McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 631
A PPE N D I X A
Tables and Graphs:
U.S. Customary Units
TABLE A.1 Values of Modulus of Elasticity for Normal-Weight Concrete
U.S. Customary Units
f c
(psi)
Ec (psi)
3,000
3,160,000
3,500
3,410,000
4,000
3,640,000
4,500
3,870,000
5,000
4,070,000
Source Notes:
Tables A.4, A.6, A.14, and A.15, as well as Graph 1, are reprinted from Design of Concrete
Structures by Winter and Nilson. Copyright © 1972 by McGraw-Hill, Inc., and with permission
of the McGraw-Hill Book Company.
Graphs 2 through 13 are reprinted from Design Handbook, Volume 2, Columns (SP-17A),
1978, with permission of American Concrete Institute.
Table A.3(A)(a) and A.3(B)(b) are reprinted from Manual of Standard Practice, 22nd ed., 1997,
second printing, Concrete Reinforcing Steel Institute, Chicago, IL.
Tables A.16 through A.20 are reprinted from Commentary on Building Code Requirements
For Reinforced Concrete (ACI 318–77) with permission of the American Concrete Institute.
TABLE A.2 Designations, Areas, Perimeters, and Weights of Standard Bars
U.S. Customary Units
Diameter (in.)
Cross-Sectional Area (in.2 )
Unit Weight (lb/ft)
3
0.375
0.11
0.376
4
0.500
0.20
0.668
5
0.625
0.31
1.043
6
0.750
0.44
1.502
7
0.875
0.60
2.044
8
1.000
0.79
2.670
Bar No.
9
1.128
1.00
3.400
10
1.270
1.27
4.303
11
1.410
1.56
5.313
14
1.693
2.25
7.650
18
2.257
4.00
13.600
631
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 632
632
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.3(A) Common Styles of Welded Wire Reinforcement Sheets
Style
Area
(in.2 /ft)
Weight
(lb/100 ft2 )
4 × 4 − W1.4 × W1.4*
0.042
31
4 × 4 − W2.0 × W2.0*
0.060
44
4 × 4 − W2.9 × W2.9*
0.087
62
4 × 4 − W3.1 × W3.1
0.093
65
4 × 4 − W4.0 × W4.0*
0.120
88
6 × 6 − W1.4 × W1.4*
0.028
21
6 × 6 − W2.0 × W2.0*
0.040
30
6 × 6 − W2.9 × W2.9*
0.058
42
6 × 6 − W4.0 × W4.0*
0.080
58
6 × 6 − W4.2 × W4.2
0.084
60
6 × 6 − W4.4 × W4.4
0.088
63
6 × 6 − W4.7 × W4.7
0.094
68
6 × 6 − W7.5 × W7.5
0.150
108
6 × 6 − W8.1 × W8.1
0.162
116
6 × 6 − W8.3 × W8.3
0.166
119
12 × 12 − W8.3 × W8.3
0.083
63
12 × 12 − W8.8 × W8.8
0.088
66
12 × 12 − W9.1 × W9.1
0.091
69
12 × 12 − W9.4 × W9.4
0.094
71
12 × 12 − W15 × W15
0.150
113
12 × 12 − W16 × W16
0.160
120
12 × 12 − W16.6 × W16.6
0.166
125
12 × 12 − W17.1 × W17.1
0.171
128
*These styles may be obtained in roll form. It is recommended that rolls be straightened
and cut to size before placement.
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 633
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.3(B) Cross-Sectional Area and Weight of Welded Wire Reinforcement (in-lb)
Wire
Size,
W or D
As (in.2 /ft) per Wire Spacing (in.)
Nom.
Diam.
(in.)
Nom.
Wt
(lb/ft)
2
3
4
6
8
10
12
45
0.757
1.530
2.700
1.800
1.350
0.900
0.675
0.540
0.450
31
0.628
1.054
1.860
1.240
0.930
0.620
0.465
0.372
0.310
30
0.618
1.020
1.800
1.200
0.900
0.600
0.450
0.360
0.300
28
0.597
0.952
1.680
1.120
0.840
0.560
0.420
0.336
0.280
26
0.575
0.884
1.560
1.040
0.780
0.520
0.390
0.312
0.260
24
0.553
0.816
1.440
0.960
0.720
0.480
0.360
0.288
0.240
22
0.529
0.748
1.320
0.880
0.660
0.440
0.330
0.264
0.220
20
0.505
0.680
1.200
0.800
0.600
0.400
0.300
0.240
0.200
18
0.479
0.612
1.080
0.720
0.540
0.360
0.270
0.216
0.180
16
0.451
0.544
0.960
0.640
0.480
0.320
0.240
0.192
0.160
14
0.422
0.476
0.840
0.560
0.420
0.280
0.210
0.168
0.140
12
0.391
0.408
0.720
0.480
0.360
0.240
0.180
0.144
0.120
11
0.374
0.374
0.660
0.440
0.330
0.220
0.165
0.132
0.110
10.5
0.366
0.357
0.630
0.420
0.315
0.210
0.158
0.126
0.105
10
0.357
0.340
0.600
0.400
0.300
0.200
0.150
0.120
0.100
9.5
0.348
0.323
0.570
0.380
0.285
0.190
0.143
0.114
0.095
9
0.339
0.306
0.540
0.360
0.270
0.180
0.135
0.108
0.090
8.5
0.329
0.289
0.510
0.340
0.255
0.170
0.128
0.102
0.085
8
0.319
0.272
0.480
0.320
0.240
0.160
0.120
0.096
0.080
7.5
0.309
0.255
0.450
0.300
0.225
0.150
0.113
0.090
0.075
7
0.299
0.238
0.420
0.280
0.210
0.140
0.105
0.084
0.070
6.5
0.288
0.221
0.390
0.260
0.195
0.130
0.098
0.078
0.065
6
0.276
0.204
0.360
0.240
0.180
0.120
0.090
0.072
0.060
5.5
0.265
0.187
0.330
0.220
0.165
0.110
0.083
0.066
0.055
5
0.252
0.170
0.300
0.200
0.150
0.100
0.075
0.060
0.050
4.5
0.239
0.153
0.270
0.180
0.135
0.090
0.068
0.054
0.045
4
0.226
0.136
0.240
0.160
0.120
0.080
0.060
0.048
0.040
3.5
0.211
0.119
0.210
0.140
0.105
0.070
0.053
0.042
0.035
3
0.195
0.102
0.180
0.120
0.090
0.080
0.045
0.036
0.030
2.9
0.192
0.099
0.174
0.116
0.087
0.058
0.044
0.035
0.029
2.5
0.178
0.085
0.150
0.100
0.075
0.050
0.038
0.030
0.025
2
0.160
0.068
0.120
0.080
0.060
0.040
0.030
0.024
0.020
1.4
0.134
0.048
0.084
0.056
0.042
0.028
0.021
0.017
0.014
Notes:
1. The above listing of plain and deformed wire sizes represents wires normally selected to manufacture welded wire
reinforcement to specific areas of reinforcement. Wire sizes other than those listed above, including larger sizes, may be
available if the quantity required is sufficient to justify manufacture.
2. The nominal diameter of a deformed wire is equivalent to the diameter of a plain wire having the same weight per foor
as the deformed wire.
3. The ACI Building Code requirements for tension development lengths and tension lap splice lengths of welded wire
reinforcement are not included in this chapter. These design requirements are covered in Reinforcing Bars: Anchorages
and Splices available from CRSI. For additional information, see Manual of Standard Practice—Structural Welded Wire
Reinforcement and Structural Detailing Manual, both published by the Wire Reinforcement Institute.
633
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 634
634
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.4 Areas of Groups of Standard Bars (in.2)—U.S. Customary Units
Number of Bars
Bar No.
2
3
4
5
6
7
8
9
10
4
0.39
0.58
0.78
0.98
1.18
1.37
1.57
1.77
1.96
5
0.61
0.91
1.23
1.53
1.84
2.15
2.45
2.76
3.07
6
0.88
1.32
1.77
2.21
2.65
3.09
3.53
3.98
4.42
7
1.20
1.80
2.41
3.01
3.61
4.21
4.81
5.41
6.01
8
1.57
2.35
3.14
3.93
4.71
5.50
6.28
7.07
7.85
9
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
10
2.53
3.79
5.06
6.33
7.59
8.86
10.12
11.39
12.66
11
3.12
4.68
6.25
7.81
9.37
10.94
12.50
14.06
15.62
14
4.50
6.75
9.00
11.25
13.50
15.75
18.00
20.25
22.50
18
8.00
12.00
16.00
20.00
24.00
28.00
32.00
36.00
40.00
Number of Bars
Bar No.
11
12
13
14
15
16
17
18
19
20
4
2.16
2.36
2.55
2.75
2.95
3.14
3.34
3.53
3.73
3.93
5
3.37
3.68
3.99
4.30
4.60
4.91
5.22
5.52
5.83
6.14
6
4.86
5.30
5.74
6.19
6.63
7.07
7.51
7.95
8.39
8.84
7
6.61
7.22
7.82
8.42
9.02
9.62
10.22
10.82
11.43
12.03
8
8.64
9.43
10.21
11.00
11.78
12.57
13.35
14.14
14.92
15.71
9
11.00
12.00
13.00
14.00
15.00
16.00
17.00
18.00
19.00
20.00
10
13.92
15.19
16.45
17.72
18.98
20.25
21.52
22.78
24.05
25.31
11
17.19
18.75
20.31
21.87
23.44
25.00
26.56
28.12
29.69
31.25
14
24.75
27.00
29.25
31.50
33.75
36.00
38.25
40.50
42.75
45.00
18
44.00
48.00
52.00
56.00
60.00
64.00
68.00
72.00
76.00
80.00
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 635
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.5 Minimum Web Width (in.) for Beams with Inside Exposure (2011 ACI Code)a,b,c —
U.S. Customary Units
Number of Bars in Single Layer of Reinforcing
Size of Bars
2
3
4
5
6
7
8
Add for Each
Additional Bar
#4
6.8
8.3
9.8
11.3
12.8
14.3
15.8
1.50
#5
6.9
8.5
10.2
11.8
13.4
15.0
16.7
1.625
#6
7.0
8.8
10.5
12.3
14.0
15.8
17.5
1.75
#7
7.2
9.0
10.9
12.8
14.7
16.5
18.4
1.875
#8
7.3
9.3
11.3
13.3
15.3
17.3
19.3
2.00
#9
7.6
9.8
12.1
14.3
16.6
18.8
21.1
2.26
#10
7.8
10.4
12.9
15.5
18.0
20.5
23.1
2.54
#11
8.1
10.9
13.8
16.6
19.4
22.2
25.0
2.82
#14
8.9
12.3
15.7
19.0
22.4
25.8
29.2
3.39
#18
10.6
15.1
19.6
24.1
28.6
33.1
37.7
4.51
a Minimum beam widths for beams were calculated using #3 stirrups.
b Maximum aggregate sizes were assumed not to exceed 3 of the clear spacing between the bars (ACI 3.3.2).
