This study is to develop simplified reliability estimation for optimum strengthening ratio of Tbeam railway bridge strengthened by CFRP strip. Until now, strengthening design has been usually proceeded to satisfy the target loadcarrying capacity by using the deterministic parameter of nominal property for concrete or FRP. For the optimum strengthening design, however, it is required that reliabilitybased strengthening design should be applied to effectively determine the amount of strengthening material and make sure of the safety of the structure. As applying the reliabilitybased strengthening ratio, more reliable strengthening design using CFRP strip is possible as well as having a structural redundancy. The reliabilitybased strengthening design methodology suggested in this study is able to contribute the optimum strengthening design for a concrete structure strengthened by CFRP strip.
FHWA 2009 [
Also NSM using FRP strips has substantially better bond resistance because it can be embedded in an adhesive entirely within the narrow groove of the substrate concrete. This advantage of FRP strips for bonding performance was previously demonstrated by Soliman et al. [
Although many studies for FRP strengthening have been conducted previously, there are few studies to suggest the strengthening guideline, such as a strengthening ratio, by applying a probabilistic and reliability analysis. This is important because of the uncertainty for material and structural point of view on the FRP strengthening method. A parametric study of GFRP rebar for design factors such as crosssectional dimension, GFRP, and concrete strength [
This study aims to propose the reliabilitybased determination procedure on the strengthening ratio of a deteriorated concrete girder with CFRP strips, which has advantages for the full composite performance with concrete members. The target bridge in this study is a double Tbeam railway bridge originally designed by LS18 (an old type of design load for a railway bridge in Korea). Therefore, it is required that the target bridge should satisfy the present design load (LS22) and enhance the design speed, with the high speed era in the future. In order to assess the optimal strengthening ratio for the target bridge in this study, the CFRP strip strengthening method was analytically applied to the target bridge.
NSM strengthening technique is more effective for enhancing flexural capacity of railway bridge in case vibration of train traffic due to its superior bond performance. The goal of this study was to calculate the reliabilitybased strengthening ratio of the concrete beams strengthened by NSM using CFRP strips by applying the reliability index for bridge design. FE analysis, on the deteriorated and then strengthened bridge, was performed using the design railway load in the Korea railway specification. FEM analysis was also used to estimate the amount of steel reinforcements of the target bridge, due to the absence of structural design information of this aged bridge. To consider the structural uncertainties of the strengthening method, the probability and reliability analysis were performed with Monte Carlo simulation (MCS). Finally, the reliabilitybased strengthening ratio which satisfies the reliability index for the structural design (
The target bridge is a simply supported railway bridge in Korea, which was built for the design load of LS18 in 1982. In Design Specification of Railway of Korea [
Standard train load in Korea (LS18, LS22).
Figures
A crosssection of the target bridge (unit: m).
A longitudinal view of the target bridge.
There are some recent diagnosis techniques [
To evaluate the requirement of strengthening amount, a loadcarrying capacity or flexural stiffness for the structural condition in service mode should be investigated. Due to the characteristics of this railway bridge, dynamic tests for structural evaluation were performed using infield monitoring data for the train loads passing the target point [
To assess the amount of adequate strengthening ratio to satisfy the requirement, the unknown reinforcement ratio of steel rebar should be reasonably estimated. Structural analysis on the target bridge without the steel rebar was initially performed to calculate external flexural moment subjected by LS18 design loads. The structural modeling and analysis were conducted by commercial FEA program [
FE analyses for dead and live loads.
Bending moment diagram for a factored dead load
Bending moment diagram for a factored live load
The design flexural moment by FE analysis was calculated as 761.0 kN·m. With this moment capacity, the area of steel rebar can be estimated as 1,718 mm^{2} by the moment equilibrium equation for the rectangular beam with an effective width of slab.
The conventional performance function for flexural capacity of the bridge crosssection consists of
where
As depicted in Figure
Compatibility diagram of strain and strength of the crosssection for strengthening.
In the case of failure mode of a concrete beam externally bonded with CFRP materials, except in a premature failure case, the following four failure modes are classified representatively:
steel yielding and concrete compressive failure before CFRP rupture;
steel yielding and CFRP rupture after concrete compressive failure;
CFRP rupture and concrete compressive failure before steel yielding;
concrete compressive failure before steel yielding and CFRP rupture.
Among the Cases 1~4, Cases 3 and 4 are typical overstrengthening failure, which leads to brittle failure of strengthening beams. Cases 1 and 2, however, would result in ductile failure, rather than that of Cases 3 and 4. For reasonable failure cases, Case 1) is more suitable for preventing the brittle failure because concrete compressive failure is less brittle than that of CFRP strip. Balanced failure means that a strengthened concrete member fails simultaneously with concrete compressive failure and CFRP rupture.
Structural safety can be conveniently calculated with respect to the reliability index
The computational uncertainties,
Cho et al. [
For reliability analyses, the statistics of random variables are defined in advance. There are three variables considered in this analysis: external load for dead and live ones, material strength of concrete, steel rebar and CFRP strip, and design of crosssection.
In the reliability analysis for structural safety, it is essential that the load effect must be considered by combining the variability of loads, dead and live loads. A related study was conducted and suggested the load and resistance factors for RC concrete design [
Result of FEM analysis for external railway load.
Probability distribution  Mean/Nominal  Mean  COV^{a}  Load factor  

