This Technical Note describes how the program calculates the moment capacity of a noncomposite steel beam, including a cover plate, if applicable.
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1 COPUTERS AND STRUCTURES, INC., BERKEEY, CAIORNIA DECEBER 001 COPOSITE BEA DESIGN AISC-RD93 Techical te This Techical te descibes how the ogam calculates the momet caacit of a ocomosite steel beam, icludig a cove late, if alicable. Oveview The ogam ol calculates the momet caacit,, if the beam is comact o ocomact. It does ot calculate if the sectio is slede. The lastic momet,, fo a ocomosite olled steel beam sectio without a cove late is calculated as Z. The exact methodolog used to comute the lastic momet caacit i the othe cases deeds o whethe the beam, icludig the cove late if it exists, is doubl o sigl smmetic, ad whethe the beam web is classified as comact o ocomact. igue 1 shows a flowchat that diects ou to the aoiate sectio i this chate fo calculatig the momet caacit of the steel sectio aloe. The figue has boxes labeled a though g; stat i the box labeled a. te that the citeia used b the ogam to detemie if a sectio is comact o ocomact fo the AISC-RD93 secificatio is descibed i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. Steel Beam Poeties If oeties fo the steel sectio aloe ae available diectl fom the ogam's sectio database, those oeties ae used to comute the momet caacit. o othe cases such as a use-defied sectio o a sectio with a cove late, the sectio oeties ae calculated i a mae simila to that descibed i Techical te Tasfomed Sectio omet of Ietia Comosite Beam Desig AISC-ASD89, excet that thee is o cocete o eifocig steel to coside. Oveview Page 1 of 13
2 Comosite Beam Desig AISC-RD93 Is sectio doubl smmetic o a chael? Is the beam web comact? Is the beam web ocomact? a b c Beam sectio is classified as slede ad is ot desiged. Go to ext tial sectio. Refe to omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio i this Techical te. Refe to omet Caacit fo a Sigl Smmetic Beam with a Comact Web i this Techical te. Refe to omet Caacit fo a Sigl Smmetic Beam with a comact Web i this Techical te. d igue 1: e f g lowchat o Detemiig Which Sectio of this Chate Alies i Calculatig Plastic omet fo Steel Sectio Aloe Afte the momet of ietia has bee calculated, the sectio moduli ad adius of gatio ae calculated usig stadad fomulas. This ocess is eeated to get oeties about both axes. The tosioal costat is detemied b summig u the tosioal costats fo the vaious comoets of the sectio. o examle it ma be detemied b summig the J's of a olled sectio ad the cove late, if alicable, o i a use-defied sectio, b summig the J's fo the to flage, web, bottom flage ad cove late, if alicable. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio igue shows a flowchat that detemies the equatios the ogam uses to calculate fo a doubl smmetic steel sectio aloe o a chael sectio aloe. The figue has boxes labeled a though k; stat i the box labeled a. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page of 13
3 Comosite Beam Desig AISC-RD93 Ae the web, flage ad cove late comact? a Is the web ocomact? Is b? g Beam sectio ot desiged. Go to ext tial sectio. k Is b? Is b? h b based o ieldig citeia i AISC- RD93 Sectio 1.1. c d Ae the flage ad cove late comact? i Beam sectio ot desiged. Go to ext tial sectio. e of ieldig citeia i AISC-RD93 Sectio 1.1 ad lateal tosioal bucklig citeia i AISC-RD93 Sectio 1.a. f of ieldig citeia i AISC-RD93 Sectio 1.1, lateal tosioal bucklig citeia i AISC-RD93 Sectio 1.a ad flage ad web local bucklig citeia i AISC- RD93 Aedix 1(b) equatio (A- 1-3). j igue : lowchat o Calculatig fo a Doubl Smmetic Steel Sectio Aloe o a Rolled Chael Steel Sectio Aloe Ifomatio elatig to how the ogam calculates the comact ad ocomact sectio equiemets is i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. The followig subsectio discusses the ubaced legth checks i the ogam that ae used to detemie how to calculate fo a doubl smmetic beam o a chael sectio. Subsequet subsectios discuss each of the code sectios metioed i igue that ae used to calculate the momet caacit. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 3 of 13
