ISO ETA062609M13 Rev. 2Version 9 Berkeley, California, USA November 2010 Concrete FrameDesign Manual Eurocode 2-2004 with Eurocode 8-2004 For ETABS COPYRIGHT Copyright Computers and Structures, Inc., 1978-2010 All rights reserved. The CSI Logo, SAP2000, and ETABS are registered trademarks of Computers and Structures, Inc. SAFETM and Watch & LearnTM are trademarks of Computers and Structures, Inc. The computer programs SAP2000 and ETABS and all associated documentation are proprietary and copyrighted products. Worldwide rights of ownership rest with Computers and Structures, Inc. Unlicensed use of these programs or reproduction of documentation in any form, without prior written authorization from Computers and Structures, Inc., is explicitly prohibited. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior explicit written permission of the publisher. Further information and copies of this documentation may be obtained from: Computers and Structures, Inc. 1995 University Avenue Berkeley, California94704USA Phone: (510) 649-2200 FAX: (510) 649-2299 e-mail: info@csiberkeley.com (for general questions) e-mail: support@csiberkeley.com (for technical support questions) web: www.csiberkeley.com DISCLAIMER CONSIDERABLETIME,EFFORTANDEXPENSEHAVEGONEINTOTHE DEVELOPMENTANDTESTINGOFTHISSOFTWARE.HOWEVER,THEUSER ACCEPTSANDUNDERSTANDSTHATNOWARRANTYISEXPRESSEDOR IMPLIEDBYTHEDEVELOPERSORTHEDISTRIBUTORSONTHEACCURACY OR THE RELIABILITY OF THIS PRODUCT. THISPRODUCTISAPRACTICALANDPOWERFULTOOLFORSTRUCTURAL DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC ASSUMPTIONSOFTHESOFTWAREMODELING,ANALYSIS,ANDDESIGN ALGORITHMSANDCOMPENSATEFORTHEASPECTSTHATARENOT ADDRESSED. THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY A QUALIFIEDANDEXPERIENCEDENGINEER.THEENGINEERMUST INDEPENDENTLYVERIFYTHERESULTSANDTAKEPROFESSIONAL RESPONSIBILITY FOR THE INFORMATION THAT IS USED. i Contents Chapter 1 Introduction 1.1Organization1-2 1.2Recommended Reading/Practice1-2 Chapter 2 Design Prerequisites2.1Design Load Combinations2-1 2.2Design and Check Stations2-3 2.3Identifying Beams and Columns 2-3 2.4Design of Beams2-3 2.5Design of Columns2-4 2.6P-Delta Effects2-5 2.7Element Unsupported Lengths2-5 2.8Choice of Input Units2-6 Chapter 3 Design Process3.1Notation3-1 Concrete Frame Design Eurocode 2-2004 ii 3.2Assumptions / Limitations3-4 3.3Design Load Combinations3-5 3.4Column Design3-8 3.4.1Generation of Biaxial Interaction Surface3-9 3.4.2Calculate Column Capacity Ratio3-12 3.4.3Design Longitudinal Reinforcement3-18 3.4.4Design Column Shear Reinforcement3-18 3.5Beam Design3-21 3.5.1Design Beam Flexural Reinforcement3-22 3.5.2Design Beam Shear Reinforcement3-30 3.5.3Design Beam Torsion Reinforcement3-33 Chapter 4 Seismic Provisions4.1Notations4-2 4.2Design Preferences4-3 4.3Overwrites4-3 4.4Supported Framing Types4-4 4.5Member Design4-4 4.5.1Ductility Class High Moment-Resisting Frames 4-4 4.5.2Ductility Class Medium Moment-Resisting Frames4-19 4.5.3Ductility Class Low Moment-Resisting Frames4-23 4.5.4Special Consideration for Seismic Design4-23 Chapter 5 Design Output 5.1Overview5-1 5.2Graphical Display of Design Information5-2 5.2.1Input/Output5-2 5.3Tabular Display of Design Output5-3 5.4Member Specific Information5-4 5.4.1Interactive Concrete Frame Design5-7 5.5Errors Messages and Warnings5-9 Contents Appendix ASecond Order P-Delta Effects Appendix BMember Unsupported Lengths and Computation of -Factors Appendix CConcrete Frame Design Preferences Appendix DConcrete Frame Overwrites Appendix E Error Messages and Warnings Appendix F Nationally Determined Parameters (NDPs) References 1 - 1 Chapter 1 Introduction The design of concrete frames is seamlessly integrated within the program. Ini-tiation of the design process, along with control of variousdesign parameters, is accomplished using the Design menu.Automateddesignattheobjectlevelisavailableforanyoneofanumberof user-selected design codes, as long as the structure has first been modeled and analyzed by the program. Model and analysis data, such as material properties andmemberforces,arerecovereddirectlyfromthemodeldatabase,andno additional user input is required if the design defaults are acceptable.The design is based on a set of user-specified loading combinations. However, theprogramprovidesdefaultloadcombinationsforeachdesigncodesup-ported.Ifthedefaultloadcombinationsareacceptable,nodefinitionofaddi-tional load combinations is required.In the design of columns, the program calculates the required longitudinal and shearreinforcement.However,theusermayspecifythelongitudinalsteel,in which case a column capacity ratio is reported. The column capacity ratio gives an indication of the load condition with respect to the capacity of the column.Thebiaxialcolumncapacitycheckisbasedonthegenerationofconsistent three-dimensionalinteractionsurfaces.Itdoesnotuseanyempiricalformula-tions that extrapolate uniaxial interaction curves to approximate biaxial action.Concrete Frame Design Eurocode 2-2004 1 - 2Organization Interactionsurfacesaregeneratedforuser-specifiedcolumnreinforcingcon-figurations. The column configurations may be rectangular, square, or circular, withsimilarreinforcingpatterns.Thecalculationofsecondordermoments, unsupported lengths, and material partial factors is automated in the algorithm.Everybeammemberisdesignedforflexure,shear,andtorsionatoutputsta-tions along the beam span.Inputandoutputdatacanbepresentedgraphicallyonthemodel,intables,or onthecalculationsheetpreparedforeachmember.Foreachpresentation method,thedataisinaformatthatallowstheengineertoquicklystudythe stressconditionsthatexistinthestructureand,intheeventthememberrein-forcing is not adequate, aids the engineer in taking appropriate remedial meas-ures,includingalteringthedesignmemberwithoutrerunningtheentire analysis.1.1Organization This manual is designed to help you quickly become productive with the con-creteframedesignoptionsofEurocode2-2004.Chapter2providesdetailed descriptionsoftheDesignPrerequisitesusedforEurocode2-2004.Chapter3 providesdetaileddescriptionsofthecode-specificprocessusedforEurocode 2-2004.Chapter4providesadetaileddescriptionofthealgorithmsrelatedto seismicprovisionsinthedesign/checkofstructuresinaccordancewithEN 1998-1:2004Eurocode8.Chapter5documentsthedesignoutputproduced by the program. The appendices provide details on certain topics referenced in this manual.1.2Recommended Reading/Practice It is strongly recommended that you read this manual and review any applica-bleWatch&LearnSeriestutorials,whichcanbefoundonourwebsite, www.csiberkeley.com,beforeattemptingtodesignaconcreteframe.Addi-tionalinformationcanbefoundintheon-lineHelpfacilityavailablefrom within the programs main menu. 2 - 1 Chapter 2 Design Prerequisites Thischapterprovidesanoverviewofthebasicassumptions,designprecondi-tions,andsomeofthedesignparametersthataffectthedesignofconcrete frames. Inwritingthismanualithasbeenassumedthattheuserhasanengineering background in the general area of structural reinforced concrete design and fa-miliaritywiththeEurocode2-2004designcodeandtheseismicprovisionsin the design/check of structures in accordance with EN 1998-1:2004 Eurocode 8. 2.1Design Load Combinations Thedesignloadcombinationsareusedfordeterminingthevariouscombina-tionsoftheloadcasesforwhichthestructureneedstobedesigned/checked. Theloadcombinationfactorstobeusedvarywiththeselecteddesigncode. Theloadcombinationfactorsareappliedtotheforcesandmomentsobtained from the associated load cases and are then summed to obtain the factored de-sign forces and moments for the load combination. For multi-valued load combinations involving response spectrum, time history, movingloadsandmulti-valuedcombinations(oftypeenveloping, square-root ofthesumofthesquaresorabsolute)whereanycorrespondencebetweenin-teractingquantitiesislost,theprogramautomaticallyproducesmultiplesub Concrete Frame Design Eurocode 2-2004 2 - 2Design Load Combinations combinationsusingmaxima/minimapermutationsofinteractingquantities. Separatecombinationswithnegativefactorsforresponsespectrumcasesare notrequiredbecausetheprogramautomaticallytakestheminimatobethe negativeofthemaximaforresponsespectrumcasesandthepreviouslyde-scribed permutations generate the required sub combinations. Whenadesigncombinationinvolvesonlyasinglemulti-valuedcaseoftime historyormovingload,furtheroptions areavailable.Theprogramhasanop-tiontorequestthattimehistorycombinationsproducesubcombinationsfor eachtimestepofthetimehistory.Also,anoptionisavailabletorequestthat movingloadcombinationsproducesubcombinationsusingmaximaandmin-ima of each design quantity but with corresponding values of interacting quan-tities.For normal loading conditions involving static dead load, live load, wind load, andearthquakeload,ordynamicresponsespectrumearthquakeload,thepro-gram has built-in default loading combinations for each design code. These are basedonthecoderecommendationsandaredocumentedforeachcodeinthe corresponding manual. Forotherloadingconditionsinvolvingmovingload,timehistory,patternlive loads,separateconsiderationofroofliveload,snowload,andsoon,theuser must define design loading combinations either in lieu of or in addition to the default design loading combinations. Thedefaultloadcombinationsassumeallstaticloadcasesdeclaredasdead load to be additive. Similarly, all cases declared as live load are assumed addi-tive.However,eachstaticloadcasedeclaredaswindorearthquake,orre-sponsespectrumcases,isassumedtobenonadditivewitheachotherand producesmultiplelateralloadcombinations.Also,windandstaticearthquake cases produce separate loading combinations with the sense (positive or nega-tive) reversed. If these conditions are not correct, the user must provide the ap-propriate design combinations. Thedefaultloadcombinationsareincludedinthedesigniftheuserrequests them to be included or if no other user-defined combinations are available for concretedesign.Ifanydefaultcombinationisincludedindesign,alldefault combinationswillautomaticallybeupdatedbytheprogramanytimethede-sign code is changed or if static or response spectrum load cases are modified.Chapter 2 - Design Prerequisites Design and Check Stations2 - 3 Live load reduction factors can be applied to the member forces of the live load case on an element-by-element basis to reduce the contribution of the live load to the factored loading. Theuseriscautionedthatifmovingloadortimehistoryresultsarenotre-quested to be recovered in the analysis for some orall of the frame members, theeffectsofthoseloadswillbeassumedtobezeroinanycombinationthat includes them. 2.2Design and Check Stations For each load combination, each element is designed or checked at a number of locationsalongthelengthoftheelement.Thelocationsarebasedonequally spacedoutputstationsalongtheclearlengthoftheelement.Thenumberof output stations in an element is requested by the user before the analysis is per-formed.Theusercanrefinethedesignalongthelengthofanelementbyre-questing more output stations. 2.3Identifying Beams and Columns In the program, all beams and columns are represented as frame elements, but designofbeamsandcolumnsrequiresseparatetreatment.Identificationfora concreteelementisaccomplishedbyspecifyingtheframesectionassignedto theelementtobeoftypebeamorcolumn.Ifanybracememberexistsinthe frame, the bracememberalso would be identified as a beam or a column ele-ment, depending on the section assigned to the brace member. 2.4Design of Beams In the design of concrete beams, in general, the program calculates and reports therequiredareasofreinforcingsteelforflexure,shear,andtorsionbasedon the beam moments, shears, load combination factors, and other criteria, which aredescribedindetailinChapter3and4(seismic).Thereinforcementre-quirements are calculated at a user-defined number of stations along the beam span. Concrete Frame Design Eurocode 2-2004 2 - 4Design of Columns Allbeamsaredesignedformajordirectionflexure,shear,andtorsiononly. Effectsresultingfromanyaxialforcesandminordirectionbendingthatmay exist in the beams must be investigated independently by the user. Indesigningtheflexuralreinforcementforthemajormomentataparticular stationofaparticularbeam,thestepsinvolvethedeterminationofthemaxi-mumfactoredmomentsandthedeterminationofthereinforcingsteel.The beamsectionisdesignedforthemaximumpositiveandmaximumnegative factored moment envelopes obtained from all of the load combinations. Nega-tivebeammomentsproducetopsteel.Insuchcases,thebeamisalwaysde-signed as a rectangular section. Positive beammoments produce bottom steel. Insuchcases,thebeammaybedesignedasarectangularbeamoraT-beam. For the design of flexural reinforcement, the beam is first designed as a singly reinforcedbeam.Ifthesinglyreinforcedbeamisnotadequate,therequired compression reinforcement is calculated. Indesigningtheshearreinforcementforaparticularbeamforaparticularset ofloadingcombinationsataparticularstationbecauseofbeammajorshear, thestepsinvolvethedeterminationof thefactoredshearforce,thedetermina-tion of the shear force that can be resisted by concrete, and the determination of any reinforcement steel required to carry the balance. 2.5Design of Columns Inthedesignofthecolumns,theprogramcalculatestherequired longitudinal steel,orifthelongitudinalsteelisspecified,thecolumnstressconditionisreportedintermsofacolumncapacityratio,whichisafactorthatgivesanindicationoftheloadconditionofthecolumnwithrespecttothecapacityof thecolumn.Thedesignprocedureforthereinforcedconcretecolumnsofthe structure involves the following steps: Generate axial force-biaxial moment interaction surfaces for all of the dif-ferent concrete section types in the model.Check the capacity of each column for the factored axial force and bending momentsobtainedfromeachloadingcombinationateachendofthecol-umn. This step is also used to calculate the required steel reinforcement (if nonewasspecified)thatwillproduceacolumncapacityratioof1.0. Chapter 2 - Design Prerequisites P-Delta Effects2 - 5 Thegenerationoftheinteractionsurfaceisbasedontheassumedstrainand stressdistributionsandothersimplifyingassumptions.Thesestressandstrain distributions and the assumptions are documented in Chapter 3. Theshearreinforcementdesignprocedureforcolumnsisverysimilartothat for beams, except that the effect of the axial force on the concrete shear capac-ity is considered.2.6P-Delta Effects TheprogramdesignprocessrequiresthattheanalysisresultsincludeP-Delta effects.Fortheindividualmemberstabilityeffects,thefirstorderanalysis moments areincreased with additional second order moments,asdocumented in Chapter 3. As an alternative, the user can turn off the calculation of second ordermomentsforindividualmemberstabilityeffects.Ifthiscalculationis turnedoff,theusershouldapplyanothermethod,suchasequivalentlateral loadingorP-Deltaanalysiswithverticalmembersdividedintoatleasttwo segments,tocapturethememberstabilityeffectsinadditiontotheglobal P-Delta effects. Users of the program should be aware that the default analysis option is turned OFF for P-Delta effect. The user can turn the P-Delta analysis ON and set the maximum number of iterations for the analysis. The default number of iteration for P-Delta analysis is 1. Further details about P-Delta analysis are provided in Appendix A of this design manual. 2.7Element Unsupported Lengths To account for column slenderness effects, the column unsupported lengths are required.Thetwounsupportedlengthsarel33andl22.Thesearethelengthsbetweensupportpointsoftheelementinthecorrespondingdirections.The length l33 corresponds to instability about the 3-3 axis (major axis), and l22 cor-responds to instability about the 2-2 axis (minor axis).Normally, the unsupported element length is equal to the length of the element, i.e.,thedistancebetweenEND-IandEND-Joftheelement.Theprogram, however,allowsuserstoassignseveralelementstobetreatedasasingle Concrete Frame Design Eurocode 2-2004 2 - 6Choice of Input Units member for design. This can be accomplished differently for major and minor bending, as documented in Appendix B.The user has options to specify the unsupported lengths of the elements on an element-by-element basis. 2.8Choice of Input Units Imperial, as well as SI and MKS metric units can be used for input and output. The codes are based on a specific system of units. The Eurocode 2-2004 design code is published in Newton-millimeter-second units and all equations and de-scriptions presented in the Design Process chapter correspond to these units. However,anysystemofunitscanbeusedtodefineanddesignastructurein the program. 3 - 1 Chapter 3 Design Process Thischapterprovidesadetaileddescriptionofthecode-specificalgorithms used in the design of concrete frames when the Eurocode 2-2004 code has been selected. For simplicity, all equations and descriptions presented in this chapter correspondtoNewton-millimeter-secondunitsunlessotherwisenoted.Italso should be noted that this section describes the implementation of the CEN De-faultversionofEurocode2-2004,withoutacountryspecificNationalAnnex.Where Nationally Determined Parameters [NDPs] are to be considered, this is highlighted in the respective section by the notation [NDP].3.1Notation The various notations used in this chapter are described herein:Ac Area of concrete used to determine shear stress, mm2

