UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Finite element modeling and parametric studies of a chevron braced frame with a vertical slotted connection… Seeton, Andrew 2005

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2005-0309.pdf [ 37.99MB ]
Metadata
JSON: 831-1.0063337.json
JSON-LD: 831-1.0063337-ld.json
RDF/XML (Pretty): 831-1.0063337-rdf.xml
RDF/JSON: 831-1.0063337-rdf.json
Turtle: 831-1.0063337-turtle.txt
N-Triples: 831-1.0063337-rdf-ntriples.txt
Original Record: 831-1.0063337-source.json
Full Text
831-1.0063337-fulltext.txt
Citation
831-1.0063337.ris

Full Text

F I N I T E E L E M E N T M O D E L I N G A N D P A R A M E T R I C S T U D I E S O F A C H E V R O N B R A C E D F R A M E W I T H A V E R T I C A L S L O T T E D C O N N E C T I O N D E T A I L b y A N D R E W S E E T O N B . A . S c , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 2002 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( C i v i l Engineer ing) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 2005 © Andrew Curtis Seeton, 2005 ABSTRACT This thesis presents the f indings o f comprehensive parametric sensi t ivi ty studies o f a series o f three-dimensional finite element models . The models considered relate to a steel chevron braced frame system w i t h an innovat ive ver t ical slotted connect ion ( V S C ) detail that idea l ly improves the seismic performance o f the system b y prec lud ing the appl ica t ion o f an unbalanced load to the mid-span o f the beam upon b u c k l i n g o f either brace. S ince s impl i f i ed models o f a prototype specimen o f the system were unsuccessful i n predic t ing the exper imental ly observed behaviour, a sophisticated three-dimensional m o d e l i n g invest igat ion was undertaken. The p r imary lateral load resistance and energy diss ipat ion qualit ies o f the chevron braced frame system are p rov ided b y the brac ing elements. Therefore, these members constituted the in i t ia l focus o f the mode l ing investigation. Stub c o l u m n simulat ions were undertaken to determine the sensi t ivi ty o f the predicted loca l b u c k l i n g behaviour to an extensive set o f model ing-rela ted parameters. N e x t , a single brace mode l was developed, to investigate the sensi t ivi ty o f g loba l b u c k l i n g predictions to the mode l ing parameters. The results f rom the parametric studies o f the stub c o l u m n and single brace models enabled a we l l - i n fo rmed approach to the m o d e l i n g o f the ful l chevron braced frame system w i t h the V S C detail , w h i c h i t se l f was also subjected to parametric study. Re f ined three-dimensional m o d e l i n g o f the V S C detail i nc lud ing material , geometric, and contact nonlineari t ies enabled the k inemat ic behaviour o f the system to be carefully scrutinized. The general purpose finite element program L S - D Y N A was used to conduct the invest igation. F i v e analysis methods were used, inc lud ing : eigenvalue v ib ra t ion analysis, eigenvalue b u c k l i n g analysis, imp l i c i t static analysis, i m p l i c i t dynamic analysis , and expl ic i t dynamic analysis. The capabil i t ies o f the analysis methods were tested, and part icular focus was p laced on the use o f the expl ic i t method. W h i l e s tabi l i ty considerations l i m i t the m a x i m u m a l lowable t ime step size for exp l ic i t t ime stepping methods, it was determined that quasi-static solutions c o u l d be obtained, so long as a sensi t ivi ty study was paired w i t h the analysis i n order to quantify the extent to w h i c h t ime scal ing procedures affect the results. A l t h o u g h an exact match between the experimental and analyt ical results was u l t imate ly not achieved, the finite element mode l was able to capture the k e y characteristics o f the lateral load ing response o f the proposed chevron braced frame system w i t h a V S C detai l . TABLE OF CONTENTS Abstract ii Table of Contents Hi List of Tables vii List of Figures viii Acknowledgements xiii 1 Introduction 1 1.1 O v e r v i e w 1 1.2 Procedures and Resul ts o f Exper imenta l Project 4 1.2.1 M a t e r i a l Tes t ing 4 1.2.2 V S C - C B F Sys tem Test P ro toco l 5 1.2.3 V S C - C B F Sys tem Performance 5 1.2.4 M a c r o M o d e l i n g o f the V S C - C B F Sys tem 6 1.3 Software and Hardware Specif icat ions for the Present Research 7 1.4 Object ives o f Research 8 1.5 Scope o f Research 9 2 Literature Review 10 2.1 Introduction.. . . . 10 2.2 F in i te E lement S imula t ion o f bol ted connections 10 2.3 U s e o f the L S - D Y N A C o d e for A n a l y s i s o f Structural and Earthquake Eng inee r ing Simula t ions 13 3 Theoretical Background Relating to the Finite Element Procedures 16 3.1 O v e r v i e w '. 16 3.2 Di rec t T i m e Integration A l g o r i t h m s 16 3.2.1 E x p l i c i t Me thods 16 3.2.2 Impl i c i t Me thods 21 3.2.3 C o m p a r i s o n o f E x p l i c i t and Impl ic i t M e t h o d s 23 3.3 U s e o f E x p l i c i t Me thods to So lve Quasi-Stat ic P rob lems 25 3.3.1 T i m e Sca l ing 25 3.3.2 M a s s S c a l i n g 25 3.3.3 Smoo th L o a d i n g 26 3.3.4 Asses s ing the Extent o f Inertial Effects i n the A n a l y s i s 26 i i i 3.4 U s e o f Under-Integrated Elements 27 3.5 Fa i lu re cri ter ia 30 4 Modeling, Part One: Stub Column Model 32 4.1 Introduct ion 32 4.2 Stub C o l u m n Compres s ion Tests 32 4.3 Desc r ip t ion o f the Base l ine M o d e l 35 4.3.1 Spat ia l Discre t iza t ion 35 4.3.2 E lement Character izat ion 36 4.3.3 B o u n d a r y Condi t ions 37 4.3.4 M a t e r i a l M o d e l 37 4.4 Ana ly se s U s i n g the Base l ine M o d e l 41 4.4.1 E x p l i c i t Di rec t Integration A n a l y s i s 41 4.4.2 Imp l i c i t D i rec t Integration Ana ly se s 43 4.4.3 Impl ic i t E igenva lue A n a l y s i s 44 4.4.4 Impl i c i t B u c k l i n g A n a l y s i s 44 4.4.5 N u m e r i c a l P r e c i s i o n 44 4.5 A n a l y s i s Resul ts for Base l ine M o d e l 45 4.5.1 Resul ts o f Di rec t Integration Ana lyses 45 4.5.2 Resul ts o f E igenva lue A n a l y s i s 50 4.5.3 Resul ts o f B u c k l i n g A n a l y s i s 52 4.6 Parametric Studies 53 4.6.1 Effect o f V a r y i n g the Spat ia l Discre t iza t ion o f the M o d e l ( Impl ic i t Solver ) 54 4.6.2 Effect o f V a r y i n g the Element Character izat ion Parameters ( Impl ic i t Solver) 55 4.6.3 Effect o f V a r y i n g the Hourglass C o n t r o l T y p e and Settings ( Impl ic i t Solver) 57 4.6.4 Effect o f V a r y i n g the M a t e r i a l M o d e l ( Impl ic i t and E x p l i c i t Solver) 59 4.6.5 Effect o f Inc lud ing Geomet r ic Imperfections ( Impl ic i t and E x p l i c i t Solver) 63 4.6.6 Effect o f V a r y i n g the A m o u n t o f D a m p i n g ( E x p l i c i t Solver) . . . . 67 4.6.7 Effect o f V a r y i n g the Prescr ibed M o t i o n C u r v e ( E x p l i c i t Solver) 69 4.6.8 Effect o f U s i n g Contact Bounda ry Condi t ions ( E x p l i c i t Solver) 73 4.6.9 Effect o f V a r y i n g the N u m e r i c a l P rec i s ion ( Impl ic i t and E x p l i c i t Solver) . . .76 4.7 Conc lus ions from the study on Stub C o l u m n M o d e l s 78 5 Modeling, Part Two: Single Brace 81 5.1 Introduct ion 81 5.2 Est imate o f Compres s ion Capac i ty 81 5.3 Desc r ip t ion o f S ing le Brace M o d e l 83 5.3.1 Brace M e m b e r 83 5.3.2 Gusset Plates , 85 5.3.3 Treatment o f welds 88 5.3.4 B o u n d a r y Cond i t ions 89 5.4 E igenva lue and B u c k l i n g Ana lyses 90 5.5 Load-Disp lacemen t Response Ana lyses 93 5.5.1 Effect o f N u m b e r o f Elements T h r o u g h the Th ickness o f the B o t t o m Gusset Plate 96 5.5.2 Effect o f Gusset Plate E las t ic M o d u l u s V a l u e 97 5.5.3 Effect o f L o a d Dura t ion 98 5.5.4 Effect o f N u m e r i c a l P rec i s ion 99 5.5.5 Effect o f M a s s Propor t ional D a m p i n g 100 5.5.6 Effect o f Geomet r ic Imperfections 102 5.6 Cons idera t ion o f Res idua l Stresses 103 5.7 D i s c u s s i o n o f S ingle Brace A n a l y s i s Resul ts 105 6 Modeling, Part Three: Full VSC-CBF System 106 6.1 Introduct ion 106 6.2 Desc r ip t ion o f the F u l l V S C - C B F M o d e l 106 6.2.1 Braces 107 6.2.2 Gusset Plates 107 6.2.3 B e a m Connec t ion Plates ( V S C Plate) 108 6.2.4 B e a m . 109 6.2.5 C o l u m n s . . . . I l l 6.2.6 Treatment o f welds I l l 6.2.7 Treatment o f B o l t e d Connect ions I l l 6.2.7.1 B o l t s i n ver t ica l slotted connect ion I l l 6.2.7.2 B o l t s Connec t ing V S C Plate to B e a m 114 6.2.7.3 P i n n e d C o l u m n - t o - B e a m Connec t ion 115 6.2.8 Contact Def in i t ions 116 6.3 A n a l y s i s M e t h o d s 117 6.4 E igenva lue A n a l y s i s 118 6.5 Pushover S imu la t i on 120 6.5.1 Pushover S imula t ion Results 122 6.5.2 Effect o f V a r y i n g the L e v e l o f M a s s Propor t ional D a m p i n g 128 6.5.3 Effect o f R e d u c i n g the Coeff ic ient o f F r i c t i o n 129 v 6.5.4 Effect o f U s i n g an Al terna t ive Contact A l g o r i t h m 131 6.5.5 Spurious Effects i n P re l imina ry M o d e l s : Hourg la s s ing and Shear L o c k i n g 134 6.6 C y c l i c L o a d i n g Simula t ions 138 6.6.1 C y c l i c S imula t ions A and B . . . . 139 6.6.2 C y c l i c S imu la t i on C . . . 143 6.6.3 C y c l i c S imu la t i on D 149 7 Discussion, Conclusions, and Recommendations 151 7.1 D i s c u s s i o n 151 7.1.1 Di sc repancy i n Peak Latera l L o a d 153 7.1.2 D i sc repancy i n Elas t ic Latera l Stiffness 154 7.1.3 Di sc repancy i n Pred ic t ion o f Brace C r a c k i n g 157 7.1.4 Pred ic t ion o f Genera l Sys tem Behav iou r 158 7.2 Conc lus ions 162 7.3 Recommendat ions 165 References 169 Appendix A: Fabrication Drawings 172 vi LIST OF TABLES Table 1.1 A p p l i e d S inuso ida l Disp lacement P ro toco l 5 Tab le 4.1 Impl i c i t eigenvalue analysis results: F i s t f ive frequencies and periods at t=0 ^1 Table 4.2 Impl i c i t b u c k l i n g analysis results: B u c k l i n g load predict ions 53 Table 4.3 Effect o f load durat ion o n run t ime for double p rec i s ion exp l i c i t analyses 72 Table 5.1 E igenva lue analysis results: Fi rs t f ive v ibra t ion frequencies and periods at t=0 91 Table 5.2 B u c k l i n g analysis results: B u c k l i n g load predict ions, t=0.005s 92 Table 6.1 E igenva lue analysis results: Fi rs t s ix v ibra t ion frequencies and periods... . . . . . 119 Table 6.2 C y c l i c s imulat ions: Displacement load ing protocols 139 LIST OF FIGURES Figure 1.1 Inelastic deformation response o f a convent ional C B F (after B u b e l a , 2003) . . . 1 F igure 1.2 Inelastic deformation response o f the V S C - C B F system (after B u b e l a , 2003). 2 F igure 1.3 V S C - C B F test specimen w i t h standard H S S braces (after B u b e l a , 2003) 3 F igure 1.4 Load-disp lacement curve for V S C C B F system w i t h standard H S S braces 6 F igure 2.1 F in i t e element m o d e l o f part ial ly-restrained connect ion studied b y C i t i p i t i o g l u et a l (2002) 12 F igure 2.2 L S - D Y N A models o f unbonded braced frames (After F i e l d , 2003) 14 F igure 3.1 Schemat ic compar i son o f operation counts for nonl inear transient analysis b y expl ic i t and i m p l i c i t methods. (After Sauve, R . G . , and Metzger , D . R . (1995), p . 170) 24 F igure 3.2: Hourg lass modes for an eight-noded so l id element. (After F lanagan & Be ly t s chko , 1981).. . 28 F igure 3.3: Four -noded plane element subjected to bending: 4-point (full) integration and 1-point (reduced) integration 29 F igure 4.1 Resul ts o f stub c o l u m n compress ion tests: A x i a l stress-strain response 33 F igure 4.2 Resul ts o f stub c o l u m n compress ion tests: A x i a l load-displacement response. 34 F igure 4.3 F in i t e element mesh for the baseline stub c o l u m n m o d e l 35 F igure 4.4 Tens i l e test results from H S S flat w a l l and corner w a l l coupons 38 F igure 4.5 M a t _ 0 2 4 f l o w stress models o f H S S flat w a l l and corner w a l l sections 39 F igure 4.6 C o m p a r i s o n o f M a t _ 0 2 4 f low stress m o d e l and tensile test results for H S S flat w a l l and co ine r w a l l sections 40 F igure 4.7 Prescr ibed ver t ica l mot ions for top edge nodes i n the exp l i c i t analysis: a) smooth ve loc i ty prof i le ; b) corresponding smooth displacement prof i le 42 F igure 4.8 Prescr ibed ver t ica l mot ions for top edge nodes i n the i m p l i c i t analyses: l inear ly increasing displacement prof i le 43 F igure 4.9 Base l ine analysis and experimental test results: A x i a l load-displacement response o f stub c o l u m n under compress ion 46 F igure 4.10 Base l ine analysis results: Defo rmed shape after 1 3 m m o f top end displacement: a) exp l ic i t analysis; b) i m p l i c i t static analysis; c) i m p l i c i t dynamic analysis 46 F igure 4.11 Expe r imen ta l test results: De fo rmed shape after 1 7 m m o f top end displacement : 47 F igure 4.12 Base l ine analysis results: Contours o f v o n M i s e s stress: a) exp l ic i t analysis; b) i m p l i c i t static analysis; c) i m p l i c i t dynamic analysis 48 F igure 4.13 Base l ine analysis results: Contours o f effective plast ic strain: a) exp l ic i t analysis; b) i m p l i c i t static analysis; c) i m p l i c i t dynamic analysis . . 4 9 Figure 4.14 Base l ine analysis results: Internal energy histories 50 F igure 4.15 Impl i c i t eigenvalue analysis results: F i r s t f ive mode shapes at t=0 51 F igure 4.16 Impl i c i t eigenvalue analysis results: V a r i a t i o n o f first f ive periods throughout i m p l i c i t static analysis 52 F igure 4.17 Impl ic i t b u c k l i n g analysis results: F i r s t f ive b u c k l i n g modes, t=0.1 52 F igure 4.18 Spat ia l discret izat ion parametric study: a) Re f ined mesh; b) baseline mesh; c) coarse mesh 54 F igure 4.19 Effect o f v a r y i n g the spatial discret izat ion: A x i a l load-displacement response 55 F igure 4.20 Effect o f va ry ing the element characterization parameters: A x i a l load-displacement response 57 F igure 4.21 Effect o f va ry ing the hourglass control type and coefficient: A x i a l load-displacement response 58 F igure 4.22 Effect o f va ry ing the hourglass control type and coefficient: Hourg lass energy 59 F igure 4.23 M a t e r i a l m o d e l parametric study: C o m p a r i s o n o f M a t _ 0 0 3 f l o w stress m o d e l and tensile test results for H S S flat w a l l and corner w a l l sections 60 F igure 4.24 Effect o f us ing b i - l inear mater ial m o d e l M a t _ 0 0 3 : A x i a l load-displacement response. 61 F igure 4.25 Effect o f us ing b i - l inear mater ial m o d e l M a t _ 0 0 3 : D e f o r m e d shape, exp l ic i t solver 61 F igure 4.26 Effect o f us ing bi - l inear mater ial m o d e l M a t _ 0 0 3 : C o m p a r i s o n o f internal energy and hourglass energy, i m p l i c i t solver 62 F igure 4.27 Effect o f us ing b i - l inear mater ial m o d e l M a t _ 0 0 3 : D e f o r m e d shape, i m p l i c i t solver: a) Hourg lass coefficient = 0.10 (baseline value) ; b) hourglass coefficient = 0.05; c) hourglass coefficient = 0.15 63 F igure 4.28 Geomet r i c imperfec t ion study: D e f i n i t i o n o f Wo/b ratio 65 F igure 4.29 Effect o f i nc lud ing geometric imperfections: D e f o r m e d shapes, i m p l i c i t analyses: a) w 0 / b = 0 .001; b) w 0 / b = 0.002; c) wo/b = 0.003; d) w 0 / b = 0.005 66 F igure 4.30 Effect o f i nc lud ing geometric imperfections: D e f o r m e d shapes, exp l i c i t analyses: a) wo/b = 0.001; b) w 0 / b = 0.002; c) w 0 / b = 0.003; d) wn/b = 0.005 66 F igure 4.31 Effect o f i n c l u d i n g geometric imperfections: A x i a l load-displacement response 67 F igure 4.32 Effect o f va ry ing the stiffness-proportional damping coefficient: D e f o r m e d shapes w i t h automat ical ly computed m i n i m u m t ime step s ize: a) no damping ; b) damping coefficient = 0.15; c) damping coefficient = 0.20 68 F igure 4.33 Effect o f v a r y i n g the stiffness-proportional damping coefficient: A x i a l load-displacement response 69 F igure 4.34 Al te rna t ive prescr ibed ver t ica l mot ions for top edge nodes i n the exp l i c i t analysis: a) constant ve loc i ty prof i le ; b) corresponding l inear ly increas ing displacement prof i le 70 F igure 4.35 Effect o f load durat ion for exp l ic i t analysis: D e f o r m e d shapes: a) durat ion = 5 T i ; b) durat ion = 10T, ; c) duration = 100T. ; d) durat ion = 2 0 0 T i 71 F igure 4.36 Effect o f load durat ion for exp l ic i t analysis: A x i a l load-displacement response 72 F igure 4.37 Contact boundary condi t ions study: Instabil i ty o f shel l edge contact 74 i x Figure 4.38 Effect o f us ing contact boundary condi t ions: D e f o r m e d shapes 75 F igure 4.39 Effect o f us ing contact boundary condi t ions: A x i a l load-displacement response 76 F igure 4.40 Effect o f us ing single p rec i s ion i n the exp l ic i t analysis: D e f o r m e d shape.... 77 F igure 4.41 Effect o f us ing single prec is ion i n the exp l ic i t analysis: A x i a l load-displacement response 78 F igure 5.1 Deta i l s o f s ingle brace mode l 83 F igure 5.2 S ing le brace m o d e l : Transi t ions i n mesh topology o f the brace 84 F igure 5.3 S ing le brace mode l : B o t t o m gusset plate 86 Figure 5.4 S ing le brace mode l : T o p gusset plate 87 F igure 5.5 C o m p a r i s o n o f M a t _ 0 2 4 f low stress m o d e l and tensile test results for top and bot tom gusset plate materials 88 F igure 5.6 M a t _ 0 2 4 f low stress models for top and bot tom gusset plate materials 88 F igure 5.7 B o u n d a r y condi t ions at top gusset plate 90 F igure 5 .8Eigenvalue analysis results: Fi rs t f ive v ibra t ion mode shapes at t=0 91 F igure 5.9 B u c k l i n g analysis results: Fi rs t eight b u c k l i n g modes, t=0.005s 92 F igure 5.10 B u c k l i n g analysis results: Close-up v i e w o f 6 t h b u c k l i n g mode 92 F igure 5.11 S ing le brace analysis: T y p i c a l effective plastic strain contours at 8 m m o f ax ia l end shortening 94 F igure 5.12 S ing le brace analysis: T y p i c a l effective stress contours at 8 m m o f ax ia l end shortening 95 F igure 5.13 Coarsened bot tom gusset plate mesh 96 F igure 5.14 E x p l i c i t analysis results: Effect o f number o f elements through the thickness o f the bot tom gusset plate: A x i a l load-displacement response 97 F igure 5.15 E x p l i c i t analysis results: Effect o f gusset plate elastic modu lus value: A x i a l load-displacement response 98 F igure 5.16 E x p l i c i t analysis results: Effect o f load ing duration: A x i a l load-displacement response 99 F igure 5.17 E x p l i c i t analysis results: Effect o f numer ica l p rec is ion : A x i a l load-displacement response, 0.70 s duration analysis 100 F igure 5.18 E x p l i c i t analysis results: Effect o f mass-proport ional damping : A x i a l load-displacement response, 0.70 s durat ion analysis 101 F igure 5.19 Impl i c i t analysis results: Effect o f in i t i a l imperfections: A x i a l load-displacement response 103 F igure 6.1 F u l l V S C - C B F M o d e l 107 F igure 6.2 F u l l V S C - C B F mode l : T o p gusset plate mesh 108 F igure 6.3 F u l l V S C - C B F mode l : V S C plate mesh 109 F igure 6.4 F u l l V S C - C B F m o d e l : B e a m mesh 110 F igure 6.5 F u l l V S C - C B F m o d e l : V S C bol t mesh 113 F igure 6.6 F u l l V S C - C B F m o d e l : Slotted bol ted connect ion 114 F igure 6.7 F u l l V S C - C B F mode l : V S C plate flange to beam flange connec t ion 115 F igure 6.8 F u l l V S C - C B F mode l : P inned column-to-beam connect ion 116 x Figure 6.9 E igenva lue analysis results: Fi rs t s ix v ibra t ion mode shapes 119 F igure 6.10 Prescr ibed hor izonta l mot ions for pushover s imula t ion : (a) smooth ve loc i ty prof i le ; (b) corresponding smooth displacement prof i le 121 F igure 6.11 Pushover s imula t ion results: Latera l load-displacement response 122 F igure 6.12 Pushover s imula t ion results: Brace ax i a l forces 123 F igure 6.13 Pushover s imula t ion results: Transla t ion o f bol ts and top gusset plate relative to V S C plate, at 5 6 m m lateral displacement 124 F igure 6.14 Pushover s imula t ion results: V e r t i c a l r i g i d b o d y displacement o f the three bolts i n the slotted connect ion 124 F igure 6.15 Pushover s imula t ion results: Contours o f effective plast ic strain 125 F igure 6.16 Pushover s imula t ion results: Contours o f effective stress: (a) p r io r to b u c k l i n g o f brace; (b) dur ing g loba l b u c k l i n g o f brace; (c) dur ing l o c a l b u c k l i n g o f brace 126 F igure 6.17 Pushover s imula t ion results: Contours o f effective stress i n the beam 127 F igure 6.18 Pushover s imula t ion results: Ene rgy levels 128 F igure 6.19 Effect o f va ry ing the l eve l o f mass propor t ional damping : La tera l load-displacement response 129 F igure 6.20 Effect o f reduc ing the coefficient o f f r ic t ion: Latera l load-displacement response 130 F igure 6.21 Effect o f reducing the coefficient o f f r ic t ion: B race ax i a l forces 131 F igure 6.22 Effect o f us ing an alternative contact a lgor i thm: T i m e his tory o f lateral load. 132 F igure 6.23 Effect o f us ing an alternative contact a lgor i thm: Penetrat ion o f bol t element edge into corner edge o f slot element 133 F igure 6.24 Effect o f us ing an alternative contact a lgor i thm: C o m p a r i s o n o f base shears. 133 F igure 6.25 Coarse mesh used i n p re l iminary models o f the V S C - C B F system 135 F igure 6.26 Hourg la s s ing i n the p re l iminary m o d e l 136 F igure 6.27 In-plane brace b u c k l i n g due to use o f fu l ly integrated elements i n s o l i d element parts 137 F igure 6.28 C y c l i c s imula t ion A : Latera l load-displacement response 140 F igure 6.29 C y c l i c s imula t ion B : Latera l load-displacement response (first 1.25 cycles on ly) : 140 F igure 6.30 C o m p a r i s o n o f c y c l i c s imulat ions A and B : La tera l load-displacement response 141 F igure 6.31 C y c l i c s imula t ion B : Latera l load-displacement response (a l l three cycles) . 142 F igure 6.32 S i m u l a t i o n C : Prescr ibed displacement pro tocol 143 F igure 6.33 C y c l i c s imula t ion C : Latera l load-displacement response 144 F igure 6.34 C y c l i c s imula t ion C : Res idua l out-of-plane brace deformations at end o f second c y c l e 145 F igure 6.35 C y c l i c s imula t ion C : Trans i t ion between curv i l inear and b i - l inear brace deformation (v i ew from base, l o o k i n g up) 146 F igure 6.36 C y c l i c s imula t ion C : Progress ion o f l o c a l b u c k l i n g : deformed shapes and corresponding contours o f effective plastic strain 147 F igure 6.37 C y c l i c s imula t ion C : Progress ion o f c r ack ing across brace sect ion 148 x i Figure 6.38 C y c l i c s imula t ion C : Contours o f effective plast ic strain i n gusset plates . . 149 F igure 6.39 C y c l i c s imula t ion D : Latera l load-displacement response 150 F igure 7.1 C o m p a r i s o n o f selected cycles from finite element predict ions and experimental results 152 F igure 7.2 Force equ i l i b r i um and kinemat ics o f C B F i n the l inear-elastic range 155 F igure 7 .3Compar i son o f experimental and analyt ical results: Trans la t ion o f bolts i n the slotted holes o f the V S C detail 159 F igure 7.4 C o m p a r i s o n o f experimental and analyt ical results: Y i e l d i n g o f the bot tom gusset plate 159 F igure 7.5 C o m p a r i s o n o f experimental and analyt ical results: G l o b a l out-of-plane b u c k l i n g o f brace 160 F igure 7.6 C o m p a r i s o n o f experimental and analyt ical results: L o c a l b u c k l i n g o f brace. 160 F igure 7.7 C o m p a r i s o n o f experimental and analyt ical results: Ini t ia t ion o f brace c rack ing 161 F igure 7.8 C o m p a r i s o n o f experimental and analyt ical results: Progress ion o f brace c rack ing 161 x i i ACKNOWLEDGEMENTS I w o u l d l ike to express m y sincere thanks to m y research supervisor, and col league, D r . Car los Ventura . D r . Ven tu ra p rov ided helpful guidance throughout the durat ion o f m y graduate studies, and always showed a genuine interest i n m y academic and professional development. I a m also indebted to m y secondary supervisor, D r . H e l m u t P r i o n , w h o p rov ided valuable feedback at cr i t ical t imes dur ing m y research. F u n d i n g for the research was p rov ided b y the Natura l Sciences and Eng inee r ing Research C o u n c i l o f Canada. I w o u l d l ike to acknowledge the mot ivat ional support o f m y classmates and officemates i n the Earthquake Eng inee r ing research group at U B C , par t icular ly C h r i s M e i s l , M a r t i n Turek, and H o u m a n Ghal ibaf ian . F i n a l l y , I owe a great thanks to m y l o v i n g fami ly . M y father L y l e , mother M e r i l y n n , and sisters R o s a l y n and M e r e d i t h are s i m p l y magnificent . Th i s thesis is dedicated to m y N a n a , H a z e l Seeton, whose k i n d words have a lways been a source o f inspira t ion for me. Thank you. x m 1 INTRODUCTION 1.1 Overview C h e v r o n braced frames are a popular lateral load resist ing system for steel framed structures that, w h e n proper ly designed, can possess adequate stiffness and energy diss ipat ion characteristics under seismic loading condi t ions. One weakness o f the system is that b u c k l i n g o f the compress ion brace can result i n a net d o w n w a r d force appl ied b y the tension brace to the mid-span o f the beam above, potent ia l ly c o m p r o m i s i n g the gravi ty load resist ing system. The inelastic deformation mode o f the convent ional chevron braced frame ( C B F ) is shown i n F igure 1.1 F igure 1.1 Inelastic deformation response o f a convent ional C B F (after B u b e l a , 2003). A n innovat ive brace-to-beam connect ion detail i n w h i c h the bolts are set w i t h i n ve r t i ca l ly oriented slots was proposed b y design engineers o f the consul t ing f i rm Fast & E p p i n Vancouve r , B r i t i s h C o l u m b i a . Since o n l y lateral loads can be transferred between the braces and beam, the proposed connect ion theoret ical ly prevents the development o f a net d o w n w a r d force o n the beam and l imi t s the brace loads to the b u c k l i n g resistance o f the compress ion brace. The inelastic deformation mode o f the ver t ica l slotted connect ion chevron braced frame ( V S C - C B F ) system is shown i n F igure 1.2. Lateral Force Plastic Hinge Tension Brace Compression Brace 1 Lateral Force Vertically Slotted Connection Tension Brace Compresion Brace Figure 1.2 Inelastic deformation response o f the V S C - C B F system (after B u b e l a , 2003). A prototype o f the V S C - C B F system was tested exper imenta l ly at the U n i v e r s i t y o f B r i t i s h C o l u m b i a i n 2002 b y B u b e l a , P r i o n , and Ven tu ra (Bube la , 2003) . T w o full-scale V S C - C B F system specimens were tested. Deta i l ed fabricat ion drawings for the test specimens can be found i n A p p e n d i x A o f this report. The specimens were ident ica l except that the first specimen featured standard h o l l o w structural section ( H S S ) steel braces, w h i l e the second specimen had H S S braces f i l l ed w i t h concrete. In both cases, the braces were square 8 9 m m x 8 9 m m x 4 . 8 m m H S S sections, 3 1 4 7 m m i n length. The purpose o f i n c l u d i n g concrete-f i l led braces i n the second spec imen was to determine the effects o f the concrete f i l l o n loca l b u c k l i n g o f the braces and the associated change i n system stabil i ty, strength, and energy diss ipat ion characteristics. B o t h specimens were tested under quasi-static c y c l i c load ing condi t ions. A photograph o f the V S C - C B F specimen w i t h standard H S S braces, and the surrounding test set-up, is s h o w n i n F igure 1.3. 2 Latera l Support Bracke ts F l o o r B e a m Tes t ing Frame H y d r a u l i c A c t u a t o r P i n - E n d e d C o l u m n s F igure 1.3 V S C - C B F test specimen w i t h standard H S S braces (after B u b e l a , 2003). F o l l o w i n g the experimental testing, B u b e l a developed s imp l i f i ed ana ly t i ca l models i n an attempt to replicate the results obtained from the experiment. H o w e v e r , the models were unable to adequately represent the system behaviour, and therefore it was recommended that a detai led finite element m o d e l be developed. It was hoped that detai led m o d e l i n g o f the three-dimensional and nonlinear nature o f the b rac ing members and slotted connect ion detai l migh t resolve the discrepancies be tween the analy t ica l and experimental results, i n bo th the l inear and nonl inear range o f the sys tem behaviour . The research reported herein outlines the detailed finite element m o d e l i n g investigations that were carr ied out, i n w h i c h the sensit ivity o f the system to a var ie ty o f mode l ing -related factors was assessed. E v e n as three-dimensional finite element m o d e l i n g o f nonl inear systems i s s l o w l y b e c o m i n g accepted b y academic and p rac t i c ing structural 3 engineering communi t ies , there is a general lack o f appreciat ion amongst researchers and engineers i n terms o f the extent to w h i c h m o d e l i n g assumptions affect the computed results. T h e present research therefore addresses this gap between analysis and design, b y systematical ly consider ing the effects o f numer ica l and p h y s i c a l parameters o n the predicted behaviour o f the finite element mode l . 1.2 Procedures and Results of Experimental Project Before deve lop ing detai led finite element models o f the V S C - C B F system, it was necessary to r ev iew the procedures fo l l owed and results obtained from the experimental project that preceded this study (Bube la , 2003). In the f o l l o w i n g sections, the k e y experimental procedures and results are out l ined, pertaining to the test spec imen w i t h empty H S S braces on ly . Further details o n the experimental study can be found i n the M . A . S c . thesis b y B u b e l a (2003). 1.2.1 Material Testing Tens i le coupon tests were performed o n samples o f the flat w a l l and corner w a l l port ions o f the square ( H S S ) braces and the two sizes o f gusset plates that were used i n the V S C -C B F system test assembly. These tests were carr ied out i n accordance w i t h Test M e t h o d E8-01 o f the A m e r i c a n Soc ie ty for Tes t ing and Mate r i a l s standard ( A S T M , 2001) . Three coupons were sampled for each o f the sections. The stress-strain curves obtained from these tests were used to develop the finite element mater ia l mode ls for the present research. S tub-co lumn compress ion tests were performed o n two 3 0 8 m m - l o n g sections o f the 8 9 m m x 8 9 m m x 4 . 8 m m H S S brace members, f o l l o w i n g the procedures out l ined b y Ga lambos (1988). These tests a l l owed invest igat ion o f the ax i a l compress ion response o f the H S S i n terms o f l o c a l b u c k l i n g , y i e l d and ult imate load capaci ty, and pos t -buck l ing behaviour. A v e r a g e y i e l d strength o f 458 M P a and ult imate strength o f 524 M P a was calculated. These tests also p rov ided a set o f we l l - con t ro l l ed and re la t ive ly s imple experiments that were s imulated w i t h finite element models i n the present invest igat ion, to assess the effects o f several numer ica l parameters. 4 1.2.2 VSC-CBF System Test Protocol The V S C - C B F system was tested under quasi-static c y c l i c load ing condi t ions . The top beam o f the test assembly was subjected to in-plane hor izonta l displacement cyc les , f o l l o w i n g the standard testing protocol as recommended b y the A p p l i e d Tech n o l ogy C o u n c i l A T C - 2 4 ( A T C , 1992). T o avo id truncation o f ampli tude peaks due to t ime lag i n the hydrau l ic system, a s inusoidal displacement control s ignal was used instead o f the suggested saw-tooth function. The s inusoidal pro tocol parameters for the test o f the spec imen w i t h empty H S S braces are l isted i n Table 1.1. Tab le 1.1 A p p l i e d S inuso ida l Disp lacement P ro toco l . Step N u m b e r o f C y c l e s C y c l e A m p l i t u d e (mm) C y c l e Pe r iod (s) A v e r a g e L o a d i n g Rate (mm/s) C u m u l a t i v e Test Dura t ion (s) 1 3 7 120 0.23 360 2 3 14 240 0.23 1080 3 3 21 360 0.23 2160 4 3 28 480 0.23 3600 5 3 56 480 0.47 5040 6 3 84 720 0.47 7200 7 ' 2 112 960 0.47 9120 8 2 140 1200 0.47 11520 1.2.3 VSC-CBF System Performance The hysteretic behaviour o f the test specimen w i t h empty H S S braces is s h o w n i n terms o f the load-displacement curve o f F igure 1.4. The peak lateral load resisted b y the system was 263 k N , w h i c h corresponds to first b u c k l i n g o f a compress ion brace. T h i s m a x i m u m load occurred at a lateral displacement o f 2 6 m m , measured at the beam leve l . The system wi ths tood "12 inelastic cycles (21 cycles i n total) un t i l a tension failure occurred i n one o f the braces (i.e. complete fracture across the brace cross-sect ion)" (Bube la , 2003) . T h e tension fracture was ini t ia ted i n earlier cyc les at the corners o f the H S S member , where l oca l b u c k l i n g associated w i t h the format ion o f a plast ic mechan i sm at the mid- length reg ion o f the brace resulted i n extreme straining and mater ia l hardening. The total energy dissipated b y the system, calculated as the area enclosed b y the 5 hysteresis loops, amounted to 106 kJ. Further qualitative assessment of the system ductility in terms of strength and stiffness degradation can be made through inspection of the load-displacement curve of Figure 1.4. i — — r • —1 4 W - | 1 1 1 I 1 1 .400 1 —I— — i — — i Displacement (mm) Figure 1.4 Load-displacement curve for VSC CBF system with standard HSS braces. 1.2.4 Macro Modeling of the VSC-CBF System Relatively simple nonlinear frame-type analytical models were developed by Bubela in an attempt to replicate the system behaviour observed in the experiment. The first model was created using the structural analysis program RUAUMOKO (Carr, 2001), which incorporated a nonlinear axial force-deformation hysteresis model for the bracing members, each of which were modeled as single beam elements. When subjected to the same loading protocol as was used in the experiment, the RUAMOKO model showed considerably higher lateral stiffness than the test specimen. The discrepancy was handled by incorporating a spring element at the connection between the beam and braces, which effectively transmitted lower displacements to the top of the braces and thereby reduced the apparent lateral stiffness of the system. A second analytical model was created using the structural analysis program SAP2000 Nonlinear (CSI, 2003), to conduct a pushover analysis for comparison with the backbone curve of the experimentally obtained hysteresis plot. The model was similar to the 6 R U A U M O K O m o d e l and it inc luded the same spr ing element descr ibed above. Instead o f a hysteretic brace b u c k l i n g mode l , plastic hinge points w i t h specif ied moment-, curvature relat ionships were assigned at the ends and mid-span o f the braces. The pushover curve obtained f rom the S A P 2 0 0 0 m o d e l was w i t h i n reasonable agreement o f the backbone curve obtained from the experiment. The fundamental def ic iency associated w i t h the two models descr ibed above is that the s imi la r i ty o f the analyt ica l results to the experimental results is attributable so le ly to the fact that the k e y m o d e l parameters (brace strength, spr ing stiffness, and plast ic hinge properties) were adjusted arbitrari ly to obtain the desired correla t ion i n results. It is diff icul t to ascertain whether the values chosen for these parameters h o l d any phys i ca l va l id i ty , thus inh ib i t ing the general use o f this approach for s imi l a r systems. Furthermore, even w h e n the parameters were adjusted to p rov ide the best match possible w i t h the exper imental results, the strength degradation and energy diss ipat ion predicted b y the models were not par t icular ly accurate. B u b e l a (2003) noted that this might have been due to the fact that "the brace bending and loca l b u c k l i n g o f tube w a l l s that occur at plast ic h inge locat ions, where a substantial part o f the energy diss ipat ion is t ak ing place, [were] not accounted for fu l ly i n the m o d e l . " She went o n to r ecommend that a more sophisticated finite element analysis w o u l d be required to capture these effects. T h i s recommendat ion was a p r imary mot ivator for the present research. 