混杂配筋(钢和FRP筋)梁正截面受弯设计方法研究

杨洋; 潘登; 吴刚; 曹大富; 陆伟刚

doi:10.6052/j.issn.1000-4750.2020.09.0703

混杂配筋(钢和FRP筋)梁正截面受弯设计方法研究

doi: 10.6052/j.issn.1000-4750.2020.09.0703

杨洋^1,,
潘登^1,,
吴刚^2, ,,
曹大富^1,,
陆伟刚^3,

1.
扬州大学建筑科学与工程学院，江苏，扬州 225127
2.
东南大学土木工程学院，江苏，南京 210096
3.
扬州大学水利科学与工程学院，江苏，扬州 225127

基金项目: 江苏省自然科学基金青年基金项目(BK20180931)；国家自然科学基金青年基金项目(51908486)；中国博士后基金项目(2020M671620)；江苏省博士后科研资助计划项目(2020Z290)

详细信息

作者简介:
杨　洋(1985−)，男，江苏盐城人，讲师，博士，硕导，主要从事结构工程研究 (E-mail: y.yang@yzu.edu.cn)

潘　登(1993−)，男，江苏宿迁人，硕士，主要从事结构工程研究 (E-mail: 360605456@qq.com)

曹大富(1964−)，男，江苏扬州人，教授，博士，博导，主要从事工程结构加固与改造研究 (E-mail: dfcao@yzu.edu.cn)

陆伟刚(1964−)，男，江苏苏州人，教授，博士，博导，主要从事水工结构及水力学研究 (E-mail: wglu@yzu.edu.cn)

通讯作者:
吴　刚(1976−)，男，浙江安阳人，教授，博士，博导，主要从事结构工程研究 (E-mail: g.wu@seu.edu.cn)

中图分类号: TU375.1
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- 被引次数: 0
出版历程
- 收稿日期: 2020-09-29
- 修回日期: 2021-04-07
- 网络出版日期: 2021-04-22
- 刊出日期: 2021-09-13

DESIGN METHOD FOR NORMAL SECTION OF HYBRID REINFORCED (STEEL AND FRP BARS) BEAMS UNDER BENDING

YANG Yang^1
,,
PAN Deng^1
,,
WU Gang^{2
, ,},
CAO Da-fu^1
,,
LU Wei-gang^3
,

1.
College of Civil Science and Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, China
2.
School of Civil Engineering, Southeast University, Nanjing, Jiangsu 210096, China
3.
College of Hydraulic Science and Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, China

摘要

摘要: 钢和纤维复合筋混杂增强混凝土(hybrid reinforced concrete，Hybrid-RC)梁具有较好的承载力、耐久性和延性。然而，基于文献调研发现，现有设计方法对Hybrid-RC梁正截面受弯的破坏形态还未能做出准确的预测。因此，该文修正了现有的设计方法，并通过数据库对比，验证了其有效性。在此基础上，对Hybrid-RC梁提出了一种新的设计思路，以减少使用成本，并得到更高的使用荷载。
- 混杂配筋(Hybrid-RC)梁 /
- 破坏形态 /
- 抗弯承载力 /
- 设计程序 /
- 数值模拟
Abstract: The hybrid reinforced concrete (Hybrid-RC) beam consisting steel bars and fiber-reinforced polymer bars can provide better bearing capacity, durability and ductility. However, literature review showed that the design methods for the normal section of Hybrid-RC beams under bending failed to predict the failure mode of these structures accurately. Therefore, it revises the existing design method, and verifies its effectiveness through database comparison. On this basis, a new design idea of Hybrid-RC beam is proposed to reduce the service cost and achieve higher service load.
- hybrid reinforced concrete (Hybrid-RC) beam /
- failure mode /
- flexural capacity /
- design procedure /
- numerical simulation

HTML全文

图 1 Hybrid-RC梁受弯破坏形态及应变分布

Figure 1. The failure mode and strain distribution of the Hybrid-RC beam under bending

