Stability of Castellated Beam Webs

STABILITY OF CASTELLATED BEAM WEBSSevak Demirdjian March 1999 Department of Civil Engineering and Applied Mechanics McGill University Montreal, Canada A thesis submitted to the faculty of Graduate Studies and Research in partial fulfilment of the requirements of the Degree of Master of Engineering O Sevak Demirdjian National tibrary Acquisitions and Bibliographie Services 395 Wellington Street OttawaON K1AûN4 du Canada BiMiiîhèque nationale Acquisitions et senrices bibliographiques 395. r w Weuhghm OctiwaON K1A ûîU4 Canada Canada The author has granted a nonexclusive licence allowing the National Library of Canada to reproduce, loan, distribute or seîi copies of this thesis in microform, paper or electronic formats. L'auteur a accordé une licence non exclusive pemiettant a la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfichelnlm, de reproduction sur papier ou sur format électronique. L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation. The author retains ownership of the copyright i this thesis. Neither the n thesis nor substantial extracts fiom it may be printed or otheMrise reproduced without the author's permission. ABSTRACT A study on the web-buckiing behavior of castellated beams is descnbed in this thesis. Both elastic and plastic methods of anaiysis are utilized to predict the faiiure modes of these beams. Interaction diagrams predicting formation of plastic mechanisrns. yietding of the horizontal weld length and etastic bucfling anaiysis using the finite element method are correlated with a number of experimentai test results fiom previous studies given in the literature. Test-to-predicted ratios for a total of 42 test beams ranging from 45" to 60" openings are computed with the plastic and elastic methods of anaiysis, and a mean of 1.086 and coefficient of variation of 0.195 are obtained. A parameter study covenng a wide range of 60" castellated bearn geomevies is perfonned to derive elastic buckling coefficients under pure shear and bending forces. An elastic buckling interaction diagram is then defined. which along with the diagrarns utilized in the plastic analysis, can be used to predict the elastic buckling and plastic failure loads under any given moment-to-shear ratio. To incorporate the effect of plasticity associated with buckling, expressions are derived to improve the previous theoretical models used, by cornbining both elastic and plastic results. This results in an irnprovement in the coefficient of variation of the test-topredicted ratios for the 60' beams considered from 0.170 to 0.137. 170 à 0. Cette addition réduit l'écart-type de 0. . Une étude paramétrique sur les coefficients de voilement Çlastique de l'âme a été effectuée pour des charges en cisaillement pur et en flexion. Ce diagramme est utilisé en combinaison avec les diagrammes pour la formation d'un mécanisme de rupture pour estimer la force de cisaillement par rapport au moment de flexion.137 sur les prédictions théoriques pour les poutres ajourées avec des ouvertures de 60'. et une moyenne de 1. pour un grand nombre de poutres ajourées avec des ouvertures de 60'.195 ont été obtenues. sont comparées aux résultats des plusieurs études antérieures. Les rapports entre les résultats expérimentaux pour 42 poutres avec 45' à 60' d'ouvertures et les prédictions par les méthodes d'analyse de plasticité et d'élasticité ont été obtenus.086 et un coefficient de variation de 0. et pour le voilement de l'âme prédit par analyse par élément finis. pour la rupture du joint de soudure horizontai par écoulement. Un diagramme d'interaction pour le voilement élastique de l'âme a été développé.RI~SUMÉ Dans la cadre de la présente thèse. L'effet de la plasticité lors du voilement de l ' h e est ensuite inclus dans les expressions théoriques. Les charges estimées par les diagrammes d'interaction pour la formation d'un mécanisme de rupture. Les modes de rupture de ces poutres et les charges correspondantes sont evalués par des anaiyses de plasticité et d'élasticité. correspondant à la formation d'un mécanisme de rupture et au voilement élastique de l'âme . une étude sur le voilement de l'âme des poutres aiourées a été effectuée. Sprcial thanks are due to Prof.yerirs. and my brother H m 9 for their intïnite support and encouragement for al1 these years. Finally 1 would like to acknowledge my uncle Joseph Bedrossian. G. R. for his valuable knowledge and help for man. The support of Fonds des Chercheurs et l'aide à la recherche (FCAR) is greatly acknowledged. McClure for al1 her help throughout the course ot'this project.ACKNOWLEDGMENTS I would like to express my sincere gratitude to Prof. encouragement and help throughout the course of this project.G. Redwood for his constant guidance. . and to al1 her guidance and advising throughout my graduate level studies. 1 uould like to thank my parents Krikor and Alice. ........................................2 Failure Modes of Castellated Beams ...............1 Genera! ..................--.6 Web Post Buckling Due To Compression ................................ 1 1 .................... ................................ 4 Rupture of Welded Joints........2-2 Flexural Mechanism....... ........................................................2...TABLE OF CONTENTS ABSTRACT ......1 Vierendeel or Shear Mechanism .... 10 1......................................................2 Outline of the Thesis ............................. ix NOTATIONS .................................................................................................... 3 Lateral Torsionid Buckling ....................................................................................x CHAPTER ONE :Introduction ............................... 5 1 ...................... -7 1 2 ......... .................. 15 CHAPTER TWO : Methods of Analysis.......................................3..... TABLE OF CONTENTS ........................ 1 1.... 16 2.............. iv .................... 16 ..... 9 1................................... LIST OF TABLES ..........1 Objective and Scope of Work ...............................3..................................................-......... vil.................................. 14 1......................2........-............................... 1 Introduction .......... 5 1......................................................................................... 14 1...................................................2.. LIST OF FIGURES ....................................................................................................................................3 Research Program ................................................................................................................ i RÉSUMÉ ................................. 6 1 2 ...... iii A C WOWLEDGMENTS .....5 Web Post Buckling .. 13 1................... .................................... 34 CHAPTER THREE :Literature Review................................... 3 ..................................................5 Loads ....... -28 2.......................................................2 Plastic Analysis .3 Mid-Post Yielding ............ 32 2............. 39 3.......................................................... 16 2....... 1 9 2......... 39 .... .3 Gaiambos.......5...........2......................................................................................................................9 Sherboume ( i 966) ................................................................... 38 3 2 5 Husain and Speirs (1971) .....5.................................................... 21 2.. 2 4 2 -5 -2 Input File Preparation ..... 3 5 3........4 Constraints .......6 Summary .............1 General ......................... 27 2.... 35 3..............................5 Finite Element Analysis ..... .........2....................................................5 -3 Model Geometry .. 4 1 ..5..........2.................................................................................... and Speirs (1975)............ 1 2......................2.........................4 Husain and Speirs ( 1 973) .............. 28 2...............................................2...... ..........2 Zaarour ( 1 996) ... 36 3....................................................... 37 3. -40 3............4 Buckling Analysis .......................................................6 Buckling Analysis .2......1 Redwood and Demirdjian ( 1 998)......... 36 3.................................... Husain.......................... 35 3........l General .................................................6 B a d e and Texier ( 1968)...............................2.......................................................2 Literature Review...5.............. 29 ....................................... 2..................2-7 Halleux ( 1967) ............................................................. .....................................3 Cornparisons .............2 Comparative Data ...................................... 23 -- 57 CHAPTER FIVE :Generalized Analysis and Design Considerations......................................................................................................3....... 43 CHAPTER FOUR :Reconciliation o f Andysis With Test Results ................................................................ I O Toprac and Cooke (1959) ................ -63 5 ...........2.................... 3................ ..................................... -78 5 -8 Effect of Inelasticity on Ultimate Strength ...........79 C HAPTER SIX :Conclusion ............................ -73 5 -6 Shear Buckling Coefficients .............4 Discussion ..................................................................2......... 67 5 -4 Parameter Smdy ........42 3........................................................7 Flexural Buckling Coefficients .............. -62 5.....................62 5....................52 4....................................3 Elastic Buckling interaction Diagram ................................ 5 2 1........................................5 Previous Parameter Study .......................... 87 APPENDIX A :Finite Element Input File AP PENDIX B :Detailed Test-To-Theory Results APPENDIX C : Elastic and Plastic Theoretical Computations ............................ 76 5..2 Loading on General Models .... -84 REFEWNCES.............................................. -52 4...................1 I Altifillisch............... Toprac and Cooke (1957) ................ 73 5...........................1 General ..............................................................1 General ....................... .4 (a) Mode1 used By Zaarour and Redwood (1 996) ................................................. ................... ............................................ 2 cHAmmnw Figure 2........................................- Figure 1-3 Castellated Beam Section Properties................................................5 Finite Element Mode1 .... ...........1 Castellated Beams .. -30 31 Figure 2.6 Lateral Torsional Buckling ..................LIST OF FIGURES ixurmwm Figure 1....................................4 (b) Non-Composite Mode1 Used by Megharief (1 997) ........ 8 Figue 1-7 Weld Joint Rupture ...............2 Interaction Diagram Demonstrating Theoretical Methods ........6 Pure Bending and SheadMoment Arrangement ......4 Castellated Beam Section Properties with Plates at Mid-Depth 4 Figure 1..................... 9 Figure 1 -8 Web Post Buckling ................................ 1 3 Figure 1.........5 ParalleIogram Mechanism ...2 Zig-Zag Cutting Dimensions of Rolled Beams .............................................1 ......... ........................3 Figure 4............................. ' -3 Figure 2....... ............1 Test Arrangement of Beam H ......... .3 Predicted Web-Post Buckling Moments ....... ....... Figure 4...................................... ..... >J ........4 ........2 0 3 Figure 2........................ 18 Figure 2...............26 Figure 2.................. 26 - Figure 2... ....2 Free-Body Diagram .....54 ............... 6 Figure 1........................1 Interaction Diagram .......... Figure 1............. ........ 69 Figure 5........1 Two Hole FEM Model Under Vertical Loads Only .. ........................3 Three Hole FEM Model Under Pure Bending Moments ..........................I0 Modified Pure Shear Buckling Coeficient Curves ....12 Elastic and Plastic Interaction Diagrams .......9 Shear Buckling Coefficient Redwood and Demirdjian ( 1998) 75 Figure 5 ........ 7 1 Figure 5...1 1 Buckling Coefficient Curves Under Pure Bending Forces ......... 72 Figure 5................. 77 Figure 5............................................. 65 Figure 5.......... Figure 5..............................66 Figure 5........... .. -70 .7 Husain and Speirs ( 1971 ) ...5 Zaarour and Redwood ( 1 996)........ Figure 5 -6 Husain and Speirs ( 1973) .. Cooke and Toprac .................4 Three and Four Hole FEM Models .....7 9 Figure 5.. 64 Figure 5......1 3 Comparison of Test Results With Proposed Expressions .............................................. 83 ... 71 Figure 5...................2 Three Hole FEM Model Under Pure Shear Forces ....8 Altifillisch.................... 80 Figure 5... ........................... ...... 67 Table 5........ Cooke and Toprac (1 957) ................... -44 Table 3...... 45 Table 3...................... 82 .. 