DESIGN AID J.1-1 Areas of Reinforcing Bars Total Areas of Bars (in.2) Bar Size 1 0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 No. 11 2 0.22 0.40 0.62 0.88 1.20 1.58 2.00 2.54 3.12 3 0.33 0.60 0.93 1.32 1.80 2.37 3.00 3.81 4.68 4 0.44 0.80 1.24 1.76 2.40 3.16 4.00 5.08 6.24 Number of Bars 5 6 7 8 9 10 0.55 0.66 0.77 0.88 0.99 1.10 1.00 1.20 1.40 1.60 1.80 2.00 1.55 1.86 2.17 2.48 2.79 3.10 2.20 2.64 3.08 3.52 3.96 4.40 3.00 3.60 4.20 4.80 5.40 6.00 3.95 4.74 5.53 6.32 7.11 7.90 5.00 6.00 7.00 8.00 9.00 10.00 6.35 7.62 8.89 10.16 11.43 12.70 7.80 9.36 10.92 12.48 14.04 15.60 Areas of Bars per Foot Width of Slab (in.2/ft) Bar Size Bar Spacing (in.) 6 7 8 9 10 11 12 13 14 15 16 17 18 No. 3 0.22 0.19 0.17 0.15 0.13 0.12 0.11 0.10 0.09 0.09 0.08 0.08 0.07 No. 4 0.40 0.34 0.30 0.27 0.24 0.22 0.20 0.18 0.17 0.16 0.15 0.14 0.13 No. 5 0.62 0.53 0.46 0.41 0.37 0.34 0.31 0.29 0.27 0.25 0.23 0.22 0.21 No. 6 0.88 0.75 0.66 0.59 0.53 0.48 0.44 0.41 0.38 0.35 0.33 0.31 0.29 No. 7 1.20 1.03 0.90 0.80 0.72 0.65 0.60 0.55 0.51 0.48 0.45 0.42 0.40 No. 8 1.58 1.35 1.18 1.05 0.95 0.86 0.79 0.73 0.68 0.63 0.59 0.56 0.53 No. 9 2.00 1.71 1.50 1.33 1.20 1.09 1.00 0.92 0.86 0.80 0.75 0.71 0.67 No. 10 2.54 2.18 1.91 1.69 1.52 1.39 1.27 1.17 1.09 1.02 0.95 0.90 0.85 No. 11 3.12 2.67 2.34 2.08 1.87 1.70 1.56 1.44 1.34 1.25 1.17 1.10 1.04 DESIGN AID J.1-2 $SSUR[LPDWH%HQGLQJ0RPHQWVDQG6KHDU)RUFHVIRU&RQWLQXRXV%HDPV DQG2QHZD\6ODEV Uniformly distributed load wu (L/D d3) Two or more spans Prismatic members ,QWHJUDOZLWK 6XSSRUW "n"nd"n "n "n wu" n wu" n wu" n wu" navg wu" n Spandrel Support 6LPSOH 6XSSRUW wu" n wu" n wu" navg Positive Moment wu" n Column Support wu " navg wu" n Note A " n avg w " u n wu " navg wu" n wu " n wu" n Negative Moment wu" n wu" n wu" n wu" n 1RWH$ $SSOLFDEOHWRVODEVZLWKVSDQV d IW DQGEHDPVZKHUHWKHUDWLRRIWKHVXPRI FROXPQVWLIIQHVVWREHDPVWLIIQHVV!DW HDFKHQGRIWKHVSDQ VSDQV " n " n wu " n wu" n IRUEHDPV Shear DESIGN AID J.1-3 9DULDWLRQRIIZLWK1HW7HQVLOH6WUDLQLQ([WUHPH7HQVLRQ6WHHO H W DQG *UDGH5HLQIRUFHPHQWDQG3UHVWUHVVLQJ6WHHO FG W ± I I H W 6SLUDO I H W 2WKHU &RPSUHVVLRQFRQWUROOHG HW 7UDQVLWLRQ 7HQVLRQFRQWUROOHG HW F GW F GW GW HW GW HW 6SLUDO I > FG W @ 2WKHU I > FG W @ DESIGN AID J.1-4 Simplified Calculation of As Assuming Tension-Controlled Section and Grade 60 Reinforcement f c′ (psi) As (in.2) 3,000 Mu 3.84d 4,000 Mu 4.00d 5,000 Mu 4.10d M u is in ft-kips and d is in inches In all cases, As = Mu can be used. 4d Notes: Mu 0.5 ρf y × d φf y 1 − 0.85 f ' c • As = • • For all values of ρ < 0.0125, the simplified As equation is slightly conservative. It is recommended to avoid ρ > 0.0125 