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
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w " u n
wu " navg
wu" n
wu " n
wu" n
Negative Moment
wu" n
wu" n
wu" n
wu" n
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VSDQV
" n " n
wu " n
wu" n
IRUEHDPV
Shear
DESIGN AID J.1-3 9DULDWLRQRIIZLWK1HW7HQVLOH6WUDLQLQ([WUHPH7HQVLRQ6WHHO H W DQG *UDGH5HLQIRUFHPHQWDQG3UHVWUHVVLQJ6WHHO
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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
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x Applicable to one-way construction not supporting or attached to partitions or other construction likely to be damaged by large deflections. 3 145 lbs/ft ) and Grade 60
x Values shown are applicable to members with normal weight concrete ( wc reinforcement. For other conditions, modify the values as follows:
For structural lightweight having wc in the range 90-120 lbs/ft3, multiply the values by 1.65 0.005wc 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
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(""2)B Minimum of "1/4 or ("2)B/4
(""2)A
½-Middle strip Column strip ½-Middle strip
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Minimum of "1/4 or ("2)A/4
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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
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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
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Two-Way Beam-Supported Slab
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Edge Beam
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