Prediction of Load Capacity Variation in FRP Bonded Concrete Specimens Using Brownian Motion

Inwet lay-up process, dry fiber sheets are saturatedwith a polymer and applied to the concrete surface by hand.This causes relatively large variation in properties of the cured FRP composite material. It is hard to know the exact mechanical properties of the FRP constructed by wet lay-up process. In addition, the stiffness of FRP changes during debonding process due to different amount of concrete attached to the debonded FRP at different locations. It is also inevitable to have considerable variations in the strength of concrete. Therefore, the behaviour of FRP bonded concrete members varies among specimens even when the same materials are used. The variation of localized FRP stiffness and concrete strength can be combined in a single parameter as variation of the localized interfacial fracture energy. In an effort to effectively model the effects of the variation of interfacial fracture energy on the load versus deflection responses of FRP bonded concrete specimens subjected toMode I andMode II loading, a randomwhite noise using a one-dimensional standard Brownian motion is added to the governing equations, yielding stochastic differential equations. By solving these stochastic equations, the bounds of load carrying capacity variation with 95% probability are found for different experimental tests.


Introduction
Extensive research has clearly shown that externally bonded fiber reinforced polymer (FRP) composites have good potential for use in strengthening of concrete members [1][2][3].The stiffness of FRP and compressive concrete strength are the most effective parameters on the responses and behaviour of strengthened concrete members [4][5][6].However, there are some variations in FRP stiffness (bending stiffness and tension stiffness; hereafter called "stiffness" only unless the tension or bending stiffness needs to be emphasized) and concrete strength, particularly the localized strength.In the following, the sources of the FRP stiffness and concrete strength variations are briefly explained, respectively.
One of the common methods to use FRP to strengthen existing concrete structures is wet lay-up bonding that consists of installation by hand using unidirectional dry fiber sheets or fabrics impregnated with a saturating resin on-site [3].Since it is hard to accurately determine Young's modulus of the cured FRP constructed by wet lay-up process and there is variation of the cured FRP thickness, it is almost impossible to obtain the accurate magnitude of the FRP stiffness.Generally, the manufacturers provide the properties of the fiber sheets, such as Young's modulus and design sheet thickness, based on the tests in laboratories.However, when it is used in field, the properties of the FRP are affected by the curing process and defects such as gouges or deep scratches which may occur through the instalment of FRP [7,8].Typically it is difficult to get the same level of quality control for wet lay-up FRP as for the FRP plates or strips precured in the factory.Therefore, this operation may cause differences between what is in the manufacturer's reports and the actual FRP stiffness achieved in situ.The FRP properties may also be dissimilar at different locations even in the same specimen because the fibers may be curved to different extents at different locations.
Existing experimental studies have shown that in the vast majority of cases, except when a weak adhesive or a high strength concrete is used, debonding failure of an FRP-concrete bonded joint is due to the fracture within the concrete at a small distance from the concrete-adhesive interface [9,10].During the debonding failure, the concrete crack path continuously changes direction and the thickness of substrate concrete layer attached to the FRP varies at different locations [11].This phenomenon causes the stiffness variation during debonding propagation.
Concrete is a mixture of water, cement, sand, aggregate, admixtures, and air.Variations in the properties or proportions of these constituents as well as variations in the transporting, placing, compaction, and curing of the concrete lead to variations in the strength of the finished concrete.In addition, discrepancies in the tests will also cause apparent differences in strength [12].The concrete near the top of concrete members tends to be weaker than the concrete lower down, probably because of increased water to cement ratio at the top due to upward water migration after the concrete is placed and by greater compaction of concrete near the bottom due to the sinking of aggregates in the form during vibration [13].Therefore, there is variation of concrete strength at different spots even in the same member.
Since FRP stiffness and concrete strength play important roles in predicting the load carrying capacity and failure modes in FRP-strengthened concrete members, it is useful to find a way to determine the range of actual values of FRP stiffness and concrete strength without testing each of them.Since the fracture energy is a function of concrete strength and FRP stiffness in FRP concrete bonds [6], FRP stiffness and concrete strength variations can combine together as the interfacial fracture energy variation.In this study, the concepts of Brownian motion and white noise from probability theory are used to find the fracture energy range instead of variations of the FRP stiffness and concrete strength separately.
Brownian motion is the macroscopic picture emerging from a particle moving randomly in -dimensional space.The term "Brownian motion" can also refer to the mathematical model used to define such random movements.If just one direction for the movements in an instant of time is assumed, then it is named one-dimensional Brownian motion.White noise generally is a random signal that can be applied to model a totally unpredictable process.It can be considered as the derivative of a Brownian motion existing in the stationary sense.In the next section, Brownian motion and white noise will be introduced briefly.
In this study, a random white noise is added to the fracture energy parameter in the governing equations to model fluctuating differences between actual and theoretical values of FRP stiffness and concrete strength.The governing equations for FRP debonding from concrete subjected to Mode I (normal stress perpendicular to the interface) and Mode II (in-plane shear stress parallel to the interface) loadings become stochastic differential equations, where the driven noise is a one-dimensional standard Brownian motion.
By solving these equations and comparing the results with the experimental data, the ranges of load carrying capacity with 95% probability are found for different experimental tests.