Dead load  Normal  1.05  180.0 kN·m  0.1  1.4 
Live load  Lognormal  1.00  580.8 kN·m  0.2–0.4  2.0 
The statistics of resistancerelated variables such as
Statistics of random design variables (I).
Design variable  Nominal value  Mean value  Standard deviation  Probability distribution 


41.34  46.16  1.94  Normal 

—  2,790  85.7  Normal 

1,900 

6.0  Normal 

1,250 

6.35  Normal 

1,200 

12.70  Normal 
Statistics of random design variables (II).
Mean/Nominal  COV  Average strength  Number of data  Standard deviation  Probability distribution  

SD 30^{a}  1.20  0.064  360.0  822  23.04  Normal 
SD 35^{a}  1.13  0.038  395.5  80  15.03  Normal 
SD 40^{a}  1.09  0.048  436.0  773  20.93  Normal 
Grade 40^{b}  1.13  0.116  317.0  —  36.72  Normal 
Grade 60^{b}  1.12  0.098  472.5  —  46.31  Normal 
Sensitive analysis for damage factor, standard deviation of damage factor, and live load effect.
Damage factor  Standard deviation  COV 

Live load moment  Standard deviation  COV  

Case 1  0.7  0.14  0.2  0.000128  290.4  87.12  0.3 
Case 2  0.8  0.16  0.2  0.000114  290.4  87.12  0.3 
Case 3  0.9  0.18  0.2  0.000102  290.4  87.12  0.3 
Case 4  0.8  0.08  0.1  0.000048  290.4  87.12  0.3 
Case 5  0.8  0.16  0.2  0.000114  290.4  87.12  0.3 
Case 6  0.8  0.24  0.3  0.000196  290.4  87.12  0.3 
Case 7  0.8  0.16  0.2  0.000095  290.4  58.08  0.2 
Case 8  0.8  0.16  0.2  0.000114  290.4  87.12  0.3 
Case 9  0.8  0.16  0.2  0.000137  290.4  116.16  0.4 
This study is to calculate the reliabilitybased strengthening ratio of CFRP strip to the existing railway bridge, which has structural uncertainties. Therefore, it is important to define how much structural safety should be acquired. This is simply determined by using the reliability index, or the target safety factor,
To evaluate the reliabilitybased strengthening ratio of the target bridge, the probability distribution between the external load and structural resistance from the limit state function was analyzed. A safety margin was used and
A procedure for determining the reliabilitybased strengthening ratio of CFRP strip with a target reliability index of 3.5.
Figure
Probability distribution curves for external load and resistance.
Critical strengthening ratio of CFRP strip strengthened by NSM.
Figure
Sensitive analysis for COV and average value of
This study suggested the reliabilitybased strengthening ratio for 30yearold railway bridge using CFRP strips. Conclusions are as follows.
In previous strengthening schemes, it has been uncertain to determine how much the strengthening effect should be required. The methodology for the reliabilitybased strengthening ratio can improve these problems of the previous strengthening method. The target reliability index for CFRP strip strengthening is considered as 3.5 according to AASHTO specification. As using the reliabilitybased strengthening ratio in this study, more effective strengthening design to concrete structure, having a specified strengthening target as well as reflecting the structural and material uncertainties, is possible.
In the result of a sensitive analysis, variation of COV for damage factor mostly affected to the reliabilitybased strengthening ratio of CFRP strip. Therefore, damage factor should be studied more properly on the target bridge. This may be possible by analyzing the database for longterm safety inspection history and its reasonable quantification. Stabilization and normalization processes of the damage factor are also required.
One of the important factors for determining the safety margin against the resistance is external load effect. In order to improve the reliabilitybased strengthening ratio of CFRP strip in this study, uncertainties for external load of a railway bridge should be analytically and experimentally verified. This can be solved by analyzing the acquired data from longterm monitoring; then the reliability of the strengthening ratio of CFRP strip will be promoted.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by Korea Institute of Energy Technology Evaluation and Planning (0000000015513) and Research grant from Gyeongnam National University of Science and Technology.