38 Table 5.................1 Surnmary of Test to Theoretical Predictions ..3 Galarnbos Husain and Speirs (1975)............1 Summary of Resuits under Pure Moment Forces ............................................... 47 Table 3 -6 Bazile and Texier ( 1968) ........... .....2 Zaarour and Redwood ( 1 996) ......................................................1 Redwood and Demirdjian ( 1998)....5 Husain and Speirs ( 1 971) ...........................................7 Halleux ( 1967) ............Table 3...............8 Sherbourne (1966) ...................................... 46 Table 3..................... 44 Table 3............................................. 51 Table 4........................ 49 Table 3.....3 Statistical Results ......... 50 cKMnEmm Table 3........... ..................................................................4 Husain and Speirs (1 973) .............................................. LlST OF TABLES Table 3........9 Toprac and Cooke (1959) .......2 Summary of Results Under Pure Shear Forces ........... 68 Table 5.......................10 Attifillisch....................................................................... 47 48 Table 3...................... NOTATIONS Ar . G GD h ho h. . area of flange area of web width of one sloping edge of the hole width of flange depth of the original beam section total depth of castellated beam section depth of bottom tee section depth of top tee section b br d '4 db 4 C compression force coefficient of variation degree of freedom cov DOF E e modulus of elasticity length of welded joint finite element analysis finite element method yield stress stiffiiess matrix differential stiffness matrix height of one sloping edge of hole height of hole height of plate FEA FEM F .4. moment of inertia depth of top tee section excluding flange buckling coeffkient flexural buckling coefficient shear buckling coefficient length of beam bending moment elastic buckling moment under pure bending forces elastic moment to cause web buckling critical moment plastic moment critical moment based on beam test results yield moment moment to fonn flexwal mechanism ultimate moment constant force elastic section modulus distance from center-line to centerline of adjacent castellation holes tension force thickness of the flange thickness of the web displacement vector . modified displacement vector Shear force elastic buckling shear under pure shear forces critical shea to cause web buckling shear obtained fiom elastic anaiysis horizontal shear force critical value of Vh cntical shear based on beam test results plastic shear shear O btained fiom plastic analysis vertical shear force to cause rnid-post yielding vertical shear force to fom plastic mechanism ultimate shear to cause web buckling applied load distance fiom top of the flange to centroid of tee-section plastic section modulus of castellated beam full section plastic modulus factor utilized in plastic analysis factor utilized in plastic analysis angle of castellation critical stress expansion ratio sii . factor appiied to shear y ield stress eigen value eigen vector poisson's ratio aspect ratio . Cnstellated béams w r e one of these solutions ( Fig. Second World L\. se\.IL incrcnsing the stiffncss of steel mcmbers w-ithout an). man>-attcmpts ha\-e been made b' structural enyineers w : i d ne\\ ways to dccreass the cost of steel structures.As 3 ~1~\3>': be result.*ar.1 1. . Eue to limitations on mrisimum cillowable deflsctions. 1. r steel 1 Fig.increasri in iteiylir rquired. the high strength pruperties of structural i t ~ t l cannut utilized to best adwxage.ernl ne\! rne~l-iodsha\. 1. I Introduction Sincc rht.CHAPTER ONE 1.e been airned O&' the .1 CastslIatzd Beams . Experirnental tests on castellated beams have shown that beam slenderness. 1. which dictate the strength and modes of failure of these beams. In the past. with the developrnent of automated cutting and welding equipment. these beams are produced in an alrnost unlimited number of depths and spans.2. Today. suitable for both light and heavy loading conditions. producing octagonal holes (Fig. castelhtion parameters and the loading type are the main parameters. The web of the section is cut by flarne along the horizontal x-x avis along a "zigzag" pattern as shown in Fig. 1. the presence of the holes in the web will change the structural behavior of the beam from that of plain webbed beams. However.2 Zig-Zag Cutting Dimensions of Rolled B e m s The two halves are then welded together to produce a beam of greater depth with hexagonal openings in the web (Fig. without an increase in weight.3). the cutting angle of castellated beams ranged from 45" to .4). 1. The resulting beam has a Iarger section modulus and greater bending rigidity than the original section. or rectangular plates may be inserted between the two parts. Casteilated bearns have been used in constmction for many years. Figure 1.Castellated (or expanded) beams are fabricated from wide flange 1-beams. 2 to 1. = height of plate b = width of sloping edge of hole - d. d = original beam depth h = depth of cut h. Their aesthetic attributes produce an attractive architectural design feature for stores.70° but currently. = d + h + h . or purlins. It should be noted that these are approxirnate values. joists.4): d. although 45" sections are also available. 1. schools and service buiidings. 60" has become a fairly standard cuning angle. Typically. In structures with ceilings. h. = depth of top tee section . the dimensions of a castellated beam are defined as follows (referring to Figs. actuai angles will vary slightly from these to accommodate other geometrical requirements.=O) 4 Expansion ratio. As roof or floor beams. (For no plates. these sections may replace solid sections or tmss members. the web openings of thesc members provide a passage for easy routing and installation of utilities and a r i conditioning ducts. y = d where. Figure 1.3 Castellated beam section properties I - 4 v Figure 1.1 Castellated beam section properties with plate rit mid depth . web post buckling in shear and compression buckling. The secondary moment. rupture of welded joints. as the horizontal length of the opening decreases. type of loading. as well as the pnmary and secondary moments. results from the action of shear force in the tee sections over the horizontal length of the opening. 1.1. 1. Shorter spans can carry higher loads leading to shear becoming the goveming loac!.5).2. This mode o f failure was first reported in the works of Altifillisch (1957). web slendemess. These modes are closely associated with beam geornetry. Under given applied transverse or coupling forces. and Toprac and Cook (1959). Bearns with relatively short spans with shdlow tee sections and longer weId lengths are susceptible to this mode of failure.2 Failure Modes of Castellated Beams To date. experimental studies on castellated beams have reported six different modes of failure (Kerdal & Nethercot 1984). lateral torsional buckling. Formation of plastic hinges at the reentrant corners of the holes deforms the tee section above the openings to a parailelogram shape (Fig. hole opening. also known as the Vierendeel moment. the tee sections above and below the openings must carry the applied shear. The primary moment is the conventional bending moment on the beam cross-section. flexurai mechanism. When a castellated beam is subjected to shear. the magnitude of the secondary moment will . and provision of lateral supports. Therefore. failure is Iikely to occur by one the following modes: Vierendeel or shear mechanism.1 Vierendeel or Sbear Mecbanism This mode of failure is associated with high shear forces acting on the beam. The location of this failure will occur at the opening under greatest shearing force. . Plastic Hinges Figure 1. This mode of failure was reported in the lcorks ot'Toprric and Cook ( 1 959) and Halleux (1967). the tee sections above and below the openings yield in tension and compression until thty becomc fully plastic. wliere Z' i tlie full s section plastic modulus taken through the vertical çenteriine of ri hole.5 ParalleIogram Mechanism 1. then the one with the greatest moment will be the critical one. provided the section is compact (at lsast Class 2 (CSA 1991)).c. Thus.2. or if several openings are subjected to the s m e maximum shear.2 Flexural Mechanism Under pure bending. They conciuded that yieiding in the tee sectioris ribo\.decrease. the maximum in-plane carrying capacity of a castellatrd beam under pure moment loading was determined to be = Z'xE. and bclow the openings of a castellated beam was similar to that of a solid beam under pure bendiny forces. 2.3 Lateral-TorsionaCBuckling As in soiid web beams. ï h e y concluded that web openings had negligible effect on the overail lateral torsional buckling behavior of the beams they tested. is usually associated with longer span beams with inadequate lateral support to the compression flange. it was suggested that design procedures to determine the lateral buckling strength of solid webbed bearns could be used for castellated beams provided reduced cross sectional properties are used.1. Nethercot and Kerdal (1982) investigated this mode of failure. as a result of relatively deeper and slender section properties. 1. conmbutes to this buckling mode.6. out of plane movement of the b a r n without any web distonions describes this mode of failure. Furthemore. Lateral torsional buckling as s h o w in Fig. . The reduced torsionai stifiess of the web. 1.Fia..6 Lateral Torsional Buckling (Redwood & Dsmirdjian 1998 ) . 1. Weld Rupture Figure 1. Dougherty (1993) found a reasonable balance of these twvo failure modes. Husain and Speirs (1971) investigated this failure mode by testing six beams with short welded joints. short weld lengths are prone to cause failure of the welded joints as the horizontal yield stress is exceeded. The horizontal length of the openings is equai to the weld length.2. formation of a Vierendeel mechanism is likely to occur in beams with long horizontal hole lengths (and hence long welds). Weld joint Rupture As mentioned in 1. by suggesting the following geometry: .1. On the other hand.2.1.7).4 Rupture of Welded Joints The mid depth weld joint of the web post between two openings rnay rupture when horizontal shear stresses exceed the yield strength of the welded joint (Fig.7. the welded throat of the web-post becomes more vulnerable to failure in this mode. This mode of failure depends upon the lengtb of the welded joint (e). and if the horizontal length is reduced to decrease secondary moments. opening pitch s = 2(6 + e ) = 24) (0. Delesque ( 1968) . sorne correlations between experimental and non-linear finite element analysis (FEM) estimations were found in the works of Zaarour and Redwood (1996). As shown in Fig. (Castetite Steel Beam Design Manual 1996).289 + 3 = 1-08h.1 h.5 Web Post buckling The horizontal shear force in the web-post is associated with double curvature bending over the height of the post. one inclined edge of the opening wili be stressed in tension. 1.. Therefore. Many analytical studies on web post buckling have also been reported to predict the webpost buckling load due to shearing force. Bazile and Texier ( 1968). Haileux (1967).8. Several cases of web post buckling have been reported in the literature: Sherbourne (1966).2. This concept has been demonstrated in many of the current available Castelite Standard Beam Geometry sections. 1. 1.Weld length hl e =3 and for a 60' cutting angle with no plates.. and the opposite edge in compression and buckling will cause a twisting effect of the web post along its height. Based on finite difference approximation for an ideally elastic-plastic-hardening material Aglan and Redwood (1976) produced sonis graphical design approximations for a wide range of beam and hole geometries. . Blodgett's method is therefore not used in this project. Zaarour and Redwood (1 996) found large differences in the results obtained frorn Blodgett's method in cornparison to their test results and finite element approximations they used. approximations of buckling loads were derived based on elastic finite eiement analysis and good correlations btmveen experimental and theoreticai estimations were found. This work showed that the results of Aglan and Redwood ( 1976) should not be used for very thin webs. However. This mode of failure and these theoretical results are discussed in greater detail in subsequent chapters. In recent works of Redwood and Demirdjian (1998).used an energy method to solve an elastic buckling problem by treating the wsb post as a variable section rectangular bearn in double curvature bending. susceptible to lateral torsional buckling. 