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图 2 等效矩形应力图

Figure 2. Equivalent rectangular stress diagram

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图 3 程序设计流程图

Figure 3. Design procedure

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图 4 ABAQUS有限元模型

Figure 4. ABAQUS finite element model

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图 5 混凝土的本构关系

Figure 5. Constitutive relation of concrete

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图 6 钢筋和FRP筋的应力-应变关系

Figure 6. Stress-strain relationship of the steel bar and FRP

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图 7 试验梁与ABAQUS模拟梁的荷载-挠度曲线

Figure 7. Load-deflection curves of test beams and ABAQUS simulated beams

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图 8 ABAQUS模拟梁的荷载-挠度曲线

Figure 8. Load-deflection curves of the ABAQUS simulated beams

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表 1 文献中混杂配筋梁的截面参数

Table 1. Cross section parameters of hybrid-RC beams in the literature

文献	梁号	b/mm	d/mm	$f_{\rm{c}}' /{\rm{MPa}}$	A_f/mm²	A_s/mm²	f_y/MPa	f_fu/MPa	E_f/GPa	破坏形态
Qu等^[20]	B3	180	206	26.48	253.23	226.08	363.0	782	45.0	A
	B4	180	204	26.48	396.91	200.96	336.0	755	41.0	A
	B5	180	206	27.52	141.69	401.92	336.0	778	37.7	A
	B6	180	204	27.52	253.23	401.92	336.0	782	45.0	A
	B7	180	208	32.52	141.69	113.04	363.0	778	37.7	A
	B8	180	197	32.52	396.91	1205.76	336.0	755	41.0	A
Aiello和Ombres^[12]	A1	150	175	45.70	88.31	100.48	465.0	1674	49.0	A
	A2	150	175	45.70	157.00	100.48	465.0	1366	50.1	A
	A3	150	175	45.70	235.50	226.08	465.0	1366	50.1	A
	C1	150	175	45.70	88.31	100.48	465.0	1674	49.0	A
LauDenvid^[22]	G10T07	280	348	39.80	981.70	628.30	597.0	582	38.0	A
LauDenvid^[22]	G06T1	280	351	44.60	567.10	981.70	550.0	588	39.5	A
Safan^[26]	B10/6S	100	136	30.00	56.60	157.00	530.0	780	41.0	A
	B10/8S	100	136	30.00	100.60	157.00	530.0	755	39.0	A
	B12/6S	100	135	30.00	56.60	226.00	470.0	780	41.0	A
	B12/8S	100	135	30.00	100.60	226.00	470.0	755	39.0	A
Leung 和 Balendran^[17]	L2	150	145	28.50	142.60	157.00	460.0	760	40.8	A
	L5	150	145	28.50	213.90	157.00	460.0	760	40.8	A
	H2	150	145	48.80	142.60	157.00	460.0	760	40.8	A
	H5	150	145	48.80	213.90	157.00	460.0	760	40.8	A
Al-Mahmoud^[27]	SC-12	150	254	35.10	113.00	226.00	600.0	1875	146.0	A
LauDenvid^[22]	G03MD1	280	351	41.30	283.50	981.70	336.0	588	39.5	B
Almusallam等^[28]	RW1F	150	170	36.60	78.50	157.00	408.0	743	40.0	B
Rahimi 和 Hutchinson^[29]	RHB3	200	135	52.30	66.00	157.00	575.0	1532	127.0	B
Tan^[16]	B2	200	375	28.24	25.12	150.70	324.4	1776	52.0	B
Nguyen等^[30]	B2	120	140	35.68	96.00	628.30	466.0	3140	181.0	C
Xiong等^[31]	CF1	120	130	15.68	8.11	226.20	569.0	3652	252.0	C
Xiong^[32]	CF3	80	110	15.68	5.11	226.20	346.7	3652	252.0	C
注：A表示钢筋屈服后混凝土压溃；B表示钢筋屈服后FRP断裂；C表示的破坏形态为混凝土压溃，钢筋未屈服(但实际出现剥离破坏)。

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表 2 文献中混杂配筋梁试验值和理论值的对比

Table 2. Comparison between experimental and theoretical values of the tested hybrid-RC beams