4 Comosite Beam Desig AISC-RD93 ateal Ubaced egth Checks The ubaced legths listed i igue ae b, ad. Defiitios of each of these items ae listed below. b ateall ubaced legth of beam; legth betwee oits which ae baced agaist lateal dislacemet of the flage, i. imitig lateall ubaced legth of beam fo full lastic bedig caacit, i. imitig lateall ubaced legth of beam fo ielastic lateal-tosioal bucklig, i. The ubaced legth of a beam, o a beam segmet, b is detemied fom the iut data. The limitig ubaced legth fo full lastic caacit,, is detemied fom Equatio 1 which is also Equatio 1-4 i AISC-RD Eq. 1 f I Equatio 1, is take fo the steel beam sectio icludig the cove late, if alicable. The f tem i Equatio 1 is fo the flage. The limitig ubaced legth fo lateal tosioal bucklig,, is detemied fom Equatio which is also Equatios 1-6 though 1-8 i AISC-RD93. X X 1 1 π S x 1+ EGJA 1+ X smalleof ( f ad, whee X ) ad C 4 I w w Sx GJ Eq. I Equatio,, the comessive esidual stess i the flage, is take as 10 ksi fo olled shaes ad 16.5 ksi fo use-defied shaes. The waig costat, C w, is based o the steel beam aloe igoig the cove late if it exists. o olled sectios, icludig chaels, the ogam takes C w fom its built-i database. o use-defied sectios C w is calculated usig Equatio 3. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 4 of 13
5 Comosite Beam Desig AISC-RD93 te that Equatio 3 actuall alies to smmetical sectios but it is also used whe the flages have diffeet dimesios. C w I d t f to 4 t f bot Eq. 3 Yieldig Citeia i AISC-RD93 Sectio 1.1 The ieldig citeia is that. The ocess fo detemiig has bee eviousl descibed i the sectio etitled "Oveview" i this techical ote. ateal Tosioal Bucklig Citeia i AISC-RD93 Sectio 1.a The lateal tosioal bucklig citeia i AISC RD 1.a is based o AISC- RD93 Equatio 1-. I this case is give b Equatio 4. b Cb ( ) Eq. 4 I Equatio 4, C b is calculated usig Equatio 5, which is also AISC-RD93 Equatio 1-3. C b max Eq. 5.5 max A B + 3 C Refe to the otatio i Techical te Geeal ad tatio Comosite Beam Desig AISC-RD93 fo a exlaatio of the tems i Equatio 5. I Equatio 4, is calculated usig Equatio, is calculated fom Equatio 1 ad comes fom Equatio 6. S Eq. 6 x whee is as descibed fo Equatio. AISC-RD Aedix 1(b) Equatio A-1-3 The limit state fo flage ad web local bucklig is based o AISC-RD93 Equatio A-1-3, which is show heei as Equatio 7. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 5 of 13
6 Comosite Beam Desig AISC-RD93 λ λ Eq. 7 ( ) λ λ Equatio 7 alies to both flage local bucklig ad web local bucklig. lage ocal Bucklig o flage local bucklig usig Equatio 7:! is calculated e Equatio 6.! λis equal to b f /(t f ) fo I-sectios ad b f /t f fo chaels. The b f ad t f tems ae fo the flage.! λ is give b Equatio 8a if the sectio is a olled o use-defied I- sectio, o Equatio 8b if the sectio is a olled chael. The f i these equatios is fo the flage. bf 65 Eq. 8a t f f bf 65 Eq. 8b t f f! λ is give b Equatio 9a if the sectio is a olled beam o chael, o Equatio 9b if it is a use-defied sectio. 141 λ, fo olled shaes Eq. 9a 16 λ, fo use-defied shaes Eq. 9b k c I Equatio 9a ad 9b, is as defied fo Equatio. I Equatio 9b, k 4 h but ot less tha 0.35 k c Equatios 9a ad 9b ae c t w take fom AISC-RD93 Table A-1.1. Web ocal Bucklig o web local bucklig usig Equatio 7: omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 6 of 13