4
c The horizontal distance from the center of the outside longitudinal bars to the inside of the stirrups was assumed to equal
the larger of two times the stirrup diameter (ACI 7.2.2) or half of the longitudinal bar diameter.
TABLE A.6 Areas of Bars in Slabs (in2 /ft)—U.S. Customary Units
Bar No.
Spacing (in.)
3
4
5
6
7
8
9
10
11
3
0.44
0.78
1.23
1.77
2.40
3.14
4.00
5.06
6.25
3 12
0.38
0.67
1.05
1.51
2.06
2.69
3.43
4.34
5.36
4
0.33
0.59
0.92
1.32
1.80
2.36
3.00
3.80
4.68
4 12
0.29
0.52
0.82
1.18
1.60
2.09
2.67
3.37
4.17
5
0.26
0.47
0.74
1.06
1.44
1.88
2.40
3.04
3.75
5 12
0.24
0.43
0.67
0.96
1.31
1.71
2.18
2.76
3.41
6
0.22
0.39
0.61
0.88
1.20
1.57
2.00
2.53
3.12
6 12
0.20
0.36
0.57
0.82
1.11
1.45
1.85
2.34
2.89
7
0.19
0.34
0.53
0.76
1.03
1.35
1.71
2.17
2.68
7 12
0.18
0.31
0.49
0.71
0.96
1.26
1.60
2.02
2.50
8
0.17
0.29
0.46
0.66
0.90
1.18
1.50
1.89
2.34
9
0.15
0.26
0.41
0.59
0.80
1.05
1.33
1.69
2.08
10
0.13
0.24
0.37
0.53
0.72
0.94
1.20
1.52
1.87
12
0.11
0.20
0.31
0.44
0.60
0.78
1.00
1.27
1.56
635
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 636
636
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.7 Values of ρ Balanced, ρ to Achieve Various t Values, and ρ Minimum for
Flexure. All Values Are for Tensilely Reinforced Rectangular Sections
f c
Grade 50
50,000 psi
(344.8 MPa)
Grade 60
60,000 psi
(413.7 MPa)
Grade 75
75,000 psi
(517.1 MPa)
4000 psi
5000 psi
6000 psi
β 1 = 0.85
β 1 = 0.85
β 1 = 0.80
β 1 = 0.75
ρ balanced
0.0371
0.0495
0.0582
0.0655
ρ when t = 0.004
0.0232
0.0310
0.0364
0.0410
ρ when t = 0.005
0.0203
0.0271
0.0319
0.0359
ρ when t = 0.0075
0.0155
0.0206
0.0243
0.0273
ρ min for flexure
0.0050
0.0050
0.0053
0.0058
ρ balanced
0.0275
0.0367
0.0432
0.0486
ρ when t = 0.004
0.0186
0.0248
0.0291
0.0328
ρ when t = 0.005
0.0163
0.0217
0.0255
0.0287
ρ when t = 0.0075
0.0124
0.0165
0.0194
0.0219
ρ min for flexure
0.0040
0.0040
0.0042
0.0046
ρ balanced
0.0214
0.0285
0.0335
0.0377
ρ when t = 0.004
0.0155
0.0206
0.0243
0.0273
ρ when t = 0.005
0.0136
0.0181
0.0212
0.0239
ρ when t = 0.0075
0.0103
0.0138
0.0162
0.0182
ρ min for flexure
0.0033
0.0033
0.0035
0.0039
ρ balanced
0.0155
0.0207
0.0243
0.0274
ρ when t = 0.004
0.0124
0.0165
0.0194
0.0219
ρ when t = 0.005
0.0108
0.0144
0.0170
0.0191
ρ when t = 0.0075
0.0083
0.0110
0.0130
0.0146
ρ min for flexure
0.0027
0.0027
0.0028
0.0031
fy
Grade 40
40,000 psi
(275.8 MPa)
3000 psi
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 637
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.8 fy = 40,000 psi; f c = 3000 psi—U.S. Customary Units
ρ
ρmin for
temp. and
shrinkage
Mu
ρ
φbd2
Mu
φbd2
ρ
Mu
φbd2
Mu
ρ
φbd2
78.74
ρmin for
0.0050
192.1
0.0080
299.9
0.0110
401.9
0.0021
82.62
flexure
0.0051
195.8
0.0081
303.4
0.0111
405.2
0.0022
86.48
0.0052
199.5
0.0082
306.8
0.0112
408.5
0.0023
90.34
0.0053
203.2
0.0083
310.3
0.0113
411.8
0.0024
94.19
0.0054
206.8
0.0084
313.8
0.0114
415.1
0.0025
98.04
0.0055
210.5
0.0085
317.3
0.0115
418.4
0.0020
0.0026
101.9
0.0056
214.1
0.0086
320.7
0.0116
421.7
0.0027
105.7
0.0057
217.8
0.0087
324.2
0.0117
424.9
0.0028
109.5
0.0058
221.4
0.0088
327.6
0.0118
428.2
0.0029
113.4
0.0059
225.0
0.0089
331.1
0.0119
431.4
0.0030
117.2
0.0060
228.7
0.0090
334.5
0.0120
434.7
0.0031
121.0
0.0061
232.3
0.0091
337.9
0.0121
437.9
0.0032
124.8
0.0062
235.9
0.0092
341.4
0.0122
441.2
0.0033
128.6
0.0063
239.5
0.0093
344.8
0.0123
444.4
0.0034
132.4
0.0064
243.1
0.0094
348.2
0.0124
447.6
0.0035
136.2
0.0065
246.7
0.0095
351.6
0.0125
450.8
0.0036
139.9
0.0066
250.3
0.0096
355.0
0.0126
454.0
0.0037
143.7
0.0067
253.9
0.0097
358.4
0.0127
457.2
0.0038
147.5
0.0068
257.4
0.0098
361.8
0.0128
460.4
0.0039
151.2
0.0069
261.0
0.0099
365.2
0.0129
463.6
0.0040
155.0
0.0070
264.6
0.0100
368.5
0.0130
466.8
0.0041
158.7
0.0071
268.1
0.0101
371.9
0.0131
470.0
0.0042
162.5
0.0072
271.7
0.0102
375.3
0.0132
473.2
0.0043
166.2
0.0073
275.2
0.0103
378.6
0.0133
476.3
0.0044
169.9
0.0074
278.8
0.0104
382.0
0.0134
479.5
0.0045
173.6
0.0075
282.3
0.0105
385.3
0.0135
482.6
0.0046
177.4
0.0076
285.8
0.0106
388.6
0.0136
485.8
0.0047
181.1
0.0077
289.3
0.0107
392.0
0.0137
488.9
0.0048
184.8
0.0078
292.9
0.0108
395.3
0.0138
492.1
0.0049
188.5
0.0079
296.4
0.0109
398.6
0.0139
495.2
(continues)
637
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 638
638
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.8 (Continued)
ρ
Mu
φbd 2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
0.0140
498.3
0.0156
547.4
0.0172
594.9
0.0188
640.8
0.0141
501.4
0.0157
550.4
0.0173
597.8
0.0189
643.6
0.0142
504.5
0.0158
553.4
0.0174
600.7
0.0190
646.4
0.0143
507.6
0.0159
556.4
0.0175
603.6
0.0191
649.2
0.0144
510.7
0.0160
559.4
0.0176
606.5
0.0192
652.0
0.0145
513.8
0.0161
562.4
0.0177
609.4
0.0193
654.8
0.0146
516.9
0.0162
565.4
0.0178
612.3
0.0194
657.6
0.0147
520.0
0.0163
568.4
0.0179
615.2
0.0195
660.3
0.0148
523.1
0.0164
571.4
0.0180
618.0
0.0196
663.1
0.0149
526.1
0.0165
574.3
0.0181
620.9
0.0197
665.9
0.0150
529.2
0.0166
577.3
0.0182
623.8
0.0198
668.6
0.0151
532.2
0.0167
580.2
0.0183
626.6
0.0199
671.4
0.0152
535.3
0.0168
583.2
0.0184
629.5
0.0200
674.1
0.0153
538.3
0.0169
586.1
0.0185
632.3
0.0201
676.9
0.0154
541.4
0.0170
589.1
0.0186
635.1
0.0202
679.6
0.0155
544.4
0.0171
592.0
0.0187
638.0
0.0203
682.3
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 639
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.9 fy = 40,000 psi; f c = 4000 psi—U.S. Customary Units
ρ
ρmin for
temp. and
shrinkage
Mu
ρ
φbd2
Mu
φbd 2
ρ
Mu
φbd2
Mu
ρ
φbd2
79.06
ρmin for
0.0050
194.1
0.0080
304.9
0.0110
411.4
0.0021
82.96
flexure
0.0051
197.9
0.0081
308.5
0.0111
414.9
0.0022
86.86
0.0052
201.6
0.0082
312.1
0.0112
418.4
0.0023
90.75
0.0053
205.4
0.0083
315.7
0.0113
421.9
0.0024
94.64
0.0054
209.1
0.0084
319.3
0.0114
425.3
0.0025
98.53
0.0055
212.9
0.0085
322.9
0.0115
428.8
0.0026
102.4
0.0056
216.6
0.0086
326.5
0.0116
432.2
0.0027
106.3
0.0057
220.3
0.0087
330.1
0.0117
435.7
0.0028
110.2
0.0058
224.1
0.0088
333.7
0.0118
439.1
0.0029
114.0
0.0059
227.8
0.0089
337.3
0.0119
442.6
0.0030
117.9
0.0060
231.5
0.0090
340.9
0.0120
446.0
0.0031
121.7
0.0061
235.2
0.0091
344.5
0.0121
449.4
0.0032
125.6
0.0062
238.9
0.0092
348.0
0.0122
452.9
0.0033
129.4
0.0063
242.6
0.0093
351.6
0.0123
456.3
0.0034
133.3
0.0064
246.3
0.0094
355.1
0.0124
459.7
0.0035
137.1
0.0065
250.0
0.0095
358.7
0.0125
463.1
0.0036
141.0
0.0066
253.7
0.0096
362.2
0.0126
466.5
0.0037
144.8
0.0067
257.4
0.0097
365.8
0.0127
469.9
0.0038
148.6
0.0068
261.1
0.0098
369.3
0.0128
473.3
0.0039
152.4
0.0069
264.8
0.0099
372.9
0.0129
476.7
0.0040
156.2
0.0070
268.4
0.0100
376.4
0.0130
480.1
0.0041
160.0
0.0071
272.1
0.0101
379.9
0.0131
483.5
0.0042
163.8
0.0072
275.8
0.0102
383.4
0.0132
486.9
0.0043
167.6
0.0073
279.4
0.0103
387.0
0.0133
490.2
0.0044
171.4
0.0074
283.1