when using the simplified As equation. DESIGN AID J.1-51 0LQLPXP1XPEHURI5HLQIRUFLQJ%DUV5HTXLUHGLQD6LQJOH/D\HU Assumptions: x *UDGHUHLQIRUFHPHQW f y SVL x &OHDUFRYHUWRWKHWHQVLRQUHLQIRUFHPHQW cc LQ x &DOFXODWHGVWUHVV f s LQUHLQIRUFHPHQWFORVHVWWRWKHWHQVLRQIDFHDW VHUYLFHORDG SVL Beam Width (in.) Bar Size 1R 1R 1R 1R 1R 1R 1R 1R Minimum number of bars, nmim: bw 2(cc 0.5db ) 1 s nmin GV where § 40,000 · ¸¸ 2.5cc s 15¨¨ © fs ¹ § 40,000 · ¸¸ d 12¨¨ f s ¹ © 1 db FF sFOHDU U ôsIRU1RVWLUUXSV sIRU1RVWLUUXSV s &OHDUVSDFHt FV FF EZ GE PD[DJJUHJDWHVL]H Alsamsam, I.M. and Kamara, M. E. (2004). Simplified Design Reinforced Concrete Buildings of Moderate Size and Heights, EB104, Portland Cement Association, Skokie, IL. DESIGN AID J.1-61 0D[LPXP1XPEHURI5HLQIRUFLQJ%DUV3HUPLWWHGLQD6LQJOH/D\HU Assumptions: x *UDGHUHLQIRUFHPHQW f y NVL x &OHDUFRYHUWRWKHVWLUUXSV cs LQ x ôLQDJJUHJDWH x 1RVWLUUXSVDUHXVHGIRU1RDQG1RORQJLWXGLQDOEDUVDQG1R VWLUUXSVDUHXVHGIRU1RDQGODUJHUEDUV Beam Width (in.) Bar Size 1R 1R 1R 1R 1R 1R 1R 1R Maximum number of bars, nmax: nmax bw 2(cs d s r ) 1 (Clear space) db GV db FF sFOHDU ôsIRU1RVWLUUXSV sIRU1RVWLUUXSV s &OHDUVSDFHt FV FF 1 U EZ GE PD[DJJUHJDWHVL]H Alsamsam, I.M. and Kamara, M. E. (2004). Simplified Design Reinforced Concrete Buildings of Moderate Size and Heights, EB104, Portland Cement Association, Skokie, IL. DESIGN AID J.1-7 0LQLPXP7KLFNQHVVhIRU%HDPVDQG2QH:D\6ODEV8QOHVV'HIOHFWLRQVDUH &DOFXODWHG Beams or Ribbed One-way Slabs h t " 1 / 18.5 h t " 2 / 21 2QHHQG FRQWLQXRXV %RWKHQGV FRQWLQXRXV "1 Solid One-way Slabs &DQWLOHYHU "2 h t " 1 / 24 h t " 2 / 28 2QHHQG FRQWLQXRXV %RWKHQGV FRQWLQXRXV "1 h t "3 /8 "3 h t " 3 / 10 &DQWLOHYHU "2 "3 x Applicable to one-way construction not supporting or attached to partitions or other construction likely to be damaged by large deflections. x Values shown are applicable to members with normal weight concrete ( wc reinforcement. For other conditions, modify the values as follows: 3 145 lbs/ft ) and Grade 60 For structural lightweight having wc in the range 90-120 lbs/ft3, multiply the values by 1.65 0.005 wc t 1.09. For f y other than 60,000 psi, multiply the values by 0.4 f y / 100,000 . x For simply-supported members, minimum h ­" / 20 for solid one - way slabs ® ¯" / 16 for beams or ribbed one - way slabs DESIGN AID J.1-8 Reinforcement Ratio ρ t for Tension-Controlled Sections Assuming Grade 60 Reinforcement f c′ (psi) ρ t when εt = 0.005 ρ t when εt = 0.004 3,000 0.01355 0.01548 4,000 0.01806 0.02064 5,000 0.02125 0.02429 Notes: 1. C = 0.85 f ' c (β1c )b T = As f y C = T ⇒ 0.85 f ' c (β1c )b = As f y a. When εt = 0.005, c/dt = 3/8. 