Concrete Frame Design Manual CSA A

Transcription

1 Conrete Frame Design Manual CSA A

2 Conrete Frame Design Manual CSA A For CSiBridge ISO BRG102816M22 Rev. 0 Proudly developed in the United States o Ameria Otober 2016

3 COPYRIGHT Copyright Computers and Strutures, In., All rights reserved. The CSI Logo and CSiBridge are registered trademarks o Computers and Strutures, In. Wath & Learn TM is a trademark o Computers and Strutures, In. The omputer program CSiBridge and all assoiated doumentation are proprietary and opyrighted produts. Worldwide rights o ownership rest with Computers and Strutures, In. Unliensed use o these programs or reprodution o doumentation in any orm, without prior written authorization rom Computers and Strutures, In., is expliitly prohibited. No part o this publiation may be reprodued or distributed in any orm or by any means, or stored in a database or retrieval system, without the prior expliit written permission o the publisher. Further inormation and opies o this doumentation may be obtained rom: Computers and Strutures, In. ino@siameria.om (or general inormation) support@siameria.om (or tehnial support)

4 DISCLAIMER CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND TESTING OF THIS SOFTWARE. HOWEVER, THE USER ACCEPTS AND UNDERSTANDS THAT NO WARRANTY IS EXPRESSED OR IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ON THE ACCURACY OR THE RELIABILITY OF THIS PRODUCT. THIS PRODUCT IS A PRACTICAL AND POWERFUL TOOL FOR STRUCTURAL DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC ASSUMPTIONS OF THE SOFTWARE MODELING, ANALYSIS, AND DESIGN ALGORITHMS AND COMPENSATE FOR THE ASPECTS THAT ARE NOT ADDRESSED. THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY A QUALIFIED AND EXPERIENCED ENGINEER. THE ENGINEER MUST INDEPENDENTLY VERIFY THE RESULTS AND TAKE PROFESSIONAL RESPONSIBILITY FOR THE INFORMATION THAT IS USED.

5 Contents Chapter 1 Introdution 1.1 Organization Reommended Reading/Pratie 1-3 Chapter 2 Design Prerequisites 2.1 Design Load Combinations Design and Chek Stations Identiying Beams and Columns Design o Beams Design o Columns Design o Joints P-Delta Eets Element Unsupported Length Choie o Input Units 2-7 i

6 Conrete Frame Design CSA A Chapter 3 Design Proess 3.1 Notation Design Load Combinations Limits on Material Strength Strength Resistane Fators Column Design Generation o Biaxial Interation Surae Calulate Column Capaity Ratio Required Reinoring Area Design Column Shear Reinorement Beam Design Design Beam Flexural Reinorement Design Beam Shear Reinorement Joint Design Determine the Panel Zone Shear Fore Determine the Eetive Area o Joint Chek Panel Zone Shear Stress Beam-Column Flexural Capaity Ratios 3-49 Appendix A Appendix B Appendix C Appendix D Appendix E Seond Order P-Delta Eets Member Unsupported Lengths and Computation o K-Fators Conrete Frame Design Preerenes Conrete Frame Overwrites Error Messages and Warnings Reerenes ii

7 Chapter 1 Introdution The design o onrete rames is seamlessly integrated within the program. Initiation o the design proess, along with ontrol o various design parameters, is aomplished using the Design menu. It should be noted that two design proesses are available in CSiBridge: superstruture design (on the Design/Rating tab) and design o the individual elements omprising the struture (the Advaned > Frame Design ommands). This manual addresses the seond design proess. Automated design at the objet level is available or any one o a number o user-seleted design odes, as long as the strutures have irst been modeled and analyzed by the program. Model and analysis data, suh as material properties and member ores, are reovered diretly rom the model database, and no additional user input is required i the design deaults are aeptable. The design is based on a set o user-speiied loading ombinations. However, the program provides deault load ombinations or eah design ode supported in the program. I the deault load ombinations are aeptable, no deinition o additional load ombinations is required. In the design o olumns, the program alulates the required longitudinal and shear reinorement. However, the user may speiy the longitudinal steel, in 1-1

8 Conrete Frame Design CSA A whih ase a olumn apaity ratio is reported. The olumn apaity ratio gives an indiation o the stress ondition with respet to the apaity o the olumn. The biaxial olumn apaity hek is based on the generation o onsistent three-dimensional interation suraes. It does not use any empirial ormulations that extrapolate uniaxial interation urves to approximate biaxial ation. Interation suraes are generated or user-speiied olumn reinoring onigurations. The olumn onigurations may be retangular, square or irular, with similar reinoring patterns. The alulation o moment magniiation ators, unsupported lengths and strength redution ators is automated in the algorithm. Every beam member is designed or lexure and shear at output stations along the beam span. All beam-olumn joints are investigated or existing shear onditions. For Dutile and Moderately Dutile moment resisting rames, the shear design o the olumns, beams and joints is based on the probable moment apaities o the members. Also, the program will produe ratios o the beam moment apaities with respet to the olumn moment apaities, to investigate weak beam/strong olumn aspets, inluding the eets o axial ore. Output data an be presented graphially on the model, in tables or both input and output data, or on the alulation sheet prepared or eah member. For eah presentation method, the output is in a ormat that allows the engineer to quikly study the stress onditions that exist in the struture and, in the event the member reinoring is not adequate, aids the engineer in taking appropriate remedial measures, inluding altering the design member without rerunning the entire analysis. 1.1 Organization This manual is designed to help you quikly beome produtive with the onrete rame design options o CSA A Chapter 2 provides detailed desriptions o the Deign Prerequisites used or CSA A Chapter 3 provides detailed desriptions o the ode-speii proess used or CSA 1-2 Organization