Ag Gross area of concrete, mm2

Ak Area enclosed by centerlines of connecting walls for torsion, mm2 As Area of tension reinforcement, mm2

A's

Area of compression reinforcement, mm2 Concrete Frame Design Eurocode 2-2004 3 - 2Notation Asl

Area of longitudinal torsion reinforcement, mm2

At/sArea of transverse torsion reinforcement (closed stirrups) per unit length of the member, mm2/mmAsw/sArea of shear reinforcement per unit length of the member, mm2/mmEc

Modulus of elasticity of concrete, MPaEs

Modulus of elasticity of reinforcement, assumed as 200 GPaIg

Moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, mm4

M01 Smaller factored end moment in a column, N-mmM02 Larger factored end moment in a column, N-mmMEd Design moment, including second order effects to be used in de-sign, N-mmM0Ed Equivalent first order end moment (EC2 5.8.8.2), N-mmM2 Second order moment from the Nominal Curvature method (EC2 5.8.8), N-mmM22 First order factored moment at a section about 2-axis, N-mmM33 First order factored moment at a section about 3-axis, N-mmNB Buckling load, N NEd Factored axial load at a section, N TEd Factored torsion at a section, N-mm VEd Factored shear force at a section, N VRd,c Design shear resistance without shear reinforcement, NVRd,max Shear force that can be carried without crushing of the notional concrete compressive struts, N aDepth of compression block, mmamax

Maximum allowed depth of compression block, mmb Width of member, mmChapter 3 - Design Process Notation3 - 3 bf

Effective width of flange (T-beam section), mmbw

Width of web (T-beam section), mmdDistance from compression face to tension reinforcement, mmd'Concrete cover-to-center of reinforcing, mmds

Thickness of slab/flange (T-beam section), mme2 Deflection due to curvature for the Nominal Curvature method (EC2 5.8.8), mm ei Eccentricity to account for geometric imperfections (EC2 5.2), mm emin Minimum eccentricity (EC2 6.1), mm fcd

Design concrete compressive strength (EC 3.1.6), MPafctm

Mean value of axial tensile strength of concrete, MPa fyd

Design yield strength of reinforcement (EC2 3.2), MPa hOverall depth of a column section, mml0 Member effective length, mm lu Member unsupported length, mm rRadius of gyration of column section, mmtef Effective wall thickness for torsion, mm uOuter perimeter of cross-section, mmuk Outer perimeter of area Ak, mm x Depth to neutral axis, mm occ

Material coefficient taking account of long-term effects on the compressive strength (EC2 3.1.6) oct Material coefficient taking account of long-term effects on the tensile strength (EC2 3.1.6) olcc Light-weight material coefficient taking account of long-termeffects on the compressive strength (EC2 11.3.5) Concrete Frame Design Eurocode 2-2004 3 - 4Assumptions / Limitations olct Light-weight material coefficient taking account of long-termeffects on the tensile strength (EC2 11.3.5) cc