Background on Brownian Motion
In 1828, the Scottish botanist Robert Brown observed irregular movement of pollen suspended in water.As explained by Einstein [14], this random movement is caused by the buffeting of the pollen by water molecules and results in dispersal or diffusion of the pollen in the water.The first mathematically rigorous construction of Brownian motion is credited to Wiener [15], and Brownian motion is sometimes called the Wiener process.Deep studies of Brownian motion and related topics can be found in Øksendal [16].
A real-valued stochastic process  = {() :  ≥ 0} defined on some probability space (Ω, , ) is called a standard Brownian motion, if (i) (0) = 0; In the above,  is the mathematical expectation with respect to the probability measure , and   := {() : 0 ≤  < } is a -algebra.One can think   as all the information available to the observer by time .Note that   ⊂   for all 0 ≤  < .The collection of -algebra {  } ≥0 is the natural filtration generated by .Also, for each  ∈ Ω, the function  → (, ) can be regarded as a sample path or a realization of the Brownian motion.In a random experiment, for example, infinite number of coin tossing, the outcome determines the sample path of the Brownian motion.Then () is the value of this path at time , and this value of course depends on which path resulted from the random experiment.Figure 1 presents five different sample paths of a 1-dimensional standard Brownian motion obtained by using MATLAB.
Even though the Brownian motion  has continuous sample paths, it can be shown that  is nowhere differentiable with probability 1, a mathematical fact explaining the high irregularity of Brownian motion.This means that the derivative of the function  → (,) does not exist in the ordinary sense for almost all  ∈ Ω.Still, we may interpret their time derivative in a distributional sense to get a generalized stochastic process called the white noise [16].White noise generally is a random signal that can be applied to model a totally unpredictable process.It can be considered as the derivative of a Brownian motion existing in the generalized sense.Figure 2 plots a numerical realization of a white noise.