9 Chapter 1 - Introdution A The appendies provide details on ertain topis reerened in this manual. 1.2 Reommended Reading/Pratie It is strongly reommended that you read this manual and review any appliable Wath & Learn Series tutorials, whih are ound on our web site, beore attempting to design a onrete rame. Additional inormation an be ound in the on-line Help aility available rom within the program s main menu. Reommended Reading/Pratie 1-3

10 Chapter 2 Design Prerequisites This hapter provides an overview o the basi assumptions, design preonditions, and some o the design parameters that aet the design o onrete rames. In writing this manual it has been assumed that the user has an engineering bakground in the general area o strutural reinored onrete design and amiliarity with CSA A odes. 2.1 Design Load Combinations The design load ombinations are used or determining the various ombinations o the load ases or whih the struture needs to be designed/heked. The load ombination ators to be used vary with the seleted design ode. The load ombination ators are applied to the ores and moments obtained rom the assoiated load ases and are then summed to obtain the atored design ores and moments or the load ombination. For multi-valued load ombinations involving response spetrum, time history, moving loads and multi-valued ombinations (o type enveloping, square-root o the sum o the squares or absolute) where any orrespondene between interating quantities is lost, the program automatially produes multiple sub ombinations using maxima/minima permutations o interating quantities. 2-1

11 Conrete Frame Design CSA A Separate ombinations with negative ators or response spetrum ases are not required beause the program automatially takes the minima to be the negative o the maxima or response spetrum ases and the above desribed permutations generate the required sub ombinations. When a design ombination involves only a single multi-valued ase o time history or moving load, urther options are available. The program has an option to request that time history ombinations produe sub ombinations or eah time step o the time history. Also an option is available to request that moving load ombinations produe sub ombinations using maxima and minima o eah design quantity but with orresponding values o interating quantities. For normal loading onditions involving stati dead load, live load, wind load, and earthquake load, or dynami response spetrum earthquake load, the program has built-in deault loading ombinations or eah design ode. These are based on the ode reommendations and are doumented or eah ode in the orresponding manuals. For other loading onditions involving moving load, time history, pattern live loads, separate onsideration o roo live load, snow load, and so on, the user must deine design loading ombinations either in lieu o or in addition to the deault design loading ombinations. The deault load ombinations assume all load ases delared as dead load to be additive. Similarly, all ases delared as live load are assumed additive. However, eah load ase delared as wind or earthquake, or response spetrum ases, is assumed to be non additive with eah other and produes multiple lateral load ombinations. Also wind and stati earthquake ases produe separate loading ombinations with the sense (positive or negative) reversed. I these onditions are not orret, the user must provide the appropriate design ombinations. The deault load ombinations are inluded in design i the user requests them to be inluded or i no other user-deined ombination is available or onrete design. I any deault ombination is inluded in design, all deault ombinations will automatially be updated by the program any time the design ode is hanged or i stati or response spetrum load ases are modiied. Live load redution ators an be applied to the member ores o the live load ase on an element-by-element basis to redue the ontribution o the live load to the atored loading. 2-2 Design Load Combinations

12 Chapter 2 - Design Prerequisites The user is autioned that i moving load or time history results are not requested to be reovered in the analysis or some or all o the rame members, the eets o those loads will be assumed to be zero in any ombination that inludes them. 2.2 Design and Chek Stations For eah load ombination, eah element is designed or heked at a number o loations along the length o the element. The loations are based on equally spaed segments along the lear length o the element. The number o segments in an element is requested by the user beore the analysis is made. The user an reine the design along the length o an element by requesting more segments. When using the CSA A design ode, requirements or joint design at the beam to olumn onnetions are evaluated at the top most station o eah olumn. The program also perorms a joint shear analysis at the same station to determine i speial onsiderations are required in any o the joint panel zones. The ratio o the beam lexural apaities with respet to the olumn lexural apaities onsidering axial ore eet assoiated with the weak-beam/strongolumn aspet o any beam/olumn intersetion are reported. 2.3 Identiying Beams and Columns In the program, all beams and olumns are represented as rame elements. But design o beams and olumns requires separate treatment. Identiiation or a onrete element is aomplished by speiying the rame setion assigned to the element to be o type beam or olumn. I any brae element exists in the rame, the brae element also would be identiied as a beam or a olumn element, depending on the setion assigned to the brae element. 2.4 Design o Beams In the design o onrete beams, in general, the program alulates and reports the required areas o steel or lexure and shear based on the beam moments, shears, load ombination ators, and other riteria, whih are desribed in detail in the ode-speii hapters. The reinorement requirements are alulated at a user-deined number o stations along the beam span. Design and Chek Stations 2-3

13 Conrete Frame Design CSA A All the beams are only designed or major diretion lexure and shear. Eets due to any axial ores, torsion and minor diretion bending that may exist in the beams must be investigated independently by the user. In designing the lexural reinorement or the major moment at a partiular setion o a partiular beam, the steps involve the determination o the maximum atored moments and the determination o the reinoring steel. The beam setion is designed or the maximum positive and maximum negative atored moment envelopes obtained rom all o the load ombinations. Negative beam moments produe top steel. In suh ases, the beam is always designed as a Retangular setion. Positive beam moments produe bottom steel. In suh ases, the beam may be designed as a Retangular beam or a T beam. For the design o lexural reinorement, the beam is irst designed as a singly reinored beam. I the beam setion is not adequate, the required ompression reinorement is alulated. In designing the shear reinorement or a partiular beam or a partiular set o loading ombinations at a partiular station due to the beam major shear, the steps involve the determination o the atored shear ore, the determination o the shear ore that an be resisted by onrete, and the determination o the reinorement steel required to arry the balane. Speial onsiderations or seismi design are inorporated into the program or CSA A Design o Columns In the design o the olumns, the program alulates the required longitudinal steel, or i the longitudinal steel is speiied, the olumn stress ondition is reported in terms o a olumn apaity ratio, whih is a ator that gives an indiation o the stress ondition o the olumn with respet to the apaity o the olumn. The design proedure or the reinored onrete olumns o the struture involves the ollowing steps: Generate axial ore-biaxial moment interation suraes or all o the dierent onrete setion types o the model. Chek the apaity o eah olumn or the atored axial ore and bending moments obtained rom eah loading ombination at eah end o the 2-4 Design o Columns

