 Open Access
Fatigue Assessment Model of Corroded RC Beams Strengthened with Prestressed CFRP Sheets
 Li Song^{1, 2} and
 Jian Hou^{3}Email author
https://doi.org/10.1007/s400690170191x
© The Author(s) 2017
 Received: 10 March 2016
 Accepted: 31 January 2017
 Published: 19 May 2017
Abstract
This paper presents a fatigue assessment model that was developed for corroded reinforced concrete (RC) beams strengthened using prestressed carbon fiberreinforced polymer (CFRP) sheets. The proposed model considers the fatigue properties of the constituent materials as well as the section equilibrium. The model provides a rational approach that can be used to explicitly assess the failure mode, fatigue life, fatigue strength, stiffness, and postfatigue ultimate capacity of corroded beams strengthened with prestressed CFRP. A parametric analysis demonstrated that the controlling factor for the fatigue behavior of the beams is the fatigue behavior of the corroded steel bars. Strengthening with one layer of nonprestressed CFRP sheets restored the fatigue behavior of beams with rebar at a low corrosion degree to the level of the uncorroded beams, while strengthening with 20 and 30%prestressed CFRP sheets restored the fatigue behavior of the beams with medium and high corrosion degrees, respectively, to the values of the uncorroded beams. Under cyclic fatigue loading, the factors for the strengthening design of corroded RC beams fall in the order of stiffness, fatigue life, fatigue strength, and ultimate capacity.
Keywords
 fatigue assessment model
 corrosion
 reinforced concrete beam
 prestressed CFRP sheets
1 Introduction
Infrastructure such as bridges and marine structures are prone to corrosion, and these structures are usually subjected to repeated loading. Corrosion reduces the steel area, worsens the steel properties, weakens the bond between the concrete and steel bars, and results in cracking and spalling of the concrete cover (Bigaud and Ali 2014; Almusallam 2001). Fatigue is a process of progressive and internal structural changes in a material that is subjected to repetitive stresses. For example, the primary girders of bridges are subjected to many traffic loading cycles every day, which can lead to failure due to these repetitive tensile stresses. Although corrosion and fatigue processes have been extensively studied, the coupled effect of these two processes has rarely been studied (BastidasArteaga et al. 2009; AiHammoud et al. 2011; Yi et al. 2010). Several experimental studies have shown that localized corrosion leading to pitting may provide sites for fatigue crack initiation and that corrosive agents increase the fatigue crack growth rate (BastidasArteaga et al. 2009; AiHammoud et al. 2010). The interactive effect of corrosion and fatigue is more damaging than the sum of the damage caused by the components individually. The loss of strength and stiffness can thus be exaggerated if corrosion is combined with fatigue loading (BastidasArteaga et al. 2009; Masoud et al. 2001).
Structural engineers face the challenge of assessing the vulnerability of deteriorated structures and deciding on appropriate strengthening methods. Externally bonded carbon fiberreinforced polymer (CFRP) sheets have been widely used to restore or increase the capacities of reinforced concrete (RC) beams due to their advantages including their low weight, high strength and stiffness, high durability, and ease of application (Kang et al. 2012; Ouezdou et al. 2009; Grelle and Sneed 2013; Ren et al. 2015). Many studies have been conducted to study the effect of CFRP sheets on the performance of RC beams (AlRousan and Issa 2011; Oudah and ElHacha 2013a, b; EISafty et al. 2014; Kang et al. 2014). However, several researchers have demonstrated that applying CFRP sheets as externally bonded strips for flexural strengthening of RC structural elements uses only 20–30% of their strength (Motavalli et al. 2011). Additionally, the serviceability of RC beams is not generally affected by CFRP sheet repair due to the relatively small increase in stiffness provided by CFRP sheets. The strips are better used when the CFRP material is prestressed. Similar to the advantages of conventional prestressed concrete compared to RC, prestressed, externally bonded reinforcement has several advantages over externally bonded reinforcement including reduced crack widths, reduced deflections, reduced stress in the internal steel, and increased fatigue resistance (Michels et al. 2013; Correia et al. 2015). Triantafillou et