梁号	Qu^[20]			Lau^[22]			提出的等刚度配筋			提出的等强度配筋			M_exp/ (kN·m)	M_s,th/ (kN·m)	M_f,th/ (kN·m)	M_exp/M_th
梁号	$\rho_{\rm{eff}}^1 $/ (%)	$\rho_{\rm{eff,b}}^1 $/ (%)	PFM	$\rho_{\rm{eff}}^2 $/ (%)	$\rho_{\rm{eff,b}}^2 $/ (%)	PFM	$\rho_{\rm{s}}^{\rm{e}} $/ (%)	($\rho_{{\rm{s,b}}{\text{-}}{\text{Ⅰ}} }^{\rm{e}} $, $\rho_{{\rm{s,b}}{\text{-}}{\text{Ⅱ}} }^{\rm{e}} $)/ (%)	PFM	$\rho_{\rm{f}}^{\rm{e}} $/ (%)	($\rho_{{\rm{f,b}}{\text{-}}{\text{Ⅰ}} }^{\rm{e}} $, $\rho_{{\rm{f,b}}{\text{-}}{\text{Ⅱ}} }^{\rm{e}} $)/ (%)	PFM	M_exp/ (kN·m)	M_s,th/ (kN·m)	M_f,th/ (kN·m)	M_exp/M_th
B3	0.76	3.57	A	0.97	0.44	A	0.76	(0.35,3.57)	A	0.97	(0.44,4.51)	A	38.38	35.59	35.59	1.08
B4	0.77	3.95	A	1.32	0.43	A	0.77	(0.25,3.95)	A	1.32	(0.43,6.80)	A	39.66	37.59	37.59	1.06
B5	1.16	4.10	A	0.85	0.40	A	1.16	(0.54,4.10)	A	0.85	(0.40,3.02)	A	36.36	35.25	35.25	1.03
B6	1.25	4.10	A	1.16	0.46	A	1.25	(0.49,4.10)	A	1.16	(0.46,3.82)	A	42.57	40.56	40.56	1.05
B7	0.37	4.21	A	0.52	0.45	A	0.37	(0.32,4.21)	A	0.52	(0.45,5.86)	A	23.55	27.48	27.48	0.86
B8	3.63	4.66	A	2.63	0.51	A	3.63	(0.71,4.66)	A	2.63	(0.51,3.38)	A	63.30	68.60	68.60	0.92
A1	0.47	3.75	A	0.44	0.17	A	0.47	(0.18,3.75)	A	0.44	(0.17,3.57)	A	25.14	20.74	20.74	1.21
A2	0.53	3.75	A	0.73	0.25	A	0.53	(0.18,3.75)	A	0.73	(0.25,5.13)	A	28.41	25.38	25.38	1.12
A3	1.09	3.75	A	1.19	0.25	A	1.09	(0.23,3.75)	A	1.19	(0.25,4.11)	A	35.55	33.59	33.59	1.06
C1	0.47	3.75	A	0.44	0.17	A	0.47	(0.18,3.75)	A	0.44	(0.17,3.57)	A	25.14	20.74	20.74	1.21
G10T07	0.85	2.38	A	1.67	0.88	A	0.85	(0.45,2.38)	A	1.67	(0.88,4.68)	A	261	224.86	224.86	1.16
G06T1	1.11	2.90	A	1.51	0.96	A	1.11	(0.71,2.90)	A	1.51	(0.96,3.93)	A	229	239.56	239.56	0.96
B10/6S	1.24	2.37	A	1.20	0.45	A	1.24	(0.47,2.37)	A	1.20	(0.45,2.29)	A	14.09	11.64	11.64	1.21
B10/8S	1.3	2.37	A	1.55	0.46	A	1.30	(0.39,2.37)	A	1.55	(0.46,2.83)	A	14.41	12.42	12.42	1.16
B12/6S	1.76	2.80	A	1.43	0.45	A	1.76	(0.56,2.80)	A	1.43	(0.45,2.27)	A	14.89	13.27	13.27	1.12
B12/8S	1.82	2.80	A	1.79	0.46	A	1.82	(0.47,2.80)	A	1.79	(0.46,2.75)	A	16.33	13.85	13.85	1.18
L2	0.86	2.78	A	1.09	0.46	A	0.86	(0.36,2.78)	A	1.09	(0.46,3.55)	A	22.23	16.25	16.25	1.37
L5	0.92	2.78	A	1.42	0.46	A	0.92	(0.30,2.78)	A	1.42	(0.46,4.28)	A	23.07	17.89	17.89	1.29
H2	0.86	3.94	A	1.09	0.65	A	0.86	(0.51,3.94)	A	1.09	(0.65,5.03)	A	21.11	19.45	19.45	1.09
H5	0.92	3.94	A	1.42	0.65	A	0.92	(0.42,3.94)	A	1.42	(0.65,6.07)	A	27.06	21.82	21.82	1.24
SC-12	0.80	2.27	A	0.49	0.29	A	0.80	(0.48,2.27)	A	0.49	(0.29,1.38)	A	65.4	61.96	61.96	1.06
G03MD1	1.06	5.30	Fail	0.86	0.92	B	1.06	(1.13,5.30)	B	0.86	(0.92,4.29)	B	147.7	164.66	164.66	0.90
RW1F	0.68	3.91	A	0.65	0.56	A	0.68	(0.59,3.91)	A	0.65	(0.56,3.73)	A	21.84	18.32	18.32	1.19
RHB3	0.73	3.04	Fail	0.46	0.47	B	0.73	(0.74,3.04)	B	0.46	(0.47,1.93)	B	20.7	23.92	23.92	0.87
B2	0.21	4.40	Fail	0.07	0.11	B	0.21	(0.34,4.40)	B	0.07	(0.11,1.47)	B	35.25	40.61	40.61	0.87
B2	4.26	3.21	C	1.13	0.14	Fail	4.26	(0.52,3.21)	C	1.13	(0.14,0.85)	C	28.6	30.33	30.33	0.94
CF1	1.52	1.14	C	0.28	0.06	Fail	1.52	(0.35,1. 14)	C	0.28	(0.06,0.21)	C	15.42	11.68	11.68	1.32
CF3	2.64	2.24	C	0.3	0.06	Fail	2.64	(0.56,2.24)	C	0.3	(0.06,0.26)	C	6.09	5.79	5.79	1.05
注：PFM表示预测的破坏形态；A表示钢筋屈服后混凝土压溃；B表示钢筋屈服后FRP断裂；C表示为混凝土压溃，钢筋未屈服；Fail表示未出现预测的破坏形态。

