Reliability-based Balanced Condition for Strengthened RC Beams Chunxia Li 1, a, Zhisheng Ding 2,b, SHilin Yan 1,c, Junming Chen 1,d

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1 Advanced Materials Research Online: ISSN: , Vols , pp doi: / Trans Tech Publications, Switzerland Reliability-based Balanced Condition for Strengthened RC Beams Chunxia Li 1, a, Zhisheng Ding 2,b, SHilin Yan 1,c, Junming Chen 1,d 1 School of Science, Wuhan University of Technology, Wuhan, Hubei province, PRC 2 Wuhan Gongda Reinforcement Engineering Co., Ltd, Wuhan, Hubei province, PRC a dingli95@126.com, b dingzhsh@126.com, c yanshl@whut.edu.cn, d jmchen1@whut.edu.cn Keywords: balanced condition, reliability indicator, flexure capability, statistical characteristic Abstract. Based on the experimental result of the flexure capability of reinforced concrete beams strengthened by carbon fiber sheets, the stress distribution changes only after steel yielding and carbon fiber sheets function better. However serious the extent of the damage is before strengthened, the tensile strain of main steel reaches about 1.6 times of the yield strain for the secondary grade of steel as failure happens. To satisfy the object reliability indicator, reliability is analyzed using the ratio of the steel strain at the balanced failure to the yield strain as variable to obtain its optimum, which is coincide with the experimental result, and maes better consistency between calculated reliability indicator and object reliability indicator. Introduction As the engineering applications of reinforced concrete beams strengthened by carbon fiber sheets (CFS of abbreviation in the following) develop increasingly, the reliability researches were made primarily on the statistical characteristics of mechanical performance of CFS and those of resistance capability, as well as their influencing factors, and reliability calibration of flexure capability [1,2], but no report about the balanced condition. To prevent failure caused by over strengthened, the ratio of the equivalent depth of compression zone at the balanced failure to the depth of the cross section is limited in current code [3]. Researchers generally consider the balanced failure occurs that the steels yield as the maximum concrete strain reaches ultimate compressive strain [4]. Then the stress and the strain of CFS will be lower because of the hysteretic strain, and the balanced beams are not generally economical. The balanced condition should be established based on reliability. Experiment for flexural capability of CFS- strengthened RC beams 8 rectangular RC beams numbered from L1 to L8 are mm mm in cross section. The beams are simply supported on a span of 1900mm. The grade of concrete is C20. The beams are subjected to two concentrated loads with the shear span of 650mm symmetrically. The position of strain and deformation test is shown in Fig.1. The load-carrying condition before strengthened and the reinforcement are shown in Table 1, where beam of L1 is used as contrast, which is loaded until flexure failure occurs with the failure load of P u. Table 1 Loading-carrying condition before strengthened type NO. reinforcement loading-carrying failed beam L1 main steel: completely unloading beams partly unloading beams L2 L3 L4 L5 L6 L7 no-unloading beam L8, HRB335 stirrups: A6@150(L2~L5), A6@100(L6~L8), HPB235 (only in shear span) tie steel: 2A6, HPB235 (only in shear span) 0 P u 0 80%P u 0 0 %P u %P u %P u 0 20%P u %P u %P u 0 %P u 40%P u 0 40%P u Fig.1 Experiment beam photo All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Par, USA-05/03/16,02:48:31)

2 26 Information Technology Applications in Industry, Computer Engineering and Materials Science All the other beams except L1 are strengthened by a layer of carbon fiber sheets (CFS--300) at the bottom of the cross section, and U-type CFS with the wide of 100mm wraps round two sides and the bottom of the cross section in the shear span of the beam to prevent peeling failure at the end of the beam, as shown in Fig.2. (a) L2~L5 (b) L6~L8 Ⅱ U-type CFS 400 Ⅱ CFS Ⅲ 50 Ⅲ Fig.3 shows the relationship curve of load-strain during loading again for strengthened beams. It is clear that the more residual strain in tensile steels before strengthened, the more hysteretic strain in tensile CFS after strengthened. At moderate loads, the strain of CFS increases with the almost same speed as that of tensile steel. With further load increase, the strain of CFS increases much faster than that of tensile steel obviously, especially after steel yielding. So the stress distribution transformation taes place, and the increasing stress is transmitted to the tensile CFS, thus CFS will function more perfectly. However serious the extent of the damage is before strengthened, the tensile strain of main steel reaches about from 2500µε to 3000µε as the failure happens after strengthened, which is approximately equal to 1.6 times of the yield strain (ε y ) of the steel for the secondary grade of the steel. So the tensile strain in the steel (ε s ) is taen as 1.6 times of ε y for ductile balanced failure. - - Fig.2 Strengthened beam diagram 2A6 Ⅱ-Ⅱ 2A6 Ⅲ-Ⅲ A6@150 A6@ 钢筋应变 steel CFRP CFS 应变 (a) L3 100 (b) L5 20 加固前钢筋应变 steel before strengthened 系列 steel after 2 strengthened 系列 CFS 3 20 (c) L6 Fig.3 Load-strain curve (d) L8 Statistical analysis of flexure capability by Monte-Carlo According to the design formula of flexure capability for strengthened beams in the current code [3], the factors influencing the flexure capability are material strength, cross section, steel area, area of CFS as well as the hysteretic strain. In order to coincide with the engineering practice, the design variables are chosen as shown in Table 2 [5].