7 Comosite Beam Desig AISC-RD93! is calculated usig Equatios 10 ad 11 fo both the to ad bottom flages seaatel. The smalle value of is used. R e f S x Eq. 10 I Equatio 10, R e is equal to 1.0 fo olled shaes ad is give b Equatio 11 fo use-defied shaes. Equatio 10 is take fom AISC-RD93 Table A R e 3 ( 3m m ) a Eq a Equatio 11 comes fom the defiitio of R e give with Equatio A-G-3 i AISC-RD93 Aedix G. I Equatio 11 the tem a is the atio of the web aea (ht w ) to the flage aea (b f t f ), but ot moe tha 10, ad m is the atio of the web ield stess to the flage ield stess.! λ is equal to h/t w.! λ is give b Equatio 5a, o 5b i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93 deedig o the axial load i the membe, if a. See the descitio accomaig these equatios fo moe ifomatio.! λ is give b oe of Equatios 6 ad 7 i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93 deedig o the te of membe ad the amout of axial, if a. See the descitio accomaig these equatios fo moe ifomatio. omet Caacit fo a Sigl Smmetic Beam with a Comact Web igue 3 shows a flowchat that detemies the equatios the ogam uses to calculate fo a sigl smmetic steel sectio aloe with a comact web. The figue has boxes labeled a though ; stat i the box labeled a. ost of the fomulas associated with this flowchat ae based o AISC- RD93 Secificatio Aedix sectio 1ad Table A-1.1. omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 7 of 13
8 Comosite Beam Desig AISC-RD93 Is web comact? a Ae the flage ad cove late comact? b This is the wog flowchat. See igue 1. e Beam sectio ot desiged. Go to ext tial sectio. te: Ae the flage ad cove late ocomact? f WB Web local bucklig B lage local bucklig TB ateal tosioal bucklig Beam sectio ot desiged. Go to ext tial sectio. Is beam comact fo TB? c g Is beam ocomact fo TB? h Is beam comact fo TB? j l Is beam ocomact fo TB? m AISC-RD93 Aedix A-1-1 fo WB A-1-1 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-1 fo WB A-1-1 fo B A-1- fo TB. AISC-RD93 Aedix A-1-1 fo WB A-1-3 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-1 fo WB A-1-3 fo B A-1- fo TB. d i k igue 3: lowchat o Calculatig fo a Sigl Smmetic Steel Sectio Aloe with a Comact Web Ifomatio elatig to how the ogam calculates the comact ad ocomact sectio equiemets is i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. The followig subsectio descibes the lateal tosioal bucklig (TB) checks i the ogam that ae used to detemie how to calculate fo a sigl smmetic beam with a comact web. Subsequet subsectios descibe each of the AISC-RD93 Secificatio Aedix equatios metioed i igue 3 that ae used to calculate the momet caacit. AISC-RD93 Equatio A-1-1 fo WB o this case is equal to, the lastic momet caacit of the sectio. omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 8 of 13
9 Comosite Beam Desig AISC-RD93 AISC-RD93 Equatio A-1-1 fo B o this case is equal to, the lastic momet caacit of the sectio. AISC-RD93 Equatio A-1-3 fo B AISC-RD93 Equatio A-1-3 fo flage local bucklig is iteeted b the ogam as show i Equatios 1a though 1f. λ λ λ λ ( ) Eq. 1a whee S λ b f t f x Eq. 1b Eq. 1c λ 65 f Eq. 1d 141 λ, olled beams ad chaels Eq. 1e λ 16, use-defied beams Eq. 1f k c I Equatio 1b, ad S x ae fo the beam flage (ot cove late). I Equatios 1c ad 1d, b f, t f ad f ae fo the beam flage (ot cove late). I Equatio 1e, is fo the beam flage (ot cove late). I Equatio 1f, is fo the beam flage (ot cove late), ad k 4 h but ot less tha 0.35 k c c t w omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 9 of 13
10 Comosite Beam Desig AISC-RD93 AISC-RD93 Equatio A-1-1 fo TB o this case is equal to, the lastic momet caacit of the sectio. AISC-RD93 Equatio A-1- fo TB AISC-RD93 Equatio A-1- fo lateal tosioal bucklig is iteeted b the ogam as show i Equatios 13a though 13d ad Equatios 14a though 14c. C b λ λ λ λ ( ) Eq. 13a whee, S S Eq. 13b xc f xt λ b c Eq. 13c λ 300 f Eq. 13d The tem λ i Equatio 13a is the value of λ fo which c as defied b Equatios 14a though 14c is equal to the smalle of S xc ad f S xt whee is the smalle of ( f - ) ad w. Whe calculatig, the tem f is the ield stess of the flage ad whe calculatig f S xt, the tem f is the ield stess of the tesio flage. whee, B 1 c ( 57000)( 1) b I J B B + B 1 Eq. 14a I c h I.5 1 Eq. 14b I b J I c I c h B 5 1 I J Eq. 14c b omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 10 of 13