0.0104
390.5
0.0134
493.6
0.0045
175.2
0.0075
286.7
0.0105
394.0
0.0135
497.0
0.0046
179.0
0.0076
290.4
0.0106
397.5
0.0136
500.3
0.0047
182.8
0.0077
294.0
0.0107
401.0
0.0137
503.7
0.0048
186.6
0.0078
297.6
0.0108
404.5
0.0138
507.0
0.0049
190.4
0.0079
301.3
0.0109
408.0
0.0139
510.4
0.0020
(continues)
639
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 640
640
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.9 (Continued)
ρ
Mu
φbd 2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
0.0140
513.7
0.0173
621.4
0.0206
723.8
0.0239
821.2
0.0141
517.1
0.0174
624.5
0.0207
726.9
0.0240
824.1
0.0142
520.4
0.0175
627.7
0.0208
729.9
0.0241
826.9
0.0143
523.7
0.0176
630.9
0.0209
732.9
0.0242
829.8
0.0144
527.1
0.0177
634.1
0.0210
735.9
0.0243
832.6
0.0145
530.4
0.0178
637.2
0.0211
738.9
0.0244
835.5
0.0146
533.7
0.0179
640.4
0.0212
741.9
0.0245
838.3
0.0147
537.0
0.0180
643.5
0.0213
744.9
0.0246
841.2
0.0148
540.3
0.0181
646.7
0.0214
747.9
0.0247
844.0
0.0149
543.6
0.0182
649.8
0.0215
750.9
0.0248
846.8
0.0150
546.9
0.0183
653.0
0.0216
753.9
0.0249
849.7
0.0151
550.2
0.0184
656.1
0.0217
756.9
0.0250
852.5
0.0152
553.5
0.0185
659.2
0.0218
759.8
0.0251
855.3
0.0153
556.7
0.0186
662.3
0.0219
762.8
0.0252
858.1
0.0154
560.0
0.0187
665.5
0.0220
765.8
0.0253
860.9
0.0155
563.3
0.0188
668.6
0.0221
768.7
0.0254
863.7
0.0156
566.6
0.0189
671.7
0.0222
771.7
0.0255
866.5
0.0157
569.8
0.0190
674.8
0.0223
774.6
0.0256
869.3
0.0158
573.1
0.0191
677.9
0.0224
777.6
0.0257
872.1
0.0159
576.3
0.0192
681.0
0.0225
780.5
0.0258
874.9
0.0160
579.6
0.0193
684.1
0.0226
783.4
0.0259
877.7
0.0161
582.8
0.0194
687.2
0.0227
786.4
0.0260
880.5
0.0162
586.1
0.0195
690.3
0.0228
789.3
0.0261
883.2
0.0163
589.3
0.0196
693.3
0.0229
792.2
0.0262
886.0
0.0164
592.5
0.0197
696.4
0.0230
795.1
0.0263
888.7
0.0165
595.7
0.0198
699.5
0.0231
798.1
0.0264
891.5
0.0166
599.0
0.0199
702.5
0.0232
801.0
0.0265
894.3
0.0167
602.2
0.0200
705.6
0.0233
803.9
0.0266
897.0
0.0168
605.4
0.0201
708.6
0.0234
806.8
0.0267
899.7
0.0169
608.6
0.0202
711.7
0.0235
809.7
0.0268
902.5
0.0170
611.8
0.0203
714.7
0.0236
812.5
0.0269
905.2
0.0171
615.0
0.0204
717.8
0.0237
815.4
0.0270
907.9
0.0172
618.2
0.0205
720.8
0.0238
818.3
0.0271
910.7
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 641
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.10 fy = 50,000 psi; f c = 3000 psi—U.S. Customary Units
ρ
ρmin for
0.0020
temp. and
shrinkage
0.0021
0.0022
Mu
φbd2
98.04
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
0.0056
264.6
0.0092
418.4
0.0128
559.4
102.8
0.0057
269.0
0.0093
422.5
0.0129
563.2
107.6
0.0058
273.5
0.0094
426.6
0.0130
566.9
0.0023
112.4
0.0059
277.9
0.0095
430.6
0.0131
570.6
0.0024
117.2
0.0060
282.3
0.0096
434.7
0.0132
574.3
0.0025
121.9
0.0061
286.7
0.0097
438.7
0.0133
578.0
0.0026
126.7
0.0062
291.1
0.0098
442.8
0.0134
581.7
0.0027
131.4
0.0063
295.5
0.0099
446.8
0.0135
585.4
0.0028
136.2
0.0064
299.9
0.0100
450.8
0.0136
589.1
0.0029
140.9
0.0065
304.2
0.0101
454.8
0.0137
592.7
0.0030
145.6
0.0066
308.6
0.0102
458.8
0.0138
596.4
0.0031
150.3
0.0067
312.9
0.0103
462.8
0.0139
600.0
0.0032
155.0
0.0068
317.3
0.0104
466.8
0.0140
603.6
0.0033
159.7
0.0069
321.6
0.0105
470.8
0.0141
607.2
0.0034
164.3
0.0070
325.9
0.0106
474.8
0.0142
610.9
0.0035
169.0
0.0071
330.2
0.0107
478.7
0.0143
614.5
0.0036
173.6
0.0072
334.5
0.0108
482.6
0.0144
618.0
0.0037
178.3
0.0073
338.8
0.0109
486.6
0.0145
621.6
0.0038
182.9
0.0074
343.1
0.0110
490.5
0.0146
625.2
ρmin for
0.0039
187.5
0.0075
347.3
0.0111
494.4
0.0147
628.7
flexure
0.0040
192.1
0.0076
351.6
0.0112
498.3
0.0148
632.3
0.0041
196.7
0.0077
355.8
0.0113
502.2
0.0149
635.8
0.0042
201.3
0.0078
360.1
0.0114
506.1
0.0150
639.4
0.0043
205.9
0.0079
364.3
0.0115
510.0
0.0151
642.9
0.0044
210.5
0.0080
368.5
0.0116
513.8
0.0152
646.4
0.0045
215.0
0.0081
372.7
0.0117
517.7
0.0153
649.9
0.0046
219.6
0.0082
376.9
0.0118
521.5
0.0154
653.4
0.0047
224.1
0.0083
381.1
0.0119
525.4
0.0155
656.9
0.0048
228.7
0.0084
385.3
0.0120
529.2
0.0156
660.3
0.0049
233.2
0.0085
389.5
0.0121
533.0
0.0157
663.8
0.0050
237.7
0.0086
393.6
0.0122
536.8
0.0158
667.3
0.0051
242.2
0.0087
397.8
0.0123
540.6
0.0159
670.7
0.0052
246.7
0.0088
401.9
0.0124
544.4
0.0160
674.1
0.0053
251.2
0.0089
406.1
0.0125
548.2
0.0161
677.5
0.0054
255.7
0.0090
410.2
0.0126
551.9
0.0162
681.0
0.0055
260.1
0.0091
414.3
0.0127
555.7
0.0163
684.4
641
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 642
642
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.11 fy = 50,000 psi; f c = 4000 psi—U.S. Customary Units
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρmin for
0.0020
98.53
0.0059
282.2
0.0098
454.6
0.0137
615.8
temp. and
shrinkage
0.0021
103.4
0.0060
286.7
0.0099
458.9
0.0138
619.8
0.0022
108.2
0.0061
291.3
0.0100
463.1
0.0139
623.7
0.0023
113.0
0.0062
295.8
0.0101
467.4
0.0140
627.7
0.0024
117.9
0.0063
300.4
0.0102
471.6
0.0141
631.7
0.0025
122.7
0.0064
304.9
0.0103
475.9
0.0142
635.6
0.0026
127.5
0.0065
309.4
0.0104
480.1
0.0143
639.6
0.0027
132.3
0.0066
313.9
0.0105
484.3
0.0144
643.5
0.0028
137.1
0.0067
318.4
0.0106
488.6
0.0145
647.5
0.0029
141.9
0.0068
322.9
0.0107
492.8
0.0146
651.4
0.0030
146.7
0.0069
327.4
0.0108
497.0
0.0147
655.3
0.0031
151.5
0.0070
331.9
0.0109
501.2
0.0148
659.2
0.0032
156.2
0.0071
336.4
0.0110
505.4
0.0149
663.1
0.0033
161.0
0.0072
340.9
0.0111
509.6
0.0150
667.0
0.0034
165.7
0.0073
345.3
0.0112
513.7
0.0151
670.9
0.0035
170.5
0.0074
349.8
0.0113
517.9
0.0152
674.8
0.0036
175.2
0.0075
354.3
0.0114
522.1
0.0153
678.7
0.0037
180.0
0.0076
358.7
0.0115
526.2
0.0154
682.5
0.0038
184.7
0.0077
363.1
0.0116
530.4
0.0155
686.4
ρmin for
0.0039
189.4
0.0078
367.6
0.0117
534.5
0.0156
690.3
flexure
0.0040
194.1
0.0079
372.0
0.0118
538.6
0.0157
694.1
0.0041
198.8
0.0080
376.4
0.0119
542.8
0.0158
697.9
0.0042
203.5
0.0081
380.8
0.0120
546.9
0.0159
701.8
0.0043
208.2
0.0082
385.2
0.0121
551.0
0.0160
705.6
0.0044
212.9
0.0083
389.6
0.0122
555.1
0.0161
709.4
0.0045
217.5
0.0084
394.0
0.0123
559.2
0.0162
713.2
0.0046
222.2
0.0085
398.4
0.0124
563.3
0.0163
717.0
0.0047
226.9
0.0086
402.7
0.0125
567.4
0.0164
720.8
0.0048
231.5
0.0087
407.1
0.0126
571.4
0.0165
724.6
0.0049
236.1
0.0088
411.4
0.0127
575.5
0.0166
728.4
0.0050
240.8
0.0089
415.8
0.0128
579.6
0.0167
732.1
0.0051
245.4
0.0090
420.1
0.0129
583.6
0.0168
735.9
0.0052
250.0
0.0091
424.5
0.0130
587.7
0.0169
327.4
0.0053
254.6
0.0092
428.8
0.0131
591.7
0.0170
743.4
0.0054
259.2
0.0093
433.1
0.0132
595.7
0.0171
747.2
0.0055
263.8
0.0094
437.4
0.0133
599.8
0.0172
750.9
0.0056
268.4
0.0095
441.7
0.0134
603.8
0.0173
754.6
0.0057
273.0
0.0096
446.0
0.0135
607.8
0.0174
758.3
0.0058
277.6
0.0097
450.3
0.0136
611.8
0.0175
762.1
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 643