1.3 Software and Hardware Specifications for the Present Research In the present invest igat ion, fu l ly three-dimensional finite element mode ls were developed i n w h i c h mater ial , geometric, and contact nonlineari t ies were inc luded . L S -D Y N A ( V e r s i o n 970) , a general-purpose finite element software package developed b y L i v e r m o r e Software Te c hno logy Corpora t ion ( L S T C ) , was used to per form the analyses. L S - D Y N A was o r ig ina l l y developed as an expl ic i t code, m a k i n g it w e l l suited to the so lu t ion o f h i g h l y nonl inear dynamic events such as impacts and blasts. T h e program features extensive capabil i t ies for m o d e l i n g contact ing interfaces o f structural components. A n i m p l i c i t solver has also been incorporated into the L S - D Y N A program, such that static and dynamic s imulat ions o f longer durat ion events can be undertaken. 7 The i m p l i c i t solver also features eigenvalue v ibra t ion analysis and eigenvalue b u c k l i n g analysis capabil i t ies . The pre-processor F e m B P C ( V e r s i o n 28) , developed b y Eng inee r ing Te c hno logy Associa tes , Inc, was used to create the finite element models and some port ions o f the L S - D Y N A input decks. The post-processor L S - P r e P o s t ( V e r s i o n 1.0), developed b y L S T C , was used to inspect the finite element results. D u r i n g the course o f the invest igat ion, over 100 s imulat ions were conducted, al though o n l y those s imulat ions that resulted i n k e y f indings are presented herein. E a c h s imula t ion was performed o n one o f three desktop personal computers, running the W i n d o w s X P Profess ional ( V e r s i o n 2002) operating system. A l l three computers had 1.0 G B o f R A M , and Pen t ium 4 processors, w i t h C P U speeds ranging from 2.53 G H z to 3.06 G H z . S ince the exact hardware configurat ion o f each machine was different, run t imes for ident ical s imulat ions o n the three machines were not equal . M o r e o v e r , background applications running o n the three machines affected the run t ime for m a n y o f the s imulat ions . Therefore, a comprehensive compar i son o f run t imes is not presented i n this report. A l t h o u g h L S - D Y N A is op t imized for use o n mass ive ly para l le l and shared m e m o r y paral le l processors, o n l y the serial vers ion o f the code was used i n the present invest igat ion. 1.4 Objectives of Research The objectives o f the finite element m o d e l i n g procedures carr ied out i n the present research were as fo l lows : • T o investigate the sensi t ivi ty o f sophisticated finite element m o d e l predict ions to numer ica l and phys i ca l parameters, and to thereby gain a thorough understanding o f the capabil i t ies and l imita t ions o f the models . • T o assess the feasibi l i ty o f s imula t ing a quasi-static structural process b y finite element mode l ing , us ing an expl ic i t t ime stepping so lu t ion method. • T o determine the extent to w h i c h sophisticated finite element mode ls are able to predict the exper imenta l ly observed behaviour o f a chevron braced frame spec imen w i t h a ver t ica l slotted connect ion detai l . 8 1.5 Scope of Research The research described i n the f o l l o w i n g chapters focuses o n the development and evaluat ion o f two sub-component models as w e l l as a fu l l m o d e l o f the complete chevron braced frame system w i t h a ver t ical slotted connect ion detai l . T h e scope o f this research includes: • T h e development o f stub c o l u m n models , to simulate the stub c o l u m n testing o f the H S S brac ing members that were used i n the V S C - C B F system. Parametr ic studies were conducted to assess the sensi t ivi ty o f the predicted stub c o l u m n response to the f o l l o w i n g parameters: spatial discret izat ion, element characterization, method o f hourglass control , mater ia l m o d e l , i n i t i a l geometric imperfect ions, damping leve l , prescribed m o t i o n characteristics, the use o f contact boundary condi t ions , and the leve l o f numer ica l prec is ion . • T h e development o f s ingle brace models , to simulate the compress ive ax i a l load response o f the H S S brace and gusset plate connect ion assembly. Parametr ic studies were conducted to assess the sensi t ivi ty o f the predicted b u c k l i n g response to the f o l l o w i n g parameters: thickness-wise mesh refinement o f the gusset plates, mater ia l stiffness o f the gusset plates, t ime scal ing, numer ica l p rec i s ion , damping leve l , and in i t i a l geometric imperfections. • The development o f a fu l l V S C - C B F m o d e l , to simulate the response o f the system to unidi rec t ional and c y c l i c lateral displacement protocols . O n l y the V S C -C B F system w i t h standard empty H S S braces was considered. Parametr ic studies were conducted to assess the sensi t ivi ty o f the predicted system response characteristics to the f o l l o w i n g parameters: damping l eve l , f r ic t ion l eve l , contact a lgor i thm specifications, t ime scal ing, and c y c l i c l oad ing p ro toco l characteristics. F i n a l l y , comparisons were made between the V S C - C B F m o d e l predict ions and the experimental results. 9 2 LITERATURE REVIEW 2.1 Introduction Bubela (2003) reviewed the literature relating to the seismic performance of conventional concentrically braced frames and some energy dissipation devices that have been proposed to enhance this performance. Because a primary aspect of the present research is the three-dimensional modeling of the bolted connection in the vertical slotted connection (VSC) detail, a survey of some relevant studies on the finite element modeling of bolted connections is presented in this chapter. In addition, a summary is provided of work by other researchers who have used the LS-DYNA program to conduct simulations relevant to the present research. 2.2 Finite Element Simulation of bolted connections The use of finite element methods to analyse connections in steel structures has been common over the past twenty years. In general, most of the literature containing reference to connection modeling is ultimately focussed on the resulting structural behaviour of the model, rather than on the modeling decisions themselves. In lieu of providing an exhaustive summary of the dozens of approaches that have been taken to modeling connections, two examples are discussed herein, as they provide some useful ideas that are relevant to the three-dimensional modeling of bolted connections that was undertaken in the present research. Researchers at Kumamoto University in. Japan conducted a thorough investigation into the three-dimensional numerical modeling of single-bolted connections (van der Vegte, Makino, & Sakimoto, 2002). Their research was set in the context of developing a reliable model of bolted beam splice connections. The problem was simplified by considering the interaction of a single bolt connecting three steel plates, in which an in-plane displacement was applied to one end of the middle plate while the opposite ends of the outer plates were held fixed, thus subjecting the bolt shaft to double shear. Van der Vegte et al noted that in order to gain a better understanding of the competing nonlinear 10 behaviours in the system, the model was built up in three stages. Specifically, the effects of friction and bolt pre-tensioning were initially omitted from the model, and then added one at a time in subsequent models. All components of the model were discretized with eight-noded solid elements, and a contact algorithm was used to treat the interfaces between the bolt, the bolthole, and the surfaces of the plates. The geometry of the bolts was simplified by omitting the washers and using identical sizes for the bolt head and nut. Van der Vegte et al used the both the implicit and explicit solvers of the ABAQUS finite element program to carry out their investigations. (A discussion of the differences between explicit and implicit methods is made in Chapter 3 of this thesis). The finite element analyses were able to capture the sliding behaviour of the plates and bolts as well as the localized yielding in the connection as the assembly was pulled to large displacements. It was determined that the incorporation of damping into the explicit analyses was helpful in reducing high frequency oscillations at the contact interfaces, thereby improving the results. An important conclusion from the work of van der Vegte et al was that the explicit solution method was successfully used to model the bolted connection, giving results that were similar to the implicit solution method, but without encountering the convergence difficulties that were observed with the implicit solver. Furthermore, their work demonstrated that a systematic parametric study approach to finite element modeling can lead to a much more thorough understanding of the computed results. Researchers at the Georgia Institute of Technology used refined three-dimensional finite element modeling techniques to investigate the moment-rotation behaviour of partially-restrained beam-column connections (Citipitioglu, Hay-Ali, & White, 2002). The modeling of partially-restrained connections is well deserving of a finite element approach because these connections contain inherent flexibilities that are overlooked if a simplified "fixed" or "pinned" idealization is used. The particular system considered consisted of a bolted seat angle connection with double web angles. A typical model that was developed and studied by Citipitioglu et al is shown in Figure 2.1. Their models 11 contained 8-noded fully-integrated so l id elements, incorporat ing three elements through the thickness o f the webs and flanges o f the beam and c o l u m n components . F igure 2.1 F in i t e element mode l o f part ial ly-restrained connec t ion studied b y C i t i p i t i o g l u et a l (2002). Contact surfaces were inc luded at a l l touching and s l i d ing interfaces, w i t h a coefficient o f fr ic t ion o f 0.33. T h e hexagonal bolt heads were mode led as cy l inders , and washers were accounted for b y averaging the diameter o f the washer and nut. A n elastic material m o d e l was used for the bol t parts, w i t h no significant differences i n the results as compared to the case w h e n an elastic-plastic bolt material was specif ied . T h e analyses were conducted u s ing the i m p l i c i t solver o f the A B A Q U S finite element program. The researchers found that the finite element models were able to predict the in i t i a l stiffness o f the moment-rotat ion response curves that were obta ined from s imi l a r experiments. In the nonl inear range o f the response, the mode l s were par t icular ly sensitive to the coefficient o f f r ic t ion at the contacting interfaces. In addi t ion , due to the influence o f c l a m p i n g o n the force transfer mechanism i n the connect ion , the computed results were sensit ive to the prescribed value o f bol t pre-tensioning. A s was the case w i t h the w o r k b y van der Veg te et a l , the f inding o f C i t i p i t i o g l u et a l con f i rmed that three-12 dimensional modeling of bolted connections can be successfully achieved, and that additional insight can be gained by using a parametric sensitivity study approach. 2.3 Use of the LS-DYNA Code for Analysis of Structural and Earthquake Engineering Simulations Documented examples of the use of LS-DYNA for structural or earthquake engineering analyses are rare, compared to other finite element codes. Examples are available in the literature on the use of LS-DYNA for simulation of sheet metal forming (Mamalis, Manolakos & Baldoukas, 1996), and crushing and collapse analysis for crashworthiness applications (Langseth, Hopperstad, & Hanssen, 1998). However these applications are not directly related to the present research, and therefore they are omitted from the literature review presented here. Field (2003) used the implicit solver of LS-DYNA Version 960 to simulate two full-scale tests of unbonded braced frames. Currently, Field's paper is the only published work in which LS-DYNA has been used to simulate braced frame systems. The studies by Field are particularly interesting because the unbonded braced frames are a relatively new design alternative to conventional concentric braced frames. Unbonded braces, also known as buckling-restrained braces or buckling inhibited braces, have shown great promise in recent tests by researchers at the University of California at Berkeley (Aiken, I.D., Mahin & Uriz, 2002) and the National Center for Research on Earthquake Engineering (NCREE) in Taipei (Tsai, Lai, Chen, Hsiao, Weng, & Lin, 2004). In the unbonded brace concept, improved cyclic loading behaviour is achieved through a decoupling of the stress-resisting and flexural-buckling-resisting aspects of compression strength (Sabelli & Lopez, 2005). Typically, a solid steel core element is located within an HSS sleeve, filled with concrete. A de-bonding agent or physical barrier separates the concrete fill and HSS sleeve from the steel core, such that the sleeve and fill provide lateral support to resist global and local buckling but do not contribute to the resistance of axial stresses. Thus a ductile bracing element is achieved in which buckling is avoided, nearly uniform axial strains are developed along the length, and compressive and tensile strengths are nearly equal. 13 F i e l d ' s analyt ical studies were focussed on deve lop ing finite element models o f the experimental specimens that were tested under quasi-static c y c l i c l oad ing condi t ions at the U n i v e r s i t y o f C a l i f o r n i a at Be rke l ey b y A i k e n et a l (2002). A s s h o w n i n F igure 2.2, two b rac ing configurat ions were considered, namely the chev ron and s ingle d iagona l configurations. Ful ly- in tegra ted shel l elements were used for a l l components o f the mode l , and a l l bol ted and welded connections were approximated b y m o n o l i t h i c a l l y mesh ing the connected parts. A bi l inear material m o d e l w i t h isot ropic strain hardening was used throughout the mode l . Rather than mode l ing the detai led interact ion between the s o l i d steel core and the concrete f i l l and steel tube sleeve, a s i m p l i f i e d approach to mode l ing the unbonded brac ing members was taken b y mesh ing beam elements w i t h bending stiffness o n l y to the so l id steel core parts. A l t h o u g h a large react ion beam was used to anchor the test assembly, this boundary member was not i nc luded i n the analysis. F igure 2.2 L S - D Y N A models o f unbonded braced frames (After F i e l d , 2003) . The finite element models o f the unbonded braced frames were s h o w n to capture the general behaviour that was observed i n the experiments. In part icular , the locations o f y i e l d i n g and plast ic s t ra ining predicted b y the finite element mode l s seemed to correlate w i t h the observed f l ak ing o f whi tewash from the test specimens, a l though quantitative compar i son o f stresses and strains was not presented. G o o d agreement was also found between the peak values o f lateral forces and displacements resisted b y the systems. Unfortunately, F i e l d ' s paper d i d not include a compar i son o f the comple te hysteretic behaviour . In general , the results presented by F i e l d suggest that m o d e l i n g o f steel braced frame systems w i t h L S - D Y N A can be achieved w i t h acceptable results. Howeve r , 14 it should be noted that a number of substantial simplifications were included in her analysis, particularly with regards to the modeling of the braces and the connection details. 15 3 THEORETICAL BACKGROUND RELATING TO THE FINITE ELEMENT PROCEDURES 3.1 Overview Effect ive use o f complex three-dimensional nonl inear finite element m o d e l i n g software cannot be achieved wi thout first appreciating the theory o n w h i c h the m o d e l i n g and so lu t ion techniques are based. In order to lend meaning to the d iscuss ion presented i n the remainder o f this thesis, the f o l l o w i n g sections p rov ide an o v e r v i e w o f some o f the relevant theory. A k e y aspect o f the present w o r k is to investigate the extent to w h i c h quasi-static s imulat ions can be achieved b y dynamic analysis techniques. The salient features o f exp l ic i t and i m p l i c i t t ime integration algori thms are compared i n this chapter, and the use o f exp l ic i t methods to solve quasi-static problems is discussed. Impl icat ions o f us ing elements w i t h reduced integration are also described, f o l l o w e d b y an ove rv i ew o f approaches to m o d e l i n g material failure i n finite element analyses. 3.2 Direct Time Integration Algorithms Direc t t ime integration algori thms are used i n finite element codes to so lve nonl inear dynamic problems b y ca lcula t ing the system response us ing stepwise integrat ion i n t ime. L S - D Y N A employs an exp l i c i t t ime integration scheme based o n the central difference method. H o w e v e r , i m p l i c i t solvers are also implemented i n the program. The characteristics o f the t ime integration method selected have a direct impact o n var ious aspects o f the m o d e l i n g process and the solutions computed. Therefore, relevant details o f exp l ic i t and i m p l i c i t t ime integration schemes are presented i n the f o l l o w i n g sections. Further in format ion o n these techniques is avai lable i n the textbooks and literature b y var ious authors i n c l u d i n g Be ly t s chko , T . , & Hughes , T . J . R . (1983); B e l y t s c h k o , T . , L i u , W . K . , & M o r a n , B . (2004); and C o o k , R . D . , M a l k u s , D . S . , P lesha , M . E . , & W i t t , R J . (2002). 3.2.1 Explicit Methods The exp l i c i t t ime integration scheme typ ica l ly employs a central difference method to approximate the noda l veloci t ies and accelerations. In L S - D Y N A , a half-step central 16 difference method is used, meaning that the difference expression for the noda l veloci t ies {v} are expressed at the midpo in t o f the n t h t ime step interval , At , i n terms o f the noda l displacements {d} as: W n + 1 ^ { d } " + 1 " { d } " (3.1) At Rearranging terms i n (3.1) gives the integration fo rmula for displacements: {d}n+1={d}n+At{v}n+1/2 (3.2) I f veloci t ies are also assumed to vary l inear ly w i t h i n a t ime step, then the noda l accelerations {a} are g iven by: {a}n = < V >,+l/2-Wn- ./2 ( 3 3 ) At Rearranging terms i n (3.3) gives the integration fo rmula for veloci t ies : M „ + i / 2 =M„-,/2+Atn{a}n (3.4) F r o m (3.4) it can be seen that the method is ca l led explicit because the condi t ions can be calculated at the future t ime state t > n A t us ing informat ion from condi t ions at p rev ious ly computed t ime states t <nAt. The discrete equations o f mot ion , subject to the relevant boundary condi t ions and constraints specif ic to the mode l , can be wri t ten as: [M]{a}n={f}n={fex,}n-{fin,}n (3.5) where: [ M ] = diagonal l umped mass matr ix , w h i c h remains constant for a Lagrang ian mesh. { f x l } = external noda l forces, w h i c h are general ly functions o f t ime but can also be functions o f noda l displacements i n geometr ica l ly nonl inear problems. 17 {f nt} = internal nodal forces, which depend on the stresses, hence, strains and strain rates, hence displacements and velocities. The internal nodal forces can also be time dependent. Rearranging (3.5), the accelerations at time tn can be expressed as: W^tMj-'lf},, (3.6) The right-hand side of the above equation is straightforward to evaluate because a lumped mass matrix is used, which is by definition a diagonal matrix and thus trivial to invert, and all other terms are computed using known displacements and velocities from time step n and earlier. Thus {a}„ can be easily evaluated and then (3.4) can be used to obtain {v}n+i/2, and finally {d}n+i can be determined from (3.2). In summary, the explicit method can be implemented as follows, as described by Belytschko et al (2004): For step n > 0, and t = nAt: 1. Get {f}n by looping over all elements e: a. Extract nodal displacements and velocities of the element from the global array. b. Compute deformation (strain) measures at each quadrature point of the element, using nodal displacements and velocities of the element. c. Compute stresses at each quadrature point using constitutive equations. d. Evaluate internal nodal forces by integrating the product of the strain-displacement matrix and the stress over the domain of the element. e. Compute the external forces on the element and set (fe}n = (f e e x t}n -f. Scatter the element nodal forces {fe}n into the global array to obtain {f}n. 2. Compute nodal accelerations: {a}„ =[M]_1{f}n 3. Update nodal velocities: M n + 1 / 2 = {v} n_ 1 / 2 + A t n {a}„ 4. Set nodal velocities equal to prescribed values on prescribed velocity boundaries. 5. Update nodal displacements: {d}n+1 ={d}n +At{v} n + 1 / 2 6. Update counter and time: n«- n+1, t<- t+At. 7. Output results for the previous time step. If t<tend, go to 1. The key feature to note in the explicit method is that the process is advanced through time without requiring the solution of any simultaneous nonlinear equations. Therefore, each time step is executed rapidly. In the implementation of explicit codes, the velocity update is typically broken into two sub-steps to allow the energy balance to be checked at integer time steps. In addition, viscous damping and hourglass resisting force terms are included in the equation of motion and contact algorithms are included in the implementation. 18 These details can be incorporated into the expl ic i t method w i t h relat ive eff ic iency, but for the sake o f c la r i ty they have been omit ted from the preceding summary . A n addi t ional detai l that was omit ted above is the t ime step size adjustment that is t yp i ca l ly performed as the mesh deforms. T h i s adjustment is required because the expl ic i t method is cond i t iona l ly stable o n l y for t ime step sizes less than the c r i t i ca l t ime step Atcrit- G r o w t h o f the solu t ion w i l l become unbounded unless: A t < A t c r i t = ^ - , (3.7) where a> m a x is the highest natural frequency o f the lumped-mass m o d e l . W h e n damping is i nc luded i n the p rob lem, the cr i t ica l t ime step is reduced to: •(J^V-Cj, (3-8) where % is the damping ratio i n the a w mode. Because the evaluat ion o f a w for the entire m o d e l w o u l d require the expensive solu t ion o f the eigenvalue p rob lem, eff ic iency can b y achieved b y r ea l i z ing that a w o f the entire mesh is necessari ly smal ler than ©max o f any i nd iv idua l element i n the mesh. Therefore the c r i t i ca l t ime step for the entire m o d e l can be found conservat ive ly as the m i n i m u m cr i t i ca l t ime step o f a l l the elements. F o r example , i n the case o f a two-noded bar element o f length L , cross-sectional area A , mass m , and Y o u n g ' s M o d u l u s E , the highest natural frequency is : „ AE 2 E 2c / 0 ,v. ©max =2, — = - , - = — , , (3.9) V mL L y p L where p is the mass density and c is the wave speed or speed o f propagat ion o f sound i n the mater ial . T h e n the c r i t i ca l t ime step for this bar element w o u l d be: A t c r i t = - ^ - = ^ (3.10) co m „ c 19 Equa t ion (3.10) was developed b y Courant , Fr iedr ichs , and L e w y (1928) and is ca l led the C F L condi t ion . T h i s expression does not str ict ly h o l d true for a l l element types, and i n fact L S - D Y N A uses a reduct ion factor o f 0.90 o n the computed c r i t i ca l t imes step to ensure stabil i ty. Nonetheless , (3.10) illustrates that the c r i t i ca l t ime step is general ly dependent o n element s ize and wave speed ( w h i c h is related to stiffness and mater ial density). F o r instance, steel has a material wave speed o f about 5050 m/s. I f steel material is used w i t h elements hav ing a characteristic length o f 1 0 m m , then the c r i t i ca l t ime step w o u l d be approximate ly 2x10~ 6 seconds. The size o f the t ime step dictates the number o f steps required to solve a g iven p rob lem, and since each step invo lves a set o f computations, it is obvious that the C P U t ime to solve the p r o b l e m is d i rec t ly dependent o n the size o f the t ime step. C o o k et a l (2002) note that the use o f lumped masses further improves the eff ic iency o f the expl ic i t method because, compared to the consistent mass element formulat ions, l umped mass element formulat ions typ i ca l ly y i e l d smaller natural frequency estimates and thus larger c r i t i ca l t ime steps. In step I d o f the exp l i c i t method summary above, the integration to evaluate the internal noda l forces is performed b y a numer ica l integration method such as Gauss integration. T h i s operat ion can be made less cos t ly b y m i n i m i z i n g the number o f integrat ion points; that is , b y us ing reduced integration. A d d i t i o n a l l y , reduced integrat ion t y p i c a l l y y ie lds more f lex ib le elements, thus a l l o w i n g a larger c r i t ica l t ime step compared to fu l ly integrated elements. Further impl ica t ions o f us ing under-integrated elements are discussed i n Sec t ion 3.4. Four -noded plane and shel l elements and eight-noded so l id elements are most c o m m o n l y used i n finite element models so lved b y expl ic i t methods. H i g h e r order elements, that is , elements w i t h larger numbers o f nodes, have more degrees o f freedom and hence larger m a x i m u m natural frequencies than elements w i t h o n l y corner nodes. T h e i r use is rare i n exp l i c i t methods because they result i n a ve ry sma l l c r i t i ca l t ime step and can, i n fact, result i n excessive osci l la t ions (noise) as stress waves propagate through the finite element mesh. 20 3.2.2 Implicit Methods The implementa t ion o f i m p l i c i t solvers i n L S - D Y N A is described i n the L S - D Y N A Theore t ica l M a n u a l b y Ha l lqu i s t (1998). H o w e v e r , a more general treatment o f the topic is p rov ided here, to a l l o w a clear understanding o f the features o f i m p l i c i t methods. The discrete equations o f m o t i o n can be wri t ten at t ime t n + 1 i n a fo rm equivalent to (3.6), as: {0} = W n + 1 =sD[M]{a}n+1 +{fm ,}n + 1 -{f e x t} n + 1 , (3.11) where SD = 0 for a static p rob lem and SD = 1 for a dynamic p rob lem. Displacements and veloci t ies at t ime tn+i are approximated us ing the difference expressions o f the one-step N e w m a r k (3 formulas: M „ + , ={v}„ +(l-y)At{a}n +yAt{a}r (3.12) {d} n + 1={d} n+At{v} n + |--P At 2 {a}" + (3 At 2 {a} 2 f„nn+l (3.13) These expressions are ca l l ed implicit because ca lcu la t ion o f { v } n + i and {d} n+i requires knowledge o f { a } n + i , w h i c h is not yet k n o w n . That is , the ca lcu la t ion o f condi t ions at future t ime states requires more than just informat ion from p rev ious ly computed t ime states. In (3.12) and (3.13), the (3 and y terms control accuracy, s tabi l i ty, and a lgor i thmic damping . (3.13) can be rearranged to g ive the accelerations at t ime t n + i : Mn + . =- 1 T ({d}n + 1 -{d}„-At{v}J-(3At 12(3 J {a}„ (3.14) P l u g g i n g the above expression into (3.12) y ie lds the veloci t ies { v } n + i as: M n + 1 = p ^ ( { d } „ + 1 - { d } n ) -vP J M „ - A t | - - i {a}„ (3.15) 21 Thus the accelerations {a} n+i and veloci t ies { v } n + i are n o w expressed i n terms o f the u n k n o w n displacements { d } n + i and the k n o w n condi t ions f rom t ime n . Therefore, b y substituting (3.14) and (3.15) into (3.11), the p rob lem becomes to f ind {d} n+i such that: {0} = s D [ M ] (3Af - ( { d } n + 1 - { d } n - A t { v } n ) - — -1 2P . {a}„ + { f m , } „ + 1 - { f e x , } „ + 1 (3.16) In the absence o f geometric and mater ial nonlineari t ies, then the g loba l stiffness matr ix [ K ] can be used to f ind { f n t } n +i = [ K ] { d } n + i , and { f x t } n + i is not dependent o n { d } n + i . In this l inear case, (3.16) can be rearranged as: p A t 2 - [M ]+ [K ] {d} n + 1 = s D [ M ] I PAt l _ ( { d } B + A t { v } J + 2p {a}„ + {f e X , }„ + , (3.17) F r o m (3.17) it can be seen that the use o f a d iagonal [ M ] provides no significant benefit i n the i m p l i c i t method, because the coefficient matr ix o f { d } n + i contains [ K ] , w h i c h is i n general non-diagonal , so the solut ion for {d}„+i w i l l s t i l l require factorizat ion or Gauss e l imina t ion operations to invert the coefficient matr ix . In the more realist ic case where mater ial nonl inear i ty is inc luded , then { f n t } n + i is found from the element stresses, w h i c h are nonl inear ly dependent o n (d} n +i . L i k e w i s e , for geometric nonl inear i ty , { f x t } n + i is nonl inear ly dependent o n {d} n +i. Therefore (3.16) becomes a set o f simultaneous nonl inear equations and an iterative so lu t ion technique such as the fu l l or m o d i f i e d N e w t o n Raphson method is used to eventual ly achieve equ i l i b r i um at each t ime step w i t h i n some specif ied tolerance. Convergence o f these iterations is not guaranteed and can be di f f icul t to achieve for some problems. The i m p l i c i t method can be used to solve a nonl inear static p r o b l e m s i m p l y b y setting SD=0 i n (3.16) to e l iminate the inert ial force component. H o w e v e r , the requirement for iterative nonl inear equation so lv ing remains for such a p rob lem. I f the static p rob lem is also l inear, then the p rob lem is.reduced to the famil iar [ K ] { d } = { f x t } . 22 3.2.3 Comparison of Explicit and Implicit Methods In the previous two sections, the expl ic i t and i m p l i c i t approaches to s o l v i n g nonl inear dynamic problems were presented. E x a m i n a t i o n o f the fundamental differences between the two methods a l lows a compar i son o f the merits o f each. The expl ic i t method a l lows the solut ion o f the nonl inear dynamic p rob lem to be advanced through t ime wi thout requi r ing the iterative solu t ion o f simultaneous nonl inear equations. The avoidance o f nonl inear equation so lv ing makes the method ve ry robust and able to handle var ious nonlineari t ies wi thout diff icul ty . M o r e o v e r , the use o f a d iagonal lumped mass matr ix w i t h the central difference method removes the need to per form any matr ix invers ion operations, so each t ime step is executed q u i c k l y w i t h m i n i m a l computer storage requirements. H o w e v e r , the condi t ional s tabi l i ty o f the exp l i c i t method necessitates the use o f a ve ry sma l l c r i t i ca l t ime step size, m a k i n g the so lu t ion o f l ong -durat ion problems ve ry expensive i n terms o f comput ing t ime. In contrast, the i m p l i c i t method is uncondi t iona l ly stable (for the appropriate choice o f (3 and y ), a l l o w i n g the use o f a t ime step size that is orders o f magni tude larger than the exp l i c i t c r i t i ca l t ime step. F o r this reason, the i m p l i c i t method is general ly chosen for long-durat ion problems. H o w e v e r , al though the s tabi l i ty o f the i m p l i c i t method is not condi t iona l o n the size o f the t ime step, a reasonably sma l l t ime step must be selected i n order to main ta in an accurate solut ion. E v e n w i t h a reasonably sma l l t ime step, the iterative nonl inear equation solvers required i n an i m p l i c i t method m a y fa i l to converge, par t icular ly for discont inuous nonlinearit ies such as changing contact condi t ions and b u c k l i n g . Sauve, R . G . , and Metzger , D . R . (1995) noted that the increase i n computat ional effort required as the size o f the finite element m o d e l increases is less for an expl ic i t procedure than for an i m p l i c i t procedure. T h i s effect is demonstrated schemat ica l ly i n F igure 3.1 be low. 23 Implicit Explicit L o g (Number o f Elements ( N ) ) F igure 3.1 Schemat ic compar i son o f operation counts for nonl inear transient analysis b y exp l i c i t and i m p l i c i t methods. (After Sauve, R . G . , and Me tzge r , D . R . (1995), p . 170). F o r events o f ve ry short durat ion such as blast or impact , the response is general ly computed for a t ime span o f less than one second and h i g h frequency components contribute s ignif icant ly . These problems are unequ ivoca l ly w e l l suited to solu t ion b y expl ic i t methods. O n the other hand, for longer durat ion events, such as earthquakes, lower frequencies dominate the response and a larger t ime step is exp lo i ted b y i m p l i c i t methods to y i e l d a more efficient solut ion. The p r imary reason for a v o i d i n g expl ic i t methods for these problems is that the C P U t ime to solve such long-durat ion events becomes unreasonably l o n g due to the smal l c r i t ica l t ime step. H o w e v e r , i n the case o f h i g h l y nonl inear problems simulated b y large models conta in ing m a n y elements, the eff ic iency o f exp l i c i t methods becomes attractive. M o r e o v e r , as compu t ing technologies continue to improve , the issue o f run-t ime is becoming less signif icant . The models o f the chevron braced frame systems that were investigated i n the present research contained over 86,000 nodes and over 68,000 elements. In addi t ion, s ignificant contact nonl inear i ty was introduced to m o d e l the slotted bol ted connect ion, and b u c k l i n g o f the braces was anticipated to dominate the system behaviour. Therefore, the exp l i c i t solver was selected for most o f the analyses. 24 3.3 Use of Explicit Methods to Solve Quasi-Static Problems Quasi-static processes, b y defini t ion, occur at a rate that is s l ow enough that inert ial effects m a y be considered to p lay an insignif icant role i n the structural response. In order to eff ic ient ly simulate a quasi-static event w i t h an expl ic i t solver, sca l ing techniques are c o m m o n l y used to overcome the restrictions o f the c r i t i ca l t ime step. These techniques are described i n the f o l l o w i n g sections, a long w i t h a d iscuss ion o f the smooth appl ica t ion o f load ing or boundary condi t ions and methods to evaluate the extent o f iner t ia l effects. 3.3.1 Time Scaling G i v e n that the size o f the expl ic i t t ime step is dictated b y stabi l i ty considerations, the simplest w a y to achieve a shorter run t ime is to speed up the t ime scale o f the p rob lem such that the s imula t ion t ime is reduced and fewer t ime steps are required. T h i s procedure is acceptable so long as the materials used are not rate-dependent and so long as inert ial effects do not become significant. In order to determine the extent to w h i c h the t ime scale can be accelerated w h i l e retaining the quasi-static nature o f the p rob lem, it is useful to consider the dynamic properties o f the load ing relat ive to the dynamic properties o f the structure. T h i s topic was investigated b y K u t t , P i f k o , N a r d i e l l o , and Papaz ian (1998), w h o recommended that for a load ing function that increases from zero to some m a x i m u m value, a quasi-static response m a y be achieved b y prescr ib ing a load durat ion either greater than or equal to f ive fundamental l inear periods o f the finite element m o d e l be ing analysed. In this context, the load m a y be i n the fo rm o f appl ied loads or prescr ibed mot ions . A l t h o u g h their guidelines were deve loped b y first cons ider ing the dynamics o f linear-elastic s ingle degree o f freedom osci l la tors , they demonstrated that their conclus ions cou ld be extended to mult i-degree-of-freedom nonlinear inelast ic structures. 