14 Chapter 2 - Design Prerequisites olumn. This step is also used to alulate the required reinorement (i none was speiied) that will produe a apaity ratio o 1.0. The generation o the interation surae is based on the assumed strain and stress distributions and some other simpliying assumptions. These stress and strain distributions and the assumptions are doumented in Chapter 3. The shear reinorement design proedure or olumns is very similar to that or beams, exept that the eet o the axial ore on the onrete shear apaity must be onsidered. For ertain speial seismi ases, the design o olumns or shear is based on the apaity shear. The apaity shear ore in a partiular diretion is alulated rom the moment apaities o the olumn assoiated with the atored axial ore ating on the olumn. For eah load ombination, the atored axial load is alulated, using the load ases and the orresponding load ombination ators. Then, the moment apaity o the olumn in a partiular diretion under the inluene o the axial ore is alulated, using the uniaxial interation diagram in the orresponding diretion, as doumented in Chapter Design o Joints To ensure that the beam-olumn joint o speial moment resisting rames possesses adequate shear strength, the program perorms a rational analysis o the beam-olumn panel zone to determine the shear ores that are generated in the joint. The program then heks this against design shear strength. Only joints having a olumn below the joint are designed. The material properties o the joint are assumed to be the same as those o the olumn below the joint. The joint analysis is done in the major and the minor diretions o the olumn. The joint design proedure involves the ollowing steps: Determine the panel zone design shear ore, Determine the eetive area o the joint, and Chek panel zone shear stress. The joint design details are doumented in Chapter 3. Design o Joints 2-5

15 Conrete Frame Design CSA A P-Delta Eets The program design proess requires that the analysis results inlude P-delta eets. The P-delta eets are onsidered dierently or braed or non-sway and unbraed or sway omponents o moments in olumns or rames. For the braed moments in olumns, the eet o P-delta is limited to individual member stability. For unbraed omponents, lateral drit eets should be onsidered in addition to individual member stability eet. The program assumes that braed or nonsway moments are ontributed rom the dead or live loads. Whereas, unbraed or sway moments are ontributed rom all other types o loads. For the individual member stability eets, the moments are magniied with moment magniiation ators, as doumented in Chapter 3 o this manual. For lateral drit eets, the program assumes that the P-delta analysis is perormed and that the ampliiation is already inluded in the results. The moments and ores obtained rom P-delta analysis are urther ampliied or individual olumn stability eet i required by the governing ode, as in the CSA A odes. Users should be aware that the deault analysis option in the program is that P-delta eets are not inluded. The user an inlude P-delta analysis and set the maximum number o iterations or the analysis. The deault number o iteration or P-delta analysis is 1. Further details on P-delta analysis are provided in Appendix A o this design manual. 2.8 Element Unsupported Lengths To aount or olumn slenderness eets, the olumn unsupported lengths are required. The two unsupported lengths are l 33 and l 22. These are the lengths between support points o the element in the orresponding diretions. The length l 33 orresponds to instability about the 3-3 axis (major axis), and l 22 orresponds to instability about the 2-2 axis (minor axis). Normally, the unsupported element length is equal to the length o the element, i.e., the distane between END-I and END-J o the element. The program, however, allows users to assign several elements to be treated as a single 2-6 P-Delta Eets

16 Chapter 2 - Design Prerequisites member or design. This an be done dierently or major and minor bending as doumented in Appendix B o this design manual. The user has options to speiy the unsupported lengths o the elements on an element-by-element basis. 2.9 Choie o Input Units English as well as SI and MKS metri units an be used or input. But the odes are based on a speii system o units. All equations and desriptions presented in the subsequent hapters orrespond to that speii system o units unless otherwise noted. For example, the CSA A ode is published in Millimeter-Newton-Seond units. By deault, all equations and desriptions presented in the hapter Design or CSA A orrespond to Millimeter- Newton-Seond units. However, any system o units an be used to deine and design the struture in the program. It should be noted that two design proesses are available in CSiBridge: superstruture design (on the Design/Rating tab) and design o the individual elements omprising the struture (the Advaned > Frame Design ommands). This manual addresses the seond design proess. Choie o Input Units 2-7

17 Chapter 3 Design Proess This hapter provides a detailed desription o the ode-speii algorithms the program uses to design onrete rames when the Canadian ode CSAA (CSA 2004) is seleted. For simpliity, all equations and desriptions presented in this hapter orrespond to Newton-Millimeter-Seond units unless otherwise noted. The program provides options to design or hek Conventional, Moderately Dutile (moderate seismi risk areas), and Dutile (high seismi risk areas) Moment Resisting rames as required or seismi design. The details o the design riteria used or the dierent raming systems are desribed in this hapter. 3.1 Notation The various notations used in this hapter are desribed herein: Av Aore Ag As A s Area o onrete used to determine shear stress, sq-mm Area o onrete ore, sq-mm Gross area o onrete, sq-mm Area o tension reinorement, sq-mm Area o ompression reinorement, sq-mm 3-1