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.11 (Continued)
ρ
Mu
φbd2
ρ
Mu
φbd 2
ρ
Mu
φbd2
ρ
Mu
φbd2
0.0176
765.8
0.0187
806.0
0.0198
845.4
0.0208
880.5
0.0177
769.5
0.0188
809.7
0.0199
849.0
0.0209
883.9
0.0178
773.2
0.0189
813.3
0.0200
852.5
0.0210
887.4
0.0179
776.8
0.0190
816.9
0.0201
856.0
0.0211
890.8
0.0180
780.5
0.0191
820.5
0.0202
859.5
0.0212
894.3
0.0181
784.2
0.0192
824.1
0.0203
863.0
0.0213
897.7
0.0182
787.8
0.0193
827.6
0.0204
866.5
0.0214
901.1
0.0183
791.5
0.0194
831.2
0.0205
870.0
0.0215
904.5
0.0184
795.1
0.0195
834.8
0.0206
873.5
0.0216
907.9
0.0185
798.8
0.0196
838.3
0.0207
877.0
0.0217
911.3
0.0186
802.4
0.0197
841.9
643
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 644
644
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.12 fy = 60,000 psi; f c = 3000 psi—U.S. Customary Units
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρmin for
0.0018
105.7
0.0048
271.7
0.0078
424.9
0.0108
565.4
temp. and
shrinkage
0.0019
111.5
0.0049
277.0
0.0079
429.8
0.0109
569.9
0.0020
117.2
0.0050
282.3
0.0080
434.7
0.0110
574.3
0.0021
122.9
0.0051
287.6
0.0081
439.5
0.0111
578.8
0.0022
128.6
0.0052
292.9
0.0082
444.4
0.0112
582.3
0.0023
134.3
0.0053
298.1
0.0083
449.2
0.0113
587.6
0.0024
139.9
0.0054
303.4
0.0084
454.0
0.0114
592.0
0.0025
145.6
0.0055
308.6
0.0085
458.8
0.0115
596.4
0.0026
151.2
0.0056
313.8
0.0086
463.6
0.0116
600.7
0.0027
156.9
0.0057
319.0
0.0087
468.4
0.0117
605.1
0.0028
162.5
0.0058
324.2
0.0088
473.2
0.0118
609.4
0.0029
168.1
0.0059
329.4
0.0089
477.9
0.0119
613.7
0.0030
173.7
0.0060
334.5
0.0090
482.6
0.0120
618.0
0.0031
179.2
0.0061
339.7
0.0091
487.4
0.0121
622.3
0.0032
184.8
0.0062
344.8
0.0092
492.1
0.0122
626.6
ρmin for
0.0033
190.3
0.0063
349.9
0.0093
496.8
0.0123
630.9
flexure
0.0034
195.8
0.0064
355.0
0.0094
501.4
0.0124
635.1
0.0035
201.3
0.0065
360.1
0.0095
506.1
0.0125
639.4
0.0036
206.8
0.0066
365.2
0.0096
510.7
0.0126
643.6
0.0037
212.3
0.0067
370.2
0.0097
515.4
0.0127
647.8
0.0038
217.8
0.0068
375.3
0.0098
520.0
0.0128
652.0
0.0039
223.2
0.0069
380.3
0.0099
524.6
0.0129
656.2
0.0040
228.7
0.0070
385.3
0.0100
529.2
0.0130
660.9
0.0041
234.1
0.0071
390.3
0.0101
533.8
0.0131
664.5
0.0042
239.5
0.0072
395.3
0.0102
538.3
0.0132
668.6
0.0043
244.9
0.0073
400.3
0.0103
542.9
0.0133
672.8
0.0044
250.3
0.0074
405.2
0.0104
547.4
0.0134
676.9
0.0045
255.7
0.0075
410.2
0.0105
551.9
0.0135
681.0
0.0046
261.0
0.0076
415.1
0.0106
556.4
0.0136
685.0
0.0047
266.4
0.0077
420.0
0.0107
560.9
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 645
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.13 fy = 60,000 psi; f c = 4000 psi—U.S. Customary Units
ρ
ρmin for
temp. and
shrinkage
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
ρ
Mu
φbd2
0.0018
106.3
0.0059
335.5
0.0100
546.9
0.0141
740.4
0.0019
112.1
0.0060
340.9
0.0101
551.8
0.0142
744.9
0.0020
117.1
0.0061
346.2
0.0102
556.7
0.0143
749.4
0.0021
123.7
0.0062
351.6
0.0103
561.7
0.0144
753.9
0.0022
129.4
0.0063
356.9
0.0104
566.6
0.0145
758.3
0.0023
135.2
0.0064
362.2
0.0105
571.5
0.0146
762.8
0.0024
141.0
0.0065
367.6
0.0106
576.3
0.0147
767.2
0.0025
146.7
0.0066
372.9
0.0107
581.2
0.0148
771.7
0.0026
152.4
0.0067
378.2
0.0108
586.1
0.0149
776.1
0.0027
158.1
0.0068
383.4
0.0109
590.9
0.0150
780.5
0.0028
163.8
0.0069
388.7
0.0110
595.7
0.0151
784.9
0.0029
169.5
0.0070
394.0
0.0111
600.6
0.0152
789.3
0.0030
175.2
0.0071
399.2
0.0112
605.4
0.0153
793.7
0.0031
180.9
0.0072
404.5
0.0113
610.2
0.0154
798.1
0.0032
186.6
0.0073
409.7
0.0114
615.0
0.0155
802.4
ρmin for
0.0033
192.2
0.0074
414.9
0.0115
619.8
0.0156
806.8
flexure
0.0034
197.9
0.0075
420.1
0.0116
624.5
0.0157
811.1
0.0035
203.5
0.0076
425.3
0.0117
629.3
0.0158
815.4
0.0036
209.1
0.0077
430.5
0.0118
634.1
0.0159
819.7
0.0037
214.7
0.0078
435.7
0.0119
638.8
0.0160
824.1
0.0038
220.3
0.0079
440.9
0.0120
643.5
0.0161
828.3
0.0039
225.9
0.0080
446.0
0.0121
648.2
0.0162
832.6
0.0040
231.5
0.0081
451.2
0.0122
653.0
0.0163
836.9
0.0041
237.1
0.0082
456.3
0.0123
657.7
0.0164
841.2
0.0042
242.6
0.0083
461.4
0.0124
662.3
0.0165
845.4
0.0043
248.2
0.0084
466.5
0.0125
667.0
0.0166
849.7
0.0044
253.7
0.0085
471.6
0.0126
671.7
0.0167
853.9
0.0045
259.2
0.0086
476.7
0.0127
676.3
0.0168
858.1
0.0046
264.8
0.0087
481.8
0.0128
681.0
0.0169
862.3
0.0047
270.3
0.0088
486.9
0.0129
685.6
0.0170
866.5
0.0048
275.8
0.0089
491.9
0.0130
690.3
0.0171
870.7
0.0049
281.2
0.0090
497.0
0.0131
694.9
0.0172
874.9
0.0050
286.7
0.0091
502.0
0.0132
699.5
0.0173
879.1
0.0051
292.2
0.0092
507.1
0.0133
704.1
0.0174
883.2
0.0052
297.6
0.0093
512.1
0.0134
708.6
0.0175
887.4
0.0053
303.1
0.0094
517.1
0.0135
713.2
0.0176
891.5
0.0054
308.5
0.0095
522.1
0.0136
717.8
0.0177
895.6
0.0055
313.9
0.0096
527.1
0.0137
722.3
0.0178
899.7
0.0056
319.3
0.0097
532.0
0.0138
726.9
0.0179
903.9
0.0057
324.7
0.0098
537.0
0.0139
731.4
0.0180
907.9
0.0058
330.1
0.0099
542.0
0.0140
735.9
0.0181
912.0
645
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 646
646
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.14 Size and Pitch of Spirals, ACI Code—U.S. Customary Units
Diameter of
Out to Out of
Column (in.)
Spiral (in.)
f c
2500
3000
4000
5000
fy = 40,000:
14, 15
11,12
16
13
17– 19
14–16
20– 23
17–20
24– 30
21–27
3
8
3
8
3
8
3
8
3
8
−2
1
4
1
4
1
4
1
4
− 1 34
−2
−
−
−
2 14
2 14
2 14
3
8
3
8
3
8
3
8
3
8
− 1 34
3
8
3
8
3
8
3
8
− 2 34
− 1 34
− 1 34
− 1 34
−2
1
2
1
2
1
2
1
2
1
2
− 2 12
3
8
3
8
3
8
3
8
−2
− 2 12
− 2 12
− 2 12
− 2 12
1
2
1
2
1
2
1
2
1
2
− 1 34
1
2
1
2
1
2
1
2
− 2 34
−2
−2
−2
−2
fy = 60,000:
14, 15
11, 12
16– 23
13–20
24– 29
21–26
30
17
− 1 34
−
−
1 34
1 34
− 2 34
−3
−3
−2
− 2 14
− 2 14
−3
−3
− 3 14
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 647
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.15 Weights, Areas, and Moments of Inertia of Circular Columns and Moments of
Inertia of Column Verticals Arranged in a Circle 5 in. Less Than the Diameter of Column: U.S.
Customary Units
gt = t – 5 in.
Diameter
of Column h (in.)
Weight per
Foot (lb)
Area (in.2 )
I (in.4 )
As Where
ρg = 0.01*
12
118
113
1,018
1.13
6.92
13
138
133
1,402
1.33
10.64
14
160
154
1,886
1.54
15.59
15
184
177
2,485
1.77
22.13
16
210
201
3,217
2.01
30.40
17
237
227
4,100
2.27
40.86
18
265
255
5,153
2.55
53.87
19
295
284
6,397
2.84
69.58
20
327
314
7,854
3.14
21
361
346
9,547
3.46
110.7
22
396
380
11,500
3.80
137.2
23
433
416
13,740
4.16
168.4
24
471
452
16,290
4.52
203.9
25
511
491
19,170
4.91
245.5
26
553
531
22,430
5.31
292.7
27
597
573
26,090
5.73
346.7
28
642
616
30,170
6.16
407.3
29
688
661
34,720
6.61
475.9
30
736
707
39,760
7.07
552.3
Is (in.4)†
88.31
∗ For other values of ρ , multiply the value by 100 ρ .
g
g
† The bars are assumed transformed into a thin-walled cylinder having the same sectional area as the bars. Then
Is = As (γ t )2/8.