11 Comosite Beam Desig AISC-RD93 To calculate λ fo Equatio 13a, the ogam detemies the value of b fo which c is equal to the smalle of S xc ad f S xt. The it divides that value of b b c to get λ. omet Caacit fo a Sigl Smmetic Beam with a comact Web igue 4 shows a flowchat that detemies the equatios the ogam uses to calculate fo a sigl smmetic steel sectio aloe with a ocomact web. The figue has boxes labeled a though ; stat i the box labeled a. Is web ocomact? a Ae the flage ad cove late comact? b This is the wog flowchat. See igue 1. e Beam sectio ot desiged. Go to ext tial sectio. te: Ae the flage ad cove late ocomact? f WB Web local bucklig B lage local bucklig TB ateal tosioal bucklig Beam sectio ot desiged. Go to ext tial sectio. Is beam comact fo TB? c g Is beam ocomact fo TB? h Is beam comact fo TB? j l Is beam ocomact fo TB? m AISC-RD93 Aedix A-1-3 fo WB A-1-1 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-3 fo WB A-1-1 fo B A-1- fo TB. AISC-RD93 Aedix A-1-3 fo WB A-1-3 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-3 fo WB A-1-3 fo B A-1- fo TB. d i k igue 4: lowchat fo Calculatig fo a Sigl Smmetic Steel Sectio Aloe with a comact Web omet Caacit fo a Sigl Smmetic Beam with a comact Web Page 11 of 13
12 Comosite Beam Desig AISC-RD93 ost of the fomulas associated with this flowchat ae based o AISC- RD93 Secificatio Aedix sectio 1ad Table A-1.1. Ifomatio elatig to how the ogam calculates the comact ad ocomact sectio equiemets is i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. The lateal tosioal bucklig checks ad all but oe of the Aedix equatios metioed i igue 4 ae descibed i the evious sectio etitled, "omet Caacit fo a Sigl Smmetic Beam with a Comact Web." Refe to that sectio fo moe ifomatio. The oe equatio that has ot bee descibed eviousl is AISC-RD93 Secificatio Aedix Equatio A-1-3. This equatio is descibed i the followig subsectio. AISC-RD93 Equatio A-1-3 fo WB AISC-RD93 Equatio A-1-3 fo web local bucklig is iteeted b the ogam as show i Equatios 15a though 15g. λ λ λ λ ( ) Eq. 15a I Equatio 15a:! is calculated usig Equatios 15b ad 15c fo both the to ad bottom flages seaatel. The smalle value of is used. R e f S x Eq. 15b I Equatio 15b, R e is give b Equatio 15c. Equatio 15b is take fom AISC-RD93 Table A-1.1. R e 3 ( 3m m ) a Eq. 15c 1 + a Equatio 15c comes fom the defiitio of R e give with Equatio A-G-3 i AISC-RD93 Aedix G. I Equatio 15c, the tem a is the atio of the web aea (ht w ) to the flage aea (b f t f ), but ot moe tha 10, ad m is the atio of the web ield stess to the flage ield stess. omet Caacit fo a Sigl Smmetic Beam with a comact Web Page 1 of 13
13 Comosite Beam Desig AISC-RD93! λ is equal to h/t w.! λ is give b Equatio 15d, o 15e deedig o the axial load i the membe, if a. λ P P 1 u u, fo P P φ b φb 0.15 Eq. 15d λ 191 Pu.33 φbp 53, Pu fo φ P b > 0.15 Eq. 15e! λ is give b eithe Equatio 15f o Equatio 15g. Equatio 15f defies λ fo beams with equal sized flages. λ P 1 φ b P u Eq. 15f I Equatio 15f, the value of used is the lagest of the values fo the beam flages ad the web. Equatio 15g defies the ocomact sectio limit fo webs i beams with uequal size flages: λ whee, h h h h c c 0.74P 1 φ b P 3 u, Eq. 15g I Equatio 15g, the value of used is the lagest of the values fo the beam flages ad the web. Equatio 15g is based o Equatio A-B5-1 i the AISC-RD93 secificatio. omet Caacit fo a Sigl Smmetic Beam with a comact Web Page 13 of 13
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