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FLEXURE DESIGN EXAMPLES These design examples have been taken from Chapter 2 of the CAC Concrete Design Handbook, 3 rd edition Example 2.1 2.1 Analys is o f a Rectangul Rectangul ar Beam with Tension Reinforcement Compute moment resistance Mr f or the rectangular section shown in the figure. f' c= 30 MPa; f y= 400 MPa.
(Example 2.1) 1.
Calculate effective depth d: d = 600 - 40 - 11.3 - (19.5/2) = 539 mm
2.
Calculate reinforcement ratio ρ :
ρ = A s/ bd = (4 x 300) / (400 x 539) = 0.0056 = 0.56% 3.
Determine Kr corresponding to ρ= 0.56% from Table 2.1: Kr = 1.8 MPa
4.
Calculate resisting moment Mr : 2
-6
2
-6
Mr = K r bd x 10 = 1.8 (400) (539) x 10 = 209 kN.m
1
Example 2.2 Design o f a Rectangu lar Beam with Tensio n Reinforcement Design the rectangular beam shown in the figure for a factored moment of M f = 415 kN.m. Use normal density concrete with f' c= 40 MPa, f y= 400 MPa, and maximum aggregate size of 25 mm. The beam is to be built in a non-corrosive environment, having interior exposure.
m m 0 0 6
m m 0 0 6
1.
Estimate effective depth d, assuming 25M bars for flexural reinforcement and 10M stirrups for transverse reinforcement. Select a clear cover of 30 mm from Table 2.6 (for noncorrosive environment, interior exposure). d = 600 - 30 - 11.3 - (25.2/2) = 546 mm
2.
Calculate resistance factor Kr : For design; M r M f = 415 kN.m 6
2
6
2
Kr = M r x 10 / bd = 415 x 10 / (400) (546) = 3.48 MPa 3.
Determine reinforcement ratio ρf rom Table 2.1:
4.
Determine required tension reinforcement: As = ρb d = 0.0113 x 400 x 546 = 2468 mm No. of 25M bars required; 2468/500 = 4.94 Select 5 - 25M bars
5.
ρ= 1.13%
2
Check minimum steel requirement: Mr 1 .2 M cr Mr = 415 kN.m for 5 - 25M bars Mcr = 0.6 λ f'c I/c t (Eq. 2-5 and 2-6) 3 3 9 4 I = bh /12 = (400)(600) / 12 = 7.2 x 10 mm ct= 600 / 2 = 300 mm; λ= 1.0 (normal density concrete) 9 6 Mcr = 0.6 (1.0) 40 (7.2 x 10 ) / (300) = 91 x 10 N .mm
2
Mr = 415 kN.m > 1.2 (91) = 109 kN.m O.K. Note: Eq. 2-7 may be used in lieu of Eq. 2-5. Asmin= 0.2 f'c b t h / f y (Eq. 2-7) 2 Asmin= 0.2 ( 40)(400)(600) / (400) = 759 mm 2 As= 5 (500) = 2500 > 759 mm O.K. 6.
Check minimum bar spacing: s = [400 - 2(30) - 2(11.3) - 5(25.2)] / 4 = 48 mm From Table 2.6; s ≥1 .4 d b = 1.4 (25.2) = 35 mm O.K. S ≥1 .4 a max = 1.4 (25) = 35 mm O.K. s ≥ 30 mm O.K.
7.
Check maximum bar spacing as governed by crack control: 1/3
Compute quantity "z" from Eq. 2-32 or Table 2.7: z = f s(dc A) Note: For calculation of A and d cc lear cover need not be taken greater than 50 mm. From Table 2.7; y = h - d = 600 - 546 = 54 mm; 2 A = 2yb/5 = 2 (54) (400) / 5 = 8640 mm f s= 0.6 f y = 0.6 (400) = 240 MPa; d c= 600 - 546 = 54 mm 1/3 z = 240 (54 x 8640) = 18,614 N/mm z = 18,614 N/mm < 30,000 N/mm (interior exposure) O.K. 8.