Strain in concrete ccu2

Ultimate strain allowed in extreme concrete fiber (0.0035 mm/mm) cs

Strain in reinforcing steel c Material partial factor for concrete (EC2 2.4.2.4) s Material partial factor for steel (EC2 2.4.2.4) Factor defining effective height of concrete stress block (EC2 3.1.7) q Factor defining effective strength of concrete stress block (EC2 3.1.7) u Angle between concrete compression strut and member axisperpendicular to the shear force ui Inclination due to geometric imperfections (EC2 5.2), ratio u0 Basic inclination for geometric imperfections (EC2 5.2), ratio 3.2Assumptions / Limitations The following general assumptions and limitations exist for the current imple-mentationofEurocode2-2004withintheprogram.Limitationsrelatedtospecific parts of the design are discussed in their relevant sections. Design of plain or lightly reinforced concrete sections is not handled. Design of prestressed or post-tensioned sections currently is not handled. The serviceability limit state currently is not handled. Design for fire resistance currently is not handled. By default, the Persistent & Transient design situation (EC2 Table 2.1N) is considered.Otherdesignsituationscanbeconsideredandmayrequire modification of some of the concrete design preference values. Chapter 3 - Design Process Design Load Combinations3 - 5 It is assumed that the structure being designed is a building type structure. Specialdesignrequirementsforspecialstructuretypes(suchasbridges, pressurevessels,offshoreplatforms,liquid-retainingstructures,andthe like) currently are not handled. It is assumed that the load actions are based on Eurocode 1. The program works with cylinder strength as opposed to cube strength. The program does not check depth-to-width ratios (EC2 5.3.1) or effective flangewidthsforT-beams(EC25.3.2).Theuserneedstoconsiderthese items when defining the sections. Itisassumedthattheuserwillconsiderthemaximumconcretestrength limit, Cmax, specified in the design code (EC2 3.1.2(3)). It is assumed that the cover distances input by the user satisfy the minimum cover requirements (EC2 4.4.1.2). The design value of the modulus of elasticity of steel reinforcement, Es, is assumed to be 200 GPa (EC2 3.2.7(4)). 3.3Design Load Combinations Thedesignloadcombinationsarethevariouscombinationsoftheprescribed load cases for which the structure is to be checked. The program creates a num-ber of default design load combinations for a concrete frame design. Users can addtheirowndesignloadcombinationsaswellasmodifyordeletethepro-gramdefaultdesignloadcombinations.Anunlimitednumberofdesignload combinations can be specified.To define a design load combination, simply specify one or more load patterns, eachwithitsownscalefactor.Thescalefactorsareappliedtotheforcesand moments from the load cases to form the design forces and moments for each design load combination. There is one exception to the preceding. For spectral analysismodalcombinations,anycorrespondencebetweenthesignsofthe moments and axial loads is lost. The program uses eight design load combina-tionsforeachsuchloadingcombinationspecified,reversingthesignofaxial loads and moments in major and minor directions.Concrete Frame Design Eurocode 2-2004 3 - 6Design Load Combinations Asanexample,ifastructureissubjectedtodeadload,D,andliveload,L, only,theEurocode2-2004designcheckwouldrequiretwodesignloadcom-binationsonly.However,ifthestructureissubjectedtowind,earthquake,or other loads, numerous additional design load combinations may be required.Theprogramallowsliveloadreductionfactorstobeappliedtothemember forces of the reducible live load case on a member-by-member basis to reduce the contribution of the live load to the factored responses.Thedesignloadcombinationsarethevariouscombinationsoftheloadcases forwhichthestructureneedstobechecked.Eurocode0-2002allowsload combinationstobedefinedbasedonEC0Eq.6.10orthelessfavorable EC0 Eqs. 6.10a and 6.10b [NDP]. > >+ + +1 1, , 0 , 1 , 1 , , ,j ii k i i Q k Q P j k j GQ Q P G (EC0 Eq. 6.10) > >+ + +1 1, , 0 , 1 , 1 , 0 1 , , ,j ii k i i Q k Q P j k j GQ Q P G (EC0 Eq. 6.10a) > >+ + +1 1, , 0 , 1 , 1 , , ,j ii k i i Q k Q P j k j G jQ Q P G (EC0 Eq. 6.10b) Loadcombinationsconsideringseismicloadingareautomaticallygenerated based on EC0 Eq. 6.12b. > >+ + +1 1, , 2 ,j ii k i Ed j kQ A P G (EC0 Eq. 6.12b) Forthiscode,ifastructureissubjectedtodead(D),live(L),wind(W),and earthquake(E)loads,andconsideringthatwindandearthquakeforcesarere-versible,thefollowingloadcombinationsneedtobeconsideredifequation 6.10 is specified for generation of the load combinations (EC0 6.4.3):Gj,sup D(EC0 Eq. 6.10) Gj,supD + Q,1 L(EC0 Eq. 6.10) Gj,inf D Q,1 WGj,sup D Q,1 W (EC0 Eq. 6.10)(EC0 Eq. 6.10)Chapter 3 - Design Process Design Load Combinations3 - 7 Gj,sup D + Q,1 L Q,i 0,i WGj,sup D Q,1 W + Q,i 0,i L (EC0 Eq. 6.10)(EC0 Eq. 6.10)D 1.0ED 1.0E + 2,i L(EC0 Eq. 6.12b) Iftheloadcombinationsarespecifiedtobegeneratedfromthemaximumof EC0Eqs.6.10aand6.10b,thefollowingloadcombinationsfrombothequa-tions are considered in the program. Gj,sup D Gj,sup D (EC0 Eq. 6.10a)(EC0 Eq. 6.10b)Gj,supD + Q,1 0,1 L Gj,supD + Q,1 L (EC0 Eq. 6.10a)(EC0 Eq. 6.10b)Gj,inf D Q,1 0,1 WGj,sup D Q,1 0,1 W Gj,inf D Q,1 W Gj,sup D Q,1 W (EC0 Eq. 6.10a)(EC0 Eq. 6.10a)(EC0 Eq. 6.10b)(EC0 Eq. 6.10b)Gj,sup D + Q,1 0,1 L Q,i 0,i WGj,sup D Q,1 0,1 W + Q,i 0,i L Gj,sup D + Q,1 L Q,i 0,i WGj,sup D Q,1 W + Q,i 0,i L (EC0 Eq. 6.10a)(EC0 Eq. 6.10a)(EC0 Eq. 6.10b)(EC0 Eq. 6.10b)D 1.0ED 1.0E + 2,i L (EC0 Eq. 6.12b) Forbothsetsofloadcombinations,thevariablevaluesfortheCENDefault version of the load combinations are defined in the list that follows. Values for other countries, as determined from their National Annex, are included in Ap-pendix E. Gj,sup = 1.35(EC0 Table A1.2(B)) Gj,inf = 1.00(EC0 Table A1.2(B)) Concrete Frame Design Eurocode 2-2004 3 - 8Column Design Q,1 = 1.5(EC0 Table A1.2(B)) 0,i = 0.7 (live load, assumed not to be storage)(EC0 Table A1.1) 0,i = 0.6 (wind load)(EC0 Table A1.1) = 0.85(EC0 Table A1.2(B)) 2,i = 0.3 (assumed to be office/residential space)(EC0 Table A1.1) Depending on the selection made in the design preferences, one of the preced-ing two sets of load combinations also makes up the default design load com-binationsintheprogramwhenevertheEurocode2-2004designcodeisused. Theusershouldapplyotherappropriatedesignloadcombinationsifrooflive load is separately treated, or if other types of loads are present. PLL is the live load multiplied by the Pattern Live Load Factor. The Pattern Live Load Factor can be specified in the Preferences.WhenusingtheEurocode2-2004designcode,theprogramdesignassumes that a P-Delta analysis has been performed.3.4Column Design The program can be used to check column capacity or to design columns. If the geometryofthereinforcingbarconfigurationoftheconcretecolumnsection hasbeendefined,theprogramcancheckthecolumncapacity.Alternatively, theprogramcancalculatetheamountofreinforcingrequiredtodesignthe columnbasedonaprovidedreinforcingbarconfiguration.Thereinforcement requirements are calculated or checked at a user-defined number of output sta-tionsalongthecolumnheight.Thedesignprocedureforreinforcedconcrete columns involves the following steps:Generate axial force-biaxial moment interaction surfaces for all of the dif-ferentconcretesectiontypesofthemodel.Atypicalbiaxialinteraction diagram is shown in Figure 3-1. For reinforcement to be designed, the pro-gramgeneratestheinteractionsurfacesfortherangeofallowablerein-forcementfromaminimumof0.2percent[NDP]toamaximumof4 percent [NDP] (EC2 9.5.2).Chapter 3 - Design Process Column Design3 - 9 Calculate the capacity ratio or the required reinforcement area for the fac-tored axial force and biaxial (or uniaxial) bending moments obtained from each load combination at each output station of the column. The target ca-pacityratioistakenastheUtilizationFactorLimitwhencalculatingthe required reinforcing area.Design the column shear reinforcement.Thefollowingfoursectionsdescribeindetailthealgorithmsassociatedwith this process.3.4.1Generation of Biaxial Interaction Surfaces The column capacity interaction volume is numerically described by a series of discretepointsthataregeneratedonthethree-dimensionalinteractionfailure surface.Inadditiontoaxialcompressionandbiaxialbending,theformulation allowsforaxialtensionandbiaxialbendingconsiderations.Atypicalinterac-tion surface is shown in Figure 3-1.The coordinates of the points on the failure surface are determined by rotating a plane of linear strain in three dimensions on the column section, as shown in Figure 3-2. The linear strain diagram limits the maximum concrete strain, cc, at the extremity of the section to 0.0035 (EC2 Table 3.1).Theformulationisbasedconsistentlyuponthegeneralprinciplesofultimate strength design (EC2 6.1).Thestressinthesteelisgivenbytheproductofthesteelstrainandthesteel modulus of elasticity, csEs, and is limited to the yield stress of the steel, fyd (EC2 3.2.7). The area associated with each reinforcing bar is assumed to be placed at the actual location of the center of the bar, and the algorithm does not assume any further simplifications with respect to distributing the area of steel over the cross-section of the column, as shown in Figure 3-2. Theconcretecompressionstressblockisassumedtoberectangular,withan effectivestrengthofq fcd(EC23.1.7)andeffectiveheightofx,asshownin Figure 3-3, where q is taken as: q = 1.0 for fck s 50 MPa(EC2 Eq. 3.21) Concrete Frame Design Eurocode 2-2004 3 - 10Column Design Figure 3-1 A typical column interaction surface q = 1.0 (fck 50)/200 for 50 < fck s 90 MPa(EC2 Eq. 3.22) and is taken as: = 0.8 for fck s 50 MPa(EC2 Eq. 3.19) = 0.8 (fck 50)/400 for 50 < fck s 90 MPa(EC2 Eq. 3.20) Theinteractionalgorithmprovidescorrectiontoaccountfortheconcretearea thatisdisplacedbythereinforcementinthecompressionzone.Thedepthof the equivalent rectangular block is further referred to as a, such that:a = x(EC2 3.1.7) where x is the depth of the stress block in compression, as shown in Figure 3-3. Chapter 3 - Design Process Column Design3 - 11 cccccccccccccccccccc Figure 3-2 Idealized strain distribution for generation of interaction surface