647
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 648
648
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.16 Moment Distribution Constants for Slabs Without Drop Panelsa
w
c1A
c1B
A
B
h
`1
CL
Column
Dimension
c1A
c1B
l1
l1
0.00
0.05
CL
Uniform Load
FEM = Coef. (wl2 l12 )
Stiffness
Factor†
Carryover
Factor
MAB
MBA
kAB
kBA
COF AB
COF BA
0.00
0.083
0.083
4.00
4.00
0.500
0.500
0.05
0.083
0.084
4.01
4.04
0.504
0.500
0.10
0.082
0.086
4.03
4.15
0.513
0.499
0.15
0.081
0.089
4.07
4.32
0.528
0.498
0.20
0.079
0.093
4.12
4.56
0.548
0.495
0.25
0.077
0.097
4.18
4.88
0.573
0.491
0.30
0.075
0.102
4.25
5.28
0.603
0.485
0.35
0.073
0.107
4.33
5.78
0.638
0.478
0.05
0.084
0.084
4.05
4.05
0.503
0.503
0.10
0.083
0.086
4.07
4.15
0.513
0.503
0.15
0.081
0.089
4.11
4.33
0.528
0.501
0.20
0.080
0.092
4.16
4.58
0.548
0.499
0.25
0.078
0.096
4.22
4.89
0.573
0.494
0.30
0.076
0.101
4.29
5.30
0.603
0.489
0.35
0.074
0.107
4.37
5.80
0.638
0.481
0.10
0.085
0.085
4.18
4.18
0.513
0.513
0.15
0.083
0.088
4.22
4.36
0.528
0.511
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 649
Tables and Graphs: U.S. Customary Units
APPENDIX A
TABLE A.16 (Continued)
Column
Dimension
c1A
c1B
l1
l1
Uniform Load
FEM = Coef. (wl2 l12 )
MAB
MBA
kAB
kBA
COF AB
COF BA
0.10
0.20
0.082
0.091
4.27
4.61
0.548
0.508
0.25
0.080
0.095
4.34
4.93
0.573
0.504
0.30
0.078
0.100
4.41
5.34
0.602
0.498
0.35
0.075
0.105
4.50
5.85
0.637
0.491
0.15
0.086
0.086
4.40
4.40
0.526
0.526
0.20
0.084
0.090
4.46
4.65
0.546
0.523
0.25
0.083
0.094
4.53
4.98
0.571
0.519
0.30
0.080
0.099
4.61
5.40
0.601
0.513
0.35
0.078
0.104
4.70
5.92
0.635
0.505
0.20
0.088
0.088
4.72
4.72
0.543
0.543
0.25
0.086
0.092
4.79
5.05
0.568
0.539
0.30
0.083
0.097
4.88
5.48
0.597
0.532
0.35
0.081
0.102
4.99
6.01
0.632
0.524
0.25
0.090
0.090
5.14
5.14
0.563
0.563
0.30
0.088
0.095
5.24
5.58
0.592
0.556
0.35
0.085
0.100
5.36
6.12
0.626
0.548
0.30
0.092
0.092
5.69
5.69
0.585
0.585
0.35
0.090
0.097
5.83
6.26
0.619
0.576
0.35
0.095
0.095
6.42
6.42
0.609
0.609
0.15
0.20
0.25
0.30
0.35
Stiffness
Factor†
Carryover
Factor
a Applicable when c /l = c /l . For other relationships between these ratios, the constants will be slightly in error.
1 1
2 2
† Stiffness is K
3
3
AB = kAB E(l2 h /12l1 ) and KBA = kBA E(l2 h /12 l1 )
649
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 650
650
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.17 Moment Distribution Constants for Slabs with Drop Panelsa
w
c1A
c1B
A
B
h
1.25h
`1 ⎜6
`1 ⎜6
2`1 ⎜3
CL
Column
Dimension
c1A
c1B
l1
l1
0.00
0.05
0.10
0.15
0.20
0.25
0.30
CL
Uniform Load
FEM = Coef. (wl2 l12 )
Stiffness
Factor†
Carryover
Factor
MAB
MBA
kAB
kBA
COFAB
COFBA
0.00
0.088
0.088
4.78
4.78
0.541
0.541
0.05
0.087
0.089
4.80
4.82
0.545
0.541
0.10
0.087
0.090
4.83
4.94
0.553
0.541
0.15
0.085
0.093
4.87
5.12
0.567
0.540
0.20
0.084
0.096
4.93
5.36
0.585
0.537
0.25
0.082
0.100
5.00
5.68
0.606
0.534
0.30
0.080
0.105
5.09
6.07
0.631
0.529
0.05
0.088
0.088
4.84
4.84
0.545
0.545
0.10
0.087
0.090
4.87
4.95
0.553
0.544
0.15
0.085
0.093
4.91
5.13
0.567
0.543
0.20
0.084
0.096
4.97
5.38
0.584
0.541
0.25
0.082
0.100
5.05
5.70
0.606
0.537
0.30
0.080
0.104
5.13
6.09
0.632
0.532
0.10
0.089
0.089
4.98
4.98
0.553
0.553
0.15
0.088
0.092
5.03
5.16
0.566
0.551
0.20
0.086
0.094
5.09
5.42
0.584
0.549
0.25
0.084
0.099
5.17
5.74
0.606
0.546
0.30
0.082
0.103
5.26
6.13
0.631
0.541
0.15
0.090
0.090
5.22
5.22
0.565
0.565
0.20
0.089
0.094
5.28
5.47
0.583
0.563
0.25
0.087
0.097
5.37
5.80
0.604
0.559
0.30
0.085
0.102
5.46
6.21
0.630
0.554
0.20
0.092
0.092
5.55
5.55
0.580
0.580
0.25
0.090
0.096
5.64
5.88
0.602
0.577
0.30
0.088
0.100
5.74
6.30
0.627
0.571
0.25
0.094
0.094
5.98
5.98
0.598
0.598
0.30
0.091
0.098
6.10
6.41
0.622
0.593
0.30
0.095
0.095
6.54
6.54
0.617
0.617
a Applicable when c /l = c /l . For other relationships between these ratios, the constants will be slightly in error.
1 1
2 2
† Stiffness is K
3
3
AB = kAB E(l2 h /12 l1 ) and KBA = kBA E(l2 h /12 l1 )
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 651
Tables and Graphs: U.S. Customary Units
APPENDIX A
TABLE A.18 Moment Distribution Constants for Slab-Beam Members with Column Capitals
c1
FEM (uniform load w) = Mw`2(`1)2
= kE`2h3
12`1
=C
K (stiffness)
Carryover factor
c1
l1
0.00
0.05
`2
w
c2
h
h
c1
c1
l2
M
k
C
c1
l1
c1
l2
M
k
C
0.00
0.083
4.000
0.500
0.10
0.25
0.087
4.449
0.530
0.05
0.083
4.000
0.500
0.30
0.087
4.535
0.535
0.10
0.083
4.000
0.500
0.35
0.088
4.618
0.540
0.15
0.083
4.000
0.500
0.40
0.088
4.698
0.545
0.20
0.083
4.000
0.500
0.45
0.089
4.774
0.550
0.25
0.083
4.000
0.500
0.50
0.089
4.846
0.554
0.30
0.083
4.000
0.500
0.00
0.083
4.000
0.500
0.35
0.083
4.000
0.500
0.05
0.084
4.132
0.509
0.40
0.083
4.000
0.500
0.10
0.085
4.267
0.517
0.45
0.083
4.000
0.500
0.15
0.086
4.403
0.526
0.50
0.083
4.000
0.500
0.20
0.087
4.541
0.534
0.00
0.083
4.000
0.500
0.25
0.088
4.680
0.543
0.05
0.084
4.047
0.503
0.30
0.089
4.818
0.550
0.10
0.084
4.093
0.507
0.00
0.083
4.000
0.500
0.15
0.084
4.138
0.510
0.05
0.085
4.170
0.511
0.20
0.085
4.181
0.513
0.10
0.086
4.346
0.522
0.25
0.085
4.222
0.516
0.15
0.087
4.529
0.532
0.30
0.085
4.261
0.518
0.20
0.088
4.717
0.543
0.35
0.086
4.299
0.521
0.25
0.089
4.910
0.554
0.40
0.086
4.334
0.523
0.30
0.090
5.108
0.564
0.45
0.086
4.368
0.526
0.35
0.091
5.308
0.574
0.50
0.086
4.398
0.528
0.40
0.092
5.509
0.584
0.00
0.083
4.000
0.500
0.45
0.093
5.710
0.593
0.05
0.084
4.091
0.506
0.50
0.094
5.908
0.602
0.10
0.085
4.182
0.513
0.00
0.083
4.000
0.500
0.15
0.085
4.272
0.519
0.05
0.085
4.204
0.512
0.20
0.086
4.362
0.524
0.10
0.086
4.420
0.525
0.15
0.20
(continues)
651
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 652
652
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.18 (Continued)
c1
l1
0.25
0.30
0.35
c1
l2
M
k
C
0.15
0.087
4.648
0.538
0.20
0.089
4.887
0.550
0.25
0.090
5.138
0.30
0.091
5.401
0.35
0.093
0.40
0.094
0.45
c1
l1
c1
l2
M
k
C
0.20
0.090
5.348
0.563
0.25
0.092
5.778
0.580
0.563
0.30
0.094
6.255
0.598
0.576
0.35
0.095
6.782
0.617
5.672
0.588
0.40
0.097
7.365
0.635
5.952
0.600
0.45
0.099
8.007
0.654
0.095
6.238
0.612
0.50
0.100
8.710
0.672
0.50
0.096
6.527
0.623
0.00
0.083
4.000
0.500
0.00
0.083
4.000
0.500
0.05
0.085
4.311
0.515
0.05
0.085
4.235
0.514
0.10
0.087
4.658
0.530
0.10
0.086
4.488
0.527
0.15
0.088
5.046
0.547
0.15
0.088
4.760
0.542
0.20
0.090
5.480
0.564
0.20
0.089
5.050
0.556
0.25
0.092
5.967
0.583
0.25
0.091
5.361
0.571
0.35
0.095
6.416
0.609
0.30
0.092
5.692
0.585
0.40
0.096
6.888
0.626
0.35
0.094
6.044
0.600
0.45
0.098
7.395
0.642
0.40
0.095
6.414
0.614
0.50
0.099
7.935
0.658
0.45
0.096
6.802
0.628
0.30
0.094
6.517
0.602
0.50
0.098
7.205
0.642
0.35
0.096
7.136
0.621
0.00
0.083
4.000
0.500
0.40
0.098
7.836
0.642
0.05
0.085
4.264
0.514
0.45
0.100
8.625
0.662
0.10
0.087
4.551
0.529
0.50
0.101
9.514
0.683
0.15
0.088
4.864
0.545
0.00
0.083
4.000
0.500
0.20
0.090
5.204
0.560
0.05
0.085
4.331
0.515
0.25
0.091
5.575
0.576
0.10
0.087
4.703
0.530
0.30
0.093
5.979
0.593
0.15
0.088
5.123
0.547
0.35
0.090
4.955
0.558
0.20
0.090
5.599
0.564
0.40
0.090
5.090
0.565
0.25
0.092
6.141
0.583
0.45
0.091
5.222
0.572
0.30
0.094
6.760
0.603
0.50
0.092
5.349
0.579
0.35
0.096
7.470
0.624
0.00
0.083
4.000
0.500
0.40
0.098
8.289
0.645
0.05
0.085
4.289
0.515
0.45
0.100
9.234
0.667
0.10
0.087
4.607
0.530
0.50
0.102
10.329
0.690
0.15
0.088
4.959
0.546
0.40
0.45
0.50
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 653
Tables and Graphs: U.S. Customary Units
APPENDIX A