al. (1992) tested beams strengthened with prestressed CFRP sheets and found that prestressed CFRP sheets significantly contributed to the improved cracking and deflection of the strengthened beams. ElHacha et al. (2001) provided a general summary of the prestressed CFRP sheet application, including feasible prestressing techniques for concrete structures. Kim et al. (2008) tested prestressed RC beams that had been strengthened with prestressed CFRP sheets and found that using prestressed CFRP sheets resulted in less localized damage and that the level of prestress in the sheets significantly contributed to the ductility and cracking behavior of the strengthened beams. Wang et al. (2012) investigated the flexural behavior and longterm prestress losses of RC beams strengthened with posttensioned CFRP sheets and noted that the prestress losses of CFRP sheets in the posttension system could be primarily attributed to the anchorage set, while the timedependent losses caused by creep, concrete shrinkage and relaxation of the CFRP sheets were relatively small. Rosenboom and Rizkalla (2006) investigated the fatigue behavior of prestressed concrete bridge girders strengthened with various CFRP systems using various strengthening levels, prestressing configurations, and fatigue loading ranges. The test results showed that CFRP strengthening could reduce crack width, crack spacing, and the induced stress ratio in the prestressed strands under service loading conditions. Although a sufficient number of studies have investigated the strength, serviceability, prestress losses, and CFRP systems in general, few studies have investigated the fatigue performance of members with prestressed CFRP sheet applications. The fatigue performance of corroded RC beams strengthened with prestressed CFRP sheets has not yet been discussed.
Additionally, the development of a fatigue assessment model (FAM) for corroded members is not an easy task due to the many relevant parameters and variables that determine the fatigue behavior. Several attempts have been made to develop an FAM for corroded beams. The most common approaches are the stresslife and strainlife models (Ma et al. 2014; Elrefai et al. 2012). These models provide the fatigue life of members but do not predict the strain and stress distributions across the beam. Song and Yu (2015) tested beams with corroded steel reinforcement (corroded beams) strengthened with CFRP sheets and proposed an analytical fatigue prediction model (FPM) to assess the fatigue behavior of the CFRPstrengthened corroded beams. In this study, the effect of prestressed CFRP systems on the fatigue behavior of corroded RC beams was not studied. None of the available FAMs are capable of accurately assessing the fatigue behavior of corroded beams strengthened with prestressed CFRP. This paper presents the second phase of a research project conducted to quantify the effect of CFRP systems on the fatigue behavior of corroded RC beams. The first phase of the project (Song and Yu 2015) aimed to examine the effect of steel corrosion and the nonprestressed CFRP systems on the fatigue behavior of corroded RC beams. The results showed that nonprestressed CFRP systems are applicable to RC beams with only low corrosion degrees (0–4.6%). The second phase intends to develop an overall assessment model capable of assessing the fatigue behavior of corroded RC beams strengthened with prestressed CFRP sheets and to discuss whether prestressed CFRP systems are applicable to RC beams with medium and high corrosion degrees.
In this study, an FAM was developed to consider the fatigue behavior of corroded beams with prestressed CFRP sheets. The model provided a rational approach that can be used to explicitly assess the failure mode, fatigue life, fatigue strength, stiffness, and postfatigue ultimate capacity of CFRPstrengthened corroded beams under fatigue loading by considering the influences of different corrosion degrees, load ranges, and prestressed levels.
2 Fatigue Assessment Model
An FAM based on the fatigue properties of the constituent materials and crosssectional stress analysis was proposed to assess the fatigue behavior of corroded beams strengthened with prestressed CFRP sheets. The fatigue properties of the constituent materials of the CFRPstrengthened corroded beams are discussed first, followed by a study of the fatigue bond properties and the crosssectional stress under cyclic loading; finally, the stepbystep procedure used to implement the developed model is presented.
