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The following example illustrates the design methods presented in the PCA book “Simplified Design - Reinforced Concrete Buildings of Moderate Size and Height” third edition. Unless otherwise noted, all referenced table, figure, and equation numbers are from that book. Example Building Below is a partial plan of a typical floor in a cast-in-place reinforced concrete building. In this example, an interior strip of a flat plate floor system is designed and detailed for the effects of gravity loads according to ACI 318-05. Design Data Materials • Concrete: normal weight (150 pcf), 3 /4-in. maximum aggregate, f’ c = 4,000 psi • Mild reinforcing steel: Grade 60 (f y = 60,000 psi) Loads • Superimposed dead loads = 30 psf • Live load = 50 psf TIME SAVING DESIGN AIDS Two-Way Slabs Page 1 of 14 The following example illustrates the design methods presented in the article “Timesaving Design Aids for Reinforced Concrete, Part 2: Two-way Slabs,” by David A. Fanella, which appeared in the October 2001 edition of Structural Engineer magazine. Unless otherwise noted, all referenced table, figure, and equation numbers are from that article. Example Building Below is a partial plan of a typical floor in a cast-in-place reinforced concrete building. In this example, an interior strip of a flat plate floor system is designed and detailed for the effects of gravity loads according to ACI 318-99. 20?-0I 20?-0I 20?-0I 24?-0I 24?-0I 20Ix 20I ( t yp . ) 24Ix 24I ( t yp . ) Design strip Minimum Slab Thickness Longest clear span l n = 24 – (20/12) = 22.33 ft From Fig. 4-3, minimum thickness h per ACI Table 9.5(c) = l n /30 = 8.9 in. Use Fig. 1-8 to determine h based on shear requirements at interior column assuming a 9 in. slab: q u = 1.2(112.5 + 30) + 1.6(50) = 251.0 psf A = 24 x 20 = 480 ft 2 A/c 1 2 = 480/2 2 = 120 From Fig. 1-8, d/c 1 ≈ 0.22 d = 0.22 x 24 = 5.3 in. h = 5.3 + 1.25 = 6.55 in. Try preliminary h = 9 in. Design for Flexure Use Fig. 4-4 to determine if the Direct Design Method of ACI Sect. 13.6 can be utilized to compute the bending moments due to the gravity loads: • 3 continuous spans in one direction, more than 3 in the other O.K. • Rectangular panels with long-to-short span ratio = 24/20 = 1.2 < 2 O.K. • Successive span lengths in each direction are equal O.K. • No offset columns O.K. • L/D = 50/(112.5 + 30) = 0.35 < 2 O.K. • Slab system has no beams N.A. Since all requirements are satisfied, the Direct Design Method can be used. Total panel moment M o in end span: M o = = 248.5 ft - kips Total panel moment M o in interior span: M o = = 244 ft - kips For simplicity, use M o = 248.5 ft-kips for all spans. Division of the total panel moment M o into negative and positive moments, and then column and middle strip moments, involves the direct application of the moment coefficients in Table 4-2. q 8 0 251 24 18 0 8 u 2 n 2 2 ll = ×× q 8 0 251 24 18 167 8 u 2 n 2 2 ll = ×× TIME SAVING DESIGN AIDS Two-Way Slabs Page 2 of 14 Slab Moments End Spans Int. Span (ft-kips) Ext. neg. Positive Int. neg. Positive Total Moment 64.6 129.2 173.9 87.0 Column Strip 64.6 77.0 131.7 52.2 Middle Strip 0 52.2 42.2 34.8 Note: All negative moments are at face of support. Required slab reinforcement. Span Location M u b * d ** As † Min. As ‡ Reinforcement + (ft-kips) (in.) (in.) (in. 2 ) (in. 2 ) End Span Ext. neg. 64.6 120 7.75 2.08 1.94 11-No. 4 Column Positive 77.0 120 7.75 2.48 1.94 13-No. 4 Strip Int. Neg. 131.7 120 7.75 4.25 1.94 22-No. 4 Ext. neg. 0.0 168 7.75 2.72 14-No. 4 Middle Positive 52.2 168 7.75 1.68 2.72 14-No. 4 Strip Int. Neg. 42.2 168 7.75 1.36 2.72 14-No. 4 Interior Span Column Positive 52.2 120 7.75 1.68 1.94 10-No. 4 Strip