1、 Critical Review of Deflection Formulas for FRP-RC MembersCarlos Mota1; Sandee Alminar2; and Dagmar Svecova31Research Assistant, Dept. of Civil Engineering, Univ. of Manitoba,Winnipeg MB, Canada R3T 5V6.2Research Assistant, Dept. of Civil Engineering, Univ. of Manitoba,Winnipeg MB, Canada R3T 5V6.3A
2、ssociate Professor, Dept. of Civil Engineering, Univ. of Manitoba,Winnipeg MB, Canada R3T 5V6 （corresponding author）. Abstract: The design of fiber-reinforced polymer reinforced concrete FRP-RC is typically governed by serviceability limit state requirements rather than ultimate limit state requirem
3、ents as conventional reinforced concrete is. Thus, a method is needed that can predict the expected service load deflections of fiber-reinforced polymer FRP reinforced members with a reasonably high degree of accuracy. Nine methods of deflection calculation, including methods used in ACI 440.1R-03,
4、and a proposed new formula in the next issue of this design guide, CSA S806-02 and ISIS M03-01, are compared to the experimental deflection of 197 beams and slabs tested by otherinvestigators. These members are reinforced with aramid FRP, glass FRP, or carbon FRP bars, have different reinforcement r
5、atios, geometric and material properties. All members were tested under monotonically applied load in four point bending configuration. The objective of the analysis in this paper is to determine a method of deflection calculation for FRP RC members, which is the most suitable for serviceability cri
6、teria. The analysis revealed that both the modulus of elasticity of FRP and the relative reinforcement ratio play an important role in the accuracy of the formulas.CE Database subject headings: Concrete, reinforced; Fiber-reinforced polymers; Deflection; Curvature; Codes; Serviceability;Statistics.I
7、ntroduction Fiber-reinforced polymer FRP reinforcing bars are currently available as a substitute for steel reinforcement in concrete structures that may be vulnerable to attack by aggressive corrosive agents. In addition to superior durability, FRP reinforcing bars have a much higher strength than
8、conventional mild steel. However, the modulus of elasticity of FRP is typically much lower than that of steel. This leads to a substantial decrease in the stiffness of FRP reinforced beams after cracking. Since deflections are inversely proportional to the flexural stiffness of the beam, even some F
9、RP over-reinforced beams are susceptible to unacceptable levels of deflection under service conditions. Hence, the design of FRP reinforced concrete （FRP-RC） is typically governed by serviceability requirements and a method is needed that can calculate the expected service load deflections of FRP re
10、inforced members with a reasonable degree of accuracy. The objective of this paper is to point out the inconsistencies in existing deflection formulas. Only instantaneous deflections will be discussed in this paper.Effective Moment of Inertia Approach ACI 318 （ACI 1999）and CSA A23.3-94 (CSA 1998) re
11、commend the use of the effective moment of inertia, Ie, to calculate the deflection of cracked steel reinforced concrete members. The procedure entails the calculation of a uniform moment of inertia throughout the beam length, and use of deflection equations derived from linear elastic analysis. The
12、 effective moment of inertia, Ie, is based on semiempirical considerations, and despite some doubt about its applicability to conventional reinforced concrete members subjected to complex loading and boundary conditions, it has yielded satisfactory results in most practical applications over the yea
