©COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001 COMPOSITE BEAM DESIGN AISC-LRFD93 Technical Note Moment Capacity for Steel Section Alone This Technical Note describes how the program calculates the moment capacity of a noncomposite steel beam, including a cover plate, if applicable. Overview The program only calculates the moment capacity, Mn, if the beam is compact or noncompact. It does not calculate Mn if the section is slender. The plastic moment, Mp, for a noncomposite rolled steel beam section without a cover plate is calculated as Mp = ZFy. The exact methodology used to compute the plastic moment capacity in the other cases depends on whether the beam, including the cover plate if it exists, is doubly or singly symmetric, and whether the beam web is classified as compact or noncompact. Figure 1 shows a flowchart that directs you to the appropriate section in this chapter for calculating the moment capacity of the steel section alone. The figure has boxes labeled a through g; start in the box labeled a. Note that the criteria used by the program to determine if a section is compact or noncompact for the AISC-LRFD93 specification is described in Technical Note Compact and Noncompact Requirements Composite Beam Design AISC-LRFD93. Steel Beam Properties If properties for the steel section alone are available directly from the program's section database, those properties are used to compute the moment capacity. For other cases such as a user-defined section or a section with a cover plate, the section properties are calculated in a manner similar to that described in Technical Note Transformed Section Moment of Inertia Composite Beam Design AISC-ASD89, except that there is no concrete or reinforcing steel to consider. Overview Page 1 of 13 Composite Beam Design AISC-LRFD93 Is section doubly No symmetric or a channel? Yes a Moment Capacity for Steel Section Alone Is the beam web compact? No Is the beam web noncompact? Yes b Yes c Refer to “Moment Capacity for a Doubly Symmetric Beam or a Channel Section” in this Technical Note. Refer to “Moment Capacity for a Singly Symmetric Beam with a Compact Web” in this Technical Note. Refer to “Moment Capacity for a Singly Symmetric Beam with a Noncompact Web” in this Technical Note. e f g Figure 1: No Beam section is classified as slender and is not designed. Go to next trial section. d Flowchart For Determining Which Section of this Chapter Applies in Calculating Plastic Moment for Steel Section Alone After the moment of inertia has been calculated, the section moduli and radius of gyration are calculated using standard formulas. This process is repeated to get properties about both axes. The torsional constant is determined by summing up the torsional constants for the various components of the section. For example it may be determined by summing the J's of a rolled section and the cover plate, if applicable, or in a user-defined section, by summing the J's for the top flange, web, bottom flange and cover plate, if applicable. Moment Capacity for a Doubly Symmetric Beam or a Channel Section Figure 2 shows a flowchart that determines the equations the program uses to calculate Mn for a doubly symmetric steel section alone or a channel section alone. The figure has boxes labeled a through k; start in the box labeled a. Moment Capacity for a Doubly Symmetric Beam or a Channel Section Page 2 of 13 Composite Beam Design AISC-LRFD93 Are the web, compression flange and compression cover plate compact? Yes Moment Capacity for Steel Section Alone Is the web noncompact? No No Yes g a Yes b Yes h Is Lb ≤ Lr? No Yes d Determine Mn based on yielding criteria in AISCLRFD93 Section F1.1. e Are the compression flange and compression cover plate compact? No Yes i c Beam section not designed. Go to next trial section. k No Is Lb ≤ Lr? No Is Lb ≤ Lp? Beam section not designed. Go to next trial section. Determine Mn based on smallest of yielding criteria in AISC-LRFD93 Section F1.1 and lateral torsional buckling criteria in AISC-LRFD93 Section F1.2a. f Determine Mn based on smallest of yielding criteria in AISC-LRFD93 Section F1.1, lateral torsional buckling criteria in AISC-LRFD93 Section F1.2a and flange and web local buckling criteria in AISCLRFD93 Appendix F1(b) equation (AF1-3). j Figure 2: Flowchart For Calculating Mn for a Doubly Symmetric Steel Section Alone or a Rolled Channel Steel Section Alone Information relating to how the program calculates the compact and noncompact section requirements is in Technical Note Compact and Noncompact Requirements Composite Beam Design AISC-LRFD93. The following subsection discusses the unbraced length checks in the program that are used to determine how to calculate Mn for a doubly symmetric beam or a channel section. Subsequent subsections discuss each of the code sections mentioned in Figure 2 that are used to calculate the moment capacity. Moment Capacity for a Doubly Symmetric Beam or a Channel Section Page 3 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone Lateral Unbraced Length Checks The unbraced lengths listed in Figure 2 are Lb, Lp and Lr. Definitions of each of these items are listed below. Lb = Laterally unbraced length of beam; length between points which are braced against lateral displacement of the compression flange, in. Lp = Limiting laterally unbraced length of beam for full plastic bending capacity, in. Lr = Limiting laterally unbraced length of beam for inelastic lateral-torsional buckling, in. The unbraced length of a beam, or a beam segment, Lb is determined from the input data. The limiting unbraced length for full plastic capacity, Lp, is determined from Equation 1 which is also Equation F1-4 in AISC-LRFD93. Lp = 300ry Eqn. 1 Fyf In Equation 1, ry is taken for the steel beam section including the cover plate, if applicable. The Fyf term in Equation 1 is for the compression flange. The limiting unbraced length for lateral torsional buckling, Lr, is determined from Equation 2 which is also Equations F1-6 through F1-8 in AISC-LRFD93. Lr = ry X 1 FL π X1 = Sx 1 + 1 + X 2 FL2 , where EGJA 2 C S and X 2 = 4 w x I y GJ 2 Eqn. 2 FL = smaller of (Fyf − Fr ) and Fyw In Equation 2, Fr, the compressive residual stress in the flange, is taken as 10 ksi for rolled shapes and 16.5 ksi for user-defined shapes. The warping constant, Cw, is based on the steel beam alone ignoring the cover plate if it exists. For rolled sections, including channels, the program takes Cw from its built-in database. For user-defined sections Cw is calculated using Equation 3. Moment Capacity for a Doubly Symmetric Beam or a Channel Section Page 4 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone Note that Equation 3 actually applies to symmetrical sections but it is also used when the flanges have different dimensions. t f − top t f − bot − I y d − 2 2 Cw = 4 2 Eqn. 3 Yielding Criteria in AISC-LRFD93 Section F1.1 The yielding criteria is that Mn = Mp. The process for determining Mp has been previously described in the section entitled "Overview" in this technical note. Lateral Torsional Buckling Criteria in AISC-LRFD93 Section F1.2a The lateral torsional buckling criteria in AISC LRFD F1.2a is based on AISCLRFD93 Equation F1-2. In this case Mn is given by Equation 4. M n = C b M p − M p − M r ( ) LL b r − L p ≤ M p − L p Eqn. 4 In Equation 4, Cb is calculated using Equation 5, which is also AISC-LRFD93 Equation F1-3. Cb = 2.5M max 12.5M max + 3M A + 4M B + 3M C Eqn. 5 Refer to the notation in Technical Note General and Notation Composite Beam Design AISC-LRFD93 for an explanation of the terms in Equation 5. In Equation 4, Lr is calculated using Equation 2, Lp is calculated from Equation 