2.1 Fatigue Properties of the Constituent Materials
To assess the fatigue behavior of a composite member, the effects of cyclic loading on each constituent component must first be understood. If any of the constituent components reaches its fatigue limit prior to the required life of the member, the member will not be capable of carrying the required loads. These loading cycles contribute to the continuous deterioration of the constitutive materials. The models of the fatigue deterioration of concrete, corroded steel bars, and CFRP sheets are detailed in the following paragraphs.
2.1.1 Concrete
2.1.2 Corroded Steel Bars
2.1.3 Carbon FiberReinforced Polymers
2.2 Fatigue Bond Properties
2.3 CrossSectional Stress Analysis of the Beam

The compressive strain variation of the concrete zone is linear from the neutral axis to the outer fiber, and the strains and stresses follow Hooke’s law.

The tensile resistance of the concrete is neglected.

The thickness of the CFRP sheets is neglected when the resisting moment is calculated.

Premature delamination failure of CFRP sheets is not considered because the anchorage prevents such a failure from occurring.

To simulate the fatigue behavior of the CFRPstrengthened corroded beams, the analysis is conducted in increments of cycles, and the concrete, steel, and CFRP fatigue properties are updated at the end of each cycle increment.
2.4 Fatigue Analysis Flowchart for the Beam
 (1)
Input the loading scheme, initial material parameters, and specimen configuration, including the fatigue load (\( M^{\text{f}} \)), the initial material parameters (\( A_{\text{s}} \), \( E_{\text{s}} \), \( f_{\text{y0}} \), \( \eta_{\text{s}} \), \( E_{\text{c}}^{{}} \), \( f_{\text{c}} \), \( A_{\text{f}}^{{}} \), \( E_{\text{f}} \), \( f_{\text{f}} \), and \( \varepsilon_{\text{f0}} \)), and the specimen configuration (b, h, and h _{0}).
 (2)
Determine the stress and strain in the concrete, rebar, and CFRP sheets at the end of each group of cycles during the fatigue life of the structure using Eqs. (25)–(28).
 (3)
Assess the fatigue states of the concrete, rebar, and CFRP sheets using Eqs. (5), (17), and (21), respectively.
 (4)
In the subsequent fatigue cycles, update the material properties, including the deformation modulus of the concrete (\( E_{\text{c}}^{\text{f}} \)), the residual crosssectional area of the corroded steel (\( A_{\text{s}}^{\text{f}} \)), and the elastic modulus of the CFRP sheet (\( E_{\text{f}}^{\text{f}} \)) using Eqs. (4), (7), and (19), respectively. Then, return to step (2).
 (5)
End the program, and output the gathered information.
2.5 Validation of the Proposed Analysis Model
The available literature provides only limited test information on the fatigue performance of corroded beams strengthened with prestressed CFRP; additionally, most published studies have presented an insufficient quantity of test results. The proposed FAM is validated by the experimental results of two uncorroded beams that were strengthened with prestressed CFRP sheets and were tested by He et al. (2011). The beams had dimensions of 150 mm × 300 mm × 2000 mm with spans of 1800 mm. The diameter of the tensioned longitudinal rebar was 14 mm, and the average compressive strength and modulus of elasticity of the concrete were 32.5 MPa and 35.6 GPa, respectively. The yield strength of the 14 mm diameter deformed bars was 335 MPa. The mechanical properties of the cured CFRP system were tested and showed an ultimate strength of 3208 MPa and a modulus of elasticity of 234 GPa. A single prestressed layer of CFRP sheet that was 140 mm wide and 0.167 mm thick was used to strengthen the beam in flexure. The initial prestressing level in the CFRP sheets was 59 and 60% of the ultimate strength of the CFRP sheets for Beam 1 and Beam 2, respectively. The maximum and minimum flexural moments in the fatigue load cycles were 21.1 and 6.3 kN m for Beam 1 and 27.3 and 8.2 kN m for Beam 2, respectively.