3.3.2 Mass Scaling The exp l i c i t c r i t i ca l t ime step was shown i n Sec t ion 3.2.3 to be dependent o n element size, but also the wave speed and therefore the density o f the mater ia l . I f mesh refinement i n a sma l l area o f the m o d e l leads a few sma l l elements to dictate the cr i t ica l t ime step, it m a y be possible to scale up the densi ty o f these few elements to achieve a 25 larger c r i t i ca l t ime step without impar t ing significant dynamic effects. H o w e v e r , for models w i t h re la t ive ly un i fo rm meshing, mass sca l ing is a less in tui t ive approach than t ime sca l ing . T h i s is because t ime sca l ing is a l inear sca l ing technique, whereas mass scal ing affects the c r i t i ca l t ime step through a square-root relat ionship. 3.3.3 Smooth Loading In their study o n the m i n i m i z a t i o n o f dynamic effects i n quasi-static s imulat ions , K u t t et a l (1998) also investigated the influence o f the fo rm o f the load h is tory prof i le . T h e y compared the response to a load function that increased w i t h constant slope to two smooth ly v a r y i n g profi les , namely, a versed sine and a c y c l o i d a l history. K u t t et a l showed that the smooth c y c l o i d a l h is tory resulted i n the fewest osc i l la t ions about the static so lu t ion due to its higher number o f continuous derivat ives at t = 0. The i r observations con f i rm that the use o f a smooth load ing prof i le is preferable for dynamic solutions o f quasi-static processes. 3.3.4 Assessing the Extent of Inertial Effects in the Analysis T o a certain extent, a l o w ratio o f k inet ic energy to internal strain energy can indicate that a quasi-static process has been achieved. These quantities can be t racked dur ing the finite element analysis and plotted for compar ison . L o w kine t ic energy i m p l i e s a quasi-static response inasmuch as it impl ies l o w veloci t ies w i t h i n the mater ial . H o w e v e r , since it is poss ible to have h i g h ve loc i ty even w i t h l o w accelerations, par t icu lar ly i n the case o f r i g i d b o d y mot ions , the energy balance cr i ter ion is not a def ini t ive assessment o f inert ial effects. T h e foregoing arguments were corroborated b y both K u t t et a l (1998), and Rus t & S c h w e i z e r h o f (2003). C h u n g , C h o , and Be ly t s chko (1998) devised an energy error estimator whereby the mesh is d i v i d e d into sub-domains to account for the loca t ion o f dynamic effect concentrations, and the entire his tory o f the k ine t ic and internal energies is considered. Unfor tunate ly their approach is not eas i ly implemented for general applicat ions. Rus t & S c h w e i z e r h o f suggested that, i f possible , a check o n static e q u i l i b r i u m is a better indicator o f the presence o f dynamic effects. T h i s can be achieved b y observ ing the magnitude o f osci l la t ions i n cross-sectional forces relative to predicted static values. O n 26 the other hand, Mat t i a sson et a l (1991) noted that osci l la t ions present i n the up-speeded results o f meta l fo rming s imulat ions c o u l d be fil tered out to g ive results that were s imi la r to the quasi-static solut ion. 3.4 Use of Under-Integrated Elements In finite element analysis , the terms " f u l l integrat ion" and "reduced in tegrat ion" refer to the approach taken for the numer ica l evaluat ion o f var ious integral expressions i n the finite element formulat ion, w i t h c o m m o n appl ica t ion be ing the evaluat ion o f the element stiffness matr ix and the integration o f stresses over an element d o m a i n to determine noda l forces. T h e numer ica l integration is carr ied out b y a quadrature rule such as Gauss quadrature, i n w h i c h the values o f the function sampled at discrete points are mu l t i p l i ed b y we igh t ing factors and summed. W h e n the integration is carr ied out over the doma in o f the element, then the posi t ions o f the sampl ing points take o n p h y s i c a l coordinates w i t h i n the element geometry. The accuracy o f a quadrature rule is i m p r o v e d b y increasing the number o f quadrature points; for example , a p o l y n o m i a l o f degree 2 n - l is integrated exact ly b y n-point Gauss quadrature. W h e n used to evaluate an element stiffness matr ix , " f u l l integrat ion" refers a quadrature rule w i t h sufficient accuracy to exact ly integrate a l l stiffness coefficients o f an undistorted element. F o r elements o f distorted geometries, fu l l integration is no longer exact, but the term is appl ied anyway. It should be noted that fu l l integration does not necessari ly i m p l y more accurate finite element results, as these are affected b y a var ie ty o f factors and are general ly stiffer than the exact so lu t ion. Integration o f less than fu l l order is ca l led "under- integrat ion" or " reduced" integration. It was indicated i n Sect ion 3.2.1 that reduced integrat ion lends improved eff ic iency to problems so lved b y expl ic i t methods. In the exp l i c i t L S - D Y N A code, frequent use is made o f the under-integrated eight-noded s o l i d element w i t h one-point quadrature, the quadrature point be ing located at the centre o f the b r i ck . The other c o m m o n element is the under-integrated four-noded shel l w i t h one-point quadrature i n -plane and mul t ip le quadrature points through the thickness. T h e m a i n disadvantage o f us ing under-integrated elements is the potential for the development o f so-cal led hourglass modes (also ca l led spurious modes , s ingular modes, 27 zero-energy modes, or k inemat ic modes). Hourg lass modes are non-phys ica l deformation modes that produce zero strain, strain energy, or stress. Ha l l qu i s t (1998) notes that hourglass modes occur i n under-integrated elements whenever d iagona l ly opposite nodes have ident ica l veloci t ies . F igure 3.2 shows the hourglass modes i n one Cartesian coordinate d i rec t ion for an eight-noded s o l i d element w i t h one-point integration. These four modes are repeated i n the other two coordinate direct ions, g i v i n g a total o f twelve hourglass modes. F igure 3.2: Hourg lass modes for an eight-noded so l id element. (Af ter F lanagan & Be ly t schko , 1981). The occurrence o f "hourg lass ing" m a y produce a softening i n the results due to the ab i l i ty o f the mesh to deform wi thout stress, and i n severe cases this effect can take o n the form o f a mechan ism. T h e mi t iga t ion o f hourglassing is therefore important and there are several ways that this can be accompl ished. Hourg la s s ing m a y be t r iggered b y point loads appl ied to i n d i v i d u a l nodes, whereas the use o f distributed loads is less l i k e l y to do so. L i k e w i s e , mesh refinement can prevent the spread o f hourglass ing through the mesh. Ano the r w a y to prevent hourglass ing is to avo id the use o f reduced integrat ion altogether. 28 H o w e v e r , fu l ly integrated four-noded plane elements and eight-noded s o l i d elements subjected to bending can exhibi t shear l o c k i n g , w h i c h leads to an excess ive ly s t i f f response. The edges o f such elements are unable to curve due to the l inear shape functions used i n their formulat ion. The locations o f the fu l l order quadrature points result i n the development o f false shear strains i n these elements w h e n subjected to pure bending, as shown i n F igure 3.3. Strain energy is expended creating shear deformation rather than pure bending deformation and the fu l ly integrated element becomes over ly stiff. T h i s effect was careful ly studied for eight-noded so l id elements, and the results are presented i n a separate report ( E E R F Report 05-02). In contrast, as can be seen i n F igure 3.3, the spurious shear strains are not computed for an under-integrated element, i n w h i c h the quadrature point is located at the midd le o f the element. M M M CFH) M Figure 3.3: Four -noded plane element subjected to bending: 4-point (full) integrat ion and 1-point (reduced) integration. Several hourglass control algori thms are implemented i n L S - D Y N A . These s tabi l iza t ion algori thms essential ly app ly hourglass resist ing forces based o n either a v i scous damping or sma l l elastic stiffness capable o f s topping the format ion o f anomalous modes but hav ing neg l ig ib le effect o n the stable g loba l modes (Hal lquis t , 1998). "Hourg lass energy" refers to the amount o f energy expended b y the s tabi l izat ion a lgor i thm to resist hourglass ing effects. T h e signif icance o f the magni tude o f hourglass energy is best determined i n the context o f the p rob lem, and w i t h compar i son to the magnitude o f the internal energy. F o r example, a l o w value o f hourglass energy can be interpreted i n two ways : either the tendency towards hourglass ing i n the m o d e l is l o w , requi r ing li t t le s tabi l izat ion; or the tendency towards hourglass ing i n the m o d e l is moderate, but the hourglass control is l o w . In the second case, hourglass ing m a y not be 29 suff ic ient ly prevented; this can be assessed b y v i sua l inspect ion o f the deformed mesh. A h igh hourglass energy general ly indicates that the tendency towards hourglass ing i n the m o d e l is h i g h and that significant energy is be ing expended to prevent its formation. In this case there is l i k e l i h o o d that the s tabi l izat ion a lgor i thm is a r t i f i c ia l ly st iffening the response o f the structure, w h i c h is c lear ly undesirable. Hourg lass energy magnitudes o f less than 10% o f the internal energy are general ly considered acceptable. 3.5 Failure criteria Steel structures subjected to tensile loads beyond their ul t imate tensile strength w i l l eventual ly exhibi t tensile fracture. T h i s aspect o f structural behaviour can be predicted i n var ious ways b y finite element models . In basic models , tensile failure can be inferred b y inspect ion o f the computed stress results, ass igning failure to any loca t ion where the load/capaci ty ratio exceeds uni ty. Ref ined three-dimensional models can take this concept one step further, a l l o w i n g the inspect ion o f stress contours plot ted o n top o f the mesh such that the locat ions and dis t r ibut ion o f stress concentrations can be identif ied. A further degree o f sophist icat ion can be achieved b y implement ing a failure cr i te r ion i n the mater ial p las t ic i ty m o d e l . C o m m o n l y w i t h this approach i n d i v i d u a l elements are deleted from the mesh w h e n a c r i t i ca l value o f equivalent plast ic strain is exceeded. T h i s raises the diff icul t task o f choos ing a cr i t ica l strain value. T h e dec i s ion is not t r i v i a l , because the delet ion o f an element has significant impact o n the surrounding elements that remain intact. W e i m a r (2001) suggests that the failure strain should general ly be set based o n reduct ion i n cross-sectional area w h e n the tensile test spec imen fai ls , w h i c h is s ignif icant ly greater than the apparent true strain i n the specimen as a w h o l e , due to l oca l effects i n the neck. B y this reasoning, detai led tensile test results are required that inc lude measurements o f the f inal cross-sectional area i n the necked reg ion o f the specimen, and the suggested failure strain formula is : failure strain = In original area necked area at failure J (3.18) There is considerable debate as to the sui tabi l i ty o f the plast ic strain failure cr i ter ion. Tornqvis t (2003) points out that such an approach often makes the fracture strain mesh-30 size dependent, and argues that "equivalent plast ic strain is not suitable as a fracture cr i ter ion where a structure is subjected to b i a x i a l stress states and even i n s imple un i ax i a l tensile tests the plast ic strain is inappropriate. B y us ing the plast ic strain the effects o f stress states o n the fracture strain w i l l not be inc luded and thus the results w i l l be less sensitive to the structural arrangements." Tornqvis t provides a summary o f advanced fracture m o d e l i n g methods i nc lud ing v o i d growth, con t inuum damage mechanics , and poros i ty based models . Several mater ia l damage models are avai lable i n L S - D Y N A . M a t _ 0 0 3 (Plastic K i n e m a t i c ) and M a t _ 0 2 4 (Piecewise L i n e a r Plas t ic i ty) a l l o w the use o f the equivalent plast ic strain cr i ter ion. M a t _ 1 2 3 ( M o d i f i e d P iecewise L i n e a r Plas t ic i ty) a l lows failure to be based o n effective plast ic strain, plast ic th inning, the major p r inc ipa l in-plane strain component, a m i n i m u m t ime step size, or a user-defined failure subroutine. M o r e sophisticated con t inuum damage mechanics models are also avai lable such as M a t _ 1 0 4 (Damage 1) and M a t _ 1 0 5 (Damage 2). In a l l cases, the implementa t ion o f a failure cr i ter ion increases the number o f computations required. Therefore, g iven the uncertainty i n the failure parameters, the models described i n this research d i d not inc lude element failure, unless spec i f ica l ly specified. 31 4 MODELING, PART ONE: STUB COLUMN MODEL 4.1 Introduction A series o f stub c o l u m n finite element models were created to s imulate the stub c o l u m n tests that were performed dur ing an early part o f the experimental phase. There were three p r imary reasons for s imula t ing the stub c o l u m n tests: 1. T o gain fami l ia r i ty w i t h the relevant analysis methods avai lable i n L S - D Y N A , namely : exp l ic i t dynamic analysis, i m p l i c i t static analysis , i m p l i c i t dynamic analysis, i m p l i c i t eigenvalue analysis, and i m p l i c i t b u c k l i n g analysis . 2. T o assess the ab i l i ty o f the finite element method to m o d e l inelast ic l oca l b u c k l i n g o f the H S S stub co lumns , as this fac i l i ty w o u l d be later required to capture brace b u c k l i n g i n the fu l l C B F - V S C mode l . 3. T o investigate the sensi t ivi ty o f the computed response to var ious m o d e l i n g parameters, thus enabl ing a rat ional choice o f parameters for the fu l l C B F - V S C m o d e l . The re la t ive ly sma l l s ize o f the stub c o l u m n models described i n this chapter made them an efficient too l for pursu ing the three objectives l is ted above, wi thout risking over-s impl i f i ca t ion o f the p rob lem. Because o f parameter interaction and the path-dependent nature o f the nonl inear p rob lem be ing studied, the o n l y rat ional w a y to achieve the th i rd objective was to start w i t h a baseline m o d e l and systematical ly m o d i f y one parameter at a t ime, to assess its influence relative to the baseline case. Before d i scuss ing the finite element m o d e l i n g , the results o f the stub c o l u m n compress ion tests are first presented and evaluated. N e x t , the development o f the baseline m o d e l is discussed, and the analyses performed o n the basel ine mode l are described. Inspection o f the results o f these analyses fo l lows . F i n a l l y , the effect o f var ious m o d e l i n g parameters is assessed through a systematic parametric study, us ing selected analysis techniques. Conc lu s ions from the stub c o l u m n analyses are presented at the end o f the chapter. 4.2 Stub Column Compression Tests In an ear ly part o f the experimental phase, stub c o l u m n compress ion tests were performed o n two n o m i n a l l y ident ica l 3 0 8 m m - l o n g specimens o f 8 9 m m x 8 9 m m x 4 .8mm H S S , cut 32 f rom the same H S S pieces that were used to fabricate the braces o f the C B F - V S C system. The stub c o l u m n compress ion tests were carr ied out accord ing to standard procedures, the details o f w h i c h were described b y B u b e l a (2003). The results o f the stub c o l u m n compress ion tests are shown i n the ax i a l stress-strain curve o f F igure 4.1 and the ax i a l load-displacement curve o f F igure 4.2. The average 0 .2% offset y i e l d strength for the two specimens was 458 M P a . The average ult imate strength was 524 M P a , and the average ult imate load was 814 k N . 600 500 H 400 CL. B -»-» 00 300 200 H 100 •Specimen 1 • Specimen 2 0 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 Strain Figure 4.1 Resul ts o f stub c o l u m n compress ion tests: A x i a l stress-strain response. 33 1000 Displacement (mm) Figure 4.2 Resul ts o f stub c o l u m n compress ion tests: A x i a l load-displacement response. In their study o f H S S stub c o l u m n behaviour, K e y and H a n c o c k (1988) described the load-displacement curves according to four zones that typ i fy the behaviour o f the stub c o l u m n . T h e four zones are label led i n F igure 4.2 and can be descr ibed as fo l lows : 1. E la s t i c Zone : y i e l d i n g has not yet occurred, and the stub c o l u m n sect ion response is l inear-elastic. 2. E las t ic -P las t ic Zone : y i e l d i n g and large displacements occur , and the ul t imate load is attained. 3. Trans i t ion Zone : the load capacity reduces i n this intermediate zone, but the spatial plast ic mechan i sm has not yet developed. 4. P las t ic M e c h a n i s m Zone : inelastic loca l b u c k l i n g becomes apparent i n the development o f a spatial plastic mechanism. It was noted that the elastic stiffness reported i n Figures 4.1 and 4.2 were less than the expected values. The tensile test results indicated that the Y o u n g ' s M o d u l u s o f the H S S material was approximate ly E = 200,000 M P a . H o w e v e r , the slope o f the elastic por t ion o f the stress-strain curve o f F igure 4.1 is o n l y 100,000 M P a . S i m i l a r l y , based o n the measured cross-sectional area ( A ) and length ( L ) o f the specimens, the elastic stiffness o f 34 the load-displacement curve was expected to be k = A E / L = 1 0 1 0 k N / m m . H o w e v e r , the slope o f the elastic por t ion o f the load-displacement curve o f F igu re 4.2 is o n l y 500 k N / m m . These observations are discussed further i n Chapter 7, but they were also kept i n m i n d w h e n c ompa r ing the test results to the finite element predic t ions discussed i n the remainder o f this chapter. 4.3 Description of the Baseline Model 4.3.1 Spatial Discretization T h e stub c o l u m n to be mode led consisted o f a 3 0 8 m m - l o n g spec imen o f 8 9 m m x 8 9 m m x 4 . 8 m m H S S . T h e mesh for the stub c o l u m n m o d e l was created i n the pre-processor p rogram F e m B , us ing an impor ted A u t o C A D l ine d r awing o f the spec imen to provide the three-dimensional geometry. T h e resul t ing mesh for the basel ine stub c o l u m n m o d e l is shown i n F igure 4.3. F igure 4.3 F in i t e element mesh for the baseline stub c o l u m n m o d e l . 35 The properties d f the four-noded shel l elements that were used i n the m o d e l are descr ibed i n Sec t ion 4.3.2. It was k n o w n that the s ize and number o f elements w o u l d have a direct impact o n the computat ional t ime required to complete the stub c o l u m n analysis. Furthermore, it was anticipated that the stub c o l u m n mesh w o u l d be repl icated i n the braces o f the fu l l V S C - C B F system mode l , for w h i c h run t ime was expected to be o f increased s ignif icance. It was therefore desirable to use the coarsest stub c o l u m n mesh poss ible w h i l e main ta in ing an adequate representation o f the structural geometry. The mesh also had to be suff ic ient ly fine to capture l oca l i zed deformations and stress concentrations, and to y i e l d results that c o u l d be considered convergent o n the exact so lu t ion to the mathemat ical approximat ion o f the p rob lem. The corner w a l l geometry was meshed w i t h three elements around the radius, and then the r ema in ing flat w a l l surfaces were meshed so as to main ta in reasonable element aspect ratios throughout the m o d e l . T h e mesh s h o w n i n F igure 4.3 contained a total o f 972 elements and 1008 nodes. The flat w a l l element dimensions were 11 .5x11 .4mm and the corner w a l l element d imensions were 5 .2x11 .4mm. The flat w a l l segments o f the actual stub c o l u m n specimens had an average measured thickness o f 4 . 77mm. A l t h o u g h the corner w a l l segments had an average thickness o f 5 .185mm, the average flat w a l l thickness o f 4 . 7 7 m m was assigned to a l l elements i n the baseline m o d e l . T h e basel ine m o d e l was mode led as a perfect structure. In other words , phys i ca l imperfect ions that w o u l d be present i n the real system were not inc luded i n the coordinates o f the mesh . 4.3.2 Element Characterization The stub c o l u m n was mode led w i t h quadrilateral Lagrang ian she l l elements. T h e element formulat ion used i n the baseline m o d e l was the default L S - D Y N A T y p e 2 B e l y t s c h k o - L i n - T s a y shel l (Be ly t schko & Tsay , 1981; and B e l y t s c h k o , L i n , & Tsay , 1984), w h i c h is based o n a combined co-rotat ional ve loc i ty strain formula t ion that incorporates M i n d l i n (1951) theory o f plates and shells (Hal lquis t , 1998, p .61) . These four-noded elements have translational and rotational degrees o f freedom at each node, and use b i l inear shape functions to interpolate the computed noda l degrees o f freedom to any point i n the element. Reduced integration is used for T y p e 2 she l l elements; they have a s ingle set o f in-plane Gauss quadrature points located at the centre o f the element. 36 M u l t i p l e through-thickness integration points can be defined to capture c o m p l e x states o f stress through the she l l thickness. Changes i n shel l thickness due to membrane straining can be computed. The general process to update element internal forces was summar ized p rev ious ly i n Sec t ion 3.2.1. The baseline m o d e l used the T y p e 2 shells described above, w i t h 3 integration points through the thickness, and shel l thickness changes were tracked. A s discussed i n Sec t ion 3.4, hourglass control is required to inh ib i t the format ion o f hourglass modes i n finite element models that conta in under-integrated elements. In the baseline m o d e l , the T y p e 4 F lanagan-Bely t schko stiffness fo rm o f hourglass control (Flanagan & B e l y t s c h k o , 1981) was used w i t h the default hourglass coeff icient o f 0.10. 4.3.3 Boundary Conditions The boundary condi t ions were specif ied i n terms o f constraints o n the degrees o f freedom o f the nodes that formed the top and bot tom edges o f the tube. A t the bo t tom edge, a l l translational and rotational nodal degrees o f freedom were constrained, g i v i n g fu l ly f ixed boundary condi t ions . A t the top edge the same constraints were appl ied, except ver t ical translation was a l lowed . These boundary restraint condi t ions were presumed to adequately represent the restraint p rov ided to the actual stub c o l u m n spec imen b y the load ing plates o f the compress ion testing machine . Ex te rna l l oad ing was specif ied i n terms o f prescr ibed ver t ica l m o t i o n o f the top edge nodes. These prescr ibed m o t i o n functions are descr ibed i n Sections 4.4.1 and 4.4.2. 4.3.4 Material Model M o s t mater ial mode ls i n L S - D Y N A require as input a f l ow stress m o d e l that defines the change i n y i e l d stress w i t h effective plast ic strain. T h i s f l ow stress m o d e l , c o m b i n e d w i t h a v o n M i s e s y i e l d cr i ter ion, an associative f l ow rule, and a hardening rule ( isotropic, k inemat ic , or a combina t ion thereof), fo rm the basis o f the plast ic i ty-based consti tutive relat ionship. F r o m the w i d e var ie ty o f mater ial models avai lable i n the L S - D Y N A mater ial l ib ra ry ( L S T C , 2003), the m o d e l ca l led M a t _ 0 2 4 (Piecewise L i n e a r Plas t ic i ty) was selected for the baseline m o d e l , due to its relative s i m p l i c i t y and ab i l i t y to represent steel mater ia l behaviour . M a t _ 0 2 4 incorporates a p iecewise l inear f l ow stress m o d e l , and assumes isotropic hardening. 37 The f low stress m o d e l parameters for M a t _ 0 2 4 were obtained f rom the results o f un i ax i a l tensile tests on the respective H S S flat w a l l and corner w a l l coupons. It was necessary to convert the engineering strain and engineering stress data f rom the tensile tests to true ( logar i thmic) strain and true (Cauchy) stress values: true strain = f— = In = ln(l + engineering strain) * L [ LA . ' L X V L o j (5.1) true stress = (engineering stress) • exp(true strain) = (engineering stress) • (l + engineering strain) (5.2) The curve for the f l o w stress mode l was then determined i n terms o f y i e l d stress versus effective plast ic strain. The effective plastic strain values were ca lcula ted from the tensile test data as the res idual true strain after elastic un loading: effective plastic strain = true strain true stress (5.3) The results o f the tensile tests on the H S S flat w a l l and corner w a l l coupons are s h o w n i n Figure 4.4, i n terms o f true stress and true strain values. T h e cor responding f low stress models used w i t h M a t _ 0 2 4 are shown i n Figure 4.5. 0. 800 700 600 500 fi | 400 oo | 300 H 200 -100 -0 0.05 Flat Wall Specimen 1 Flat Wall Specimen 2 Flat Wall Specimen 3 Comer Wall Specimen 1 Comer Wall Specimen 2 Comer Wall Specimen 3 —\—i— i— i— i—i—i— i—r 0.1 0.15 True Strain 0.2 0.25 Figure 4.4 Tens i l e test results from H S S flat w a l l and corner w a l l coupons. 38 700 600 i C Flat Wall Section > 200 *— Comer Wall Section 100 H 0 0.00 0.05 0.10 0.15 0.20 0.25 Effective Plastic Strain Figure 4.5 M a t _ 0 2 4 f low stress models o f H S S flat w a l l and corner w a l l sections. F r o m F igure 4.4 the effects o f c o l d forming, namely increased y i e l d strength and decreased mater ial duc t i l i ty relative to v i r g i n steel, can be observed, par t icu la r ly i n the corner w a l l specimens. It was noted that the stress and strain data obta ined from the tensile tests contained some scatter, and the interpretation o f the Y o u n g ' s modu lus and in i t i a l y i e l d stress was therefore somewhat subjective. A value o f E=200 ,000 M P a was used for the H S S flat w a l l and corner w a l l materials, s ince this is a t yp i ca l va lue o f Y o u n g ' s modulus for steel, and was shown to give a reasonable fit to the tensile test data. A l t h o u g h B u b e l a reported y i e l d strength values based on the 0 . 2 % offset method, different in i t i a l y i e l d stress values that a l lowed a closer fit to the tensile test results were used for the M a t _ 0 2 4 f l o w stress models . B y adding the l inear elastic por t ion to the y i e l d stress vs. effective plast ic strain input data, the f low stress mode ls can be compared w i t h the true stress vs. true strain data from the tensile tests, as s h o w n i n F igu re 4.6. 39 700 600 „ 500 od a. 400 oo 300 5 H 200 100 Flat Wall Specimen 1 Flat Wall Specimen 2 - Flat Wall Specimen 3 •~|—r-i—I—i—r—r-i—r—i—i—i—i—r—i—r—i—i—I—)—i— -•—Mat_024 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 1 1 — 1 — 1 — r — 1 — 1 — 1 — 1 — | — 1 — r — ! — l — r — l — 1 — 1 — 1 — 0.05 0.1 0.15 True Strain 0.2 0.25 ~\ 1 1 1 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 True Strain Figure 4.6 C o m p a r i s o n o f M a t _ 0 2 4 f low stress mode l and tensile test results for H S S flat w a l l and corner w a l l sections. M a t _ 0 2 4 has the ab i l i ty to treat material failure b y delet ing elements w h e n a prescr ibed failure effective plast ic strain is exceeded. Howeve r , as discussed i n Sec t ion 3.5, the increased computat ional expense associated w i t h this failure cr i te r ion was decided to be unjustifiable i n v i e w o f the debate surrounding the va l id i t y o f the approach. The m o d e l can also incorporate rate effects, but these were not inc luded because steel does not 40 exhibi t strain rate sensi t ivi ty w h e n subjected to quasi-static load ing , as was appl ied i n the stub c o l u m n compress ion tests. 4.4 Analyses Using the Baseline Model The baseline stub c o l u m n m o d e l was analysed us ing five analysis methods: exp l ic i t direct integration analysis , i m p l i c i t direct integration analysis (both static and dynamic) , i m p l i c i t eigenvalue analysis, and i m p l i c i t b u c k l i n g analysis . Deta i l s o f each o f these analyses are described i n the f o l l o w i n g sections, i nc lud ing a descr ip t ion o f the differences i n the load ing condi t ions that were used for each analysis approach. 4.4.1 Explicit Direct Integration Analysis The exp l i c i t direct integration analysis w i l l be referred to hereafter s i m p l y as the expl ic i t analysis . W i t h this analysis , the nonl inear dynamic response h is tory o f the stub c o l u m n m o d e l was determined us ing the expl ic i t solver. A l t h o u g h the actual stub c o l u m n compress ion tests were quasi-static processes, the cr i t ica l t ime step o f the exp l i c i t method necessitated the use o f t ime scal ing. The stub c o l u m n was compressed b y 1 3 m m over a per iod o f 4 seconds i n the s imula t ion , whereas the same displacement was imposed over a per iod o f 850 seconds i n the actual test. In order to s imulate the d o w n w a r d movement o f the top plate o f the testing machine used i n the stub c o l u m n compress ion test, the load ing was specif ied i n terms o f prescr ibed ver t ica l m o t i o n o f the top edge nodes. F o l l o w i n g the recommendat ions o f K u t t et a l (1998), a smooth ly ramped prescribed displacement curve was sought, i n order to m i n i m i z e the inert ial effects. S u c h a load ing curve is not d i rec t ly implemented i n L S -D Y N A , but a smooth ve loc i ty curve is avai lable, w h i c h can be integrated to g ive an equivalent smooth displacement curve. T h e prescr ibed smooth v e l o c i t y curve is shown i n F igure 4.7a. The rise t ime o f this curve was 1.0 s and the magni tude o f the constant ve loc i ty por t ion was chosen such that the area under the curve gave a total ver t ical displacement o f 13mm. The equivalent displacement prof i le s h o w n i n F igu re 4.7b. 41 0 1 2 3 4 Time (s) 0 1 2 3 4 Time (s) Figure 4.7 Prescr ibed ver t ica l mot ions for top edge nodes i n the exp l i c i t analysis: a) smooth ve loc i ty prof i le ; b) corresponding smooth displacement prof i le . In exp l ic i t s imulat ions , smal l numer ica l effects and vibrat ions o f the mesh at the element leve l m a y induce high-frequency vibrat ions. These vibrat ions are reso lved i n the analysis due to the ve ry s m a l l t ime step size, and result i n numer ica l noise i n the solut ion. Furthermore, ana ly t ica l results containing such vibrat ions cannot be compared d i rec t ly w i t h the experimental results due to the f i l ter ing o f the experimental s ignal that general ly occurs i n electronic data acquis i t ion systems. Stiffness-proportional R a y l e i g h damping was therefore used to damp out the high-frequency vibrat ions i n the numer ica l solut ion. A l t h o u g h stiffness-proportional R a y l e i g h damping general ly invo lves a damping matr ix 42 that is proportional to the stiffness matrix, the global stiffness matrix is not formed in the explicit solution process. The damping terms are instead applied as corrections to the element stresses. Although the details of the stiffness-proportional damping formulation in L S - D Y N A are proprietary, the L S - D Y N A Keyword Manual does state that the stiffness-proportional damping coefficient "is defined such that a value of 0.10 roughly corresponds to 10% damping in the high frequency domain." (LSTC, 2003, 8.8). This 0.10 value was adopted in the explicit analysis of the baseline model. 4.4.2 Implicit Direct Integration Analyses As explained in Section 3.3, the implicit method can be used as an incremental response history analysis technique for both nonlinear static and nonlinear dynamic simulations. Both approaches were used to analyse the baseline stub column model. As in the explicit analysis, a duration of 4 seconds was chosen for the implicit simulations. In the case of the static analysis, "time" is not physical time; it is simply a measure in which to break the simulation into a number of increments. As in the explicit analysis, the loading was specified in terms of prescribed vertical motion of the top edge nodes. However, for the implicit analyses, this prescribed motion was a displacement curve that increased linearly from 0 to 13mm, as shown in Figure 4.8. 0 1 2 3 4 Time (s) Figure 4.8 Prescribed vertical motions for top edge nodes in the implicit analyses: linearly increasing displacement profile. 43 T h e i m p l i c i t dynamic analysis o f the baseline m o d e l was formulated near ly iden t ica l ly to the i m p l i c i t static analysis , the o n l y difference be ing that a N e w m a r k t ime integration sheme was specif ied, w i t h y=0.50 and (3=0.25. These values o f t ime integrat ion constants y=0.50 and P=0.25 y i e l d a N e w m a r k method w i t h no a lgor i thmic damping . Stiffness-propor t ional damping was not inc luded i n the i m p l i c i t dynamic analysis o f the baseline m o d e l . A n in i t i a l t ime step size o f l x l O " 5 seconds was specif ied for the i m p l i c i t analyses. A n automatic t ime step control ler was used to automat ical ly adjust the t ime step size depending o n the number o f equ i l i b r ium iterations required at each t ime step. T h e t ime step was a l l owed to be adjusted to a m a x i m u m size o f 1x10" seconds and a m i n i m u m size o f 1x10" seconds. 4.4.3 Implicit Eigenvalue Analysis A n i m p l i c i t eigenvalue analysis was carr ied out to determine the natural mode shapes and frequencies o f the basel ine mode l . D u r i n g the i m p l i c i t static analysis , the i m p l i c i t e igenvalue analysis was performed intermittently, to extract the lowest 10 eigenvalues and corresponding eigenvectors. The eigenvalue analysis was performed at 0.5 second intervals to a l l o w invest igat ion o f h o w the v ibra t ion characteristics o f the m o d e l change i n response to changes i n geometry and stress. 4.4.4 Implicit Buckling Analysis A nonl inear i m p l i c i t b u c k l i n g analysis was performed w i t h the static i m p l i c i t s imula t ion . Af te r loads were appl ied to the mode l , the b u c k l i n g e igenproblem was so lved to determine b u c k l i n g mode shapes and eigenvalues that represent mul t ip l ie r s o f the appl ied loads that w o u l d cause the structure to buck le i n the g iven mode. T h e b u c k l i n g analysis was carr ied out before and after y i e ld ing o f the stub c o l u m n , i n order to observe whether any change occurred i n the predicted b u c k l i n g modes or b u c k l i n g loads. 4.4.5 Numerical Precision Single p rec i s ion f loating-point numbers are stored i n computer m e m o r y w i t h 4 bytes (32 bits) whereas double p rec i s ion floating-point numbers are stored w i t h 8 bytes (64 bits). 