8 U'eb Post Buckling (Redwood rP: Demirdjian 1998 > .Fig. 1. .2. This mode was reported in the expenrnents conducted by Toprac and Cook (1959). A strut approach was proposed in the works of (Dougherty 1993). Such a failure mode could be prevented if adequate web reinforcing stiffeners are provided.6 Web Post Buckling Due to Compression A concentrated load or a reaction point applied directly over a web-post causes this failure mode. Buckling of the web post under large compression forces is not accornpanied by twisting of the post. which suggests that standard column equations could be used to determine the strength of the web post located at a load or a reaction point. Husain and Speirs (1973).1. as it would be under shearing force. and thus validation of the suggested methods described. are developed to predict eiastic buckling n ' loads under a M N ratios. based on pure shear and pure bending forces to cause web buckling. Thc second part of' the thesis tocuses on general design considerations and thus is aimed at the principal objective of the research. and derive expressions that will predict critical shear force causing web-post buckling.3 RESEARCH PROGRAM 1.1 Objective and Scope of Work The objective of the current research is to study failure of castellated b e m s \cith panicular emphasis on web post buckiing. Correlations between experimental and theoretical results are then made. Results of elastic buckling and mechanism yielding loads are then combined and fitted curves are derived to predict ultimate shearing forces causins . Well-defined relationships. T h e first part of the research program focuses on the theoretical methods of analysis to be used to predict failure loads of castellated beams. The finite element method is used to perform elastic buckling analysis and predict critical loads of al1 test beams. The goal is to make use of the availabie elastic and plastic analysis methods.3. Elastic buckling modes are investigated under different moment to shear ( M N ) ratios. This thesis uses many previous experimental results to provide cornparisons \\-ith theoretical approximations. A thorough literature search then follows to list al1 relevant experimental data to be compared with theoretical methods.1. These methods include plastic analysis of the Vierendeel mechanism and for yielding of the mid-post joints. Chapter 5 focuses on design considerations for castellated beams. Concluding rem& are summarized in Chapter 6. Suçgested predictions are then tested against actual test results. Chapter 3 contains a surnmary of relevant test data provided by previous testing and available in the Merature. as welI as expressions estimating shear force causing buckling are derived. Results of suggested methods are tested against actual experimental test results. and reconciliation of analysis with test results is the topic covered in Chapter 4. Relationships defining elastic buckling under any M N ratio are developed.2 Outiine of the Thesis The thesis is divided into six sections. A parametric study.web-post buckiing.3. Theoretical approaches described in Chapter 2 are tested against actual experimental test beams. Relevant information on each test beam is tabulated. M e r a bnef introduction to castellated beams and their modes of failure of Chapter 1. 1. buckiing analysis. and correlations between tests and theories are made. and good correlations are obtained. web-post yielding at mid-height. and finite element approximations. . a parametric study investigating the behavior of a wide range of casteltated beam geometries is developed. Chapter 2 focuses on several theoretical methods of analysis to predict modes of failure of castellated beams. and buckiing coefficients under pure shear and bending forces are derived. To apply these expressions in a more general fashion. ï h e s e methods include plastic analysis. 1 General Several theoretical approaches are considered to analyze the yielding and buckling fidure modes of castellated bearns. This diagram can be used to study failure caused by the formation of a Vierendeel mechanisrn formed by the development of four plastic hinges at the re-entrant comers of the tee section. Finite element mode1 generation as well as buc kling analysis in the MSCMASTRAN finite element package are described. primary and secondary stresses resulting fiom combined effect of shear and moment forces lead to complete yield at the four corners thus forming plastic hinges. the web and flanges are assumed CO be stable and withstand the high shear load until plastic hinges are fonned at the reentrant corners of an opening in high shear region. Elastic finite element buckiing analysis is used to predict buckling loads. This analysis is based on the assurnption of perfectly plastic matenal behavior with yielding according to Von .CHAPTER TWO METHODS OF ANALYSE 2. above and below the hole. Plastic anaiysis of the Vierendeel mechanism failure. For the beam to anain this plastic failure. ris rvell as anaiysis of mid web post yielding are sumrnaiized. 2.2 Plastic Analysis The construction of an interaction diagram relating shear force and bending moment at mid-length of an opening has been described by Redwood (1983). As the load increases. The diagram can be constructed using the following results: v =JS A.Y Where shear area A.Mises criterion.1. To generate the curve. is varied between O and 1. = dg&. Below the value 1. The shear and moment values have been non-dirnensionalized by division of the section's tùi1y plastic shear and moment capacities. the curve becomes vertical. k. For given beam characteristics and hole location subjected to a load. The horizontal and vertical coordinates of the intercepted point then predict the shear and moment values to cause yield nlechanism failure. 2. a radial line can be drawn fiom the origin to intercept the interaction diagram for the corresponding shear-to-moment ratio (VIM). I. A typical interaction dia- is shown in Fig. . 1 Interaction Diagram (Redwood and Demirdjian ( 1 998)) .Interaction Diramm Specimen 10-5a I j -YieidTheory A Test Resuit Figure 2. The vertical shear force to cause mid-post yielding is defined through.2. It is possible for yielding of the web-pst at mid-height to occur before failure due to formation of shear mechanism takes place. the yielding is contained. and it cm .(d x . The horizontal shear force. ) where. 2. and the basic approach to define this relationship (Hosain and Speirs 1971 ) is derived by using equilibrium equations from the free body diagram of castellated beam section as shown in Fig. are equal. The web post will yield when the minimum weld-post area is subjected to the shear yidd Due to the maximum shear stress k i n g at the throat. and V. . V. This equation is based on the assurnption that the line of action of forces C I and C->are acting at the centroid of the tee section above the openings.2 y . This mode of failure occurs particularly to beams with closely spaced openings with low moment-to-shear ratio. can be expressed as when the vertical shear force V. is defined as the difference between the two horizontal forces C and C7.23 iMid-Post Yielding . then v x s . V. 2 Free-body diagram of castellated bearn .be expected that strain-hardening will develop leading to a significantly higher failure load than that given by Eqn. later increased by a factor P. V.Q Figure 2. for this mode of Mure only.1. 2. In the work of Husain and Speirs ( 197 1 ) the sliear y ield stress has been measwed directly and is significantly higher than the expected value based on F~ 4 3In view of this the yield stress used. will be . as discussed in Chapter 4. However. For different hole height to minimum width ratios. where yield on the smallest web-pst cross-section - fi is an imposed upper limit on V.3. 2.a given beam. The materiai was considered to be an elastic-perfectly plastic linear strain-hardening material. ) ~h. in more recent work (Redwood and Demirdjian 1998).4 Buckling Analysis Based on a finite difference bifùrcation analysis of the web post ueated as a beam spanning between the top and bottom of the openings.. Vh diagram of Figure 2. In the work of Zaarour and Redwood (1996).= et. critical moments in the post at the level of the top and bottom of the opening. the vertical shear force to cause buckling in the web-pst is then derived as VL. divided by that ( s section's plastic moment capacity. .2 y . the Vfl ratio is given by . the horizontal shear ' r acting at the minimum weld length is calculated as Vh = -..(d.2. Therefore. 2 5 ~ .~. as shown in is first read from Fig. = 0 ..3..ef Fig. the value of MJM.l. graphical results relating critical moments in the p s t to different beam opening geometries were developed by Aglan and Redwood (1 976).. A(.. were presented.= S d$y. 2 . satisfactory predictions were obtained with the Aglan and Redwood (1976) approach.2. tests of very thin .. From the free body h. By multiplying the given ratio by the section's plastic capacity M. as given above. 2. F0r. who tested 12 castellated beams. w.. webbed castellated beams showed that the graphical results such as shown in Fig.3 provided unsafe predictions. a result that was believed to be due to the assumed restraint conditions at the top and bottom of the web-pst. . 2. The method of Aglan and Redwood ( 1976) is therefore not considered M e r in this study. 3 Predicted Web-Post Buckling Moments for 4=60U (Aglan and R e d ~ o o d 1976) .#? .=lO 20 > @ - i 0.4 O. 0.0 1 dt.? O 30 Figure 2. Their mode1 consisted of full flanges. The model used in the . web and transverse stiffeners and the model comprised two complete web openings as shown in Fig. 2. This larger model was needed in order to incorporate the shear c o ~ e c t i o n between steel section and slab. 1996 and Megharief and Redwood. Mesh refmement was based on the convergence of web p s t buckling loads in cornparison to several experimental test results.2. and hence the composite action on the bearn. 1997) FE AM studies of the buckling of web-posts in composite and non-composite beams were found to give good approximations of test resuits (2-10% variations).5 FINITE ELEMENT ANALYSIS 2. This section therefore describes the sofhare used and the specifics of the application to castellated beams.5.4(b). Megharief and Redwood (1997) investigated the behavior of web-post buckiing of composite castellated beams. 2. The same package is used in the current research with the objective to utilize FEM as a reiiable tool to simulate experimental tests and generate web p s t buckling loads.1 General The finite element method has previously been used to pertorm buckling analyses on castellated beams and is also used in this project.4(a). Both studies utilized the finite element package MSCRVASTRAN developed by the MacNeal SchwindIer Corporation (Caffiey and Lee 1994). as show in Fig. Zaarour and Redwood (1996) studied buckling of thin webbed castellated beams based on a single web-post model. In previous work (Zaarour and Redwood. 2. . based on the different needs in the current work. however. The following sections describe the particular steps necessary to use the MSCNASTRAN system and the details of the generation o f the models. as discussed subsequently.current research is similar to the non-composite beam mode1 utilized by Megharief and Redwood (1997) as shown in Fig. more refmed meshes and a greater number of openings are used.4(b). Fig. 2.4(b) Non-Composite Model used by Megharief and Redwood ( 1997) . 2.4(a) Model used by Zaarour and Redwood ( 1996) Fig. Case control and %ulkdata. Case Control Section: specifies a co~lection grid point numbers or element numbers of to be used in the analysis.Material properties: definition of Young's modulus and Poisson's ratio.5. constraints. SampIe input file is given in Appendix A. Includes geometric locations of grid points.2 Input File Prepamtion Elastic finite element bifurcation analysis was carried out for al1 test beams. and the bending properties of each element. element connections. which consists of three major sections: Executive control. Requests output selections and loading subcases.2. Bulk Data Entry: contains al1 necessary data for describing the structural model. time allocation and system diagnostics. element properties and loads. hanalysis in MSCNASTRAPJ is submitted in an input file. Efement properties: definition of the thickness. the following classes of input data must be provided: Gromerry: locations of gnd points and the orientations of the coordinate system Elrrneni connectivity: identification number of grid points to which each dernent is co~ected. . Executive Control Sectioa: is the first required group of statements to detïne the type of analysis. C'onsrrninis: specifications of boundary and symmetry conditions to constrain free-body motion that will cause the anaiysis to fail. To prepare a detailed description of a model. . Louds: definition of extemally applied loads at grid points. y. the grid points are connecteci by finite elements. linear elastic properties of the material.2. T2.5. modulus of elasticity. y.3 Mode1 Geomety and Type of Elernents A skeleton model based on a given beam geometry is first developed through defining the x.R3).Similady. bending. Poisson's ratio are defined in the MAT1 data entry input card by assigning a property identification number in the PSHELL entry card. T3) and three rotations (RI. tinear elastic (MAT 1) membranebending quadrilateral plate elements were chosen to define the finite elements of the model. to which finite elements are attached. membrane.5. ail essential material properties. Grid points are used to define the geometry of a structure. Each gnd point possesses six possible degrees of fieedom (DOF) about the x. W. and z coordinates of each grid point. thickness. shear and coupling effects of the elements are defined in the shell element input property card (PSHELL). three translations (T 1.4 Constrainta Single point constraints (SPC) are used to enforce a prescribed displacement (components of translation or rotation) on a grid point. As the geometry of the structure is defined. Two-dimensional CQUAD4 isotropic. By assigning a material identification number in the CQUAD4 input card. The degrees of fieedom in MSC/NASTEWN . and z-axes. 2. which constrain the grids to displace with the loaded structure. CQUAD4 element input card is defined through four grid points whose physical location determines the length and width of the element. Moment loads were applied by applying two equal and opposite (x-direction) concentrated horizontal loads at the lefi- . flanges and the stiffeners had zero normal twisiting stiffness. R2. 