The effect of the material partial factors, c and s [NDPs], and the material co-efficients, occ, oct, olcc, and olct [NDPs], are included in the generation of the in-teraction surface (EC2 3.1.6).Defaultvaluesforc,s,occ,oct,olcc,andolctareprovidedbytheprogrambut can be overwritten using the Preferences.Concrete Frame Design Eurocode 2-2004 3 - 12Column Design Concrete Section Strain Diagram Stress Diagramcd'1sc2sc3sc4sca x =cdf q3 cuc4sT3sT2sC1sCCConcrete Section Strain Diagram Stress Diagramcd'1sc2sc3sc4sca x =cdf q3 cuc4sT3sT2sC1sCC Figure 3-3 Idealization of stress and strain distribution in a column section 3.4.2Calculate Column Capacity Ratio Thecolumncapacityratioiscalculatedforeachdesignloadcombinationat each output station of each column. The following steps are involved in calcu-latingthecapacityratioofaparticularcolumn,foraparticulardesignload combination, at a particular location:Determinethefactoredfirstordermomentsandforcesfromtheanalysis cases and the specified load combination factors to give Ned, M22, and M33.DeterminethesecondordermomentbasedonthechosenSecondOrder Method[NDP],whichcanbeeitherNominalStiffness(EC25.8.7)or NominalCurvature(EC25.8.8)andcanbechangedusingthePrefer-ences. Thereis also aNone option if the user wants to explicitly ignore secondorder effectswithinthedesigncalculations.Thismaybedesirable if the second order effects have been simulated with equivalent loads or if a P-Delta analysis has been undertaken and each column member is divided intoatleasttwoelements,suchthatM22andM33alreadyaccountforthe second order effects.Chapter 3 - Design Process Column Design3 - 13 Add the second order moments to the first order moments if the column is determinedtobeslender(EC25.8.3.1).Determinewhetherthepoint,de-fined by the resulting axial load and biaxial moment set, lies within the in-teraction volume.Thefollowingthreesectionsdescribeindetailthealgorithmsassociatedwith this process.3.4.2.1Determine Factored Moments and Forces The loads for a particular design load combination are obtained by applying the correspondingfactorstoalloftheanalysiscases,givingNEd,M22,andM33. Thesefirstorderfactoredmomentsarefurtherincreasedtoaccountforgeo-metric imperfections (EC2 5.2). The eccentricity to account for geometric im-perfections, ei, is defined as: 02i ie l u = (EC2 Eq. 5.2) where l0 is the effective length of the member and ui is an inclination, defined as a ratio as: ui = u0 oh om (EC2 Eq. 5.1) whereomisareductionfactorforthenumberofmembers,takenas1inthe programforisolatedmembers,ohisareductionfactorforlength,takenas , 2 landu0[NDP]isthebasicinclination,definedasaratio,andcanbe overwritteninthePreferences.Theresultingfirstordermoments,including geometric imperfections, are calculated as: M22 = M22 + ei2 NEd M33 = M33 + ei3 NEd whereeishallbetakengreaterthanorequaltothecodespecifiedminimum eccentricity emin (EC2 6.2). The minimum eccentricity, emin, is defined as: min30 20 mm e h = > (EC2 6.1)Themomentgeneratedbecauseofthegeometricimperfectioneccentricity,or the min eccentricity if greater, is considered only in a single direction at a time. Concrete Frame Design Eurocode 2-2004 3 - 14Column Design 3.4.2.2Second Order Moments The design algorithm assumes that the moments M22 and M32 are obtained from a second order elastic (P-A) analysis or by applying fictitious, magnified hori-zontalforcesfollowingtherecommendationsofEC2AnnexH.Formorein-formation about P-A analysis, refer to Appendix A.Thecomputedmomentsarefurtherincreasedforindividualcolumnstability effects,P-o(EC25.8.5),bycomputingamomentmagnificationfactorbased ontheNominalStiffnessmethod(EC25.8.7) orcomputinganominalsecond order moment based on the Nominal Curvature method (EC2 5.8.8). 3.4.2.2.1Nominal Stiffness Method Theoveralldesignmoment,MEd,basedontheNominalStiffnessmethodis computed as: MEd = M0Ed (factor)(EC2 Eq. 5.31) where M0Ed is defined as: M0Ed = 0.6 M02 + 0.4 M01 > 0.4 M02(EC2 Eq. 5.32) M02 and M01 are the moments at the ends of the column, and M02 is numerically largerthanM01. 1 2 O OM Mispositiveforsinglecurvaturebendingandnega-tive for double curvature bending. The preceding expression of M0Ed is valid if there is no transverse load applied between the supports. The moment magnification factor associated with the major or minor direction of the column is defined as:( ) 1 1B Edfactor N N |( = + (EC2 Eq. 5.28) The factor | depends on the distribution of the first and second order moments and is defined as: 20c , | t = (EC2 Eq. 5.29) where c0 is a coefficient that depends on the distribution of the first order mo-mentandistakenequalto8,whichisconsistentwithaconstantfirstorder moment. The term NB is the buckling load and is computed as:Chapter 3 - Design Process Column Design3 - 15 22) (u lBlEIN|t=where |l is conservatively taken as 1; however, the program allows the user to overwrite this value. The unsupported length of the column for the direction of bending considered is defined as lu. The two unsupported lengths are l22 and l33, correspondingtoinstabilityintheminorandmajordirectionsoftheobject,respectively,asshowninFigureB-1inAppendixB.Thesearethelengthsbetween the support points of the object in the corresponding directions.RefertoAppendixBformoreinformationabouthowtheprogramautomati-callydeterminestheunsupportedlengths.Theprogramallowstheuserto overwrite the unsupported length ratios, which are the ratios of the unsupported lengths for the major and minor axis bending to the overall member length.Whenusingthestiffnessmethod,theEIassociatedwithaparticularcolumn direction is considered in the design as:EI = 0.3EcIg Thisvalueneglectscreepeffectsandmaybeunconservativeifsignificant creep effects exist. The additional moment from the Nominal Stiffness method must be a positive number. Therefore, NEd must be greater than NB. If NEd is found to be less than or equal to NB, a failure condition is declared.3.4.2.2.2Nominal Curvature Method Theoveralldesignmoment,MEd,basedontheNominalCurvaturemethodis computed as: MEd = M0Ed + M2(EC2 Eq. 5.31) where M0Ed is defined as: M0Ed = 0.6 M02 + 0.4 M01 > 0.4 M02(EC2 Eq. 5.32) M02 and M01 are the moments at the ends of the column, and M02 is numerically larger than M01. 01 02M Mis positive for single curvature bending and negative Concrete Frame Design Eurocode 2-2004 3 - 16Column Design fordoublecurvaturebending.TheprecedingexpressionofM0Edisvalidifno transverse load is applied between the supports. The additional second order moment associated with the major or minor direc-tion of the column is defined as: M2 = NEd e2(EC2 Eq. 5.33) where NEd is the design axial force, and e2, the deflection due to the curvature, is defined as: ( )221oe r l c = (EC2 5.8.8.2(3)) The effective length, lo, is taken equal to |l lu. The factor c depends on the cur-vature distribution and is taken equal to 8, corresponding to a constant first or-der moment. The terml ris the curvature and is defined as: 01rl r K K r|=(EC2 Eq. 5.34) The correction factor, Kr, depends on the axial load and is conservatively taken as 1.The factor K| is also taken as 1, which represents the situation of negligi-ble creep. The term 01 ris defined as: ( )01 0 45ydr . d c = (EC2 5.8.8.3(1)) The preceding second order moment calculations are performed for major and minor directions separately.3.4.2.3Determine Capacity Ratio As a measure of the load condition of the column, a capacity ratio is calculated. Thecapacityratioisafactorthatgivesanindicationoftheloadconditionof the column with respect to the load capacity of the column.Before entering the interaction diagram to check the column capacity, the sec-ondordermomentsareaddedtothefirstorderfactoredloadstoobtainNEd, MEd2, and MEd3. The point (NEd, MEd2, MEd3) is then placed in the interaction space shown as point L in Figure 3-4. If the point lies within the interaction volume, the column capacity is adequate. However, if the point lies outside the interac-tion volume, the column is overloaded.Chapter 3 - Design Process Column Design3 - 17 Figure 3-4 Geometric representation of column capacity ratio This capacity ratio is achieved by plotting the point L and determining the lo-cation of point C. Point C is defined as the point where the line OL (if extended outwards)willintersectthefailuresurface.Thispointisdeterminedby three-dimensional linear interpolation between the points that define the failure surface,asshowninFigure3-4.Thecapacityratio,CR,isgivenbytheratio OL OC. If OL = OC (or CR = 1), the point lies on the interaction surface and the column is loaded to capacity. If OL < OC (or CR < 1), the point lies within the interaction volume and the column capacity is adequate. If OL > OC (or CR > 1), the point lies outside the interaction volume and the column is overloaded. Concrete Frame Design Eurocode 2-2004 3 - 18Column Design ThemaximumofallthevaluesofCRcalculatedfromeachdesignloadcom-binationisreportedforeachcheckstationofthecolumn,alongwiththecon-trolling NEd, MEd2, and MEd3 set and associated design load combination name.3.4.3Design Longitudinal Reinforcement If the reinforcing area is not defined, the program computes the required rein-forcement that will give a column capacity ratio equal to the Utilization Factor Limit, which is set to 0.95 by default.3.4.4Design Column Shear Reinforcement The shear reinforcement is designed for each design combination in the major andminordirectionsofthecolumn.Thefollowingstepsareinvolvedinde-signingtheshearreinforcingforaparticularcolumn,foraparticulardesign load combination resulting from shear forces in a particular direction:Determinethedesignforcesactingonthesection,NEdandVEd.Notethat NEd is needed for the calculation of VRd,c.Determinethedesignshearresistanceofthememberwithoutshearrein-forcement, VRd,c.Determinethemaximumdesignshearforcethatcanbecarriedwithout crushing of the notional concrete compressive struts, VRd,max. Determine the required shear reinforcement as area per unit length, swA s.Thefollowingfoursectionsdescribeindetailthealgorithmsassociatedwith this process. 3.4.4.1Determine Design Shear Force In the design of the column shear reinforcement of concrete frames, the forces foraparticulardesignloadcombination,namely,thecolumnaxialforce,NEd, andthecolumnshearforce,VEd,inaparticulardirectionareobtainedbyfac-toring the load cases with the corresponding design load combination factors.Chapter 3 - Design Process Column Design3 - 19 3.4.4.2Determine Design Shear Resistance Given the design force set NEd and VEd, the shear force that can be carried with-out requiring design shear reinforcement, VRd,c, is calculated as:VRd,c = [CRd,c k (100 l fck)1/3 + k1 ocp] bwd(EC2 Eq. 6.2.a) with a minimum of: VRd,c = (vmin + k1 ocp) bwd(EC2 Eq. 6.2.b) where fck is in MPa, and k, l, and ocp are calculated as: dk2001+ = s 2.0 (d is in mm)(EC2 6.2.2(1)) l = d bAwss 0.02(EC2 6.2.2(1)) 0 2 (in MPa)cp Ed c cdN A . f o = < (EC2 6.2.2(1)) Theeffectivesheararea,Ac,isshownshadedinFigure3-6.Forcircularcol-umns,Acistakentobeequaltothegrossareaofthesection.Thefactork1= 0.15 [NDP] and the values of CRd,c [NDP] and vmin [NDP] are determined as: 0 18Rd,c cC . = (EC2 6.2.2(1)) vmin = 0.035 k3/2 fck1/2(EC2 Eq. 6.3N) 3.4.4.3Determine Maximum Design Shear Force To prevent crushing of the concrete compression struts, the design shear force VEd is limited by the maximum sustainable design shear force, VRd,max. If the de-signshearforceexceedsthislimit,afailureconditionoccurs.Themaximum sustainable shear force is defined as: ( )max 1cot tanRd, cw w cdV b z v f o u u = + (EC2 Eq. 6.9) Concrete Frame Design Eurocode 2-2004 3 - 20Column Design RECTANGULARdd'bdd'bcvASQUARE WITH CIRCULAR REBARcvAcvACIRCULARdd'DIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCERECTANGULARdd'bdd'bcvASQUARE WITH CIRCULAR REBARcvAcvAcvACIRCULARdd'DIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCERECTANGULARdd'bdd'bcvASQUARE WITH CIRCULAR REBARcvAcvACIRCULARdd'DIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCERECTANGULARdd'bdd'bcvASQUARE WITH CIRCULAR REBARcvAcvAcvACIRCULARdd'DIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCEDIRECTION OF SHEAR FORCE Figure 3-6 Shear stress area, Ac Thecoefficientocw[NDP]takesaccountofthestateofstressinthecompres-sion chord and is taken equal to 1, which is recommended for non-prestressed structures.Thestrengthreductionfactorforconcretecrackedinshear,v1 [NDP], is defined as:| |10 6 1 250ckv . f = (fck is in MPa)(EC2 Eq. 6.6N) The inner lever arm distance, z, is approximated as 0.9d. The angle between the concretecompressionstrutandthecolumnaxisperpendiculartotheshear forceisdefinedasuandtakenas45degrees.EC2allowsutobetakenbe-tween 21.8 and 45 degrees.The program assumes the conservative value of 45 degrees. Chapter 3 - Design Process Beam Design3 - 21 3.4.4.4Determine Required Shear Reinforcement If VEd is greater than VRd,c and less than VRd,max, the required shear reinforcement in the form of stirrups or ties per unit spacing, swA s ,is calculated as: u cotywdEd swzfVsA= (EC2 Eq. 6.8) Intheprecedingexpressions,forarectangularsection,bwisthewidthofthe column,distheeffectivedepthofthecolumn,andAcistheeffectiveshear area, which is equal to bwd. For a circular section, bw is replaced with D, which istheexternaldiameterofthecolumn,disreplacedwith0.8D,andAcisre-placed with the gross area 24 D . t The maximum of all of the calculated swA svalues, obtained from each design load combination, is reported for the major and minor directions of the column, along with the controlling combination name.Thecolumnshearreinforcementrequirementsreportedbytheprogramare basedpurelyonshearstrengthconsideration.Anyminimumstirruprequire-ments to satisfy spacing considerations or transverse reinforcement volumetric considerations must be investigated independently by the user.3.5Beam Design Inthedesignofconcretebeams,theprogramcalculatesandreportsthere-quiredareasofsteelforflexureandshearbasedonthebeammoments,shear forces, torsions, design load combination factors, and other criteria described in thetextthatfollows.Thereinforcementrequirementsarecalculatedata user-defined number of output stations along the beam span.Allbeamsaredesignedformajordirectionflexure,shear,andtorsion only.Effects resultingfromanyaxialforcesandminordirection bendingthat may exist in the beams must be investigated independently by the user.The beam design procedure involves the following steps:Design flexural reinforcementConcrete Frame Design Eurocode 2-2004 3 - 22Beam Design Design shear reinforcementDesign torsion reinforcement3.5.1Design Beam Flexural Reinforcement The beam top and bottom flexural reinforcement is designed at output stations along the beam span. The following steps are involved in designing the flexural reinforcement for the major moment for a particular beam, at a particular sec-tion:Determine the maximum factored momentsDetermine the required reinforcing steel3.5.1.1Determine Factored Moments Inthedesignofflexuralreinforcementofconcretebeams,thefactoredmo-mentsforeachdesignloadcombinationataparticularbeamsectionareob-tained by factoring the corresponding moments for different load cases with the corresponding design load combination factors.Thebeamsectionisthendesignedforthefactoredmomentsobtainedfrom each of the design load combinations. Positive moments produce bottom steel. In such cases, the beam may be designed as a rectangular or a T-beam section. Negativemomentsproducetopsteel.Insuchcases,thebeamisalwaysde-signed as a rectangular section.3.5.1.2Determine Required Flexural Reinforcement Intheflexuralreinforcementdesignprocess,theprogramcalculatesboththe tensionandcompressionreinforcement.Compressionreinforcementisadded when the applied design moment exceeds the maximum moment capacity of a singlyreinforcedsection.Theusercanavoidtheneedforcompressionrein-forcement by increasing the effective depth, the width, or the grade of concrete.Thedesignprocedureisbasedonasimplifiedrectangularstressblock,as showninFigure3-7(EC23.1.7(3)).Whentheappliedmomentexceedsthe momentcapacity,theareaofcompressionreinforcementiscalculatedonthe Chapter 3 - Design Process Beam Design3 - 23 assumption that the additional moment will be carried by compression and ad-ditional tension reinforcement.Thedesignprocedureusedbytheprogramforbothrectangularandflanged sections(T-beams)issummarizedinthefollowingsubsections.Itisassumed that the design ultimate axial force is negligible, hence all beams are designed ignoring axial force.Beam Section Strain Diagram Stress Diagramd'd hxbsc3 cucsA'sAsTsf 'sCa x =cdf qcTBeam Section Strain Diagram Stress Diagramd'd hxbsc3 cucsA'sAsTsf 'sCa x =cdf qcT Figure 3-7 Rectangular beam design Indesigningforafactorednegativeorpositivemoment,MEd(i.e.,designing top or bottom steel), the effective strength and depth of the compression block are given by qfcd and x (see Figure 3-7) respectively, where: = 0.8 for fck s 50 MPa,(EC2 Eq. 3.19) = 0.8 ( ) 50 400ckf( for 50 < fck s 90 MPa,(EC2 Eq. 3.20) q = 1.0 for fck s 50 MPa,(EC2 Eq. 3.21) q = 1.0 ( ) 50 200ckf( for 50 < fck s 90 MPa,(EC2 Eq. 3.22) Concrete Frame Design Eurocode 2-2004 3 - 24Beam Design where x is the depth of the neutral axis, is a factor defining the effective height of the compression zone, and q is a factor defining the effective strength. Thelimitingvalueoftheratiooftheneutralaxisdepthattheultimatelimit state to the effective depth, (x/d)lim, is expressed as a function of the ratio of the redistributed moment to the moment before redistribution, o, as follows: ( ) ( )1 2limx d d k k = for fck s 50 MPa(EC2 Eq. 5.10a) ( ) ( )3 4limx d d k k = for fck > 50 MPa(EC2 Eq. 5.10b) No redistribution is assumed, such that o is assumed to be 1. The four factors, k1, k2, k3, and k4 [NDPs], are defined as: k1 = 0.44(EC2 5.5(4)) ( )2 21 25 0 6 0 0014cuk . . . c = + (EC2 5.5(4)) k3 = 0.54(EC2 5.5(4)) ( )4 21 25 0 6 0 0014cuk . . . c = + (EC2 5.5(4)) where the ultimate strain, ccu2 [NDP], is determined from EC2 Table 3.1 as: ccu2 = 0.0035 for fck < 50 MPa(EC2 Table 3.1) ccu2 = 2.6 + 35 ( )490 100ckf( for fck > 50 MPa(EC2 Table 3.1) 3.5.1.2.1Rectangular Beam Flexural Reinforcement For rectangular beams, the normalized moment, m, and the normalized section capacity as a singly reinforced beam, mlim, are determined as: q=cdMmbd f2 ((

