appendix b flexural resistance of members with reinforcing...
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Appendix B
Flexural Resistance of Members with Reinforcing Bars Lacking Well-Defined Yield
Plateau
B-1
B.1 Introduction The nominal moment capacity (Mn) for non-prestressed members is commonly calculated by assuming a constant yield stress for the steel. For bars without a well-defined yield plateau, several approaches may be used to define the yield stress. In order to examine these methods, parametric studies were performed to assess the flexural resistance of members reinforced with various grades of steel reinforcement that do not have a well-defined yield plateau. The moment capacity was calculated by a number of methods ranging from simple design oriented procedures to complex fiber analysis. In fiber analysis, a cross section is divided into layers (fibers). The cross-sectional and material properties for each layer are defined, and strain compatibility between the layers is enforced. Realistic complete stress-strain relationships for concrete and steel layers are employed as opposed to simplified relationships typically used in the strain compatibility method. Therefore, complex analyses can be performed by fiber analysis technique. Comparing the results from the range of models made it possible to evaluate whether approximate methods are appropriate for members reinforced with reinforcing bars with no clear yield plateau and what material properties to use in these cases. B.2 Members and Parameters Sections modeled were deck slabs, rectangular beams, and T-beams with varying steel types, amounts of steel, and concrete compressive strengths. The variables considered are summarized in Table B1. A total of 286 cross sections were analyzed.
Table B1 Variables for Parametric Studies Parameter Deck Slab Rectangular Beam T-Beam Dimensions 7” and 10” thick 12”!16”, 12”!28”,
12”!36”, 16”!28”, 16”!36”, and 16”!40”
12”!28”, 12”!36”, 16”!36”, and 16”!40”
with 96” effective flange width and 7” flange
thickness Concrete Strength, f’c 5 ksi 5, 10, and 15 ksi Reinforcement Grades A706, A496 & A82*, A955 (3 grades), and A1035
Bar Sizes #4, #5 and #6 #6 for 12” wide beams; and #8 for 16” wide beams. All beams are assumed to have #4 stirrups with 2 inches
of clear cover Tension Reinforcement Based on AASHTO
spacing limitations As,min, As,max,
0.5(As,min+As,max) As,max from corresponding
rectangular beams * The properties established for A496 and A82 wire reinforcement were applied to #4, #5, #6, and #8 bars even though the bar diameters and areas are different from those for wire reinforcement.
The cross-sectional dimensions for the rectangular beams were chosen from the PCI Design Manual (2004). These sections were chosen to given representative cases of what may be used in design. The four largest sections were then used as the T-Beam dimensions. The flange thickness of the T-Beam was taken as 7 inches, a thickness commonly used for bridge decks. The effective flange width was determined from AASHTO §4.6.2.6. In this analysis, the effective span length was assumed to be 50 feet and the center-to-center spacing of the beams was taken as 8 feet. The controlling effective flange width was found to be 8 feet. Seven-inch
B-2
and ten-inch thick slabs were used in the analysis. The slab cross-sections were analyzed on a per foot basis. Three different amounts of tensile reinforcement were incorporated in the rectangular beams. A maximum area of steel, As,max was determined based on the ACI 318-08 (2008) imposed minimum steel strain of 0.004. For a solid rectangular section, As,max is calculated from Eq. (B-1).
As,max =
! fc'"1
37
dt
#$%
&'(
b
fy
(B-1)
where dt is the distance from the extreme compression fiver to the centroid of the extreme layer
of longitudinal tension steel, b is the width of the beam, fy is the nominal yield strength, f’c is the
concrete compressive strength,
! =0.85 for f 'c " 10ksi
0.85# 0.02 f 'c#10( ) $ 0.75 for f 'c > 10ksi
%&'
(' and
!1 =0.85 for f 'c " 4ksi
0.85# 0.05 f 'c# 4( ) $ 0.65 for f 'c > 10ksi
%&'
('
A minimum area of steel, As,min, was established to satisfy AASHTO §5.7.3.3.2, i.e., to ensure that the flexural resistance with As,min is at least 1.2Mcr, where Mcr is the cracking moment of the section. The average of As,min and As,max was also considered. Rectangular beams with As,min are in the tension-controlled region. Rectangular beams reinforced with As,max have the lowest steel strains allowed by ACI 318. The average of As,min and As,max results in cross-sections with strains between these limits. Because of the additional compression strength provided by the flanges of the T-beams, the calculated amount of steel required to provide As,max (i.e., to ensure a minimum strain of 0.004) was found to be excessive and impractical. Therefore, the values of As,max determined for the rectangular beams were provided in the corresponding T-beams. Nonetheless, the selected values of As,max resulted in members that fell well into the tension-controlled region. Providing more steel to obtain members in the transition region was impractical; hence, only one amount of reinforcement was used for the T-beams. The amount of steel provided in the slabs was determined based on spacing limitations prescribed in AASHTO LRFD Bridge Design Specifications §5.10.3.1, §5.10.3.2, and §5.10.8 (AASHTO 2008). B.3 Capacity Calculation Procedures The nominal moment capacity of each section was calculated by a number of methods ranging from simple design oriented procedures to complex analysis techniques. The analyses included
B-3
strain compatibility using different methods for modeling the steel stress-strain relationships, and fiber analysis. B.3.1 Fiber Analysis In fiber analysis, a cross section is divided into layers (fibers). The cross-sectional and material properties for each layer are defined, and strain compatibility between the layers is enforced. A commercial computer program XTRACT (Imbsen, 2007) was used to perform the fiber analyses. The concrete was modeled using the unconfined concrete model proposed by Razvi and Saatcioglu (1999). The measured stress-strain data for each type of reinforcing steel were input directly into the XTRACT program. By using the experimentally obtained data, a more accurate capacity can be determined. Moment-curvature analyses were run in which the concrete strain was limited to 0.003, the level of strain used in the strain compatibility analyses. The results from fiber analyses are deemed to predict the most accurate flexural capacity. Figure B1 shows fiber models of representative rectangular and T-beams.
(a) Rectangular Beam
(b) T Beam
Figure B1 Representative Fiber Models
B-4
B.3.2 Strain Compatibility An Excel program (Shahrooz 2010) was used to compute flexural capacities based on strain compatibility analysis. The constitutive relationship of the reinforcing bars was modeled (a) as elastic-perfectly plastic with the yield point obtained by the 0.2% offset method and the stress at both strain = 0.0035 and strain = 0.005; and (b) by the Ramberg-Osgood function (Ramberg and Osgood, 1943) determined to best fit the experimentally obtained data (refer to Appendix A). The analyses utilized data from the measured stress-strain relationships of 98 samples of A706, A496 and A82, A955, and A1035 reinforcing bars. The yield strengths obtained from each method are summarized in Figure B2, and the averages and standard deviations are provided in Table B2.
(a) A82 & A496
(b) A706
(c) A955
(d) A1035
Figure B2 Measured Yield Strengths
B-5
Table B2 Average and Standard Deviations of fy (ksi) Methodology for establishing the yield strength
0.2% offset method Strain @ 0.005 Strain @ 0.0035 Bar
Avg. (ksi)
Std. Dev. (ksi)
Avg. (ksi)
Std. Dev. (ksi)
Avg. (ksi)
Std. Dev. (ksi)
A82 & A496 93 5.82 92 5.55 88 5.93 A706 68 3.30 68 3.83 67 3.05 A995 80 6.20 80 6.20 73 3.69 A1035 127 6.35 115 4.13 93 3.62
The values of the parameters for the Ramberg-Osgood function were calibrated to produce an analytical stress-diagram that would approximately represent the “average” of the measured stress-strain diagrams. The selected values of these parameters are summarized in Table B3.