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表 3 设计混杂梁的截面参数

Table 3. Cross section parameters of the designed hybrid-RC beams

梁号	b/mm	d/mm	$ f_{\rm{c}}' $/MPa	A_f/mm²	A_s/mm²	f_y/MPa	f_fu/MPa	E_f/GPa	M_th/(kN·m)	A_f /A_s
B4	180	204	26.48	396.91	200.96	336	755	41.0	37.59	1.98
B4-1	180	204	26.48	221.01	452.39	336	755	41.0	39.66	0.49
B4-2	180	204	26.48	163.52	508.94	336	755	41.0	39.66	0.32
B4-3	180	204	26.48	42.19	628.32	336	755	41.0	39.66	0.07
G03MD1	280	351	41.30	283.50	981.70	336	588	39.5	−	−
G03MD1-1	280	351	20.00	74.10	1407.40	336	588	39.5	147.70	0.05
G03MD1-2	280	280	41.30	133.60	1608.50	336	588	39.5	147.70	0.08
G03MD1-3	280	351	41.30	31.20	1256.60	336	1675	49.0	147.70	0.02
B2	120	140	35.68	96.00	628.30	466	3140	181.0	−	−
B2-1	120	140	60.00	30.66	452.39	466	3140	181.0	28.60	0.07
B2-2	120	180	35.68	21.55	339.29	466	3140	181.0	28.60	0.06
B2-3	120	140	35.68	85.10	804.25	300	3140	181.0	28.60	0.11