|.|

\||.|

\|=lim limlim21dxdxmChapter 3 - Design Process Beam Design3 - 25 Thereinforcingsteelareaisdeterminedbasedonwhethermisgreaterthan, less than, or equal to mlim. If m s mlim, a singly reinforced beam will be adequate.Calculate the normal-ized steel ratio, e, and the required area of tension reinforcement, As, as: e = 1m 2 1As = e q ( ( ( cdydf bdf This area of reinforcing steel is to be placed at the bottom if MEd is positive, or at the top if MEd is negative. If m > mlim, compression reinforcement is required.Calculate the normalized steel ratios, e', elim, and e, as: elim =lim|.|

\|dx= 1 lim2 1 m e' = ' m md dlim1 e = elim +e' where d' is the depth to the compression steel, measured from the concrete compression face. Calculate the required area of compression and tension reinforcement, As' and As, as: As' = e' q ( (' ( cdsf bdf As = e q ( ( ( cdydf bdf

where,sf ' the stress in the compression steel, is calculated as: Concrete Frame Design Eurocode 2-2004 3 - 26Beam Design 'sf=Es cc '( ( limdx1s fyd As is to be placed at the bottom and As' is to be placed at the top if MEd is posi-tive, and As' is to be placed at the bottom and As is to be placed at the top if MEd is negative. 3.5.1.2.2T-Beam Flexural Reinforcement IndesigningaT-beam,asimplifiedstressblock,asshowninFigure3-8,isassumediftheflangeisincompression,i.e.,ifthemomentispositive.Ifthe moment is negative, the flange is in tension, and therefore ignored. In that case, a simplified stress block, similar to that shown in Figure 3-8, is assumed on the compression side.Flanged Beam Under Negative Moment In designing for a factored negative moment, MEd (i.e., designing top steel), the calculationofthereinforcingsteelareaisexactlythesameasdescribedfora rectangular beam, i.e., no specific T-beam data is used.Flanged Beam Under Positive Moment Indesigningforafactoredpositivemoment,MEd,theprogramanalyzesthe sectionbyconsideringthedepthofthestressblock.Ifthedepthofthestress block is less than or equal to the flange thickness, the section is designed as a rectangular beam with a width bf. If the stress block extends into the web, addi-tional calculation is required. For T-beams, the normalized moment, m, and the normalized section capacity as a singly reinforced beam, mlim, are calculated as: q=f cdMmb d f2 ((

|.|

\||.|

\|=lim limlim21dxdxmChapter 3 - Design Process Beam Design3 - 27 Calculate the normalized steel ratios elim and e, as: elim =lim|.|

\|dx e = 1 m 2 1Calculate the maximum depth of the concrete compression block, amax, and the effective depth of the compression block, a, as: amax = elim d a = ed Thereinforcingsteelareaisdeterminedbasedonwhethermisgreaterthan, lessthan,orequaltomlim.Themaximumallowabledepthoftherectangular compression block, amax, is given by: Ifa shf ,the subsequentcalculationsforAsareexactlythesame as previ-ously defined for rectangular beam design. However, in this case, the width ofthebeamistakenasbf ,asshowninFigure3-8.Compressionrein-forcement is required if a > amax.If a > hf , the calculation for As has two parts. The first part is for balancing the compressive force from the flange, and the second part is for balancing the compressive force from the web, as shown in Figure 3-8.Therequiredreinforcingsteelarea,As2,andcorrespondingresistivemo-ment,M2,forequilibratingcompressionintheflangeoutstandsarecalcu-lated as: ( )q =f w f cdsydb b h fAf2 ||.|

\| =22 2fyd shd f A MConcrete Frame Design Eurocode 2-2004 3 - 28Beam Design d'sCa x =cdf q cdf qwcfCdxsf 'sc sTwTfT3 cucsA'sABeam Section Strain Diagram Stress Diagramwbfhfbd'sCa x =cdf q cdf qwcfCdxsf 'sc sTwTfT3 cucsA'sABeam Section Strain Diagram Stress Diagramwbfhfb Figure 3-8 T-beam design NowcalculatetherequiredreinforcingsteelareaAs1fortherectangular sectionofwidthbwtoresisttheremainingmomentM1=MEdM2.The normalized moment, m1 is calculated as: q=w cdMmb d f112 Thereinforcingsteelareaisdeterminedbasedonwhetherm1isgreater than, less than, or equal to mlim. If m1 s mlim, a singly reinforced beam will be adequate. Calculate the normalizedsteelratio,e1,andtherequiredareaoftensionrein-forcement, As1, as: e1 = 1m 2 1As1 = e1 q ( ( ( cdydf bdf Ifm1>mlim,compressionreinforcementisrequired.Calculatethe normalized steel ratios, e', elim, and e, as: Chapter 3 - Design Process Beam Design3 - 29 elim =lim|.|

\|dx e'= ' m md dlim1 e1= elim +e' where d' is the depth to the compression steel, measured from the concrete compression face. Calculatetherequiredareaofcompressionandtensionreinforce-ment, As' and As, as: As'= e' q ( (' ( cdsf bdf As1 = e1 q ( ( ( cdydf bdf where fs', the stress in the compression steel, is calculated as: 'sf=Escc