Table B3 Summary of Analysis Input Values Steel A B C Es (ksi) fu (ksi)
A1035 0.0150 190 2.5 29000 160 A955 0.0120 350 7.20 29000 100 A706 0.0138 358 2.10 29000 100
A82 & A496 0.0150 310 3.90 29000 100 The equivalent stress block for high strength concrete, developed as part of NCHRP 12-64 (Rizkalla et al., 2007), was used to compute the concrete contribution to section behavior. The stress block parameters and modulus elasticity are summarized in Eq. (B-2).
!1 =0.85 for f 'c " 4ksi
0.85# 0.05 f 'c# 4( ) $ 0.65 for f 'c > 4ksi
%&'
('f 'c in ksi( )
) =0.85 for f 'c " 10ksi
0.85# 0.02 f 'c#10( ) $ 0.75 for f 'c > 10ksi
%&'
('
Ec = 310,000K1 Wc( )2.5f 'c( )0.33
where K1 = 1;Wc = 0.15 kcf
(B-2)
B.4 Results The moment capacity for each section computed based on the aforementioned methods was normalized with the corresponding capacity calculated from the fiber analyses. The results for rectangular beams are presented in Table B4 for various grades of reinforcing bars. The corresponding values for T-beams and slabs are summarized in Tables B5 and B6, respectively.
B-6
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(a) A
496-
A82
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
5 12
16
19
2 3.
08
0.01
60
2436
24
86
2406
25
74
0.94
64
0.94
64
0.93
48
5 12
16
19
2 2.
20
0.01
15
1945
20
46
2098
21
78
0.89
28
0.89
28
0.96
33
5 12
16
19
2 1.
32
0.00
69
1277
13
49
1458
14
94
0.85
45
0.85
45
0.97
60
5 12
28
33
6 5.
72
0.01
70
8460
86
72
8370
89
49
0.94
54
0.94
54
0.93
52
5 12
28
33
6 3.
52
0.01
05
6051
63
78
6692
68
90
0.87
82
0.87
82
0.97
12
5 12
28
33
6 1.
32
0.00
39
2543
26
95
2996
30
48
0.83
45
0.83
45
0.98
30
5 12
36
43
2 7.
48
0.01
73
1450
2 14
864
1433
2 15
270
0.94
97
0.94
97
0.93
86
5 12
36
43
2 4.
40
0.01
02
9952
10
493
1105
9 11
380
0.87
45
0.87
45
0.97
18
5 12
36
43
2 1.
32
0.00
31
3388
35
93
3999
40
02
0.84
66
0.84
66
0.99
93
5 16
28
44
8 7.
90
0.01
76
1223
1 12
539
1201
9 12
780
0.95
70
0.95
70
0.94
04
5 16
28
44
8 5.
53
0.01
23
9368
98
57
1013
6 10
410
0.89
99
0.89
99
0.97
37
5 16
28
44
8 3.
16
0.00
71
5850
61
84
6712
68
11
0.85
89
0.85
89
0.98
54
5 16
36
57
6 10
.27
0.01
78
2074
2 21
302
2042
3 21
660
0.95
76
0.95
76
0.94
29
5 16
36
57
6 6.
32
0.01
10
1442
5 15
202
1593
0 16
270
0.88
66
0.88
66
0.97
91
5 16
36
57
6 3.
16
0.00
55
7872
83
33
9234
92
60
0.85
01
0.85
01
0.99
72
5 16
40
64
0 11
.06
0.01
73
2533
7 26
311
2526
9 26
690
0.94
93
0.94
93
0.94
68
5 16
40
64
0 7.
11
0.01
11
1816
0 19
138
2004
3 20
480
0.88
67
0.88
67
0.97
87
5 16
40
64
0 3.
16
0.00
49
8883
94
07
1044
4 10
490
0.84
68
0.84
68
0.99
56
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-7
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(a
) A49
6-A
82 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
10
12
16
192
4.40
0.
0229
34
84
3608
35
34
3742
0.
9310
0.
9464
0.
9445
10
12
16
19
2 2.
64
0.01
38
2430
25
67
2727
27
93
0.87
00
0.89
28
0.97
64
10
12
16
192
1.32
0.
0069
13
31
1411
15
69
1592
0.
8362
0.
8545
0.
9853
10
12
28
33
6 8.
36
0.02
49
1226
9 12
667
1235
6 13
080
0.93
80
0.94
54
0.94
47
10
12
28
336
5.28
0.
0157
89
99
9503
10
029
1027
3 0.
8760
0.
8782
0.
9763
10
12
28
33
6 1.
76
0.00
52
3440
36
48
4064
41
27
0.83
36
0.83
45
0.98
46
10
12
36
432
11.0
0 0.
0255
21
064
2172
3 21
175
2243
0 0.
9391
0.
9497
0.
9441
10
12
36
43
2 6.
16
0.01
43
1402
4 14
821
1581
4 16
170
0.86
73
0.87
45
0.97
80
10
12
36
432
1.76
0.
0041
45
66
4846
54
00
5472
0.
8344
0.
8466
0.
9869
10
16
28
44
8 11
.85
0.02
65
1849
0 19
054
1839
3 19
410
0.95
26
0.95
70
0.94
76
10
16
28
448
7.90
0.
0176
13
699
1445
7 15
115
1538
0 0.
8907
0.
8999
0.
9828
10
16
28
44
8 3.
16
0.00
71
6085
64
49
7173
71
94
0.84
58
0.85
89
0.99
71
10
16
36
576
15.8
0 0.
0274
31
921
3281
9 31
619
3343
0 0.
9549
0.
9576
0.
9458
10
16
36
57
6 9.
48
0.01
65
2177
4 22
995
2427
2 24
660
0.88
30
0.88
66
0.98
43
10
16
36
576
3.16
0.
0055
81
06
8597
95
74
9628
0.
8419
0.
8501
0.
9944
10
16
40
64
0 17
.38
0.02
72
3936
8 40
638
3923
1 41
400
0.95
09
0.94
93
0.94
76
10
16
40
640
10.2
7 0.
0160
26
525
2801
8 29
672
3012
0 0.
8807
0.
8867
0.
9851
10
16
40
64
0 3.
16
0.00
49
9117
96
73
1077
5 10
830
0.84
18
0.84
68
0.99
49
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-8
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(a
) A49
6-A
82 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
15
12
16
192
5.72
0.
0298
42
29
4346
42
80
4447
0.
9511
0.
9464
0.
9624
15
12
16
19
2 3.
52
0.01
83
3156
33
33
3531
35
81
0.88
12
0.89
28
0.98
62
15
12
16
192
1.32
0.
0069
13
45
1426
15
88
1606
0.
8373
0.
8545
0.
9885
15
12
28
33
6 10
.56
0.03
14
1461
5 15
069
1482
7 15
350
0.95
21
0.94
54
0.96
59
15
12
28
336
6.16
0.
0183
10
373
1096
1 11
677
1184
0 0.
8761
0.
8782
0.
9863
15
12
28
33
6 1.
76
0.00
52
3463
36
75
4097
41
43
0.83
59
0.83
45
0.98
88
15
12
36
432
14.0
8 0.
0326
25
100
2586
2 25
427
2634
0 0.
9529
0.
9497
0.
9653
15
12
36
43
2 8.
80
0.02
04
1901
9 20
086
2121
0 21
510
0.88
42
0.87
45
0.98
60
15
12
36
432
2.64
0.
0061
67
06
7114
79
27
7957
0.
8428
0.
8466
0.
9963
15
16
28
44
8 15
.01
0.03
35
2244
0 23
185
2251
8 23
240
0.96
56
0.95
70
0.96
90
15
16
28
448
9.48
0.
0212
16
224
1713
1 18
059
1816
0 0.
8934
0.
8999
0.
9944
15
16
28
44
8 3.
