Calculation method of flexural capacity of ultra-high performance concrete beams reinforced with FRP rebars
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摘要: 目前国内外尚未提出纤维增强树脂复合材料(FRP)筋超高性能混凝土(UHPC)梁受弯承载力的显式计算公式。基于ABAQUS中的塑性损伤模型,建立了FRP筋UHPC梁的受弯性能非线性有限元分析模型,通过已有试验结果验证了该模型的有效性。开展了40根UHPC梁的参数分析,重点研究了截面尺寸、UHPC强度、FRP筋抗拉强度和配筋率等因素对梁受弯性能的影响规律。基于目前国际上最常用的法国NF P 18-710规范,对UHPC受拉本构模型进行了简化,推导了UHPC受压区和受拉区等效矩形应力图块系数,提出了FRP筋UHPC梁平衡配筋率的计算方法,建立了受压破坏和受拉破坏模式下梁截面受弯承载力的理论计算公式。该公式计算值与试验结果、有限元分析结果和条带法计算结果均吻合良好。Abstract: Till now, no closed-form theoretical equation of flexural capacity of ultra-high performance concrete (UHPC) beams reinforced with fiber reinforced polymer (FRP) rebars has been proposed worldwide. Based on the concrete damage plasticity model in ABAQUS, a nonlinear finite element analysis model was developed to simulate the flexural performance of UHPC beams reinforced with FRP rebars. The model was verified with available test results. A parametric analysis of 40 UHPC beams was performed and the influence of parameters including cross section dimension, strength of UHPC, tensile strength of FRP rebars and reinforcement ratio on flexural performance were studied. Based on the most widely used standard worldwide French NF P 18-710, the constitutive model of UHPC was simplified and the parameters for equivalent rectangular stress block in compression zone and tension zone were deducted, then the calculation method of balanced reinforcement ratio and the theoretical equations for flexural capacity under compression failure and tension failure were developed for UHPC beams reinforced with FRP rebars. The calculated results from the theoretical equations are in a good agreement with results from tests, finite element analysis and strip method.
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图 2 有限元分析和试验结果得到的纤维增强树脂复合材料(FRP)筋UHPC梁受弯性能对比[17]
FEM—Finite element analysis; BC12—Reinforced with φ12 carbon fiber reinforced composite (CFRP) rebar; BG20—Reinforced with φ20 glass fiber reinforced composite (GFRP) rebar; φ—Diameter
Figure 2. Comparison of flexural performance of fiber reinforced composite (FRP)-UHPC beams obtained from tests and finite element analysis[17]
图 8 基于法国NF P 18-710的UHPC本构模型[8]
Figure 8. Constitutive model of UHPC based on NF P 18-710[8]
εu,el, fctk,el—Cracking strain and cracking strength; εu,pic, fctfk—Peak tensile strain and post-cracking strength; εu,1%, fctfk,1%—Characteristic tensile strain and characteristic tensile strength; εu,lim—Ultimate tensile strain; εc0d, fcd —Design compressive strain and design compressive strength; εcud—Ultimate compressive strain; Ec—Elastic modulus; γcf—Partial factor; K—Orientation factor
图 9 条带法计算流程
Figure 9. Computation procedure of the strip method
i—Number of iterations; Δε—Increment of strain; εf—Tensile strain of FRP rebar; εfu—Ultimate tensile strain of FRP rebar; εcu—Ultimate compressive strain of UHPC; F—Sum of the force of each element; Mi—Sum of the moment from each element towards the neutral axis during the ith loop
图 10 FRP筋UHPC梁受弯时截面应力应变的分布与简化
Figure 10. Distribution and simplification of strain and stress of FRP-UHPC beam cross-section under flexure
ft—Tensile strength of UHPC; εfu, ffu—Ultimate strain and tensile strength of FRP rebar; εc,top, σc,top—Strain and stress of UHPC at the top of the section; εt,bot—Strain of UHPC at the bottom of the section; εf, σf—Strain and stress of FRP rebar; x—Height of the compression zone; α1, β1—Equivalent rectangular stress block parameters for UHPC compression zone under compressive failure; α1', β1'—Equivalent rectangular stress block parameters for UHPC compression zone under tensile failure; α2, β2—Equivalent rectangular stress block parameters for UHPC tension zone
图 11 FRP筋UHPC梁受弯承载力理论公式计算流程
k—Number of iterations; ρb—Balanced reinforcement ratio; ρf—Reinforcement ratio; x—Depth of compression zone; x(k)—Value of x in kth iteration; η—Parameter; η(k)—Value of η in kth iteration.