Check if skin reinforcement is needed: h ≤7 50 mm no skin reinforcement is needed.
9.
Final design: Use 5 -25M bars as longitudinal tension reinforcement with 10M stirrups and 30 mm clear cover for the stirrup steel.
3
Example 2.3 Analys is of a Rectangul ar Beam with Tension and Compression Reinforc ement Calculate the flexural resistance of the beam shown in the figure. f'c = 30 MPa; f y= 400 MPa.
m m 0 0 4
(Example 2.3) 1.
Compute moment resistance provided by steel couple M'r f rom Table 2.2 : Note : Total moment resistance = Mr + M' r
From Table 2.2; M'r = K' r bd x 10 k N.m, and for 2
ρ' = 0.71 %,
-6
K'r = 1.93 MPa; M' r = 1.93 (600) (330) x 10 = 126 kN.m Note: M' r = 126 kN.m is found assuming that the compression steel is yielding. 2.
Compute moment resistance provided by tension reinforcement (As - A' s) from Table 2.1: = (As - A's) / bd = (8 x 500 - 2 x 700) / (600) (330) = 0.0131 From Table 2.1; Mr = Kr bd2 x 10-6 kN.m, and for
Total moment resistance of section: Mr + M'r = 250 + 126 = 376 kN.m
4
Example 2.4 Design o f a Rectangu lar Beam with Tensio n and Compression Reinforc ement Design the rectangular beam section shown in the figure for a factored moment of M f = 700 kN.m. The beam is to be built as part of a parking structure located in Ottawa, with a limited cross-sectional size due to functional requirements. f'c = 30 MPa; f y= 400 MPa; maximum aggregate size = 25 mm.
m m 0 6 5
m m 0 6 5
a) Given Section
1.
b) Final Design (Example 2.4)
Determine concrete cover for corrosive environment from Table 2.6: Select 60 mm clear cover to stirrups.
2.
Estimate effective depth d assuming 30M bars for longitudinal reinforcement and 10M stirrups. d = 560 - 60 - 11.3 - (29.9/2) = 474 mm
3.
Determine the required tension reinforcement from Table 2.1 : 700 kN.m. For design; Mr ≥ M f = 6 2 6 2 For Mr = 700 kN.m; K r = M r x 10 /(bd ) = 700x10 / [(460)(474) ] = 6.77 MPa There is no ρv alue given in Table 2.1 for K r = 6.77 MPa indicating that the section can not be designed to behave in a ductile manner (under-reinforced) unless compression reinforcement is used. Try using compression reinforcement.
4.
Determine required compression reinforcement from Table 2.2: Provide maximum tension reinforcement given in Table 2.1. For f'c= 30 MPa; select ρ= 2.63 %, and read corresponding K r = 6.4 MPa 2 -6 Moment resistance provided by this reinforcement; M r = K r bd x 10 2 -6 Mr = 6.4 (460)(474) x 10 = 661 kN.m Remaining moment resistance to be provided by compression reinforcement; M'r = M f - M r = 700 - 661 = 39 kN.m
5
6
2
6
2
K'r = M' r x 10 / bd = 39 x 10 / (460)(474) = 0.38 MPa Compute d' based on assumed compression bar size of 15M; d' = 60 + 11.3 + (16/2) = 79 mm d' / d = 79 / 474 = 0.17; from Table 2.2; ρ' = 0.14 % 2 A's = ρ' bd = 0.0014(460)(474) = 305 mm 2 Use 2-15M with A' s= 400 mm a nd d' = 79 mm. 5.
Use 9 – 30M bars with A s= 6300 mm . Note: 9 – 30M bars can not be placed within b = 460 mm in a single row without violating the cover and/or minimum spacing limitations specified in Table 2.6. Therefore, use double layers of reinforcement and revise the design. 6.
Revise d: Revise "d" based on double layers of 30M bars and 45 mm clear spacing between the two layers: d = 560 - 60 - 11.3 - 29.9 - 45/2 = 436 mm Note: More reinforcement will be needed since "d" is reduced.