3. Displacements in the x and z directions at the upper and lower flange to web intersecring nodes at the right end are restricted by constraints 1 and 3.5 Loads Shearing forces were applied to the models by assigning two transverse (negative y direction) loads at the right hand end. these prevent movement in the vertical and out of plane directions. Values of 100. These constraints simulate symmetry of half the span of a simply supported beam geometry.5.are defined as numbers 1.4. to prevent t-igid body rotation about the z-axis. this one comprising two openings. as shown in Fig.000 was tested to provide acceptable results. 2. T2. 5. as recommended in the manuals.5. T3.000 are recommended by the manuals. and three rotational degrees of freedom. This value is a fictitious number assigned to suppress singularities associated with the normal degrees of freedom. One way to ensure non-singularity in the stifiess matrix and to account for the out of plane rotational stiffhess or the sixth degree of fieedom (R3)is through AUTOSPC and KGROT commands in the Bulk Data Entry. In ail models K6ROT was taken as 10. 2.000 or 100.5 shows a typical mesh. a value of 10. 2. 10. The model is supported at the bottom lefi-hand corner where constraints 2 and 3 are applied. corresponding to three translation. and 6. RI. RX The propenies of CQUAD4 elements used in modeling the web.000. T l . Out of plane displacements are prevented on the perirneter of the web. Fig. however. 2. iland end at the flange-to-web intersections (Fiy.6). Fig. Thus sliear and moment could be assigned in any desired combination. 2. 2.5 Finite Elernent Mode1 . Fig.6 Pure Bending and SheadMoment Arrangement . 2. 2. [Go] the differential stifiess matrix is introduced that results f o including higher-order ternis of the strain-displacement im relations (these relations are assumed to be independent of the displacements of the structure associated with an arbitrary intensity of load). To include the differential stiffbess effects. and P the applied load vector. ( [G] q [ G D ] ){u*) = (qP) where u* is the modified displacernent vector resulting fiom displacements under an intensity of Ioad. by introducing q as an arbitrary scalar multiplier for another "intensity" of Ioad. Subcase 1 will define the static load condition applied to the system. the equilibrium equation becomes. u the displacement vector. Two loading conditions must be defined in the case Control section. The equilibriurn equations for a structure subjected to a constant force may be written as P l {u) = {Pl where G is the stifniess rnatrix. By perturbing the structure slightly at a variety of Ioad intensities.5. the "intensity" factor q . and subcase 2 selects the method of eigen value extraction method. Linear buckiing analysis is defined through SOL 105 command. and fiom differentiaI sti ffness effects. Hence.6 Buckling Analysis The type of analysis to be performed in MSCINASTRAN is specified in the Executive Case Control section in the input file using the SOL command with the CEND delimiter to represent the end of this section. The product of the first load intensity factor or the first eigenvalue q with the applied load would give the fïrst buckling load of the rnodel.4. PARAM entry is another statement used to account for AUTOSPC command to constrain dl singularities on the stifiess matrix as described in Section 2. (q different fiom zero) only for specific values of q that would make the matrix [G -qGD]singular.I{v) = 0 . the range of interests of eigenvalue limits is determined. will be the factor to c a w buckling. By using the EIGB entry. The requirements for an eigen value solution in MSCMASTRAN are defined in the Bulk Data Entry. (163 t r l l G ~ l ) W = 0. Limitations of SOLLOS required small deflections in the prebuckied c ~ ~ g u r a t i o n and stresses to be elastic and linearly related to strains. The solution is nontrivial. and hence is used in al1 computations. The S W method is an enhanced version of the iNV method. and the eigenvector <p. the bucicied shape. . I t is suggested that S W is a more reliable and more efficient method than the INV method. ) This requires the solution of an eigenvalue problern: [G -tlG. Lt uses Snirm sequence techniques to ensure that ail roots in the specified range have k e n found. The two conditions were fully satisfied.5. Two methods of eigenvalue extraction methods are avaiiable in the software invoked by the cornmands: INV and S M . and specifying a set identification number for the model.to create unstable equilibriurn conditions. and in each such case the associated eigenvalue was negative. . Unrealistic buckling modes were sometimes O btained. The lowest value was accepted.6 Summary In this chapter the severai methods of analysis used later in this thesis have been described. Further details. 2. the two identical symmetric modes were associated with positive and negative eigenvdues of almost equal magnitude. are described when particular applications are discussed in the following chapters. Under pure shear. for example buc kling on the tension side of the beam under pure bending. and in some cases the negative one was msrrginally iower than the positive one.Buckling modes resulting from the analyses were examined carefully in each case. and was rejected. especially of the FEM applications. An outline of previous expenmental work on castellated beams is reported here with the objective of describing only the main features of each investigation. 3. The test programs are described in reverse chtonological order. For each test beam.CHAPTER THREE LITERATURE: REVIEW 3. The data and test results for the beams described are the subject of detailed analysis in subsequent chapters of this thesis. the section properties. . generally accepted design methods have not been established due to the complexity of castellated beams and their associated modes of failure.1 General An investigation of previous literature on non-composite castellated beam tests was conducted fiom which data was obtained in order to make comparisons between experimental and theoretical results in later chapters.2 Literature Review Reviews on non-composite castellated beams have been extensively reported in the Li terature. geometry and experimental arrangements were studied and relevant data are summarized in tables at the end of this chapter. However. 8 mm) high plates welded between the two beam halves at the web-post mid-depth. Ultimate load values were given as the peak test loads. and 14 inch light beams (Bantam sections manufactured by Chaparral Steel Company) were tested. Mean flange and web yield stress values were obtained from tensile coupon tests.2. except beam 10-7. (50. Based on the experimental ultimate loads. The buckling mode involved twisting of the post in opposite directions above and below the mid-depth. which failed by lateral torsional buckling.2 Zaarour (1995) Fourteen castellated beams fabncated €rom 8.3. IO-5(b). al1 with identical cross sectional properties. The objective of the experiments was to study the buckling of the web post between . were tested.14% variations). two identical ones with four openings 10-5(a). Six of these had 2 in.10.2. Al1 beams were provided with bearing stiffeners at support and at load points. and good predictions of the buckling loads were reported (4.12. 3. since interest is in web buckling only. The main focus of the experiment was to investigate the buckling of the web post between holes and to study any effects of moment-to-shear ratio on the mode of failure. Test conditions were then sirnulated by elastic finite element analysis. Simple supports and a central single concenuated load were used for al1 specimens. a third with six openings (10-6) and a fourth with eight openings (10-7).1 Redwood and Demirdjiaa (1998) Four castellated beams. Bearn 10-7 is omitted fiom further consideration in this project. buckling of the web post was the observed mode of failure of al1 these bearns. 1 for dimensions)) were tested to validate a numerical analysis approach to determine the optimum expansion ratio based on both elastic and plastic methods of analysis. local buckling of the tee-section above the openings subjected to greatest bending moments occurred. Ultimate loads were recorded. Al1 beams were provided with bearing stiffeners at support and at Ioad points. FEM analysis was also used to predict web-pst buckling load. Simple supports and a central single concentrated load were used for al1 specimens. but the depths were varied based on different expansion ratios. Husain and Speirs (1975) . Four castellated beams fabrïcated fiom W 10x 1 5 sections ( 10 in deep. but no further discussion about the modes of failure was given. Two lateral torsional buckling modes were also observed.. .openings. 15 pounds per foot (see Table 3. The reponed ultimate strengths were based on peak load capacities of the bearns. and in two cases. these have been omitted tiom further consideration shce interest is in web buckling only. The span and weld lengths were kept constant. Al1 beams were simply supported and were subjected to a concentrated load at mid-span. Web post buckling was observed in the failure of 10 cases. Average flange and web yield stresses were obtained from tensile coupon tests for each size of beam. 3 2 3 Galambos. beams B-1.2. and B-3.4 Husain and Speirs (1973) Bearns fabricated fiom twelve 10%15 beams (alternative designation for W 1OX i 5 ) were tested to investigate the effect of hole geometry on the mode of failure and ultimate strength of castellated beams. G-2. C and D were subjected to two concentrated point loads. C. Specimens A-2. C. yielding of the flanges in the region of high bending moment iead to flexural failure. and D failed prematurely due to web buckling directly under the point of load application. . and D were omitted fiom m e r study. B-1. A-2. a class 2 section. As for Specimens G.1. ïhus. and G3.1-94 class 1 section properties. Bearns 8-2. Similar failure was exhibited by Beam B-1 that failed by web buckling under the concentrated load before a Vierendeel mechanism had fonned.3. and the rest of the beams had a single concentrated load at midspan. with flanges of Canadian Standard S 16. Specimens A-1. Al1 beams were simply supponed and adequate lateral bracing and full depth bearing stiffeners were provided (except for beams C and D where partial depth stiffeners were used). B-2. failed by the formation of plastic hinges at the re-entrant corners of the opening where both shear and moment forces are acting. The class section properties were calculated for some beams in an anempt to investigzte if any local buckiing possibilities were present. The loads were based on the ultirnate load values obtained during the experiments. The measured shcar stresses \iwe significantly higher than values which wvould have been expected tiom tsnsile coupon tests.6 Husain and Speirs (1971) The main focus of this experiment was to study the yielding and rupture of urlded joints of castellated beams. The experimental investigation consisted of testing sis simply supported beams under various load systems. consenative. E-3.1 ) kvere tested to failure. wrp.Etrt: pro~ided to pre\. Loads PI .3. measured yield and ultimate shear stress values. Three test loads.2. The reponed final results were caiculated on the b a i s of directI). four HEAS60 and three IPE270 sections (for dimensions see Table 3.3) can therefore be espected ro bc. Full depth-bearing stiffeners and sufficient lateral bracings wr. The prediction of' ultimate strengtli based on web-post yield (see Section 2.7 Bazile and Texier (1968) T~vo series of beams. The objective of the experiment was to de\. probably as a result of strain hardening.ent premature buckling.elop a tùrther understanding of different beam characteristics and properties. P L and P3 wrtt reponed to describe the different phases of the load-deflection diagrarn of each beam. P l .reIease \vas the common mode of failure for all beams.2. A single concentrated point load \vas applied to b e m s E-2. geometn and espansion ratios of castellated beams. The simply supported beams were tested under eight unifonnly distributed concentrated loads. 3. F-1 and F-3 and two concentrated loads were used for beams E-1 and F-2. Sudden weld rupture accornpanied by violent strain energj. . It is therefore evident that these were compression buckling failures under the action of the concentrated loads acting directly above the unstiffened webposts. Therefore. these two beams were not considered M e r . The experimental faiiure load was based on the intersection of the tangent to the tinear part of the load vs. Five types of beams with different geometricai properties. Since this mode is not k i n g studied. bracket the uitimate test value of the concentrated load.and P2 define sudden changes in slope and P3 was the ultimate load. The beams F and G failed by lateral torsional buckling and were thus omitted from M e r study herein. were tested to destruction under two equal concentrated loads applied at the third-span points. and it is later stated that yield stresses determined from umeponed tensile tests were significantly higher than the above-mentioned value. Measured yield stresses are not reported. Calculations in the reference are based on the yield stress of the material. 