TABLE A.19 Moment Distribution Constants for Slab-Beam Members with Column Capitals and
Drop Panels
c1
w
FEM (uniform load w)
Mw (
K (stiffness)
kE 2h3
12
2
1)
`2
h1
h2
c2
h
c1
1
`1⎜6
Constants for
h2 = 1.25h1
c1
l1
0.00
0.05
0.10
Constants for
h2 = 1.5h2
c1
l2
M
k
C
M
k
C
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.088
4.795
0.542
0.093
5.837
0.589
0.10
0.088
4.795
0.542
0.093
5.837
0.589
0.15
0.088
4.795
0.542
0.093
5.837
0.589
0.20
0.088
4.795
0.542
0.093
5.837
0.589
0.25
0.088
4.795
0.542
0.093
5.837
0.589
0.30
0.088
4.797
0.542
0.093
5.837
0.589
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.088
4.846
0.545
0.093
5.890
0.591
0.10
0.089
4.896
0.548
0.093
5.942
0.594
0.15
0.089
4.944
0.551
0.093
5.993
0.596
0.20
0.089
4.990
0.553
0.094
6.041
0.598
0.25
0.089
5.035
0.556
0.094
6.087
0.600
0.30
0.090
5.077
0.558
0.094
6.131
0.602
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.088
4.894
0.548
0.093
5.940
0.593
0.10
0.089
4.992
0.553
0.094
6.042
0.598
0.15
0.090
5.039
0.559
0.094
6.142
0.602
0.20
0.090
5.184
0.564
0.094
6.240
0.607
0.25
0.091
5.278
0.569
0.095
6.335
0.611
0.30
0.091
5.368
0.573
0.095
6.427
0.615
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.089
4.938
0.550
0.093
5.986
0.595
0.10
0.090
5.082
0.558
0.094
6.135
0.602
(continues)
653
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 654
654
A P P E N D I X A Tables and Graphs: U.S. Customary Units
TABLE A.19 (Continued)
Constants for
h2 = 1.25h1
Constants for
h2 = 1.5h2
c1
l1
c1
l2
M
k
C
M
k
C
0.15
0.15
0.090
5.228
0.565
0.095
6.284
0.608
0.20
0.091
5.374
0.573
0.095
6.432
0.614
0.25
0.092
5.520
0.580
0.096
6.579
0.620
0.30
0.092
5.665
0.587
0.096
6.723
0.626
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.089
4.978
0.552
0.093
6.027
0.597
0.10
0.090
5.167
0.562
0.094
6.221
0.605
0.15
0.091
5.361
0.571
0.095
6.418
0.613
0.20
0.092
5.558
0.581
0.096
6.616
0.621
0.25
0.093
5.760
0.590
0.096
6.816
0.628
0.30
0.094
5.962
0.590
0.097
7.015
0.635
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.089
5.015
0.553
0.094
6.065
0.598
0.10
0.090
5.245
0.565
0.094
6.300
0.608
0.15
0.091
5.485
0.576
0.095
6.543
0.617
0.20
0.092
5.735
0.587
0.096
6.790
0.626
0.25
0.094
5.994
0.598
0.097
7.043
0.635
0.30
0.095
6.261
0.600
0.098
7.298
0.644
0.00
0.088
4.795
0.542
0.093
5.837
0.589
0.05
0.089
5.048
0.554
0.094
6.099
0.599
0.10
0.090
5.317
0.567
0.095
6.372
0.610
0.15
0.092
5.601
0.580
0.096
6.657
0.620
0.20
0.093
5.902
0.593
0.097
6.953
0.631
0.25
0.094
6.219
0.605
0.098
7.258
0.641
0.30
0.095
6.550
0.618
0.099
7.571
0.651
0.20
0.25
0.30
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 655
APPENDIX A
Tables and Graphs: U.S. Customary Units
TABLE A.20 Stiffness Factors and Carryover Factors for Columns
B
B
b
B
b
`u
`c
A
b
`u
`c
A
`u
`c
A
a
a
a
lu
ln
a
b
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.95
0.90
0.85
0.80
0.75
kAB
4.32
4.70
5.33
5.65
6.27
CAB
0.57
0.64
0.71
0.80
0.89
kAB
4.40
4.89
5.45
6.15
7.00
CAB
0.56
0.61
0.68
0.74
0.81
kAB
4.46
5.02
5.70
6.54
7.58
CAB
0.55
0.60
0.65
0.70
0.76
kAB
4.51
5.14
5.90
6.85
8.05
CAB
0.54
0.58
0.63
0.67
0.72
kAB
4.55
5.23
6.06
7.11
8.44
CAB
0.54
0.57
0.61
0.65
0.68
kAB
4.58
5.30
6.20
7.32
8.77
CAB
0.53
0.57
0.60
0.63
0.66
kAB
4.61
5.36
6.31
7.51
9.05
CAB
0.53
0.56
0.59
0.61
0.64
kAB
4.63
5.42
6.41
7.66
9.29
CAB
0.53
0.55
0.58
0.60
0.62
kAB
4.65
5.46
6.49
7.80
9.50
CAB
0.53
0.55
0.57
0.59
0.60
kAB
4.67
5.51
6.56
7.92
9.68
CAB
0.52
0.54
0.56
0.58
0.59
Notes:
1. Values computed by column analogy method.
2. kc = kAB from table EI0 /ln .
655
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 656
60
⎜6
=f
y ⎜f
c′
A P P E N D I X A Tables and Graphs: U.S. Customary Units
Values of Mu ⎜Ábd 2, psi
656
Mu
Ábd
2
= ρfy (1 – 0.59p
ρmax = 0.85 × β1
fy
fc′
)
fc′
0.003
×
0.003 + 0.004
fy
ρmin = 200
fy
Ratio of tension reinforcement ρ = As⎜bd
G R A P H 1 Moment capacity of rectangular sections.
(Note: The upper ends of the curves shown here for 40 ksi and 50 ksi bars
correspond to ρ values for which t < 0.004 in the steel.)
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 657
APPENDIX A
Tables and Graphs: U.S. Customary Units
2.0
z
h
h
INTERACTION DIAGRAM L4–60.6
fc' = 4 ksi
= 0.08
fy = 60 ksi
1.8
= 0.6
0.07
1.6
Kmax
0.06
e
1.4
Pn
0.05
0.04
Kn = Pn ⎜ fc' Ag
1.2
fs ⎜fy = 0
0.03
1.0
0.02
0.25
0.01
0.8
0.50
0.6
0.75
0.4
t
0.2
0.0
0.00
t
0.05
= 0.0035
1.0
= 0.0050
0.10
0.15
0.20
0.25
Rn = Pne⎜ fc' Agh
0.30
0.35
0.40
0.45
G R A P H 2 Column interaction diagrams for rectangular tied columns with bars on end faces only.
(Published with the permission of the American Concrete Institute.)
657
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 658
A P P E N D I X A Tables and Graphs: U.S. Customary Units
2.0
z
fy = 60 ksi
= 0.7
0.07
1.6
1.4
h
h
INTERACTION DIAGRAM L4–60.7
fc' = 4 ksi
= 0.08
1.8
Kmax
0.06
e
0.05
0.04
Pn
fs⎜ fy = 0
1.2
Kn = Pn ⎜ fc' Ag
658
0.03
1.0
0.02
0.25
0.01
0.8
0.50
0.6
0.75
0.4
1.0
t
0.2
0.0
0.00
t
0.05
= 0.0035
= 0.0050
0.10
0.15
0.20
0.25
0.30
Rn = Pne⎜ fc' Agh
0.35
0.40
0.45
0.50
0.55
G R A P H 3 Column interaction diagrams for rectangular tied columns with bars on end faces only.
(Published with the permission of the American Concrete Institute.)
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 659
APPENDIX A
Tables and Graphs: U.S. Customary Units
2.0
z
fy = 60 ksi
1.8
0.07
1.6
1.4
= 0.8
Kmax
0.06
Kn = Pn ⎜ fc' Ag
1.0
Pn
e
0.05
0.04
1.2
h
h
INTERACTION DIAGRAM L4–60.8
fc' = 4 ksi
= 0.08
fs ⎜fy = 0
0.03
0.02
0.25
0.01
0.8
0.50
0.6
0.75
0.4
1.0
= 0.0035
t = 0.0050
t
0.2
0.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Rn = Pne⎜ fc' Agh
0.40
0.45
0.50
0.55
0.60
G R A P H 4 Column interaction diagrams for rectangular tied columns with bars on end faces only.
(Published with the permission of the American Concrete Institute.)
659
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 660
A P P E N D I X A Tables and Graphs: U.S. Customary Units
2.0
z
fy = 60 ksi
0.07
1.6
h
h
INTERACTION DIAGRAM L4–60.9
fc' = 4 ksi
= 0.08
1.8
= 0.9
Kmax
0.06
e
0.05
Pn
1.4
0.04
1.2
Kn = Pn ⎜ fc' Ag
660
1.0
fs ⎜ fy = 0
0.03
0.02
0.25
0.01
0.50
0.8
0.6
0.75
1.0
0.4
t
t
0.2
0.0
0.00
0.05
= 0.0035
= 0.0050
0.10
0.15
0.20
0.25
0.30 0.35 0.40
Rn = Pne⎜ fc' Agh
0.45
0.50
0.55
0.60
G R A P H 5 Column interaction diagrams for rectangular tied columns with bars on end faces only.
(Published with the permission of the American Concrete Institute.)
0.65
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 661
APPENDIX A
Tables and Graphs: U.S. Customary Units
2.0
z
1.8
1.6
h
h
INTERACTION DIAGRAM R4–60.6
fc' = 4 ksi
= 0.08
fy = 60 ksi
0.07
= 0.6
0.06
Kmax
0.05
e
Pn
1.4
0.04
Kn = Pn ⎜ fc' Ag
1.2
0.03
fs ⎜ fy = 0
0.02
1.0
0.01
0.25
0.8
0.50
0.6
0.75
0.4
t
0.2
0.0
0.00
= 0.0035
1.0
t = 0.0050
0.05
0.10
0.15
0.20
Rn = Pne⎜ fc' Agh
0.25
0.30
G R A P H 6 Column interaction diagrams for rectangular tied columns with bars on all four faces.
(Published with the permission of the American Concrete Institute.)
0.35
661
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 662
A P P E N D I X A Tables and Graphs: U.S. Customary Units
2.0
z
1.8
h
h
INTERACTION DIAGRAM R4–60.7
fc' = 4 ksi
= 0.08
fy = 60 ksi
0.07
= 0.7
0.06
Kmax
1.6
0.05
1.4
1.2
Kn = Pn ⎜ fc' Ag
662
e
Pn
0.04
fs⎜ fy = 0
0.03
0.02
0.25
1.0
0.01
0.8
0.50
0.6
0.75
0.4
1.0
t
0.2
0.0
0.00
t
0.05
= 0.0035
= 0.0050
0.10
0.15
0.20
Rn = Pne⎜ fc' Agh
0.25
0.30
0.35
G R A P H 7 Column interaction diagrams for rectangular tied columns with bars on all four faces.