Figure 11. Computation procedure of the theoretical equations for the flexural capacity of FRP-UHPC beams
表 1 混凝土塑性损伤(CDP)模型参数取值
Table 1. Values of parameters for the concrete damage plasticity (CDP) model
Parameter Poisson’s ratio Dilation angle/(°) Eccentricity Kc fb0/fc0 Viscosity parameter Value 0.2 35 0.1 2/3 1.05 0.005 Notes: Kc—Ratio of the second stress invariant on the tensile meridian to the second stress invariant on the compressive meridian; fb0/fc0—Ratio of initial equibiaxial compressive yield strength to the initial uniaxial compressive yield strength. 表 2 FRP筋UHPC梁受弯承载力的有限元模拟值、条带法计算值和理论公式计算值与试验结果的对比[17]
Table 2. Comparison of flexural capacity of FRP-UHPC beams obtained from finite element analysis, strip method and theoretical equations with test results[17]
Resource Specimen Experiment results FEM analysis Strip method Theoretical equations Mexp/
(kN·m)Failure
modeMFEM/
(kN·m)MFEM/
MexpMu,1/
(kN·m)Mu,1/
MexpMu,2/
(kN·m)Mu,2/
Mexp[17] BC12 131.3 T 126.5 0.96 137.1 1.04 134.6 1.02 [17] BG20 167.1 C 169.5 1.01 171.0 1.02 176.2 1.05 Notes: Mexp, MFEM, Mu,1, Mu,2—Flexural capacities obtained from experiment, finite element analysis, strip method and theoretical equations, respectively; T—Tensile failure (FRP rebar rupture); C—Compression failure (UHPC crushing). 表 3 FRP筋UHPC梁受弯性能的有限元分析结果、条带法计算结果和理论公式计算结果
Table 3. Flexural performance of FRP-UHPC beams, obtained from finite element analysis, strip method and theoretical equations
Specimens for parametric study FEM analysis Strip method Theoretical equations Specimen Parameter studied Values MFEM/
(kN·m)Δmax/mm Failure
modeMu,1/
(kN·m)Failure
modeMu,2/
(kN·m)Failure
modeBC12 – – 126.5 72.2 T 137.1 T 136.2 T BC12-C120 fcu 120 MPa 123.5 73.3 T 132.3 T 134.6 T BC12-C180 180 MPa 129.0 69.3 T 141.4 T 137.9 T BC12-C200 200 MPa 128.2 66.3 T 143.5 T 138.8 T BC12-EP1 εfu 1% 106.9 54.6 T 116.5 T 114.8 T BC12-EP1.5 1.5% 144.6 89.1 T 158.4 T 158.2 T BC12-EP2 2% 154.3 107.3 C 163.4 C 168.2 C BC12-EP2.5 2.5% 154.3 107.3 C 163.4 C 168.2 C BC12-H150 h 150 mm 56.4 144.5 C 60.7 C 62.8 C BC12-H200 200 mm 93.4 98.2 T 100.9 T 101.6 T BC12-H300 300 mm 157.2 56.4 T 176.0 T 172.4 T BC12-R0.5 ρf 0.5% 103.4 66.5 T 113.2 T 112.0 T BC12-R1 1% 186.5 88.7 T 197.5 C 205.4 T BC12-R2 2% 236.4 59.5 C 256.5 C 266.1 C BC12-R3 3% 277.0 51.4 C 293.7 C 305.7 C BC12-R4 4% 306.5 46.9 C 320.9 C 334.6 C BC12-R5 5% 328.8 44.2 C 342.1 C 357.1 C BG20 – – 169.5 108.2 C 171.0 C 176.2 C BG20-C120 fcu 120 MPa 142.6 99.2 C 145.3 C 147.4 C BG20-C180 180 MPa 188.4 107.9 C 199.3 C 208.9 C BG20-C200 200 MPa 185.8 116.4 T 217.0 C 213.2 T BG20-EP1 εfu 1% 123.5 59.8 T 132.1 T 130.7 T BG20-EP1.5 1.5% 164.8 101.0 T 171.0 C 176.2 C BG20-EP2 2.0% 169.5 108.2 C 171.0 C 176.2 C BG20-EP2.5 2.5% 169.5 108.2 C 171.0 C 