24 kglmm'(235 MPa). deflection diagram with the tangent to the h o s t horizontal part of the curve. Flange and web yield stresses were obtained f o beam coupon tests and full depth stitTeners were im provided at support reaction points. Beams A. and were reported as faiiing by web-pst buckling. dl fabricated fiom the IPE300 rolled steel sections. Beams C and D had deep (200rnm) plates at mid- depth. using the colwnn strength formula of CSA (1994) assuming widths equal to the maximum and minimum actual widths. B and E failed under web buckling in the zone of maximum shear. 3 2 8 Halleux (1967) . that is. Estimated strengths of the posts of these two bems. due to . Load-deflection curves are given in the paper. Failure of this beam however. under the two point loading system. Seven tests were perfonned which ranged from pure shear to pure bending loading conditions. . The first two were reported to fail by flexurai mechanisms. Specimen E4 was designed to study the effect of pure shear across the central opening. The hole closest to the load was the most severely damaged. From these the ultimate loads and Ioads obtained fiom the intersection of tangents to the initiai linear part and to the almost linear pst-yield part were obtained. Beams L 1. was outside the central control section and was associated with extensive yielding in the end zones experiencing both shear and moment forces. 3.2. The deflection curve demonstrates considerable strain-hardening. L3 was also reported to fail by fiexural mechanism. and web buckling was the observed mode of faiiure. The test arrangement consisted of simply supported beams with fidl depth bearing stiffeners under load and reaction points. subjected to a single concentrated load at mid-span.9 Sherbourne (1966) This test program was designed to investigate the interaction of shear and moment forces on the behavior of castellated beams under varying load conditions. L2.the uncertainty in the yield stresses the reported results must be treated circumspectly. Beam E2 was designed to investigate the effect of pure moment. Beam El. and was subjected to two concentrated point loads. Web buckling was the mode of failure of specimen E3 in the zone of maximum shear. and L3 were tested under pure bending moments. lateral torsional buckling was also associated with the failure mode. however. failed through extensive yielding of the b o a t at middepth of the post between the first and second hole opening. Well-defined yield stress values were obtained through coupon tests and adequate bearing stiffeners were provided under reaction points. to compare observed results with theoretical calculations. to study load carrying capacity and modes of failure. but no fùrther details were given. Specimen 1. Local buckling of the compression flange in the constant moment region was d s o the observed failure mode of specimen F. Specimen H.2. Specimens A and C failed through excessive lateral buckling and are omitted from further study. failed through buckling of the compression fiange in the constant moment region.3. the beam buckled laterally. with a class 2 îlange section. with a class 1 web tee stem section failed through a Vierendeel rnechanism in the highest shear region. Loads were applied at four concentrated points and failure loads were reported as the ultimate loads.10 Toprac and Cooke (1959) Nine castellated beams fabricated fiom 8B10 rolled sections were tested to destruction. web throat. A Vierendeel mechanism in the region of highest shear was the mode of failure of specimen G. as the load was hrther increased. The objectives of the investigation were to study the stniciural behavior in elastic and plastic ranges. tee section and compression flange yielding progressed in the shear span. yield at the top low moment hole corner and at web-post mid-depth was evident. Yielding and buckling of the compression flange in the pure bending region was the failure mode of Bearn E. As for specimen D which had a ciass 2 web tee stem section. and to determine an optimum expansion ratio for such bearns. however. . As the maximum load was reached. The iiltimate load of specimen B was recorded. B1 and B2. . followed by local buckling of the compression flange at the other end of the opening. The fàilure mode of this beam involved yielding of the web at the top low-moment corner of the opening in the shear span nearest the load application point. The first two tests were in the elastic range and the third was loaded to destruction. hole and web-post geometries were studied for each of em these tests. Yielding of the throat was also noticed. Beam A was provided with full bearing stiffeners under each load. but was omitted fiom M e r study because of the inadequacy of lateral bracing system. Bearn C was provided with shon bearing stiffeners. Test loads were reported as the ultimate loads obtained during the experiments. The flange to width ratio of beam A corresponded to a class 2 section. The flange had a Class 2 section properties.1 1 ~ltfilliscb. It failed through extensive yielding of the tee section and local compression flange buckling in the region of constant moment. (approximately half beam depth) below the load points. Cooke and Toprac (1957) The objective of the investigation was to study the structural behavior of castellated beams both in the elastic and plastic ranges.2.3. Three joists fabricated from 10B 1 1. loads were in the elastic range in order to venQ theoretical stress and deflection analyses. Bearn B consisted of three tests. B3. Varying expansion ratio. involved loading to destruction. The third test. b a depths.5 shapes with equal spans and simple supports and with varying positions of two symmetrical concentrated loads were used. and to study their strength and mode of faiIure. In the first two. 2.1 Redwood & Demirdiian ( 1998) TABLE 3.2.TABLE 3. .b Zaarour & Redwood (1 996) (continued) For ' refer to description of footnotes on page 5 1. .a Zaarour & Redwood ( 1996) TABLE 3. A. " .A.3 333.89 5.68 68 -3 Oh 338.65 101-60 354.5 1 ljo" sa 425.58 101-60 101.60 5.86 153.3." 101.9 303-59 425.84 6.35 425 -45 55.67 333.86 N.40 152.6 1 403 -35 d.75 302.84 tu 6.43 F. N.60 1 1 -60 0 bi 5 -83 5.86 6.86 4 il e 152.TABLE 3.58 302.2. N.86 1 53.3.30 176.c Zaarour & Redwood ( 1996) (continued) TABLE 3.b Galambos Husain & Speirs (1975) (continued) BEAiM H-3 P H-4 340. N.13 I l TABLE 3.A.45 59.40 335-45 39.a Galambos Husain & Speirs ( 1975) 253. 5.A.40 100.84 6.43 333 -43 333.84 6. TABLE 3.4.a Husain & Speirs ( 1973) TABLE 3.4.b Husain & Speirs (1 973) (continued) TABLE 3.4.c Husain & S ~ e i r ( 1973) (continuedl s TABLE 3.5.a Husain & S~eirs 197 1 1 ( TABLE 3.5.b Husain & S ~ e i r s 197 1 , (continued 1 ( ) . BEAM F- 1 F-2 38 1 .O0 38 1 .O0 d2 a - F-3 38 1 .O0 101.60 5 -33 6.83 50.55 254.00 347.65 60.00 248.2 1 1 b. a ; 10 1.60 L tfil 5.33 6.83 50.55 254.00 247.65 60.00 248.2 1 e h,," sa Oh F, 101.60 5.33 6.83 50.55 254.00 217.65 60.00 248.2 1 TABLE 3.6.a Bazile & Tesier ( 1968) TABLE 3.6.b B a d e & Texier ( 1968) (continued) TABLE 3.7.a Halleus ( 1967) Series 1 TABLE 3.7.b Halieux ( 1967) Series 1 (continued) .TABLE 3.7.8 Sherbourne ( 1 966) .c Hallei (1 967) Series 2 (continued) TABLE 3 7 6 Halleux ( 1967) Series 2 (continued) . TABLE 3. 9.9 1 d.4 1 296.4 1 309.4 1 N.63 385. N.56 45 2 9 6 . 354. 4.3 1 370.9 1 100.A.3 3 100.70 5.9.. S." a (Ph.72 3.20 264. Fyb F.10 eil h.b Toprac & Cooke ( 1 959) (continued) : TABLE 3.a Tovrac & Cooke (1959) TABLE 3.A.33 1 00.A K.5 1 45 296.20 295.35 1.4 1 296.TABLE 3.33 4.i\. il .A. tianaç ' 5.c Toprac gi Cooke ( 1959) (continued) G BEAM H 330.1 O 194.83 45 296.18 76.4 1 5. 41 296. t..33 4. .1 1 N. N.16 4 16..il 1 J 200. " 100.Y.16 38..33 b.70 5-13 38.A.. Angle in degrees.TABLE 3. in Mpa. ' Yield Stress F.10 Altfillisch. Cooke & Toprac (1957) . . A B C I b " Al1 dimensions are in mm. and a governing mode of failure is predicted.1 General The results of the previous research work on castellated beams described in Chapter 3 are compared in this chapter with the methods of anaiysis described in Chapter 2.1. more of these beams had to be removed fiom consideration. Detailed computations for each of the four predicted failure modes (Vierendeel and horizontal web-pst yield mechanisms. flexural mechanism and FEM buckiing analysis) are given for each beam in Appendix B. A summaxy of these results is given in Table 4.10.CHAPTER FOUR RECONCILIATION OF ANALYSIS W H TEST RESULTS T 1. 21 were eliminated fiom fiuther consideration because they failed by modes other than those k i n g considered in this project. The complete set of data for al1 78 bearns tested in the references of Chapter 3 are given in Tables 3. Because of the varying moment-ta-shear ratios at each hole in a beam. The remaining 57 beams are considered in this chapter.1 to 3. Correlations between test results to theory are then reported. and the most critical one . 1 2 Comparative Data . For the remainder the predicted and measured ultimate loads are compared. For reasons discussed below. al1 holes must be considered independently. Al1 shear and bending moment loads are non-dimensionalized by dividing by the plastic shear or moment capacity of the section to facilitate numerical cornparirom. Of these. 4. l ) . 4. m-DLIQn Figure 4. Construction of the interaction diagrams representing plastic failure mechanisms was first carried out.1 Test Arrangement of Beam Fi (Toprac & Cooke 1959) .for each failure mode must be identified. For the given beam arrangement shown below (Fig.2. such a diagram is demonstrated in Fig. The M N ratio at the centerline of each opening is used.2 Interaction Diapram Demonstrating Theoretical Methods of Analyses. theoretical predictions of V N . For each opening. The first of these is based on .. and the buckling load predicted by FEM..N.. are obtained from the intersections of the radial lines with the interaction diagrarn representing Vierendeel and flexural plastic mec hanisms. with the two holes under pure bending being represented on the vertical axis as holes 7 and 8.P " hob 1 Figure 4. On the diagram are also plotted the predicted îàilure loads corresponding to mid-post yielding (V.. and M/M. The radial lines represent the M N ratios for each of the openings in one-half of the span. Eqn 2. These FEM results are plotted on the interaction diagram as two points with ordinates representing the moments at the two holes used in the model. As load is applied to the beam. and initially the simplest sotution was sought. and was subject to the restraints and other details outlined in Section 2. Based on the typical FEM model arrangements of Section 2. This has a constant value for al1 webposts. The influence of plasticity is considered in Chapter 5. this assumption is examined in detail in Chapter 5. 2. 4. 4.2. a two-hole model with 8 16 elements.5. Elastic FEM results are given. and plots on Fig. although it is recognized that this buckling usually involves inelastic action. The triangies represent the loading (V and M) at each hole for a given load on the beam (values given in fact correspond to the failure load). Thus it is implicitly assumed that moment has negligible effect.3 Cornparisons Al1 modes of failure for each hole in a beam are identifiable on a diagram such as Fig. Only vertical loads were used and the model is subjected to constant shear force with some small bending forces which were considered to be negligible insofar as they wodd affect the buckling load (see Redwood and Demirdjian 1998). and is neglected at this stage as good results with elastic analysis have been reported by Redwood and Demirdjian (1 W8). was chosen to sirnulate the behavior of a web-post under high shearing force. 4.5. as shown in Fig.1 with the shear yield stress taken as f3F443.2 as a vertical Iine (WO lines corresponding to two values of P are shown).5. This represents a half-span of a beam with four holes. . and are far below the elastic buckiing load. 2 and 3 would have been criticai (with both predicted failure loads lower than observed). and so some uncertainty .6 ksi (287 MPa). . 4. The critical hole is the one for which the plotted point first reaches the failure envelope.') Husain and Speirs ( 1971 ) directly measured the shear yield stress of notched specimens fabricated from ASTM A36 steel (nominal F. If these results (i. If the horizontai yield mode had been considered holes 1. F There is some evidence that the effective shear stress at rnid-depth of the post at failure is very high compared with the expected value F//. The results shown in Fig. 2 and 3 are loaded well below the Vierendeel mechanism load. in effect the verticai line should be shifted to the right to reflect a higher shear yield stress than 1. The observed failure mode was that of pure bending. and holes 5 and 4 in the same shear span are farther fiom the failure surface. Holes 1. It seems clear that in this case. The tensile yield stress was not reported. is known to be quite conservative. the vertical dashed lines) are ignored it can be seen that a flexural mechanism failure is predicted at holes 7 and 8: hole 6 is almost at the point of failure in a Vierendeel mechanism mode. ~ S443. This may alternatively be interpreted as identiQing the failure hole as that one for which the ratio of test load to predicted load is a maximum.e.2 are affected by the analysis for the horizontal web-post shear yield mode which. as predicted by the above reasoning. the horizontal yield mode was not relevant.these points c m be considered as expanding proportionally outward from the origin.