1 and Mr comes from Equation 6. M r = FL S x Eqn. 6 where FL is as described for Equation 2. AISC-LRFD Appendix F1(b) Equation A-F1-3 The limit state for flange and web local buckling is based on AISC-LRFD93 Equation A-F1-3, which is shown herein as Equation 7. Moment Capacity for a Doubly Symmetric Beam or a Channel Section Page 5 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone λ − λp Mn = Mp − Mp − Mr λr − λ p ( ) Eqn. 7 Equation 7 applies to both flange local buckling and web local buckling. Flange Local Buckling For flange local buckling using Equation 7: ! Mr is calculated per Equation 6. ! λis equal to bf /(2tf) for I-sections and bf/tf for channels. The bf and tf terms are for the compression flange. ! ! λp is given by Equation 8a if the section is a rolled or user-defined Isection, or Equation 8b if the section is a rolled channel. The Fyf in these equations is for the compression flange. bf 65 ≤ 2t f Fyf Eqn. 8a bf 65 ≤ tf Fyf Eqn. 8b λr is given by Equation 9a if the section is a rolled beam or channel, or Equation 9b if it is a user-defined section. λr = λr = 141 , for rolled shapes FL 162 FL , for user-defined shapes Eqn. 9a Eqn. 9b kc In Equation 9a and 9b, FL is as defined for Equation 2. In Equation 9b, kc = 4 h t w but not less than 0.35 ≤ kc ≤ 0.763. Equations 9a and 9b are taken from AISC-LRFD93 Table A-F1.1. Web Local Buckling For web local buckling using Equation 7: Moment Capacity for a Doubly Symmetric Beam or a Channel Section Page 6 of 13 Composite Beam Design AISC-LRFD93 ! Moment Capacity for Steel Section Alone Mr is calculated using Equations 10 and 11 for both the top and bottom flanges separately. The smaller value of Mr is used. Mr = ReFyfSx Eqn. 10 In Equation 10, Re is equal to 1.0 for rolled shapes and is given by Equation 11 for user-defined shapes. Equation 10 is taken from AISC-LRFD93 Table AF1.1. ( ) 12 + a r 3m − m 3 ≤ 1.0 Re = 12 + 2a r Eqn. 11 Equation 11 comes from the definition of Re given with Equation A-G2-3 in AISC-LRFD93 Appendix G. In Equation 11 the term ar is the ratio of the web area (htw) to the flange area (bftf), but not more than 10, and m is the ratio of the web yield stress to the flange yield stress. ! λ is equal to h/tw. ! λp is given by Equation 5a, or 5b in Technical Note Compact and Noncompact Requirements Composite Beam Design AISC-LRFD93 depending on the axial load in the member, if any. See the description accompanying these equations for more information. ! λr is given by one of Equations 6 and 7 in Technical Note Compact and Noncompact Requirements Composite Beam Design AISC-LRFD93 depending on the type of member and the amount of axial compression, if any. See the description accompanying these equations for more information. Moment Capacity for a Singly Symmetric Beam with a Compact Web Figure 3 shows a flowchart that determines the equations the program uses to calculate Mn for a singly symmetric steel section alone with a compact web. The figure has boxes labeled a through n; start in the box labeled a. Most of the formulas associated with this flowchart are based on AISCLRFD93 Specification Appendix F section F1and Table A-F1.1. Moment Capacity for a Singly Symmetric Beam with a Compact Web Page 7 of 13 Composite Beam Design AISC-LRFD93 Is web compact? No Yes a Are the compression flange and compression cover plate compact? Moment Capacity for Steel Section Alone This is the wrong flowchart. See Figure 1. e No Note: WLB = Web local buckling FLB = Flange local buckling LTB = Lateral torsional buckling Are the compression flange and compression cover plate noncompact? No Yes b Beam section not designed. Go to next trial section. Is beam compact for LTB? Is beam compact for LTB? c g No Is beam noncompact for LTB? Yes h Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-1 for WLB A-F1-1 for FLB A-F1-1 for LTB. Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-1 for WLB A-F1-1 for FLB A-F1-2 for LTB. Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-1 for WLB A-F1-3 for FLB A-F1-1 for LTB. Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-1 for WLB A-F1-3 for FLB A-F1-2 for LTB. d i k n Yes Figure 3: No Yes Beam section not designed. Go to next trial section. f Yes j No l No Is beam noncompact for LTB? Yes m Flowchart For Calculating Mn for a Singly Symmetric Steel Section Alone with a Compact Web Information relating to how the program calculates the compact and noncompact section requirements is in Technical Note Compact and Noncompact Requirements Composite Beam Design AISC-LRFD93. The following subsection describes the lateral torsional buckling (LTB) checks in the program that are used to determine how to calculate Mn for a singly symmetric beam with a compact web. Subsequent subsections describe each of the AISC-LRFD93 Specification Appendix F equations mentioned in Figure 3 that are used to calculate the moment capacity. AISC-LRFD93 Equation A-F1-1 for WLB For this case Mn is equal to Mp, the plastic moment capacity of the section. Moment Capacity for a Singly Symmetric Beam with a Compact Web Page 8 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone AISC-LRFD93 Equation A-F1-1 for FLB For this case Mn is equal to Mp, the plastic moment capacity of the section. AISC-LRFD93 Equation A-F1-3 for FLB AISC-LRFD93 Equation A-F1-3 for flange local buckling is interpreted by the program as shown in Equations 12a through 12f. λ − λp Mn = Mp − Mp − Mr λr − λ p ( ) ≤ Mp Eqn. 12a where M r = FL S x Eqn. 12b bf 2t f Eqn. 12c λ= λp = 65 Fyf λr = 141 λr = 162 FL FL kc Eqn. 12d , rolled beams and channels Eqn. 12e , user-defined beams Eqn. 12f In Equation 12b, FL and Sx are for the beam compression flange (not cover plate). In Equations 12c and 12d, bf, tf and Fyf are for the beam compression flange (not cover plate). In Equation 12e, FL is for the beam compression flange (not cover plate). In Equation 12f, FL is for the beam compression flange (not cover plate), and kc = 4 h t w but not less than 0.35 ≤ kc ≤ 0.763. Moment Capacity for a Singly Symmetric Beam with a Compact Web Page 9 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone AISC-LRFD93 Equation A-F1-1 for LTB For this case Mn is equal to Mp, the plastic moment capacity of the section. AISC-LRFD93 Equation A-F1-2 for LTB AISC-LRFD93 Equation A-F1-2 for lateral torsional buckling is interpreted by the program as shown in Equations 13a through 13d and Equations 14a through 14c. M n = C b M p − M p − M r ( ) λλ −−λλ p r p ≤ M p Eqn. 13a where, M r = FL S xc ≤ Fyf S xt λ= Eqn. 13b Lb ryc λp = Eqn. 13c 300 Eqn. 13d Fyf The term λr in Equation 13a is the value of λ for which Mcr as defined by Equations 14a through 14c is equal to the smaller of FLSxc and FyfSxt where FL is the smaller of (Fyf - Fr) and Fyw. When calculating FL, the term Fyf is the yield stress of the compression flange and when calculating FyfSxt, the term Fyf is the yield stress of the tension flange. M cr = (57000)(1) Lb I y J B1 + 1 + B 2 + B12 Eqn. 14a where, I yc h − 1 B1 = 2.25 2 I y L b Iy I yc I yc h B 2 = 25 1 − J Lb I y J Eqn. 14b 2 Moment Capacity for a Singly Symmetric Beam with a Compact Web Eqn. 14c Page 10 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone To calculate λr for Equation 13a, the program determines the value of Lb for which Mcr is equal to the smaller of FLSxc and FyfSxt. Then it divides that value of Lb by ryc to get λr. Moment Capacity for a Singly Symmetric Beam with a Noncompact Web Figure 4 shows a flowchart that determines the equations the program uses to calculate Mn for a singly symmetric steel section alone with a noncompact web. The figure has boxes labeled a through n; start in the box labeled a. No Is web noncompact? Yes a Are the compression flange and compression cover plate compact? This is the wrong flowchart. See Figure 1. Note: WLB = Web local buckling FLB = Flange local buckling LTB = Lateral torsional buckling e No Are the compression flange and compression cover plate noncompact? No Yes b Beam section not designed. Go to next trial section. Is beam compact for LTB? Is beam compact for LTB? Yes c g No Is beam noncompact for LTB? Yes h Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-3 for WLB A-F1-1 for FLB A-F1-1 for LTB. Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-3 for WLB A-F1-1 for FLB A-F1-2 for LTB. Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-3 for WLB A-F1-3 for FLB A-F1-1 for LTB. Determine Mn based on smallest of the following AISC-LRFD93 Appendix F equations: A-F1-3 for WLB A-F1-3 for FLB A-F1-2 for LTB. d i k n Figure 4: No Yes Beam section not designed. Go to next trial section. f No Yes j l No Is beam noncompact for LTB? Yes m Flowchart for Calculating Mn for a Singly Symmetric Steel Section Alone with a Noncompact Web Moment Capacity for a Singly Symmetric Beam with a Noncompact Web Page 11 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone Most of the formulas associated with this flowchart are based on AISCLRFD93 Specification Appendix F section F1and Table A-F1.1. Information relating to how the program calculates the compact and noncompact section requirements is in Technical Note Compact and Noncompact Requirements Composite Beam Design AISC-LRFD93. The lateral torsional buckling checks and all but one of the Appendix F equations mentioned in Figure 4 are described in the previous section entitled, "Moment Capacity for a Singly Symmetric Beam with a Compact Web." Refer to that section for more information. The one equation that has not been described previously is AISC-LRFD93 Specification Appendix F Equation A-F1-3. This equation is described in the following subsection. AISC-LRFD93 Equation A-F1-3 for WLB AISC-LRFD93 Equation A-F1-3 for web local buckling is interpreted by the program as shown in Equations 15a through 15g. λ − λp Mn = Mp − Mp − Mr λr − λ p ( ) ≤ Mp Eqn. 15a In Equation 15a: ! Mr is calculated using Equations 15b and 15c for both the top and bottom flanges separately. The smaller value of Mr is used. Mr = ReFyfSx Eqn. 15b In Equation 15b, Re is given by Equation 15c. Equation 15b is taken from AISC-LRFD93 Table A-F1.1. Re = ( ) 12 + a r 3m − m 3 ≤ 1.0 12 + 2a r Eqn. 15c Equation 15c comes from the definition of Re given with Equation A-G2-3 in AISC-LRFD93 Appendix G. In Equation 15c, the term ar is the ratio of the web area (htw) to the flange area (bftf), but not more than 10, and m is the ratio of the web yield stress to the flange yield stress. Moment Capacity for a Singly Symmetric Beam with a Noncompact Web Page 12 of 13 Composite Beam Design AISC-LRFD93 Moment Capacity for Steel Section Alone ! λ is equal to h/tw. ! λp is given by Equation 15d, or 15e depending on the axial load in the member, if any. ! λp = 2.75Pu 640 1− φ b Py Fy , for Pu ≤ 0.125 φ b Py λp = P 191 2.33 − u φ b Py Fy 253 ≥ , Fy P for u > 0.125 φ b Py Eqn. 15d Eqn. 15e λr is given by either Equation 15f or Equation 15g. Equation 15f defines λr for beams with equal sized flanges. λr = 0.74Pu 970 1− φ b Py Fy Eqn. 15f In Equation 15f, the value of Fy used is the largest of the Fy values for the beam flanges and the web. Equation 15g defines the noncompact section limit for webs in beams with unequal size flanges: h 253 1 + 2.83 Fy hc where, λr = 0.74Pu 1 − φ b Py , Eqn. 15g 3 h 3 ≤ ≤ 4 hc 2 In Equation 15g, the value of Fy used is the largest of the Fy values for the beam flanges and the web. Equation 15g is based on Equation A-B5-1 in the AISC-LRFD93 specification. Moment Capacity for a Singly Symmetric Beam with a Noncompact Web Page 13 of 13