16
0.00
71
6142
65
13
7254
72
96
0.84
18
0.85
89
0.99
42
15
16
36
576
19.7
5 0.
0343
38
614
3982
8 38
676
3995
0 0.
9666
0.
9576
0.
9681
15
16
36
57
6 11
.85
0.02
06
2689
0 28
406
3009
9 30
270
0.88
83
0.88
66
0.99
44
15
16
36
576
3.16
0.
0055
81
63
8663
96
54
9678
0.
8435
0.
8501
0.
9976
15
16
40
64
0 22
.12
0.03
46
4813
8 49
635
4824
0 49
850
0.96
57
0.94
93
0.96
77
15
16
40
640
12.6
4 0.
0198
32
298
3412
8 36
316
3653
0 0.
8842
0.
8867
0.
9941
15
16
40
64
0 3.
16
0.00
49
9174
97
35
1085
6 10
870
0.84
40
0.84
68
0.99
88
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-9
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(b
) A70
6 f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
5 12
16
19
2 3.
52
0.01
83
2423
25
02
2297
25
90
0.93
56
0.96
60
0.88
68
5 12
16
19
2 1.
76
0.00
92
1468
15
62
1650
17
73
0.82
81
0.88
08
0.93
05
5 12
16
19
2 0.
88
0.00
46
771
824
941
1045
0.
7379
0.
7881
0.
9008
5
12
28
336
7.04
0.
0210
85
42
8740
80
54
9151
0.
9334
0.
9551
0.
8801
5
12
28
336
3.52
0.
0105
53
80
5718
59
72
6418
0.
8383
0.
8910
0.
9306
5
12
28
336
0.88
0.
0026
15
10
1615
20
01
2159
0.
6994
0.
7481
0.
9270
5
12
36
432
7.92
0.
0183
13
626
1430
8 13
245
1479
0 0.
9213
0.
9674
0.
8955
5
12
36
432
4.40
0.
0102
88
40
9401
98
84
1062
0 0.
8324
0.
8852
0.
9307
5
12
36
432
0.88
0.
0020
20
03
2143
27
70
2887
0.
6937
0.
7422
0.
9595
5
16
28
448
9.48
0.
0212
12
262
1263
6 11
528
1300
0 0.
9432
0.
9720
0.
8868
5
16
28
448
5.53
0.
0123
83
54
8866
90
25
9645
0.
8662
0.
9193
0.
9357
5
16
28
448
1.58
0.
0035
26
75
2859
34
01
3692
0.
7244
0.
7744
0.
9212
5
16
36
576
11.8
5 0.
0206
20
654
2146
1 19
578
2196
0 0.
9405
0.
9773
0.
8915
5
16
36
576
7.90
0.
0137
15
446
1637
8 16
400
1760
0 0.
8776
0.
9306
0.
9318
5
16
36
576
2.37
0.
0041
52
71
5633
65
99
7180
0.
7342
0.
7845
0.
9191
5
16
40
640
13.4
3 0.
0210
25
942
2683
2 24
485
2751
0 0.
9430
0.
9753
0.
8900
5
16
40
640
7.90
0.
0123
17
659
1874
7 19
188
2049
0 0.
8618
0.
9149
0.
9364
5
16
40
640
2.37
0.
0037
59
36
6343
75
34
8226
0.
7216
0.
7711
0.
9159
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
0
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(b
) A70
6 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
10
12
16
192
5.28
0.
0275
34
93
3601
33
53
3672
0.
9512
0.
9806
0.
9131
10
12
16
19
2 2.
64
0.01
38
2150
22
91
2435
25
53
0.84
22
0.89
73
0.95
39
10
12
16
192
0.88
0.
0046
79
0 84
5 10
24
1102
0.
7166
0.
7666
0.
9293
10
12
28
33
6 8.
80
0.02
62
1146
0 12
081
1130
4 12
110
0.94
64
0.99
76
0.93
34
10
12
28
336
4.40
0.
0131
68
42
7295
78
81
8132
0.
8414
0.
8971
0.
9691
10
12
28
33
6 1.
32
0.00
39
2280
24
39
3062
32
27
0.70
64
0.75
58
0.94
88
10
12
36
432
11.4
4 0.
0265
19
501
2054
2 19
271
2086
0 0.
9349
0.
9847
0.
9238
10
12
36
43
2 6.
16
0.01
43
1240
1 13
217
1414
9 14
770
0.83
96
0.89
49
0.95
80
10
12
36
432
1.32
0.
0031
30
19
3230
42
33
4300
0.
7020
0.
7513
0.
9845
10
16
28
44
8 14
.22
0.03
17
1837
8 19
016
1744
3 19
100
0.96
22
0.99
56
0.91
33
10
16
28
448
7.90
0.
0176
12
147
1292
9 13
423
1393
0 0.
8720
0.
9281
0.
9636
10
16
28
44
8 2.
37
0.00
53
4046
43
27
5207
55
14
0.73
37
0.78
47
0.94
43
10
16
36
576
22.1
2 0.
0384
31
789
3344
2 30
852
3297
0 0.
9642
1.
0143
0.
9358
10
16
36
57
6 11
.06
0.01
92
2190
6 23
302
2388
5 24
830
0.88
22
0.93
85
0.96
19
10
16
36
576
3.16
0.
0055
71
18
7614
97
50
1168
0 0.
6094
0.
6519
0.
8348
10
16
40
64
0 21
.33
0.03
33
3977
6 40
948
3757
6 41
250
0.96
43
0.99
27
0.91
09
10
16
40
640
11.8
5 0.
0185
26
501
2820
1 29
143
3024
0 0.
8764
0.
9326
0.
9637
10
16
40
64
0 3.
16
0.00
49
8004
85
61
1044
7 10
920
0.73
30
0.78
40
0.95
67
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-1
1
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(b
) A70
6 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
15
12
16
192
6.16
0.
0321
39
44
3923
37
98
4116
0.
9582
0.
9532
0.
9227
15
12
16
19
2 3.
52
0.01
83
2793
29
76
3150
32
19
0.86
78
0.92
44
0.97
85
15
12
16
192
0.88
0.
0046
79
4 85
0 10
73
1112
0.
7142
0.
7643
0.
9652
15
12
28
33
6 12
.32
0.03
67
1400
5 14
521
1367
9 14
440
0.96
98
1.00
56
0.94
73
15
12
28
336
7.04
0.
0210
10
188
1084
7 11
339
1161
0 0.
8775
0.
9343
0.
9767
15
12
28
33
6 1.
32
0.00
39
2290
24
51
3220
32
39
0.70
70
0.75
66
0.99
40
15
12
36
432
15.8
4 0.
0367
23
876
2477
5 23
356
2464
0 0.
9690
1.
0055
0.
9479
15
12
36
43
2 8.
80
0.02
04
1684
2 17
938
1889
8 19
330
0.87
13
0.92
80
0.97
77
15
12
36
432
1.76
0.
0041
40
24
4307
56
38
5726
0.
7028
0.
7521
0.
9845
15
16
28
44
8 18
.17
0.04
06
2222
4 22
924
2124
5 22
420
0.99
12
1.02
25
0.94
76
15
16
28
448
11.0
6 0.
0247
16
251
1728
6 17
663
1801
0 0.
9023
0.
9598
0.
9808
15
16
28
44
8 3.
16
0.00
71
5393
57
68
6934
72
12
0.74
78
0.79
98
0.96
14
15
16
36
576
23.7
0 0.
0411
38
127
3934
1 36
431
3847
0 0.
9911
1.
0226
0.
9470
15
16
36
57
6 16
.59
0.02
88
2744
0 32
729
3040
0 30
530
0.89
88
1.07
20
0.99
57
15
16
36
576
3.95
0.
0069
89
11
9531
11
540
1195
0 0.
7457
0.
7975
0.
9657
15
16
40
64
0 26
.07
0.04
07
4730
4 48
934
4531
0 47
850
0.98
86
1.02
27
0.94
69
15
16
40
640
14.2
2 0.
0222
31
422
3345
7 35
013
3560
0 0.
8826
0.
9398
0.
9835
15
16
40
64
0 3.
95
0.00
62
1001
6 10
717
1317
5 13
600
0.73
65
0.78
80
0.96
88
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-1
2
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(c
) A99
5 f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
5 12
16
19
2 2.
64
0.01
38
1935
22
28
2195
23
26
0.83
21
0.95
78
0.94
37
5 12
16
19
2 1.
76
0.00
92
1430
16
67
1745
18
36
0.77
90
0.90
82
0.95
02
5 12
16
19
2 0.
88
0.00
46
750
883
978
1058
0.
7088
0.
8345
0.
9237
5
12
28
336
5.28
0.
0157
71
28
8158
79
42
8509
0.
8378
0.
9588
0.
9334
5
12
28
336
3.52
0.
0105
52
42
6101
63
32
6656
0.
7876
0.
9167
0.
9514
5
12
28
336
1.32
0.
0039
21
76
2567
28
96
3153
0.
6902
0.
8141
0.
9187
5
12
36
432
7.04
0.
0163
12
354
1410
1 13
679
1465
7 0.
8429
0.
9621
0.
9333
5
12
36
432
4.4
0.01
02
8614
10
035
1046
5 10
970
0.78
52
0.91
48
0.95
40
5 12
36
43
2 1.
32
0.00
31
2894
34
19
3955
42
82
0.67
58
0.79
86
0.92
36
5 16
28
44
8 7.
9 0.
0176
10
771
1217
9 11
640
1294
7 0.
8319
0.
9407
0.
8990
5
16
28
448
3.95
0.
0088
61
84
7227
76
37
8160
0.
7579
0.
8856
0.
9359
5
16
28
448
1.58
0.
0035
26
00
3070
35
11
3815
0.
6816
0.
8046
0.
9202
5
16
36
576
10.2
7 0.
0178
18
277
2065
9 19
771
2162
0 0.
8454
0.
9556
0.
9145
5
16
36
576
6.32
0.
0110
12
501
1454
4 15
032
1553
0 0.
8049
0.
9365
0.
9680
5
16
36
576
1.58
0.
0027
34
60
4760
54
64
5990
0.
5775
0.
7946
0.
9122
5
16
40
640
11.0
6 0.
0173
22
270
2538
9 24
410
2653
3 0.
8393
0.
9569
0.
9200
5
16
40
640
7.11
0.
0111
15
739
1831
0 18
914
2015
7 0.
7808
0.
9084
0.
9384
5
16
40
640
2.37
0.
0037
57
71
6812
77
81
8446
0.
6833
0.
8066
0.
9213
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
3
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(c
) A99
5 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
10
12
16
192
4.4
0.02
29
3069
34
83
3408
35
85
0.85
62
0.97
16
0.95
08
10
12
16
192
2.64
0.
0138
20
93
2451
25
77
2652
0.
7893
0.
9242
0.
9715
10
12
16
19
2 1.
32
0.00
69
1138
13
43
1498
15
65
0.72
73
0.85
82
0.95
71
10
12
28
336
8.8
0.02
62
1118
4 12
460
1210
6 12
867
0.86
93
0.96
84
0.94
09
10
12
28
336
5.28
0.
0157
77
60
9076
94
75
9739
0.
7968
0.
9319
0.
9729
10
12
28
33
6 1.
76
0.00
52
2937
34
72
3990
41
96
0.69
99
0.82
76
0.95
11
10
12
36
432
10.5
6 0.
0244
18
116
2075
0 20
260
2118
3 0.
8552
0.
9795
0.
9564
10
12
36
43
2 6.
16
0.01
43
1207
2 14
146
1494
5 15
320
0.78
80
0.92
34
0.97
55
10
12
36
432
1.76
0.
0041
38
94
4609
54
44
5725
0.
6802
0.
8050
0.
9509
10
16
28
44
8 11
.85
0.02
65
1614
5 18
442
1772
8 18
880
0.85
52
0.97
68
0.93
90
10
16
28
448
7.11
0.
0159
10
797
1223
4 12
805
1413
0 0.
7641
0.
8658
0.
9062
10
16
28
44
8 2.
37
0.00
53
3933
46
50
5363
56
90
0.69
12
0.81
73
0.94
25
10
16
36
576
15.8
0.
0274
27
987
3176
5 30
506
3251
3 0.
8608
0.
9770
0.
9383
10
16
36
57
6 9.
48
0.01
65
1877
8 19
161
2013
6 23
843
0.78
76
0.80
36
0.84
45
10
16
36
576
2.37
0.
0041
52
23
6181
73
26
7715
0.
6769
0.
8011
0.
9496
10
16
40
64
0 17
.38
0.02
72
3450
8 39
288
3781
0 40
240
0.85
76
0.97
63
0.93
96
10
16
40
640
10.2
7 0.
0160
22
863
2675
5 27
960
2913
7 0.
7847
0.
9183
0.
9596
10
16
40
64
0 3.
16
0.00
49
7780
92
02
1040
0 10
519
0.73
96
0.87
48
0.98
87
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
4
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(c
) A99
5 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
15
12
16
192
5.28
0.
0275
36
06
4146
40
78
4163
0.
8664
0.
9960
0.
9798
15
12
16
19
2 3.
52
0.01
83
2720
31
83
3336
33
85
0.80
35
0.94
04
0.98
56
15
12
16
192
1.32
0.
0069
11
48
1357
15
48
1614
0.
7115
0.
8410
0.
9596
15
12
28
33
6 10
.56
0.03
14
1299
0 14
578
1429
9 14
710
0.88
31
0.99
11
0.97
21
15
12
28
336
6.16
0.
0183
89
31
1046
3 11
035
1118
3 0.
7986
0.
9356
0.
9867
15
12
28
33
6 1.
76
0.00
52
2953
34
96
4131
43
17
0.68
41
0.80
97
0.95
68
15
12
36
432
13.2
0.
0306
21
616
2459
8 24
187
2469
7 0.
8753
0.
9960
0.
9794
15
12
36
43
2 7.
92
0.01
83
1508
9 17
682
1868
3 18
883
0.79
90
0.93
64
0.98
94
15
12
36
432
2.64
0.
0061
57
21
6769
78
77
8138
0.
7030
0.
8317
0.
9679
15
16
28
44
8 15
.01
0.03
35
1972
1 22
382
2167
0 22
450
0.87
84
0.99
70
0.96
53
15
16
28
448
7.9
0.01
76
1209
1 14
181
1505
6 15
493
0.78
04
0.91
53
0.97
17
15
16
28
448
2.37
0.
0053
39
56
4683
55
52
5852
0.
6761
0.
8002
0.
9487
15
16
36
57
6 19
.75
0.03
43
3393
6 38
494
3723
1 37
743
0.89
91
1.01
99
0.98
64
15
16
36
576
11.8
5 0.
0206
23
175
2712
2 28
368
2912
0 0.
7959
0.
9314
0.
9742
15
16
36
57
6 3.
16
0.00
55
6963
82
40
9759
10
153
0.68
58
0.81
16
0.96
12
15
16
40
640
22.1
2 0.
0346
42
383
4800
7 46
438
4806
3 0.
8818
0.
9988
0.
9662
15
16
40
64
0 11
.85
0.01
85
2639
7 30
951
3277
1 33
663
0.78
41
0.91
94
0.97
35
15
16
40
640
3.16
0.
0049
78
20
9260
11
099
1152
0 0.