18 Conrete Frame Design CSA A As(required) Ast Av a ab b b bw Cm b d Area o steel required or tension reinorement, sq-mm Total area o olumn longitudinal reinorement, sq-mm Area o shear reinorement, sq-mm Depth o ompression blok, mm Depth o ompression blok at balaned ondition, mm Width o member, mm Eetive width o lange (T-beam setion), mm Width o web (T-beam setion), mm Coeiient, dependent upon olumn urvature, used to alulate moment magniiation ator Depth to neutral axis, mm Depth to neutral axis at balaned onditions, mm Distane rom ompression ae to tension reinorement, mm d Conrete over to enter o reinoring, mm ds E Es y yh h Ig Ist k L Thikness o slab (T-beam setion), mm Modulus o elastiity o onrete, MPa Modulus o elastiity o reinorement, assumed as 200,000 MPa Speiied ompressive strength o onrete, MPa Speiied yield strength o lexural reinorement, MPa Speiied yield strength o shear reinorement, MPa Dimension o beam or olumn, mm Moment o inertia o gross onrete setion about entroidal axis, negleting reinorement, mm 4 Moment o inertia o reinorement about entroidal axis o member ross-setion, mm 4 Eetive length ator Clear unsupported length, mm 3-2 Notation

19 Chapter 3 - Design Proess M1 M2 M Mns Ms M Mx My N Pb P Pr,max Po P V VD+L Vp V Vs α α1 β β1 βd θ Smaller atored end moment in a olumn, N-mm Larger atored end moment in a olumn, N-mm Fatored moment to be used in design, N-mm Non-sway omponent o atored end moment, N-mm Sway omponent o atored end moment, N-mm Fatored moment at setion, N-mm Fatored moment at setion about X-axis, N-mm Fatored moment at setion about Y-axis, N-mm Fatored axial load at setion (tension positive), N Axial load apaity at balaned strain onditions, N Critial bukling strength o olumn, N Maximum axial load strength allowed, N Axial load apaity at zero eentriity, N Fatored axial load at setion (ompression positive), N Shear resisted by onrete, N Shear ore rom span loading, N Shear ore omputed rom probable moment apaity, N Fatored shear ore at a setion, N Shear ore at a setion resisted by steel, N Reinoring steel overstrength ator Fator or obtaining average ompressive stress in onrete blok Fator aounting or shear resistane o raked onrete Fator or obtaining depth o ompression blok in onrete Absolute value o the ratio o the maximum atored axial dead load moment to the maximum atored total load moment Angle o inlination o diagonal ompressive stresses with the longitudinal axis o beam or olumn Notation 3-3

20 Conrete Frame Design CSA A δb δs ε εs φ φs φm λ Moment magniiation ator or non-sway moments Moment magniiation ator or sway moments Strain in onrete Strain in reinoring steel Strength redution ator or onrete Strength redution ator or steel Strength redution ator or member Shear strength ator 3.2 Design Load Combinations The design load ombinations are the various ombinations o the presribed load ases or whih the struture is to be heked. The program reates a number o deault design load ombinations or a onrete rame design. Users an add their own design load ombinations as well as modiy or delete the program deault design load ombinations. An unlimited number o design load ombinations an be speiied. To deine a design load ombination, simply speiy one or more load ases, eah with its own sale ator. The sale ators are applied to the ores and moments rom the load ases to orm the atored design ores and moments or eah design load ombination. There is one exeption to the preeding. For spetral analysis modal ombinations, any orrespondene between the signs o the moments and axial loads is lost. The program uses eight design load ombinations or eah suh loading ombination speiied, reversing the sign o axial loads and moments in major and minor diretions. As an example, i a struture is subjeted to dead load, DL, and live load, LL, only, the CSA A design hek may need only one design load ombination, namely, 1.25 DL +1.5 LL. However, i the struture is subjeted to wind, earthquake or other loads, numerous additional design load ombinations may be required. The program allows live load redution ators to be applied to the member ores o the reduible live load ase on a member-by-member basis to redue the ontribution o the live load to the atored responses. 3-4 Design Load Combinations

21 Chapter 3 - Design Proess The design load ombinations are the various ombinations o the presribed load ases or whih the struture is to be heked. For this ode, i a struture is subjeted to dead load (DL), live load (LL), wind (WL), and earthquake (EL) loads, and onsidering that wind and earthquake ores are reversible, the ollowing load ombinations should be onsidered (CSA ): 1.4 DL (CSA 8.3.2, Table C.1, Case 1) 1.25 DL LL (CSA 8.3.2, Table C.1, Case 2) 1.25 DL ± 1.40 WL (CSA 8.3.2, Table C.1, Case 4) 0.90 DL ± 1.40 WL (CSA 8.3.2, Table C.1, Case 4) 1.25 DL LL ± 1.40 WL (CSA 8.3.2, Table C.1, Case 4) 1.00 DL ± 1.00 EL (CSA 8.3.2, Table C.1, Case 5) 1.00 DL LL ± 1.00 EL (CSA 8.3.2, Table C.1, Case 5) These are also the deault design load ombinations in the program whenever the CSA A ode is used. In generating the preeding deault loading ombinations, the importane ator is taken as 1. The user should use other appropriate design load ombinations i roo live load is separately treated, or i other types o loads are present. PLL is the live load multiplied by the Pattern Live Load Fator. The Pattern Live Load Fator an be speiied in the Preerenes. When using the CSA A ode, the program design assumes that a P-delta analysis has been perormed. 3.3 Limits on Material Strength The upper and lower limits o should be 80 MPa and 20 MPa respetively, or all raming types (CSA , ). 80MPa (CSA , ) 20MPa (CSA ) The upper limit o y should be 500 MPa or all rames (CSA 8.5.1, ) Limits on Material Strength 3-5

