75 □ Flexure of reinforced concrete beams钢筋混凝土梁的性能
Example 3-10 (Design of a reinforced concrete beam with tension reinforcement only)==A reinforced concrete beam , with an effective depth of 16 inches and a width of 12 inches ,is reinforced with Grade 60 bars and has a concrete cylinder strength of 4000 pounds per square inch. The beam carries a superimposed dead load ,including its self weight, of 2.5 kips per foot run and superimposed live load of 2.7 kips per foot run over on effective span of 15 feet .Determine the area of tension reinforcement required.
例3-10 （在拉力作用下钢筋混凝土梁的设计）一个钢筋混凝土梁，其有效高度为16英寸，宽度为12英寸，钢筋的强度等级为60，混凝土的强度等级为4000磅每平方英寸，计算跨度为15英尺，该梁承受的恒载为2.5千磅每英尺，活载为2.7千磅每英尺，求受拉钢筋的截面面积。
Solution 解：
The applied dead load moment is given by恒载作用下的弯矩为：
MD?wD?2/8?2.5?152/8
?70.13 ki pfeetThe applied live load moment is given by活载作用下的弯矩为：
MD?wD?2/8?2.7?152/8
?75.94 ki pfeetThe factored moment at mid span is obtained from ACI Equation (9-1) as 由ACI方程式得到的在跨中的弯矩为：
Mu?1.4MD?1.7ML
?1.4?70.31?1.7?75.95 ?227.53 ki pfeetThe maximum allowable factored moment for a singly reinforced beam is obtained from Table 3.3 as
由表3.3得出，单筋混凝土梁的最大允许弯矩为：
2Mu?Ku(mabxw)d
?936?12?162/12,000
?239.62 ki pfeet? MuHence the section is adequate.从而这个弯矩是满足的。
76 □ Flexure of reinforced concrete beams钢筋混凝土梁的性能
The design moment factor is弯矩设计系数为
Ku?Mu/bWd2?227.53?12,000/(12?162)
?888.79 poun pdsesrq ua irnechKu/fC'?888.79/4000and 和
?0.2222From Table 3-2, the corresponding tension reinforced index is从表3-2可知，相应的张拉系数是
29 9 ? ? 0.The required tension reinforcement ratio is given by所需受拉钢筋的配筋率是：
? ??f'c/fy
?0.299?4000/60,000 ?0.020The required area of tension reinforcement is受拉钢筋的截面面积为
As ??bwd
?0.020?12?16?3.8 s3qua irnechesProvide three No. 10 bars which, from Table 1-7, given an area of 3.81 square inches. ======== 由表1-7查得3根10元钢筋的面积是3.81平方英寸。
Examples 3-11 (Analysis of a reinforced concrete slab) ============================== A one-way, reinforced concrete slab is continuous over four, equal, clear spans of 15 feet. The slab has an overall depth of 7 inches with concrete cylinder strength of 3000 pounds per square inch. The reinforcement at the top of the slab at the first interior support consists of No. 4,Grade 60,bars at 11 inches on center. In addition to its own weight, the slab carries a superimposed dead load of 26 pounds per square foot. Determine the maximum uniformly distributed superimposed live load which the slab can support.