(Published with the permission of the American Concrete Institute.)
0.40
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 663
APPENDIX A
Tables and Graphs: U.S. Customary Units
2.0
z
fy = 60 ksi
1.8
= 0.8
0.07
1.6
1.4
h
h
INTERACTION DIAGRAM R4–60.8
fc' = 4 ksi
= 0.08
Kmax
0.06
e
0.05
Pn
fs ⎜ fy = 0
0.04
Kn = Pn ⎜ fc' Ag
1.2
0.03
1.0
0.25
0.02
0.01
0.50
0.8
0.75
0.6
1.0
0.4
t
0.2
0.0
0.00
t
0.05
= 0.0035
= 0.0050
0.10
0.15
0.20
0.25
Rn = Pne⎜ fc' Agh
0.30
0.35
0.40
G R A P H 8 Column interaction diagrams for rectangular tied columns with bars on all four faces.
(Published with the permission of the American Concrete Institute.)
0.45
663
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 664
A P P E N D I X A Tables and Graphs: U.S. Customary Units
2.0
z
fy = 60 ksi
= 0.9
0.07
1.6
1.4
h
h
INTERACTION DIAGRAM R4–60.9
fc' = 4 ksi
= 0.08
1.8
Kmax
0.06
e
0.05
Pn
fs⎜fy = 0
0.04
1.2
Kn = Pn ⎜ fc' Ag
664
0.03
0.25
1.0
0.02
0.01
0.50
0.8
0.75
0.6
1.0
0.4
t = 0.0035
=
t 0.0050
0.2
0.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Rn = Pne⎜ fc' Agh
0.35
0.40
0.45
G R A P H 9 Column interaction diagrams for rectangular tied columns with bars on all four faces.
(Published with the permission of the American Concrete Institute.)
0.50
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 665
APPENDIX A
Tables and Graphs: U.S. Customary Units
2.0
z
fy = 60 ksi
1.8
= 0.6
0.07
1.6
1.4
h
h
INTERACTION DIAGRAM C4–60.6
fc' = 4 ksi
= 0.08
Kmax
0.06
e
0.05
Pn
0.04
Kn = Pn ⎜ fc' Ag
1.2
fs ⎜fy = 0
0.03
0.02
1.0
0.25
0.01
0.8
0.50
0.6
0.75
0.4
0.2
t
= 0.0035
1.0
t = 0.0050
0.0
0.000
0.025
0.050
0.075
0.100
0.125
0.150
Rn = Pne⎜ fc' Agh
G R A P H 10 Column interaction diagrams for circular spiral columns.
(Published with the permission of the American Concrete Institute.)
0.175
0.200
0.225
0.250
665
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 666
A P P E N D I X A Tables and Graphs: U.S. Customary Units
2.0
z
1.8
1.6
h
h
INTERACTION DIAGRAM C4–60.7
fc' = 4 ksi
= 0.08
fy = 60 ksi
0.07
= 0.7
0.06
Kmax
0.05
e
Pn
1.4
fs⎜fy = 0
0.04
1.2
Kn = Pn ⎜ fc' Ag
666
0.03
0.02
0.25
1.0
0.01
0.8
0.50
0.6
0.75
0.4
0.2
0.0
0.00
1.0
t = 0.0035
=
0.0050
t
0.05
0.10
0.15
Rn = Pne⎜ fc' Agh
0.20
G R A P H 11 Column interaction diagrams for circular spiral columns.
(Published with the permission of the American Concrete Institute.)
0.25
0.30
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 667
APPENDIX A
Tables and Graphs: U.S. Customary Units
2.0
z
1.8
1.6
h
h
INTERACTION DIAGRAM C4–60.8
fc' = 4 ksi
= 0.08
fy = 60 ksi
0.07
= 0.8
0.06
Kmax
0.05
e
1.4
fs⎜fy = 0
0.04
Kn = Pn ⎜ fc' Ag
1.2
Pn
0.03
0.25
0.02
1.0
0.01
0.50
0.8
0.75
0.6
1.0
0.4
t
0.2
0.0
0.00
= 0.0035
t = 0.0050
0.05
0.10
0.15
0.20
Rn = Pne⎜ fc' Agh
G R A P H 12 Column interaction diagrams for circular spiral columns.
(Published with the permission of the American Concrete Institute.)
0.25
0.30
0.35
667
McCormac b01.tex V2 - January 10, 2013 5:36 P.M. Page 668
A P P E N D I X A Tables and Graphs: U.S. Customary Units
2.0
z
1.8
h
h
INTERACTION DIAGRAM C4–60.9
fc' = 4 ksi
= 0.08
fy = 60 ksi
0.07
= 0.9
0.06
Kmax
1.6
0.05
1.4
1.2
Kn = Pn ⎜fc' Ag
668
e
f s ⎜ fy = 0
Pn
0.04
0.03
0.25
0.02
1.0
0.50
0.01
0.8
0.75
0.6
1.0
0.4
t
0.2
0.0
0.00
= 0.0035
t = 0.0050
0.05
0.10
0.15
0.20
Rn = Pne⎜ fc' Agh
0.25
G R A P H 13 Column interaction diagrams for circular spiral columns.
(Published with the permission of the American Concrete Institute.)
0.30
0.35
0.40
McCormac b02.tex V2 - January 10, 2013 1:42 P.M. Page 669
A PPE N D I X B
Tables in SI Units
TABLE B.1 Values of Modulus of
Elasticity for Normal-Weight Concrete
f c (MPa)
Ec (MPa)
17
17 450
21
21 500
24
23 000
28
24 900
35
27 800
42
30 450
TABLE B.2 Designations, Diameters, Areas, Perimeters, and
Masses of Metric Bars
Nominal Dimensions
Area (mm2 )
Bar No.
Diameter (mm)
10
9.5
71
Mass (kg/m)
0.560
13
12.7
129
0.994
16
15.9
199
1.552
19
19.1
284
2.235
22
22.2
387
3.042
25
25.4
510
3.973
29
28.7
645
5.060
32
32.3
819
6.404
36
35.8
1006
43
43.0
1452
11.38
57
57.3
2581
20.24
7.907
669
McCormac b02.tex V2 - January 10, 2013 1:42 P.M. Page 670
670
APPENDIX B
Tables in SI Units
TABLE B.3 Grades of Reinforcing Bars and Metric Bar
Sizes Available for Each
ASTM No.
Steel Grade (MPa)
Bar Sizes
300
#10– #19
420
#10– #57
520
#19– #57
350
#10– #36
420
#10– #36
300
#10– #36
420
#10– #36
420
#10– #57
A615M Billet
A616M Rail
A617M Axle
A706M Low-Alloy
TABLE B.4 Areas of Groups of Standard Metric Bars (mm2 )
Number of Bars
Bar Designation
2
3
4
5
6
7
8
9
#10
142
#13
10
213
284
355
426
497
568
639
710
258
387
516
645
774
903
1032
1161
1290
#16
398
597
796
995
1194
1393
1592
1791
1990
#19
568
852
1136
1420
1704
1988
2272
2556
2840
#22
774
1161
1548
1935
2322
2709
3096
3483
3870
#25
1020
1530
2040
2550
3060
3570
4080
4590
5100
#29
1290
1935
2580
3225
3870
4515
5160
5805
6450
#32
1638
2457
3276
4095
4914
5733
6552
7371
8190
#36
2012
3018
4024
5030
6036
7042
8048
9054
10 060
#43
2904
4356
5808
7260
8712
10 162
11 616
13 068
14 520
#57
5162
7743
10 324
12 905
15 486
18 067
20 648
23 229
25 810
Number of Bars
Bar Designation
15
16
17
18
19
20
#10
11
781
12
852
13
923
14
994
1065
1136
1207
1278
1349
1420
#13
1419
1548
1677
1806
1935
2064
2193
2322
2451
2580
#16
2189
2388
2587
2786
2985
3184
3383
3582
3781
3980
#19
3124
3408
3692
3976
4260
4544
4828
5112
5396
5680
#22
4257
4644
5031
5418
5805
6192
6579
6966
7353
7740
#25
5610
6120
6630
7140
7650
8160
8670
9180
9690 10 200
#29
7095
7740
8385
9030
9675 10 320 10 965 11 610 12 255 12 900
#32
9009
9828 10 647 11 466 12 285 13 104 13 913 14 742 15 561 16 380
#36
11 066 12 072 13 078 14 084 15 090 16 096 17 102 18 108 19 114 20 120
#43
15 972 17 424 18 876 20 328 21 780 23 232 24 684 26 136 27 588 29 040
#57
28 391 30 972 33 553 36 134 38 715 41 296 43 877 46 458 49 039 51 620
McCormac b02.tex V2 - January 10, 2013 1:42 P.M. Page 671
APPENDIX B
TABLE B.5 Minimum Beam Width (mm) for Beams with Inside Exposure (1995
ACI Metric Code)a,b,c
Number of Bars in Single Layer of Reinforcement
Size of
Bars
2
3
4
5
6
7
8
Add for Each
Additional Bar
#13
175
213
251
288
326
364
401
37.7
#16
178
219
260
301
342
383
424
40.9
#19
182
226
270
314
358
402
446
44.1
#22
185
232
279
326
373
421
468
47.2
#25
188
239
290
341
391
442
493
50.8
#29
195
252
310
367
424
482
539
57.4
#32
202
267
331
396
460
525
590
64.6
#36
209
281
353
424
496
567
639
71.6
#43
228
314
400
486
572
658
744
86.0
#57
271
386
501
615
730
844
959
114.6
a Minimum beam widths for beams were calculated using #10 stirrups.
b Maximum aggregate sizes were assumed not to exceed 3 of the clear spacing between the bars (ACI
4
3.3.2).
c The horizontal distance from the center of the outside longitudinal bars to the inside of the stirrups
was assumed to equal the larger of two times the stirrup diameter (ACI 7.2.2) or half the longitudinal bar
diameter.