22 Conrete Frame Design CSA A When the ompression strength o onrete used in design is beyond the given limits or when the yield strength o steel used in design exeeds the given limits, the ode does not over suh ases. The ode allows use o and y beyond the given limits, provided speial are is taken regarding the detailing and dutility (CSA , ). The program does not enore any o these limits or olumn P-M-M interation hek or design and lexure design o beam. The speiied strengths are used or design. The user is responsible or using the proper strength values while deining the materials. For shear design, speial limits are appliable (CSA , , ) and enored in the program as desribed herein. The ode allows the use o reinorement with y less than 400 MPa in members resisting earthquake indued ores without any restrition. The program also allows the use o y greater than 400 MPa. However, i y is between 400 MPa and 500 MPa, the ode requires that inreased strain be taken into aount beause higher-yield-strength steel, in general, redues dutility (CSA ). The program also onsiders the inreased strain through strain-ompatibility relations during P-M-M interation apaity alulations. 3.4 Strength Resistane Fators The strength resistane ator, ϕ, is material dependent and is deined as ϕ = 0.65or onrete and (CSA ) ϕ = 0.85or steel. (CSA 8.4.3a) s In some speial ases, a member resistane ator, ϕm, is used as an additional redution ator in addition to ϕ and ϕ s (CSA 8.4.1). In onnetion with bukling resistane, ϕm is taken as 0.75 (CSA ). The values o ϕ and ϕ s given herein are the deault values. These values an be modiied in the Preerenes. For strutural onrete manuatured in pre-qualiied manuaturing plants, ϕ an be taken as 0.7 (CSA 8.4.2, ). 3-6 Strength Resistane Fators

23 Chapter 3 - Design Proess 3.5 Column Design The user may deine the geometry o the reinoring bar oniguration o eah onrete olumn setion. I the area o reinoring is provided by the user, the program heks the olumn apaity. However, i the area o reinoring is not provided by the user, the program alulates the amount o reinoring required or the olumn. The design proedure or the reinored onrete olumns o the struture involves the ollowing steps: Generate axial ore/biaxial moment interation suraes or all o the dierent onrete setion types o the model. A typial biaxial interation surae is shown in Figure 3-1. When the steel is undeined, the program generates the interation suraes or the range o allowable reinorement: 1 to 8 perent or Conventional, Moderately Dutile Resisting rames (CSA and CSA ) and 1 to 6 perent or Dutile Moment Resisting rames (CSA ). Calulate the apaity ratio or the required reinoring area or the atored axial ore and biaxial (or uniaxial) bending moments obtained rom eah loading ombination at eah station o the olumn. The target apaity ratio is taken as one when alulating the required reinoring area. Design the olumn shear reinorement. The ollowing three subsetions desribe in detail the algorithms assoiated with this proess Generation o Biaxial Interation Suraes The olumn apaity interation volume is numerially desribed by a series o disrete points that are generated on the three-dimensional interation ailure surae. In addition to axial ompression and biaxial bending, the ormulation allows or axial tension and biaxial bending onsiderations. A typial interation surae is shown in Figure 3-1. The oordinates o these points are determined by rotating a plane o linear strain in three dimensions on the setion o the olumn. See Figure 3-2. The linear strain diagram limits the maximum onrete strain, ε, at the extremity o the setion to (CSA ). Column Design 3-7

24 Conrete Frame Design CSA A Figure 3-1 A typial olumn interation surae The ormulation is based onsistently on the general priniples o ultimate strength design (CSA 10.1), and allows or any doubly symmetri retangular, square, or irular olumn setion. The stress in the steel is given by the produt o the steel strain and the steel modulus o elastiity, εses, and is limited to the yield stress o the steel, y (CSA ). The area assoiated with eah reinoring bar is assumed to be plaed at the atual loation o the enter o the bar, and the algorithm does not assume any urther simpliiations in the manner in whih the area o steel is distributed over the ross-setion o the olumn (suh as an equivalent steel tube or ylinder), as shown in Figure Column Design

25 Chapter 3 - Design Proess Figure 3-2 Idealized strain distribution or generation o interation surae The onrete ompression stress blok is assumed to be retangular, with a stress value o α 1 (CSA ). See Figure 3-3. The depth o the stress blok is β 1, where α = , α 0.67 (CSA ) 1 1 β = , β (CSA ) 1 1 Column Design 3-9

26 Conrete Frame Design CSA A Figure 3-3 Idealization o stress and strain distribution in a olumn setion The interation algorithm provides a orretion to aount or the onrete area that is displaed by the reinorement in the ompression zone. The eets o the strength redution ators are inluded in the generation o the interation suraes. The maximum ompressive axial load is limited to P where the maximum atored axial load resistane is given by r,max, ( ) Pr,max = Ag Ast s y A ϕ α +ϕ st (tied olumn) ( ) Pr,max = Ag Ast s y A ϕ α +ϕ st (spiral olumn). (CSA ) (CSA ) Calulate Column Capaity Ratio The olumn apaity is heked or eah loading ombination at eah hek station o eah olumn. In heking a partiular olumn or a partiular loading ombination at a partiular loation, the program uses the ollowing steps: Determine the atored moments and ores rom the load ases and the speiied load ombination ators to give P, M x, and M y Column Design