176.2 C BG20-H150 h 150 mm 57.1 129.8 C 61.7 C 63.8 C BG20-H200 200 mm 106.9 123.9 C 111.1 C 115.1 C BG20-H300 300 mm 213.6 88.6 C 240.2 C 245.9 C BG20-R0.5 ρf 0.5% 62.5 73.8 T 70.0 T 67.3 T BG20-R1 1% 116.9 101.6 T 126.3 T 124.4 T BG20-R2 2% 173.2 106.8 C 179.4 C 185.0 C BG20-R3 3% 194.6 81.0 C 210.6 C 217.8 C BG20-R4 4% 208.9 63.7 C 234.6 C 242.9 C BG20-R5 5% 244.6 60.0 C 253.6 C 263.3 C Notes: Specimen BC12 and BG20 are in the control group. For BC12, fcu=148 MPa, εfu=1.25%, h=250 mm, ρf=0.63%. For BG20, fcu=148 MPa, εfu=1.78%, h=250 mm, ρf=1.78%. The other specimens are in the experimental group. They are designed by changing one parameter compared to the corresponding specimen in the control group. i.e., BC12-C120 is designed based on BC12 by setting fcu=120 MPa, BG20-R2 is designed based on BG20 by setting ρf=2%. fcu—Cubic compressive strength of UHPC; εfu—Ultimate tensile strain of FRP rebar; h—Height of the beam section; ρf—Reinforcement ratio of FRP rebar, obtained from the reinforcement area divided by bh0, where b is the width of the beam section and h0 is the depth of FRP rebar; Δmax—Displacement of mid-span at the peak moment. 表 4 筋材基本参数
Table 4. Material parameters of rebars
Rebar type Elastic modulus
/GPaYielding
strength
/MPaUltimate strength
/MPaCFRP rebar 144 ─ 1100 GFRP rebar 62 ─ 1800 HRB500 500 500 500 1860 MPa steel strand wire 1581 1581 1860 Note: HRB500—Deformed steel bar. 表 5 钢筋UHPC梁和钢绞线UHPC梁受弯承载力的有限元模拟值
Table 5. Flexural capacity of UHPC beams reinforced with steel rebars and steel strands obtained from finite element analysis
Specimen Reinforcement ρ*/% ρ/% MFEM
/(kN·m)Δmax
/mmFailure mode G-R0.5 HRB
5000.5 1.10 82.2 19.9 F G-R1 1.0 2.20 133.3 22.4 F G-R2 2.0 4.40 233.6 29.5 F C-R0.5 1860 MPa steel strand wire 0.5 0.57 112.9 47.7 F C-R1 1.0 1.14 200.3 60.4 F C-R2 2.0 2.28 285.5 52.3 C C-R3 3.0 3.42 299.3 39.9 C C-R4 4.0 4.55 349.8 41.2 C Notes: ρ*—Reinforcement ratio of FRP-UHPC comparative specimens; ρ—Actual reinforcement ratio designed according to the rule of equivalent tensile force; F—Flexural failure (failure after rebar yielding); C—Compression failure (UHPC crushing before rebar yielding). -
[1] AFGC/SETRA. Ultra high performance fibre-reinforced concretes[S]. Bagneux: French Civil Engineering Association, 2013. [2] HABER Z B, DE LA VARGA I, GRAYBEAL B A, et al. Properties and behavior of UHPC-class materials, FHWA-HRT-18-036[R]. Virginia: Federal Highway Administration, 2018. [3] 彭飞, 薛伟辰. 基于可靠度的GFRP筋混凝土梁抗弯承载力设计方法[J]. 