44 D o u b l e p rec i s ion numbers a l l o w an increased number o f digi ts i n the s ign i f icand as w e l l as an increased range o f numbers to be represented. F o r m a n y finite element computations, s ingle p rec i s ion is sufficiently accurate. H o w e v e r the s m a l l t ime step used w i t h exp l ic i t calculat ions can mean that m i l l i o n s o f computat ional cycles are required and numer ica l round o f f errors can accumulate. It is for this reason that L S - D Y N A suppliers r ecommend us ing the double prec is ion executable o f L S - D Y N A for p rob lems i n v o l v i n g more than 500,000 cycles ( C A D - F E M , 2004). The drawback o f us ing double p rec i s ion is that the associated run t ime can increase b y a factor o f two or more depending o n the comput ing pla t form, whereas the increased p rec i s ion m a y have l i t t le effect o n the solu t ion computed. The baseline i m p l i c i t analyses were run w i t h single p rec i s ion because o f the l o w number o f cycles i nvo lved , w h i l e the exp l i c i t analysis was run w i t h double prec i s ion . In order to achieve the most accurate p red ic t ion o f v ibra t ion characteristics and b u c k l i n g modes, double p rec i s ion was used for the i m p l i c i t eigenvalue analysis and b u c k l i n g analysis. 4.5 Analysis Results for Baseline Model 4.5.1 Results of Direct Integration Analyses The results o f the expl ic i t analysis, i m p l i c i t static analysis, and i m p l i c i t d y n a m i c analysis are s h o w n i n terms o f the ax i a l load-displacement curves o f F igu re 4.9. In these curves the ax ia l load is the top edge noda l force react ion i n the ver t ica l (Z) d i rec t ion , and the displacement is the corresponding Z-d i rec t ion displacement o f the top edge nodes. T h e response o f the experimental stub c o l u m n specimen is also s h o w n i n F igure 4.9 for compar ison . T h e exp l i c i t analysis predicted an ult imate load o f 836 k N , w h i c h is 2 . 7% higher than the test average o f 814 k N . The i m p l i c i t analyses predicted an ult imate load o f 844 k N , w h i c h is 3 .7% higher than the test average. B o t h the exp l i c i t and i m p l i c i t analysis results showed the same four zones o f stub c o l u m n response descr ibed i n Sec t ion 4.2, namely the elastic, elastic-plastic, transition, and plast ic mechan i sm zones. The finite element models predicted an elastic load-displacement stiffness o f 1055 k N / m m , w h i c h was w i t h i n 1% o f the expected value based o n the cross sect ional area, length, and 45 Y o u n g ' s M o d u l u s o f the baseline mode l . Howeve r , as noted i n Sec t ion 4.2, the test results contained an unexpectedly l o w elastic stiffness o f 500 k N / m m . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Displacement (mm) Figure 4.9 Base l ine analysis and experimental test results: A x i a l load-displacement response o f stub c o l u m n under compress ion . The f inal deformed shape from each analysis is shown i n F igure 4 .10. A photograph o f the f inal deformed shape o f the experimental specimen is s h o w n i n F igu re 4 .11 . F igure 4.10 Base l ine analysis results: Deformed shape after 1 3 m m o f top end displacement: a) exp l i c i t analysis; b) imp l i c i t static analysis; c) i m p l i c i t d y n a m i c analysis. 46 Figure 4.11 Exper imenta l test results: De fo rmed shape after 1 7 m m o f top end displacement. It is apparent that the finite element analyses predicted a symmet r ic extensional mode o f loca l b u c k l i n g , whereas i n the experiment an asymmetr ic mode was observed. The asymmetr ic mode has also been observed i n experiments b y other researchers ( K e y & Hancock , 1988, R o n d a l & M a q u o i , 1985, Y u & T a l l , 1971) and is genera l ly presumed to be the expected failure mode. A l t h o u g h the extensional mode s h o w n i n F igure 4.10 is mathemat ical ly poss ible for a perfect tubular structure, this m o d e requires the development o f membrane stresses i n the hoop direct ion. In contrast, the asymmetr ic mode, i n w h i c h two oppos ing wal l s fo ld inward and the other two oppos ing wa l l s fo ld outward, does not require the same development o f hoop membrane stress. A s a result, the asymmetr ic mode is a l o w energy mode that should be " f avoured" b y the structure. T h e larger area under the load-displacement curve i n the plast ic m e c h a n i s m zone o f the finite element results c o u l d be interpreted as i m p l y i n g that more energy was absorbed through the extensional deformation mode compared to the asymmetr ic m o d e observed i n the experiment. X u e , L i n , and J iang (2000) conducted finite element studies o f the dynamic crushing behaviour o f th in-wal led m i l d steel square tubes. T h e col lapse behaviour o f steel tubes is different under impact load ing condi t ions compared to the quasi-static compress ion tests considered i n the present study. Nevertheless , it was 47 interesting to note their f ind ing that the finite element mode ls exhib i ted extensional mode b u c k l i n g unless i n i t i a l geometric imperfections were inc luded i n the m o d e l . Nodal -averaged contour plots o f v o n M i s e s stress i n the f inal conf igura t ion for each analysis are shown i n F igure 4.12. The invariant v o n M i s e s stresses, or effective stresses, provide a convenient measure o f stress i n three dimensions . V o n M i s e s stress, Ce, is defined i n terms o f the p r inc ipa l stress components as: a e ~ [ ( a , - a 2 ) 2 + ( a 2 - a 3 ) 2 + ( a 3 - a 1 ) 2 f / 2 (5.4) It can be seen from Equa t ion 5.4 that the von M i s e s stress is equivalent to the y i e l d stress w h e n the load ing condi t ions are un iax ia l (o"i>0; 02=03=0). T h e v o n M i s e s stress measure is attractive because its magnitude is independent o f the coordinate sys tem i n w h i c h it is computed. S i m i l a r l y , effective plastic strain is a convenient three-dimensional strain measure. Contours o f effective plastic strain i n the f inal conf igura t ion for each analysis are shown i n F igure 4 .13. 48 Figure 4.13 Base l ine analysis results: Contours o f effective plas t ic strain: a) exp l ic i t analysis; b) imp l i c i t static analysis; c) i m p l i c i t d y n a m i c analysis . F r o m F igure 4.13, effective plastic strains o f approximate ly 2 0 % can be seen i n the region o f the spatial plast ic mechanism. The associated effective stresses are 640 M P a i n the corner w a l l sections o f the H S S and 610 M P a i n the flat w a l l sections, w h i c h correspond to the m a x i m u m material stresses i n the f low stress mode ls f rom F igure 4.5. C o m p a r i n g the i m p l i c i t static analysis results to the i m p l i c i t dynamic analysis results, it was not possible to discern any significant differences i n the load-displacement curves, the deformed shapes, the strain contours, or the stress contours. T h i s was taken as an ind ica t ion that the p r o b l e m had been successfully mode led as quasi-static. C o m p a r e d to the i m p l i c i t analysis results, the expl ic i t analysis predicted a s imi l a r extensional spatial plastic mechanism, but it was formed i n the upper por t ion o f the stub c o l u m n rather than the lower por t ion. Plots o f internal strain are shown i n F igure 4.14 for both the exp l i c i t and i m p l i c i t analyses. The strain energy profiles mir ror the prescribed displacement profi les for each analysis. The i m p l i c i t static and i m p l i c i t dynamic analyses showed ident ica l internal energy histories, w h i c h indicated that they shared ident ica l deformat ion histories. T h e m a x i m u m internal energies for the expl ic i t and imp l i c i t analyses were 10.1 k J and 9.85 k J respectively. The m a x i m u m kinet ic energies for the exp l i c i t analysis and the i m p l i c i t 49 dynamic analysis were 5.4xl0"6 kj and 2.5xlO"3 kJ, respectively. The low kinetic/internal energy ratios for both dynamic analyses suggests that inertial effects were sufficiently minimized. Time (s) Figure 4.14 Baseline analysis results: Internal energy histories. It is somewhat difficult to compare the run times of the explicit, implicit static, and implicit dynamic analyses, because each of these were solved on a different computer with different processing power. The explicit analysis took approximately 9.5 hours, whereas both implicit analyses were executed in approximately 40 minutes. The implicit static and dynamic analyses were completed in approximately the same time because each had the same step size and the majority of the computational effort for implicit analyses is attributed to the storing and inverting of matrices and solving nonlinear equations at each step, which is required for both static and dynamic analyses. 4.5.2 Results of Eigenvalue Analysis The lowest five eigenvalues and eigenvectors were extracted at 0.5s intervals from the implicit static analysis of the baseline model. The mode shapes from t = 0 are shown in Figure 4.15, and the corresponding frequencies and periods are listed in Table 5.1. The variation of modal periods throughout the implicit static analysis is shown in Figure 4.16, which shows how the periods changed in response to changing geometry and stress conditions. The periods lengthened as the material yielded during the elastic-plastic 50 phase, but then shortened due to the change i n geometry i n the t ransi t ion and plast ic mechanism phases. C o m p a r i n g the lowest per iod value to the largest pe r iod value for each mode throughout the duration o f the s imulat ion, variat ions o f be tween 2 5 % - 6 8 % are observed. F igure 4.15 Impl i c i t eigenvalue analysis results: F i rs t f ive mode shapes at t=0. Table 4.1 Impl ic i t e igenvalue analysis results: F is t f ive frequencies and per iods at t=0. M O D E Eigenva lue C i r c . F req . (rad/sec) Freq . ( H z ) P e r i o d (sec) 1 9.86E+07 9.93E+03 1.58E+03 6 .33E-04 2 1.58E+08 1.26E+04 2.00E+03 4 .99E-04 3 1.96E+08 1.40E+04 2.23E+03 4 . 4 9 E - 0 4 4 1.96E+08 1.40E+04 2.23E+03 4 . 4 9 E - 0 4 5 2 .86E+08 1.69E+04 2.69E+03 3 .71E-04 51 •o o "C Q. -•— 1st Mode 2nd Mode -©— 3rd Mode -*—4th Mode 5th Mode 1.00E-03 9.00E-04 8.00E-04 7.00E-04 6.00E-04 5.00E-04 4.00E-04 3.00E-04 2.00E-04 1.00E-04 O.OOE+00 0 1 2 3 4 Simulation Time Figure 4.16 Implicit eigenvalue analysis results: Variation of first five periods throughout implicit static analysis. 4.5.3 Results of Buckling Analysis The first five buckling modes were obtained at t=0.1 and t=0.5, corresponding to times of pre- and post-yield conditions in the model. The fist five buckling modes predicted at t=0.1 are shown in Figure 4.17. The buckling modes predicted at t=0.5 were identical, except that the order of modes 3 and 4 were switched. 1 2 3 4 5 Figure 4.17 Implicit buckling analysis results: First five buckling modes, t=0.1. 52 The eigenvalues obtained b y the b u c k l i n g analysis represent mul t ip l ie r s to the current load ing that w o u l d cause bifurcat ion. The loads, load mul t ip l ie r s , and resul t ing b u c k l i n g load predict ions are l is ted i n Table 5.2 for both t=0.1 and t=0.5. Tab le 4.2 Impl i c i t b u c k l i n g analysis results: B u c k l i n g load predict ions . t = 0.10s ( L o a d = 3 4 3 k N ) t = 0.50s ( L o a d = 759 k N ) B u c k l i n g M o d e B u c k l i n g L o a d M u l t i p l i e r Predicted B u c k l i n g L o a d [kN] B u c k l i n g L o a d M u l t i p l i e r Predic ted B u c k l i n g L o a d [kN] 1 10.51 3605 4.60 3489 2 10.83 3716 4.68 3554 3 13.22 4534 5.37 4080 4 . 13.36 4584 5.71 4332 5 14.96 5131 6.35 4816 F r o m Tab le 5.2 it is clear that the b u c k l i n g loads predicted b y the b u c k l i n g analysis are m u c h higher than the ult imate load observed i n the fu l l finite element analyses and the actual stub c o l u m n compress ion test. T h i s is l i k e l y because the b u c k l i n g analysis is s o l v i n g for bifurcat ion, whereas the stub c o l u m n fails b y loca l plast ic mechan i sm. 4.6 Parametric Studies A suite o f mod i f i ed models was created, w i t h each mod i f i ed m o d e l conta in ing a modi f i ca t ion to a single parameter from the basel ine m o d e l . In the f o l l o w i n g sections, the formulat ion o f these mod i f i ed models is described a long w i t h the analysis results. Because the i m p l i c i t analyses o f the baseline m o d e l were comple ted i n s ign i f ican t ly less run t ime than the expl ic i t analyses, the i m p l i c i t solver was used for the major i ty o f the parametric studies. The static i m p l i c i t approach was taken instead o f the dynamic i m p l i c i t approach, since both approaches gave s imi la r results for the basel ine m o d e l . S o m e parameters were investigated w i t h the expl ic i t solver. T h e sect ion titles indicate w h i c h solver was used for each parametric study. 53 4.6.1 Effect of Varying the Spatial Discretization of the Model (Implicit Solver) A s shown i n F igure 4 .3 , the flat w a l l portions o f the baseline m o d e l were meshed w i t h 6 co lumns b y 27 rows o f elements, w h i l e the corner w a l l port ions conta ined 3 co lumns b y 27 rows o f elements. Thus the flat w a l l element d imensions were 11 .5x11 .4mm and the corner w a l l element d imensions were 5 .2x11.4mm, and the m o d e l contained a total o f 972 elements and 1008 nodes. A mod i f i ed mode l was created i n w h i c h the mesh was refined b y a factor o f four, g i v i n g 3888 elements and 3960 nodes. A second m o d i f i e d m o d e l was created i n w h i c h the flat w a l l port ions o f the mesh were coarsened to 5 co lumns b y 27 rows (each element be ing 13.8mmx 11.4mm), w h i l e the corner w a l l portions were unchanged. The coarsened mode l contained 864 elements and 896 nodes. The refined mesh m o d e l , baseline mode l , and coarsened mesh m o d e l are s h o w n i n F igure 4.18. In a third mod i f i ed mode l , the corner w a l l element thicknesses were changed f rom the baseline value o f 4 . 7 7 m m to the average measured corner w a l l th ickness o f 5 .185mm. F igure 4.18 Spatial discret izat ion parametric study: a) Re f ined mesh; b) baseline mesh; c) coarse mesh. 54 The load-displacement response for the three mod i f i ed mode ls descr ibed above is shown together w i t h the baseline m o d e l and experimental results i n F igure 4.19. T h e refined mesh gave ve ry s imi la r results to the baseline m o d e l , w i t h s l igh t ly earl ier format ion o f the spatial plast ic mechan i sm and corresponding drop i n the load ca r ry ing capacity. T h e coarse mesh resulted i n nearly ident ica l response to the basel ine m o d e l . In the m o d e l w i t h increased corner w a l l thickness, the load-displacement curve showed an upward shift as expected, ref lect ing the increased capacity due to the addi t ional mater ial i n the corners. 1000 -| 1 Displacement (mm) Figure 4.19 Effect o f va ry ing the spatial d iscret izat ion: A x i a l load-displacement response. 4.6.2 Effect of Varying the Element Characterization Parameters (Implicit Solver) T h e baseline i m p l i c i t m o d e l used T y p e 2 B e l y t s c h k o - L i n - T s a y shells , w i t h 3 integration points through the thickness, and shel l thicknesses were updated i n response to membrane straining. A mod i f i ed m o d e l us ing reduced-integration T y p e 1 H u g h e s - L i u shells was created, to investigate the sensi t ivi ty o f the p r o b l e m to the choice o f element formulat ion. Ano the r m o d e l was created us ing T y p e 16 fully-integrated elements, to 55 compare the cases o f reduced and fu l l integration. Sens i t iv i ty to the number o f through-thickness integration points was investigated w i t h two models that used 5 and 7 Gauss quadrature points , respectively. F i n a l l y , the importance o f shel l thickness updates was investigated b y creating a m o d e l i n w h i c h these updates were not performed. The load-displacement curves o f the f ive mod i f i ed models descr ibed above are shown together w i t h the baseline m o d e l and experimental results i n F i g 5.20. A s can be seen i n F igure 4.20, neither changing to T y p e 1 shells nor increasing the number o f through-thickness integration points had a noticeable effect o n the load-displacement response o f the stub c o l u m n m o d e l . A l t h o u g h the g loba l response o f the m o d e l is not not iceably affected b y increas ing the number o f through-thickness integrat ion points , cross-sectional stresses and force resultants at the element l eve l cou ld be better reso lved i n the case o f a more complex through-thickness stress dis tr ibut ion. U s i n g ful ly-integrated elements postponed the format ion o f the plast ic mechan i sm b y a sma l l amount. T h i s was l i k e l y due to the fact that fully-integrated elements are general ly stiffer than reduced-integration elements. W h e n shel l thicknesses were not updated due to membrane straining, the load-displacement curve showed an earlier format ion o f the plast ic mechan i sm, w h i c h was poss ib ly due to the fact that the shel l thicknesses were not increased w i t h compress ion . H o w e v e r , i n a l l f ive o f the mod i f i ed models described above the f ina l deformat ion mode was the same as the basel ine mode l . 56 1000 900 H Displacement (mm) Figure 4.20 Effect of varying the element characterization parameters: Axial load-displacement response. 4.6.3 Effect of Varying the Hourglass Control Type and Settings (Implicit Solver) The implicit baseline models used Type 4 Flanagan-Belytschko stiffness form of hourglass control with the default hourglass coefficient of 0.10. In order to determine if the hourglass coefficient of 0.10 was unnecessarily high, resulting in excessive artificial stiffening of the response, a model was created with a reduced hourglass coefficient of 0.01. In addition, two models were created with Type 1 standard LS-DYNA viscous hourglass control, using hourglass coefficients of 0.10 and 0.01, respectively. The load-displacement curves of the three modified models described above are shown together with the baseline model and experimental results in Fig 5.21. Varying the hourglass control type and coefficient magnitude did not have a noticeable effect on the load-displacement response of the stub column model. In fact, the curves in Figure 4.21 lie directly on top of each other. Likewise, the deformation modes of the three modified models were the same as the baseline model. 57 1000 900 •] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Displacement (mm) Figure 4.21 Effect o f va ry ing the hourglass control type and coefficient: A x i a l load-displacement response. F igure 4.22 shows the t ime-var ia t ion o f hourglass energy for each o f the four models studied. A s ment ioned previous ly , hourglass energy is an ind ica t ion o f the amount o f energy expended b y the s tabi l izat ion algori thm to resist hourglass ing effects. The models us ing T y p e 4 hourglass control y ie lded higher hourglass energies, and for bo th types o f hourglass control considered hourglass energy was reduced for l o w e r hourglass control coefficients. H o w e v e r , the m a x i m u m hourglass energy i n a l l cases was less than 1% o f the total internal strain energy (10.06 kJ) , w h i c h suggests that the tendency towards hourglass ing i n this p rob lem was adequately contro l led b y a l l o f the s tabi l izat ion algori thms investigated, and ar t i f ic ia l stiffening o f the response d i d not occur to any significant extent. 58 0 1 2 3 Time (s) Figure 4.22 Effect of varying the hourglass control type and coefficient: Hourglass energy. 4.6.4 Effect of Varying the Material Model (Implicit and Explicit Solver) The baseline model used the Mat_024 Piecewise-Linear material model. A modified model was created with the Mat_003 Plastic-Kinematic material model, which uses a bilinear flow stress model defined simply in terms of the Young's modulus, the yield stress, and a tangent plastic modulus. Isotropic, kinematic, or a combined hardening rule can be assigned, allowing substantial control of hysteretic material behaviour. In Figure 4.23, the Mat_003 flow stress model is compared with the true stress vs. true strain data from the tensile tests. It was of interest to determine whether this simpler bi-linear model would be adequate to capture the nonlinear response of the stub-column. 59 700 0 0.05 0.1 0.15 0.2 0.25 True Strain 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 True Strain Figure 4.23 M a t e r i a l m o d e l parametric study: C o m p a r i s o n o f M a t _ 0 0 3 flow stress m o d e l and tensile test results for H S S flat w a l l and corner w a l l sections. A s shown i n F igure 4.24, the elastic-plastic por t ion o f the stub c o l u m n load-displacement curve was crudely approximated when the M a t _ 0 0 3 m o d e l was used w i t h the exp l ic i t solver. H o w e v e r , the pred ic t ion o f the ultimate load and pos t -buck l ing behaviour o f the s tub-column were s imi l a r to the baseline case. The deformation mode resul t ing from the exp l i c i t analysis is s h o w n i n Figure 4.25. Interestingly, the spatial p las t ic mechan i sm was 60 triggered i n the l ower por t ion o f the stub co lumn , g i v i n g a f inal deformed shape that was s imi la r to the i m p l i c i t analysis o f the baseline mode l (wi th M a t _ 0 2 4 ) . 1000 T 900 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Displacement (mm) Figure 4.24 Effect o f us ing bi- l inear mater ial m o d e l M a t _ 0 0 3 : A x i a l load-displacement response. F igure 4.25 Effect o f us ing bi- l inear mater ial m o d e l M a t _ 0 0 3 : Deformed shape, expl ic i t solver. 61 W h e n the M a t _ 0 0 3 m o d e l was run w i t h the imp l i c i t solver , unsatisfactory results were obtained. The i m p l i c i t solver experienced convergence d i f f icu l t ies , and some discontinuit ies can be seen i n the load-displacement results o f F igu re 4.24. M o r e o v e r , as s h o w n i n F igure 4.26, the hourglass control energy was several orders o f magnitude greater than the internal energy; another indicator that a poor so lu t ion was achieved. The deformed shape o f the M a t _ 0 0 3 m o d e l shown i n F igure 4 .27a l o o k e d reasonable at first, un t i l it was rea l ized that the asymmetric b u c k l i n g mode had actual ly been triggered b y numer ica l divergence and excessive hourglassing. At tempts were made to mit igate the p rob lem b y adjusting the hourglass control coefficients, but the d ivergence problems persisted and the resul t ing deformed shapes were not improved , as seen i n F igure 4.27 b and c. Al te rna t ive attempts to mitigate the p rob lem b y reduc ing the i m p l i c i t t ime step size were also unsuccessful . 900 800 700 c r 600 f 500 BP « 400 B LU 300 200 100 -j 0 2 Time (s) Hourglass Energy • Internal Energy Figure 4.26 Effect o f us ing bi- l inear mater ia l m o d e l M a t _ 0 0 3 : C o m p a r i s o n o f internal energy and hourglass energy, i m p l i c i t solver . 62 Figure 4.27 Effect o f us ing bi- l inear mater ial m o d e l M a t _ 0 0 3 : De fo rmed shape, i m p l i c i t solver: a) Hourglass coefficient = 0.10 (baseline value); b) hourglass coefficient = 0.05; c) hourglass coefficient = 0.15. 4.6.5 Effect of Including Geometric Imperfections (Implicit and Explicit Solver) B u c k l i n g is k n o w n to be an imperfection-sensit ive p rob lem. The f indings o f X u e et a l (2000) suggest that geometr ic imperfections must be inc luded i n the finite element mode l i n order to achieve the exper imental ly observed asymmetr ic b u c k l i n g mode . Therefore, the baseline m o d e l was mod i f i ed to include vary ing levels o f i n i t i a l imperfect ions. Rus t and S c h w e i z e r h o f (2003) discussed four c o m m o n methods o f in t roduc ing imperfect ions into finite element models for th in-wal led structures, i n the absence o f imperfec t ion measurement data. These include imperfect ion patterns from eigenvalue or b u c k l i n g analyses, random noda l coordinate perturbations, and s inusoida l funct ion generators. Rus t and S c h w e i z e r h o f also discussed the so-cal led " d y n a m i c imper fec t ion . " D y n a m i c imperfections result from h i g h frequency osci l la t ions i n transient analyses, w h i c h act to trigger b u c k l i n g . T h i s latter type w o u l d not be appl icable i n the case o f static i m p l i c i t 63 analyses but might appear i n expl ic i t analyses, par t icular ly those i n v o l v i n g h i g h veloci t ies or l o w damping . L S - D Y N A and its compat ib le pre-processors do not p rov ide a b u i l t - i n u t i l i ty to introduce geometric imperfect ions into the finite element m o d e l . Therefore a cus tomized spreadsheet was created to calculate the imperfect noda l coordinates, w h i c h were then incorporated into the m o d e l input deck. Impl ic i t eigenvalue and b u c k l i n g analyses were performed to obtain the v ibra t ional and b u c k l i n g mode shapes s h o w n i n F igures 4.15 and 5.17, respect ively. F r o m Figure 4.17 it can be seen that the b u c k l i n g mode shapes consisted o f s inusoidal patterns o f decreasing wavelengths, and that the 1 s t b u c k l i n g mode shape was near ly ident ica l to the 5 v ibra t ional mode shape. A l t h o u g h an imperfect ion pattern based o n the 1 s t b u c k l i n g mode shape w o u l d l i k e l y tr igger b u c k l i n g i n the desired mode, the 1 s t v ibra t ional mode shape was closer to the imperfec t ion pattern reported b y K e y and H a n c o c k (1988) from their measurements o n H S S real stub co lumns . K e y and H a n c o c k quantif ied imperfect ions i n stub co lumns b y the ratio wn/b, where wo is the m a x i m u m devia t ion from the perfect geometry and b is the general ized plate w i d t h (outside face d imens ion minus two times the w a l l thickness), as s h o w n i n F igure 4.28. T h e y considered w 0 / b ratios o f 0 .0001, 0.0005, 0.001, and 0.005. K e y and H a n c o c k used the 0.005 ratio for most o f their analyt ical work , but noted that the 0.001 ratio was the most representative o f the measured imperfect ion magnitudes. A s imi l a r approach was taken for the present parametric study, i n w h i c h the noda l coordinates o f the 1 s t vibra t ional mode shape were scaled to give imperfect ion patterns w i t h m a x i m u m wo/b ratios o f 0.001, 0.002, 0.003 and 0.005. 64 Figure 4.28 Geomet r ic imperfect ion study: D e f i n i t i o n o f Wn /b ratio. The results o f i nc lud ing geometric imperfections are most read i ly seen b y observing the deformation modes shown i n Figures 4.29 and 4.30. W h e n the static i m p l i c i t solver was used, the deformation mode was s imi la r to the exper imental ly observed asymmetr ic mode for imperfec t ion ratios w 0 / b o f 0.002 and larger. W h e n the exp l i c i t solver was used, the expected deformation mode was achieved even for a wn / b ratio o f 0 .001. It is possible that the dynamic imperfect ion phenomenon helped to trigger the asymmetr ic mode at the lower imperfec t ion ratio. A s discussed later i n Sect ion 4.5.12, the exp l i c i t models that were run w i t h the single prec is ion executable often showed the asymmetr ic deformation mode even wi thout the i nc lu s ion o f geometric imperfect ions. T h i s c o u l d have been the result o f combined dynamic imperfections and round-of f error induced imperfect ions. F o r both the i m p l i c i t and expl ic i t analyses, the locat ion o f the spatial p las t ic mechan i sm changed to the mid-height o f the stub c o l u m n w h e n the wn /b ratio was 0.005. 65 Figure 4.29 Effect o f i nc lud ing geometric imperfections: D e f o r m e d shapes, i m p l i c i t analyses: a) w 0 / b = 0.001; b) w 0 / b = 0.002; c) wrVb = 0.003; d) wo/b = 0.005. F igure 4.30 Effect o f i nc lud ing geometric imperfections: D e f o r m e d shapes, exp l ic i t analyses: a) w 0 / b = 0.001; b) wo/b = 0.002; c) w 0 / b = 0.003; d) wo/b = 0.005. 66 The format ion o f the asymmetr ic b u c k l i n g mode w h e n suff ic ient ly large geometric imperfections were inc luded resulted i n ax ia l load-displacement response that dropped o f f more dras t ical ly i n the plastic mechanism range, as s h o w n i n F i g u r e 4 .31 . 1000 i — i 900 A 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Displacement (mm) Figu re 4.31 Effect o f i nc lud ing geometric imperfect ions: A x i a l load-displacement response. 4.6.6 Effect of Varying the Amount of Damping (Explicit Solver) In the expl ic i t analysis o f the baseline mode l , 10% stiffness-proportional damping was inc luded. In order to assess the effect o f the damping l eve l on the response o f the stub c o l u m n mode l , mod i f i ed models were created wi th stiffness d a m p i n g coefficients o f 0.00 (no damping) , 0.02, 0.05, 0.15, and 0.20. The models w i t h d a m p i n g coefficients o f 0.02 and 0.05 produced results that were near ly ident ica l to the basel ine case, suggesting that, for the load ing condi t ions considered, damping levels as l o w as 2 % can effectively damp out any h igh frequency osci l la t ions that might occur i n the stub c o l u m n mode l . However , w h e n d a m p i n g was e l iminated, the load-displacement response was erratic and the tube deformed i n an unusual shape, as shown i n F igure 4.32(a). S i m i l a r effects were observed i n the mode ls w i t h h i g h damping 67 values (0.15 and 0.20), as shown i n F igure 4.32(b) and (c). A s exp la ined i n Sec t ion 3.2, damping has the effect o f l ower ing the c r i t i ca l t ime step i n exp l i c i t analyses. Furthermore, the L S - D Y N A documentat ion suggests that the stiffness-proportional damping a lgor i thm is k n o w n to destabil ize for large d a m p i n g values. Therefore, these models were re-run w i t h a reduced m a x i m u m time step. The t ime step was l imi t ed to 2/3 o f the automat ica l ly-computed c r i t i ca l t ime step. W i t h this mod i f i ca t ion , the mode ls w i t h 0.15 and 0.20 damping coefficients produced smooth load-displacement responses and deformed shapes s imi l a r to the baseline model . H o w e v e r , reduc ing the c r i t i ca l t ime step d i d not improve the zero damping case. The load-displacement curves for the damping study are shown i n F igu re 4.33. DAMPINGCOEFFEQ0 DAMPINGCOEFFEQ0.15 DAMPINGCOEFFEQ0.20 Time = 2.5321 T i m e = A Time = A Figure 4.32 Effect o f v a r y i n g the stiffness-proportional damping coeff icient : De fo rmed shapes w i t h automatical ly computed m i n i m u m t ime step s ize: a) no damping ; b) damping coefficient = 0.15; c) damping coeff icient = 0.20. 68 1000 5 6 7 8 9 10 11 12 13 14 Displacement (mm) Figure 4.33 Effect o f va ry ing the stiffness-proportional d a m p i n g coefficient: A x i a l load-displacement response. 4.6.7 Effect of Varying the Prescribed Motion Curve (Explicit Solver) A smooth ly va ry ing ve loc i ty curve was used i n the exp l i c i t analysis o f the basel ine mode l , i n an attempt to m i n i m i z e dynamic effects. In order to investigate the sensi t ivi ty o f the response to the shape o f the load ing function, the constant v e l o c i t y curve shown i n F igure 4.34a was tested. T h e corresponding l inear ly increasing d isp lacement curve was the same as the displacement curve prescribed i n the i m p l i c i t analyses o f the baseline mode l , and is shown i n F igure 4.34b. 69 0 1 2 3 4 0 1 2 3 4 Time (s) Time (s) Figure 4.34 Al te rna t ive prescribed ver t ical mot ions for top edge nodes i n the expl ic i t analysis: a) constant ve loc i ty prof i le ; b) corresponding l inear ly increasing displacement prof i le . There were no noticeable differences i n the expl ic i t analysis results w h e n the ve loc i ty prof i le o f F igure 4.34a was adopted. The deformed shapes, load-displacement curves, strain results, and stress results were a l l unchanged. T h i s outcome was l i k e l y due to the fact that the load ing durat ion o f 4 seconds was suff icient ly l o n g to simulate quasi-static condi t ions . The effect o f load ing durat ion i n the expl ic i t analysis was investigated b y m o d i f y i n g the t ime scale o f the smooth ve loc i ty curve shown i n F igure 4.5, w h i c h o r i g i n a l l y terminated at t = 4.0 s. F r o m the eigenvalue analysis results presented i n Sec t ion 4.4.2, the fundamental pe r iod o f the stub c o l u m n m o d e l was determined to be T i = 6.3x10" 4 s. F o u r mod i f i ed ve loc i ty curves were created, i n w h i c h the load ing durat ion was adjusted to be equal to 5 T , = 0.0031 s, 10Ti = 0.0063 s, 100T, = 0.0630 s, and 2 0 0 T i = 0.126 s. In each case the rise t ime o f the ve loc i ty curve was mainta ined at 2 5 % o f the total load ing duration. The deformed shapes that resulted from the increased load ing rates can be seen i n the elevat ion profi les shown i n F igure 4.35. F o r load ing durations o f 5 T i and 1 0 T i , the beginnings o f two spatial plast ic mechanisms can be seen - one near the top o f the stub c o l u m n and one near the bot tom. A s the loading durat ion was lengthened to l O O T i and 70 200Ti, the deformed shape approached the shape obtained from the o r ig ina l exp l i c i t analysis o f the baseline m o d e l . F igure 4.35 Effect o f load duration for expl ic i t analysis : D e f o r m e d shapes: a) durat ion = 5 T i ; b) durat ion = 10T, ; c) duration = l O O T i ; d) dura t ion = 2 0 0 T i . The effects o f load durat ion can also be seen i n terms o f the load-displacement curves shown i n F igure 4.36. The spatial plastic mechanism zones o f the curves approach the or ig ina l baseline analysis as the load ing duration is increased. 71 1000 900 j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Displacement (mm) Figure 4.36 Effect o f load duration for exp l i c i t analysis : A x i a l load-displacement response. Decreas ing the durat ion o f the load ing made a direct and s ignif icant impact on the run times for the expl ic i t analyses. A s discussed above, the results for the analysis w i t h duration o f 100T] = 0.0630 s gave essentially the same results as the o r ig ina l analysis w i t h durat ion o f 4 seconds, but the run t ime for the up-speeded m o d e l was reduced from 9.5 hours to 10 minutes . The run times for a l l the load ing durations are compared i n Table 5.3. Table 4.3 Effect o f load duration o n run t ime for double p rec i s ion exp l i c i t analyses. L o a d i n g Dura t ion R u n T i m e (hrs.:min.:sec.) 5 T i = 0.0031 s 0:00:47 10Ti = 0.0063 s 0:01:07 100T] = 0.0630 s 0:10:35 2 0 0 T , = 0.1260 s 0:20:57 Base l ine , E x p l i c i t A n a l y s i s 4.0000 s 9:20:50 72 4.6.8 Effect of Using Contact Boundary Conditions (Explicit Solver) Constraints and prescribed motions on the boundary nodal degrees of freedom were used in the analyses of the baseline model to approximate the boundary conditions of the actual stub-column test. In order to model the actual boundary conditions more precisely, the contact options available in LS-DYNA were investigated. Several contact options are available in LS-DYNA, most of which are based on algorithms that check for penetration of "slave" nodes or segments against "master" nodes or segments. If penetration is detected then a stiffness-proportional opposing force is automatically applied to counteract the penetration. A flat top plate part was created using shell elements to model the top bearing plate of the compression testing machine. Contact was defined between the plate surface and the top edge nodes of the stub column using the *Contact_Nodes_to_Surface keyword. By assigning rigid material properties to the plate part, the run time was optimized because computation of rigid part deformations are not performed by LS-DYNA. A corollary of this approach was that the plate mesh could be set arbitrarily fine, to maximize the accuracy of the contact calculations, without affecting the critical time step. Two separate approaches were taken to modeling the bottom bearing plate. In the first approach, the bottom bearing plate was defined by a stationary rigid horizontal plane of infinite dimensions, using the *Contact_Entity keyword. In the second approach, the bottom bearing plate was defined by a flat rigid plate part, just as was done for the top plate, using the *Contact_Nodes_to_Surface keyword to define the contact. In both models described above, the slave components were defined as the edge nodes of the stub column while the master components were the top and bottom plates. Friction was defined at the contact interfaces by a static and dynamic coefficient of friction value of 0.4. A prescribed rigid body motion curve, identical to the curve shown in Figure 4.7, was assigned to the top plate. Initially, all degrees of freedom of the stub column edge nodes were unconstrained. However, it was found that contact of the shell edges of the stub column against the 73 hor izonta l surfaces o f the plates was unstable, w i t h the stub c o l u m n edge elements tending to overturn as s h o w n i n F igure 4.37. T o prevent this effect, the rotat ional degrees o f freedom o f the edge nodes were constrained. The translational degrees o f freedom remained unconstrained, thus a l l o w i n g transverse expans ion o f the tube under compress ion . Z Figure 4.37 Contact boundary condit ions study: Instabi l i ty o f she l l edge contact. W i t h the rotational degrees o f freedom constrained, the deformat ion modes for the contact ing plate mode ls were s imi la r to the baseline case, as s h o w n i n F igu re 4.38. T h e spatial plast ic mechan i sm formed near the bot tom o f the c o l u m n w h e n *Contac t_Ent i ty was used for the bo t tom plate interface. 