. 2007. "Stability Analys is and the 2005 AISC Specification." Modern Steel Construction Newton, Matthew, and Jason Ericksen. 2010. The Direct Analysis Method Made Simple. edited by CSC Inc.: CSC Inc. Salmon, Charles G., John Edwin Johnson, and Faris Amin Malhas. 2009. Steel structures : design and behavior : em phasizing load and resistance factor design 5th ed. Upper Saddle River, NJ: Pearson/Prentice Hall. Shanley, F. R. 1946. "The Column Paradox." Journal of the Aeronautical Sciences (Institute of the Aeronautical Sciences) 13 (12):678678. doi: 10.2514/8.11478.
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143 Shanley, F. R. 1950. "Applied Column Theory." ASCE Transactions 115:54. Singer, J., Johann Arbocz, and T. Weller. 1998. Buckling experiments : experimental methods in buckli ng of thinwalled structures 2 vols. Chichester ; New York: Wiley. Singer, Josef, Johann Arbocz, and Tanchum Weller. 2002. Buckling Experiments: Experimental Methods in Buck ling of ThinWalled Structures 2 vols. Vol. 2. New York, NY: John Wiley and Sons, Inc. Structural Engineering Institute. Technical Committee on Load and Resistance Factor Design. Task Committee on Effective Length. 1997. Effective length and notional load approaches fo r assessing frame stability : implications for American steel design New York: The Society. Timoshenko, Stephen. 1953. History of strength of mate rials, with a brief account of the history of theory of el asticity and theory of structures New York,: McGrawHill. Timoshenko, Stephen. 1961. Theory of elastic stability 2d ed, Engineering societies monographs New York,: McGrawHill. Truesdell, C. 1960. "Summary: EulerÂ’s heritage." In The Rational Mechanics of Flexible or Elastic Bodies 1638Â–1788 136141. Birkhuser Basel. Vable, Madhukar. 2002. Mechanics of materials New York: Oxford University Press. Weng, C. C., and Teoman Pekoz. 1988. "Res idual Stresses in ColdFormed Steel Members." Ninth International Spec ialty Conference on ColdFormed Steel Structures, St. Louse, Missouri, USA. White, Donald, Andrea Surovek, Bulent Alemdar, ChingJen Chang, Yoon Duk Kim, and Garret Huckenbecker. 2006. "S tability Analysis and Design of Steel Building Frames Using th e 2005 AISC Specification." Steel Structures 91 (6):21. Yura, Joseph A. 2011. Five Useful Stability Concepts. The American Institute of Steel Construction. Ziemian, Ronald D. 2010. Guide to stability design cr iteria for metal structures 6th ed. Hoboken, N.J.: John Wiley & Sons.
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