土木工程学报, 2018, 51(5):64-71.PENG Fei, XUE Weichen. Reliability-based design method for ultimate load-bearing capacity of GFRP reinforced concrete beams under flexure[J]. China Civil Engineering Journal,2018,51(5):64-71(in Chinese). [4] YANG I H, JOH C, KIM B S. Structural behavior of ultra high performance concrete beams subjected to bending[J]. Engineering Structures,2010,32(11):3478-3487. doi: 10.1016/j.engstruct.2010.07.017 [5] ACI Committee 440. Guide for the design and construction of concrete reinforced with FRP bars: ACI 440.1 R-15[S]. Michigan: American Concrete Institute, 2015. [6] 中华人民共和国住房和城乡建设部, 中华人民共和国国家质量监督检验检疫总局. 纤维增强复合材料建设工程应用技术规范: GB/T 50608—2010[S]. 北京: 中国计划出版社, 2010.Ministry of Housing and Urban-Rural Development of the People's Republic of China, General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China. Technical code for infrastructure application of FRP composites: GB/T 50608—2010[S]. Beijing: China Planning Press, 2010(in Chinese). [7] 中华人民共和国住房和城乡建设部. 纤维增强复合材料筋混凝土桥梁技术标准: CJJ/T 280—2018[S]. 北京: 建筑工业出版社, 2018.Ministry of Housing and Urban-Rural Development of the People’s Republic of China. Technical standard for concrete bridge with FRP reinforcements: CJJ/T 280—2018[S]. Beijing: China Architecture & Building Press, 2018(in Chinese). [8] AFNOR. National addition to Eurocode 2–Design of concrete structures: Specific rules for ultra-high performance fiber-reinforced concrete (UHPFRC): NF P 18-710[S]. Paris: Association Française de Normalisation, 2016. [9] MCS-EPFL. Recommendation: Ultra-high performance fibre reinforced cement-based composites (UHPFRC) construction material, dimensioning und application[S]. Zurich: MCS-EPFL, 2016. [10] AALETI S, PETERSEN B, SRITHARAN S. Design guide for precast ultra-high-performance concrete waffle deck panel system, including connections: FHWA-HIF-13-032[R]. Virginia: Federal Highway Administration, 2013. [11] 湖南省住房和城乡建设厅. 湖南省活性粉末混凝土结构技术规程: DBJ43/T 325—2017[S]. 北京: 中国建筑工业出版社, 2017.Ministry of Housing and Urban-Rural Development of the Hunan Province. Technical standard for concrete bridge with FRP reinforcements: DBJ43/T 325—2017[S]. Beijing: China Architecture & Building Press, 2017(in Chinese). [12] 邓宗才, 王义超, 肖锐, 等. 高强钢筋 UHPC 梁抗弯性能试验研究与理论分析[J]. 应用基础与工程科学学报, 2015, 23(1):68-78.DENG Zongcai, WANG Yichao, XIAO Rui, et al. Flexural test and theoretical analysis of UHPC beams with high strength rebars[J]. Journal of Basic Science and Engineering,2015,23(1):68-78(in Chinese). [13] 郑文忠, 卢姗姗, 李莉. GFRP筋活性粉末混凝土梁受力性能试验研究[J]. 建筑结构学报, 2011, 32(6):115-124.ZHENG Wenzhong, LU Shanshan, LI Li. Experimental research on mechanical performance of reactive powder concrete beams reinforced with GFRP bars[J]. Journal of Building Structures,2011,32(6):115-124(in Chinese). [14] FERRIER E, MICHEL L, ZUBER B, et al. Mechanical behaviour of ultra-high-performance short-fibre-reinforced concrete beams with internal fibre reinforced polymer bars[J]. Composites Part B: Engineering,2015,68:246-258. doi: 10.1016/j.compositesb.2014.08.001 [15] YOO