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表 4 模拟混杂梁的截面参数

Table 4. Cross section parameters of the Abaqus simulated hybrid-RC beams

梁号	b/mm	d/mm	$f_{\rm{c}}' $/MPa	A_f/mm²	A_s/mm²	f_y/MPa	f_fu/MPa	E_f/GPa	A_f /A_s	Δ_u/Δ_y
B3	180	206	26.48	253.23	226.08	363	782	45.0	1.12	5.18
B3-1	180	206	26.48	73.20	508.68	363	782	45.0	0.14	3.11
B6	180	204	27.52	253.23	401.92	336	782	45.0	0.63	4.38
B6-1	180	204	27.52	93.25	628.32	336	782	45.0	0.15	3.07
B7	180	208	32.52	141.69	113.04	363	778	37.7	1.25	6.51
B7-1	180	208	32.52	54.62	200.96	363	778	37.7	0.27	4.93
C1	150	175	45.70	88.31	100.48	465	1674	49.0	0.88	6.35
C1-1	150	175	45.70	49.50	226.19	465	1674	49.0	0.22	4.69

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参考文献(39)

[1]	Ruan X J, Lu C H, Xu K, et al. Flexural behavior and serviceability of concrete beams hybrid-reinforced with GFRP bars and steel bars [J]. Composite Structures, 2020, 235.
[2]	黄华, 郝润奇, 黄敏. FRP筋混凝土结构的研究现状分析[J]. 玻璃钢/复合材料, 2018, 7: 108 − 116. Huang Hua, Hao Runqi, Huang Min. Research status analysis of FRP reinforced concrete structures [J]. Fiber Reinforced Plastics/Composites, 2018, 7: 108 − 116. (in Chinese)
[3]	吴智深, 汪昕, 史健喆. 玄武岩纤维复合材料性能提升及其新型结构[J]. 工程力学, 2020, 37(5): 1 − 14. Wu Zhishen, Wang Xin, Shi Jianzhe. Advancement of basalt fiber-reinforced polymer (BFRPs) and the novel structures reinforced with BFRPs [J]. Engineering Mechanics, 2020, 37(5): 1 − 14. (in Chinese)
[4]	叶列平, 冯鹏. FRP在工程结构中的应用与发展[J]. 土木工程学报, 2006, 39(3): 24 − 36. doi: 10.3321/j.issn:1000-131X.2006.03.004 Ye Lieping, Feng Peng. Applications and development of fiber-reinforced polymer in engineering structures [J]. China Civil Engineering Journal, 2006, 39(3): 24 − 36. (in Chinese) doi: 10.3321/j.issn:1000-131X.2006.03.004
[5]	Sun Z Y, Fu L C, Feng D C, et al. Experimental study on the flexural behavior of concrete beams reinforced with bundled hybrid steel/FRP bars [J]. Engineering Structures, 2019, 197. doi: 10.1016/j.engstruct.2019.109443
[6]	梅葵花, 徐进. FRP简介及其在桥梁工程中的应用综述[J]. 中外公路, 2010, 30(4): 247 − 250. Mei Kuihua, Xu Jin. Introduction of FRP and the application in bridge engineering [J]. Journal of China & Foreign Highway, 2010, 30(4): 247 − 250. (in Chinese)
[7]	ACI 222R-19, Protection of metals in concrete against corrosion [S]. Farmington Hills, MI: American Concrete Institute, 2019.
[8]	徐新生, 纪涛, 郑永峰. FRP筋混凝土梁挠度的特点及计算方法[J]. 工程力学, 2009, 26(增刊 1): 171 − 175. Xu Xinsheng, Ji Tao, Zheng Yongfeng. Characteristics and calculation methods of concrete beams reinforced with FRP bars [J]. Engineering Mechanics, 2009, 26(Suppl 1): 171 − 175. (in Chinese)
[9]	Yail J Kim. Flexural response of concrete beams prestressed with AFRP tendons: Numerical investigation [J]. Journal of Composites for Construction, 2010, 14(6): 647 − 658. doi: 10.1061/(ASCE)CC.1943-5614.0000128
[10]	Gu X Y, Dai Y Q, Jiang J W. Flexural behavior investigation of steel-GFRP hybrid-reinforced concrete beams based on experimental and numerical methods [J]. Engineering Structures, 2020, 206. doi: 10.1016/j.engstruct.2019.110117