6788
0.
8038
0.
9634
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
5
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(d
) A10
35
f' c
b h
Ag
As
Mom
ent C
apac
ity (k
ips-
in)
EPP/
FA
EPP/
FA
EPP/
FA
(ksi
) (in
) (in
) (in
2 ) (in
2 ) ! g
0.
0035
0.
005
0.20
%
RO
FA
0.
0035
0.
005
0.2%
of
fset
R
O/F
A
5 12
16
19
2 2.
64
0.01
4 24
52
2523
25
23
2444
27
05
0.90
66
0.93
28
0.93
28
0.90
33
5 12
16
19
2 2.
20
0.01
1 22
42
2448
24
72
2253
26
03
0.86
12
0.94
05
0.94
96
0.86
55
5 12
16
19
2 1.
76
0.00
9 19
20
2173
23
35
2111
24
37
0.78
80
0.89
18
0.95
80
0.86
62
5 12
28
33
6 4.
40
0.01
3 82
04
8660
87
02
8334
91
98
0.89
19
0.94
15
0.94
61
0.90
61
5 12
28
33
6 3.
08
0.00
9 62
92
7132
76
69
7062
81
39
0.77
31
0.87
63
0.94
23
0.86
77
5 12
28
33
6 1.
76
0.00
5 39
26
4496
48
69
5151
59
45
0.66
04
0.75
63
0.81
89
0.86
65
5 12
36
43
2 5.
28
0.01
2 13
271
1456
5 14
763
1401
5 15
430
0.86
01
0.94
40
0.95
68
0.90
83
5 12
36
43
2 3.
52
0.00
8 96
88
1101
7 11
872
1138
9 13
140
0.73
73
0.83
84
0.90
35
0.86
68
5 12
36
43
2 1.
76
0.00
4 52
64
6045
65
58
7238
83
16
0.63
30
0.72
69
0.78
86
0.87
04
5 16
28
44
8 6.
32
0.01
4 11
893
1225
5 12
255
1182
4 12
690
0.93
72
0.96
57
0.96
57
0.93
18
5 16
28
44
8 3.
95
0.00
9 83
45
9474
10
197
1008
1 11
470
0.72
76
0.82
60
0.88
90
0.87
89
5 16
28
44
8 3.
16
0.00
7 68
42
7802
84
22
8899
92
72
0.73
79
0.84
14
0.90
83
0.95
97
5 16
36
57
6 7.
90
0.01
4 19
874
2100
0 21
000
2016
6 22
160
0.89
68
0.94
77
0.94
77
0.91
00
5 16
36
57
6 5.
53
0.01
0 15
006
1700
4 18
281
1763
4 19
230
0.78
04
0.88
43
0.95
06
0.91
70
5 16
36
57
6 3.
16
0.00
5 92
44
1058
2 11
455
1294
4 13
830
0.66
84
0.76
52
0.82
83
0.93
59
5 16
40
64
0 8.
69
0.01
4 24
483
2604
9 26
090
2500
2 27
470
0.89
13
0.94
83
0.94
98
0.91
02
5 16
40
64
0 5.
53
0.00
9 17
108
1943
8 20
936
2096
3 22
780
0.75
10
0.85
33
0.91
90
0.92
02
5 16
40
64
0 3.
16
0.00
5 10
444
1197
1 12
972
1499
5 15
930
0.65
56
0.75
15
0.81
43
0.94
13
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
6
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(d
) A10
35 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
EP
P/FA
(ksi
) (in
) (in
) (in
2 ) (in
2 ) ! g
0.
0035
0.
005
0.20
%
RO
FA
0.
0035
0.
005
0.2%
of
fset
R
O/F
A
10
12
16
192
3.52
0.
018
3548
38
04
3884
36
86
4113
0.
8627
0.
9248
0.
9444
0.
8963
10
12
16
19
2 2.
64
0.01
4 28
37
3229
34
69
3172
37
05
0.76
57
0.87
14
0.93
63
0.85
62
10
12
16
192
1.76
0.
009
2057
23
57
2553
26
20
3050
0.
6745
0.
7728
0.
8371
0.
8589
10
12
28
33
6 6.
60
0.02
0 12
274
1313
5 13
360
1271
8 14
140
0.86
81
0.92
89
0.94
48
0.89
94
10
12
28
336
4.40
0.
013
9060
10
334
1116
0 10
560
1230
0 0.
7366
0.
8402
0.
9073
0.
8585
10
12
28
33
6 1.
76
0.00
5 40
64
4680
50
88
5770
66
08
0.61
49
0.70
82
0.76
99
0.87
32
10
12
36
432
8.80
0.
020
2127
3 22
615
2297
0 21
892
2433
0 0.
8743
0.
9295
0.
9441
0.
8998
10
12
36
43
2 5.
28
0.01
2 14
503
1656
8 17
911
1740
6 20
270
0.71
55
0.81
74
0.88
36
0.85
87
10
12
36
432
1.76
0.
004
5400
62
28
6777
79
21
8963
0.
6025
0.
6949
0.
7561
0.
8838
10
16
28
44
8 9.
48
0.02
1 18
122
1920
0 19
332
1846
2 19
910
0.91
02
0.96
43
0.97
10
0.92
72
10
16
28
448
6.32
0.
014
1323
4 15
081
1627
6 16
009
1697
0 0.
7798
0.
8887
0.
9591
0.
9434
10
16
28
44
8 3.
16
0.00
7 71
73
8245
89
51
1054
8 10
870
0.65
99
0.75
86
0.82
35
0.97
04
10
16
36
576
12.6
4 0.
022
3150
6 33
087
3327
1 31
845
3531
0 0.
8923
0.
9370
0.
9422
0.
9019
10
16
36
57
6 7.
90
0.01
4 21
944
2503
0 27
030
2706
1 29
630
0.74
06
0.84
48
0.91
23
0.91
33
10
16
36
576
3.16
0.
005
9574
11
025
1198
4 14
712
1538
0 0.
6225
0.
7169
0.
7792
0.
9565
10
16
40
64
0 13
.43
0.02
1 38
068
4082
6 41
255
3925
0 43
460
0.87
59
0.93
94
0.94
93
0.90
31
10
16
40
640
7.90
0.
012
2494
6 28
506
3082
3 32
006
3488
0 0.
7152
0.
8173
0.
8837
0.
9176
10
16
40
64
0 3.
16
0.00
5 10
775
1241
6 13
500
1681
1 17
480
0.61
64
0.71
03
0.77
23
0.96
17
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
7
Tab
le B
4 M
omen
t Cap
aciti
es –
Rec
tang
ular
Sec
tions
(con
t.)
(d
) A10
35 (c
ont.)
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
EP
P/FA
(ksi
) (in
) (in
) (in
2 ) (in
2 ) ! g
0.
0035
0.
005
0.20
%
RO
FA
0.
0035
0.
005
0.2%
of
fset
R
O/F
A
15
12
16
192
4.40
0.
023
4228
45
79
4683
44
70
4819
0.
8773
0.
9502
0.
9718
0.
9276
15
12
16
19
2 3.
08
0.01
6 33
04
3767
40
68
3834
43
77
0.75
48
0.86
07
0.92
93
0.87
60
15
12
16
192
1.76
0.
009
2091
24
02
2607
28
16
3226
0.
6481
0.
7446
0.
8080
0.
8728
15
12
28
33
6 8.
36
0.02
5 14
887
1598
2 16
344
1559
9 16
750
0.88
88
0.95
42
0.97
58
0.93
13
15
12
28
336
5.28
0.
016
1079
3 12
326
1332
3 12
901
1471
0 0.
7337
0.
8379
0.
9057
0.
8770
15
12
28
33
6 1.