=36 ksi (248 MPa)) and for a number of specimens the average value was 41. as discussed in Chapter 2. and the mode would be identified by the part of the envelope attained. Certain tests had reported maximum test loads. while others derived their failure loads fiom the intersection of tangents of the two curves of load vs.6+(53/'13)) times that expected value of FJ J ~ Greater enhancement would occur if the estimate of the tensile . In the example of Fig. and nominal values have been used.Most of the cases with poor correlation. as indicated in Table 4. These beams are identified by asterisks. test-to-predicted Ioad ratios were computed for each test beam. Following the above procedure. However. and are noted in the literature review of Section 3. Excluding the identified beams for which F.exists as to the enhancement above F443 that this represents. the numencd results indicate good correlation with test results.225.127 and 0. see Redwood and McCutcheon (1969)) then the measured shear yield is 1.2. the mean and the coefficient of variation (COV) of the test-to-predicted ratios for al1 other beams are 1. is not known. . both reported loads are used for comparisons.35. it has been asswned throughout that the effective shear yield stress at the mid-depth of the posts is 1. 4. Whenever applicable. if it is assumed that the A36 web material had a real t e n d e yield stress of about 53 ksi (365 MPa) (such hi& values have been measured for A36 steels in the 1960-70 period.1. deflection diagrarn.35 (=41.4 Discussion In generai. 1.2.35 times ~ 4 d 3Thus the factor P is taken as . it appears that even this enhancement is insufficient to reflect the effective shear in the test beam. 4. . On this basis. are those for which yield stress values were not given. yield was too high. 1 1.These are based on the ultimate loads. Redwood & Demirdjian (1998) 10-5a 10-Sb 10-6 Web Buckling Web Buckiing Flange and Tee Buckling Web Buckling Web Buckling Web Buckling Zaarour & 8.105 Redwood (1996) 8-2 8-3 8-4 10. in others.1 Summary of Test and Theoretical Predictions Reference Beam Tedtheory Ul timate Loads 1 .967 0.O86 and 0.646 0. there were only small differences between the failure load for the predicted mode and that of an alternative mode.132 Testhheory Tangential Loads Mode of failure Test Theory - .847 Shear Mechanism Web Buckling Shear iMechanism Web Buckling Web Buckling Web Buckling Shear -Mechanisrn Shear Mechanisrn Shear iMechanism Web Buckling Shear ~Mechanism Web Buckling .O43 1.1 10-2 0. in others modes are identified as flange buckling when a yield mechanism may have been imminent or already developing ( S ) . if the tangential Ioad is used where available. some test modes were not defined (4).793 0. the uncertainty concerning the shear capacity of the web-post affects the prediction. Of the 57 beams listed. Table 4. these numbers becorne 1. approximately half (29) had the mode of failure predicted correctly.137 1.195.9 15 0. Of the others. and for most of the remaining cases. 953 Buck1ing Shear Mechanism Shear .960* 1. N.A.137 G.Re ference Beam ' Testltheory Ultimate Loads 10-3 10-4 Test'theory Tangentiai Loads Mode of failure Test TheonWeb Buckling Web Buckling Web BuckIing LVeb 4 0.146 1.8 13 Cb-sb Buckling L* b !e 12-1 0.O5 1 1. N.314 1.A.136 1.950 0.A.Meclianisrn U'c b - 12-2 O -966 Buckling 13-3 0. . Husain & Speirs (1975) H-2 \Vsb Buckling Web Buckling N.158 A-1 A-2 1.208 1.% 1 .196 1.O46 Husain & Speirs (1971) E-1 E-2 Mid-Pest Yielding Mid-Post Yielding Sliear Mschanism Mid-Post Yielding Mid-Post Y iclding .840 1.81 l* 1.Mecliariisrn Sh e u Mechanism B-3 1.186 N.001 12-4 Calam bos.1 G-2 G-3 1.O62 1.990 1 .O87 1 .259 Shear Mechanism Shear Mechanism Sheor Mechanisrn Shear Mechanisni Shear Mec hanism Shear Mechanism Mid-Post Y ielding Mid-Post Yielding Buckling Sliear blechanism Sliear .857 0.Meclianism Shsar kleciianisrn Shear Mechanism Shear iLlechanisrn Shear . Buckling L\'e b H-3 H-3 P H-4 Husain & Speirs ( 1973) 1 .173 0. 854** 2.397* 2.Mecfianism Sfierir ~Mechanisrn Sliear blechanism Sliear Meclianisin SIierir LMechrinism Shear iblechrinism .090* * 1.l E-2 1. .576** 1.Reference Beam Testltheory Ultimate Loads 1.1 F-2 F-3 1.181** 2. ~Mid-Post Yielding [Mid-Post Yielding Mid-Post Irit.OOO* * 2.504** Web Buckling We b Buckling We b Bucklitig Slierir blechanism Shear Mechanism Flesural Mechanism FlesuraI blschanism - -.630 Sherir h4echanisni Shear Meciianism Shear Mechanism Flesural Meclianism hl id-Post Y ielding Mid-Post Y ielding SI i a r .1 16 0.3 14 -- B E Halleux (1967) Series 1 1 .530* Sliear blcchanism Sliear Mechanisin Shear Mechanism Flesural h~lèclianism Sliear Mechrinism Slierir klcchanism Shear iMechanism Shear iMechanism - Bazile & Texier ( 1968) A - 1. Series 2 1 2.226 12 8 4 1.lditig E-3 F.942 2.809* 1.135* TesU'theory Tangent ial Loads Mode O!' fai 1ure T Test Mid-Post Y ielding Mid-Post Y ielding $1 id-Post Y ielding Flesural Mec tianisii~ Theor?.058** 1..82 i ** IB 3 5 3 .503 3 3B 5 Sherbourne (1965) E.- 1. 808 H 1 Altfillisch.T.1 13 Flexural Mechanism Flexwal Mechanism Shear Mecha~sm Shear Mechanism Mid-Post Yielding Shear Mechanisrn Flexural Mechanisrn Shear Mec hanism (L. ** Actual yield stress values of these beams were not reported.O63 Testltheory Tangential Loads 1.B?) Toprac & Cooke D 0.1 Shear Mechanism Shear hlechanism Fiexural Mechanism Flexural .700 1.442 1 . Minimum yield stress value of 235 MPa (24kg/rnm2)was used to compute these ratios.1 13 1.122 C *Minimum yield stress values of the conesponding beams were defined.887 1.423 1.277 1.613 1.O63 Mode of failure Test Theory Web Buckling Web Buckling Flexural Mechanism Flexurai Mechanism Flexural Mechanism E4 L. Toprac & Cooke (1957) A 0. .218 1.425 1.O43 1. The nominal yield stress of 248 MPa (36ksi) was used to compute these ratios.O43 L-3 1.Mechankm L-2 1.956 (1959) Flange Buckling Flange Buckling Shear Mechanism Flange Buckiing Shear Mechanism Flange Buckling Flange Buckling E G 1.Reference Beam E-3 Testhheory Ultimate Loads 1. and the impact of inelasticit). in conjunction uith the plastic analyses is examined in section 5.a~-ing from pure shear to pure bsnding are considered.3 modeIs containing up to four openings are considered under pure shear as tvell as pure bending.7 a parameter study deriving web buckling coefiicients c o ~ e r i n g wide a range of geometries is performed. These results are used to establish a generai forrn of interaction diagram to define elastic buckling loads of castellated beams under any shear to moment ratio.1 General In the FEM analyses considered so far. the analysis has dealt only with elastic buckling behavior. . the only loading condition treated approximates pure shear. Having established this form. is esamined.8 with the aim of developing inelastic buckling equations. The effect of moment-CO-shear ratio is then considered for four test beams representative of a wide range of castellated beam geometries. In section 5.CHAPTER FIVE GENERALIZED ANALYSIS AND DESIGN CONSIDERATIONS 5. and furthemore the mode1 has been limited to one cornpx-ising only two openings. in sections 2. These are then compared with relevant test results. In this chapter more cornplete models are examined and moment-to-shear ratios \. In addition. The use of these elastic results.2. the loading used to create any moment-to-shear ratio is described and in section 5.3-5. I) used in the analyses described in section 2. 5.5. 5.3. the region above the middle opening resisting the compression force is buckled. 5. Under pure bending conditions however.g. Similarly. a clockwise couple applied by such horizontal forces on the lefi end of the beam was used to simulate pure moment conditions. In order to produce pure shear force conditions at any point within the length of the model. To create pure shear and pure bending forces. with the vertical loads removed. with large twisting of the flange to accomodate the buckled shape. This couple was created by applying equal and opposite horizontal forces at the top and bonom web-to-flange intersection points at the left hand end of the model.2 Loading on General Models To study the behavior of models under various shear to moment ratios. different loading patterns had to be irnposed on the finite element model descnbed in Chapter 2. as shown in Fig. . the hole centerlines).2. as shown in Fig. the two vertically concentrated static loads (Fig.1 and 5. The deformed shapes under vertical loads and under pure shear conditions as shown in Figs.4 must be supplemented with forces producing a counter-clockwise couple. Any combination of shear and moment forces could be generated by cornbining these verticai and horizontal loads in m y desired proportion. demonstrate the same buckling pattern of the post.2.5. In the several models considered below these forces could be adjusted to provide pure shear at any desired point (e. several MSCMASTRAN elastic fmite element buckling anaiysis nins were necessary. with slight rwisting of the flange to accomodate the double curvanue bending efiect over the hieght of the post. as well as various V M ratios. Fig.1 Two Hole FEM Mode1 under Vertical loads onl?. 5. . Fig. 5.2 Three Hole FEM Mode1 Under Pure Shear Forces . 3 Three Hole FEM Model Under pure Bending Moments . 5.Fig. 3. Similar numerical simulations were conducted to investigate the behavior of 2.1. three and four hole models. To ensure there is no such restmint. Summary of Results Under Pure Moment Forces. Both pure bending and pure shear forces were considered for two. al1 under the same boundary and loading conditions.5. These four beams were found to have the diverse properties representing a wide range of castellated beam geometries.1.3 Elastic Buckling Interaction Diagram Due to the presence of the stiffener on the left end and the applied constraints on the right of the model.4). beam B-1 fiom Altifillisch Toprac and Cook (1957). These analyses were canied out for four of the test beams described in the literature. The chailenge here was to determine at which hole zero moment forces should be edorced to produce the pure shear condition. Resuits for pure bending are expressed as the beam buckling moment as a ratio of the plastic moment and are given in Table 5. it was thought that the stiffened web posts adjacent to these ends mi@ provide restraint to the rotations of the imer web-post of the two hole model. . The three and four hole models produce similar buckling moments and these were lower than for the two hole rnodel. beam F-3 Husain and Speirs (1971)and b a r n 10-3 from Zaarour and Redwood (1996). 1 Beams 1 2 Hole Mode1 1 3 Hole Mode1 1 4 Hole Mode1 I 1 I I Table 5. models consisting of three and four holes were also investigated. These were beam G2 from Husain and Speirs (1973). 5. and 4 hole models under pure shear conditions (Fig. with no trend discernible between the rnodels with different numbers of holes.2.25 3 Hole at hole 1 Vcr (kW 39. For the four hoie model only the two interior holes had imposed the zero moment conditions and again only minor differences are evident.at hole 2 Vcr Vcr (W 39.5 8 (W 38. The differences in the critical buckling shear loads of 2.09 83.84 8 1.36 Table 5.25 OcN) 39.00 83 .2. In view of these results and to be consistent in subsequent analyses. and 4 hole rnodeIs were less than 3%.99 82.40 Mode1 at hole 2 Vcr (kW 40. Al1 the holes of the two and three hole models were tested. Surnmary of Results Under Pure Shear Forces. several analyses were done to create the zero moment force condition at different holes. 3.42 (W 38.62 82. and only minor difierences in the results were obtained. . It sbould be noted that under pure shear loading the different models produced only marginally different results and the two hole mode1 utilized for the analysis of Chapter 4 was thus conftnned to be satisfactory for that application. M =O Beam 10-3 B-1 2 Hole Mode1 at hole 1 .As indicated in Table 5.46 at hole 3 Vcr 4 Hole at hole 2 Vcr Mode1 at hole 3 Vcr (W 39.09 82. the three hole model was chosen to represent al1 further FEM analyses in this study.66 81. Fig. 5.4 Three and Four Hole FEM Models. A complete interaction diagram for elastic buckling was obtained for each of the four selected beams using the three hole model. The results are shown in Figures 5.5. 5.6. 