7.
Determine required compression reinforcement from Table 2.2: Provide maximum tension reinforcement given in Table 2.1. For f'c= 30 MPa; select ρ= 2.63 %, and read corresponding K r = 6.4 MPa 2 -6 Moment resistance provided by this reinforcement; M r = K r bd x 10 2 -6 Mr = 6.4 (460)(436) x 10 = 560 kN.m Remaining moment resistance to be provided by compression reinforcement; M'r = M f - M r = 700 - 560 = 140 kN.m 6 2 6 2 K'r = M' r x 10 / bd = 140 x 10 / (460)(436) = 1.60 MPa d' / d = 79 / 436 = 0.18; from Table 2.2 read ρ' = 0.57 % 2 A's = ρ' bd = 0.0057(460)(436) = 1143 mm (required) 2 Use 2-30M with A' s= 1400 mm ( provided) Note: The effect of the change in d’ from 79 mm to 86 mm, because of the use of 30M top reinforcement instead of 15M initially assumed, is negligible and is compensated in the extra steel area provided.
Use 10-30M bars in two layers (5-30M in each layer), with A s= 7000 mm 9.
2
Check spacing of tension reinforcement from Table 2.6: s = [460 - 2(60) - 2(11.3) - 5(29.9)] / 4 = 42 mm 42 mm = 1.4 bd= 1.4(29.9) = 42 mm > 1.4 amax= 1.4(25) = 35 mm > 30 mm Note: This section is heavily reinforced in the tension region and hence is not likely to violate minimum steel and crack control requirements. Final design is illustrated in the figure.
6
Table 2.1 M r
=
Reinforcement ratio (%) for rectangular sections with tension reinforcement f y = 400 MPa As ρφ c f y ⎤ ⎡ ρ = K r bd 2 x 10 −6 kN.m; ρφ K r = ⎢1 − f ; ⎥ sy bd 2α φ f '
⎣
f' c (MPa) α1 : β1 : ρbal : K r 0.5 0.6 0.7 0.8 0.9
20 25 0.82 0.81 0.92 0.91 1.83 2.24
30 0.81 0.90 2.63
35 0.80 0.88 3.00
0.15 0.18 0.21 0.24 0.28
0.15 0.18 0.21 0.24 0.27
0.15 0.18 0.21 0.24 0.27
1.0 1.1 1.2 1.3 1.4
0.31 0.34 0.38 0.41 0.44
0.31 0.34 0.37 0.40 0.44
1.5 1.6 1.7 1.8 1.9
0.48 0.51 0.55 0.58 0.62
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
1 c
c
⎦
45 50 0.78 0.78 0.86 0.85 3.67 3.98
55 0.77 0.83 4.27
60 0.76 0.82 4.55
0.15 0.18 0.21 0.24 0.27
40 0.79 0.87 3.34 (%) 0.15 0.18 0.21 0.24 0.27
0.15 0.18 0.21 0.24 0.27
0.15 0.18 0.21 0.24 0.27
0.15 0.18 0.21 0.24 0.27
0.15 0.18 0.21 0.24 0.27
0.30 0.34 0.37 0.40 0.43
0.30 0.33 0.37 0.40 0.43