D Y, BANTHIA N, YOON Y S. Flexural behavior of ultra-high-performance fiber-reinforced concrete beams reinforced with GFRP and steel rebars[J]. Engineering Structures,2016,111:246-262. doi: 10.1016/j.engstruct.2015.12.003 [16] YOO D Y, BANTHIA N, YOON Y S. Predicting service deflection of ultra-high-performance fiber-reinforced concrete beams reinforced with GFRP bars[J]. Composites Part B: Engineering,2016,99:381-397. doi: 10.1016/j.compositesb.2016.06.013 [17] SINGH M, ALI M S M, SHEIKH A, et al. Structural behaviour of ultra high performance fibre reinforced concrete beams with steel and polymer bar reinforcement[C]//Proceedings of the 11th Fib International PhD Symposium in Civil Engineering. Tokyo: University of Tokyo, 2016: 287-294. [18] YOKOTA H, ROKUGO K, SAKATA N. Recommendations for design and construction of high performance fiber reinforced cement composite with multiple fine cracks[S]. Tokyo: Japan Society of Civil Engineers, 2008. [19] SINGH M, SHEIKH A, ALI M M, et al. Experimental and numerical study of the flexural behaviour of ultra-high performance fibre reinforced concrete beams[J]. Construction and Building Materials,2017,138:12-25. doi: 10.1016/j.conbuildmat.2017.02.002 [20] GRAYBEAL B, DAVIS M. Cylinder or cube: Strength testing of 80 to 200 MPa (11.6 to 29 KSI) ultra-high-performance fiber-reinforced concrete[J]. ACI Materials Journal,2008,105(6):603-609. [21] 郭晓宇, 亢景付, 朱劲松. 超高性能混凝土单轴受压本构关系[J]. 东南大学学报(自然科学版), 2017, 47(2):170-177.GUO Xiaoyu, KANG Jingfu, ZHU Jinsong. Constitutive relationship of ultrahigh performance concrete under uni-axial compression[J]. Journal of Southeast University (Natural Science Edition),2017,47(2):170-177(in Chinese). [22] RUSSELL H G, GRAYBEAL B A, RUSSELL H G. Ultra-high performance concrete: A state-of-the-art report for the bridge community: FHWA-HRT-13-060[R]. McLean: Federal Highway Administration, 2013. [23] BIRTEL V, MARK P. Parameterised finite element modelling of RC beam shear failure[C]//Proceedings of the 19th Annual International ABAQUS Users’ Conference. Massachusetts: In ABAQUS Users’ Conference, 2006: 95-108. [24] 单波. 活性粉末混凝土基本力学性能的试验与研究[D]. 长沙: 湖南大学, 2002.SHAN Bo. Experiment and research on basic mechanical properties of reactive powder concrete[D]. Changsha: Hunan University, 2002(in Chinese). [25] 马亚峰. 活性粉末混凝土(RPC200)单轴受压本构关系研究[D]. 北京: 北京交通大学, 2006.MA Yafeng. Study on constitutive relationship of 200 MPa reactive powder concrete under uni-axial compression[D]. Beijing: Beijing Jiaotong University, 2006(in Chinese). [26] 闫光杰. 200 MPa级活性粉末混凝土(RPC200)的破坏准则与本构关系研究[D]. 北京: 北京交通大学, 2005.YAN Guangjie. Study on failure criterion and constitutive relationship of 200 MPa reactive powder concrete[D]. Beijing: Beijing Jiaotong University, 2005(in Chinese).