0.85 f ' c β1 3 d t b = As f y 8 ( ρt = ) 0.85β1 f c′( 3 ) As 8 = bd t fy b. When εt = 0.004, c/dt = 3/7. 0.85 f ' c β1 3 d t b = As f y 7 ( ) 0.85β1 f c′( 3 ) As 7 = ρt = bd t fy 2. β1 is determined according to 10.2.7.3. DESIGN AID J.1-9 Simplified Calculation of bw Assuming Grade 60 Reinforcement and ρ = 0.5ρ max f c′ (psi) 3,000 4,000 5,000 bw (in.)* 31.6 M u d2 23.7 M u d2 20.0 M u d2 * M u is in ft-kips and d is in inches In general: bw = 36,600 M u ρ β1 f c′ (1 − 0.2143ρ β1 )d 2 where ρ = ρ / ρ max , f c′ is in psi, d is in inches and M u is in ft-kips and ρ max = 0.85β1 f c′ 0.003 (10.3.5) fy 0.004 + 0.003 DESIGN AID J.1-10 T-beam Construction 8.12 be 2 be1 h = hf bw1 bw3 bw2 s1 Span length bw1 + 12 be1 ≤ bw1 + 6h 3b b s w1 − w2 + 1 4 2 4 s2 Span length 4 be 2 ≤ bw2 + 16h b b +b s +s w2 − w1 w3 + 1 2 4 2 2 be ≤ 4bw h = hf ≥ bw Isolated T-beam bw 2 DESIGN AID J.1-11 Values of φVs = Vu − φVc (kips) as a Function of the Spacing, s* s d/2 No. 3 U-stirrups 19.8 No. 4 U-stirrups 36.0 No. 5 U-stirrups 55.8 d/3 29.7 54.0 83.7 d/4 39.6 72.0 111.6 * Valid for Grade 60 ( f yt = 60 ksi) stirrups with 2 legs (double the tabulated values for 4 legs, etc.). In general: φVs = φAv f yt d s (11.4.7.2) where f yt used in design is limited to 60,000 psi, except for welded deformed wire reinforcement, which is limited to 80,000 psi (11.4.2). DESIGN AID J.1-12 Minimum Shear Reinforcement Av,min / s * f c′ (psi) Av,min in.2 in. s ≤ 4,500 0.00083bw 5,000 0.00088bw * Valid for Grade 60 ( f yt = 60 ksi) shear reinforcement. In general: Av,min s = 0.75 f c′ bw 50bw ≥ f yt f yt Eq. (11-13) where f yt used in design is limited to 60,000 psi, except for welded deformed wire reinforcement, which is limited to 80,000 psi (11.4.2). DESIGN AID J.1-13 Torsional Section Properties Section* Edge bw Acp pcp Aoh ph bwh + behf 2(h + bw + be) x1y1 2(x1 + y1) bw(h - hf) + behf 2(h + be) x1y1 2(x1 + y1) b1h1 + b2h2 2(h1 + h2 + b2) x1y1 + x2y2 2(x1 + x2 + y1) b1h1 + b2h2 2(h1 + h2 + b2) x1y1 + 2x2y2 2(x1 + 2x2 + y1) be = h - hf ≤ 4hf h hff yyo hh 1 x1 = bw - 2c - ds y1 = h - 2c - ds x1 Interior be = bw + 2(h - hf) ≤ bw + 8hf xxo1 hhf f yyo h 1 x1 = bw - 2c - ds y1 = h - 2c - ds bw L-shaped bb11 x1 x2 h1 yy11 y2 h2 x1 = b1 - 2c - ds y1 = h1 + h2 - 2c - ds x2 = b2 - b1 y2 = h2 - 2c - ds b2 Inverted tee b1 x1 x2 h1 y1 y2 h2 x1 = b1 - 2c - ds y1 = h1 + h2 - 2c - ds x2 = (b2 - b1)/2 y2 = h2 - 2c - ds b2 * Notation: xi, yi = center-to-center dimension of closed rectangular stirrup c = clear cover to closed rectangular stirrup(s) ds = diameter of closed rectangular stirrup(s) 2 Note: Neglect overhanging flanges in cases