13、rs. In North American codes, deflection calculation of flexural members are mainly based on equations derived from linear elastic analysis, using the effective moment of inertia, Ie, given by Bransons formula (1965) （1）=cracking moment;=moment of inertia of the gross section; =moment of inertia of t
14、he cracked section transformed to concrete; and =effective moment of inertia. Research by Benmokrane et al. (1996)suggested that in order to improve the performance of the original equation, Eq.(1) will need to be further modified. Constants to modify the equation were developed through a comprehens
15、ive experimental program. The effective moment of inertia was defined according to Eq.(2) if the reinforcement was FRP （2） Further research has been done in order to define an effective moment of inertia equation which is similar to that of Eq.(1), and converges to the cracked moment of inertia quic
16、ker than the cubic equation. Many researchers (Benmokrane et al. 1996; Brown and Bartholomew 1996; Toutanji and Saafi 2000) argue that the basic form of the effective moment of inertia equation should remain as close to the original Bransons equation as possible, because it is easy to use and design
17、ers are familiar with it.The modified equation is presented in the following equation: （3） A further investigation of the effective moment of inertia was performed by Toutanji and Saafi (2000). It was found that the order of the equation depends on both the modulus of elasticity of the FRP, as well
18、as the reinforcement ratio. Based on their research, Toutanji and Saafi (2000) have recommended that the following equations be used to calculate the deflection of FRPreinforced concrete members: （4）WhereIf Otherwise （5） m =3where =reinforcement ratio; =modulus of elasticity of FRP reinforcement; an
19、d =modulus of elasticity of steel reinforcement. The ISIS Design Manual M03-01 （Rizkalla and Mufti 2001） has suggested the use of an effective moment of inertia which is quite different in form compared to the previous equations. It suggests using the modified effective moment of inertia equation de
20、fined by the following equation to be adopted for future use: （6）where=uncracked moment of inertia of the section transformed to concrete. Eq. （6） is derived from equations given by the CEB-FIP MC-90 (CEB-FIP 1990). Ghali et al. (2001) have verified that Ie calculated by Eq.（6） gives good agreement
21、with experimental deflection of numerous beams reinforced with different types of FRP materials. According to ACI 440.1R-03 (ACI 2003), the moment of inertia equation for FRP-RC is dependent on the modulus of elasticity of the FRP and the following expression for Ie is proposed to calculate the defl
22、ection of FRP reinforced beams: (7) where （8）where=reduction coefficient; =bond dependent coefficient（until more data become available, =0.5）; and =modulus of elasticity of the FRP reinforcement. Upon finding that the ACI 440.1R-03 （ACI 2003） equation often underpredicted the service load deflection
23、 of FRP reinforced concrete members, several attempts have been made in order to modify Eq.（7）. For instance, Yost et al. （2003） claimed that the accuracy of Eq.（7） primarily relied on the reinforcement ratio of the member. It was concluded that the formula could be of the same form, but that the bo
24、nd dependent coefficient, , had to be modified. A modification factor, , was proposed in the following form: （9）where =balanced reinforcement ratio. The ACI 440 Committee (ACI 2004) has also proposed revisions to the design equation in ACI 440.1R-03 (ACI 2003). The moment of inertia equation has ret