0.30 0.33 0.36 0.40 0.43
0.30 0.33 0.36 0.39 0.43
0.30 0.33 0.36 0.39 0.42
0.30 0.33 0.36 0.39 0.42
0.30 0.33 0.36 0.39 0.42
0.47 0.50 0.54 0.57 0.61
0.46 0.50 0.53 0.56 0.60
0.46 0.49 0.53 0.56 0.59
0.46 0.49 0.52 0.55 0.59
0.46 0.49 0.52 0.55 0.58
0.46 0.49 0.52 0.55 0.58
0.45 0.49 0.52 0.55 0.58
0.45 0.48 0.52 0.55 0.58
0.66 0.69 0.73 0.77 0.81
0.64 0.68 0.71 0.75 0.79
0.63 0.67 0.70 0.73 0.77
0.62 0.66 0.69 0.73 0.76
0.62 0.65 0.69 0.72 0.75
0.62 0.65 0.68 0.71 0.75
0.61 0.65 0.68 0.71 0.74
0.61 0.64 0.68 0.71 0.74
0.61 0.64 0.67 0.70 0.74
0.85 0.89 0.93 0.98 1.02
0.82 0.86 0.90 0.94 0.98
0.81 0.84 0.88 0.91 0.95
0.79 0.83 0.86 0.90 0.93
0.79 0.82 0.85 0.89 0.92
0.78 0.81 0.85 0.88 0.92
0.78 0.81 0.84 0.88 0.91
0.77 0.80 0.84 0.87 0.90
0.77 0.80 0.83 0.87 0.90 7
Table 2.1 (Cont’d )
f' c( (MPa)
20
25
30
35
40
45
50
55
60
0.95 0.98 1.02 1.05 1.09
0.94 0.98 1.01 1.04 1.08
0.94 0.97 1.00 1.04 1.07
0.93 0.97 1.00 1.03 1.07
K r
3.0 3.1 3.2 3.3 3.4
1.06 1.11 1.15 1.20 1.25
1.02 1.06 1.10 1.14 1.18
0.99 1.03 1.06 1.10 1.14
0.97 1.01 1.04 1.08 1.12
(%) 0.96 0.99 1.03 1.06 1.10
3.5 3.6 3.7 3.8 3.9
1.30 1.35 1.40 1.46 1.51
1.22 1.26 1.31 1.35 1.40
1.18 1.22 1.26 1.30 1.34
1.15 1.19 1.23 1.27 1.31
1.14 1.17 1.21 1.25 1.28
1.12 1.16 1.19 1.23 1.27
1.11 1.15 1.18 1.22 1.25
1.11 1.14 1.17 1.21 1.24
1.10 1.13 1.17 1.20 1.23
4.0 4.1 4.2 4.3 4.4
1.57 1.63 1.69 1.76 1.83
1.45 1.49 1.54 1.59 1.64
1.38 1.43 1.47 1.51 1.56
1.35 1.39 1.43 1.47 1.51
1.32 1.36 1.40 1.44 1.47
1.30 1.34 1.38 1.41 1.45
1.29 1.32 1.36 1.40 1.43
1.28 1.31 1.35 1.38 1.42
1.27 1.30 1.34 1.37 1.41
4.5 4.6 4.7 4.8 4.9
1.69 1.75 1.80 1.85 1.91
1.60 1.65 1.69 1.74 1.79
1.55 1.59 1.63 1.67 1.72
1.51 1.55 1.59 1.63 1.67
1.49 1.53 1.56 1.60 1.64
1.47 1.51 1.54 1.58 1.62
1.45 1.49 1.53 1.56 1.60
1.44 1.48 1.51 1.55 1.59
5.0 5.1 5.2 5.3 5.4
1.97 2.03 2.09 2.16 2.23
1.84 1.88 1.93 1.99 2.04
1.76 1.81 1.85 1.90 1.94
1.71 1.75 1.80 1.84 1.88
1.68 1.72 1.76 1.80 1.84
1.66 1.69 1.73 1.77 1.81
1.64 1.67 1.71 1.75 1.79
1.62 1.66 1.69 1.73 1.77
5.5 5.6 5.7 5.8 5.9
2.09 2.15 2.20 2.26 2.32
1.99 2.04 2.08 2.13 2.18
1.92 1.97 2.01 2.06 2.10
1.88 1.92 1.96 2.00 2.05
1.85 1.89 1.93 1.97 2.01