3 Advanced Materials Research Vols variable Table 2 Design variables for statistical analysis of flexure capability δ δ δ distribution function f c [Mpa] normal f y [Mpa] normal f f [Mpa] normal b[mm] normal h[mm] normal h 0 [mm] normal ρ s 0.5% % % normal ρ f ρ fb ρ fb ρ fb normal ε f % % logarithmic- normal where and δ are average and variation coefficient of the ratio of variable to its respectively, f c is the compressive strength of concrete, f y is the yield strength of steel, f f is the ultimate tensile strength of CFS, b and h are the width and depth of cross section respectively, h 0 is effective depth, ρ s is the reinforcement ratio, ρ f is a ratio of the area of CFS to the area of section, ρ fb is the ratio of CFS that would produce the balanced failure, ε f0 is the hysteretic strain. Giving a certain ratio of ε s to ε y for the balanced failure, 324 simulated members would be obtained, and the statistical analysis is made on random samples of for every simulated member by Monte-Carlo method [6], the statistical characteristics of flexure capability for strengthened beams would be acquired using statistical toolbox [7], with the ratio of ε s to ε y varying from 1.0 to 2.2, as shown in Table3. Table 3 Statistical characteristics of flexure capability at balanced failure ε s /ε y R δ R distribution function logarithmic- normal where R and δ R are average and variation coefficient of resistance capability at the balanced failure. Reliability-based balanced condition At the balanced failure, the reliability indicator (β) would vary with the ratio of ε s to ε y. So the of β would be calculated when giving a certain ratio of ε s to ε y, and then compared with the object reliability indicator (β T ) [8]. The ratio of ε s to ε y would be achieved to mae β matching β T much better when the of H is the minimum by the following equation: H = ω ( β β ) 2 i i T (1) where ω i is the weighting coefficient taen as the same. For the secondary structures, considering the simple load combination such as dead load with live load of office building or with wind load, giving the ratio of live load or wind load to dead load varying from 0.1 to 2.0 with the dispersion of 0.1, the of β with a certain ratio of ε s to ε y is calculated by JC method [6], and then the curve of H varying with the ratio of ε s to ε y by Eq.1 is shown in Fig.4. For the combination the dead load with the live load of office buildings, it is clear that the of H reaches lower as the ratio of ε s to ε y at For the combination the dead load with the wind load, the of H fluctuates with the increasing of the ratio of ε s to ε y, and troughs occurs as the ratio of ε s to ε y at 1.2, 1.6, 2.0 reapectively in the dashed curve. So the of H reaches the minimum as the ratio of ε s to ε y equal to 1.6 in both aboved combinations. Taing ε s /ε y as 1.6 for the balanced failure is coincide with the experiment, and maes better consistency between calculated reliability indicator and object reliability indicator.

4 28 Information Technology Applications in Industry, Computer Engineering and Materials Science H ε s /ε y Fig.4 H-ε s /ε y curve at balanced failure dead load with live load 恒载 + 活载 ( 办公楼 ) (office building) 恒载 + 风载 dead load with wind load Conclusion The experimental result of flexure capability of RC beams strengthened by CFS in the different load-carrying conditions shows that the stress distribution changes only after steel yielding. The increasing stress is transmitted to the tensile CFS which will mae it woring perfectly. However serious the extent of the damage is before strengthened, the tensile strain of main steel reaches about 1.6 times of the yield strain of the secondary grade as failure happens. The reliability analysis on the ratio of the steel strain at the balanced failure to the yield strain is made to obtain its optimum, which is coincide with the experiment, and maes better consistency between calculated reliability indicator and object reliability indicator. Acnowledgment This research is supported by National Science foundation of PRC ( ). References [1] Yongxin Yang. Reliability analysis of RC concrete Beams Strengthened with CFRP Sheets [J]. Journal of Building Structures, pp.88-91, (8 S1) (in Chinese). [2] Shihua He. Discussion on mechanical properties of index for carbon fiber sheet [J]. Industry Construction, Vol. 42, pp , (2012) (in Chinese). [3] National Standard of PRC. Code for design of Strengthened concrete structures (GB ) [S]. Beijing: Building industry Press, 6 (in Chinese). [4] Kongguo Hu, Xiaobin Chen, Qingrui Yue et al. Calculating method for bending strength of concrete bending member strengthened with CFRP considering the secondary load [J]. Building Structure, Vol. 31, pp.63-65, (1) (in Chinese). [5] Chunxia Li. Analysis on reliability and flexural capacity of concrete beams strengthened by CFRP under secondary loading [D]. Wuhan: Wuhan University of Technology, (2012) (in Chinese). [6] Jinxin Gong. Calculation methods for reliability of engineering structures [M]. Dalian: Dalian University of Technology Press, 3 (in Chinese). [7] Zhiyong Zhang. Mastering Matlab (6.5 edition) [M]. Beijing: Beijing University of Aeronautics and Astronautics Press, 2010 (in Chinese). [8] National Standard of PRC. United of design for reliability of building structures (GB ) [S]. Beijing: Building industry Press, 1 (in Chinese).

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