25、ained the same familiar form as that of Eq. (7) in these revisions. However, the form of the reduction coefficient, , to be used in place of Eq. (8) was modified. The new reduction coefficient has changed the key variable in the equation from the modulus of elasticity to the relative reinforcement r
26、atio as shown in the following equation: （10）MomentCurvature Approach The momentcurvature approach for deflection calculation is based on the first principles of structural analysis. When a momentcurvature diagram is known, the virtual work method can be used to calculate the deflection of structura
27、l members under any load as （11）where L=simply supported length of the section; M/EI=curvature of the section; and m=bending moment due to a unit load applied at the point where the deflection is to be calculated. A momentcurvature approach was taken by Faza and GangaRao (1992), who defined the mids
28、pan deflection for fourpoint bending through the integration of an assumed moment curvature diagram. Faza and GangaRao (1992) made the assumption that for four-point bending, the member would be fully cracked between the load points and partially cracked everywhere else. A deflection equation could
29、thus be derived by assuming that the moment of inertia between the load points was the cracked moment of inertia, and the moment of inertia elsewhere was the effective moment of inertia defined by Eq.(1). Through the integration of the moment curvature diagram proposed by Faza and GangaRao (1992), t
30、he deflection for four-point loading is defined according to the following equation: （12）where =shear span. Eq.(12) has limited use because it is not clear what assumptions for the application of the effective moment of inertia should be used for other load cases. However, it worked quite accurately
31、 for predicting the deflection of the beams tested by Faza and GangaRao (1992). The CSA S806-02 (CSA 2002) suggests that the momentcurvature method of calculating deflection is well suited for FRP reinforced members because the momentcurvature diagram can be approximated by two linear regions: one b
32、efore the concrete cracks, and the second one after the concrete cracks (Razaqpur et al. 2000). Therefore, there is no need for calculating curvature at different sections along the length of the beam as for steel reinforced concrete. There are only three pairs of moments with corresponding curvatur
33、e that define the entire momentcurvature diagram: at cracking, immediately after cracking, and at ultimate. With this in mind, simple formulas were derived for deflection calculation of simply supported FRP reinforced beams and are used in CSA S806-02 (CSA 2002). The deflection due to fourpoint bend
34、ing can be found using the following equation: （13）Verification of Proposed Methods The nine methods of deflection calculation presented in this paper were used to analyze 197 simply supported beams and slabs tested by other investigators. Material and geometric properties of the beams used in this
35、investigation could not be published due to the extent of the statistical sample but can be found in Mota(2005). Table 1 shows the range of some of the important properties of the members in the database. All information used in the analysis, such as cracking moment and modulus of elasticity of conc
36、rete, was calculated using CSA A23.3-94 (CSA 1998)based on input given by researchers. To check the accuracy of formulas developed by other investigators, a statistical analysis has been performed on each of the equations comparing the calculated deflection to the experimental deflection at several