FRP-to-Concrete Interface under Mode II Loading Condition
Figure 3 shows an FRP/concrete interface under a single shear pull-out action; that is, the specimen is subjected to Mode II loading (in-plane shear stress).In this figure,  0 is the initial crack length between the FRP sheet/plate and the concrete substrate, and  is the new crack length during debonding propagation.As it is already mentioned, experimental studies suggest that the failure of FRP/concrete joints generally occurs in concrete at a few millimetres from the FRP/concrete interface.The ultimate load of the joint therefore depends strongly on concrete failure behaviour.For a brittle material, the debonding region does not transfer any stresses.Theoretically, there should not be further increasing in load carrying after debonding initiation [4].For a quasi-brittle material like concrete, the new crack surfaces formed by debonding may be in contact and are tortuous in nature.This leads to toughening mechanisms at the tip of the crack that is taken into account by a conceptual fracture process zone (FPZ).As a result of FPZ, the newly formed crack surfaces may continue to sustain more stresses after debonding initiation, which is characterized by a softening branch in a traction-separation relationship [19].The traction-separation relationship in FRP bonded concrete joints in a single shear pull-out test is known as bond-slip behaviour.Figure 4 shows typical bond-slip behaviour of a single shear pull-out test specimen.Area under the bondslip curve is considered as required interfacial Mode II fracture energy,  II , for debonding propagation of FRP from concrete surface subjected to shear stress.
Existing studies indicate that the interfacial fracture energy of FRP/concrete joint is a function of concrete strength and FRP stiffness.Therefore, it is assumed that variation of the interfacial fracture energy can present the variations of concrete strength and FRP stiffness together.
The theoretical ultimate interfacial pull-out force in the FRP sheet,   , which is equal to the axial force in FRP, can be expressed as where   is the elastic modulus of FRP;   is the thickness of FRP;   is the width of FRP sheet; and  max is the maximum strain of FRP sheets corresponding to the maximum pull-out force.On the other hand, the interfacial fracture energy can be calculated by [6] This relation is independent of bond-slip relation and suitable for any other type of bond-slip models [20].By substituting (2) into (1), the maximum bearing load is a function of the interfacial fracture energy as shown in the following: Regarding Figure 3, FRP strain at the tip of the initial crack is equal to  = ( 1 −  2 )/, where  1 is the interfacial slip at the tip of the initial crack and  2 is the interfacial slip at the end of the new crack.When "" reaches a sufficiently long value " max ",  1 , which is the elongation of FRP over the fully debonded zone, will be much larger than  2 , which is the elongation of FRP over the bonded zone.In other words,  2 is negligible compared to  1 .Using this assumption, the FRP strain at the tip of the initial crack is equal to  max =  max / max , where  max is the maximum interfacial slip at the tip of the initial crack and  max is the maximum crack length in front of the initial crack corresponding to the ultimate pullout force,   , during the interface crack propagation (i.e., macro interface debonding).
By substituting  max =  max / max into (3), it can be rewritten as By setting  = 1/ max and  0 = 2 II    max , (4) in differential equation form is Interfacial fracture energy,  II , is a material property [19] and theoretically it has a constant value.Previous research has shown that " max " has a fixed relationship with the interfacial fracture energy  II [21] and therefore is assumed to be constant in this study.The width of FRP,   , can also be considered as a constant.Therefore,  0 = 2     max is assumed as a constant in this study.
The interfacial fracture energy is a function of the FRP stiffness and the concrete strength.As discussed in the previous section, the actual FRP tension stiffness achieved in situ is not the same as the calculated value.It is inevitably subjected to random fluctuations, resulting from the wet layup process and different amount of concrete attached to the debonded FRP at different locations.Also there are some sources for concrete compressive strength variation.In order to find the effect of the FRP tension stiffness and concrete strength variations on the ultimate load carrying capacity (i.e., the maximum pullout load) of the specimens, a white noise is added to the parameter  0 as in the following: where  is a constant positive value ( > 0) that presents the distribution or variation of experimental data, and Ẇ is a one-dimensional white noise.Since the measured value of  is a parameter which varies due to the wet lay-up construction process and different amount of concrete attached to the debonded FRP at different locations, Ẇ is a function of  ( here).By substituting ( 6) into ( 5), the ultimate axial force can be written as Since white noise can be formally considered as the derivative of a Brownian motion, (7) can be rewritten as where () is a one-dimensional standard Brownian motion.In this application, a Brownian motion is introduced to present the effect of totally unpredictable interfacial fracture energy variation on the load versus deflection responses of the specimens.Equation ( 8) is a stochastic differential equation and the solution is It is well known that if the random variable  has normal distribution with mean  and variance  2 , then ( − )/ has a standard normal distribution with mean 0 and variance 1 [22].Recall that if a stochastic process  = {(), 0 ≤  < ∞} is a Brownian motion, then  has independent and stationary increments [16], and for each , () is a normal random variable with mean 0 and variance .Therefore, ()/ √  has a standard normal distribution.Then, See Figure 5 for an illustration.By substituting (9) into (10), with 95% probability, the upper and lower bounds can be found for the ultimate axial force,   , as In (12), the value of constant , which represents the distribution of experimental data, is needed to determine ×10 4 K 0 (1/ max ) 0.5 (N) the lower and upper bounds of the ultimate load.This value has been obtained in this study by using the experimental data from Dai et al. [6].
Table 1 includes the material properties and experimental results of the single shear pullout tests conducted by Dai et al. [6].The width of the FRP sheets,   , is 100 mm.In order to use data from different tests with different materials, normalized  0   against  0 √  curves are drawn in Figure 6.In this figure, the thin line represents the theoretical response calculated from (4), the thick lines are the results from (12), and * points are the experimental data.value of  in (12), band the lower upper   can to all experimental data.test results,  = 1200 is a good estimation to experimental data.
Because   is between  and lower , and Δ max / , and bounds of interfacial Mode II fracture energy, ( II ) , 95% be expressed = 1200 (13), the of Mode II energy bounds determined presented Table 1.On average, fracture for this group specimens are within plus or minus 25.4% of the values.
To give an instance, the ultimate load versus maximum slip for the first specimen in Table 1 (CR1L1) is shown in Figure 7.The thin line is the theoretical one from (4), and the thick lines are the upper and lower bounds of ultimate load based on (12) with  = 1200.Due to the Brownian motion, (9) represents a stochastic process, which has many different sample paths.The dashed line in Figure 7 plots one of such sample paths.Figure 8 shows the experimental ultimate loads ( ex ) against the theoretical ones ( th ).Because the relationship between the ultimate load and Mode II interfacial fracture energy,  II , is linear (4), the ultimate bearing load changes 25.4% when the fracture energy changes 25.4%.In Figure 8, the thick lines are the bounds of experimental maximum load versus theoretical one; that is,  ex = (1 ± 0.254) th .This may explain why the debonding strength of FRP always exhibits quite large scatter [23,24].
As it can be seen in Figure 8, all experimental ultimate loads are in the bounds.Therefore, it confirms that the proposed method is valid to predict the load capacity variation of the FRP bonded concrete specimens under Mode II loading.