76
0.00
5 40
97
4725
51
41
6011
67
94
0.60
30
0.69
54
0.75
66
0.88
48
15
12
36
432
11.4
4 0.
026
2593
3 27
537
2811
0 26
864
2879
0 0.
9008
0.
9565
0.
9764
0.
9331
15
12
36
43
2 6.
60
0.01
5 17
776
2031
7 21
970
2154
9 24
540
0.72
44
0.82
79
0.89
53
0.87
81
15
12
36
432
1.76
0.
004
5434
62
73
6830
82
28
9120
0.
5958
0.
6879
0.
7489
0.
9022
15
16
28
44
8 11
.85
0.02
6 22
371
2391
9 24
259
2308
6 24
080
0.92
90
0.99
33
1.00
74
0.95
87
15
16
28
448
7.11
0.
016
1501
8 17
158
1854
8 19
175
1971
0 0.
7619
0.
8705
0.
9411
0.
9729
15
16
28
44
8 3.
16
0.00
7 72
54
8353
90
80
1115
0 11
380
0.63
74
0.73
40
0.79
79
0.97
98
15
16
36
576
15.8
0 0.
027
3873
4 41
104
4159
5 39
650
4239
0 0.
9138
0.
9697
0.
9813
0.
9354
15
16
36
57
6 9.
48
0.01
6 26
115
2982
6 32
238
3312
8 35
350
0.73
88
0.84
37
0.91
20
0.93
72
15
16
36
576
3.16
0.
005
9654
11
133
1211
3 15
373
1582
0 0.
6103
0.
7037
0.
7657
0.
9718
15
16
40
64
0 17
.38
0.02
7 47
713
5098
6 51
733
4925
6 52
650
0.90
62
0.96
84
0.98
26
0.93
55
15
16
40
640
10.2
7 0.
016
3180
1 36
341
3929
3 40
787
4346
0 0.
7317
0.
8362
0.
9041
0.
9385
15
16
40
64
0 3.
16
0.00
5 10
856
1252
4 13
629
1751
7 17
940
0.60
52
0.69
81
0.75
97
0.97
64
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
8
Tab
le B
5 M
omen
t Cap
aciti
es –
T-b
eam
Sec
tions
(a) A
496-
A82
f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
5 12
28
92
4 5.
72
0.00
62
1031
6 10
942
1219
0 12
800
0.80
59
0.85
48
0.95
24
5 12
36
10
20
7.48
0.
0073
17
658
1872
7 20
867
2191
0 0.
8059
0.
8547
0.
9524
5
16
36
1136
10
.27
0.00
90
2488
8 26
393
2937
2 30
841
0.80
70
0.85
58
0.95
24
5 16
40
12
00
11.0
6 0.
0092
30
127
3194
8 35
564
3734
2 0.
8068
0.
8555
0.
9524
10
12
28
92
4 8.
36
0.00
90
1430
7 15
185
1693
1 17
777
0.80
48
0.85
42
0.95
24
10
12
36
1020
11
.00
0.01
08
2460
2 25
378
2829
0 28
720
0.85
66
0.88
37
0.98
50
10
16
36
1136
15
.80
0.01
39
3693
5 39
176
4364
4 45
826
0.80
60
0.85
49
0.95
24
10
16
40
1200
17
.38
0.01
45
4545
1 48
208
5371
1 56
396
0.80
59
0.85
48
0.95
24
15
12
28
924
10.5
6 0.
0114
17
192
1824
5 20
350
2136
7 0.
8046
0.
8539
0.
9524
15
12
36
10
20
14.0
8 0.
0138
29
814
3044
5 33
942
3469
0 0.
8594
0.
8776
0.
9784
15
16
36
11
36
19.7
5 0.
0174
44
662
4737
0 52
776
5541
5 0.
8060
0.
8548
0.
9524
15
16
40
12
00
22.1
2 0.
0184
55
808
5918
9 65
946
6924
4 0.
8060
0.
8548
0.
9524
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-1
9
Tab
le B
5 M
omen
t Cap
aciti
es –
T-b
eam
Sec
tions
(con
t.)
(b
) A70
6 f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
5 12
28
92
4 6.
16
0.00
67
9633
10
307
1284
7 13
340
0.72
21
0.77
26
0.96
30
5 12
36
10
20
8.80
0.
0086
17
781
1901
2 23
395
2468
0 0.
7204
0.
7703
0.
9479
5
16
36
1136
11
.85
0.01
04
2486
5 26
583
3159
8 33
020
0.75
30
0.80
50
0.95
69
5 16
40
12
00
13.4
3 0.
0112
31
371
3353
4 39
784
4144
0 0.
7570
0.
8092
0.
9600
10
12
28
92
4 9.
68
0.01
05
1404
3 15
028
1868
5 19
120
0.73
45
0.78
60
0.97
73
10
12
36
1020
13
.20
0.01
29
2479
7 26
531
3273
5 34
160
0.72
59
0.77
67
0.95
83
10
16
36
1136
18
.96
0.01
67
3762
2 40
232
4766
0 50
110
0.75
08
0.80
29
0.95
11
10
16
40
1200
21
.33
0.01
78
4723
2 50
511
5978
4 62
490
0.75
58
0.80
83
0.95
67
15
12
28
924
12.3
2 0.
0133
16
789
1796
3 22
322
2257
0 0.
7439
0.
7959
0.
9890
15
12
36
10
20
15.8
4 0.
0155
28
386
3037
3 37
851
3876
0 0.
7324
0.
7836
0.
9765
15
16
36
11
36
23.7
0 0.
0209
45
171
4829
6 57
397
5959
0 0.
7580
0.
8105
0.
9632
15
16
40
12
00
26.0
7 0.
0217
55
687
5953
9 70
919
7316
0 0.
7612
0.
8138
0.
9694
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-2
0
Tab
le B
5 M
omen
t Cap
aciti
es –
T-b
eam
Sec
tions
(con
t.)
(c
) A99
5 f' c
b
h A
g A
s M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
(k
si)
(in)
(in)
(in2 )
(in2 )
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
5 12
28
92
4 5.
28
0.00
57
8233
84
69
9866
10
290
0.80
01
0.82
31
0.95
88
5 12
36
10
20
7.04
0.
0069
14
315
1472
7 17
095
1803
0 0.
7940
0.
8168
0.
9482
5
16
36
1136
10
.27
0.00
90
2126
1 21
870
2429
8 26
280
0.80
90
0.83
22
0.92
46
5 16
40
12
00
11.0
6 0.
0092
25
729
2818
9 31
208
3169
0 0.
8119
0.
8895
0.
9848
10
12
28
92
4 8.
8 0.
0095
12
718
1308
4 15
060
1564
0 0.
8132
0.
8366
0.
9629
10
12
36
10
20
10.5
6 0.
0104
20
328
2091
5 24
413
2524
0 0.
8054
0.
8287
0.
9672
10
16
36
11
36
15.8
0.
0139
31
516
3242
2 36
097
3895
0 0.
8091
0.
8324
0.
9267
10
16
40
12
00
17.3
8 0.
0145
38
779
3989
3 44
493
4751
0 0.
8162
0.
8397
0.
9365
15
12
28
92
4 10
.56
0.01
14
1465
9 15
085
1752
3 17
870
0.82
03
0.84
41
0.98
06
15
12
36
1020
13
.2
0.01
29
2422
1 24
925
2917
2 29
830
0.81
20
0.83
56
0.97
80
15
16
36
1136
19
.75
0.01
74
3810
4 39
197
4378
9 46
410
0.82
10
0.84
46
0.94
35
15
16
40
1200
22
.12
0.01
84
4761
2 48
978
5469
2 57
380
0.82
98
0.85
36
0.95
32
0.00
35 =
yie
ld st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.003
5 0.