37、given load levels. It must be noted that the deflection is typically only checked at the service load level. However, since the service load criteria is only explicitly stated in the ISIS M03-01 (Rizkalla and Mufti 2001), it is unclear at this point what the service load level for each code is. Thus
38、, a statistical analysis was carried out at both low loads and at elevated loads to encompass the entire loaddeflection curve, as well as at the service level given by ISIS M03-01 （Rizkalla and Mufti 2001）. This will allow the designers to choose an accurate formula, based on the results of the anal
39、ysis, at the load level which most closely resembles their service load criteria. The statistical analysis was performed by applying a log transformation to the ratios of the experimental to calculated deflection ratios. A log transformation was employed to give equal weight to those ratios which we
40、re below one and those which were above one. When considering long-term deflection, perhaps only the accuracy of short-term deflection equation is required since this number will be further modified by other coefficients. However, since only short-term deflection has been considered here, the predic
41、ted deflection should be also consistently conservative. Journal of Composites for Construction, Vol. 10, No.3, June 1, 2006. ASCE, ISSN 1090-0268/2006/3-183194.FRP-RC构件的挠度计算公式的评论卡洛斯.莫塔1;桑德.阿尔米纳尔2;达格玛.斯维克瓦31.加拿大曼尼托巴大学土木工程学院研究员2.加拿大曼尼托巴大学土木工程学院研究员2.加拿大曼尼托巴大学土木工程学院副教授（通讯作者）摘 要： 纤维复合材料包覆钢筋混凝土（FRP-RC）的设
42、计通常是由正常使用极限状态的要求控制,而不是像由传统的钢筋混凝土极限状态要求控制。因此,需要一种可以预测FRP-RC构件正常使用的负载变形量的精确度的方法。计算挠度的九种方法,包括被测试人员用于下一期拟议的ACI 440.1 R-03和CSA S806-02和ISIS M03-01中的新公式设计指南中的实验197个梁和板的挠度进行测试的方法。这些构件与芳纶玻璃钢钢筋、玻璃玻璃钢或碳纤维塑料筋配筋率、几何和材料属性不同。所有构件在四点弯曲加载配置下进行测试单调递增的应用荷载。本文分析的目的是确定FRP-RC构件挠度的计算方法,也是确定最适用的可靠性的准则。分析表明,FRP的弹性模量和相对配筋率在
43、公式的准确性中发挥重要作用。关键词: 钢筋混凝土,纤维增强聚合物;挠度弯曲;规范;适用性;统计数据。介 绍：纤维复合材料钢筋目前可用来代替容易受到侵蚀性腐蚀破坏的钢筋混凝土结构。除了优越的耐用性,FRP钢筋强度远高于传统的低碳钢。然而,玻璃钢的弹性模量通常比钢低得多。这导致开裂后大量减少FRP加固的梁的刚度。由于变形量和梁的抗弯刚度是成反比的,甚至一些纤维复合材料超钢筋加固梁在使用情况下容易受到不可接受的水平偏转。因此,纤维复合材料包覆钢筋混凝土的设计通常是由正常使用极限状态的要求控制，需要一个方法,计算维复合材料构件的预期工作负载挠度的合理精确度。 本文的目的是指出现有的挠度公式和论证所有的
44、通用方程在计算FRP-RC构件有局限性。本文只讨论瞬时挠度。1.有效惯性矩法 ACI 318 （ACI 1999）和CSA a23.3 - 94 （CSA 1998）推荐使用有效惯性矩计算钢筋混凝土构件破坏时的挠度。这个过程需要一个适用于整个梁长的惯性矩计算,并使用由线性弹性分析所得的挠度方程。 有效惯性矩是基于半经验的考虑,虽然当它受到复杂的加载和边界条件时，与传统钢筋混凝土构件有适用性问题,但是它在大多数实际应用中取得了令人满意的结果。 在北美的规范中,构件的挠度计算公式主要是由线性弹性分析所得的方程,即使用由1965年的布兰森公式所得的有效惯性矩， (1)=开裂弯矩;=毛截面惯性矩,=破
45、坏截面混凝土惯性矩;=有效截面惯性矩 1996年Benmokrane 的研究表明,为了提高起始方程的性能，需要进一步修改方程（1）。可以通过综合实验修改方程的常量。如果使用FRP加固构件，此时有效惯性矩可由方程（2）所得 (2) 研究人员做了进一步的研究,以便于确定一个类似于方程式（1）但更快捷的有效惯性矩方程。许多研究人员（Benmokrane等1996;布朗和巴塞洛缪 1996;Toutanji和萨菲2000）认为有效惯性矩方程的基本形式应尽可能接近于原始的布兰森方程,为了它容易被使用而且设计师对它比较熟悉。修改后的方程如下: (3)Toutanji 和萨菲（2000） 对有效惯性矩进行了
46、进一步的研究。他们发现方程的顺序取决于FRP的弹性模量和配筋率。根据Toutanji 和萨菲（2000）的研究,他们建议使用下面的方程来计算FRP构件的挠度: (4)当 若 （5） 则 m=3=配筋率;=FRP的弹性模量;=钢筋的弹性模量。 ISIS Design Manual M03-01 （里兹卡拉和穆夫提 2001）建议使用完全不同于先前方程式的形式计算有效惯性矩。它建议今后使用进行修改过的有效惯性矩方程，如下所示: （6）=未破坏截面处的惯性矩 方程式（6）取自CEB-FIP MC-90（CEB-FIP 1990）加利等（2001）。通过用大量的梁来进行挠度试验，这些梁是由不同类型的F
47、RP材料制作的，大量试验所得的，与方程式（6）所得的相同。 根据ACI 440.1R-03 (ACI 2003),FRP-RC的惯性矩方程取决于FRP的弹性模量和由计算的FRP加固梁的挠度方程，可由如下方程式得: (7) （8） =换算系数; =相关系数; =FRP的弹性模量。根据ACI 440.1R-03（ACI 2003）中的方程可得，在工作荷载作用下，FRP构件的挠度通常是可预测的,经过几次尝试对方程修改从而得到方程式（7）。例如,约斯特等（2003）认为,方程式（7）的准确性主要依赖于其构件的配筋率，并得出结论，方程式的形式不变,但应对进行修改，由此得出的方程式： （9）=平均配筋率ACI 440委员会（ACI 2004)对ACI 440.1R-03 (ACI 2003)中的方程也进行了修改。惯性矩公式延用公式（7）,但对降低系数进行了修改，降低系数取决于弹性模量相对配筋率，见以下方程: （10）2.弯矩-曲率法弯矩-曲率法是进行结构分析中计算挠度的首选。在负载情况下，当弯矩-曲率图已知,虚拟的工作法可以用来计算结构构件的挠度
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