27 Chapter 3 - Design Proess Determine the moment magniiation ators or the olumn moments. Apply the moment magniiation ators to the atored loads obtained in the irst step. Determine whether the point, deined by the resulting axial load and biaxial moment set, lies within the interation volume. The ollowing three setions desribe in detail the algorithms assoiated with this proess Determine Fatored Moments and Fores The atored loads or a partiular load ombination are obtained by applying the orresponding load ators to all the load onditions, giving P, M x, and M y. The omputed moments are urther ampliied by using Moment Magniiation Fators to allow or Lateral Drit Eet and Member Stability Eet Determine Moment Magniiation Fators The moment magniiation ators are applied in two stages. First the moments are separated into their sway and non-sway omponents. The sway omponents are ampliied or lateral drit eet (CSA 10.16). Although aording to the ode, this ampliiation is signiiant or unbraed rames only and may be avoided or braed rames, the program treats all rames uniormly to ampliy non-sway omponents o moments. These ampliied moments are urther ampliied or individual member stability eet. Aording to the ode, the individual member stability eet is only signiiant i: kl r 10M 25 M2 > P A 1 g, or non-sway rames, and (CSA ) kl r > P 35 A g, or sway rames. (CSA ) However, the program onsiders individual member stability eet or all ompression olumn elements. Column Design 3-11

28 Conrete Frame Design CSA A Lateral Drit Eet For all rames, the moment magniiation ator or lateral drit eet is applied only to the sway moment in the program. M = M + δ M (CSA ) ns s s The moment obtained rom analysis is separated into two omponents: the sway (M s) and the non-sway (M ns) omponents. The non-sway or braed omponents, whih are identiied by ns subsripts are predominantly aused by gravity load. The sway omponents are identiied by s subsripts. The sway moments are predominantly aused by lateral loads and are related to the ause o sidesway. The moment magniiation ators in the major and minor diretions an, in general, be dierent. The moment magniiation ators or moments ausing sidesway, δ sx and δ sy an be taken as 1.0 i a P-delta analysis is arried out (CSA ). The program assumes that a P-delta analysis has been perormed and, thereore, moment magniiation ators or moments ausing sidesway are taken as unity. For the P-delta analysis the load should orrespond to a load ombination o (1.25 DL LL)/ϕ m (CSA ), where ϕ m is the strength redution ator or stability and is equal to 0.75 (CSA ). See also White and Hajjar (1991). The user is reminded o the speial analysis requirements, espeially those related to the value o EI used in analysis (CSA ). The program analysis uses the EI o the gross ross-setional area in onjuntion with a multipliation ator. The multipliation ators are deined on a setion-by-setion basis, so that the eet o axial ore and raking an be modeled in a irst order analysis. I the program assumptions are not satisatory or a partiular member, the user an expliitly speiy values o δ sx and δ sy Member Stability Eets All ompression members are designed using the atored axial load, P, rom the analysis and a magniied atored moment, M. The magniied moment is omputed as, 3-12 Column Design

29 Chapter 3 - Design Proess M =δ M M (CSA ) b 2 2, where M 2 is the olumn maximum end moment obtained rom elasti analysis ater onsidering minimum eentriity and lateral drit eet, and M is the maximum moment ourring at the end or at an interior point within the span o the olumn. The moment magniiation ator, δ b, or moments not ausing sidesway assoiated with the major or minor diretion o the olumn is given by Cm δ b = 1.0, where (CSA ) P 1 ϕ P m ϕ = 0.75, (CSA ) m 2 π EI P =, (CSA ) ( kl) 2 k is onservatively taken as 1, however the user an overwrite the value, EI is assoiated with a partiular olumn diretion given by 0.4EI EI = 1+β d g, and (CSA ) C m Ma = , (CSA ) M b M a and M b are the moments at the ends o the olumn, and M b is numerially larger than M a. M a / M b is positive or single urvature bending and negative or double urvature bending. The preeding expression o C m is valid i there is no transverse load applied between the supports. I transverse load is present on the span, or the length is overwritten, or or any other ase, C m = 1 (CSA ). C m an be overwritten by the user on an element- by-element basis. The pre-magniied atored moments are inreased, i required, to obtain minimum eentriities suh that M /P is at least ( h) mm about eah axis separately, where h is the dimension o the olumn in mm units in the orresponding diretion (CSA ). Column Design 3-13

30 Conrete Frame Design CSA A ( ) M P + h (CSA ) The moment magniiation ator, δ b, must be a positive number and greater than one. Thereore P m must be less than ϕ mp. I P is ound to be greater than or equal to ϕ mp, a ailure ondition is delared. δ b is taken as 1 or tension members. The preeding alulations use the unsupported length o the olumn. The two unsupported lengths are l 22 and l 33 orresponding to instability in the minor and major diretions o the element, respetively. These are the lengths between the support points o the element in the orresponding diretions. I the program assumptions are not satisatory or a partiular member, the user an expliitly speiy values o δ s and δ b Determine Capaity Ratio As a measure o the stress ondition o the olumn, a apaity ratio is alulated. The apaity ratio is basially a ator that gives an indiation o the stress ondition o the olumn with respet to the apaity o the olumn. Beore entering the interation diagram to hek the olumn apaity, the moment magniiation ators are applied to the atored loads to obtain P, M x, and M y. The point (P, M x, M y) is then plaed in the interation spae, shown as point L in Figure 3-4. I the point lies within the interation volume, the olumn apaity is adequate; however, i the point lies outside the interation volume, the olumn is overstressed. This apaity ratio is ahieved by plotting the point L and determining the loation o point C. The point C is deined as the point where the line OL (i extended outwards) will interset the ailure surae. This point is determined by three-dimensional linear interpolation between the points that deine the ailure OL surae. See Figure 3-4. The apaity ratio, CR, is given by the ratio. OC I OL = OC (or CR = 1) the point lies on the interation surae and the olumn is stressed to apaity Column Design