3 Model Application
3.1 Corroded RC Beams
The fatigue behavior of beams was modeled using the FAM, and the results produced were verified using He’s experimental data. He’s test beams were used to study the effect of the steel corrosion degree, the fatigue load range, and the prestress level on the fatigue life, fatigue strength, stiffness, and postfatigue ultimate capacity. The test beams were assumed to be corroded at a low level (i.e., 0–5% average mass loss), a medium level (i.e., 5–10% average mass loss) and a high level (i.e., 10–20% average mass loss) and were subsequently strengthened with a single layer of a prestressed CFRP sheet. The ACI Committee 440 (2000) recommends that the maximum prestressed tensile stress of CFRP sheets is limited to 0.55 of their ultimate strength to avoid brittle fracture. Previous studies have also shown that the loss of prestress in CFRP sheets averages 10–30% of the control tensile stress (Yang and Li 2010). Thus, in the model, the maximum tensile prestress of the CFRP sheets was limited to 0.55 of the ultimate strength, and the effective prestress levels in the CFRP sheets were 0, 2742, and 4113 με, which were approximately 0, 20, and 30% of the ultimate strain of the CFRP sheets, respectively. The minimum fatigue flexural moment was determined to be 8.2 kN m, and the maximum fatigue flexural moments were 13.4, 14.9, 16.4, and 17.9 kN m, which represented 45.0, 50.0, 55.0, and 60.0% of the ultimate strength of the uncorroded, unstrengthened beam, respectively.
3.2 Fatigue Life
3.3 S–N Curves
3.4 Stiffness Degradation
The stiffness after each fatigue loading cycle was calculated to provide a quantitative measurement of the stiffness degradation in the beams. The calculation method involved recording the strain in the steel rebar \( \varepsilon_{\text{s}} \), the strain at the top fiber of the concrete at the midspan \( \varepsilon_{\text{c}} \), the effective beam depth beam \( h_{0} \), and the predicted curvature radius; the stiffness was then calculated from the curvature radius and the fatigue load for each loading cycle. For each beam, the obtained stiffness values were normalized with respect to the initial stiffness of the uncorroded unstrengthened control beam before fatigue loading was applied.
The change in stiffness is related to the opening and propagation of flexural cracks, which implies a relative slip between the concrete, steel reinforcement, and the CFRP sheet as well as fatigue damage of the constituent materials and tensile failure of the concrete. The prestress in the CFRP sheets is the primary reason for the increase in stiffness because prestressed CFRP sheets confine the tensioned face of the beams, reduce the widening of flexural cracks and delay the damage evolution of the steel bars during fatigue cyclic loading.
3.5 PostFatigue Ultimate Strength
4 Conclusions
 (1)
The controlling factor for the fatigue behavior of the beam is the fatigue behavior of the corroded steel bars. The failures of the unstrengthened beams, the beams strengthened with nonprestressed CFRP sheets and the beams strengthened with prestressed CFRP sheets were caused by rupturing of the steel reinforcement.
 (2)
Using one layer of nonprestressed CFRP sheets to repair beams with rebar at low corrosion degrees increased the beam fatigue life and strength to approximately the values of the uncorroded beams. Using prestressed CFRP sheets restored the beam fatigue life and strength with rebar at medium and high corrosion degrees to the values of the uncorroded beams.
 (3)
Steel rebar corrosion was shown to reduce the initial stiffness and ultimate strength and to significantly increase the stiffness degradation rates and the ultimate strength degradation rates of the structure. Using one layer of nonprestressed CFRP sheets increased the initial stiffness and strength and decreased the stiffness degradation rates and the strength degradation rates; however, the stiffness and strength of the CFRPstrengthened beams after the loading cycles were still less than the stiffness and strength of the uncorroded control beam. Strengthening with 20 and 30%prestressed CFRP sheets restored the stiffness and strength of the beams with rebar at medium and high corrosion degrees, respectively, to the levels of the uncorroded beams.
 (4)
Similar to strengthening with nonprestressed CFRP sheets, the factors for the design of prestressed CFRP strengthening of corroded RC beams fall in the order of stiffness, fatigue life, fatigue strength, and ultimate capacity.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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