lim1dx'( ( s fyd The total tensile reinforcement is As = As1 + As2, and the total compression reinforcement is As'.As is to be placed at the bottom and As' is to be placed at the top of the section.3.5.1.3Minimum and Maximum Tensile Reinforcement Theminimumflexuraltensilesteelreinforcement,As,min[NDP],requiredina beam section is given as the maximum of the following two values:As,min = 0.26 ( ) ctm ykf f bt d(EC2 Eq. 9.1N) As,min = 0.0013bt d Concrete Frame Design Eurocode 2-2004 3 - 30Beam Design wherebtisthemeanwidthofthetensionzone,equaltothewebwidthfor T-beams,andfctmisthemeanvalueofaxialtensilestrengthoftheconcrete, calculated as: fctm = 0.30fck(2/3) for fck s 50 MPa(EC2 Table 3.1) fctm = 2.12 ln( ) 1 10cmf + for fck > 50 MPa(EC2 Table 3.1) fcm = fck + 8 MPa Themaximumflexuralsteelreinforcement,As,max[NDP],permittedaseither tension or compression reinforcement is defined as: As,max = 0.04Ac(EC2 9.2.1.1(3)) where Ac is the gross cross-sectional area. 3.5.2Design Beam Shear Reinforcement The required beam shear reinforcement is calculated for each design load com-binationateachoutputstationalongthebeamlength.Thefollowingassump-tions are made for the design of beam shear reinforcement: Thebeamsectionisassumedtobeprismatic.Theshearcapacityisbased onthebeamwidthattheoutputstationandthereforeavariationinthe width of the beam is neglected in the calculation of the concrete shear ca-pacity at each particular output station.All shear reinforcement is assumed to be perpendicular to the longitudinal reinforcement. Inclined shear steel is not handled. Thefollowingstepsareinvolvedindesigningtheshearreinforcementfora particular output station subjected to beam major shear:Determine the design value of the applied shear force, VEd.Determinethedesignshearresistanceofthememberwithoutshearrein-forcement, VRd,c.Determinethemaximumdesignshearforcethatcanbecarriedwithout crushing of the notional concrete compressive struts, VRd,max. Chapter 3 - Design Process Beam Design3 - 31 Determine the required shear reinforcement as area per unit length, swA s.Thefollowingfoursectionsdescribeindetailthealgorithmsassociatedwith this process. 3.5.2.1Determine Design Shear ForceInthedesignofthebeamshearreinforcement,theshearforcesandmoments foraparticulardesignloadcombinationataparticularbeamsectionareob-tainedbyfactoringtheassociatedshearforcesandmomentswiththecorre-sponding design load combination factors.3.5.2.2Determine Design Shear Resistance Theshearforcethatcanbecarriedwithoutrequiringdesignshearreinforce-ment, VRd,c, is calculated as:VRd,c =( )1 31100Rd,c l ck cpC k f k o (+ bwd(EC2 Eq. 6.2.a) with a minimum of: VRd,c = (vmin + k1 ocp) bwd(EC2 Eq. 6.2.b) where fck is in MPa, and k, l, and ocp are calculated as: dk2001+ = s 2.0 (d is in mm)(EC2 6.2.2(1)) l = d bAwss 0.02(EC2 6.2.2(1)) ocp = NEd /Ac < 0.2fcd (in MPa) (EC2 6.2.2(1)) The effective shear area, Ac, is taken as bwd. The factor k1 = 0.15 [NDP] and the values of CRd,c [NDP] and vmin [NDP] are determined as: CRd,c =0 18c. (EC2 6.2.2(1)) vmin = 0.035 k3/2 fck1/2(EC2 Eq. 6.3N)Concrete Frame Design Eurocode 2-2004 3 - 32Beam Design 3.5.2.3Determine Maximum Design Shear Force To prevent crushing of the concrete compression struts, the design shear force VEd is limited by the maximum sustainable design shear force, VRd,max. If the de-signshearforceexceedsthislimit,afailureconditionoccurs.Themaximum sustainable shear force is defined as: VRd,max = ( )1cot tancw w cdb z v f o u u + (EC2 Eq. 6.9) Thecoefficientocw[NDP]takesaccountofthestateofstressinthecompres-sion chord and is taken equal to 1, which is recommended for non-prestressed structures.Thestrengthreductionfactorforconcretecrackedinshear,v1 [NDP] is defined as:v1 =| | 0 6 1 250ck. f (fck is in MPa)(EC2 Eq. 6.6N) The inner lever arm distance, z, is approximated as 0.9d.Theanglebetweentheconcretecompressionstrutandthebeamaxisperpen-dicular to the shear force is defined as u.If torsion is significant i.e., TEd> Tcr where Tcr is defined as: ,,1Edcr Rd cRd cVT TV| |= | |\ .(EC2 Eq. 6.31) andiftheloadcombinationincludeseismic,thevalueofuistakenas45. However, for other cases u is optimized using the following relationship: ( )1cot tan 0.9cw cd Edv f v u u o + =where21.8 45 u s s 3.5.2.4Determine Required Shear Reinforcement If VEd is greater than VRd,c and less than VRd,max, the required shear reinforcement in the form of stirrups or ties per unit spacing, swA s,is calculated as:Chapter 3 - Design Process Beam Design3 - 33 u cotywdEd swzfVsA= (EC2 Eq. 6.8) The computation of u is given in preceding section. Themaximumofallthecalculated swA svalues,obtainedfromeachdesign loadcombination,isreportedforthebeam,alongwiththecontrollingcombi-nationname.Thecalculatedshearreinforcementmustbegreaterthanthe minimum reinforcement ratio of: ( ) min0 08w, ck yk. f f = (EC2 Eq. 9.5N) The beam shear reinforcement requirements reported by the program are based purely on shear strength considerations. Any minimum stirrup requirements to satisfyspacingandvolumetricconsiderationmustbeinvestigatedindepend-ently of the program by the user.3.5.3Design Beam Torsion Reinforcement Thetorsionreinforcementisdesignedforeachdesignloadcombinationata user-definednumberofoutputstationsalongthebeamspan.Thefollowing stepsareinvolvedindesigningthelongitudinalandshearreinforcementfora particular station due to beam torsion:Determine the factored torsion, TEd.Determine torsion section properties.Determine the torsional cracking moment.Determine the reinforcement steel required.3.5.3.1Determine Factored Torsion Inthedesignoftorsionreinforcementofanybeam,thefactoredtorsionsfor eachdesignloadcombinationataparticulardesignstationareobtainedby factoringthecorrespondingtorsionfordifferentloadcaseswiththecorre-sponding design load combination factors.Concrete Frame Design Eurocode 2-2004 3 - 34Beam Design 3.5.3.2Determine Torsion Section Properties For torsion design, special torsion section properties, including Ak, tef, u, uk, and zi, are calculated. These properties are described as follows (EC2 6.3).Ak =Area enclosed by centerlines of the connecting walls, where the centerline is located a distance of2eft from the outer surface tef=Effective wall thickness u=Outer perimeter of the cross-section uk=Perimeter of the area Ak zi =Side length of wall i, defined as the distance between the intersec-tion points of the wall centerlines FortorsiondesignofT-beamsections,itisassumedthatplacing torsionrein-forcementintheflangeareaisinefficient.Withthisassumption,theflangeis ignoredfortorsionreinforcementcalculation.However,theflangeisconsid-ered during calculation of the torsion section properties. With this assumption, the special properties for a Rectangular beam section are given as follows:A =bh Ak=(b tef)(h tef) u =2b + 2h uk=2(b tef) + 2(h tef) where,thesectiondimensionsb,h,andtefareshowninFigure 3-9.Similarly, the special section properties for a T-beam section are given as follows:A = ( )+ w f w sb h b b dAk=(bf tef)(h tef) u=2bf+ 2h uk =2(bf tef) + 2(h tef) wherethesectiondimensionsbf,bw,h,anddsforaT-beamareshowninFigure 3-9.Chapter 3 - Design Process Beam Design3 - 35 3.5.3.3Determine Torsional Cracking Moment The torsion in the section can be ignored with only minimum shear reinforce-ment (EC2 9.2.1.1) necessary if the following condition is satisfied:1 0Ed Rd,c Ed Rd,cT T V V . + s (EC2 Eq. 6.31) where TRd,c is the torsional cracking moment, determined as: TRd,c = fctd tef 2Ak where tef, and fctd, the design tensile strength, are defined as: c bw2 ccc cccc b 2 hsdClosed Stirrup in Rectangular BeamClosed Stirrup in T-Beam Sectionc h 2 hbc h 2 wbbfc bw2 ccc cccc b 2 hsdClosed Stirrup in Rectangular BeamClosed Stirrup in T-Beam Sectionc h 2 hbc h 2 wbbf Figure 3-9 Closed stirrup and section dimensions for torsion design tef =A u (EC2 6.3.2(1)) 0 05 ctd ct ctk . cf f o =(EC2 Eq. 3.16) whereAisthegrosscross-sectionarea,uistheoutercircumferenceofthe cross-section,oct[NDP]isacoefficient,takenas1.0,takingaccountoflong- term effects on the tensile strength, and fctk0.05 is defined as:fctk0.05 = 0.7fctm(EC2 Table 3.1) Concrete Frame Design Eurocode 2-2004 3 - 36Beam Design 3.5.3.4Determine Torsion Reinforcement IfEC2Eq.6.31,aspreviouslydefined,issatisfied,torsioncanbesafelyig-nored(EC26.3.2(5)).Inthatcase,theprogramreportsthatnotorsionshear reinforcementisrequired.However,iftheequationisnotsatisfied,itisas-sumedthatthetorsionalresistanceisprovidedbyclosedstirrups,longitudinal bars, and compression diagonals.If torsion reinforcement in the form of closed stirrups is required, the shear due to this torsion, Vt, is first calculated, followed by the required stirrup area, as: ( )( )con2 2t ef Ed kV h t T T A( = u cotywdt tzfVsA=The required longitudinal reinforcement area for torsion is defined as: ydkkEdslfuATA u cot2= (EC2 Eq. 6.28) where u is the angle of the compression struts, as previously defined for beam shear.Intheprecedingexpressions,uistakenas45.Thecodeallowsany valuebetween21.8and45(EC26.2.3(2)),whiletheprogramassumesthe conservative value of 45.An upper limit of the combination of VEd and TEd that can be carried by the sec-tion without exceeding the capacity of the concrete struts also is checked using the following equation.0 . 1max , max ,s +RdEdRdEdVVTT(EC2 Eq. 6.29) where, TRd,max, the design torsional resistance moment is defined as: TRd,max = 2vocw fcd Ak tef sinu cosu(EC2 Eq. 6.30) The value v is defined by EC2 Eq. 6.6N. If the combination of VEd and TEd ex-ceeds this limit, a failure message is declared. In that case, the concrete section should be increased in size.Chapter 3 - Design Process Beam Design3 - 37 The maximum of all the calculated Asl and tA svalues obtained from each de-signloadcombinationisreportedalongwiththecontrollingcombination names.Thebeamtorsionreinforcementrequirementsreportedbytheprogramare basedpurelyonstrengthconsiderations.Anyminimumstirruprequirements andlongitudinalrebarrequirementstosatisfyspacingconsiderationsmustbe investigated independently of the program by the user. 4 - 1 Chapter 4 Seismic Provisions This chapter provides a detailed description of the algorithms related to seismic provisions in the design/check of structures in accordance with the "EN 1998-1:2004 Eurocode 8: Design of Structures for Earthquake Resistance" (EC 8 2004).Thecodeoption"Eurocode2-2004"coverstheseprovisions.Theim-plementation covers load combinations from "Eurocode 2-2004," which is de-scribedinthesection"DesignLoading Combination"ofChapter3.Theload-ing based on "Eurocode 8-2004" has been described in a separate document en-titled "CSI Lateral Load Manual" (CSI 2009).For referring to pertinent sections of the corresponding code, a unique prefix is assigned for each code. Reference to the Eurocode EN 1990:2001 code is identified with the prefix "EC0." ReferencetotheEurocodeEN1992-1-1:2004codeisidentifiedwiththe prefix "EC2." Reference to the Eurocode EN 1998-1:2004 code is identified with the pre-fix "EC8."Concrete Frame Design Eurocode 2-2004 4 - 2Notations 4.1Notations The following notations are used in this chapter. AshTotal area of horizontal hoops in a beam-column joint, mm2 Asv,iTotal area of column vertical bars between corner bars in one direc-tion through a joint, mm2 EMRbSum of design values of moments of resistance of the beams fram-ing into a joint in the direction of interest, N-mm EMRcSum of design values of the moments of resistance of the columns framing into a joint in the direction of interest, N-mm Mi,dEnd moment of a beam or column for the calculation of its capacity design shear, N-mm MRb,iDesign value of beam moment of resistance at end i, N-mm MRc,iDesign value of column moment of resistance at end i, N-mm VEd,maxMaximum acting shear force at the end section of a beam from ca-pacity design calculation, N VEd,minMinimum acting shear force at the end section of a beam from ca-pacity design calculation, N hCross-section depth, mm hcCross-sectional depth of the column in the direction of interest, mm hjcDistance between extreme layers of the column reinforcement in a beam-column joint, mm hjwDistance between beam top and bottom reinforcement, mm lclClear length of a beam or a column, mm qoBasic value of the behavior factor ooPrevailing aspect ratio of the walls of the structural design Chapter 4 - Seismic Provisions Design Preferences4 - 3 o1Multiplier of the horizontal design seismic action at formation of the first plastic hinge in the systemouMultiplier of the horizontal seismic design action at formation of the global plastic mechanism RdModel uncertainty factor on the design value of resistance in the estimation of the capacity design action effects, accounting for vari-ous sources of overstrength Ratio, min max Ed, Ed,V V , between the minimum and maximum acting shear forces at the end section of a beam |Curvature ductility factor oDisplacement ductility factor Tension reinforcement ratio 4.2Design Preferences TheconcreteframedesignPreferencesarebasicassignmentsthatapplytoall oftheconcreteframemembers.ThefollowingsteelframedesignPreferences are relevant to the special seismic provisions. Framing Type Ignore Seismic Code? 4.3Overwrites The concrete frame design Overwrites are basic assignments that apply only to thoseelementstowhichtheyareassigned.Thefollowingsteelframedesign overwrite is relevant to the special seismic provisions. Frame Type Concrete Frame Design Eurocode 2-2004 4 - 4Supported Framing Types 4.4Supported Framing Types Thecode(Eurocode8:2004)nowrecognizesthefollowingtypesofframing systems(EC85.2.1(4)).Withregardtotheseframingtypes,theprogramhas implemented specifications for the types of framing systems listed. Framing TypeReferences DCH MRF (Ductility Class High Moment-Resisting Frame)EC8 5.5 DCM MRF (Ductility Class Medium Moment-Resisting Frame)EC8 5.4 DCL MRF (Ductility Class Low Moment-Resisting Frame)EC8 5.3 Secondary The program default frame type is Ductility Class High Moment-Resisting Frame (DCHMRF).However,thatdefaultcanbechangedinthePreferenceforall framesorintheOverwritesonamember-by-memberbasis.Ifaframetype Preference is revised in an existing model, the revised frame type does not ap-plytoframesthathavealreadybeen assignedaframetypethroughtheOver-writes; the revised Preference applies only to new frame members added to the model after the Preference change and to the old frame members that were not assigned a frame type though the Overwrites.4.5Member Design This section describes the special requirements for designing a member. 4.5.1Ductility Class High Moment-Resisting Frames (DCH MRF) Forthisframingsystem( )1 14 5 1 1 1 3o u uq . , . . o o oo = = (EC85.2.2.2(5), Table 5.1), the following additional requirements are checked or reported (EC8 5.5). NOTE:ThegeometricalconstraintsandmaterialrequirementsgiveninEC8 Section5.5.1shouldbecheckedindependentlybytheuserbecausethepro-gram does not perform those checks.Chapter 4 - Seismic Provisions Member Design4 - 5 4.5.1.1Design Forces 4.5.1.1.1Beams The design values of bending moments and axial forces are obtained from the analysisofthestructurefortheseismicdesignsituationinaccordancewith EC0section6.4.3.4,takingintoaccountsecondordereffectsinaccordance withEC04.4.2.2andthecapacitydesignrequirementsof5.2.3.3(2)(EC8 5.5.2).Thedesignvaluesforshearforcesofprimaryseismicbeamsandcol-umns are determined in accordance with EC8 5.5.2.1 and EC8 5.5.2.2, respec-tively. Inprimaryseismicbeams,thedesignshearforcesaredeterminedinaccor-dance with the capacity design rule, on the basis of the equilibrium of the beam under (a) the transverse load acting on it in the seismic design situation and (b) endmomentsMi,d (withi=1,2denotingtheendsectionsofthebeam),corre-spondingtoplastichingeformationforpositiveandnegativedirectionsof seismic loading. The plastic hinges shouldbe taken to form at the ends of the beamsor(iftheyformtherefirst)intheverticalelementsconnectedtothe jointsintowhichthebeamendsframe(seeFigure4-1)(EC85.5.2.1, 5.4.2.2(1)P). 2g q +2 Rd Rh,M +( )1 Rd Rh, Rc RbM M M RdMRcM RcMRcMRb RcM M < Rb RcM M > 1 21 cl2g q +2 Rd Rh,M +( )1 Rd Rh, Rc RbM M M RdMRcM RcMRcMRb RcM M < Rb RcM M > 1 21 cl Figure 4-1Capacity Design Shear Force for beams Theaboveconditionhasbeenimplementedasfollows(EC85.5.2.1, 5.4.2.2(2)P,5.4.2.2(3)P). Concrete Frame Design Eurocode 2-2004 4 - 6Member Design a) At end section i, two values of the acting shear force should be calculated, i.e.themaximumVEd,max,iandtheminimumVEd,min,icorrespondingtothe maximumpositiveandthemaximumnegativeendmomentsMi,dthatcan develop at ends 1 and 2 of the beam. 2, ,, ,, ,, ,min1, min1,max ++ (| | | | (+|| || (\ . \ . = + Rdc RdcRd Rdbi RdbjRdb Rdbi ji d g q oclM MM MM MV Vl 2, ,, ,, ,, ,min1, min1,min+ + (| | | | (+|| || (\ . \ . = + Rdc RdcRd Rdbi RdbjRdb Rdbi ji d g q oclM MM MM MV Vl Asthemomentsandshearsontheright-handsideofequationspresented abovearepositive,theoutcomemaybepositiveornegative.Ifitisposi-tive, the shear at any section will not change the sense of action despite the cyclicnatureofseismicloading;ifitisnegative,thesheardoeschange sense. The ratio ,,minmax =i dii dVV isusedinthedimensioningofshearreinforcementofDCHbeamsasa measure of reversal of the shear force at end i (similarly at end j). b) End moments Mi,d is determined as follows: , ,min1, | | | = |\ .Rci d Rd Rb iRbMM MM(EC8 Eq. 5.8) where Rd isthefactoraccountingforpossibleoverstrengthbecauseofsteel strain hardening, which in the case of DCH beams is taken as equal to 1.2; Chapter 4 - Seismic Provisions Member Design4 - 7 MRb,iisthedesignvalueofthebeammomentofresistanceatendiinthe senseoftheseismicbendingmomentundertheconsideredsenseofthe seismic action; MRc and MRb are the sum of the design values of the moments of resis-tanceofthecolumnsandthesumofthedesignvaluesofthemomentsof resistanceofthebeamsframingintothejoint,respectively.Thevalueof MRc should correspond to the column axial force(s) in the seismic design situation for the considered sense of the seismic action. c) At a beam end where the beam is supported indirectly by another beam in-steadofframingintoaverticalmember,thebeamendmoment,Mi,d,may be taken as equal to the acting moment at the beam end section in a seismic design situation. 4.5.1.1.2Columns Inprimaryseismiccolumns,thedesignvaluesofshearforcesaredetermined in accordance with the capacity design rule, on the basis of the equilibrium of the column under end moments Mi,d (with i=1.2 denoting the end stations of the column), corresponding to plastic hinge formation for positive and negative di-rectionsofseismicloading.Theplastichingesshouldbetakentoformatthe ends of the beams connected to the joints into which the column end frames, or (if they form there first) at the ends of the columns (see Figure 4-2). , ,, 1 , 2, ,,min 1, min 1,maxRdb RdbRd Rdc RdcRdc Rdci jCD cclM MM MM MVl (| | | | (+ || || (\ . \ . = End moments Mi,d is determined as follows: , ,min1,Rbi d Rd Rc iRcMM MM| |= | |\ .(EC8 Eq. 5.9) Concrete Frame Design Eurocode 2-2004 4 - 8Member Design ( )2 Rd Rb Rc Rc,M M M RcM1 clRb RcM M > 12RbM1 Rd Rc,M RbMRcMRb RcM M < ( )2 Rd Rb Rc Rc,M M M RcM1 clRb RcM M > 12RbM1 Rd Rc,M RbMRcMRb RcM M < Figure 4-2Capacity Design Shear Force for Columns where Rd is the factor accounting for possible overstrength due to steel strain harden-ing, which in the case of DCH beams is taken as equal to 1.3; MRc,iisthedesignvalueofthecolumnmomentofresistanceatendiinthe sense of the seismic bending moment under the considered sense of the seismic action; Chapter 4 - Seismic Provisions Member Design4 - 9 MRc and MRb are the sum of the design values of the moments of resistance of the columns and the sum of the design values of the moments of resistance of the beams framing into the joint, respectively. The value of MRc should correspond to the column axial force(s) in the seis-mic design situation for the considered sense of the seismic action. 4.5.1.2Design Resistance The beam and column bending resistance is computed in accordance with EN 1992-1-1:2004 (EC8 5.5.3.1.1(1)P). The beam shear resistance is computed in accordance with EN 1992-1-1:2004 with the following exceptions (EC8 5.5.3.1.1): (1)In the critical regions of primary seismic beams, the strut inclination, u, in the truss model is 45. (2)Withregardtothearrangementofshearreinforcementwithinthecritical region at an end of a primary seismic beam where the beam frames into a column, the following cases should be distinguished, depending on the al-gebraic value of the ratio: =VEd,min/VEd,maxratiobetweentheminimumandmaximumacting shear forces, as derived in accordance with EC8 section 5.5.2.1(3). (a) If 0.5, the shear resistance provided by the reinforcement should be computed in accordance with EN 1992-1-1:2004. (b)IfcT T(EC8 5.2.3.4(3), Eqn. 5.4) ( )11 2 1 = + o cq T Tif 1 Concrete Frame Design > Start Design/Check command has been used, access the de-tailed design information by right clicking a frame member to display the Con-crete Column Design Information form if a column member was right clicked ortheConcreteBeamDesignInformationformifabeammemberwasright clicked. Table 5-3 identifies the types of data provided by the forms. Thelongitudinalandshearreinforcingareaarereportedintheircurrentunits, which are displayed in the drop-down list in the lower right corner of the pro-gramwindow.Typically,thelongitudinalreinforcingareaisreportedinin2, mm2,cm2andsoon.Shearreinforcingareastypicallyarereportedinin2/in, mm2/mm, cm2/cm and so on. Table 5-3 Member Specific Data for Columns and Beams ColumnBeam Load combination ID Station locations Longitudinal reinforcement area Major shear reinforcement areas Minor shear reinforcement areas Load combination ID Station location Top reinforcement areas Bottom reinforcement areas Longitudinal reinforcement for torsion design Shear reinforcement area for shearShear reinforcement area for torsion design Buttons on the forms can be used to access additional forms that provide the following data Overwrites Element section ID Element framing type Code-dependent factors Live load reduction factors Effective length factors, , for major and minor direction bending Overwrites Element section ID Element framing type Code-dependent factors Live load reduction factors Effective length factors, , for major and minor direction bending Concrete Frame Design Eurocode 2-2004 5 - 6Member Specific Information Table 5-3 Member Specific Data for Columns and Beams Summary design data Geometric data and graphical representation Material properties Minimum design moments Moment factors Longitudinal reinforcing areas Design shear forces Shear reinforcing areas Shear capacities of steel and concrete Interaction diagram, with the axial force and biaxial moment showing the state of stress in the column Summary design data Geometric data and graphical representation Material properties Design moments and shear forces Minimum design moments Top and bottom reinforcing areas Shear capacities of concrete and steel Shear reinforcing area Torsion reinforcing area The load combination is reported by its name, while station data is reported by itslocationasmeasuredfromtheI-endofthecolumn.Thenumberofline items reported is equal to the number of design combinations multiplied by the numberofoutputstations.Onelineitemwillbehighlightedwhentheform firstdisplays.Thislineitemwillhavethelargestrequiredlongitudinalrein-forcing,unlessanydesignoverstressorerroroccursforanyoftheitems.In thatcase,thelastitemamongtheoverstresseditemsoritemswitherrorswill be highlighted. In essence, the program highlights the critical design item. If a column has been selected and the column has been specified to be checked by the program, the form includes the same information as that displayed for a designedcolumn,exceptthatthedataforacheckedcolumnincludestheca-pacityratioratherthanthetotallongitudinalreinforcingarea.Similartothe design data, the line item with the largest capacity ratio is highlighted when the form first displays, unless an item has an error or overstress, in which case, that itemwillbehighlighted.Inessence,theprogramhighlightsthecriticalcheck item. Theprogramcanbeusedtocheckandtodesignrebarinacolumnmember. Whentheuserspecifiesthattheprogramistochecktherebarinthecolumn, the program checks the rebar as it is specified. When the user specifies that the program design the rebar configuration, the program starts with the data speci-fiedforrebarandthenincreasesordecreasestherebarinproportiontothe relative areas of rebar at the different locations of rebar in the column.Chapter 5 - Design Output Member Specific Information5 - 7 5.4.1Interactive Concrete Frame Design Theinteractiveconcreteframedesignandreviewisapowerfulmodethatal-lows the user to review the design results for any concrete frame design, to re-visethedesignassumptionsinteractively,andtoreviewtherevisedresults immediately.Before entering the interactive concrete frame design mode, the design results mustbeavailableforatleastonemember.Thatmeansthedesignmusthave been run for all the members or for only selected members. If the initial design has not been performed yet, run a design by clicking the Design menu > Con-creteFrameDesign>StartDesign/CheckofStructure command. Alterna-tively,clicktheDesignmenu>ConcreteFrameDesign>DisplayDesign Info command to access the Display Design Results form and select a type of result or (after design) the Designmenu>ConcreteFrameDesign>Inter-active Concrete Frame Design command.After using any of the three commands, right click on a frame member to enter the interactive Concrete Frame Design Mode and access the Concrete Column DesignInformationformifacolumnmemberwasrightclickedortheCon-creteBeamDesignInformationformifabeammemberwasrightclicked. TheseformshaveOverwritesbuttonsthataccessestheConcreteFrameDe-signOverwritesform.Theformcanbeusedtochangethedesignsections, element type, live load reduction factor for reducible live load, and many other designfactors.SeeAppendixDforadetaileddescriptionoftheoverwrite items.WhenchangestothedesignparametersaremadeusingtheOverwrites form,theConcreteBeamorColumnDesignInformationformsupdateimme-diately to reflect the changes. Other buttons on the Concrete Beam or Column DesignInformationformscanthenbeusedtodisplayadditionalformsshow-ing the details of the updated design. See the Member Specific Information sec-tion of this chapter for more information.In this way, the user can change the overwrites any number of times to produce a satisfactory design. After an acceptable design has been produced by chang-ing the section or other design parameters, click the OK button on the Concrete Beam or Column Design Information forms to permanently change the design sections and other overwrites for that member. However, if the Cancel button isused,allchangesmadetothedesignparametersusingtheConcreteFrame Design Overwrites form are temporary and do not affect the design.Concrete Frame Design Eurocode 2-2004 5 - 8Error Messages and Warnings 5.5Error Messages and Warnings In many places of concrete frame design output, error messages and warnings are displayed. Appendix E provides a list of those messages and warnings. The messages are numbered and assumed to be self-explanatory. APPENDICES