Middle Positive 34.8 168 7.75 1.12 2.72 14-No. 4 Strip * Column strip width b = (20 x 12)/2 = 120 in. * Middle strip width b = (24 x 12) – 120 = 168 in. ** Use average d = 9 – 1.25 = 7.75 in. † A s = M u /4d where M u is in ft-kips and d is in inches ‡ Min. A s = 0.0018bh = 0.0162b; Max. s = 2h = 18 in. or 18 in. (Sect. 13.3.2) + For maximum spacing:120/18 = 6.7 spaces, say 8 bars 168/18 = 9.3 spaces, say 11 bars Design for Shear Check slab shear and flexural strength at edge column due to direct shear and unbalanced moment transfer. Check slab reinforcement at exterior column for moment transfer between slab and column. Portion of total unbalanced moment transferred by flexure = γ f M u b 1 = 20 + (7.75/2) = 23.875 in. b 2 = 20 + 7.75 = 27.75 in. b 1 /b 2 = 0.86 From Fig. 4-16, γ f = 0.62 * * The provisions of Sect. 13.5.3.3 may be utilized; however, they are not in this example. TIME SAVING DESIGN AIDS Two-Way Slabs Page 3 of 14 γ f M u = 0.62 x 64.6 = 40 ft-kips Required A s = 40/(4 x 7.75) = 1.29 in. 2 Number of No. 4 bars = 1.29/0.2 = 6.5, say 7 bars Must provide 7-No. 4 bars within an effective slab width = 3h + c 2 = (3 x 9) + 20 = 47 in. Provide the required 7-No. 4 bars by concentrating 7 of the column strip bars (11-No. 4) within the 47 in. slab width over the column. Check bar spacing: For 7-No. 4 within 47 in. width: 47/7 = 6.7 in. < 18 in. O.K. For 4-No. 4 within 120 – 47 = 73 in. width: 73/4 = 18.25 in. > 18 in. Add 1 additional bar on each side of the 47 in. strip; the spacing becomes 73/6 = 12.2 in. < 18 in. O.K. Reinforcement details at this location are shown in the figure on the next page. Check the combined shear stress at the inside face of the critical transfer section. TIME SAVING DESIGN AIDS Two-Way Slabs Page 4 of 14 v u = Factored shear force at edge column: V u = 0.251[(24 x 10.83) – (1.99 x 2.31)] = 64.1 kips When the end span moments are determined from the Direct Design Method, the fraction of unbalanced moment transferred by eccentricity of shear must be 0.3M o = 0.3 x 248.5 = 74.6 ft-kips (Sect. 13.6.3.6). γ v = 1 – γ f = 1 – 0.62 = 0.38 c 2 /c 1 = 1.0 c 1 /d = 20/7.75 = 2.58 From Table 4.9 A c = (2b 1 + b 2 ) d = 585.1 in. 2 J c = [2b 1 d(b 1 + 2b 2 ) + d 3 (2b 1 + b 2 )/b 1 ]/6 = 5,141 in. 3 v u = v u = 109.6 + 57.6 = 167.2 psi Determine allowable shear stress φv c from Fig. 4-13: b o /d = (2b 1 + b 2 )/d b o /d = [(2 x 23.875) + 27.75]/7.75 = 9.74 β c = 1 φv c = 189.7 psi > v u = 168.2 psi OK 9” slab is OK Reinforcement Details The figures below show the reinforcement details for the column and middle strips. The bar lengths are determined from Fig. 13.3.8 of ACI 318-05. 64 105 585.1 0 38 64 6 12 000 5 141 , , + ×× , V A M Jc u c vu + γ / TIME SAVING DESIGN AIDS Two-Way Slabs Page 5 of 14 TIME SAVING DESIGN AIDS Two-Way Slabs Page 6 of 14 The PCA computer program pcaSlab can be used to expedite the design of different slab systems. The program covers wide range of two-way slab systems and can be used for more complex slab layouts. The output of the program for the slab in the example is shown in the following pages. Please note that the Equivalent Frame Method is used by the pcaSlab program. TIME SAVING DESIGN AID Page 7 of 14 Two-Way Slabs X Y Z pcaSlab v2.00. Licensed to: pcaStructurePoint. License ID: 12345-1234567-4-2D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Two-Way Slabs.slb Project: Time Saving Design Aids Frame: Two-Way Slab Engineer: PCA TIME SAVING DESIGN AID Page 8 of 14 Two-Way Slabs pcaSlab v2.00. Licensed to: pcaStructurePoint. License ID: 12345-1234567-4-2D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Two-Way Slabs.slb Project: Time Saving Design Aids Frame: Two-Way Slab Engineer: PCA Column