31 Chapter 3 - Design Proess Figure 3-4 Geometri representation o olumn apaity ratio I OL < OC (or CR < 1) the point lies within the interation volume and the olumn apaity is adequate. I OL > OC (or CR > 1) the point lies outside the interation volume and the olumn is overstressed. The maximum o all the values o CR alulated rom eah load ombination is reported or eah hek station o the olumn, along with the ontrolling P, M x, and M y set and assoiated load ombination number. I the reinoring area is not deined, the program omputes the reinorement that will give an interation ratio o unity. Column Design 3-15

32 Conrete Frame Design CSA A Required Reinoring Area I the reinoring area is not deined, the program omputes the reinorement that will give a olumn apaity ratio equal to the Utilization Fator Limit, whih is set to 0.95 by deault Design Column Shear Reinorement The shear reinorement is designed or eah loading ombination in the major and minor diretions o the olumn. In designing the shear reinoring or a partiular olumn or a partiular loading ombination due to shear ores in a partiular diretion, the program uses the ollowing steps: Determine the atored ores ating on the setion, M, P, and V. Note that M and P are needed or the alulation o v. Determine the shear stress, v, that an be resisted by onrete alone. Calulate the reinorement steel required to arry the balane. For Dutile and Moderately Dutile moment resisting onrete rames, the shear design o the olumns is also based on the probable and nominal moment apaities, respetively, o the members in addition to the atored moments (CSA (b), (b)). Eets o the axial ores on the olumn moment apaities are inluded in the ormulation. The ollowing three setions desribe in detail the algorithms assoiated with this proess Determine Setion Fores In the design o the olumn shear reinorement o a Conventional moment resisting onrete rame, the ores or a partiular load ombination, namely, the olumn axial ore, P, the olumn moment, M, and the olumn shear ore, V, in a partiular diretion are obtained by atoring the program load ases with the orresponding load ombination ators. In the shear design o Dutile moment resisting rames (seismi design), the ollowing are heked in addition to the requirement or Conventional moment 3-16 Column Design

33 Chapter 3 - Design Proess resisting rames. In the design o Dutile moment resisting onrete rames, the design shear ore, V, in a partiular diretion is also alulated rom the probable moment apaities o the olumn assoiated with the atored axial ore ating on the olumn (CSA (b)). For eah load ombination, the atored axial load, P, is alulated. Then, the positive and negative moment apaities, M + and M j, o the olumn in a partiular diretion under the inluene o the axial ore P are alulated using the uniaxial interation diagram in the orresponding diretion. The design shear ore, V, is then given by V= Vp+ V D + L (CSA (b)) where, VP is the shear ore obtained by applying the alulated probable ultimate moment apaities at the two ends o the olumn ating in two opposite diretions. Thereore, VP is the maximum o V 1 and VP 2, where P V V + MI + MJ P1 =, L + MI + MJ P2 =, L and where M +, I M +, L J M I M J = Positive and negative moment apaities at end I o the olumn using a steel yield stress value o α y and no ϕ ϕ =ϕ = 1.0, ators ( ) s = Positive and negative moment apaities at end J o the olumn using a steel yield stress value o α y and no ϕ ϕ =ϕ = 1.0, and ators ( ) = Clear span o olumn. s For Dutile moment resisting rames, α is taken as 1.25 (CSA 2.2, ). is the ontribution o shear ore rom the in-span distribution o gravity V D + L loads. For most o the olumns, it is zero. Column Design 3-17

34 Conrete Frame Design CSA A The Vp is enored not to be greater than the atored shear ore omponent rom speiied gravity load inreased by RR(CSA d o ). In the design o Moderately Dutile moment resisting rames (seismi), the shear apaity o the olumn is also heked or the nominal shear due to the nominal ( ϕ s =ϕ = 1.0) moment apaities and the atored gravity load (CSA ), in addition to the design heks required or Conventional moment resisting rames. The proedure or alulating nominal moment apaity is the same as that or omputing the probable moment apaity or Dutile moment resisting rames, exept thatα is taken equal to 1 (CSA 2.2, a) rather than The design shear strength is enored not to be greater than the atored shear ore with the seismi load inreased by RR(CSA d o b). See also Table 3-1 or details Determine Conrete Shear Capaity Given the design ore set M, N, and V, the shear apaity provided by the onrete alone, V, is alulated as ollows: V = ϕ λβ bd, (CSA ) w ϕ is the resistane ator or onrete. By deault, it is taken as 0.65 (CSA8.4.2). For onrete produed in a pre-qualiied manuaturing plants, its value an be taken as 0.70 (CSA ). This value an be overwritten in the Preerenes. λ is the strength redution ator to aount or low density onrete (CSA 2.2). For normal density onrete, its value is 1 (CSA 8.6.5), whih is the program deault value. For onrete using lower density aggregate, the user an hange the value o λ in the material properties. The reommended values or λ is as ollows (CSA 8.6.5) Column Design