[11]	董志强, 吴刚. FRP筋增强混凝土结构耐久性能研究进展[J]. 土木工程学报, 2019, 52(10): 1 − 19, 29. Dong Zhiqiang, Wu Gang. Research progress on durability of FRP bars reinforced concrete structures [J]. China Civil Engineering Journal, 2019, 52(10): 1 − 19, 29. (in Chinese)
[12]	Aiello M A, Ombres L. Structural performances of concrete beams with hybrid (fiber-reinforced polymer-steel) reinforcements [J]. Journal of Composites Construction, 2002, 6(2): 133 − 140. doi: 10.1061/(ASCE)1090-0268(2002)6:2(133)
[13]	Ilker Fatih Kara, Ashraf F Ashour, Mehmet Alpaslan Koroglu. Flexural behavior of hybrid FRP/steel reinforced concrete beams [J]. Composite Structures, 2015, 129: 111 − 121. doi: 10.1016/j.compstruct.2015.03.073
[14]	朱海堂, 程晟钊, 高丹盈, 等. 玄武岩纤维增强聚合物筋钢纤维高强混凝土梁受弯试验及裂缝宽度计算方法研究[J]. 建筑结构学报, 2020, 41(6): 133 − 142. Zhu Haitang, Cheng Shengzhao, Gao Danying, et al. Bending experimental study and crack width calculation method of high-strength concrete beams reinforced with BFRP bars and steel fiber [J]. Journal of Building Structures, 2020, 41(6): 133 − 142. (in Chinese)
[15]	Goldston M, Remennikov A, Neaz Sheikh M. Experimental investigation of the behaviour of concrete beams reinforced with GFRP bars under static and impact loading [J]. Engineering Structures, 2016, 113: 220 − 232. doi: 10.1016/j.engstruct.2016.01.044
[16]	Tan K H. Behaviour of hybrid FRP-steel reinforced concrete beams [C]. Proceedings of the third international symposium on non-metallic (FRP) reinforcement for concrete structures (FRPRCS-3). Sapporo: Japan Concrete Institute, 1997: 487 − 494.
[17]	Leung H Y, Balendran R V. Flexural behaviour of concrete beams internally reinforced with GFRP rods and steel rebars [J]. Structural Survey, 2003, 21(4): 146 − 157. doi: 10.1108/02630800310507159
[18]	Qin R Y, Zhou A, Lau D. Effect of reinforcement ratio on the flexural performance of hybrid FRP reinforced concrete beams [J]. Composites Part B: Engineering, 2017, 108: 200 − 209. doi: 10.1016/j.compositesb.2016.09.054
[19]	Pang L, Qu W J, Zhu P, et al. Design propositions for hybrid FRP-steel reinforced concrete beams [J]. Journal of Composites for Construction, 2015, 20(4): 04015086.
[20]	Qu W J, Zhang X L, Huang H Q. Flexural behavior of concrete beams reinforced with hybrid (GFRP and steel) bars [J]. Journal of Composites for Construction, 2009, 13(5): 350 − 359. doi: 10.1061/(ASCE)CC.1943-5614.0000035
[21]	ACI 318–19, Building code requirements for structural concrete [S]. Farmington Hills, MI, USA: American Concrete Institute, 2019.
[22]	Lau D, Pam H J. Experimental study of hybrid FRP reinforced concrete beams [J]. Engineering Structures, 2010, 32(12): 3857 − 3865. doi: 10.1016/j.engstruct.2010.08.028
[23]	ACI 440·1R–15, Guide for the design and construction of concrete reinforced with FRP bars [S]. Farmington Hills, MI, USA: American Concrete Institute, 2015.
[24]	王勃, 程思源. FRP-钢筋混杂配筋混凝土梁正截面界限破坏分析[J]. 北方建筑, 2020, 5(1): 17 − 20. Wang Bo, Cheng Siyuan. The boundary failure mode of FRP-steel hybrid reinforced concrete beam [J]. Northern Architecture, 2020, 5(1): 17 − 20. (in Chinese)