74 Figure 4.38 Effect o f us ing contact boundary condi t ions : D e f o r m e d shapes. The load-displacement curves for the two contacting plate mode l s were somewhat different, as shown i n F igure 4.39. F o r the mode l w i t h the *Contac t_Ent i ty plane as the bot tom plate, the load-displacement curve was nearly ident ica l to the basel ine case. In contrast, the mode l w i t h the *Contact_Nodes_to_Surface plate part as the bot tom plate showed a softer load-displacement response i n the elastic zone, match ing the experimental curve more c lose ly i n this zone than any o f the other mode l s . It i n i t i a l l y seemed as though this approach to mode l ing the boundary condi t ions was g i v i n g a more realist ic response. H o w e v e r , it was noted that the edge nodes o f the stub c o l u m n i n this mode l were penetrating the bot tom plate part, w i t h a m a x i m u m penetrat ion o f 0.42 m m . T h i s penetration was an indica t ion that the contact interface was softer than the case o f a r ig id bear ing plate, and as a result a shift i n the load-displacement curve was achieved w h i c h approached the experimental curve on ly b y coinc idence . In the m o d e l w i t h the *Contac t_Ent i ty plane as the bot tom plate, the m a x i m u m penetration o f the bot tom edge nodes was o n l y 0 .02mm, and as a result the load-displacement curve from this m o d e l matched the baseline m o d e l results more closely. 75 1000 -900 Displacement (mm) Figure 4.39 Effect o f us ing contact boundary condi t ions : A x i a l load-displacement response. It should be noted that attempts were made to solve the contact boundary cond i t ion models w i t h the i m p l i c i t solvers, but convergence diff icul t ies prevented the s imulat ions from comple t ing successful ly. 4.6.9 Effect of Varying the Numerical Precision (Implicit and Explicit Solver) In the analyses o f the baseline mode l , the double p rec i s ion executable was used for the expl ic i t analysis, w h i l e the single prec is ion executable was used for the i m p l i c i t analyses. W h e n the i m p l i c i t analyses o f the baseline mode l were re-run w i t h double prec is ion , no differences were d iscernib le i n the results as compared to the s ingle p rec i s ion case. The deformed shapes, load-displacement curves, strain results, and stress results were a l l unchanged. The results were unchanged because the i m p l i c i t analyses were able to exploi t a large t ime step (as large as 1x10" seconds) and thus required o n l y a few thousand cycles to solve the p rob lem, meaning that round-of f errors d i d not accumulate. 76 In contrast, when the exp l i c i t analysis o f the baseline m o d e l was re-run w i t h single prec is ion , significant differences were observed i n the results as compared to the double prec i s ion case. In the exp l i c i t analysis, the cr i t ica l t ime step was approximate ly 8.8x10" 7 seconds. A s a result, the 4-second duration o f the s imula t ion required approximate ly 4.5 m i l l i o n cycles to comple te . F o r such a large number o f cyc les , i t i s poss ib le that s m a l l round-of f errors c o u l d have accumulated. The f inal deformed shape f rom the expl ic i t analysis w i t h single p rec i s ion is shown i n Figure 4.40. It can be seen that the asymmetr ic b u c k l i n g mode was predic ted w h e n s ingle prec is ion was used. It i s poss ib le that this mode was triggered b y a smal l round-off error i n a h i g h frequency component o f the response, leading to a dynamic imperfect ion effect. T h e asymmetr ic b u c k l i n g mode resulted i n a reduced load-displacement response i n the plast ic m e c h a n i s m zone o f the curve, as shown i n F igu re 4.41. The run t ime for the exp l i c i t analysis w i t h single prec is ion was 2 hours and 54 minutes. Compared to the double p rec i s ion run t ime o f 9 hours and 21 minutes, a r un t ime savings o f 69% was achieved w h e n s ing le p rec i s ion was used. F igure 4.40 Effect o f us ing single prec is ion i n the exp l i c i t analysis: D e f o r m e d shape. 77 1000 900 H Displacement (mm) Figure 4.41 Effect o f us ing single prec is ion i n the exp l i c i t analysis : A x i a l load-displacement response. 4.7 Conclusions from the study on Stub Column Models W i t h regards to the objectives set out i n the introductory sect ion o f this chapter, the f o l l o w i n g general conc lus ions can be made as a result o f the stub c o l u m n analyses: 1. The f ive analysis techniques considered each p roved to be useful tools so l o n g as their l imi ta t ions were respected. The expl ic i t analysis me thod was effective for s o l v i n g prob lems i n v o l v i n g contact. The i m p l i c i t methods a l l o w e d fast so lu t ion o f the quasi-static s imula t ion w i t h on ly modest t ime sca l ing . T h e eigenvalue and b u c k l i n g analyses p rov ided useful information for understanding the dynamics o f the p rob lem and gu id ing the selection o f geometric imperfect ions . 2. The finite element analyses were able to predict the ul t imate stub c o l u m n load capaci ty as w e l l as the four characteristic zones o f the stub c o l u m n load-displacement response w i t h reasonable accuracy compared to the exper imental results. The analyses predicted elastic stiffness values that matched the expected values, a l though these expected values were not seen i n the exper imenta l results. 3. The finite element analyses conf i rmed that the type o f spatial plast ic mechan i sm that is formed determines the shape o f the stub c o l u m n load-displacement curve i n the pos t -buck l ing range. A m o n g the various m o d e l i n g parameters studied, the 78 spatial plast ic mechan i sm was most sensitive to geometric imperfec t ion magni tude and, i n the case o f exp l ic i t analysis, numer ica l p rec i s ion . The parametric studies o f the stub c o l u m n models showed that the predicted stub c o l u m n response is indeed inf luenced b y a number o f model ing-rela ted factors. A s such, it is important to note that any conclus ions d rawn f rom the stub c o l u m n study are o n l y str ict ly appl icable for the part icular condi t ions considered i n the study. W i t h that caveat i n m i n d , the f o l l o w i n g observations were made to assist i n the development o f the fu l l V S C - C B F system m o d e l : 1. T h e coarsened mesh described i n Sec t ion 4.6.1 w i t h constant 4 . 7 7 m m w a l l thickness should be considered for implementa t ion i n m o d e l i n g the H S S brac ing members o f the V S C - C B F system. 2. The element characterizat ion parameters and mater ial parameters from the basel ine stub c o l u m n m o d e l gave acceptable results. These parameters were: T y p e 1 shells w i t h 3 integration points through the thickness, T y p e 4 hourglass control w i t h a coefficient o f 0.10, and the M a t _ 0 2 4 mater ia l m o d e l . 3. S m a l l geometric imperfect ions m a y be required to tr igger a b u c k l i n g mode that matches experimental observations. Geomet r ic imperfect ions can be incorporated as noda l coordinates from scaled v ibra t ion mode shapes. In transient analyses, dynamic imperfect ions m a y trigger b u c k l i n g . N u m e r i c a l round-of f error m a y also act as a fo rm o f imperfect ion, al though this is not recommended as a rat ional approach to m o d e l i n g imperfections. 4. F o r exp l i c i t analyses the use o f stiffness-proportional damping higher than 10% is not recommended, due to the detrimental impact o n the c r i t i ca l t ime step required for s tabil i ty. A n a l y s e s w i t h zero damping m a y lead to undesirable results, the reason for w h i c h was not determined b y the stub c o l u m n study. 5. T h e use o f a smooth load ing prof i le i n the expl ic i t analyses is r ecommended w h e n quasi-static condi t ions are simulated. H o w e v e r , the stub c o l u m n models d i d not prove that a smooth load ing prof i le is essential. 6. F o r exp l i c i t analyses, significant run-t ime savings can be achieved b y t ime scal ing . H o w e v e r , the results m a y be affected i f t ime sca l ing is excessive. M o d a l informat ion from an eigenvalue analysis can prov ide useful in format ion to guide the choice o f t ime scale factors and assess their effect. 7. Contact a lgori thms are capable o f s imula t ing contact ing surfaces, but care should be taken to assess the effect o f the contact options selected. T h e exp l i c i t solver handles contact more easi ly than the i m p l i c i t solver. 79 For explicit analyses involving millions of computational cycles, double precision should be used to minimize round-off error. The effects of potential round-off errors should be weighted against the significant run-time savings associated with single precision. 80 5 MODELING, PART TWO: SINGLE BRACE 5.1 Introduction The experimental research phase conf i rmed that the response o f the V S C - C B F system was la rge ly governed b y the b u c k l i n g behaviour o f the H S S braces. Therefore, a finite element m o d e l o f a s ingle brace subjected to ax ia l load was created, to investigate the predic t ion o f brace b u c k l i n g . The brace member and gusset plate connections were mode led w i t h the same dimensions and geometry as was used i n the fu l l V S C - C B F specimen. E igenva lue and b u c k l i n g analyses o f the brace m o d e l were performed, fo l l owed b y exp l i c i t and i m p l i c i t analyses to determine the fu l l range o f ax i a l load-displacement response. T h e sensi t ivi ty o f the ax ia l load-displacement response to several model ing-re la ted parameters was investigated. T h e parameters invest igated b y expl ic i t analyses inc luded: thickness-wise mesh refinement o f the bo t tom gusset plate, mater ial stiffness o f the gusset plates, load ing duration, numer ica l p rec i s ion and mass-proport ional damping levels . Impl i c i t analyses were used to investigate the effects o f geometric imperfect ions. 5.2 Estimate of Compression Capacity A l t h o u g h direct ax ia l compress ion testing was not carr ied out o n the brace and gusset plate assemblies dur ing the experimental phase o f the research, it is poss ible to infer the m a x i m u m compress ion capaci ty from the experimental data o f the V S C - C B F system tests. D u e to the geometry o f the chevron brac ing configurat ion, the lateral load, V , at first b u c k l i n g o f a brace can be expressed as: V = 2PcosG (5.1) where P is the ax ia l load i n one brace, and 0 = 6 6 ° is the acute angle between the brace and the beam. Thus the ax ia l load can be inferred as: P = — — (5.2) 2cos0 81 The lateral load at first b u c k l i n g o f a brace was measured as 263 k N . Therefore, accord ing to (5.2), the m a x i m u m compressive ax ia l load i n the brace was about 330 k N . The b u c k l i n g load can al ternatively be interpreted us ing the average recorded strains from the strain gages that were mounted o n each face o f the H S S member at mid- leng th (see B u b e l a [2003] for a complete descript ion o f the specimen instrumentation). T h e strain recordings were l inear up to the point o f first b u c k l i n g . F r o m the strain measurements the b u c k l i n g load can be computed as: P = EAs (5.3) U s i n g (5.3) w i t h an average strain reading o f 0.00119 gives a m a x i m u m compress ive ax ia l load o f approximate ly 370 k N . The c o l u m n strength curves used i n current des ign practice can be used to predict the compress ion capaci ty as an alternative to the back-ca lcu la t ion approach discussed above. The Canad ian steel standard C A N / C S A - S 1 6 - 0 1 provides an express ion based o n the Structural S tab i l i ty Research C o u n c i l ( S S R C ) c o l u m n curves, from w h i c h the unfactored compress ive resistance C u can be obtained as: where n = 1.34 for co ld- formed non-stress-relieved H S S , and the slenderness parameter X is defined as: where K L is the effective length o f the member and r is the least radius o f gyra t ion o f the cross-section. F y m a y be taken as the 0.2%-offset y i e l d stress o f the stub c o l u m n test for the member , w h i c h i n the present case is 458 M P a . U p o n compar ing the results o f 76 compress ion tests from 9 exper imental programs that inc luded var ious steel b rac ing member sizes, shapes, and configurat ions, T r e m b l a y noted |l/n (5.4) (5.5) 82 that the S S R C curves are generally conservative, w i t h a mean test-to-predicted ratio o f 1.16 (Tremblay , 2002) . Incorporat ing this mean 16% increase into the values predicted b y (5.4), w i t h 0 . 8 L < K L < 1.0L, gives expected compress ion capacit ies o f between 322 k N and 448 k N . T h i s range encompasses the values obtained from (5.2) and (5.3). 5.3 Description of Single Brace Model The finite element m o d e l o f the single brace was extracted from the fu l l V S C - C B F mode l that w i l l be later d iscussed i n Chapter 6. A s shown i n F igu re 5.1, the s ingle brace m o d e l consisted o f the H S S brace and the top and bot tom connect ion gusset plates. The details o f the i nd iv idua l components o f the mode l are discussed i n the f o l l o w i n g sections. T Figure 5.1 Deta i l s o f single brace m o d e l 5.3.1 Brace Member The m o d e l o f the 3 1 4 7 m m - l o n g 89x89x4.8 H S S brace member was created us ing the same approach as was used for the stub c o l u m n mode l . T h e central 4 9 5 m m length o f the tube, where curvatures and strains were expected to be greatest, was meshed w i t h the same mesh density as described i n the coarse stub c o l u m n mesh descr ibed i n Sec t ion 4.6.1, i n accordance w i t h the recommendations i n Sec t ion 4.7. A s imi l a r mesh density 83 was used at the ends o f the brace, inc lud ing the areas surrounding the 1 8 0 m m - l o n g slots that were cut to accommodate the we lded gusset plate connect ions. In the r emain ing regions o f the brace, a larger mesh size was deemed acceptable because stress concentrations were not expected there. In these regions the element widths were unchanged around the perimeter o f the brace, but the element lengths i n the longi tudina l di rect ion o f the brace were increased from 10mm to 19mm. T h e transitions i n mesh topology are h ighl ighted i n F igure 5.2. Wtfft I I I t | | | | |) | | | | fTYj j | | 1 1 Figure 5.2 S ing le brace mode l : Transi t ions i n mesh topo logy o f the brace. The parametric studies o f the stub c o l u m n mode l suggested that the f o l l o w i n g m o d e l i n g details should be implemented i n the brace mode l : -Type 2 shel l elements w i t h 3 integration points through the thickness -Type 4 hourglass cont ro l , w i t h a coefficient o f 0.10 -The M a t _ 0 2 4 mater ia l models described i n Sect ion 4.3.4 Geomet r ic imperfect ions were not in i t i a l ly inc luded i n the brace member , but they were g iven specia l considerat ion i n the i m p l i c i t analyses described i n Sec t ion 5.5.6. 84 5.3.2 Gusset Plates Fin i t e element m o d e l i n g o f gusset plate connections has been the focus o f numerous studies i n the past 20 years. Chambers and Ernst (2005) c o m p i l e d a comprehensive literature r ev i ew o f braced frame gusset plate research, and noted that the finite element method o f analysis has p roven to be a valuable too l i n understanding the behaviour o f gusset plate connections. The bot tom gusset plates o f the test specimen were 12 .7mm th ick and were w e l d e d to base plates that were bol ted to the boundary test frame. The base plates were omit ted from the finite element mode l . The bot tom gusset plate was mode led w i t h eight-noded T y p e 1 constant stress hexahedron so l id (brick) elements, w i t h 4 layers o f elements through the thickness. These under-integrated elements have a s ingle integration point at the centre o f the element. A study o f the performance o f cant i levered beam models us ing var ious types o f s o l i d elements showed that good performance i n bend ing c o u l d be achieved b y us ing T y p e 1 so l id elements, w i t h T y p e 6 hourglass control and an hourglass coefficient o f 1.0. The study also showed that s o l i d element performance was i m p r o v e d w h e n element aspect ratios o f 5 were not exceeded. The details o f this s tudy are inc luded i n a separate report ( E E R F Repor t 05-02). These f indings were used to guide the creat ion o f the bo t tom gusset plate part, w h i c h is shown i n F igure 5.3. 85 Figure 5.3 S ing le brace mode l : B o t t o m gusset plate. The top gusset plate was 19 m m thick. D u r i n g the design stage o f the V S C - C B F system the thickness o f the top gusset plate was increased relative to the thickness o f the bot tom gusset plate, i n an attempt to m i n i m i z e the plastic h inge rotations o f the top gusset plate w h i c h c o u l d cause b i n d i n g o f the bolts a long the edges o f the slotted holes . The top gusset plate was mode l ed us ing the same approach as was used for the bo t tom plate, w i t h 4 through-thickness layers o f T y p e 1 so l id elements, s tabi l ized b y T y p e 6 hourglass control w i t h an hourglass coefficient o f 1.0. A s shown i n F igure 5.4, the geometry o f the top gusset plate part was obtained from a h a l f section o f the complete top gusset plate that was used i n the fu l l V S C - C B F system, w i t h the upper por t ion o f the plate conta in ing the boltholes removed. 86 Figure 5.4 S ing le brace mode l : T o p gusset plate. The M a t _ 0 2 4 P iecewise L inea r P las t ic i ty material m o d e l was used for bo th gusset plate parts. The f low stress m o d e l for each part was adapted to the tensile test results from the respective plate samples, w h i c h are shown i n F igure 5.5 i n terms o f true stress and true strain. T h e typ ica l characteristics o f hot ro l l ed steel plate can be seen i n F igure 5.5, namely a distinct y i e l d point , a plastic plateau, and strain hardening. T h e f low stress m o d e l for each gusset plate part is superimposed on the tensile test results o f F igure 5.5, and shown separately i n F igure 5.6. The elastic modu lus for each gusset plate was determined b y averaging the elastic slopes from the tensile test data. T h i s procedure gave a value o f E = 256,000 M P a and E = 242,000 M P a for the top and bo t tom gusset plates, respectively. A t a later stage i n the fu l l system m o d e l i n g phase, these values were adjusted to the more t yp i ca l value o f E = 200,000 M P a . H o w e v e r , for the m o d e l described i n the present chapter, the higher values were used. 87 800 -r 700 -600 -(MPa) 500 -tress 400 -1 s 300 -H 200 -100 -19mm Plate Specimen 1 19mm Plate Sepecimen 2 19mm Plate Sepecimen 3 Mat_024: Top Gusset Plate 12.7mm Plate Specimen 1 12.7 mm Plate Specimen 2 12.7 mm Plate Specimen 3 Mat_024: Bottom Gusset Plate 0.05 0.1 0.15 True Strain 0.2 0.25 Figure 5.5 C o m p a r i s o n o f M a t _ 0 2 4 f low stress m o d e l and tensile test results for top and bot tom gusset plate materials. TJ 300 200 100 H 0 0.00 • Mat_024: Top Gusset Plate Mat_024: Bottom Gusset Plate 0.05 0.10 0.15 Effective Plastic Strain 0.20 0.25 Figure 5.6 M a t _ 0 2 4 f l o w stress models for top and bo t tom gusset plate materials. 5.3.3 Treatment of welds In the V S C - C B F test specimen, the braces were j o i n e d to the top and bo t tom gusset plates b y 6 m m fillet w e l d around 180mm-long slots cut i n the braces. In the finite element mode l , the brace was mode l ed w i t h shel l elements, w h i l e the gusset plates were mode led 88 w i t h s o l i d elements. M e r g i n g the nodes o f the brace and gusset parts was not a t r iv i a l dec i s ion because the noda l degrees o f freedom o f s o l i d and she l l elements are not compat ib le ; the so l id element nodes do not have rotat ional degrees o f freedom. The shel l edge / b r i c k interface can be handled b y the constraint equations contained w i t h i n the L S -D Y N A keywords *constrained_shel l_to_sol id or *constra ined_nodal_r igid_body. H o w e v e r , i f the shel l and so l id parts are meshed such that the interfacing nodes are coincident , then these nodes can s i m p l y be merged together and L S - D Y N A ignores the incompat ib le rotational degrees o f freedom, treating the connect ions as p ins . Because o f the three d imens iona l nature o f the brace-gusset connect ion and the l i m i t e d strength o f the one-sided fi l let w e l d , it was decided that the merged node technique was acceptable. Other methods for m o d e l i n g welds are possible , however the added l eve l o f c o m p l e x i t y inherent i n such detailed w e l d mode l ing was not considered to be wor thwh i l e . F i l l e t w e l d models are avai lable i n L S - D Y N A , w h i c h treat the w e l d as a noda l r i g i d body , ass igning noda l constraint equations combined w i t h failure cr i ter ia that are based o n exceedence o f a c r i t i ca l stress or plastic strain value. Al te rna t ive ly , three-dimensional m o d e l i n g o f the w e l d mater ial is possible, us ing s o l i d elements, as descr ibed b y S l e c k z a (2004). The use o f s o l i d elements to mode l the weldment w o u l d necessitate a large number o f ve ry s m a l l elements, and such an approach w o u l d be infeasible for the large m o d e l considered i n the present study. 5.3.4 Boundary Conditions F i x e d noda l boundary condi t ions were appl ied to the nodes l y i n g o n the top hor izonta l edge and the top inside edge o f the top gusset plate, as shown i n F igure 5.7. A s described i n Sec t ion 5.3.2, the top gusset plate used i n the single brace m o d e l corresponded to a section o f the larger top gusset plate used i n the fu l l V S C - C B F assembly. It was expected that the f ixed edge noda l boundary condi t ions for the top gusset plate i n the single brace m o d e l w o u l d result i n a top gusset plate w i t h s l igh t ly greater out-of-plane rotational stiffness as compared to the fu l l gusset plate i n the V S C - C B F system. H o w e v e r , the gusset plate i n the single brace m o d e l d i d inc lude the free length section where the major i ty o f the elastic and plast ic h inge rotations were expected to occur , so it 89 was deemed to be an adequate m o d e l to investigate the b u c k l i n g behav iour o f the single brace. Fixed Nodal Degrees of Freedom Figure 5.7 Bounda ry condit ions at top gusset plate. Compress ive ax ia l load was appl ied to the brace assembly b y a p p l y i n g prescr ibed noda l displacements to the bo t tom edge nodes o f the bot tom gusset plate, i n a d i rec t ion defined b y a unit vector a l igned w i t h the longi tudinal axis o f the brace. T h e prescr ibed ax ia l displacements were a mono ton ica l ly increased f rom O m m to 8 m m . T h e l oad ing durations for the expl ic i t dynamic and i m p l i c i t static analyses were 0.35 seconds and 1.0 "seconds," respectively. 5.4 Eigenvalue and Buckling Analyses Impl ic i t e igenvalue and b u c k l i n g analyses were performed o n the s ingle brace m o d e l to determine the v ib ra t ion modes and frequencies as w e l l as the b u c k l i n g modes and 90 b u c k l i n g load mul t ip l ie rs . The first five vibra t ion mode shapes f rom t = 0 are shown i n F igure 5.8, and the corresponding frequencies and periods are l is ted i n T a b l e 6.1. F igure 5 .8Eigenvalue analysis results: First f ive v ib ra t ion m o d e shapes at t=0. Table 5.1 E igenva lue analysis results: Firs t five v ibra t ion frequencies and periods at t=0. M O D E Eigenva lue C i r c . F r e q . (rad/sec) F req . ( H z ) Per iod (sec) 1 39979 200 32 0.031 2 183710 429 68 0.015 3 461040 679 108 0.009 4 1227171 1108 176 0.006 5 1724762 1313 209 0.005 The first f ive b u c k l i n g modes predicted 0.005 s after the start o f the exp l i c i t analysis are shown i n F igure 5.9. F o r modes 6 and higher, the b u c k l i n g mode shapes corresponded to loca l b u c k l i n g modes. A s an example, a close-up v i e w o f the 6 t h b u c k l i n g mode is shown i n F igure 5.10. The loads, load mul t ip l iers , and resul t ing b u c k l i n g l oad predict ions for the first eight modes are l is ted i n Table 5.2. The predicted first mode b u c k l i n g load was 559 k N , w h i c h is 2 5 % higher than the m a x i m u m expected b u c k l i n g l oad o f 448 k N from the calculat ions i n Sec t ion 5.2. 91 Figure 5.9 B u c k l i n g analysis results: First eight b u c k l i n g modes , t=0.005s Tab le 5.2 B u c k l i n g analysis results: B u c k l i n g load predict ions , t=0.005s 92 t = 0.0015s ( L o a d = 3.5 k N ) B u c k l i n g M o d e B u c k l i n g L o a d M u l t i p l i e r (Eigenvalue) Predic ted B u c k l i n g L o a d [kN] 1 160 559 2 482 1688 3 496 1737 4 940 3290 5 953 3334 6 970 3394 7 970 3395 8 974 • 3411 5.5 Load-Displacement Response Analyses E x p l i c i t and i m p l i c i t analyses were performed to determine the ax ia l load-displacement response o f the single brace m o d e l , and the sensi t ivi ty o f this response to a var ie ty o f parameters. A l t h o u g h the m a x i m u m ax ia l load resisted b y the brace was different for each o f these analyses, the deformation mode and pattern o f strain and stress concentrations were s imi la r i n nearly a l l cases. The brace remained straight un t i l sudden elastic b u c k l i n g occurred, after w h i c h point the brace vibrated about its b u c k l e d configurat ion, w h i c h corresponded to the first b u c k l i n g mode shape s h o w n i n F igure 5.9. The typ ica l d is t r ibut ion o f effective plastic strains and effective (von M i s e s ) stresses are shown i n the contour plots o f F igures 5.11 and 5.12, respect ively, for the m a x i m u m considered ax ia l displacement o f 8 m m . F r o m F igure 5.11 it can be seen that some plast ic straining occurred at the mid- length region o f the brace, however , for the displacement l eve l considered, plas t i f icat ion o f the entire cross-section d i d not occur . A s a result, a fu l l plast ic hinge and l o c a l b u c k l i n g at the mid- length o f the brace were not observed i n these analyses. L o c a l b u c k l i n g might have been observed i f the brace was subjected to a larger ax ia l displacement or to c y c l i c loading , w h i c h w o u l d have l oca l i zed the b u c k l i n g pattern i n the plast ic hinge. F o r the displacement leve l considered, elastic bend ing o f the gusset plate connections was observed, w i t h l imi t ed plast ic straining. F r o m F igu re 5.11 it can be seen that plast ic strain concentrations were developed i n the area o f the brace surrounding the ends o f the gusset plate slots. H o w e v e r , i n F igure 5.12, the effective stress levels i n these areas are less than the y i e l d stress o f the H S S flat w a l l material . 93 Y i e l d i n g and plast ic s t raining developed i n these areas o f the brace before b u c k l i n g occurred. U p o n b u c k l i n g , a redistr ibution o f forces occur red throughout the brace assembly, and the areas o f the brace surrounding the gusset plate slots were unloaded, such that the f inal effective stress i n the buck led conf igura t ion was b e l o w the y i e l d l eve l , but res idual plast ic strains remained. z Fringe Levels 1.000e-03 9.000e-04 8.000e 04 7.000e-04 6.000e-04 5.000e-04 4.000e-04 3.000e-04 2.000e-04 I.OOOe 04 0.000e+00 Figure 5.11 S ing le brace analysis: T y p i c a l effective plast ic strain contours at 8 m m o f ax ia l end shortening. 94 Fringe Levels 0.000e+00 4.500e+01 9.000e+01 1.350e+02 2.250e+02 1.800e+02 2.700e+02 3.150e+02 4.050e+02 3.600e+02 4.500e+02 Figure 5.12 S ing le brace analysis: T y p i c a l effective stress contours at 8 m m o f ax ia l end shortening. E x p l i c i t analyses o f the s ingle brace mode l were performed to determine the sensi t ivi ty o f the b u c k l i n g load to the f o l l o w i n g parameters: 1. The number o f elements through the thickness o f the gusset plates, 2. The value o f elastic modulus (E) assigned to the gusset plates, 3. The durat ion o f the loading, 4. T h e numer ica l p rec i s ion (single or double), and 5. The level o f mass propor t ional damping Items 1 and 2 above relate to the computed stiffness o f the gusset plates, and therefore g ive an assessment o f the effect o f end condit ions o n the b u c k l i n g l o a d o f the brace. Items 3, 4, and 5 relate to parameters that must be considered i n the exp l i c i t analysis o f a quasi-static process. In the f o l l o w i n g sections, the modif ica t ions to the mode l s are br ie f ly described, and the results are presented i n terms o f the ax ia l load-displacement response. Impl ic i t static analyses were performed on the single brace m o d e l to determine the sensi t ivi ty o f the ax ia l load-displacement response to geometric imperfect ions. 95 5.5.1 Effect of Number of Elements Through the Thickness of the Bottom Gusset Plate The response o f the brace mode l described i n Sec t ion 5.3 was compared to a s imi la r m o d e l i n w h i c h the number o f elements through the thickness o f the bo t tom gusset plate was reduced to 2, w h i l e the top gusset plate mesh was unchanged. H a v i n g o n l y 2 elements through the thickness o f the bottom gusset plate a l l o w e d the mesh to be coarsened overa l l w h i l e main ta in ing element aspect ratios o f less than 5. T h e coarsened bot tom gusset plate mesh is shown i n F igure 5.13. F igure 5.13 Coarsened bot tom gusset plate mesh. The ax ia l load-displacement responses o f the brace mode ls i n v o l v i n g 4 and 2 elements through the thickness o f the bot tom gusset plate are compared i n F igu re 5.14. It should be noted that i n the pos t -buck l ing region o f this figure, the j agged appearance o f the curves is due to the use o f a re la t ively l o w output sampl ing rate o f the stored results. The m a x i m u m load for the m o d e l w i t h 4 elements through the thickness o f the bot tom gusset plate was 604 k N . In contrast, the m a x i m u m load for the m o d e l w i t h 2 elements through the thickness o f the bo t tom gusset plate was 639 k N ; an increase o f 6%. 96 700 £ 300 * 500 1 400 a; c o 600 100 0 0 2 3 4 5 6 7 8 Displacement (mm) Figure 5.14 E x p l i c i t analysis results: Effect o f number o f elements through the thickness o f the bo t tom gusset plate: A x i a l load-displacement response. 5.5.2 Effect of Gusset Plate Elastic Modulus Value A s noted i n Sec t ion 5.3, the elastic modulus for the top and bot tom gusset plates was specif ied as E = 256,000 M P a and E = 242,000 M P a , respect ively, based o n the averaging o f the elastic slopes o f the tensile test data. F o r the sake o f compar i son , a mod i f i ed mode l was created i n w h i c h a lower bound value o f E = 120,000 M P a was assigned to both gusset plate parts. The axia l load-displacement responses o f the two brace models are compared i n F igure 5.15. The use o f E = 120,000 M P a i n the gusset plates resulted i n a sl ight reduct ion o f the overa l l stiffness o f the brace assembly, and the m a x i m u m load was o n l y reduced b y 1%, from 604 k N to 598 k N . 97 700 600 * 500 o u. e u CO <D 400 300 200 100 0 / i ' A / Gusset Plate E= 256,000 Mpa, ~¥ 242,000 MPa 1——-Gusset Plate E - 120,000 MPa 1 1 I 1 1 1 i 0 1 8 2 3 4 5 6 7 Displacement (mm) Figure 5.15 E x p l i c i t analysis results: Effect o f gusset plate elastic modu lus value: A x i a l load-displacement response. 5.5.3 Effect of Load Duration A s noted i n Sec t ion 5.3, the prescribed ax ia l end shortening i n the exp l i c i t analysis was ramped from 0 m m to 8 m m over 0.35 seconds. Based o n the e igenvalue analysis results, the durat ion o f 0.35 seconds corresponds to 11 t imes the fundamental per iod o f the structure. The response for this analysis was compared to a second analysis i n w h i c h the load ing durat ion was reduced by a factor o f ten (the 8 m m ax i a l d isplacement was ramped over 0.035 seconds), and to a third analysis i n w h i c h the load ing dura t ion was doubled (the 8 m m ax ia l displacement was ramped over 0.70 seconds). T h e ax i a l load-displacement responses f rom the three analyses are compared i n F i g u r e 5.16. In the case o f the 0.035 s analysis , the brace d i d not show a lateral b u c k l i n g def lec t ion , al though the ax ia l load capaci ty d i d begin to reduce as the ax i a l displacement o f 8 m m was approached. The load-displacement curve for the 0.035 s case showed osc i l l a t i on about the expected solu t ion i n the elastic range, w h i c h was an ind ica t ion that the load durat ion was too short to capture the quasi-static response o f the brace. W h e n the load ing durat ion was increased to 0.70 s, the m a x i m u m load was reduced b y 14% compared to the 0.35 s 98 case, from 604 k N to 521 k N . The run times for the 0.035s, 0.35s, and 0.70s durat ion analyses were 44 minutes , 428 minutes, and 772 minutes, respect ively. 700 0 1 2 3 4 5 6 7 Displacement (mm) Figure 5.16 E x p l i c i t analysis results: Effect o f l oad ing durat ion: A x i a l load-displacement response. 5.5.4 Effect of Numerical Precision The expl ic i t analysis us ing a loading duration o f 0.35 s required approx imate ly 930,000 cycles to complete. W h e n the loading duration was increased to 0.70 s, the cyc l e count also doub led to approximate ly 1.86 m i l l i o n cycles . In order to assess the effects o f potential round-of f error for such a large number o f cyc les , the s ingle p rec i s ion 0.70 s analysis was compared to the equivalent analysis us ing double p rec i s ion . T h e ax ia l load-displacement responses from the two analyses are compared i n F igu re 5.17. W h e n double p rec i s ion was used, the m a x i m u m load was increased b y 2 % , from 521 k N to 534 k N . T h e run t imes for the 0.7 s durat ion analyses w i t h s ingle and doub le p rec i s ion were 772 minutes and 3216 minutes, respectively. 99 700 600 0 1 2 3 4 5 6 7 8 Displacement (mm) Figure 5.17 E x p l i c i t analysis results: Effect o f numer ica l p rec i s ion : A x i a l load-displacement response, 0.70 s durat ion analysis . 