1.82 1.86 1.90 1.94 1.98
1.80 1.84 1.88 1.92 1.95
6.0 6.1 6.2
2.38 2.23 2.44 2.28 2.50 2.33
2.15 2.19 2.24
2.09 2.05 2.13 2.09 2.17 2.13
2.02 2.06 2.10
1.99 2.03 2.07
8
6.3 6.4
2.57 2.39 2.63 2.44
2.29 2.33
2.22 2.17 2.26 2.21
2.14 2.18
2.11 2.15
Table 2.1 (Cont’d )
f' c (MPa) K r 6.5 6.6 6.7 6.8 6.9
35
40
45
50
2.50 2.55 2.61 2.67 2.73
(%) 2.38 2.43 2.48 2.53 2.58
2.31 2.35 2.40 2.44 2.49
2.25 2.30 2.34 2.38 2.43
2.22 2.26 2.30 2.34 2.38
2.19 2.23 2.26 2.30 2.34
2.63 2.68 2.74 2.79 2.85
2.54 2.58 2.63 2.68 2.73
2.47 2.52 2.56 2.61 2.65
2.42 2.46 2.51 2.55 2.59
2.39 2.43 2.47 2.51 2.55
7.5 7.6 7.7 7.8 7.9
2.90 2.96 3.02 3.08 3.14
2.78 2.83 2.88 2.93 2.99
2.70 2.74 2.79 2.84 2.89
2.64 2.68 2.72 2.77 2.81
2.59 2.63 2.68 2.72 2.76
8.0 8.1 8.2 8.3 8.4
3.20 3.26 3.33
3.04 3.09 3.15 3.20 3.26
2.93 2.98 3.03 3.08 3.13
2.86 2.91 2.95 3.00 3.05
2.80 2.85 2.89 2.94 2.98
8.5 8.6 8.7 8.8 8.9
3.32 3.38 3.44 3.50 3.56
3.18 3.24 3.29 3.34 3.40
3.09 3.14 3.19 3.24 3.29
3.02 3.07 3.12 3.16 3.21
9.0 9.1 9.2 9.3 9.4
3.62
3.45 3.51 3.56 3.62 3.68
3.34 3.39 3.44 3.49 3.54
3.25 3.30 3.35 3.40 3.45
3.74 3.80 3.86
3.59 3.65 3.70
3.49 3.54 3.59
7.0 7.1 7.2 7.3 7.4
9.5 9.6 9.7
20
25
30
2.79 2.85 2.91 2.98
55
60
9
9.8 9.9 10.0 10.5 11.0
3.92 3.98
3.76 3.81
3.64 3.69
3.87 4.16
3.75 4.01 4.29
10
Table 2.2 Compress ion reinfor cement ratio ’ (%); f y = 400 MPa M' r = K' r bd 2 x 10 −6 kN.m
d'/d:
⎡ ⎣
K' r = ⎢1 −
d' ⎤ ρ ' φ s f ' y d ⎥⎦
ρ '
=
0.05
0.10
0.15 ' (%)
0.20
0.25
0.20 0.40 0.60 0.80 1.00
0.06 0.12 0.19 0.25 0.31
0.07 0.13 0.20 0.26 0.33
0.07 0.14 0.21 0.28 0.35
0.07 0.15 0.22 0.29 0.37
0.08 0.16 0.24 0.31 0.39
1.20 1.40 1.60 1.80 2.00
0.37 0.43 0.50 0.56 0.62
0.39 0.46 0.52 0.59 0.65
0.42 0.48 0.55 0.62 0.69
0.44 0.51 0.59 0.66 0.74
0.47 0.55 0.63 0.71 0.78
2.20 2.40 2.60 2.80 3.00
0.68 0.74 0.80 0.87 0.93
0.72 0.78 0.85 0.92 0.98
0.76 0.83 0.90 0.97 1.04
0.81 0.88 0.96 1.03 1.10
0.86 0.94 1.02 1.10 1.18
3.20 3.40 3.60 3.80 4.00
0.99 1.05 1.11 1.18 1.24
1.05 1.11 1.18 1.24 1.31
1.11 1.18 1.25 1.31 1.38
1.18 1.25 1.32 1.40 1.47
1.25 1.33 1.41 1.49 1.57
K' r
A' s bd
11
Table 2.6 Spacing and cover requirements f or b eam reinforc ement