35 Chapter 3 - Design Proess 1.00, or normal density onrete, 0.85, or semi-low-density onrete λ= in whih all o the ine aggregate is natural sand, 0.75, or semi-low-density onrete in whih none o the ine aggregate is natural sand. β is the ator or aounting or the shear resistane o raked onrete (CSA 2.2). Its value is normally between 0.1 and 0.4. It is determine aording to setion o the Code, whih is desribed in the setions that ollow. b w is the width o the ross-setion resisting the shear perpendiular to the shear ore diretion. For olumns with retangular ross-setions, bw is taken as the width o the setion perpendiular to the shear diretion. For olumns with irular ross-setions, bw is taken as the average width appropriate or shear diretion. dv is the eetive shear depth. It is taken as the greater o 0.9d or 0.72h, { } d = max 0.9 d,0.72h (CSA 2.2) v where d is the distane rom the extreme ompression iber to the entroid o tension reinorement, and h is the overall depth o the ross-setion in the diretion o shear ore. The value o the β ator is preerably determine using the Simpliied method (CSA ) i appliable. When the onditions o Simpliied method do not apply, the General method (CSA ) is used. For designing onrete olumn setions or shear ore, the speial value o β does not apply (CSA ) When the axial ore is ompressive, the speiied yield strength o the longitudinal reinoring y does not exeed 400 MPa, and the speiied onrete strength does not exeed 60 MPa,β is determined in aordane with the Simpliied method, as ollows (CSA ): When the setion ontains at least the minimum transverse reinorement, β is taken as 0.18 (CSA a). Column Design 3-19

36 Conrete Frame Design CSA A β = 0.18 (CSA a) When the setion ontains no transverse reinorement, β is determined based on the speiied maximum nominal size o oarse aggregate, a g. For maximum-sized oarse aggregate not less than 20 mm, β is taken as: 230 β= dv (CSA b) where dv is the eetive shear depth expressed in millimeter, as desribed in preeding setions. For maximum sized oarse aggregate less than 20 mm, β is taken as: 230 β= sze (CSA ()) where, 35 Sze = Sz 0.85 Sz a g (CSA ) In the preeding expression, the rak spaing parameter, Sze, shall be taken as the minimum o d v and the maximum distane between layers o distributed longitudinal reinorement. However, S ze is onservatively taken as equal to d v. In summary, or simpliied ases, β an be expressed as ollows: 0.18, i minimum transverse reinorement is provided, 230 β=, i no transverse reinorement provided, and ag 20mm, 1000+dv 230, i no transverse reinorement provided, and ag < 20mm Sze 3-20 Column Design

37 Chapter 3 - Design Proess When the axial ore is tensile, or the speiied yield strength o the longitudinal reinoring y > 400 MPa, or the speiied onrete strength > 60 MPa, β is determined in aordane with the General method, as ollows (CSA , ): β= ( ε ) ( S ) x ze (CSA ) In the preeding expression, the equivalent rak spaing parameter, S ze, is taken as equal to 300 mm i minimum transverse reinorement is provided (CSA ). Otherwise it is determines as stated in simpliied method. S ze 300, i minimum transverse = reinorement is provided, 35 Sz 0.85 Sz, otherwise ag The longitudinal strain, ε x, at mid-depth o the ross-setion is omputed rom the ollowing equation: M dv + V + 0.5N ε x = (CSA ) s EA ( ) s s In evaluating the εx, the ollowing onditions apply: V and M are taken as positive quantities (CSA a) A s is taken as the total area o longitudinal reinorement in the olumn setion. For the olumn setion hek option, the program uses the sum o user-deined reinorement in the setion. For the olumn setion design option, the longitudinal reinorement area is taken as the envelope o reinorement required or all design load ombinations. Atual provided reinorement might be slightly higher than this quantity. The reinorement should be developed to ahieve ull strength (CSA b). Column Design 3-21

38 Conrete Frame Design CSA A I the value o ε x alulated rom the preeding equation is negative, it is realulated as ollows: M d ε x = 2 v + V + 0.5N ( EA+ EA ) s s t (CSA d) For setions loser than d v rom the ae o the support, ε x is alulated based on M, V, and N at a setion at a distane d v rom the ae o the support (CSA d). I the axial tension is large enough to rak the lexural ompression ae o the setion, the value o ε x is inreased by a ator o 2 (CSA e). The program uses a linear elasti stress distribution to hek this ondition. An upper limit on ε x is imposed as ollows: e (CSA ) x ε x is positive or tensile ation. N is positive or tensile ation. For Dutile moment resisting rames, an upper limit on β o 0.10 is enored (CSA ) as ollows: β 0.10 (CSA ) In both the Simpliied method and the General method, the shear strength o the setion due to onrete, v, depends on whether the minimum transverse reinorement is provided. To hek this ondition, the program perorms the design in two passes. In the irst pass, it is assumes that no transverse shear reinorement is needed. When the program determines that shear reinorement is need, the program perorms the seond pass with the assumption that at least minimum shear reinorement is provided Determine Required Shear Reinorement The minimum possible atored shear resistane, V r,max is omputed as ollows: 3-22 Column Design

39 Chapter 3 - Design Proess V = 0.25 ϕ bd. (CSA ) r,max w The shear reinorement per unit spaing, A s, is omputed as ollows: I V V, v A v s = 0 (CSA ) else i V < V Vr,max A ( V V)tan θ v = s ϕ d s yh v (CSA , ) else i V > V r,max a ailure ondition is delared. (CSA ) A minimum area o shear reinorement is provided in the ollowing regions (CSA ): (a) in regions o lexural members where the atored shear ore V exeeds V, (b) in regions o beams and olumns with an overall depth greater than 750 mm, and () in regions o beams and olumns where the atored torsion T exeeds 0.25 T r. Where the minimum shear reinorement is required by setion CSA , or by alulations, the minimum area o shear reinorement per unit spaing is taken as: A s v bw (CSA ) y Column Design 3-23