5.5.5 Effect of Mass Proportional Damping A l t h o u g h 10% stiffness-proportional damping was used i n a l l o f the exp l i c i t analyses o f the single brace m o d e l , this was o n l y effective at damping h i g h frequency vibrat ions that w o u l d result i n noise i n the numer ica l solution. The exp l i c i t analyses therefore s t i l l predicted v ibra t ion o f the brace near its fundamental pe r iod after the sudden elastic b u c k l i n g . The ampli tudes o f such vibrat ions w o u l d l i k e l y be m i n i m a l i n a true quasi -static test. Furthermore, it was speculated that the presence o f such vibrat ions i n the expl ic i t analysis o f the fu l l V S C - C B F system cou ld lead to di f f icul t ies w i t h the contact interfaces i n the c y c l i c load ing simulat ions. Therefore, mass-propor t ional R a y l e i g h damping , w h i c h damps m a i n l y the lower frequency modes, was in t roduced into the single brace m o d e l to obtain the steady-state response. In L S - D Y N A , mass-proport ional damping can be assigned b y a d a m p i n g constant D , w h i c h is defined such that c r i t ica l damping i n the fundamental mode i s approached w h e n D = 2coi, where ©i is the c i rcular frequency o f the fundamental mode . T h e n mass-propor t ional damping can be assigned for a g iven fraction o f c r i t i c a l damping b y 100 assigning a value o f D that is equal to 2coi t imes a d a m p i n g rat io %. T h e eigenvalue analysis results predic ted that the fundamental c i rcu lar frequency o f the single brace structure was 200 rad/s. Mass-propor t iona l d a m p i n g constants corresponding to 2 5 % , 5 0 % and 1 0 0 % o f c r i t i ca l damping were tr ied, i n order to observe any resul t ing change i n the predic ted m a x i m u m loads relat ive to the undamped case. The ax ia l load-displacement responses f rom these four analyses are compared i n F igure 5.18. A l l four analyses were performed w i t h a load ing durat ion o f 0.7 s, and single precis ion. The m a x i m u m load i n the undamped case was 521 k N . F o r d a m p i n g constants corresponding to 2 5 % , 5 0 % , and 100% o f cr i t ica l damping , the m a x i m u m loads were 538 k N , 551 k N , and 569 k N , corresponding to increases o f 3%, 6%, and 9%, respect ively, compared to the undamped case. Whereas h igh values o f st iffness-proport ional d a m p i n g were s h o w n i n Sec t ion 5.6.6 to reduce the stable exp l i c i t t ime step, this effect was not observed i n the present study for large values o f mass-proport ional damping . 1 2 3 4 5 6 7 Displacement (mm) Figure 5.18 E x p l i c i t analysis results: Effect o f mass-propor t ional damping : A x i a l load-displacement response, 0.70 s durat ion analysis . 101 5.5.6 Effect of Geometric Imperfections Measurements o n H S S co lumns b y other researchers have s h o w n that fabricat ion and handl ing processes t yp i ca l ly result i n an in i t i a l crookedness that can be described b y single curvature that is re la t ively symmetr ic about the mid- length o f the tube ( D a v i s o n & B i r k e m o e , 1983). The c o l u m n strength curves suggested b y the Structural S tabi l i ty Research C o u n c i l (Ga lambos , 1998) are based o n an in i t i a l mid- leng th deflect ion o f L / 1 0 0 0 . D a v i s o n & B i r k e m o e (1983) noted that N o r t h A m e r i c a n manufacturers o f c o l d -formed products t yp i ca l l y specify a m a x i m u m crookedness o f L / 4 8 0 , but this imperfec t ion magni tude w o u l d be reduced w h e n a l ong length o f H S S is cut to the length that is used i n construct ion. Salvarinas (1977) computed a mean crookedness o f L / 5 8 0 0 from measurements o f 20 H S S specimens. K e y and H a n c o c k (1988) reported that a crookedness o f L / 5 0 0 0 corresponded to the mean value from their laboratory measurements o f H S S co lumns . It is important to note that geometric imperfect ions exist due to the in i t i a l camber i n the member, but also due to load eccentr ic i ty that m a y occur as a result o f misa l ignment o f members i n the test assembly. Ini t ia l imperfect ions were incorporated into the single brace m o d e l b y sca l ing noda l displacements o f the first b u c k l i n g mode shape ( shown i n F igure 5.9) to g ive a m a x i m u m mid- length crookedness o f L / 5 0 0 , L / 1 0 0 0 , L / 2 5 0 0 , and L / 5 0 0 0 . T h e resul t ing ax ia l - load displacement responses from the i m p l i c i t static analysis o f each o f these mode ls and the perfectly straight m o d e l are shown i n F igure 5.19. T e m a x i m u m load for the perfect ly straight m o d e l was 609 k N . The m a x i m u m loads for the mode ls i n c l u d i n g geometric imperfect ions o f L / 5 0 0 , L / 1 0 0 0 , L / 2 5 0 0 , and L / 5 0 0 0 were 406 k N , 443 k N , 498 k N , and 518 k N , respect ively. These loads correspond to decreases o f 3 3 % , 2 7 % , 18%, and 1 5 % relative to the perfect ly straight case. 102 700 600 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Axial Displacement (mm) Figure 5.19 Impl ic i t analysis results: Effect o f i n i t i a l imperfect ions: A x i a l load-displacement response. 5.6 Consideration of Residual Stresses The co ld - fo rming process b y w h i c h H S S members are manufactured sections results i n the format ion o f c o m p l e x residual stress gradients i n the section. T h e re la t ive ly large w a l l thicknesses o f t yp ica l H S S sections cause the residual stress gradients contained i n an H S S member to be somewhat different than c o m m o n co ld - fo rmed th in-wal led profi les. The residual stress gradients are not easy to measure accurately, as they are typ i ca l ly obtained from careful sect ioning o f sma l l coupons instrumented w i t h strain gages. D a v i s o n and B i r k e m o e (1983) modeled the state o f res idual stress i n an H S S member b y superposi t ion o f longi tudinal membrane and through-thickness bend ing components. K e y and H a n c o c k (1983) proposed an addi t ional through-thickness " l aye r ing" component , w h i c h was obtained b y a spark-erosion technique. A l t h o u g h the res idual stresses were not measured i n the H S S members used i n the V S C -C B F test specimen, the adoption o f the residual stress gradients p roposed b y other researchers was considered, as a w a y to gage the sensi t iv i ty o f the finite element 103 predict ions to the magnitude o f residual stress. In L S - D Y N A it is poss ible to assign in i t i a l stress values to i nd iv idua l elements. The in i t i a l stresses are defined i n the g loba l coordinate system and m a y be specif ied at a number o f points through the thickness o f the shel l element. The approach taken was to select an assumed res idual stress pattern based o n the w o r k b y D a v i s o n and B i r k e m o e (1983), compute the stress components at discrete points through the thickness o f each element i n the l oca l element coordinate system, transform the stress components into the g loba l coordinate system, and create an automated process b y w h i c h to format the results into the appropriate L S - D Y N A input deck syntax. Sample calculat ions for this approach are presented i n a separate report (EEPvF Repor t 05-02). T h e res idual stresses i n any cross-section must be self-equi l ibrat ing. H o w e v e r , the bending stress gradient results i n a lack o f equ i l i b r ium at the free ends o f the member unless end moments are appl ied. E q u i l i b r i u m iterations must take place at the beg inn ing o f the s imula t ion to a l l o w the residual stresses to come into equ i l i b r ium. T h i s results i n deformation o f the member as w e l l as a different state o f in i t i a l res idual stresses than what was first intended, and as a result the cont ro l over the in i t i a l res idual stress state is lost. F o r these reasons the attempts at i nc lud ing residual stresses d i rec t ly i n the finite element m o d e l were deemed f lawed and therefore the results are not presented here. K e y and H a n c o c k (1988) showed that the longi tudina l bending and l aye r ing components o f res idual stress have a more significant influence o n the ax i a l compress ion response o f H S S members than the other membrane and transverse components res idual stress components. W i l k i n s o n (1999) noted that tensile test coupons cut from an H S S member typ i ca l ly deflect i n a curved shape as these bending and l ayer ing residual stress components are released. H e argued that because the tensile test coupons are bent back into the straight conf igurat ion dur ing the tensile test, the effects o f the bend ing and layer ing components o f res idual stress are accounted for i n the resul t ing mater ia l stress-strain curves. Therefore neglect ing the residual stresses i n the finite element m o d e l m a y be jus t i f iable so long as a mater ial f l ow stress m o d e l is used that is based o n actual tensile test data. H a d d a d (2004) reported excellent correlat ion between test results and predict ions o f a finite element analysis o f the c y c l i c behaviour o f an H S S brace w i t h 104 w e l d e d gusset plate connections. In his paper H a d d a d made no ment ion o f the i nc lu s ion o f residual stresses. 5.7 Discussion of Single Brace Analysis Results F r o m the analyses described i n the previous sections, the k e y observations can be summar ized as fo l lows : 1. The exp l i c i t and i m p l i c i t analyses o f the perfectly straight brace member predicted s imi la r m a x i m u m loads o f 609 k N and 604 k N , respect ively. These values are approximate ly 3 5 % higher than the m a x i m u m estimated b u c k l i n g load o f 4 4 8 k N . 2. F o r the range o f parameters and condit ions considered, the i n c l u s i o n o f geometric imperfect ions had the most significant influence o n the b u c k l i n g load . Inc lud ing an in i t i a l crookedness i n the form o f the first b u c k l i n g mode shape w i t h a magnitude o f L / 1 0 0 0 resulted i n a 2 7 % reduct ion o f the b u c k l i n g load to 443 k N . 3. In the exp l i c i t analyses, the load ing durat ion was s h o w n to s igni f icant ly influence the predicted b u c k l i n g load o f the perfectly straight brace. A 1 4 % reduct ion i n the b u c k l i n g load was achieved b y changing the load ing durat ion from 1 l T i to 2 2 T i . 4. T h e use o f four elements through the thickness o f the bo t tom gusset plate resulted i n a s l igh t ly l ower b u c k l i n g load compared to the case o f two elements through the thickness. 5. The value o f elastic modulus for the gusset plates had neg l ig ib le effect o n the computed b u c k l i n g load, and o n l y a smal l effect o n the ax i a l stiffness o f the system. 6. The use o f double p rec i s ion resulted i n a 2 % increase i n the predicted b u c k l i n g load and a 4 1 7 % increase i n the run t ime required to solve the p rob lem. F o r the analyses considered i n this chapter, the sma l l loss i n accuracy associated w i t h single p rec i s ion seemed just i f iable i n v i e w o f the large savings i n run t ime. 7. Mass -p ropor t iona l damping was used effect ively to obtain the steady state solut ion, a l though it resulted i n an increase i n the b u c k l i n g load . A 9 % increase i n the b u c k l i n g load was observed w h e n cr i t i ca l damping o f the fundamental mode was prescribed. 8. T h e direct i nc lu s ion o f in i t i a l residual stresses was not successful ly achieved. 105 6 MODELING, PART THREE: FULL VSC-CBF SYSTEM 6.1 Introduction A three-dimensional finite element m o d e l o f the V S C - C B F system was developed, to investigate the predicted response o f the system to lateral load ing . In part icular, it was desired to see to what extent the f o l l o w i n g response characteristics w o u l d be captured b y the analysis: • T h e overa l l lateral load-displacement response o f the system. • T h e g loba l and loca l deformation modes o f the braces and components o f the V S C detai l . • T h e dis t r ibut ion o f y i e ld ing throughout the system. The major i ty o f the analysis effort was focused o n two sets o f s imulat ions . In the first set o f s imulat ions , the m o d e l was subjected to increasing story drift i n a s ingle direct ion. Parametric studies were performed to determine the sensi t ivi ty o f these pushover s imulat ions to the value o f mass proport ional damping prescr ibed i n the m o d e l and the coefficient o f f r ic t ion and contact a lgor i thm employed at s l i d ing interfaces. In the second set o f s imulat ions , the m o d e l was subjected to four separate c y c l i c l oad ing protocols . The protocols selected a l l owed the invest igat ion o f the effects o f v a r y i n g the load ing rate, repeating cyc les o f constant ampli tude, and increasing the cyc l e ampl i tude to large displacements. The results presented i n this chapter pertain o n l y to the finite element m o d e l predict ions. These predict ions are compared to the experimental results i n Chapter 7. 6.2 Description of the Full VSC-CBF Model The fu l l V S C - C B F M o d e l , shown i n F igure 6.1, contained a total o f 69,302 elements and 86,129 nodes. In the f o l l o w i n g sections, the m o d e l i n g considerations for each component and connect ion are described i n detail and the use o f analyt ica l contact surfaces is described. T h e geometry for each component o f the m o d e l was adopted f rom the fabrication drawings for the test specimen ( A p p e n d i x A ) . 106 Figure 6.1 F u l l V S C - C B F M o d e l 6.2.1 Braces In Sec t ion 5.3.1, the m o d e l i n g o f the H S S brace member for the s ingle brace analysis was described. The same brace mesh as shown i n F igure 5.2 was used for bo th braces i n the V S C - C B F mode l . T y p e 2 shell elements were again used w i t h 5 Gauss integrat ion points through the thickness, a l though for some o f the analyses 3 integrat ion points were used without any discernible impact o n the results. T y p e 4 hourglass cont ro l was used, and since the tendency towards hourglassing i n the single brace models had been observed to be l o w , the hourglass coefficient was reduced from 0.10 to 0 .01. T h e same M a t _ 0 2 4 P iecewise L inea r P las t i c i ty models that were described i n Sec t ion 4.3.4 were again used for the brace flat w a l l and corner w a l l materials. A failure plast ic strain o f 0.35 was specif ied for both the flat w a l l and corner w a l l materials. 6.2.2 Gusset Plates The 12 .7mm-thick bo t tom gusset plates and the 19mm-th ick top gusset plates were mode led i n a s imi la r manner as was described for the s ingle brace m o d e l i n Sec t ion 5.3.2, 107 except that the elastic modulus for both parts was specif ied as E = 200 ,000 M P a . F o u r T y p e 1 s o l i d elements were used through the thickness o f the gusset plates. The degrees o f freedom o f the bo t tom edge nodes o f the bot tom gusset plates were f ixed , s imula t ing the we lded connect ion to the base plates that were bol ted to the test ing frame. (The base plates were omit ted f rom the finite element model) . T h e mesh for the 19mm-th ick top gusset plate is s h o w n i n F igure 6.2. The mesh o f the top gusset plate was refined i n the area surrounding the three boltholes, i n order to approximate the round geometry o f the boltholes and to adequately resolve the anticipated stress concentrations i n this region. F igure 6.2 F u l l V S C - C B F mode l : T o p gusset plate mesh . 6.2.3 Beam Connection Plates (VSC Plate) The mesh for the beam connect ion plates is shown i n F igure 6.3. T h e beam connect ion plate assembly, hereafter co l l ec t ive ly referred to as the V S C plate, consisted o f two ver t ical 12 .7mm-thick plates (the "webs") , each conta in ing three ver t ica l slots, and a 25mm-th ick hor izonta l plate (the "flange") containing 2 rows o f 6 bol tholes . The V S C plate was mode led w i t h T y p e 1 so l id elements and T y p e 6 hourglass cont ro l , w i t h an hourglass coefficient o f 1.0. Four elements were used through the thickness o f the web plates, w h i l e three elements were used through the thickness o f the flange plate. The 108 mesh was refined i n the area surrounding the vert ical slots, i n order to adequately resolve the concentrated stresses due to bear ing o f the bolts against the edges o f the slots and to improve the performance o f the contact a lgor i thm that was used to m o d e l the bolt - slot interface. The mater ia l properties o f the 12 .7mm plate sample were assigned to a l l elements i n the V S C plate, w i t h the same M a t _ 0 2 4 P iecewise L i n e a r P las t i c i ty mater ial mode l that was used for the 12.7mm-thick bottom gusset plate. T h e m o d e l i n g o f the bolts i n the slotted bol ted connect ion and the V S C plate-to-beam connec t ion is discussed later i n Sec t ion 6.2.7, and the use o f contact interfaces at these connect ions is discussed i n Sec t ion 6.2.8. F igure 6.3 F u l l V S C - C B F mode l : V S C plate mesh . 6.2.4 Beam T h e mesh for the W 2 0 0 x 8 6 beam is shown i n F igure 6.4. The beam was mode led w i t h T y p e 1 s o l i d elements and T y p e 6 hourglass control , w i t h an hourglass coefficient o f 1.0. The number o f elements used through the thickness o f the flange and w e b port ions o f the 109 beam was l imi t ed b y the need to mainta in reasonable element aspect ratios w h i l e o p t i m i z i n g the total number o f elements used to mode l the member . Three elements were used through the thickness o f the flanges, w h i l e two elements were used through the thickness o f the web . T h e mesh was refined i n the central r eg ion o f the beam, where interaction w i t h the V S C plate occurred through a s imulated bo l t ed connect ion and a contact surface between the two parts. (The details o f the V S C plate-to-beam connect ion are discussed later i n Sect ions 6.2.7.2 and 6.2.8). The exper imental phase o f the research showed that the ver t ica l deflections at the mid-span o f the b eam were s m a l l dur ing c y c l i c loading. Therefore, it was deemed appropriate to assign elastic mater ia l properties to the beam, us ing the M a t _ 0 0 1 Elas t ic material mode l w i t h an elastic m o d u l u s o f E = 200,000 M P a . W h e n evaluat ing the analysis results, the computed element stresses i n the beam were examined for excurs ions beyond the assumed y i e l d stress o f 350 M P a . F igure 6.4 F u l l V S C - C B F mode l : B e a m mesh. 110 In the V S C - C B F test specimen, rol lers were used near the mid- span o f the beam to provide lateral support against out-of-plane movement o f the beam. T h i s restraint was s imulated i n the m o d e l b y us ing single point constraints to constrain the Y- t rans la t iona l degree o f freedom for sets o f nodes o n either side o f the beam, i n the regions that were contacted b y the rol lers dur ing the test. 6.2.5 Columns The co lumns o f the test assembly consisted o f pin-ended 1682mm- long 89x89x4 .8 H S S members ( 1 9 1 0 m m i n length between the p i n holes). These members carr ied ax i a l load o n l y and d i d not participate i n the lateral resistance o f the frame. It was decided that no benefits w o u l d be gained b y us ing a detailed m o d e l i n g approach for the co lumns , since ax i a l load was the o n l y quantity o f interest for these members . A single p in- jo in ted truss element was used to m o d e l each c o l u m n , w i t h cross sect ional area equal to that o f the H S S c o l u m n members . F o r each truss element, the degrees o f freedom o f the lower end node were f ixed against translation, thus creating the p inned support condi t ions . T h e top end node o f each truss element was connected to the beam trough a noda l r i g i d b o d y def ini t ion, w h i c h is described i n detail i n Sec t ion 6.2.7.3. A n elastic mater ia l m o d e l (Mat_001 Elas t i c ) was assigned to the c o l u m n parts, w i t h an elastic modu lus o f 200,000 M P a . 6.2.6 Treatment of welds The w e l d e d connect ions between the braces and gusset plates were treated b y merg ing the interfacing nodes, w h i c h were strategically zoned to be coincident . T h e ra t ional izat ion for this approach was described i n Sec t ion 5.3.3. T h e same approach was used to simulate the w e l d e d connect ion between the three plate components o f the V S C plate assembly. 6.2.7 Treatment of Bolted Connections 6.2.7.1 Bolts in vertical slotted connection T h e unique feature o f the V S C - C B F system is the ver t ical slotted connec t ion assembly between the top ends o f the braces and the beam above. The i m p r o v e d performance o f 111 the system under lateral load ing is due to the behaviour o f this connect ion: w h e n the compress ion brace buckles and suffers a reduct ion i n load ca r ry ing capacity, the appl ica t ion o f an unbalanced ver t ical load to the mid-span o f the beam b y the tension brace is prevented because the ver t ica l ly slotted bol ted connect ion assembly does not p rov ide resistance to ver t ica l force. A s the frame continues to translate lateral ly, the k inemat ics o f the system are such that the bolts translate d o w n w a r d i n the ver t ica l slots o f the V S C plate. I f the bolts were to come to bear against the bo t tom end o f the slots, then the tension brace w o u l d beg in to p i c k up addi t ional load and a net d o w n w a r d force w o u l d be exerted o n the beam. Therefore, the p r o v i s i o n o f sufficient slot length was an important des ign parameter for the test specimen. A d d i t i o n a l l y , the poss ib i l i t y o f b i n d i n g o f the bolts against the slot edges was o f concern. In order to accurately m o d e l the interaction between the bolts , the slots, the top gusset plate, the V S C plate, and the beam, a fu l ly three-dimensional finite element mode l o f a l l parts was developed that inc luded considerat ion o f contact between cr i t ica l touching surfaces. T h e three bolts i n the ver t ical slotted connect ion were 1-inch (25 .4mm) diameter A 3 2 5 bolts , 3.25 inches (82.6mm) i n length. The bolt mesh for the finite element m o d e l is shown i n F igu re 6.5. In order to reduce the number o f contact planes considered, the washer, bol t head, bol t shaft, and nut were a l l mode led as a s ingle s o l i d entity o f symmetr ic proport ions for each bolt . Thus the term "bo l t " herein refers to the complete bol t assembly. A n average diameter o f 46 m m was used for the bol t head and nut/washer port ions. The bolts were o f higher y i e l d strength than the steel plate materials that they connected, and deformat ion o f the bolts was not observed i n the experiments b y B u b e l a (2003). Therefore, the bol t parts o f the m o d e l were assigned r i g i d mater ia l properties. T h i s assumption a l l o w e d the bol t mesh to be arbitrari ly refined because the element s ize o f non-deformable parts does not affect the expl ic i t c r i t i ca l t ime step. Thus it was possible to use a fine mesh o f eight-noded s o l i d elements to discret ize the c y l i n d r i c a l shape o f the bol t shaft w i t h reasonable accuracy, w h i c h enabled a more accurate contact treatment between the bolts and the edges o f the slotted connect ion. E a c h bol t was defined w i t h a unique part ED, such that the equations o f rigid b o d y dynamics were appl ied to each bol t i n d i v i d u a l l y rather than be ing appl ied to a l l three as a group. 112 Figure 6.5 F u l l V S C - C B F mode l : V S C bol t mesh . In the V S C - C B F test assembly, each o f the three V S C bolts was centered i n a 1-1/16 i n c h (27mm) hole i n the top gusset plate. The interaction between the bo l t shaft and the edge o f the hole i n the top gusset plate was deemed to be o f l i t t le importance to the performance o f the connect ion, and it was decided that the addi t iona l numer ica l complex i ty (and noise) that w o u l d be invo lved w i t h i n c l u d i n g this interface was not jus t i f ied . Therefore the nodes o f the bolt shaft were merged w i t h the adjacent nodes o f the bol thole edge i n the top gusset plate. The interaction between the bolts and surfaces o f the V S C plate was mode led w i t h a contact surface def in i t ion , as descr ibed i n Sec t ion 6.2.8. In the actual experiment the bolts i n the ver t ica l slotted connec t ion were o n l y tightened "f inger t ight ," and therefore no pre-stress was assigned to the bol ts i n the finite element m o d e l . A close-up v i e w o f the slotted bol ted connect ion po r t ion o f the m o d e l is shown i n F igure 6.6. 113 Figure 6.6 F u l l V S C - C B F mode l : Slotted bol ted connect ion . 6.2.7.2 Bolts Connecting VSC Plate to Beam In the V S C - C B F test specimen, the flange o f the V S C plate was connected to the bot tom flange o f the beam b y two rows o f 1-inch (25.4mm) diameter A 3 2 5 bol ts . D u e to the fact that no s l ip was anticipated i n this bol ted connect ion, it was not deemed necessary to mode l the connect ion w i t h the same level o f detail as was used for the slotted bol ted connect ion described previous ly . Howeve r , a r i g i d connec t ion obtained b y merg ing adjacent nodes o f the two parts w o u l d not capture the l o c a l stresses due to force transfer through the bol ted connect ion. The approach that was adopted was to include the discrete boltholes i n the so l id element mesh o f the beam and V S C plate parts, and to connect the inside surfaces o f each bol thole pai r b y a noda l r i g i d b o d y that w o u l d approximate the bolt . T h i s concept is illustrated i n F igure 6.7. A n o d a l r i g i d b o d y is a r i g i d entity that is def ined b y r i g i d l y l i n k i n g two or more nodes. A s ingle node was defined at the centre o f the bol thole diameter, i n the hor izonta l p lane l y i n g coincident w i t h the top surface o f the V S C plate flange and the bo t tom surface o f the beam flange. F r o m this central node, noda l r i g i d bodies extended to a l l the nodes o f the ins ide surfaces 114 o f the boltholes above and be low. W i t h this approach, the transfer o f shear and ax ia l forces through the bol ted connect ion c o u l d be adequately mode led , wi thou t the need to use a so l id element m o d e l o f the bolts. A s described later i n Sec t i on 6.2.8, a contact surface was defined between the V S C plate flange and the bo t tom beam flange, al though penetration o f these two surfaces i n the presence o f the noda l r i g i d b o d y connect ion mode l was un l ike ly . F igure 6.7 F u l l V S C - C B F mode l : V S C plate flange to beam flange connect ion . 6.2.7.3 Pinned Column-to-Beam Connection The p inned connect ion o f the top o f each c o l u m n member to the beam was also mode led w i t h the nodal r i g i d b o d y approach, as shown i n F igure 6.8. T h e top end node o f the c o l u m n truss element served as an anchor point, from w h i c h n o d a l r i g i d bodies were defined to the nodes o n the underside surface o f the bo t tom beam flange, corresponding to the area that was supported i n the test specimen. The lack o f momen t resistance at the end node o f the p in- jo in ted truss member enabled the connect ion to behave as a p i n . 115 Figure 6.8 F u l l V S C - C B F mode l : P inned column- to-beam connect ion . 6.2.8 Contact Definitions It was anticipated that out-of-plane b u c k l i n g o f the braces w o u l d cause the top gusset plate to deform and contact the inside surfaces o f the V S C plate webs . In order to prevent penetration o f the top gusset plate through the V S C plate, surface-to-surface contact was defined between the two parts us ing the L S - D Y N A k e y w o r d *Contact_Automat ic_Surface_to_Surface . A s imi l a r contact def in i t ion was made between the top surface o f the V S C plate flange and the underside surface o f the bot tom flange o f the beam. In both cases, the standard surface-to-surface penalty-based contact a lgor i thm was used, i n w h i c h the slave segment nodes are checked for penetration o f the master segment surface, fo l lowed b y a reverse check for penetrat ion o f the master segment nodes through the slave segment surface. Contact was also def ined between the three-dimensional bol ts and the V S C plate. The b o l t - V S C plate interact ion was expected to invo lve the contact o f n o m i n a l l y para l le l element surfaces, but also the contact o f element corner edges. W i t h the standard surface-to-surface penalty-based contact a lgori thm, w h i c h is based o n the penetration o f 116 i n d i v i d u a l nodes through segment surfaces, it is poss ible for element edge-to-edge contact to go undetected. In contrast, the segment-based penal ty formula t ion contact a lgor i thm checks for penetration o f segment planes from both bodies, such that element edge-to-edge contact is detectable. The s l id ing behaviour a long element edges can be further i m p r o v e d b y us ing an element connect iv i ty approach, i n w h i c h the segments that surround the current ly contacted segment are moni tored for potential contact as the s l i d ing proceeds. In order to accurately capture the surface-to-surface and edge-to-edge s l i d ing nature o f the b o l t - V S C plate interact ion, the *Contact_Automat ic_Surface_to_Surface k e y w o r d was used, a long w i t h the S O F T = 2 , S B O P T = 4 , and D E P T H = 5 options to activate segment-based contact. The coefficients o f static and dynamic friction for a l l contact surfaces described above were set to 0.3, w h i c h was deemed to be a reasonable value based o n the results o f tests o f bol ted steel jo in ts reported b y Vasa rhe ly i and C h i a n g (1967). 6.3 Analysis Methods The m o d e l i n g approach carr ied w i t h it certain impl ica t ions for the methods o f analysis. D u e to the in i t i a l gap between the bolts and the edges o f the slots i n the V S C plate, the m o d e l i n i t i a l l y consisted o f two uncoupled systems: 1) the beam and V S C plate, supported b y the p in-ended co lumns , and 2) the top gusset plate and bol t assembly, connected to the braces and bot tom gusset plates. S ince the first sys tem constituted a mechan i sm, the m o d e l c o u l d not be analysed w i t h L S D Y N A b y i m p l i c i t static analysis or b y l inear b u c k l i n g analysis . It was , however , s t i l l possible to conduct an i m p l i c i t e igenvalue analysis to gain u n understanding o f the v ib ra t ion characteristics o f the structure. It was recognized that us ing the expl ic i t solver for the t ime stepping analyses i n the pushover and c y c l i c load ing s imulat ions c o u l d require s ignif icant run t ime, due to the exp l i c i t c r i t i ca l t ime step. L S - D Y N A offers an automatic i m p l i c i t - e x p l i c i t swi tch ing technique, i n w h i c h the i m p l i c i t dynamic solver can be used for the major i ty o f the s imula t ion , passing control to the expl ic i t solver for a number o f computa t ional cycles 117 whenever the i m p l i c i t solver encounters convergence diff icul t ies . The automatic swi tch ing technique was tr ied for the pushover and c y c l i c l oad ing s imulat ions , but it was found that contact-related nonlineari t ies caused significant convergence diff icul t ies for the i m p l i c i t solver, such that the expl ic i t solver ended up treating the major i ty o f the s imula t ion . Furthermore, even w h e n the i m p l i c i t solver d i d converge, the large size o f the m o d e l caused the i m p l i c i t solver to be less efficient than the exp l i c i t solver. Therefore, the expl ic i t solver was used for the t ime stepping analyses described i n sections 6.5 and 6.6. 6.4 Eigenvalue Analysis A n i m p l i c i t eigenvalue analysis was performed to obtain the v ib ra t ion frequencies and mode shapes for the V S C - C B F mode l . T h e first s ix v ibra t ion mode shapes are shown i n F igure 6.9, and the corresponding frequencies and periods are l is ted i n Tab le 7.1. D u e to the phys i ca l gaps between some parts o f the m o d e l (specif ica l ly , be tween the bolts and the slots i n the V S C plate, and between the top gusset plate and the V S C plate) some o f the v ib ra t ion modes i n v o l v e d loca l v ib ra t ion o f certain parts independent from the rest o f the m o d e l . 118 Figure 6.9 E igenva lue analysis results: Fi rs t s ix v ib ra t ion mode shapes. Table 6.1 E igenva lue analysis results: Firs t six v ibra t ion frequencies and periods. M o d e E igenva lue C i r c . F r e q . (rad/sec) F req . ( H z ) P e r i o d (sec) 1 19731 140 22 0.045 2 33729 184 29 0.034 3 56158 237 38 0.027 4 137376 371 59 0.017 5 165910 407 65 0.015 6 244607 495 79 0.013 119 6.5 Pushover Simulation Before attempting a complete c y c l i c load ing s imula t ion o f the V S C - C B F system, a pushover s imula t ion was conducted. The term "pushover" is used i n this context to denote the unid i rec t ional in-plane lateral m o t i o n that was appl ied to the beam; it should not be confused w i t h the "pushover analys is" methods used i n the nonl inear static analysis o f structures for se ismic design, al though the end result is s imi la r . F o r the pushover s imula t ion , the complete response o f the system to increas ing lateral displacement (storey drift) was assessed. Thus , the load ing condi t ions m a y be l ikened to the first quarter cyc l e o f a c y c l i c load ing pro tocol . Geomet r ic imperfect ions were introduced i n the compress ion brace on ly . The imperfect ions were appl ied b y impor t ing the imperfect noda l coordinates f rom the single brace m o d e l w i t h a m a x i m u m mid- length crookedness o f L / 1 0 0 0 i n the out-of-plane direct ion. Stiffness propor t ional damping o f 10% was used to reduce h i g h frequency vibrat ions , and mass-proport ional damping was used w i t h a damping constant o f D = 100 to damp the l ower modes o f deformation. The eigenvalue analysis predicted a fundamental c i rcu lar frequency o f 140 rad/s for the V S C - C B F system. H o w e v e r , it was not anticipated that b u c k l i n g o f the brace dur ing the pushover s imula t ion w o u l d cause deformat ion i n the fundamental mode from that analysis, i n w h i c h both braces vibrate out-of-plane together. Rather, it was anticipated that o n l y the buck l ed brace w o u l d vibrate substantially, and therefore the fundamental mode f rom the eigenvalue analysis o f the s ingle brace m o d e l was deemed to be a more appropriate parameter for ass igning the mass propor t ional damping l eve l . U s i n g the fundamental c i rcular frequency o f 200 rad/s from the single brace analysis , the damping constant D = 100 was interpreted as p r o v i d i n g 2 5 % mass propor t ional damping to the V S C - C B F system. T h e s ingle brace analyses showed that the use o f s ingle p rec i s ion made o n l y a 2 % difference i n the predicted b u c k l i n g load, but caused the runt ime to increase b y more than a factor o f four. S ing le p rec i s ion was therefore employed i n order to complete a number o f pushover s imulat ions i n a reasonable amount o f t ime. 120 T h e lateral load ing for the pushover s imula t ion was app l ied b y p resc r ib ing a v e l o c i t y curve to a group o f nodes o n the top surface o f the top flange o f the beam, i n the region equivalent to the attachment point o f the load actuator to the test spec imen. The shape o f the ve loc i ty curve was chosen so as to give a total lateral d isplacement o f 5 6 m m , w h i c h corresponded to t w o t imes the y i e l d displacement observed i n the quasi-static c y c l i c load ing test. A load ing duration o f 0.70s was specif ied, w h i c h corresponded to approximate ly 15 t imes the fundamental per iod o f the entire system, or 22 t imes the fundamental per iod o f the single brace. The prescribed v e l o c i t y curve is s h o w n i n F igure 6.10, a long w i t h the corresponding displacement curve. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Time (s) (a) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Time (s) (b) F igure 6.10 Prescr ibed hor izonta l motions for pushover s imula t ion : (a) smooth ve loc i ty prof i le ; (b) corresponding smooth displacement prof i le . 