A - 1 Appendix A Second Order P-Delta Effects Typically, design codes require that second order P-Delta effects be considered when designing concrete frames. These effects are the global lateral translation of the frame and the local deformation of members within the frame.Consider the frame object shown in Figure A-1, which is extracted from a story level of a larger structure. The overall global translation of this frame object is indicated by A. The local deformation of the member is shown as o. The total secondorderP-DeltaeffectsonthisframeobjectarethosecausedbybothA and o . TheprogramhasanoptiontoconsiderP-Deltaeffectsintheanalysis.When P-Delta effects are considered in the analysis, the program does a good job of capturing the effect due to the A deformation shown in Figure A-1, but it does nottypicallycapturetheeffectofthe odeformation(unless,inthemodel,the frame object is broken into multiple elements over its length). ConsiderationofthesecondorderP-Deltaeffectsisgenerallyachievedby computing the flexural design capacity using a formula similar to that shown in the following equation.MCAP=aMnt + bMlt where, MCAP=Flexural design capacity required Concrete Frame Design Eurocode 2-2004 A - 2Appendix A Figure A-1 The total second order P-delta effects on aframe element caused by both A and o Mnt =Required flexural capacity of the member assuming there is no joint translation of the frame (i.e., associated with the o deformation in Figure A-1) Mlt=Required flexural capacity of the member as a result of lateral translation of the frame only (i.e., associated with the Adeformation in Figure A-1) a=Unitless factor multiplying Mnt b=Unitless factor multiplying Mlt (assumed equal to 1 by the program; see the following text) Whentheprogramperformsconcreteframedesign,itassumesthatthefactorbisequalto1andcalculatesthefactora.Thatb=1assumesthatP-Deltaeffects have been considered in the analysis, as previously described. Thus, in general,whenperformingconcreteframedesigninthisprogram,consider P-Delta effects in the analysis before running the program. B - 1 Appendix B Member Unsupported Lengths andComputation of |-FactorsThecolumnunsupportedlengthsarerequiredtoaccountforcolumnslender-nesseffects.Theprogramautomaticallydeterminestheunsupportedlengthratios, which are specified as a fraction of the frame object length. Those ratios timestheframeobjectlengthgivetheunbracedlengthsforthemembers. Thoseratiosalsocanbeoverwrittenbytheuseronamember-by-memberbasis, if desired, using the overwrite option.There are two unsupported lengths to consider. They are l33 and l22, as shown in Figure B-1. These are the lengths between support points of the member in the corresponding directions. The length l33 corresponds to instability about the 3-3 axis(majoraxis),andl22correspondstoinstabilityaboutthe2-2axis(minor axis).In determining the values for l22 and l33 of the members, the program recognizes variousaspectsofthestructurethathaveaneffectonthoselengths,suchas memberconnectivity,diaphragmconstraintsandsupportpoints.Theprogram automaticallylocatesthemembersupportpointsandevaluatesthecorre-sponding unsupported length.Concrete Frame Design Eurocode 2-2004 B - 2Member Unsupported Lengths and Computation of K-Factors Itispossiblefortheunsupportedlengthofaframeobjecttobeevaluatedby theprogramasgreaterthanthecorrespondingmemberlength.Forexample, assume a column has a beam framing into it in one direction, but not the other, at a floor level. In that case, the column is assumed to be supported in one di-rection only at that story level, and its unsupported length in the other direction will exceed the story height. Figure B-1 Axis of bending and unsupported length