FRP-to-Concrete Interface under Mode I Loading Condition
The relationship between the load and displacement in a Mode I loading condition can be obtained in a similar way as Mode II loading case.Figure 9 shows the schematic model around the crack tip subjected to a Mode I load, .If the value of the crack (debonded) length, , is small, the debonded FRP can be assumed as a cantilever beam subjected to load , and the bonded parts of FRP can be assumed as an Euler-Bernoulli beam on an elastic 26].
In this study, the Mode I of bonded parts FRP, Δ , 9, is assumed equal to zero because very small the displacement at the crack tip, Δ 2 .Therefore, the relationship the load,   , the I Δ , can be obtained as where  the bending of  max is crack length to ultimate load.Based the compliance method, Mode I fracture be expressed as 26] where the compliance of the unbonded/debonded FRP and can expressed by By substituting (15) and ( 16) into ( 14) and setting  0 = 2   I  and  = 1/Δ max , the ultimate Mode I load,   , can be expressed as To model the unpredictable changes in the Mode I interfacial fracture energy,  I , a white noise is added to the parameter  0 , where  0 is calculated by the manufacturer reported FRP material properties and concrete strength.The process is similar to what has been done for the Mode II loading case.The upper and lower bounds for ultimate Mode I loading,   , with probability 95% are To find the magnitude of , experimental data from Wan et al. [17] and Ouyang and Wan [18] are used.
Wan et al. [17] used modified double cantilever beam (MDCB) specimens to release rate of debonding concrete subjected to Mode I CFRP were applied the substrates by wet layup process.Ouyang Wan [18] MDCB test measure the interfacial fracture energy of the CFRP plate debonding under I loading.The of tests are in Tables 2 and 3, The normalized  0   against  0 √  curves are drawn in Figure 10 for the specimens in  data in Wan et al. [17].The  value for experimental data in Ouyang and Wan [18] is equal to 240 as shown in Figure 11.
Because   (peak load) is between  upper and  lower at Δ max , using the bounds of interfacial energy, ( I , with 95% probability can be expressed as By using (19) and the obtained values of , the range of the bounds of the Mode I interfacial fracture energy has been determined for the experimental data, and they are presented in Tables 2 and 3.It can be seen that, on average, the variations of the actual fracture energies of the specimens tested by Wan et al. [17] are within plus or minus 15.7% of the theoretical values, while it is 17.4% for the specimens tested by Ouyang and Wan [18].
According to (17), the relationship between ultimate bearing load and interfacial Mode I fracture energy is linear.Figure 12 shows the experimental ultimate load against analytical ones for specimens in Wan et al. [17] and Ouyang and Wan [18].In this figure, the thick lines are bounds of experimental maximum load versus theoretical one; that is,  ex = (1 ± 0.174) th .As it can be seen, all experimental ultimate loads are within the bounds.Therefore, it confirms that the proposed method can predict the load capacity variation of the specimens subjected to Mode I loading.

Conclusion
In this study, white noise and Brownian motion are introduced to model the variation of interfacial fracture energy in FRP bonded concrete specimens subjected to Mode I and Mode II loadings.A systematic method is developed to determine the range of interfacial fracture energy and load carrying capacity.Experimental data from literatures are used to demonstrate the validity of this methodology.For the experimental data used in this research, Mode I and Mode II load carrying capacities are within ±17.4% and ±25.4%, respectively, of their theoretical values.

Figure 5 :
Figure 5: Probability density function for standard Brownian motion.

Figure 6 :
Figure 6: Application of experimental data from Dai et al. [6] to find .

Figure 9 :
Figure 9: Simulation of FRP on the concrete substrate as a cantilever beam and a beam on elastic foundation for Mode I loading.

Figure 10 :
Figure 10: Application of experimental data from Wan et al. [17] to find .

Figure 12 :
Figure12: Experimental ultimate load versus theoretical ones for the tests by Wan et al.[17] and Ouyang and Wan[18].