TABLE B.6 Areas of Bars in Slabs (mm2 /m)
Spacing
(mm)
Bar Number
10
13
16
19
22
25
29
32
36
75
947
1720
2653
3787
5160
6800
8600
10 920
13 413
90
789
1433
2211
3156
4300
5667
7167
9100
11 178
100
710
1290
1990
2840
3870
5100
6450
8190
10 060
115
617
1122
1730
2470
3365
4435
5609
7122
8748
130
546
992
1531
2185
2977
3923
4962
6300
7738
140
507
921
1421
2029
2764
3643
4607
5850
7186
150
473
860
1327
1893
2580
3400
4300
5460
6707
165
430
782
1206
1721
2345
3091
3909
4964
6097
180
394
717
1106
1578
2150
2833
3583
4550
5589
190
374
679
1047
1495
2037
2684
3395
4311
5295
200
355
645
995
1420
1935
2550
3225
4095
5030
225
316
573
884
1262
1720
2267
2867
3640
4471
250
284
516
796
1136
1548
2040
2580
3276
4024
300
237
430
663
947
1290
1700
2150
2730
3353
Tables in SI Units
671
McCormac b02.tex V2 - January 10, 2013 1:42 P.M. Page 672
672
APPENDIX B
Tables in SI Units
TABLE B.7 Values of ρ Balanced, ρ to Achieve Various t Values, and ρ
Minimum for Flexure. All Values are for Tensilely Reinforced Rectangular
Sections
f c (MPa)
21
β1 = 0.85
28
β1 = 0.85
35
β1 = 0.814
42
β1 = 0.764
ρ balanced
0.0337
0.0450
0.0538
0.0606
ρ when t = 0.004
0.0217
0.0289
0.0346
0.0390
ρ when t = 0.005
0.0190
0.0253
0.0303
0.0341
ρ when t = 0.075
0.0144
0.0193
0.0231
0.0260
ρ min for flexure
0.0047
0.0047
0.0049
0.0054
ρ balanced
0.0274
0.0365
0.0437
0.0492
ρ when t = 0.004
0.0186
0.0248
0.0297
0.0334
ρ when t = 0.005
0.0163
0.0217
0.0259
0.0292
ρ when t = 0.0075
0.0124
0.0165
0.0198
0.0223
ρ min for flexure
0.0040
0.0040
0.0042
0.0046
ρ balanced
0.0212
0.0283
0.0339
0.0382
ρ when t = 0.004
0.0155
0.0206
0.0247
0.0278
ρ when t = 0.005
0.0135
0.0181
0.0216
0.0244
ρ when t = 0.0075
0.0103
0.0138
0.0165
0.0186
ρ min for flexure
0.0033
0.0033
0.0035
0.0039
ρ balanced
0.0156
0.0208
0.0249
0.0281
ρ when t = 0.004
0.0125
0.0167
0.0200
0.0225
ρ when t = 0.005
0.0109
0.0146
0.0175
0.0197
ρ when t = 0.0075
0.0083
0.0111
0.0133
0.0150
ρ min for flexure
0.0027
0.0027
0.0028
0.0031
fy (MPa)
300
350
420
520
McCormac b02.tex V2 - January 10, 2013 1:42 P.M. Page 673
APPENDIX B
TABLE B.8 fy = 420 MPa; f c = 21 MPa—SI Units
ρ
Mu
φbd 2
ρ
Mu
φbd 2
ρ
Mu
φbd 2
ρ
Mu
φbd 2
ρmin for
0.0018
0.740
0.0048
1.902
0.0078
2.975
0.0107
3.928
temp. and
0.0019
0.780
0.0049
1.939
0.0079
3.010
0.0108
3.960
shrinkage
0.0020
0.820
0.0050
1.976
0.0080
3.044
0.0109
3.991
0.0021
0.860
0.0051
2.013
0.0081
3.078
0.0110
4.022
0.0022
0.900
0.0052
2.050
0.0082
3.112
0.0111
4.053
0.0023
0.940
0.0053
2.087
0.0083
3.146
0.0112
4.084
0.0024
0.980
0.0054
2.124
0.0084
3.179
0.0113
4.115
0.0025
1.019
0.0055
2.161
0.0085
3.213
0.0114
4.146
0.0026
1.059
0.0056
2.197
0.0086
3.247
0.0115
4.177
0.0027
1.098
0.0057
2.233
0.0087
3.280
0.0116
4.207
0.0028
1.137
0.0058
2.270
0.0088
3.313
0.0117
4.238
0.0029
1.176
0.0059
2.306
0.0089
3.347
0.0118
4.268
0.0030
1.216
0.0060
2.342
0.0090
3.380
0.0119
4.298
0.0031
1.255
0.0061
2.378
0.0091
3.413
0.0120
4.328
0.0032
1.293
0.0062
2.414
0.0092
3.446
0.0121
4.359
ρmin
0.0033
1.332
0.0063
2.450
0.0093
3.479
0.0122
4.389
flexure
0.0034
1.371
0.0064
2.486
0.0094
3.511
0.0123
4.418
0.0035
1.409
0.0065
2.521
0.0095
3.544
0.0124
4.448
0.0036
1.448
0.0066
2.557
0.0096
3.577
0.0125
4.478
0.0037
1.486
0.0067
2.592
0.0097
3.609
0.0126
4.508
0.0038
1.525
0.0068
2.628
0.0098
3.641
0.0127
4.537
0.0039
1.563
0.0069
2.663
0.0099
3.674
0.0128
4.566
0.0040
1.601
0.0070
2.698
0.0100
3.706
0.0129
4.596
0.0041
1.639
0.0071
2.733
0.0101
3.738
0.0130
4.625
0.0042
1.677
0.0072
2.768
0.0102
3.770
0.0131
4.654
0.0043
1.715
0.0073
2.803
0.0103
3.802
0.0132
4.683
0.0044
1.752
0.0074
2.837
0.0104
3.834
0.0133
4.712
0.0045
1.790
0.0075
2.872
0.0105
3.865
0.0134
4.741
0.0046
1.827
0.0076
2.907
0.0106
3.897
0.0135
4.769
0.0047
1.865
0.0077
2.941
Tables in SI Units
673
McCormac b02.tex V2 - January 10, 2013 1:42 P.M. Page 674
674
APPENDIX B
Tables in SI Units
TABLE B.9 fy = 420 MPa; f c = 28 MPa—SI Units
ρ
Mu
φbd 2
ρ
Mu
φbd 2
ρ
Mu
φbd 2
ρ
Mu
φbd 2
ρmin for
0.0018
0.744
0.0059
2.349
0.0100
3.829
0.0141
5.185
temp. and
0.0019
0.785
0.0060
2.387
0.0101
3.864
0.0142
5.217
shrinkage
0.0020
0.825
0.0061
2.424
0.0102
3.898
0.0143
5.248
0.0021
0.866
0.0062
2.462
0.0103
3.933
0.0144
5.280
0.0022
0.906
0.0063
2.499
0.0104
3.967
0.0145
5.311
0.0023
0.946
0.0064
2.536
0.0105
4.001
0.0146
5.342
0.0024
0.987
0.0065
2.573
0.0106
4.036
0.0147
5.373
0.0025
1.027
0.0066
2.611
0.0107
4.070
0.0148
5.404
0.0026
1.067
0.0067
2.648
0.0108
4.104
0.0149
5.435
0.0027
1.107
0.0068
2.685
0.0109
4.138
0.0150
5.466
0.0028
1.147
0.0069
2.722
0.0110
4.172
0.0151
5.497
0.0029
1.187
0.0070
2.758
0.0111
4.205
0.0152
5.528
0.0030
1.227
0.0071
2.795
0.0112
4.239
0.0153
5.558
0.0031
1.266
0.0072
2.832
0.0113
4.273
0.0154
5.589
0.0032
1.306
0.0073
2.869
0.0114
4.306
0.0155
5.620
ρmin
0.0033
1.346
0.0074
2.905
0.0115
4.340
0.0156
5.650
flexure
0.0034
1.385
0.0075
2.942
0.0116
4.373
0.0157
5.681
0.0035
1.424
0.0076
2.978
0.0117
4.407
0.0158
5.711
0.0036
1.464
0.0077
3.014
0.0118
4.440
0.0159
5.741
0.0037
1.503
0.0078
3.051
0.0119
4.473
0.0160
5.771
0.0038
1.542
0.0079
3.087
0.0120
4.506
0.0161
5.801
0.0039
1.582
0.0080
3.123
0.0121
4.539
0.0162
5.831
0.0040
1.621
0.0081
3.159
0.0122
4.572
0.0163
5.861
0.0041
1.660
0.0082
3.195
0.0123
4.605
0.0164
5.891
0.0042
1.699
0.0083
3.231
0.0124
4.638
0.0165
5.921
0.0043
1.737
0.0084
3.267
0.0125
4.671
0.0166
5.951
0.0044
1.776
0.0085
3.302
0.0126
4.704
0.0167
5.980
0.0045
1.815
0.0086
3.338
0.0127
4.736
0.0168
6.010
0.0046
1.854
0.0087
3.374
0.0128
4.769
0.0169
6.040
0.0047
1.892
0.0088
3.409
0.0129
4.801
0.0170
6.069
0.0048
1.931
0.0089
3.444
0.0130
4.834
0.0171
6.098
0.0049
1.969
0.0090
3.480
0.0131
4.866
0.0172
6.128
0.0050
2.007
0.0091
3.515
0.0132
4.898
0.0173
6.157
0.0051
2.046
0.0092
3.550
0.0133
4.930
0.0174
6.186
0.0052
2.084
0.0093
3.585
0.0134
4.963
0.0175
6.215
0.0053
2.122
0.0094
3.621
0.0135
4.995
0.0176
6.244
0.0054
2.160
0.0095
3.656
0.0136
5.027
0.0177
6.273
0.0055
2.198
0.0096
3.690
0.0137
5.058
0.0178
6.302
0.0056
2.236
0.0097
3.725
0.0138
5.090
0.0179
6.331
0.0057
2.274
0.0098
3.760
0.0139
5.122
0.0180
6.359
0.0058
2.311
0.0099
3.795
0.0140
5.154
0.0181
6.388
34
Appendix C: European Rebar Sizes
20.0
BAR AREAS
Table A: Sectional areas of groups of bars (mm2)
Bar
size
(mm)
Number of bars
1
2
3
4
5
6
7
8
9
10
6
8
10
28.3
50.3
78.6
56.6
101
157
84.8
151
236
113
201
314
141
251
393
170
302
471
198
352
550
226
402
628
255
452
707
283
503
786
12
16
20
113
201
314
226
402
628
339
603
943
452
804
1257
566
1005
1571
679
1207
1885
792
1408
2199
905
1609
2514
1018
1810
2828
1131
2011
3142
25
32
40
491
804
1257
982
1609
2514
1473
2413
3770
1964
3217
5027
2455
4022
6284
2946
4826
7541
3437
5630
8798
3928
6435
10054
4418
7239
11311
4909
8044
12568
Table B: Sectional area per meter width for various bar spacing (mm2/m)
Bar
size
(mm)
Spacing of bars
50
75
100
125
150
175
200
225
250
300
6
8
10
566
1005
1571
377
670
1047
283
503
786
226
402
628
189
335
524
162
287
449
141
251
393
126
223
349
113
201
314
94
168
262
12
16
20
2262
4022
6284
1508
2681
4189
1131
2011
3142
905
1609
2514
754
1341
2095
646
1149
1795
566
1005
1571
503
894
1396
452
804
1257
377
670
1047
25
32
40
9819
16087
25136
6549
10725
16757
4909
8044
12568
3928
6435
10054
3273
5362
8379
2805
4596
7182
2455
4022
6284
2182
3575
5586
1964
3217
5027
1636
2681
4189