121 6.5.1 Pushover Simulation Results T h e response o f the V S C - C B F system to in-plane lateral l oad is s h o w n i n terms o f the lateral load-displacement curve o f F igure 6.11. In this curve, the lateral l oad is measured as the total X - c o m p o n e n t o f the noda l reaction force at the loca t ion o f the prescr ibed ve loc i ty o n the beam, and the lateral displacement i s the X - c o m p o n e n t o f the beam displacement, where the x -ax i s is a l igned w i t h the in-plane lateral d i rec t ion . A lateral displacement o f about 1 m m can be seen i n F igure 6.11 before the l o a d begins to increase i n the l inear range. T h i s 1 m m o f displacement corresponds to the lateral displacement o f the beam that was required to cause the bolts to come into contact w i t h the edges o f the slots i n the V S C plate. O n c e the bolts were "set" i n the connect ion , the response o f the system was essential ly l inear un t i l g loba l out-of-plane b u c k l i n g o f the compress ion brace occurred. T h e m a x i m u m lateral load resisted b y the sys tem was 372 k N , at a lateral displacement o f 1 4 m m . F o l l o w i n g the attainment o f the m a x i m u m load , the lateral load leve l dropped, s h o w i n g a l oca l m i n i m u m point o n the load-displacement curve near 1 7 m m o f displacement. T h i s osc i l l a t ion was deemed to be due to the fact that the system was not c r i t i c a l l y damped. 400 - i 350 - / \ 300 - / \ g 200 - / 150 - / ^ - ^ ^ 100 - / 50 - / 0 -IT- 1 1 1 1 1 0 10 20 30 40 50 60 Displacement (mm) Figure 6.11 Pushover s imula t ion results: Lateral load-displacement response. 122 F r o m Equa t ion 6.2, the m a x i m u m lateral load o f 372 k N w o u l d i m p l y a m a x i m u m brace ax i a l load o f 4 5 7 k N . Indeed, a plot o f the no rma l cross-sect ional forces i n the compress ion and tension braces (Figure 6.12) shows that the m a x i m u m ax i a l forces were 464 k N and 452 k N , respect ively. Af t e r b u c k l i n g o f the compress ion brace, the ax ia l load i n the tension brace decreased, but not to the same extent as the l oad i n the compress ion brace. It was speculated that the presence o f fr ic t ion at the slotted connec t ion might have been responsible for the unbalanced brace loads i n the pos t -buck l ing range. 500 0 10 20 30 40 50 60 System Lateral Displacement (mm) Figure 6.12 Pushover s imula t ion results: B race ax i a l forces. The effective shortening o f the compress ion brace due to b u c k l i n g and the cont inued lateral displacement o f the frame necessitated that the bolts translate d o w n w a r d i n the ver t ical slotted connect ion . T h e translation o f the bolts was captured i n the analysis , as shown i n F igure 6.13 for the final lateral displacement o f 5 6 m m . T h e ver t ica l translations o f the centres o f mass o f each o f the three bolts are plot ted i n F igure 6.14. The b u c k l i n g o f the compress ion brace caused a s l ight in-plane rotat ion o f the top gusset plate, w h i c h led the bol t nearest the compress ion brace to translate farther than the centre bol t and the bolt nearest the tension brace. The total ver t ica l t ranslat ion o f the bol t near 123 the compress ion brace dur ing the s imula t ion was 26 m m , w i t h a ver t i ca l gap o f 44 m m remain ing between the bot tom edge o f the bolt shaft and the bot tom edge o f the slot. « § § z Lx Figure 6.13 Pushover s imula t ion results: Translat ion o f bolts and top gusset plate relat ive to V S C plate, at 5 6 m m lateral displacement. -5 6 -10 J2 8-S -15 o 1 -20 > -25 H -30 Bolt Near Tension Brace Bolt in Middle Position Bolt Near Compression Brace 10 20 30 40 System Lateral Displacement (mm) 50 60 Figure 6.14 Pushover s imula t ion results: V e r t i c a l r i g i d b o d y displacement o f the three bolts i n the slotted connect ion. 124 Loca t ions o f y i e l d i n g throughout the mode l can be v i s u a l l y ident i f ied b y p lo t t ing contours o f effective plast ic strain and us ing a l o w m a x i m u m value for the range o f fringe levels . Contours o f effective plastic strain are shown i n F igure 6.15, i n w h i c h the range o f fringe levels has been set from 0 to 0.002. Y i e l d i n g occurred through the mid- length region o f the compress ion brace, and i n the top and bot tom gusset plates. The development o f a plast ic hinge at the mid- length o f the compress ion brace resulted i n the formation o f l oca l b u c k l i n g o f the member as the lateral d isplacement o f the beam reached approximate ly 4 5 m m . (The loca l buck l ing effects are discussed i n further detail i n the context o f the c y c l i c load ing s imula t ion o f Sec t ion 6.6.2). T h e redis t r ibut ion o f loads throughout the compress ion brace as the member deformed from the unbuck led conf igurat ion to the g loba l ly -buck led configurat ion and f ina l ly to the l o c a l l y b u c k l e d configurat ion can be seen b y compar ing the effective stress contours s h o w n i n F igure 6.16. Fringe Levels 2.000e-03_ 1.800e-03_ 1 1.600e-03 _ 1.400e-03_ 1.200e-03_ 1.000e-03_ 8.000e-04_ 6.000e-rj4 _ 4.000e-04 _ 2.000e-04 _ O.OOOe+00 _ max ipt. value min= 0.000 max=0.350001 Figure 6.15 Pushover s imula t ion results: Contours o f effective plas t ic strain. 125 Figure 6.16 Pushover s imula t ion results: Contours o f effective stress: (a) p r io r to b u c k l i n g o f brace; (b) dur ing g loba l b u c k l i n g o f brace; (c) dur ing l o c a l b u c k l i n g o f brace. S ince an elastic mater ial mode l was used for the beam part, the contours o f effective plast ic strain shown i n F igure 6.15 do not have any mean ing for this part. H o w e v e r , the contour plots o f effective stress i n F igure 6.17 show that the m a x i m u m effective stress d i d not exceed 350 M P a for any element i n the beam, w h i c h indicates that y i e l d i n g o f the beam d i d not occur du r ing the pushover s imulat ion. T h e m a x i m u m effective stress i n the beam was concentrated i n a few elements surrounding the bol tholes i n the bo t tom flange. These elements reached a m a x i m u m effective stress o f about 320 M P a , at the t ime o f peak lateral load transfer between the beam and the V S C plate. T h e m a x i m u m mid-span ver t ical def lect ion o f the beam was 3.2 m m . 126 ] Fringe Levels (MPa) 3.207e+02 2.886e+02 2.566e+02 2.245e+02 1.924e+02 1.604e+02 1.283e+B2 9.622e + U1 6.416e + 01 3.209e+01 . 2.305e-02 Time = 0.27 Contours of Effective Stress (v-m) max ipt. value min=0.0230526, at elem# 21982 max=320.688, at elem# 27560 Figure 6.17 Pushover s imula t ion results: Contours o f effective stress i n the beam. The g loba l energy levels for the pushover s imula t ion are s h o w n i n F i g u r e 6.18. T h e total energy dissipated b y the system, w h i c h is equal to the external w o r k appl ied to the system, reached a m a x i m u m value o f 11.21 k J . The internal strain energy reached a m a x i m u m o f 8.73 k J , account ing for 7 8 % o f the m a x i m u m total energy. A s the translation o f the bol ts i n the ver t ical slots proceeded du r ing the s imula t ion , the energy dissipated due to s l i d i n g at the contact interfaces increased to a m a x i m u m value o f 1.79 k J , w h i c h accounted for 16% o f the m a x i m u m total energy. T h e mass-proport ional damping energy reached a m a x i m u m o f 0.56 k J , account ing for 5 % o f the m a x i m u m total energy. The m a x i m u m kinet ic energy was negl ig ib le compared to the m a x i m u m internal energy (it is bare ly v i s i b l e i n F igure 6.18), suggesting that a quasi-static process was achieved. 127 Time (s) Figure 6.18 Pushover s imula t ion results: Ene rgy levels . It should be noted that the run t ime to complete the pushover s imu la t ion was 76 hours and 32 minutes (about 3.2 days). T h e lengthy run t ime p laced prac t i ca l restrict ions o n the number o f s imulat ions that cou ld be carr ied out, so it was important to care fu l ly select the parameters that were investigated i n the sensi t ivi ty studies descr ibed i n the f o l l o w i n g sections. 6.5.2 Effect of Varying the Level of Mass Proportional Damping The pushover s imula t ion was repeated w i t h two mod i f i ed models , i n w h i c h the mass proport ional damping constant was modi f i ed from the o r ig ina l va lue o f D = 100 to D = 200 and D = 0, corresponding to 5 0 % o f cr i t ica l and zero damping , respect ively . T h e general deformat ion responses o f the two mod i f i ed mode ls were s imi l a r to the or ig ina l m o d e l : g loba l b u c k l i n g o f the compress ion brace was f o l l o w e d b y l o c a l b u c k l i n g o f the brace and d o w n w a r d translation o f the bolts as the system was pushed to large lateral displacements. In the mode l w i t h no mass-proport ional damping , the brace was seen to vibrate about the g loba l l y buck l ed configurat ion. 128 The effect o f chang ing the damping leve l is shown i n terms o f the resu l t ing lateral load-displacement curves o f F igure 6.19. F o r the case o f D = 200 , the m a x i m u m lateral load resisted b y the system was 398 k N , w h i c h was 7 % higher than the o r ig ina l case w i t h D = 100. In the pos t -buck l ing range, the response was s imi la r to the o r ig ina l case, al though the load osc i l l a t ion f o l l o w i n g the in i t ia l b u c k l i n g was reduced due to the higher l eve l o f damping . F o r the m o d e l without mass-proportional damping , the m a x i m u m lateral load resisted b y the sys tem was 340 k N , w h i c h was a 9 % reduct ion compared to the o r ig ina l case. In the pos t -buck l ing range, the v ibra t ion o f the undamped brace induced osci l la t ions o f the lateral load-displacement curve. 10 20 30 40 50 60 Displacement (mm) Figure 6.19 Effect o f v a r y i n g the leve l o f mass propor t iona l d a m p i n g : Latera l load-displacement response. 6.5.3 Effect of Reducing the Coefficient of Friction In order to consider an ideal case o f zero fr ict ion at the bol t-s lot interface, a mod i f i ed mode l was analysed, i n w h i c h the coefficient o f fr ict ion was changed from the or ig ina l value o f 0.3 to zero. 129 The g loba l deformation response for the modi f i ed mode l was again s i m i l a r to the o r ig ina l m o d e l . T h e ver t ica l translat ion o f the bolts increased i n magni tude b y approx imate ly 1.5 m m i n the absence o f fr ic t ion. The response o f the m o d e l w i t h zero f r ic t ion at the bolt-slot interface is compared to the or ig ina l mode l i n terms o f the lateral load-displacement curves o f F igure 6.20. T h e two systems showed s imi l a r response up to the point o f g loba l b u c k l i n g o f the compress ion brace. In the pos t -buckl ing range, the sys tem w i t h the frictionless connect ion showed lower lateral load resistance for a g i v e n l eve l o f displacement. T h i s behaviour m a y be explained b y invest igat ing the var ia t ion i n brace ax ia l forces, as s h o w n i n F igure 6.21. F r o m Figure 6.21 it is apparent that, i n the absence o f f r ic t ion, the forces i n the two braces remained essential ly iden t ica l even i n the post-b u c k l i n g range. T h e tension brace d i d not provide greater pos t -buck l ing resistance than the compress ion brace, as was the case i n the or ig ina l system, and thus the resul t ing post-b u c k l i n g lateral load resistance p rov ided b y the system w i t h the fr ict ionless connect ion was reduced compared to the or ig ina l system. H o w e v e r , the equa l iza t ion o f brace forces that was achieved i n the system w i t h the frictionless connec t ion ef fec t ive ly e l imina ted the appl ica t ion o f unbalanced load to the beam above; the m a x i m u m mid- span displacement o f the beam was o n l y 1.4 m m . 0 10 20 30 40 50 60 Displacement (mm) Figure 6.20 Effect o f reducing the coefficient o f f r ic t ion : Latera l load-displacement response. 130 0 10 20 30 40 50 60 System Lateral Displacement (mm) Figure 6.21 Effect o f reducing the coefficient o f friction: B race ax i a l forces. 6.5.4 Effect of Using an Alternative Contact Algorithm A s descr ibed i n Sec t ion 6.2.7.1, a segment-based contact a lgo r i thm w i t h specia l capabil i t ies for hand l ing edge-to-edge contact was used for the contact interface between the bolts and the slotted connect ion. In some pre l iminary models , the standard surface-to-surface penalty-based contact a lgor i thm was used at this interface. A sample plot o f the lateral load-displacement response o f such a m o d e l is s h o w n i n F i g u r e 6.22. In the pos t -buck l ing range, a pecul iar step-wise response was observed. T h e reason for this g loba l behaviour was determined b y observing the l oca l behaviour o f the slotted connect ion i n the pos t -buck l ing range. A s the bolts translated i n the slot, s m a l l penetrations o f the bol t elements into the elements o f the V S C plate were observed, as shown i n F igure 6.23. These penetrations were seen to occur as the edges o f the bol t elements attempted to s l ide past the corner edge nodes o f the slot. T h e bol t elements were temporar i ly "caught" at the edge nodes o f the slot, caus ing the bol ts to d i g into the edges o f the slot. Even tua l ly the bolt elements w o u l d break free and continue to sl ide downward , unt i l be ing caught at the next corner node o f the slot mesh . D u r i n g the t ime that the bolts were caught b y the slot corner edge nodes, ver t ical load was resisted b y the 131 slotted connect ion, and thus the system behaved s i m i l a r l y to a conven t iona l braced frame. The base shear reactions at the bot tom gusset plates are compared i n F igu re 6.24. T h e base shear at the gusset plate connected to the tension brace showed an increase dur ing the t ime that the bol ts were caught b y slot corner edge nodes, i nd ica t ing that the tension brace was p i c k i n g up load . The summat ion o f the two base shear components shown i n F igure 6.24 is equivalent to the total lateral load resistance that was measured at the beam leve l , shown i n F igure 6.22. 0 10 20 30 40 50 60 Displacement (mm) Figure 6.22 Effect o f us ing an alternative contact a lgor i thm: T i m e h is tory o f lateral load. 132 Figure 6.23 Effect o f us ing an alternative contact a lgor i thm: Penetrat ion o f bol t element edge into corner edge o f slot element. System Lateral Displacement (mm) Figure 6.24 Effect o f us ing an alternative contact a lgor i thm: C o m p a r i s o n o f base shears. 133 When the contact algorithm was changed to the segment-based type, the problem was remedied, allowing the bolt edges to slide smoothly along the corner edges of the slot. Although the peculiar behaviour described above was due to modeling-related factors rather than physical behaviour, it may be interpreted as illustrating the hypothetically consequences if real binding of the bolts in the slotted connection were to occur. 6.5.5 Spurious Effects in Preliminary Models: Hourglassing and Shear Locking It is instructive to report the analysis results from some preliminary models, in order to appreciate the numerical effects of hourglassing and shear locking. In the first attempt to model the full VSC-CBF system, the meshes for the VSC plate, the bolts, and both gusset plates were coarser than the corresponding meshes described previously. Specifically, the webs of the VSC plate and bottom gusset plate contained only two solid elements through the thickness, while the top gusset plate contained only three solid elements through the thickness. The height and width dimensions of the elements in these parts were also larger than those in later models. Perhaps the most significant difference in the preliminary mesh was the relative lack of refinement in the bolt mesh and the areas surrounding the edges of the vertical slotted connection. Figure 6.25 shows the coarse mesh used in the preliminary models. The solid elements used in the preliminary models were the same reduced-integration Type 1 solid elements, but the default LS-DYNA viscous hourglass control (Type 1) was used, rather than the Type 6 Belytschko-Bindeman assumed strain co-rotational stiffness form of hourglass control that was later adopted. 134 Figure 6.25 Coarse mesh used i n p re l iminary models o f the V S C - C B F system. The pushover s imula t ions w i t h the pre l iminary m o d e l resulted i n the anticipated g loba l behaviour, w i t h out-of-plane b u c k l i n g o f the compress ion brace. H o w e v e r , the loca l element deformations were severely distorted due to hourglass ing o f the mesh. The hourglass ing was par t icular ly significant i n the V S C plate, as s h o w n i n F i g u r e 6.26. 135 Figure 6.26 Hourg lass ing i n the p re l imina ry m o d e l . In an attempt to e l iminate the hourglassing o f the s o l i d element parts, the under-integrated so l id elements were replaced w i t h ful ly integrated s o l i d elements. A l t h o u g h the hourglass ing p r o b l e m was solved b y this approach, the compress ion brace was seen to buck le in-plane rather than out-of-plane, as shown i n F igure 6.27. T h e unexpected i n -plane deformation mode resulted due to shear l o c k i n g o f the ful ly-integrated elements i n the top and bot tom gusset plates, w h i c h caused the gusset plates to be ext remely s t i f f i n bending, thus prevent ing out-of-plane b u c k l i n g o f the brace. 136 Figure 6.27 In-plane brace b u c k l i n g due to use o f fu l ly integrated elements i n so l id element parts. The behaviour o f the m o d e l was eventually improved through two k e y adjustments to the mode l . The first adjustment was to revert to the under-integrated T y p e 1 s o l i d elements, combined w i t h the T y p e 6 B e l y t s c h k o - B i n d e m a n assumed strain co-rotat ional stiffness form o f hourglass cont ro l . The detailed formulat ion o f T y p e 6 hourglass control was discussed i n a paper b y Be ly t schko and B i n d e m a n (1993), i n w h i c h it was s h o w n that the formulat ion gives excel lent results for under-integrated s o l i d elements subjected to elastic bending. In order to achieve acceptable performance i n inelast ic bend ing , the second adjustment to the m o d e l was to refine the mesh i n the through-thickness di rect ion. The improved behaviour o f T y p e 1 so l id elements w i t h T y p e 6 hourglass control was conf i rmed through a study o f s imple canti levered beam models . T h e details o f the canti levered beam study are inc luded i n a separate report ( E E R F 05-02) . In addi t ion to i m p r o v i n g the performance o f the so l id elements i n bending, the ref inement o f the mesh i n the v i c i n i t y o f the slotted connect ion improved the d is t r ibut ion o f contact forces and prevented element penetrations at this interface. 137 6.6 Cyclic Loading Simulations The full VSC-CBF model described in Section 6.2 was subjected to a variety of cyclic loading protocols in order to investigate the prediction of the inelastic cyclic behaviour of the system. In the quasi-static cyclic loading test of the VSC-CBF system, the top beam of the test assembly was subjected to in-plane horizontal cycles dictated by a sinusoidal displacement control signal, the parameters for which are listed in Table 1.1. To allow eventual comparison of experimental and analytical results, it was desired to simulate a similar loading protocol in the finite element model. However, it was noted that the run time for the pushover simulations, which included only a quarter cycle of 56mm amplitude over 0.7s duration, was typically greater than 3 days. This suggested that a simulation of 3 full cycles at that same loading rate would have taken at least 36 days, and this would not even consider cycles at other displacement amplitudes. It was clear that the loading rate would have to be reconsidered, and that complete replication of the loading protocol listed in Table 1.1 might not be feasible with the current modeling approach and available computing hardware. Four separate cyclic simulations were performed, and are designated herein by the simulation codes A, B, C, and D. All four simulations were based on the same model described in Section 6.2, with the exception of the treatment of geometric imperfections. In simulations A and B, the geometric imperfections (of magnitude L/1000) were only included in the brace that was subjected to compression first. The imperfections were omitted in the other brace, to allow the investigation of whether the general deformations of the system in the first half-cycle would cause sufficient eccentricities to trigger buckling in the initially "perfect" brace during the reversed loading of the second half-cycle. In simulations C and D, geometric imperfections (of magnitude L/1000) were included in both braces, in opposite out-of-plane directions. The loading protocol parameters used for the five cyclic simulations are listed in Table 7.2. The protocols were implemented in the simulations by prescribing smooth velocity curves, but it is the corresponding displacement curve parameters that are listed in Table 138 7.2. The average rate for each cycle was computed as the total lateral displacement in each cycle divided by the cycle period. Table 6.2 Cyclic simulations: Displacement loading protocols. Simulation Step Amplitude (mm) Period (s) Average Rate (mm/s) No. of Cycles A 1 14 1.2 47 1 2 21 1.2 70 0.25 B 1 14 0.12 467 1 2 21 0.12 700 1 3 28 0.12 933 1 C 1 14 0.4 140 1 2 28 0.8 140 1 3 56 1.6 140 1 D 1 14 0.8 70 3 Double precision was used in all five cyclic simulations, to reduce the possibility of round-off error accumulation over the large number of computational cycles that were expected. It was recognized that the decision to use double precision would carry a significant run time penalty. In the following sections, the results from simulations A and B are compared first, followed by separate discussion of the results of simulation C and simulation D. 6.6.1 Cyclic Simulations A and B The input for simulations A and B were identical except that the cycle period for simulation B was reduced by a factor of 10 compared to simulation A. Also, due to the higher loading rate in simulation B, it was possible to complete an additional cycle (28mm amplitude) in a reasonable length of run time. The first 1.25 cycles from simulation B included the same displacement amplitudes as the first 1.25 cycles from simulation A. The lateral load-displacement curves for the first 1.25 cycles from 139 s imulat ions A and B are shown separately i n Figures 6.28 and 6.29, respect ively. In F igure 6.30, the two curves are compared directly. Displacement (mm) Figure 6.28 C y c l i c s imula t ion A : Lateral load-displacement response. -6ee-1 =666^ Displacement (mm) Figure 6.29 C y c l i c s imula t ion B: Lateral load-displacement response (first 1.25 cycles on ly) . 140 me L _ -600 1 1 Displacement (mm) Figure 6.30 Comparison of cyclic simulations A and B : Lateral load-displacement response. During the first quarter cycle of simulation A , the compression brace buckled globally at a lateral load level of 348 kN and lateral displacement of 13mm. Upon reversal of the loading, the other brace (in which geometric imperfections were not included) buckled globally at a lateral load level of 416 kN and lateral displacement of -14mm. During the first quarter of the second cycle, degradation of the system stiffness and strength were observed, as the compression brace buckled globally at a lower lateral load level of 300 kN. Local buckling of the braces was not observed for the lateral displacement levels considered in this simulation. During the first cycle of simulation B , lateral load levels of 400 kN and lateral displacements of +/- 14mm were achieved without buckling of either brace. During the first quarter of the second cycle, the system resisted additional lateral load up to a maximum of 517 kN, at a lateral displacement of 20mm, at which point global buckling was finally triggered in the compression brace. 141 The significant differences between simulat ions A and B h ighl igh t the inf luence o f t ime scal ing o n the response o f the system, part icular ly w i t h regard to b u c k l i n g o f the braces. W h i l e the results o f s imula t ion A were i n l ine w i t h the results o f the pushover s imula t ion , s imula t ion B showed quite different behaviour. The response o f the m o d e l at h igh loading rates can be further invest igated b y observ ing the complete lateral- load displacement curve for s imula t ion B , s h o w n i n F igure 6.31, w h i c h includes the addi t ional cyc le to lateral displacements o f +/- 2 8 m m . The pecul iar rounded shape o f the hysteretic loops, combined w i t h the j agged response as the system passed through zero displacement, seemed to suggest that a quasi-static so lu t ion was not been achieved b y this s imula t ion . Displacement (mm) Figure 6.31 C y c l i c s imula t ion B : Lateral load-displacement response (a l l three cycles) . The load ing rates i n s imula t ion A were such that the run t ime was approx imate ly 351 hours (14.6 days) to complete the 1.25 cycles . The load ing rates i n s imula t ion B were such that the run t ime was approximately 125 hours (5.2 days) to comple te the 3 cycles . In each o f these s imulat ions , the cyc le per iod was he ld constant w h i l e the displacement 142 amplitude was increased for successive cycles. A s a result, the average loading rate increased with each cycle. 6.6.2 Cyclic Simulation C In simulation C, lateral displacement cycles with 14mm, 28mm, and 56mm amplitudes were prescribed. For these three cycles, it was estimated that the run time would approach 35 days i f the 1.2 sec cycle period from simulation A were adopted for simulation C . This estimated run time was considered to be the maximum acceptable run time, but it was noted that the cycle periods should be adjusted to give a constant average loading rate for each cycle. The average loading rate selected was 140mm/s for all cycles, which was a faster rate than the one used for simulation A , but slower than the one used for simulation B . The prescribed displacement protocol from simulation C is shown in Figure 6.32. 60 -60 Time (s) Figure 6.32 Simulation C: Prescribed displacement protocol. The results of simulation C are shown in terms of the lateral load-displacement curve o f Figure 6.33. K e y points of the system behaviour are numbered on Figure 6.33, and reference is made to these points in the following paragraphs. In the first quarter o f the 143 first cyc le , the compress ion brace buck l ed g loba l ly out-of-plane at a lateral load o f 372 k N , and a lateral displacement o f 1 3 m m (Point 1, F igu re 6.33). U p o n load reversal, the second brace buck l ed g loba l ly out-of-plane at a lateral load o f 400 k N , and a lateral displacement o f - 1 3 m m (Point 2, F igure 6.33). The peak load resisted b y the system was higher i n the second h a l f o f the first cyc le , even though both braces i nc luded geometric imperfections. H o w e v e r , the dispari ty between the two m a x i m u m loads was o n l y 28 k N , as compared to the 68 k N dispari ty observed in s imula t ion A . D u r i n g the second cyc le , degradation o f the sys tem strength and stiffness can be seen i n F igu re 6.33 (Points 3 and 4). A t the end o f the second cycle , significant residual out-of-plane brace deformations due to y i e ld ing o f the braces were observed as the sys tem passed through zero storey drift, as shown i n F igure 6.34. 40& Displacement (mm) Figure 6.33 C y c l i c s imula t ion C : Lateral load-displacement response. 144 z Figure 6.34 C y c l i c s imula t ion C : Res idua l out-of-plane brace deformations at end o f second cyc le . D u r i n g the first quarter o f the third cyc le , the b u c k l i n g pattern l o c a l i z e d to the mid- leng th region o f the brace, as the lateral displacement o f the system reached 4 0 m m (Figure 6.33, Po in t 5). A s the l o c a l b u c k l i n g progressed at the mid- leng th o f the brace, the overa l l profi le o f the buck l ed member transitioned from a smooth curv i l inea r shape to a b i - l inear shape (Figure 6.35). The loca l b u c k l i n g pattern was characterized b y an indentat ion o f the face o f the tube w a l l corresponding to the compress ion flange, and simultaneous outward b u l g i n g o f the two adjacent wa l l s (webs). The progress ive development o f the loca l b u c k l i n g pattern is shown i n F igure 6.36, a long w i t h cor responding contours o f effective plastic strain. 145 Time = 1.29 F igure 6.35 C y c l i c s imula t ion C : Transi t ion between curv i l inea r and b i - l inear brace deformation (v iew from base, l o o k i n g up). 146 T ime= 1.45 Z max ipt. value min= 0 max=0.120511 Tlme = 1.6 7 max ipt. value min=-0. max=O.3500O1 Figure 6.36 C y c l i c s imula t ion C : Progression o f l oca l b u c k l i n g : deformed shapes and corresponding contours o f effective plast ic strain. 147 A s the lateral displacement cyc le was reversed, the l o c a l l y b u c k l e d member began to straighten out under tensile load (Figure 6.33, Po in t 6). H o w e v e r , the l o c a l b u c k l i n g o f the tube wa l l s had induced severe strain concentrations at the corners o f the section j o i n i n g the compress ion flange to the adjacent webs, and as the a x i a l l oad i n the brace was reversed, the m o d e l predicted the ini t ia t ion o f c r ack ing at these corners (Figure 6.33, Po in t 7). T h e c rack ing was predicted i n the form o f element fai lure, b y w h i c h the corner elements were deleted once their average integration point va lue o f effective plast ic strain reached the specif ied failure value o f 0.35. A s the lateral d isplacement o f the frame continued, the increas ing tensile load across the reduced net sect ion o f the brace precipitated a spread o f the c rack ing pattern across the tube w a l l . T h e delet ion o f elements caused sudden changes i n the load carrying capaci ty o f the system, as evidenced b y the j agged nature o f the lateral load displacement curve (F igure 6.33, Po in t 8). T h e formation and progress ion o f the c rack ing pattern is s h o w n i n F igure 6.37. Time = 2.09 Time = 2.16 Time = 2.25 Figure 6.37 C y c l i c s imula t ion C : Progression o f c r ack ing across brace section. A s the d i rec t ion o f load ing was reversed dur ing the last quarter o f the th i rd cyc le , the extensive damage to the cracked brace member resulted i n a complete deterioration o f the lateral strength and stiffness o f the system (Figure 6.33, Po in t 9). The out-of-plane b u c k l i n g o f the braces was accompanied b y extensive y i e l d i n g i n the top and bot tom gusset plates. Plast ic hinges were eventual ly deve loped i n the free length regions o f the gusset plates, di rect ly beyond the ends o f the braces. Con tours o f effective plast ic strain i n the top and bot tom gusset plates are s h o w n i n F i g u r e 6.38. The strain states s h o w n i n F igu re 6.38 correspond to the t ime at w h i c h the out-of-plane rotations o f the plates were greatest. 148 Fringe Levels Time = 1.61 max ipt. value o.nnoe+nn 2.000e 04 8.000e 04 4.000e-04 6.000e-04 1.200e-03 1.000e-03 1.600e-03 1.400e-03 2.000e-03 1.800e-03 Figure 6.38 C y c l i c s imula t ion C : Contours o f effective plast ic strain i n gusset plates. 6.6.3 Cyclic Simulation D In s imula t ions A , B , and C , o n l y a s ingle cyc l e at each ampl i tude l e v e l was considered. T o investigate the complete c y c l i c behaviour o f the system, a more r igorous analysis w o u l d inc lude increas ing cyc le amplitudes, w i t h mul t ip le cyc les at each ampli tude leve l . H o w e v e r , such an analysis was not possible to complete w i t h i n a reasonable run t ime without increasing the load ing rate beyond the point at w h i c h iner t ia l effects r emain insignif icant . In s imula t ion D , 2.5 cyc les were conducted at an ampl i tude l e v e l o f 14mm, to observe the predicted response due to repeated cyc les at constant ampl i tude . T h e results o f s imula t ion D are shown i n terms o f the lateral load-displacement curve o f F igure 6.39. F o r the 1 4 m m displacement ampli tude considered, some degradation i n stiffness and strength were observed i n the successive cycles , howeve r it appears that greater insight might have been made i f the cyc le ampl i tude had been s l igh t ly larger, since the in i t ia t ion o f b u c k l i n g just bare ly occurred at the 1 4 m m ampl i tude l eve l . A 2 0 % reduct ion i n strength and an 1 1 % reduct ion i n stiffness were observed i n the first quarter o f the second cyc le , compared to the first quarter o f the first cyc le . A 2 5 % reduct ion i n strength and a 2 1 % reduct ion i n stiffness were observed i n the first quarter o f the third cyc le , compared to the first quarter o f the first cyc le . 149 Displacement (mm) Figure 6.39 C y c l i c s imula t ion D : Lateral load-displacement response. 150 7 DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS 7.1 Discussion H i g h - l e v e l finite element mode l i ng o f nonl inear systems requires a careful and systematic approach. It is not reasonable to expect that the results o f a first attempt at m o d e l i n g a complex structure w i l l guarantee any insight into the true behaviour o f the structure, since the finite element predict ions w i l l l i k e l y be sensitive to a number o f model ing-re la ted factors. It is for this reason that the w o r k presented i n this thesis was focussed o n a parametric sensi t ivi ty study approach. The parametric studies resulted i n a number o f useful f indings, w h i c h are summar ized i n the Conc lus ions sect ion o f this chapter. H a v i n g first gained an understanding o f the abil i t ies , l imi ta t ions , and sensit ivit ies o f the V S C - C B F finite element m o d e l , it is o f pract ical interest to u l t imate ly compare the predicted c y c l i c response o f the m o d e l to the actual response observed i n the quasi-static c y c l i c testing program. A s discussed i n Chapter 6, it was not feasible to replicate the entire exper imental load ing pro tocol i n the finite element s imulat ions . H o w e v e r , the results f rom c y c l i c s imula t ion C , i n w h i c h single cycles at three ampli tude levels were prescribed, can be compared w i t h selected cyc les from the experiment, as s h o w n i n F igure 7.1. In F igure 7.1, the cycles w i t h the f o l l o w i n g ampli tudes, from both the finite element m o d e l and the experiment, are compared: • the first cyc l e to +/- the y i e l d displacement • the first cyc l e to +/- 2 8 m m • the first cyc l e to +/- 5 6 m m In the case o f the finite element results, the y i e l d displacement occurred dur ing the cyc l e w i t h 1 4 m m ampli tude. In the case o f the experimental data, the y i e l d displacement cyc l e and the 2 8 m m ampli tude cyc le were one and the same, and therefore o n l y two experimental cyc les (the first cycles to +/- 2 8 m m and +/- 56mm) are s h o w n i n F igure 7.1. 151 gAA J\AJ 400 -1 =599^  Displacement (mm) Figure 7.1 C o m p a r i s o n o f selected cycles from finite element predict ions and experimental results. It is apparent from F igu re 7.1 that an exact match between the finite element predict ions and the exper imental data was not achieved. Howeve r , the general shape and backbone o f the finite element and experimental hysteresis curves are fa i r ly s imi la r , par t icular ly i n the inelastic cycles (i.e., the pos t -y ie ld displacement cyc les ) . T h e finite element hysteresis curve shows the same " p i n c h i n g " effect, due to pos t -buck l ing stiffness degradation, as was observed i n the experiment. Three p r ima ry differences exist between the finite element predict ions and the experimental results: 1. T h e lateral load at first b u c k l i n g i n the mode l was 372 k N , w h i c h is 4 1 % higher than the exper imental result o f 263 k N . 2. The in i t i a l elastic stiffness o f the mode l was 30.8 k N / m m , w h i c h is 2.9 t imes higher than the experimental result o f 10.5 k N / m m . 