3.7 and 5.8 (the two ordinates of the elastic FEM results plotted for each V N p ratio refer to the M N ratio for the first two holes of the model). It c m be seen that under pure bending, plastic failure occurs at much lower loads than the buckling loads. Under pure shear, buckling loads may range from much lower to much higher vatues than the plastic failure load. The resuits shown on these diagrams will be discussed below. Interaction Diagmm Beam 103 Figure 5.5 Zaarour and Redwood (1 996) Interaction Diagram Beam G-2 - , , Yield , Theory BasUCFRd n =2 Figure 5.6 Husain and Speirs ( 1973) Interaction Diagnm Beam F 3 ,,Y RH Theory ,, Bastic FEM n = 2 Figure 5.7 Husain and Speirs ( 197 1 ) Interaction Oiagnm Beam B Figure 5. The curve found to best represent the FEA results and for the fuIl range of M N was fouid to correspond to n = 2. V.8 Altifillisch. given -Mc. Cook and Toprac ( 1 957) The interaction buckIing relationships c m be approximated by a curve defined by: with Mo and V. corresponding to pure shear and bending conditions respectively. In this way.. a relationship defining the buckling behavior under any M N ratio is established. values. . Several dif'ferent values o f n were exarnined. but conservative estimates of flange dimensions were used for the general case. The flange was included in the mode! because of its importance in restraining web rotations.. The relevant parameters were considered to be the ratio of hole height to minimum web-post width. 5. The beams were designed and selected to present various ratios and proportions of castellated bearn geometries.1 Previous Parameter Study In a previous study by Redwood and Demirdjian (1998). These assumed that the flange was only as thick as the web. The study assumed elastic behavior throughout. and thus the following computations are restricted to castellations witli a hole edge siope of 60° to the horizontal.ide range of bearn characteristics.5. eh. Elastic finite element analysis was performed on 27 beams to derive elastic web buckling coeftkients under pure shear and pure bending conditions. and the f'lange width was that of a Canadian Standard S 16. the pararneter study had to be of limited scope. a pararneter study to tind the elastic buckling loads under high shear loading was carried out that incorporated a \\. a pararneter study relating the behavior of beams with different geometries under pure shear and pure bending conditions was carried out. Because of the wide range of possible beam and castellated hole geornetries. Thus: .. and the ratio of minimum web-post width to web thickness. hJe.1-94 class 3 section.1 Parameter Study Having established a generai expression defining the elastic buckling beha~iorof castellated beams under any M N ratio. The midpost weld was assumed to be Full penetration and had the sarns thickness and material properties as the web. hJe. For each senes. each with a constant hole height-to-beam depth ratio. the relevant parameters were selected to be the hole height to minimum web-post width. The castellations had hole edge slopes of 60" to the horizontal. Ln the study the critical horizontal web-post shear force dong the welded joint was found using FEM. with tu-O vertical loads applied at one end at the level of the flanges. e/t. The FEM model consisted of two holes and was identical to that used for the analyses described above in Chapter 4. lncorporating the principal parameters by writing horizontal shear force at buckling as a non dimensional shear buckling coefficient k was derived as X: = The tlnite elernent analysis gave the ratio of shearing force in the web post to the vertical .V' Two series of beams were considered. without intermediate plates at mid-height. with the model supported venically by a point load at the other end. and the ratio of minimum web-post tvidth to web thickness. This angle is representative of present industry standard cutting angles. and then the corresponding vertical shearing force on the beam was found. Thus loading was primarily a sliear load. 983 with coefficient of variation of 0. -= " S dx-2~$ . dalues of k obtained from the parameter study are shown in Figure 5.2. defines the line of action of the longitudinal force resultant acting in the the tee section...shear on the beam.9.... (Eqn 2. suggesting that the centroid provided a close approximation.02 was found. where y. V. The product of the ratio V. which was taken as being at the crntroid. This was verified by comparing this value with that given by the FEM for the 27 beams used in the parameter study. in the web-post at the welded joint. An average ratio of 0. k curves Figure 5.9 . was then related to vertical shear V through 4. Shear Buckling Coefficient Redwood & Demirdjian (1998) The vertical shear that will cause web post buckling can therefore be obtained by -reading the value of k fiom Figure 5. VJV.N and the vertical shear force to cause buckling gave the criticai horizontal shear force V.9.3) derived from the free body diagram of Figure 2. 2 to find the horizontal shear in the web post -using equation 2 -3 to uansform V. the buckling coefficient for a wide range of beam geometries c m be detemined.e. which was found to be slightly unconservative for some compact sections. \vas taken as 350 MPa. 6. two vertical forces were applied on the nglit end at the level of the flanges and two horizontal counter clockwise coupling forces were applied on the . and to make the flange modelling slightly more conservative. For the current research the selected mode1 consisted of three holes as FEM results revealed its better performance under bending moment: although no irnprovement was noted for pure shear. Canadian Standard S 16.-using equation 5. Under pure shear conditions. to vertical shear V These curves cover a wide range of castellated beam geometries with 60" openings.. The flange dimensions assumed were modified so that the width was based on the assumption of Class 1 section. 5. Narrower flange widths would make al1 cases consenfative as compared to class 3 section. = ri 2(l45). i.6. making the tlange restraint slightly more conservative. E ..5 Shear Buckling Coeff~cients (k. A new study to incorporate pure bending is descnbed in section 5. This reduced the flange widths. consistency between models for the two load cases was considered desirable. Through linear interpolation between the two series of curves.1-94) ~vhere F.) The previous parameter study was refined in the current research to correspond to buckling under pure shear. (Clause 1 1 2. From the FEM studies of different models in Section 5. 5.S. Modified Pure Shear buckling coefficent Curves . There were minor variations between the results of the new and the previous studp due to the minor modelling changes.10 shows the results of the analyses for web-buckling coefficient k. and this together with the flange modelling change explains the differences between the results shown in Figs. than evident in the previous study.2). Thus there were no bending moments at the centre o f the span (Fig.3 differences up to 3% can be expected between the hvo and three hole models. there are minor differences between the shape of the c u n e s for h.lefi end at the flange to web intersecting nodes to counter the overturning etlèct of the vcrtically applied forces. Furthemore. = 15 is plotted on the curves to demonstrate the slightly greater dependency of e/4./d.5. For beams with hJdZ = 0. due to pure shear.10.10. k" curves Figure 5. Fig. =O. 5.9 and 5. el\. 5. While camparing the two . indicating that the flexural buckling load is not . The k values Vary less than the k curves.P This tlexural buckling coefficient. sensitive to the ratio of hole height to minimum width (hJe). indicating that hole height to minimum width (h.where S is the section rnodulus of the unperforated section. .r Taking a . and . S assuming that the area of the web resisting the compression force is jt.5. is given in Fig.1 1 for a given variety of castellated beam geometries. = .74 unti1 the lines curve downward. M./e) ratio has very little effect on the overall beam buckling behavior under pure bending forces. the same series of beams under the sarne conditions were subjected to two horizontal clockwise coupling forces. 5.7 Flexural Buckling Coefficients (kb) To derive an expression for web buckling due to pure bending moment forces. a coefficient k is defined by S impli @ing by incorporating [2(1 kx ' ')] into kb . Almost constant kb values are maintained in the hJdo = 0. kb. the critical moment to cause elastic buckling is simply caiculated using equation (5.11. I Figure 5. . where beams with lower tee sections with hJd. larger buckling coefficients under pure moment conditions were found for the senes of bearns Mth larger tee sections hJd.series of bearns.7 Effect of Inelasticity on Ultimate Strengîb Since buckling usually involves inelastic action.=0.=û.5). BuckIing Coefficient Curves Under Pure Bending Forces 5. the influence of plasticity is considered in this section to improve the already mentioned methods of analysis and derive general expressions incorporating both elastic and inelastic buckling actions. Thus based on a given beam geometry. but the behavior was reversed under pure shear conditions. .74 had higher k coefficients.5. 3. For each test beam.) and moment (M. computed from the k and kb curves. the plastic mechanism and elastic buckling shears. (see Fig. and V. such a diagram can be plotted on the sarne a.12 Elastic and Plastic Interaction Diagrams On this diagram.12) Interaction Diagnm 6eam 6-2 Huaain & Speim (t973) Figure 5.The construction of interaction diagrarns for elastic buckling can now be performed for any beam with 60" openings.. V. the two governing shear values were thus obtained. and follows the procedure used for the four bearns as discussed in Section 5.. 5.) can be . Elastic buckling values of shear (V.xes as the yield mechanism interaction diagram.. radial lines fiom the origin for each hole of the test bearns were then drawn and from each line a plastic and elastic buckling shear capacity is obtained at the intersection points. By dividing the resuits by the plastic shear and moment capacities of the section. . it was convenient to divide Vu by V . . fiom colvmn strength equations (CS& 1994) C .VI.as indicated in Fig. = 1.7. Based on the results of 17 test beams with 60' holes and relevant failure modes.13. where h. we c m w ï t e Alternarively.15 M. y.. is replaceci by V. following expression is proposed. = A F . The equations were then plotted and compared against acnial test results for the 60" casteilated beams (summary of results is given in Appendix C). the . = y. both equations 5. If M.6. (1 + k2")-. 1. the following two cases were considered: From equations for inelastic lateral buckling of beams (Clause 13.6 and 5.. CSA 1994).To obtain an estimate of the ultimate shear load of the test beams which incorporates the possible interaction of elastic buckling and yielding failure modes.):O ( . were found to provide similar . 5. + A" ' 1. is now interpreted as and n is a coefficient based on fitting to test results. To plot the results in a non dimensional form wtiile maintaining consistency. c . with n taken as 4. . and My by V ..0 in the latter. As s h o w in Fig.O96 and COV of 0. but it may be noted that the actual beam cross-section dimensions were not given. produced a mean of 1.6 and 5.148 For these 17 beams the simplified approach taken in Chapter 4.137 0.13. The increased mean value for the two equations is expected.predictions of the test results. in which the predicted strength was taken as the lower of the yield strength and the elastic buckiing (FEM) strength. The reason for this is not clear. The following statistics apply to the two predictor equations TesVPredicted Eqa.3 Statistical Results COV 0.166 Table 5.7 are used. the elastic buckling strengths were computed using the generalized buckling interaction equation 5.6 Eqn.5 reported by Sherbourne (1 968) show significant overstrength compared with the predictions. since both will predict a lower value than the lowest of the yield and elastic buckling strengths.O and 5. whereas the computations in Chapter 4 were based on exact rnodeiing of each bearn. 5. 5. The lower COVs represent an improvement in the prediction if equations 5 . the four bearns with of about 0.1. and nominal values have been used in the calculations.7 Mean 1. 5.170. It should also be noted that for use in equations 5. .7.1 13 1. 13.VuNpl. Cornparison of Test Results With Proposed Expressions . MuIMpl Vs Lambda 1 Lambda - Figure 5. These results established elastic buckling interaction diagrams. yielding at mid-depth o f web-posts and eiastic buckling anafyses u-ere correlated with the results of a number of physical tests of castellated beams reported in the Iiterature. Several theoretical rnethods predicting fom~ation of' plastic mechanisms. to modify the theoretical nlodeIs used initially. For any given M N ratio.Results obtained from the interaction diagrams based on plastic anaiysis used to predict 84 . The following remarks on the behavior of castellated beams are based on the several theoretical models used incorporating both elastic and plastic analyses. Since web buckling usually involved inelastic action. results obtained from elastic and plastic interaction diagrams were established.0 Conclusion The objective o f this research prograrn was to study the failure of castellated b e a n ~ s with particular emphasis on web-buckling.CHAPTER SIX CONCLUSION 6. and their cornparisons with physical test results.4 parameter study for a wide range of castellated beam geornetries rias performed to deri\*eelristic web buckling coefficients under pure shear and pure bending forces. . the effect of plasticity was considered in conjunction with elastic FEM results. . A factor of P = 1.shear or flexural mechanisms were found to give generaily satisfactory predictions. loads.Given the eiastic criticai buckling loads under pure shear and pure bending ( . b) V. it was considered necessary to take into account the effect of plasticity on the buckling loads. and this led to more realistic resuits. it does not account for yielding of the web-pst. This diagram is designed based on the properties of a given beam. This study in conjunction with the elastic . . a curve of shape (MMO)" (VN. However. or web-buckiing. was found to be conservative. . and therefore was used to perfonn various parameter studies.A parameter study was performed to denve the buckling coefficients under pure shear and pure bending conditions covering a wide range of castellated beaxn geometries. Much higher failure loads were then obtained compared with those given by the initiai stress limit equation. - Elastic buckling analysis with FEM models could be correlated with experimental results.Y ield stress developed at the minimum horizontal width of the mid-post. However. the following steps were taken: . equation 2.3. To do this.35 was applied to the sheaf yield stress to account for the sttain hardening eEect expected to be developed at this section.)" 