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Beams

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T II M E S A V II N G D E S II G N A II D S T MESAV NG DES GN A DS Beams and One-Way Slabs Portland Cement Association Page 1 of 6 The following example illustrates the design methods presented in the article “Timesaving Design Aids for Reinforced Concrete, Part 1: Beams and One-way Slabs,” by David A. Fanella, which appeared in the August 2001 edition of Structural Engineer magazine. Unless otherwise noted, all referenced table, figure, and equation numbers are from that article. Example Buil
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   TTT IIIMMMEEESSSAAAVVVIIINNNGGG  DDD EEESSSIIIGGGNNN  AAA IIIDDDSSS   Beams and One-Way Slabs Portland Cement Association Page 1 of 6 The following example illustrates thedesign methods presented in the article“Timesaving Design Aids for ReinforcedConcrete, Part 1: Beams and One-waySlabs,” by David A. Fanella, whichappeared in the August 2001 edition of Structural Engineer magazine. Unlessotherwise noted, all referenced table,figure, and equation numbers are fromthat article. Example Building Below is a partial plan of a typical floor in acast-in-place reinforced concrete building.The floor framing consists of wide-modulejoists and beams. In this example, thebeams are designed and detailed for thecombined effects of gravity and lateral(wind) loads according to ACI 318-99. 30 ′ -0 ″  30 ′ -0 ″ 30 ′ -0 ″      3   2            ′   -   6            ″     3   2            ′   -   6            ″   18 ″ x18 ″ (typ.)24 ″ x 24 ″ (typ.)   TTT IIIMMMEEESSSAAAVVVIIINNNGGG  DDD EEESSSIIIGGGNNN  AAA IIIDDDSSS   Beams and One-Way Slabs Portland Cement Association Page 2 of 6 Design Data Materials ã   Concrete: normal weight (150 pcf), ¾-in. maximum aggregate, f ′ c = 4,000 psi ã   Mild reinforcing steel: Grade 60 (f y =60,000 psi)Loads ã   Joists (16 + 4½ x 6 + 66) = 76.6 psf ã   Superimposed dead loads = 30 psf ã   Live load = 100 psf ã   Wind loads: per ASCE 7-98 Gravity Load Analysis The coefficients of ACI Sect. 8.3 areutilized to compute the bending momentsand shear forces along the length of thebeam. From preliminary calculations, thebeams are assumed to be 36 x 20.5 in.Live load reduction is taken per ASCE 7-98.psf3.7eighteam 32.515014420.56 == ××  Live load reduction per ASCE 7-98 Sect.4.8.1:      += TL150.25 o  From Table 4.2 of ASCE 7-98, K LL = liveload element factor = 2 for interior beamsA T = tributary area = 32.5 x 30 = 975 ft 2  K LL A T = 2 x 975 = 1,950 ft 2 > 400 ft 2  o.59L1,950150.25o =     +=  Since the beams support only one floor, Lshall not be less than 0.50L o .Therefore, L = 0.59 x 100 = 59 psf.Total factored load w u :w u = 1.4(76.6 + 23.7 + 30) + 1.7(59)= 282.7 psf= 282.7 x 32.5/1,000 = 9.19 klfFactored reactions per ACI Sect. 8.3:Neg. M u at ext. support = w u l n2 /16= 9.19 x 28.25 2 /16= 458.4 ft-kips   TTT IIIMMMEEESSSAAAVVVIIINNNGGG  DDD EEESSSIIIGGGNNN  AAA IIIDDDSSS   Beams and One-Way Slabs Portland Cement Association Page 3 of 6 Pos. M u at end span = w u l n2 /14= 9.19 x 28.25 2 /14= 523.9 ft-kipsNeg. M u at int. col. = w u l n2 /10*= 9.19 x 28.125 2 /10= 726.9 ft-kipsPos. M u at int. span = w u l n2 /16= 9.19 x 28 2 /16= 450.3 ft-kipsV u at exterior col. = w u l n /2= 9.19 x 28.25/2= 129.8 kipsV u at interior col. = 1.15w u l n /2= 1.15 x 129.8= 149.3 kips Wind Load Analysis As noted above, wind forces are computedper ASCE 7-98. Calculations yield thefollowing reactions:M w = ±90.3 ft-kipsV w = 6.0 kips *Average of adjacent clear spans   Design for Flexure Sizing the cross-sectionPer ACI Table 9.5(a), minimum thickness = l /18.5 = (30 x 12)/18.5 = 19.5 in.Since joists are 20.5 in. deep, use 20.5-in.depth for the beams for formwork economy.With d = 20.5 – 2.5 = 18 in., solve Eq. (2)for b using maximum M u along span (note:gravity moment combination governs):bd 2 = 20M u  b = 20 x 726.9/18 2 = 44.9 in. > 36 in.This implies that using a 36-in. wide beam, ρ will be greater than 0.5 ρ max .Check minimum width based on ρ = ρ max  (see Chapter 3 of the PCA publication Simplified Design of Reinforced ConcreteBuildings of Moderate Size and Height forderivation):bd 2 = 13M u  b = 13 x 726.9/18 2 = 29.2 in. < 36 in.This implies that ρ will be less than ρ max .Use 36 x 20.5 in. beam.    TTT IIIMMMEEESSSAAAVVVIIINNNGGG  DDD EEESSSIIIGGGNNN  AAA IIIDDDSSS   Beams and One-Way Slabs Portland Cement Association Page 4 of 6 Required ReinforcementBeam moments along the span aresummarized in the table below.Load Case LocationEnd Span(ft-kips)Interior span(ft-kips)Exterior negative -211.2     Positive 241.4 207.5ead (D)Interior negative -335.0 -301.8Exterior negative -95.6     Positive 109.3 94.0ive (L)Interior negative -151.7 -136.7Exterior negative ±90.3     Positive          ind (W)Interior negative ±90.3 ±90.3No. Load CombinationExterior negative -458.4     Positive 523.9 450.31.4D + 1.7LInterior negative -726.9 -654.9-228.5     xterior negative-458.8Positive 392.8 337.7-660.3 -376.12 0.75(1.4D + 1.7L + 1.7W)Interior negative-430.0 -606.3-72.7     xterior negative-307.5Positive 217.3 186.8-418.9 -154.23 0.9D + 1.3WInterior negative-184.1 -389.0
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