3. Des tab i l i za t ion o f the finite element hysteresis curve occurred due to tensile c rack ing o f a brace dur ing the second quarter o f the first 5 6 m m ampli tude cyc l e , whereas c r ack ing was not observed i n the experiment un t i l the second quarter o f the second 8 4 m m ampli tude cyc le . 152 E a c h o f the three i tems l is ted above w i l l be separately discussed i n the f o l l o w i n g sections. 7.1.1 Discrepancy in Peak Lateral Load C y c l i c s imula t ion C o f the V S C - C B F m o d e l predicted a peak lateral load o f 372 k N , w h i c h corresponded to first b u c k l i n g o f a brace. T h i s peak strength was 4 1 % higher than the exper imental result o f 263 k N . The parametric sensi t ivi ty studies conducted o n the single brace mode ls and the fu l l V S C - C B F models p rov ided some insight as to the factors affecting the peak load predicted b y the finite element m o d e l . F o r example , the parametric studies showed that the predicted b u c k l i n g load decreases as the magni tude o f in i t i a l geometric imperfect ions is increased. It was also s h o w n that, for exp l i c i t dynamic s imulat ions , the predicted b u c k l i n g load increases as t ime sca l ing and mass propor t ional damping levels are increased. In the first cyc l e o f c y c l i c s imula t ion A (Sect ion 6.6.1), the s imulated average load ing rate o f 47mm/s l ed to a peak lateral load predic t ion o f 348 k N . The parametric studies o f the pushover s imulat ions (Sect ion 6.5.2) showed that a 9 % decrease i n peak lateral load was achieved w h e n mass propor t ional damping was e l iminated from the m o d e l . These observations suggest that i f c y c l i c s imula t ion C had been conducted at the load ing rate from c y c l i c s imula t ion A , and i f mass propor t ional damping had been e l iminated, the peak lateral load might have been about 317 k N . T h i s w o u l d leave a 1 7 % discrepancy between the finite element predic t ion and the experimental result, w h i c h c o u l d poss ib ly be resolved b y a combina t ion o f a further decrease i n the s imulated load ing rate and a slight increase i n the magni tude o f assumed in i t i a l geometric imperfect ions. A s a matter o f due di l igence , an invest igat ion was conducted into the data acquis i t ion procedures that were fo l l owed i n the experiment, as they relate to the recorded load values. A thorough descr ipt ion o f the instrumentation used i n the experiment was described b y B u b e l a (2003), i n w h i c h she noted that the capaci ty o f the load c e l l that was used to measure the lateral load i n the quasi-static c y c l i c test was 450 k N (100 k ip ) . D u r i n g the r ev iew o f the data acquis i t ion procedures, it was noted that the part icular load c e l l that was used i n the experiment was marked w i t h a non-manufacturer 's label that 153 indicated a capaci ty o f 450 k N . Howeve r , a check o f the manufacturer 's catalogue for the M T S 661.22 m o d e l number showed that the load c e l l was i n fact o n l y rated to 250 k N , w i t h a static over load capaci ty o f 150% at the rated load ( M T S , 1996). T w o independent re-calibrations o f the load ce l l were therefore conducted, w h i c h showed that the load c e l l readings remained l inear for loads up to 340 k N , and w h i c h conf i rmed the cal ibra t ion factor that was used i n the experiment. The poss ib i l i t y o f a ga in setting error was also considered, s ince the ga in levels were set manua l ly and not automat ical ly recorded dur ing the testing. T h e ga in setting o n the load c e l l channel o f the data acquis i t ion system should have been 200. It was determined that, based o n the load c e l l ca l ibra t ion factor and a standard exci ta t ion voltage o f 1 0 V , i f the load c e l l ga in had been erroneously set at the next highest setting (500), the fu l l scale reading o f the system w o u l d have been 195 k N , w h i c h is less than the m a x i m u m 263 k N reading that was obtained. It was therefore conc luded that a ga in setting error was not made for the load c e l l channel . 7.1.2 Discrepancy in Elastic Lateral Stiffness A l t h o u g h differences i n the peak lateral load were observed between the finite element predict ions and the experimental result, the more s t r ik ing discrepancy was i n the value o f in i t i a l elastic stiffness. The elastic lateral stiffness o f the fu l l V S C - C B F mode ls was consistently i n the range o f 30-31 k N / m m . In contrast, the exper imental data showed an elastic lateral stiffness o f 10.5 k N / m m . It is important to emphasize that the elastic lateral stiffness predicted b y the finite element models was not affected b y any o f the parameter variat ions considered i n the parametric sensi t ivi ty studies. T h e parametric studies showed effects o n the peak l oad l eve l and the response i n the pos t -buck l ing nonl inear range, but the elastic predict ions were unaffected. In addi t ion, it was observed that relative ver t ica l m o t i o n i n the slotted connect ion assembly d i d not occur un t i l b u c k l i n g o f the braces was ini t iated. Therefore, the elastic lateral stiffness o f the system should be easi ly predicted b y s imple arguments based o n geometry and elastic mechanics o f materials theory, as expla ined next. 154 In the l inear-elastic range, a steel brace subjected to ax ia l e longat ion or shortening w i l l experience an ax ia l load, P , of: P = A E A (7.1) L where: A = cross-sectional area o f the brace E = elastic modu lus o f the brace A = ax ia l e longat ion or shortening o f the brace L = length o f the brace In the l inear-elastic range, the force equ i l ib r ium and kinemat ics o f the C B F are as shown i n F igure 7.2, i n w h i c h u is the lateral displacement at the top o f the braces, V is the lateral load appl ied to the system, and 0 is the acute angle between the brace and the beam above or be low. F igure 7.2 Force equ i l i b r i um and kinemat ics o f C B F i n the l inear-elast ic range. It is apparent f rom F igure 7.2 that the lateral load resistance o f the system is : A l s o , the lateral drift o f the system and the elongat ion or shortening o f the brace are . related by : V u B 7 \ / A V = 2Pcos0 (7.2) A = u cos 0 (7.3) 155 Therefore, the elastic lateral stiffness o f the system can be expressed as: K = V = 2Pcos0 u [ A j vcos0, Subst i tut ing (7.1) into (7.4) gives the elastic lateral stiffness as: 2AEcos2 0 /n c\ K = — (7.5) F o r the d imensions and mater ial properties o f the system under considerat ion, the elastic lateral stiffness is predicted b y (7.5) as K = 31.4 k N / m m . S ince the finite element models predicted an elastic stiffness between 30-31 k N / m m , it can be conc luded that the finite element m o d e l agrees w i t h fundamental mechanics pr inc ip les , and is not g i v i n g an ove r ly s t i f f result due to any numer ica l effects such as the leve l o f mesh refinement. It was noted i n Chapter 4 that the stub c o l u m n models also predicted the expected elastic ax ia l stiffness based o n fundamental theory, but that the exper imental stub c o l u m n response was o n l y h a l f as s t i f f as expected. G i v e n that the cross-sect ional areas o f the specimens were careful ly measured, and that an elastic modu lus o f 100,000 M P a is considered h i g h l y u n l i k e l y for steel, it is probable that the l o w stiffness reported i n the stub c o l u m n tests was due to inaccuracies i n measurement over the first 2 m m o f end shortening. Therefore it was conc luded that the l o w lateral stiffness o f the fu l l V S C - C B F test spec imen was not related to the l o w stiffness measured i n the stub c o l u m n tests. The braces o f the test spec imen were instrumented w i t h strain gages at their mid- length . The strain gage data was examined, to check the correla t ion between the measured brace strains and the measured lateral displacements. T h e lateral displacement at the top end o f the braces was predicted i n terms o f the average brace strain, s, by : A e L (7.6) cos 0 cos 0 U s i n g the average recorded strain measurements w i t h equation (7.6) gave predict ions o f lateral displacements that were rough ly 5 0 % o f the measured displacement values. T h i s 156 suggested that either some inaccuracy was present i n the lateral displacement measurements, or the deformations o f the system were not consistent w i t h the assumed kinemat ics expressed b y (7.3). In the experimental testing program, a second spec imen was tested i n w h i c h the braces were f i l l ed w i t h concrete. F o r this specimen on ly , an addi t ional set o f displacement measurements were made b y an act ive-opt ical m o t i o n capture system. B u b e l a (2003) reported that the lateral displacements captured b y this system were up to 2 6 % greater than the displacements measured b y the string potentiometers. T h e act ive-opt ica l mo t ion capture system was not used i n the test o f the h o l l o w tube specimen, but B u b e l a ' s observat ion suggests that the string potentiometer displacement measurements o f the h o l l o w tube spec imen m a y have inc luded some error. It is ent irely poss ible that the l o w elastic stiffness exhibi ted b y the test spec imen m a y be due to a combina t ion o f factors. S m a l l f lex ib i l i t ies at var ious jo in ts throughout the system m a y have contr ibuted to the overa l l f l ex ib i l i t y o f the system. A d d i t i o n a l l y , there was some f l e x i b i l i t y i n the permanent react ion frame, i n c l u d i n g the f loor beam to w h i c h the spec imen was anchored, and these components were not inc luded i n the finite element mode l . The finite element m o d e l d i d however inc lude detailed m o d e l i n g o f the ver t ica l slotted connec t ion detai l . A l t h o u g h translations and rotations o f the V S C detai l were captured b y the analysis , the magnitudes o f these deformations were o n l y s ignif icant outside o f the elastic range. 7.1.3 Discrepancy in Prediction of Brace Cracking In c y c l i c s imula t ion C o f the V S C - C B F mode l , l oca l b u c k l i n g was predicted i n the compress ion brace as the lateral displacement approached the m a x i m u m ampli tude o f 5 6 m m . C r a c k i n g was predicted i n the h i g h l y strained reg ion o f the l o c a l b u c k l i n g pattern as the load ing o f the brace was reversed. The progressive failure o f elements across the cross section led to a s ignif icant deterioration o f system strength and stiffness b y the end o f the cyc le . S i m i l a r behaviour was observed i n the experiment, but c r ack ing was not ini t iated unt i l later i n the load ing protocol . In this regard, the finite element m o d e l under-predicted the duc t i l i ty o f the braces. 157 The early pred ic t ion o f c rack ing i n the finite element m o d e l is p r i m a r i l y due to the somewhat arbitrary element failure cr i ter ion that was adopted. A s discussed i n Sec t ion 4.5, other researchers have noted that the use o f a plast ic failure strain cr i ter ion is an approx imat ion at best. A more robust approach to m o d e l i n g failure w o u l d l i k e l y have i m p r o v e d the correla t ion between the finite element predict ions and the experimental results. H o w e v e r , it is wor th not ing that the loca t ion o f crack in i t ia t ion (at the corners o f the tube w a l l i n the l oca l l y b u c k l e d region) was correct ly predicted, w h i c h indicates that the dis t r ibut ion o f plast ic strains was w e l l captured b y the m o d e l . 7.1.4 Prediction of General System Behaviour A l t h o u g h some signif icant discrepancies were observed between the analyt ica l and experimental results, the finite element m o d e l d i d successful ly capture the k e y characteristics o f the system behaviour. In general, these characteristic behaviours were observed at d i f fer ing t imes i n the mode l and experiment, w h i c h is l a rge ly attributable to the discrepancies i n system stiffnesses discussed above. In F igures 7.3-7.8, the predic t ion o f some the f o l l o w i n g system characteristics are demonstrated b y a side-by-side compar i son o f the finite element output plots and photographs from the test: • In-plane and ou t -of plane deformations o f the V S C detai l , i n c l u d i n g ver t ica l translation o f the bolts i n the slotted holes (Figure 7.3). • Y i e l d i n g and plast ic hinge formation i n the top and bot tom gusset plates (Figure 7.4). • G l o b a l out-of-plane b u c k l i n g o f the braces (Figure 7.5). • L o c a l b u c k l i n g o f the braces at large system displacements (F igure 7.6). • Ini t ia t ion o f c r ack ing o f the brace at the corners o f the tube w a l l (Figure 7.7). • Progress ive c rack ing o f the brace across the H S S sect ion (Figure 7.8). 158 Figure 7.3Comparison o f experimental and analy t ica l results: Trans la t ion o f bolts i n the slotted holes o f the V S C detai l . F igure 7.4 C o m p a r i s o n o f experimental and analy t ica l results: Y i e l d i n g o f the bottom gusset plate. 159 Figure 7.5 C o m p a r i s o n o f experimental and ana ly t ica l results: G l o b a l out-of-plane b u c k l i n g o f brace. F igure 7.6 C o m p a r i s o n o f experimental and analyt ical results: L o c a l b u c k l i n g o f brace. 160 re anson of experimental and analy t ica l results: Init iat ion o f brace c rack ing . Progress ion o f brace c rack ing . 161 7.2 Conclusions The w o r k discussed i n this thesis was centred o n the general theme o f deve lop ing a sophisticated finite element m o d e l o f a chevron braced frame w i t h an innovat ive ver t ica l slotted connect ion detai l . Rather than immedia te ly t ack l ing this p r o b l e m i n its entirety, a systematic approach was taken whereby cr i t i ca l components o f the system were mode led ind iv idua l l y , and the sensi t ivi ty o f these sub-models to a var ie ty o f m o d e l i n g and phys i ca l parameters was assessed. The lessons learned through the parametric studies o f the component models enabled we l l - in fo rmed decis ions to be made w h i l e deve lop ing the fu l l V S C - C B F m o d e l . T h e fu l l V S C - C B F m o d e l was also subjected to parametric study, such that the influence o f the m o d e l i n g details o n the fu l ly nonl inear behaviour o f the m o d e l was determined. Interpretation o f the results o f every parametric study was based o n the impl ica t ions o f these results for the predicted structural behaviour . The major results and conclus ions o f the finite element m o d e l i n g invest igat ion are summar ized i n the f o l l o w i n g paragraphs. The b u c k l i n g behaviour o f the stub co lumns and braces was s igni f icant ly inf luenced b y the magni tude o f in i t i a l geometric imperfect ions that were inc luded i n the models . In the stub c o l u m n models , the imperfect ion magnitude determined the type o f spatial plast ic mechan i sm (symmetr ic extensional or asymmetr ic) that was predicted, w h i c h i n turn determined the ax i a l load-displacement response i n the pos t -buck l ing range. In the s ingle brace models , the i nc lu s ion o f an in i t i a l crookedness i n the fo rm o f the first b u c k l i n g mode shape w i t h a magni tude o f L / 1 0 0 0 resulted i n a 2 7 % reduct ion i n the b u c k l i n g load as compared to the i n i t i a l l y "perfect" case. The effect o f geometric imperfect ions was carr ied over to the fu l l V S C - C B F m o d e l , since the load at first b u c k l i n g o f a brace determined the peak lateral load resisted b y the system. T h e use o f an exp l i c i t solver for the t ime stepping analyses, i n w h i c h quasi-static solutions were desired, necessitated certain considerations b e y o n d those typ i ca l ly required for structural problems. The sma l l c r i t i ca l t ime step o f the exp l i c i t method 162 required that t ime sca l ing be employed i n order to achieve solut ions i n an acceptable run t ime. In the stub c o l u m n simulat ions, the over-predic t ion o f the pos t -buck l ing strength increased as the t ime sca l ing was increased (i.e., l oad ing durat ion reduced). In the s ingle brace m o d e l , the b u c k l i n g load was reduced b y 14% w h e n the load ing durat ion was changed from U T i to 2 2 T i , whereas b u c k l i n g was not even tr iggered for the end shortening considered w h e n the load ing durat ion was changed to l . l T i . In the c y c l i c s imulat ions o f the fu l l V S C - C B F m o d e l , vas t ly different hysteretic results were obtained depending o n the average load ing rate. S ince the expl ic i t method yie lds dynamic solutions, the use o f d a m p i n g was investigated to obtain the static response. In the stub c o l u m n s imulat ions , the use o f 2 % - 1 0 % stiffness-proportional damping was shown to effect ively damp h i g h frequency vibrat ions, w h i l e higher values were shown to reduce the stable exp l i c i t t ime step. In addi t ion, the use o f 0 % stiffness-proportional damping led to unstable solutions. Mass -propor t iona l damping was used i n the single brace and fu l l V S C - C B F models to damp the v ib ra t ion o f the brace about its b u c k l e d configurat ion. In the single brace m o d e l , the use o f c r i t i ca l mass-proport ional damping i n the fundamental mode o f the brace cause the b u c k l i n g load to increase b y 9%. S i m i l a r effects were observed i n the pushover s imulat ions o f the fu l l V S C - C B F m o d e l . A c c u m u l a t i o n o f round-of f errors due to the use o f the s ingle p rec i s ion executable o f the exp l i c i t solver was observed to affect the computed results for p rob lems i n v o l v i n g m i l l i o n s o f computat ional cycles . In the case o f the stub c o l u m n models , the round-of f errors were sufficient to trigger a different b u c k l i n g mode. In the s ingle brace models , the effect was less significant: a decrease i n the b u c k l i n g load o f o n l y 2 % was observed w h e n single p rec i s ion was used. The run t ime for s ingle p rec i s ion s imulat ions was typ i ca l ly 7 5 % less than equivalent s imula t ion w i t h double prec is ion . Under- integrated 4-noded she l l elements w i t h the L S - D Y N A stiffness f o r m o f hourglass control and 3 or 5 Gauss integration points through the thickness were shown to be suitable for m o d e l i n g the stub c o l u m n and brace members . Under- integrated 8-noded s o l i d elements w i t h an assumed strain fo rm o f hourglass cont ro l were s h o w n to be 163 suitable for m o d e l i n g the plate parts o f the m o d e l , par t icular ly w h e n four or more o f these elements were used through the thickness o f the plate i n order to capture inelastic bending. T h e exp l i c i t solver was able to handle contact nonl inear i ty wi thout d i f f icu l ty , whereas the i m p l i c i t solver was unsuccessful. The parametric studies o f contact i n the stub c o l u m n m o d e l showed that the approach to i nc lud ing contact surfaces c o u l d influence the computed results s ignif icant ly . A segment-based contact a lgor i thm was appropriate for treating the edge-to-edge element contact at the slotted connect ion interface i n the V S C -C B F m o d e l . T h e coefficient o f f r ic t ion prescr ibed at the contact ing surfaces o f the V S C detail had an effect o n the dis t r ibut ion o f loads throughout the system i n the post-b u c k l i n g range. W h e n the coefficient o f friction was specif ied as 0.3, the fr ic t ion resistance i n the connect ion was sufficient to develop unbalanced pos t -buck l ing loads i n the braces. In contrast, w h e n the ideal case o f zero friction was considered, the brace loads were l im i t ed b y the compressive strength, and the appl ica t ion o f a net d o w n w a r d load at the beam mid-span was largely avoided. The finite element m o d e l o f the fu l l V S C - C B F system was able to capture the general lateral l oad ing response characteristics that were observed i n the quasi-static c y c l i c load ing tests performed b y B u b e l a (2003). G l o b a l and loca l b u c k l i n g deformat ion modes o f the braces were predicted, i n addi t ion to c rack ing o f the l o c a l l y b u c k l e d brace upon load reversal . T h e m o d e l also correct ly ident i f ied locat ions o f y i e l d i n g and plast ic h ing ing i n the top and bot tom gusset plates and mid- leng th o f the braces. M o s t important ly , the m o d e l captured the three-dimensional translations and rotations o f the V S C detai l . Comple te rep l ica t ion o f the experimental c y c l i c load ing pro toco l was not feasible w i t h the proposed m o d e l i n g approach and avai lable computer hardware. C o m p a r i s o n o f selected load ing cyc les showed that the m o d e l predict ions and the exper imental results had general ly s imi l a r hysteretic curves, a l though an exact match was not obtained. The p r imary difference i n behaviour was attributable to the elastic lateral stiffness o f the system, w h i c h was s ignif icant ly l ower i n the experiment than what was expected based 164 o n fundamental pr inc ip les o f mechanics and the finite element predict ions . A l t h o u g h the force dis t r ibut ion and kinemat ic behaviour o f the V S C detai l was captured b y the finite element analysis , the m o d e l was not able to reconci le the differences i n the linear-elastic range. 7.3 Recommendations The results o f the parametric studies have conf i rmed that the computed results o f the nonl inear finite element m o d e l are indeed s ignif icant ly inf luenced b y a number o f model ing-re la ted factors. Therefore the most important recommendat ion that arises from this w o r k is that parametric sensi t ivi ty analysis must absolutely be inc luded i n any comprehensive finite element mode l i ng study o f complex systems. S ince the sensi t ivi ty to any part icular parameter is general ly model-dependent, the parameter-related conclus ions presented i n the previous section should be taken as indicators as to poss ib ly inf luent ia l parameters, rather than as un iversa l ly appl icable rules. W i t h regards to the further development o f the V S C - C B F m o d e l , the f o l l o w i n g recommendat ions can be made: • T h e exp l i c i t method o f analysis is a robust t ime stepping method that can handle nonl inear problems easily. H o w e v e r , the c r i t i ca l t ime step o f the exp l i c i t method forces prac t ica l l imita t ions o n the s imula t ion durations that can be achieved, mean ing that s imula t ion o f quasi-static processes is not straightforward. In future m o d e l i n g efforts, a greater effort should be p laced o n ach iev ing convergent solut ions w i t h an i m p l i c i t t ime stepping method. T h i s approach m a y also require a d o w n s i z i n g i n the m o d e l complex i ty , so that the i m p l i c i t stiffness matr ix invers ion process m a y be completed efficiently. • I f exp l i c i t methods are retained as the analyt ical so lu t ion method, the analyses should be carr ied out o n paral le l comput ing systems (clusters), i n order to s igni f icant ly reduce the run t ime for each s imula t ion . • The response o f the V S C - C B F m o d e l to se ismic ground mot ions shou ld be investigated. T h e ult imate purpose o f the V S C - C B F system is to improve se ismic performance, so this performance should be evaluated i n the context o f appropriate earthquake excitations. A l t h o u g h the durat ion o f earthquake strong mot ions m a y be as l i t t le as 5 seconds, it is noted that t ime sca l ing c o u l d s t i l l be required for an expl ic i t analysis o f such events, and therefore t ime sca l ing sensi t ivi ty studies w o u l d s t i l l be necessary. 165 • A n opportuni ty to improve the V S C - C B F m o d e l exists b y incorpora t ing a more r igorous element failure m o d e l , to better predict the duc t i l i ty and fracture l i fe o f the H S S brac ing members . S u c h a failure cr i ter ion shou ld be developed i n the context o f a s ingle brace m o d e l subjected to c y c l i c load ing , and w i l l require experimental data for ca l ibra t ion purposes. • The effects o f residual stresses i n the co ld- formed H S S members shou ld be further studied. Res idua l stresses c o u l d be incorporated d i rec t ly i n the finite element m o d e l b y s imula t ing the c o l d fo rming o f an H S S member , and then compar ing the results to measured values. • Parametr ic studies o f the assumed boundary condi t ions i n the m o d e l should be conducted. T h e beam and c o l u m n elements o f the permanent react ion frame to w h i c h the specimen was anchored should be mode led , either d i rec t ly b y s o l i d m o d e l i n g or ind i rec t ly b y spring elements. • Future m o d e l i n g efforts should attempt to include the effects o f concrete f i l l i n the H S S tubes. Exper imenta l data obtained b y B u b e l a (2003) is avai lable for compar i son w i t h the finite element m o d e l predict ions. B u b e l a (2003) made a number o f recommendations as to future exper imental studies that should be performed i n order to better understand the behaviour o f the V S C - C B F system. In part icular, she recommended component testing o f the V S C detai l under c y c l i c quasi-static load ing condi t ions , addi t ional full-scale quasi-static c y c l i c testing o f the V S C - C B F system, and dynamic shake table testing o f the system. Recommenda t ions for future testing, further to those proposed b y B u b e l a , inc lude : • Fu l l - sca le quasi-static testing o f the V S C - C B F system i n the l inear-elastic range on ly , w i t h the goal o f determining the source o f the l o w stiffness observed i n earlier tests. S u c h testing w i l l require extensive instrumentat ion o f the entire test specimen and also the boundary react ion frame. • M o n o t o n i c and c y c l i c ax ia l load testing o f a s ingle brace w i t h top and bot tom gusset plate connections inc luded. These tests can be compared w i t h the s ingle brace mode ls developed i n the present invest igat ion. • Repeated stub c o l u m n testing to ver i fy the stub c o l u m n stiffness i n the elastic range. These tests can be compared w i t h the stub c o l u m n mode ls developed i n the present invest igat ion. 166 All future experimental work should include careful measurement of initial geometric imperfections, particularly in the brace members. These measurements can be made of each component individually, but the most important measurements will be the imperfections and misalignment in the fully assembled test specimen. Real geometric imperfection data can then be incorporated into the finite element models. 167 REFERENCES A i k e n , I .D . , M a h i n , S . A . , & U r i z , P . (2002) Large-scale testing o f b u c k l i n g restrained braced frames. Proceedings, Japan Passive Control Symposium, Y o k o h a m a , Japan: T o k y o Institute o f Techno logy . A S T M . (2001). Standard test methods for tension testing of metallic materials. ( M e t h o d E8-01 ) . W e s t Conshohocken , P A : A m e r i c a n Soc ie ty for Tes t ing and Mate r i a l s . A T C . (1992). Guidelines for cyclic seismic testing of components of steel structures. ( A T C - 2 4 ) . R e d w o o d C i t y , C A : A p p l i e d Techno logy C o u n c i l . Be ly t s chko , T . , & B i n d e m a n , L . P . (1993). A s s u m e d strain s tabi l iza t ion o f the eight node hexahedral element. Computer Methods in Applied Mechanics and Engineering, 105, 225-260. B e l y t s c h k o , T . , & Hughes , T . J . R . (Eds.) (1983). Computational methods for transient analysis. N o r t h - H o l l a n d , Ams te rdam: E l sev i e r Science Publ ishers . B e l y t s c h k o , T . , L i n , J . , & Tsay, C . S . (1984). E x p l i c i t a lgori thms for nonl inear dynamics o f shells. Computer Methods in Applied Mechanics and Engineering, 42, 225-251 . B e l y t s c h k o , T . , L i u , W . K . , & M o r a n , B . (2004). Nonlinear finite elements for continua and structures. Chichester , U K : John W i l e y & Sons. B e l y t s c h k o , T . B . , & Tsay , T . S . (1981). E x p l i c i t a lgori thms for nonl inear dynamics o f shells. In Hughes , T . J . R , P i f k o , A . , Jay, A . (Eds.) , Nonlinear finite element analysis of plates and shells: presented at the Winter Annual Meeting of the American Society of Mechanical Engineering, Washington, D.C, November 15-20, 1981; sponsored by Applied Mechanics Division (AMD (Series); Vol. 48), N e w Y o r k , N Y : A S M E , 209-231 . B u b e l a , R . (2003). An experimental and analytical study of chevron braced fames with vertical slotted connections. M . A . S c . Thesis , Facu l ty o f Graduate Studies (Department o f C i v i l Engineer ing) , U n i v e r s i t y o f B r i t i s h C o l u m b i a , Vancouve r , B C , Canada . C A D - F E M . (2004). LS-DYNA support and service: Tips and tricks for solver. Re t r ieved January 12, 2004, f rom http: / /portal .ecadfem.eom/Solver.2464.0.html. Carr , A . J . (2001). RUAUMOKO: The Maori God of Volcanoes and Earthquakes. Cantebury, N e w Zea land : U n i v e r s i t y o f Cantebury. Chambers , J .J , & Ernst , C . J . (2005). Brace frame gusset plate research: Phase 1 literature review, V o l u m e 1. Salt L a k e C i t y : Department o f C i v i l & E n v i r o n m e n t a l Engineer ing , U n i v e r s i t y o f U tah . 168 C i t i p i t i o g l u , A . M . , H a y - A l i , R . M . , & W h i t e , D . W . (2002) Re f ined 3 D finite element m o d e l i n g o f par t ia l ly restrained connections i nc lud ing s l ip . Journal of Constructional Steel Research, 58, 995-1013. C o o k , R . D . , M a l k u s , D . S . , P lesha , M . E . , & W i t t , R J . (2002). Concepts and applications of finite element analysis. N e w Y o r k , N Y : John W i l e y & Sons. Courant , R . , Fr iedr ichs , K . O . , & L e w y , H . (1928). U b e r die par t ie l len Dif ferenzensle ichungen der Mathemat i schen P h y s i k , Mathematische Annalen, 100, 32. C S I . (2003). SAP2000 Nonlinear Version 8.1.2. B e r k e l y , C A : Compute rs and Structures, Inc. F i e l d , C . J . (2003). S imu la t i on o f full-scale seismic-restraint structural frame tests us ing L S - D Y N A 960 Impl ic i t Solver . In 4th European LS-DYNA Users Conference ( H - I - 1 7 -30). U L M , Germany : D Y N A / w o r e , G m b H . F lanagan, D . P . , & B e l y t s c h k o , T . (1981). A un i fo rm strain hexahedron and quadrilateral and or thogonal hourglass control . International Journal for Numerical Methods in Engineering, 17, 679-706. Ga lambos , T . V . (1988). Guide to stability design criteria for metal structures, 4 t h E d i t i o n . N e w Y o u k , N Y : John W i l e y & Sons. Ha l lqu i s t , J . O . (Ed.) . (1998). LS-DYNA - Theoretical Manual. L i v e r m o r e , C A : L i v e r m o r e Software T e c h n o l o g y Corpora t ion . K u t t , L . M . , P i f k o , A . B . , N a r d i e l l o , J . A . , & Papazian , J . M . (1998). S l o w - d y n a m i c finite element s imula t ion o f manufactur ing processes. Computers & Structures, 66 (1), 1-17. Langseth , M . , Hoppers tad , O . S . , & Hanssen, A . G . (1998). C ra sh behaviour o f th in -wal led a l u m i n i u m members . Thin-Walled Structures, 32, 127-150. L S T C . (2003). LS-DYNA Keyword User's Manual. ( V e r s i o n 970). L i v e r m o r e , C A : L i v e r m o r e Software T e c h n o l o g y Corpora t ion . M a m a l i s , A . G . , M a n o l a k o s , D . E . , & Ba ldoukas , A . K . (1996) O n the finite-element m o d e l l i n g o f the deep-drawing o f square sections o f coated steels. Journal of Materials Processing Technology, 58, 153-159. Mat t iasson , K . , Bernspang , L . , Honecker , A . , Sched in , E . , H a m m a m , T . , & Melande r , A . (1991). O n the use o f exp l i c i t t ime integration i n finite element s imula t ion o f industr ia l sheet fo rming processes. In FE-Simulation of 3-D Sheet Metal Forming Processes in Automotive Industry (pp.479-497). Dusseldorf : V e r l a g des Ve re in s Deutscher Ingenieure. 169 M i n d l i n , R . D . (1951). inf luence o f rotary inert ia and shear o n f lexura l mot ions o f isotropic , elastic plates. Journal of Applied Mechanics, 18, 31-38. M T S Systems Corpora t ion . (1996). Series 661 h i g h capaci ty force transducers. E d e n Pra i r ie , M N : M T S Systems Corpora t ion . Rust , W . , & Schweizerhof , K . (2003). F in i t e element l i m i t load analysis o f th in -wa l l ed structures b y A N S Y S ( impl i c i t ) , L S - D Y N A (expl ic i t ) and i n combina t ion . Thin-Walled Structures, 41, 227-244. Sabe l l i , R . , & L o p e z , W . (2005). D e s i g n o f buckl ing-res t ra ined braced frames. Proceedings of the 2005 North American Steel Construction Conference. C h i c a g o : A m e r i c a n Institute o f Steel Const ruct ion . Sauve, R . G . , & Metzge r , D . R . (1995). Advances i n D y n a m i c Re l axa t i on Techniques for N o n l i n e a r F in i t e E lement A n a l y s i s . Journal of Pressure Vessel Technology, Transactions oftheASME, 117(2), 170-176. S l eckza , L . (2004). L o w cyc le fatigue strength assessment o f butt and fi l let w e l d connections. Journal of Constructional Steel Research, 60, 701-712. Tornqvis t , R . (2003). Design of Crashworthy Ship Structures. P h D . Thesis . Techn i ca l U n i v e r s i t y o f Denmark , M a r i t i m e Engineer ing . T s a i , K . C . , L a i , J . W . , C h e n , C . H . , H s i a o , B . C . , W e n g , Y . T . , & L i n , M . L . (2004) Pseudo D y n a m i c Tests o f a F u l l Scale C F T / B R B Compos i t e Frame. In G . E . B l a n d f o r d (Ed.) , Structures 2004 - Building On The Past: Securing The Future (Proceedings O f T h e 2004 Structures Congress , M a y 22.26, 2004, N a s h v i l l e , Tennessee; Sponsored b y The Structural Eng inee r ing Institute o f the A m e r i c a n Socie ty o f C i v i l Engineers) . Res ton , V A : A S C E . van der Veg te , G . J . , M a k i n o , Y . , & Sakimoto , T . (2002) N u m e r i c a l research o n single-bol ted connect ions us ing i m p l i c i t and expl ic i t solut ion techniques. Memoirs of the Faculty of Engineering, Kumamoto University, 47 (1), 19-44. K u m a m o t o , Japan: The Facu l ty o f Eng ineer ing , K u m a m o t o Un ive r s i ty . V a s a r h e l y i , D . D . , & C h i a n g , K . C . (1967). Coeff ic ient o f friction i n jo in ts o f var ious steels. Journal of the Structural Division, A S C E , 93 (ST4) , 227-243 . W e i m a r , K . (Ed. ) . (2001). LS-DYNA User's Guide, R e v . 1.19. C A D - F E M G m b H : M u n i c h , Germany . X u e , L . , L i n , Z . , & J iang , Z . (2000). Effects o f in i t i a l geometric imperfec t ion o n square tube col lapse. In 6th International LS-DYNA User's Conference (Sess ion 9-4, pp. 9 .31-9.46). L i v e r m o r e , C a l i f o r n i a : L i v e r m o r e Software T e c h n o l o g y Corpora t ion . 170 APPENDIX A - FABRICATION DRAWINGS TcST FRAME peTAiUS OCT. n /«> o+- o O - i - O 4>*2S vv •z S SP. €>3» -72 if TEST FfeAMs ELfevATIo^ ^Tt#M A-A 172 2127 '•' — — ONE ~ n ONE - 01 ONE ~ HI LOOSE PfECES FOR TESTING MATERIAL FIELD BOLTS: 24 ~ f » A325 * 3 \ 24 ~ f » A335 » 2 \ 8 ~-V» A32S x 3 H SOLID R O C K i ^ ^ ^ CUSTOMER: PROJECT: WATTffft , 1 yAwcniVER tme fBfrmnw DRAWWB. DRAIN Bft NAV MOV. 8000 CHECKED DK 1} ME 8)00 1353 - L B l t - U O F M A T E R I A L MARK QTY MATERIAL LENGTH 2 ' A2 ASSEMBLIES 4 PL 38 X 341 364 P' 4 PL 12.7 X 319 376 Pi 2 PL 19 X 5 t 0 525 IIM 4 HSS 89 x 89 x 4.8 3147 torn*-) U- a- U . LB 4 ~SA2 — SUB—ASSEMBLY 4 ~ ma M M * X « « U I » r > g ~ A 2 ASSEMBLIES soup ROCK Aaa^k HUE) DRAW BYi MAV NOV. 2000 CHECKED BY: SJ DEC 2000 1353 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0063337/manifest

Comment

Related Items