005
= yi
eld
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
05
0.2%
= y
ield
stre
ss d
efin
ed b
y th
e 0.
2% o
ffse
t met
hod
RO
= st
eel s
tress
-stra
in d
iagr
am is
mod
eled
by
the
Ram
berg
-Osg
ood
func
tion
EPP
= el
astic
-per
fect
ly p
last
ic a
naly
sis
FA =
fibe
r ana
lysi
s
B-2
1
Tab
le B
5 M
omen
t Cap
aciti
es –
T-b
eam
Sec
tions
(con
t.)
(d
) A10
35
f' c
b
h A
g A
s M
omen
t Cap
acity
(k-in
.) EP
P/FA
EP
P/FA
EP
P/FA
(ksi
) (in
) (in
) (in
2 ) (in
2 ) ! g
0.
0035
0.
005
0.20
%
RO
FA
0.
0035
0.
0050
0.
2%
offs
et
RO
/FA
5 12
28
92
4 5.
28
0.00
57
1141
3 13
162
1431
4 18
475
1877
0 0.
6080
0.
7013
0.
7626
0.
9843
5
12
36
1020
5.
28
0.00
52
1542
3 17
807
1938
2 25
604
2562
0 0.
6020
0.
6950
0.
7565
0.
9994
5
16
36
1136
7.
9 0.
0070
23
325
2687
9 29
228
3734
5 38
400
0.60
74
0.70
00
0.76
12
0.97
25
5 16
40
12
00
7.9
0.00
66
2631
9 30
352
3302
0 42
742
4342
0 0.
6062
0.
6990
0.
7605
0.
9844
10
12
28
92
4 7.
04
0.00
76
1476
9 17
054
1856
5 24
386
2560
6 0.
5768
0.
6660
0.
7250
0.
9524
10
12
36
10
20
7.04
0.
0069
20
118
2324
6 25
320
3357
0 35
248
0.57
08
0.65
95
0.71
83
0.95
24
10
16
36
1136
11
.85
0.01
04
3411
4 39
359
4283
9 55
147
5629
0 0.
6060
0.
6992
0.
7610
0.
9797
10
16
40
12
00
11.8
5 0.
0099
38
618
4457
6 48
522
6328
2 63
770
0.60
56
0.69
90
0.76
09
0.99
23
15
12
28
924
8.8
0.00
95
1774
8 20
496
2231
4 29
297
3076
2 0.
5770
0.
6663
0.
7254
0.
9524
15
12
36
10
20
8.8
0.00
86
2443
2 28
242
3075
8 40
783
4282
2 0.
5705
0.
6595
0.
7183
0.
9524
15
16
36
11
36
15.8
0.
0139
43
977
5073
0 55
213
7077
5 72
260
0.60
86
0.70
20
0.76
41
0.97
94
15
16
40
1200
15
.8
0.01
32
4998
1 57
230
6225
9 78
429
8225
0 0.
6077
0.
6958
0.
7569
0.
9535
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-2
2
Tab
le B
6 M
omen
t Cap
aciti
es –
Slab
s
(a) A
496-
A82
M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
Sl
ab
As=
As'
As+
As'
Ag
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
7" S
lab
(#4
bars
) 0.
52
1.05
84
0.
012
207
207
222
303
0.68
50
0.68
50
0.73
28
7" S
lab
(#5
bars
) 0.
92
1.84
84
0.
022
300
300
304
439
0.68
30
0.68
30
0.69
36
7" S
lab
(# 6
bar
s)
1.33
2.
65
84
0.03
2 37
0 37
0 35
3 52
9 0.
6991
0.
6991
0.
6677
10
" Sl
ab(#
4 b
ars)
0.
39
0.79
12
0 0.
007
266
278
300
402
0.66
35
0.69
32
0.74
75
10"
Slab
(#5
bars
) 0.
61
1.23
12
0 0.
010
377
394
421
618
0.60
93
0.63
83
0.68
19
10"
Slab
(#6
bars
) 0.
88
1.77
12
0 0.
015
501
526
556
820
0.61
06
0.64
07
0.67
73
(b
) A70
6 M
omen
t Cap
acity
(kip
s-in
) EP
P/FA
EP
P/FA
Sl
ab
As=
As'
As+
As'
Ag
! g
0.00
35
0.00
5/0.
2%
RO
FA
0.
0035
0.
005/
0.2%
R
O/F
A
7" S
lab
(#4
bars
) 0.
34
0.67
84
0.
008
143
150
152
161
0.89
03
0.93
56
0.94
64
7" S
lab
(#5
bars
) 0.
57
1.13
84
0.
013
202
210
209
225
0.89
76
0.93
38
0.93
26
7" S
lab
(# 6
bar
s)
0.88
1.
77
84
0.02
1 26
6 27
7 26
4 29
2 0.
9115
0.
9512
0.
9043
10
" Sl
ab(#
4 b
ars)
0.
26
0.52
12
0 0.
004
170
181
201
211
0.80
52
0.85
78
0.95
14
10"
Slab
(#5
bars
) 0.
43
0.87
12
0 0.
007
262
275
295
311
0.84
26
0.88
43
0.95
08
10"
Slab
(#6
bars
) 0.
62
1.25
12
0 0.
010
345
363
386
406
0.85
02
0.89
43
0.95
13
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-2
3
Tab
le B
6 M
omen
t Cap
aciti
es –
Slab
s
(c) A
995
Mom
ent C
apac
ity (k
ips-
in)
EPP/
FA
EPP/
FA
Slab
A
s=A
s' A
s+A
s' A
g ! g
0.
0035
0.
005/
0.2%
R
O
FA
0.00
35
0.00
5/0.
2%
RO
/FA
7" S
lab
(#4
bars
) 0.
52
1.05
84
0.
012
189
192
192
201
0.93
77
0.95
33
0.95
24
7" S
lab
(#5
bars
) 0.
92
1.84
84
0.
022
270
275
274
288
0.93
85
0.95
59
0.95
24
7" S
lab
(# 6
bar
s)
1.33
2.
65
84
0.03
2 33
9 34
6 33
3 35
6 0.
9529
0.
9713
0.
9346
10
" Sl
ab(#
4 b
ars)
0.
39
0.79
12
0 0.
007
238
243
249
261
0.91
11
0.92
93
0.95
24
10"
Slab
(#5
bars
) 0.
61
1.23
12
0 0.
010
334
341
348
365
0.91
27
0.93
23
0.95
24
10"
Slab
(#6
bars
) 0.
88
1.77
12
0 0.
015
441
453
462
485
0.91
06
0.93
50
0.95
24
(d
) A10
35
Mom
ent C
apac
ity (k
ips-
in)
EPP/
FA
EPP/
FA
EPP/
FA
Slab
A
s=A
s' A
s+A
s' A
g ! g
0.
0035
0.
005
0.20
%
RO
FA
0.
0035
0.
0050
0.
2%
offs
et
RO
/FA
7" S
lab
(#4
bars
) 1.
18
2.36
84
0.
028
382
382
382
373
403
0.94
85
0.94
85
0.94
85
0.92
67
7" S
lab
(#5
bars
) 1.
84
3.68
84
0.
044
434
434
434
436
487
0.89
14
0.89
14
0.89
14
0.89
46
7" S
lab
(# 6
bar
s)
2.65
5.
30
84
0.06
3 54
1 48
0 48
0 48
9 57
2 0.
9443
0.
8391
0.
8391
0.
8544
10
" Sl
ab(#
4 b
ars)
0.
67
1.35
12
0 0.
011
466
525
565
606
634
0.73
54
0.82
90
0.89
11
0.95
54
10"
Slab
(#5
bars
) 1.
05
2.10
12
0 0.