where Acp / pcp calculated for a beam with flanges is less than that computed for the same beam ignoring the flanges (11.5.1.1). DESIGN AID J.1-14 Moment of Inertia of Cracked Section Transformed to Concrete, I cr Gross Section Cracked Transformed Section b Cracked Moment of Inertia, I cr b I cr = kd As nAs b b where d n.a. h kd = I cr = d′ kd A′s n.a. h As (n – 1)A′s b(kd )3 + nAs (d − kd ) 2 3 2dB + 1 − 1 B b(kd ) 3 + nAs (d − kd ) 2 3 + (n − 1) As′ (kd − d ′) 2 d where nAs kd = rd ′ 2 2dB + 1 + + (1 + r ) − (1 + r ) d B ---continued next page--I g = bh 3 / 12 n = E s / Ec B = b /(nAs ) r = (n − 1) As′ /(nAs ) DESIGN AID J.1-14 Moment of Inertia of Cracked Section Transformed to Concrete, I cr (continued) Cracked Transformed Section Gross Section b Cracked Moment of Inertia, I cr b hf I cr = kd h nAs As 12 b (kd ) 3 + w 3 hf + (b − bw )h f kd − 2 d n.a. (b − bw )h 3f 2 + nAs (d − kd ) 2 bw where kd = b b hf A′s yt d′ I cr = kd h As (n – 1)A′s nAs bw n.a. d C (2d + h f f ) + (1 + f ) 2 − (1 + f ) C (b − bw )h 3f 12 b (kd ) 3 + w 3 hf + (b − bw )h f kd − 2 2 + nAs (d − kd ) 2 + (n − 1) As′ (kd − d ′) 2 where kd = C (2d + h f f + 2rd ′) + (1 + r + f ) 2 − (1 + r + f ) C yt = h − {0.5[(b − bw )h 2f + bw h 2 ] /[(b − bw )h f + bw h]} I g = (b − bw )h 3f / 12 + bw h 3 / 12 + (b − bw )h f (h − 0.5h f − yt ) 2 + bw h( yt − 0.5h) 2 n = E s / Ec C = bw /(nAs ) f = h f (b − bw ) /(nAs ) r = (n − 1) As′ /(nAs ) DESIGN AID J.1-15 Approximate Equation to Determine Immediate Deflection, ∆ i , for Members Subjected to Uniformly Distributed Loads ∆i = 5 KM a 2 48 Ec I e where M a = net midspan moment or cantilever moment = span length (8.9) Ec = modulus of elasticity of concrete (8.5.1) = w1c.5 33 f c′ for values of wc between 90 and 155 pcf wc = unit weight of concrete I e = effective moment of inertia (see Flowchart A.1-5.1) K = constant that depends on the span condition Span Condition Cantilever* 2.0 Simple 1.0 Continuous * K 1.2 − 0.2( M o / M a )** Deflection due to rotation at supports not included ** M o = w 2 / 8 (simple span moment at midspan) DESIGN AID J.2-1 &RQGLWLRQVIRU$QDO\VLVE\WKH'LUHFW'HVLJQ0HWKRG )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHV(T PXVWDOVREHVDWLVILHG d D f " D f " d Ecb I b (T Ecs I s ZKHUH Df Ec wc f cc IRUYDOXHVRI wc EHWZHHQDQGSFI Ib I s PRGXOXVRIHODVWLFLW\RIFRQFUHWH PRPHQWRILQHUWLDRIEHDPDQGVODEUHVSHFWLYHO\ VHH'HVLJQ$LG- Page 1 of 11 DESIGN AID J.2-2 'HILQLWLRQVRI&ROXPQ6WULSDQG0LGGOH6WULS (""2)B Minimum of "1/4 or ("2)B/4 (""2)A ½-Middle strip Column strip ½-Middle strip "1 Minimum of "1/4 or ("2)A/4 Page 2 of 11 DESIGN AID J.2-3 'HILQLWLRQRI&OHDU6SDQ " n K K K K K K K K " n t " " Page 3 of 11 DESIGN AID J.2-4 'HVLJQ0RPHQW&RHIILFLHQWVXVHGZLWKWKH'LUHFW'HVLJQ0HWKRG ± Flat Plate or Flat Slab Flat Plate or Flat Slab with Spandrel Beams 6HH 'HVLJQ $LG - IRU GHWHUPLQDWLRQ RI E t Page 4 of 11 DESIGN AID J.2-4 'HVLJQ0RPHQW&RHIILFLHQWVXVHGZLWKWKH'LUHFW'HVLJQ0HWKRG ± FRQWLQXHG Flat Plate or Flat Slab with End Span Integral with Wall Flat Plate or Flat Slab with End Span Simply Supported on Wall Page 5 of 11 DESIGN AID J.2-4 'HVLJQ0RPHQW&RHIILFLHQWVXVHGZLWKWKH'LUHFW'HVLJQ0HWKRG ± FRQWLQXHG Two-Way Beam-Supported Slab 6HH 'HVLJQ $LGV - DQG - IRU GHWHUPLQDWLRQ RI D f DQG E t UHVSHFWLYHO\ 1RWHV x 0RLVGHILQHGSHU Page 6 of 11 DESIGN AID J.2-5 (IIHFWLYH%HDPDQG6ODE6HFWLRQVIRU&RPSXWDWLRQRI6WLIIQHVV5DWLR D f Interior Beam a C C "2 "2 CL Slab, Is Slab, Is h h a Beam, Ib Beam, Ib b b beff = b + 2(a – h) d b + 8h beff = b + (a – h) d b + 4h Ecb I b (T Ecs I s Df Ec Edge Beam PRGXOXVRIHODVWLFLW\RIFRQFUHWH wc f cc IRUYDOXHVRI wc EHWZHHQDQGSFI " h Is Ib ah· h · § § b a h b a h ¨ yb ¸ beff h beff h¨ a yb ¸ ¹ ¹ © © ZKHUH yb h· b § beff h¨ a ¸ a h ¹ © beff h b a h Page 7 of 11 DESIGN AID J.2-6 &RPSXWDWLRQRI7RUVLRQDO6WLIIQHVV)DFWRU E t IRU7DQG/6HFWLRQV Interior Beam C C "2 h a b beff = b + 2(a – h) d b + 8h Case A y2 y2 x2C A x1 § x ·x y ¨¨ ¸¸ y ¹ © § x · x y ¨¨ ¸¸ y ¹ © y1 Case B y2 x2C B x1 § x · x y ¨¨ ¸¸ y ¹ © § x · x y ¸ ¨ ¨ ¸ y ¹ © y1 C PD[LPXPRI C A DQG C B Ecb C Et (T Ecs I s ZKHUH I s " h DQG E wc f cc IRUYDOXHVRI wc EHWZHHQDQGSFI Page 8 of 11 DESIGN AID J.2-6 &RPSXWDWLRQRI7RUVLRQDO6WLIIQHVV)DFWRU E t IRU7DQG/6HFWLRQV FRQWLQXHG CL "2 Edge Beam h a b beff = b + (a – h) d b + 4h Case A y2 x2 x1 CA § x · x y ¨¨ ¸¸ y ¹ © § x · x y ¨¨ ¸¸ y ¹ © y1 Case B y2 x2 x1 CB § x ·x y ¨¨ ¸¸ y ¹ © § x · x y ¨¨ ¸¸ y ¹ © y1 C PD[LPXPRI C A DQG C B Ecb C Et (T Ecs I s ZKHUH I s " h DQG E wc f cc IRUYDOXHVRI wc EHWZHHQDQGSFI Page 9 of 11 DESIGN AID J.2-7 0RPHQW'LVWULEXWLRQ&RQVWDQWVIRU6ODE%HDP0HPEHUVZLWKRXW'URS3DQHOV " cN cF c N " cF c N " cN " 6WLIIQHVV)DFWRU k NF &DUU\RYHU)DFWRU C NF )L[HGHQG0RPHQW &RHIILFLHQW m NF 6ODEEHDPVWLIIQHVV K sb k NF Ecs I sb " )L[HGHQGPRPHQW FEM m FN qu " " $SSOLFDEOHZKHUH c N c F DQG c N c F DQG DXQLIRUPO\GLVWULEXWHGORDG qu DFWVRYHUWKHHQWLUHVSDQOHQJWK6HHPCA Notes on ACI 318-11IRURWKHUFDVHVLQFOXGLQJ FRQVWDQWVIRUPHPEHUVZLWKGURSSDQHOV Page 10 of 11 DESIGN AID J.2-8 6WLIIQHVVDQG&DUU\2YHU)DFWRUVIRU&ROXPQV $ H "c % H "c 6WLIIQHVV)DFWRU k AB &DUU\RYHU)DFWRU C AB ­ K c AB k AB Ecc I c " c ° &ROXPQVWLIIQHVV ® °K ¯ c BA k BA Ecc I c " c 6HHPCA Notes on ACI 318-11IRURWKHUFDVHVLQFOXGLQJIDFWRUVIRUPHPEHUVZLWKGURS SDQHOVDQGFROXPQFDSLWDOV Page 11 of 11