1 with n=2 + = was fitted to define the buckling loads under any VA4 ratio. t 90 to O.The design considerations and computations incorporating the effect of elasticity and plasticity on the buckling loads is limited to 60Ucastellated beam geometries.137. . gave the elastic buckling loads of a variety of castellated beams under any M / V ratio. with coefficient of variations fiom O. . Extension to other beam geometries is desirable.FEM buckling curves. .Expressions incorporating both elastic and inelastic behavior of web buckling gave better approximations of the buckling loads. 47. and Miller. Raymond. D. 1971.A. Altifillisch. Bazile. Welding Research Supplement. 57.G. and Speirs.G.. No. M. J. Beam tests with unreinforced web openings. and Lee. D. Megharief... Failure of castellated beams due to rupture of welded joints. Failwe modes for castellated beams. Husain. Eng. V68. Halleu. Universal Off'set Limited.. California. 1994. Instn. R. 1967. Vol. Part 2.R. An investigation of open web expanded beams. Department of Civil Engineering and Applied Mechanics. American Welding Society. Civ. Handbook of steel construction.REFERENCES Aglan. Engineering Optimization.. Canada Galambos. Vol.U.. M. 1957. Paris. Journal of Constructional Steel Research. J.K. U. 1975. Engrs. Series No. Vo1.. M.. ASCE. The Macneal-Schwendler Corporation. W. 1-1 7. pp 295-3 15. Ontario.3. MSC/NASTRAN: Quick reference guide.. 1995.. 1973. California. Kerdal. 2nd edition. Vo1. M. Los Angeles. Cooke. R. 1994. Husain. Proc. . J.U.P.. Redwood. 1997.G.O. J. 1984. A. B. Experiments on castellated steel beams. Great Britain.U. A. Constr. P. Essais de poutres ajourées (Tests on castellated beams). M. A. 325. 1. Markham. Los Angeles. J. London. and Speirs. 1. No. Acier-Stahl-Steel. Limit anaiysis of castellated steel beams. 4. and McCutcheon. The Macneal-Schwendler Corporation.1968. MSCMASTRAN: Linear static analysis user's guide.. 5. pp 307-320. Acier-Stahl-Steel. Journal of the Structural Division.M. Optimum expansion ratio of castellated steel beams.R. 1969. and S p i n W. pp 21322s. USA. 1974. pp 329s-3423. Métallique.G. W. M. and Redwood.ST1.D. Thesis. 133144.D. M. Caffiey. pp 77s-88s. Behavior of composite castellated beams. and Nethercot. Web buckling in castellated beams. Husain.. France. Vol. USA Canadian Institute of Steel Construction.94. and Texier.A. 52:8. and Toprac. McGill University. Welding Research Council Bulletin.A. pp 12-25. London. A. V68.G. and Demirdjian S. Pittsburgh. New York. Pa. Redwood R. The Steel Cofistniction Institute. C2.Redwood. . and Cho. The plastic behavior of castellated bearns. Sherboume.G. Journal of Stmc?ural Engineering. Castellated beam web buckling in Shear. J. W. An experimental investigation of open-web beams. American Society of Civil Engineers.Eng. R. London. Of Welding.23-42.G. Journal of Constructionai Steel Research. 1995..N. Thesis.. 1998.H. 124(8): 12021207. 1990.S. McGill University. - Ward. 1993.47. Zaarour. A.K. pp 1-5.. Redwood. ASCE Annuai Meetings and National Meeting on Structural Engineering. and Cooke.R. Web buckling in thin webbed castellated beams. pp 1 10.. Toprac. Inst. 1966.J. 1959. Ultimate strength design of beams with multiple openings. S. 757. Welding Research Council Bulletin.G. Design of steel composite beams with web openings. Department of Civil Engineering and Applied Mechanics. No. Preprint No. Series No. R.A.A. U. 25: 1&2. Proc. A.1968. 2"" Commonwealth Welding Conference. B. Design of composite and non-composite cellular beams. M. .APPENDIX A Finite Element Input File This Appendis contains a sample input file to construct the 2 hole Finitr: Element mode1 and perforrn Elastic Buckling Analysis. ...769.233. EXECUTIVE CONTROL SECTION S !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! S s Elastic Buckling analysis of "Castellated Beam" s 2 Hole !Mode1 of reference Beam 10-3 (Zaarour and Redivood ( 1996)) S S S SOL 105 Tl M E=9OO CEND S S ..683.0..5 14....803 S S ECHO = NONE FORCE = I SPC = I O SPCFORCE = ALL STRESS(PL0T) = ALL DISPLACEMENT(PL0T) = ALL S S SUBCASE I SPC = 10 LOAD = I O DlSP = ALL FORCE = ALL S SUBCASE 2 SPC = I O METHOD = 100 FORCE = ALL DlSP = ALL S S BEGIN BULK PARA M... 3 73..428...83..16 1.. IOO..799.93.....509. 10000............238..499. 774..238.78.......S1NV..430.+EIGB +EIGB..0 PARAM............0 PARAM.674...73...164.....162.....MAX S S THIS SECTION CONTAINS BULK DATA FOR SE O ..43 1...163... C A S E C O N T R O L S E C T I O N S .POST...3.. S TITLE = beam 10-3 S SET 1 = 1..-5......253...........5....YES EIGB..0..88..504..427..333.378..........$-:Ar S ...3....669...679........AUTOSPC.......165.258. S ........ B:.........664....429.5 19.348.....KfiROT. 0 O 368.1.56 12.0 S S SPC 10 17 235 SPC 10 609 13 SPC 10 617 13 9 3 SPC I O SPC IO 10 3 SPC I O 115 3 SPC 10 116 3 SPC 10 117 3 .0.956 0.3 1 O s************************************ CQUADJ 816 Z S 8 16 elements are thus defined through grid points 867 617 1033 1057 S S THIS SECTION CONTAINS THE LOADS..6068.200000 0.1000 FORCE.2 PSHELL.0 0..0 O 28. 069.9.3 0.. A N D CONTROL BULK DATA SENTRIES S s S MATI.....00.34 0. 5 0 .0.3.1 PSHELL.333 333 S S PSHELL.4. 10..0.6 17.. GRlD IO57 O 705.S S GRID 1 GR!D 2 GRID 3 GRlD 4 O O O O 0.0 0.5000.333333 MAT I . 1 .0 0.185.. . 1.2.. s*************************************** S .99 -35.0 O S s*************************************** s The coordinates for 1057 grid points are defined.333333 M A T 1 .3. 0 .O. s S FORCE. CONSTRAINTS.2700000.0 0.-1.0 0.3 . . 1. .0.2..0 O 339.0.525. 0 0 ..2OOOOO.445.3. . SPC SPC SPC SPC SPC SPC IO IO 10 10 IO SPC SPC SPC 10 10 IO 10 IO 10 IO 10 1 0 10 1 0 SPC SPC SPC SPC SPC SPC sec SPC SPC SPC SPC SPC SPC SPC SPC IO IO IO IO 10 10 10 10 10 IO 10 SPC SPC SPC S ENDDATA APPENDIX B Detai led Test-to-Theory Results This Appendix contains detailed Test-to-Theory computations for al1 the beams Iisted in Table 4.1. For each test beam. each hole until mid-span is studied. Al1 results are transformed to shear and moment forces, and are non-dimensional. Reported ultimate test and (M,,,/M,), elastic FEM buckling (VJV,). Shear mechanism (Vl,N,). load (V ,, N p ) yielding of the horizontal joint (VyhNJV,), flexural mechanism (M,,,/M,) and ratios are al1 calculated. Ratios of test results to the predicted failure modes are then computed. aiid maximum ratio on each row is calculated. The predicted failure mode is derived based on the ratio selected by the maximum of ail the ratios of Test-to-Theory on each row. s i 3; , 3E: E,i : * , 1 ' , 81i 8 I l ! . :oip , kib! ! si SI 1 : El :.ai"/ % 1 , ' Wi ? !W ! ~ h O a. Pi m'cv F i :o. ; di 2: 00 O . k k ?, hl'-: Q D I P . i >-: El I , ,g ~ j m m q q I 1 -1 ; '0,OI . . $ E! si , ! Cr) . 9 '9 1 Cr)I i ' (O O O ' > El c3-h! 0 "!'" o1 O . r ~ " > - 0 0 0 0 0 - CD'* a (O t )I Q a 5 0) 0'0 ' 1 0 0 (9:q4<4'Cc1 g a > - O 0 % % 8 %0 .r .o O r : ?i q * C V I ' O IqO il 0 1 1 * r" s' 2 QD ! 9 1 l b O. m b V) (DJb l h *'?? a1 raO hl! e 0 0 0 0 F . ~ 0 a 0.b r a)aDaqaoQ 0 0 0 0 0 0 0 0 0 0 N <D (O b v O O ~ c q c o b m q ( V Q o Q .o' - - 0 cwm F .5.o O F . = I si 21 $! .a0 o o l m a01 "'v)!? rr.fliq:U?i o \ o l0 : o C ~ .-!FJ"sF/ * 0 : Oi ..' v n SI P' . h l l mC U I .r . O1OiO'Ol 5 . 0 0 0 0 P -5: aQS&CZ :ha O ~ o O cumU-)S O h l .V ) U 3 r n O p ~ ' -9 q O 00. X. Q ~ . g si' r .- C I 5 w b a l m ' l n in. .* W O C V orna- 3' a. a D m ( D 0 0 0 0 0 . * ml*0 * Of0 O aX CnQiCnaQ) 0 0 0 0 0 * 0 1 1 r-r-oco C? Y . $ Ei aE I a--v . ? - 5 0 0 0 0 0 L " I c > ~ I ~ ? ~ Q a u D h l 0 0 0 m e C D wm(Oaoa 0 h .F - - 9' > O . O ! o '3"' pi cr) < .7 F 0 0 0. 847 2.415 0.817 buckling -0.847 0.847 2.023 O.847 1.eam iole Max ratio -.813 1 744 2 897 4 051 Max 1 2 3 4 buckling - .--.812 O 812 O 812 O 577 0.245 0. 950 4 235 hole test ~redictior ratio over over VirstNp theoq O 812 0..- 0.& 1 2 3 4 Bar( 0.e.95(1 3.4 0 s m p over VtrslNp test )tediction over heoq .950 buckling 0.915 0 847 3..- 1 2 3 4 .- Max M.745 . .95C O.-- lean .085 0.95(1 1 818 0.vole -- ratio -.4Mp test mdictior --avec over :heoq vu. 5 -.606 0. . . ove1 heor buckling .e.Max MtesdM.662 1.. -- over v t.641 5. 0.s.840 1. . test predictioi ratio --. 840 . .827 0.. . 840 3.609 0....592 3.338 0..O01 1.986 j.310 1.409 1. ratio over over L / V p heoq I 1. J .966 shear mech.785 4.055 . . .- O.M-P test predictioi . Max k -. f l p .. .84( 0. 1 ml F i b 1 .-( E r3 a -1 . z: =? QO a ) O . E z 5 1 -' (q z F a 5. . -1 ad . > a S. y. as q %1 a). 136 . .976 0.259 shear mech.474 .91C 1... O.l86 -.052 -. Max ratio --.- 0. .976 1.259 0.976 0. .862 0.. 1.106 shear mech. 1. 1.. shear mech. Max ratio test predictior over heory 1.136 .Max ratio . test predictior over heory 1. test predictioi ovet heory . . . . . 283 or000 Max tesUM test irediction ratio over ove1 ItesUV heor) 1 2 3 1.-- -- Y?" 0.oz0 '2.804 shear mech.283 0.412 1 . 7 1. ref: lean hole -..020 1.484 1 .O20 3. test ~redictior over :heor 1 2' 3 -- 5 6 8 4- 1.#.O20 0.3ear holt Max ratio over V.O39 infinite .484 1 .020 .%O 1..020 1.268 1.O46 4.876 2. .020 1.0.960 0.948 1 .O20 1.960 mid-post 1 960 1.495 1.O16 . 7 hole tesW VcrN tesWy tesuMy .1 1 .. . 109....k.- Mp 105.. . . ..- WasilMp over v.w -..m .-- 109..-.809 mid-posl .. test .-.. -.esfl. .-.predictioi over heory -. test predictioi ove1 heory 1. ml! 1.4.* ..: -.n . .1 Csflv. ..- Max ratio . .8l mid-post 1 -- kN.497 mid-post kN. .- kN..-..- Max tesUM test predictioi ratio over over iheory -1. . n l -nl OD 0 O s . h LD CO y si >a. 'm'm'm9. N i rC iV tb .. O ' . .O . d b. T .8/8iO.O . ! 1 8 W 1 N 1 C V i N . . o .b -~ .c ~ o J -. 7 ' - W : ( D (D W ( D ' C D (O (D . * I 9 l I m ! < D i b : < .i . . g ! ~: ! S l ~ :i . C V CV W -r'Wa*. : . Y y T ' y 0 : o '0 0 O O O CV hl w 9 O r3 Q) ! ! . : T: i ~ ~ m rn m 'O . E: a: . I I . . ! ' P -. ~ ~ ~ ~ j F C0.' IO 10 .0 a Q> Q ) . Ob 0. a N in 0 ! O4 rN .a0 ep * IoD :al1 Q i*! . I I . Y ài ~ l ca.o ! I ! I . w i m ' F i a. % ! aa~ O rC) w . ! ~ I l I C Y I i ~ . O . i b !! : ~j 8~8 .' o ! >i 1 I .. 0 [ 0 1 0 ~ ~. I I : . * ri -2 m.NS N FNi /S . o . C V : r n ! O (O'bai ! I . . j l : . OI O I O . 7 .I C V .i ' . . . O ! O O O O O . O O . . o : o . ~ ~ ~ 8 ! ~ ~ ~ l ~ : $: ~E i . CV. ! s! . O O ! O j / T 1 '+ 7 5 .- .1818181Pi~i a) I Q D aD joD :a) !a3 tQD . ~ C VI ~ Ci V ~! -? ~ ~ .! ? ïüa Cu *. SN SP.O .O O ~ o 4 0 N C ~V .w.O O ~ . . - shear .test predictior .838 infinite .v.. ratio over over .. - 3.090 1.296 1.795 2.bear holt Max M. mech.510 0.00( 2.845 O 433 2. 4 5 6 l 1 .00c O 360 3..943 1. 090 3 027 1.830 infinite D.. . theo~ -1. . .fl.845 2 162 !.4MF.742 infinite 1. theor 1 23 4 5 -6-.hols Max M.es4M1 test predictioi ratio over ove1 VI*. bean -.fl.945 1.742 infinite 1 2 3 shear mech.075 2.845 1.953 2. . . ." 2.636 2r729 infinite shear mech.854 .728 2.822 2.821 0.854 2.-. .j .Max Mt@ S m .181 1. test aredictioi ratio over ovet theor Vtes& -.907 0.. 907 infinite Max ratio 4e*mp over VieriNp test ~redictior over heoq s hear mech 1.574 2..181 1 725 1 364 0. .359 2 866 infinite 1.364 infinite . O.547 2. .603 1S i 6 1.479 2.051 1.461 1.136 .bea ho1 ratic 1 *-- -over Vte.ove1 heory 1 2 3 4 0.051 0. OS( 2. rok - Max rtio . --. lear.O43 ./v? .576 .- shear mech.493 2.809 3.547 .- 2.547 test vedictioi ...015 shear mech 5 infinite infinite .08( infinite I.1N test predictia over theor] 23 4 5 .058 ..O& infinite .- Ove1 v!. 2. . 850 226 1416 225 shear mech .113 flexural .226 1 2 3 4 .283 0.000 1. 1.O63 flexural 1.4 . L2 L3 ential Loads 106 infinite 1 . test predictio over heoq 12 --3 -. ean - !oie- Max rtio k S m f ovec test ~redictioi ovec ieory 1.- ~ear holr 1 Max ratio M. mech.606 0.21C 1.613 shear mech.M ISI over vte. .225 O 283 .567 0.fl.214 1-61 3 1..11 infinite 1.O43 f~exura~ 1.O4 infinite 1 . ~i' .850 0.225 0. . . . 429 t .211 1.211 mid-post . Max MtedM.214 1.87: j. I .134 5.281 .s/M.ooi 0.550 6.oor i.8Oîl L9OO 1 953 1.808 .Man Mt. -. .404 7. -.257 infin~te infinite . - ..211 3. ratic over Vtesflp 1 test predictio ove1 :heoi 1. .580 1.821 1. 2.259 infinite shear mech.099 6. 1.771 3. 1 .. test predlctioi ratio over over heoq v l e p p .- l.741 5.8O8 0.99. . .APPENDIX C Elastic and Plastic Theoretical Computations This Appendix contains al1 caIculations in deriving the Buckling loads under the effect of Inelasticity on the Ultimate Strength. . m u .5 s r -: Ccs gi d! a 12 ):St1 -. -1 i a'-Y) Y ) ' Stm1 .0-Q a Qt Qt a D D D if. p n . bbf' hlmm ? ? - . 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