Strip Flexural Reinforcement Middle Strip Flexural Reinforcement 10-#4(81.9) 12-#4(83.9) 11-#4(55.6) 16-#4(240.0)c 12-#4(83.3) 11-#4(55.2) 12-#4(83.3) 11-#4(55.2) 10-#4(240.0)c 12-#4(83.9) 11-#4(55.6) 10-#4(81.9) 16-#4(240.0)c 14-#4(73.4) 14-#4(240.0)c 14-#4(82.5) 14-#4(82.5) 14-#4(240.0)c 14-#4(73.4) 14-#4(240.0)c TIME SAVING DESIGN AID Page 9 of 14 Two-Way Slabs pcaSlab v2.00. Licensed to: pcaStructurePoint. License ID: 12345-1234567-4-2D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Two-Way Slabs.slb Project: Time Saving Design Aids Frame: Two-Way Slab Engineer: PCA Column Strip Moment Capacity - k-ft -300.0 300.0 Middle Strip Moment Capacity - k-ft -300.0 300.0 LEGEND: Envelope Curve Capacity Curve Support Centerline Face of Support TIME SAVING DESIGN AID Page 10 of 14 Two-Way Slabs pcaSlab v2.00 © Portland Cement Association 05-03-2007, 11:27:26 AM Licensed to: pcaStructurePoint, License ID: 12345-1234567-4-2D2DE-2C8D0 c:\Work\Time Saving Design Aids\318-05\Rev 2\Data\Two-Way Slabs.slb Page 1 ooooooo oooooo ooooo oooooooo oooooooo ooooooo oo oo oo oo oo oo oo oo oo oo oo oooooooo oo ooooooo ooooo ooooooo oo oo ooooooo ooooo oo oo oo oo oo oo oooooooo oo oo oo oooooo oo oo oooooo o o oooooooo oo ooooo oo oo oo o oo oo oooo oo o oo oo oooooo oo oooooo oooooo oooo oo oo oo oo oo oo oo oo oo oo oo oooooooo oo o oo oo oo oo oooooo ooo ooooo o ooooo ============================================================================= pcaSlab v2.00 (TM) A Computer Program for Analysis, Design, and Investigation of Reinforced Concrete Beams, One-way and Two-way Slab Systems ============================================================================= Copyright © 2003-2006, Portland Cement Association All rights reserved Licensee stated above acknowledges that Portland Cement Association (PCA) is not and cannot be responsible for either the accuracy or adequacy of the material supplied as input for processing by the pcaSlab computer program. Furthermore, PCA neither makes any warranty expressed nor implied with respect to the correctness of the output prepared by the pcaSlab program. Although PCA has endeavored to produce pcaSlab error free the program is not and cannot be certified infallible. The final and only responsibility for analysis, design and engineering documents is the licensees. Accordingly, PCA disclaims all responsibility in contract, negligence or other tort for any analysis, design or engineering documents prepared in connection with the use of the pcaSlab program. ============================================================================================= [1] INPUT ECHO ============================================================================================= General Information: ==================== File name: c:\Work\Time Saving Design Aids\318-05\Rev 2\Data\Two-Way Slabs.slb Project: Time Saving Design Aids Frame: Two-Way Slab Engineer: PCA Code: ACI 318-02 Reinforcement Database: ASTM A615 Mode: Design Number of supports = 4 Floor System: Two-Way Live load pattern ratio = 75% Minimum free edge for punching shear = 10 times slab thickness Deflections are based on cracked section properties. In negative moment regions, Ig and Mcr DO NOT include flange/slab contribution (if available) Long-term deflections are calculated for load duration of 60 months. 0% of live load is sustained. Compression reinforcement calculations NOT selected. Material Properties: ==================== Slabs|Beams Columns wc = 150 150 lb/ft3 f'c = 4 4 ksi Ec = 3834.3 3834.3 ksi fr = 0.47434 0.47434 ksi fy = 60 ksi, Bars are not epoxy-coated fyv = 60 ksi Es = 29000 ksi Reinforcement Database: =============== Units: Db (in), Ab (in^2), Wb (lb/ft) Size Db Ab Wb Size Db Ab Wb