[25]	董坤, 郝建文, 李鹏, 等. 环境温差下FRP-混凝土界面粘结行为分析[J]. 工程力学, 2020, 37(11): 117 − 126. doi: 10.6052/j.issn.1000-4750.2019.12.0783 Dong Kun, Hao Jianwen, Li Peng, et al. Studies on the bond performance of FRP-to-concrete interfaces under environmental temperature difference [J]. Engineering Mechanics, 2020, 37(11): 117 − 126. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.12.0783
[26]	Safan M A. Flexural behavior and design of steel-GFRP reinforced concrete beams [J]. ACI Materials Journal, 2013, 110(6): 677 − 685.
[27]	Al-Mahmoud F, Castel A, Francois R, Tourneur C. Strengthening of RC members with near-surface mounted CFRP rods [J]. Composite Structures, 2009, 91: 138 − 147. doi: 10.1016/j.compstruct.2009.04.040
[28]	Almusallam H T, Elsanadedy M H, Al-Salloum Y A, et al. Experimental and numerical investigation for the flexural strengthening of RC beams using near-surface mounted steel or GFRP bars [J]. Construction and Building Materials, 2013, 40: 145 − 161. doi: 10.1016/j.conbuildmat.2012.09.107
[29]	Rahimi H, Hutchinson A. Concrete beams strengthened with externally bonded FRP plates [J]. Journal of Composites for Construction, 2001, 5: 44 − 56. doi: 10.1061/(ASCE)1090-0268(2001)5:1(44)
[30]	Nguyen D M, Chan T K, Cheong H K. Brittle failure and bond development length of CFRP-concrete beams [J]. Journal of Composites for Construction, 2001, 5: 12 − 17. doi: 10.1061/(ASCE)1090-0268(2001)5:1(12)
[31]	Xiong G J, Jiang H, Huang J Z. Behavior of reinforced concrete beams strengthened with externally bonded hybrid carbon fiber–glass fiber sheets [J]. Journal of Composites for Construction, 2004, 8(3): 275 − 278. doi: 10.1061/(ASCE)1090-0268(2004)8:3(275)
[32]	Lu X Z, Teng J G, Ye L P, et al. Intermediate crack debonding in FRP-strengthened RC beams: FE analysis and strength model [J]. Journal of Composites for Construction, 2007, 11(2): 161 − 174. doi: 10.1061/(ASCE)1090-0268(2007)11:2(161)
[33]	Hibbett, Karlsson, Sorensen. ABAQUS/standard: User's manual [M]. Hibbett: Karlsson & Sorensen, 1998.
[34]	庄茁. 基于ABAQUS的有限元分析和应用[M]. 北京: 清华大学出版社, 2009. Zhuang Zhuo. Finite element analysis and application of ABAQUS [M]. Beijing: Tsinghua University Press, 2009. (in Chinese)
[35]	李宝磊, 宋力, 樊成, 等. FRP筋混凝土梁受弯性能影响因素的有限元分析[J]. 水利与建筑工程学报, 2015, 13(3): 100 − 104, 115. doi: 10.3969/j.issn.1672-1144.2015.03.020 Li Baolei, Song Li, Fan Cheng, et al. Finite element analysis of the bending performance influencing factors of FRP reinforced concrete beams [J]. Journal of Water Resources and Architectural Engineering, 2015, 13(3): 100 − 104, 115. (in Chinese) doi: 10.3969/j.issn.1672-1144.2015.03.020
[36]	Hognestad Eivind, Hanson N W, McHenry Douglas. Concrete stress distribution in ultimate strength design [J]. Journal Proceedings, 1955, 12: 455 − 480.
[37]	过镇海. 混凝土的强度和本构关系-原理与应用 [M]. 北京: 中国建筑工业出版社, 2004. Guo Zhenhai. Constitutive relation and strength of concrete -Principle and application [M]. Beijing: China Building Industry Press, 2004. (in Chinese)
[38]	GB 50010−2010, 混凝土结构设计规范 [S]. 北京: 中国建筑工业出版社, 2010. GB 50010−2010, Code for design of concrete structures [S]. Beijing: China Building Industry Press, 2010. (in Chinese)
[39]	过镇海. 钢筋混凝土原理 [M]. 北京: 清华大学出版社, 2013. Guo Zhenhai. Principle of reinforced concrete [M]. Beijing: Tsinghua University Press, 2013. (in Chinese)