C - 1 Appendix C Concrete Frame Design Preferences The Concrete Frame Design Preferences are basic assignments that apply to all oftheconcreteframemembers.TableC-1liststheConcreteFrameDesign PreferencesfortheEurocode2-2004codeDefaultvaluesareprovidedforall preference items. Thus, it is not necessary to specify or change any of the pref-erences. However, at least review the default values to ensure they are accept-able.Someofthepreferenceitemsalsoareavailableasmemberspecific overwriteitems.TheOverwritesaredescribedinAppendixD.Overwritten values take precedence over the preferences. Table C-1 Preferences Item Possible Values Default Value Description CountryCEN DefaultUnited KingdomCEN DefaultToggle to choose the country for which the Nationally Determined Parameters [NDPs] should be used. The various NDPs and their associated values for each country are described in Appendix F.Combinations Equation Eq. 6.10 Max of Eq. 6.10a/6.10b Eq. 6.10Toggle to choose which equation is used to generate automatic load combinations. Concrete Frame Design Eurocode 2-2004 C - 2Appendix C Item Possible Values Default Value Description Second Order Method NominalStiffness NominalCurvature None NominalCurvature Toggle to choose the method to be used to calculate isolated member P-o effects. If None is chosen, these effects should be captured by alternative methods. Time HistoryDesign Envelopes,Step-by-Step EnvelopesToggle for design load combinations that include a time history designed for the envelope of the time history, or designed step-by-step for the entire time history. If a single design load combination has more than one time history case in it, that design load combination is designed for the en-velopes of the time histories, regardless of what is specified here.Number ofInteraction Curves Multiple of 4 > 4 24Number of equally spaced interaction curves used to create a full 360-degree interaction surface (this item should be a multiple of four). We recommend 24 for this item. Number ofInteractionPoints Any odd value > 5 11Number of points used for defining a single curve in a concrete frame; should be odd.ConsiderMinimumEccentricity No, YesYesToggle to specify if minimum eccentricity is considered in design. Theta0 (ratio)> 00.005Inclination angle for imperfections.GammaS (steel)> 01.15Partial safety factor for steel. GammaC(concrete) > 01.5Partial safety factor for concrete. AlphaCC(compression) > 01Material coefficient taking account of long- term effects on the concrete compressive strength and unfavorable effects from the load application. Appendix C Concrete Frame Design Preferences PreferencesC - 3 Item Possible Values Default Value Description AlphaCT(tension) > 01Material coefficient taking account of long- term effects on the concrete tensile strength and unfavorable effects from the load application. AlphaLCC (Light-weight compres-sion) > 00.85Light-weight material coefficient taking account of long-term effects on the con-crete compressive strength and unfavor-able effects from the load application. AlphaLCT (Lightweighttension) > 00.85Light-weight material coefficient taking account of long-term effects on the con-crete tensile strength and unfavorableeffects from the load application. Pattern LiveLoad Factor > 00.75The live load factor for automatic genera-tion of load combinations involving pattern live loads and dead loads. Utilization Factor Limit > 00.95Stress ratios that are less than or equal to this value are considered acceptable. D - 1 Appendix D Concrete Frame Overwrites The concrete frame design overwrites are basic assignments that apply only to thoseelementstowhichtheyareassigned.Table D-1listsconcreteframedesignoverwritesforEurocode2-2004.Defaultvaluesareprovidedforall overwrite items. Thus, it is not necessary to specify or change any of the over-writes.However,atleastreviewthedefaultvaluestoensuretheyareaccept-able.Whenchangesaremadetooverwriteitems,theprogramappliesthe changes only to the elements to which they are specifically assigned.Table D-1 Overwrites Item Possible Values Default Value Description CurrentDesignSection Any defined con-cretesection Analysissection The design section for the selected frame objects. ElementType SwayNonsway FromReference Frame type defined as sway (unbraced) or nonsway (braced). Live LoadReduction Factor > 0CalculatedThe reduced live load factor. A reducible live load is multiplied by this factor to obtain the reduced live load for the frame object. Speci-fying 0 means the value is program deter-mined. Concrete Frame Design Eurocode 2-2004 D - 2Appendix D Item Possible Values Default Value Description Unbraced Length Ratio (Major) > 0CalculatedUnbraced length factor for buckling about the frame object major axis. This it