018
669
757
814
799
856
0.78
16
0.88
38
0.95
11
0.93
25
10"
Slab
(#6
bars
) 1.
77
3.53
12
0 0.
029
1024
11
02
1102
10
54
1182
0.
8660
0.
9319
0.
9319
0.
8914
0.
0035
= y
ield
stre
ss is
the
stre
ss c
orre
spon
ding
to st
rain
equ
al to
0.0
035
0.00
5 =
yiel
d st
ress
is th
e st
ress
cor
resp
ondi
ng to
stra
in e
qual
to 0
.005
0.
2% =
yie
ld st
ress
def
ined
by
the
0.2%
off
set m
etho
d R
O =
stee
l stre
ss-s
train
dia
gram
is m
odel
ed b
y th
e R
ambe
rg-O
sgoo
d fu
nctio
n EP
P =
elas
tic-p
erfe
ctly
pla
stic
ana
lysi
s FA
= fi
ber a
naly
sis
B-24
Table B7 summarizes the average, minimum, maximum, and standard deviation of the results of the strain compatibility analyses conducted using the Ramberg-Osgood function for the rectangular beams, T-beams, and slabs for all the steel types and the selected concrete strengths considered. The computed capacities are below or nearly equal to those calculated based on fiber analysis, i.e., the ratios are close to or slightly less than unity. The exceptional estimates of the expected capacity based on the Ramberg-Osgood function in conjunction with strain compatibility analysis should be expected as this function closely replicates the measured stress-strain curves that were used in the fiber analyses. Additionally, the good correlation suggests that well-established procedures can be used to calculate the flexural capacity of members reinforced with bars that do not have a well-defined yield plateau so long as the stress-strain relationship is modeled accurately.
Table B7 Ratios of Flexural Capacity Determined from Ramberg-Osgood Strain Compatibility Analysis to that Determined from Fiber Model!
Section Average Minimum Maximum Standard Deviation
Rectangular 0.944 0.835 0.999 0.037 T-beam 0.962 0.925 0.999 0.017 Slab 0.875 0.668 0.955 0.107
Note: Ratio less than 1 is conservative. In spite of its success, the use of Ramberg-Osgood functions is not appropriate for routine design. Most designers are familiar with using a single value of reinforcing bar yield, fy, i.e., an elastic-plastic analysis with a value of yield strength. For this reason, the results from strain compatibility analyses with the yield strength taken at strain = 0.0035, strain = 0.005, and 0.2% offset method are examined further. The average, minimum, maximum, and standard deviation of the ratio of the computed capacities to the capacity based on fiber analysis are summarized in Table B8.
Table B8 Ratios of Rectangular Beam Flexural Capacity Calculated from Elastic-Plastic Analyses to that from Fiber Model
Yield Point f'c (ksi) Average Minimum Maximum Standard Deviation
5 0.820 0.578 0.958 0.094 10 0.815 0.603 0.964 0.100 @ Strain = 0.0035 15 0.825 0.596 0.991 0.108 5 0.884 0.727 0.977 0.070 10 0.880 0.652 1.014 0.084 @ Strain = 0.005 15 0.891 0.688 1.072 0.092 5 0.909 0.789 0.966 0.057 10 0.884 0.756 0.971 0.075 0.2% offset 15 0.890 0.749 1.007 0.092
Note: Ratio less than 1 is conservative. For the beams having 5 ksi concrete, the ratios from any of the values of yield strength are less
B-25
than unity, i.e., the flexural strength can be conservatively computed based on any of three methods used to establish the yield strength. The same conclusion cannot be drawn for the beams with 10 and 15 ksi concrete. For a limited number of cases (given in Table B9) involving relatively large longitudinal reinforcement ratios (!g), the strength ratio exceeds unity if the capacity is based on an idealized elastic-perfectly plastic stress-strain relationship with the yield strength taken as the stress at a strain of 0.005 or determined based on the 0.2% offset method. That is, the yield strengths based on these two methods may result in slightly unconservative estimates of the expected capacity in cases with large reinforcement ratios and high-strength concrete.
Table B9 Cases where Elastic-Plastic Analysis Overestimate Flexural Capacity Steel Type f'c (ksi) !g Ratio
10 3.84% 1.014 15 3.67% 1.006 15 3.67% 1.005 15 4.06% 1.022 15 4.11% 1.023 15 2.88% 1.072
A706
15 4.07% 1.023 A995 15 3.43% 1.020 A1035 15 2.65% 1.007
The aforementioned behavior can be understood with reference to Figure B3, which depicts a measured stress-strain curve for an A706 bar along with the idealized elastic-perfectly plastic model based on the yield strength taken as the value determined from the 0.2% offset method and the stress at strain equal to 0.005. Note that in this case, these two methods result in the same values of yield strength. Between points “a” and “b” (see Figure B3) the elastic-perfectly plastic model deviates from the measured stress-strain diagram. The stresses based on this model exceed the actual values. For strains below point “a” and strains above “b”, the stresses from the idealized model are equal to or less than the measured values. As the reinforcement ratio increases, i.e., as the amount of longitudinal steel becomes larger, the strain in the reinforcing bars at any given applied moment will become less. For the cases involving the large reinforcement ratios shown in Table B9, the steel strains fall between points “a” and “b” when the extreme concrete compressive stress of 0.003 is reached. Thus, the higher yield strength from the elastic-perfectly plastic model overestimates the actual flexural capacity. In the case of T-beams and slabs, any of the aforementioned methods for establishing the yield strength result in acceptable, conservative flexural capacities. As evident from Table B10, the ratios of the flexural capacity based on simple elastic-perfectly plastic models to the corresponding values from fiber analysis are less than one. The trend of data is expected, as the longitudinal strain in a T-beam will be higher than that in an equivalent rectangular beam because of the additional compressive force that can be developed in the flange. The smaller depths of the slabs will also increase the strain in the longitudinal bars. In both these cases, the larger strains will correspond to cases beyond the strain at point “b” in Figure B3, where the elastic-plastic assumption underestimates the real stress developed in the steel.
B-26
Figure B3 Typical Measured Stress-Strain Diagram and Elastic-Perfectly Plastic Model
Table B10 Ratios of T-Beam and Slab Flexural Capacity Calculated from Elastic-Plastic Analyses to that from Fiber Model!
T-Beams
Yield Point Average Minimum Maximum Standard Deviation
@ Strain =0.0035 0.741 0.571 0.859 0.091 @ Strain =0.005 0.795 0.659 0.890 0.069 0.2% offset 0.748 0.718 0.764 0.019
Deck Slabs
Yield Point Average Minimum Maximum Standard Deviation
@ Strain =0.0035 0.828 0.609 0.953 0.115 @ Strain =0.005 0.854 0.638 0.971 0.113 0.2% offset 0.909 0.839 0.951 0.043 Note: Ratio less than 1 is conservative.
B.5 Summary
Considering the presented results, the use of Ramberg-Osgood functions for defining the stress-strain characteristics of reinforcing bars without a well-defined yield plateau will produce the
B-27
most accurate estimate of the actual flexural capacity. The use of the strain compatibility approach assuming an elastic-perfectly plastic steel stress strain relationship having a yield stress defined at either a strain of 0.0035 or 0.005 ensures that the flexural capacity is computed conservatively and reliably for the range of reinforcement ratios and concrete compressive strengths encountered in practice. However, for beams with reinforcement ratios exceeding 3%, the definition of the yield stress at a strain of 0.0035 is more appropriate. The latter approach is consistent with the currently prescribed ACI 318 (ACI 318-08, 2008) approach. The use of the stress at 0.0035 strain effectively ensures that the steel strain under the design condition is beyond point “b”, shown in Figure B3. Recall that this is enforced in the design approach through the definition of As max as the steel content that allows a steel strain of 0.004 to be achieved.