例 3-11（钢筋混凝土板的分析） 一个四边连续，相等的混凝土单向板，其净跨为15英尺，板的全高为7英寸，其混凝土强度等级为3000磅每平方英寸，板的上部配筋为间距11英寸4根强度等级60的钢筋，包括板的自重，板上的恒载为26磅每平方英尺。求板所能承受的最大均布活载。 Solution 解:
The minimum allowable concrete cover to the reinforcement in the slab, in accordance with ACI section 7.7.1, is根据ACI 7.7.1小节知，混凝土最小保护层厚度为：
77 □ Flexure of reinforced concrete beams钢筋混凝土梁的性能
cc ? 0 . 7 i5nch esThe effective depth provides is 有效高度为
d ?h-cc?db/2
?7?0.75?0.5/2 ?6 inchesThe area of tension reinforcement in a 12 inches width of slab with No. 4 bars at 11inches on center is obtained form Table 1-8 as
由表1-8可得，宽12英寸配置元4间距为11英寸的受拉钢筋截面面积为： As ?0.2 s2qua irnech esThe tension reinforcement ratios provide is given by受拉钢筋的配筋率为：
? ?As/bwd
?0.22/(12?6)?0.00306??min?(?0.0018 from ACI Section 7.12.2)??max?(?0.0160 fro mAC SIecti1o0n. 3).3
Hence the reinforcement provide is satisfactory.因此配置的钢筋是满足要求的。 The tension reinforcement index is 受拉钢筋系数为：
? ??fy/fc'
?0.00306?60,000/3000 ?0.0612From Table 3-2, the corresponding value of由表3-2可得，其相应的值为： Ku/fc'?0.0531
The allowable factored moment is given by允许弯矩设计值为：
Mu ?Kubwd2?0.0531?3000?12?62/12,000+ ?5.73 ki pfeet
The self weight of the slab is板的自重为：
WSD?7?12?0.15/144 ?0.087 k5i ppefro otThe total dead load is given by 恒载的总值为：
wD?wSD?wAD
?0.087?50.026 ?0.113 k5i ppefro ot78 □ Flexure of reinforced concrete beams钢筋混凝土梁的性能
Using the approximate moment coefficient given in ACI Section 8.3.3, the negative moment produced by the dead load at the first interior support is
利用由ACI 8.3.3小节给出弯矩近似法，由恒载产生的负弯矩为：
MD?wD?2/10
?0.1135?152/10 ?2.55 ki pfeetThe factored dead load moment is obtained from ACI Section 9.2.1 as 由ACI 9.2.1小节得到恒载弯矩设计值为：
MuD?1.4?MD
?1.4?2.55 ?3.57 ki pfeetThe allowable factored live moment is given by 活载弯矩设计值为：
MuL?Mu?MuD
?5.73?3.57 ?2.16 ki pfeetThe allowable non-factored live load moment is obtained from ACI Section 9.2.1 as 由ACI 9.2.1小节得到活载弯矩标准值为;
ML?MuL/1.7
?2.16/1.7?1.27 ki pfeetThe allowable non-factored live load is obtained from ACI Section 8.3.3 as 由ACI 8.3.3 小节得到的活载标准值为：
wL?10ML/?2n
?10?1.27?1000/152 ?56.4 poun pdsefro otThe slab can support a uniformly distributed live load of 56.4 pounds per square foot. ========= 此板所能承受的均布活载为56.4磅每平方英尺。
Figure 3-6 Reinforced concrete beam with compression reinforcement 图3-6 在压力作用下的钢筋混凝土梁
3.2.2 Rectangular beam with compression reinforcement 3.2.2在压力作用下的钢筋混凝土梁
When the applied factored moment exceeds the maximum design strength of a singly reinforced member with the maximum allowable reinforcement ratio, compression reinforcement and
79 □ Flexure of reinforced concrete beams钢筋混凝土梁的性能
additional tension reinforcement must be provide, as shown in Figure 3-6.The difference between the applied factored moment and the maximum design moment strength of a singly reinforced section is
在最大配筋率下，当单筋混凝土梁的施加弯矩值大于最大弯矩设计值时，其受压钢筋和附加受拉钢筋由图3-6所示，单筋混凝土梁的施加弯矩值与最大弯矩设计值之差为 Mr?Mu?Mmax? resid umaolme nThe required area of compression reinforcement is given by所需受压钢筋的截面面积为：
' As?Mr/?fs'(d-d')
Where 式中 fs'?stress in the compression reinforcement受压钢筋的应力强度 The additional area of tension reinforcement required is所需附加受拉钢筋的截面面积为：
AT?Mr/?fs'(d-d')?Af/fy''ss
The total required area of tension reinforcement is受拉钢筋的总面积为：
As?Amax?AT?Amax?Mr/?fy(d-d')
The depth of the stress block is obtained by equating forces and is given by 由水平方向的平衡方程可得，受压区的高度为： a ?fyAmax/0.85fc'bw
The neutral axis depth is given by ACI Section 10.2.7 as由ACI 10.2.7 给出的中性轴高度为： c=a/?1
The strain in the compression reinforcement is obtained from the strain diagram in Figure 3-6 and is 由图3-6应力图得到，受压钢筋的应变为： ?'??c(1?d'/c) Where 式中 ?c?0.003
The stress in the compression reinforcement is受压钢筋的应力强度为：
fs'??cEs(1?d'/c)
?0.003?29,000,000(1?d'/c)?87,00(01?d/c)?fy'